On the OAI mapping for a j = 132 shell

Nuclear Physics ONorth-Holland

A451 (1986) 91-103 Publishing Company

ON THJ3 OAI MAPPING FOR A j = y SHELL* P. HALSE Department of Physics and Atmospheric Science, College of Science, Drexel University, Philadelphia, PA 19104, USA Received

16 August

1985

Abstract: An SD pair subspace is constructed for identical particles in a j = y shell, and overlaps with the eigenstates of H = - Q. Q are evaluated. The lowest levels lie almost entirely within the SD space, and are well described by states corresponding to SU(3) combinations of the associated bosons. Procedures for mapping the hamiltonian are compared.

1. Introduction

The success of the interacting boson model (IBM-l)‘), which treats nuclei with 2N valence nucleons as systems of N bosons each with angular momentum 0 (s) or 2 (d), has prompted many investigations into its possible microscopic interpretation. Otsuka et al. *) have suggested a simple correspondence, the so-called OAI mapping, between IBM and shell-model states, and thence operators. The OAI procedure, in which the s- and d-bosons are associated with S- and D-nucleon pairs, is generally accepted for systems where seniority is approximately conserved3-5), but has been expected to fail when seniority is strongly mixed, such as in deformed nuclei, and other schemes have been postulated6). Studies of deformed nuclei have indeed shown that S- and D-pairing alone does not allow a good approximation to the intrinsic ground state7-9), b u t since this will generally involve levels of higher angular momentum than those produced by the IBM, it is not clear that this result is of direct relevance to the present discussion. In contrast, the OAI procedure has been shown to provide a good approximation to the most favoured SU(3) states of four particles in an sd shell”). It would clearly be more satisfying to have a single scheme for all situations, rather than several ad hoc procedures, particularly as an attractive feature of the IBM is its unified description of various phenomena; if the SD subspace is indeed also relevant to deformed nuclei, these pairs would be candidates for the “elementary building blocks” of collective motion suggested by Rowe”). In the present work, states with good angular momentum in a j = y shell are constructed from S- and D-pairs, and compared to eigenfunctions of a quadrupole * Work supported

by NSF grant PHY-844-1891. 91

P. Hulse / OA I mapping

92

force. Although it is not clear whether a single j-shell can adequately simulate rotational motion, where configuration mixing is important, the effect of seniority breaking on the relevance of the OAI mapping may certainly be conveniently studied. In addition, this system has recently been discussed using two other approaches. Faessler and Morrison “) claim that the “seniority-conserving mapping” leads to errors of a factor of ten in the ground-band energies, but in obtaining this result a particular ansatz is made ~nceming the IBM h~lto~~. In response, Wu et al. 13) argue that comparison with the composite particle representation indicates that the IBM states are spurious. In neither study are the wave functions themselves analysed. In contrast, the present work follows the philosophy of Evans et al. 14) and of Halse’O) in examining the structure of the pair-space states (using which, all operators may be mapped) but for the simpler case of identical particles. The emphasis is placed on determining the validity of the SD truncation for “rotational” states and on examining the relevance of the various IBM dynamical symmetries. Using these results it is straightforward to map the hamiltonian by equating matrix elements, as suggested by Otsuka et al. ‘). This work can also be seen as a pair-based shell-model truncation scheme, suggested by the IBM, but not to be followed by any boson appro~mation. In sect. 2, the OAI state mapping procedure is briefly reviewed. An application to the case of a single j = 9 shell is made in sect. 3, where overlaps with the eigenstates of a hamiltonian H = -Q. Q are evaluated, and the relevance of the IBM SU(3) and O(6) chains is discussed. Various techniques for mapping the hamiltonian are compared in sect. 4.

2. The OAI state mapping procedure The IBM l) treats nuclei with 2N valence nucleons as systems of N bosons which may each have angular momentum 0 (s) or 2 (d), s+, s; d;, d”:

[PJ$]

=I$&,;

0)

hence the model states are described by a U(6) group, which allows the three well-known subgroup chains (eq. (2)): U(6) ZI U(5) 2 O(5) 2 O(3),

[Nl

Pdl

(4

J

U(6) f) O(6) 3 O(5) z, O(3), (~00)

U(6) 3 SU(3) 1 O(3).

