a phenomenological test of the independent quasi-particle model

[ Nuclear Physics A103 (1967) 257--299; (~) North-Holland Publishing Co., Amsterdam 1.D.1

I Not to be reproduced by photoprint or microfilm without written permission from the publisher

A P H E N O M E N O L O G I C A L TEST

OF THE INDEPENDENT QUASI-PARTICLE MODEL

V I N C E N T GIL L E T , B E R T R A N D G I R A U D and M A N N Q U E R H O

Service de Physique Th~orique, Centre d'~tudes Nucl~aires de Saclay, BP. No. 2 91-Gif-sur-Yvette

Received 29 June 1967

A b s t r a c t : The analysis o f the exper imental low-lying spectra o f single-closed shell nuclei by means o f the inverse gap equa t ions me t hod yields uniquely in the pair ing mode l the effective nucleon- nucleon force s trength, the Har t ree-Fock single-particle energies and the BCS wave functions. The results are free f rom any adjustable parameter . The cons tancy in a given nuclear region o f the force s t rength and o f the H F energies thus provides an u n a m b i g u o u s test o f the pure one- quasi-part icle descript ion for such nuclei. The analysis is carried out with a central, finite-range force for all spherical single-closed shell nuclei for which enough data are available (i.e. the isotones N : 28 and 50, and the isotopes Z = 28, 50 and 82). The results show the impor tance o f including more than one major shell for the pair ing effects. Inclusion o f remote shells renor- malizes the force s t rength appreciably. However , it does not affect the wave funct ions o f quasi- particle states near the Fermi level. W h e n the n u m b e r o f nucleons is varied, the extracted force s t rength and the Har t ree-Fock energies remain nearly cons tan t (within :J= 1 MeV and ±0 .5 MeV, respectively), which is close to their average value in mos t o f the s tudied nuclear regions.

1. Introduction

The main effect of the nucleon-nucleon residual interaction in single-closed shell (s.c.s.) nuclei is pairing t). Neglecting all other effects, the Hartree-Bogoliubov- Valatin method or the BCS method leads to an interpretation of the ground state and first few excited states of s.c.s, odd-mass nuclei as independent single quasi-particle excitations.

Such an approximate description is very appealing, for it gives a simple under- standing of the structure of low-lying s.c.s, excited states and provides a convenient first step towards more sophisticated calculations. The further refinements consist in treating the residual interaction between quasi-particles, which are supposedly weaker than the interaction between particles in pure Hartree-Fock orbits. Among them, the most currently studied are the description of collective vibrations in s.c.s. doubly-even nuclei in terms of two interacting quasi-particles 2) and the treatment of the effect of three-quasi-particle configurations on excitations in odd-mass nuclei 3). The latter work shows that the mixing of three-quasi-particle states to the first few low-excited states is in general small, and hence the pure single quasi-particle description for these states is indeed quite appropriate.

Unfortunately the application of the BCS method in nuclear physics has been plagued by major ambiguities stemming from too many parameters in the theory.

257

2 5 8 V. GILLET e t al.

Not only the effective residual two-body force constants but also all Hartree-Fock single-particle energies e~ (which are not attainable at the present stage of computa- tional capabilities) appear as parameters. The situation is here in marked contrast with the one met in the calculations of double-closed shell regions 4). Here the success of the spectroscopic studies is due in large part to the use of experimental single- particle energies extracted directly from the low-lying excitations of the nuclei with double-closed shell plus or minus one nucleon. On the other hand, the Hartree-Fock single-particle energies for the s.c.s, nuclei can only be obtained by adjusting them in a :g2 search so that the BCS quasi-particle spectrum fits the observed spectrum of a s.c.s, odd-mass nucleus 2). In this procedure, one also has at disposal the effective residual interaction parameters to improve the fit. The obtained sets of single-particle energies and force parameters are far from being unique.

The uncertainties inherent to the numerous parameters in the usual application of the Bogoliubov-Valatin method not only obscure the interpretation of further configuration mixing calculations, but even worse, render difficult an appreciation of the goodness of the pure quasi-particle picture in connection with experimental data.

Such difficulties are alleviated by the phenomenological analysis of experimental data in the framework of the inverse gap equation (IGE) method which was proposed in ref. s). The IGE method is based solely on the hypothesis that a few low-lying states in the s.c.s, odd-mass nuclei, namely the states which are most strongly excited in one-nucleon transfer experiments, can be identified to be pure one-quasi-particle states. It yields unambiguously the renormalized Hartree-Fock energies g~ as well as the force strength V o in addition to the energy gaps A~, and the occupation probabili-

2 ties v~ of Hartree-Fock orbitals. The only parameter left in the theory is the ratio of triplet (T = 1, S = 1) to singlet (T = 1, S = 0) force strengths which plays a minor part as we shall see.

The advantage of the present method of theoretical analysis of experimental data is the immediate insight it gives into the over-all validity of the pure one-quasi-particle hypothesis. Without the intermediate steps of dubious least-squares fits with many variables, one can check immediately whether or not the calculated g~ behave smoothly when filling a shell, and whether the calculated force strength Vo remains practically constant from a nucleus to the neighbouring ones. Furthermore the behaviour of the calculated occupation probabilities v~ can be directly verified through the extraction of spectroscopic factors.

The calculated g~ and V o depend on the size of the configuration space taken into account; a new result of the present paper is that the pairing interaction in the closed- core shells plays an important role in the determination of the effective force strength, but it does not affect much the values and behaviour of the occupation amplitudes and the Hartree-Fock energies of the states near the Fermi level.

In sect. 2 we recall the well-known results of the usual BCS method and give the definitions of the quantities used subsequently. In sect. 3, we develop the procedures of analysing the experimental data by means of the IGE method. The methcd is to

INDEPENDENT QUASI-PARTICLE MODEL 259

invert the gap equations in which the quasi-particle energies are replaced by experimental values.

The IGE method is applied here to all s.c.s, stable spherical nuclei of the periodic table, except for the N = 82 region where relevant experimental data are almost completely lacking. We analyse the isotopes of nickel, tin and lead and the isotones N = 28 (49Sc, 51V, 53Mn and 55C0) and N = 50 (89Rb, 89y, 91Nb and 93Tc).

In sect. 4 we discuss the experimental data which have been used, viz. the experimental levels of the s.c.s, odd-mass nuclei with assigned spin and parity, which may be identified as predominantly independent one-quasi-particle excitations. Al- though experimentalists have devoted many efforts to some regions (such as the tin isotopes), the over-all knowledge of the single quasi-particle states is still very scarce to the whim of theoreticians. This is particularly true of high-lying quasi-particle and quasi-hole states. In the present paper the available data had to be supplemented with reasonable guesses. For this reason it must be emphasized that all results obtained here be taken as indicative of the scope of the method. They may undergo some significant modifications when more complete experimental data become available. In sect. 5, we give and discuss the tables and figures which summarize the extracted values for the gv, v~, Av and for the force strength V o. The role of the size of the configuration space on these values is discussed by comparing the results obtained with and without the contributions from shells far from the Fermi level.

In a forthcoming paper these uniquely determined values of the parameters of the theory are used for calculating the excitations in the s.c.s, doubly even nuclei in terms of interacting quasi-particles. Thus in a latter stage, the IGE method links directly (with no free parameters) the properties of doubly-even nuclei to the experimental spectra of the neighbouring odd-mass ones, and it permits an unambiguous discussion of the success or failure of the interacting quasi-particle picture.

2. The gap equations

We briefly review here the main features of the quasi-particle formalism with the purpose to define the quantities which will be used in the following sections and to show the relationships between the input experimental data of the IGE method and the theory.

In second quantized form, the nuclear Hamiltonian in the Hartree-Fock representation is,

H = ~, (em - ~'m)l~2~m .3ff ¼ 2 (FS] ]/]mrl~rl+rl:rlnrlm. (l) m rsmn

The indices r, s, m and n denote all the quantum numbers defining Hartree-Fock single-particle states. The creation and annihilation fermion operators 17 + and q are defined relative to the true vacuum 10) (absence of nucleons) such that

~/~10) = 0 for all v, (2)

260 v. GILLET et al.

and the matrix elements of the nucleon-nucleon force V are antisymmetrized. The first term of eq. ( I ) is just the kinetic energy T, since the Hartree-Fock energies

ev and the self-consistent potential ¢~ are defined by

= <viZlv>+ Z 2 J + l <(vs)JI vl(vs)J), (3)

~'~ = Z 2J + l ( (vs)J[Vl(vs)J) , (4) s,~ 2j~+ 1

in a coupled angular momentum representation for the matrix elements. The summation index s extends over all the quantum numbers, except the magnetic ones, of the single-particle orbitals which are occupied in the reference Hartree-Fock state.

