Nuclear Theory at Sussex 1961-2011 Tony Evans

There was close collaboration between the Nuclear Theory group and the experimental group headed by Dennis Hamilton, who was always anxious for his graduate students to understand as much about nuclear structure theory as possible. Due to the large number of students in the experimental group, it was a case of ‘standing room only’ in joint meetings of the two groups. Sometimes this collaboration resulted in joint publications{1}

Although Phil was still developing his SU(3) work {2}, the main thrust of the group’s work for several years after it was formed related to the effective interactions between nucleons in the nucleus. Oscillator-basis relative matrix elements of a ‘smooth’ phenomenological potential were calculated directly from experimental nucleon-nucleon phase shifts{3} These ‘Sussex matrix elements’ (SME) were widely used throughout the 1970s, by members of the local group as well as researchers in other centres. For some features, e.g. single-nucleon spin-orbit splittings, good agreement with empirical data was obtained in first order perturbation theory if the nuclear density was constrained to a realistic value{4}. However, the interaction did not exhibit realistic saturation properties, i.e. the nuclei collapsed to too high a density on removal of the constraint. In later work the original set of SME was modified in order to improve the saturation properties by introducing a simple hard-core interaction with variable radius in relative s-states, chosen so as to maintain the fit to the experimental two-nucleon phase-shifts {4}. Brueckner theory was used in order to obtain finite matrix elements. Realistic values of the nuclear rms radii were obtained by choosing a core radius of around 0.3fm. The modified interaction was used to calculate, up to second order in perturbation theory, binding energies and spectra of some light nuclei having closed shells or with one or two valence nucleons or holes {5}. Eric Sanderson and his Brazilian student, Coraci Malta carried out Hartree-Fock calculations, together with second-order corrections, of binding energy and single particle level spectra in the light doubly-magic nuclei ⁴He, ¹⁶O, ⁴⁰Ca {6}. Although the inclusion of a hard-core potential improved the results generally, there was still a binding energy deficiency of about 3MeV per nucleon and the results did not give the correct saturation properties for nuclear matter. For this reason, a simple density-dependent addition to the original SME was then considered, the parameters of which were fitted empirically to the binding energies of nuclear matter and ¹⁶O {7}. In this form the interaction was subsequently used in a successful Hartree-Fock calculation of the deformed 0⁺state at 6.05MeV in ¹⁶O{8}.

The modified SME were also used in a shell-model calculation across the whole 0p-shell{9}. This calculation was performed in a new and unusual way, avoiding the use of fractional parentage coefficients. Following a suggestion by BR Judd that atomic structure calculations could be simplified by working with quasi-particles which were 50% particle and 50% hole, the idea was applied to nuclear structure by members of the Sussex group {10} and, after further development by KT Hecht and S Szpikowsky{11}, proved to be a very efficient and convenient method for small nuclear shells. The only drawback was that the energy matrices were slightly larger than those in a conventional shell-model calculation. Although the states of the quasi-particle basis were good eigenstates of J (angular momentum) and T (isotopic spin) the same was not true of particle number n in the configuration (0p)ⁿ. For example, in the case of J=3/2, T=1/2 the 38 quasi-particles states could be divided into two sets of 19, the first of these containing one state with n=11, thirteen with n=7 and five with n=3, while the other furnished similar numbers of states in the ‘hole’configurations (0p)¹²^ⁿ. Perhaps the most significant feature of this work was the inclusion of the effective three-nucleon interaction which arises in second order. This was found to have little effect on the spectrum of ⁷Li (n=3), but was much more significant in ¹⁰B(n=6) and ¹⁴N(n=10) where it produced level shifts of order of a few MeV. In all these nuclei the agreement with experimental spectra was comparable to those obtained in earlier work by other groups using simple 2-body interactions with empirically determined parameters. The contribution to the total binding energy of the interaction between the valence nucleons in the 0p-shell was found to be in good agreement with experiment in all the nuclei from ⁶Li to ¹⁶O.