(2)

93

P. Halse / OAI mapping

The states of a single j-shell of identical nucleons (eq. (3)) may be partially classified using the concept of seniority u [ref. 15)] (eq. (4)): {am, u;,}

ai,am:

= 6”,

-jgmmj,

(3)

U(2j+l)XSp(2j+ml;XSU(2).

PI

(1”)

J

(4)

Otsuka et al. ‘) ~tab~sh~ a co~esponden~ between N-boson IBM states and those of 2N particles in a single j-shell by making the associations s+-,S+=A+(0), d++Ds=pA+(2) 3 (5) where A+(J)=; ,/$-[a+a+](J) and P is a projection normalisation,

operator

]sv-hdkyJ)

(6)

onto states of maximum seniority. Thus, after -+ ]j 2N~=2Nd(D)Ndc’y”J).

(7) The shell-model analogue states clearly vanish if 2N, > (j + 4). For smaller values of j, this may also happen in other cases: for instance, there is no u = 4 J = 0 state if j < $ [ref. 16)], indicating that the corresponding IBM state ](dd)O) would then be spurious. IBM states classified in some other way, for instance by the SU(3) or O(6) chains (eq. (Z)), may be mapped indirectly using the OAI procedure (eq. (7)) by first making a ~~sfo~ation to the U(5) z) O(5) chain (eq. (8)): ]N/3J) = c CgNdr]sN-h dNdyJ) +Ni c?YIj 2N~=2Nd(D)Nd”yV). (8) NdY This state truncation can be made the only appro~mation to the full shell-model problem by defining the boson operators to reproduce all matrix elements in the SD-pair space. However, such a mapping to all orders (the n th-order approximation to a k-body operator consists of k’-boson operators where k’ G k + n) would be cumbersome, and it has been claimed2) that the zeroth-order approximation (for example, the zeroth-order image of a two-body h~lto~~ consists only of oneand two-boson operators) is usually reasonable. 3. OAI for a j = y shell 3.1. CONSTRUCTION

OF THE PAIR STATES

A j = 9 shell is half-filled with seven particles, so we need consider only n = 2,4,6 (N = 1,2,3). Larger numbers of particles would be treated in terms of holes. The available IBM and shell-model states 16) are enumerated in table 1, where

P. Hake / UAI mapping

94

TABLE1 The number of states with angular momentum J (< 6) arising from each (1"07-") subspace of a j = \I shell and from the [ Nd] representation of U(5) J 0

{~)

1

2

4

3

5

6

1

[ I

1

(11~) Ul

1 1

(llllooo) PI

1 1

(1111110) f31

2 1

1

1

I

3 1

1

4 1

3

4

3 1

4 1

6 1

4

8 1

O-

=s *8”

-l-

-¶ 14

-2 -2 6 -0

-0

4

-2

E -3-

-4

-8 -

-0

-2

-!4 -2-

-4

-10 -6

-4

-4 -2 -0

-2 -0

-2

-0 -2

-0 -4-

faf

EXACT

(b) S-D

(4

(df

Fig. 1. Spectra for H = -Q - & in (y)” (see text, sect. 4): (a) obtained using the full shell-model space; (b) obtained by diagonalising the matrices of H in the SD subspace (eq. (7)); (c) IBM spectrum obtained using the assumed form (eq. (IO)) for the image of H; (d) IBM spectrum obtained using the assumed form (eq. (11)) for the image of H.

95

P. Hake / OA I mapping

,0, iV = fn, and A$ = iv. Thus the IBM states for a given n, v=n,n-2,... IN, N,, = 2, J = 2) (N = 2,3) correspond to some particular combination of the three shell-model states with n = 2N, u = 4, and J = 2, the IBM state IN = 3, Nd = 3, J = 6) corresponds to some particular combination of the eight shell-model states with n = 6, v = 6, and J = 6, while the shell-model states with J = 5 have no IBM analogues. No IBM states are necessarily spurious, in contrast to the cases j 6 $ discussed above, i.e. for each J and u = 2N, the number of shell-model states is greater than or equal to that in the IBM. Explicit construction of the OAI analogues (eq. (7)) indeed shows that none vanish. However, if g-bosons were included in the IBM l’), there would be, for instance, three J = 0 states for N = 2 where there are only two shell-model analogues, and seven J = 0 states for N = 3 where only four such states exist. The existence of such spurious states represents a clear ~gument against the inclusion of a g-boson in applications of the IBM to a single j = y shell; the