The ground state of a system of identical nucleons in the same subshell can be ap- proximately described by a configuration where all the nucleons are grouped into pairs of angular momentum J = 0. Such a configuration gives a maximum overlap of occupied orbitals resulting in maximum binding energy. This pairing effect is taken into account if we use the variational principle with the Bardeen-Cooper-Schrieffer trial wave function,

U j r - m y + + IBCS) = I-[ ( ~ + ( - ) v~r/~ r/_,)lO), v > O

(s)

where u v and vv are related by the normalization condition of the wave function

2+v2 1. (6) U v

The u and v are chosen to be real and positive. The choice yields the best solution (i.e. the lowest energy) with the phase choice for the pairing matrix elements all real and negative. The BCS wave function is the sum of all Slater determinants which contain only the pairing correlations between nucleons, i.e. if the state v(v -- n, l, j, m) is occupied with the probability v 2, the state - v ( - v = n, l, j , - r n ) is also occupied with the same probability.

The "occupation' 'probabilities v 2, which are the variational parameters of the theory, are independent of magnetic quantum numbers because of rotational invariance.

The simple trial wave function ofeq. (5) is not an eigenfunction of the particle number operator. However one retains a simple variational problem if one imposes the constraint that the number of particles is conserved on the average:

(BCSIJV'I BCS) = Z (2j~+ 1)v 2 = N, (7) v

where JU is the number operator. For this purpose one introduces a Lagrange multi- plier 2. The BCS ground state is now determined by the stationarity condition

- - <BCSlU- XIBCS> = 0, (Sa ) ~vv


which yields when substituting eqs. (1) and (5),

~(vv - uZ)Av = 0, (8b) ( ~ v _ , ~ ) u v v v + ~ 2

where we have defined the "self-energies" G

.~ 2 J + 1 vZ ( (v s )d l V l ( v s ) d ) , (9) gv -= G - 19~+ ~ 2jr+ 1

and the "energy gaps" 2Av

Av= -½ ~ ( 2 j s + l ] 4- \2 j r+ 11 <(vv)01Vl(ss)0)u~v~. (10)

The summation over s extends over all the single-particle quantum numbers except the magnetic ones which are already summed over. The self-energies %v appear here as a generalization of the Hartree-Fock energies G of eq. (3), where now all the states s give a contribution to the self-consistent field proportional to their occupation probability v 2.

From the gap equations (Sb) and the normalization condition eq. (6), we have the following useful relations:

G = ~ 1 - ( l la )

i Av u v G - ( l lb)

2 ( (g- 2) 2 + A 2)&"

We turn now to the "quasi-particle" language, where the approximate ground state IBCS), defined in eq. (5) and solution of eq. (8), is used as a reference state. It is now considered as a "vacuum" not of true particles, but of "quasi-particles" whose annihilation operators ~v are defined by

~vIBCS> = 0 for all v. (12)

In order to satisfy eq. (12) and the fermion commutation rule, these operators must be of the form

iv = UvI~v--(-- 1)Jv-mVVv•+---v,

= - 1 ) v n_v, (13)

where the u and v are determined by the stationarity condition (8). Substituting eq. (13) into eq. (1) and using eq. (8b), one obtains the nuclear Hamiltonian in the "quasi-particle" representation,

H ' = H-2,A/" = EBcs+ Z Ev~+~v+'V'Qp. (14) v

The first term is the reference energy of the BCS ground state. The second term, diagonal in this representation, describes independent quasi-particles of energy E v and


wave function Iv>

ev = (15)

Iv> = ~+IBCS) = t/+ H (u~'+v¢(-)Jv'-mv'q~q-+¢)10). (16) v,>o, ~1~1

The energy gaps A~ derive their name from the fact that E~ cannot be smaller than A~. The third term is a residual interaction between quasi-particles, whose explicit form is not needed in the present paper.

In principle the exact solution of the nuclear problem is obtained by diagonalizing eFOp over all quasi-particle configurations. However due to the energy gaps Av, the creation of a quasi-particle requires a fairly large energy E~. Therefore for the first few excited states of the nucleus one may in first approximation consider the configurations with the minimum number of quasi-particles. Thus we take into account only one-quasi-particle excitations, neglecting three-quasi-particle ones, to describe low- lying excitations of odd-mass, s.c.s, nuclei.

The two-body force entering the nuclear Hamiltonian is usually chosen as purely central of the form

V = Vo e-(r/u)2{PsE + flPTo + 7Pso + 6PTE}, (17)

where we have factored out the over-all strength V o. The range # is generally taken to be 1.7 fm for a Gaussian form factor; PsE, Pso, PTE and PTO are, respectively, the singlet-even (T = 1, S = 0), triplet-even (T = 0, S = 1), singlet-odd (T = 0, S = 0) and triplet-odd (T = 1, S = 1) projection operators. For a system of identical nucleons, only the singlet-even and triplet-odd terms contribute.

In the usual numerical procedures, not only the force strength V o and the triplet- to-singlet strength ratio ~ but also the many self-energies g~ for all v are parameters.

The solutions of the gap equations are obtained by an iterative method. The lowest quasi-particle energy E~o obtained thus corresponds in the independent quasi-particle model to the odd-even mass difference P~

PN = {2BN--BN+I--BN-1} , (18)

where B u is the total energy of the ground state of nucleus N. We have

if we assume Evo ~ ½en, (19a)

BN+I = ~BCSI~'N+I'-}-"~N+I(N-[-I),

BN_1 ~-~ 1 = EBCS +•N- ( N - l ) ,

BN I{I:N + 1 = 2, .cs + + ½N(k ' +' + , ) + e , ,

where the last equation implies a linear variation of the BCS ground-state energy and of the chemical potential with the number of particles. The other quasi-particle energies are associated to the excited spectrum E* of the odd-mass nucleus

E~-E~o = E*. (19b)


In ordinary calculations, the parameters of the problem (V o, ~, gv) are varied until the solutions E~ come as close as possible to the experimental data, e.g. eqs. (19a) and (19b). Such procedure is lengthy, the fit is never very satisfactory, and the set of values for the parameters is not unique.

3. The inverse gap equation method

We describe here the inverse gap equation method proposed in ref. 5). It removes completely the uncertainties of the independent quasi-particle model associated to the unknown parameters V o and gv.

We use as input to the problem experimental energies E* and the odd-even mass difference PN. We have the quasi-particle energies Ev according to the hypothesis of the independent quasi-particle picture of eqs. (19a) and (19b). Truly speaking this correspondence is only approximate, and there exists some energy shift 6E v

E~* = (E~- Evo)- 6 <

due to the coupling to three- or more-quasi-particle configurations. However theo- retically 6Ev turns out to be small [~ 200 keV for tin isotopes, ref. 4)], and the wave functions of low-lying excitations of the s.c.s, odd-mass nuclei remain in general practically of a pure one-quasi-particle nature. Accordingly in most cases the effect of higher configurations will not affect significantly the results of the present method. The only exception to this feature comes from the case where there is a strong coupling of one quasi-particle to a phonon; we shall point out such cases in sect. 4 where we describe the collection of the experimental energies for input data.

The Ev are known from experiment, and they are introduced in the gap equation [eq. (10)] using the definitions (1 lb) and (15). If we define the matrix

1 (2jr, + 1] & ((vv)O117l(v'v')O> (20a) M~, = -- 4 \ 2 j ~ + l / E,. '

where 17 is the two-body force of eq. (17) with a strength V o = - 1, then the gap equation becomes a real non-symmetric eigenvalue problem

MA'(m) = o~A'(~o). (20b)

The only physically significant solution is that of eigenvalue m whose eigenvector A'(m) has positive components

A~(m) > 0 for all v,

corresponding to a positive energy gap with the phase choice Mv~. > 0. By Frobenius theorem, a positive matrix M always has such an eigenvector and it has only one. The corresponding eigenvalue ~v is simple; it is the largest, real, positive. The eigenvalue yields the force strength V o

1 Vo . . . . . ( 2 0

W

264 v. GILLET et aL

Not ice tha t the eigenvector A ' (w) is defined by eq. (20) within a normal iza t ion

constant . Thus one more condi t ion is necessary to relate its componen t A~(t~) to the

physical energy gaps A~. The condi t ion

A~ = ~A' (w) (22)

is p rov ided by the number equa t ion * (7)

E (2j~+ l).0v(¢) _ ½ Z (2j~+ 1 ) - N, (23) 2 E ~ v

where we have subst i tuted in eq. (7) the definit ion (1 l a ) wri t ten as

v ~ = [~ ( 1 - - g ~ ( ~ ) ~ - , (24) E v / J

with

~(~) = g ~ - 2 = & ( E Z - ( ¢ A ' ~ ) : ) ~, (25)

2 "hole- l ike state"~ s ~ = s i g n ( 9 0 = - 1 i f v~ > ½ or

= + 1 it" v~ < ½ or "par t ic le- l ike s ta te" . (26)

The sign funct ion s~ has to be known before solving eq. (23) for the normal iza t ion 4.

The re la t ion of s~ to the exper imenta l da t a will be discussed in sect. 4.