In 1980 the research led by Phil Elliott changed direction. This was due to the appearance, in 1974, of a new model of nuclear structure: the interacting boson model (IBM). Before describing the work done at Sussex on the IBM during the 1980s and 1990s, it will be useful to give some historical and technical background. As first described by A.Arima and F.Iachello{12}, the IBM attempts a unified description of low lying collective states of medium and heavy even-even nuclei well away from closed shells in terms of bosons carrying no intrinsic spin which may occupy a lower state of zero orbital angular momentum or an upper state of orbital angular momentum 2. These bosons, referred to as ‘s-’ and ‘d-’ bosons respectively, are subject to a mutual interaction with seven phenomenological parameters, and the resulting spectrum is determined, as in the shell model, by computing the eigenvalues of the energy matrix. The total number of bosons, N, is taken to be half the number of nucleons (or holes) in the open shell. The model clearly possesses a simple symmetry group, viz. the group U_sd(6) of unitary transformations in the 6-dimensional boson space, and, for bosons, all the states must belong to the maximally symmetric irrep (irreducible representation), [N], of this group. The U_sd(6) symmetry group possesses certain sub-group chains which underlie the authors’ claim to a unified treatment of collective states. When the interaction matrix elements are all relatively weak compared to the spacing between the d and s levels, the states are accurately characterized by the chain U_sd(6)U_d(5)O(5)O(3). The irreps of the group U_d(5) of unitary transformations among the d-bosons are [n_d] where n_d=0, 1, 2,…,N is the number of d-bosons. The O(3) irrep at the bottom of the chain gives the angular momentum, L, of the states. This is the vibrational limit of the model, in which the main contribution to the excitation energy is proportional to n_d. Alternatively, if the interaction is comparable to the s-d spacing and contains a strong quadrupole-quadrupole component, the states are well characterised by the chain U_sd(6)SU(3)O(3), which represents the rotational limit of the model. In this dynamical limit each irrep, (,), of SU(3) comprises a set of rotation bands, with excitation energies proportional to L(L+1) analogous to those in the harmonic oscillator shell-model as shown by Phil Elliott in his seminal work of 1958. In its original version, known as IBM1, the model took no account of the distinction between neutrons and protons. This drastic simplification was abandoned with the advent of IBM2 which uses proton bosons, s__,,d_and neutron bosons s_, d_, the number, N_,of the former being half the number of valence protons (or proton holes) and similarly for N_, the number of neutron bosons. This was an important difference because in many applications the open proton shell is less than half full and the open neutron shell more than half full or vice versa. Another significant difference between IBM1 and IBM2 is that the U_sd(6) irrep is not necessarily the fully symmetric [N], but can be ‘mixed symmetry’, i.e. [f₁,f₂] with f1>f2>0 and f₁+f₂=N. Half the difference F=( f₁f₂)/2 is called the ‘F-spin’, and measures the departure from full symmetry by having a value less than its possible maximum N/2. The reason for this difference is that the symmetry group of IBM2 is U_sd(6)U_(2) where the second factor is the group of unitary transformations between the and states of a boson. Overall symmetry requires that the irreps of the two groups should be the same. Thus IBM2 describes a whole class of states with F<N/2 which are entirely absent from IBM1. These mixed symmetry states in IBM2 can be very different from IBM1 states. In the rotational limit, for example, with five bosons, the SU(3) irreps contained in [5] include (10,0) so that there would be a low-lying K=0 band with angular momenta L=0,2,4,6,8,10. However, the mixed symmetry states [41] include (,) = (8,1) giving rise to a K=1 band with L=1,2,3,4,5,6,7,8,9.

The IBM may be thought of as a useful intermediary between the macroscopic collective model and the shell model. In the Sussex group the main focus of interest was on establishing links between the IBM and the shell model. The hope here was to construct a microscopic foundation for the IBM and calculate the boson interaction from the inter-nucleon interaction. The first step in this direction was taken by Arima, Iachello and T. Otsuka{13} and was based on the well-known ‘seniority scheme’ for nuclear states. The basic idea of this is that any configuration of n like nucleons with the same angular momemtum j may be thought of as (nv)/2 pairs of nucleons, exchange-coupled to total spin zero, together with v unpaired nucleons, v being the seniority. The addition of further pairs does not change either the angular momentum or the seniority of the state. The Otsuka, Arima, Iachello (OAI) scheme maps the J=0 pairs of nucleons onto s-bosons. This is reasonable as it is easy to show that the nucleon pairs obey the boson commutation rule in the limit of very low occupancy of the shell. The d-bosons cannot be incorporated by a simple mapping from J=2 pairs of the unpaired nucleons, because adding such a pair to a state of seniority v produces states of seniority v and v2 as well as v+2. Thus, for example, two J=2 pairs coupled to J=0 would not be orthogonal to the v=0 state of 4 nucleons which is mapped onto two s-bosons. The OAI scheme resolves this conflict by mapping onto the d-boson a J=2 nucleon pair which onlyincreases the seniority by 2, the other two possibilities being eliminated explicitly by a projection operator. Thus the boson states are mapped from a truncated shell-model space formed by successively adding J=2 pairs, and at each stage selecting v+2 as the resultant seniority. The mapping is made separately for protons and neutrons in IBM2, so that the number of d_bosons is v_n/2 where v_nis the neutron seniority and the number of s_bosons is (nv_n)/2 for n neutrons. The numbers of s_and d_bosons are determined analogously.

