Exact pairing calculations in a large Nilsson-model basis: comparison with BCS and Lipkin-Nogami methods

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 602 (1996) 21-40

Exact pairing calculations in a large Nilsson-model basis: comparison with BCS and Lipkin-Nogami methods O. Burglin a, N. Rowley a,b a Department of Physics, University of Surrey, Guildford, Surrey GU2 5XH, UK b Department of Physics, University of Manchester, Manchester M13 9PL, UK Received 9 October 1995; revised 9 February 1996

Abstract Full many-body calculations of the spectra of heavy deformed nuclei become prohibitively large and one is forced to employ approximate mean-field methods. The deformation is accounted for by a deformed single-particle potential (e.g. Nilsson model), the rotation by a one-body cranking term and the pairing forces by the introduction of non-interacting quasiparticles. Some properties, however, [e.g. the moments of inertia of multi-quasiparticle (MQP) bands] are rather sensitive to details of the pairing correlations. In this case it would be preferable to account for the pairing exactly and evaluate the moments of inertia in a configuration-mixingcalculation or a Harris-type perturbation theory. We show that by exploiting the symmetries of the general state-dependent pairing force of the form ~ Gu~a~at_a~a~ the pairing correlations in the band heads of these /x states, and any states with which they will strongly mix, can be calculated exactly in a large Nilsson-model basis. For 176Hf, where many MQP bands are known and where the neutron and proton level densities are relatively low, a detailed comparison is made with the results of the BCS and Lipkin-Nogami approximations. While a reasonably good representation of the spectra can be obtained with a renormalisation of the pairing strength, there remain significant discrepancies in the single-particle occupations produced by these methods. PACS: 21.60.Ev; 21.60.Jz; 27.70.+q Keywords: Pairing;Exact calculations;BCS; Lipkin-Nogami;Deformed nuclei

1. Introduction The simplest and perhaps most widely used treatment of pairing correlations in nuclei is the BCS approximation [ 1 ]. The aim of this method is to regain the simplicity of independent-particle models by applying a mean-field approximation to the pairing force 0375-9474/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved PII S0375-9474(96) 00096-6

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O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

through the introduction of non-interacting quasiparticles. The BCS model is known to produce good results if the level density close to the Fermi energy is large enough for many single-particle states to participate in the pairing correlations. This condition is, however, not always well fulfilled in nuclei. This is especially true for deformed systems, where the degeneracy of the spherical shells is broken by the deformed mean field. We have, for example, recently shown that in the mass-180 region, both the neutron and proton pairing correlations may prematurely collapse in many of the corresponding twoquasiparticle excited states [2]. In other words, the level densities are already sufficiently low that the blocking of two levels near to the Fermi energy completely destroys the correlations, yielding a BCS pair gap d which is zero in many of these excited states. An interesting feature of this mass region is that many multi-quasiparticle (MQP) states have been identified experimentally. Many of these are high-K isomeric states which may be strongly populated in heavy-ion-xn reactions. Each of these high-K isomers is the lowest member of a different rotational band, and the moments of inertia J of these bands have been observed [3] to vary according to which (and how many) single-particle levels are blocked in the band-head configuration. It has long been known that pairing correlations can produce a significant reduction in the moment of inertia from its rigid-body value ~rigid [4]. Since the level density in these nuclei is such that A = 0 in the first two-quasiparticle state, it is clear that this theory will not be able to describe the differential changes in ,7" which take place with increasing quasiparticle number. It is, therefore, essential to calculate the pairing correlations in a more exact fashion to describe this phenomenon. The aim of this paper is to show that by exploiting simple symmetries of the general two-body pairing interaction, an exact diagonalisation of this force, in a large Nilsson-model basis, is feasible. We confine ourselves in the present paper mainly to the calculation of the vacuum and the MQP band heads of the above problem and to a comparison of our results with the standard BCS and Lipkin-Nogami models [5-9]. We also show, however, that an exact diagonalisation for any specified MQP state is straightforward and thus that we can generate a suitable many-particle basis in which to perform a configuration-mixing calculation of the effects of rotations on the system, from which f f can be extracted. We shall also confine our exact calculations to the usual monopole form of the pairing force, though the symmetries that we shall discuss in Section 3 are equally valid for the more general state-dependent pairing interaction of Eq. (1). Investigations of state-dependent effects and configuration-mixing are in progress [ 10].

