The Shell Model as Unified View of Nuclear Structure

ar X i v : n u c l - t h / 0 4 0 2 0 4 6 v 1 1 3 F e b 2 0 0 4 The Shell Model as Unified View of Nuclear Structure E. Caurier, 1, ∗ G. Mart´ınez-Pinedo, 2, 3, † F. Nowacki, 1, ‡ A. Poves, 4, § and A. P. Zuker 1, ¶ 1 Institut de Recherches Subatomiques, IN2P3-CNRS, Université Louis Pasteur, F-67037 Strasbourg, France 2 Institut d’Estudis Espacials de Catalunya, Edifici Nexus, Gran Capità 2, E-08034 Barcelona, Spain 3 Institució Catalana de Recerca i Estudis Avan¸cats, Llu´ıs Companys 23, E-08010 Barcelona, Spain 4 Departamento de F´ısica Teórica, Universidad Autónoma, Cantoblanco, 28049, Madrid, Spain (Dated: February 8, 2008) The last decade has witnessed both quantitative and qualitative progresses in Shell Model studies, which have resulted in remarkable gains in our understanding of the structure of the nucleus. Indeed, it is now possible to diagonalize matrices in determinantal spaces of dimensionality up to 10 9 using the Lanczos tridiagonal construction, whose formal and numerical aspects we will analyze. Besides, many new approximation methods have been developed in order to overcome the dimensionality limitations. Furthermore, new effective nucleon-nucleon interactions have been constructed that contain both two and three-body contributions. The former are derived from realistic potentials (i.e., consistent with two nucleon data). The latter incorporate the pure monopole terms necessary to correct the bad saturation and shell-formation properties of the realistic two-body forces. This combination appears to solve a number of hitherto puzzling problems. In the present review we will concentrate on those results which illustrate the global features of the approach: the universality of the effective interaction and the capacity of the Shell Model to describe simultaneously all the manifestations of the nuclear dynamics either of single particle or collective nature. We will also treat in some detail the problems associated with rotational motion, the origin of quenching of the Gamow Teller transitions, the double β-decays, the effect of isospin non conserving nuclear forces, and the specificities of the very neutron rich nuclei. Many other calculations—that appear to have “merely” spectroscopic interest—are touched upon briefly, although we are fully aware that much of the credibility of the Shell Model rests on them. Contents I. Introduction 2 A. The three pillars of the shell model 2 B. The competing views of nuclear structure 3 1. Collective vs. Microscopic 3 2. Mean field vs Diagonalizations 4 3. Realistic vs Phenomenological 4 C. The valence space 5 D. About this review: the unified view 6 II. The interaction 7 A. Effective interactions 8 1. Theory and calculations 8 B. The monopole Hamiltonian 9 1. Bulk properties. Factorable forms 10 2. Shell formation 11 C. The multipole Hamiltonian 13 1. Collapse avoided 14 D. Universality of the realistic interactions 15 III. The solution of the secular problem in a finite space 16 A. The Lanczos method 16 B. The choice of the basis 17 C. The Glasgow m-scheme code 18 D. The m-scheme code ANTOINE 18 E. The coupled code NATHAN. 19 F. No core shell model 20 G. Present possibilities 20 ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] IV. The Lanczos basis 21 A. Level densities 21 B. The ground state 22 1. The exp(S) method 23 2. Other numerical approximation methods 24 3. Monte Carlo methods 24 C. Lanczos Strength Functions 25 V. The 0ω calculations 26 A. The monopole problem and the three-body interaction 26 B. The pf shell 29 1. Spectroscopic factors 29 2. Isospin non conserving forces 30 3. Pure spectroscopy 30 C. Gamow Teller and magnetic dipole strength. 32 1. The meaning of the valence space 32 2. Quenching 33 VI. Spherical shell model description of nuclear rotations 34 A. Rotors in the pf shell 35 B. Quasi-SU(3). 36 C. Heavier nuclei: quasi+pseudo SU(3) 38 D. The 36 Ar and 40 Ca super-deformed bands 39 E. Rotational bands of unnatural parity 40 VII. Description of very neutron rich nuclei 42 A. N=8; 11 Li: halos 42 B. N=20; 32 Mg, deformed intruders 43 C. N=28: Vulnerability 44 D. N=40; from “magic” 68 Ni to deformed 64 Cr 45 VIII. Other regions and themes 47 A. Astrophysical applications 47 B. ββ-decays 50 C. Radii isotope shifts in the Calcium isotopes 51 D. Shell model calculations in heavier nuclei 52 E. Random Hamiltonians 53 2 IX. Conclusion 53 Acknowledgments 54 A. Basic definitions and results 54 B. Full form of Hm 55 1. Separation of Hmnp and H m0 56 2. Diagonal forms of Hm 57 C. The center of mass problem 57 References 58 I. INTRODUCTION In the early days of nuclear physics, the nucleus, composed of strongly interacting neutrons and protons confined in a very small volume, didn’t appear as a system to which the shell model, so successful in the atoms, could be of much relevance. Other descriptions—based on the analogy with a charged liquid drop—seemed more natural. However, experimental evidence of independent particle behavior in nuclei soon began to accumulate, such as the extra binding related to some precise values of the number of neutrons and protons (magic numbers) and the systematics of spins and parities. The existence of shell structure in nuclei had already been noticed in the thirties but it took more than a decade and numerous papers (see Elliott and Lane, 1957, for the early history) before the correct prescription was found by Mayer (1949) and Axel, Jensen, and Suess (1949). To explain the regularities of the nuclear properties associated to “magic numbers”—i.e., specific values of the number of protons Z and neutrons N—the authors proposed a model of independent nucleons confined by a surface corrected, isotropic harmonic oscillator, plus a strong attractive spin-orbit term 1 , U(r) = 1 2 ω r 2 +D l 2 +C l s. (1) In modern language this proposal amounts to assume that the main effect of the two-body nucleon-nucleon interactions is to generate a spherical mean field. The wave function of the ground state of a given nucleus is then the product of one Slater determinant for the protons and another for the neutrons, obtained by filling the lowest subshells (or “orbits”) of the potential. This primordial shell model is nowadays called “independent particle model” (IPM) or, “naive” shell model. Its foundation was pioneered by Brueckner (1954) who showed how the short range nucleon-nucleon repulsion combined with the Pauli principle could lead to nearly independent particle motion. 1 We assume throughout adimensional oscillator coordinates, i.e., r −→(mω/) 1/2 r. As the number of protons and neutrons depart from the magic numbers it becomes indispensable to include in some way the “residual” two-body interaction, to break the degeneracies inherent to the filling of orbits with two or more nucleons. At this point difficulties accumulate: “Jensen himself never lost his skeptical attitude towards the extension of the single-particle model to include the dynamics of several nucleons outside closed shells in terms of a residual interaction” (Weidenm¨ uller, 1990). Nonetheless, some physicists chose to persist. One of our purposes is to explain and illustrate why it was worth persisting. The keypoint is that passage from the IPM to the interacting shell model, the shell model (SM) for short, is conceptually simple but difficult in practice. In the next section we succinctly review the steps involved. Our aim is to give the reader an overall view of the Shell Model as a sub-discipline of the Many Body problem. “Inverted commas” are used for terms of the nuclear jargon when they appear for the first time. The rest of the introduction will sketch the competing views of nuclear structure and establish their connections with the shell model. Throughout, the reader will be directed to the sections of this review where specific topics are discussed. A. The three pillars of the shell model The strict validity of the IPM may be limited to closed shells (and single particle –or hole– states built on them), but it provides a framework with two important components that help in dealing with more complex situations. One is of mathematical nature: the oscillator orbits define a basis of Slater determinants, the m-scheme, in which to formulate the Schrödinger problem in the occupation number representation (Fock space). It is important to realize that the oscillator basis is relevant, not so much because it provides an approximation to the individual nucleon wave-functions, but because it provides the natural quantization condition for self-bound systems. Then, the many-body problem becomes one of diagonalizing a simple matrix. We have a set of determinantal states for A particles, a † i1 . . . a † iA [0) = [φ I ), and a Hamil- tonian containing kinetic / and potential energies 1 H = ij / ij a † i a j − i≤j k≤l 1 ijkl a † i a † j a k a l (2) that adds one or two particles in orbits i, j and removes one or two from orbits k, l, subject to the Pauli principle (¦a † i a j ¦ = δ ij ). The eigensolutions of the problem [Φ α ) = I c I,α [φ I ), are the result of diagonalizing the matrix ¸φ I [H[φ I ′ ) whose off-diagonal elements are either 0 or ±1 ijkl . However, the dimensionalities of the matrices—though not infinite, since a cutoff is inherent to a non-relativistic approach—are so large as to make the problem intractable, except for the lightest nuclei. Stated in these terms, the nuclear many body problem 3 does not differ much from other many fermion problems in condensed matter, quantum liquids or cluster physics; with which it shares quite often concepts and techniques. The differences come from the interactions, which in the nuclear case are particularly complicated, but paradoxically quite weak, in the sense that they produce sufficiently little mixing of basic states so as to make the zeroth order approximations bear already a significant resemblance with reality. This is the second far-reaching –physical– component of the IPM: the basis can be taken to be small enough as to be, often, tractable. Here some elementary definitions are needed: The (neutron and proton) major oscillator shells of principal quantum number p = 0, 1, 2, 3 . . ., called s, p, sd, pf . . . respectively, of energy ω(p + 3/2), contain orbits of total angular momentum j = 1/2, 3/2 . . . p + 1/2, each with its possible j z projections, for a total degeneracy D j = 2j + 1 for each subshell, and D p = (p + 1)(p + 2) for the major shells (One should not confuse the p = 1 shell —p-shell in the old spectroscopic notation—with the generic harmonic oscillator shell of energy ω(p + 3/2)). When for a given nucleus the particles are restricted to have the lowest possible values of p compatible with the Pauli principle, we speak of a 0ω space. When the many-body states are allowed to have components involving basis states with up to N oscillator quanta more than those pertaining to the 0ω space we speak of a Nω space (often refered as “no core”) For nuclei up to A ≈ 60, the major oscillator shells provide physically meaningful 0ω spaces. The simplest possible example will suggest why this is so. Start from the first of the magic numbers N = Z = 2; 4 He is no doubt a closed shell. Adding one particle should produce a pair of single-particle levels: p 3/2 and p 1/2 . Indeed, this is what is found in 5 He and 5 Li. What about 6 Li? The lowest configuration, p 2 3/2 , should produce four states of angular momentum and isospin JT = 01, 10, 21, 30. They are experimentaly observed. There is also a JT = 20 level that requires the p 3/2 p 1/2 configuration. Hence the idea of choosing as basis p m (p stands generically for p 3/2 p 1/2 ) for nuclei up to the next closure N = Z = 8, i.e., 16 O. Obviously, the general Hamiltonian in (2) must be transformed into an “effective” one adapted to the restricted basis (the “valence space”). When this is done, the results are very satisfactory (Cohen and Kurath, 1965). The argument extends to the sd and pf shells. By now, we have identified two of the three “pillars” of the Shell Model: a good valence space and an effective interaction adapted to it. The third is a shell model code capable of coping with the secular problem. In Sec- tion I.C we shall examine the reasons for the success of the classical 0ω SM spaces and propose extensions capable of dealing with more general cases. The interaction will touched upon in Section I.B.3 and discussed at length in Section II. The codes will be the subject of Section III. B. The competing views of nuclear structure Because Shell Model work is so computer-intensive, it is instructive to compare its history and recent developments with the competing—or alternative—views of nuclear structure that demand less (or no) computing power. 1. Collective vs. Microscopic The early SM was hard to reconcile with the idea of the compound nucleus and the success of the liquid drop model. With the discovery of rotational motion (Bohr, 1952; Bohr and Mottelson, 1953) which was at first as surprising as the IPM, the reconciliation became apparently harder. It came with the realization that collective rotors are associated with “intrinsic states” very well approximated by deformed mean field determinants (Nilsson, 1955), from which the exact eigenstates can be extracted by projection to good angular momentum (Peierls and Yoccoz, 1957); an early and spectacular example of spontaneous symmetry breaking. By further introducing nuclear superfluidity (Bohr et al., 1958), the unified model was born, a basic paradigm that remains valid. Compared with the impressive architecture of the unified model, what the SM could offer were some striking but isolated examples which pointed to the soundness of a many-particle description. Among them, let us mention the elegant work of Talmi and Unna (1960), the f n 7/2 model of McCullen, Bayman, and Zamick (1964)— probably the first successful diagonalization involving both neutrons and protons—and the Cohen and Kurath (1965) fit to the p shell, the first of the classical 0ω regions. It is worth noting that this calculation involved spaces of m-scheme dimensionalities, d m , of the order of 100, while at fixed total angular momentum and isospin d JT ≈ 10. The microscopic origin of rotational motion was found by Elliott (1958a,b). The interest of this contribution was immediately recognized but it took quite a few years to realize that Elliott’s quadrupole force and the underlying SU(3) symmetry were the foundation, rather than an example, of rotational motion (see Section VI). It is fair to say that for almost twenty years after its inception, in the mind of many physicists, the shell model still suffered from an implicit separation of roles, which assigned it the task of accurately describing a few specially important nuclei, while the overall coverage of the nuclear chart was understood to be the domain of the unified model. A somehow intermediate path was opened by the well known Interacting Boson Model of Arima and Iachello (1975) and its developments, that we do not touch here. The interested reader can find the details in (Iachello and Arima, 1987). 4 2. Mean field vs Diagonalizations The first realistic 2 matrix elements of Kuo and Brown (1966) and the first modern shell model code by French, Halbert, McGrory, and Wong (1969) came almost simultaneously, and opened the way for the first generation of “large scale” calculations, which at the time meant d JT ≈ 100-600. They made it possible to describe the neighborhood of 16 O (Zuker, Buck, and McGrory, 1968) and the lower part of the sd shell (Halbert et al., 1971). However, the increases in tractable dimensionalities were insufficient to promote the shell model to the status of a general description, and the role-separation mentioned above persisted. Moreover the work done exhibited serious problems which could be traced to the realistic matrix elements themselves (see next Section I.B.3). However, a fundamental idea emerged at the time: the existence of an underlying universal two-body interaction, which could allow a replacement of the unified model description by a fully microscopic one, based on mean-field theory. The first breakthrough came when Baranger and Kumar (1968) proposed a new form of the unified model by showing that Elliott’s quadrupole force could be somehow derived from the Kuo-Brown matrix elements. Adding a pairing interaction and a spherical mean field they proceeded to perform Hartree Fock Bogoliubov calculations in the first of a successful series of papers (Kumar and Baranger, 1968). Their work could be described as shell model by other means, as it was restricted to valence spaces of two contiguous major shells. This limitation was overcome when Vautherin and Brink (1972) and Dechargé and Gogny (1980) initiated the two families of Hartree Fock (HF) calculations (Skyrme and Gogny for short) that remain to this day the only tools capable of giving microscopic descriptions throughout the periodic table. They were later joined by the relativistic HF approach (Serot and Walecka, 1986). (For a review of the three variants see Bender, Hee- nen, and Reinhard, 2003b). See also Péru, Girod, and Berger (2000) and Rodriguez-Guzmán, Egido, and Rob- ledo (2000) and references therein for recent work with the Gogny force, the one for which closest contact with realistic interactions and the shell model can be established (Sections VI.A and II.D). Since single determinantal states can hardly be expected accurately to describe many body solutions, everybody admits the need of going beyond the mean field. Nevertheless, as it provides such a good approximation to the wavefunctions (typically about 50%) it suggests efficient truncation schemes, as will be explained in Sec- tions IV.B.1 and IV.B.2. 2 Realistic interactions are those consistent with data obtained in two (and nowadays three) nucleon systems. 3. Realistic vs Phenomenological Nowadays, many regions remain outside the direct reach of the Shell Model, but enough has happened in the last decade to transform it into a unified view. Many steps—outlined in the next Section I.D—are needed to substantiate this claim. Here we introduce the first: A unified view requires a unique interaction. Its free parameters must be few and well defined, so as to make the calculations independent of the quantities they are meant to explain. Here, because we touch upon a different competing view, of special interest for shell model experts, we have to recall the remarks at the end of the first paragraph of the preceding Section I.B.2: The exciting prospect that realistic matrix elements could lead to parameter-free spectroscopy did not materialize. As the growing sophistication of numerical methods allowed to treat an increasing number of particles in the sd shell, the results became disastrous. Then, two schools of thought on the status of phenomenological corrections emerged: In one of them, all matrix elements were considered to be free parameters. In the other, only average matrix elements (“centroids”)—related to the bad saturation and shell formation properties of the two-body potentials—needed to be fitted (the monopole way). The former lead to the “Universal sd” (USD) interaction (Wildenthal, 1984), that enjoyed an immense success and for ten years set the standard for shell model calculations in a large valence space (Brown and Wilden- thal, 1988), (see Brown, 2001, for a recent review). The second, which we adopt here, was initiated in Eduardo Pasquini PhD thesis (1976), where the first calculations in the full pf shell, involving both neutrons and protons were done 3 . Twenty years later, we know that the minimal monopole corrections proposed in his work are sufficient to provide results of a quality comparable to those of USD for nuclei in f 7/2 region. However, there were still problems with the monopole way: they showed around and beyond 56 Ni, as well as in the impossibility to be competitive with USD in the sd shell (note that USD contains non-monopole two-body corrections to the realistic matrix elements). The solution came about only recently with the introduction of three-body forces: This far reaching development will be explained in detail in Sections II, II.B and V.A. We will only anticipate on the conclusions of these sections through two syllogisms. The first is: The case for realistic two-body interactions is so strong that we have to accept them as they are. Little is known about three- body forces except that they exist. Therefore problems with calculations involving only two body forces must be blamed on the absence of three body forces. This argu- 3 Two months after his PhD, Pasquini and his wife “disappeared” in Argentina. This dramatic event explains why his work is only publicly available in condensed form (Pasquini and Zuker, 1978). 5 ment raises (at least) one question: What to make of all calculations, using only two-body forces, that give satisfactory results? The answer lies in the second syllogism: All clearly identifiable problems are of monopole origin. They can be solved reasonably well by phenomenological changes of the monopole matrix elements. Therefore, fitted interactions can differ from the realistic ones basically through monopole matrix elements. (We speak of R-compatibility in this case). The two syllogisms become fully consistent by noting that the inclusion of three-body monopole terms always improves the performance of the forces that adopt the monopole way. For the 0ω spaces, the Cohen Kurath (Cohen and Kurath, 1965), the Chung Wildenthal (Wildenthal and Chung, 1979) and the FPD6 (Richter et al., 1991) interactions turn out to be R-compatible. The USD (Wilden- thal, 1984) interaction is R-incompatible. It will be of interest to analyze the recent set of pf-shell matrix elements (GXPF1), proposed by Honma et al. (2002), but not yet released. C. The valence space The choice of the valence space should reflect a basic physical fact: that the most significant components of the low lying states of nuclei can be accounted for by many-body states involving the excitation of particles in a few orbitals around the Fermi level. The history of the Shell Model is that of the interplay between experiment and theory to establish the validity of this concept. Our present understanding can be roughly summed up by saying that “few” mean essentially one or two contiguous major shells 4 . For a single major shell, the classical 0ω spaces, exact solutions are now available from which instructive conclusions can be drawn. Consider, for instance,the spectrum of 41 Ca from which we want to extract the single- particle states of the pf shell. Remember that in 5 Li we expected to find two states, and indeed found two. Now, in principle, we expect four states. They are certainly in the data. However, they must be retrieved from a jungle of some 140 levels seen below 7 MeV. Moreover, the f 7/2 ground state is the only one that is a pure single particle state, the other ones are split, even severely in the case of the highest (f 5/2 ) (Endt and van der Leun, 1990). (Section V.B.1 contains an interesting example of the splitting mechanism). Thus, how can we expect the pf shell to be a good valence space? For few particles above 40 Ca it is certainly not. However, as we report in this work, it turns out that above A = 46, the lowest (pf) m configurations 5 are sufficiently detached from all 4 Three major shells are necessary to deal with super-deformation. 5 A configuration is a set of states having fixed number of particles in each orbit. others so as to generate wavefunctions that can evolve to the exact ones through low order perturbation theory. The ultimate ambition of Shell Model theory is to get exact solutions. The ones provided by the sole valence space are, so to speak, the tip of the iceberg in terms of number of basic states involved. The rest may be so well hidden as to make us believe in the literal validity of the shell model description in a restricted valence space. For instance, the magic closed shells are good valence spaces consisting of a single state. This does not mean that a magic nucleus is 100% closed shell: 50 or 60% should be enough, as we shall see in Section IV.B.1. In section V.C.1 it will be argued that the 0ω valence spaces account for basically the same percentage of the full wavefunctions. Conceptually the valence space may be thought as defining a representation intermediate between the Schrödinger one (the operators are fixed and the wavefunction contains all the information) and the Heisenberg one (the reverse is true). At best, 0ω spaces can describe only a limited number of low-lying states of the same (“natural”) parity. Two contiguous major shells can most certainly cope with all levels of interest but they lead to intractably large spaces and suffer from a “Center of Mass problem” (analyzed in Appendix C). Here, a physically sound pruning of the space is suggested by the IPM. What made the success of the model is the explanation of the observed magic numbers generated by the spin-orbit term, as opposed to the HO (harmonic oscillator) ones (for shell formation see Section II.B.2). If we separate a major shell as HO(p)= p > ⊕ r p , where p > is the largest subshell having j = p + 1/2, and r p is the “rest” of the HO p- shell, we can define the EI (extruder-intruder) spaces as EI(p)= r p ⊕ (p + 1) > : The (p + 1) > orbit is expelled from HO(p + 1) by the spin-orbit interaction and intrudes into HO(p). The EI spaces are well established standards when only one “fluid” (proton or neutron) is active: The Z or N = 28, 50, 82 and 126 isotopes or isotones. These nuclei are “spherical”, amenable to exact diagonalizations and fairly well understood (see for example Abzouzi et al., 1991). As soon as both fluids are active, “deformation” effects become appreciable, leading to “coexistence” between spherical and deformed states, and eventually to dominance of the latter. To cope with this situation we propose “Extended EI spaces” defined as EEI(p)= r p ⊕∆ p+1 , where ∆ p = p > ⊕(p > −2)⊕. . ., i.e., the ∆j = 2 sequence of orbits that contain p > , which are needed to account for rotational motion as explained in Section VI. The EEI(1) (p 1/2 , d 5/2 , s 1/2 ) space was successful in describing the full low lying spectra for A = 15-18 (Zuker et al., 1968, 1969). It is only recently that the EEI(2) (s 1/2 , d 3/2 , f 7/2 , p 3/2 ) space (region around 40 Ca) has become tractable (see Sections VI.D and VIII.C). EEI(3) is the natural space for the proton rich region centered in 80 Zr. For heavier nuclei exact diagonalizations are not possible but the EEI spaces provide simple (and excellent) estimates of quadrupole moments at the beginning 6 of the well deformed regions (Section VI.C). As we have presented it, the choice of valence space is primarily a matter of physics. In practice, when exact diagonalizations are impossible, truncations are introduced. They may be based on systematic approaches, such as approximation schemes discussed in Sections IV.B.1 and IV.B.2 or mean field methods mentioned in Section I.B.2, and analyzed in Section VI.A. D. About this review: the unified view We expect this review to highlight several unifying aspects common to the most recent successful shell model calculations: a) An effective interaction connected with both the two and three nucleon bare forces. b) The explanation of the global properties of nuclei via the monopole Hamiltonian. c) The universality of the multipole Hamiltonian. d) The description of the collective behavior in the laboratory frame, by means of the spherical shell model. e) The description of resonances using the Lanczos strength function method. Let us see now how this review is organized along these lines. References are only given for work that will not be mentioned later. Section II. The basic tool in analyzing the interaction is the monopole-multipole separation. The former is in charge of saturation and shell properties; it can be thought as the correct generalization of Eq. (1). Monopole theory is scattered in many references. Only by the inclusion of three-body forces could a satisfactory formulation be achieved. As this is a very recent and fundamental development which makes possible a unified viewpoint of the monopole field concept, this section is largely devoted to it. The “residual” multipole force has been extensively described in a single reference which will be reviewed briefly and updated. The aim of the section is to show how the realistic interactions can be characterized by a small number of parameters. Section III. The ANTOINE and NATHAN codes have made it possible to evolve from dimensionalities d m ∼ 5 10 4 in 1989 to d m ∼ 10 9 , d J ∼ 10 7 nowadays. Roughly half of the–order of magnitude–gains are due to increases in computing power. The rest comes from algorithmic advances in the construction of tridiagonal Lanczos matrices that will be described in this section. Section IV. The Lanczos construction can be used to eliminate the “black box” aspect of the diagonalizations, to a large extent. We show how it can be related to the notions of partition function, evolution operator and level densities. Furthermore, it can be turned into a powerful truncation method by combining it to coupled cluster theory. Finally, it describes strength functions with maximal efficiency. Section V. After describing the three body mechanism which solves the monopole problem that had plagued the classical 0ω calculations, some selected examples of pf shell spectroscopy are presented. Special attention is given to Gamow Teller transitions, one of the main achievements of modern SM work. Section VI. Another major recent achievement is the shell model description of rotational nuclei. The new generation of gamma detectors, Euroball and Gamma- sphere has made it possible to access high spin states in medium mass nuclei for which full 0ω calculations are available. Their remarkable harvest includes a large spectrum of collective manifestations which the spherical shell model can predict or explain as, for instance, deformed rotors, backbending, band terminations, yrast traps, etc. Configurations involving two major oscillator shells are also shown to account well for the appearance of superdeformed excited bands. Section VII. A conjunction of factors make light and medium-light nuclei near the neutron drip line specially interesting: (a) They have recently come under intense experimental scrutiny. (b) They are amenable to shell model calculations, sometimes even exact no-core ones (c) They exhibit very interesting behavior, such as halos and sudden onset of deformation. (d) They achieve the highest N/Z ratios attained. (e) When all (or most of) the valence particles are neutrons, the spherical shell model closures, dictated by the isovector channel of the nuclear interaction alone, may differ from those at the stability valley. These regions propose some exacting tests for the theoretical descriptions, that the conventional shell model calculations have passed satisfactorily. For nearly unbound nuclei, the SM description has to be supplemented by some refined extensions such as the shell model in the continuum, which falls outside the scope of this review. References to the subject can be found in Bennaceur et al. (1999, 2000), and in Id Betan et al. (2002); Michel et al. (2002, 2003) that deal with the Gamow shell model. Section VIII. There is a characteristic of the Shell Model we have not yet stressed: it is the approach to nuclear structure that can give more precise quantitative information. SM wave functions are, in particular, of great use in other disciplines. For example: Weak decay rates are crucial for the understanding of several astrophysical processes, and neutrinoless ββ decay is one of the main sources of information about the neutrino masses. In both cases, shell model calculations play a central role. The last section deal with these subjects, and some others... Appendix B contains a full derivation of the general form of the monopole field. Two recent reviews by Brown (2001) and Otsuka et al. (2001b) have made it possible to simplify our task and avoid redundancies. However we did not feel dispensed from quoting and commenting in some detail important work that bears directly on the subjects we treat. Sections II, IV and the Appendices are based on unpublished notes by APZ. 7 II. THE INTERACTION The following remarks from Abzouzi, Caurier, and Zuker (1991) still provide a good introduction to the subject: “The use of realistic potentials (i.e., consistent with NN scattering data) in shell-model calculation was pioneered by Kuo and Brown (1966). Of the enormous body of work that followed we would like to extract two observations. The first is that whatever the forces (hard or soft core, ancient or new) and the method of regularization (Brueckner G matrix (Kahana et al., 1969a; Kuo and Brown, 1966), Sussex direct extraction (Elliott et al., 1968) or Jastrow correlations (Fiase et al., 1988)) the effective matrix elements are extraordinarily similar (Pasquini and Zuker, 1978; Rutsgi et al., 1971). The most recent results (Jiang et al., 1989) amount to a vindication of the work of Kuo and Brown. We take this similarity to be the great strength of the realistic interactions, since it confers on them a model-independent status as direct links to the phase shifts. The second observation is that when used in shell- model calculations and compared with data these matrix elements give results that deteriorate rapidly as the number of particle increases (Halbert et al., 1971) and (Brown and Wildenthal, 1988). It was found (Pasquini and Zuker, 1978) that in the pf shell a phenomenological cure, confirmed by exact diagonalizations up to A=48 (Caurier et al., 1994), amounts to very simple modifications of some average matrix elements (centroids) of the KB interaction (Kuo and Brown, 1968).” In Abzouzi et al. (1991) very good spectroscopy in the p and sd shells could be obtained only through more radical changes in the centroids, involving substantial three- body (3b) terms. In 1991 it was hard to interpret them as effective and there were no sufficient grounds to claim that they were real. Nowadays, the need of true 3b forces has become irrefutable: In the 1990 decade several two-body (2b) potentials were developed—Nijmegen I and II (Stoks et al., 1993), AV18 (Wiringa et al., 1995), CD-Bonn-(Machleidt et al., 1996)—that fit the ≈ 4300 entries in the Ni- jmegen data base (Stoks et al., 1994) with χ 2 ≈ 1, and none of them seemed capable to predict perfectly the nucleon vector analyzing power in elastic (N, d) scattering (the A y puzzle). Two recent additions to the family of high-precision 2b potentials—the charge-dependent “CD-Bonn” (Machleidt, 2001) and the chiral Idaho-A and B (Entem and Machleidt, 2002)—have dispelled any hopes of solving the A y puzzle with 2b-only interactions (Entem et al., 2002). Furthermore, quasi-exact 2b Green Function Monte Carlo (GFMC) results (Pudliner et al., 1997; Wiringa et al., 2000) which provided acceptable spectra for A ≤ 8 (though they had problems with binding energies and spin-orbit splittings), now encounter serious trouble in the spectrum of 10 B, as found through the No Core Shell Model (NCSM) calculations of Navr´ atil and Ormand (2002) and Caurier et al. (2002), confirmed by Pieper, Varga, and Wiringa (2002), who also show that the problems can be remedied to a large extent by introducing the new Illinois 3b potentials developed by Pieper et al. (2001). It is seen that the trouble detected with a 2b-only description—with binding energies, spin-orbit splittings and spectra—is always related to centroids, which, once associated to operators that depend only on the number of particles in subshells, determine a “monopole” Hamiltonian H m that is basically in charge of Hartree Fock selfconsistency. As we shall show, the full H can be separated rigorously as H = H m + H M . The multipole part H M includes pairing, quadrupole and other forces responsible for collective behavior, and—as checked by many calculations—is well given by the 2b potentials. The preceding paragraph amounts to rephrasing the two observations quoted at the beginning of this section with the proviso that the blame for discrepancies with experiment cannot be due to the—now nearly perfect— 2b potentials. Hence the necessary “corrections” to H m must have a 3b origin. Given that we have no complaint with H M , the primary problem is the monopole contribution to the 3b potentials. This is welcome because a full 3b treatment would render most shell model calculations impossible, while the phenomenological study of monopole behavior is quite simple. Furthermore it is quite justified because there is little ab initio knowledge of the 3b potentials 6 . Therefore, whatever information that comes from nuclear data beyond A=3 is welcome. In Shell Model calculations, the interaction appears as matrix elements, that soon become far more numerous than the number of parameters defining the realistic potentials. Our task will consist in analyzing the Hamilto- nian in the oscillator representation (or Fock space) so as to understand its workings and simplify its form. In Sec- tion II.A we sketch the theory of effective interactions. In Section II.B it is explained how to construct from data a minimal H m , while Section II.C will be devoted to extract from realistic forces the most important contributions to H M . The basic tools are symmetry and scaling arguments, the clean separation of bulk and shell effects and the reduction of H to sums of factorable terms (Dufour and Zuker, 1996). 6 Experimentally there will never be enough data to determine them. The hope for a few parameter description comes from chiral perturbation theory (Entem and Machleidt, 2002). 8 A. Effective interactions The Hamiltonian is written in an oscillator basis as H = / + r≤s, t≤u, Γ 1 Γ rstu Z + rsΓ Z tuΓ + r≤s≤t, u≤v≤w, Γ 1 Γ rstuvw Z + rstΓ Z uvwΓ , (3) where / is the kinetic energy, 1 Γ rr ′ the interaction matrix elements, Z + rΓ ( Z rΓ ) create (annihilate) pairs (r ≡ rs) or triples (r ≡ rst) of particles in orbits r, coupled to Γ = JT. Dots stand for scalar products. The basis and the matrix elements are large but never infinite 7 . The aim of an effective interaction theory is to reduce the secular problem in the large space to a smaller model space by treating perturbatively the coupling between them, thereby transforming the full potential, and its repulsive short distance behavior, into a smooth pseudopotential. In what follows, if we have to distinguish between large (Nω) and model spaces we use H, /and 1 for the pseudo-potential in the former and H, K and V for the effective interaction in the latter. 1. Theory and calculations The general procedure to describe an exact eigenstate in a restricted space was obtained independently by Suzuki and Lee (1980) and Poves and Zuker (1981a) 8 . It consists in dividing the full space into model ([i)) and external ([α)) determinants, and introducing a transformation that respects strict orthogonality between the spaces and decouples them exactly: [ı) = [i) + α A iα [α) [α) = [α) − i A iα [i) (4) ¸ı[α) = 0, A iα is defined through ¸ı[H[α) = 0. (5) The idea is that the model space can produce one or several starting wavefunctions that can evolve to exact eigenstates through perturbative or coupled cluster evaluation of the amplitudes A iα , which can be viewed as matrix elements of a many body operator A. In coupled cluster theory (CCT or expS) (Coester and K¨ ummel (1960), K¨ ummel et al. (1978)) one sets A = exp S, where 7 Nowadays, the non-relativistic potentials must be thought of as derived from an effective field theory which has a cutoff of about 1 GeV (Entem and Machleidt, 2002). Therefore, truly hard cores are ruled out, and Hshould be understood to act on a sufficiently large vector space, not over the whole Hilbert space. 8 The notations in both papers are very different but the perturbative expansions are probably identical because the Hermitean formulation of Suzuki (1982) is identical to the one given in Poves and Zuker (1981a). This paper also deals extensively with the coupled cluster formalism. S = S 1 + S 2 + + S k is a sum of k-body (kb) operators. The decoupling condition ¸ı[H[α) = 0 then leads to a set of coupled integral equations for the S i amplitudes. When the model space reduces to a single determinant, setting S 1 = 0 leads to Hartree Fock theory if all other amplitudes are neglected. The S 2 approximation contains both low order Brueckner theory (LOBT) and the random phase approximation (RPA) . In the presence of hard-core potentials, the priority is to screen them through LOBT, and the matrix elements contributing to the RPA are discarded. An important implementation of the theory was due to Zabolitzky, whose calculations for 4 He, 16 O and 40 Ca, included S 3 (Bethe Fad- deev) and S 4 (Day Yacoubovsky) amplitudes (K¨ ummel, L¨ uhrmann, and Zabolitzky, 1978). This “Bochum truncation scheme” that retraces the history of nuclear matter theory has the drawback that at each level terms that one would like to keep are neglected. The way out of this problem (Heisenberg and Mi- haila, 2000; Mihaila and Heisenberg, 2000) consists on a new truncation scheme in which some approximations are made, but no terms are neglected, relying on the fact that the matrix elements are finite. The calculations of these authors for 16 O (up to S 3 ) can be ranked with the quasi-exact GFMC and NCSM ones for lighter nuclei. In the quasi-degenerate regime (many model states), the coupled cluster equations determine an effective interaction in the model space. The theory is much simplified if we enforce the decoupling condition for a single state whose exact wavefunction is written as [ref) = (1 +A 1 +A 2 +. . . )(1 +B 1 +B 2 +. . . )[ref) = (1 +C 1 +C 2 +. . . )[ref), C = exp S, (6) where [ref) is a model determinant. The internal amplitudes associated with the B operators are those of an eigenstate obtained by diagonalizing H eff = H(1 + A) in the model space. As it is always possible to eliminate the A 1 amplitude, at the S 2 level there is no coupling i.e., the effective interaction is a state independent G- matrix (Zuker, 1984), which has the advantage of providing an initialization for H eff . Going to the S 3 level would be very hard, and we examine what has become standard practice. The power of CCT is that it provides a unified framework for the two things we expect from decoupling: to smooth the repulsion and to incorporate long range correlations. The former demands jumps of say, 50ω, the latter much less. Therefore it is convenient to treat them separately. Standard practice assumes that G-matrix elements can provide a smooth pseudopotential in some sufficiently large space, and then accounts for long range correlations through perturbation theory. The equation to be solved is G ijkl = 1 ijkl − αβ 1 ijαβ G αβkl ǫ α +ǫ β −ǫ i −ǫ j + ∆ , (7) where ij and kl now stand for orbits in the model space while in the pair αβ at least one orbit is out of it, ǫ x 9 is an unperturbed (usually kinetic) energy, and ∆ a free parameter called the “starting” energy. Hjorth-Jensen, Kuo, and Osnes (1995) describe in detail a sophisticated partition that amounts to having two model spaces, one large and one small. In the NCSM calculations an Nω model space is chosen with N ≈ 6-10. When initiated by Zheng et al. (1993) the pseudo-potential was a G-matrix with starting energy. Then the ∆ dependence was eliminated either by arcane perturbative maneuvers, or by a truly interesting proposal: direct decoupling of 2b elements from a very large space (Navr´ atil and Barrett, 1996), further implemented by Navr´ atil, Vary, and Barrett (2000a), and extended to 3b effective forces (Navr´ atil et al., 2000), (Navr´ atil and Ormand, 2002). It would be of interest to compare the resulting effective interactions to the Brueckner and Bethe Faddeev amplitudes obtained in a full exp S approach (which are also free of arbitrary starting energies). A most valuable contribution of NCSM is the proof that it is possible to work with a pseudo-potential in Nω spaces. The method relies on exact diagonalizations. As they soon become prohibitive, in the future, CCT may become the standard approach: going to S 3 for N = 50 (as Heisenberg and Mihaila did) is hard. For N = 10 it should be much easier. Another important NCSM indication is that the excitation spectra converge well before the full energy, which validates formally the 0ω diagonalizations with rudimentary potentials 9 and second order corrections. The 0ω results are very good for the spectra. How- ever, to have a good pseudopotential to describe energies is not enough. The transition operators also need dressing. For some of them, notably E2, the dressing mechanism (coupling to 2ω quadrupole excitations) has been well understood for years (see Dufour and Zuker, 1996, for a detailed analysis), and yields the, abundantly tested and confirmed, recipe of using effective charges of ≈ 1.5e for the protons and ≈ 0.5e for neutrons. For the Gamow Teller (GT) transitions, mediated by the spin-isospin στ ± operator the renormalization mechanism involves an overall “quenching” factor of 0.7–0.8 whose origin is far subtler. It will be examined in Sections V.C.1 and V.C.2. The interested reader may wish to consult them right away. B. The monopole Hamiltonian A many-body theory usually starts by separating the Hamiltonian into an “unperturbed” and a “residual” part, H = H 0 + H r . The traditional approach consists in choosing for H 0 a one body (1b) single-particle field. 9 Even old potentials fit well the low energy NN phase shifts, the only that matter at 0ω. Since H contains two and three-body (2b and 3b) components, the separation is not mathematically clean. There- fore, we propose the following H = H m +H M , (8) where H m , the monopole Hamiltonian, contains / and all quadratic and cubic (2b and 3b) forms in the scalar products of fermion operators a † rx a sy 10 , while the multipole H M contains all the rest. Our plan is to concentrate on the 2b part, and introduce 3b elements as the need arises. H m has a diagonal part, H d m , written in terms of number and isospin operators (a † rx a ry ). It reproduces the average energies of configurations at fixed number of particles and isospin in each orbit (jt representation) or, alternatively, at fixed number of particles in each orbit and each fluid (neutron-proton, np, or j representation). In jt representation the centroids 1 T st = J 1 JT stst (2J + 1)[1 −(−) J+T δ st ] J (2J + 1)[1 −(−) J+T δ st ] , (9a) a st = 1 4 (31 1 st +1 0 st ), b st = 1 1 st −1 0 st , (9b) are associated to the 2b quadratics in number (m s ) and isospin operators (T s ), m st = 1 1 +δ st m s (m t −δ st ), (10a) T st = 1 1 +δ st (T s T t − 3 4 m st δ st ), (10b) to define the diagonal 2b part of the monopole Hamilto- nian (Bansal and French (1964), French (1969)) H d mjt = / d + s≤t (a st m st +b st T st ) +1 d3 m , (11) The 2b part is a standard result, easily extended to include the 3b term 1 d3 m = s t u (a stu m stu +b stu T stu ), (12) where m stu ≡ m st m u , or m s (m s − 1)(m s − 2)/6 and T stu ≡ m s T tu or (m s −2)T su . The extraction of the full H m is more complicated and it will be given in detail in Appendix B. Here we only need to note that H m is closed under unitary transformations of the underlying fermion operators, and hence, under spherical Hartree- Fock (HF) variation. This property explains the interest of the H m +H M separation. 10 r and s are subshells of the same parity and angular momentum, x and y stand for neutrons or protons 10 1. Bulk properties. Factorable forms H m must contain all the information necessary to produce the parameters of the Bethe Weiszäcker mass formula, and we start by extracting the bulk energy. The key step involves the reduction to a sum of factorable forms valid for any interaction (Dufour and Zuker, 1996). Its enormous power derives from the strong dominance of a single term in all the cases considered so far. For clarity we restrict attention to the full isoscalar centroids defined in Eqs. (B18a) and (B18c) 11 1 st = a st − 3δ st b st 4(4j s + 1) . (13) Diagonalize 1 st U −1 1 d m U = c =⇒1 st = k U sk U tk c k , ∴∴∴ (14) 1 d m = k c k s U sk m s t U tk m t − s 1 ss m s . (15) For the 3b interaction, the corresponding centroids 1 stu are treated as explained in (Dufour and Zuker, 1996, Appendix B1b): The st pairs are replaced by a single index x. Let L and M be the dimensions of the x and s arrays respectively. Construct and diagonalize an (L +M) (L +M) matrix whose non-zero elements are the rectangular matrices 1 xs and 1 sx . Disregarding the contractions, the strictly 3b part 1 (3) can then be written as a sum of factors 1 d3 m = k c (3) k s U (1) sk m s x≡tu U (2) tu,k m t m u . (16) Full factorization follows by applying Eqs. (14,15) to the U (2) tu,k matrices for each k. 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 U p , U s p U s U p FIG. 1 The Us terms in Eq. (17). Arbitrary scale. 11 They ensure the right average energies for T = 0 closed shells, which is not the case in Eq. (11) if one simply drops the bst terms. In Eq. (15) a single term strongly dominates all others. For the KLS interaction (Kahana et al., 1969b) and including the first eight major shells, the result in Fig. 1 is roughly approximated by U s ≈ 4. −0.5 (l −¸l)) + (j −l) _ D p , (17) where p is the principal quantum number of an oscillator shell with degeneracy D p = s (2j s +1) = (p+1)(p+2), ¸l) = l l(2l +1)/ l (2l +1). The (unitary) U matrices have been affected by an arbitrary factor 6 to have numbers of order unity. Operators of the form (D s = 2j s +1) ˆ Ω = s m s Ω s with s D s Ω s = 0 (18) vanish at closed shells and are responsible for shell effects. As l −¸l)) and j −l are of this type, only the the U p part contributes to the bulk energy. To proceed it is necessary to know how interactions depend on the oscillator frequency ω of the basis, related to the observed square radius (and hence to the density) through the estimate (Bohr and Mottelson, 1969, Eq. (2- 157)): ω (A) 1/3 = 35.59 ¸r 2 ) =⇒ω ≈ 40 A 1/3 MeV (19) A δ force scales as (ω) 3/2 . A 2b potential of short range is essentially linear in ω; for a 3b one we shall tentatively assume an (ω) 2 dependence while the Coulomb force goes exactly as (ω) 1/2 . To calculate the bulk energy of nuclear matter we average out subshell effects through uniform filling m s ⇒ m p D s /D p . Though the Ω-type operators vanish, we have kept them for reference in Eqs. (21,22) below. The latter is an educated guess for the 3b contribution. The eigenvalue c 0 for the dominant term in Eq. 15 is replaced by ω1 0 defined so as to have U p = 1. The subindex m is dropped throughout. Then, using Boole’s factorial powers, e.g., p (3) = p(p − 1)(p − 2), we obtain the following asymptotic estimates for the leading terms (i.e., 11 disregarding contractions) / d = ω 2 p m p (p + 3/2) =⇒ ω 4 (p f + 3) (3) (p f + 2) (20) 1 d ≈ ω1 0 _ p m p _ D p + ˆ Ω _ 2 =⇒ω1 0 [p f (p f + 4)] 2 , (21) 1 d3 ≈ (ω) 2 β1 0 _ p m p D p + ˆ Ω 1 __ p m p _ D p + ˆ Ω 2 _ 2 =⇒(ω) 2 β1 0 p 3 f (p f + 4) 2 . (22) Finally, relate p f to A : p m p = p f p=0 2(p + 1)(p + 2) =⇒A = 2(p f ) (3) 3 . (23) Note that in Eq. (22) we can replace (ω) 2 by (ω) 1+κ and change the powers of D p in the denominators accordingly. Assuming that non-diagonal / d and 1 d terms cancel we can vary with respect to ω to obtain the saturation energy E s = / d + (1 −βω κ p f )1 d , ∂E ∂ω = 0 ⇒ βω κ e p f = / d +1 d (κ + 1) 1 d ∴∴∴ E s = κ κ + 1 (/ d +1 d ) = ω e 4 κ κ + 1 (1 −41 0 ) _ 3 2 A _ 4/3 . (24) The correct saturation properties are obtained by fixing 1 0 so that E s /A ≈ 15.5 MeV. The ω e ≈ 40A −1/3 choice ensures the correct density. It is worth noting that the same approach leads to V C ≈ 3e 2 Z(Z−1)/5R c (Duflo and Zuker, 2002). It should be obvious that nuclear matter properties— derived from finite nuclei—could be calculated with techniques designed to treat finite nuclei: A successful theoretical mass table must necessarily extrapolate to the Bethe Weiszäcker formula (Duflo and Zuker, 1995). It may be surprising though, that such calculations could be conducted so easily in the oscillator basis. The merit goes to the separation and factorization properties of the forces. Clearly, Eq. (24) has no (or trivial) solution for κ = 0, i.e., without a 3b term. Though 2b forces do saturate, they do it at the wrong place and at a heavy price because their short-distance repulsion prevents direct HF variation. The crucial question is now: Can we use realistic (2+3)b potentials soft enough to do HF? Probably we can. The reason is that the nucleus is quite dilute, and nucleons only “see” the low energy part of the 2b potential involving basically s wave scattering, which has been traditionally well fitted by realistic potentials. Which in turn explains why primitive versions of such potentials give results close to the modern interactions for G-matrix elements calculated at reasonable ω values. This is a recurrent theme in this review (see especially Section II.D) and Fig. 2 provides an example of particular relevance for the study of shell effects in next Section II.B.2. Before we move on, we insist on what we have learned so far: It may be possible to describe nuclear structure with soft (2+3)b potentials consistent with the low energy data coming from the A = 2 and 3 systems. 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 1 2 3 4 5 6 7 8 9 10 11 U s s Bonn KLS FIG. 2 The KLS (Kahana et al., 1969b) shell effects in Fig. 1 for the first 10 orbits compared to those produced by a Bonn potential (Hjorth-Jensen, 1996) (to within an overall factor). Points in the same major shell are connected by lines and ordered by increasing j. 2. Shell formation Fig. 2 provides a direct reading of the expected single particle sequences produced by realistic interactions above T = 0 closed shells. Unfortunately, they do not square at all with what is seen experimentally. In particular, the splitting between spin-orbit partners appears to be nearly constant, instead of being proportional to l. As it could no longer be claimed that the 2b realistic forces are “wrong”, the solution must come from the 3b operators. The “educated guess” in Eq. (22) may prove of help in this respect but it has not been implemented yet. What we propose instead is to examine an existing, strict (1+2)b, model of H d m that goes a long way in explaining shell formation. Though we know that 3b ingredients are necessary, they may be often mocked by lower rank ones, in analogy with the kinetic energy, usually represented as 1b, when in reality it is a 2b operator, as explained in Appendix C. Something similar seems to happen with the spin-orbit force which must have a 3b origin, while a 1b picture is phenomenologically quite satisfactory, as our study will confirm. We shall also find out that some mechanisms have an irreducible 3b character. 12 H d m manifests itself directly in nuclear spectra through the doubly magic closures and the particle and hole states built on them. The members of this set, which we call cs ±1 are well represented by single determinants. Con- figuration mixing may alter somewhat the energies but, most often, enough experimental evidence exists to correct for the mixing. Therefore, a minimal characterization, ˜ H d m , can be achieved by demanding that it reproduce all the observed cs ±1 states. The task was undertaken by Duflo and Zuker (1999). With a half dozen parameters a fit to the 90 available data achieved an rmsd of some 220 keV. The differences in binding energies (gaps) 2BE(cs) −BE(cs +1) −BE(cs −1), not included in the fit, also came out quite well and provided a test of the reliability of the results. The model concentrates on single particle splittings which are of order A −1/3 . The valence spaces involve a number of particles of the order of their degeneracy, i.e., ≈ (p f + 3/2) 2 ≈ (3A/2) 2/3 according to Eq. (23). As a consequence, at midshell, the addition of single particle splittings generates shell effects of order A 1/3 . To avoid bulk contributions all expressions are written in terms of ˆ Ω-type operators [Eq. (18)] and the model is defined by ˜ H d m ≡ (W −4K) + ˆ Ω(ls) + ˆ Ω(ll) + ˆ Ω(ffi) + ˆ Ω(zni) + ˆ Ω(ffc) + ˆ Ω(znc), (25) Where: [I] W −4K = _ p m p _ D p _ 2 −2 p m p (p + 3/2). (26) comes from / d and 1 d . Both terms lead to the same gaps, and the chosen combination has the advantage of canceling exactly to orders A and A 2/3 . [II] ˆ Ω(ls) ≈ −22 ls/A 2/3 (very much the value in (Bohr and Mottelson, 1969, Eq. (2-132)) and ˆ Ω(ll) ≈ −22[l(l + 1)−p(p+3)/2]/A 2/3 (in MeV), are ˆ Ω-type operators that reproduce the single-particle spectra above HO closed shells. [III] ˆ Ω(ffi), ˆ Ω(zni), ˆ Ω(ffc) and ˆ Ω(znc) 12 are ˆ Ω-type 2b operators whose effect is illustrated in Fig. 3: In going from 40 Ca to 48 Ca the filling of the f 7/2 (f for short) neutron orbit modifies the spectra in 49 Ca, 49 Sc, 47 Ca and 47 K. As mentioned, realistic 2b forces will miss the particle spectrum in 41 Ca. Once it is phenomenologically corrected by ls + ll such forces account reasonably well for the zni and eei mechanism, within one major exception: they fail to depress the f 7/2 orbit with respect to the others. As a consequence, and quite generally, they fail 12 ff stands for “same fluid”, zn for “proton-neutron”, i for “intrashell” and c for “cross-shell” 41 Sc 41 Ca p p f d 1/2 3/2 7/2 3/2 s 1/2 d 5/2 Ca Sc 49 49 Ca 47 47 K π ν or π ν f 5/2 ffc zni ffi 39 K Ca 39 znc FIG. 3 Evolution of cs ±1 states from 40 Ca to 48 Ca to produce the EI closures. Though zni and eei can be made to cure this serious defect, it is not clear how well they do it, since the correct mechanism necessarily involves 3b terms, as will be demonstrated in Section V.A. Another problem that demands a three body mechanism is associated with the znc and ffc operators which always depress orbits of larger l [as in the hole spectra of 47 Ca and 47 K in Fig. 3], which is not the case for light nuclei as seen from the spectra of 13 C and 29 Si. Only 3b operators can account for this inversion. In spite of these caveats, ˜ H d m should provide a plausible view of shell formation, illustrated in Figure 4 for t = -180 -160 -140 -120 -100 -80 -60 -40 -20 0 0 10 20 30 40 50 60 E ( t = 0 , M e v ) N W-4K W-4K+ll+ls Hm FIG. 4 The different contributions to ˜ H d m for N=Z nuclei 13 -160 -140 -120 -100 -80 -60 -40 10 20 30 40 50 60 E ( t = 0 . . . 8 , M e v ) Z Hm0 Hm2 Hm4 Hm6 Hm8 -160 -150 -140 -130 -120 -110 -100 -90 -80 10 20 30 40 50 60 E ( t = 1 0 . . . 1 8 , M e v ) Z Hm10 Hm12 Hm14 Hm16 Hm18 FIG. 5 ˜ H d m for even t = N −Z = 10 . . . 