Once the gaps A~ are known, one obta ins direct ly the relative self-energies

g ~ - g~o = 9 ~ - 0~o. (27)

Thus we have ob ta ined in a unique way f rom the exper imenta l da t a the force strength

V o [eq. (21)], the gaps A~ [eqs. (20) and (23)] the relat ive self-energies gv [eqs. (25)

and (27)] and the occupa t ion probabi l i t ies v z [eq. (24)].

We shall discuss in sect. 5 how these quant i t ies depend on the number o f Har t ree-

TABLE 1

a) Vo ~ --34.0 MeV 6<N> = 0 b) V0 = --32.64 6<N> = 0.82

input computed computed input

I g_~ 0.20 2.19 0.98 0.82 0.28 1,90 0.98 0.68 2d~r 0. 2.47 0.97 1.08 0. 2,21 0.98 0.83 2d~ 1 3.20 1.47 0.41 1.10 3.04 1,34 0.36 0.90 3s{_ 2.10 1.29 0.74 1.28 1.48 1,18 0.86 1.04 1 h ~t 3.20 1.40 0.39 1.01 3.49 1.50 0.23 0.69

a) Solution of the ordinary gap equation for lxTSn. The input ~'v and V0 are parameters. The exchange force mixture is assumed to be 0~ = 1, V0 ~ --34 MeV, for a Gaussian shaped potential of range

/~ = 1.85 fm. The error in particle number 6<N) is zero. b) Solution of the inverse gap quation for 117Sn. The input values Ev are true experimental values. The error 6(N) is non-zero.

* Although the nucleus is an odd-mass one and is therefore described by eq. (16) with "blocking" corrections, for the sake of simplicity we solve eq. (23) with the actual odd value of N.


Fock states v taken into account, which is the dimension of the eigenvalue problem of eq. (20).

As an illustration of the difference between the results given by the solution of the usual gap equations (sect. 2) and the solution of the inverse gap equations, we consider the case of l l7Sn. The experimental spectrum is given in table lb. The results of the ordinary method are given in table la, where g~ and V o were varied in a lengthy least-squares search such that the calculated/iv come out as close as possible to the experimental values. The error in the average particle number is zero, but the fit is not satisfactory. The results of the IGE method are given in table lb, where the experimental Ev are used as input to the problem. The error in the average particle number is not zero (it is equal to 6<N) = 0.82 for 17 nucleons), indicating that the experimental spectrum may not be a pure one-quasi-particle one. Although the occupation

2 (i.e. the wave functions) are similar in both methods, there are sig- probabilities v~ nificant differences in the gap values and the obtained ~. Although many sets of gv and Vo could give similar fits in the case of the usual method (table la), the values obtained for these quantities with the IGE method (table lb) are unique.

4. Experimental data

4.1. PRELIMINARY REMARKS

We apply the IGE method to the sequences of single, closed-shell odd-mass nucle i Z = 28 (nickel isotopes), Z = 50 (tin isotopes), Z = 82 (lead isotopes) and the isotones of N = 28 (49Sc, 51V, 53Mn and 55Co) and N = 50 (SVRb, 89y, 91Nb and

93To). The experimental data required by the IGE method are the energies E~ of low- lying states and the sign functions sv of eq. (26) with assigned spin and parity in these nuclear regions. Such information is far from complete at the present moment, and the available data have been supplemented by reasonable guesses for those states which are not known experimentally. Our results may he modified when more data become available. In spite of these uncertainties, we expect to obtain some information on the consistency of the independent quasi-particle model. The main purpose of our work is, however, more limited; it is twofold (i) illustrate the results one may expect from the phenomenological analysis of data with the IGE method and (ii) discuss the sensitivity of the IGE results to the number of quasi-particle states, which determines the dimension of the matrix in eq. (20). Thus we test the familiar concept of renormalization effect associated with the shell-model choice of the contributing subshells.

We shall present two sets of results denoted in the figures by A and B, respectively. Case A is obtained with nl quasi-particle states situated in the vicinity of the Fermi level 2 (corresponding to the traditional choice of all the previous works). The other set of results, case B, is obtained with n2 quasi-particle states (n2 > nl) including quasi-hole and quasi-particle states lying deep below or high above the Fermi level 2.

The shell model provides a rough estimate of the position of the Fermi level 2 in each of the nuclear regions considered. In general the space nl is chosen as the major


shell containing 2. The space n 2 includes in part or in totality a second major shell. In the case of the lead isotopes and the N = 50 isotones, it is the next upper major shell above the space nl, since for these nuclei the subshells in the space n 1 are almost filled. In the nickel and tin isotopes and the N = 28 isotones, it is the major shell just below the space nl. In a particular calculation for 12 3Sn and 2°spb aimed at showing the influence of extremely remote shells on the force strength V o, the number of subshells was extended step by step up to 29 (from the ls~ subshell up to the lj~e).

4.2. E X P E R I M E N T A L I D E N T I F I C A T I O N OF QUASI-PARTICLE STATES

The levels of s.c.s, odd-mass nuclei which may be reasonably identified as mostly pure one-quasi-particle states, should show the following characteristics:

(i) They should be preferentially excited in one-nucleon transfer reactions on doubly even target (stripping or pick-up).

(ii) There should be only one level E~ of spin Jv and parity rcv in a rather extended energy region ( ~ 2-3 MeV) corresponding to each subshell v, as one expects in general a small mixing with higher configurations. If several levels of the same spin and parity are excited, indicating that a single quasi-particle state is split by mixing with higher configurations, one may use, as a prescription, the calculated energy average of these levels weighted by the respective spectroscopic factors if available. The ground states of the s.c.s, odd-mass nuclei are associated with the lowest quasi-particle energy Evo according to eq. (19a).

(iii) A check of correct identification consists in ascertaining that the experimental quasi-particle energies E~ = E* +Evo follow regular trends when the mass number increases: a) if the subshell v is above 2 and mostly empty, the values of E~ decrease; the sign function sv [eq. (26)] is then positive; b) when the quasi-particle state v becomes a ground state, the E~ remains practically constant and the sign function changes from positive to negative; c) when the subshell v is below 2 and mostly filled, Ev increases and the sign function s~ remains negative.

In practice we have encountered several difficulties: (a) A low-lying three-quasi-particle state (vlv2v3) s sometimes occurs due to a large

diagonal matrix element of ~:Qp. Such phenomenon happens in two cases as shown by Kisslinger 6).

(i) j~ = J~2 = J~3, J = Jr, - 1. In this instance the identification of the three-quasi- particle state is not difficult. The energy shift is proportional to jr,, and hence only a state with large total angular momentum J > 25- may lie sufficiently low. The example is the theoretical observation of a low-lying J = 29-- level coming from the configuration (h~)~ in the Sn isotopes 3).

(ii)j~, ¢ Jw In this case, it is less easy to identify the low-lying three-quasi-particle level. One has to rely on theoretical calculations in order to identify them. An exam- pie is the low-lyingj = ½- level in the Pb isotopes (A < 207) which is believed to be a three-quasi-particle state of this type. Experimentally such states should be ruled out by their weak excitation probability in one-nucleon transfer experiments.


( b ) F o r t h e l eve l s o f t h e s p a c e n l , m o s t o f t h e m w e r e k n o w n e x p e r i m e n t a l l y . H o w -

,ever , s o m e w e r e m i s s i n g m a i n l y a t t h e u n s t a b l e e n d s o f t h e c o n s i d e r e d s e q u e n c e s o f

n u c l e i . I n s u c h a c a s e , w e h a v e m a d e r e a s o n a b l e g u e s s e s f o r t h e i r p o s i t i o n s a s s u m i n g

t h e i r s m o o t h d i s p l a c e m e n t in m o v i n g a w a y f r o m n u c l e i w h e r e t h e y a r e o b s e r v e d .