The first Sussex contribution to IBM research was an undergraduate project{14} supervised by Phil Elliott. Its aims were twofold: to construct an isispin invariant form of the IBM and apply this qualitatively to the nuclear sd-shell. IBM2 had been applied to heavy nuclei in which neutrons and protons are filling different shells. In lighter nuclei where the valence shells are the same for both types of nucleon it is to be expected that neutron-proton pairing is equally as important as pairing of like nucleons. This suggests that neutron-proton pairs should be mapped onto a new type of boson, the -boson. The bosons now form an isospin triplet with T=1 and M_T=+1, -1, 0 for ,, and bosons respectively, and this new model, known as IBM3, can be made isispin invariant, which is not possible for IBM2. . The mapping of the nucleon pairs onto bosons follows the generalized seniority scheme, based on ther group O(5) or, equivqlently, Sp(4) of which the irreps are labelled by seniority v and reduced isospin t. Here again the seniority of the state is the number of nucleons not coupled in J=0 pairs and t is the isospin of these unpaired nucleons. As in IBM2, the J=0 pairs are mapped onto s-bosons, and the d-bosons are incorporated in a manner analogous to that in IBM2. The extra bosons in IBM3 imply that the orbital U_sd(6) irrep can have up to three rows, i.e. [f₁,f₂,f₃] with f₁_≥f₂_f₃>0 and f₁+f₂+f₃=N. The group U_(3) of unitary transformtions between , , and must have the same irrep [f₁,f₂,f₃] to achieve overall symmetry and the irreps of its O(3) subgroup are labelled by T. A reasonable application of IBM3 would be to the sd-shell, where the isobaric triplet of 0⁺states in ¹⁸O, ¹⁸F, and ¹⁸Ne provide the single s__,s_, s_bosons respectively, while the upper isobaric triplet of 2⁺states provides the d-bosons. The two boson states represent A=20 nuclei. The U(6) irrep [2] contains two J=0 states (s²,d²), two J=2 states (sd,d²) and one J=4 state (d²), while the corresponding U(3) irrep gives T=0,2. The lowest states in ²⁰F have T=1 and belong to the antisymmetric irrep [11] containing one state each of J=1,2,3. The boson interaction depends on the isospin, so that there are 7 matrix elements (analogous to IBM1) for T=0,2 and 3 for T=1 giving a total of 17 interaction parameters. A still more elaborate model followed {15} in which a T=0 boson having an intrinsic spin S=1, was added to the three bosons of IBM3. This model (IBM4), thus employs 6 different bosons. The symmetry group of IBM4 is U_sd(6)U_TS(6) where the second factor is the group of unitary transformations between the 6 spin-isospin states. Its irreps can be reduced under the subgroup chain U_TS(6)U(4)O_T(3)O_S(3) where the U(4) group is the ‘Wigner-supermultiplet’ group familiar from the L-S coupled shell-model. Again, overall symmetry requires the same irrep for both U(6) groups. This means that the Young Tableau specifying the irrep of U_sd(6) can have up to four rows. The appearance of the Wigner-supermultiplet group in this scheme is welcome because this label is known to be fairly pure for low-lying states in light nuclei. A more immediate reason for introducing the extra bosons of IBM4 was the fact that IBM3 provides no description whatever for the T=0 states in ¹⁸F, not even its ground state. Because of its mass, this nucleus should be described by a single boson, which must therefore have T=0. Moreover, as the ground state spin is J=1, a spinless s or d boson would not work. The difficulty is resolved by recalling that two nucleons in a symmetric orbital state must belong to the Wigner U(4) symmetry [11], and therefore have T=1,S=0 or T=0,S=1. For this reason, in IBM4, the isovector-scalar bosons of IBM3 are supplemented by the inclusion of vector-isoscalar bosons having intrinsic spin 1. Thus the ground state of ¹⁸F is given by some combination of ¹³s₁and ¹³d₁and the orthogonal combination represents an excited J=1 state. The new bosons also provide J=2 and 3 states with T=0, and there are candidates for all of these among the low lying states of ¹⁸F. A detailed test and comparison of all the models, IBM2,3 and 4, was carried out by Philip Halse{16} in the even nuclei filling the lower half of the sd-shell (A=18 – 28) involving up to 6 bosons. This region is interesting because some of the nuclei exhibit collective rotational features and it has been the subject of extensive shell-model calculations. Only fully symmetric states in the boson orbital space were included in the calculation so that a standard IBM1 program could be used. This involved the use of an effective IBM1 interaction, different for each of the extended models, defined so that the IBM1 calculation would reproduce as nearly as possible the result of an exact calculation in the extended model within the restricted spce of the symmetric orbital states. The single boson energies were taken from experimental data in the A=18 nuclei, although use was made of quadrupole matrix elements in establishing the correct admixture of ¹³s₁and ¹³d₁in the ¹⁸F ground state. Details the of the boson-boson interaction were obtained from energy levels in ²⁰O and ²⁰Ne. Care was taken to exclude states believed to contain large intruder components from outside the sd-shell configuration and published shell model work was useful in this regard. Again quadrupole matrix elements were used to determine s-d boson mixing, but shell-model rather than experimental values were used. The results show that IBM4 agrees very well with the experimental spectra and binding energies, while IBM3 and, to a greater extent, IBM2 agree much less well. These latter models seriously overbind all the nuclei and show major distortions in the spectra, especially with regard to the relative positions of the second 0⁺and 2⁺states and the first 4⁺. These discrepancies increase with atomic mass, A.