2. The BCS and Lipkin-Nogami approximations In order to highlight some of the problems of an approximate quasiparticle approach we briefly outline here the BCS and Lipkin-Nogami methods. Consider first a timereversal-invariant number-conserving Hamiltonian /~ containing a single-particle term and a state-dependent pairing interaction. In a model space where the single-particle term is diagonal, we may write

O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

~ f/sp q" /~pair = Z

8v

(at~av ÷ a~a~) - Z

v>0

G~z~ata~a~av,

23 (1)

p.,v>0

where the index v stands for all quantum numbers specifying a given single-particle orbital and I~) denotes the time-reversed partner of Iv). In the present paper,/-'/describes either a neutron or a proton system separately and the single-particle states are taken to be Nilsson-model states, i.e. v = [ N, n3, A] 12. The standard form of the BCS ground-state wave function is ]BCS) --- I ) = I-I (u~ + v~at~a~) 10),

(2)

P

where 10) denotes the particle vacuum and v~ (u~) are the amplitudes for the corresponding single-particle states Iv) and their time-reversed partners 1~) to be occupied (unoccupied). In the monopole pairing approximation, all the two-body matrix elements G~, are taken to be equal to a single G. In this case, the pair field which replaces the full interaction is

^t = -- A ~f',(ata t "+"a~av ) [?/pair ---4Hpair v>0

(3)

in which the pair gap A is given by

A=G( I ~--~a~a~l ) =G~-~u~vv. v>0

(4)

v>0

The Hamiltonian of the system is then replaced by

I)'= Z ( e v

- A) (a~a~ +a~a~) - Z

v>0

A(a~a~ +a~a~).

(5)

v>0

Since the pair field does not conserve particle number it is necessary to introduce the Fermi energy ,~ which must be varied in order to obtain the correct average particle number. A Hamiltonian of the simple form (5), with the addition of a one-body cranking term - ~ ( / z [ j x Iv)a~a~ is frequently used to describe the rotational properties (especially band crossings) of deformed heavy nuclei [11]. For the Hamiltonian (5), the quasiparticles are defined by the Bogolyubov-Valatin translormation [12], i.e. through the quasiparticle creation operators

ce~ = u,at~ - v~a~,

ce~ = u~a~ + v~a~

(6)

and their Hermitian conjugate annihilation operators. The wave function (2) may be shown to be the quasiparticle vacuum (i.e. a~ I ) = 0) and its energy may be minimised with respect to the variational parameters u, v. The variational condition is such that terms of the form oztozt v /z and o~,o~u vanish from H', yielding an effective quasiparticle Hamiltonian 12/qp----" EBCS -t- Z v>0

Ev(°t~o'v -1:-Ol;Ol,j),

(7)

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O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

where EBCS is the energy of the quasiparticle vacuum and E~ are the usual singlequasiparticle energies (see e.g. Refs. [ 13,14] for details). Excited states of the system are given by acting on I ) with the appropriate quasiparticle creation operators. It is well known [ 13,14] that the above procedure works well if all the single-particle states are degenerate. For example, for N particles in 12 degenerate levels (i.e. 212 degenerate single-particle states), the ground-state energy is given to order N/122, and the energy of the first two-quasiparticle excited state is given exactly. Kerman et al. [15] have studied the problem of two non-degenerate spherical shells and shown that results are reliable to around 500 keV for the ground state with a typical error of around 200 keV for the excitation energy of the first two-quasiparticle state. Moreover, they showed that by projecting out the component of the vacuum wave function (2) which has the appropriate particle number, they obtain an overlap of around 98% with the exact ground state. Despite these successes of the BCS theory, problems arise when one tries to go beyond such problems. In particular, in deformed systems, the single-particle levels are only doubly degenerate (time-reversal symmetry) and it is possible to have regions of rather low level density. In such a situation, there may be no non-trivial ("superconducting") solution of the BCS equations, i.e. solutions exist only with A = 0. This means (see Eq. (5)) that the pairing correlations are then completely ignored. This situation becomes particularly acute when studying MQP states. If the level density is low, the BCS theory gives poor energies for states with quasiparticle number greater than 2, and one may correct for this by blocking [ 16,17] the already occupied levels. This clearly leads to a further reduction of level density and a greater likelihood of pairing collapse. The Lipkin-Nogami (LN) approach is a method of avoiding this collapse [5-9]. It is still very much in the spirit of the BCS method in that it simply introduces an extra constraint into the number non-conserving Hamiltonian (5). Thus in addition to the term -A/V, where N is the particle-number operator, one also has the term _,~t/Q2. This approximation always gives configuration mixing no matter how weak the pairing strength and provides a more reliable description of the correlations without losing the simplicity underlying the quasiparticle picture. It should, however, be emphasized (see Section 5) that the LN method does not suppress particle-number fluctuations. Indeed it enhances them - this being the reason that the pairing correlations may survive when they might otherwise vanish. Only for spherical multi-shell problems [5,15,18-20] have exact calculations been carried out (using the quasi-spin algebra [21]) and compared with the above approximation schemes. We wish, in the present paper, to extend such comparisons to calculations in a large deformed shell-model space. Our principal motivation for this comparison is that it is in such systems that low level densities may occur. The true (physical) reduction of pairing correlations is accelerated by the Coriolis forces present when such deformed systems rotate, and our ultimate goal is to use the exact manyparticle eigenstates of (1) in a configuration-mixing calculation of the moments of inertia of such systems. Such calculations could also be performed to take account of residual neutron-proton interactions.