18. N−Z = 0 nuclei. ˜ H d m is calculated by filling the oscillator orbits in the order dictated by ls + ll. The W − 4K term produces enormous HO closures (showing as spikes). They are practically erased by the 1b contributions, and it takes the 2b ones to generate the EI 13 spikes. Note that the HO magicity is strongly attenuated but persists. The drift toward smaller binding energies that goes as A 1/3 is an artifact of the W −4K choice and should be ignored. In Figs. 5 we show the situation for even t = N−Z = 0 to 18. Plotting along lines of constant t has the advantage of detecting magicity for both fluids. In other words, the spikes appear when either fluid is closed, and are reinforced when both are closed. The spikes are invariably associated either to the HO magic numbers (8, 20, 40) or to the EI ones (14, 28, 50), but the latter always show magicity while the former only do it at and above the double closures (Z, N)=(8,14),(20,28) and (40,50). For example, at Z = 20 40 Ca shows as a weak closure. 42, 44, 46 Ca are not closed (which agrees with experiment), 48 Ca is definitely magic, and spikes persist for the heavier isotopes. At Z = 40, 80 Zr shows a nice spike, a bad pre- 13 Remember: EI valence spaces consist in the HO orbits shell p, except the largest (extruder), plus the largest (intruder) orbit from shell p + 1. diction for a strongly rotational nucleus. However, there are no (or weak) spikes for the heavier isotopes except at 90 Zr and 96 Zr, both definitely doubly magic. There are many other interesting cases, and the general trend is the following: No known closure fails to be detected. Con- versely, not all predicted closures are real. They may be erased by deformation—as in the case of 80 Zr—but this seldom happens, thus suggesting fairly reliable predictive power for ˜ H d m . C. The multipole Hamiltonian The multipole Hamiltonian is defined as H M =H-H m . As we are no longer interested in the full H, but its restriction to a finite space, H M will be more modestly called H M , with monopole-free matrix elements given by W JT rstu = V JT rstu −δ rs δ tu V T rs . (27) We shall describe succinctly the main results of Du- four and Zuker (1996), insisting on points that were not stressed sufficiently in that paper (Section II.C.1), and adding some new information (Section II.D). There are two standard ways of writing H M : H M = r≤s,t≤u,Γ W Γ rstu Z † rsΓ Z tuΓ , or (28) H M = rstuΓ [γ] 1/2 f γ rtsu (S γ rt S γ su ) 0 , (29) where f γ rtsu = ω γ rtsu _ (1 +δ rs )(1 +δ tu )/4. The matrix elements ω γ rtsu and W Γ rstu are related through Eqs. (A9) and (A10). Replacing pairs by single indices rs ≡ x, tu ≡ y in eq. (28) and rt ≡ a, su ≡ b in eq. (29), we proceed as in Eqs. (14) and (15) to bring the matrices W Γ xy ≡ W Γ rstu and f γ ab ≡ f γ rtsu , to diagonal form through unitary transformations U Γ xk , u γ ak to obtain factorable expressions H M = k,Γ E Γ k x U Γ xk Z † xΓ y U Γ yk Z yΓ , (30) H M = k,γ e γ k _ a u γ ak S γ a b u γ bk S γ b _ 0 [γ] 1/2 , (31) which we call the E (or normal or particle-particle or pp) and e (or multipole or particle-hole or ph) representations. Since H m contains all the γ = 00 and 01 terms, for H M , ω 00 rstu = ω 01 rstu = 0 (see Eq. (B11). There are no one body contractions in the e representation because they are all proportional to ω 0τ rstu . The eigensolutions in eqs. (30) and (31) using the KLS interaction (Kahana et al., 1969b; Lee, 1969), yield the density of eigenvalues (their number in a given interval) 14 0 2000 -10 -8 -6 -4 -2 0 2 4 n u m b e r o f s t a t e s 2 0 + 0 1 + 0 0 - 1 0 - 2 0 - 1 0 + Energy (MeV) E-eigenvalue density FIG. 6 E-eigenvalue density for the KLS interaction in the pf+sdg major shells ω = 9. Each eigenvalue has multiplicity [Γ]. The largest ones are shown by arrows. in the E representation that is shown in Fig. 6 for a typical two-shell case. It is skewed, with a tail at negative energies which is what we expect from an attractive interaction. The e eigenvalues are plotted in fig. 7. They are very symmetrically distributed around a narrowcentral group, but a few of them are neatly detached. The strongest have γ π = 1 − 0, 1 + 1, 2 + 0, 3 − 0, 4 + 0. “If the corresponding eigenvectors are eliminated from H in” eq. (31) and the associated H in eq. (30) is recalculated 14 , the E distribution becomes quite symmetric, as expected for a random interaction. If the diagonalizations are restricted to one major shell, negative parity peaks are absent, but for the positive parity ones the results are practically identical to those of Figs. 6 and 7, except that the energies are halved. This point is crucial: If u p1 and u p2 are the eigenvectors obtained in shells p 1 and p 2 , their eigenvalues are approximately equal e p1 ≈ e p2 = e. When diagonalizing in p 1 +p 2 , the unnormalized eigenvector turns out to be u p1 +u p2 with eigenvalue e. In the figures the eigenvalues for the two shell case are doubled, because they are associated with normalized eigenvectors. To make a long story short: The contribution to H M associated to the large Γ = 01, and γ = 20 14 Inverted commas are meant to call attention on an erratum in (Dufour and Zuker, 1996). 0 5000 -4 -3 -2 -1 0 1 2 3 4 5 n u m b e r o f s t a t e s e-eigenvalue density 2 0 + 4 0 + 1 0 + 3 0 - 1 1 + 1 0 - Energy (MeV) FIG. 7 e-eigenvalue density for the KLS interaction in the pf+sdg major shells. Each eigenvalue has multiplicity [γ]. The largest ones are shown by arrows. terms, H¯ P = − ω ω 0 [E 01 [(P † p +P † p+1 ) (P p +P p+1 ) (32) H ¯ q = − ω ω 0 [e 20 [(¯ q p + ¯ q p+1 ) (¯ q p + ¯ q p+1 ), (33) turns out to be the usual pairing plus quadrupole Hamil- tonians, except that the operators for each major shell of principal quantum number p are affected by a normalization. E 01 and e 20 are the one shell values called generically e in the discussion above. To be precise ¯ P † p = r∈p Z † rr01 Ω 1/2 r /Ω 1/2 p , (34) ¯ q p = rs∈p S 20 rs q rs /^ p , (35) where Ω r = j r + 1, q rs = _ 1 5 ¸r|r 2 Y 2 |s) (36) and Ω p = 1 2 D p , ^ 2 p = Σq 2 rs ∼ = 5 32π (p + 3/2) 4 (37) 1. Collapse avoided The pairing plus quadrupole (P +Q) model has a long and glorious history (Baranger and Kumar, 1968; Bes and Sorensen, 1969), and one big problem: as more shells are added to a space, the energy grows, eventually leading to collapse. The only solution was to stay within limited spaces, but then the coupling constants had to be 15 readjusted on a case by case basis. The normalized versions of the operators presented above are affected by universal coupling constants that do not change with the number of shells. Knowing that ω 0 = 9 MeV, they are [E 01 [/ω 0 = g ′ = 0.32 and [e 20 [/ω 0 = κ ′ = 0.216 in Eqs. (32) and (33). Introducing A mf ≈ 2 3 (p f + 3/2) 3 , the total number of particles at the middle of the Fermi shell p f , the relationship between g ′ , κ ′ , and their conventional counterparts (Baranger and Kumar, 1968) is, for one shell 15 0.32ω Ω p ∼ = 19.51 A 1/3 A 2/3 mf = G ≡ G 0 A −1 , 0.216ω ^ 2 p ∼ = 1 2 216 A 1/3 A 4/3 mf = χ ′ 2 ≡ χ ′ 0 2 A −5/3 . (38) To see how collapse occurs assume m = O(D f ) = O(A 2/3 ) in the Fermi shell, and promote them to a higher shell of degeneracy D. The corresponding pairing and quadrupole energies can be estimated as E P = −[G[4m(D −m+ 2) = −[G[O(mD), (39) and E q ≈ −[χ ′ [Q 2 0 = −[χ ′ [O(m 2 D), (40) respectively, which become for the two possible scalings E P (old) = O( mD A ) =⇒E P (new) = O( m A 1/3 ) (41) E q (old) = O( m 2 D A 5/3 ) =⇒E q (new) = O( m 2 A 1/3 D ). (42) If the m particles stay at the Fermi shell, all energies go as A 1/3 as they should. If D grows both energies grow in the old version. For sufficiently large D the gain will become larger than the monopole loss O(mD 1/2 ω) = O(D 1/2 A 1/3 ). Therefore the traditional forces lead the system to collapse. In the new form there is no collapse: E P stays constant, E q decreases and the monopole term provides the restoring force that guarantees that particles will remain predominantly in the Fermi shell. As a model for H M , P +Q is likely to be a reasonable first approximation for many studies, provided it is supplemented by a reasonable H m , to produce the P +Q+m model. D. Universality of the realistic interactions To compare sets of matrix elements we define the overlaps O AB = V A V B = rstuΓ V Γ rstuA V Γ rstuB (43) O AB = O AB √ O AA O BB . (44) 15 According to Dufour and Zuker (1996) the bare coupling constants, should be renormalized to 0.32(1+0.48) and 0.216(1+0.3) Similarly, O T AB is the overlap for matrix elements with the same T. O AA is a measure of the strength of an interaction. If the interaction is referred to its centroid (and a fortiori if it is monopole-free), O AA ≡ σ 2 A . The upper TABLE I The overlaps between the KLS(A) and Bonn(B) interactions for the first ten oscillator shells (2308 matrix elements), followed by those for the pf shell (195 matrix elements) for the BonnC (C) and KB (K) interactions. O 0 AA = 17.56 O 1 AA = 2.61 OAA = 6.49 O 0 BB = 11.84 O 1 BB = 2.31 OBB = 4.78 O 0 AB = 0.98 O 1 AB = 0.99 OAB = 0.98 O 0 CC = 3.71 O 1 CC = 0.56 OCC = 1.41 O 0 KK = 3.02 O 1 KK = 0.46 OKK = 1.15 O 0 CK = 0.99 O 1 CK = 0.97 OCK = 0.99 set of numbers in Table I contains what is probably the most important single result concerning the interactions: a 1969 realistic potential (Kahana et al. (1969b); Lee (1969)) and a modern Bonn one (Hjorth-Jensen, 1996) differ in total strength(s) but the normalized O(AB) are very close to unity. Now: all the modern realistic potentials agree closely with one another in their predictions except for the binding energies. For nuclear matter the differences are substantial (Pieper et al., 2001), and for the BonnA,B,C potentials studied by Hjorth-Jensen et al. (1995), they become enormous. However, the matrix elements given in this reference for the pf shell have normalized cross overlaps of better than 0.998. At the moment, the overall strengths must be viewed as free parameters. At a fundamental level the discrepancies in total strength stem for the degree of non-locality in the potentials and the treatment of the tensor force. In the old interactions, uncertainties were also due to the starting energies and the renormalization processes, which, again, affect mainly the total strength(s), as can be gathered from the bottom part of Table I. The dominant terms of H M are central and table II collects their strengths (in MeV) for effective interactions in the pf-shell: Kuo and Brown (1968) (KB), the potential fit of Richter et al. (1991) (FPD6), the Gogny force—successfully used in countless mean field studies— (Dechargé and Gogny, 1980) and BonnC (Hjorth-Jensen et al., 1995). There is not much to choose between the different forces which is a nice indication that overall nuclear data (used in the FPD6 and Gogny fits) are consistent with the NN data used in the realistic KB and BonnC G-matrices. The splendid performance of Gogny deserves further study. The only qualm is with the weak quadrupole strength of KB, which can be understood by looking again at the bottom part of Table I. In assessing the JT = 01 and JT = 10 pairing terms, it should be borne in mind that their renormalization remains an 16 open question (Dufour and Zuker, 1996). Conclusion: Some fine tuning may be in order and 3b forces may (one day) bring some multipole news, but as of now the problem is H m , not H M . TABLE II Leading terms of the multipole Hamiltonian Interaction particle-particle particle-hole JT=01 JT=10 λτ=20 λτ=40 λτ=11 KB -4.75 -4.46 -2.79 -1.39 +2.46 FPD6 -5.06 -5.08 -3.11 -1.67 +3.17 GOGNY -4.07 -5.74 -3.23 -1.77 +2.46 BonnC -4.20 -5.60 -3.33 -1.29 +2.70 III. THE SOLUTION OF THE SECULAR PROBLEM IN A FINITE SPACE Once the interaction and the valence space ready, it is time to construct and diagonalize the many body secular matrix of the Hamiltonian describing the (effective) interaction between valence particles in the valence space. Two questions need now consideration: which basis to take to calculate the non-zero many-body matrix elements (NZME) and which method to use for the diagonalization of the matrix. A. The Lanczos method FIG. 8 m-scheme dimensions and total number of non-zero matrix elements in the pf-shell for nuclei with M = Tz = 0 In the standard diagonalization methods (Wilkinson, 1965) the CPU time increases as N 3 , N being the dimension of the matrix. Therefore, they cannot be used in large scale shell model (SM) calculations. Nuclear SM calculations have two specific features. The first one is that, in the vast majority of the cases, only a few (and very often only one) eigenstates of a given angular momentum (J) and isospin (T) are needed. Secondly, the matrices are very sparse. As can be seen in figure 8, the number of NZME varies linearly instead of quadratically with the size of the matrices. For these reasons, iterative methods are of general use, in particular the Lanczos method (Lanczos, 1950). As an alternative, the Davidson method (Davidson, 1975) has the advantage of avoiding the storage of a large number of vectors, however, the large increase in storage capacity of modern computers has somehow minimized these advantages. The Lanczos method consists in the construction of an orthogonal basis in which the Hamiltonian matrix (H) is tridiagonal. A normalized starting vector (the pivot) [1) is chosen. The vector [a 1 ) = H[1) has necessarily the form [a 1 ) = H 11 [1) +[2 ′ ), with ¸1[2 ′ ) = 0. (45) Calculate H 11 = ¸1[H[1) through ¸1[a 1 ) = H 11 . Nor- malize [2) = [2 ′ )/¸2 ′ [2 ′ ) 1/2 to find H 12 = ¸1[H[2) = ¸2 ′ [2 ′ ) 1/2 . Iterate until state [k) has been found. The vector [a k ) = H[k) has necessarily the form [a k ) = H k k−1 [k −1) +H k k [k) +[(k + 1) ′ ). (46) Calculate ¸k[a k ) = H k k . Now [(k + 1) ′ ) is known. Nor- malize it to find H k k+1 = ¸(k + 1) ′ [(k + 1) ′ ) 1/2 . The (real symmetric) matrix is diagonalized at each iteration and the iterative process continues until all the required eigenvalues are converged according to some criteria. The number of iterations depends little on the dimension of the matrix. Besides, the computing time is directly proportional to the number of NZME and for this reason it is nearly linear (instead of cubic as in the standard methods) in the dimension of the matrix. It depends on the number of iterations, which in turn depends on the number of converged states needed, and also on the choice of the starting vector. The Lanczos method can be used also as a projection method. For example in the m-scheme basis, a Lanc- zos calculation with the operator J 2 will generate states with well defined J. Taking these states as starting vectors, the Lanczos procedure (with H) will remain inside a fixed J subspace and therefore will improve the convergence properties. When only one converged state is required, it is convenient to use as pivot state the solution obtained in a previous truncated calculation. For instance, starting with a random pivot the calculation of the ground state of 50 Cr in the full pf space (dimension in m-scheme 14,625,540 Slater determinants) needs twice more iterations than if we start with the pivot obtained as solution in a model space in which only 4 particles are allowed outside the 1f 7/2 shell (dimension 1,856,720). The overlap between these two 0 + states is 0.985. When more eigenstates are needed (n c > 1), the best choice for the pivot is a linear combination of the n c lower states in the truncated space. Even if the Lanczos method is very efficient in the SM framework, there may be numerical problems. Mathematically the Lanczos vectors 17 are orthogonal, however numerically this is not strictly true due to the limited floating point machine precision. Hence, small numerical precision errors can, after many iterations, produce catastrophes, in particular, the states of lowest energy may reappear many times, rendering the method inefficient. To solve this problem it is necessary to orthogonalize each new Lanczos vector to all the preceding ones. The same precision defects can produce the appearance of non expected states. For example in a m- scheme calculation with a J = 4, M = 0 pivot, when many iteration are performed, it may happen that one J = 0 and even one J = 2 state, lower in energy than the J = 4 states, show up abruptly. This specific problem can be solved by projecting on J each new Lanczos vector. B. The choice of the basis Given a valence space, the optimal choice of the basis is related to the physics of the particular problem to be solved. As we discuss later, depending on what states or properties we want to describe (ground state, yrast band, strength function,. . . ) and depending on the type of nucleus (deformed, spherical,. . . ) different choices of the basis are favored. There are essentially three possibilities depending on the underlying symmetries: m-scheme, J coupled scheme, and JT coupled scheme. TABLE III Some dimensions in the pf shell A 4 8 12 16 20 M = Tz = 0 4000 2 ×10 6 1.10 ×10 8 1.09 ×10 9 2.29 ×10 9 J = Tz = 0 156 41355 1.78 ×10 6 1.54 ×10 7 3.13 ×10 7 J = T = 0 66 9741 3.32 ×10 5 2.58 ×10 6 5.05 ×10 6 As the m-scheme basis consists of Slater Determinants (SD) the calculation of the NZME is trivial, since they are equal to the decoupled two body matrix element (TBME) up to a phase. This means that, independently of the size of the matrix, the number of possible values of NZME is relatively limited. However, the counterpart of the simplicity of the m-scheme is that only J z and T z are good quantum numbers, therefore all the possible (J, T) states are contained in the basis and as a consequence the dimensions of the matrices are maximal. For a given number of valence neutrons, n v , and protons, z v the number of different Slater determinants that can be built in the valence space is : d = _ D n n v _ _ D p z v _ (47) where D n and D p are the degeneracies of the neutron and proton valence spaces. Working at fixed M and T z the bases are smaller (d = M,Tz d (M, T z )). The J or JT coupled bases, split the full m-scheme matrix in boxes whose dimensions are much smaller. This is especially spectacular for the J = T = 0 states (see table III). It is often convenient to truncate the space. In the particular case of the pf shell, calling f the largest subshell (f 7/2 ), and r, generically, any or all of the other subshells of the p = 3 shell, the possible t-truncations involve the spaces f m−m0 r m0 +f m−m0−1 r m0+1 + +f m−m0−t r m0+t , (48) where m 0 ,= 0 if more than 8 neutrons (or protons) are present. For t = m − m 0 we have the full space (pf) m for A = 40 +m. In the late 60’s, the Rochester group developed the algorithms needed for an efficient work in the (J, T) coupled basis and implemented them in the Oak-Ridge Rochester Multi-Shell code (French et al., 1969). The structure of the calculation is as follows: In the first place, the states of n i particles in a given j i shell are defined: [γ i ) = [(j i ) ni v i J i x i ) (v i is the seniority and x i any extra quantum number). Next, the states of N particles distributed in several shells are obtained by successive angular momentum couplings of the one-shell basic states: _ _ [[γ 1 )[γ 2 )] Γ2 [γ 3 ) _ Γ3 . . . [γ k ) _ Γ k . (49) Compared to the simplicity of the m-scheme, the calculation of the NZME is much more complicated. It involves products of 9j symbols and coefficients of fractional parentage (cfp), i.e., the single-shell reduced matrix elements of operators of the form (a † j1 a † j2 ) λ , (a † j1 a j2 ) λ , a † j1 , ((a † j1 a † j2 ) λ a j3 ) j4 . (50) This complexity explains why in the OXBASH code (Brown et al., 1985), the JT-coupled basis states are written in the m-scheme basis to calculate the NZME. The Oxford-Buenos Aires shell model code, widely distributed and used has proven to be an invaluable tool in many calculations. Another recent code that works in JT-coupled formalism is the Drexel University DUPSM (Novoselsky and Vallières, 1997) has had a much lesser use. In the case of only J (without T) coupling, a strong simplification in the calculation of the NZME can be achieved using the Quasi-Spin formalism (Ichimura and Arima, 1966; Kawarada and Arima, 1964; Lawson and Macfarlane, 1965), as in the code NATHAN described below. The advantages of the coupled scheme decrease also when J and T increase. As an example, in 56 Ni the ratio dim(M = J)/dim(J) is 70 for J = 0 but only 5.7 for J = 6. The coupled scheme has another disadvantage compared to the m-scheme. It concerns the percentage of NZME. Let us give the example of the state 4 + in 50 Ti (full pf space); the percentages of NZME are respectively 14% in the JT-basis (see Novoselsky et al., 1997), 5% in the J-basis and only 0.05% in m-scheme. For all these 18 reasons and with the present computing facilities, we can conclude that the m-scheme is the most efficient choice for usual SM calculations, albeit with some notable exceptions that we shall mention below. C. The Glasgow m-scheme code The steady and rapid increase of computer power in recent years has resulted in a dramatic increase in the dimensionality of the SM calculations. A crucial point nowadays is to know what are the limits of a given computer code, their origin and their evolution. As far as the NZME can be calculated and stored, the diagonalization with the Lanczos method is trivial. It means that the fundamental limitation of standard shell model calculations is the capacity to store the NZME. This is the origin of the term “giant” matrices, that we apply to those for which it is necessary to recalculate the NZME during the diagonalization process. The first breakthrough in the treatment of giant matrices was the SM code developed by the Glasgow group (Whitehead et al., 1977). Let us recall its basic ideas: It works in the m-scheme and each SD is represented in the computer by an integer word. Each bit of the word is associated to a given individual state [nljmτ). Each bit has the value 1 or 0 depending on whether the state is occupied or empty. A two-body operator a † i a † j a k a l will select the words having the bits i, j, k, l in the configuration 0011, say, and change them to 1100, generating a new word which has to be located in the list of all the words using the bi-section method. D. The m-scheme code ANTOINE The shell model code ANTOINE 16 (Caurier and Nowacki, 1999) has retained many of these ideas, while improving upon the Glasgow code in the several aspects. To start with, it takes advantage of the fact that the dimension of the proton and neutron spaces is small compared with the full space dimension, with the obvious exception of semi-magic nuclei. For example, the 1,963,461 SD with M = 0 in 48 Cr are generated with only the 4,865 SD (corresponding to all the possible M values) in 44 Ca. The states of the basis are written as the product of two SD, one for protons and one for neutrons: [I) = [i, α). We use, I, J, capital letters for states in the full space, i, j, small case Latin letters for states of the first subspace (protons or neutrons), α, β, small case Greek letters for states of the second subspace (neutrons or protons). The Slater determinants i and α can be classified by their M values, M 1 and M 2 . The total M being fixed, the SD’s of 16 A version of the code can be downloaded from the URL http: //sbgat194.in2p3.fr/~theory/antoine/main.html the 2 subspaces will be associated only if M 1 +M 2 = M. A pictorial example is given in fig. 9. a + m a n 7 8 9 10 1 2 3 4 5 6 7 8 10 11 12 13 14 −M−1 i α . . . J zn . . . J zp −M M M+2 M+1 −M−2 a + a µ λ 15 1 2 3 4 5 6 9 11 FIG. 9 Schematic representation of the basis It is clear that for each [i) state the allowed [α) states run, without discontinuity, between a minimum and a maximum value. Building the basis in the total space by means of an i-loop and a nested α-loop, it is possible to construct numerically an array R(i) that points to the [I) state 17 : I = R(i) +α (51) For example, according to figure 9, the numerical values of R are: R(1) = 0, R(2) = 6, R(3) = 12,. . . The relation (51) holds even in the case of truncated spaces, provided we define sub-blocks labeled with M and t (the truncation index defined in (48)). Before the diagonalization, all the calculations that involve only the proton or the neutron spaces separately are carried out and the results stored. For the proton-proton and neutronneutron NZME the numerical values of R(i), R(j), W ij and α, β, W αβ , where ¸i[H[j) = W ij and ¸α[H[β) = W αβ , are pre-calculated and stored. Therefore, in the Lanc- zos procedure a simple loop on α and i generates all the proton-proton and neutron-neutron NZME, W I,J = ¸I[H[J). For the proton-neutron matrix elements the situation is slightly more complicated. Let’s assume that the [i) and [j) SD are connected by the one-body operator a † q a r (that in the list of all possible one-body operators appears at position p), with q = nljm and r = n ′ l ′ j ′ m ′ and m ′ − m = ∆m. Equivalently, the [α) and [β) SD are connected by a one-body operator whose position is denoted by µ. We pre-calculate the numerical values of R(i), R(j), p and α, β, µ. Conservation of the total M implies that the proton operators with ∆m must be associated to the neutron operators with −∆m. Thus we could draw the equivalent to figure 9 for the proton and neutron one-body operators. In the same way as we did before for I = R(i) + α, we can now define an index K = Q(p) + µ that labels the different two-body matrix 17 We use Dirac’s notation for the quantum mechanical state |i, the position of this state in the basis is denoted by i. 19 elements (TBME). Then, we denote V (K) the numerical value of the proton-neutron TBME that connects the states [i, α) and [j, β). Once R(i), R(j), Q(p) and α, β, µ are known, the non-zero elements of the matrix in the full space are generated with three integer additions: I = R(i) +α, J = R(j) +β, K = Q(p) +µ. (52) The NZME of the Hamiltonian between states [I) and [J) is then: ¸I[H[J) = ¸J[H[I) = V (K). (53) The performance of the code is optimal when the two subspaces have comparable dimensions. It becomes less efficient for asymmetric nuclei (for semi-magic nuclei all the NZME must be stored) and for large truncated spaces. No-core calculations are typical in this respect. If we consider a N = Z nucleus like 6 Li, in a valence space that comprises up to 14ω configurations, there are 50000 Slater determinants for protons and neutrons with M = 1/2 and t = 14. Their respective counterparts can only have M = −1/2 and t = 0 and have dimensionality 1. This situation is the same as in semi-magic nuclei. As far as two Lanczos vectors can be stored in the RAM memory of the computer, the calculations are straightforward. Until recently this was the fundamental limitation of the SM code ANTOINE. It is now possible to overcome it dividing the Lanczos vectors in segments: Ψ f = k Ψ (k) f . (54) The Hamiltonian matrix is also divided in blocks so that the action of the Hamiltonian during a Lanczos iteration can be expressed as: Ψ (k) f = q H (q,k) Ψ (q) i (55) The k segments correspond to specific values of M (and occasionally t) of the first subspace. The price to pay for the increase in size is a strong reduction of performance of the code. Now, ¸I[H[J) and ¸J[H[I) are not generated simultaneously when [I) and [J) do not belong at the same k segment of the vector and the amount of disk use increases. As a counterpart, it gives a natural way to parallelize the code, each processor calculating some specific Ψ (k) f . This technique allows the diagonalization of matrices with dimensions in the billion range. All the nuclei of the pf-shell can now be calculated without truncation. Other m-scheme codes have recently joined ANTOINE in the run, incorporating some of its algorithmic findings as MSHELL (Mizusaki, 2000) or REDSTICK (Ormand and Johnson, 2002). E. The coupled code NATHAN. As the mass of the nucleus increases, the possibilities of performing SM calculations become more and more limited. To give an order of magnitude the dimension of the matrix of 52 Fe with 6 protons and 6 neutrons in the pf shell is 10 8 . If now we consider 132 Ba with also 6 protons and 6 neutron-holes in a valence space of 5 shells, four of them equivalent to the pf shell plus the 1h 11/2 orbit, the dimension reaches 210 10 . Furthermore, when many protons and neutrons are active, nuclei tend to get deformed and their description requires even larger valence spaces. For instance, to treat the deformed nuclei in the vicinity of 80 Zr, the ”normal” valence space, 2p 3/2 , 1f 5/2 , 2p 1/2 , 1g 9/2 , must be complemented with, at least, the orbit 1d 5/2 . This means that our span will be limited to nuclei with few particles outside closed shells ( 130 Xe remains an easy calculation) or to almost semi-magic, as for instance the Tellurium isotopes. These nuclei are spherical, therefore the seniority can provide a good truncation scheme. This explains the interest of a shell model code in a coupled basis and quasi-spin formalism. In the shell model code NATHAN (Caurier and Nowacki, 1999) the fundamental idea of the code AN- TOINE is kept, i. e. splitting the valence space in two parts and writing the full space basis as the product of states belonging to these two parts. Now, [i) and [α) are states with good angular momentum. They are built with the usual techniques of the Oak-Ridge/Rochester group (French et al., 1969). Each subspace is now partitioned with the labels J 1 and J 2 . The only difference with the m-scheme is that instead of having a one-to-one association (M 1 +M 2 = M), for a given J 1 we now have all the possible J 2 , J min ≤ J 2 ≤ J max , with J min = [J 0 − J 1 [ and J max = J 0 + J 1 . The continuity between the first state with J min and the last with J max is maintained and consequently the fundamental relation I = R(i) +α still holds. The generation of the proton-proton and neutronneutron NZME proceeds exactly as in m-scheme. For the proton-neutron NZME the one-body operators in each space can be written as O λ p = (a † j1 a j2 ) λ . There exists a strict analogy between ∆m in m-scheme and λ in the coupled scheme. Hence, we can still establish a relation K = Q(p) +ω. The NZME read now: ¸I[H[J) = ¸J[H[I) = h ij h αβ W(K), (56) with h ij = ¸i[O λ p [j), h αβ = ¸α[O λ ω [β), and W(K) ∝ V (K) _ ¸ _ ¸ _ i α J j β J λ λ 0 _ ¸ _ ¸ _ , (57) where V (K) is a TBME. We need to perform –as in the m-scheme code– the three integer additions which generate I, J, and K, but, in addition, there are two floating point multiplications to be done, since h ij and h αβ , which in m-scheme were just phases, are now a product of cfp’s and 9j symbols (see formula 3.10 in French et al. (1969)). Within this formalism we can introduce seniority truncations, but the problem of the semi-magic nuclei and the asymmetry between the protons and neutrons spaces remains. To overcome this difficulty in equalizing the 20 dimension of the two subspaces we have generalized the code as to allow that one of the two subspaces contains a mixing of proton and neutron orbits. For example, for heavy Scandium or Titanium isotopes, we can include the neutron 1f 7/2 orbit in the proton space. For the N = 126 isotones we have only proton shells. We then take the 1i 13/2 and 1h 9/2 orbits as the first subspace, while the 2f 7/2 , 2f 5/2 , 3p 3/2 and 3p 1/2 form the second one. The states previously defined with J p and J n are now respectively labelled A 1 , J 1 and A 2 , J 2 , A 1 and A 2 being the number of particles in each subspace. In h ij and h αβ now appear mean values of all the operators in Eq.(50) The fact that in the coupled scheme the dimensions are smaller and that angular momentum is explicitly enforced makes it possible to perform a very large number of Lanczos iterations without storage or precision problems. This can be essential for the calculation of strength functions or when many converged states are needed, for example to describe non-yrast deformed bands. The coupled code has another important advantage. It can be perfectly parallelized without restriction on the number of processors. The typical working dimension for large matrices in NATHAN is 10 7 (10 9 with ANTOINE). It means that there is no problem to define as many final vectors as processors are available. The calculation of the NZME is shared between the different processors (each processor taking a piece of the Hamiltonian, H = k H (k) ) leading to different vectors that are added to obtain the full one: Ψ (k) f = H (k) Ψ i , Ψ f = k Ψ (k) i . (58) F. No core shell model The ab initio no-core shell model (NCSM) (Navr´ atil et al., 2000a,b) is a method to solve the nuclear many body problem for light nuclei using realistic inter-nucleon forces. The calculations are performed using a large but finite harmonic-oscillator (HO) basis. Due to the basis truncation, it is necessary to derive an effective interaction from the underlying inter-nucleon interaction that is appropriate for the basis size employed. The effective interaction contains, in general, up to A-body components even if the underlying interaction had, e.g. only two-body terms. In practice, the effective interaction is derived in a sub-cluster approximation retaining just two- or three-body terms. A crucial feature of the method is that it converges to the exact solution when the basis size increases and/or the effective interaction clustering increases. In a first phase, their applications were limited to the use of realistic two-nucleon interactions, either G-matrixbased two-body interactions (Zheng et al., 1994), or derived by the Lee-Suzuki procedure (Suzuki and Lee, 1980) for the NCSM (Navr´ atil and Barrett, 1996). This resulted in the elimination of the purely phenomenological parameter ∆ used to define G-matrix starting energy. A truly ab initio formulation of the formalism was presented by Navr´ atil and Barrett (1998), where convergence to the exact solutions was demonstrated for the A = 3 system. The same was later accomplished for the A = 4 system (Navr´ atil and Barrett, 1999), where it was also shown that a three-body effective interaction can be introduced to improve the convergence of the method. The capability to derive a three-body effective interaction and apply it in either relative-coordinate (Navr´ atil et al., 2000) or Cartesian-coordinate formalism (Navr´ atil and Ormand, 2002) together with the ability to solve a three-nucleon system with a genuine three-nucleon force in the NCSM approach (Marsden et al., 2002) now opens the possibility to include a realistic three-nucleon force in the NCSM Hamiltonian and perform calculations using two- and three-nucleon forces for the p-shell nuclei. Among successes of the NCSM approach was the first published result of the binding energy of 4 He with the CD-Bonn NN potential (Navr´ atil and Barrett, 1999), the near-converged results for A = 6 using a non-local Hamil- tonian (Navr´ atil et al., 2001), the first observation of the incorrect ground-state spin in 10 B predicted by the realistic two-body nucleon-nucleon interactions (Caurier et al., 2002). This last result, together with the already known problems of under-binding, confirms the need of realistic three-nucleon forces, a conclusion that also stems from the Green Function Monte Carlo calculations of Wiringa and Pieper (2002) (see also Pieper and Wiringa, 2001, for a review of the results of the Urbana-Argonne collaboration for nuclei A≤10). Recently, a new version of the shell model code AN- TOINE has been developed for the NCSM applications (Caurier et al., 2001b). This new code allows to perform calculations in significantly larger basis spaces and makes it possible to reach full convergence for the A = 6 nuclei. Besides, it opens the possibility to investigate slowly converging intruder states and states with unnatural parity. The largest bases reached with this code so far are, according to the number of HO excitations, the 14ω space for 6 Li and, according the matrix dimension, the 10ω calculations for 10 C. In the latter case, the m-scheme matrix dimension exceeds 800 millions. G. Present possibilities The combination of advances in computer technology and in algorithms has enlarged the scope of possible SM studies. The rotational band of 238 U seems to be still very far from the reach of SM calculations, but predictions for 218 U are already available. Let us list some of the present opportunities. i) No-core calculations. One of the major problems of the no-core SM is the convergence of the results with the size of the valence space; For 6 Li we can handle excitations until 14ω and at least 8ω for all the p-shell nuclei. 21 ii) The pf-shell. This where the shell model has been most successful and exact diagonalizations are now possible throughout the region Beyond 56 Ni, as the 1f 7/2 orbit becomes more and more bound, truncated calculations are close to exact, for instance, in 60 Zn (Mazzocchi et al., 2001) the wave-functions are fully converged when 6p-6h excitations are included. iii) The r 3 g valence space. We use the notation r p for the set of the nlj orbits with 2(n − 1) + l = p excluding the orbit with maximum total angular momentum j = p + 1/2. This space describes nuclei in the region 28 < N, Z < 50; 56 Ni is taken as inert core. Most of them are nearly spherical, and can be treated without truncations. The ββ decay (with and without neutrinos) of 76 Ge and 82 Se are a prime example. The deformed nuclei (N ∼ Z ∼ 40) are more difficult because they demand the inclusion of the 2d 5/2 orbit to describe prolate states and oblate-prolate shape coexistence. iv) The pfg space. To enlarge the pf valence space to include the 1g 9/2 orbit is hopeless nowadays. Furthermore, serious center of mass spuriousness is expected in the 1f −k 7/2 1g k 9/2 configurations (see Ap- pendix C). For neutron rich nuclei in the Nickel region, a tractable valence space that avoids the center of mass problem can be defined: A 48 Ca core on top of which pf protons and r 3 g neutrons are active. v) Heavy nuclei. All the semi-magic nuclei, for instance the N = 126 isotones, can be easily studied and the addition of few particles or holes remains tractable. Some long chains of Tellurium and Bis- muth isotopes have been recently studied (Caurier et al., 2003). IV. THE LANCZOS BASIS There exists a strong connection between the Lanczos algorithm, the partition function, Z(β) = i ¸i[ exp(−βH)[i), and the evolution operator exp(iHt). In the three cases, powers of the Hamiltonian determine the properties of the systems. The partition function can be written as Z(β) = E ρ(E) exp(−βE) i.e., the Laplace transform of the density of states, a quantity readily accessible once the Hamiltonian has been fully diagonalized. The evolution operator addresses more specifically the problem of evolving from a starting vector into the exact ground state. We shall discuss both questions in turn. A. Level densities For many, shell model is still synonymous with diagonalizations, in turn synonymous with black box. One still hears questions such as: Who is interested in diagonalizing a large matrix? As an answer we propose to examine Fig. 10 showing the two point-functions that define the diagonal, H ii , and off-diagonal elements, H i i+1 in a typical Lanczos construction. The continuous lines -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 T r i d i a g o n a l e l e m e n t s i/dim H i i+1 (exact) H i i+1 (nib) H i i (exact) H i i (log) FIG. 10 Tridiagonal matrix elements for a 6579 dimensional matrix, and the logarithmic and inverse binomial approximations (nib) (Zuker et al., 2001). are calculated knowing the first four moments of the matrix (i.e. of the 1+2-body Hamiltonian H it represents). These results hold at fixed quantum numbers, i.e., when the matrix admits no block decomposition. When the matrix is diagonalized the level density is well reproduced by a continuous binomial ρ b (x, N, p, S) = p xN q ¯ xN d Γ(N + 1) Γ(xN + 1)Γ(¯ xN + 1) N S , (59) where x is an a-dimensional energy, ¯ x = 1 − x, N the number of valence particles, p an asymmetry parameter, p + q = 1, and S the span of the spectrum (distance between lowest and highest eigenstates). Introducing the energy scale ε, S, the centroid E c , the variance σ 2 and x are given by S = Nε, E c = Npε, σ 2 = Np(1 −p)ε 2 , x = E S . (60) Note that ρ b (x, N, p, S) reduces to a discrete binomial, _ N n _ if x = n/N = nε/S, with integer n. N, p and S are calculated using the moments of the Hamiltonian H, i.e., averages given by the traces of H K , to be equated to the corresponding moments of ρ b (x, N, p, S), which for low K are the same as those of a discrete binomial. The necessary definitions and equalities follow. d −1 tr(H K ) = ¸H K ), E c = ¸H 1 ), / K = ¸(H−E c ) K ) σ 2 = / 2 , / K = / K σ K , γ 1 = / 3 = q −p √ Npq γ 2 = / 4 −3 = 1 −6pq Npq ; d = d 0 (1 +p/q) N . (61) 22 These quantities also define the logarithmic and inverse binomial (nib) forms of H ii and H i i+1 in Fig. 10. Note that the corresponding lines are almost impossible to distinguish from those of the exact matrix. The associated level densities are found in Fig. 11. 0 200 400 600 800 1000 1200 0 2 4 6 8 10 12 14 16 ρ ( S t a t e s p e r M e V ) E(MeV) ρ(fpgd J=6) ρ(bin) FIG. 11 Exact and binomial level densities for the matrix in Fig. 10. The mathematical status of these results is somewhat mixed. Mon and French (1975) proved that the total density of a Hamiltonian system is to a first approximation a Gaussian. Zuker (2001) extended the approximation to a binomial, but the result remains valid only in some neighborhood of the centroid. Furthermore, one does not expect it to hold generally because binomial thermodynamics is trivial and precludes the existence of phase transitions. In the example given above, we do not deal with the total density, which involves all states, ρ = JT (2J + 1)(2T + 1)ρ JT but with a partial ρ JT at fixed quantum numbers. In this case Zuker et al. (2001) conjectured that the tridiagonal elements given by the logarithmic and nib forms are valid, and hence describe the full spectrum. The conjecture breaks down if a dynamical symmetry is so strong as to define new (approximately) conserved quantum numbers. Granted that a single binomial cannot cover all situations we may nonetheless explore its validity in nuclear physics, where the observed level densities are extremely well approximated by the classical formula of Bethe (1936) with a shift ∆, ρ B (E, a, ∆) = √ 2π 12 e √ 4a(E+∆) (4a) 1/4 (E + ∆) 5/4 . (62) Obviously, if binomials are to be useful, they must reproduce—for some range of energies—Eq. (62). They do indeed, as shown in Fig. 12, where the experimental points for 60 Ni (Iljinov et al., 1992) are also given. Though Bethe and binomial forms are seen to be equivalent, the latter has the advantage that the necessary parameters are well defined and can be calculated, while in the former, the precise meaning of the a parameter is elusive. The shift ∆ is necessary to adjust the ground -5 0 5 10 15 20 25 30 0 10 20 30 40 50 L e v e l D e n s i t y Energy Binomial Bethe 60 Ni (exp) FIG. 12 Binomial, Bethe and experimental level densities for 60 Ni. state position. The problem also exists for the binomial 18 and the result in the figure (Zuker, 2001) solves it phenomenologically. The Shell Model Monte Carlo method provides the only parameter-free approach to level densities (Dean et al., 1995; Langanke, 1998; Nakada and Alhassid, 1997) whose reliability is now established (Al- hassid et al., 1999). The problem is that the calculations are hard. As the shape of the level density is well reproduced by a binomial except in the neighborhood of the ground state, to reconcile simplicity with full rigor we have to examine the tridiagonal matrix at the origin. B. The ground state Obviously, some dependence on the pivot should exist. We examine it through the J = 1 T = 3 pf states in 48 Sc 19 . The matrix is 8590-dimensional, and we calculate the ground state with two pivots, one random (homogeneous sum of all basic states), the other variational (lowest eigenstate in f 8 7/2 space). A zoom on the first matrix elements in Fig. 13 reveals that they are very different for the first few iterations, but soon they merge into the canonical patterns discussed in the preceding section. The ground state wavefunction is unique of course, but it takes the different aspects shown in Fig. 14. In both cases the convergence is very fast, and it is not difficult to show in general that it occurs for a number of iterations of order N log N for dimensionality d = 2 N . However, the variational pivot is clearly better if we are interested in the ground state: If its overlap with the exact solution exceeds 50%, all other contributions are bound to 18 The energies are referred to the mean (centroid) of the distribution, while they are measured with respect to the ground state. 19 They are reached in the 48 Ca(p, n) ( 48)Sc reaction, which will be our standard example of Gamow Teller transitions. 23 -5 0 5 10 15 20 25 30 0 20 40 60 80 100 T r i d i a g o n a l e l e m e n t s i H ii , Variational H ii Random H ii+1 , Variational H ii+1 , Random FIG. 13 Tridiagonal matrix elements after 100 iterations for a random and a variational pivot. be smaller and in general they will decrease uniformly 20 . We shall see how to exploit this property of good pivots to simplify the calculations. 1. The exp(S) method 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 5 10 15 20 25 30 35 40 W a v e f u n c t i o n ( s ) i Random Variational FIG. 14 The ground state wavefunction in the Lanczos basis for a random, and a variational pivot. The plotted values are the squared overlaps of the successive Lanczos vectors with the final wave-function In Section II.A.1 we have sketched the coupled-cluster (or exp(S)) formalism. The formulation in the Lanczos basis cannot do justice to the general theory, but it is a good introduction to the underlying ideas. Furthermore, it turns out to be quite useful. The construction is as in Eqs. (45,46), but the succeeding vectors—except the pivot—are not normalized. Then, 20 There are some very interesting counter-examples. One is found in Fig. 24, where the natural pivot is heavily fragmented. the full wavefunction takes the form [ ¯ 0) = (1 +c 1 P 1 +c 2 P 2 + +c I P I )[0), (63) where P m is a polynomial in H that acting on the pivot produces orthogonal unnormalized vectors in the Lanczos basis: P m [0) = [m). Eq. (46) becomes H[m) = V m [m−1) +E m [m) +[m+ 1). (64) To relate with the normalized version (m ⇒ ¯ m); Divide by ¸m[m) 1/2 , then multiply and divide [m−1) and [m+1) by their norms to recover Eq. (46), and obtain E m = H ¯ m ¯ m , V m = ¸m[m)/¸m − 1[m − 1) = H 2 ¯ m ¯ m−1 . The secular equation (H−E)[ ¯ 0) = 0 leads to the recursion c m−1 + (E m −E) c m +V m+1 c m+1 = 0, (65) whose solution is equivalent to diagonalizing a matrix with V m+1 in the upper diagonal and 1 in the lower one, which is of course equivalent to the symmetric problem. However, here we shall solve the recursion by transforming it into a set of non-linear coupled equations for the c m amplitudes. The profound reason for using an unnormalized basis is that first term in Eq. (65), E = E 0 +V 1 c 1 implies that once c 1 is known the problem is solved. Call- ing ε m = E m −E 0 and replacing E = E 0 +V 1 c 1 in Eq (65) leads to c m−1 + (ε m −V 1 c 1 ) c m + V m+1 c m+1 = 0. (66) Now introduce c m P m = exp( S m P m ), (67) Expanding the exponential and equating terms c 1 = S 1 , c 2 = S 2 + 1 2 S 2 1 , (68) c 3 = S 3 +S 1 S 2 + 1 3! S 3 1 , etc. (69) Note that we have used the formal identification P m P n = P n+m as a heuristic way of suggesting Eqs. (68). In- serting in Eq. (66) and regrouping, a system of coupled equations obtains. The first two are 0 =S 2 V 2 +S 2 1 ( 1 2 V 2 −V 1 ) +S 1 ε 1 + 1 (70) 0 =S 3 V 3 +S 2 ε 2 +S 1 S 2 (V 3 −V 1 )+ 1 2 S 2 1 ε 2 + S 3 1 ( 1 3! V 3 − 1 2 V 1 ) +S 1 . (71) Hence, instead of the usual c m -truncations we can use S m -truncations, which have the advantage of providing a model for the wavefunction over the full basis. At the first iteration Eq. (70) is solved neglecting S 2 . The resulting value for S 1 is inserted in Eq. (71) which is solved by neglecting S 3 . The resulting value for S 2 is reinserted in Eq. (70) and the process is repeated. Once S 1 and S 2 are known, the equation for S 3 (not shown) may be 24 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 2 4 6 8 10 12 14 16 E n e r g i e s truncation t -E c (J=0) -E c (J=2) E(J=2)-E(J=0) exp. fits FIG. 15 Energy convergence for ground and first excited state in 56 Ni as a function of the truncation level t (Caurier et al., 1999c). incorporated, and so on. Since the energy depends only on S 1 , convergence is reached when its value remains constant from step to step. With a very good pivot, Eq. (70) should give a fair approximation, improved by incorporating Eq. (71), and checked by the equation for S 3 . If all amplitudes except S 1 are neglected, the difference scheme (65) is the same that would be obtained for an harmonic H ≡ εS 0 + V (S + + S − ). Successive approximations amount to introduce anharmonicities. An equivalent approach—numerically expedient—consists in diagonalizing the matrices at each c m truncation level. Then, after some iteration, the energies converge exponentially: Introduce conv(i, a, e 0 , i 0 ) = e 0 exp(−ai) −1 exp(−ai 0 ) −1 , (72) which equals e 0 at point i = i 0 . Choose a so as to yield conv(i = i 0 + 1) = e(i 0 + 1) i.e., the correct energy at the next point . Check that e(i 0 + 2) is well reproduced. Fig. 15 provides an example of the efficiency of the method for the lowest two states in 56 Ni. The index i is replaced by the truncation level i ≡ t/2 [Eq. (48, with m = 16, m 0 = 0]: We set i 0 = 1, fix a so as to reproduce the energy at the second iteration and check that the curve gives indeed the correct value at the third point. The results reproduce those of the S 2 truncation, confirming that “exponential convergence” and exp(S) are very much the same thing. In this example, the closed shell pivot is particularly good, and the exponential regime sets in at the first iteration. In general, it will always set in for some value of i 0 that may be quite large (Fig. 14 suggests i 0 ≈ 25 for the random pivot). Fortu- nately, it is almost always possible to find good pivots, and the subject deserves a comment. In the case of 56 Ni the good pivot is a closed shell. As a consequence, the first iterations are associated to truncated spaces of much smaller dimensionalities than the total one (d m ≈ 10 9 ). This also happens for lighter pf shell nuclei, for which the f m 7/2 [or eventually (f 7/2 p 3/2 ) m ] subspaces provide a good pivot. The same argument applies for other regions. For well deformed nuclei Hartree Fock should provide good determinantal pivots, and hence enormous gains in dimensionality, once projection to good angular momentum can be tackled efficiently. As we shall see next, the Lanczos and exp S procedures provide a convenient framework in which to analyze other approaches. 2. Other numerical approximation methods The exponential convergence method introduced in the previous section was first described by Horoi et al. (1999), under a different but equivalent guise. In later work, a hierarchy of configurations determined by their average energy and width was proposed. Successive diagonalizations make it possible to reach the exact energy by exponential extrapolation. The method has been successfully applied to the calculation of the binding energies of the pf shell nuclei (Horoi et al., 2002) and to calculation of the excitation energies of the deformed 0 + states of 56 Ni and 52 Cr (Horoi et al., 2003). The work of Mizusaki and Imada (2002, 2003), is based on the fact that the width of the total Hamiltonian in a truncated space tends to zero as the solution approaches the exact one. They have devised different extrapolation methods and applied them to some pf-shell nuclei. Andreozzi, Lo Iudice, and Porrino (2003) have recently proposed a factorization method that allows for importance sampling to approximate the exact eigenvalues and transition matrix elements. A totally different approach which has proven its power in other fields, the density matrix renormalization group, has been proposed (Dukelsky and Pittel, 2001; Dukelsky et al., 2002), and Papenbrock and Dean (2003) have developed a method based in the optimization of product wavefunctions of protons and neutrons that seems very promising. Still, to our knowledge, the only prediction for a truly large matrix that has preceded the exact calculation remains that of 56 Ni, J = 2 in Fig. 15. It was borne out when the diagonalization became feasible two years later. 3. Monte Carlo methods Monte Carlo methods rely on the possibility to deal with the imaginary-time evolution operator acting on some trial wavefunction exp(−βH)[0) which tends to the exact ground state [ ¯ 0) as β ⇒∞. This is very much what the Lanczos algorithm does, but no basis is constructed. Instead, the energy (or some observable ˆ Ω) is calculated through ¸0[e −βH/2 ˆ Ω e −βH/2 [0) ¸0[e −βH [0) β→∞ =⇒ ¸ ¯ 0[ ˆ Ω[ ¯ 0) ¸ ¯ 0[ ¯ 0) (73) 25 which is transformed into a quotient of multidimensional integrals evaluated through Monte Carlo methods with importance sampling. A “sign problem” arises because the integrands are not definite positive, leading to enormous precision problems. The Green Function Monte Carlo studies mentioned at the beginning of Section II are conducted in coordinate space. The Shell Model Monte Carlo (SMMC) variant is formulated in Fock space, and hence directly amenable to comparisons with standard SM results (see Koonin, Dean, and Langanke, 1997a, for a review). The approach relies on the Hubbard- Stratonovich transformation, and the sign problem is circumvented either by an extrapolation method or by choosing Hamiltonians that are free of it while remaining quite realistic (e.g. pairing plus quadrupole). At present it remains the only approach that can deal with much larger valence spaces than the standard shell model. It should be understood that SMMC does not lead to detailed spectroscopy, as it only produces ground state averages, but it is very well suited for finite temperature calculations. The introduction of Monte Carlo techniques in the Lanczos construction is certainly a tempting project. . . . The quantum Monte Carlo diagonalization method (QMCD) of Otsuka, Honma, and Mizusaki (1998) consists in exploring the mean field structure of the valence space by means of Hartree-Fock calculations that break the symmetries of the Hamiltonian. Good quantum numbers are enforced by projection techniques (Peierls and Yoccoz, 1957). Then the authors borrow from SMMC to select an optimal set of basic states and the full Hamil- tonian is explicitly diagonalized in this basis. More basic states are iteratively added until convergence is achieved. For a very recent review of the applications of this method see (Otsuka et al., 2001b). A strong connection between mean field and shell model techniques is also at the heart of the Vampir approach (Petrovici et al., 1999), (Schmid, 2001) and of the projected shell model (PSM) (Hara and Sun, 1995). C. Lanczos Strength Functions The choice of pivot in the Lanczos tridiagonal construction is arbitrary and it can be adapted to special problems. One of the most interesting is the calculation of strength functions (Bloom, 1984; Whitehead, 1980): If U i j is the unitary matrix that achieves diagonal form, its first column U i 0 gives the amplitudes of the ground state wavefunction in the tridiagonal basis while the first row U 0 j determines the amplitude of the pivot in the j-th eigenstate. U 2 0 j plotted against the eigenenergies E j is called the “strength function” for that pivot. In practice, given a transition operator T , act with it on a target state [t) to define a pivot [0 ′ ) = T [t) that exhausts the sum rule ¸0 ′ [0 ′ ) for T . Normalize it. Then, FIG. 16 Evolution of the Gamow-Teller strength function of 48 Ca as the number of Lanczos iterations on the doorway state increases. 26 by definition [0) = T [t) _ ¸0 ′ [0 ′ ) = j U 0j [j), whose moments, (74) ¸0[H k [0) = j U 2 0j E k j , are those of the (75) strength function S(E) = j δ(E −E j )U 2 0j , (76) As the Lanczos vector [I) is obtained by orthogonaliz- FIG. 17 Strength functions convoluted with gaussians: Upper panel; 50 iterations, Bottom panel; 1000 iterations. ing H I [0) to all previous Lanczos vectors [i), i < I, the tridiagonal matrix elements are linear combinations of the moments of the strength distribution. Therefore the eigensolutions of the I I matrix define an approximate strength function S I (E) = i=1,I δ(E − E i ) ¸i[T [0) 2 , whose first 2I − 1 moments are the exact ones. The eigenstates act as “doorways” whose strength will be split until they become exact solutions for I large enough, as illustrated in Fig. 16 that retraces the fragmentation process of the sum rule pivot; in this case a 48 Sc “doorway” obtained by applying the Gamow-Teller operator to the 48 Ca ground state. The term doorway applies to vectors that have a physical meaning but are not eigenstates. Af- ter full convergence is achieved for all states in the resonance region, the strength function has the aspect shown at the bottom of the figure. In practice, all the spikes are affected by an experimental width, and assuming a perfect calculation, the observed profile would have the aspect shown at the bottom of Fig. 17, after convoluting with gaussians of 150 keV width. The upper panel shows the result for 50 iterations, and 250 keV widths for the non converged states. The profiles become almost identical. V. THE 0ω CALCULATIONS In this section we first revisit the p, sd, and pf shells to explain how a 3b monopole mechanism solves problems hitherto intractable. Then we propose a sample of pf shell results that will not be discussed elsewhere. Fi- nally Gamow Teller transitions and strength functions are examined in some detail. A. The monopole problem and the three-body interaction The determinant influence of the monopole interaction was first established through the mechanism of Bansal and French (1964). Its efficiency in cross-shell calculations was further confirmed by Zamick (1965), and the success of the ZBM model (Zuker, Buck, and McGrory, 1968) is implicitly due to a monopole correction to a realistic force. Zuker (1969) identified the main shortcoming of the model as due to what must be now accepted as a 3b effect 21 . Trouble showed in 0ωcalculations somewhat later simply because it takes larger matrices to detect it in the sd shell than in the ZBM space (dimensionalities of 600 against 100 for six particles). Up to 5 particles the results of Halbert et al. (1971) with a realistic interaction were quite good, but at 22 Na they were so bad that they became the standard example of the unreliability of the realistic forces (Brown and Wildenthal, 1988) and lead to the titanic 22 Universal SD (USD) fit of the 63 matrix elements in the shell by Wildenthal (1984). Though the pf shell demands much larger dimensionalities, it has the advantage of containing two doubly magic nuclei 48 Ca and 56 Ni for which truncated calculations proved sufficient to identify very early the core of the monopole problem: The failure to produce EI closures. 