O-

-0 -24

-2

-l-

-2 -18 -2-

-2 -4

2 -1

-0

-0 0

-2

_

-2

-2

-6

-3

0

3

-0

-3

-2

-2

E

-0

-0

-6 -4 -2

--

-2 -0 -4 6

-2

---r-2 -4 -2 -0

-4 -2 -0

--4

-4

-2

-4-

-0 -2

-0 -6

i

((11EXACT

(bl S-D

(c) ZEROTH ORDER

(d)

(9)

Fig. 2. Spectra for H = - Q. Q in (y)6 (see text, sect. 4): (a) obtained using the full shell-model space; (b) obtained by diagonalising the matrices of H in the SD subspace (eq. (7)); (c) IBM spectrum obtained using the zero&order image of H defined by matrix elements between states with ~44 in the SD subspace; (d) IBM spectrum obtained using the assumed form (cq. (10)) for the image of H, (e) IBM spectrum obtained using the assumed form (eq. (11)) for the image of H.

96

P. Hake

/ OA I mapping

separate question of whether explicit G-pairing is needed, even in the lowest states, will be addressed below.

3.2. SD-PAIR

ANALYSIS

OF Q - Q EIGENSTATES

For a given set of shell-model eigenfunctions, the degree to which the SD subspace (eq. (7)) exhausts th e 1owest levels may now be evaluated, and the relevance of the various dynamical symmetries (eq. (2)) determined. Both these properties will depend on the shell-model interaction chosen. A “realistic” interaction would be inappropriate because it is the ability of the SD truncation to cope with the mixing of seniority (as a simulation of rotational motion) which is of primary interest, and it is known that “realistic” like-particle interactions do not strongly break seniority; indeed such forces are typically dominated by a pairing component, the effects of which have already been studied2). However, any product of even-rank multipoles does have the desired property, and a quadrupole force would seem to be the most suitable. We will take H=

(9)

-Q-Q,

where Q = - [a+~#~). Shell-model calculations were performed using the Oxford-Buenos Aires program OXBASH I*), and the resulting spectra are displayed in figs. la and 2a for n = 4 and TABLE 2

Overlaps

of the OAI analogues

of two-boson

from H=

-Q*Q

states with the eigenstates

arising

for j=y Level number

J

0

(ss) (dd)

2

0.8184 0.5746

IBM

100

(sd) (dd)

0.8652 - 0.4589

IBM 4

2

1

(dd) IBM

The sum of the squared is given as a percentage.

95.9 0.9189 84.4 overlaps

3

4

0.3259 0.2356

- 0.0037 0.1577

5

- 0.5746 0.8184 100 0.3810 0.8421 85.4 - 0.0515 0.3

16.2 0.2901 8.4

for each level, being its total occupancy

2.5 - 0.2619

- 0.0141

6.9

0.0

of the IBM (or SD) subspace,

97

P. Ha/se / OAI mapping TABLE3 Overlaps of the OAI analogues of three-boson states with the eigenstates arising from H= -Q.Qfor

j=y Level number

J 0

(sss) (sdd) (ddd)

2

W)

IBM (W Wd) IBM 3

(ddd)

4

(W (ddd)

6

(dW

IBM

IBM

IBM

1

2

3

4

0.8268 0.5121 - 0.2218

0.3212 -0.1110 0.9359

- 0.4098 0.6795 0.1632

- 0.2128 0.5136 0.2134

99.8 0.8506 - 0.4150 0.3172 99.6 0.9189 84.4 0.8924 - 0.4155 96.9 0.9432 89.0

99.1 0.4076 0.8855 0.0913 95.9 0.3643 13.3 0.3488 0.6831 58.8

65.6 - 0.1492 0.0791 0.3849 17.7 0.0032 0.0 - 0.0513 0.0621 0.6

5

35.5 -0.1712 0.1051 0.5621 35.6 0.1231

- 0.0168 0.0478 0.1768 3.4 0.0882

1.5

0.8

0.1955 0.4769

0.0701 0.1782

23.4

3.7

0.1299

0.1444

0.1663

0.0864

1.7

2.1

2.8

0.7

The sum of the squared overlaps for each level, being its total occupancy of the IBM (or SD) subspace, is given as a percentage.