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25

Of course a further way of overcoming the premature pairing collapse is to use a particle-number-projected theory. The type of projection performed by Kerman et al. [15] is done after the variational problem is solved. Thus, if there is a collapse of pairing, this projection method does not remedy it. Indeed if A = 0, the wave function one obtains already has good particle number. To remedy this problem one should project good particle number from Eq. (2) before variation [14,22-24]. This has the advantage that one needs to make no approximations to the original Hamiltonian of the system, i.e. one may minimise the ground-state energy in the exact many-particle space using the exact Hamiltonian (1). The only approximation in this method comes from the fact that the full many-body space containing

c~" =

(2~)! N! ( 2 n - N)! '

(8)

distinct configurations (SIater determinants), is described using only .(2 variational parameters, so the variation is significantly restricted. The full diagonalisations which we shall perform here contain, of course, the full space of C2a many-particle states and an equal number of independent amplitudes. The emphasis throughout this paper will be on a comparison of our exact results with the corresponding results of the quasiparticle theories, i.e. BCS and Lipkin-Nogami, which are frequently used in studies of heavy deformed systems. In addition to the spectra that we shall obtain, we shall also provide a comparison of the relevant wave functions. However, since the exact wave functions have good N and the approximate ones do not, this will be done at the level of the respective occupations of the different single-particle orbitals. This is a meaningful comparison since, within the quasiparticle theories, one may calculate matrix elements of one-body operators in terms of the corresponding single-particle matrix elements and the u, v. For example the matrix element of a single-particle operator Q between a two-quasiparticle state and the BCS vacuum may be written [ 13]

( I Q d ~ l )=Q~(v~u~ +~uuv~),

(9)

where Qu~ is a single-particle matrix element and r = + l , depending on whether Q is even or odd under time reversal. It is, therefore, essential to obtain reasonable values of u, u in the quasiparticle theories if such matrix elements are to be reproduced reliably. It is of course partly the presence of r = - 1 for the cranking term -Wjx which is responsible for the reduction of J in the ground-state rotational band. Similar considerations apply to the decay of high-K isomers [25], where a Coriolis matrix element is involved for each degree of K-forbiddeness. In what follows, we discuss in Section 3 the symmetry of the general pairing force of Eq. ( 1 ) which permits an exact diagonalisation of the problem in a realistic singleparticle space. In Section 4 we discuss a set of exact eigenstates of the problem which have no obvious counterpart in the approximate theories, i.e. configurations in which all the single-particle states and their time-reversed partners are equally occupied but the particles are nonetheless not fully paired. In Section 5 we discuss how the interaction

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O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

G may be renormalised in such a way that the approximate calculations yield good energies for a large set of excited states. Section 6 concerns the dependence of the effective interaction strength on the size of the single-particle basis employed. Finally we present a summary of the problem in Section 7.

3. Symmetries of the general pairing force and basis sizes The total number of many-particle states for a system of N nucleons in s2 doubly degenerate levels is given by Eq. (8), and for large systems the exact treatment of the problem becomes numerically intractable. For example, a system where N = S2 = 18 leads to 9075 135300 many-particle basis states. Clearly it is impossible to handle matrices of this size and one has to exploit the disconnected subspaces of the full Hilbert space. To achieve this, consider the quasi-spin [21] raising operators

s + - ata t

~, = 1

, s2.

(10)

These operators each create two particles in time-reversed orbits. We refer to such particles as accompanied. For even N, the basis states may be written /N/2

,+o>=

ins:)0> s .+

=

(11)

(12)

a <,, t a* I0),

l~lx =#f=O"1,0' 2

and in general

1/rV..... >= ( (Nl_~)/2s+)

10).

(13)

This latter wave function has N - V accompanied particles and V (even) unaccompanied particles which occupy the levels al . . . . . av. (These particles may be either in the state ]c~) or ]~).) For an odd-particle-number system, the above wave functions each contain an extra single-particle creation operator a t and V must be odd. Clearly V is a good quantum number for the eigenstates of the Hamiltonian (1). In other words, /2/commutes with the operator t t ~'= Z ( a ~ a ~ + a~a~) - 2 Za~a~a~a~

~'>0

v>0

(14)

which gives the number of unaccompanied particles, i.e. it is just the number of particles minus the number of accompanied particles. It should not be confused with the seniority v [26] which, as for the total quasi-spin, is not a good quantum number unless the singleparticle states are all degenerate and Gu~ = G. The seniority v denotes the number of unpaired particles. Note, for example, that in a degenerate single-j shell containing N