21 The ZBM model describes the region around 16 O through a p 1/2 s 1/2 d 5/2 space. The French Bansal parameters b pd and bps (see Eq. (9b)) must change when going from 14 N to 16 O; which demands a 3b mechanism 22 It took two years on a Vax. Nowadays it would take an afternoon on a laptop. 27 0 1 2 3 4 0 1 2 3 4 3/2 − 1/2 − 7/2 − 5/2 − 5/2 − 3/2 − 7/2 − 9/2 − 3/2 − 1/2 − 7/2 − 5/2 − 5/2 − 3/2 − 7/2 − 9/2 − 3/2 − 1/2 − 7/2 − 5/2 − 5/2 − 3/2 − 7/2 − 9/2 − 3/2 − 1/2 − 9/2 + 5/2 − (1/2 − ,3/2 − ) 5/2 − 7/2 + 3/2 − ∆ E ( M e V ) Expt. KB KB’ KB3 FIG. 18 The spectrum of 49 Ca for the KB, KB’ and KB3 interactions compared to experiment Fig. 18 gives an idea of what happens in 49 Ca with the KB interaction (Kuo and Brown, 1968): there are six states below 3 MeV, where only one exists. In Pasquini (1976) and Pasquini and Zuker (1978) the following modifications were proposed (f ≡ f 7/2 , r ≡ f 5/2 , p 3/2 , p 1/2 ), V T fr (KB1) = V T fr (KB) −(−) T 300 keV, V 0 ff (KB1) = V 0 ff (KB) −350 keV, V 1 ff (KB1) = V 1 ff (KB) −110 keV. (77) The first line defines KB’ in Fig. 18. The variants KB2 and KB3 in Poves and Zuker (1981b) keep the KB1 centroids and introduce very minor multipole modifications. KB3 was adopted as standard 23 in successful calculations in A = 47-50 that will be described in the next sections (Caurier et al., 1994; Mart´ınez-Pinedo et al., 1996b, 1997). For higher masses, there are some problems, but nothing comparable to the serious ones encountered in the sd shell, where modifications such as in Eq. (77) are beneficial but (apparently) insufficient. The effective single particle energies in Fig. 19 are just the monopole values of the particle and hole states at the subshell closures. They give an idea of what happens in a 2b description. At the origin, in 41 Ca, the spectrum is the experimental one. At 57 Ni there is a bunching of the upper orbits that the realistic BonnC (BC) and KB describe reasonably well, but they fail to produce a substantial gap. Hence the need of KB1-type corrections. At the end of the shell BC and KB reproduce an expanded version of the 41 Ca spectrum, which is most certainly incorrect. The indication from ˜ H d m in Eq. (II.B.2) is that the bunching of the upper orbits should persist. By incorporating this hint—which involves the V rr ′ centroids—and fine-tuning 23 The multipole changes—that were beneficial in the perturbative treatment of Poves and Zuker (1981b)—had much less influence in the exact diagonalizations. 40 50 60 70 80 A -25 -20 -15 -10 -5 0 E ( i n M e v ) 1f7/2 BC 2p3/2 2p1/2 1f5/2 40 50 60 70 80 A -25 -20 -15 -10 -5 0 E ( i n M e v ) 1f7/2 KB3 2p3/2 2p1/2 1f5/2 FIG. 19 Effective single particle energies in the pf-shell along the N = Z line, computed with the BC and KB3 interactions. the V fr ones Poves et al. (2001) defined a KB3G interaction that brings interesting improvements over KB3 in A = 50-52, but around 56 Ni there are still some problems, though not as severe as the ones encountered in the sd and p shells. Compared to KB3, the interaction FPD6 has a better gap in 56 Ni, however, the orbit 1f 5/2 is definitely too low in 57 Ni. This produces problems with the description of Gamow-Teller processes (see Borcea et al., 2001, for a recent experimental check in the beta decay of 56 Cu). The classic fits of Cohen and Kurath (1965) (CK) defined state of the art in the p shell for a long time. As they preceded the realistic G-matrices, they also contributed to hide the fact that the latter produce in 10 B a catastrophe parallel to the one in 22 Na. The work of Navr´ atil and Ormand (2002), and Pieper, Varga, and Wiringa (2002) acted as a powerful reminder that brought to the fore the 3b nature of the discrepancies. Once this is understood, the solution follows: Eq. (77) is assumed to be basically sound but the corrections are taken to be linear in the total number of valence particles m (the simplest form that a 3b term can take). Using f ≡ (p 3/2 , d 5/2 , f 7/2 ) generically in the (p, sd, pf) shells respectively, and r = p 1/2 and r ≡ d 3/2 , s 1/2 for the p and sd shells, Zuker (2003) proposes (κ = κ 0 + (m−m 0 ) κ 1 ) 28 -2 -1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 E ( M e V ) J NO KLS CK KLS κ=1.1 EXP FIG. 20 The excitation spectrum of 10 B for different interactions. V T fr (R) =⇒V T fr (R) −(−) T κ V T ff (R) =⇒V T ff (R) −1.5 κδ T0 (78) where R stands for any realistic 2b potential. The results for 10 B are in Fig. 20. The black squares (NO) are from (Navr´ atil and Ormand, 2002, Fig 4, 6Ω). -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 E ( M e V ) J KB USD KB κ=0.90 EXP FIG. 21 The excitation spectrum of 22 Na for different interactions. The black circles correspond to the bare KLS G-matrix (ω =17 MeV), the white squares to the same with κ = 1.1, and the pentagons to the CK fit. NO and KLS give quite similar spectra, as expected from the discussion in Sections II.A and II.D. The κ correction eliminates the severe discrepancies with experiment and give values close to CK. In 22 Na the story repeats itself: BC and KB are very close to one another, the ground state spin is again J = 1 instead of J = 3 and the whole spectrum is awful. The κ correction restores the levels to nearly correct positions, though USD still gives a better fit. This simple cure was not discovered earlier because κ is not a constant. In 24 Mg we could still do with the value for 22 Na but around 28 Si it must be substantially smaller. The spectra in Fig. 22 are obtained with κ(m) = 0.9 − 0.05(m− 6), and now the spectra are as good as those given by USD. 0 1 2 3 4 5 6 7 0 1 2 3 4 E ( M e V ) J KB BC USD KB κ=0.80 EXP 0 1 2 3 4 5 1 3 5 E ( M e V ) 2J KB USD KB κ=0.55 EXP FIG. 22 The excitation spectra of 24 Mg and 29 Si for different interactions. To determine a genuine 3b effect (i.e., the linearity of κ) a sufficiently large span of A values is necessary. In the p shell, corrections to V rr ′ become indispensable very soon and must be fitted simultaneously, as done successfully by Abzouzi et al. (1991), so we do not dwell on the subject. As we have seen, in the pf shell KB3, which is nearly perfect in A = 47-50, must be turned into KB3G for A = 50-52, which also does well at the lower masses. To find a real problem with a 2b R-interaction (i.e., compatible with NN data) 24 we have to move to 56,58 Ni. In particular the first B(E2)(2 −→0) transition in 58 Ni falls short of the observed value (140 e 2 fm 4 ) by a factor ≈ 0.4 with any monopole corrected KB interaction. The problem can be traced to weak quadrupole strength and it is not serious: as explained in Section II.D it is due to a normalization uncertainty, and Table I shows that with equal normalizations BC and KB in the pf shell are 24 It may be possible to fit the data with a purely 2b set of matrix elements: USD is the prime example, but it is R-incompatible (Du- four and Zuker, 1996). 29 as close as in the sd shell. In (Zuker, 2003) BC was adopted. Small modifications of the V fr matrix were made to improve the (already reasonable) r-spectrum in 57 Ni (see Fig. 19). The particular mixture in Eq. (78) was actually chosen to make possible, in the simplest way, a good gap in 48 Ca and a good single-particle spectrum in 49 Sc. It was found that for A = 48, κ(m = 8) ≈ 0.43. For A = 56, truncated calculations yielded κ(16) 0.28. The B(E2)(2 → 0) in 58 Ni indicated convergence to the right value. 0 2 4 6 8 10 12 14 16 500 1000 1500 2000 2500 3000 3500 J ∆(E J ) (keV) BC κ=0.43 BC κ=0.28 KB3 EXP FIG. 23 Backbending in 48 Cr. See text. With κ(m = 8), BC reproduces the yrast spectrum of 48 Cr almost as well as KB3, but not with κ(m = 16) (Fig. 23), indicating that the three-body drift is needed, though not as urgently as in the sd shell. Another indication comes from the T=0 spectrum of 46 V, the counterpart of 22 Na and 10 B in the pf-shell. The realistic interactions place again the J=1 and J=3 states in the wrong order, the correct one is re-established by the three-body monopole correction. B. The pf shell Systematic calculations of the A=47-52 isobars can be found in (Caurier et al., 1994, A = 48, KB3), (Mart´ınez- Pinedo et al., 1997, A = 47 and 49, KB3) and (Poves et al., 2001, A = 50-52, KB3G). Among the other full 0ω calculations let’s highlight the following: The SMMC studies using either the FPD6 interaction (Al- hassid et al., 1994) or the KB3 interaction (Langanke et al., 1995). A comparison of the exact results with the SMMC can be found in (Caurier et al., 1999b). For a recollection of the SMMC results in the pf shell see also (Koonin et al., 1997b). The recent applications of the exponential extrapolation method (Horoi et al., 2002, 2003). The calculations of Novoselsky et al. (1997) for 51 Sc and 51 Ti using the DUPSM code (see also the erratum in Novoselsky et al., 1998b) and for 52 Sc and 52 Ti (Novoselsky et al., 1998a). The extensive QMCD calculation of the spectrum of 56 Ni have been able to reproduce the exact result for the ground state binding energy within 100∼200 keV and to give a fairly good description of the highly deformed excited band of this doubly magic nucleus (Mizusaki et al., 2002, 1999; Otsuka et al., 1998). Other applications to the pf shell can be found in (Honma et al., 1996). The existence of excited collective bands in the N=28 isotones is studied in (Mizusaki et al., 2001). Very recently, a new interaction for the pf shell (GXPF1) has been produced by a Tokyo-MSU collaboration (Honma et al., 2002), following the fitting procedures that lead to the USD interaction. The fit starts with the G-matrix obtained from the Bonn-C nucleon nucleon potential (Hjorth-Jensen et al., 1995). The fit privileges the upper part of the pf shell, as seen by the very large difference in single particle energies between the 1f 7/2 and 12p 3/2 orbits (3 MeV instead of the standard 2 MeV). In the calcium isotopes i.e., when only the T=1 neutron neutron interaction is active) this new interaction retains the tendency of the bare G-matrices (KB, Bonn-C, KLS) to produce large gaps at N=32 and N=34, contrary to FPD6 or KB3G that only predict a large gap at N=32. Some early spectroscopic applications of GXPF1 to the heavy isotopes of Titanium, Vanadium and Chromium can be found in (Janssens et al., 2002) and (Mantica et al., 2003), where the N=32-34 gaps are explored. Each nucleus has its interest, sometimes anecdotic, sometimes fundamental. The agreement with experiment for the energies, and for the quadrupole and magnetic moments and transitions is consistently good, often excellent. These results have been conclusive in establishing the soundness of the minimally-monopole-modified realistic interaction(s). There is no point in reproducing them here, and we only present a few typical examples concerning spectroscopic factors, isospin non-conserving forces and “pure spectroscopy”. 1. Spectroscopic factors The basic tenet of the independent particle model is that addition of a particle to a closed shell a † r [cs) produces an eigenstate of the [cs +1) system. Nowadays we know better: it produces a doorway that will be fragmented. If we choose [cs) = [ 48 Ca) and [cs +1) = [ 49 Sc), the four pf orbits provide the doorways. The lowest, f 7/2 , leads to an almost pure eigenstate. The middle ones, p 3/2, 1/2 , are more fragmented, but the lowest level still has most of the strength and the fragments are scattered at higher energies. Fig. 24 shows what happens to the f 5/2 strength: it remains concentrated on the doorway but splits locally. The same evolution with energy of the quasi-particles (Landau’s term for single particle doorways) was later shown to occur generally in finite systems (Altshuler et al., 1997). 30 0.0 5.0 10.0 15.0 20.0 Energy (MeV) −2.0 −1.0 0.0 1.0 2.0 S Th. Exp. FIG. 24 Spectroscopic factors, (2j +1)S(j, tz), corresponding to stripping of a particle in the orbit 1f 5/2 (Mart´ınez-Pinedo et al., 1997). 2. Isospin non conserving forces Recent experiments have identified several yrast bands in mirror pf nuclei (Bentley et al., 1998, 1999, 2000; Brandolini et al., 2002b; Lenzi et al., 2001; O’Leary et al., 1997, 2002) for A = 47, 49, 50 and 51. The naive view that the Coulomb energy should account for the MDE (mirror energy differences) turns out to be untenable. The four pairs were analyzed in (Zuker et al., 2002) where it was shown that three effects should be taken into account. A typical result is proposed in Fig. 25, where V CM and V BM stand for Coulomb an nuclear isospin breaking multipole contributions, while V Cm is a monopole Coulomb term generated by small differences of radii between the members of the yrast band. The way these disparate contributions add to reproduce the observed pattern is striking. -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 5 7 9 11 13 15 17 19 21 23 25 27 M E D ( M e V ) 2J A=49 Exp V Cm +V CM +V BM V Cm V CM V BM FIG. 25 Experimental (O’Leary et al., 1997) and calculated (Zuker et al., 2002) MED for the pair 49 Cr- 49 Mn. Another calculation of the isospin non conserving effects, and their influence in the location of the proton drip line, is due to (Ormand, 1997). He has also analyzed the A=46 isospin triplet in (Garrett et al., 2001). 3. Pure spectroscopy The first nucleus we have chosen is also one that has been measured to complete the mirror band (Bentley et al., 2000) in A = 51, whose MED are as well described as those of A = 49 in Fig. 25. Such calculations need very good wavefunctions, and the standard test they have to pass is the “purely spectroscopic” one. Work on the subject usually starts with a litany. Such as: Quadrupole effective charges for neutrons q ν =0.5 and protons q π =1.5 and bare g-factors g s π =5.5857 µ N , g s ν =- 3.3826 µ N , g l π =1.0 µ N and g l ν =0.0 µ N in M1 transitions and moments are used. Except when the M1 transitions are fully dominated by the spin term, the use of effective g-factors does not modify the results very much due to the compensation between the spin and orbital modifications. Then some spectrum: The yrast band of 51 Mn, calculated in the full pf-shell space, is compared in Fig. 26 with the experimental data. The first part of the test is passed. At most the examiner will complain about slightly too high, high spin states. 0 1 2 3 4 5 6 7 8 E ( M e V ) 0 1 2 3 4 5 6 7 8 E ( M e V ) Exp. 7/2 − 17/2 − 19/2 − 9/2 − 5/2 − 11/2 − 21/2 − 23/2 − 27/2 − 15/2 − 13/2 − 5/2 − 7/2 − 9/2 − 11/2 − 13/2 − 15/2 − 17/2 − 19/2 − 21/2 − 23/2 − 27/2 − 25/2 − 3/2 − ,5/2 − ,7/2 − 1/2 − 3/2 − KB3G 1/2 − FIG. 26 Yrast band of 51 Mn; experiment vs shell model calculation in the full pf-shell space. Finally the transitions: The transitions in Table IV are equally satisfactory. Note the abrupt change in both B(M1) and B(E2) for J = (17/2) − , beautifully reproduced by the calculation. The origin of this isomerism is in the sudden alignment of two particles in the 1f 7/2 orbit, which provides an intuitive physical explanation for the abrupt change in the MED (Bentley et al., 2000). At the end one can add some extras: The electromagnetic moments of the ground state are also known: Their values µ exp =3.568(2)µ N and Q exp =42(7) efm 2 (Fire- stone, 1996) compare quite well with the calculated µ th =3.397µ N and Q th =35 efm 2 . 31 TABLE IV Transitions in 51 Mn. Exp. Th. B(M1) (µ 2 N ) (µ 2 N ) 7 2 − → 5 2 − 0.207(34) 0.177 9 2 − → 7 2 − 0.16(5) 0.116 11 2 − → 9 2 − 0.662(215) 0.421 17 2 − → 15 2 − 0.00012(4) 0.00003 19 2 − → 17 2 − >0.572 0.797 B(E2) (e 2 fm 4 ) (e 2 fm 4 ) 7 2 − → 5 2 − 528(146) 305 9 2 − → 5 2 − 169(67) 84 9 2 − → 7 2 − 303(112) 204 11 2 − → 7 2 − 236(67) 154 11 2 − → 9 2 − 232(75) 190 17 2 − → 13 2 − 1.236(337) 2.215 We complete this section with an even-even and an odd-odd nucleus. In figure 27 we show the yrast states of 52 Cr, up to the band termination, calculated in the full pf-shell space. The agreement of the KB3G results with the experiment is again excellent. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 E ( M e V ) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 E ( M e V ) Exp. 0 + 2 + 4 + 8 + 10 + 12 + 6 + 8 + 10 + 12 + 14 + 16 + KB3G 0 + 2 + 4 + 6 + FIG. 27 Yrast band of 52 Cr; experiment vs shell model calculation in the full pf-shell space. The electromagnetic moments of the first J = 2 + state are known: µ exp =2.41(13) µ N (Speidel et al., 2000) and Q exp = −8.2(16) e fm 2 (Firestone, 1996) and they are in very good agreement with the theoretical predictions: µ th =2.496 µ N and Q th =–9.4 e fm 2 In table V the experimental values for the electromagnetic transitions between the states of the yrast sequence are given. The calculated values are in very good correspondence with the experiment. The yrast states of the odd-odd nucleus 52 Mn, calculated in the full pf-shell, are displayed in fig. 28. No- TABLE V Transitions in 52 Cr. (a) from (Brown et al., 1974). Exp. Th. B(M1) (µ 2 N ) (µ 2 N ) 9 + →8 + 0.057(38) 0.040 B(E2) (e 2 fm 4 ) (e 2 fm 4 ) 2 + →0 + 131(6) 132 3 + →2 + 7 +7 −5 5 4 + 1 →2 + 83(17) a 107 4 + 2 →2 + 69(19) 26 6 + →4 + 59(2) 68 8 + →6 + 75(24) 84 9 + →8 + 0.5(20) 0.6 0 1 2 3 4 5 6 7 8 9 10 11 12 E ( M e V ) 0 1 2 3 4 5 6 7 8 9 10 11 12 E ( M e V ) Exp. 6 + 2 + 1 + 4 + 3 + 7 + 5 + (8 + ) (9 + ) (11 + ) (10 + ) 6 + 2 + 1 + 4 + 3 + 7 + 5 + 8 + 9 + 11 + 10 + 12 + 13 + 14 + 15 + 16 + KB3G 12 + 13 + 14 + 15 + 16 + FIG. 28 Yrast band of 52 Mn; experiment vs shell model calculation in the full pf-shell space. tice the perfect correspondence between theory and experiment for the states belonging to the multiplet below 1 MeV. The experimental data for the spins beyond the band termination (11 + to 16 + ) come from a recent experiment (Axiotis, 2000). The agreement between theory and experiment is spectacular. There are only three experimental states with spin assignment not drawn in the figure: a second J = (5 + ) at 1.42 MeV, a third J = (5 + ) at 1.68 MeV and a second J = (6 + ) at 1.96 MeV. The calculation places the second J = 5 + at 1.37 MeV, the third J = 5 + at 1.97 MeV and the second J = 6 + at 1.91 MeV. The experimental magnetic and quadrupole moments for the 6 + ground state are also known; µ exp =3.062(2) µ N and Q exp =+50(7) e fm 2 (Firestone, 1996). The calculated values, µ th =2.952 µ N and Q th =+50 e fm 2 reproduce them nicely. Some electromagnetic transitions along the yrast sequence have been measured. The calculations are in very good agreement with them (see table VI). 32 TABLE VI Transitions in 52 Mn. Exp. Th. B(M1) (µ 2 N ) (µ 2 N ) 7 + →6 + 0.501(251) 0.667 8 + →7 + >0.015 0.405 9 + →8 + 1.074 +3.043 −0.537 0.759 B(E2) (e 2 fm 4 ) (e 2 fm 4 ) 7 + →6 + 92 +484 −81 126 8 + →6 + >1.15 33 8 + →7 + >4.15 126 9 + →7 + 104 +300 −46 66 11 + →9 + 54(6) 53 C. Gamow Teller and magnetic dipole strength. Out of the approximately 2500 known nuclei that are bound with respect to nucleon emission, only 253 are stable. The large majority of the rest decay by β emission or electron capture, mediated by the weak interaction. When protons and neutrons occupy the same orbits, as in our case, the dominant processes are allowed Fermi and Gamow-Teller transitions. The information obtained from the weak decays has been complemented by the (p, n) and (n, p) reactions in forward kinematics, that make it possible to obtain total GT strengths and strength functions that cannot be accessed by the decay data because of the limitations due to the Q β windows. From a theoretical point of view, the comparison of calculated and observed strength functions provide invaluable insight into the meaning of the valence space and the nature of the deep correlations detected by the “quenching” effect. The half-life for a transition between two nuclear states is given by (Behrens and B¨ uhring, 1982; Schopper, 1966): (f A +f ǫ )t = 6144.4 ±1.6 (f V /f A )B(F) +B(GT) . (79) The value 6144.4 ± 1.6 is obtained from the nine bestknown superallowed beta decays (Towner and Hardy, 2002) (see Wilkinson, 2002a,b, for an alternative study). f V and f A are the Fermi and Gamow-Teller phase-space factors, respectively (Chou et al., 1993; Wilkinson and Macefield, 1974). f ǫ is the phase space for electron capture (Bambynek et al., 1977), that is only present in β + decays, If t 1/2 is the total lifetime, the partial lifetime of a level with branching ratio b r is t = t 1/2 /b r . B(F) and B(GT) are defined as B(F) = _ ¸f | k t k ± | i) √ 2J i + 1 _2 (80a) B(GT) = __ g A g V _ ¸f | k σ k t k ± | i) √ 2J i + 1 _2 . (80b) Matrix elements are reduced (A3) with respect to spin only, ± refers to β ± decay, σ = 2S and (g A /g V ) = −1.2720(18) (Hagiwara et al., 2002) is the ratio of the weak interaction axial-vector and vector coupling constants. For states of good isospin the value of B(F) is fixed. It can be altered only by a small isospin-symmetry-breaking δ C correction (Towner and Hardy, 2002). Shell model estimates of this quantity can be also found in (Ormand and Brown, 1995). B(F) = _ T(T + 1) −T zi T z f ¸ δ if (1 −δ C ), (81) where δ if allows only transitions between isobaric analog states. Superallowed decays may shed light on the departures from unitarity of the Cabibbo Kobayashi Maskawa matrix. The total strengths S ± are related by the sum rules S − (F) −S + (F) = N −Z, (82a) S − (GT) −S + (GT) = 3(N −Z), (82b) where N and Z refer to the initial state and S ± (GT) does not contain the g A /g V factor. The comparison of the (p, n) and (n, p) data with the Gamow Teller sum rule (Ikeda et al., 1963) led to the long standing “quenching” problem; only approximately one half of the sum rule value was found in the experiments. The GT strength is not protected by a conservation principle and depends critically on the wavefunctions used. Full 0ω calculations already show a large quenching with respect to the independent particle limit as seen in Table VII where the result of a full pf calculation is compared with that obtained with an uncorrelated Slater determinant having the same occupancies: S = i,k n p i n h k (2j i + 1)(2j k + 1) ¸i[[σ[[k) 2 (83) the sum on i runs over the proton (neutron) orbits in the valence space and k over the proton (neutron) orbits for S + (S − ). n p and n h denote the number of particles and holes, respectively. The determinantal state demands a quenching factor that is almost twice as large as the standard quenching factor Q 2 = (0.74) 2 ) that brings the full calculation in line with experiment. 1. The meaning of the valence space Before moving to the explanation of the quenching of the GT strength, it is convenient to recall the meaning 33 TABLE VII Comparison of GT+ strengths. For 54 Fe, 55 Mn, 58 Ni and 59 Co the calculations are truncated to t = 8, t = 4, t = 6 and t = 4 respectively. The data are from (Alford et al., 1993; El-Kateb et al., 1994; Vetterli et al., 1990; Williams et al., 1995). Nucleus Uncorrelated Correlated Expt. Unquenched Q = 0.74 51 V 5.15 2.42 1.33 1.2 ±0.1 54 Fe 10.19 5.98 3.27 3.3 ±0.5 55 Mn 7.96 3.64 1.99 1.7 ±0.2 56 Fe 9.44 4.38 2.40 2.8 ±0.3 58 Ni 11.9 7.24 3.97 3.8 ±0.4 59 Co 8.52 3.98 2.18 1.9 ±0.1 62 Ni 7.83 3.65 2.00 2.5 ±0.1 of the valence space as discussed in Section II.A.1. The 48 Ca(p, n) 48 Sc reaction (Anderson et al., 1990) provides an excellent example. In Fig. 29, adapted from Caurier et al. (1995b), the experimental data are compared with the strength function produced by a calculation in the full pf-shell using the interaction KB3. The peaks have all J = 1, T = 3. In the pf shell there are 8590 of them and the calculation has been pushed to 700 iterations in the Lanczos strength function to ensure fully converged eigenstates below 11 MeV. Of these eigenstates 30 are below 8 MeV: They are at the right energy and have the right strength profile. At higher energies the peaks are much too narrow compared with experiment. It means that they may well be eigenstates of the effective Hamil- tonian in the pf shell, but not eigenstates of the full system. Therefore, they should be viewed as doorways, subject to further mixing with the background of intruders which dominates the level density after 8 MeV, as corroborated by the experimental tail that contains only intruders and can be made to start naturally at that energy. The KB3 effective interaction is doing a very good job, but it is certainly not decoupling 8590 pf states from the rest of the space. If the fact is not explicitly recognized we end up with the often raised (Hjorth-Jensen et al., 1995) “intruder problem”: Decoupling cannot be enforced perturbatively when intruders are energetically close to model states. Figure 29 indicates that although few eigenstates are well decoupled, it is possible to make sense of many others if one interprets them as doorways. The satisfactory description of the lowest states indicates that the model space makes sense. It means that it ensures good decoupling at the S 2 level, or simply in second order perturbation theory, which guarantee a stateindependent interaction. Energetically we are in good shape, and we concentrate on the renormalization of the GT operator: the calculated strength has been quenched by a factor ≈ (0.74) 2 . Why? Why this factor ensures the right detailed strength 5 10 15 20 25 30 35 0 5 10 15 20 25 30 G a m o w T e l l e r S t r e n g t h B ( G T ) E (MeV) Calculated Experimental FIG. 29 Gamow Teller strength in 48 Ca(p, n) 48 Sc from (An- derson et al., 1990)—after elimination of the Fermi peak at around 6 MeV—compared with the calculated peaks (KB3 interaction) after 700 Lanczos iterations (Caurier et al., 1995b). The peaks have been smoothed by gaussians having the instrumental width of the first measured level. for about 30 states below 8 MeV? 2. Quenching To understand the quenching problem it is best to start by the (tentative) solution. The dressed states in Eq. (4) are normalized to unity in the model space. This trick is essential in the formulation of linked cluster or expS theories (hence Eq. (63)). It makes possible the calculation of the energy—and some transitions, such as the E2— without knowledge of the norm of the exact wavefunction. In general though, we need an expectation value between exact, normalized states: ¸ ˆ f | T | ˆ i) 2 . If we write [ ˆ i) = α[0ω) + n=0 β n [nω), (84) and a similar expression in α ′ , β ′ for ¸ ˆ f[, we find ¸ ˆ f | T | ˆ i) 2 = _ _ αα ′ T 0 + n=0 β n β ′ n T n _ _ 2 , (85) since the GT operator does not couple states with different number of ω excitations. If we make two assumptions: (a) neglect the n ,= 0 contributions; (b) α ≈ α ′ ; it follows that if the projection of the physical wavefunction in the 0ω space is Q ≈ α 2 , its contribution to the transition will be quenched by Q 2 . Exactly the same arguments apply to transfer reactions—for which T = a s (or a † s )—but are simpler because T n=0 = 0. The transition strength is given by the spectroscopic factor, which can be identified with Q when one particle is removed and 1−Q when it is added. The assumption that the model amplitudes in the exact wavefunctions are approximately constant is borne 34 out by systematic calculations of Q in the p shell (Chou et al., 1993, Q = 0.820(15)), the sd shell (Wildenthal et al., 1983, Q = 0.77(2)) and the pf shell (illustrated in Fig. 30, Mart´ınez-Pinedo et al., 1996b, Q = 0.744(15)). 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 T(GT) Th. T ( G T ) E x p . 0.77 0.744 FIG. 30 Comparison of the experimental and theoretical values of the quantity T(GT) in the pf shell (Mart´ınez-Pinedo et al., 1996b). The x and y coordinates correspond to theoretical and experimental values respectively. The dashed line shows the “best-fit” for Q = 0.744. The solid line shows the result obtained in the sd-shell nuclei (Wildenthal et al., 1983) These numbers square well with the existing information on spectroscopic factors from (d, p) (Vold et al., 1978, Q ≈ 0.7) and (e, e ′ p) (Cavedon et al., 1982, Q = 0.7) data (see also Pandharipande et al., 1997). This consistency is significant in that it backs assumption (a) above, which is trivially satisfied for spectroscopic factors. It opens the perspective of accepting the GT data as a measure of a very fundamental quantity that does not depend on particular processes. The proposed “solution” to the quenching problem amounts to reading data. FIG. 31 Effective spin g-factor of the M1 operator deduced from the comparison of shell model calculations and data for the total B(M1) strengths in the stable even mass N=28 isotones (from von Neumann-Cosel et al. (1998)) Experimentally, the challenge is to locate all the strength, constrained by the Ikeda sum rule that relates the direct and inverse processes. The careful analysis of Anderson et al. (1985) suggests, but does not prove, that the experimental tail in Fig. 29 contains enough strength to satisfy approximately the sum rule. A similar result is obtained for 54 Fe(p, n) (Anderson et al., 1990). More recent experiments by the Tokyo group establish that the strength located at accessible energies exhausts 90(5)% of the sum rule in the 90 Zr(n, p) and 84(5)% in 27 Al(p, n) (Wakasa et al., 1998, 1997). The theoretical problem is to calculate Q. It has been compounded by a sociological one: the full GT operator is (g A /g V )στ, where g A /g V ≈ −1.27 is the ratio of weak axial and vector coupling constants. The hotly debated question was whether Q was due to non nucleonic renormalization of g A or nuclear renormalization of στ (Arima, 2001; Osterfeld, 1992). We have sketched above the nuclear case, along the lines proposed by Caurier, Poves, and Zuker (1995b), but under a new guise that makes it easier to understand. The calculations of Bertsch and Hamamoto (1982), Dro˙ zd˙ z et al. (1986) and Dang et al. (1997) manage to place significant amounts of strength beyond the resonance region, but they are based on 2p-2h doorways which fall somewhat short of giving a satisfactory view of the strength functions. No-core calculations are under way, that should be capable of clarifying the issue. The purely nuclear origin of quenching is borne out by (p, p ′ ), (γ, γ ′ ) and (e, e ′ ) experiments that determine the spin and convection currents in M1 transitions, in which g A /g V play no role (see Richter, 1995, for a complete review). An analysis of the data available for the N = 28 isotones in terms of full pf-shell calculations concluded that agreement with experiment was achieved by quenching the στ operator by a factor 0.75(2), fully consistent with the value that explains the Gamow-Teller data (von Neumann-Cosel et al., 1998) (see Fig. 31). These results rule out the hypothesis of a renormalization of the axial-vector constant g A : it is the στ operator that is quenched. VI. SPHERICAL SHELL MODEL DESCRIPTION OF NUCLEAR ROTATIONS Theoretical studies in the pf shell came in layers. The first, by McCullen, Bayman, and Zamick (1964), restricted to the f 7/2 space, was a success, but had some drawbacks: the spectra were not always symmetric by interchange of particles and holes, and the quadrupole moments had systematically the wrong sign. The first diagonalizations in the full shell [(Pasquini, 1976), (Pasquini and Zuker, 1978)] solved these problems to a large extent, but the very severe truncations necessary at the time made it impossible to treat on the same footing the pairing and quadrupole forces. The situation improved very much by dressing perturbatively the pure two-body f 7/2 part of the Hamiltonian H 2 , with a three body term H R1 , 35 mostly due to the quadrupole force (Poves and Zuker, 1981b). The paper ended with the phrase: “It may well happen, that in some cases, not in the pf shell but elsewhere, H R1 will overwhelm H 2 . Then, and we are only speculating, we shall speak, perhaps, of the rotational coupling scheme.” Indeed, some nuclei were indicating a willingness to become rotational but could not quite make it, simply because the perturbative treatment was not enough for them. The authors missed prophecy by one extra conditional: “not in the pf shell should have been “not necessarily...” At the time it was thought impossible to describe rotational motion in a spherical shell-model context. The glorious exception, discovered by Elliott (1958a,b), was (apparently) associated to strict SU(3) symmetry, approximately realized only near 20 Ne and 24 Mg. A. Rotors in the pf shell TABLE VIII 48 Cr; quadrupole properties of the yrast band J B(E2)exp B(E2) th Q0(t) Q0(s) Q0(t)[f7/2,p3/2] 2 321(41) 228 107 103 104 4 330(100) 312 105 108 104 6 300(80) 311 100 99 103 8 220(60) 285 93 93 102 10 185(40) 201 77 52 98 12 170(25) 146 65 12 80 14 100(16) 115 55 13 50 16 37(6) 60 40 15 40 The fourth layer was started when the ANTOINE code (Section III.D) came into operation: 48 Cr was definitely a well deformed rotor (Caurier et al., 1994), as can be surmised by comparing with the experimental spectrum from Lenzi et al. (1996) in Fig. 32 (see also Cameron et al., 1993) and the transition properties in Table VIII from Brandolini et al. (1998) where we have used Q 0 (s) = (J + 1) (2J + 3) 3K 2 −J(J + 1) Q spec (J), K ,= 1 (86) B(E2, J → J −2) = 5 16π e 2 [¸JK20[J −2, K)[ 2 Q 0 (t) 2 K ,= 1/2, 1; (87) to establish the connection with the intrinsic frame descriptions: A good rotor must have a nearly constant Q 0 , which is the case up to J = 10, then 48 Cr backbends. This is perhaps the most striking result to come out of our pf calculations, because such a behaviour had been thought to occur only in much heavier nuclei. Fig. 33 compares the experimental patterns with those obtained with KB3, and with the Gogny force. The latter, when diagonalized gives surprisingly good results. 48 Cr 0 + 0 0 + 0 2 + 752 2 + 806 4 + 1858 4 + 1823 6 + 3445 6 + 3398 8 + 5189 8 + 5128 10 + 7063 10 + 6978 12 + 8406 12 + 8459 14 + 10277 14 + 10594 16 + 13306 16 + 13885 (16 + ) 15727 16 + 16268 752 1106 1587 1744 1874 1343 1871 3029 5450 806 1017 1575 1730 1850 1481 2135 3291 5674 exp. theor. FIG. 32 48 Cr level scheme; experiment vs. theory When treated in the cranked Hartree Fock Bogoliubov (CHFB) approximation, the results are not so good. The discrepancy is more apparent than real: The predictions for the observables are very much the same in both cases (see Caurier et al., 1995a, for the details). The reason is given in Fig 34, where exact KB3 diagonalizations are done, subtracting either of the two pairing contributions, JT = 01 and 10 (Poves and Mart´ınez-Pinedo, 1998). It is apparent that the JT = 01-subtracted pattern is quite close to the CHFB one in Fig. 33, especially in the rotational regime before the backbend. The JT = 10 subtraction goes in the same direction. The interpretation is transparent: CHFB does not “see” proton-neutron pairing at all, and it is not very efficient in the low pairing regime. As it does everything else very well, the inevitable conclusion is that pairing can be treated in first order perturbation theory, i.e., the energies are very sensitive to it, but not the wavefunctions. Floods of ink have gone into “neutron-proton” pairing, which is a problem 36 0.0 1.0 2.0 3.0 4.0 E γ (MeV) 2 4 6 8 10 12 14 16 J Exp SM-KB3 CHFB SM-GOGNY FIG. 33 The yrast band of 48 Cr; experiment vs. the shell model calculations with KB3 and the Gogny force and the CHFB results also with the Gogny force. 0.0 1.0 2.0 3.0 4.0 E γ (MeV) 2 4 6 8 10 12 14 16 J KB3 KB3-P10 KB3-P01 FIG. 34 Gamma-ray energies along the yrast band of 48 Cr (in MeV), full interaction (KB3), isoscalar pairing retired (KB3– P01) and isovector pairing retired (KB3–P01). for mean field theories but neither for the Gogny force nor for the shell model. Furthermore, the results show that ordinary pairing is also a mean field problem when nuclei are not superfluids. 48 Cr has become a standard benchmark for models (Cranked Hartree Fock Bogoliuvov (Caurier et al., 1995a), Cranked Nilsson Strutinsky (Juodagalvis and ˚ Aberg, 1998), Projected Shell Model (Hara et al., 1999), Cluster Model (Descouvemont, 2002), etc.) Nuclei in the vicinity also have strong rotational features. The mirror pairs 47 V- 47 Cr and 49 Cr- 49 Mn closely follow the semiclassical picture of a particle or hole hole strongly coupled to a rotor (Mart´ınez-Pinedo et al., 1997) in full agreement with the experiments (Bentley et al., 1998; Cameron et al., 1991, 1994; O’Leary et al., 1997; Tonev et al., 2002). 50 Cr was predicted to have a second backbending by Mart´ınez-Pinedo et al. (1996a), confirmed experimentally (Lenzi et al., 1997) (see figure 35). When more particles or holes are added, the collective behaviour fades, though even for 52 Fe, a rotor like band appears at low spin with an yrast trap at J=12 + , both are accounted 0 1000 2000 3000 4000 5000 6000 E_gamma (in keV) 0 2 4 6 8 10 12 14 16 18 20 J EXP KB3 FIG. 35 The yrast band of 50 Cr; experiment vs. the shell model calculations with the KB3 interaction for by the shell model calculations (Poves and Zuker, 1981b)(sic), (Ur et al., 1998). For spectroscopic comparisons with the odd-odd nuclei, see Brandolini et al. (2001); Lenzi et al. (1999) for 46 V and Svensson et al. (1998) for 50 Mn. Recently, a highly deformed excited band has been discovered in 56 Ni (Rudolph et al., 1999). It is dominated by the configuration (1f 7/2 ) 12 (2p 3/2 , 1f 5/2 , 2p 1/2 ) 4 . The calculations reproduce the band, that starts at about 5 MeV excitation energy and has a deformation close to β=0.4. B. Quasi-SU(3). To account for the appearance of backbending rotors, a theoretical framework was developed by Zuker et al. (1995), and made more precise by Mart´ınez-Pinedo et al. (1997). Here we give a compact overview of the scheme that will be shown to apply even to the classic examples of rotors in the rare-earth region. Let us start by considering the quadrupole force alone, taken to act in the p-th oscillator shell. It will tend to maximize the quadrupole moment, which means filling the lowest orbits obtained by diagonalizing the operator Q 0 = 2q 20 = 2z 2 − x 2 − y 2 . Using the cartesian representation, 2q 20 = 2n z − n x − n y , we find eigenvalues 2p, 2p−3,. . . , etc., as shown in the left panel of Fig. 36, where spin has been included. By filling the orbits orderly we obtain the intrinsic states for the lowest SU(3) representations (Elliott, 1958a,b): (λ, 0) if all states are occupied up to a given level and (λ, µ) otherwise. For instance: putting two neutrons and two protons in the K = 1/2 level leads to the (4p,0) representation. For four neutrons and four protons, the filling is not complete and we have the (triaxial) (8(p − 1),4) representation for which we expect a low lying γ band. In jj coupling the angular part of the quadrupole op- 37 0 3 6 9 12 1/2 3/2 5/2 7/2 9/2 1/2 3/2 5/2 7/2 9/2 Q + k 0 SU3 quasi.SU3 FIG. 36 Nilsson orbits for SU(3) (k = −2p) and quasi-SU(3) (k = −2p + 1/2)). erator q 20 = r 2 C 20 has matrix elements ¸j m[C 2 [j + 2 m) ≈ 3[(j + 3/2) 2 −m 2 ] 2(2j + 3) 2 , (88) ¸j m[C 2 [j + 1 m) = − 3m[(j + 1) 2 −m 2 ] 1/2 2j(2j + 2)(2j + 4) (89) The ∆j = 2 numbers in Eq. (88) are—within the approximation made—identical to those in LS scheme, obtained by replacing j by l. The ∆j = 1 matrix elements in Eq. (88) are small, both for large and small m, corresponding to the lowest oblate and prolate deformed orbits respectively. If the spherical j-orbits are degenerate, the ∆j = 1 couplings, though small, will mix strongly the two ∆j = 2 sequences (e.g., (f 7/2 p 3/2 ) and (f 5/2 p 1/2 )). The spin-orbit splittings will break the degeneracies and favour the decoupling of the two sequences. Hence the idea (Zuker et al., 1995) of neglecting the ∆j = 1 matrix elements and exploit the correspondence l −→j = l + 1/2 m −→m+ 1/2 sign(m). which is one-to-one except for m = 0. The resulting “quasi SU(3)” quadrupole operator respects SU(3) relationships, except for m = 0, where the correspondence breaks down. The resulting spectrum for quasi-2q 20 is shown in the right panel of Fig 36. The result is not exact for the K = 1/2 orbits but a very good approximation. To check the validity of the decoupling, a Hartree calculation was done for H = εH sp + H q , where H sp is the observed single particle spectrum in 41 Ca (essentially equidistant orbits with 2MeV spacings) and H q is the quadrupole force in Eq. (33) with a properly renormalized coupling. The result is exactly a Nilsson (1955) calculation (Mart´ınez-Pinedo et al., 1997), H Nilsson = ω _ εH sp − δ 3 2q 20 _ , (90) where δ 3 = 1 4 ¸2q 20 ) ¸r 2 ) = ¸2q 20 ) (p + 3/2) 4 . (91) In the right panel of Fig. 37 the results are given in the usual form. 0.0 0.5 1.0 1.5 2.0 ε −10.0 −5.0 0.0 5.0 10.0 ∆ E ( M e V ) 1/2[330] 1/2[321] 1/2[310] 1/2[301] 3/2[321] 3/2[312] 3/2[301] 5/2[312] 5/2[303] 7/2[303] −0.30 −0.10 0.10 0.30 δ 1/2[330] 1/2[321] 1/2[310] 1/2[301] 3/2[321] 3/2[312] 3/2[301] 5/2[312] 5/2[303] 7/2[303] FIG. 37 Nilsson diagrams in the pf shell. Energy vs. single particle splitting ε (left panel), energy vs. deformation δ (right panel). In the left panel we have turned the representation around: since we are interested in rotors, we start from perfect ones (SU(3)) and let ε increase. At a value of ≈ 0.8 the four lowest orbits are in the same sequence as the right side of fig. 36 (Remember here that the real situation corresponds to ε ≈ 1.0). The agreement even extends to the next group, although now there is an intruder (1/2[310] orbit). The suggestion is confirmed by an analysis of the wavefunctions: For the lowest two orbits, the overlaps between the pure quasi-SU(3) wavefunctions calculated in the restricted ∆j = 2 space (fp from now on) and the ones in the full pf shell exceeds 0.95 throughout the interval 0.5 < ε < 1. More interesting still: the contributions to the quadrupole moments from these two orbits vary very little, and remain close to the values obtained at ε = 0 i.e., from fig. 36). We have learned that –for the rotational features– calculations in the restricted (fp) n spaces account remarkably well for the results in the full major shell (pf) n (see last column of Table VIII). Let us move now to larger spaces. In Fig. 38 we have yrast transition energies for different configurations of 8 particles in ∆j = 2 spaces. The force is KLS, ω = 9 MeV, the single particle splittings uniform at ε = 1 MeV, and gds, say, is the lower sequence in the sdg, p = 4, shell. Rotational behavior is fair to excellent at low J. As expected from the normalization property of the realistic quadrupole force [Eq. (35)] the moments of inertia in the rotational region go as (p + 3/2) 2 (p ′ + 3/2) 2 , i.e. if we multiply all the E γ values by this factor the lines become parallel. The intrinsic quadrupole moment Q 0 [Eq. (86)] remains constant to within 5% up to a critical J value at which the bands backbend. Why and how do the bands backbend? We have no simple answer, but Fig. 39 from (Velázquez and Zuker, 2002) shows the behavior of the (gds) 8 space under the influence of a symmetric random interaction, gradually 38 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 J E(J+2)-E(J) fp8 fp4.gds4 gds8 gds4.hfp4 FIG. 38 Yrast transition energies Eγ = E(J + 2) − E(J) for different configurations, KLS interaction. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 E ( J + 2 ) - E ( J ) J KLS a=-0.1 a=-0.2 a=-0.3 a=-0.4 FIG. 39 Backbending patterns in (gds) 8 T = 0, with the KLS interaction and with a random interaction plus a constant W JT rstu = −a. made more attractive by an amount a. There is no need to stress the similarity with Fig. 38. The appearance of backbending rotors seems to be a general result of the competition between deformation and alignment, characteristic of nuclear processes. The group theoretical aspects of quasi-SU3 have been recently discussed and applied to the description of the sd-shell nuclei by Vargas, Hirsch, and Draayer (2001, 2002a). C. Heavier nuclei: quasi+pseudo SU(3) We have seen that quasi-SU(3) is a variant of SU(3) that obtains for moderate spin-orbit splittings. For other forms of single particle spacings, the pseudo-SU(3) scheme (Arima et al. (1969); Hecht and Adler (1969); 0 3 6 9 12 [ [ ] ] h g 50 82 r 4 5 r h f p i g d s MeV Positive Negative FIG. 40 Schematic single particle spectrum above 132 Sn. rp is the set of orbits in shell p excluding the largest. For the upper shells the label l is used for j = l + 1/2 Raju et al. (1973), see also Vargas et al. (2002b) for more recent applications) will be favored (in which case we have to use the left panel of Fig. 36, with pseudo- p = p −1). Other variants of SU(3) may be possible and are well worth exploring. In cases of truly large deformation SU(3) itself may be valid in some blocks. To see how this works, consider Fig. 40 giving a schematic view of the single particle energies in the space of two contiguous major shells—in protons (π) and neutrons (ν)—adequate for a SM description of the rare earth region. We want to estimate the quadrupole moments for nuclei at the onset of deformation. We shall assume quasi- SU(3) operates in the upper shells, and pseudo-SU(3) in the lower ones. The number of particles in each shell for which the energy will be lowest will depend on a balance of monopole and quadrupole effects, but Nilsson diagrams suggest that when nuclei acquire stable deformation, two orbits K=1/2 and 3/2—originating in the upper shells of Fig. 36—become occupied, i.e., the upper blocks are precisely the 8-particle configurations we have studied at length. Their contribution to the electric quadrupole moment is then Q 0 = 8[e π (p π −1) +e ν (p ν −1)], (92) with p π = 5, p ν = 6; e π and e ν are the effective charges. Consider even-even nuclei with Z=60-66 and N=92- 98, corresponding to 6 to 10 protons with pseudo-p = 3, and 6 to 10 neutrons with pseudo-p = 4 in the lower shells. From the left part of Fig. 36 we obtain easily their contribution to Q 0 , which added to that of Eq. (92) yields a total Q 0 = 56e π + (76 + 4n)e ν , (93) 39 TABLE IX B(E2) ↑ in e 2 b 2 compared with experiment (Ra- man et al., 1989) N Nd Sm Gd Dy 92 4.47 4.51 4.55 4.58 2.6(7) 4.36(5) 4.64(5) 4.66(5) 94 4.68 4.72 4.76 4.80 5.02(5) 5.06(4) 96 4.90 4.95 4.99 5.03 5.25(6) 5.28(15) 98 5.13 5.18 5.22 5.26 5.60(5) for 152+2n Nd, 154+2n Sm, 156+2n Gd and 158+2n Dy respectively. At fixed n, the value is constant in the four cases because the orbits of the triplet K=1/2, 3/2, 5/2 in Fig. 36 have zero contribution for p=3. Q 0 (given in dimensionless oscillator coordinates, i.e., r → r/b with b 2 ≈ 1.01A −1/3 fm 2 ), is related to the E2 transition probability from the ground state by B(E2) ↑= 10 −5 A 2/3 Q 2 0 . The results, using effective charges of e π = 1.4, e ν = 0.6 calculated in (Dufour and Zuker, 1996) are compared in table IX with the available experimental values . The agreement is quite remarkable and no free parameters are involved. Note in particular the quality of the prediction of constancy (or rather A 2/3 dependence) at fixed n, which does not depend on the choice of effective charges. The discrepancy in 152 Nd is likely to be of experimental origin, since systematics indicate, with no exception, much larger rates for a 2 + state at such low energy (72.6 keV). It is seen that by careful analysis of exact results one may come to very simple computational strategies. In the last example on B(E2) rates, the simplicity is such that the computation reduces to a couple of sums. D. The 36 Ar and 40 Ca super-deformed bands The arguments sketched above apply to the region around 16 O, where a famous 4-particles 4-holes (4p-4h) band starting at 6.05 MeV was identified by Carter et al. (1964), followed by the 8p-8h band starting at 16.75 MeV (Chevallier et al., 1967). Shell model calculations in a very small space, p 1/2 d 5/2 s 1/2 , could account for the spectroscopy in 16 O, including the 4p-4h band (Zuker et al., 1968, ZBM), but the 8p-8h one needs at least three major shells and was tackled by an α-cluster model (Abgrall et al., 1967a,b). It is probably the first superdeformed band detected and explained. In 40 Ca, the first excited 0 + state is the 4p-4h bandhead. It is only recently that another low-lying highly deformed band has been found (Ideguchi et al., 2001), following the discovery of a similar structure in 36 Ar (Svens- son et al., 2000). 0 1000 2000 3000 4000 5000 E γ (keV) 0 2 4 6 8 10 12 14 16 18 20 J Expt. 4p−4h SDPF FIG. 41 The superdeformed band of 36 Ar, experiment vs. SM By applying the quasi+pseudo-SU(3) recipes of Sec- tion VI.C we find that the maximum deformations attainable are of 8p-8h character in 40 Ca and 4p-4h in 36 Ar with Q 0 = 180 efm 2 and Q 0 = 136 efm 2 respectively. The natural generalization of the ZBM space consists in the d 3/2 s 1/2 fp orbits (remember: fp ≡ f 7/2 p 3/2 ), which is relatively free of center of mass spuriousness. And indeed, it describes well the rotational regime of the observed bands. However, to track them beyond backbend, it is convenient to increase the space to d 3/2 s 1/2 pf. The adopted interaction, sdfp.sm is the one originally constructed by Retamosa et al. (1997), and used in (Cau- rier et al., 1998) (USD, KB3 for intra shell and KLS for cross shell matrix elements), with minor monopole adjustments dictated by new data on the single particle structure of 35 Si (Nummela et al., 2001a). The calculations are conducted in spaces of fixed number of particles and holes. In figure 41 the calculated energy levels in 36 Ar are compared to the data. The agreement is excellent, except at J = 12 where the data show a clear backbending while the calculation produces a much smoother upbending pattern. In table X the calculated spectroscopic quadrupole moments (Q s ) and the B(E2)’s are used to compute the intrinsic Q as in Eqs. (86,87), using standard effective charges δq π =δq ν =0.5. As expected both Q 0 (s) and Q 0 (t) are nearly equal and constant—and close to the quasi+pseudo SU(3) estimate—up to the backbend. The calculated B(E2)’s agree well with the experimental ones (Svensson et al., 2001). The value of Q 0 corresponds to a deformation β ≈ 0.5. Now we examine the 8p-8h band in 40 Ca (Ideguchi et al., 2001). The valence space adopted for 36 Ar is truncated by limiting the maximum number of particles in the 1f 5/2 and 2p 1/2 orbits to two. The experimental (Ideguchi et al., 2001) and calculated yrast gamma-ray energies are compared in fig. 42. The patterns agree reasonably well but the change of slope at J = 10 —where the backbend in 48 Cr starts (Fig. 33)—is missed by the calculation, which only backbends at J = 20, the band termination for the configura- 40 TABLE X Quadrupole properties of the 4p-4h configuration’s yrast-band in 36 Ar (in e 2 fm 4 and efm 2 ) B(E2)(J→J-2) J EXP TH Qspec Q0(s) Q0(t) 2 315 -36.0 126 126 4 372(59) 435 -45.9 126 124 6 454(67) 453 -50.7 127 120 8 440(70) 429 -52.8 125 114 10 316(72) 366 -52.7 121 104 12 275(72) 315 -53.0 119 96 14 232(53) 235 -54.3 120 82 16 >84 131 -56.0 122 61 0 1000 2000 3000 4000 5000 E γ (keV) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 J Expt. 8p−8h, SDPF FIG. 42 The superdeformed band in 40 Ca; exp. vs. 8p-8h calculation tion f 8 7/2 (d 3/2 s 1/2 ) −8 . The extra collectivity induced by the presence of sd particles in pseudo-SU(3) orbitals is responsible or the delay in the alignment, but apparently too strong to allow for the change in slope at J = 10. The experimental Q 0 (t)=180 +39 −29 obtained from the fractional Doppler shifts corresponds to a deformation β ≈ 0.6 (Ideguchi et al., 2001). It is extracted from an overall fit that assumes constancy for all measured values, and corresponds exactly to the quasi+pseudo SU(3) estimate. It also squares well with the calculated 172 efm 2 in Table XI where the steady decrease in collectivity remains consistent with experiment within the quoted uncertainties. A reanalysis of the experimental lifetimes in the superdeformed band of 40 Ca (Chiara et al., 2003) suggest that for low spins the deformation would be smaller due to the mixing with less deformed states of lower np- nh rank. This calculation demonstrates that a detailed description of very deformed bands is within reach of the shell model. The next step consists in remembering that states at fixed number of particles are doorways that will fragment in an exact calculation that remains to be done. The band in 36 Ar has been also described by the projected shell model in (Long and Sun, 2001). Beyond mean TABLE XI Quadrupole properties of the 8p-8h configuration’s yrast-band in 40 Ca (in e 2 fm 4 and efm 2 ), calculated in the sdpf valence space J B(E2)(J → J-2) Qspec Q0(t) Q0(s) 2 589 -49.3 172 172 4 819 -62.4 170 172 6 869 -68.2 167 171 8 860 -70.9 162 168 10 823 -71.6 157 164 12 760 -71.3 160 160 14 677 -71.1 149 157 16 572 -72.2 128 158 18 432 -75.0 111 162 20 72 -85.1 22 8 -79.1 24 7 -81.5 field methods using the Skyrme interaction have been recently applied to both the 36 Ar and 40 Ca superdeformed bands by Bender, Flocard, and Heenen (2003a). E. Rotational bands of unnatural parity The occurrence of low-lying bands of opposite parity to the ground state band is very frequent in pf shell nuclei. Their simplest characterization is as particle-hole bands with the hole in the 1d 3/2 orbit. In a nucleus whose ground state is described by the configurations (pf) n the opposite parity intruders will be (1d 3/2 ) −1 (pf) n+1 . The promotion of a particle from the sd to the pf shell costs an energy equivalent to the local value of the gap, that in this region is about 7 MeV. On the other side, the presence of one extra particle in the pf-shell may produce an important increase of the correlation energy, that can compensate the energy lost by the particle hole jump. For instance, the very low-lying positive parity band of 47 V, can be interpreted as (1d 3/2 ) −1 (a proton hole) coupled to (pf 8 ) T=0. Indeed, the correlation energy of this pseudo- 48 Cr is larger than the correlation energy of the ground state of 47 V and even larger than the correlation energy of the real 48 Cr, explaining why the band starts at only 260 keV of excitation energy (Poves and S´ anchez Solano, 1998). The most extreme case is 45 Sc, where the intruder band based in the configuration (1d 3/2 ) −1 coupled to 46 Ti, barely miss (just by 12 keV) to become the ground state. Many bands of this type have been experimentally studied in recent years, mainly at the Gasp and Euroball detectors (Brandolini et al., 1999), and explained by shell model calculations. We present now some recent results in 48 V (Brandolini et al., 2002a). The positive parity levels of this odd- odd nucleus are very well reproduced by the shell model calculation using the interactions KB3 (or KB3G) as can 41 FIG. 43 The ground state (K=4 + ) and the side band (K=1 + ) of 48 V, experiment vs. theory. be seen in Fig. 43, another example of good quality “pure spectroscopy”. The levels are grouped in two bands, one connected to the 4 + ground state, that would correspond to K=4 + in a Nilsson context (the aligned coupling of the last unpaired proton K=3/2 − and the last neutron K=5/2 − ) and a second one linked to the 1 + member of the ground state multiplet that results of the anti-aligned coupling. The negative parity states can be interpreted as the result of the coupling of a hole in the 1d 3/2 orbit to 49 Cr. As we have mentioned earlier, the ground state band of 49 Cr can be understood in a particle plus rotor picture, and assigned K=5/2 − . Now we have to couple the 1d 3/2 hole to our basic rotor 48 Cr. Therefore the structure of the negative parity states of 48 V is one particle and one hole coupled to the rotor core of 48 Cr. Hence we expect that the lowest lying bands would have K=1 − and K=4 − (aligned and anti-aligned coupling of the quasiparticle and the quasihole). These two bands have been found experimentally at about 500 keV of excitation energy. Their structure is in very good agreement with the SM predictions (see Fig. 44) except for a small shift of the K=4 − and K=8 − bandheads relative to the K=1 − state. A more stringent test of the theoretical predictions is provided by the electromagnetic properties. In this particular nucleus the experimental information is very rich and a detailed comparison can be made for both the positive and negative bands and for E2 and M1 transitions. The results for the ground state band are collected in FIG. 44 The negative parity K=1 − and K=4 − bands of 48 V, experiment vs. theory. FIG. 45 Theoretical and experimental B(M1) and B(E2) transition probabilities in the ground state band of 48 V Fig. 45. Notice that the calculation reproduces even the tiniest details of the experimental results, as, for instance the odd-even staggering of the B(M1)’s. The B(E2)’s correspond to a deformation β=0.2 that is clearly smaller than that of 48 Cr. The transition probabilities in the negative parity bands are presented in Fig. 46. Now the experimen- 42 FIG. 46 Theoretical and experimental B(M1) and B(E2) transition probabilities in the negative parity bands of 48 V tal results have larger uncertainties. Despite that, the agreement in absolute values and trends is very good for the lowest band (K=1 − ). The quantitative agreement is worse for the other two bands, nevertheless, the main features of the data are well reproduced. The deformation of the K=1 − band extracted from the B(E2) properties turns out to be larger than the deformation of the ground state band and close to that of 48 Cr. VII. DESCRIPTION OF VERY NEUTRON RICH NUCLEI The study of nuclei lying far from the valley of stability is one of the most active fields in today’s experimental nuclear physics. What it specific about these nuclei, from a shell model point of view? As everywhere else: the model space and the monopole behavior of the interaction. They go together because the effective single particle energies (ESPE, see Section V) depend on occupancies, which in turn depend on the model space. In broad terms, the specificity of light and medium neutron rich nuclei is that the EI closures (corresponding to the filling of the p + 1/2 orbit of each oscillator shell) take over as boundaries of the model spaces. As discussed in Section II.B.2, the oscillator closures (HO) may be quite solid for double magic nuclei, but they become vulnerable in the semi-magic cases. For instance the d 3/2 -f 7/2 neutron gap in 40 Ca of ≈ 7 MeV goes down to ≈ 2.5 MeV around 28 O. Since the d 5/2 orbit is well below its sd partners, now quite close to f 7/2 , the natural model space is no longer the sd shell, but the EI space bounded by the N = 14 and 28 closures, supplemented by the p 3/2 subshell whenever the p 3/2 -f 7/2 gap becomes small: from ≈ 6 MeV in 56 Ni, it drops to ≈ 4.5 MeV in 48 Ca, then ≈ 2 MeV in 40 Ca, and finally ≈ 0 MeV in 28 Si. As explained at the end of Section II.B.2 this monopole drift provides direct evidence for the need of three-body mechanisms. For the p shell, the situation is similar: The imposing 11.5 MeV p 1/2 -d 5/2 gap in 16 O is down to some 3 MeV in 12 C. The monopole drift of the s 1/2 -d 5/2 gap brings it from about 8 MeV in 28 Si to nearly -1 MeV in 12 C. According to Otsuka et al. (2001a) the drift may well continue: in 8 He, the s 1/2 -p 1/2 bare gap is estimated at 0.8 MeV. All the numbers above (except the last) are experimental. The bare monopole values are smaller because correlations increase substantially the value of the gaps, but they