From the mid 1980s to the mid 1990s a series of papers explored the relations between (i) the IBM and the shell model{17,22,24,25},(ii) the IBM and the collective model{20,21} and (iii) the different versions of the IBM{34}; an isospin invariant form of the IBFM (interacting boson-fermion model) applicable to odd nuclei was developed{26,27,28,29}, and it was shown that in cases where the open neutron shell is nearly empty and the open proton shell nearly full, or vice versa, IBM2 rather than IBM3 should be used. Much of this work was very ably contributed to by graduate students attached to the group: Mourad Abdelaziz(Algeria?), Philip Halse(UK), Suheil Suleiman(Syria), John Thompson(UK) and Andreas Williams(Germany?).

In the OAI paper {13} the authors used their mapping from the shell-model to the boson space to construct a boson hamiltonian and other operators to agree with the corresponding shell-model operators within the truncated space up to two d-pairs. This required a knowledge of the nucleon-number, n, dependence of shell model matrix elements,which in general, requires the Wigner coefficients of the relevant quasi-spin algebra. For a jⁿconfiguration of neutrons this is O(3) i.e. rotations in 3 dimensions, the most familiar of all Lie groups. In contrast, for a mixed configuration of neutrons and protons the quasi-spin group is O(5)_ rotations in 5 dimensions, or equivalently, Sp(4)_ simplectic transformations in 4 dimensions. These Wigner coefficients,which would be required to construct an IBM3 hamiltonian from shell-model parameters, were not generally available. The necessary formulae for the n and T dependence of shell-model matrix elements were obtained, for both the single-j {31} and multi-j cases {33}, using a vector coherent state technique {41,42}. By equating the IBM3 matrix elements involving zero, one and two d-bosons to their counterparts in the shell model through the seniority mapping, the IBM3 parameters were obtained directly from a simple shell-model potential in numerical form {32,35}. The N and T dependence of E2 and M1 operators in IBM3 was also treated by the same method{36}. Based on this work, an empirical IBM3 analysis of several nuclei beyond the magic nucleus ⁵⁶Ni (N=Z=28) was carried out {37}. Wave functions from this calculation were used subsequently to test a new coupling scheme suggested by J. Ginocchio{45}. As there are 18 bosons in IBM3, the states of N bosons must belong to the irrep [N] of U(18). Starting with the orthogonal subgroup of U(18), the new coupling scheme is based on the chain O(18)O(3)O(15) O(3) O(3) O_sd(5), where the irreps of the two O(3)s give T_sand T_d. The conclusion was that the computed eigenstates were slightly closer to the O(18) scheme than to the usual U(6) {38}. The O(18) scheme suggested a new method of performing exact IBM3 calculations {39}.