TABLE 2

The quasi-particle energies E v (input data) for each isotone N = 28 and the resulting effective strength V0, gaps Av, occupation amplitudes vv, self-energies 7v 2 and Hart ree-Fock energies e v )~

~

nl j E v A v Vv gv gv

49Sc

Vo = - 3 2 . 3 l

lf@ 1.27 0.83 0.35 0.96 0.25 2p~ (4.81) 0.55 0.06 4.78 4.06 lf~ (5.96) 0.99 0.08 5.88 6.61 2p~ (7.31) 0.59 0.04 7.29 7.95 lg~: (8) 0.60 0.04 7.98 7.37 ld~r (7) 1.12 1.00 -- 6.91 -- 6.38 2s½ (8) 0.89 1.00 -- 7.95 -- 9.93 ld~ (12) 0.98 1.00 --11.96 --13.12

zx V V0 = --30.92

lf~ 1.22 1.18 0.60 0.33 -- 0.16 2p~_ (4.22) 0.79 0.09 4.15 3.64 lf~ (5.52) 1.39 0.13 5.34 5.85 2p{_ (6.52) 0.84 0.06 6.47 6.93 lg~ (7.5) 0.86 0.06 7.45 7.02 ldk (7) 1.58 0.99 -- 6.82 -- 6.45 2s~ (8) 1.26 1.00 -- 7.90 9.27 ld~_ (12) 1.38 1.00 --11.92 --12.73

aaMn Vo = 29.17

lf~ 1.15 1.12 0.78 - - 0.25 -- 0.54 2pk (3.64) 0.77 0.11 3.55 3.26 lf~ (5.16) 1.33 0.13 4.98 5.28 2p~ (5.85) 0.82 0.07 5.79 6.06 lg~ (7) 0.82 0.06 6.95 6.70 ld~_ (7) 1.50 0.99 -- 6.84 -- 6.62 2s.~ (8) 1.20 1.00 -- 7.91 -- 8.69 ld~ (12) 1.30 1.00 --11.93 --12.40

~5Co 1/"o = --27.36

lf~_ 1.07 0.73 0.93 -- 0.78 -- 0.88 2p k (3.50) 0.50 0.07 3.46 3.37 lf~r (4.65) 0.86 0.09 4.57 4.66 2Ok (5.03) 0.53 0.05 5.00 5.09 lg~_ (6.5) 0.53 0.04 6.48 6.40 ld~_ (7) 0.97 1.00 -- 6.93 -- 6.86 2s~ (8) 0.78 1.00 7.96 -- 8.21 ld~ (12) 0.84 1.00 --11.97 --12.12

The sign functions s v are the signs of the self-energies 7v--2. The calculation was performed with a Gaussian force o f range 1.7 frn and triplet-to-singlet strength ratio ~ = --0.4. The configuration space included all the subshells v indicated in this table (case B).

(c) T h e l eve l s o f t h e s p a c e n z a r e a l m o s t c o m p l e t e l y u n o b s e r v e d , e s p e c i a l l y t h e

o n e s p r o d u c e d in p i c k - u p e x p e r i m e n t s w i t h l a r g e n e g a t i v e Q - v a l u e s . T h e q u a s i -

268 v. GILLET e t at .

particle energies of the space n 2 are set equal to shell-model Hartree-Fock energies (relative to 2) with the neglect of A 2 in eq. (15). Although the results of the I G E method were strongly affected by the introduction of the r/2 space, it was found that

TABLE 3

T h e i n p u t d a t a a n d the r e su l t s o f t h e I G E m e t h o d a p p l i e d to t h e n i cke l i s o t o p e s

nO Ev d,, vv ~ a~

5VNi Vo = - - 2 4 . 5

2 p k 0 .97 0 .78 0 .45 0 .58 0 .15 l f~ 1.73 0.61 0 .18 1.62 1 .80 2 p ~ 2 .05 0 .82 0 .20 1.88 2 .17 l g . k (4 .40) 0 .49 0 .06 4 .37 4 . 1 4 ld~: (10 .8) 0 .69 1.00 - - 1 0 . 7 8 - - 1 1 . 0 9

2s½ (6 .7) 0 .86 1 .00 - - 6 .64 - - 7 .77 l d k (5 .7) 0 .67 1.00 - - 5 .66 - - 5 .44 lf~_ (3 .10) 0 .62 0 .99 - - 3 .04 - - 3.31

5~Ni Vo = - - 2 5 . 0 3

2p~ 1.19 1.19 0.71 0 .00 - - 0 .34 lf~_ 1.53 1.02 0 .36 1.14 1.36 2p~ 1.66 1.21 0 .40 1.13 1.46 l g ~ (4 .25) 0 .83 0 .10 4 .17 3 . 9 4 l d ~ (10 .9) 1.17 1.00 - - 1 0 . 8 4 - - 1 1 . 1 3

2s k (6 .9) 1.36 1.00 - - 6 .76 - - 7 .59 ld~ z (5 .9) 1.12 1.00 - - 5.79 - - 5.55 l f÷ (3 .15) 1.05 0 .99 - - 2 .97 - - 3.23

elNi Vo = - - 2 5 . 7 5

2p~ 1.39 1.38 0 .76 - - 0 .19 - - 0 . 6 6 1 f~ 1.46 1.27 O. 50 0 .72 1.16 2 p ~ 1.67 1.39 0 .88 - - 0 .93 - - 0 . 2 6 l g k (4 .25) 1 .04 0 .12 4 .12 3 . 7 6 l d k (10 .9 ) 1.49 1.00 - - 10.8 - - 11.27 2s~ (6 .9) 1.64 0 .99 - - 6 .70 - - 7.63 l d ~ (6 .0) 1.41 0 .99 - - 5.83 - - 5 . 3 6 l f k (3 .16) 1.33 0 .98 - - 2 .86 - - 3 .28

n3Ni Vo = - - 2 6 . 1 7

2 p k (1 .57) 1.42 0 .84 - - 0 .67 - - 1.16 l f ~ (1 .50) 1.32 0 .86 - - 0.71 - - 0 . 1 0 2p~_ ( 1 . 4 1 ) 1.40 0 .67 0 .14 1.01 l g ~ (4 .20) 1.08 0 .13 4 .06 3 .55 l d i_ (11 .0) 1.55 1 .00 - - 10.9 - - 11.49 2s~_ (7 .0) 1.69 0 .99 - - 6 .79 - - 7 .72 l d k (6 .0) 1.46 0 .99 - - 5.81 - - 5.18 l f~ (3 .17) 1.38 0 .97 - - 2 .85 - - 3 . 43

~SNi Vo ~ - - 2 6 . 6 9

2p~_ 1.76 0 .95 0 .96 - - 1.48 - - 1.84 l f~ 1.44 0 .94 0 .94 - - 1.09 - - 0 .53 2 p ÷ 1.50 0 .94 0 .33 1.17 1.94 lg~_ (4 .10) 0 .77 0 .09 4 .03 3.55 l d ~ (11 .2) 1.11 1.00 - - 1 1 . 1 4 - - 1 1 . 6 5 2s k (7 .6) 1.16 1.00 - - 7.51 - - 8 .12 l d k (6 .2) 1.05 1.00 - - 6.11 - - 5 .54 l f~ (3 .20) 1 .00 0 .99 - - 3 .04 - - 3 . 5 6

See c o m m e n t s to t a b l e 2.


t h i s i n f l u e n c e w a s a s s o c i a t e d w i t h t h e d e g e n e r a c i e s o f t h e s e c o m p l e m e n t a r y s t a t e s b u t

w a s h a r d l y d e p e n d e n t o n t h e e x a c t p o s i t i o n o f t h e r e m o t e leve ls .

TABLE 4

The input data and the results o f the IGE method applied to the isotones N = 50

. 0 E~ A ~ v~ ~v a~

87Rb

V0 = --23.13

1 g~_ (1.84) 0.64 0.18 1.72 1.69 2p~ (2.0) 0.94 0.24 1.76 1.81 lf~_ 1.39 0.68 0.97 -- 1.21 -- 1.20 2p~ r 0.99 0.91 0.83 -- 0.38 -- 0.49 lg÷ (3) 0.62 0.10 2.94 2.94 2d~ (3.5) 0.57 0.08 3.45 3.39 2dk (4) 0.58 0.07 3.96 3.98 3sk (4.5) 0.40 0.04 4.48 4.31 l h ¥ (5) 0.50 0.05 4.97 4.95

8~y

110 = --24.66

lg~: 1.57 0.45 0.15 1.50 1.47 2p~ 0.65 0.65 0.66 0.09 0.24 lf÷ (2.40) 0.49 0.99 -- 2.35 -- 2.27 2p~. 2.18 0.70 0.99 -- 2.07 -- 2.12 lg k (3) 0.45 0.08 2.97 3.04 2d~_ (3.5) 0.43 0.06 3.47 3.43 2d~ (4) 0.41 0.05 3.98 4.07 3s~. (4.5) 0.30 0.03 4.49 4.44 l h ~ (5) 0.36 0.04 4.99 4.96

9aNb V0 = --25.02

lg~_ 1.03 0.70 0.36 0.75 0.75 2p~ 1.13 0.86 0.91 -- 0.74 -- 0.49 lf~_ (3.4) 0.83 0.99 -- 3.30 -- 3.19 2p~ 2.53 0.88 0.98 -- 2.37 -- 2.41 lg~_ (3) 0.76 0.13 2.90 3.00 2d~_ (3.5) 0.57 0.08 3.45 3.43 2d~ (4) 0.57 0.07 3.96 4.11 3s~ (4.5) 0.42 0.05 4.48 4.48 l h ~ (5) 0.55 0.06 4.97 4.96

93Tc

Vo = --26.30

lg~_ (1.0) 0.93 0.57 0.36 0.46 2p~ (1.4) 1.10 0.90 -- 0.86 -- 0.69 lf~_ (4.4) 1.13 0.99 -- 4.25 -- 4.26 2P~r (3.0) 1.11 0.98 -- 2.79 -- 2.71 lg~ (3.0) 1.04 0.18 2.81 2.83 2dk (3.5) 0.73 0.11 3.42 3.51 2d~ (4.0) 0.74 0.09 3.93 4.03 3s½ (4.5) 0.56 0.06 4.46 4.64 l h ¥ (5.0) 0.74 0.07 4.94 5.03

See comments to table 2.