do not change qualitatively the strong monopole drifts. Their main consequence is that “normal” states, i.e., those described by 0ω p or sd calculations often “coexist” with “intruders” that involve promotion to the next oscillator shell. Let us examine how this happens. A. N=8; 11 Li: halos The 1/2 + ground state in 11 Be provided one of the first examples of intrusion. The expected 0ω normal state lies 300 keV higher. The explanation of this behavior has varied with time (Aumann et al., 2001; Sagawa et al., 1993; Suzuki and Otsuka, 1994; Talmi and Unna, 1960), see also (Brown, 2001) for a recent review and (Suzuki et al., 2003) for a new multi-ω calculation, but the idea has remained unchanged. The normal state corresponds to a hole on the N = 8 closure. The monopole loss of promoting a particle to the sd shell is compensated by a pairing gain for the p −2 1/2 holes. A quadrupole gain due to the interaction of the sd neutron with the p 2 protons is plausible, but becomes questionable when we note that the same phenomenon occurs in 9 He which has no p 2 particles, suggesting the need for a further reduction of the monopole loss (Otsuka et al., 2001a). The interest in 11 Be—as a “halo” nucleus—was revived by the discovery of the remarkable properties of 11 Li, which sits at the drip line (≈ 200 keV two-neutron separation energy) and has a very large spatial extension, due to a neutron halo (Hansen and Jonson, 1987; Tanihata et al., 1985). The shell model has no particular problem with halo nuclei, whose large size is readily attributed to the large size of the s 1/2 orbit. As shown by (Kahana et al., 1969a) the use of Woods Saxon wavefunctions affect the matrix elements involving this orbit, but the uncertainties involved are easily absorbed by the monopole field. As a consequence, the whole issue hinges on the s 1/2 contribution to the wavefunctions, which is sensitively detected by the β decay to the first excited 1/2 − state in 11 Be. In Borge et al. (1997) it was shown that calculations producing a 50% split between the neutron closed shell and the s 2 1/2 p −2 1/2 configuration lead to the right lifetime. This result was confirmed by Simon et al. (1999). Further confirmation comes from Navin et al. (2000), with solid indications that the supposedly semi-magic 12 Be ground 43 state is dominated by the same s 2 1/2 p −2 1/2 configuration. B. N=20; 32 Mg, deformed intruders In the mid 1970’s it was the sd shell that attracted the most attention. Nobody seemed to remember 11 Be, and everybody (including the authors of this review active at the time) were enormously surprised when a classic experiment Thibault et al. (1975) established that the mass and β-decay properties of the 31 Na ground state— expected to be semimagic at N = 20—could not possibly be that of a normal state (Wildenthal and Chung, 1979). The next example of a frustrated semi-magic was 32 Mg (Detraz et al., 1979). Early mean field calculations had interpreted the discrepancies as due to deformation (Campi et al., 1975) but the experimental confirmation took some time (Guillemaud-Mueller et al., 1984; Klotz et al., 1993; Motobayashi et al., 1995). Exploratory shell model calculations by Storm et al. (1983), including the 1f 7/2 orbit in the valence space, was able to improve the mass predictions, however, deformation was still absent. To obtain deformed solutions demanded the inclusion of the 2p 3/2 orbit as demonstrated by Poves and Retamosa (1987). These calculations were followed by many others (Fukunishi et al., 1992; Heyde and Woods, 1991; Otsuka and Fukunishi, 1996; Poves and Retamosa, 1994; Siiskonen et al., 1999; Warburton et al., 1990), that mapped an “island of inversion”, i.e. the region where the intruder configurations are dominant in the ground states. The detailed contour of this “island” depends strongly on the behavior of the effective single particle energies (ESPE), in turn dictated by the monopole hamiltonian. Let’s recall, that according to what we have learned in Section VI.B, the configurations (d 5/2 s 1/2 ) 2−4 p (f 7/2 p 3/2 ) 2−4 n ,corresponding to N = 20, Z = 10-12, have a “quasi-SU(3)” quadrupole coherence close to that of SU(3) (i.e., maximal). The ESPE in Fig. 47 represent H m for the sdfp.sm and Tokyo group interactions (Utsuno et al., 1999). Within details they are quite close, and such as efficiently to favor deformation. Both interactions lead to an “island of inversion” for Z=10, 11 and 12; N=19, 20 and 21. Consider now some detailed information obtained with sdpf.sm by Caurier et al. (2001c). The S 2N values of Fig. 48 locate the neutron drip line, consistent with what is known for oxygen and fluorine, where the last bound isotopes are 24 O and 31 F (Sakurai et al., 1999). Note the kink due to deformed correlations in the latter. For the other chains, the behavior is smoother and the last predicted bound isotopes are 34 Ne, 37 Na and 40 Mg. Some results for the even Mg isotopes (N=18, N=20 and N=22) are gathered in table XII. In 30 Mg the normal configuration is the one that agrees with the existing experimental data (Pritychenko et al., 1999). In 32 Mg the situation is the opposite as the experimental data (the 2 + excitation energy (Guillemaud-Mueller et al., 1984) and the 0 + → 2 + B(E2) (Motobayashi et al., 1995)) clearly 28 30 32 34 36 38 40 A -25 -20 -15 -10 -5 0 5 10 2p3/2 1f7/2 1d3/2 2s1/2 1d5/2 28 30 32 34 36 38 40 A -25 -20 -15 -10 -5 0 5 10 2p3/2 1f7/2 2d3/2 2s1/2 2d5/2 FIG. 47 Effective single particle energies (in MeV) at N=20 with the sdpf.sm interaction (upper panel) and the Tokyo group interaction (bottom panel) 16 18 20 22 24 26 28 30 N -2 0 2 4 6 O F Ne Na Mg FIG. 48 Two neutron separation energies (in MeV) calculated with the sdpf.sm interaction. prefer the intruder. A preliminary measure of the 4 + excitation energy reported by Azaiez (1999) goes in the same direction. Data and calculations suggest prolate deformation with β ≈ 0.5. In 34 Mg the normal configuration that contains two pf neutrons, is already quite collective. It can be seen in the table that it resembles the 32 Mg ground state. The 4p-2h intruder is even more 44 TABLE XII Properties of the even magnesium isotopes. N stands for normal and I for intruder. Energies in MeV, B(E2)’s in e 2 fm 4 and Q’s in efm 2 30 Mg 32 Mg 34 Mg N I EXP N I EXP N I EXP ∆E(0 + I ) +3.1 -1.4 +1.1 0 + 0.0 0.0 0.0 0.0 0.0 0.0 2 + 1.69 0.88 1.48 1.69 0.93 0.89 1.09 0.66 0.67 4 + 4.01 2.27 2.93 2.33 (2.29) 2.41 1.86 2.13 6 + 6.82 3.75 9.98 3.81 3.52 3.50 B(E2) 2 + → 0 + 53 112 59(5) 36 98 90(16) 75 131 126(25) 4 + → 2 + 35 144 17 123 88 175 6 + → 4 + 23 140 2 115 76 176 Qspec(2 + ) -12.4 -19.9 -11.4 -18.1 -15.4 -22.7 deformed (β ≈ 0.6) and a better rotor. Results from the Riken experiments of Yoneda et al. (2000) and Iwasaki et al. (2001) seem to favor the intruder option. In the QMCD calculations of Utsuno et al. (1999), the ground state band is dominantly 4p-2h; the 2 + comes at the right place, but the 4 + is too high. Clearly, 34 Mg is at the edge of the “island of inversion”. Another manifestation of the intruder presence in the region has been found at Isolde (Nummela et al., 2001b): The decay of 33 Na indicates that the ground state of 33 Mg has J π =3/2 + instead of the expected J π =3/2 − or J π =7/2 − . This inversion is nicely reproduced by the sdpf.sm calculation. C. N=28: Vulnerability Let us return for a while on Fig. 47. The scale does not do justice to a fundamental feature: the drift of the p 3/2 −f 7/2 gap, which decreases as protons are removed. As explained at the end of Section II.B.2, this behavior is contrary to a very general trend in heavier nuclei, and demands a three-body mechanism to resolve the contradiction. Next we remember that the oscillator closures are quite vulnerable even at the strict monopole level. As we have seen, quadrupole coherence takes full advantage of this vulnerability. On the contrary, the EI closures are very robust. However, because of the drift of the p 3/2 − f 7/2 gap even the N = 28 closure becomes vulnerable. The sdpf.sm interaction leads to a remarkable result summarized in Table XIII: the decrease of the gap combined with the gain of correlation energy of the 2p-2h lead to a breakdown of the N = 28 closure for 44 S and 40 Mg. The double magic 42 Si resists. Towards N=Z, 52 Cr and 54 Fe exhibit large correlation energies associated to the prolate deformed character of their 2p-2h neutron configurations (β ∼ 0.3). When the 2p-2h bandheads of 40 Mg and 52 Cr are allowed to mix in the full 0ω space, using them as pivots in the LSF procedure, they keep their identity to a large extent. This can be seen in Fig. 49, where we have plotted the strength functions of the 2p-2h 0 + states in the full space. In 40 Mg the 2p-2h state represents the 60% of the ground state while in 52 Cr it is the dominant component (70%) of the first excited 0 + . This is a very interesting illustration of the mechanism of intrusion; the intruder state is present in both nuclei, but it is only in the very neutron rich one that it becomes the ground state. FIG. 49 LSF of the 2p-2h 0 + bandheads of 40 Mg (upper panel) and 52 Cr (bottom panel) in the full 0ω space Table XIV gives an idea of the properties of the iso- 45 TABLE XIII N=28 isotones: quasiparticle neutron gaps, difference in correlation energies between the 2p-2h and the 0p- 0h configurations and their relative position 40 Mg 42 Si 44 S 46 Ar 48 Ca 50 Ti 52 Cr 54 Fe 56 Ni gap 3.35 3.50 3.23 3.84 4.73 5.33 5.92 6.40 7.12 ∆ECorr 8.45 6.0 6.66 5.98 4.08 7.59 10.34 10.41 6.19 E ∗ 2p−2h -1.75 1.0 -0.2 1.7 5.38 3.07 1.50 2.39 8.05 tones in which configuration mixing is appreciable, but no clearcut deviation from N=28 magicity can be detected. In Fig. 50 we have plotted the low energy spectra of the heaviest known sulphur isotopes. The agreement with the accumulated experimental results (see Glasmacher, 1998, for a recent review) is excellent and extends to the new data of Sohler et al. (2002). Analyzing their proton occupancies we conclude that the rise in collectivity along the chain is correlated with the equal filling of the d 3/2 and s 1/2 orbitals, (the d 5/2 orbital remains always nearly closed). The maximum proton collectivity is achieved when both orbitals are degenerate, which corresponds to the Pseudo-SU(3) limit. For the neutrons, the maximum collectivity occurs at N = 24, the f 7/2 mid-shell. Ac- cording to the calculation, 42 S is a prolate rotor, with an incipient γ band. In 44 S the spherical and deformed configurations mix equally. The N=27 isotones also reflect the regular transition from sphericity to deformation in their low-lying spectrum. The excitation energy of the 3/2 − state should be sensitive to the correlations and to the neutron gap. While in 47 Ca, it lies quite high (at around 2 MeV) due to the strong f 7/2 closure, in 45 Ar it appears at about 0.4 MeV. Concerning 43 S, the information comes from two recent experiments: the mass measurements at Ganil by Sarazin et al. (2000) that observed a low-lying isomer around 400 keV excitation energy and the MSU Coulex experiment of Ibbotson et al. (1999) that detected a strong E2 transition from the ground state to an excited state around 940 keV. According to our calculations the ground state corresponds to the deformed configuration and has spin 3/2 − . The spherical single hole state 7/2 − would be the first excited state and his lifetime is consistent with that of the experimental isomer. The third known state is short lived and it should correspond to the 7/2 − member of the ground state band. D. N=40; from “magic” 68 Ni to deformed 64 Cr Systematics is a most useful guide, but it has to be used with care. The p and sd shells can be said to be “full” 0ω spaces, in the sense that the region boundaries are well defined by the N = 4, 8 and 20 closures, with the exception for the very neutron-rich halo nuclei and the island of inversion discussed previously. In the pf-shell it is already at N = Z ≈ 36 that the 0ω model TABLE XIV N=28 isotones: Spectra, quadrupole properties and occupancies 40 Mg 42 Si 44 S 46 Ar E ∗ (2 + )(MeV) 0.81 1.49 1.22 1.51 E ∗ (4 + ) 2.17 2.68 2.25 3.46 E ∗ (0 + 2 ) 1.83 1.57 1.26 2.93 Q(2 + )(e fm 2 ) -21 16 -17 20 B(E2)(e 2 fm 4 ) 108 71 93 93 n 7/2 5.54 6.16 6.16 6.91 (f 7/2 ) 8 % 3 28 24 45 TABLE XV 2 + 1 energies and g 9/2 intruder occupation in the Nickel isotopic chain, from (Sorlin et al., 2002) 62 Ni 64 Ni 66 Ni 68 Ni 70 Ni 72 Ni 74 Ni E(2 + ) calc. 1.11 1.24 1.49 1.73 1.50 1.42 1.33 E(2 + )exp. 1.173 1.346 1.425 2.033 1.259 B(E2↑) calc. 775 755 520 265 410 505 690 n 9/2 0.24 0.43 0.67 1.07 0.84 0.55 0.45 space collapses under the invasion of deformed intruders. Naturally, we expect something similar in the neighborhood of N = 40 isotones. Recent experiments involving the Coulomb excitation of 66−68 Ni (Sorlin et al., 2002), the β decay of 60−63 V (Sorlin et al., 2003), and the decay and the spectroscopy of the 67m Fe (Sawicka et al., 2003), support this hypothesis. In addition, the β decays of the neutron-rich isotopes 64 Mn and 66 Mn, investigated at Isolde (Hannawald et al., 1999), that show a sudden drop of the 2 + energies in the daughter nuclei 64 Fe and 66 Fe, point also in the same direction. In defining the new model spaces, the full pf results are quite useful: 56 Ni turns out to be an extraordinarily good pivot (refer to Fig. 15), and as soon as a few particles are added it becomes a good core. Therefore, for the first time we are faced with a genuine EI space r 3 g 9/2 (remember r 3 is the “rest” of the p = 3 shell). Its validity is restricted to nearly spherical states. Deformation will demand the addition of the d 5/2 subshell. The effective interaction is based on three blocks: i) the TBME from KB3G effective interaction (Poves et al., 2001), ii) the G-matrix of ref. (Hjorth-Jensen et al., 1995) with the modifications of ref. (Nowacki, 1996) and iii) the Kahana, Lee and Scott G-matrix (Kahana et al., 1969b) for the remaining matrix elements. We have computed all the nickel isotopes and followed their behavior at and beyond N=40. Let’s first focus on 68 Ni. The calculated low energy spectrum, an excited 0 + at 1.85 MeV, 2 + at 2.15 MeV, 5 − at 2.72 MeV and 4 + at 3.10 MeV, fit remarkably well with the experimental results. The ground state wave function is 50% closed shell. This number 46 FIG. 50 Predicted level schemes of the heavy sulphur isotopes, compared with the experiment. Energies in keV, B(E2)’s in e 2 fm 4 is very sensitive to modifications of the pf-g gap, and smaller values of the closed shell probability can be obtained without altering drastically the rest of the properties. The SM results have been compared with those of other methods in (Langanke et al., 2003b). In table XV TABLE XVI Spectroscopic properties of 60−64 Cr in the fpgd valence space 60 Cr 62 Cr 64 Cr E ∗ (2 + ) (MeV) 0.67 0.65 0.51 Qs(e.fm 2 ) -23 -27 -31 B(E2)↓(e 2 .fm 4 ) 288 302 318 Qi(e.fm 2 ) from Qs 82 76 109 Qi(e.fm 2 ) from B(E2) 101 103 106 E ∗ (4 + ) (MeV) 1.43 1.35 1.15 Qs(e.fm 2 ) -37 -30 -43 B(E2)↓(e 2 .fm 4 ) 426 428 471 Qi(e.fm 2 ) from Qs 102 84 119 Qi(e.fm 2 ) from B(E2) 117 117 123 we compare the excitation energy of the 2 + states in the Nickel chain with the available experimental data (including the very recent 70 Ni value (Sorlin et al., 2002)). The agreement is quite good, although the experiment gives a larger peak at N=40. Similarly the B(E2)’s in Fig. 51 follow the experimental trends including the drop at 68 Ni recently measured (Sorlin et al., 2002). Note that for a strict closure the transition would vanish. The β decays of 60 V and 62 V indicate very low energies for the 2 + 1 states in 60 Cr and 62 Cr (Sorlin et al., 26 28 30 32 34 36 38 40 42 44 Neutron number 0 200 400 600 800 1000 1200 B ( E 2 ) [ e 2 f m 4 ] FIG. 51 B(E2) (0 + → 2 + ) in e 2 fm 4 for the Nickel isotopes, from (Sorlin et al., 2002) 2000, 2003). If N=40 were a magic number, one would expect higher 2 + energies as the shell closure approaches, particularly in 64 Cr. As seen in Fig. 52, exactly the opposite seems to happen experimentally. The three possible model spaces tell an interesting story: pf ≡ r 3 is acceptable at N = 32, 34. The addition of g 9/2 does more harm than good because the gaps have been arbitrarily reduced to reproduce the N = 36 experimental point by allowing 2p-2h jumps including d 5/2 . But then, in N = 38 the 2 + is too high, and the experimental trend promises no improvement at N = 40. Properties of the 2 + 1 energies given by the pfgd calculation are collected in table XVI. They point to prolate structures with deformation β ≈ 0.3. 47 30 32 34 36 38 40 42 Neutron number 0 0.5 1 1.5 2 E ( 2 + ) ( M e v ) exp. fp fpg fpgd FIG. 52 Experimental and calculated 2 + energies in the pf, pfg and pfgd spaces. VIII. OTHER REGIONS AND THEMES A. Astrophysical applications Astrophysical environments involve conditions of temperature and densities that are normally not accessible in laboratory experiments. The description of the different nuclear processes requires then theoretical estimates. As discussed in previous sections, shell-model calculations are able to satisfactorily reproduce many experimental results so that it should be possible to obtain reliable predictions for nuclei and/or conditions not yet accessible experimentally. As a first example, consider the decay properties of nuclei totally stripped of the atomic electrons at typical cosmic rays energies of 300 MeV/A. A nucleus such as 53 Mn, unstable in normal conditions, becomes stable. Another isotope, 54 Mn, could be used as a chronometer to study the propagation of iron group nuclei on cosmic-rays (Duvernois, 1997) provided its half-life under cosmic-rays conditions is known. The necessary calculation involves decay by second-forbidden unique transitions to the ground states of 54 Cr and 54 Fe. Due to phase space arguments the β − decay to 54 Fe is expected to dominate. In two difficult and elegant experiments the very small branching ratio for the β + decay to the ground state of 54 Cr has been measured: (1.8±0.8)10 −9 by Zaerpoor et al. (1997) and (1.20±0.26)10 −9 by Wu- osmaa et al. (1998). Taking the weighted mean of this values, (1.26 ± 0.25) 10 −9 , and knowing that of 54 Mn, 312.3(4) d, a partial β + half-life of (6.8 ± 1.3) 10 8 yr is obtained compared with 5.6 10 8 yr from the shell model calculation of Mart´ınez-Pinedo and Vogel (1998), which yields 5.0 10 5 yr for the dominant β − branch. Both theoretical results are sensitive to uncertainties in the renormalization of the unique second-forbidden operators that should be removed taking the ratio of the β − and β + half-lives. Multiplying the theoretical ratio by the experimental value of the β + half-life yields a value of (6.0 ±1.2) 10 5 yr for the β − branch. Using a similar argument Wuosmaa et al. (1998) estimate the partial β − half-life to be (6.0 ±1.3[stat] ±1.1[theor]) 10 5 . The influence of this half-life on the age of galactic cosmic-rays has been discussed recently by Yanasak et al. (2001). An- other possible cosmic ray chronometer, 56 Ni, could measure the time between production of iron group nuclei in supernovae and the accelaration of part of this matterial to form cosmic rays (Fisker et al., 1999). Before acceleration, the decay of 56 Ni proceeds by electron capture to the 1 + state in 56 Co with a half-live of 6.075(20) days. After acceleration 56 Ni is stripped of its electrons, the transition to the 1 + state is no longer energetically allowed and the decay proceeds to the 3 + state at 158 keV via a second forbidden unique transition. Currently only a lower limit for the half-life of totally ionized 56 Ni could be established (2.9 10 4 y) (Sur et al., 1990). A recent shell-model calculation by (Fisker et al., 1999) predict a half-live of 4 10 4 y that is too short for 56 Ni to serve as a cosmic ray chronometer. Nuclear beta-decay and electron capture are important during the late stages of stellar evolution (see Langanke and Mart´ınez-Pinedo, 2003, for a recent review). At the relevant conditions in the star electron capture and β decay are dominated by Gamow-Teller (and Fermi) transitions. Earlier determinations of the appropriate weak interaction rates were based in the phenomenological work of Fuller, Fowler, and Newman (1980, 1982a,b, 1985). The shell model makes it possible to refine these estimates. For the sd shell nuclei, important in stellar oxygen and silicon burning, we refer to Oda et al. (1994). More recently, it has been possible to extend these studies to pf shell nuclei relevant for the pre-supernova evolution and collapse (Caurier et al., 1999a; Langanke and Mart´ınez- Pinedo, 2000; Langanke and Mart´ınez-Pinedo, 2001). The astrophysical impact of the shell-model based weak interaction rates have been recently studied by Heger et al. (2001a,b) The basic ingredient in the calculation of the different weak interaction rates is the Gamow-Teller strength distribution. The GT + sector directly determines the electron capture rate and also contributes to the beta- decay rate through the thermal population of excited states (Fuller et al., 1982a). The GT − strength contributes to the determination of the β-decay rate. To be applicable to calculating stellar weak interaction rates the shell-model calculations should reproduce the available GT + (measured by (n, p)-type reactions) and GT − (measured in (p, n)-type reactions). Figure 53 compares the shell-model GT + distributions with the pioneering measurements performed at TRIUMF. These measurements had a typical energy resolution of ≈ 1 MeV. Recently developed techniques, involving (d, 2 He) charge-exchange reactions at intermediate energies (Rak- ers et al., 2002), have improved the energy resolution by an order of magnitude or more. Figure 54 compares the shell-model GT + distribution computed using the KB3G interaction (Poves et al., 2001) with a recent experimental measurement of the 51 V(d, 2 He) performed at 48 −2 0 2 4 6 8 10 12 E (MeV) 0.0 0.5 1.0 0.0 0.2 0.4 0.6 G T S t r e n g t h 0.0 0.2 0.4 0.6 0.8 1.0 Exp SM −2 0 2 4 6 8 10 12 E (MeV) 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.0 0.1 0.2 0.3 0.4 0.5 56 Fe 54 Fe 58 Ni 51 V 55 Mn 59 Co FIG. 53 Comparison of shell model GT+ distributions with experimental data (Alford et al., 1993; El-Kateb et al., 1994; Rönnqvist et al., 1993) for selected nuclei. The shell model results (discrete lines) have been folded with the experimental resolution (histograms) and include a quenching factor of (0.74) 2 (adapted from Caurier et al., 1999a). KVI (B¨ aumer et al., 2003). Figure 55 compares the shell-model GT − distributions with the data obtained in (p, n) charge-exchange reaction measurements for 54,56 Fe and 58,60 Ni (Anderson et al., 1990; Rapaport et al., 1983). The GT − operator acting on a nucleus with neutron excess and ground state isospin T can lead to states in the daughter nucleus with three different isospin values (T −1, T, T +1). As a consequence, the GT − strength distributions have significantly more structure and extend over a larger excitation energy interval than the GT + distributions, making their theoretical reproduction more challenging. Nevertheless, the agreement with the experimental data is quite satisfactory. The shell-model results for 58 Ni have been recently compared with high-resolution data (50 keV) obtained using the ( 3 He, t) reaction (Fujita et al., 2002). Shell-model diagonalization techniques have been used to determine astrophysically relevant weak interaction rates for nuclei with A ≤ 65. Nuclei with higher masses are relevant to study the collapse phase of core-collapse supernovae (Langanke and Mart´ınez-Pinedo, 2003). The calculation of the relevant electron-capture rates is currently beyond the possibilities of shell-model diagonal- B ( G T + ) SM 0.2 0.3 0.4 B ( G T + ) 0 1 2 3 4 5 6 7 E x [MeV] 0.1 0.1 0.2 0.3 0.4 51 V(d, 2 He) 51 Ti FIG. 54 Comparison of the shell-model GT+ distribution (lower panel) for 51 V with the high resolution (d, 2 He) data (from Bäumer et al., 2003). The shell-model distribution includes a quenching factor of (0.74) 2 . ization calculations due to the enormous dimensions of the valence space. However, this dimensionality problem does not apply to Shell-Model Monte-Carlo methods (SMMC, see section IV.B.3). Moreover, the high temperatures present in the astrophysical environment makes necessary a finite temperature treatment of the nucleus, this makes SMMC methods the natural choice for this type of calculations. Initial studies by Langanke et al. (2001) showed that the combined effect of nuclear correlations and finite temperature was rather efficient in unblocking Gamow-Teller transitions on neutron rich germanium isotopes. More recently this calculations have been extended to cover all the relevant nuclei in the range A = 65–112 by Langanke et al. (2003a). The resulting electron-capture rates have a very strong influence in the collapse (Langanke et al., 2003a) and post-bounce (Hix et al., 2003) The astrophysical r-process is responsible for the synthesis of at least half of the elements heavier than A ≈ 60 (Wallerstein et al., 1997). Simulations of the r-process require the knowledge of nuclear properties far from the valley of stability (Kratz et al., 1998; Pfeiffer et al., 2001). As the relevant nuclei are not experimentally accessible, theoretical predictions for the relevant quantities (i.e., neutron separation energies and half-lives) are needed. The calculation of β decay half-lives usually requires two ingredients: the Gamow-Teller strength distribution in the daughter nucleus and the relative energy scale between parent and daughter (i.e. the Q β value). Due to the huge number of nuclei relevant for the r pro- 49 0 5 10 15 E (MeV) 0.0 1.0 2.0 G T 0.0 2.0 4.0 6.0 8.0 10.0 σ ( m b / ( s r M e V ) ) IAS 0 5 10 15 E (MeV) 0.0 1.0 2.0 3.0 G T 0.0 2.0 4.0 6.0 8.0 10.0 σ ( m b / ( s r M e V ) ) IAS 0 5 10 15 E (MeV) 0.0 0.5 1.0 1.5 0.0 2.0 4.0 6.0 IAS 0 5 10 15 E (MeV) 0.0 0.5 1.0 0.0 2.0 4.0 6.0 8.0 IAS 54 Fe 56 Fe 58 Ni 60 Ni FIG. 55 Comparison of (p, n) L = 0 forward angle cross section data (Rapaport et al., 1983) (upper panels) with the calculated GT− strength distributions (lower panels). For 54 Fe the solid curve in the lower panel shows the experimental GT− data from (Anderson et al., 1990), while the shell model results are given by the dashed line. The Fermi transition to the isobaric analog state (IAS) is included in the data (upper panels) but not in the calculations. A quenching factor of (0.74) 2 is included in the calculations. (from Caurier et al., 1999a). cess, the estimates of the half-lives are so far based on a combination of global mass models and the quasi particle random-phase approximation (see Langanke and Mart´ınez-Pinedo, 2003, for a description of the different models). However, recently shell-model calculations have become available for some key nuclei with a magic neutron number N = 50 (Langanke and Mart´ınez-Pinedo, 2003), N = 82 (Brown et al., 2003; Mart´ınez-Pinedo and Langanke, 1999), and N = 126 (Mart´ınez-Pinedo, 2001). All this calculations suffer from the lack of spectroscopic information on the regions of interest that is necessary to fine tune the effective interactions. This situation is improving at least for N = 82 thanks to the recent spectroscopic data on 130 Cd (Dillmann et al., 2003). Nuclear reaction rates are the key input data for simulations of stellar burning processes. Experiment-based reaction rates for the simulation of explosive processes such as novae, supernovae, x-ray bursts, x-ray pulsars, and merging neutron stars are scarce because of the experimental difficulties associated with radioactive beam measurements (K¨ appeler et al., 1998). Most of the reaction rate tables are therefore based on global model predictions. The most frequently used model is the statistical Hauser Feshbach approach (Rauscher and Thiele- mann, 2000). For nuclei near the drip-lines or near closed shell configurations, the density of levels is not high enough for the Hauser Feshbach approach to be applicable. For these cases alternative theoretical approaches such as the nuclear shell model need to be applied. Shell- model calculations were used for the determination of the relevant proton capture reaction rates for sd-shell nuclei necessary for rp process studies (Herndl et al., 1995). This calculations have been recently extended to include pf-shell nuclei (Fisker et al., 2001). Knowledge of neutrino nucleus reactions is necessary for many applications, e.g. the neutrino oscillation studies, detection of supernova neutrinos, description of the neutrino transport in supernovae and nucleosynthesis studies. Most of the relevant neutrino reactions have not been studied experimentally so far and their cross sections are typically based on nuclear theory (see Kolbe et al., 2003, for a recent review). The model of choice for the theoretical description of neutrino reactions depends of the energy of the neutrinos that participate in the reaction. For low neutrino energies, comparable to the nuclear excitation energy, neutrino-nucleus reactions are very sensitive to the appropriate description of the nuclear response that is very sensitive to correlations among nucleons. The model of choice is then the nuclear shell- model. 0ω calculations have been used for the calculation of neutrino absorption cross sections (Sampaio et al., 2001) and scattering cross sections (Sampaio et al., 2002) for selected pf shell nuclei relevant for supernovae evolution. For lighter nuclei complete diagonalizations can be performed in larger model spaces, e.g. 4ω calculations for 16 O (Haxton, 1987; Haxton and Johnson, 1990) and 6ω calculations for 12 C (Hayes and Towner, 2000; Volpe et al., 2000). Other examples of shell-model calculations of neutrino cross sections are the neutrino absorption cross sections on 40 Ar of Ormand et al. (1995) for solar neutrinos (see Bhattacharya et al., 1998, for an experimental evaluation of the same cross section), this cross section have been recently evaluated by (Kolbe et al., 2003) for supernova neutrinos. And the evaluation by Haxton (1998) of the solar neutrino absorption cross section on 71 Ga relevant for the GALLEX and SAGE solar neutrino experiments. For higher neutrino energies the standard method of choice is the random phase approximation as the neutrino reactions are sensitive mainly to the total strength and energy centroids of the different multipoles contributing to the cross section. In some selected cases, the Fermi and Gamow-Teller contribution to the cross section could be determined from shell-model calculation that is supplemented by RPA calculations for higher multipoles. This type mix calculation has been carried out for several iron isotopes (Kolbe et al., 1999; Toivanen et al., 2001) and 50 for 20 Ne (Heger et al., 2003). B. ββ-decays The double beta decay is the rarest nuclear weak process. It takes place between two even-even isobars, when the decay to the intermediate nucleus is energetically forbidden or hindered by the large spin difference between the parent ground state and the available states in the intermediate nuclei. It comes in three forms: The twoneutrino decay ββ 2ν : A Z X N −→ A Z+2 X N−2 +e − 1 +e − 2 + ¯ ν 1 + ¯ ν 2 is just a second order process mediated by the Standard model weak interaction It conserves the lepton number and has been already observed in a few nuclei. The second mode, the neutrinoless decay ββ 0ν : A Z X N −→ A Z+2 X N−2 +e − 1 +e − 2 needs an extension of the standard model of electroweak interactions as it violates lepton number. A third mode, ββ 0ν,χ is also possible A Z X N −→ A Z+2 X N−2 +e − 1 +e − 2 +χ in some extensions of the standard model and proceeds via emission of a light neutral boson, a Majoron χ. The last two modes, not yet experimentally observed, require massive neutrinos –an issue already settled by the recent measures by Super-Kamiokande (Fukuda et al., 1998), SNO (Ahmad et al., 2002) and KamLAND (Eguchi et al., 2003). Interestingly, the double beta decay without emission of neutrinos would be the only way to sign the Majo- rana character of the neutrino and to distinguish between the different scenarios for the neutrino mass differences. Experimentally, the three modes show different electron energy spectra ((see figure 2 in Zdesenko, 2002)), the ββ 2ν and ββ 0ν,χ are characterized by a continuous spectrum ending at the maximum available energy Q ββ , while the ββ 0ν spectrum consists in a sharp peak at the end of the Q ββ spectrum. This should, in principle, make it easier the signature of this mode. In what follows we shall concentrate in the ββ 2ν and ββ 0ν modes. The theoretical expression of the half-life of the 2ν mode can be written as: [T 2ν 1/2 ] −1 = G 2ν [M 2ν GT [ 2 , (94) with M 2ν GT (J) = m ¸J + [[ σt − [[1 + m )¸1 + m [[ σt − [[0 + ) E m + E 0 (J) (95) (there is an implicit sum over all the nucleons). G 2ν contains the phase space factors and the axial coupling constant g A . The calculation of M 2ν GT requires the precise knowledge of the ground state of the parent nuclei and the ground state and occasionally a few excited states of the grand daughter, both even-even. Besides, it is necessary to have a good description of the Gamow- Teller strength functions of both of them, which implies a detailed description of the odd-odd intermediate nucleus. This is why this calculation is a challenge for the nuclear models, and why agreement with the experiment in this channel is taken as a quality factor to be applied to the predictions of the models for the neutrinoless mode. It is also a show-case of the use of the Lanczos strength function (LSF) method. It works as follows: Once the relevant wave functions of parent and grand daughter are obtained, it is straightforward to build the doorway states σt − [0 + initial ) and σt + [J + final ). In a second step, one of them is fragmented using LSF, producing, at iteration N, N 1 + states in the intermediate nucleus, with excitation energies E m . Overlapping these vectors with the other doorway, entering the appropriate energy denominators and adding up the N contributions gives an approximation to the exact value of M 2ν GT (N=1 is just the closure approximation). Finally, the number of iterations is increased until full convergence is reached. The method is very efficient, for instance, in the A=48 case, 20 iterations suffice largely. The contributions of the different intermediate states to the final matrix element are plotted in Fig. 56. FIG. 56 LSF for 48 Ca → 48 Ti 2ν decay. Each bar corresponds to a contribution to the matrix element. Notice the interfering positive and negative contributions. In the pf-shell there is only one double beta emitter, namely 48 Ca. For many years, the experimental information was limited to a lower limit on the 2ν-ββ half-life T 2ν 1/2 > 3.610 19 yr (Bardin et al., 1970). The calculation of the 48 Ca half-life was one of the first published results stemming from the full pf-shell calculations using the code ANTOINE. The resulting matrix elements are: M 2ν GT (0 + )=0.083 and M 2ν GT (2 + )=0.051. 51 For the ground state to ground state decay 48 Ca→ 48 Ti, G 2ν =1.110 −17 yr −1 (Tsuboi et al., 1984). The phase space factor hinders the transition to the 2 + that represents only about 3% of the total probability. Putting everything together, and using the Gamow- Teller quenching factor, already discussed, the resulting half-life is T 2ν 1/2 = 3.710 19 yr, (Caurier et al., 1990) (see also erratum in Caurier et al., 1994). The prediction was a success, because a later measure gave T 2ν 1/2 = 4.3 +2.4 −1.1 [stat]±1.4[syst] 10 19 yr (Balysh et al., 1996). Among the other ββ emitters in nature (around 30), only a few are potentially interesting for experiment since they have a Q ββ value, sufficiently large (≥ 2.5MeV ) for the 0ν signal not to be drowned in the surrounding natural radioactivity. With the exception of 150 Nd, all of them can be described within a shell model approach. The results for the 2ν mode of the lightest emitters 48 Ca, 76 Ge and 82 Se, for which full space calculations are doable, are gathered in table XVII (Caurier et al., 1996). The SMMC method has also been applied to the calculation of 2ν double beta decays in (Radha et al., 1996). TABLE XVII Calculated T 2ν 1/2 half-lives for several nuclei and 0 + →0 + transitions Parent 48 Ca 76 Ge 82 Se T 2ν 1/2 th.(y) 3.7 ×10 19 2.6 ×10 21 3.7 ×10 19 T 2ν 1/2 exp.(y) 4.3 ×10 19 1.8 ×10 21 8.0 ×10 19 The expression for the neutrinoless beta decay half-life, in the 0 + →0 + case, can be brought to the following form (Doi et al., 1985; Takasugi, 1981): [T (0ν) 1/2 (0 + − > 0 + ] −1 = G 0ν _ M (0ν) GT − _ g V g A _ 2 M (0ν) F _ 2 _ ¸m ν ) m e _ 2 where ¸m ν ) is the effective neutrino mass, G 0ν the kine- matic space factor and M (0ν) GT and M (0ν) F the following matrix elements (m and n sum over nucleons): M (0ν) GT = _ 0 + f _ _ _ n,m h(r)( σ n σ m )t n t m _ _ 0 + i _ (96) M (0ν) F = _ 0 + f _ _ _ n,m h(r)t n t m _ _ 0 + i _ , (97) where, due to the presence of the neutrino propagator, the “neutrino potential” h(r) is introduced. In this case, the matrix elements are just expectation values of two body operators, without a sum over intermediate states. This was believed to make the results less dependent of the nuclear model employed to obtain the wave function. An assumption that has not survived to the actual calculations. Full details of the calculations as well as predictions for other 0ν and 2ν decays in heavier double beta emitters can be found in Retamosa et al. (1995) and Caurier et al. (1996) (see also Suhonen and Civitarese, 1998, for a recent and very comprehensive review of the nuclear aspects of the double beta decay). The upper bounds on the neutrino mass resulting of our SM calculations, assuming a reference half-life T 0ν 1/2 ≥ 10 25 y. are collected in table XVIII. TABLE XVIII 0ν matrix elements and upper bounds on the neutrino mass for T 0ν 1/2 ≥ 10 25 y. mν in eV. Parent 48 Ca 76 Ge 82 Se M 0ν GT 0.63 1.58 1.97 M 0ν F -0.09 0.19 -0.22 mν 0.94 1.33 0.49 For the heavier emitters, some truncation scheme has to be employed and the seniority truncation seem to be the best, since the dimensions are strongly reduced in particular for 0 + states. An interesting feature in the calculation of the double beta decay matrix elements is shown in Fig. 57 for the 76 Ge case : the convergence of the M 0ν GT matrix element is displayed as a function of the truncation of the valence space, either with the seniority v or with the configurations t (t is the maximum number of particles in the g9 2 orbital). The t = 16 and v = 16 values correspond to the full space calculation and are consequently equal. The two truncation schemes show very distinct patterns, with the seniority truncation being more efficient. Such patterns have been also observed in the Tellurium and Xenon isotopes. This seems to indicate that the most favorable nuclei for the theoretical calculation of the 0ν mode would be the spherical emitters, where seniority is an efficient truncation scheme. C. Radii isotope shifts in the Calcium isotopes We have already seen in section VI.E that deformed np-nh configurations can appear at very low excitation energy around shell closures and even to become yrast in the case of neutron rich nuclei. This situation simply reflects the limitation of the spherical mean field description of the nucleus, and shows that even in magic cases, the correlations produce a sizeable erosion of the Fermi surface. This is the case in the Calcium isotopic chain. Experiments based on optical isotope shifts or muonic atom data, reveal that the nuclear charge radii ¸r 2 c ) follow a characteristic parabolic shape with a pronounced odd-even staggering (Fricke et al., 1995; Palmer, 1984). 52 0 5 10 15 Truncation 0 1 2 3 4 < m > v truncation t truncation FIG. 57 Variation of the mν limit as a function of seniority truncations (circles) and configuration truncations (diamonds) For the description of the nuclei around N=Z=20, a model space comprising the orbits 1d3 2 , 2s 1 2 , 1f 7 2 and 2p3 2 is a judicious choice (see Caurier et al., 2001a). The interaction is the same used to describe the neutron rich nuclei in the sd-pf valence space, called sdpf.sm in section VII, with modified single particle energies to reproduce the spectrum of 29 Si. Several important features of the nuclei at the sd/pf interface are reproduced by the calculation, among others, the excitation energies of the intruder 0 + 2 states in the Calcium isotopes as well as the location of the 3 2 + states in the Scandium isotopes (see Fig. 58), the excitation energies of the 2 + and 3 − and the B(E2)’s between the 2 + 1 and the 0 + 1 states in the Calcium isotopes. 40 42 44 46 48 A −1 0 1 2 3 4 5 E n e r g y ( M e V ) 3/2 + in Sc 0 2 + in Ca exp sm exp sm FIG. 58 Comparison between calculated and experimental intruder excitation energies in the Calcium and Scandium isotopic chains Due to the cross shell pairing interaction, protons and neutrons are lifted from the sd to the fp orbitals. The former produce an increase of ¸r 2 c ) that, using HO wave functions, can be expressed as: δr 2 c (A) = 1 Z n π pf (A)b 2 (98) where Z=20 for the calcium chain, b is the oscillator parameter, and n π pf is extracted from the calculated wave functions. The charge radii isotope shifts of the Calcium isotopes relative to 40 Ca are shown in Fig. 59, together with the experimental values. The global trends are very well reproduced, although the calculated shifts are a bit smaller than the experimental ones. This is probably due to the limitations in the valence space, that excludes the 1d5 2 , the 1f 5 2 and the 2p1 2 orbits. 40 42 44 46 48 A −0.1 0 0.1 0.2 0.3 I s o t o p e s h i f t [ r 2 ( A ) − r 2 ( 4 0 ) ] ( f m 2 ) FIG. 59 Isotopes shifts in Calcium chain: experimental (circles) versus shell model calculations (stars) D. Shell model calculations in heavier nuclei There are some regions of heavy nuclei in which physically sound valence spaces can be designed that are at the same time tractable. A good example is the space comprising the neutron orbits between N=50 and N=82 for the tin isotopes (Hjorth-Jensen et al., 1995; Nowacki, 1996). When protons are allowed the dimensions grow rapidly and the calculations have been limited until now to nuclei with few particle or holes on the top of the closed shells (see Covello et al., 1997, for a review of the work of the Napoli group). The SMMC, that can overcome these limitations, has been applied in this region to 128 Te and 128 Xe that are candidates to γ soft nuclei by Alhassid et al. (1996) and also to several Dysprosium isotopes A=152-162 in the Kumar-Baranger space by White et al. (2000). The QMCD method has been also applied to the study of the spherical to deformed transition in the even Barium isotopes with A=138-150 by Shimizu et al. (2001). Hints on the location of the hypothetical islands of super heavy elements are usually inspired in mean-field calculations of the single particle structure. The predictions 53 for nuclei far from stability can be tested also in nuclei much closer to stability, where shell model calculations are now feasible. In particular, recent systematic mean field calculations suggest a substantial gap for Z = 92 and N = 126 ( 218 U) corresponding to the 1h9 2 shell closure. On the other hand, the single quasiparticle energies extrapolated from lighter N=126 isotones up to 215 Ac, do not support this conclusion. To shed light on these controversial predictions, shell model calculations (as well as experimental spectroscopic studies of 216 Th (Hauschild et al., 2001)) were undertaken in the N = 126 isotones up to 218 U. Shell model calculations were performed in the 1h9 2 , 2f 7 2 , 1i 13 2 , 3p3 2 , 2f 5 2 , 3p1 2 proton valence space, using the realistic Kuo-Herling interaction (Kuo and Herling, 1971) as modified by Brown and Warburton (Warburton and Brown, 1991). The calculation reproduces nicely the ground state energies, the 2 + energy systematics as well as the high-spin trends. The cases of 214 Ra, 216 Th and 218 U are shown in figure 60. The only deviations between theory and experiment are in the 3 − energy and reflect the particular nature of theses states which are known to be very collective and corresponding to particle hole excitations of the 208 Pb core. No shell gap for Z=92, corresponding to the 1h9 2 closure, is predicted. On the contrary, the ground state of the N=126 isotones is characteristic of a superfluid regime with seniority zero components representing more than 95% of the wave functions for all nuclei. E. Random Hamiltonians The study of random Hamiltonians is a vast interdisciplinary subject that falls outside the scope of this review (see Porter, 1965, for a collection of the pioneering papers). Therefore here we shall only give a bibliographical guide to work that has recently attracted wide attention and may have consequences in future shell model studies. Johnson et al. (1998, 1999) noticed that random interactions had a strong tendency to produce J = 0 ground states. This is an empirical fact that was hitherto attributed to the pairing force. Bijker and Frank (2000a,b); Bijker et al. (1999) showed that in an interacting boson context this also occurred, and was associated to the typical forms of collectivity found in the IBM. For the fermion problem no collectivity occurs for purely random interactions (Horoi et al., 2001). The origin of J = 0 ground state-dominance was attributed to “geometric chaoticity” (Mulhall et al., 2000). The geometric aspects of the phenomenon were investigated in some simple cases by Zhao and Arima (2001); Zhao et al. (2001) and Chau Huu-Tai et al. (2003). Velázquez and Zuker (2002) argued that the general cause of J = 0 ground state-dominance was to be found in time-reversal invariance, and showed that when the random matrix elements were displaced to have a negative centroid, well developed rotational motion appeared in the valence spaces where the realistic interactions would also produce it (as 0 + 2 + 4 + 6 + 8 + 8 + 11 – 10 + 12 + 12 + 14 + 13 – 14 + 15 – 17 – 15 – 16 + (16 – ) 17 – 16 + 18 + 19 – 17 – 18 – 19 – 20 + 0 + 2 + 4 + 8 + 6 + 8 + 11 – 3 – 10 + 10 + 12 + 12 + 14 + 13 + 13 – 15 – 17 – 14 + 16 – 15 – 17 – 18 – 16 + 16 + 19 – 20 + 18 + 19 – 0 + (2 + ) (3 – ) (4 + ) (6 + ) (8 + ) (11 – ) (12 + ) (14 + ) 0 + 2 + 4 + 8 + 6 + 8 + 11 – 3 – 10 + 12 + 12 + 14 + 15 – 17 – 20 + 18 + 19 + 0 + 2 + 4 + 8 + 6 + 8 + 3 – 11 – 10 + 12 + 14 + EX SM2 EX SM2 SM2 214 88 Ra 126 216 90 Th 126 218 92 U 126 0 1000 2000 3000 4000 5000 E* [keV] FIG. 60 Experimental vs theoretical spectrum of 214 Ra, 216 Th, and 218 U in Fig. 39). This is very much in line with what was found by Cortes et al. (1982), and the interesting point is that the collective ingredient induced by the displacement of the matrix elements is the quadrupole force. It is unlikely that we shall learn much more from purely random Hamiltonians. However, the interplay of random and collective interactions may deserve further study. In particular: We know that a monopole plus pairing Hamil- tonian is a good approximation. Would it be a good idea to replace the rest of the interaction by a random one, instead of neglecting it? IX. CONCLUSION Nuclei are idiosyncratic, especially the lighter ones, accessible to shell model treatment. Energy scales that are well separated in other systems (as vibrational and rotational states in molecules) do overlap here, leading to strong interplay of collective and single particle modes. Nonetheless, secular behavior—both in the masses and the spectra—eventually emerges and the pf shell is the boundary region were rapid variation from nucleus to nucleus is replaced by smoother trends. As a consequence, larger calculations become associated with more transparent physics, and give hints on how to extend the shell model philosophy into heavier regions where exact diag- 54 onalizations become prohibitive. In this review we have not hesitated to advance some ideas on how this could be achieved, by suggesting some final solution to the monopole problem, and exploiting the formal properties of the Lanczos construction. The shell model has been craft and science: one invented model spaces and interactions and forced them on the spectra. Sometimes it worked very well. Then one wondered why such a phenomenology succeeded, to discover that there was not so much phenomenology after all. It is to be hoped that this state of affairs will persist. Acknowledgments This review owes much to many colleagues; in the first place to Joaqu´ın Retamosa who has been a member of the Strasbourg-Madrid Shell Model collaboration for many years, then to Guy Walter, Mike Bentley, Franco Brandolini, David Dean, Jean Duflo, Marianne Dufour, Luis Egido, José Mar´ıa Gómez, Hubert Grawe, Mar´ıa José Garc´ıa Borge, Morten Hjorth-Jensen, Karl- heinz Langanke, Silvia Lenzi, Peter Navr´ atil, Peter von Neumann-Cosel, Achim Richter, Luis Miguel Robledo, Dirk Rudolph, Jorge S´ anchez Solano, Olivier Sorlin, Carl Svensson, Petr Vogel to name only a few. This work has been partly supported by the IN2P3-France, CICyT- Spain agreements and by a grant of the MCyT-Spain, BFM2000-030. Many thanks are given to Benoit Speckel for his continuous computational support. APPENDIX A: Basic definitions and results • p is the principal oscillator quantum number • D p = (p +1)(p +2) is the (single fluid) degeneracy of shell p • Orbits are called r, s, etc. D r = 2j r + 1 • ˆ δ rs = δ jrjs but r ,= s, and both have the same parity. • m r is the number of particles in orbit r, T r is used for both the isospin and the isospin operator. In neutron-proton (np) scheme, m rx specifies the fluid x. Alternatively we use n r and z r . • V Γ rstu or 1 Γ rstu are two body matrix elements. W Γ rstu is used after the monopole part has been subtracted. A few equations have to exhibit explicitly angular momentum (J), and isospin (T) conservation. We use Bruce French’s notations (French, 1966): Γ stands for JT. Then (−1) Γ = (−1) J+T , [Γ] = (2J + 1)(2T + 1), and in general F(Γ) = F(J)F(T). Also(−1) r = (−1) jr+1/2 , [r] = 2(2j r + 1). Expressions carry to neutron-proton formalism simply by dropping the isospin factor. The one particle creation and anhilation operators A rrz = a † rrz B rrz = ˜ a rrz = (−1) r+rz a r−rz . (A1) can be coupled to quadratics in A and B, X † ΓΓz (rs) = (A r A s ) Γ Γz , X ΓΓz (rs) = (B r B s ) Γ Γz , S γ γz (rt) = (A r B t ) γ γz . (A2) Z † ΓΓz (rs) = (1 +δ rs ) −1/2 X † ΓΓz (rs) is the normalized pair operator and Z ΓΓz (rs) its Hermitian conjugate. For reduced matrix elements we use Racah’s definition < αα z [P γ γz [ββ z >= (A3) (−1) α−αz _ α γ β −α z γ z β z _ < α | P γ | β > . The normal and multipole representations of H are obtained through the basic recoupling −(X † Γ (rs)X Γ (tu)) 0 = −(−1) u+t−Γ _ Γ r _ 1/2 δ st S 0 ru + γ [Γγ] 1/2 (−1) s+t−γ−Γ _ r s Γ u t γ _ (S γ rt S γ su ) 0 , (A4) whose inverse is (S γ rt S γ su ) 0 = (−1) u−t+γ _ γ r _ 1/2 δ st S 0 ru − (A5) Γ [Γγ] 1/2 (−1) s+t−γ−Γ _ r s Γ u t γ _ (X † Γ (rs)X Γ (tu)) 0 . The “normal” representation of 1 is then 1 = r≤s t≤u,Γ 1 Γ rstu Z † rsΓ Z tuΓ = − (rstu)Γ ξ rs ξ tu [Γ] 1/2 1 Γ rstu (X † rsΓ X tuΓ ) 0 , (A6) where we have used ξ rs = _ (1 +δ rs ) −1/2 if r≤ s, (1 +δ rs ) 1/2 /2 if no restriction, (A7) so as to have complete flexibility in the sums. Accord- ing to (A4), 1 can be transformed into the “multipole” representation 1 = (rstu)γ ξ rs ξ tu _ [γ] 1/2 ω γ rtsu (S γ rt S γ su ) 0 +δ st ˆ δ ru [s] 1/2 ω 0 russ S 0 ru _ , (A8) where (sum only over Pauli-allowed Γ) 55 ω γ rtsu = Γ (−1) s+t−γ−Γ _ r s Γ u t γ _ W Γ rstu [Γ], (A9) W Γ rstu = γ (−1) s+t−γ−Γ _ r s Γ u t γ _ ω γ rtsu [γ]. (A10) Eq.(A5) suggests an alternative to (A8) 1 = (rstu)γ ξ rs ξ tu [γ] 1/2 ω γ rstu (A11) _ (S γ rt S γ su ) 0 −(−1) γ+r−s _ γ r _ 1/2 δ st ˆ δ ru S 0 ru _ , where each term is associated with a pure two body operator. APPENDIX B: Full form of Hm If we were interested only in extracting the diagonal parts of the monopole Hamiltonian in jt scheme, H mT , the solution would consist in calculating traces of H, and showing that they can be written solely in terms of number and isospin operators. Before we describe the technique for dealing with the non diagonal parts of H mT i.e. for generalizing to them the notion of centroid, some remarks on the interest of such an operation may be in order. The full H m contains all that is required for Hartree Fock (HF) variation, but it goes beyond. Minimizing the energy with respect to a determinantal state will invariably lead to an isospin violation because neutron and proton radii tend to equalize (Duflo and Zuker, 2002), which demands different orbits of neutrons and protons. Therefore, to assess accurately the amount of isospin violation in the presence of isospin-breaking forces we must ensure its conservation in their absence. More generally, a full diagonalization of H m is of great intrinsic interest. Now, the technical details. Define the generalized number and isospin operators S rs = ˆ δ rs [r] 1/2 S 00 rs , T rs = 1 2 ˆ δ rs [r] 1/2 S 01 rs (B1) which, for δ rs = 1, reduce to S rr = n r , T rr = T r . By definition H m contains the two body quadratic forms in S rs and T rs : S rtsu = ζ rs ζ tu (S rt S su −δ st S ru ), (B2a) T rtsu = ζ rs ζ tu (T rt T su − 3 4 δ st S ru ), (B2b) which in turn become for δ rt = δ su = 1, m rs and T rs in Eqs. (10a) and (10b) It follows that the form of H mT must be H mT = /+ all (a rtsu S rtsu −b rtsu T rtsu ) ˆ δ rt ˆ δ su , (B3) where the sum is over all possible contributions. This is a special Hamiltonian containing only λ = 0 terms. Trans- forming to the normal representation through Eq. (A10) ω 00 = δ λ0 δ τ0 =⇒1 J0 rstu = 1 J1 rstu = [rs] −1/2 ω 01 = δ λ0 δ τ1 =⇒1 J0 rstu = 3[rs] −1/2 , 1 J1 rstu = −[rs] −1/2 and therefore S rtsu = Γ Z † rsΓ Z tuΓ = (B4a) = J Z † rsJ0 Z tuJ0 + J Z † rsJ1 Z tuJ1 −T rtsu = 3 4 Z † rsJ0 Z tuJ0 − 1 4 Z † rsJ1 Z tuJ1 (B4b) and inverting J Z † rsJ0 Z tuJ0 = 1 4 (S rtsu −4T rtsu ) (B5a) J Z † rsJ1 Z tuJ1 = 1 4 (3S rtsu + 4T rtsu ). (B5b) When j r = j s = j t = j u and r ,= s and t ,= u, both S rust , T rust and S rtsu , T rtsu are present. They can be calculated from (B5a) and (B5b) by exchanging t and u, Z † rsJ0 Z utJ0 = − (−1) J Z † rsJ0 Z tuJ0 = = 1 4 (S rust −4T rust ) (B6a) Z † rsJ1 Z utJ1 = (−1) J Z † rsJ1 Z tuJ1 = = 1 4 (3S rust + 4T rust ) (B6b) and by combining (B5a), (B5b), (B6a) and (B6b) J Z † rsJ0 Z tuJ0 (1 ±(−1) J ) 2 = (B7a) 1 8 ((S rtsu −4T rtsu ) ∓(S rust −4T rust )) J Z † rsJ1 Z tuJ1 (1 ±(−1) J ) 2 = (B7b) 1 8 ((3S rtsu + 4T rtsu ) ±(3S rust + 4T rust )) 56 To write H mT we introduce the notations Φ(P) = 1 −(1 −δ rs )(1 −δ tu ), (B8) Φ(e) ≡ ( ˆ δ rs −δ rs )( ˆ δ tu −δ tu ), (B9) So for r = s or t = u, Φ(P) = 1 and for j r = j s = j t = j u and Φ(P) = 0, Φ(e) = 1. Then H mT = /+ r≤s t≤u T,ρ=± ˆ δ rt ˆ δ su _ (1 −Φ(e))1 T rstu Ω T rstu + Φ(e)1 ρT rstu Ω ρT rstu _ ρ = sign (−1) J , Ω T rstu = J Z † rsJT Z tuJT Ω ±T rstu = J Z † rsJT Z tuJT (1 ±(−1) J ) 2 . (B10) The values of the generalized centroids 1 T rstu and 1 ρT are determined by demanding that H−H mT = H M contain no contributions with λ = 0. In other words W JT rstu = 1 JT rstu − − ˆ δ rt ˆ δ su _ (1 −Φ(e))1 T rstu + Φ(e)1 ρT rstu _ , (B11) must be such that ω 0τ rtsu = 0, and from Eq. (A10) (J) [J]W JT rstu = 0 ∴ 1 T rstu = (J) [J]1 JT rstu / (J) [J]. (B12) Applying this prescription to all the terms leads to (obviously ˆ δ rt ˆ δ su = 1 in all cases) 1 T rstu = (J) 1 JT rstu [J]/ (J) [J] (J) [J] = 1 4 D r (D s + 2Φ(P)(−1) T ) 1 + Φ(P) 1 ±T rstu = J 1 JT rstu [J](1 ±(−1) J )/ J [J](1 ±(−1) J ) J [J](1 ±(−1) J ) = 1 4 D r (D r ∓2) D r = [r], Φ(e) = 1, for 1 ±T rstu . (B13) Through eqs.(B5a), (B5b), (B7a), and (B7b) we can obtain the form of H mT in terms of the monopole operators by regrouping the coefficients affecting each of them. To simplify the presentation we adopt the following convention _ α ≡ rstu r ≤ s, t ≤ u, ˆ δ rt ˆ δ su = 1, BUT S α = S rtsu , S ¯ α = S rust , T α = T rtsu , T ¯ α = T rust , then H mT = / + α (1 −Φ(e))(a α S α +b α T α ) +Φ(e)(a d α S α +b d α T α +a e α S ¯ α +b e α T ¯ α ), with (B14a) a α = 1 4 (3 ¯ 1 1 α + ¯ 1 0 α ) b α = 1 4 ( ¯ 1 1 α − ¯ 1 0 α ) (B14b) a d α = 1 8 (3 ¯ 1 +1 α + 3 ¯ 1 −1 α + ¯ 1 +0 α + ¯ 1 −0 α ) a e α = 1 8 (3 ¯ 1 +1 α −3 ¯ 1 −1 α − ¯ 1 +0 α + ¯ 1 −0 α ) b d α = 1 2 (3 ¯ 1 +1 α + 3 ¯ 1 −1 α − ¯ 1 +0 α − ¯ 1 −0 α ) b e α = 1 2 (3 ¯ 1 +1 α − ¯ 1 −1 α + ¯ 1 +0 α − ¯ 1 −0 α ) (B14c) 1. Separation of Hmnp and Hm0 In j formalismH mnp is H mT under another guise: neutron and proton shells are differentiated and the operators T rs and S rs are written in terms of four scalars S rxsy ; x, y = n or p. We may also be interested in extracting only the purely isoscalar contribution to H mT , which we call H m0 . The power of French’s product notation becomes particularly evident here, because the form of both terms is identical. It demands some algebraic manipulation to find H mnp or H m0 = / + α _ ¯ 1 α S α (1 −Φ(e))+ + 1 2 (( ¯ 1 + α + ¯ 1 − α )S α + ( ¯ 1 − α − ¯ 1 + α )S ¯ α )Φ(e) _ (B15) with 1 rstu = (Γ) 1 Γ rstu [Γ]/ (Γ) [Γ] (Γ) [Γ] = D r (D s −Φ(P))/(1 + Φ(P)) 1 ± rstu = Γ 1 rstu [Γ](1 ±(−1) Γ+2r ) Γ [Γ](1 ±(−1) Γ+2r ) = D r (D r ∓(−1) 2r ) (B16) Of course we must remember that for H mnp , D r = 2j r +1, (−1) 2r = −1, Γ ≡ J, etc., while for H m0 D r = 2(2j r + 1), (−1) 2r = +1, Γ ≡ JT; etc. It should be noted that H m0 is not obtained by simply discarding the b coefficients in eqs.(B14), because we can extract some γ =00 contribution from the T α operators. The point will become quite clear when considering the diagonal contributions. 