O. Burglin, N. R o w l e y / N u c l e a r Physics A 602 (1996) 2 1 - 4 0

27

(even) particles in s2 = j + 1/2 levels, there a r e Cff/2fully accompanied (V = 0) states but only one fully paired (v = 0) state. The Hamiltonian (1) cannot scatter an unaccompanied particle from the state it occupies and, due to the Pauli principle, the corresponding level is not available for pair correlations. It corresponds to a blocked level and may be excluded from all further considerations. Also, within a given V subspace, r/ has vanishing matrix elements between wave functions where the unaccompanied particles occupy different singleparticle states. It is clear from the above results that all the eigenstates of the Hamiltonian (1) can be expressed as a number of unaccompanied particles coupled to a "core" of fully accompanied ones. The core wave functions may be obtained by diagonalising /t/ in a space in which the unaccompanied particles and the particular states which they occupy have been removed. This is a very similar structure to the usual blocking calculations. The important difference is that the core states will be calculated exactly. The core vacuum will be analogous to the BCS one but will have good particle number. Core excited states although fully accompanied are not fully paired. Let us examine the largest possible system for a fixed number of levels: a half-filled system N = ~ = 2n. According to the above symmetries, the corresponding model space, spanned by C4~] many-particle wave functions, may be split into disconnected subspaces with V = 2m. There are C22'~ ways of choosing the blocked levels and since each is doubly degenerate, '~ 2,,c. 2, ways of choosing the blocked single-particle states. The remaining particles are fully accompanied and thus give a core space of dimension C2.-2m The block decomposition of/2/is, therefore, summarized by the mathematical 11--171

"

identity

C4, 2n z

)~

2. t" 2n-2m 2 2m t" "~ 2m " - ' . - - I l l "

(15)

This means that the largest block which one has to diagonalise is the single block with V = 0 and dimensionality C~". Thus, for n = 9, the exact treatment of/-~/, which requires a basis including 9 075 135 300 many-particle states, may be performed in significantly smaller subspaces, the largest having a dimension of 48 620. If one is interested only in the lowest few eigenstates in each block, as is almost always the case, matrices with such dimensions may be readily treated using the Lanczos method. This method is employed for all the exact calculations in the rest of this paper. A sufficient number of iterations have been used to obtain both the wave functions and the eigenvalues of the levels displayed to a precision better than 10 -m. The exact calculations are carried out in the model space outlined above, i.e. 36 active single-particle states containing 18 valence nucleons. Extending to a larger number of active levels is difficult for obvious practical reasons. Our calculations will, however, yield realistic results for nuclei in the mass-180 region, where the level densities of both neutrons and protons near the corresponding Fermi energies are relatively low. This means that only a limited number of levels play a determining role in the structure of

28

O. Burglin, N. Rowley / Nuclear Physics A 602 (1996) 21-40

6

[642] 3/2 [505] 9/2 [651] 1/2

5

[615] 11/2 [503] 7/2 [512] 3/2

4

[510] 1/2 [624] 9/2 [514] 7/2

3

2

1

[512] 5/2 [521] 1/2 [633] 7/2 [642] 5/2 [523] 5/2 .......[651] 3/2 [505] 11/2

0

"111[521] 3/2 ' [660] 1/2

Fig. I. The Nilsson levels for neutron numbers around 104 which have been used in all the calculations presented in the text. The energies are taken from Ref. [2] where they were slightly adjusted from their Nilsson-modelvalues in order to fit experimentaldata in the region of 176Hf, the yrast and near-yrast states. We shall study in particular, neutron excitations in 176Hf using the neutron levels of Ref. [2]. The corresponding Nilsson orbitals are shown in Fig. 1. Each V-configuration gives rise to 2 v states which may be distinguished by their projections K = :tzS21 4 - . . . 4S2v of the total spin I along the symmetry axis. All of these states (for each distinct V-multiplet) are, of course, degenerate in the present model and we simply label the resulting multiplet by its most yrast band-head spin I = K =/2j 4-. • .4-J2v. (The inclusion of spin-dependent residual interactions will split the levels as discussed in Ref. [2],

O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

Table 1 Multi-quasineutron configurations in

29

176Hf

V

K ~"

otl @... @Otv

2 2 2 2 2 2 2 2 2 2 4 4 4

6+ a 4+ a 7- a 753+ 8+ 1+ 3+ 811- b 14+ b 12+ h

[51215/2@ [51417/2 [52111/2@ 151417/2 [63317/2@ [51417/2 [51215/2@[62419/2 ]52111/2@ [624]9/2 1 5 1 2 1 5 / 2 @[51011/2 [633]7/2@ [624]9/2 [52111/2@ [51011/2 [52111/2@ [51215/2 [51417/2@ [624]9/2 [52111/2@ [51215/2@ [51417/2@ [624]9/2 [633]7/2@ [51215/2@ [51417/2@ [624]9/2 [633]7/2@ [52111/2@ [51417/2@ [624]9/2