6, respectively (some levels with J-values not contained in the IBM have been omitted). Overlaps of the OAI states (eq. (7)) and the Q - Q (eq. (9)) eigenvectors are displayed in tables 2 and 3, together with the correspond~g occupancy of the SD-pair space. The seniority label is generally well mixed, as indicated by the importance of OAI states with different numbers of D-pairs (eq. (7)); nevertheless, the SD subspace accounts for 9%100% of the lowest levels. Using these overlaps, those corresponding to any combinations of the IBM states may be evaluated (eq. (8)). For n = 4, the J = 0 states are reproduced exactly because there are as many (2) in the IBM as in the shell model (table 1). However, the lowest J = 2 and J = 4 levels also lie largely within the SD subspace; for the latter this indicates that the u = 2 component is much less important than those with u = 4, and hence that inclusion of the single-g-boson analogue would not lead to a significant improvement (the state SG actually accounts for only 4.5% of the lowest J = 4 level).

98

P. Hake / OAI mapping

0

i

2 V

Fig. 3. Projections of IBM SU(3) OAI analogues, and - Q f Q shell-model eigenstates, onto a seniority basis for n = 4 (see text, subsect. 3.2).

For n = 6, the lowest two J = 0 and J = 2 levels lie almost entirely within the SD subspace, as do the lowest J = 4,6 and 3 levels. Again, the states omitted in the OAI description appear not to play a significant part in the lowest eigenstates (the first .I = 4 level contains only 0.9% of SSG, the first f = 6 level only 3.5% of SSI, and 4.6% of states with u = 4). This apparent correlation of the states important for a description of the lowest Q - Q eigenstates and those generated by the OAI procedure is emphasised in figs. 3 and 4, which show the projections of the OAI analogues of the most favoured SU(3) IBM states (eq. (8)), and of the lowest eigenfunctions of the hamiltonian (eq. (9)), onto a basis partially defined using seniority (eq. (4), table 1). Where a given seniority does not appear in the SD construction, for instance no components with u = 2 or 4 occur for .I = 6, such states are virtually absent from the lowest eigenstates. We also note that the particular combination of states of a given seniority which enters into these levels is very close to that obtained by taking products of D-pairs. Squared overlaps, expressed as percentages, corresponding to IBM states defined by the SU(3) and O(6) 3 O(5) subgroups (eqs. (2), (8)) are compared for n = 4 and n = 6 in tables 4 and 5, respectively. For n = 4, the f = 2 levels are described rather better by SU(3), the others displaying little or no difference. For y1= 6, the SU(3)

P. Hake / OAI mapping 1

99

J=6

1

SM

J=4 SM

01 1

1-_&+__&, , , I

1

I

I

I

,

i

II

OAI

0

1

J=2

0I 1

SM

I+&_,, OAI

0 1

0 1

I? /? J=O

SM

OAI 0 I

,\

,

0

2

/?

e

i V

Fig. 4. Projections

of IBM W(3)

OAI analogues, and - Q. Q shell-model basis for n = 6 (see text, subsect. 3.2).

eigenstates,

onto a seniority

symmetry seems more appropriate to the J = 2 [where the two states occurring for (22) are distinguished by the K-label 19), 0’ denoting the state orthogonal to that with K = 2, which has approximately K = 0] and J = 4 levels, while O(6) appears better for those with J = 0.