57 2. Diagonal forms of Hm We are going to specialize to the diagonal terms of H mT and H mnp , which involve only 1 Γ rsrs matrix elements whose centroids will be called simply 1 rs and 1 T rs (the overline in 1 T rstu was meant to avoid confusion with possible matrix elements 1 1 rstu or 1 0 rstu , it can be safely dropped now). Then (B14) becomes Eq. (11) and (B15) becomes H d mnp or H d m0 = /+ r≤s 1 rs n r (n s −δ rs )/(1+δ rs ) (B17) We rewrite the relevant centroids incorporating explicitly the Pauli restrictions 1 rs = Γ 1 Γ rsrs [Γ](1 −(−1) Γ δ rs ) D r (D s −δ rs ) (B18a) 1 T rs = 4 J 1 JT rsrs [J](1 −(−1) J+T δ rs ) D r (D s + 2δ rs (−1) T ) (B18b) a rs = 1 4 (31 1 rs +1 0 rs ) = 1 rs + 3 4 δ rs D r −1 b rs b rs = 1 1 rs −1 0 rs (B18c) The relationship between a rs and 1 rs makes it possible to combine eqs. (11) and (B17) in a single form H d m = / + 1 (1 +δ rs ) _ 1 rs n r (n s −δ rs ) + +b rs _ T r T s − 3n r ¯ n r 4(D r −1) δ rs __ (B19) in which now the b rs term can be dropped to obtain H d m0 or H d mnp . In the np scheme each orbit r goes into two r n and r p and the centroids can be obtained through (x, y = n or p, x ,= y) 1 rxsy = 1 2 _ 1 1 rs _ 1 − 2δ rs D r _ +1 0 rs _ 1 + 2δ rs D r __ 1 rxsx = 1 1 rs (B20) Note that the diagonal terms depend on the representation: H d mnp ,= H d mT in general. APPENDIX C: The center of mass problem What do you do about center of mass? is probably the standard question most SM practitioners prefer to ignore or dismiss. Even if we may be tempted to do so, there is no excuse for ignoring what the problem is, and here we shall explain it in sufficient detail, so as to dispel some common misconceptions. The center of mass (CM) problem arises because in a many body treatment it is most convenient to work with A coordinates and momenta while only A−1 of them can be linearly independent since the solutions cannot depend on the center of mass coordinate R = ( i r i )/ √ A or momentum P = ( i p i )/ √ A. The way out is to impose a factorization of the wavefunctions into relative and CM parts: Φ(r 1 r 2 . . . r A ) = Φ rel φ CM . The potential energy is naturally given in terms of relative values, and for the kinetic energy we should do the same by referring to the CM momentum, i (p i − P √ A ) 2 = i p 2 i −P 2 = 1 A ij (p i −p j ) 2 , (C1) and change accordingly / ij in Eq. (2). As we are only interested in wavefunctions in which the center of mass is at rest (or in its lowest possible state), we can add a CM operator to the Hamiltonian H =⇒H+λ(R 2 +P 2 ), and calling r ij = r i −r j we have R 2 +P 2 = i (r 2 i +p 2 i ) − 1 A ij (r 2 ij +p 2 ij ), (C2) so that upon diagonalization the eigenvalues will be of the form E = E rel + λ(N CM + 3/2), and it only remains to select the states with N CM =0. We are taking for granted separation of CM and relative coordinates. Unfortunately, this happens only for spaces that are closed under the CM operator (C2): for N CM to be a good quantum number the space must include all possible states with N CM oscillator quanta. The problem was raised by Elliott and Skyrme (1955) who initiated the study of “particle-hole” (ph) excitations on closed shells. They noted that acting with R−iP on the IPM ground state of 16 O ([0)) leads to _ 1 18 (¯ p 1 s 1 − √ 2¯ p 3 s 1 − √ 5¯ p 1 d 3 + ¯ p 3 d 3 +3¯ p 3 d 5 )[0), (C3) where ¯ p 2j removes a particle and s 1 or d 2j add a particle on [0). As R−iP has tensorial rank J π T = 1 − 0, Eq. (C3) is telling us that out of five possible 1 − 0 excitations, one is “spurious” and has to be discarded. Assume now that we are interested in 2p2p excitations. They involve jumps of two oscillator quanta (2ω ) and the CM eigenstates (R − iP) 2 [0) involve the operator in Eq. (C3), but also jumps to other shells, of the type ¯ s p¯ p (sd) ≡ ¯ s (sd) or ( ¯ sd) (pf)¯ p (sd) ≡ ¯ p (pf). Therefore, as anticipated, relative-CM factorization can be achieved only by including all states involving a given number of oscillator quanta. The clean way to proceed is through complete Nω spaces, discussed in Sec- tions II.A.1 and III.F. The 0ω and EI spaces are also free of problems (the latter because no 1ω 1 − 0 states exist). It remains to analyze the EEI valence spaces, where CM spuriousness is always present but strongly suppressed because the main contributors to R −iP –of the type pj =⇒ p + 1j ± 1, with the largest j– are always excluded. Consider EEI(1)=p 1/2 , d 5/2 , s 1/2 . The 58 only possible 1ω 1 − 0 state is ¯ p 1 s 1 , which according to Eq. (C3) accounts for (1/18)% = 5.6% of the spurious state. This apparently minor problem was unduly transformed into a serious one through a proposal by Gloeck- ner and Lawson (1974). It amounts to project CM spuriousness through the H =⇒H+λ(R 2 +P 2 ) prescription in the EEI(1) space by identifying R−iP ≡ ¯ p 1 s 1 . The procedure is manifestly incorrect, as was repeatedly pointed out (see for instance Whitehead et al., 1977) but the misconception persists. An interesting and viable alternative was put forward by Dean et al. 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Polls (Springer, Berlin), volume 209 of Lec- ture Notes in Physics, p. 157. Zuker, A. P., 2001, Phys. Rev. C 64, 021303(R). Zuker, A. P., 2003, Phys. Rev. Lett. 90, 042502. Zuker, A. P., B. Buck, and J. B. McGrory, 1968, Phys. Rev. Lett. 21, 39. Zuker, A. P., B. Buck, and J. B. McGrory, 1969, Shell Model Wavefunctions for A=15-18 Nuclei, Technical Report PD- 99 BNL 14085, Brookhaven National Laboratory. Zuker, A. P., S. Lenzi, G. Mart´ınez-Pinedo, and A. Poves, 2002, Phys. Rev. Lett. 89, 142502. Zuker, A. P., J. Retamosa, A. Poves, and E. Caurier, 1995, Phys. Rev. C 52, R1741. Zuker, A. P., L. Waha-Ndeuna, F. Nowacki, and E. Caurier, 2001, Phys. Rev. C 64, 021304(R). 2 IX. Conclusion Acknowledgments A. Basic definitions and results B. Full form of Hm 1. Separation of Hmnp and Hm0 2. Diagonal forms of Hm C. The center of mass problem References 53 54 54 55 56 57 57 58 I. INTRODUCTION In the early days of nuclear physics, the nucleus, composed of strongly interacting neutrons and protons confined in a very small volume, didn’t appear as a system to which the shell model, so successful in the atoms, could be of much relevance. Other descriptions—based on the analogy with a charged liquid drop—seemed more natural. However, experimental evidence of independent particle behavior in nuclei soon began to accumulate, such as the extra binding related to some precise values of the number of neutrons and protons (magic numbers) and the systematics of spins and parities. The existence of shell structure in nuclei had already been noticed in the thirties but it took more than a decade and numerous papers (see Elliott and Lane, 1957, for the early history) before the correct prescription was found by Mayer (1949) and Axel, Jensen, and Suess (1949). To explain the regularities of the nuclear properties associated to “magic numbers”—i.e., specific values of the number of protons Z and neutrons N —the authors proposed a model of independent nucleons confined by a surface corrected, isotropic harmonic oscillator, plus a strong attractive spin-orbit term1 , U (r) = 1 ω r2 + D l 2 + C l · s. 2 (1) As the number of protons and neutrons depart from the magic numbers it becomes indispensable to include in some way the “residual” two-body interaction, to break the degeneracies inherent to the filling of orbits with two or more nucleons. At this point difficulties accumulate: “Jensen himself never lost his skeptical attitude towards the extension of the single-particle model to include the dynamics of several nucleons outside closed shells in terms of a residual interaction” (Weidenm¨ ller, u 1990). Nonetheless, some physicists chose to persist. One of our purposes is to explain and illustrate why it was worth persisting. The keypoint is that passage from the IPM to the interacting shell model, the shell model (SM) for short, is conceptually simple but difficult in practice. In the next section we succinctly review the steps involved. Our aim is to give the reader an overall view of the Shell Model as a sub-discipline of the Many Body problem. “Inverted commas” are used for terms of the nuclear jargon when they appear for the first time. The rest of the introduction will sketch the competing views of nuclear structure and establish their connections with the shell model. Throughout, the reader will be directed to the sections of this review where specific topics are discussed. A. The three pillars of the shell model In modern language this proposal amounts to assume that the main effect of the two-body nucleon-nucleon interactions is to generate a spherical mean field. The wave function of the ground state of a given nucleus is then the product of one Slater determinant for the protons and another for the neutrons, obtained by filling the lowest subshells (or “orbits”) of the potential. This primordial shell model is nowadays called “independent particle model” (IPM) or, “naive” shell model. Its foundation was pioneered by Brueckner (1954) who showed how the short range nucleon-nucleon repulsion combined with the Pauli principle could lead to nearly independent particle motion. The strict validity of the IPM may be limited to closed shells (and single particle –or hole– states built on them), but it provides a framework with two important components that help in dealing with more complex situations. One is of mathematical nature: the oscillator orbits define a basis of Slater determinants, the m-scheme, in which to formulate the Schr¨dinger problem in the oco cupation number representation (Fock space). It is important to realize that the oscillator basis is relevant, not so much because it provides an approximation to the individual nucleon wave-functions, but because it provides the natural quantization condition for self-bound systems. Then, the many-body problem becomes one of diagonalizing a simple matrix. We have a set of determinantal states for A particles, a† . . . a† |0 = |φI , and a Hamili1 iA tonian containing kinetic K and potential energies V H= ij Kij a† aj − i i≤j k≤l Vijkl a† a† ak al i j (2) 1 We assume throughout adimensional oscillator coordinates, i.e., r −→ (mω/ )1/2 r. that adds one or two particles in orbits i, j and removes one or two from orbits k, l, subject to the Pauli principle ({a† aj } = δij ). The eigensolutions of the probi lem |Φα = I cI,α |φI , are the result of diagonalizing the matrix φI |H|φI ′ whose off-diagonal elements are either 0 or ±Vijkl . However, the dimensionalities of the matrices—though not infinite, since a cutoff is inherent to a non-relativistic approach—are so large as to make the problem intractable, except for the lightest nuclei. Stated in these terms, the nuclear many body problem 3 does not differ much from other many fermion problems in condensed matter, quantum liquids or cluster physics; with which it shares quite often concepts and techniques. The differences come from the interactions, which in the nuclear case are particularly complicated, but paradoxically quite weak, in the sense that they produce sufficiently little mixing of basic states so as to make the zeroth order approximations bear already a significant resemblance with reality. This is the second far-reaching –physical– component of the IPM: the basis can be taken to be small enough as to be, often, tractable. Here some elementary definitions are needed: The (neutron and proton) major oscillator shells of principal quantum number p = 0, 1, 2, 3 . . ., called s, p, sd, pf . . . respectively, of energy ω(p + 3/2), contain orbits of total angular momentum j = 1/2, 3/2 . . . p + 1/2, each with its possible jz projections, for a total degeneracy Dj = 2j + 1 for each subshell, and Dp = (p + 1)(p + 2) for the major shells (One should not confuse the p = 1 shell —p-shell in the old spectroscopic notation—with the generic harmonic oscillator shell of energy ω(p + 3/2)). When for a given nucleus the particles are restricted to have the lowest possible values of p compatible with the Pauli principle, we speak of a 0 ω space. When the many-body states are allowed to have components involving basis states with up to N oscillator quanta more than those pertaining to the 0 ω space we speak of a N ω space (often refered as “no core”) For nuclei up to A ≈ 60, the major oscillator shells provide physically meaningful 0 ω spaces. The simplest possible example will suggest why this is so. Start from the first of the magic numbers N = Z = 2; 4 He is no doubt a closed shell. Adding one particle should produce a pair of single-particle levels: p3/2 and p1/2 . Indeed, this is what is found in 5 He and 5 Li. What about 6 Li? The lowest configuration, p2 , should produce four states of angular 3/2 momentum and isospin JT = 01, 10, 21, 30. They are experimentaly observed. There is also a JT = 20 level that requires the p3/2 p1/2 configuration. Hence the idea of choosing as basis pm (p stands generically for p3/2 p1/2 ) for nuclei up to the next closure N = Z = 8, i.e., 16 O. Obviously, the general Hamiltonian in (2) must be transformed into an “effective” one adapted to the restricted basis (the “valence space”). When this is done, the results are very satisfactory (Cohen and Kurath, 1965). The argument extends to the sd and pf shells. By now, we have identified two of the three “pillars” of the Shell Model: a good valence space and an effective interaction adapted to it. The third is a shell model code capable of coping with the secular problem. In Section I.C we shall examine the reasons for the success of the classical 0 ω SM spaces and propose extensions capable of dealing with more general cases. The interaction will touched upon in Section I.B.3 and discussed at length in Section II. The codes will be the subject of Section III. B. The competing views of nuclear structure Because Shell Model work is so computer-intensive, it is instructive to compare its history and recent developments with the competing—or alternative—views of nuclear structure that demand less (or no) computing power. 1. Collective vs. Microscopic The early SM was hard to reconcile with the idea of the compound nucleus and the success of the liquid drop model. With the discovery of rotational motion (Bohr, 1952; Bohr and Mottelson, 1953) which was at first as surprising as the IPM, the reconciliation became apparently harder. It came with the realization that collective rotors are associated with “intrinsic states” very well approximated by deformed mean field determinants (Nilsson, 1955), from which the exact eigenstates can be extracted by projection to good angular momentum (Peierls and Yoccoz, 1957); an early and spectacular example of spontaneous symmetry breaking. By further introducing nuclear superfluidity (Bohr et al., 1958), the unified model was born, a basic paradigm that remains valid. Compared with the impressive architecture of the unified model, what the SM could offer were some striking but isolated examples which pointed to the soundness of a many-particle description. Among them, let us mention the elegant work of Talmi and Unna (1960), the n f7/2 model of McCullen, Bayman, and Zamick (1964)— probably the first successful diagonalization involving both neutrons and protons—and the Cohen and Kurath (1965) fit to the p shell, the first of the classical 0 ω regions. It is worth noting that this calculation involved spaces of m-scheme dimensionalities, dm , of the order of 100, while at fixed total angular momentum and isospin dJT ≈ 10. The microscopic origin of rotational motion was found by Elliott (1958a,b). The interest of this contribution was immediately recognized but it took quite a few years to realize that Elliott’s quadrupole force and the underlying SU(3) symmetry were the foundation, rather than an example, of rotational motion (see Section VI). It is fair to say that for almost twenty years after its inception, in the mind of many physicists, the shell model still suffered from an implicit separation of roles, which assigned it the task of accurately describing a few specially important nuclei, while the overall coverage of the nuclear chart was understood to be the domain of the unified model. A somehow intermediate path was opened by the well known Interacting Boson Model of Arima and Iachello (1975) and its developments, that we do not touch here. The interested reader can find the details in (Iachello and Arima, 1987). This limitation was overcome when Vautherin and Brink (1972) and Decharg´ and Gogny (1980) initie ated the two families of Hartree Fock (HF) calculations (Skyrme and Gogny for short) that remain to this day the only tools capable of giving microscopic descriptions throughout the periodic table. The former lead to the “Universal sd” (USD) interaction (Wildenthal. Then. Here. Moreover the work done exhibited serious problems which could be traced to the realistic matrix elements themselves (see next Section I. involving both neutrons and protons were done3 . but enough has happened in the last decade to transform it into a unified view. as it was restricted to valence spaces of two contiguous major shells.. Girod. was initiated in Eduardo Pasquini PhD thesis (1976). 2001. and Wong (1969) came almost simultaneously.1 and IV. Heenen. Therefore problems with calculations involving only two body forces must be blamed on the absence of three body forces. They made it possible to describe the neighborhood of 16 O (Zuker. They were later joined by the relativistic HF approach (Serot and Walecka. II. the results became disastrous. Egido. 1968). there were still problems with the monopole way: they showed around and beyond 56 Ni. 1968) and the lower part of the sd shell (Halbert et al. The first is: The case for realistic two-body interactions is so strong that we have to accept them as they are. We will only anticipate on the conclusions of these sections through two syllogisms. 2003b). The second. for a recent review). Adding a pairing interaction and a spherical mean field they proceeded to perform Hartree Fock Bogoliubov calculations in the first of a successful series of papers (Kumar and Baranger.A. 1971). the one for which closest contact with realistic interactions and the shell model can be established (Sections VI. 1988). which at the time meant dJT ≈ 100-600. where the first calculations in the full pf shell.2: The exciting prospect that realistic matrix elements could lead to parameter-free spectroscopy did not materialize. as it provides such a good approximation to the wavefunctions (typically about 50%) it suggests efficient truncation schemes.D). This dramatic event explains why his work is only publicly available in condensed form (Pasquini and Zuker. Buck. 1984). Its free parameters must be few and well defined.B. Here we introduce the first: A unified view requires a unique interaction.B. 1986). which we adopt here. and the role-separation mentioned above persisted. and Roba ledo (2000) and references therein for recent work with the Gogny force. Little is known about threebody forces except that they exist. However. Many steps—outlined in the next Section I. which could allow a replacement of the unified model description by a fully microscopic one. because we touch upon a different competing view. that enjoyed an immense success and for ten years set the standard for shell model calculations in a large valence space (Brown and Wildenthal.B and V. In the other. many regions remain outside the direct reach of the Shell Model. Realistic vs Phenomenological The first realistic2 matrix elements of Kuo and Brown (1966) and the first modern shell model code by French. as well as in the impossibility to be competitive with USD in the sd shell (note that USD contains non-monopole two-body corrections to the realistic matrix elements). (see Brown.B. Nevertheless. However. Two months after his PhD.4 2. 1978). Pasquini and his wife “disappeared” in Argentina. Their work could be described as shell model by other means.B. . (For a review of the three variants see Bender.3). Mean field vs Diagonalizations 3. everybody admits the need of going beyond the mean field. all matrix elements were considered to be free parameters. The solution came about only recently with the introduction of three-body forces: This far reaching development will be explained in detail in Sections II. and e Berger (2000) and Rodriguez-Guzmń. McGrory. Nowadays. However. See also P´ru. two schools of thought on the status of phenomenological corrections emerged: In one of them. only average matrix elements (“centroids”)—related to the bad saturation and shell formation properties of the two-body potentials—needed to be fitted (the monopole way). so as to make the calculations independent of the quantities they are meant to explain.2. the increases in tractable dimensionalities were insufficient to promote the shell model to the status of a general description. based on mean-field theory. As the growing sophistication of numerical methods allowed to treat an increasing number of particles in the sd shell. Since single determinantal states can hardly be expected accurately to describe many body solutions. and McGrory. we have to recall the remarks at the end of the first paragraph of the preceding Section I. This argu- 3 2 Realistic interactions are those consistent with data obtained in two (and nowadays three) nucleon systems. as will be explained in Sections IV. of special interest for shell model experts. The first breakthrough came when Baranger and Kumar (1968) proposed a new form of the unified model by showing that Elliott’s quadrupole force could be somehow derived from the Kuo-Brown matrix elements.D—are needed to substantiate this claim. and Reinhard. Twenty years later. we know that the minimal monopole corrections proposed in his work are sufficient to provide results of a quality comparable to those of USD for nuclei in f7/2 region. a fundamental idea emerged at the time: the existence of an underlying universal two-body interaction. Halbert.A and II. and opened the way for the first generation of “large scale” calculations. 5 ment raises (at least) one question: What to make of all calculations. 1991). d5/2 .. For a single major shell. the other ones are split. and indeed found two. fitted interactions can differ from the realistic ones basically through monopole matrix elements.C). In section V. d3/2 . The EI spaces are well established standards when only one “fluid” (proton or neutron) is active: The Z or N = 28. proposed by Honma et al. where p> is the largest subshell having j = p + 1/2. so to speak. the classical 0 ω spaces. Remember that in 5 Li we expected to find two states. exact solutions are now available from which instructive conclusions can be drawn. 1991) interactions turn out to be R-compatible. 1984) interaction is R-incompatible. (2002). for instance.B. others so as to generate wavefunctions that can evolve to the exact ones through low order perturbation theory. leading to “coexistence” between spherical and deformed states. For heavier nuclei exact diagonalizations are not possible but the EEI spaces provide simple (and excellent) estimates of quadrupole moments at the beginning C. it turns out that above A = 46. Now. . 1965). These nuclei are “spherical”. 1979) and the FPD6 (Richter et al. It is only recently that the EEI(2) (s1/2 . i. For the 0 ω spaces.1.2). . Here. At best. we can define the EI (extruder-intruder) spaces as EI(p)= rp ⊕ (p + 1)> : The (p + 1)> orbit is expelled from HO(p + 1) by the spin-orbit interaction and intrudes into HO(p). as we shall see in Section IV. in principle.. “deformation” effects become appreciable. The history of the Shell Model is that of the interplay between experiment and theory to establish the validity of this concept. However. 1990). which are needed to account for rotational motion as explained in Section VI. A configuration is a set of states having fixed number of particles in each orbit.e. the f7/2 ground state is the only one that is a pure single particle state. the tip of the iceberg in terms of number of basic states involved. They can be solved reasonably well by phenomenological changes of the monopole matrix elements. but not yet released. 1968. amenable to exact diagonalizations and fairly well understood (see for example Abzouzi et al. Consider. The two syllogisms become fully consistent by noting that the inclusion of three-body monopole terms always improves the performance of the forces that adopt the monopole way. .1 contains an interesting example of the splitting mechanism). The USD (Wildenthal. Conceptually the valence space may be thought as defining a representation intermediate between the Schr¨dinger one (the operators are fixed and o the wavefunction contains all the information) and the Heisenberg one (the reverse is true). even severely in the case of the highest (f5/2 ) (Endt and van der Leun. To cope with this situation we propose “Extended EI spaces” defined as EEI(p)= rp ⊕∆p+1 . p3/2 ) space (region around 40 Ca) has become tractable (see Sections VI. The ones provided by the sole valence space are. how can we expect the pf shell to be a good valence space? For few particles above 40 Ca it is certainly not. and rp is the “rest” of the HO pshell. Two contiguous major shells can most certainly cope with all levels of interest but they lead to intractably large spaces and suffer from a “Center of Mass problem” (analyzed in Appendix C). As soon as both fluids are active. However. 50. where ∆p = p> ⊕(p> −2)⊕. the Chung Wildenthal (Wildenthal and Chung. The valence space The choice of the valence space should reflect a basic physical fact: that the most significant components of the low lying states of nuclei can be accounted for by many-body states involving the excitation of particles in a few orbitals around the Fermi level. as we report in this work.B.D and VIII. as opposed to the HO (harmonic oscillator) ones (for shell formation see Section II. that give satisfactory results? The answer lies in the second syllogism: All clearly identifiable problems are of monopole origin. we expect four states.C. If we separate a major shell as HO(p)= p> ⊕ rp . f7/2 . EEI(3) is the natural space for the proton rich region centered in 80 Zr. s1/2 ) space was successful in describing the full low lying spectra for A = 15-18 (Zuker et al. Therefore. Our present understanding can be roughly summed up by saying that “few” mean essentially one or two contiguous major shells4 . For instance. the lowest (pf )m configurations 5 are sufficiently detached from all 4 5 Three major shells are necessary to deal with super-deformation. The ultimate ambition of Shell Model theory is to get exact solutions. Moreover. a physically sound pruning of the space is suggested by the IPM. 1969).B. (We speak of R-compatibility in this case). What made the success of the model is the explanation of the observed magic numbers generated by the spin-orbit term. They are certainly in the data. they must be retrieved from a jungle of some 140 levels seen below 7 MeV. 82 and 126 isotopes or isotones. using only two-body forces.the spectrum of 41 Ca from which we want to extract the singleparticle states of the pf shell. This does not mean that a magic nucleus is 100% closed shell: 50 or 60% should be enough. Thus. 0 ω spaces can describe only a limited number of low-lying states of the same (“natural”) parity.1 it will be argued that the 0 ω valence spaces account for basically the same percentage of the full wavefunctions... The rest may be so well hidden as to make us believe in the literal validity of the shell model description in a restricted valence space. the magic closed shells are good valence spaces consisting of a single state. It will be of interest to analyze the recent set of pf -shell matrix elements (GXPF1).. the ∆j = 2 sequence of orbits that contain p> . The EEI(1) (p1/2 . the Cohen Kurath (Cohen and Kurath. and eventually to dominance of the latter. (Section V. Roughly half of the–order of magnitude–gains are due to increases in computing power. A conjunction of factors make light and medium-light nuclei near the neutron drip line specially interesting: (a) They have recently come under intense experimental scrutiny. the classical 0 ω calculations. which falls outside the scope of this review.2. Special attention is given to Gamow Teller transitions. References are only given for work that will not be mentioned later. Configurations involving two major oscillator shells are also shown to account well for the appearance of superdeformed excited bands. 2000). Euroball and Gammasphere has made it possible to access high spin states in medium mass nuclei for which full 0 ω calculations are available.1 and IV. The new generation of gamma detectors. c) The universality of the multipole Hamiltonian. Section III. (d) They achieve the highest N/Z ratios attained. (e) When all (or most of) the valence particles are neutrons. d) The description of the collective behavior in the laboratory frame. In practice. The former is in charge of saturation and shell properties.B. Finally. dictated by the isovector channel of the nuclear interaction alone. such as halos and sudden onset of deformation. The Lanczos construction can be used to eliminate the “black box” aspect of the diagonalizations. and neutrinoless ββ decay is one of the main sources of information about the neutrino masses. truncations are introduced. Two recent reviews by Brown (2001) and Otsuka et al. such as approximation schemes discussed in Sections IV. of great use in other disciplines. for instance. Only by the inclusion of three-body forces could a satisfactory formulation be achieved. backbending. dJ ∼ 107 nowadays. The last section deal with these subjects. evolution operator and level densities.. Section II. yrast traps. may differ from those at the stability valley. Section VII. some selected examples of pf shell spectroscopy are presented. IV and the Appendices are based on unpublished notes by APZ. (2002). e) The description of resonances using the Lanczos strength function method.B. For nearly unbound nuclei. and in Id Betan et al. These regions propose some exacting tests for the theoretical descriptions. Section V. The basic tool in analyzing the interaction is the monopole-multipole separation. it can be thought as the correct generalization of Eq. deformed rotors. that the conventional shell model calculations have passed satisfactorily. b) The explanation of the global properties of nuclei via the monopole Hamiltonian. and some others. Their remarkable harvest includes a large spectrum of collective manifestations which the spherical shell model can predict or explain as. There is a characteristic of the Shell Model we have not yet stressed: it is the approach to nuclear structure that can give more precise quantitative information. They may be based on systematic approaches. References to the subject can be found in Bennaceur et al. As this is a very recent and fundamental development which makes possible a unified viewpoint of the monopole field concept. The ANTOINE and NATHAN codes have made it possible to evolve from dimensionalities dm ∼ 5 × 104 in 1989 to dm ∼ 109 .2 or mean field methods mentioned in Section I.A. Another major recent achievement is the shell model description of rotational nuclei. etc. Monopole theory is scattered in many references. The “residual” multipole force has been extensively described in a single reference which will be reviewed briefly and updated. to a large extent. in particular. (b) They are amenable to shell model calculations. band terminations. and analyzed in Section VI.. However we did not feel dispensed from quoting and commenting in some detail important work that bears directly on the subjects we treat. Appendix B contains a full derivation of the general form of the monopole field. this section is largely devoted to it. Section VI. As we have presented it. by means of the spherical shell model. it describes strength functions with maximal efficiency. (2002.C). We show how it can be related to the notions of partition function. After describing the three body mechanism which solves the monopole problem that had plagued . Furthermore. the spherical shell model closures. (1999. 2003) that deal with the Gamow shell model. when exact diagonalizations are impossible. it can be turned into a powerful truncation method by combining it to coupled cluster theory.B. shell model calculations play a central role. sometimes even exact no-core ones (c) They exhibit very interesting behavior. Let us see now how this review is organized along these lines. the SM description has to be supplemented by some refined extensions such as the shell model in the continuum. (1). The aim of the section is to show how the realistic interactions can be characterized by a small number of parameters. The rest comes from algorithmic advances in the construction of tridiagonal Lanczos matrices that will be described in this section. one of the main achievements of modern SM work. Section VIII. For example: Weak decay rates are crucial for the understanding of several astrophysical processes. D.6 of the well deformed regions (Section VI. (2001b) have made it possible to simplify our task and avoid redundancies. SM wave functions are. the choice of valence space is primarily a matter of physics. Michel et al. Sections II. Section IV. About this review: the unified view We expect this review to highlight several unifying aspects common to the most recent successful shell model calculations: a) An effective interaction connected with both the two and three nucleon bare forces. In both cases. 1971). Sussex direct extraction (Elliott et al. once associated to operators that depend only on the number of particles in subshells. which. consistent with N N scattering data) in shell-model calculation was pioneered by Kuo and Brown (1966). the primary problem is the monopole contribution to the 3b potentials. amounts to very simple modifications of some average matrix elements (centroids) of the KB interaction (Kuo and Brown.. 2000) which provided acceptable spectra for A ≤ 8 (though they had problems with binding energies and spin-orbit splittings). In 1991 it was hard to interpret them as effective and there were no sufficient grounds to claim that they were real. now encounter serious trouble in the spectrum of 10 B..7 II. and Wiringa (2002). This is welcome because a full 3b treatment would render most shell model calculations impossible. 1966). the full H can be separated rigorously as H = Hm + HM . 1995). 1997.. 1993). The most recent results (Jiang et al. In Section II. (2001). AV18 (Wiringa et al. the clean separation of bulk and shell effects and the reduction of H to sums of factorable terms (Dufour and Zuker.. 1978. 1996)—that fit the ≈ 4300 entries in the Nijmegen data base (Stoks et al. The preceding paragraph amounts to rephrasing the two observations quoted at the beginning of this section with the proviso that the blame for discrepancies with experiment cannot be due to the—now nearly perfect— 2b potentials. It is seen that the trouble detected with a 2b-only description—with binding energies. Given that we have no complaint with HM . 1969a.e. 6 Experimentally there will never be enough data to determine them. Varga. Therefore.. and—as checked by many calculations—is well given by the 2b potentials.. 2001) and the chiral Idaho-A and B (Entem and Machleidt. the need of true 3b forces has become irrefutable: In the 1990 decade several two-body (2b) potentials were developed—Nijmegen I and II (Stoks et al. and Zuker (1991) still provide a good introduction to the subject: “The use of realistic potentials (i. The multipole part HM includes pairing. The second observation is that when used in shellmodel calculations and compared with data these matrix elements give results that deteriorate rapidly as the number of particle increases (Halbert et al. The basic tools are symmetry and scaling arguments.. 2002). In Section II. while the phenomenological study of monopole behavior is quite simple.A we sketch the theory of effective interactions. 1988). Two recent additions to the family of high-precision 2b potentials—the charge-dependent “CD-Bonn” (Machleidt.B it is explained how to construct from data a minimal Hm . 2002)—have dispelled any hopes of solving the Ay puzzle with 2b-only interactions (Entem et al. . It was found (Pasquini and Zuker.” In Abzouzi et al... Of the enormous body of work that followed we would like to extract two observations. The first is that whatever the forces (hard or soft core. 1994). whatever information that comes from nuclear data beyond A=3 is welcome. 1971) and (Brown and Wildenthal. the interaction appears as matrix elements.. Caurier. Kuo and Brown.. 1996). since it confers on them a model-independent status as direct links to the phase shifts. quasi-exact 2b Green Function Monte Carlo (GFMC) results (Pudliner et al. 1989) amount to a vindication of the work of Kuo and Brown. involving substantial threebody (3b) terms. (1991) very good spectroscopy in the p and sd shells could be obtained only through more radical changes in the centroids. while Section II. Nowadays. confirmed by exact diagonalizations up to A=48 (Caurier et al. and none of them seemed capable to predict perfectly the nucleon vector analyzing power in elastic (N.. CD-Bonn-(Machleidt et al. Furthermore it is quite justified because there is little ab initio knowledge of the 3b potentials6 . as found through the No Core Shell Model (NCSM) calculations of Navr´til and Ormand a (2002) and Caurier et al. The hope for a few parameter description comes from chiral perturbation theory (Entem and Machleidt. We take this similarity to be the great strength of the realistic interactions. 1988)) the effective matrix elements are extraordinarily similar (Pasquini and Zuker. that soon become far more numerous than the number of parameters defining the realistic potentials. 2002). Wiringa et al. quadrupole and other forces responsible for collective behavior. confirmed by Pieper.. ancient or new) and the method of regularization (Brueckner G matrix (Kahana et al. As we shall show. Rutsgi et al. spin-orbit splittings and spectra—is always related to centroids. 1978) that in the pf shell a phenomenological cure.C will be devoted to extract from realistic forces the most important contributions to HM . 1968) or Jastrow correlations (Fiase et al... 1968). Furthermore. 1994) with χ2 ≈ 1. (2002). d) scattering (the Ay puzzle). Hence the necessary “corrections” to Hm must have a 3b origin. THE INTERACTION The following remarks from Abzouzi. Our task will consist in analyzing the Hamiltonian in the oscillator representation (or Fock space) so as to understand its workings and simplify its form. who also show that the problems can be remedied to a large extent by introducing the new Illinois 3b potentials developed by Pieper et al. determine a “monopole” Hamiltonian Hm that is basically in charge of Hartree Fock selfconsistency. In Shell Model calculations. 8 A. Effective interactions The Hamiltonian is written in an oscillator basis as H = K+ + r≤s≤t, u≤v≤w, Γ + Γ Vrstu ZrsΓ · ZtuΓ + Γ Vrstuvw ZrstΓ · ZuvwΓ , r≤s, t≤u, Γ (3) Γ where K is the kinetic energy, Vrr′ the interaction matrix + elements, ZrΓ ( ZrΓ ) create (annihilate) pairs (r ≡ rs) or triples (r ≡ rst) of particles in orbits r, coupled to Γ = JT . Dots stand for scalar products. The basis and the matrix elements are large but never infinite7 . The aim of an effective interaction theory is to reduce the secular problem in the large space to a smaller model space by treating perturbatively the coupling between them, thereby transforming the full potential, and its repulsive short distance behavior, into a smooth pseudopotential. In what follows, if we have to distinguish between large (N ω) and model spaces we use H, K and V for the pseudo-potential in the former and H, K and V for the effective interaction in the latter. 1. Theory and calculations The general procedure to describe an exact eigenstate in a restricted space was obtained independently by Suzuki and Lee (1980) and Poves and Zuker (1981a)8. It consists in dividing the full space into model (|i ) and external (|α ) determinants, and introducing a transformation that respects strict orthogonality between the spaces and decouples them exactly: |ı = |i + Aiα |α |α = |α − Aiα |i (4) (5) S = S1 + S2 + · · · + Sk is a sum of k-body (kb) operators. The decoupling condition ı|H|α = 0 then leads to a set of coupled integral equations for the Si amplitudes. When the model space reduces to a single determinant, setting S1 = 0 leads to Hartree Fock theory if all other amplitudes are neglected. The S2 approximation contains both low order Brueckner theory (LOBT) and the random phase approximation (RPA) . In the presence of hard-core potentials, the priority is to screen them through LOBT, and the matrix elements contributing to the RPA are discarded. An important implementation of the theory was due to Zabolitzky, whose calculations for 4 He, 16 O and 40 Ca, included S3 (Bethe Faddeev) and S4 (Day Yacoubovsky) amplitudes (K¨ mmel, u L¨ hrmann, and Zabolitzky, 1978). This “Bochum trunu cation scheme” that retraces the history of nuclear matter theory has the drawback that at each level terms that one would like to keep are neglected. The way out of this problem (Heisenberg and Mihaila, 2000; Mihaila and Heisenberg, 2000) consists on a new truncation scheme in which some approximations are made, but no terms are neglected, relying on the fact that the matrix elements are finite. The calculations of these authors for 16 O (up to S3 ) can be ranked with the quasi-exact GFMC and NCSM ones for lighter nuclei. In the quasi-degenerate regime (many model states), the coupled cluster equations determine an effective interaction in the model space. The theory is much simplified if we enforce the decoupling condition for a single state whose exact wavefunction is written as |ref = (1 + A1 + A2 + . . . )(1 + B1 + B2 + . . . )|ref = (1 + C1 + C2 + . . . )|ref , C = exp S, (6) where |ref is a model determinant. The internal amplitudes associated with the B operators are those of an eigenstate obtained by diagonalizing Hef f = H(1 + A) in the model space. As it is always possible to eliminate the A1 amplitude, at the S2 level there is no coupling i.e., the effective interaction is a state independent Gmatrix (Zuker, 1984), which has the advantage of providing an initialization for Hef f . Going to the S3 level would be very hard, and we examine what has become standard practice. The power of CCT is that it provides a unified framework for the two things we expect from decoupling: to smooth the repulsion and to incorporate long range correlations. The former demands jumps of say, 50 ω, the latter much less. Therefore it is convenient to treat them separately. Standard practice assumes that G-matrix elements can provide a smooth pseudopotential in some sufficiently large space, and then accounts for long range correlations through perturbation theory. The equation to be solved is Vijαβ Gαβkl Gijkl = Vijkl − , (7) ǫα + ǫβ − ǫi − ǫj + ∆ αβ α i ı|α = 0, Aiα is defined through ı|H|α = 0. The idea is that the model space can produce one or several starting wavefunctions that can evolve to exact eigenstates through perturbative or coupled cluster evaluation of the amplitudes Aiα , which can be viewed as matrix elements of a many body operator A. In coupled cluster theory (CCT or exp S) (Coester and K¨ mmel u (1960), K¨ mmel et al. (1978)) one sets A = exp S, where u 7 8 Nowadays, the non-relativistic potentials must be thought of as derived from an effective field theory which has a cutoff of about 1 GeV (Entem and Machleidt, 2002). Therefore, truly hard cores are ruled out, and H should be understood to act on a sufficiently large vector space, not over the whole Hilbert space. The notations in both papers are very different but the perturbative expansions are probably identical because the Hermitean formulation of Suzuki (1982) is identical to the one given in Poves and Zuker (1981a). This paper also deals extensively with the coupled cluster formalism. where ij and kl now stand for orbits in the model space while in the pair αβ at least one orbit is out of it, ǫx 9 is an unperturbed (usually kinetic) energy, and ∆ a free parameter called the “starting” energy. Hjorth-Jensen, Kuo, and Osnes (1995) describe in detail a sophisticated partition that amounts to having two model spaces, one large and one small. In the NCSM calculations an N ω model space is chosen with N ≈ 6-10. When initiated by Zheng et al. (1993) the pseudo-potential was a G-matrix with starting energy. Then the ∆ dependence was eliminated either by arcane perturbative maneuvers, or by a truly interesting proposal: direct decoupling of 2b elements from a very large space (Navr´til and Barrett, 1996), further ima plemented by Navr´til, Vary, and Barrett (2000a), and a extended to 3b effective forces (Navr´til et al., 2000), a (Navr´til and Ormand, 2002). It would be of intera est to compare the resulting effective interactions to the Brueckner and Bethe Faddeev amplitudes obtained in a full exp S approach (which are also free of arbitrary starting energies). A most valuable contribution of NCSM is the proof that it is possible to work with a pseudo-potential in N ω spaces. The method relies on exact diagonalizations. As they soon become prohibitive, in the future, CCT may become the standard approach: going to S3 for N = 50 (as Heisenberg and Mihaila did) is hard. For N = 10 it should be much easier. Another important NCSM indication is that the excitation spectra converge well before the full energy, which validates formally the 0 ω diagonalizations with rudimentary potentials9 and second order corrections. The 0 ω results are very good for the spectra. However, to have a good pseudopotential to describe energies is not enough. The transition operators also need dressing. For some of them, notably E2, the dressing mechanism (coupling to 2 ω quadrupole excitations) has been well understood for years (see Dufour and Zuker, 1996, for a detailed analysis), and yields the, abundantly tested and confirmed, recipe of using effective charges of ≈ 1.5e for the protons and ≈ 0.5e for neutrons. For the Gamow Teller (GT) transitions, mediated by the spin-isospin στ± operator the renormalization mechanism involves an overall “quenching” factor of 0.7–0.8 whose origin is far subtler. It will be examined in Sections V.C.1 and V.C.2. The interested reader may wish to consult them right away. Since H contains two and three-body (2b and 3b) components, the separation is not mathematically clean. Therefore, we propose the following H = Hm + HM , (8) where Hm , the monopole Hamiltonian, contains K and all quadratic and cubic (2b and 3b) forms in the scalar products of fermion operators a†x · asy 10 , while the mulr tipole HM contains all the rest. Our plan is to concentrate on the 2b part, and introduce 3b elements as the need arises. d Hm has a diagonal part, Hm , written in terms of num† ber and isospin operators (arx · ary ). It reproduces the average energies of configurations at fixed number of particles and isospin in each orbit (jt representation) or, alternatively, at fixed number of particles in each orbit and each fluid (neutron-proton, np, or j representation). In jt representation the centroids T Vst = J JT Vstst (2J + 1)[1 − (−)J+T δst ] , J+T δ ] st J (2J + 1)[1 − (−) 1 0 bst = Vst − Vst , (9a) (9b) ast = 1 1 0 (3Vst + Vst ), 4 are associated to the 2b quadratics in number (ms ) and isospin operators (Ts ), mst = Tst 1 ms (mt − δst ), 1 + δst 1 3 = (Ts · Tt − mst δst ), 1 + δst 4 (10a) (10b) to define the diagonal 2b part of the monopole Hamiltonian (Bansal and French (1964), French (1969)) d Hmjt = Kd + d3 (ast mst + bst Tst ) + Vm , (11) s≤t The 2b part is a standard result, easily extended to include the 3b term d3 Vm = (astu mstu + bstu Tstu ), stu (12) B. The monopole Hamiltonian A many-body theory usually starts by separating the Hamiltonian into an “unperturbed” and a “residual” part, H = H0 + Hr . The traditional approach consists in choosing for H0 a one body (1b) single-particle field. where mstu ≡ mst mu , or ms (ms − 1)(ms − 2)/6 and Tstu ≡ ms Ttu or (ms − 2)Tsu . The extraction of the full Hm is more complicated and it will be given in detail in Appendix B. Here we only need to note that Hm is closed under unitary transformations of the underlying fermion operators, and hence, under spherical HartreeFock (HF) variation. This property explains the interest of the Hm + HM separation. 9 Even old potentials fit well the low energy N N phase shifts, the only that matter at 0 ω. 10 r and s are subshells of the same parity and angular momentum, x and y stand for neutrons or protons 10 1. Bulk properties. Factorable forms Hm must contain all the information necessary to produce the parameters of the Bethe Weisz¨cker mass fora mula, and we start by extracting the bulk energy. The key step involves the reduction to a sum of factorable forms valid for any interaction (Dufour and Zuker, 1996). Its enormous power derives from the strong dominance of a single term in all the cases considered so far. For clarity we restrict attention to the full isoscalar centroids defined in Eqs. (B18a) and (B18c)11 Vst = ast − Diagonalize Vst d U −1 Vm U = E =⇒ Vst = d Vm = k In Eq. (15) a single term strongly dominates all others. For the KLS interaction (Kahana et al., 1969b) and including the first eight major shells, the result in Fig. 1 is roughly approximated by Us ≈ 4. − 0.5 (l − l ) + (j − l) , Dp (17) 3δst bst . 4(4js + 1) (13) Usk Utk Ek , s ∴ Vss ms . (14) (15) where p is the principal quantum number of an oscillator shell with degeneracy Dp = s (2js + 1) = (p + 1)(p + 2), l = l l(2l + 1)/ l (2l + 1). The (unitary) U matrices have been affected by an arbitrary factor 6 to have numbers of order unity. Operators of the form (Ds = 2js + 1) k Ek Usk ms s t Utk mt − For the 3b interaction, the corresponding centroids Vstu are treated as explained in (Dufour and Zuker, 1996, Appendix B1b): The st pairs are replaced by a single index x. Let L and M be the dimensions of the x and s arrays respectively. Construct and diagonalize an (L + M ) × (L + M ) matrix whose non-zero elements are the rectangular matrices Vxs and Vsx . Disregarding the contractions, the strictly 3b part V (3) can then be written as a sum of factors d3 Vm = k ˆ Ω= s m s Ωs with s D s Ωs = 0 (18) vanish at closed shells and are responsible for shell effects. As l − l ) and j − l are of this type, only the the Up part contributes to the bulk energy. To proceed it is necessary to know how interactions depend on the oscillator frequency ω of the basis, related to the observed square radius (and hence to the density) through the estimate (Bohr and Mottelson, 1969, Eq. (2157)): Ek (3) s Usk ms x≡tu (1) Utu,k mt mu . (2) (16) Full factorization follows by applying Eqs. (14,15) to the (2) Utu,k matrices for each k. 3 2.5 2 Up, Us 1.5 1 0.5 0 0 1 2 3 p 4 5 6 7 Us Up ω 35.59 40 = =⇒ ω ≈ 1/3 MeV r2 (A)1/3 A (19) A δ force scales as ( ω)3/2 . A 2b potential of short range is essentially linear in ω; for a 3b one we shall tentatively assume an ( ω)2 dependence while the Coulomb force goes exactly as ( ω)1/2 . To calculate the bulk energy of nuclear matter we average out subshell effects through uniform filling ms ⇒ mp Ds /Dp . Though the Ω-type operators vanish, we have kept them for reference in Eqs. (21,22) below. The latter is an educated guess for the 3b contribution. The eigenvalue E0 for the dominant term in Eq. 15 is replaced by ωV0 defined so as to have Up = 1. The subindex m is dropped throughout. Then, using Boole’s factorial powers, e.g., p(3) = p(p − 1)(p − 2), we obtain the following asymptotic estimates for the leading terms (i.e., FIG. 1 The Us terms in Eq. (17). Arbitrary scale. 11 They ensure the right average energies for T = 0 closed shells, which is not the case in Eq. (11) if one simply drops the bst terms. 8 0. .2. when in reality it is a 2b operator. 1969b) shell effects in Fig.8 1. Which in turn explains why primitive versions of such potentials give results close to the modern interactions for G-matrix elements calculated at reasonable ω values. 2 The KLS (Kahana et al. 1995). that such calculations could be conducted so easily in the oscillator basis.4 1. In particular. Assuming that non-diagonal Kd and V d terms cancel we can vary with respect to ω to obtain the saturation energy Es = Kd + (1 − βω κ pf )V d . This is a recurrent theme in this review (see especially Section II. Finally. while a 1b picture is phenomenologically quite satisfactory. instead of being proportional to l. Bonn KLS 1 2 3 4 5 s 6 7 8 9 10 11 Kd + V d ∂E κ = 0 ⇒ βωe pf = ∂ω (κ + 1) V d κ (Kd + V d ) ∴ Es = κ+1 = ωe κ (1 − 4V0 ) 4 κ+1 3 A 2 4/3 FIG.. Unfortunately. Though 2b forces do saturate. Clearly. 2 provides a direct reading of the expected single particle sequences produced by realistic interactions above T = 0 closed shells.e. Shell formation The correct saturation properties are obtained by fixing V0 so that Es /A ≈ 15. Before we move on.2 1 0.5 MeV. 1996) (to within an overall factor). The crucial question is now: Can we use realistic (2+3)b potentials soft enough to do HF? Probably we can.6 0. 2002). What we propose instead is to examine an existing. 1. Points in the same major shell are connected by lines and ordered by increasing j. It is worth noting that the same approach leads to VC ≈ 3e2 Z(Z −1)/5Rc (Duflo and Zuker. the splitting between spin-orbit partners appears to be nearly constant. 1 for the first 10 orbits compared to those produced by a Bonn potential (Hjorth-Jensen. The merit goes to the separation and factorization properties of the forces. in analogy with the kinetic energy. . as explained in Appendix C. As it could no longer be claimed that the 2b realistic forces are “wrong”. the solution must come from the 3b operators. i. we insist on what we have learned so far: It may be possible to describe nuclear structure with soft (2+3)b potentials consistent with the low energy data coming from the A = 2 and 3 systems. as our study will confirm. they do not square at all with what is seen experimentally. relate pf to A : pf βV0 p3 (pf f (22) 2(pf )(3) . they do it at the wrong place and at a heavy price because their short-distance repulsion prevents direct HF variation. (22) we can replace ( ω)2 by ( ω)1+κ and change the powers of Dp in the denominators accordingly. (22) may prove of help in this respect but it has not been implemented yet. It should be obvious that nuclear matter properties— derived from finite nuclei—could be calculated with techniques designed to treat finite nuclei: A successful theoretical mass table must necessarily extrapolate to the Bethe Weisz¨cker formula (Duflo and Zuker. It a may be surprising though. Eq. usually represented as 1b. Something similar seems to happen with the spin-orbit force which must have a 3b origin. which has been traditionally well fitted by realistic potentials. they may be often mocked by lower rank ones. without a 3b term. strict d (1+2)b. 2 provides an example of particular relevance for the study of shell effects in next Section II. The ωe ≈ 40A−1/3 choice ensures the correct density.B.D) and Fig.. and Fig. Though we know that 3b ingredients are necessary.11 disregarding contractions) Kd = =⇒ d ω 2 mp (p + 3/2) p ω (pf + 3)(3) (pf + 2) 4 mp ˆ +Ω Dp 2 (20) V ≈ ωV0 p =⇒ ωV0 [pf (pf + 4)]2 . (24) has no (or trivial) solution for κ = 0.6 p =⇒ ( ω) + 4) . (24) 2. We shall also find out that some mechanisms have an irreducible 3b character. V d3 (21) mp ˆ + Ω2 Dp 2 ≈ ( ω) βV0 2 2 p mp ˆ + Ω1 Dp 2 nucleons only “see” the low energy part of the 2b potential involving basically s wave scattering.4 0 mp = p p=0 2(p + 1)(p + 2) =⇒ A = Note that in Eq. model of Hm that goes a long way in explaining shell formation. The “educated guess” in Eq. (23) 3 Us 1. The reason is that the nucleus is quite dilute. e. and the chosen combination has the advantage of canceling exactly to orders A and A2/3 . Both terms lead to the same gaps. realistic 2b forces will miss the particle spectrum in 41 Ca. Once it is phenomenologically corrected by ls + ll such forces account reasonably well for the zni and eei mechanism. a minimal characteriza˜d tion. As a consequence. ˆ [II] Ω(ls) ≈ −22 l·s/A2/3 (very much the value in (Bohr ˆ and Mottelson. The differences in binding energies (gaps) 2BE(cs) − BE(cs + 1) − BE(cs − 1). it is not clear how well they do it. Configuration mixing may alter somewhat the energies but. illustrated in Figure 4 for t = [I] W − 4K = p mp Dp −2 mp (p + 3/2). Hm . zn for “proton-neutron”. Eq. Mev) -60 -80 -100 -120 -140 -160 -180 0 W-4K W-4K+ll+ls Hm 10 20 30 N 40 50 60 12 f f stands for “same fluid”. since the correct mechanism necessarily involves 3b terms. 47 Ca and 47 K. 4 The different contributions to Hd for N=Z nuclei . Only 3b operators can account for this inversion. at midshell. As a consequence. 3].. Though zni and eei can be made to cure this serious defect. (23). as will be demonstrated in Section V. i for “intrashell” and c for “cross-shell” ˜m FIG.A. 49 Sc. 3: In going from 40 Ca to 48 Ca the filling of the f7/2 (f for short) neutron orbit modifies the spectra in 49 Ca. they fail 0 -20 -40 E(t=0. With a half dozen parameters a fit to the 90 available data achieved an rmsd of some 220 keV. within one major exception: they fail to depress the f7/2 orbit with respect to the others. also came out quite well and provided a test of the reliability of the results. (2-132)) and Ω(ll) ≈ −22[l(l + ˆ 1)−p(p+3)/2]/A2/3 (in MeV). As mentioned. Ω(f f c) and Ω(znc)12 are Ω-type 2b operators whose effect is illustrated in Fig. which is not the case for light nuclei as seen from the spectra of 13 C and 29 Si. The model concentrates on single particle splittings which are of order A−1/3 . i. which we call cs ± 1 are well represented by single determinants. Therefore. ˜d In spite of these caveats. The valence spaces involve a number of particles of the order of their degeneracy. most often.12 d Hm manifests itself directly in nuclear spectra through the doubly magic closures and the particle and hole states built on them. (26) p comes from Kd and V d . To avoid bulk contributions all expressions are written in terms of ˆ Ω-type operators [Eq. can be achieved by demanding that it reproduce all the observed cs ± 1 states. (18)] and the model is defined by f 5/2 p1/2 p 3/2 f 7/2 zni 49 ffi 49 41 Sc 41 Ca Sc Ca 39 K 39 Ca 47 K 47 Ca d 3/2 s 1/2 π or znc π ffc ν ν d 5/2 FIG. Ω(zni). Another problem that demands a three body mechanism is associated with the znc and f f c operators which always depress orbits of larger l [as in the hole spectra of 47 Ca and 47 K in Fig. ˆ ˆ ˆ ˆ ˆ [III] Ω(f f i). not included in the fit. are Ω-type operators that reproduce the single-particle spectra above HO closed shells. and quite generally. Where: 2 Ca to 48 Ca (25) to produce the EI closures. ≈ (pf + 3/2)2 ≈ (3A/2)2/3 according to Eq. the addition of single particle splittings generates shell effects of order A1/3 . 3 Evolution of cs ± 1 states from 40 ˜d ˆ ˆ Hm ≡ (W − 4K) + Ω(ls) + Ω(ll) ˆ ˆ ˆ ˆ +Ω(f f i) + Ω(zni) + Ω(f f c) + Ω(znc). The members of this set. 1969. Hm should provide a plausible view of shell formation. enough experimental evidence exists to correct for the mixing. The task was undertaken by Duflo and Zuker (1999). (28) and rt ≡ a. 00 01 for HM .13 -40 -60 E(t=0. (B11). The spikes are invariably associated either to the HO magic numbers (8. 5 we show the situation for even t = N −Z = 0 to 18. 40) or to the EI ones (14. su ≡ b in eq. 80 Zr shows a nice spike.. 48 Ca is definitely magic. or (28) ˜m FIG. The matrix γ Γ elements ωrtsu and Wrstu are related through Eqs. The eigensolutions in eqs. there are no (or weak) spikes for the heavier isotopes except at 90 Zr and 96 Zr. (A9) and (A10). Hm is calculated by filling the oscillator orbits in the order dictated by ls + ll.t≤u. . C. plus the largest (intruder) orbit from shell p + 1. 28. In Figs. They are practically erased by the 1b contributions. both definitely doubly magic. 