a Observed states. h These configurations form part of the structure of observed six-quasiparticle states in conjunction with two-quasiproton excitations (see Ref. [2] ). though the states IK) and IR) = exp(iTra~)lK ) remain degenerate in the absence of rotations.) The relevant blocked levels for the configurations under investigation are listed in Table 1. Although all o f the theoretical analysis that we have presented in this section is valid for any values o f the Gu~, we shall in the rest of this paper, perform calculations using the m o n o p o l e pairing interaction. An example of our exact results is given in Fig. 2 which shows energy levels arising from the lowest V = 0 states as well as from the three lowest V = 2 and V = 4 configurations. The pairing strength G = 0.181 MeV was chosen to reproduce the lowest experimental V = 2 state in 176Hf ( K ~" = 6+). The spectrum exhibits the characteristic energy gaps for an even-N system with a pairing force. The ground state belongs to the V = 0 subspace. This is to be expected since in this space all o f the nucleons can interact via the pairing force. The lowest excitation energy in each V-configuration increases with V, since this determines the number Of blocked levels. As V increases, there are less levels available for the pairing correlations and this leads to a decrease in the pairing energy along with an increase of the single-particle energy contributions. It should be emphasized though that the pairing correlations, although lower than those for V = 0, do not vanish. Indeed, one finds (/'?'/pair)= 5.013, 3.538 and 2.803 MeV for the 0 +, 6 + and 1 1 - states (i.e. the lowest V = 0, 2 , 4 states) respectively. Note also that the three observed [2] two-quasiparticle high-K isomers K ~" = 6 +, 4 +, 7 - , are the only V = 2 states which lie below the first excited V = 0, K ~ = 0 + state. In other words, the states which may loosely be referred to as two-quasiparticle states or seniority-2 states include states from the V = 0 block as well as from V = 2. For V = 0, however, one must have K = 0 and so these states will be less important experimentally if the nucleus is populated in a heavy-ion-xn reaction, where the principal intensity will find its way

O. Burglin, N. Rowley/Nuclear Physics A 602 (1996)21-40

30

7 6 5 12+

4 ......

:"-14 + ',\

117

2

:'

......

"JI-

-----4

1 o+

0 V=0

V=2

V=4

Fig. 2. The energy spectrum resulting from an exact diagonalisation of the Hamiltonian (1) with G~,, = G = 0.181 MeV in the single-particle space of Fig. 1. We display the lowest 19 levels for V = 0, 5 levels for each V = 2 configuration shown and 2 for each V = 4 configuration shown. The configurations are given in Table I and are labelled by K 7r (projection of the total spin along the symmetry axis and parity). into the near-yrast states.

4. Excited V = 0 states Our p u r p o s e in this section is to study the structure o f the excited V = 0 states, though we note that similar results are valid for the fully a c c o m p a n i e d excited core states o f any V-configuration. We wish in particular to underline the importance o f an exact treatment o f these states by c o m p a r i n g our results to those obtained with the B C S [ 1] or the L N [ 5 - 9 ] methods. We shall also present results in the b l o c k i n g f o r m a l i s m [ 16,17] for both o f these m e t h o d s ( b l o c k e d B C S ( B B C S ) , blocked L N ( B L N ) ) . All o f our a p p r o x i m a t e calculations were p e r f o r m e d in the same m o d e l space as the exact ones. We should e m p h a s i s e here that, unlike the approximate approaches, the exact treatment leads to proper o r t h o n o r m a l eigensolutions with g o o d particle n u m b e r and makes no prior

O. Burglin, N. Rowley / Nuclear Physics A 602 (1996) 21-40

31

1.0 0.8 eq

0.6 -

'..,

/v

A G.S

0.4 0.2 0 1.0 0.8 eq

0.6

o---o 1 Ex ©------o G . S

0.4 )

0.2 0 0

1

2

3

4

5

6

Energy (MeV) Fig. 3. Occupation probabilities for the ground state and the first excited V = 0 state of Fig. 2 (G LN = G Exact = 0.181 MeV). Filled circles indicate BLN results (blocked level: [51417/2).

assumption about the form of these solutions. Any one of the physical states may be calculated exactly by an appropriate choice of blocked levels and no spurious states exist. In the exact calculations, the lowest state in any given V subspace corresponds to the vacuum of the remaining accompanied particles and the full state is, therefore, completely analogous to one containing V quasiparticles. However, excited states in a given V-configuration are more complex. At best they can be thought of as V + 2, V + 4 . . . . quasiparticle states in which the additional quasiparticles are placed in timereversed orbits. Such states are generally not considered in a quasiparticle theory. Since all the states in the same V-block have the same K, it is clear that the yrast states are the simplest ones corresponding to V quasiparticles. The higher-energy states are, however, entirely physical. In Fig. 3 we show the level occupations v 2 as a function of the single-particle energies for the lowest two states of the V = 0 block. The lower of these is of course the neutron ground state. The exact structure of the first excited V = 0 state is fundamentally different from what is assumed in LN-type calculations in that the [51417/2 level

O. Burglin, N. Rowley /Nuclear Physics A 602 (1996) 21-40

32

'

I

'

I

'

I

'

I

i

I

i

1.0 0.8 eq

v

0.6

T3 Ex

~7--V 2 E x

0.4 0.2 ,

0 0

I

1

,

I

2

,

I

3

,

/ ~ 4

5

6

Energy (MeV) Fig. 4. As for Fig. 3 but for the second and third excited V = 0 states. Note that the enhanced l: 2 for the second excited state occurs in the same single-particle level as for the first excited state (see Fig. 3).