4. Mapping of the hamiltonian To elevate the OAI mapping from being a means of understanding the structure of low-lying states to one enabling calculations to be performed, it is necessary to construct operators from their shell-model equivalents, using if necessary a systematic procedure for renormalisation to compensate for the intrusion of non-SD

loo

P. Hake / OA I mapping TABLE 4

Percentage W(3)

occupancies of OAI analogues of IBM states classified according to the (labelled by (Xp)) and O(6) > O(5) (labelled by UT) group chains in eigenstates of the four-particle system Level number

J

0

2

4

1

2

(40) (02)

98.6 1.4

1.4 98.6

20 00

96.4 3.6

3.6 96.4

(40) (02)

95.9 0.0

21 22

3

4

0.4 85.0

3.1 13.1

0.6 1.9

14.9 21.0

14.5 70.9

10.6 5.6

0.0 2.5

(40)

84.4

0.3

8.4

6.9

0.0

22

84.4

0.3

8.4

6.9

0.0

5

components. Here we shall not consider such modifications, concentrating only on information which may be extracted from the SD subspace. The spectrum resulting from diagonalising the matrices of the hamiltonian (eq. (9)) in the two-SD-pair states (eq. (7)) constructed above (subsect. 3.2, table 2) is shown in fig. lb. The energies of the J = 0 states are reproduced exactly because of the equivalence of the full and SD spaces discussed above. However, this unrenormalised hamiltonian leads to a factor-of-two error in the ground-band excitation energies; this happens in spite of the accuracy with which the wave functions are constructed because of the large ratio of inter-“band” to intra-“band” energies displayed by this particular case (the small admixtures of excited levels carry a relatively large energy contribution). Fig. lc shows the spectrum corresponding to the adoption of a quadrupolequadrupole boson hamiltonian (eq. (10)): H IBM

=

- KQIBM

’ QIBM

3

00)

where

Q IBM= (s+d+d+s) +X(d+dy2’, and the values of K and x are set by the matrix elements (SSl H( SS) and (SD(HISD). This procedure has great simplicity in that products of D-pairs need not be constructed; however, the spectrum shows large deviations from both the exact and SD calculations.

P. Hake / OAI mapping

101

TABLE 5 Percentage occupancies of OAI analogues of IBM states classified according to the W(3) (labelled by (Xp)) and O(6) 3 O(5) (labelled by UT) group chains in eigenstates of the six-particle system Level number

J

1

2

3

4

(22) (00)

81.6 16.3 1.9

6.0 56.1 37.0

1.4 22.1 36.1

5.0 5.5 25.0

30 33 00

93.1 5.2 0.9

3.5 81.6 8.0

0.1 2.1 62.2

2.1 4.5 28.9

(60) (22)2 (22)O’

92.0 0.9 6.8

1.7 91.1 3.0

0.3 1.6 15.8

1.4 3.8 30.4

0.3 0.8 2.3

31 32 11

82.4 17.2 0.0

17.1 78.5 0.3

0.0 0.6 17.1

0.2 1.1 34.3

0.3 0.2 2.9

3

(22)

84.4

13.3

0.0

1.5

0.8

33

84.4

13.3

0.0

1.5

0.8

4

(60) (22)

95.1 1.2

0.3 58.5

0.6 0.1

0.6 25.9

0.1 3.5

32 33

19.6 17.3

12.2 46.1

0.1 0.4

3.8 22.1

0.5 3.2

(60)

89.0

1.7

2.1

2.8

0.7

33

89.0

1.7

2.1

2.8

0.7

0

2

6

(60)

5

The spectrum obtained using the ansatz suggested by Faessler and Morrison12) (es. (W),

-Q.Q+

-QON.QOAI~

(11)

where QoAI is the (N-dependent) zeroth-order image of the shell-model quadrupole operator Q [ref.2)], is shown in fig. Id. The ground-band excitation energies show errors of a factor of ten, as has been noted 12), but it is now clear that this is not due to any inadequacy of the SD subspace (subsect. 3.2, table 2) nor to the zeroth-order mapping2) (fig. lb), but rather is introduced by the ansatz eq. (11). The SD-space spectrum for six particles is displayed in fig. 2b, and reproduces well the results of the full shell-model calculation. A boson hamiltonian defined to reproduce the corresponding matrix elements exactly would now, at least in principle, require three-boson operators.