( d ) T h e s i g n f u n c t i o n s sv c a n n o t b e c h o s e n s t r a i g h t f o r w a r d l y f o r t h e h a l f - f i l l e d l eve l s

(gv v e r y c l o s e t o 2) . S u c h a m b i g u i t i e s w h i c h a f f e c t a t m o s t t w o o r t h r e e l eve l s a r e re -

m o v e d b y c h o o s i n g t h e s e t o f s v w h i c h c a n s a t i s f y a t b e s t t h e n u m b e r e q u a t i o n (23)

270 V. GILLET et al.

TABLE 5

T h e i npu t d a t a a n d the resul ts o f the I G E m e t h o d ap p l i ed to the t in i so topes

nlj Ev Av vv gv gv

lxaSn 1Io = - -33 .39

lg~ 1.31 1.18 0.85 - - 0.58 - - 0.73

2dff (1.84) 1.17 0.94 - - 1.42 - - 1.20

2d~ 1.74 1.16 0.36 1.30 1.16

3s~ 1.24 1.22 0.64 0.21 0.63

l h ~ 1.98 1.05 0.28 1.68 1.84

lg{_ (6) 1.23 0.99 - - 5.87 - - 5.69

2p_ I_ (9) 1.36 1.00 - - 8.90 - - 9.13 l f~ (9) 1.23 1.00 - - 8.91 -- 9.14

2p~ (9) 1.37 1.00 - - 8.89 - - 8.69

115Sn Vo = - 3 3 . 9 3

l g~ 1.60 0.84 0.96 - - 1.36 - - 1.45

2d~ (2.44) 0.85 0.98 - - 2.29 - - 2.09

2d~ 1.49 0.83 0.29 1.24 1.16

3s~ 1.00 1.00 0.67 0.10 0.55

l h ~ 1.72 0.74 0.22 1.56 1.67

l g~ (6) 0.85 1.00 5.94 -- 5.81

2p_ I_ (9) 0.96 1.00 - - 8.95 - - 9.05

l f~ (9) 0.87 1.O0 - 8.96 - - 9.07

2p~ (9) 0.98 1.00 - - 8.95 - - 8.73

1175n 1Io = - -34 .21

lg½ 1.90 1.10 0.95 - - 1.55 - 1.65

2d~ (2.48) 1.10 0.97 - - 2.22 - - 1.98

2d~: 1.34 1.07 0.45 0.80 0.73

3s~_ 1.18 1.18 0.71 0.00 0.52

l h ~ 1.50 0.94 0.33 1.17 1.31 l g k (6) 1.08 1.00 - - 5.90 -- 5.74

2p~ (9) 1.25 1.00 - - 8.91 - - 9.00

l f~ (9) 1.13 1.00 - - 8.93 - - 9.07

2p~ (9) 1.26 1.00 - - 8.91 - - 8.65

l~gSn 11o = - -34 .57

lg~_ 2.09 1.30 0.94 - - 1.63 -- 1.69

2d~ (2.70) 1.26 0.97 - - 2.39 - - 2.13 2d~_ 1.32 1.23 0.83 - - 0.49 -- 0.47

2s½ 1.30 1.30 0.71 0.00 0.48

l h ~ 1.38 1.08 0.44 0.85 0.99

lg~ (6) 1.25 0.99 - - 5.87 - - 5.71

2p~: (9) 1.44 1.00 - - 8.88 - - 8.75 l fk (9) 1.33 1.00 - - 8.90 - - 9.01

2pk (9) 1.46 1.00 - - 8.88 - - 8.63

121Sn 1Io = - - 3 4 . 8 0

lg~ 2.23 1.37 0.95 - - 1.76 - - 1.87

2d~ (2.70) 1.32 0.97 - - 2.35 - - 2 .06 2d½ 1.30 1.29 0.74 - - 0.13 - - 0.16

3s_4_ 1.35 1.35 0.71 0.00 0.55 l h ~ 1.35 1.13 0.48 0.74 0.90 lg~_ (6) 1.31 0.99 - - 5.86 -- 5.66 2p~ (9) 1.52 1.00 - - 8.87 - - 8.83

l f~ (9) 1.40 1.00 - - 8.89 - - 9.05 2p~: (9) 1.53 1.00 - - 8.87 - - 8.58

INDEPENDENT QUASI-PARTICLE MODEL

TABLE 5 (continued)

271

nO E~ A ~ ~ k~ a~

12~5t- 1 Vo = --34.51

lg~ 2.24 1.42 0.94 -- 1.74 -- 2.18 2d~ (2.60) 1.34 0.96 -- 2.23 1.48 2d k 1.32 1.32 0.71 -- 0.00 -- 0.25 3s~ 1.45 1.33 0.84 -- 0.57 0.71 lh~ k 1.30 1.16 0.85 -- 0.59 -- 0.11 lg~ (6) 1.34 0.99 -- 5.85 -- 5.24 2p~ (9) 1.55 1.00 -- 8.87 -- 9.19 lf~ (9) 1.44 1.00 -- 8.88 -- 9.45 2pk (9) 1.56 1.00 -- 8.86 -- 8.13

125Sn V0 = --34.16

lg.~ 2.21 1.39 0.94 -- 1.72 -- 2.15 2d~ (2.62) 1.31 0.97 -- 2.27 -- 1.52 2d k 1.30 1.28 0.77 -- 0.22 -- 0.46 3s~_ 1.49 1.28 0.87 -- 0.77 0.54 lh~ 1.27 1.13 0.85 -- 0.57 -- 0.10 lg~ (6) 1.31 0.99 -- 5.86 -- 5.25 2p~ (9) 1.51 1.00 8.87 -- 9.15 lf~_ (9) 1.41 1.00 -- 8.89 -- 9.43 2pk (9) 1.52 1.00 -- 8.87 -- 8.13

See comments to table 2.

and yields the gv and v 2 wi th the s m o o t h e s t b e h a v i o u r as a f unc t i on o f the mass n u m b e r s .

T h e i npu t d a t a Ev and sv chosen fo r the I G E m e t h o d are g iven in tables 2-6. W h e n

the Ev va lues are the resul ts o f r e a sonab l e guesses in t he absence o f e x p e r i m e n t a l evi-

dence o r c o m e f r o m she l l -mode l es t imates , they are g iven be tween brackets . T h e

l owes t quas i -pa r t i c l e energy [equal to h a l f o f the o d d - e v e n mass di f ference eq. (19a)]

is unde r l i ned . W e n o w discuss briefly the d a t a for each o f the nuc l ea r regions .

4.2.1. The N = 28 isotones. In t he nucle i 49Sc, 53Mn and 55Co, t he a s s u m e d

" q u a s i - p a r t i c l e e n e r g i e s " fo r the shells lf~, 2p~, lf~ and 2p~ were t a k e n f r o m the mass

t ab le 7) a n d ref. 8). T h e va lues we h a v e chosen for these levels in 51V were ju s t in te r -

po la t ed . W e h a v e a lso inc luded the Ig~ shell in the ca l cu la t ion wi th an e s t i m a t e d

quas i -pa r t i c l e energy o f 8 M e V in a9sc dec reas ing to 6.5 M e V in 55Co. T h e second

m a j o r shell was t he sd shell wi th a 5 M e V spl i t t ing be tween ld~ and l d a as for usua l

she l l -mode l es t imates . T h e 2s~ shell was e s t ima ted to lie 1 M e V be low the ld~ shell.