has an enhanced occupation for both the state and its time reverse but is not fully occupied. The excitation energy E~LN = 1.826 MeV is, however, surprisingly close to the exact value E~xact = 1.856 MeV whereas the unblocked method gives E~N = 2.090 MeV. Similar results hold within the BCS framework where E~Bcs = 1.836 MeV and

EBCs = 1.540 MeV. The BCS and LN approximations, due to their variational nature, are not well adapted to determine excited V = 0 states and do not lead to realistic wave functions. The occupations of the first three exact excited states are shown in Figs. 3 and 4. They clearly have no obvious counterparts in a quasiparticle theory. In particular, one can see that the second excited V = 0 state (Fig. 4) has an enhanced occupation for the same single-particle level as the first excited state (Fig. 3). Clearly by blocking this single level, a quasiparticle theory could not yield two states of different energy. Of course, the BCS and LN vacuum wave functions, although linear combinations of states with different N, have occupation probabilities resembling the exact solution, as expected fi'om the variational nature of these methods.

5. E f f e c t i v e interaction

We now turn our attention to the yrast states of the low-lying V = 2 configurations. Their excitation energies, evaluated with the same G-value in all calculations, are shown in Fig. 5. Apart from the BBCS method, all the pair-field calculations lead to higher energies than the exact treatment. The largest errors occur for the unblocked LN formalism (up to 600 keV) which significantly compresses the V = 2 spectrum. The ordering of the exact level is, however, well reproduced, except for the last two states (K 7r = 3 ÷, 8 - ) . These both stem from configurations where the unaccompanied nucleons both lie on

O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

33

2.6 \3 +

2.4 \

1+

', \

2.2

.

~,,.,...

~ ,

8 +

3+

\

2.0 /

1.8

/

~

//

7"

//

/ / ~ /

1.6

4+ 6+

/

1.4 LN

BCS

BLN

BBCS

Exact

Fig. 5. Excitation energies for the 10 lowest V = 2 states. Only the lowest state of each configuration is plotted. The configurations are listed in Table 1. All calculations were performed with the same G = 0.181 MeV.

the same side of the Fermi energy (see Table 1) and come out in the wrong positions especially in the unblocked results. The blocking formalism gives much better results, since it accounts for the large shift in A in such excited states. As expected for a system with low level density, blocking is also important for other levels and generally leads to a lowering of the energy. There exists no a priori method for estimating the pairing strength and one must rely on experimental data by using effective pairing fields, i.e. interactions with phenomenological strengths which fit the data. We show in Fig. 6 the effects of such a procedure. The spectra produced by the BCS, BLN, BBCS and exact calculations become much more similar if the pairing strengths are adjusted to fit the lowest experimental twoquasiparticle state, though the LN formalism still yields a compressed spectrum. Without blocking, the problems related to the 3 + and 8 - state remain. Clearly, the inclusion of blocking effects is crucial for determining states where the unaccompanied nucleons lie all on the same side of the Fermi energy. In Fig. 7, we compare the pair correlations obtained with the above renormalisation. They were estimated by means of the BCS gap defined in Eq. (4) calculated with the appropriate v, u values. These quantities can of course be defined in an exact calculation, where no other natural definition of A exists. The BLN and exact calculations give almost identical results. Despite the fact that the BBCS describes the energies rather well, it exhibits a premature collapse of the correlations. This sets a serious question mark against any attempt to draw, from a correct description of energies, conclusions about corresponding wave functions or matrix elements. Let us now return to the ground state of the system. In order to evaluate the accuracy of the BCS and LN vacuum-occupation probabilities, we show in Fig. 8 their deviations

O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

34

2.6

_

3+

2.4 i~

2.2

,,

1+ 8+ 3+

~i~

5-

2.0

77-

1.8

4+

1.6

6+

1.4 LN

BCS

BLN

BBCS

Exact

Fig. 6. As Fig. 5 but with pairing strengths fitted to the lowest observed V = 2 state ( K 7r = 6+). The pairing strengths used are G LN = 0.145, G BCs = 0.176, G BLN = 0.168, G BI3cs = 0.197 and G Exact -- 0.181 MeV.

from those of the exact solution. For states below the Fermi energy we display the percentage deviation D,, in v 2. For states above the Fermi energy this quantity may become very large, so in this case we display the analogous quantity for u 2. The errors are shown as a function of the pairing strength G. The well known result that below a critical strength (Go ~ 0.16 MeV in this case) the BCS theory fails to give nontrivial solutions is readily seen. On the other hand, this does not occur in the LN

'

I

'

I

'

I

'

[

i

I

f

1.0 0.8 e.q

,"r "r 3 Ex ~--7 2 Ex

0.6

0.4 -

0.2 g. i

0 0

I

1

i

2

3

4

5

Energy (MeV) Fig. 7. The pairing gap (4) for the states involved in Fig. 6.

6

35

O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

20

-

~

L 15

= =

.