1969). For example. (31) 13 Remember: EI valence spaces consist in the HO orbits shell p. thus suggesting fairly reliable predictive ˜d power for Hm . The multipole Hamiltonian The multipole Hamiltonian is defined as HM =H-Hm . Mev) -110 -120 -130 -140 -150 -160 10 20 30 Z 40 50 60 20 30 Z Hm10 Hm12 Hm14 Hm16 Hm18 40 50 60 Hm0 Hm2 Hm4 Hm6 Hm8 diction for a strongly rotational nucleus. which we call the E (or normal or particle-particle or pp) and e (or multipole or particle-hole or ph) representations. ωrstu = ωrstu = 0 (see Eq. Lee. They may be erased by deformation—as in the case of 80 Zr—but this seldom happens. and the general trend is the following: No known closure fails to be detected.1). As we are no longer interested in the full H.Γ † Γ Wrstu ZrsΓ · ZtuΓ .Γ Γ Ek x † Γ Uxk ZxΓ · Γ Uyk ZyΓ .50). 50). 18..28) and (40. and are reinforced when both are closed. at Z = 20 40 Ca shows as a weak closure. 44.(20. HM will be more modestly called HM . There are no one body contractions in the e representation because 0τ they are all proportional to ωrstu . Plotting along lines of constant t has the advantage of detecting magicity for both fluids. insisting on points that were not stressed sufficiently in that paper (Section II. Mev) -80 -100 -120 -140 -160 10 -80 -90 -100 E(t=10. but the latter always show magicity while the former only do it at and above the double closures (Z. to diagonal form through unitary Γ transformations Uxk . Since Hm contains all the γ = 00 and 01 terms. we proceed Γ as in Eqs. Note that the HO magicity is strongly attenuated but persists. There are many other interesting cases. (29). In other words.γ eγ k a γ u γ Sa ak b γ u γ Sb bk [γ]1/2 . At Z = 40. (27) We shall describe succinctly the main results of Dufour and Zuker (1996). but its restriction to a finite space. (30) and (31) using the KLS interaction (Kahana et al. y (30) 0 HM = k. (14) and (15) to bring the matrices Wxy ≡ γ γ Γ Wrstu and fab ≡ frtsu . Conversely.8. the spikes appear when either fluid is closed. 5 Hd for even t = N − Z = 10 . . 42. rstuΓ (29) γ γ where frtsu = ωrtsu (1 + δrs )(1 + δtu )/4. uγ to obtain factorable expressions ak HM = k. with monopole-free matrix elements given by JT JT T Wrstu = Vrstu − δrs δtu Vrs . The W − 4K term produces enormous HO closures (showing as spikes).. and it takes the 2b ones to generate the EI13 spikes. tu ≡ y in eq. yield the density of eigenvalues (their number in a given interval) . except the largest (extruder)...18. 46 Ca are not closed (which agrees with experiment). HM = ˜d N −Z = 0 nuclei. However.C. and spikes persist for the heavier isotopes. The drift toward smaller binding energies that goes as A1/3 is an artifact of the W − 4K choice and should be ignored.D). 1969b. and adding some new information (Section II. a bad pre- γ γ γ [γ]1/2 frtsu (Srt Ssu )0 .14). There are two standard ways of writing HM : HM = r≤s. not all predicted closures are real. 20. Replacing pairs by single indices rs ≡ x. N )=(8. because they are associated with normalized eigenvectors. in the E representation that is shown in Fig.14 2000 E-eigenvalue density number of states 5000 e-eigenvalue density number of states 0+ 1 + 1 0 2+0 0 0-0 1. In the figures the eigenvalues for the two shell case are doubled. Collapse avoided 14 Inverted commas are meant to call attention on an erratum in (Dufour and Zuker. Each eigenvalue has multiplicity [Γ].0 2-0 4 1. (31) and the associated H in eq. The e eigenvalues are plotted in fig. 2 qrs = 2 2 Np = Σqrs 1 r r2 Y 2 s (36) 5 ∼ 5 (p + 3/2)4 (37) = 32π 1. The strongest have γ π = 1− 0. but then the coupling constants had to be . the unnormalized eigenvector turns out to be up1 + up2 with eigenvalue e. 1969). 1+ 1. 6 E-eigenvalue density for the KLS interaction in the pf+sdg major shells ω = 9. The only solution was to stay within limited spaces. with a tail at negative energies which is what we expect from an attractive interaction. and γ = 20 HP = − ¯ Hq ¯ ω 01 |E |(P † + P † ) · (P p + P p+1 ) (32) p p+1 ω0 ω 20 = − |e |(¯p + qp+1 ) · (¯p + qp+1 ). the E distribution becomes quite symmetric. 2+ 0. r p 20 Srs qrs /Np . 4+ 0. terms. 1996). They are very symmetrically distributed around a narrow central group. The pairing plus quadrupole (P + Q) model has a long and glorious history (Baranger and Kumar. 7 e-eigenvalue density for the KLS interaction in the pf+sdg major shells. E 01 and e20 are the one shell values called generically e in the discussion above. rs∈p (34) (35) ¯ qp = where Ωr = jr + 1. (30) is recalculated14 . except that the energies are halved. The largest ones are shown by arrows. 6 for a typical two-shell case. 6 and 7. as expected for a random interaction. and one big problem: as more shells are added to a space.0 1+1 -10 -8 -6 -4 -2 0 2 Energy (MeV) FIG. and Ωp = 1 Dp . To make a long story short: The contribution to HM associated to the large Γ = 01. 3− 0. their eigenvalues are approximately equal ep1 ≈ ep2 = e. 7. negative parity peaks are absent. When diagonalizing in p1 +p2 . except that the operators for each major shell of principal quantum number p are affected by a normalization. “If the corresponding eigenvectors are eliminated from H in” eq. Each eigenvalue has multiplicity [γ]. the energy grows.0 1+0 + 20 + 40 0 -4 -3 -2 -1 0 1 2 3 4 5 Energy (MeV) 3. Bes and Sorensen. FIG. The largest ones are shown by arrows. To be precise ¯† Pp = r∈p † Zrr01 Ω1/2 /Ω1/2 . This point is crucial: If up1 and up2 are the eigenvectors obtained in shells p1 and p2 . but for the positive parity ones the results are practically identical to those of Figs. q ¯ q ¯ (33) ω0 turns out to be the usual pairing plus quadrupole Hamiltonians. eventually leading to collapse. but a few of them are neatly detached. It is skewed. If the diagonalizations are restricted to one major shell. 1968. 99 1 OCC = 0. for one shell15 0. There is not much to choose between the different forces which is a nice indication that overall nuclear data (used in the FPD6 and Gogny fits) are consistent with the N N data used in the realistic KB and BonnC G-matrices. uncertainties were also due to the starting energies and the renormalization processes. (1995). 2 Introducing Amf ≈ 3 (pf + 3/2)3 . = 1/3 A2/3 Ωp A mf 216 χ′ χ′ 0. provided it is supplemented by a reasonable Hm . OAA is a measure of the strength of an interaction. as can be gathered from the bottom part of Table I.32 ω ∼ 19. the Gogny force—successfully used in countless mean field studies— (Decharg´ and Gogny.B.32(1+0. OAB is the overlap for matrix elements with the same T . the relationship between g ′ . which become for the two possible scalings mD m ) =⇒ EP (new) = O( 1/3 ) (41) A A m2 D m2 Eq (old) = O( 5/3 ) =⇒ Eq (new) = O( 1/3 ). OAA OBB 15 According to Dufour and Zuker (1996) the bare coupling constants.98 OCC = 1. In assessing the JT = 01 and JT = 10 pairing terms.97 OAA = 6. Therefore the traditional forces lead the system to collapse.46 1 O CK = 0. Lee (1969)) and a modern Bonn one (Hjorth-Jensen. 1980) and BonnC (Hjorth-Jensen e et al. Eq decreases and the monopole term provides the restoring force that guarantees that particles will remain predominantly in the Fermi shell. The upper TABLE I The overlaps between the KLS(A) and Bonn(B) interactions for the first ten oscillator shells (2308 matrix elements). (32) and (33).. = = 2 Np 2 A1/3 A4/3 2 2 mf (38) T Similarly.56 0 OBB = 11. The dominant terms of HM are central and table II collects their strengths (in MeV) for effective interactions in the pf -shell: Kuo and Brown (1968) (KB).32 and |e20 |/ ω0 = κ′ = 0. (1969b). affect mainly the total strength(s). However. κ′ .48) and 0. and their conventional counterparts (Baranger and Kumar. 1996) differ in total strength(s) but the normalized O(AB) are very close to unity. the total number of particles at the middle of the Fermi shell pf . 1995). the potential fit of Richter et al. In the new form there is no collapse: EP stays constant.61 1 OBB = 2. which. the matrix elements given in this reference for the pf shell have normalized cross overlaps of better than 0.84 0 O AB = 0.02 0 O CK = 0. The corresponding pairing and quadrupole energies can be estimated as EP = −|G|4m(D − m + 2) = −|G|O(mD).15 readjusted on a case by case basis. 0 OAA = 17.216 ω ∼ 1 ≡ 0 A−5/3 .216 in Eqs. they are |E 01 |/ ω0 = g ′ = 0.C potentials studied by Hjorth-Jensen et al. and Eq ≈ −|χ ′ (39) (40) |Q2 0 = −|χ |O(m D). The splendid performance of Gogny deserves further study. The only qualm is with the weak quadrupole strength of KB. followed by those for the pf shell (195 matrix elements) for the BonnC (C) and KB (K) interactions. P + Q is likely to be a reasonable first approximation for many studies. Universality of the realistic interactions To compare sets of matrix elements we define the overlaps OAB = VA · VB = O AB = √ Γ Γ VrstuA VrstuB rstuΓ (43) (44) OAB . 2001). which can be understood by looking again at the bottom part of Table I. In the old interactions.3) set of numbers in Table I contains what is probably the most important single result concerning the interactions: a 1969 realistic potential (Kahana et al. If the interaction is referred to its centroid (and 2 a fortiori if it is monopole-free).56 1 OKK = 0. OAA ≡ σA .216(1+0. EP (old) = O( D. and promote them to a higher shell of degeneracy D.99 To see how collapse occurs assume m = O(Df ) = O(A2/3 ) in the Fermi shell. (42) A A D If the m particles stay at the Fermi shell. Now: all the modern realistic potentials agree closely with one another in their predictions except for the binding energies.31 1 O AB = 0. to produce the P +Q+m model. the overall strengths must be viewed as free parameters. it should be borne in mind that their renormalization remains an .15 O CK = 0. 1968) is. At a fundamental level the discrepancies in total strength stem for the degree of non-locality in the potentials and the treatment of the tensor force.98 0 OCC = 3.71 0 OKK = 3. again..99 1 OAA = 2.998. For nuclear matter the differences are substantial (Pieper et al. all energies go as A1/3 as they should. As a model for HM . If D grows both energies grow in the old version.78 O AB = 0. Knowing that ω0 = 9 MeV. they become enormous.41 OKK = 1. At the moment.49 OBB = 4.51 = G ≡ G0 A−1 . ′ 2 respectively. The normalized versions of the operators presented above are affected by universal coupling constants that do not change with the number of shells. and for the BonnA. should be renormalized to 0. (1991) (FPD6). For sufficiently large D the gain will become larger than the monopole loss O(mD1/2 ω) = O(D1/2 A1/3 ). the Lanczos procedure (with H) will remain inside a fixed J subspace and therefore will improve the convergence properties.33 λτ =40 -1.985. As an alternative. the Davidson method (Davidson.07 -4. For these reasons.16 open question (Dufour and Zuker.60 particle-hole λτ =20 -2. which in turn depends on the number of converged states needed. Mathematically the Lanczos vectors . N being the dimension of the matrix. THE SOLUTION OF THE SECULAR PROBLEM IN A FINITE SPACE Once the interaction and the valence space ready.70 very often only one) eigenstates of a given angular momentum (J) and isospin (T ) are needed. the number of NZME varies linearly instead of quadratically with the size of the matrices. For example in the m-scheme basis. they cannot be used in large scale shell model (SM) calculations. Conclusion: Some fine tuning may be in order and 3b forces may (one day) bring some multipole news. the large increase in storage capacity of modern computers has somehow minimized these advantages.79 -3. the best choice for the pivot is a linear combination of the nc lower states in the truncated space. A. and also on the choice of the starting vector. The vector |a1 = H|1 has necessarily the form |a1 = H11 |1 + |2′ .29 λτ =11 +2. As can be seen in figure 8. with 1|2′ = 0.23 -3.720). however. The Lanczos method consists in the construction of an orthogonal basis in which the Hamiltonian matrix (H) is tridiagonal.17 +2. The = H|k has necessarily the form (46) III.11 -3. iterative methods are of general use. The Lanczos method |ak = Hk k−1 |k − 1 + Hk k |k + |(k + 1)′ . in particular the Lanczos method (Lanczos. Taking these states as starting vectors.77 -1. The Lanczos method can be used also as a projection method. Besides. The number of iterations depends little on the dimension of the matrix. When only one converged state is required. only a few (and Calculate k|ak = Hk k . Secondly.08 -5. Normalize it to find Hk k+1 = (k + 1)′ |(k + 1)′ 1/2 .06 -4.39 -1. Therefore. but as of now the problem is Hm . starting with a random pivot the calculation of the ground state of 50 Cr in the full pf space (dimension in m-scheme 14. The first one is that. Even if the Lanczos method is very efficient in the SM framework. When more eigenstates are needed (nc > 1). not HM . The overlap between these two 0+ states is 0. Two questions need now consideration: which basis to take to calculate the non-zero many-body matrix elements (NZME) and which method to use for the diagonalization of the matrix. the matrices are very sparse. in the vast majority of the cases. the computing time is directly proportional to the number of NZME and for this reason it is nearly linear (instead of cubic as in the standard methods) in the dimension of the matrix.856.67 -1. a Lanczos calculation with the operator J 2 will generate states with well defined J.625. A normalized starting vector (the pivot) |1 is chosen.46 -5. Now |(k + 1)′ is known. it is time to construct and diagonalize the many body secular matrix of the Hamiltonian describing the (effective) interaction between valence particles in the valence space.540 Slater determinants) needs twice more iterations than if we start with the pivot obtained as solution in a model space in which only 4 particles are allowed outside the 1f7/2 shell (dimension 1. 1996). 8 m-scheme dimensions and total number of non-zero matrix elements in the pf -shell for nuclei with M = Tz = 0 In the standard diagonalization methods (Wilkinson.74 -5. 1965) the CPU time increases as N 3 . Nuclear SM calculations have two specific features. (45) Calculate malize |2 2′ |2′ 1/2 . The (real symmetric) matrix is diagonalized at each iteration and the iterative process continues until all the required eigenvalues are converged according to some criteria. there may be numerical problems. For instance. it is convenient to use as pivot state the solution obtained in a previous truncated calculation.75 -5. It depends on the number of iterations.46 +3. 1975) has the advantage of avoiding the storage of a large number of vectors. vector |ak H11 = 1|H|1 through 1|a1 = H11 .46 +2. 1950). Nor= |2′ / 2′ |2′ 1/2 to find H12 = 1|H|2 = Iterate until state |k has been found.20 JT=10 -4. FIG. TABLE II Leading terms of the multipole Hamiltonian Interaction particle-particle JT=01 KB FPD6 GOGNY BonnC -4. To solve this problem it is necessary to orthogonalize each new Lanczos vector to all the preceding ones.09 × 109 2. when many iteration are performed. (49) Compared to the simplicity of the m-scheme. Let us give the example of the state 4+ in 50 Ti (full pf space).e. . The same precision defects can produce the appearance of non expected states. However. In the late 60’s. . the number of possible values of NZME is relatively limited.10 × 108 1. The coupled scheme has another disadvantage compared to the m-scheme. Kawarada and Arima. The choice of the basis Given a valence space. There are essentially three possibilities depending on the underlying symmetries: m-scheme. Tz )). For example in a mscheme calculation with a J = 4. In the particular case of the pf shell.78 × 106 1. The structure of the calculation is as follows: In the first place. Next. ) and depending on the type of nucleus (deformed. strength function. and protons. rendering the method inefficient. 1969).58 × 106 5. (a†1 aj2 )λ . and r.7 for J = 6. depending on what states or properties we want to describe (ground state. independently of the size of the matrix. j j j j j j (50) As the m-scheme basis consists of Slater Determinants (SD) the calculation of the NZME is trivial. i. M = 0 pivot. whose dimensions are much smaller. 1965).05 × 106 J = T = 0 66 |γ3 Γ3 Γk . For t = m − m0 we have the full space (pf )m for A = 40 + m. Another recent code that works in JT-coupled formalism is the Drexel University DUPSM (Novoselsky and Valli`res..05% in m-scheme.32 × 105 2. 5% in the J-basis and only 0. split the full m-scheme matrix in boxes This complexity explains why in the OXBASH code (Brown et al. In the case of only J (without T ) coupling. small numerical precision errors can. This means that. after many iterations. The advantages of the coupled scheme decrease also when J and T increase. . lower in energy than the J = 4 states. This is especially spectacular for the J = T = 0 states (see table III). The J or JT coupled bases. the states of N particles distributed in several shells are obtained by successive angular momentum couplings of the one-shell basic states: [|γ1 |γ2 ] Γ2 B. the percentages of NZME are respectively 14% in the JT -basis (see Novoselsky et al. TABLE III Some dimensions in the pf shell A 4 8 12 16 20 M = Tz = 0 4000 2 × 106 1. J coupled scheme. 1964. The Oxford-Buenos Aires shell model code. 1966.. a strong simplification in the calculation of the NZME can be achieved using the Quasi-Spin formalism (Ichimura and Arima. zv the number of different Slater determinants that can be built in the valence space is : d= Dn nv · Dp zv (47) where Dn and Dp are the degeneracies of the neutron and proton valence spaces. widely distributed and used has proven to be an invaluable tool in many calculations. ((a†1 a†2 )λ aj3 )j4 . It involves products of 9j symbols and coefficients of fractional parentage (cfp). As we discuss later. This specific problem can be solved by projecting on J each new Lanczos vector. in 56 Ni the ratio dim(M = J)/dim(J) is 70 for J = 0 but only 5. the JT-coupled basis states are written in the m-scheme basis to calculate the NZME. nv . the counterpart of the simplicity of the m-scheme is that only Jz and Tz are good quantum numbers... Working at fixed M and Tz the bases are smaller (d = M. For all these . ) different choices of the basis are favored.13 × 107 9741 3. calling f the largest subshell (f7/2 ).54 × 107 3. T ) coupled basis and implemented them in the Oak-Ridge Rochester Multi-Shell code (French et al. . therefore all the possible (J.Tz d (M. the states of ni particles in a given ji shell are defined: |γi = |(ji )ni vi Ji xi (vi is the seniority and xi any extra quantum number). For a given number of valence neutrons. |γk . . produce catastrophes. 1985).17 are orthogonal. As an example. and JT coupled scheme.29 × 109 J = Tz = 0 156 41355 1. 1997) has had a much e lesser use. . T ) states are contained in the basis and as a consequence the dimensions of the matrices are maximal. generically. It concerns the percentage of NZME. the states of lowest energy may reappear many times. the Rochester group developed the algorithms needed for an efficient work in the (J. Lawson and Macfarlane. show up abruptly. the optimal choice of the basis is related to the physics of the particular problem to be solved. however numerically this is not strictly true due to the limited floating point machine precision. It is often convenient to truncate the space. a†1 . the calculation of the NZME is much more complicated. Hence. it may happen that one J = 0 and even one J = 2 state. in particular. any or all of the other subshells of the p = 3 shell. as in the code NATHAN described below. (48) where m0 = 0 if more than 8 neutrons (or protons) are present.. yrast band. the single-shell reduced matrix elements of operators of the form (a†1 a†2 )λ . spherical.. 1997). the possible t-truncations involve the spaces f m−m0 rm0 +f m−m0 −1 rm0 +1 +· · ·+f m−m0 −t rm0 +t . since they are equal to the decoupled two body matrix element (TBME) up to a phase. are pre-calculated and stored. A crucial point nowadays is to know what are the limits of a given computer code. the numerical values of R are: R(1) = 0. We pre-calculate the numerical values of R(i). A pictorial example is given in fig. Therefore. p and α. To start with. in the Lanczos procedure a simple loop on α and i generates all the proton-proton and neutron-neutron NZME. The first breakthrough in the treatment of giant matrices was the SM code developed by the Glasgow group (Whitehead et al.. The m-scheme code ANTOINE The shell model code ANTOINE16 (Caurier and Nowacki.18 reasons and with the present computing facilities. Before the diagonalization.J = I|H|J . with the obvious exception of semi-magic nuclei. where i|H|j = Wij and α|H|β = Wαβ . I. albeit with some notable exceptions that we shall mention below. β. it takes advantage of the fact that the dimension of the proton and neutron spaces is small compared with the full space dimension. Each bit of the word is associated to a given individual state |nljmτ . one for protons and one for neutrons: |I = |i. 9 Schematic representation of the basis It is clear that for each |i state the allowed |α states run. . R(j).html 17 We use Dirac’s notation for the quantum mechanical state |i . a+ a m n M i 1 2 3 4 5 6 7 8 M+1 9 10 11 12 13 14 15 M+2 . the 2 subspaces will be associated only if M1 + M2 = M . Jzn −M −M−1 a+ a λ µ −M−2 FIG.865 SD (corresponding to all the possible M values) in 44 Ca. the SD’s of For example. say.in2p3. l in the configuration 0011. R(2) = 6. all the calculations that involve only the proton or the neutron spaces separately are carried out and the results stored. β. As far as the NZME can be calculated and stored. the position of this state in the basis is denoted by i. and change them to 1100. Wαβ . Let’s assume that the |i and |j SD are connected by the one-body operator a† ar (that in the list of all possible one-body operators q appears at position p). The total M being fixed. This is the origin of the term “giant” matrices. Conservation of the total M implies that the proton operators with ∆m must be associated to the neutron operators with −∆m. For the proton-proton and neutronneutron NZME the numerical values of R(i). with q = nljm and r = n′ l′ j ′ m′ and m′ − m = ∆m. capital letters for states in the full space. R(j). i. generating a new word which has to be located in the list of all the words using the bi-section method. between a minimum and a maximum value. Jzp C. small case Greek letters for states of the second subspace (neutrons or protons). For the proton-neutron matrix elements the situation is slightly more complicated. . it is possible to construct numerically an array R(i) that points to the |I state17 : I = R(i) + α (51) D. M1 and M2 . A two-body operator a† a† ak al will select the words having the bits i j i. .963.fr/~theory/antoine/main. Each bit has the value 1 or 0 depending on whether the state is occupied or empty. We use. . The relation (51) holds even in the case of truncated spaces.. the diagonalization with the Lanczos method is trivial. j. β. without discontinuity. 1977). their origin and their evolution. k. according to figure 9. The states of the basis are written as the product of two SD. the 1. It means that the fundamental limitation of standard shell model calculations is the capacity to store the NZME. . Building the basis in the total space by means of an i-loop and a nested α-loop. that we apply to those for which it is necessary to recalculate the NZME during the diagonalization process. WI. α . provided we define sub-blocks labeled with M and t (the truncation index defined in (48)). Let us recall its basic ideas: It works in the m-scheme and each SD is represented in the computer by an integer word. α 1 2 3 4 5 6 7 8 9 10 11 . α. . Equivalently. j. In the same way as we did before for I = R(i) + α. The Glasgow m-scheme code The steady and rapid increase of computer power in recent years has resulted in a dramatic increase in the dimensionality of the SM calculations. we can conclude that the m-scheme is the most efficient choice for usual SM calculations. the |α and |β SD are connected by a one-body operator whose position is denoted by µ. Wij and α. while improving upon the Glasgow code in the several aspects. µ. J. 1999) has retained many of these ideas. R(3) = 12. . small case Latin letters for states of the first subspace (protons or neutrons). Thus we could draw the equivalent to figure 9 for the proton and neutron one-body operators. we can now define an index K = Q(p) + µ that labels the different two-body matrix 16 A version of the code can be downloaded from the URL http: //sbgat194.461 SD with M = 0 in 48 Cr are generated with only the 4. The Slater determinants i and α can be classified by their M values. For example. 9. Furthermore. β . e. This technique allows the diagonalization of matrices with dimensions in the billion range. Each subspace is now partitioned with the labels J1 and J2 . the dimension reaches 2×1010. I|H|J and J|H|I are not generated simultaneously when |I and |J do not belong at the same k segment of the vector and the amount of disk use increases. Jmin ≤ J2 ≤ Jmax . K = Q(p) + µ. (1969)). matrix of 52 Fe with 6 protons and 6 neutrons in the pf shell is 108 . the calculations are straightforward. Then. The only difference with the m-scheme is that instead of having a one-to-one association (M1 + M2 = M ). are now a product of cfp’s and 9j symbols (see formula 3. For instance. In the shell model code NATHAN (Caurier and Nowacki. It becomes less efficient for asymmetric nuclei (for semi-magic nuclei all the NZME must be stored) and for large truncated spaces. at least. Their respective counterparts can only have M = −1/2 and t = 0 and have dimensionality 1. for a given J1 we now have all the possible J2 . in a valence space that comprises up to 14 ω configurations. The coupled code NATHAN. and K. the ”normal” valence space. since hij and hαβ . No-core calculations are typical in this respect. we denote V (K) the numerical value of the proton-neutron TBME that connects the states |i. (k) (54) The Hamiltonian matrix is also divided in blocks so that the action of the Hamiltonian during a Lanczos iteration can be expressed as: Ψf (k) = q H (q. If we consider a N = Z nucleus like 6 Li. but.k) Ψi (q) (55) The k segments correspond to specific values of M (and occasionally t) of the first subspace. This means that our span will be limited to nuclei with few particles outside closed shells (130 Xe remains an easy calculation) or to almost semi-magic.. These nuclei are spherical. 2p1/2 . To overcome this difficulty in equalizing the and  J  J . α and |j. |i and |α are states with good angular momentum. there are 50000 Slater determinants for protons and neutrons with M = 1/2 and t = 14. there are two floating point multiplications to be done. four of them equivalent to the pf shell plus the 1h11/2 orbit. If now we consider 132 Ba with also 6 protons and 6 neutron-holes in a valence space of 5 shells. 1g9/2 . Once R(i). therefore the seniority can provide a good truncation scheme. must be complemented with. (52) The NZME of the Hamiltonian between states |I and |J is then: I|H|J = J|H|I = V (K). This situation is the same as in semi-magic nuclei. J = R(j) + β. incorporating some of its algorithmic findings as MSHELL (Mizusaki. β. 1f5/2 . The NZME read now: I|H|J = J|H|I = hij · hαβ · W (K). the possibilities of performing SM calculations become more and more limited. Q(p) and α. 1969). µ are known. 1999) the fundamental idea of the code ANTOINE is kept. There exists j a strict analogy between ∆m in m-scheme and λ in the coupled scheme. They are built with the usual techniques of the Oak-Ridge/Rochester group (French et al. when many protons and neutrons are active. R(j). nuclei tend to get deformed and their description requires even larger valence spaces.  0 (57) . Within this formalism we can introduce seniority truncations. λ λ with hij = i|Op |j . As a counterpart. For the proton-neutron NZME the one-body operators in each λ space can be written as Op = (a†1 aj2 )λ . E. 2p3/2 . hαβ = α|Oω |β . 2002). It is now possible to overcome it dividing the Lanczos vectors in segments: Ψf = k Ψf .10 in French et al. J. Until recently this was the fundamental limitation of the SM code ANTOINE. Now. with Jmin = |J0 − J1 | and Jmax = J0 + J1 . in addition. As far as two Lanczos vectors can be stored in the RAM memory of the computer. This explains the interest of a shell model code in a coupled basis and quasi-spin formalism. as for instance the Tellurium isotopes. The continuity between the first state with Jmin and the last with Jmax is maintained and consequently the fundamental relation I = R(i) + α still holds. it gives a natural way to parallelize the code. each processor calculating some specific (k) Ψf . i. We need to perform –as in the m-scheme code– the three integer additions which generate I. splitting the valence space in two parts and writing the full space basis as the product of states belonging to these two parts. Other m-scheme codes have recently joined ANTOINE in the run. which in m-scheme were just phases. to treat the deformed nuclei in the vicinity of 80 Zr. the non-zero elements of the matrix in the full space are generated with three integer additions: I = R(i) + α. 2000) or REDSTICK (Ormand and Johnson. we can still establish a relation K = Q(p) + ω. Hence. The price to pay for the increase in size is a strong reduction of performance of the code. Now.19 elements (TBME). but the problem of the semi-magic nuclei and the asymmetry between the protons and neutrons spaces remains.   i α  W (K) ∝ V (K) · j β  λ λ (53) The performance of the code is optimal when the two subspaces have comparable dimensions. To give an order of magnitude the dimension of the where V (K) is a TBME. All the nuclei of the pf -shell can now be calculated without truncation. (56) As the mass of the nucleus increases. The generation of the proton-proton and neutronneutron NZME proceeds exactly as in m-scheme. the orbit 1d5/2 . 1980) for the NCSM (Navr´til and Barrett. 1994). This last result. i) No-core calculations. Due to the basis truncation. In a first phase. 2001). 2000) or Cartesian-coordinate formalism (Navr´til a and Ormand. together with the already known problems of under-binding. We then take the 1i13/2 and 1h9/2 orbits as the first subspace. One of the major problems of the no-core SM is the convergence of the results with the size of the valence space.. The typical working dimension for large matrices in NATHAN is 107 (109 with ANTOINE). according to the number of HO excitations. Let us list some of the present opportunities. we can include the neutron 1f7/2 orbit in the proton space. while the 2f7/2 . it is necessary to derive an effective interaction from the underlying inter-nucleon interaction that is appropriate for the basis size employed. 3p3/2 and 3p1/2 form the second one. either G-matrixbased two-body interactions (Zheng et al.20 dimension of the two subspaces we have generalized the code as to allow that one of the two subspaces contains a mixing of proton and neutron orbits. for heavy Scandium or Titanium isotopes. the a near-converged results for A = 6 using a non-local Hamiltonian (Navr´til et al. 1996). This new code allows to perform calculations in significantly larger basis spaces and makes it possible to reach full convergence for the A = 6 nuclei. The states previously defined with Jp and Jn are now respectively labelled A1 . The coupled code has another important advantage.. according the matrix dimension. The rotational band of 238 U seems to be still very far from the reach of SM calculations. the first observation of the a incorrect ground-state spin in 10 B predicted by the realistic two-body nucleon-nucleon interactions (Caurier et al. The calculations are performed using a large but finite harmonic-oscillator (HO) basis. e.. In hij and hαβ now appear mean values of all the operators in Eq.b) is a method to solve the nuclear many body problem for light nuclei using realistic inter-nucleon forces. For example. J1 and A2 . the m-scheme matrix dimension exceeds 800 millions. 2002) now opens the possibility to include a realistic three-nucleon force in the NCSM Hamiltonian and perform calculations using two. This can be essential for the calculation of strength functions or when many converged states are needed. only two-body terms. the 10 ω calculations for 10 C. where convera gence to the exact solutions was demonstrated for the A = 3 system. 1999). No core shell model The ab initio no-core shell model (NCSM) (Navr´til a et al. J2 . but predictions for 218 U are already available. A1 and A2 being the number of particles in each subspace. Present possibilities The combination of advances in computer technology and in algorithms has enlarged the scope of possible SM studies. In the latter case. 2002). for example to describe non-yrast deformed bands. For the N = 126 isotones we have only proton shells.g. (k) (58) F. A crucial feature of the method is that it converges to the exact solution when the basis size increases and/or the effective interaction clustering increases. 2000a. It can be perfectly parallelized without restriction on the number of processors. For 6 Li we can handle excitations until 14 ω and at least 8 ω for all the p-shell nuclei. . 2001. it opens the possibility to investigate slowly converging intruder states and states with unnatural parity. the effective interaction is derived in a sub-cluster approximation retaining just twoor three-body terms.and three-nucleon forces for the p-shell nuclei. confirms the need of realistic three-nucleon forces. a conclusion that also stems from the Green Function Monte Carlo calculations of Wiringa and Pieper (2002) (see also Pieper and Wiringa. H = k H (k) ) leading to different vectors that are added to obtain the full one: Ψf = H (k) Ψi . a new version of the shell model code ANTOINE has been developed for the NCSM applications (Caurier et al. in general. where it was a also shown that a three-body effective interaction can be introduced to improve the convergence of the method. 2001b). up to A-body components even if the underlying interaction had. 2f5/2 . the 14 ω space for 6 Li and. Among successes of the NCSM approach was the first published result of the binding energy of 4 He with the CD-Bonn NN potential (Navr´til and Barrett.. The largest bases reached with this code so far are. The same was later accomplished for the A = 4 system (Navr´til and Barrett. This rea sulted in the elimination of the purely phenomenological parameter ∆ used to define G-matrix starting energy. their applications were limited to the use of realistic two-nucleon interactions. 2002) together with the ability to solve a three-nucleon system with a genuine three-nucleon force in the NCSM approach (Marsden et al.(50) The fact that in the coupled scheme the dimensions are smaller and that angular momentum is explicitly enforced makes it possible to perform a very large number of Lanczos iterations without storage or precision problems. Recently. Besides. 1999).. (k) Ψf = k Ψi . The capability to derive a three-body effective interaction and apply it in either relative-coordinate (Navr´til a et al. G. or derived by the Lee-Suzuki procedure (Suzuki and Lee. It means that there is no problem to define as many final vectors as processors are available. In practice.. The effective interaction contains.. for a review of the results of the Urbana-Argonne collaboration for nuclei A≤10). A truly ab initio formulation of the formalism was presented by Navr´til and Barrett (1998). The calculation of the NZME is shared between the different processors (each processor taking a piece of the Hamiltonian. can be easily studied and the addition of few particles or holes remains tractable. p. N the ¯ number of valence particles. 56 Ni is taken as inert core. as the 1f7/2 orbit becomes more and more bound. iii) The r3 g valence space. and can be treated without truncations. a quantity readily accessible once the Hamiltonian has been fully diagonalized. Some long chains of Tellurium and Bismuth isotopes have been recently studied (Caurier et al. powers of the Hamiltonian determine the properties of the systems. and S the span of the spectrum (distance between lowest and highest eigenstates). (59) Γ(xN + 1)Γ(¯N + 1) S x where x is an a-dimensional energy. for instance the N = 126 isotones. v) Heavy nuclei. S) = pxN q xN d N Γ(N + 1) . the partition function. We shall discuss both questions in turn. MK = K . x = E . IV. A. N. truncated calculations are close to exact. S). p an asymmetry parameter.e. shell model is still synonymous with diagonalizations. in turn synonymous with black box. and off-diagonal elements. The continuous lines 4 3 Tridiagonal elements 2 1 0 -1 -2 -3 0 0. N. 2001). σ 2 = N p(1 − p)ε2 . THE LANCZOS BASIS still hears questions such as: Who is interested in diagonalizing a large matrix? As an answer we propose to examine Fig. the Laplace transform of the density of states. Z(β) = i i| exp(−βH)|i . All the semi-magic nuclei.2 0.. which for low K are the same as those of a discrete binomial.. the variance σ 2 and x are given by S = N ε. This space describes nuclei in the region 28 < N. with integer n. For neutron rich nuclei in the Nickel region. Ec = H1 . We use the notation rp for the set of the nlj orbits with 2(n − 1) + l = p excluding the orbit with maximum total angular momentum j = p + 1/2. The ββ decay (with and without neutrinos) of 76 Ge and 82 Se are a prime example. and the logarithmic and inverse binomial approximations (nib) (Zuker et al. N. 2003). to be equated to the corresponding moments of ρb (x. averages given by the traces of HK .21 ii) The pf -shell. (60) S There exists a strong connection between the Lanczos algorithm.. a tractable valence space that avoids the center of mass problem can be defined: A 48 Ca core on top of which pf protons and r3 g neutrons are active.. Level densities Note that ρb (x. The necessary definitions and equalities follow.8 1 Hi i+1(exact) Hi i+1(nib) Hi i(exact) Hi i(log) FIG. of the 1+2-body Hamiltonian H it represents). for instance.e. (61) γ2 = M4 − 3 = N pq For many. i. When the matrix is diagonalized the level density is well reproduced by a continuous binomial ¯ ρb (x. Ec = N pε. iv) The pf g space. p + q = 1. This where the shell model has been most successful and exact diagonalizations are now possible throughout the region Beyond 56 Ni. 10 showing the two point-functions that define the diagonal. are calculated knowing the first four moments of the matrix (i. 10 Tridiagonal matrix elements for a 6579 dimensional matrix.4 i/dim 0. N . To enlarge the pf valence space to include the 1g9/2 orbit is hopeless nowadays. Hi i+1 in a typical Lanczos construction. x = 1 − x. Furthermore. serious center of mass spuriousness is −k k expected in the 1f7/2 1g9/2 configurations (see Appendix C). N n if x = n/N = nε/S. γ1 = M3 = √ σ N pq 1 − 6pq . d = d0 (1 + p/q)N . 2001) the wave-functions are fully converged when 6p-6h excitations are included. S. in 60 Zn (Mazzocchi et al..e. p. when the matrix admits no block decomposition. In the three cases. The deformed nuclei (N ∼ Z ∼ 40) are more difficult because they demand the inclusion of the 2d5/2 orbit to describe prolate states and oblate-prolate shape coexistence. The evolution operator addresses more specifically the problem of evolving from a starting vector into the exact ground state. Most of them are nearly spherical. Hii .. One . i. The partition function can be written as Z(β) = E ρ(E) exp(−βE) i. the centroid Ec . MK = (H − Ec )K MK q−p σ 2 = M2 .e. S) reduces to a discrete binomial. and the evolution operator exp(iHt). d−1 tr(HK ) = HK . These results hold at fixed quantum numbers. Z < 50. p and S are calculated using the moments of the Hamiltonian H.6 0. Introducing the energy scale ε. p. the latter has the advantage that the necessary parameters are well defined and can be calculated. ρ = JT (2J + 1)(2T + 1)ρJT but with a partial ρJT at fixed quantum numbers. They are reached in the 48 Ca(p. In both cases the convergence is very fast. all other contributions are bound to 18 19 The energies are referred to the mean (centroid) of the distribution. while they are measured with respect to the ground state. The shift ∆ is necessary to adjust the ground state position. Furthermore.22 These quantities also define the logarithmic and inverse binomial (nib) forms of Hii and Hi i+1 in Fig. to reconcile simplicity with full rigor we have to examine the tridiagonal matrix at the origin. 1200 1000 ρ(fpgd J=6) ρ(bin) 30 25 20 Level Density 15 10 5 0 -5 0 10 20 Binomial Bethe Ni (exp) ρ(States per MeV) 60 800 600 400 200 0 0 2 4 6 8 10 12 14 16 30 Energy 40 50 FIG. n)( 48)Sc reaction. which will be our standard example of Gamow Teller transitions. They do indeed. and it is not difficult to show in general that it occurs for a number of iterations of order N log N for dimensionality d = 2N . Granted that a single binomial cannot cover all situations we may nonetheless explore its validity in nuclear physics. Zuker (2001) extended the approximation to a binomial. 1995. the variational pivot is clearly better if we are interested in the ground state: If its overlap with the exact solution exceeds 50%. which involves all states. one random (homogeneous sum of all basic states). ∆) = . Though Bethe and binomial forms are seen to be equivalent. 12 Binomial. B. . a. The conjecture breaks down if a dynamical symmetry is so strong as to define new (approximately) conserved quantum numbers. The ground state wavefunction is unique of course. 1999). Nakada and Alhassid. E(MeV) FIG. 10. The problem is that the calculations are hard. but the result remains valid only in some neighborhood of the centroid. As the shape of the level density is well reproduced by a binomial except in the neighborhood of the ground state. as shown in Fig. 14. if binomials are to be useful. The problem also exists for the binomial18 and the result in the figure (Zuker. and we calculate the ground state with two pivots. Mon and French (1975) proved that the total density of a Hamiltonian system is to a first approximation a Gaussian. 11 Exact and binomial level densities for the matrix in Fig. and hence describe the full spectrum. 13 reveals that they are very different for the first few iterations. 2001) solves it phenomenologically. In the example given above.. 11. but it takes the different aspects shown in Fig. Note that the corresponding lines are almost impossible to distinguish from those of the exact matrix. The mathematical status of these results is somewhat mixed. The matrix is 8590-dimensional.. some dependence on the pivot should exist. 10. they must reproduce—for some range of energies—Eq. The associated level densities are found in Fig. while in the former. where the observed level densities are extremely well approximated by the classical formula of Bethe (1936) with a shift ∆. one does not expect it to hold generally because binomial thermodynamics is trivial and precludes the existence of phase transitions. The Shell Model Monte Carlo method provides the only parameter-free approach to level densities (Dean et al. but soon they merge into the canonical patterns discussed in the preceding section. 12. the precise meaning of the a parameter is elusive. 1997) whose reliability is now established (Alhassid et al.. A zoom on the first matrix elements in Fig. we do not deal with the total density. (2001) conjectured that the tridiagonal elements given by the logarithmic and nib forms are valid. √ √ e 4a(E+∆) 2π ρB (E. The ground state Obviously. We examine it through the J = 1 T = 3 pf states in 48 19 Sc . (62) 12 (4a)1/4 (E + ∆)5/4 Obviously. Langanke. 1992) are also given. the other variational 8 (lowest eigenstate in f7/2 space). In this case Zuker et al. Bethe and experimental level densities for 60 Ni. where the experimental points for 60 Ni (Iljinov et al. However. 1998. (62). However. (67) (66) FIG. Now introduce cm Pm = exp( Sm Pm ). (68). a system of coupled equations obtains. We shall see how to exploit this property of good pivots to simplify the calculations. but it is a good introduction to the underlying ideas. Variational Hii+1. etc. and a variational pivot. 2 1 3 c3 = S3 + S1 S2 + S1 . The first two are 2 1 0 =S2 V2 + S1 ( V2 − V1 ) + S1 ε1 + 1 2 0 =S3 V3 + S2 ε2 + S1 S2 (V3 − V1 )+ 1 1 2 3 1 S1 ε2 + S1 ( V3 − V1 ) + S1 . but the succeeding vectors—except the pivot—are not normalized. 20 1. At the first iteration Eq. (65) whose solution is equivalent to diagonalizing a matrix with Vm+1 in the upper diagonal and 1 in the lower one. (46).6 Wavefunction(s) 0. 0 (63) where Pm is a polynomial in H that acting on the pivot produces orthogonal unnormalized vectors in the Lanczos basis: Pm |0 = |m . E = E0 +V1 c1 implies that once c1 is known the problem is solved. (45.8 0.23 30 25 Tridiagonal elements 20 15 10 5 0 -5 0 20 40 i Hii. Then. where the natural pivot is heavily fragmented.A. Inserting in Eq. it turns out to be quite useful. (66) and regrouping. (70) and the process is repeated. 2 3! 2 (70) In Section II. Eq. (65). then multiply and divide |m−1 and |m+1 by their norms to recover Eq. Vm = m|m / m − 1|m − 1 = Hm m−1 . which have the advantage of providing a model for the wavefunction over the full basis. One is found in Fig. (64) ¯ To relate with the normalized version (m ⇒ m). 14 The ground state wavefunction in the Lanczos basis for a random.3 0. Variational Hii Random Hii+1. The resulting value for S2 is reinserted in Eq. the equation for S3 (not shown) may be .1 0 0 5 10 15 20 i Random Variational Expanding the exponential and equating terms 1 2 c2 = S 2 + S 1 .1 we have sketched the coupled-cluster (or exp(S)) formalism. The profound reason for using an unnormalized basis is that first term in Eq. The exp(S) method 0. Furthermore. (71) 20 There are some very interesting counter-examples. (46) becomes H|m = Vm |m − 1 + Em |m + |m + 1 . The ¯ ¯ ¯ ¯ secular equation (H − E)|¯ = 0 leads to the recursion 0 cm−1 + (Em − E) cm + Vm+1 cm+1 = 0. The plotted values are the squared overlaps of the successive Lanczos vectors with the final wave-function Note that we have used the formal identification Pm Pn = Pn+m as a heuristic way of suggesting Eqs. The construction is as in Eqs. The resulting value for S1 is inserted in Eq. Once S1 and S2 are known. and obtain Em = 2 Hm m . here we shall solve the recursion by transforming it into a set of non-linear coupled equations for the cm amplitudes. 3! c1 = S 1 . The formulation in the Lanczos basis cannot do justice to the general theory. be smaller and in general they will decrease uniformly . Hence.4 0. 13 Tridiagonal matrix elements after 100 iterations for a random and a variational pivot. (71) which is solved by neglecting S3 . Calling εm = Em −E0 and replacing E = E0 +V1 c1 in Eq (65) leads to cm−1 + (εm − V1 c1 ) cm + Vm+1 cm+1 = 0. (70) is solved neglecting S2 .5 0. which is of course equivalent to the symmetric problem. instead of the usual cm -truncations we can use Sm -truncations. Random 60 80 100 the full wavefunction takes the form |¯ = (1 + c1 P1 + c2 P2 + · · · + cI PI )|0 .7 0. Divide by m|m 1/2 . 24.46). 25 30 35 40 (68) (69) FIG.2 0. 5 6 5. Monte Carlo methods Monte Carlo methods rely on the possibility to deal with the imaginary-time evolution operator acting on some trial wavefunction exp(−βH)|0 which tends to the exact ground state |¯ as β ⇒ ∞. 2002) and to calculation of the excitation energies of the deformed 0+ states of 56 Ni and 52 Cr (Horoi et al. As we shall see next. In general. to our knowledge. the closed shell pivot is particularly good. confirming that “exponential convergence” and exp(S) are very much the same thing. but no basis is constructed. 2. It was borne out when the diagonalization became feasible two years later. for which the f7/2 [or eventually (f7/2 p3/2 )m ] subspaces provide a good pivot. the energy (or some observable Ω) is calculated through ˆ 0|e−βH/2 Ω e−βH/2 |0 0|e−βH |0 β→∞ =⇒ ¯ Ω|¯ 0| ˆ 0 ¯¯ 0|0 (73) . a. the first iterations are associated to truncated spaces of much smaller dimensionalities than the total one (dm ≈ 109 ). Successive diagonalizations make it possible to reach the exact energy by exponential extrapolation. and Porrino (2003) have recently proposed a factorization method that allows for importance sampling to approximate the exact eigenvalues and transition matrix elements. and the exponential regime sets in at the first iteration. once projection to good angular momentum can be tackled efficiently. 2003). and so on. and hence enormous gains in dimensionality. the correct energy at the next point . The same argument applies for other regions. In later work. Other numerical approximation methods FIG. exp(−ai0 ) − 1 (72) which equals e0 at point i = i0 . 2001. If all amplitudes except S1 are neglected. fits m shell nuclei. Still. the only prediction for a truly large matrix that has preceded the exact calculation remains that of 56 Ni. 2003). 1999c). m0 = 0]: We set i0 = 1.. Fortunately. has been proposed (Dukelsky and Pittel. i0 ) = e0 exp(−ai) − 1 . Then. This also happens for lighter pf The exponential convergence method introduced in the previous section was first described by Horoi et al.5 3 2. incorporated.e. The method has been successfully applied to the calculation of the binding energies of the pf shell nuclei (Horoi et al. Choose a so as to yield conv(i = i0 + 1) = e(i0 + 1) i. (70) should give a fair approximation. (1999).. 15. (48. 3. and Papenbrock and Dean (2003) have developed a method based in the optimization of product wavefunctions of protons and neutrons that seems very promising. J = 2 in Fig. improved by incorporating Eq.24 7 6. Dukelsky et al.5 4 3. and the subject deserves a comment. In this example. with m = 16. For well deformed nuclei Hartree Fock should provide good determinantal pivots. 15 provides an example of the efficiency of the method for the lowest two states in 56 Ni.5 Energies 5 4. The work of Mizusaki and Imada (2002. A totally different approach which has proven its power in other fields. the difference scheme (65) is the same that would be obtained for an harmonic H ≡ εS 0 + V (S + + S− ). As a consequence. e0 . Eq. The index i is replaced by the truncation level i ≡ t/2 [Eq. They have devised different extrapolation methods and applied them to some pf -shell nuclei. it is almost always possible to find good pivots. the Lanczos and exp S procedures provide a convenient framework in which to analyze other approaches. With a very good pivot. The results reproduce those of the S2 truncation. An equivalent approach—numerically expedient—consists in diagonalizing the matrices at each cm truncation level. ˆ Instead. is based on the fact that the width of the total Hamiltonian in a truncated space tends to zero as the solution approaches the exact one. Andreozzi. Fig. 15 Energy convergence for ground and first excited state in 56 Ni as a function of the truncation level t (Caurier et al. it will always set in for some value of i0 that may be quite large (Fig. fix a so as to reproduce the energy at the second iteration and check that the curve gives indeed the correct value at the third point. Check that e(i0 + 2) is well reproduced.. after some iteration. convergence is reached when its value remains constant from step to step. 2002). 14 suggests i0 ≈ 25 for the random pivot). (71).5 2 4 6 8 10 truncation t 12 14 16 -Ec(J=0) -Ec(J=2) E(J=2)-E(J=0) exp. In the case of 56 Ni the good pivot is a closed shell. under a different but equivalent guise. and checked by the equation for S3 . a hierarchy of configurations determined by their average energy and width was proposed.. the density matrix renormalization group. the energies converge exponentially: Introduce conv(i. This is very much what 0 the Lanczos algorithm does. Lo Iudice.. Successive approximations amount to introduce anharmonicities. Since the energy depends only on S1 . . pairing plus quadrupole). 1957). The Green Function Monte Carlo studies mentioned at the beginning of Section II are conducted in coordinate space.g. 1984. At present it remains the only approach that can deal with much larger valence spaces than the standard shell model. A “sign problem” arises because the integrands are not definite positive. It should be understood that SMMC does not lead to detailed spectroscopy. for a review). Then. leading to enormous precision problems. Whitehead. . and Mizusaki (1998) consists in exploring the mean field structure of the valence space by means of Hartree-Fock calculations that break the symmetries of the Hamiltonian. The approach relies on the HubbardStratonovich transformation. its first column Ui 0 gives the amplitudes of the ground state wavefunction in the tridiagonal basis while the first row U0 j determines the amplitude of the pivot in the j-th 2 eigenstate. 2001b). act with it on a target state |t to define a pivot |0′ = T |t that exhausts the sum rule 0′ |0′ for T . 1995). C. Honma. One of the most interesting is the calculation of strength functions (Bloom. as it only produces ground state averages. Good quantum numbers are enforced by projection techniques (Peierls and Yoccoz. In practice. The introduction of Monte Carlo techniques in the Lanczos construction is certainly a tempting project. given a transition operator T . A strong connection between mean field and shell model techniques is also at the heart of the Vampir approach (Petrovici et al. Lanczos Strength Functions The choice of pivot in the Lanczos tridiagonal construction is arbitrary and it can be adapted to special problems. 1997a. More basic states are iteratively added until convergence is achieved. 1980): If Ui j is the unitary matrix that achieves diagonal form. For a very recent review of the applications of this method see (Otsuka et al. .. FIG. 16 Evolution of the Gamow-Teller strength function of 48 Ca as the number of Lanczos iterations on the doorway state increases. . and Langanke. Then the authors borrow from SMMC to select an optimal set of basic states and the full Hamiltonian is explicitly diagonalized in this basis. 1999). The Shell Model Monte Carlo (SMMC) variant is formulated in Fock space. but it is very well suited for finite temperature calculations.25 which is transformed into a quotient of multidimensional integrals evaluated through Monte Carlo methods with importance sampling. Dean. and the sign problem is circumvented either by an extrapolation method or by choosing Hamiltonians that are free of it while remaining quite realistic (e. 2001) and of the projected shell model (PSM) (Hara and Sun. (Schmid. Normalize it. U0 j plotted against the eigenenergies Ej is called the “strength function” for that pivot. and hence directly amenable to comparisons with standard SM results (see Koonin. The quantum Monte Carlo diagonalization method (QMCD) of Otsuka.. and McGrory. the observed profile would have the aspect shown at the bottom of Fig. but at 22 Na they were so bad that they became the standard example of the unreliability of the realistic forces (Brown and Wildenthal. Trouble showed in 0 ω calculations somewhat later simply because it takes larger matrices to detect it in the sd shell than in the ZBM space (dimensionalities of 600 against 100 for six particles). it has the advantage of containing two doubly magic nuclei 48 Ca and 56 Ni for which truncated calculations proved sufficient to identify very early the core of the monopole problem: The failure to produce EI closures. and 250 keV widths for the non converged states. After full convergence is achieved for all states in the resonance region. Its efficiency in cross-shell calculations was further confirmed by Zamick (1965). 16 that retraces the fragmentation process of the sum rule pivot. 1000 iterations. A. THE 0 ω CALCULATIONS As the Lanczos vector |I is obtained by orthogonaliz- In this section we first revisit the p. ing HI |0 to all previous Lanczos vectors |i . whose first 2I − 1 moments are the exact ones. and the success of the ZBM model (Zuker. all the spikes are affected by an experimental width. Nowadays it would take an afternoon on a laptop. Though the pf shell demands much larger dimensionalities. which demands a 3b mechanism It took two years on a Vax. The profiles become almost identical. as illustrated in Fig. Therefore the eigensolutions of the I × I matrix define an approximate 2 strength function SI (E) = i=1. The upper panel shows The determinant influence of the monopole interaction was first established through the mechanism of Bansal and French (1964).I δ(E − Ei ) i|T |0 . 1988) and lead to the titanic22 Universal SD (USD) fit of the 63 matrix elements in the shell by Wildenthal (1984). The French Bansal parameters bpd and bps (see Eq. Up to 5 particles the results of Halbert et al. after convoluting with gaussians of 150 keV width. the strength function has the aspect shown at the bottom of the figure. sd. Zuker (1969) identified the main shortcoming of the model as due to what must be now accepted as a 3b effect21 . The monopole problem and the three-body interaction FIG. 21 22 The ZBM model describes the region around 16 O through a p1/2 s1/2 d5/2 space. 17. 17 Strength functions convoluted with gaussians: Upper panel. Then we propose a sample of pf shell results that will not be discussed elsewhere. Buck. i < I. and pf shells to explain how a 3b monopole mechanism solves problems hitherto intractable. (9b)) must change when going from 14 N to 16 O. In practice. 0|Hk |0 = j 2 δ(E − Ej )U0j . are those of the (74) (75) (76) the result for 50 iterations. 2 k U0j Ej . in this case a 48 Sc “doorway” obtained by applying the Gamow-Teller operator to the 48 Ca ground state. The term doorway applies to vectors that have a physical meaning but are not eigenstates. the tridiagonal matrix elements are linear combinations of the moments of the strength distribution.26 by definition |0 = T |t = j 0′ |0′ U0j |j . whose moments. . Bottom panel. Finally Gamow Teller transitions and strength functions are examined in some detail. strength function S(E) = j V. (1971) with a realistic interaction were quite good. 50 iterations. 1968) is implicitly due to a monopole correction to a realistic force. The eigenstates act as “doorways” whose strength will be split until they become exact solutions for I large enough. and assuming a perfect calculation. and r = p1/2 and r ≡ d3/2 . At 57 Ni there is a bunching of the upper orbits that the realistic BonnC (BC) and KB describe reasonably well. for a recent experimental check in the beta decay of 56 Cu). p1/2 ). The effective single particle energies in Fig. The classic fits of Cohen and Kurath (1965) (CK) defined state of the art in the p shell for a long time. the solution follows: Eq. As they preceded the realistic G-matrices. Once this is understood. (2001) defined a KB3G interaction that brings interesting improvements over KB3 in A = 50-52.27 9/2− 7/2− 3/2− 5/2− 5/2− 0 3/2− 5/2− 5/2− 9/2+ 7/2+ (1/2−. In Pasquini (1976) and Pasquini and Zuker (1978) the following modifications were proposed (f ≡ f7/2 . 2001. Mart´ ınez-Pinedo et al. The work of Navr´til and a Ormand (2002). At the end of the shell BC and KB reproduce an expanded version of the 41 Ca spectrum. Zuker (2003) proposes (κ = κ0 + (m − m0 ) κ1 ) . 