F ina l ly the d i s t ance be tween the F e r m i level lf~ and the ld~ shell was r o u g h l y esti-

m a t e d f r o m the mass di f ferences be tween 39K, 4 0Ca and 41 Sc. The smal l energy shifts

fo r the sd shell as t he lf~ shell gets m o r e filled are neg lec ted ent irely. (They r e m a i n at

t he s a m e energy wi th respect to the lf~_ in the th ree nuclei , )

4.2.2. The Z = 28 isotopes. The low- ly ing levels wi th J = ~ - , 2 s - and ½- are

ass igned to the single quas i -pa r t i c l e con f igu ra t i ons 2p~, lf~ and 2p~, respect ive ly . T h e

ev idence shows tha t in the S9Ni, ~ - and ~ - states are re la t ive ly pure , whi le the s t reng th

o f 1 - s ta te is spl i t in to several levels. This m a y be pa r t i cu l a r ly the case fo r 61Ni as can

be seen in the a n o m a l y in the theore t i ca l results . T h e ~ - level in b o t h 57Ni and 59Ni

272 v. 'GILLET et aL

TABLE 6

The inpu t da ta and the resul ts o f the I G E m e t h o d appl ied to the lead isotopes

nt/ E~ A~ v, ~ g,

l~Tpb

Vo = --30.37

28~_ (4.1) 0.60 0.07 4.06 4.04 l i ~ 1.05 0.77 0.92 - - 0.71 -- 0.70 3p½ (1.3) 0.79 0.32 1.04 0.85 2f~ 0.81 0.80 0.78 -- 0.18 - - 0.32 3p~ 0.90 0.78 0.50 0.45 0.35 2fg_ (2.0) 0.83 0.98 -- 1.82 - - 1.84 lh~_ (1.2) 0.85 0.92 -- 0.85 - - 0.92 3d~ (5.0) 0.41 0.04 4.98 4.96 li~. (4.15) 0.76 0.09 4.08 4.01 2g~ (4.5) 0.56 0.06 4.47 4.36 4s~ (6.5) 0.29 0.02 6.49 6.30 3d/r (6.0) 0.40 0.03 5.99 5.92 l j ~ (5.5) 0.63 0.06 5.46 5.47

199pb

Vo = --28.83

2g~_ (3.9) 0.47 0.06 3.87 3.86 l i 9 1.05 0.55 0.96 - - 0.89 -- 0.89 3p~ r (1.1) 0.63 0.30 0.90 0.74 3p~_ (0.8) 0.63 0.43 0.50 0.40 2f~_ 0.63 0.61 0.79 - - 0.15 - - 0.28 2f k (2.1) 0.66 0.99 - - 2.00 -- 2.02 lh~ (1.3) 0.61 0.97 - - 1.15 -- 1.22 3d{ (4.8) 0.32 0.03 4.79 4.77 li~/, (3.95) 0.55 0.07 3.91 3.84 2g~ (4.2) 0.44 0.05 4.18 4.07 4s~ (6.3) 0.22 0.02 6.30 6.10 3dk (5.7) 0.31 0.03 5.69 5.63 l j ~ (5.2) 0.45 0.04 5.18 5.19

soipb

1Io = --32.69

2g~. (3.7) 0.62 0.08 3.65 3.68 l i ~ 1.44 0.71 0.97 -- 1.26 -- 1.21 3p~ (0.9) 0.87 0.61 0.23 0.02 3p~ (0.9) 0.88 0.78 -- 0.20 - - 0.21 2f~_ 0.81 0.80 0.77 -- 0.16 -- 0.32 2fg_ (2.2) 0.85 0.98 -- 2.03 - - 1.99 lh~ (1.4) 0.76 0.96 -- 1.18 - - 1.28 3d~ (4.6) 0.45 0.05 4.58 4.61 li~t (3.75) 0.68 0.09 3.69 3.59 28~ (3.9) 0.58 0.07 3.86 3.74 4S~ (6.5) 0.30 0.02 6.49 6.44 3d~ (5.4) 0.43 0.04 5.38 5.30 l j ~ (4.9) 0.58 0.06 4.86 4.90

~o3pb Vo = --30.91

2g~ (3.5) 0.48 0.06 3.47 3.52 l i ~ 1.48 0.51 0.98 - - 1.39 -- 1.34 3p~ k 0.78 0.68 0.50 0.39 0.16 3pk 0.84 0.69 0.89 -- 0.49 -- 0.45 2f~ 0.66 0.61 0.83 -- 0.25 - - 0.40 2f~, (2.30) 0.66 0.99 -- 2.20 -- 2.15


TABLE 6 ( c o n t i n u e d )

nlj E v d v v v g v gv

l h ] 1.48 0.54 0.98 - - 1.37 - - 1.48 3d~ (4.4) 0.35 0.04 4.39 4.44 l i~ k (3.55) 0.49 0.07 3.52 3.42 2g~ (3.6) 0.44 0.06 3.57 3.45 4s~ (6.9) 0.23 0.02 6.90 6.92 3d~ (5.1) 0.33 0.03 5.09 4.98 l j ~ (4.6) 0.42 0.05 4.58 4.62

~oapb

Vo = - -30 .91

2ospb

Vo = - -31 .98

2g~ (3.24) 0.35 0.05 3.22 3.27 l i ¥ 1.69 0.36 0.99 - - 1.65 -- 1.60 3p~_ 0.68 0.50 0.40 0.46 0.26 3p~ z 0 .94 0.51 0.96 -- 0.78 - - 0.73 2 f ] 0.67 0.44 0.94 -- 0.51 - - 0 .64 2fg_ 2.44 0.48 1.00 -- 2.39 - - 2.33 lh~_ 1.67 0.38 0.99 -- 1.62 - - 1.71 3d~_ (4.2) 0.26 0.03 4.19 4.26 l i ~ (3.25) 0.34 0.05 3.23 3.15 2g~r (3.28) 0.32 0.05 3.27 3.16 4s~ (6.7) 0.17 0.01 6.70 6.75 3d½ (4.8) 0 .24 0.03 4.79 4.69 l j ~ (4.3) 0.30 0.03 4.29 4.33

to7pb

Vo = - -32 .28

2g~ (2.0) 0.19 0.05 1.99 2.01 l i ~ 1.95 0.16 1 . 0 0 - - 1 . 9 5 - - 1.93 3p~ r 0.32 0.31 0.77 - - 0 .06 -- 0.15 3p~: 1.21 0 . 3 5 0.99 - - 1.16 -- 1.13 2f~r 0.89 0.24 0.99 -- 0.86 -- 0.91 2f÷ 2.66 0.25 1.00 -- 2.65 -- 2.62 l h ~ 3.79 0.18 1.00 -- 3.79 - - 3.82 3d l_ (4.0) O. 17 0.02 4.00 4.03 l i ¥ (3.0) 0.16 0.03 3.00 2.96 2g~_ (3.2) 0.18 0.03 3.20 3.15 4sI~ (4.0) 0.10 0.01 4.00 4.03 3d~_ (4.5) 0.15 0.02 4.50 4.45 l j ~ (4.0) 0 .14 0.02 4.00 4.01

2o~pb

Vo = - -42 .35

2g~ 0.62 0.33 0.27 0.52 0.55 l i ~ (6.0) 0 .34 1.00 -- 5.99 -- 5.97 3p~_ (2.5) 0.37 1.00 - - 2.47 -- 2.49 3p~ r (4.9) 0.37 1.00 - - 4.89 -- 4.85 2f~r (5 .4) 0 .44 1.00 -- 5.38 -- 5.40 2f~ (5.5) 0.41 1.00 -- 5.48 -- 5.46 l h ~ (6.5) 0.37 1.00 -- 6.49 -- 6.51 3d~ 2.18 0.21 0.05 2.17 2.19 l i~t 1.41 0.34 0.12 1.37 1.35 2g~ 3.09 0.37 0.06 3.07 3.05 4s~ 2.65 O. 16 0.03 2.64 2.68 3d~ 3.16 0.21 0.03 3.15 3.14 l j ~ 2.03 0.29 0.07 2.01 2.02

See c o m m e n t s to t a b l e 2.


was taken from the paper of Brussel et al. 9). The quasi-particle energies for lg~, ld~, 2s~ and ld~ are estimated from Cosman et al. i 0) for 59Ni and extrapolated for others. The shift for these levels from one nucleus to another should be small ( ~ 0.4 MeV) compared with the E v values themselves. There is some doubt as to the experimental order of the ~- , -~- and ½- levels in 63Ni. We have taken the order quoted by Auer- bach 11). Finally the odd-even mass differences are estimated from the mass table 7).

4.2.3. The N = 50 isotones. For the four nuclei 87Rb, 89y, 91Nb and 93Tc, we have used the nuclear data sheet 12). The known levels J~ = ~+, ½-, ~- and 3 - in 89y are assumed to be quasi-particle states of the nlj assignments given in table 4. The spins of the ground states are well identified, and the spins and excitation energies of at least two other levels in each nucleus are reasonably known except for 93Tc. We choose to take for the extra states of the/72 space those relative energies consistent with shell model and to keep arbitrarily those levels constant along the sequence. The rest of the energy assignments are reasonable guesses. Once again the odd-even mass differences were taken from the mass table, except for 93Tc, where it was estimated at 1 MeV.

4.2.4. The Z = 50 (Sn) isotopes. The quasi-particle energies are taken from the works of Cohen et al. 13). The spectroscopic factors are available for these nuclei; the 2d~ quasi-particle energy corresponds to a weighted average, since in all Sn isotopes, several 5+ levels are excited in (d, p) or (d, t) reactions.

For the nz configuration space, the lg , hole state is taken at ~ 6 MeV, estimated from the Nilsson diagram. There is no experimental evidence for this. Odd-even mass differences are taken from the mass table 7) and from Q-values for (d, t) and (d, p) reactions ~ a). The sign functions s,, however, differ somewhat from Cohen's analysis.