=[51215/2 " [521] 1/2

t" ¢

~ \

,L

~. [510] 1/2

=

= [624] 9/2

=

=[51417/2

/

~ , 10

¢~

5

-5 0

0.1

0.2

0.3

0.4 0

0.1

0.2

0.3

0.4

G (MeV)

Fig. 8. Accuracy of the ground-state wave function in the BCS and LN methods. The quantity plotted is D x = 100 X [ (x~,Exact)2 _ Xuj/~X vl2/¢ Exact,~2with x = v for the first three levels below the Fermi energy and x = u for the first three levels above. Open (filled) symbols indicate BCS (LN) results. method which systematically overestimates the smearing out of the Fermi surface. An interesting result is that in the regime G > Gc, where the BCS method does lead to superconducting solutions, it gives significantly better results than the LN calculations. This is particularly obvious in the range of realistic G-values (G = 0.16-0.20 MeV). Note also that, independent of G, D,, and D, become larger as the levels get closer to the Fermi energy. In the strong pairing limit, the discrepancies tend to vanish but, as shown in Fig. 9, the uncertainty in the particle number AN = 2

u~G

(16)

then becomes quite large. An important conclusion is, therefore, that for realistic pairing strengths, neither the BCS nor the LN method describe the vacuum wave function very accurately even at the level of the occupation probabilities. It is also important to realise that although the LN formalism provides a method of avoiding the collapse of pairing correlations, it does not suppress particle-number fluctuations. Indeed it enhances them. For both BCS and LN methods, particle-number fluctuations are a necessary condition for obtaining superconducting solutions. While expressions such as (9) for the corresponding matrix elements accept this fact, it is clear that significant errors in the u, v will still lead to poor results. This is especially true when the uv factors are of the opposite sign. In order to eliminate number fluctuations one could, of course, perform particle-number projection before variation [ 14,22-24]. However, it is clear that to describe properties such as

O. Burglin, N. Rowley / Nuclear Physics A 602 (1996) 21-40

36

4

~---eLN

3

'

'

~

'

I

'

'

~

,

,

'

I

'

~

J

~

,

,

,

'

'

'

'

I

'I

1

0

,

,

,

,

0 Fig.

9. Particle-number

fluctuations

I

,

0.1 for

the

,

I

,

,

,

,

0.2 0.3 G (MeV) BCS

and

LN

ground-state

f

[

0.4 wave

functions

(N

= s2 =

18).

moments of inertia, many intermediate states may be required. The exact calculations we have presented here may be readily extended to such states.

6. Basis size The question arises of how many levels one should include in the model space for such calculations. Of course, as one increases the number of levels s2, the phenomenological pairing strengths must be readjusted. We have performed our calculations for various basis sizes, always adjusting G to reproduce the lowest experimental excited state. The resulting variations in the energies of the other V = 2 states are shown in Figs. 10 and 11. While the lower excited states remain almost unchanged, we note that, for higher states, the energies obtained from the exact treatment are much more stable than those resulting from the BLN calculations. The corresponding G-values are plotted in Fig. 12. They have roughly a l/s2 dependence. This can be understood from the degenerate j-shell results, where the excitation energy of the lowest V = 2 configuration is simply GS2. Thus, if we had kept the same G-value irrespective of the basis size, energies as well as pairing gaps would have diverged. Even with non-degenerate levels, this behaviour clearly persists to some extent. The ideal situation would be to include all the Nilsson levels in the exact calculations. This, however, is neither possible for practical reasons nor does it seem necessary on

O. Burglin, N. Rowley / Nuclear Physics A 602 (1996) 21-40

37

2.6 \ \ \,

/1+_ 3-t-

2.4 2.2

3+

\8+ 577-

2.0 m

1.8

4+ 1.6 1.4

6+ 6 lev.

12 lev.

18 lev.

64 lev.

Fig. 10. BLN results for the ten lowest V = 2 configurations are shown for single-particle spaces of different size, i,e. spaces including respectively 6, 12, 18 and 64 single-particle levels. The corresponding pairing strengths are shown in Fig. 12.

physical grounds, since pairing correlations are generally supposed to be restricted to the neighborhood of the Fermi energy. We believe that this behaviour arises from the use of a state-independent G which almost certainly introduces too much collectivity [10]. If one took more realistic two-body matrix elements Gu~, coming for example from a short-range force, the matrix elements for scattering from levels close to the

2.6

j8~3 +

2.4

jl + . . . . .

2.2

~

-~8+

~3 + 5-

2.0

7 7-

1.8

4+ 1.6

1.4

6+

6 lev.

12 lev.

18 lev.

Fig. 1 I. As Fig. 10 but for the exact results. Note the greater stability of the higher-lying levels in this exact calculation (cf. Fig. 10).