19 are just the monopole values of the particle and hole states at the subshell closures.2) is that the bunching of the upper orbits should persist. (II. 1997). p3/2 .. -5 A 1f7/2 2p3/2 2p1/2 1f5/2 KB3 FIG. By incorporating this hint—which involves the Vrr′ centroids—and fine-tuning FIG. Using f ≡ (p3/2 . where modifications such as in Eq. they also contributed to hide the fact that the latter produce in 10 B a catastrophe parallel to the one in 22 Na. r ≡ f5/2 . which is most certainly incorrect. 1996b. 1994. Compared to KB3. there are some problems. This produces problems with the description of Gamow-Teller processes (see Borcea et al. 23 The multipole changes—that were beneficial in the perturbative treatment of Poves and Zuker (1981b)—had much less influence in the exact diagonalizations. (77) are beneficial but (apparently) insufficient. They give an idea of what happens in a 2b description. and Pieper. 18.B. The indi˜d cation from Hm in Eq. At the origin.3/2−) 4 ∆E (MeV) 5/2− 9/2− 7/2− 3/2− 5/2− 4 -5 3 2 1 0 9/2− 5/2− 7/2− 1/2− 5/2− 3/2− 7/2− 7/2 − 1f7/2 2p3/2 2p1/2 1f5/2 BC 3 E (in Mev) -10 1/2− 7/2− 1/2− 1/2− 2 1 -15 -20 3/2− 3/2− 3/2− 3/2− 0 -25 40 0 50 60 70 80 KB KB’ KB3 Expt.. KB3 was adopted as standard23 in successful calculations in A = 47-50 that will be described in the next sections (Caurier et al. the orbit 1f5/2 is definitely too low in 57 Ni. Hence the need of KB1-type corrections. (77) is assumed to be basically sound but the corrections are taken to be linear in the total number of valence particles m (the simplest form that a 3b term can take). sd. KB’ and KB3 interactions compared to experiment Fig. pf ) shells respectively. 19 Effective single particle energies in the pf -shell along the N = Z line. Varga. the Vf r ones Poves et al. 18 gives an idea of what happens in 49 Ca with the KB interaction (Kuo and Brown. the spectrum is the experimental one. For higher masses. Vf1f (KB1) = Vf1f (KB) − 110 keV. (77) E (in Mev) -10 -15 -20 -25 40 50 60 70 80 A The first line defines KB’ in Fig. s1/2 for the p and sd shells. the interaction FPD6 has a better gap in 56 Ni. where only one exists. and Wiringa (2002) acted as a powerful reminder that brought to the fore the 3b nature of the discrepancies. though not as severe as the ones encountered in the sd and p shells. but around 56 Ni there are still some problems. but they fail to produce a substantial gap. however. The variants KB2 and KB3 in Poves and Zuker (1981b) keep the KB1 centroids and introduce very minor multipole modifications. d5/2 . but nothing comparable to the serious ones encountered in the sd shell. 18 The spectrum of 49 Ca for the KB. 1968): there are six states below 3 MeV. f7/2 ) generically in the (p. r r Vf0f (KB1) = Vf0f (KB) − 350 keV.. in 41 Ca. VfT (KB1) = VfT (KB) − (−)T 300 keV. computed with the BC and KB3 interactions. . In the p shell. which also does well at the lower masses. in the pf shell KB3. a 4.5 4 3. which is nearly perfect in A = 47-50. As we have seen. must be turned into KB3G for A = 50-52. 22 The excitation spectra of interactions. 1996). and the pentagons to the CK fit. 20.5 0 -0. Na for different inter- The black circles correspond to the bare KLS G-matrix ( ω =17 MeV). In 24 Mg we could still do with the value for 22 Na but To determine a genuine 3b effect (i.28 8 7 6 5 E (MeV) 4 3 2 1 0 E (MeV) -1 -2 0 1 2 J 3 4 4 3 2 10 NO KLS CK KLS κ=1.80 EXP FIG. The spectra in Fig.05(m − 6).5 3 E (MeV) 2. The κ correction eliminates the severe discrepancies with experiment and give values close to CK. In 22 Na the story repeats itself: BC and KB are very close to one another. the ground state spin is again J = 1 instead of J = 3 and the whole spectrum is awful. The κ correction restores the levels to nearly correct positions.90 EXP 0 1 3 2J 24 5 FIG. The results for 10 B are in Fig. This simple cure was not discovered earlier because κ is not a constant. Fig 4. and now the spectra are as good as those given by USD.9 − 0. NO and KLS give quite similar spectra.55 EXP where R stands for any realistic 2b potential. and Table I shows that with equal normalizations BC and KB in the pf shell are 24 It may be possible to fit the data with a purely 2b set of matrix elements: USD is the prime example. To find a real problem with a 2b R-interaction (i. the white squares to the same with κ = 1. 6 Ω). though USD still gives a better fit. as expected from the discussion in Sections II. 21 The excitation spectrum of actions.5 κ δT0 f f (78) E (MeV) 4 KB USD KB κ=0.4 with any monopole corrected KB interaction.D. but it is R-incompatible (Dufour and Zuker.5 0 1 2 3 J 22 3 2 1 KB USD KB κ=0. so we do not dwell on the subject.D it is due to a normalization uncertainty. (1991). the linearity of κ) a sufficiently large span of A values is necessary. B for different interac- 1 0 0 5 1 2 J 3 4 VfT (R) =⇒ VfT (R) − (−)T κ r r VfT (R) =⇒ VfT (R) − 1.5 2 1. The black squares (NO) are from (Navr´til and Ormand.5 1 0. Mg and 29 Si for different 4 5 6 FIG. 22 are obtained with κ(m) = 0.1 EXP around 28 Si it must be substantially smaller.A and II. corrections to Vrr′ become indispensable very soon and must be fitted simultaneously. The problem can be traced to weak quadrupole strength and it is not serious: as explained in Section II. .. 20 The excitation spectrum of tions.e.58 Ni.e. 2002. 7 6 5 KB BC USD KB κ=0.1. In particular the first B(E2)(2 −→ 0) transition in 58 Ni falls short of the observed value (140 e2 fm4 ) by a factor ≈ 0. compatible with N N data)24 we have to move to 56. as done successfully by Abzouzi et al. Some early spectroscopic applications of GXPF1 to the heavy isotopes of Titanium. KLS) to produce large gaps at N=32 and N=34. when only the T=1 neutron neutron interaction is active) this new interaction retains the tendency of the bare G-matrices (KB. often excellent. but the lowest level still has most of the strength and the fragments are scattered at higher energies. A = 48. 1999b). Vanadium and Chromium can be found in (Janssens et al. 2002. The same evolution with energy of the quasi-particles (Landau’s term for single particle doorways) was later shown to occur generally in finite systems (Altshuler et al. The realistic interactions place again the J=1 and J=3 states in the wrong order. If we choose |cs = |48 Ca and |cs + 1 = |49 Sc . κ(m = 8) ≈ 0. Among the other full 0 ω calculations let’s highlight the following: The SMMC studies using either the FPD6 interaction (Alhassid et al. The recent applications of the exponential extrapolation method (Horoi et al.. 1998). These results have been conclusive in establishing the soundness of the minimally-monopole-modified realistic interaction(s). 2001). The pf shell Systematic calculations of the A=47-52 isobars can be found in (Caurier et al. In (Zuker. There is no point in reproducing them here. Fig. Another indication comes from the T=0 spectrum of 46 V.. 2001. 1997. 1995). See text. It was found that for A = 48. The particular mixture in Eq. A = 47 and 49. f7/2 . The lowest. A = 50-52. BC reproduces the yrast spectrum of Cr almost as well as KB3. 1994) or the KB3 interaction (Langanke et al. but not with κ(m = 16) (Fig. 2002) and (Mantica et al.. 1/2 .. Other applications to the pf shell can be found in (Honma et al. 23 Backbending in Cr. . sometimes fundamental... leads to an almost pure eigenstate. 2003). 2002). the four pf orbits provide the doorways. 1997). 19). In the calcium isotopes i. 1994.. With κ(m = 8). 23). Otsuka et al. 1998b) and for 52 Sc and 52 Ti (Novoselsky et al. sometimes anecdotic. 2003) BC was adopted. Each nucleus has its interest. in the simplest way. where the N=32-34 gaps are explored... The calculations of Novoselsky et al. 48 B.. The fit starts with the G-matrix obtained from the Bonn-C nucleon nucleon potential (Hjorth-Jensen et al. Very recently. (78) was actually chosen to make possible. 1995)..e. 16 14 12 10 J 8 6 4 2 0 500 1000 1500 2000 ∆(EJ) (keV) 48 BC κ=0. a good gap in 48 Ca and a good single-particle spectrum in 49 Sc. (1997) for 51 Sc and 51 Ti using the DUPSM code (see also the erratum in Novoselsky et al. 1999. A comparison of the exact results with the SMMC can be found in (Caurier et al. 1997b). KB3G). p3/2. Nowadays we know better: it produces a doorway that will be fragmented..29 as close as in the sd shell.28 KB3 EXP 2500 3000 3500 within 100∼200 keV and to give a fairly good description of the highly deformed excited band of this doubly magic nucleus (Mizusaki et al. a new interaction for the pf shell (GXPF1) has been produced by a Tokyo-MSU collaboration (Honma et al. following the fitting procedures that lead to the USD interaction. KB3) and (Poves et al. (Mart´ ınezPinedo et al. 2002. Bonn-C. 1998a). The agreement with experiment for the energies... isospin non-conserving forces and “pure spectroscopy”. are more fragmented. The existence of excited collective bands in the N=28 isotones is studied in (Mizusaki et al. FIG. Small modifications of the Vf r matrix were made to improve the (already reasonable) r-spectrum in 57 Ni (see Fig. KB3).43 BC κ=0.43. the correct one is re-established by the three-body monopole correction.. For A = 56. The extensive QMCD calculation of the spectrum of 56 Ni have been able to reproduce the exact result for the ground state binding energy 1.. indicating that the three-body drift is needed. The middle ones. and we only present a few typical examples concerning spectroscopic factors. truncated calculations yielded κ(16) 0. contrary to FPD6 or KB3G that only predict a large gap at N=32. 1996). The B(E2)(2 → 0) in 58 Ni indicated convergence to the right value. as seen by the very large difference in single particle energies between the 1f7/2 and 12p3/2 orbits (3 MeV instead of the standard 2 MeV). 2003). 24 shows what happens to the f5/2 strength: it remains concentrated on the doorway but splits locally..28. Spectroscopic factors The basic tenet of the independent particle model is that addition of a particle to a closed shell a† |cs pror duces an eigenstate of the |cs + 1 system. though not as urgently as in the sd shell. and for the quadrupole and magnetic moments and transitions is consistently good. For a recollection of the SMMC results in the pf shell see also (Koonin et al.. the counterpart of 22 Na and 10 B in the pf -shell.. The fit privileges the upper part of the pf shell.. 19/2 17/2 15/2 13/2 − − − − 19/2 − 17/2 − 15/2 − 13/2 − 1/2 − − − 3/2 .0 A=46 isospin triplet in (Garrett et al.30 2. gν =l l 3. 1997) and calculated (Zuker et al. 1997. Exp. Th. is compared in Fig. Pure spectroscopy 0..0 −1. experiment vs shell model calculation in the full pf -shell space.5/2 . 1996) compare quite well with the calculated µth =3.0 µN in M 1 transitions and moments are used.0 10.5 and s s protons qπ =1.02 -0. A typical result is proposed in Fig. 2000. The first part of the test is passed. He has also analyzed the Finally the transitions: The transitions in Table IV are equally satisfactory. 2002) MED for the pair 49 Cr-49 Mn. which provides an intuitive physical explanation for the abrupt change in the MED (Bentley et al. Except when the M1 transitions are fully dominated by the spin term. The way these disparate contributions add to reproduce the observed pattern is striking.0 5. Another calculation of the isospin non conserving effects.08 0. where VCM and VBM stand for Coulomb an nuclear isospin breaking multipole contributions. Work on the subject usually starts with a litany.5 and bare g-factors gπ =5.0 µN and gν =0. 2. 2001). The four pairs were analyzed in (Zuker et al. 26 Yrast band of 51 Mn. the use of effective g-factors does not modify the results very much due to the compensation between the spin and orbital modifications. 0.0 20. Note the abrupt change in both B(M 1) and B(E2) for J = (17/2)− . ...04 -0.. (2j + 1)S(j.0 −2. 1997).06 MED (MeV) 0. 49. Brandolini et al. Such calculations need very good wavefunctions. beautifully reproduced by the calculation. 2002b.0 Energy (MeV) FIG. 2002) for A = 47.. 2000).397µN and Qth =35 efm2 . while VCm is a monopole Coulomb term generated by small differences of radii between the members of the yrast band. corresponding to stripping of a particle in the orbit 1f5/2 (Mart´ ınez-Pinedo et al. The naive view that the Coulomb energy should account for the MDE (mirror energy differences) turns out to be untenable. 24 Spectroscopic factors. is due to (Ormand..7/2 11/2 9/2 − − − 1/2 − 3/2 − 11/2 9/2 − − 7/2 5/2 − − 7/2 − 5/2 − KB3G FIG. gπ =1.04 0. 25.0 3. 1999. Lenzi et al. 1998. O’Leary et al. 2001. 25. Such as: Quadrupole effective charges for neutrons qν =0. and the standard test they have to pass is the “purely spectroscopic” one. 25 Experimental (O’Leary et al. 1997). 2002) where it was shown that three effects should be taken into account.3826 µN . tz ).0 0. high spin states..1 0. 2000) in A = 51. 26 with the experimental data. The origin of this isomerism is in the sudden alignment of two particles in the 1f7/2 orbit.5857 µN . 8 7 6 21/2 27/2 23/2 − S 25/2 27/2 − − − 23/2 21/2 − − − E(MeV) 5 4 3 2 1 0 Exp. At the end one can add some extras: The electromagnetic moments of the ground state are also known: Their values µexp =3. Then some spectrum: The yrast band of 51 Mn..568(2)µN and Qexp =42(7) efm2 (Firestone. At most the examiner will complain about slightly too high. Isospin non conserving forces Recent experiments have identified several yrast bands in mirror pf nuclei (Bentley et al. and their influence in the location of the proton drip line.0 15. 50 and 51. FIG. calculated in the full pf -shell space.. 1... whose MED are as well described as those of A = 49 in Fig.06 5 7 9 11 13 15 17 19 21 23 25 27 2J A=49 Exp VCm+VCM+VBM VCm VCM VBM The first nucleus we have chosen is also one that has been measured to complete the mirror band (Bentley et al.02 0 -0. .236(337) 12 11 16 + 16 + E(MeV) We complete this section with an even-even and an odd-odd nucleus. There are only three experimental states with spin assignment not drawn in the figure: a second J = (5+ ) at 1. a third J = (5+ ) at 1.797 (e2 f m4 ) 305 84 204 154 190 2. The agreement of the KB3G results with the experiment is again excellent. 6 4 2 + + 6 4 2 + + + + 0 + 0 + KB3G FIG.68 MeV and a second J = (6+ ) at 1.96 MeV. the third J = 5+ at 1.421 0. (µ2 ) N 0. calculated in the full pf -shell space.496 µN and Qth =–9. KB3G 8 + 8 + FIG. (µ2 ) N 0.91 MeV.42 MeV. experiment vs shell model calculation in the full pf -shell space.00012(4) >0.062(2) µN and Qexp =+50(7) e fm2 (Firestone. 28 Yrast band of 52 Mn. Exp. B(M 1) 9+ → 8+ B(E2) 2+ → 0+ 3+ → 2+ 4+ → 2+ 1 4+ → 2+ 2 6+ → 4+ 8+ → 6+ 9+ → 8+ (µ2 ) N 0. Exp. µth =2. The calculations are in very good agreement with them (see table VI). are displayed in fig. The calculated values are in very good correspondence with the experiment. (a) from (Brown et al. calculated in the full pf -shell. up to the band termination.. In figure 27 we show the yrast states of 52 Cr.37 MeV. 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Exp. 2000). The calculation places the second J = 5+ at 1. 1996) and they are in very good agreement with the theoretical predictions: µth =2.040 (e2 f m4 ) 132 5 107 26 68 84 0.572 (e2 f m4 ) 528(146) 169(67) 303(112) 236(67) 232(75) 1.00003 0.952 µN and Qth =+50 e fm2 reproduce them nicely. 16 + 10 9 8 7 6 5 15 14 13 12 + 15 14 13 12 + + + + + + + (10 ) + + 4 14 + (11 ) (9 ) (8 ) 5 + + + + 10 + 11 9 8 + + + 3 2 E(MeV) 12 + 12 + 1 0 7+ + 4 3+ + 2 1 + 6 +5 7 + + 4+ 3 1 2+ + 6 + 10 + 10 + Exp.215 TABLE V Transitions in 52 Cr. The calculated values.116 0. The agreement between theory and experiment is spectacular.41(13) µN (Speidel et al. Some electromagnetic transitions along the yrast sequence have been measured. 28. The yrast states of the odd-odd nucleus 52 Mn. The experimental data for the spins beyond the band termination (11+ to 16+ ) come from a recent experiment (Axiotis.97 MeV and the second J = 6+ at 1.6 B(M 1) 5− 7− → 2 2 9− 7− → 2 2 − 11 − → 9 2 2 15 − 17 − → 2 2 − 19 − → 17 2 2 B(E2) 7− → 2 9− → 2 9− → 2 11 − → 2 11 − → 2 17 − → 2 5− 2 5− 2 7− 2 7− 2 9− 2 13 − 2 (µ2 ) N 0. The electromagnetic moments of the first J = 2+ state are known: µexp =2.2(16) e fm2 (Firestone. µexp =3. 27 Yrast band of 52 Cr. 1974).31 TABLE IV Transitions in 51 Mn.057(38) (e2 f m4 ) 131(6) 7+7 −5 83(17) a 69(19) 59(2) 75(24) 0. Th.5(20) Th.16(5) 0. 1996). . 2000) and Qexp = −8.4 e fm2 In table V the experimental values for the electromagnetic transitions between the states of the yrast sequence are given. No- tice the perfect correspondence between theory and experiment for the states belonging to the multiplet below 1 MeV.662(215) 0.177 0.207(34) 0. experiment vs shell model calculation in the full pf -shell space. The experimental magnetic and quadrupole moments for the 6+ ground state are also known. Superallowed decays may shed light on the departures from unitarity of the Cabibbo Kobayashi Maskawa matrix. that is only present in β + decays.2720(18) (Hagiwara et al. 1966): u where δif allows only transitions between isobaric analog states.405 0. 1993. the sum on i runs over the proton (neutron) orbits in the valence space and k over the proton (neutron) orbits for S+ (S− ). 1963) led to the long standing “quenching” problem..4 ± 1. Th. 2002) is the ratio of the weak interaction axial-vector and vector coupling constants. the partial lifetime of a level with branching ratio br is t = t1/2 /br . 2002) (see Wilkinson. The information obtained from the weak decays has been complemented by the (p.32 TABLE VI Transitions in 52 Mn. The large majority of the rest decay by β emission or electron capture. From a theoretical point of view.15 >4.537 (e2 f m4 ) 92+484 −81 >1.667 0. The meaning of the valence space Before moving to the explanation of the quenching of the GT strength.043 −0. 2002). The determinantal state demands a quenching factor that is almost twice as large as the standard quenching factor Q2 = (0. B(M 1) 7+ → 6+ 8+ → 7+ 9+ → 8+ B(E2) 7 → 6+ 8+ → 6+ 8+ → 7+ 9+ → 7+ 11+ → 9+ + (80a) i 2 (µ2 ) N 0. p) reactions in forward kinematics. (µ2 ) N 0. for an alternative study).501(251) >0. as in our case. The total strengths S± are related by the sum rules S− (F ) − S+ (F ) = N − Z. For states of good isospin the value of B(F ) is fixed. The comparison of the (p.074+3.. only approximately one half of the sum rule value was found in the experiments. 2002a. The GT strength is not protected by a conservation principle and depends critically on the wavefunctions used. (82a) (82b) where N and Z refer to the initial state and S± (GT ) does not contain the gA /gV factor.759 (e2 f m4 ) 126 33 126 66 53 B(F ) and B(GT ) are defined as B(F ) = B(GT ) = gA gV f f tk i √ k ± 2Ji + 1 σ k tk ± √k 2Ji + 1 2 Exp. that make it possible to obtain total GT strengths and strength functions that cannot be accessed by the decay data because of the limitations due to the Qβ windows. f ǫ is the phase space for electron capture (Bambynek et al. The half-life for a transition between two nuclear states is given by (Behrens and B¨ hring. np and nh denote the number of particles and holes.b. S− (GT ) − S+ (GT ) = 3(N − Z). (80b) Matrix elements are reduced (A3) with respect to spin only. (81) C. the dominant processes are allowed Fermi and Gamow-Teller transitions. σ = 2S and (gA /gV ) = −1.k np nh i k i||σ||k (2ji + 1)(2jk + 1) 2 (83) (fA + f ǫ )t = 6144.6 is obtained from the nine bestknown superallowed beta decays (Towner and Hardy. p) data with the Gamow Teller sum rule (Ikeda et al. respectively.15 104+300 −46 54(6) .6 .. (fV /fA )B(F ) + B(GT ) (79) The value 6144. 1982.4 ± 1. Schopper. the comparison of calculated and observed strength functions provide invaluable insight into the meaning of the valence space and the nature of the deep correlations detected by the “quenching” effect. mediated by the weak interaction. Gamow Teller and magnetic dipole strength. fV and fA are the Fermi and Gamow-Teller phase-space factors. n) and (n. Wilkinson and Macefield. respectively (Chou et al. n) and (n. It can be altered only by a small isospin-symmetry-breaking δC correction (Towner and Hardy. Shell model estimates of this quantity can be also found in (Ormand and Brown. 1995). 1974). only 253 are stable. 1977).015 1. If t1/2 is the total lifetime. Full 0 ω calculations already show a large quenching with respect to the independent particle limit as seen in Table VII where the result of a full pf calculation is compared with that obtained with an uncorrelated Slater determinant having the same occupancies: S= i. ± refers to β ± decay.74)2 ) that brings the full calculation in line with experiment. When protons and neutrons occupy the same orbits.. it is convenient to recall the meaning . B(F ) = T (T + 1) − Tzi Tzf δif (1 − δC ). 1. Out of the approximately 2500 known nuclei that are bound with respect to nucleon emission. Williams et al. 1995). The dressed states in Eq. (1995b). (85) .4 1. In general though.1 3. 1990)—after elimination of the Fermi peak at around 6 MeV—compared with the calculated peaks (KB3 interaction) after 700 Lanczos iterations (Caurier et al. The 48 Ca(p.40 3.8 ± 0. The KB3 effective interaction is doing a very good job. t = 6 and t = 4 respectively. β ′ for f |. Nucleus Uncorrelated V 54 Fe 55 Mn 56 Fe 58 Ni 59 Co 62 Ni 51 35 Gamow Teller Strength B(GT) 30 25 20 15 10 5 0 5 10 Calculated Experimental Correlated Unquenched Q = 0. for about 30 states below 8 MeV? 2. it is possible to make sense of many others if one interprets them as doorways. but it is certainly not decoupling 8590 pf states from the rest of the space. such as the E2— without knowledge of the norm of the exact wavefunction. 1995) “intruder problem”: Decoupling cannot be enforced perturbatively when intruders are energetically close to model states. normalized states: f T i write |ˆ = α|0 ω + i n=0 βn |n ω .9 ± 0. If we between exact. In the pf shell there are 8590 of them and the calculation has been pushed to 700 iterations in the Lanczos strength function to ensure fully converged eigenstates below 11 MeV.5 1. Exactly the same arguments apply to transfer reactions—for which T = as (or a† )—but are simpler s because Tn=0 = 0..38 7. It means that it ensures good decoupling at the S2 level.24 3. n)48 Sc from (Anderson et al.00 Expt. subject to further mixing with the background of intruders which dominates the level density after 8 MeV. At higher energies the peaks are much too narrow compared with experiment.1.2 2.99 2. Quenching To understand the quenching problem it is best to start by the (tentative) solution. The satisfactory description of the lowest states indicates that the model space makes sense.98 3. which guarantee a stateindependent interaction.A.3 ± 0. we need an expectation value ˆ ˆ 2 . It makes possible the calculation of the energy—and some transitions.18 2. 1. 1995b).98 3. It means that they may well be eigenstates of the effective Hamiltonian in the pf shell. as corroborated by the experimental tail that contains only intruders and can be made to start naturally at that energy. The data are from (Alford et al.33 3.5 ± 0.. 1994.74)2 .. Energetically we are in good shape. 29 Gamow Teller strength in 48 Ca(p.44 11. which can be identified with Q when one particle is removed and 1 − Q when it is added. The assumption that the model amplitudes in the exact wavefunctions are approximately constant is borne = αα′ T0 + n=0 ′ β n βn T n  . we find  2 ˆ f T ˆ i 2 since the GT operator does not couple states with different number of ω excitations.64 4. For 54 Fe. it follows that if the projection of the physical wavefunction in the 0 ω space is Q ≈ α2 . 1993. In Fig.65 1. or simply in second order perturbation theory. they should be viewed as doorways. The peaks have all J = 1. (84) ˆ and a similar expression in α′ .1 2. 55 Mn..33 TABLE VII Comparison of GT+ strengths.96 9.7 ± 0. the experimental data are compared with the strength function produced by a calculation in the full pf -shell using the interaction KB3. The peaks have been smoothed by gaussians having the instrumental width of the first measured level.. and we concentrate on the renormalization of the GT operator: the calculated strength has been quenched by a factor ≈ (0.2 ± 0. n)48 Sc reaction (Anderson et al.27 1. T = 3.9 8.52 7.. Why? Why this factor ensures the right detailed strength FIG. but not eigenstates of the full system. (b) α ≈ α′ .. adapted from Caurier et al.1 5. Figure 29 indicates that although few eigenstates are well decoupled.97 2. If the fact is not explicitly recognized we end up with the often raised (Hjorth-Jensen et al. This trick is essential in the formulation of linked cluster or exp S theories (hence Eq. (4) are normalized to unity in the model space. t = 4. 1990) provides an excellent example. The transition strength is given by the spectroscopic factor. Of these eigenstates 30 are below 8 MeV: They are at the right energy and have the right strength profile. 58 Ni and 59 Co the calculations are truncated to t = 8.3 3. If we make two assumptions: (a) neglect the n = 0 contributions. (63)).74 2. Therefore.. 29. 1990. Vetterli et al.83 15 E (MeV) 20 25 30 of the valence space as discussed in Section II.8 ± 0. El-Kateb et al.19 7.42 5. its contribution to the transition will be quenched by Q2 .15 10. 1992). along the lines proposed by Caurier. and Zamick (1964). 1995. We have sketched above the nuclear case.7) and (e. (γ.2 0.8 1. Q = 0.4 0.744(15)). 1993. fully consistent with the value that explains the Gamow-Teller data (von Neumann-Cosel et al. The hotly debated question was whether Q was due to non nucleonic renormalization of gA or nuclear renormalization of στ (Arima..75(2). 1997). 30 Comparison of the experimental and theoretical values of the quantity T (GT ) in the pf shell (Mart´ ınez-Pinedo et al. It opens the perspective of accepting the GT data as a measure of a very fundamental quantity that does not depend on particular processes. but the very severe truncations necessary at the time made it impossible to treat on the same footing the pairing and quadrupole forces.77(2)) and the pf shell (illustrated in Fig. restricted to the f7/2 space. (1986) and Dang et al. with a three body term HR1 . The proposed “solution” to the quenching problem amounts to reading data. by McCullen. 0. .744.77 0. Dro˙ d˙ et al. Q = 0. but under a new guise that makes it easier to understand. The purely nuclear origin of quenching is borne out by (p.0 0. Mart´ ınez-Pinedo et al.0 0.820(15)). Q ≈ 0. 1990). Q = 0. 1978. The solid line shows the result obtained in the sd-shell nuclei (Wildenthal et al.7) data (see also Pandharipande et al. 1998) (see Fig. The situation improved very much by dressing perturbatively the pure two-body f7/2 part of the Hamiltonian H2 . SPHERICAL SHELL MODEL DESCRIPTION OF NUCLEAR ROTATIONS FIG. for a complete review). VI. 0.8 T(GT) Exp.. Bayman. The careful analysis of Anderson et al. z z (1997) manage to place significant amounts of strength beyond the resonance region. More recent experiments by the Tokyo group establish that the strength located at accessible energies exhausts 90(5)% of the sum rule in the 90 Zr(n. p′ ). (Pasquini and Zuker. 30.6 0.. A similar result is obtained for 54 Fe(p. but they are based on 2p-2h doorways which fall somewhat short of giving a satisfactory view of the strength functions. e′ ) experiments that determine the spin and convection currents in M1 transitions. It has been compounded by a sociological one: the full GT operator is (gA /gV )στ . e′ p) (Cavedon et al..0 0. the challenge is to locate all the Theoretical studies in the pf shell came in layers. where gA /gV ≈ −1. and the quadrupole moments had systematically the wrong sign. No-core calculations are under way. 31). and Zuker (1995b).. (1998)) Experimentally. that the experimental tail in Fig. constrained by the Ikeda sum rule that relates the direct and inverse processes. but does not prove. The x and y coordinates correspond to theoretical and experimental values respectively. (1985) suggests. n) (Wakasa et al. 1997). 1976). in which gA /gV play no role (see Richter... 1998. which is trivially satisfied for spectroscopic factors. 2001. The calculations of Bertsch and Hamamoto (1982).. 1978)] solved these problems to a large extent. Osterfeld. the sd shell (Wildenthal et al. The theoretical problem is to calculate Q. n) (Anderson et al. These results rule out the hypothesis of a renormalization of the axial-vector constant gA : it is the στ operator that is quenched. Q = 0.27 is the ratio of weak axial and vector coupling constants. strength. p) (Vold et al..4 0.2 0. p) and 84(5)% in 27 Al(p. This consistency is significant in that it backs assumption (a) above. was a success. The first diagonalizations in the full shell [(Pasquini. An analysis of the data available for the N = 28 isotones in terms of full pf -shell calculations concluded that agreement with experiment was achieved by quenching the στ operator by a factor 0. 1996b).0 0. 29 contains enough strength to satisfy approximately the sum rule. that should be capable of clarifying the issue. 31 Effective spin g-factor of the M1 operator deduced from the comparison of shell model calculations and data for the total B(M1) strengths in the stable even mass N=28 isotones (from von Neumann-Cosel et al. The first. γ ′ ) and (e.744 FIG. The dashed line shows the “best-fit” for Q = 0. 1982... 1983) These numbers square well with the existing information on spectroscopic factors from (d.6 T(GT) Th. Poves. 1996b. 1.34 out by systematic calculations of Q in the p shell (Chou et al. but had some drawbacks: the spectra were not always symmetric by interchange of particles and holes. 1983. .D) came into operation: 48 Cr was definitely a well deformed rotor (Caurier et al. simply because the perturbative treatment was not enough for them.. experiment vs. Fig. 33 compares the experimental patterns with those obtained with KB3. subtracting either of the two pairing contributions. Then. (1996) in Fig. As it does everything else very well. HR1 will overwhelm H2 . JT = 01 and 10 (Poves and Mart´ ınez-Pinedo. we shall speak. discovered by Elliott (1958a. then 48 Cr backbends. especially in the rotational regime before the backbend. 32 48 Cr level scheme.. quadrupole properties of the yrast band B(E2)th 228 312 311 285 201 146 115 60 Q0 (t) 107 105 100 93 77 65 55 40 Q0 (s) 103 108 99 93 52 12 13 15 Q0 (t)[f7/2.. some nuclei were indicating a willingness to become rotational but could not quite make it. when diagonalized gives surprisingly good results. This is perhaps the most striking result to come out of our pf calculations. was (apparently) associated to strict SU(3) symmetry. Floods of ink have gone into “neutron-proton” pairing. but not the wavefunctions. as can be surmised by comparing with the experimental spectrum from Lenzi et al. J → J − 2) = K =1 (86) FIG. 48 theor. The interpretation is transparent: CHFB does not “see” proton-neutron pairing at all.” At the time it was thought impossible to describe rotational motion in a spherical shell-model context. where exact KB3 diagonalizations are done. The latter. 1994). Rotors in the pf shell 1874 8+ 5189 1744 6+ 1587 4+ 1106 2+ 752 0+ 1858 752 0 4+ 2+ 0+ 1017 806 806 0 3445 6+ 1575 1823 8+ 1730 3398 1850 5128 16 + (16 + ) 15727 16268 16 + 16 + 5450 3291 3029 14 + 10277 2135 8406 12 + 1481 7063 10 + 13306 5674 13885 14 + 1871 12 + 1343 10 + 10594 8459 6978 TABLE VIII J 2 4 6 8 10 12 14 16 B(E2)exp 321(41) 330(100) 300(80) 220(60) 185(40) 170(25) 100(16) 37(6) 48 Cr. (87) 16π to establish the connection with the intrinsic frame descriptions: A good rotor must have a nearly constant Q0 . and it is not very efficient in the low pairing regime. 3K 2 − J(J + 1) B(E2. The discrepancy is more apparent than real: The predictions for the observables are very much the same in both cases (see Caurier et al. The authors missed prophecy by one extra conditional: “not in the pf shell should have been “not necessarily. A. When treated in the cranked Hartree Fock Bogoliubov (CHFB) approximation. and with the Gogny force. which is the case up to J = 10.” Indeed.p3/2] 104 104 103 102 98 80 50 40 exp. theory 5 2 e | JK20|J − 2. 1995a. the inevitable conclusion is that pairing can be treated in first order perturbation theory. It is apparent that the JT = 01-subtracted pattern is quite close to the CHFB one in Fig. 1981b). of the rotational coupling scheme. K |2 Q0 (t)2 K = 1/2. for the details). (1998) where we have used Q0 (s) = (J + 1) (2J + 3) Qspec (J). that in some cases. 1. the energies are very sensitive to it. Cr The fourth layer was started when the ANTOINE code (Section III. i. which is a problem . 1993) and the transition properties in Table VIII from Brandolini et al. perhaps. and we are only speculating.b). The glorious exception. 33. approximately realized only near 20 Ne and 24 Mg.. 32 (see also Cameron et al. 1998).35 mostly due to the quadrupole force (Poves and Zuker. because such a behaviour had been thought to occur only in much heavier nuclei. The JT = 10 subtraction goes in the same direction.. not in the pf shell but elsewhere. The paper ended with the phrase: “It may well happen. the results are not so good. The reason is given in Fig 34.e. 1994. . (1996a). For spectroscopic comparisons with the odd-odd nuclei. .. Cameron et al. .0 KB3 KB3-P10 KB3-P01 1. that starts at about 5 MeV excitation energy and has a deformation close to β=0. 2p1/2 )4 .. Furthermore. 36. Let us start by considering the quadrupole force alone. B. When more particles or holes are added. for mean field theories but neither for the Gogny force nor for the shell model. 16 14 12 10 FIG. the results show that ordinary pairing is also a mean field problem when nuclei are not superfluids. 1998. In jj coupling the angular part of the quadrupole op- .. 2002).0 3. the shell model calculations with the KB3 interaction J 8 6 4 2 0. 1998). 50 Cr was predicted to have a second backbending by Mart´ ınez-Pinedo et al. and made more precise by Mart´ ınez-Pinedo et al. 2q20 = 2nz − nx − ny . the shell model calculations with KB3 and the Gogny force and the CHFB results also with the Gogny force. Quasi-SU(3). (1995).0 for by the shell model calculations (Poves and Zuker. etc. Recently. Lenzi et al. a rotor like band appears at low spin with an yrast trap at J=12+ . 1997) in full agreement with the experiments (Bentley et al. 1f5/2 . which means filling the lowest orbits obtained by diagonalizing the operator Q0 = 2q20 = 2z 2 − x2 − y 2 . The mirror pairs 47 V-47 Cr and 49 Cr-49 Mn closely follow the semiclassical picture of a particle or hole hole strongly coupled to a rotor (Mart´ ınez-Pinedo et al. 2002). the filling is not complete and we have the (triaxial) (8(p − 1).. 34 Gamma-ray energies along the yrast band of 48 Cr (in MeV). both are accounted To account for the appearance of backbending rotors. we find eigenvalues 2p.. 1998). Eγ (MeV) FIG. (Ur et al.36 16 14 12 10 20 18 16 14 12 EXP KB3 J 8 6 4 2 0.0 2. as shown in the left panel of Fig. (1997). By filling the orbits orderly we obtain the intrinsic states for the lowest SU(3) representations (Elliott.. (1999) for 46 V and Svensson et al. Cranked Nilsson Strutinsky (Juodagalvis and ˚berg. where spin has been included. 1981b)(sic). Here we give a compact overview of the scheme that will be shown to apply even to the classic examples of rotors in the rare-earth region. 1958a. taken to act in the p-th oscillator shell.. a theoretical framework was developed by Zuker et al. 33 The yrast band of 48 Cr. experiment vs.4) representation for which we expect a low lying γ band. 1991. A Cluster Model (Descouvemont. O’Leary et al. Using the cartesian representation. 35 The yrast band of 50 Cr. see Brandolini et al. etc. It will tend to maximize the quadrupole moment.0 J 10 8 6 4 2 0 0 1000 2000 3000 4000 5000 6000 Eγ (MeV) E_gamma (in keV) FIG.. 0) if all states are occupied up to a given level and (λ. experiment vs. For instance: putting two neutrons and two protons in the K = 1/2 level leads to the (4p. Tonev et al. 1995a).. 2p−3. 1999).0 Exp SM-KB3 CHFB SM-GOGNY 1. The calculations reproduce the band. µ) otherwise.. (2001).0 4.. 1999). 1997. isoscalar pairing retired (KB3– P01) and isovector pairing retired (KB3–P01).0) representation. (1998) for 50 Mn. Projected Shell Model (Hara et al. It is dominated by the configuration (1f7/2 )12 (2p3/2 .) Nuclei in the vicinity also have strong rotational features..0 3. 1997) (see figure 35). a highly deformed excited band has been discovered in 56 Ni (Rudolph et al. confirmed experimentally (Lenzi et al.b): (λ. the collective behaviour fades. though even for 52 Fe. 48 Cr has become a standard benchmark for models (Cranked Hartree Fock Bogoliuvov (Caurier et al.0 2. For four neutrons and four protons.0 4.4. full interaction (KB3). 0 1/2[310] 5/2[303] 3/2[301] 1/2[301] 5/2[303] 1/2[301] 3/2[301] 3/2[312] 1/2[321] 7/2[303] 3/2[312] 1/2[310] 7/2[303] 5/2[312] 1/2[321] 3/2[321] 1/2[330] 0.0 −5.5 < ε < 1. Rotational behavior is fair to excellent at low J. Why and how do the bands backbend? We have no simple answer.30 −0. The suggestion is confirmed by an analysis of the wavefunctions: For the lowest two orbits.0 0. a 2002) shows the behavior of the (gds)8 space under the influence of a symmetric random interaction. although now there is an intruder (1/2[310] orbit). but Fig. The agreement even extends to the next group. 10. a Hartree calculation was done for H = εHsp + Hq . obtained by replacing j by l. gradually . from fig. we start from perfect ones (SU(3)) and let ε increase. The result is not exact for the K = 1/2 orbits but a very good approximation. 3 HN ilsson = ω εHsp − (90) In the left panel we have turned the representation around: since we are interested in rotors.0). (33) with a properly renormalized coupling.0 −0.0 5. say. 36 (Remember here that the real situation corresponds to ε ≈ 1..g. At a value of ≈ 0. shell. (88) are small. ω = 9 MeV.37 where Q0 + k 12 9 6 3 0 1/2 3/2 1/2 5/2 3/2 7/2 5/2 ∆E (MeV) SU3 quasi. energy vs. will mix strongly the two ∆j = 2 sequences (e. both for large and small m. As expected from the normalization property of the realistic quadrupole force [Eq. 36). δ 2q20 . and gds.. 39 from (Vel´zquez and Zuker. Let us move now to larger spaces.SU3 9/2 7/2 9/2 1 2q20 2q20 δ .5 2. The ∆j = 1 matrix elements in Eq. if we multiply all the Eγ values by this factor the lines become parallel.10 δ 0. which is one-to-one except for m = 0. where the correspondence breaks down. The spin-orbit splittings will break the degeneracies and favour the decoupling of the two sequences. If the spherical j-orbits are degenerate. Hence the idea (Zuker et al.e. 38 we have yrast transition energies for different configurations of 8 particles in ∆j = 2 spaces. (88) are—within the approximation made—identical to those in LS scheme. More interesting still: the contributions to the quadrupole moments from these two orbits vary very little. (86)] remains constant to within 5% up to a critical J value at which the bands backbend. 1997). the ∆j = 1 couplings.. the single particle splittings uniform at ε = 1 MeV. the overlaps between the pure quasi-SU(3) wavefunctions calculated in the restricted ∆j = 2 space (f p from now on) and the ones in the full pf shell exceeds 0. where Hsp is the observed single particle spectrum in 41 Ca (essentially equidistant orbits with 2MeV spacings) and Hq is the quadrupole force in Eq. In Fig. deformation δ (right panel).8 the four lowest orbits are in the same sequence as the right side of fig. The resulting spectrum for quasi-2q20 is shown in the right panel of Fig 36. 2(2j + 3)2 (88) (89) FIG.0 5/2[312] 3/2[321] 1/2[330] −10. is the lower sequence in the sdg. The result is exactly a Nilsson (1955) calculation (Mart´ ınez-Pinedo et al. ε 1.10 0. 37 Nilsson diagrams in the pf shell. To check the validity of the decoupling.30 erator q 20 = r2 C 20 has matrix elements j m|C 2 |j + 2 m ≈ j m|C 2 |j + 1 m = − 3[(j + 3/2)2 − m2 ] . though small. (35)] the moments of inertia in the rotational region go as (p + 3/2)2 (p′ + 3/2)2 . (f7/2 p3/2 ) and (f5/2 p1/2 )). corresponding to the lowest oblate and prolate deformed orbits respectively.5 1. The resulting “quasi SU(3)” quadrupole operator respects SU(3) relationships. 3m[(j + 1)2 − m2 ]1/2 2j(2j + 2)(2j + 4) The ∆j = 2 numbers in Eq. 37 the results are given in the usual form.0 0. The intrinsic quadrupole moment Q0 [Eq. = = 3 4 r2 (p + 3/2)4 (91) In the right panel of Fig.e.0 FIG. 1995) of neglecting the ∆j = 1 matrix elements and exploit the correspondence l −→ j = l + 1/2 m −→ m + 1/2 × sign(m). except for m = 0. single particle splitting ε (left panel).. 36 Nilsson orbits for SU(3) (k = −2p) and quasi-SU(3) (k = −2p + 1/2)). Energy vs. p = 4. and remain close to the values obtained at ε = 0 i. We have learned that –for the rotational features– calculations in the restricted (f p)n spaces account remarkably well for the results in the full major shell (pf )n (see last column of Table VIII).95 throughout the interval 0. i. The force is KLS. 6 0. consider Fig.2 0.1 a=-0. 38. (93) .4 0.2 a=-0. Their contribution to the electric quadrupole moment is then Q0 = 8[eπ (pπ − 1) + eν (pν − 1)]. 36. two orbits K=1/2 and 3/2—originating in the upper shells of Fig. Raju et al. The appearance of backbending rotors seems to be a general result of the competition between deformation and alignment. Hirsch.3 0. (1973).gds4 gds8 gds4. with the KLS interaction and with a random interaction plus a constant JT Wrstu = −a. (2002b) for more recent applications) will be favored (in which case we have to use the left panel of Fig. There is no need to stress the similarity with Fig. The number of particles in each shell for which the energy will be lowest will depend on a balance of monopole and quadrupole effects.8 E(J+2)-E(J) 0. We want to estimate the quadrupole moments for nuclei at the onset of deformation. 40 Schematic single particle spectrum above 132 Sn.e. 40 giving a schematic view of the single particle energies in the space of two contiguous major shells—in protons (π) and neutrons (ν)—adequate for a SM description of the rare earth region.2 0 0 5 10 15 20 25 KLS a=-0. with pπ = 5. eπ and eν are the effective charges. In cases of truly large deformation SU(3) itself may be valid in some blocks. Heavier nuclei: quasi+pseudo SU(3) We have seen that quasi-SU(3) is a variant of SU(3) that obtains for moderate spin-orbit splittings.4 fp8 fp4. and 6 to 10 neutrons with pseudo-p = 4 in the lower shells. Consider even-even nuclei with Z=60-66 and N=9298. The group theoretical aspects of quasi-SU3 have been recently discussed and applied to the description of the sd-shell nuclei by Vargas. made more attractive by an amount a.38 0.hfp4 12 MeV s d g i r 5 Positive Negative 9 6 r 4 3 p f h 50 0 g[ ] h[ 82 ] FIG.1 J 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 FIG. For the upper shells the label l is used for j = l + 1/2 J FIG.7 0. (92) yields a total Q0 = 56eπ + (76 + 4n)eν . 1. rp is the set of orbits in shell p excluding the largest.3 a=-0. We shall assume quasiSU(3) operates in the upper shells.2 1 E(J+2)-E(J) 0. KLS interaction. (1969). Hecht and Adler (1969). and Draayer (2001.. pν = 6.5 0.8 0. characteristic of nuclear processes. For other forms of single particle spacings.6 0. Other variants of SU(3) may be possible and are well worth exploring. see also Vargas et al. From the left part of Fig. the pseudo-SU(3) scheme (Arima et al. with pseudop = p − 1).4 1. 38 Yrast transition energies Eγ = E(J + 2) − E(J) for different configurations. and pseudo-SU(3) in the lower ones. To see how this works. but Nilsson diagrams suggest that when nuclei acquire stable deformation. the upper blocks are precisely the 8-particle configurations we have studied at length. which added to that of Eq. (92) C. i. 39 Backbending patterns in (gds)8 T = 0. corresponding to 6 to 10 protons with pseudo-p = 3. 36 we obtain easily their contribution to Q0 . 36—become occupied.4 0. 2002a). 1967a.6 calculated in (Dufour and Zuker..13 Sm 4. it describes well the rotational regime of the observed bands.03 5. the simplicity is such that the computation reduces to a couple of sums. 33)—is missed by the calculation. but the 8p-8h one needs at least three major shells and was tackled by an α-cluster model (Abgrall et al. 2001) and calculated yrast gamma-ray energies are compared in fig. the value is constant in the four cases because the orbits of the triplet K=1/2... could account for the spectroscopy in 16 O.80 5. the band termination for the configura- . KB3 for intra shell and KLS for cross shell matrix elements).. since systematics indicate. is related to the E2 transition probability from the ground state by B(E2) ↑= 10−5 A2/3 Q2 . Q0 (given in dimensionless oscillator coordinates.26 5... It is probably the first superdeformed band detected and explained. In table X the calculated spectroscopic quadrupole moments (Qs ) and the B(E2)’s are used to compute the intrinsic Q as in Eqs. SM for 152+2n Nd. The adopted interaction.18 Gd 4. The calculations are conducted in spaces of fixed number of particles and holes. 5/2 in Fig. 36 have zero contribution for p=3. The agreement is excellent.90 5. 41 The superdeformed band of 36 Ar.02(5) 4.72 4.. By applying the quasi+pseudo-SU(3) recipes of Section VI. it is convenient to increase the space to d3/2 s1/2 pf . 4p−4h SDPF J 10 8 6 4 2 0 0 1000 2000 3000 4000 5000 Eγ (keV) FIG.5. D. At fixed n. sdfp. Shell model calculations in a very small space.51 4. including the 4p-4h band (Zuker et al.28(15) 5. 2001). experiment vs. 42. with no exception.4. The value of Q0 corresponds to a deformation β ≈ 0. The natural generalization of the ZBM space consists in the d3/2 s1/2 f p orbits (remember: f p ≡ f7/2 p3/2 ). Now we examine the 8p-8h band in 40 Ca (Ideguchi et al. The agreement is quite remarkable and no free parameters are involved.60(5) 18 16 14 12 Expt.6(7) 4. In 40 Ca. The results.58 4. In figure 41 the calculated energy levels in 36 Ar are compared to the data. eν = 0. 2001). And indeed..75 MeV (Chevallier et al.06(4) 5. where a famous 4-particles 4-holes (4p-4h) band starting at 6. Note in particular the quality of the prediction of constancy (or rather A2/3 dependence) at fixed n.99 5.C we find that the maximum deformations attainable are of 8p-8h character in 40 Ca and 4p-4h in 36 Ar with Q0 = 180 efm2 and Q0 = 136 efm2 respectively. i. However. 1968. 2001a). 154+2n Sm..5. 1996) are compared in table IX with the available experimental values . 156+2n Gd and 158+2n Dy respectively.55 4. with minor monopole adjustments dictated by new data on the single particle structure of 35 Si (Nummela et al. which does not depend on the choice of effective charges.36(5) 4. followed by the 8p-8h band starting at 16. (1997).. and used in (Caurier et al. r → r/b with b2 ≈ 1.b).sm is the one originally constructed by Retamosa et al. 3/2.76 5. following the discovery of a similar structure in 36 Ar (Svensson et al.95 5.05 MeV was identified by Carter et al. In the last example on B(E2) rates. The calculated B(E2)’s agree well with the experimental ones (Svensson et al. The valence space adopted for 36 Ar is truncated by limiting the maximum number of particles in the 1f5/2 and 2p1/2 orbits to two. ZBM).87). p1/2 d5/2 s1/2 . 1967). to track them beyond backbend.6 keV). (86. (1964).25(6) 5.01A−1/3 f m2 ). The patterns agree reasonably well but the change of slope at J = 10 —where the backbend in 48 Cr starts (Fig. It is seen that by careful analysis of exact results one may come to very simple computational strategies. which is relatively free of center of mass spuriousness. The experimental (Ideguchi et al.66(5) 4.47 2. the first excited 0+ state is the 4p-4h bandhead.22 Dy 4. 2001). except at J = 12 where the data show a clear backbending while the calculation produces a much smoother upbending pattern. As expected both Q0 (s) and Q0 (t) are nearly equal and constant—and close to the quasi+pseudo SU(3) estimate—up to the backbend.. using standard effective charges δqπ =δqν =0. The discrepancy in 152 Nd is likely to be of experimental origin.. The 36 Ar and 40 Ca super-deformed bands The arguments sketched above apply to the region around 16 O. 2000). much larger rates for a 2+ state at such low energy (72. 1989) N 92 94 96 98 Nd 4.e.64(5) 4. It is only recently that another low-lying highly deformed band has been found (Ideguchi et al. which only backbends at J = 20.39 20 TABLE IX B(E2) ↑ in e2 b2 compared with experiment (Raman et al. using effective 0 charges of eπ = 1.68 4. 1998) (USD. 8p−8h.3 -62.9 -71. and corresponds exactly to the quasi+pseudo SU(3) estimate. 2003) suggest that for low spins the deformation would be smaller due to the mixing with less deformed states of lower npnh rank. Flocard.3 -71.2 -75. Many bands of this type have been experimentally studied in recent years. barely miss (just by 12 keV) to become the ground state. that in this region is about 7 MeV..7 -53. The positive parity levels of this oddodd nucleus are very well reproduced by the shell model calculation using the interactions KB3 (or KB3G) as can .2 -70. 1999). Beyond mean The occurrence of low-lying bands of opposite parity to the ground state band is very frequent in pf shell nuclei. calculated in the sdpf valence space J 2 4 6 8 10 12 14 16 18 20 22 24 B(E2)(J → J-2) 589 819 869 860 823 760 677 572 432 72 8 7 Qspec -49. Their simplest characterization is as particle-hole bands with the hole in the 1d3/2 orbit.9 -50.40 TABLE X Quadrupole properties of the 4p-4h configuration’s yrast-band in 36 Ar (in e2 fm4 and efm2 ) B(E2)(J→J-2) J 2 4 6 8 10 12 14 16 EXP 372(59) 454(67) 440(70) 316(72) 275(72) 232(53) >84 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0 TABLE XI Quadrupole properties of the 8p-8h configuration’s yrast-band in 40 Ca (in e2 fm4 and efm2 ). It is extracted from an overall fit that assumes constancy for all measured values.5 Q0 (t) 172 170 167 162 157 160 149 128 111 Q0 (s) 172 172 171 168 164 160 157 158 162 TH 315 435 453 429 366 315 235 131 Qspec -36. and explained by shell model calculations. A reanalysis of the experimental lifetimes in the superdeformed band of 40 Ca (Chiara et al. and Heenen (2003a). The next step consists in remembering that states at fixed number of particles are doorways that will fragment in an exact calculation that remains to be done.6 (Ideguchi et al. can be interpreted as (1d3/2 )−1 (a proton hole) coupled to (pf 8 ) T=0.. E. the presence of one extra particle in the pf -shell may produce an important increase of the correlation energy. In a nucleus whose ground state is described by the configurations (pf )n the opposite parity intruders will be (1d3/2 )−1 (pf )n+1 . where the intruder band based in the configuration (1d3/2 )−1 coupled to 46 Ti. The experimental Q0 (t)=180+39 obtained from the −29 fractional Doppler shifts corresponds to a deformation β ≈ 0. 1998). 42 The superdeformed band in calculation 40 Ca.8 -52. the very low-lying positive parity band of 47 V.1 -72.3 -56.4 -68.0 -45. Rotational bands of unnatural parity 1000 2000 3000 4000 5000 Eγ (keV) FIG. that can compensate the energy lost by the particle hole jump..0 Q0 (s) 126 126 127 125 121 119 120 122 Q0 (t) 126 124 120 114 104 96 82 61 Expt.7 -52. The most extreme case is 45 Sc. We present now some recent results in 48 V (Brandolini et al. Indeed. the correlation energy of this pseudo-48 Cr is larger than the correlation energy of the ground state of 47 V and even larger than the correlation energy of the real 48 Cr. exp.0 -54. vs. explaining why the band starts at only 260 keV of excitation energy (Poves and Sńchez a Solano. 8p-8h 8 tion f7/2 (d3/2 s1/2 )−8 .6 -71. but apparently too strong to allow for the change in slope at J = 10. SDPF J field methods using the Skyrme interaction have been recently applied to both the 36 Ar and 40 Ca superdeformed bands by Bender. The band in 36 Ar has been also described by the projected shell model in (Long and Sun.1 -81. 2001).1 -79. 2001).. For instance. It also squares well with the calculated 172 efm2 in Table XI where the steady decrease in collectivity remains consistent with experiment within the quoted uncertainties. On the other side.0 -85. 2002a). This calculation demonstrates that a detailed description of very deformed bands is within reach of the shell model. mainly at the Gasp and Euroball detectors (Brandolini et al. The promotion of a particle from the sd to the pf shell costs an energy equivalent to the local value of the gap. The extra collectivity induced by the presence of sd particles in pseudo-SU(3) orbitals is responsible or the delay in the alignment. The transition probabilities in the negative parity bands are presented in Fig. FIG. Hence we expect that the lowest lying bands would have K=1− and K=4− (aligned and anti-aligned coupling of the quasiparticle and the quasihole). Now the experimen- . Now we have to couple the 1d3/2 hole to our basic rotor 48 Cr. 43 The ground state (K=4+ ) and the side band (K=1+ ) of 48 V. 48 V. Notice that the calculation reproduces even the tiniest details of the experimental results. theory. the ground state band of 49 Cr can be understood in a particle plus rotor picture. and assigned K=5/2−. In this particular nucleus the experimental information is very rich and a detailed comparison can be made for both the positive and negative bands and for E2 and M1 transitions. A more stringent test of the theoretical predictions is provided by the electromagnetic properties. Therefore the structure of the negative parity states of 48 V is one particle and one hole coupled to the rotor core of 48 Cr. theory. that would correspond to K=4+ in a Nilsson context (the aligned coupling of the last unpaired proton K=3/2− and the last neutron K=5/2−) and a second one linked to the 1+ member of the ground state multiplet that results of the anti-aligned coupling. 46. As we have mentioned earlier. The B(E2)’s correspond to a deformation β=0.41 FIG. 44 The negative parity K=1− and K=4− bands of experiment vs. The levels are grouped in two bands. 45 Theoretical and experimental B(M1) and B(E2) transition probabilities in the ground state band of 48 V Fig. The results for the ground state band are collected in FIG. one connected to the 4+ ground state. These two bands have been found experimentally at about 500 keV of excitation energy. experiment vs. 44) except for a small shift of the K=4− and K=8− bandheads relative to the K=1− state. 45. 43. be seen in Fig. for instance the odd-even staggering of the B(M1)’s. Their structure is in very good agreement with the SM predictions (see Fig. another example of good quality “pure spectroscopy”. The negative parity states can be interpreted as the result of the coupling of a hole in the 1d3/2 orbit to 49 Cr.2 that is clearly smaller than that of 48 Cr. as. with solid indications that the supposedly semi-magic 12 Be ground . 11 Li: halos tal results have larger uncertainties. but they become vulnerable in the semi-magic cases. What it specific about these nuclei. (2000). 1987. The shell model has no particular problem with halo nuclei. which in turn depend on the model space. but they do not change qualitatively the strong monopole drifts. it drops to ≈ 4. N=8. the s1/2 -p1/2 bare gap is estimated at 0... The quantitative agreement is worse for the other two bands. which sits at the drip line (≈ 200 keV two-neutron separation energy) and has a very large spatial extension. 2003) for a new multi. DESCRIPTION OF VERY NEUTRON RICH NUCLEI The study of nuclei lying far from the valley of stability is one of the most active fields in today’s experimental nuclear physics. Their main consequence is that “normal” states. the main features of the data are well reproduced. The monopole loss of promoting a particle to the sd shell is compensated by a pairing gain for the p−2 holes.. but the uncertainties involved are easily absorbed by the monopole field. According to Otsuka et al.5 MeV p1/2 -d5/2 gap in 16 O is down to some 3 MeV in 12 C.2 The 1/2+ ground state in 11 Be provided one of the first examples of intrusion. The explanation of this behavior has varied with time (Aumann et al. As explained at the end of Section II.. due to a neutron halo (Hansen and Jonson. but becomes questionable when we note that the same phenomenon occurs in 9 He which has no p2 particles.B. 46 Theoretical and experimental B(M1) and B(E2) transition probabilities in the negative parity bands of 48 V A. those described by 0 ω p or sd calculations often “coexist” with “intruders” that involve promotion to the next oscillator shell.e. (2001a) the drift may well continue: in 8 He. suggesting the need for a further reduction of the monopole loss (Otsuka et al.. then ≈ 2 MeV in 40 Ca. They go together because the effective single particle energies (ESPE. 1993. the situation is similar: The imposing 11.ω calculation. the agreement in absolute values and trends is very good for the lowest band (K=1− ).42 this monopole drift provides direct evidence for the need of three-body mechanisms. 2001) for a recent review and (Suzuki et al. see Section V) depend on occupancies.5 MeV around 28 O. see also (Brown. All the numbers above (except the last) are experimental. For instance the d3/2 -f7/2 neutron gap in 40 Ca of ≈ 7 MeV goes down to ≈ 2. The bare monopole values are smaller because correlations increase substantially the value of the gaps. whose large size is readily attributed to the large size of the s1/2 orbit. from a shell model point of view? As everywhere else: the model space and the monopole behavior of the interaction.5 MeV in 48 Ca.2. 1985). As shown by (Kahana et al. For the p shell. The normal state corresponds to a hole on the N = 8 closure. Since the d5/2 orbit is well below its sd partners. The deformation of the K=1− band extracted from the B(E2) properties turns out to be larger than the deformation of the ground state band and close to that of 48 Cr. VII.. This 1/2 1/2 result was confirmed by Simon et al. 2001a). nevertheless. supplemented by the p3/2 subshell whenever the p3/2 -f7/2 gap becomes small: from ≈ 6 MeV in 56 Ni.B. Sagawa et al. which is sensitively detected by the β decay to the first excited 1/2− state in 11 Be. 1960). 1969a) the use of Woods Saxon wavefunctions affect the matrix elements involving this orbit. (1997) it was shown that calculations producing a 50% split between the neutron closed shell and the s2 p−2 configuration lead to the right lifetime. A quadrupole gain due to 1/2 the interaction of the sd neutron with the p2 protons is plausible. now quite close to f7/2 . In Borge et al. Let us examine how this happens.8 MeV. (1999).. As a consequence. Tanihata et al. Talmi and Unna. Despite that. The interest in 11 Be—as a “halo” nucleus—was revived by the discovery of the remarkable properties of 11 Li. Suzuki and Otsuka. the specificity of light and medium neutron rich nuclei is that the EI closures (corresponding to the filling of the p + 1/2 orbit of each oscillator shell) take over as boundaries of the model spaces. the natural model space is no longer the sd shell. Further confirmation comes from Navin et al. the whole issue hinges on the s1/2 contribution to the wavefunctions. 