4.2.5. The Z = 82 (Pb) isotopes. The particle levels f o r / ° 9 p b and hole levels for 2°Tpb are well known. Their energies come from both the nuclear data sheet and ref. 14) (where one can find references to the experiments). It is plausible that the observed low-lying levels in A < 205 are also quasi-particle states except for the ~-- states. The two low-lying ~- levels at 0.70 and 1.04 MeV in 2°SPb appear to be three- quasi-particle states of the second kind discussed above, and hence the third 7- observed at 1.77 MeV is instead chosen to be the quasi-particle 2f~ state. Although a 7 - level is likewise observed at low energy in / °3pb , this does not appear to be a quasi- particle state either. Thus a reasonable extrapolation based on the trend seen in 2°Spb and 2°7Pb is made for the rest of the Pb isotopes. The assignments in A < 201 are extrapolations except for the ~a_+ level and the sz+ ground states.

5. Results and discussions of the IGE analysis

5.1. PARAMETERS

All calculations are performed with the finite range force of Gaussian shape of eq. (17). The range # is set equal to 1.7 fm.

Since the depth V 0 is uniquely determined by solving the IGE secular problem,


eqs. (20) and (21), and since we deal with systems of identical nucleons, the only parameter of the IGE analysis is the triplet-to-singlet strength ratio ~ (i.e. the ratio between the T = 1, S = 1 and T = 1, S = 0 components). In rn'ost of the usual shell- model effective forces, ~ turns out to be negative and equal to about - 0.4. It has been varied here between the extreme values 0 and - 1.0, but the results as shall be shown in fig. 6 do not depend sensitively on the precise value used. This insensitivity of the I G E method to the only parameter of the theory is due to the fact that the pairing matrix elements are mostly determined by the singlet part of the force alone. We shall adopt, unless otherwise indicated, the value ~ = -0 .4 .

The radial single-particle wave functions are given by a harmonic oscillator (HO) potential. The HO length b = ~/fi/mo9 is roughly determined by comparing the root mean square radius as calculated with HO wave functions and its value deduced from electron scattering analysis using Saxon-Wood continuous charge distribution. The b-values are taken as constant within each nuclear region and are taken to be 2 fm for the N = 28 isotones, Z = 28 isotopes, 2.13 fm for the N = 50 isotones, 2.27 fm for the Z = 50 isotopes and 2.50 fm for the Z = 82 isotopes.

5.2. DEPENDENCE OF THE RESULTS ON THE INPUT DATA

The results of the IGE method are directly dependent on the goodness of the independent quasi-particle description of the odd s.c.s, spectra. The model is only approximate and in practically all cases it applies fairly well only to a part of the data, the remaining part requiring a more refined description. Accordingly it is necessary to have a preliminary but thorough discussion on the dependence of the IGE results on the uncertainties in input data.

We shall investigate the influence on the results of the following sources of uncertainty.

(i) the uncertainties in the odd-even mass difference P. These arise either from experimental errors in the Q-values or from the lack of experimental data which can only be supplemented by reasonable guesses. Since P defines the reference level of the quasi-particle spectrum, the uncertainty in P affects all the E~ by a constant energy;

(ii) the uncertainties in the energies of the second major shell not containing 2. Such energies are not yet known experimentally and in many cases may not even be determined at all;

(iii) the uncertainties in the experimental independent quasi-particle spectrum. They may arise (a) if a low-lying three-quasi-particle level is falsely taken to be a one- quasi-particle state, and/or (b) if there are several close-lying levels with same j which arise from the splitting caused by the coupling of a single quasi-particle configuration to higher quasi-particle configuration and whose spectroscopic factors are not known.

With the development of experimental techniques, the uncertainties of the type (i) and (iiib) should disappear. However, we have to learn how to live with those of the type (ii) and (iiia).

The discussion is displayed in graphic form on figs. 1-3, which show the behaviour

V o

50.

40.

30.

c~ (N),

~ P 1 - -

l

3.

2.

1.

0.

Vo ~(N),

e . . . . - . ~

O.

, , , ~ E (h)

0'.~-o'.2 ~;. 0'.2 0.~- _ J . - & o. o.5 1;

50. /.0.

2. B 2.

/ ,

I I [ I - - 0 . E ~,E3~v2 -1. -0.5 0. 0.5 1. -1. -0.5 0. 0.5 1.

Fig. 1. Dependence of the force strength V0 in l~lSn to the input data. Curves A and B are obtained respectively with one and two major shells nearest to the Fermi level. The heavy line curves represent the error 6 ( N ) left after solving at best the number equation for case B. In case a) the odd-even mass difference is varied by 6P (in MeV) around its central values P = 1.3 MeV. In case b) the positions of the levels of the second major shell are varied by an energy 6E ~) around their central value E ~h) = 8 MeV. In cases c) and d) the quasi-particle energies for the levels 2d{ and 3s~ are varied

around their central values of 2.7 MeV and 1.35 MeV, respectively.


3.

2.

O.

£v- ~ 2d5/2 s!

4. ,h11/2 3sl/2

3.

~ 2 0 3 / 2 2.

elgT/2

(]

I &P

-0.4 -0.2 0. 0.2 0.~.

-1.

E%~~2dS/2

= : J : .lh11/2

_ _ ~ 3 s 1 / 7 : : 2d3/2

- - : : =197/2

b bE (h}

-1. -0.5 O. 05 1.

5!

3.

2

1

-1.

Ev-E2d5/2

lh11/2 /..

3sl/2 2 d 3/2 3

I Q

2

~ 197/2 1

O.

C -1.

~ E 2 d s / 2

-1, -0,5 O. 0,5 !.

A~'~'2 d 5/2 5.7:

~ 3sl/2

lh11/2

~ 2 d 3/2

: :197/2

d ~Essl/2

J ~ L.. _.~._d__t__.lb. -1. -0.5 0. 05 1.

Fig. 2. Dependence of the self-energies ~v to the input data in ~lSn with two major shells (case B). Cases a)-d) are those of fig. 1. The self-energy of the state 2d{ is taken as the reference energy. Here

the ordinate scale is used for both the ~v (in MeV) and O(N) (dimensionless).

278 v. G[LLET et al,

of V 0, the self-energies g, and the occupat ion amplitudes v~ against variations o f the input data corresponding to the uncertainties (i), (ii) and (iii) above. We have chosen

0.~

0.5

0.2E

0.

q7~

~ . -2d5/2 - - ~ l g ?/2

_~ 2d 3/2

: ~ 3 s l / 2

lh 11/2

0?5

0.5

0.25

(3

~P n I i I j .~_ 0 ,

-0.4 -0.2 O. 0.2 0.4

. 2 d 5 / 2 _; -~ z -197/2

*'-- -'L"'-......4------'2 d 3/2 j - 43 s 1/2

~_ . lh11/2 w

b 6E (h)

I I I I I - 1 . -0 .5 0. 0.5 1.

~'qYV 1.

%% --".--...-- ;

%%

025

0.5

0.25

O.

2d 5/2 e ® 1 g?/2

1.

2d 3/2 ,, 4,- 4. 07E

- - • 3sl/2

lh11/2 0.5

0.2. =

] I ' ' I ~ E 2 d 5 / L O,

-1. -0.5 O. 0.5 1.

~ 2d 5/2 lg?/2

\ ~.~...,2d 3/2 \ /

- ~ 1 h 1 1 / 2

d ~ 3sl/2

~E3s1/2 I t I I I

-1. -0.5 O. 0.5 1.

Fig. 3. Influence of the input data on the occupation amplitudes in azxSn, See legend of fig. 2.


for this discussion the case of 121Sn whose spectrum is fairly well known and for which the number equation could not be exactly solved. We consider here the best solution, i.e. the one with smallest 6(N) and with non-vanishing gaps. The corresponding error in the particle number varies around 2.5.

Each of figs. 1-3 contains four discussions denoted by a), b), c) and d), respectively. Two major shells, one containing the Fermi level and the other one just below, are included for these calculations. For the discussion of V0 we consider both cases A (one major shell containing 2) and B (two major shells). The four discussions are as follows:

a) The odd-even mass difference is shifted by a quantity 6P from - 0 . 4 MeV to +0.4 MeV around its value of 1.30 MeV.

b) The positions of the levels of the closed-core neutron shell, which can only be guessed, are varied by 6 E (h) -~ + 1 MeV from the values as given in table 5.

c) The energy of the quasi-particle level 2d~ is varied by + 1 MeV from its value of table 5. It is chosen as an example of a level which is split among several peaks and where the extraction of the spectroscopic factors can yield only a rough estimate of the single-particle distribution.

d) The energy of the quasi-particle level 3s~ is varied by + 1 MeV from its value of table 5.

On all the figures, 6(N) represents the error in the average number of particles. One can observe that this approximate IGE solution cannot be improved significantly by such variations of the input data as shown by the moderate variations of 6 (N) .