An excursion into nuclear reactions theory was inspired by the presence of Ernesto Maqueda from the Tandar accelerator laboratory of the CNEA, Buenos Aires. This related to the fact that measured cross-sections for - transfer reactions were generally poorly reproduced by theory and -clustering in the nuclear surface had been mooted as a possible explanation for the discrepancies. The specific reaction considered was ¹²C(⁶Li,d)¹⁶O in which a carbon target, bombarded with ⁶Li ions emits a deuteron leaving an oxygen residual nucleus. Calculations based on the standard spherical shell-model configurations; (0s)⁴(op_3/2)⁸,(0s)⁴(0p)¹²for ¹²C and ¹⁶O respectively give cross-sections too small compared to experiment by factors of 215,29,94 and 11 for reactions leaving the oxygen in its 0⁺ground state, the 3^at 6.13 MeV and two 4⁺states at 10.35 MeV and 11.09 MeV respectively.

The method used was to add a tetrahedral deformation proportional to xyz to the harmonic oscillator potential defining the single nucleon orbits in ¹⁶O {18}. This deforming potential is invariant under the symmetry group, T_d, of a tetrahedron with its vertices at (1/3, 1/3, 1/3),(1/3, 1/3, 1/3),(1/3, 1/3, 1/3), (1/3, 1/3, 1/3). The deformation breaks up the SU(3) symmetry of the oscillator levels and each level was allowed to mix with those up to three oscillator shells above it. The lowest two deformed orbitals calculated in this way belong to the irreps A₁and F₂of T_d. When these deformed levels are filled with 6 protons and 6 neutrons the resulting intrinsic stateis tetrahedral in shape with an - particle at each vertex. States of good angular momentum are obtained by projection and include the 0⁺ground state and the rotation band based on it in which the next two states are 3^and 4⁺. These two states were identified with the experimental ones at 6.13 MeV and 11.09 MeV and the cross-sections for -transfer to all three states computed. It was found that for a particular strength of the tetrahedral deformation, all three cross-sections were close to 1/6 of the experimental values. Not only is this better agreement with experiment than that obtained with the spherical shell-model, but it successfully reproduces the ratios of the cross-sections for -transfer to the three final states.

The tetrahedral deformation also strongly enhances the E3 -decay from the 3^state to the ground state, which has a measured lifetime of about half that given by the p^-1sd “particle-hole” transition in the spherical shell model.

Although the work of the group concerned mainly microscopic models, the collective model was not entirely neglected. In 1986 a soluble -unstable hamiltonian was presented{19} and its results compared to those of the IBM. This led, in 2005, to some new solutions being found for a very old problem, viz. the determination of the -dependent part of the wave functions representing -unstable surface vibrations{42}.

In 1998 the Lewes Conference commemorated 40 years of SU(3) in nuclear structure theory. This may have been a trigger for Phil Elliott to consider the possibility of direct mappings from shell-model SU(3) to boson-model SU(3) {40,41}. This would be especially appropriate for deformed nuclei, where the seniority scheme, on which all previous mappings were based, is strongly mixed.

References

{1} S A A Eid, W D Hamilton, J P Elliott, Phys Lett. 166B,(1986),267; D S Mosbah, J A Evans,W D Hamilton, J.Phys. G20,(1994),787

{2} J P Elliott, M Harvey, Proc.Roy. Soc.A272, (1963)557; J P Elliott, C E Wilsdon, Proc.Roy.Soc. A302, (1968),509

{3} J P Elliott, A D Jackson, ,H A Mavromatis, E A Sanderson, B Singh, Nucl.Phys. A121, (1968)241.