102

P. H&e / OA I mapping

To test the zeroth-order approximation, an IBM hamiltonian containing only oneand two-boson terms can be constructed to reproduce the matrix elements between SD-space states having u Q 4; thus it is not necessary to consider products of three D-pairs. The responding spectrum is shown in fig. 2c, and clearly demonstrates that the zeroth-order appro~mation generally introduces only small errors. The spectra obtained by using the restricted hamiltonians of eqs. (9) and (11) are displayed in figs. 2d and 2e, respectively, and, as above, show large deviations from the exact, SD, and zeroth-order calculations. 5. Conclusion

The OAI state mapping procedure has been used to construct an SD-pair subspace for a j = .ai”shell, and overlaps with the eigenstates of H = - Q. Q, which leads to strong breaking of seniority, have been evaluated. No IBM states are spurious. Moreover, the lowest shell-model levels lie almost entirely within the SD subspace; for this restricted set of levels there is little need for explicit G-pairing. In addition, IBM wave functions classified by the SU(3) subgroup, in particular all those of the most favoured r~resentation (2N 0), are mapped onto approximate eigenstates of the shell-model system, which has of course no SU(3) symmetry itself. Thus it appears that an approximately conserved SU(3) symmetry can be defined, in terms of S- and D-pairs, for such “rotational” states of a j-shell; the effect on this property of including additional pairs, for j sufficiently large that none of the new states are spurious, is being investigated. Previous studies of the SD-pair content of “rotational” wave functions have yielded rather lower overlaps, typically 60%-80%. However, in contrast to the present work where states with good angular momentum are constructed exactly, these have considered intrinsic shell-model states involving all levels of the ground band, many of which have higher angular momenta than those contained in the IBM and necessarily require G, and even higher-rank, pairs. Various procedures for mapping the hamiltonian have been compared. Explicit evaluation of all SD-space matrix elements, and of those involving states with seniority 4 or less (the zeroth-order appro~mation)~ both lead to spectra whose lowest levels show only small deviations from those of the exact calculation. Further approximations, involving the restriction to a quadrupole-quadrnpole boson hamiltonian whose parameters are determined by matrix elements involving states with seniority 2 or less, introduced significant errors [however, this procedure has been shown to work well when mapping a proton-neutron quadrupole force’)]. The SD subspace generated by the OAI state mapping procedure is adequate for a description of the lowest j = y shell eigenstates arising from a quad~pole-quadspole hamiltonian, in addition to those with good seniority, and so appears to be capable of providing a microscopic justification for in both cases the IBM.

P. Hake / OAI mapping

103

The author would like to thank D.H. Feng for drawing his attention to the work of Faessler and Morrison, A. Etchegoyen for demonstrating the use of the OXBASH shell-model program, and M. Vallieres for suggestions concerning the manuscript. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

A. Arima and F. Iachello, Ann. Rev. Nucl. Part. Sci. 31 (1981) 75 T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) 1 0. Scholten, Phys. Rev. C28 (1983) 1783 A. van Egmond, K. Allaart and G. Bonsignori, Nucl. Phys. A436 (1985) 458 B.R. Barrett, S. Pittel and P.D. Duval, Nucl. Phys. A3% (1983) 267~ D. Bonatsos and A. Klein, Phys. Rev. C31 (1985) 992 A. Arima, Prop. Part. Nucl. Phys. 9 (1983) 51 A. Faessler, Nucl. Phys. .43% (1983) 291~ T.D. Cohen, Nucl. Phys. A434 (1985) 165 P. Halse, Phys. Lett. 156B (1985) 1 D.J. Rowe, Nucl. Phys. A347 (1980) 409 A. Faessler and I. Morrison, Nucl. Phys. A423 (1984) 320 CL. Wu, W.X. Tang and D.H. Feng, Phys. Lett. 155B (1985) 208 J.A. Evans, J.P. Elliott and S. Szpikowski, Nucl. Phys. A435 (1985) 31’7 B.H. Flowers, Proc. Roy. Sot. A212 (1952) 245 P.H. Butler, Appendix of tables, in B.G. Wyboume, Symmeny principles and atomic spectroscopy (Wiley, New York, 1970) 17) R.D. Ratna Raju, Phys. Rev. C23 (1981) 518 18) OXBASH-MSU, The Oxford-Buenos Aires-Mich. State Univ. Shell Model Code: A. Etchegoyen, W.D.M. Rae and N.S. Godwin [MSU version: B.A. Brown] 19) J.P. Elliott, Proc. Roy. Sot. A245 (1958) 562