1994. and finally ≈ 0 MeV in 28 Si. As discussed in Section II. 2001. but the idea has remained unchanged. i. The expected 0 ω normal state lies 300 keV higher. In broad terms. FIG. the oscillator closures (HO) may be quite solid for double magic nuclei. The monopole drift of the s1/2 -d5/2 gap brings it from about 8 MeV in 28 Si to nearly -1 MeV in 12 C. but the EI space bounded by the N = 14 and 28 closures. In 30 Mg the normal configuration is the one that agrees with the existing experimental data (Pritychenko et al.sm interaction (upper panel) and the Tokyo group interaction (bottom panel) 6 4 O F Ne Na Mg 2 0 -2 16 18 20 22 24 26 28 30 N FIG. The 4p-2h intruder is even more . and everybody (including the authors of this review active at the time) were enormously surprised when a classic experiment Thibault et al. 47 Effective single particle energies (in MeV) at N=20 with the sdpf. 1990)..sm and Tokyo group interactions (Utsuno et al. 1999). 1/2 1/2 10 5 2p3/2 1f7/2 1d3/2 2s1/2 1d5/2 B.B. is already quite collective. that mapped an “island of inversion”..e. 48 locate the neutron drip line. 11 and 12. (1975) established that the mass and β-decay properties of the 31 Na ground state— expected to be semimagic at N = 20—could not possibly be that of a normal state (Wildenthal and Chung.corresponding p to N = 20. deformed intruders 0 -5 -10 -15 -20 -25 10 5 0 -5 -10 -15 -20 -25 In the mid 1970’s it was the sd shell that attracted the most attention. deformation was still absent. N =19. It can be seen in the table that it resembles the 32 Mg ground state. however. Warburton et al. To obtain deformed solutions demanded the inclusion of the 2p3/2 orbit as demonstrated by Poves and Retamosa (1987). Some results for the even Mg isotopes (N=18... 37 Na and 40 Mg. Consider now some detailed information obtained with sdpf. 1999. the region where the intruder configurations are dominant in the ground states. (1983).sm interaction. Nobody seemed to remember 11 Be. N=20. Otsuka and Fukunishi. 1991. including the 1f7/2 orbit in the valence space. Early mean field calculations had interpreted the discrepancies as due to deformation (Campi et al. Siiskonen et al.. 1992. 20 and 21. 47 represent Hm for the sdfp. Let’s recall. 1975) but the experimental confirmation took some time (Guillemaud-Mueller et al.5. the 2−4 configurations (d5/2 s1/2 )2−4 (f7/2 p3/2 )n . 1984) and the 0+ → 2+ B(E2) (Motobayashi et al. Data and calculations suggest prolate deformation with β ≈ 0.sm by Caurier et al. The next example of a frustrated semi-magic was 32 Mg (Detraz et al.. The ESPE in Fig. Heyde and Woods.e. The S2N values of Fig. 48 Two neutron separation energies (in MeV) calculated with the sdpf. 1993. 1979).. In 32 Mg the situation is the opposite as the experimental data (the 2+ excitation energy (Guillemaud-Mueller et al.. (2001c). In 34 Mg the normal configuration that contains two pf neutrons. 1984. Poves and Retamosa. and such as efficiently to favor deformation. Z = 10-12. consistent with what is known for oxygen and fluorine.. 32 Mg. Klotz et al. Motobayashi et al. These calculations were followed by many others (Fukunishi et al. 1994..43 state is dominated by the same s2 p−2 configuration. maximal). N=20 and N=22) are gathered in table XII. prefer the intruder. A preliminary measure of the 4+ excitation energy reported by Azaiez (1999) goes in the same direction. Both interactions lead to an “island of inversion” for Z=10. i. Exploratory shell model calculations by Storm et al. 1999). 1979). 1996. Within details they are quite close. have a “quasi-SU(3)” quadrupole coherence close to that of SU(3) (i. in turn dictated by the monopole hamiltonian. Note the kink due to deformed correlations in the latter. For the other chains. where the last bound isotopes are 24 O and 31 F (Sakurai et al. was able to improve the mass predictions. 1995)) clearly 28 30 32 34 A 36 38 40 2p3/2 1f7/2 2d3/2 2s1/2 2d5/2 28 30 32 34 A 36 38 40 FIG.. 1995). the behavior is smoother and the last predicted bound isotopes are 34 Ne. The detailed contour of this “island” depends strongly on the behavior of the effective single particle energies (ESPE). that according to what we have learned in Section VI. 1999).... The sdpf. tent. Energies in MeV.09 2.82 53 35 23 -12. When the 2p-2h bandheads of 40 Mg and 52 Cr are allowed to mix in the full 0 ω space. the ground state band is dominantly 4p-2h. the intruder state is present in both nuclei. but it is only in the very neutron rich one that it becomes the ground state. this behavior is contrary to a very general trend in heavier nuclei. C. N stands for normal and I for intruder. 2001b): The decay of 33 Na indicates that the ground state of 33 Mg has Jπ =3/2+ instead of the expected Jπ =3/2− or Jπ =7/2−. Towards N=Z.69 4.4 0. 49. where we have plotted the strength functions of the 2p-2h 0+ states in the full space.33 3. As we have seen. In the QMCD calculations of Utsuno et al.86 3.sm calculation.2. This inversion is nicely reproduced by the sdpf.50 131 175 176 -22. the EI closures are very robust.0 0. N=28: Vulnerability Let us return for a while on Fig.1 0.0 1. On the contrary. (2000) and Iwasaki et al. B(E2)’s in e2 fm4 and Q’s in efm2 30 Mg I EXP N 0. (1999).98 36 17 2 -11.52 75 88 76 -15.01 6.7 0.1 0.88 2.sm interaction leads to a remarkable result summarized in Table XIII: the decrease of the gap combined with the gain of correlation energy of the 2p-2h lead to a breakdown of the N = 28 closure for 44 S and 40 Mg. Clearly. The double magic 42 Si resists.29) +1. Results from the Riken experiments of Yoneda et al.93 9. Next we remember that the oscillator closures are quite vulnerable even at the strict monopole level.0 0.81 98 123 115 -18.93 2.0 1.69 2.44 TABLE XII Properties of the even magnesium isotopes.3).41 3. This is a very interesting illustration of the mechanism of intrusion. However.27 3. which decreases as protons are removed. the 2+ comes at the right place.4 +3.75 112 144 140 -19.13 59(5) 90(16) 126(25) deformed (β ≈ 0. but the 4+ is too high. In 40 Mg the 2p-2h state represents the 60% of the ground state while in 52 Cr it is the dominant component (70%) of the first excited 0+ . quadrupole coherence takes full advantage of this vulnerability. and demands a three-body mechanism to resolve the contradiction.9 1.6) and a better rotor. 49 LSF of the 2p-2h 0+ bandheads of 40 Mg (upper panel) and 52 Cr (bottom panel) in the full 0 ω space Table XIV gives an idea of the properties of the iso- .4 32 Mg I EXP N 0.66 1.B. because of the drift of the p3/2 − f7/2 gap even the N = 28 closure becomes vulnerable. As explained at the end of Section II.67 2. This can be seen in Fig. Another manifestation of the intruder presence in the region has been found at Isolde (Nummela et al.1 0. The scale does not do justice to a fundamental feature: the drift of the p3/2 − f7/2 gap. 47. they keep their identity to a large ex- FIG.89 (2. 52 Cr and 54 Fe exhibit large correlation energies associated to the prolate deformed character of their 2p-2h neutron configurations (β ∼ 0..0 1. 34 Mg is at the edge of the “island of inversion”. (2001) seem to favor the intruder option.4 34 Mg I EXP N 0 2+ 4+ 6+ B(E2) 2+ → 0+ 4+ → 2+ 6+ → 4+ Qspec (2+ ) ∆E(0+ ) I + 0.0 0. using them as pivots in the LSF procedure.48 -1. The third known state is short lived and it should correspond to the 7/2− member of the ground state band. For the neutrons. the maximum collectivity occurs at N = 24.54 3 1.45 6. (1999) that detected a strong E2 transition from the ground state to an excited state around 940 keV.16 28 1.19 E∗ 1. but it has to be used with care. B(E2↑)calc. In addition. n9/2 + 1. in 45 Ar it appears at about 0. In 44 S the spherical and deformed configurations mix equally. The N=27 isotones also reflect the regular transition from sphericity to deformation in their low-lying spectrum.66 5.05 2p−2h -1. point also in the same direction.98 4.85 MeV. Naturally. (Nowacki. with an incipient γ band.346 1.50 1. quadrupole properties and occupancies 40 Mg 42 Si 44 S 46 Ar 48 Ca 50 Ti 52 Cr 54 Fe 56 Mg 42 Si 44 S 46 Ar Ni gap 3. The maximum proton collectivity is achieved when both orbitals are degenerate.25 1. (the d5/2 orbital remains always nearly closed).033 1.4 MeV. Deformation will demand the addition of the d5/2 subshell. from (Sorlin et al.72 MeV and 4+ at 3.. 1969b) for the remaining matrix elements. the β decay of 60−63 V (Sorlin et al..10 MeV.07 410 0.34 10. but no clearcut deviation from N=28 magicity can be detected.84 505 0.57 16 71 6. While in 47 Ca. fit remarkably well with the experimental results.84 4.7 5. the full pf results are quite useful: 56 Ni turns out to be an extraordinarily good pivot (refer to Fig.26 -17 93 6..35 3.259 775 0.0 6.40 7.0 -0. This number .23 3..38 3.45 TABLE XIII N=28 isotones: quasiparticle neutron gaps. The effective interaction is based on three blocks: i) the TBME from KB3G effective interaction (Poves et al. 50 we have plotted the low energy spectra of the heaviest known sulphur isotopes. support this hypothesis. Analyzing their proton occupancies we conclude that the rise in collectivity along the chain is correlated with the equal filling of the d3/2 and s1/2 orbitals. (2000) that observed a low-lying isomer around 400 keV excitation energy and the MSU Coulex experiment of Ibbotson et al.2 tones in which configuration mixing is appreciable. (2002). for the first time we are faced with a genuine EI space r3 g9/2 (remember r3 is the “rest” of the p = 3 shell).12 ∆ECorr 8. 15). 42 S is a prolate rotor.93 20 93 6.33 5. in the sense that the region boundaries are well defined by the N = 4. investigated at Isolde (Hannawald et al.92 6.49 1. The agreement with the accumulated experimental results (see Glasmacher.43 520 0. 2001). with the exception for the very neutron-rich halo nuclei and the island of inversion discussed previously. Recent experiments involving the Coulomb excitation of 66−68 Ni (Sorlin et al.22 2.91 45 TABLE XV 2+ energies and g9/2 intruder occupation in the 1 Nickel isotopic chain.07 1. The p and sd shells can be said to be “full” 0 ω spaces.46 2. Lee and Scott G-matrix (Kahana et al. and the decay and the spectroscopy of the 67m Fe (Sawicka et al. (Hjorth-Jensen et al. from “magic” 68 Ni to deformed 64 Cr Systematics is a most useful guide.45 D.17 1. 2003)..59 10. ii) the G-matrix of ref. it lies quite high (at around 2 MeV) due to the strong f7/2 closure. In the pf -shell it is already at N = Z ≈ 36 that the 0 ω model space collapses under the invasion of deformed intruders. and as soon as a few particles are added it becomes a good core. the information comes from two recent experiments: the mass measurements at Ganil by Sarazin et al. We have computed all the nickel isotopes and followed their behavior at and beyond N=40. According to our calculations the ground state corresponds to the deformed configuration and has spin 3/2−. 1999). difference in correlation energies between the 2p-2h and the 0p0h configurations and their relative position 40 TABLE XIV N=28 isotones: Spectra. N=40.11 1. 1998. which corresponds to the Pseudo-SU(3) limit.08 7. Concerning 43 S..50 3. for a recent review) is excellent and extends to the new data of Sohler et al.55 690 0. E (2 )(MeV) E∗ (4+ ) E∗ (0+ ) 2 Q(2+ )(e fm2 ) B(E2)(e2 fm4 ) n7/2 (f7/2 )8 % ∗ + 0. the β decays of the neutron-rich isotopes 64 Mn and 66 Mn. 1995) with the modifications of ref.73 1. 2002) 62 Ni 64 Ni 66 Ni 68 Ni 70 Ni 72 Ni 74 Ni E(2+ )calc. The calculated low energy spectrum. 2+ at 2.24 755 0. The ground state wave function is 50% closed shell. E(2 )exp. According to the calculation. that show a sudden drop of the 2+ energies in the daughter nuclei 64 Fe and 66 Fe. The excitation energy of the 3/2− state should be sensitive to the correlations and to the neutron gap.39 8.73 5. In defining the new model spaces..81 2.68 1.75 1.49 2. 8 and 20 closures. In Fig. 1996) and iii) the Kahana.41 6..51 3.83 -21 108 5. an excited 0+ at 1. The spherical single hole state 7/2− would be the first excited state and his lifetime is consistent with that of the experimental isomer.33 1. Therefore. 2003).42 1. 2002).425 2. Its validity is restricted to nearly spherical states.67 265 1.24 1. 5− at 2. the f7/2 mid-shell.173 1.15 MeV. we expect something similar in the neighborhood of N = 40 isotones.50 2.16 24 1. Let’s first focus on 68 Ni. They point to prolate structures with deformation β ≈ 0. and smaller values of the closed shell probability can be obtained without altering drastically the rest of the properties. 2002). compared with the experiment. and the experimental trend promises no improvement at N = 40..fm2 ) B(E2)↓(e2 ..65 -27 302 76 103 1. 1 2000. 52. 51 follow the experimental trends including the drop at 68 Ni recently measured (Sorlin et al. The SM results have been compared with those of other methods in (Langanke et al. B(E2)’s in e2 fm4 B(E2) [e fm ] is very sensitive to modifications of the pf -g gap..fm2 ) B(E2)↓(e2 . 34. in N = 38 the 2+ is too high. Properties of the 2+ energies given by the pf gd calcu1 lation are collected in table XVI.. . If N=40 were a magic number. 2002) we compare the excitation energy of the 2+ states in the Nickel chain with the available experimental data (including the very recent 70 Ni value (Sorlin et al.15 -43 471 119 123 200 0 26 28 30 Neutron number 32 34 36 38 40 42 44 FIG. Energies in keV.35 -30 428 84 117 0. Note that for a strict closure the transition would vanish.fm2 ) from Qs Qi (e.46 FIG. 2003b). from (Sorlin et al.3. But then. The addition of g9/2 does more harm than good because the gaps have been arbitrarily reduced to reproduce the N = 36 experimental point by allowing 2p-2h jumps including d5/2 . exactly the opposite seems to happen experimentally.fm2 ) from B(E2) ∗ + 0. Similarly the B(E2)’s in Fig. 50 Predicted level schemes of the heavy sulphur isotopes.fm2 ) from Qs Qi (e. particularly in 64 Cr. As seen in Fig.67 -23 288 82 101 1.43 -37 426 102 117 0.51 -31 318 109 106 1. one would expect higher 2+ energies as the shell closure approaches. 2003). although the experiment gives a larger peak at N=40. The β decays of 60 V and 62 V indicate very low energies for the 2+ states in 60 Cr and 62 Cr (Sorlin et al.fm4 ) Qi (e. 2002)). In table XV TABLE XVI Spectroscopic properties of valence space 60 60−64 1200 1000 4 2 800 Cr in the f pgd 64 600 400 Cr 62 Cr Cr E∗ (2+ ) (MeV) Qs (e. 51 B(E2) (0+ → 2+ ) in e2 fm4 for the Nickel isotopes.. The agreement is quite good. The three possible model spaces tell an interesting story: pf ≡ r3 is acceptable at N = 32.fm2 ) from B(E2) E (4 ) (MeV) Qs (e.fm4 ) Qi (e. Fowler. 1997) provided its half-life under cosmic-rays conditions is known. Another possible cosmic ray chronometer.b. Both theoretical results are sensitive to uncertainties in the renormalization of the unique second-forbidden operators that should be removed taking the ratio of the β − and β + half-lives. 2 He) performed at E(2+) (Mev) .3) × 108 yr is obtained compared with 5. Figure 53 compares the shell-model GT+ distributions with the pioneering measurements performed at TRIUMF.6 × 108 yr from the shell model calculation of Mart´ ınez-Pinedo and Vogel (1998). Due to phase space arguments the β − decay to 54 Fe is expected to dominate. 1982a. Astrophysical applications Astrophysical environments involve conditions of temperature and densities that are normally not accessible in laboratory experiments. important in stellar oxygen and silicon burning.2) × 105 yr for the β − branch. 2003. Currently only a lower limit for the half-life of totally ionized 56 Ni could be established (2. A recent shell-model calculation by (Fisker et al. 1990). 56 Ni. Figure 54 compares the shell-model GT+ distribution computed using the KB3G interaction (Poves et al. (1998).0 ± 1. 1999) predict a half-live of 4 × 104 y that is too short for 56 Ni to serve as a cosmic ray chronometer. Nuclear beta-decay and electron capture are important during the late stages of stellar evolution (see Langanke and Mart´ ınez-Pinedo. At the relevant conditions in the star electron capture and β decay are dominated by Gamow-Teller (and Fermi) transitions.8)×10−9 by Zaerpoor et al. (2001a.075(20) days.8 ± 1. 2001). More recently. Another isotope.8±0. could be used as a chronometer to study the propagation of iron group nuclei on cosmic-rays (Duvernois. 1999a..1[theor]) × 105 . Before acceleration. Earlier determinations of the appropriate weak interaction rates were based in the phenomenological work of Fuller. (1997) and (1. 2001) with a recent experimental measurement of the 51 V(d. unstable in normal conditions. n)-type reactions). the decay of 56 Ni proceeds by electron capture to the 1+ state in 56 Co with a half-live of 6. we refer to Oda et al. consider the decay properties of nuclei totally stripped of the atomic electrons at typical cosmic rays energies of 300 MeV/A. it has been possible to extend these studies to pf shell nuclei relevant for the pre-supernova evolution and collapse (Caurier et al. 1999). 2002). The description of the different nuclear processes requires then theoretical estimates. 1982a). 2000. (1998) estimate the partial β − half-life to be (6. 2 He) charge-exchange reactions at intermediate energies (Rakers et al. (1994).3(4) d. These measurements had a typical energy resolution of ≈ 1 MeV.9 × 104 y) (Sur et al.25) × 10−9 .. VIII. (1. After acceleration 56 Ni is stripped of its electrons.. For the sd shell nuclei. As a first example. and knowing that of 54 Mn. for a recent review).26)×10−9 by Wuosmaa et al. In two difficult and elegant experiments the very small branching ratio for the β + decay to the ground state of 54 Cr has been measured: (1. involving (d. The astrophysical impact of the shell-model based weak interaction rates have been recently studied by Heger et al.0 × 105 yr for the dominant β − branch. pf g and pf gd spaces. and Newman (1980.5 1 0.. Recently developed techniques.. have improved the energy resolution by an order of magnitude or more.. 1985). 54 Mn..b) The basic ingredient in the calculation of the different weak interaction rates is the Gamow-Teller strength distribution. As discussed in previous sections.26 ± 0. a partial β + half-life of (6.20±0. Using a similar argument Wuosmaa et al. Langanke and Mart´ ınez-Pinedo.5 0 30 32 34 36 38 40 42 Neutron number FIG. A nucleus such as 53 Mn.47 2 exp. The shell model makes it possible to refine these estimates.3[stat] ± 1. the transition to the 1+ state is no longer energetically allowed and the decay proceeds to the 3+ state at 158 keV via a second forbidden unique transition. 312. Multiplying the theoretical ratio by the experimental value of the β + half-life yields a value of (6. fp fpg fpgd 1. The GT− strength contributes to the determination of the β-decay rate. p)-type reactions) and GT− (measured in (p. The GT+ sector directly determines the electron capture rate and also contributes to the betadecay rate through the thermal population of excited states (Fuller et al. The necessary calculation involves decay by second-forbidden unique transitions to the ground states of 54 Cr and 54 Fe. To be applicable to calculating stellar weak interaction rates the shell-model calculations should reproduce the available GT+ (measured by (n. (2001). The influence of this half-life on the age of galactic cosmic-rays has been discussed recently by Yanasak et al. which yields 5. shell-model calculations are able to satisfactorily reproduce many experimental results so that it should be possible to obtain reliable predictions for nuclei and/or conditions not yet accessible experimentally. Langanke and Mart´ ınezPinedo. becomes stable. 52 Experimental and calculated 2+ energies in the pf .0 ± 1. Taking the weighted mean of this values. OTHER REGIONS AND THEMES A. could measure the time between production of iron group nuclei in supernovae and the accelaration of part of this matterial to form cosmic rays (Fisker et al. .. The calculation of the relevant electron-capture rates is currently beyond the possibilities of shell-model diagonalization calculations due to the enormous dimensions of the valence space. 1998.0 0. The calculation of β decay half-lives usually requires two ingredients: the Gamow-Teller strength distribution in the daughter nucleus and the relative energy scale between parent and daughter (i..2 0. Nuclei with higher masses are relevant to study the collapse phase of core-collapse supernovae (Langanke and Mart´ ınez-Pinedo.48 54 51 0. 2003) The astrophysical r-process is responsible for the synthesis of at least half of the elements heavier than A ≈ 60 (Wallerstein et al.5 0. 1997).e. making their theoretical reproduction more challenging. the agreement with the experimental data is quite satisfactory.6 0..4 59 Co 0. Due to the huge number of nuclei relevant for the r pro- . Simulations of the r-process require the knowledge of nuclear properties far from the valley of stability (Kratz et al. 2001).1 0. Initial studies by Langanke et al. Moreover.4 0 1 2 3 4 5 6 7 Ex [MeV] 1.1 0. 2 He) data (from Bümer et al.3 SM 0.2 0..8 0. The resulting electron-capture rates have a very strong influence in the collapse (Langanke et al. R¨nnqvist et al... T..2 0. KVI (Bümer et al. The shell-model distribution ina cludes a quenching factor of (0. 1993. 2003). 1983).4 0.3 0. the high temperatures present in the astrophysical environment makes necessary a finite temperature treatment of the nucleus.5 0. The shell-model results for 58 Ni have been recently compared with high-resolution data (50 keV) obtained using the (3 He.4 55 Mn 0.0 0. T +1). 2003). Pfeiffer et al. see section IV. 53 Comparison of shell model GT+ distributions with experimental data (Alford et al.. 1994.3 0.3).56 Fe and 58.B. neutron separation energies and half-lives) are needed.0 0.0 V B(GT+) 1. El-Kateb et al. the GT− strength distributions have significantly more structure and extend over a larger excitation energy interval than the GT+ distributions. Rapaport et al. More recently this calculations have been extended to cover all the relevant nuclei in the range A = 65–112 by Langanke et al. (2001) showed that the combined effect of nuclear correlations and finite temperature was rather efficient in unblocking Gamow-Teller transitions on neutron rich germanium isotopes. Nevertheless.2 0.6 0.6 0. this dimensionality problem does not apply to Shell-Model Monte-Carlo methods (SMMC.. 1990.2 0. (2003a)...e. 54 Comparison of the shell-model GT+ distribution (lower panel) for 51 V with the high resolution (d. However.0 B(GT+) Ni 0. 1993) for selected nuclei. The shell model o results (discrete lines) have been folded with the experimental resolution (histograms) and include a quenching factor of (0. The GT− operator acting on a nucleus with neutron excess and ground state isospin T can lead to states in the daughter nucleus with three different isospin values (T −1.4 0. As a consequence.2 0.1 51V(d.4 0. t) reaction (Fujita et al... theoretical predictions for the relevant quantities (i. FIG.60 Ni (Anderson et al. Shell-model diagonalization techniques have been used to determine astrophysically relevant weak interaction rates for nuclei with A ≤ 65. this makes SMMC methods the natural choice for this type of calculations.4 0. a Figure 55 compares the shell-model GT− distributions with the data obtained in (p. 2003a) and post-bounce (Hix et al.74)2 . 2003). 2002).2He)51Ti 56 GT Strength Fe 0.0 58 0.. As the relevant nuclei are not experimentally accessible.2 0.0 Fe Exp SM 0. 1999a).74)2 (adapted from Caurier et al. the Qβ value).0 −2 0 2 4 6 8 10 12 −2 0 2 4 6 8 10 12 E (MeV) E (MeV) FIG. n) charge-exchange reaction measurements for 54. For nuclei near the drip-lines or near closed shell configurations. 2003) for supernova neutrinos. 1990) and 6 ω calculations for 12 C (Hayes and Towner. description of the neutrino transport in supernovae and nucleosynthesis studies. e. for a description of the different models). 4 ω calculations for 16 O (Haxton. Haxton and Johnson. N = 82 (Brown et al..0 1.g. This calculations have been recently extended to include pf -shell nuclei (Fisker et al. For low neutrino energies. 2003. 2003. For these cases alternative theoretical approaches such as the nuclear shell model need to be applied.0 54 8.0 6. the density of levels is not high enough for the Hauser Feshbach approach to be applicable.. 55 Comparison of (p.5 0. And the evaluation by Haxton (1998) of the solar neutrino absorption cross section on 71 Ga relevant for the GALLEX and SAGE solar neutrino experiments. (1995) for solar neutrinos (see Bhattacharya et al.0 IAS 60 Ni IAS 1. Volpe et al. and N = 126 (Mart´ ınez-Pinedo. neutrino-nucleus reactions are very sensitive to the appropriate description of the nuclear response that is very sensitive to correlations among nucleons. 0 ω calculations have been used for the calculation of neutrino absorption cross sections (Sampaio et al.0 2.. 2000). x-ray bursts. this cross section have been recently evaluated by (Kolbe et al. The model of choice for the theoretical description of neutrino reactions depends of the energy of the neutrinos that participate in the reaction. e...0 0. 2001).49 10. 1990). 1987. 2001) and scattering cross sections (Sampaio et al. This situation is improving at least for N = 82 thanks to the recent spectroscopic data on 130 Cd (Dillmann et al. 2003).5 0. detection of supernova neutrinos. 1999. 2003.0 2. The model of choice is then the nuclear shellmodel.0 4. 1998. Knowledge of neutrino nucleus reactions is necessary for many applications. A quenching factor of (0..0 IAS Fe IAS 6..0 2. In some selected cases.0 0 5 10 15 E (MeV) E (MeV) FIG. the estimates of the half-lives are so far based on a combination of global mass models and the quasi particle random-phase approximation (see Langanke and Mart´ ınez-Pinedo. the Fermi and Gamow-Teller contribution to the cross section could be determined from shell-model calculation that is supplemented by RPA calculations for higher multipoles.. 1999). For lighter nuclei complete diagonalizations can be performed in larger model spaces. 1998). (from Caurier et al. 2000. This type mix calculation has been carried out for several iron isotopes (Kolbe et al. the neutrino oscillation studies. for an experimental evaluation of the same cross section).g.0 2.0 1. 2003).0 Ni 4.0 2. Mart´ ınez-Pinedo and Langanke. 1999a). 2001). 1995).0 0.0 6.0 0 5 10 15 0 5 10 15 E (MeV) 10.. Toivanen et al.0 58 E (MeV) 6.. Most of the rea action rate tables are therefore based on global model predictions... cess.0 2. 1983) (upper panels) with the calculated GT− strength distributions (lower panels)... for a recent review)..0 0. The Fermi transition to the isobaric analog state (IAS) is included in the data (upper panels) but not in the calculations.. The most frequently used model is the statistical Hauser Feshbach approach (Rauscher and Thielemann.0 0. supernovae. Nuclear reaction rates are the key input data for simulations of stellar burning processes. All this calculations suffer from the lack of spectroscopic information on the regions of interest that is necessary to fine tune the effective interactions. n) L = 0 forward angle cross section data (Rapaport et al.74)2 is included in the calculations. For higher neutrino energies the standard method of choice is the random phase approximation as the neutrino reactions are sensitive mainly to the total strength and energy centroids of the different multipoles contributing to the cross section. Most of the relevant neutrino reactions have not been studied experimentally so far and their cross sections are typically based on nuclear theory (see Kolbe et al. 2000). x-ray pulsars. Experiment-based reaction rates for the simulation of explosive processes such as novae. while the shell model results are given by the dashed line.5 1.0 0.0 σ (mb/(sr MeV)) 8.0 3. comparable to the nuclear excitation energy. 2002) for selected pf shell nuclei relevant for supernovae evolution.0 1. For 54 Fe the solid curve in the lower panel shows the experimental GT− data from (Anderson et al.0 4.0 56 Fe 4.0 σ (mb/(sr MeV)) 8.0 0 5 10 15 0.0 0. However. 2001) and GT GT . Shellmodel calculations were used for the determination of the relevant proton capture reaction rates for sd-shell nuclei necessary for rp process studies (Herndl et al. recently shell-model calculations have become available for some key nuclei with a magic neutron number N = 50 (Langanke and Mart´ ınez-Pinedo.0 0. and merging neutron stars are scarce because of the experimental difficulties associated with radioactive beam measurements (K¨ppeler et al. Other examples of shell-model calculations of neutrino cross sections are the neutrino absorption cross sections on 40 Ar of Ormand et al. namely 48 Ca. It works as follows: Once the relevant wave functions of parent and grand daughter are obtained. 2003). Overlapping these vectors with the other doorway. 56 LSF for 48 Ca → 48 Ti 2ν decay. N 1+ states in the intermediate nucleus. not yet experimentally observed. while the ββ0ν spectrum consists in a sharp peak at the end of the Qββ spectrum.051. The 1/2 calculation of the 48 Ca half-life was one of the first published results stemming from the full pf -shell calculations using the code ANTOINE. in the A=48 case. the ββ2ν and ββ0ν. for instance.χ are characterized by a continuous spectrum ending at the maximum available energy Qββ . make it easier the signature of this mode. the number of iterations is increased until full convergence is reached. 2003). It comes in three forms: The twoneutrino decay ββ2ν : A Z XN −→A XN −2 + e− + e− + ν1 + ν2 ¯ ¯ Z+2 1 2 is just a second order process mediated by the Standard model weak interaction It conserves the lepton number and has been already observed in a few nuclei. require massive neutrinos –an issue already settled by the recent measures by Super-Kamiokande (Fukuda et al. the double beta decay without emission of neutrinos would be the only way to sign the Majorana character of the neutrino and to distinguish between the different scenarios for the neutrino mass differences. 1970). it is necessary to have a good description of the Gamow. the neutrinoless decay ββ0ν : A Z XN −→A XN −2 + e− + e− Z+2 1 2 needs an extension of the standard model of electroweak interactions as it violates lepton number. the experimental information was limited to a lower limit on the 2ν-ββ half-life T2ν > 3.χ is also possible A Z XN − −→A XN −2 + e1 + e− + χ Z+2 2 the ground state and occasionally a few excited states of the grand daughter. Besides. The contributions of the different intermediate states to the final matrix element are plotted in Fig. For many years... ββ0ν. which implies a detailed description of the odd-odd intermediate nucleus. at iteration N. both even-even.. Notice the interfering positive and negative contributions. The calculation of MGT requires the precise knowledge of the ground state of the parent nuclei and In the pf -shell there is only one double beta emitter. This should. 1998).. (94) FIG. in principle. It is also a show-case of the use of the Lanczos strength function (LSF) method. A third mode. SNO (Ahmad et al. in some extensions of the standard model and proceeds via emission of a light neutral boson. The last two modes. producing. 2002)). B.083 and MGT (2+ )=0. with excitation energies Em . . The second mode. ββ-decays The double beta decay is the rarest nuclear weak process. Each bar corresponds to a contribution to the matrix element. 20 iterations suffice largely. the three modes show different electron energy spectra ((see figure 2 in Zdesenko. when the decay to the intermediate nucleus is energetically forbidden or hindered by the large spin difference between the parent ground state and the available states in the intermediate nuclei. with 2ν MGT (J) = m J + ||σt− ||1+ 1+ ||σt− ||0+ m m Em + E0 (J) (95) (there is an implicit sum over all the nucleons). In a second step. The resulting ma2ν 2ν trix elements are: MGT (0+ )=0. a Majoron χ. In what follows we shall concentrate in the ββ2ν and ββ0ν modes. Finally. It takes place between two even-even isobars. G2ν contains the phase space factors and the axial coupling 2ν constant gA . it is straightforward to build the doorway + states σt− |0+ initial and σt+ |Jf inal . The theoretical expression of the half-life of the 2ν mode can be written as: 2ν 2ν [T1/2 ]−1 = G2ν |MGT |2 .6×1019 yr (Bardin et al. Interestingly. Experimentally. one of them is fragmented using LSF.Teller strength functions of both of them.50 for 20 Ne (Heger et al. 2002) and KamLAND (Eguchi et al. The method is very efficient. 56. This is why this calculation is a challenge for the nuclear models. entering the appropriate energy denominators and adding up the N contributions gives an 2ν approximation to the exact value of MGT (N=1 is just the closure approximation).. and why agreement with the experiment in this channel is taken as a quality factor to be applied to the predictions of the models for the neutrinoless mode. 1996). With the exception of 150 Nd. 76 Ge and 82 Se. (1996) (see also Suhonen and Civitarese.09 0. either with the seniority v or with the configurations t (t is the maximum number of particles in the g 9 orbital). Full details of the calculations as well as predictions for other 0ν and 2ν decays in heavier double beta emitters can be found in Retamosa et al.. The results for the 2ν mode of the lightest emitters 48 Ca. in the 0+ → 0+ case.3 × 1019 48 Ge 2. The upper bounds on the neutrino mass resulting of our SM calcu0ν lations.8 × 1021 76 Se 3. the resulting half-life is T2ν = 3. where mν is the effective neutrino mass.22 0. 1994). already discussed. Such patterns have been also observed in the Tellurium and Xenon isotopes. sufficiently large (≥ 2. only a few are potentially interesting for experiment since they have a Qββ value.4 2ν T1/2 = 4.. . 1981): [T1/2 (0+ − > 0+ ]−1 = gV gA 2 2 (0ν) G0ν MGT − (0ν) MF (0ν) mν me 2 For the heavier emitters.(y) This was believed to make the results less dependent of the nuclear model employed to obtain the wave function.19 1. i (97) where. mν in eV. for a recent and very comprehensive review of the nuclear aspects of the double beta decay). The t = 16 and v = 16 2 values correspond to the full space calculation and are consequently equal.1×10−17 yr−1 (Tsuboi et al. for which full space calculations are doable. G2ν =1. without a sum over intermediate states. An interesting feature in the calculation of the double beta decay matrix elements is shown in Fig. TABLE XVIII 0ν matrix elements and upper bounds on the neutrino mass for T0ν ≥ 1025 y. 1998. the matrix elements are just expectation values of two body operators. some truncation scheme has to be employed and the seniority truncation seem to be the best. TABLE XVII Calculated T2ν half-lives for several nuclei and 1/2 0+ → 0+ transitions Parent 2ν T1/2 th. The two truncation schemes show very distinct patterns. reveal that the nuclear charge radii rc follow a characteristic parabolic shape with a pronounced odd-even staggering (Fricke et al. due to the presence of the neutrino propagator.0 × 1019 82 The expression for the neutrinoless beta decay half-life. This is the case in the Calcium isotopic chain. 1984). 1996). because a later measure gave +2.m h(r)tn tm 0+ .1 [stat]±1. 1996). Palmer... and shows that even in magic cases. (Caurier et al.7 × 1019 4. 1984). This situation simply reflects the limitation of the spherical mean field description of the nucleus.7 × 1019 8.. In this case.7×1019 yr.m h(r)(σn · σm )tn tm 0+ i (96) MF (0ν) + = 0f n.97 -0. 1990) 1/2 (see also erratum in Caurier et al. Radii isotope shifts in the Calcium isotopes = 0+ f n. Putting everything together. The prediction was a success. Takasugi..3−1.94 76 Ge 1.. are gathered in table XVII (Caurier et al.5M eV ) for the 0ν signal not to be drowned in the surrounding natural radioactivity. We have already seen in section VI.49 82 Ca 3. all of them can be described within a shell model approach. This seems to indicate that the most favorable nuclei for the theoretical calculation of the 0ν mode would be the spherical emitters. assuming a reference half-life T1/2 ≥ 1025 y. An assumption that has not survived to the actual calculations. can be brought to the following form (Doi et al.. 1/2 Parent 0ν MGT 0ν MF mν 48 Ca 0. 1995.4[syst] ×1019 yr (Balysh et al.E that deformed np-nh configurations can appear at very low excitation energy around shell closures and even to become yrast in the case of neutron rich nuclei.51 For the ground state to ground state decay 48 Ca→48 Ti. with the seniority truncation being more efficient. the correlations produce a sizeable erosion of the Fermi surface. and using the GamowTeller quenching factor. 1985. G0ν the kine(0ν) (0ν) the following matic space factor and MGT and MF matrix elements (m and n sum over nucleons): (0ν) MGT C. The phase space factor hinders the transition to the 2+ that represents only about 3% of the total probability.(y) 2ν T1/2 exp. where seniority is an efficient truncation scheme. Among the other ββ emitters in nature (around 30).63 -0.33 Se 1.6 × 1021 1.58 0. the “neutrino potential” h(r) is introduced. Experiments based on optical isotope shifts or muonic 2 atom data. are collected in table XVIII. The SMMC method has also been applied to the calculation of 2ν double beta decays in (Radha et al. (1995) and Caurier et al. 57 for the 76 Ge case : the convergence of 0ν the MGT matrix element is displayed as a function of the truncation of the valence space. since the dimensions are strongly reduced in particular for 0+ states. The SMMC. 2s 2 . that can overcome these limitations. When protons are allowed the dimensions grow rapidly and the calculations have been limited until now to nuclei with few particle or holes on the top of the closed shells (see Covello et al. 1f 7 and 2p 2 2 is a judicious choice (see Caurier et al. has been applied in this region to 128 Te and 128 Xe that are candidates to γ soft nuclei by Alhassid et al. (1996) and also to several Dysprosium isotopes A=152-162 in the Kumar-Baranger space by White et al. 1995. (2000). the excitation energies of the 2+ and 3− and the B(E2)’s between + the 21 and the 0+ states in the Calcium isotopes. This is probably due to the limitations in the valence space. that excludes the 5 1 1d 2 . The QMCD method has been also applied to the study of the spherical to deformed transition in the even Barium isotopes with A=138-150 by Shimizu et al. 58). 2 2 δrc (A) = 0. 1996).1 40 42 44 46 48 A FIG. Hints on the location of the hypothetical islands of super heavy elements are usually inspired in mean-field calculations of the single particle structure.. 59. The charge radii isotope shifts of the Calcium isotopes relative to 40 Ca are shown in Fig.. a 1 3 3 model space comprising the orbits 1d 2 . the 1f 5 and the 2p 2 orbits. among others.1 2 0 −0. can be expressed as: 1 π n (A)b2 (98) Z pf where Z=20 for the calcium chain.3 3 <m> 2 1 0 0 5 10 Truncation 15 Isotope shift [r (A) − r (40)] (fm ) FIG. The predictions . called sdpf.. together with the experimental values. 59 Isotopes shifts in Calcium chain: experimental (circles) versus shell model calculations (stars) D.2 For the description of the nuclei around N=Z=20. for a review of the work of the Napoli group). b is the oscillator parameter. (2001).52 4 v truncation t truncation functions. Several important features of the nuclei at the sd/pf interface are reproduced by the calculation. 57 Variation of the mν limit as a function of seniority truncations (circles) and configuration truncations (diamonds) 2 0. 1 5 exp 4 3 2 sm 1 exp 0 −1 3/2 in Sc + 2 0. Shell model calculations in heavier nuclei Energy (MeV) 02 in Ca sm + 40 42 44 46 48 A FIG. The 2 former produce an increase of rc that. the excitation energies of the intruder 0+ states in 2 + the Calcium isotopes as well as the location of the 3 2 states in the Scandium isotopes (see Fig.sm in section VII. with modified single particle energies to reproduce the spectrum of 29 Si. 58 Comparison between calculated and experimental intruder excitation energies in the Calcium and Scandium isotopic chains Due to the cross shell pairing interaction. and nπ is extracted from the calculated wave pf functions. although the calculated shifts are a bit smaller than the experimental ones. A good example is the space comprising the neutron orbits between N=50 and N=82 for the tin isotopes (Hjorth-Jensen et al. 1997. using HO wave There are some regions of heavy nuclei in which physically sound valence spaces can be designed that are at the same time tractable. protons and neutrons are lifted from the sd to the f p orbitals. The interaction is the same used to describe the neutron rich nuclei in the sd-pf valence space. Nowacki. The global trends are very well reproduced. 2001a). shell model calculations (as well as experimental spectroscopic studies of 216 Th (Hauschild et al. However. In particular. and showed that when the random matrix elements were displaced to have a negative centroid.53 for nuclei far from stability can be tested also in nuclei much closer to stability. E* [keV] 5000 20 + 19 – 18 – 17 – 19 – 18 + 16 + 17 –– (16 ) 16 + 15 – 17 – 15 – 14 + 13 – 14 + 12 + 12 + 19 – 20 + 19 – 18 + 16 + 18 – 17 – 15 – 16 + 16 – 17 – 15 – 13 – 14 + 13 + 14 + 12 + 12 + 10 + 10 + 3– 11 – 19 + 18 + 20 + 17 – 15 – 12 + 14 + 10 + 4000 (14 + ) (12 + ) 12 + 14 + 12 + 10 + 3000 10 + 11 – (11 – ) 3 –– 11 11 – 3– 8+ 6+ 8+ 4+ 2+ 2000 8+ 8+ 6+ 4+ 2+ 8+ 6+ 8+ 4+ 2+ (8 + ) (6 + ) (4 + ) (3 – ) (2 + ) 8+ 6+ 8+ 4+ 2+ 1000 0 EX 0+ SM2 0+ EX 0+ SM2 0+ SM2 218 92U126 0+ 214 88Ra126 216 90Th126 FIG. E. As a consequence.. Zhao et al. Energy scales that are well separated in other systems (as vibrational and rotational states in molecules) do overlap here. (2003). 39). is predicted. Vel´zquez and a Zuker (2002) argued that the general cause of J = 0 ground state-dominance was to be found in time-reversal invariance. especially the lighter ones. Random Hamiltonians The study of random Hamiltonians is a vast interdisciplinary subject that falls outside the scope of this review (see Porter. and 218 U 214 Ra. corresponding to the 1h 9 clo2 sure. (2001) and Chau Huu-Tai et al. where shell model calculations are now feasible. 1991). secular behavior—both in the masses and the spectra—eventually emerges and the pf shell is the boundary region were rapid variation from nucleus to nucleus is replaced by smoother trends. On the contrary. do not support this conclusion. 3p 3 . and was associated to the typical forms of collectivity found in the IBM. This is very much in line with what was found by Cortes et al. 2f 2 . The cases of 214 Ra. the ground state of the N=126 isotones is characteristic of a superfluid regime with seniority zero components representing more than 95% of the wave functions for all nuclei. using 2 2 2 the realistic Kuo-Herling interaction (Kuo and Herling. No shell gap for Z=92. 216 Th and 218 U are shown in figure 60. larger calculations become associated with more transparent physics. The origin of J = 0 ground state-dominance was attributed to “geometric chaoticity” (Mulhall et al. recent systematic mean field calculations suggest a substantial gap for Z = 92 and N = 126 (218 U) corresponding to the 1h 9 shell clo2 sure. (1982). CONCLUSION Nuclei are idiosyncratic. well developed rotational motion appeared in the valence spaces where the realistic interactions would also produce it (as in Fig. Bijker et al. Johnson et al. 1965. and give hints on how to extend the shell model philosophy into heavier regions where exact diag- .b). the interplay of random and collective interactions may deserve further study. 1971) as modified by Brown and Warburton (Warburton and Brown. The geometric aspects of the phenomenon were investigated in some simple cases by Zhao and Arima (2001). 3p 2 proton valence space. the 2+ energy systematics as well as the high-spin trends. 60 Experimental vs theoretical spectrum of 216 Th. leading to strong interplay of collective and single particle modes. accessible to shell model treatment. Bijker and Frank (2000a. Would it be a good idea to replace the rest of the interaction by a random one. It is unlikely that we shall learn much more from purely random Hamiltonians.. 2001)) were undertaken in the N = 126 isotones up to 218 U. 1999) noticed that random interactions had a strong tendency to produce J = 0 ground states. This is an empirical fact that was hitherto attributed to the pairing force. (1999) showed that in an interacting boson context this also occurred. Therefore here we shall only give a bibliographical guide to work that has recently attracted wide attention and may have consequences in future shell model studies. for a collection of the pioneering papers). 2001). On the other hand. 2f 5 . To shed light on these controversial predictions. For the fermion problem no collectivity occurs for purely random interactions (Horoi et al. In particular: We know that a monopole plus pairing Hamiltonian is a good approximation. and the interesting point is that the collective ingredient induced by the displacement of the matrix elements is the quadrupole force. 2000). The only deviations between theory and experiment are in the 3− energy and reflect the particular nature of theses states which are known to be very collective and corresponding to particle hole excitations of the 208 Pb core. (1998. Nonetheless. instead of neglecting it? IX. The calculation reproduces nicely the ground state energies.. 1i 13 . Shell model calculations were performed in the 9 7 1 1h 2 . the single quasiparticle energies extrapolated from lighter N=126 isotones up to 215 Ac. Luis Miguel Robledo. (A6) − (rstu)Γ where we have used ξrs = (1 + δrs )−1/2 if r≤ s. γ XΓΓz (rs) = (Br Bs )Γz . r s Γ † (XΓ (rs)XΓ (tu))0 . This review owes much to many colleagues. Then one wondered why such a phenomenology succeeded. Γ Γ Γ • Vrstu or Vrstu are two body matrix elements. Marianne Dufour. Peter Navr´til. Dirk Rudolph. (A1) can be coupled to quadratics in A and B.Γ † Γ Vrstu ZrsΓ · ZtuΓ = † Γ ξrs ξtu [Γ]1/2 Vrstu (XrsΓ XtuΓ )0 . The shell model has been craft and science: one invented model spaces and interactions and forced them on the spectra. Γ (A2) † † ZΓΓz (rs) = (1 + δrs )−1/2 XΓΓz (rs) is the normalized pair operator and ZΓΓz (rs) its Hermitian conjugate. (1 + δrs )1/2 /2 if no restriction. V can be transformed into the “multipole” representation V= γ γ γ ξrs ξtu [γ]1/2 ωrtsu (Srt Ssu )0 (rstu)γ 0 0 ˆ +δst δru [s]1/2 ωruss Sru . Jorge Sńchez Solano. APPENDIX A: Basic definitions and results (−1)α−αz α γ β −αz γz βz <α The normal and multipole representations of H are obtained through the basic recoupling 1/2 0 δst Sru + † − (XΓ (rs)XΓ (tu))0 = −(−1)u+t−Γ Γ r [Γγ]1/2 (−1)s+t−γ−Γ γ r s Γ γ γ (Srt Ssu )0 . † XΓΓz (rs) = (Ar As )Γz . Luis Egido. (A8) A few equations have to exhibit explicitly angular momentum (J). Sometimes it worked very well. (A4) u t γ whose inverse is γ γ (Srt Ssu )0 = (−1)u−t+γ γ r 1/2 0 δst Sru − (A5) [Γγ]1/2 (−1)s+t−γ−Γ Γ • p is the principal oscillator quantum number • Dp = (p + 1)(p + 2) is the (single fluid) degeneracy of shell p • Orbits are called r. Hubert Grawe. in the first place to Joaqu´ Retamosa who has been a memın ber of the Strasbourg-Madrid Shell Model collaboration for many years. Tr is used for both the isospin and the isospin operator. Silvia Lenzi. Franco Brandolini. This work has been partly supported by the IN2P3-France. Karlıa e ıa heinz Langanke. Alternatively we use nr and zr . • mr is the number of particles in orbit r. 1966): Γ stands for JT . David Dean. Many thanks are given to Benoit Speckel for his continuous computational support. e ıa o Mar´ Jos´ Garc´ Borge. CICyTSpain agreements and by a grant of the MCyT-Spain. Wrstu is used after the monopole part has been subtracted. In neutron-proton (np) scheme. Peter von a Neumann-Cosel. BFM2000-030. Expressions carry to neutron-proton formalism simply by dropping the isospin factor. and exploiting the formal properties of the Lanczos construction. then to Guy Walter. It is to be hoped that this state of affairs will persist.54 onalizations become prohibitive. to discover that there was not so much phenomenology after all. Morten Hjorth-Jensen. Achim Richter. Petr Vogel to name only a few. Acknowledgments The one particle creation and anhilation operators Arrz = a† z rr ˜ Brrz = arrz = (−1)r+rz ar−rz . mrx specifies the fluid x. and both have the same parity. etc. and in general F (Γ) = F (J)F (T ). (A7) so as to have complete flexibility in the sums. Mike Bentley. Olivier Sorlin. Γ γ Sγz (rt) = (Ar Bt )γz . We use Bruce French’s notations (French. by suggesting some final solution to the monopole problem. For reduced matrix elements we use Racah’s definition γ < ααz |Pγz |ββz >= (A3) Pγ β> . Jean Duflo. [r] = 2(2jr + 1). In this review we have not hesitated to advance some ideas on how this could be achieved. Also(−1)r = (−1)jr +1/2 . Then (−1)Γ = (−1)J+T . s. and isospin (T ) conservation. Carl a Svensson. [Γ] = (2J + 1)(2T + 1). According to (A4). Jos´ Mar´ G´mez. where (sum only over Pauli-allowed Γ) . u t γ The “normal” representation of V is then V= r≤s t≤u. Dr = 2jr + 1 ˆ • δrs = δjr js but r = s. which demands different orbits of neutrons and protons. † ZrsJ0 · ZutJ0 = − † (−1)J ZrsJ0 · ZtuJ0 = = 1 (Srust − 4Trust ) 4 (B6a) † ZrsJ1 · ZutJ1 = = Trs 1ˆ 01 = δrs [r]1/2 Srs 2 1 (3Srust + 4Trust ) 4 † (−1)J ZrsJ1 · ZtuJ1 = (B6b) (B1) and by combining (B5a). Before we describe the technique for dealing with the non diagonal parts of HmT i. the solution would consist in calculating traces of H. 3 Trtsu = ζrs ζtu (Trt · Tsu − δst Sru ). Define the generalized number and isospin operators 00 ˆ Srs = δrs [r]1/2 Srs . 4 (B2a) J (1 ± (−1)J ) = 2 (B7a) 1 ((Srtsu − 4Trtsu ) ∓ (Srust − 4Trust )) 8 † ZrsJ1 · ZtuJ1 (B2b) J (1 ± (−1)J ) = 2 (B7b) which in turn become for δrt = δsu = 1. 2002). for generalizing to them the notion of centroid. The full Hm contains all that is required for Hartree Fock (HF) variation. APPENDIX B: Full form of Hm = J J −Trtsu = 3 4 † ZrsJ0 · ZtuJ0 − 1 4 † ZrsJ1 · ZtuJ1 (B4b) If we were interested only in extracting the diagonal parts of the monopole Hamiltonian in jt scheme. Minimizing the energy with respect to a determinantal state will invariably lead to an isospin violation because neutron and proton radii tend to equalize (Duflo and Zuker. to assess accurately the amount of isospin violation in the presence of isospin-breaking forces we must ensure its conservation in their absence. Vrstu = −[rs]−1/2 and therefore Srtsu = Γ † ZrsΓ · ZtuΓ = † ZrsJ0 · ZtuJ0 + † ZrsJ1 · ZtuJ1 ˆ 0 δst δru Sru . 4 (B5b) When jr = js = jt = ju and r = s and t = u. This is a special Hamiltonian containing only λ = 0 terms. reduce to Srr = nr . (B6a) and (B6b) † ZrsJ0 · ZtuJ0 which. Trust and Srtsu . (A9) HmT = K + all ˆ ˆ (artsu Srtsu − brtsu Trtsu )δrt δsu . HmT . Trtsu are present. By definition Hm contains the two body quadratic forms in Srs and Trs : Srtsu = ζrs ζtu (Srt Ssu − δst Sru ). mrs and Trs in Eqs.(A5) suggests an alternative to (A8) V = γ ξrs ξtu [γ]1/2 ωrstu (rstu)γ γ γ × (Srt Ssu )0 − (−1)γ+r−s (A11) γ r 1/2 J0 J1 ω 01 = δλ0 δτ 1 =⇒ Vrstu = 3[rs]−1/2 . Transforming to the normal representation through Eq. and inverting † ZrsJ0 · ZtuJ0 = J 1 (Srtsu − 4Trtsu ) 4 (B5a) J † ZrsJ1 · ZtuJ1 = 1 (3Srtsu + 4Trtsu ). for δrs = 1. the technical details. They can be calculated from (B5a) and (B5b) by exchanging t and u. (B3) Γ Wrstu = γ (−1)s+t−γ−Γ r s Γ u t γ γ ωrtsu [γ]. (B4a) where each term is associated with a pure two body operator. Therefore. both Srust .55 It follows that the form of HmT must be γ ωrtsu = Γ (−1)s+t−γ−Γ r s Γ u t γ Γ Wrstu [Γ]. (10a) and (10b) 1 ((3Srtsu + 4Trtsu ) ± (3Srust + 4Trust )) 8 . and showing that they can be written solely in terms of number and isospin operators. (A10) where the sum is over all possible contributions. some remarks on the interest of such an operation may be in order. (A10) J0 J1 ω 00 = δλ0 δτ 0 =⇒ Vrstu = Vrstu = [rs]−1/2 Eq. More generally. Now. but it goes beyond. a full diagonalization of Hm is of great intrinsic interest.e. (B5b). Trr = Tr . (B8) (B9) then HmT = K + d +Φ(e)(aα Sα + So for r = s or t = u. BUT Sα = Srtsu . which we call Hm0 . Tα = Trust . Φ(P ) = 1 and for jr = js = jt = ju and Φ(P ) = 0. (B5b). and from Eq. . because we can extract some γ =00 contribution from the Tα operators. x. ±T V rstu = (B13) Γ Γ Dr = [r]. Tα = Trtsu .ρ=± T ρT ˆ ˆ δrt δsu (1 − Φ(e))V rstu ΩT + Φ(e)V rstu ΩρT rstu rstu ad = α ae = α ρ = sign (−1)J . (−1)2r = −1. 2 T + ae Sα + be Tα ). Then HmT = K+ r≤s t≤u T. (A10) JT [J]Wrstu = 0 ∴ V rstu = T JT [J]Vrstu / (J) (J) (B11) [J]. (B7a). Separation of Hmnp and Hm0 T ρT ˆ ˆ −δrt δsu (1 − Φ(e))V rstu + Φ(e)V rstu . t ≤ u. etc. Φ(e) = 1. Γ ≡ J.56 To write HmT we introduce the notations Φ(P ) = 1 − (1 − δrs )(1 − δtu ). 0τ must be such that ωrtsu = 0.(B14). and (B7b) we can obtain the form of HmT in terms of the monopole operators by regrouping the coefficients affecting each of them. Γ ≡ JT . Dr = 2jr + 1. ¯ ¯ Of course we must remember that for Hmnp . We may also be interested in extracting only the purely isoscalar contribution to HmT . In other words JT JT Wrstu = Vrstu − 1.. The point will become quite clear when considering the diagonal contributions. ±T Ωrstu = J ΩT = rstu J † ZrsJT · ZtuJT bd = α (B10) be = α † ZrsJT · ZtuJT (1 ± (−1)J ) . for V rstu . ˆ ˆ Φ(e) ≡ (δrs − δrs )(δtu − δtu ). The power of French’s product notation becomes particularly evident here. It demands some algebraic manipulation to find Hmnp or Hm0 = K + ¯ Vα Sα (1 − Φ(e))+ (B15) (J) Applying this prescription to all the terms leads to (obˆ ˆ viously δrt δsu = 1 in all cases) V rstu = [J] = (J) ±T V rstu T JT Vrstu [J]/ α 1 ¯+ ¯− ¯− ¯+ ¯ + ((Vα + Vα )Sα + (Vα − Vα )Sα )Φ(e) 2 with V rstu = Γ Vrstu [Γ]/ [J] (J) (J) 1 Dr (Ds + 2Φ(P )(−1)T ) 4 1 + Φ(P ) JT Vrstu [J](1 ± (−1)J )/ J [Γ] (Γ) (Γ) = J J [J](1 ± (−1)J ) (Γ) ± [Γ] = Dr (Ds − Φ(P ))/(1 + Φ(P )) Vrstu [Γ](1 ± (−1)Γ+2r ) (B16) J 1 [J](1 ± (−1) ) = Dr (Dr ∓ 2) 4 Φ(e) = 1. because the form of both terms is identical. [Γ](1 ± (−1)Γ+2r ) = Dr (Dr ∓ (−1)2r ) Through eqs. while for Hm0 Dr = 2(2jr + 1). (B12) In j formalism Hmnp is HmT under another guise: neutron and proton shells are differentiated and the operators Trs and Srs are written in terms of four scalars Srx sy . (−1)2r = +1. δrt δsu = 1. It should be noted that Hm0 is not obtained by simply discarding the b coefficients in eqs. Sα = Srust . To simplify the presentation we adopt the following convention ˆ ˆ α ≡ rstu r ≤ s. y = n or p. with (B14a) α ¯ α ¯ 1 ¯1 ¯0 aα = (3Vα + Vα ) 4 1 ¯1 ¯0 bα = (Vα − Vα ) (B14b) 4 1 ¯ +1 −1 ¯α + Vα + Vα ) ¯ +0 ¯ −0 (3Vα + 3V 8 1 ¯ +1 ¯ −1 ¯ +0 ¯ −0 (3Vα − 3Vα − Vα + Vα ) 8 1 ¯ +1 ¯ −1 ¯ +0 ¯ −0 (3Vα + 3Vα − Vα − Vα ) 2 1 ¯ +1 ¯ −1 ¯ +0 ¯ −0 (3Vα − Vα + Vα − Vα ) (B14c) 2 α bd T α α (1 − Φ(e))(aα Sα + bα Tα ) The values of the generalized centroids V rstu and V ρT are determined by demanding that H − HmT = HM contain no contributions with λ = 0.(B5a). etc. F. The way out is to impose a factorization of the wavefunctions into relative and CM parts: Φ(r1 r2 . The center of mass (CM) problem arises because in a many body treatment it is most convenient to work with . it can be safely dropped now). . this happens only for spaces that are closed under the CM operator (C2): for NCM to be a good quantum number the space must include all possible states with NCM oscillator quanta. relative-CM factorization can be achieved only by including all states involving a given number of oscillator quanta. (11) and (B17) in a single form d Hm = K + +brs 1 Vrs nr (ns − δrs ) + (1 + δrs ) 3nr nr ¯ Tr · Ts − (B19) δrs 4(Dr − 1) so that upon diagonalization the eigenvalues will be of the form E = Erel + λ(NCM + 3/2). x = y) Vrx sy = 2δrs 1 V1 1 − 2 rs Dr 0 + Vrs 1 + 2δrs Dr (B20) 1 Vrx sx = Vrs Note that the diagonal terms depend on the represend d tation: Hmnp = HmT in general. The in which now the brs term can be dropped to obtain d d Hm0 or Hmnp .A. The 0 ω and EI spaces are also free of problems (the latter because no 1 ω 1− 0 states exist). and here we shall explain it in sufficient detail. (C1) or d Hm0 = K+ r≤s Vrs nr (ns −δrs )/(1+δrs ) (B17) i i ij We rewrite the relevant centroids incorporating explicitly the Pauli restrictions Vrs = T Vrs = Γ Γ Vrsrs [Γ](1 − (−1)Γ δrs ) (B18a) Dr (Ds − δrs ) JT J+T δrs ) J Vrsrs [J](1 − (−1) (B18b) T) Dr (Ds + 2δrs (−1) and change accordingly Kij in Eq. but also jumps to other shells. one is “spurious” and has to be discarded. As R−iP has tensorial rank J π T = 1− 0. of ¯ the type s p¯ (sd) ≡ s (sd) or (sd) (pf )¯ (sd) ≡ p (pf ). and it only remains to select the states with NCM =0. Then (B14) becomes Eq. Consider EEI(1)=p1/2 . Eq. ij ij (C2) ars = brs 1 3 δrs 1 0 (3Vrs + Vrs ) = Vrs + brs 4 4 Dr − 1 1 0 = Vrs − Vrs (B18c) The relationship between ars and Vrs makes it possible to combine eqs. and calling rij = ri − rj we have R2 + P 2 = i 2 (ri + p2 ) − i 4 1 A 2 (rij + p2 ). Even if we may be tempted to do so. d5/2 . (2). (11) and (B15) becomes d Hmnp A coordinates and momenta while only A−1 of them can be linearly independent since the solutions cannot √ depend on the center of mass coordinate R = ( i ri )/ A √ or momentum P = ( i pi )/ A. Assume now that we are interested in 2p2p excitations. The problem was raised by Elliott and Skyrme (1955) who initiated the study of “particle-hole” (ph) excitations on closed shells. ¯ p ¯ p ¯ Therefore. so as to dispel some common misconceptions. as anticipated. APPENDIX C: The center of mass problem What do you do about center of mass? is probably the standard question most SM practitioners prefer to ignore or dismiss. there is no excuse for ignoring what the problem is.57 2. It remains to analyze the EEI valence spaces. we can add a CM operator to the Hamiltonian H =⇒ H + λ(R2 + P 2 ). (C3) is telling us that out of five possible 1− 0 excitations. with the largest j– are always excluded. (C3) p 18 where p2j removes a particle and s1 or d2j add a particle ¯ on |0 . where CM spuriousness is always present but strongly suppressed because the main contributors to R − iP –of the type pj =⇒ p + 1j ± 1. Unfortunately. They involve jumps of two oscillator quanta (2 ω ) and the CM eigenstates (R − iP )2 |0 involve the operator in Eq. and for the kinetic energy we should do the same by referring to the CM momentum. . Diagonal forms of Hm We are going to specialize to the diagonal terms of Γ HmT and Hmnp . (C3). The clean way to proceed is through complete N ω spaces. P (pi − √ )2 = A p2 − P 2 = i 1 A (pi − pj )2 . discussed in Sections II. They noted that acting with R − iP on the IPM ground state of 16 O (|0 ) leads to √ √ 1 p p ¯ p (¯1 s1 − 2¯3 s1 − 5¯1 d3 + p3 d3 + 3¯3 d5 )|0 . As we are only interested in wavefunctions in which the center of mass is at rest (or in its lowest possible state). 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The Shell Model as Unified View of Nuclear Structure

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