{4} J Dey, J P Elliott, A D Jackson, H A Mavromatis, E A Sanderson, B Singh, Nucl Phys. A134,(1969),385

{5} E A Sanderson, J P Elliott, H A Mavromatis, B.Singh, Nucl.Phys. A219,(1974),190

{6} C P Malta, E A Sanderson, Nucl.Phys.A225,(1974),189

{7}R K Tripathi, J P Elliott, E A Sanderson, Nucl. Phys.A380, (1982), 483

(8) K Halkia, N I Kassis,E A Sanderson, J P Elliott, Nucl.Phys.A378(1982),461

{9} H Dirim, J P Elliott, J A Evans, Nucl.Phys.A244, (1975),301

{10} J P Elliott, J A Evans, Phys. Lett.31B,(1970),157

{11} K T Hecht, S Szpikowski, Nucl.Phys. A158, (1970),449

{12} F Iachello, A Arima, Phys Lett.53B, (1974)309

{13}T Otsuka,A Arima, F Iachello, Nucl.Phys.A309, (1978),1

{14} J P Elliott, A P White, Phys. Lett.,97B, (1980),169

{15} J P Elliott, J A Evans, Phys.Lett. 101B, (1981),216

{16}P Halse, J P Elliott, J A Evans, Nucl.Phys.A417, (1984) 301

[14] J A Evans, J P Elliott, S.Szpikowski, Nucl.Phys.A435,(1985),217

{17} J P Elliott, J A Evans,E E Maqueda, Nucl.Phys.A437,(1985),208

{18} J P Elliott, J A Evans,P Park, Phys.Lett.169B,(1986),309

{19} J P Elliott, P Park,J A Evans, Phys.Lett.171B,(1986),145

{20} J P Elliott, J A Evans, P Van Isacker,Phys.Rev.Lett.57,(1986),1124

{21} J P Elliott, J A Evans, A P Williams,Nucl.Phys.A469,(1987),51

{22} J P Elliott, J A Evans, Phys. Lett. 195B(1987)1

{23} M J Thompson, J P Elliott, J A Evans,Phys. Lett. 195B(1987)511

{24} M Abdelaziz, M J Thompson, J P Elliott, J A Evans, J. Phys. G14 (1988)219

{25} J P Elliott, J A Evans, P Van Isacker, Nucl. Phys. A481 (1988)245

{26} P Van Isacker, J P Elliott, J A Evans, J. Phys. G14 (1988)L201

{27} J A Evans, P Van Isacker, J P Elliott, Nucl. Phys. A489 (1988)269

{28} M Abdelaziz, J P Elliott, M J Thompson, J A Evans, Nucl. Phys A503 (1989)452

{29} M J Thompson, J P Elliott, J A Evans, Nucl. Phys. A504 (1989)436

{30} J P Elliott, J A Evans, G L Long, J. Phys. A25 (1992)4633

{31} J A Evans, G L Long, J P Elliott, Nucl. Phys. A561 (1993)201

{32} J P Elliott, J A Evans, G L Long, V-S Lac, J. Phys. A27 ,(1994) 4465

{33} V-S Lac, J P Elliott,J A Evans,G L Long, Nucl. Phys. A587,(1995)101

{34} J A Evans, J P Elliott, V-S Lac, G L Long, Nucl. Phys. A593,(1995)85

{35}.J P Elliott, J A Evans,V-S Lac, Nucl. Phys. A597,(1996)341

{36} J P Elliott, J A Evans, V S Lac, G L Long, Nucl. Phys. A609,(1996)1

{37} V S Lac, J P Elliott, J A Evans, Phys. Letters B394,(1997)231

{38} J A Evans, V S Lac, J P Elliott,Nucl. Phys. A627,(1997)71

{39} J P Elliott, J A Evans, P Halse, Lewes Conference, J.Phys.G25(1998)577

{40} JPElliott,J A Evans, J. Phys. G25, (1999)2071

{41} J P Elliott,J A Evans, J.Phys. A38 (2005) 5507

{42} K T Hecht, J P Elliott, Nucl.Phys.A438,(1984),29

{43} K T Hecht, Nucl.Phys. A493,(1989),23

{44} J N Ginocchio, Phys. Rev. Lett. 77,(1996),28

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Nuclear Theory at Sussex 1961-2011 Tony Evans

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