38

O. Burglin, N. R o w l e y / N u c l e a r Physics A 602 (1996) 2 1 - 4 0

0.4 []x

• - - • BLN ,x

o --o

\

eA~.... " ~ .

zx

BBCS

A Exact

0.3

"''l_

0.2

-12-_ --0

,1

i

i

i

i

i

i

i

6

8

10

12

14

16

18

Number

of levels

Fig. 12. Pairing strengths, adjusted to the lowest V = 2 state ( K ~" = 6 + ) , as a function o f the n u m b e r o f single-particle levels included in the model space.

Fermi energy to states a long way from it would be significantly attenuated, due to the different radial wave functions involved. In this case the calculations should converge more rapidly as a function of the basis size. On the other hand, if one really did need to take account of distant levels, this would pose problems since the basic assumptions underlying the shell model would then be thrown back into question, as would the Nilsson-Strutinsky technique [27] which assumes that the nuclear energy comprises a smooth macroscopic term plus a shell correction arising only from those levels close to the Fermi energy.

7. Summary By exploiting the symmetries of the general pairing force ~ Gu~a~at-a~a~, we have /z pedbrmed the exact solution of the corresponding many-body problem. For an even number N of particles in s2 doubly degenerate levels the maximum matrix size which one has to deal with i s CN/Z,'Q as opposed to the full Hilbert space of dimension C 2a. This property does not require additional degeneracies in the single-particle spectrum. This large reduction of basis size permits the exact evaluation of many-particle configurations and allows a stringent test of the BCS and Lipkin-Nogami approximations. The method presented in this paper is capable of obtaining exact excited states in a fashion equally simple to that for the vacuum. Of course the basis used is on the limit of our computational capabilities and we have, therefore, investigated the convergence of our results as a function of the basis size. These results have proved to be very stable, although the effective pairing force

O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40

39

must be renormalised as the number of single-particle levels is changed. The same is, however, true of the approximate theories and this poses some questions on the validity of the constant-pairing approximation. We believe that this convergence would be significantly improved with the use of state-dependent pairing. It has been shown that the approximate particle-number-non-conserving theories can yield relatively good spectra if the pairing strength is appropriately renormalised. For some states, however, it is essential to take explicit account of blocking effects. Of course the overlap of the exact wave functions with those of these approximate theories will be relatively small, since the latter contain large components corresponding to N ± 2, N ~ 4, etc. In order to give a more meaningful comparison of the usefulness of such wave functions we have compared the occupation probabilities v2 rather than overlaps. Approximate calculations appear to reproduce even this aspect of the wave function relatively poorly. Not surprisingly, this is especially true for excited states. One- and two-body matrix elements will be explicitly compared elsewhere [ 10]. It is, of course, well known that for spherical nuclei, the particle-number projected BCS with variation after projection works rather well. This may, however, be because of the relatively high level densities one has in spherical systems. It would, therefore, be interesting to compare this method with exact results in the kind of systems studied in the present work, i.e. deformed nuclei with relatively low level densities [ 10]. The major problem with the present method is that it requires the pairwise degeneracy of the Nilsson single-particle levels. This symmetry is, of course, broken by the nuclear rotation, usually represented by a cranking term -WJx. In this case, however, we may use the present formalism to set up a many-particle basis in which one may account for this operator in a configuration-mixing calculation. This technique is being studied through an investigation of the moments of inertia of the rotational bands based on higb-K isomeric states for nuclei in the region of 176Hf [3,28], where the experimental data show an interesting dependence on the degree of blocking.

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O. Burglin, N. Rowley/Nuclear Physics A 602 (1996) 21-40 P. Ring and P. Schuck, The nuclear many-body problem (Springer-Verlag, New York, 1980). A.K. Kerman, R.D. Lawson and M.H. Macfarlane, Phys. Rev. 124 (1961) 162. V.G. Soloviev, Sov. Phys. Dokl. 5 (1961) 778. V.G. Soloviev, Theory of complex nuclei (Pergamon, Oxford, 1976). E Andreozzi, A. Covello and A. Porrino, Phys. Rev. C 21 (1980) 1094. M. Rho and J.O. Rasmussen, Phys. Rev. 135 (1964) B1295. H. Chen, T. Song and D.J. Rowe, Nucl. Phys. A 582 (1995) 181. A.K. Kerman, Ann. Phys. (N.Y.) 12 (1961) 300. K. Dietrich, H.J. Mang and J.H. Pradal, Phys. Rev. 135 (1964) B22. H.J. Mang, J.K. Poggenburg and J.O. Rasmussen, Nucl. Phys. 64 (1965) 353. Ma and J.O. Rasmussen, Phys. Rev. C 16 (1977) 1179. PM. Walker, J. Phys. G 16 (1990) L233. 1. Talmi, Simple models of complex nuclei (Harwood, Chur, 1993). V.M. Strutinsky, Sov. J. Nucl. Phys. 3 (1966) 449. C.S. Purry, P.M. Walker, G.D. Dracoulis, T. Kibddi, S. Bayer, A.M. Bruce, A.P. Byrne, M. Dasgupta, W. Gelletly, E Kondev, P.H. Regan and C. Thwaites, Phys. Rev. Lett. 75 (1995) 406, and private communication.