The conclusions we draw from these figures are as follows: The effective force strength Vo is exceedingly sensitive to the values of the even-odd

mass difference P. It varies by about 20 MeV when P is varied by +0.4 MeV. Such sensitivity is easily understood by considering that the even-odd mass difference is roughly equal to the energy gap, which in turn is proportional to the strength V 0. A 40 To variation of P around its input value of 1.3 MeV yields a 40 ~ variation of V o around the value 35 MeV. On the other hand, V o is rather insensitive to the uncertainties in the energies of the low-lying core levels (case b) and of levels near the Fermi level 2 (cases c and d) except in case d when the quasi-particle energy of the 3s~ level is lowered.

The self-energy spectrum shown in fig. 2 is practically unaffected by the variations 6P and 6E (h). In the first case the variation of P entails a shift of all quasi-particle energies by a constant energy 6P. In the second case the contributions of distant levels to those near 2 are small, since in the pairing matrix they are weighted down by large energy denominators. In cases c) and d) one sees an important feature of the IGE solution, namely that the uncertainty in the quasi-particle energy Ev of a level (here the 2d~ and the 3s~ levels, respectively) affects only the self-energy gv of this level. In case c) this is reflected by the nearly constant slope of the curves. In case d) it is reflected by the weak slope of all gv except g3s~_ a s E3s~_ varies. Obviously Igv- 2] for

v = 3s~ increases when the corresponding input energy E, increases.


The occupation amplitude v~ and the self-energy gv are directly related, so that all the conclusions reached above for g~ apply equally well to the vv. They are insensitive to f P and fiE (h) and sensitive only to variations in the corresponding input quasi- particle energy E~.

We summarize our discussion by pointing out two important features: (i) error in even-odd mass difference affects only the value of the extracted strength

constant and leaves unaffected the HF single-particle energy spectrum and the associated occupation amplitudes;

(ii) error in one input quasi-particle energy E~ (due to either experimental uncertainties or due to the wrong identification of a state as a pure independent quasi- particle one) affects only the extracted theoretical quantities associated with that particular level. It leaves unaffected the remaining self-energy spectrum g~ and the occupation amplitudes v~ as well as the force strength constant.

5.3. DEPENDENCE OF THE RESULTS ON THE DIMENSION OF THE CONFIGURATION SPACE

An important question which arises in every spectroscopic study of a nucleus is the choice of configurations to be included in the calculation. The choice necessarily requires a truncation of the configuration space. The question arises then where to truncate the configuration space without losing important physical informations. The ordinary shell model has no unique criterion to make a consistent correlation between the truncated space and the renormalizable parameters (i.e. the force strength). To give an example, it would be a misleading practice to use the pairing force strength G = c/A ~ where c is a constant without regard to the number of H F single-particle levels used.

The IGE method provides a means of ensuring the consistency of all the parameters of such a calculation. We have already seen that the method exhibits the strong correlation which exists between the effective strength and the quasi-particle energies. We shall now discuss, as one other result of the IGE method, the relation which exists between the parameters of a configuration mixing calculation and the dimension of the configuration space spanned by the single particle states taken into account.

5.3.1. Convergence o f the effective force strength V o. The influence of the dimension of the configuration space in which the pairing matrix is constructed is demonstrated in fig. 4, which gives the variation of the extracted V o in xZ3Sn and 2°SPb with the number of single-particle states included (here up to 180). The behaviour of V o as a function of the number D of single-particle states may be roughly understood from a schematic model which assumes all the matrix elements Mvv, [eq. (20)] to be equal to a constant C. Then Vo becomes a hyperbolic function of D

V o = 1/DC,

which goes to zero for D --+ ~ . Naturally in a realistic case Vo decreases to a finite value when D increases, since the matrix elements corresponding to remote configurations are reduced by their large energy denominators and by the diminishing overlap


in the radial integrals. The effect is demonstrated in fig. 4 which shows that V o in

123Sn and 2°5pb converges effectively to some finite value when one includes up to

180 single-particle states from the ls~ shell to the lj,~ shell. In fact, in order to reach

approximate convergence, it is sufficient to include two extra major shells besides the

one conta in ing the Fermi level 2, namely the ones immediately above and below the

latter.

v0

50

40

30

20

123 Sn 205 Pb

\,\

Io D

o 40 5 '0 - - 19'o Fig. 4. The effective strength Vo as a function of the dimension D of the configuration space for 12aSn (b -- 2.27 fm) and 2°~Pb (b -- 2.50 fm). The triplet-to-singlet strength ratio was chosen as c~ = --0.4. The configuration space was enlarged by adding successively more subshells further above and further below the Fermi level until all subshells from lsz~ to lja d were included. Counting the m degeneracy of a subshell nlj, the final dimension of the configuration space was then D = 184. Both curves begin to flatten when the two major shells nearest to the Fermi level are included. They have

a similar asymptotic behaviour and converge roughly towards Vo ~ --27 MeV.

It is also quite interesting to note that in the examples chosen in fig. 4, the renor-

malized values of V o are roughly identical in Sn and Pb (about - 2 7 MeV) provided

that enough configurat ions are taken into account.

5.3.2. Stability of the gaps, self-energies and occupation amplitudes. The calcula-

t ion which was performed in 123Sn and 2°spb [fig. 4)] gave also the behaviour of the

gaps Av, self-energies gv = g v - 2 and occupat ion ampli tudes vv as functions of the

increasing dimension D of the configurat ion space. These results for 123Sn are dis-

played in fig. 5. One easily sees at once that, in contrast with the case of the effective

force strength V o, no significant renormalizat ion effect takes place. All quanti t ies

LY (

Mcv

) ~.

(Mev

) t

“J”

2d S

/2

2d5/2

.

0 A_

D

0.

D

10

50

100

150

50

100

150

0

50

100

150

Fig.

5.

E

ffec

t of

th

e di

men

sion

D

of

th

e co

nfig

urat

ion

spac

e on

th

e ga

ps

d,,

the

self

-ene

rgie

s 5,

=

‘&-I

. an

d th

e oc

cupa

tion

ampl

itude

s uy

for

12

3Sn.

T

he

enla

rgem

ent

of

this

sp

ace

is c

arri

ed

out

as

in

fig.

4.

T

he

forc

e pa

ram

eter

s ar

e al

so

the

sam

e.

7he

2d+

stat

e ha

s a

stro

ng

pair

ing

mat

rix

elem

ent

with

the

2d+.

sta

te.

Its

intr

oduc

tion

has

a vi

sibl

e ef

fect

on

A

X6 +

. T

here

afte

r,

all

quan

titie

s A

,, 5,

an

d L

’” ar

e sm

ooth

fu

nctio

ns

of

D,

exce

pt

for

the

low

de

- ge

nera

cy

leve

l 3s

).

,*-_

.’

, -w

.

:

2d5/2

<’

:197

/Z

a.51

lg7/2

!

0.25 t

s ,1

23


A,, j, and v, are extremely stable against the enlargement of the configuration space.

The only exception lies in the case of the level 3s,, whose instability remains anyhow

very small and could be attributed to its low degeneracy and therefore to the sensitivity

of its occupation amplitude to any perturbation.

The jump observed in Azd3 at the inclusion of the state 2d, in the configuration

space (fig. 5) can be attributed to the strong coupling of these spin-orbit partner states.

A further enlargement of the space does not give rise to any new marked jump in any of

the quantities A, lj or v. One may draw from this the conclusion that the stability of

the gaps, self-energies and occupation amplitudes is ensured as long as all dominant

matrix elements of M are taken in account or equivalently, all the levels in the major

shell containing the Fermi levels are included. Such conclusion is a little too optimistic,

since the number equation requires in some cases (see fig. 7) more than one major

shell to be solved exactly. We shall say, on the average, that the use of a configuration

space extended to the two major shells nearest to the Fermi level is necessary to give

well defined and non-renormalizable gaps, self-energies and occupation amplitudes.

5.3.3. Consequence for the true Hartree-Fock energies. We have shown that the

enlargement of the configuration space induces a strong renormalization on the strength

V, but not such great effect on the self-energies EIy and the occupation amplitudes u,.

Turning now to the Hartree-Fock energies E, defined in subsect. 5.6 below, we see

that since the E”, and the v, rapidly become near constant when the dimension D of

the configuration space increases, the E, will depend on D through the interaction V,.

The interaction strength being sensitive to D, the true Hartree-Fock energies will

depend on D. The situation is then somewhat ambiguous for the following two reasons:

(i) When going to very remote shells it is not consistent anymore to neglect the inter-

action with the nucleons of the magic shell (viz. the proton shells in Sn isotopes) which

in the BCS approximation is considered as inert.

(ii) The concept of a well-behaved, purely central effective interaction with a simple

radial dependence, which is used for describing the pairing correlations, is inadequate

for the construction of a realistic self-consistent nuclear field. Accordingly the field

forming matrix elements which enter into the definition of the true Hartree-Fock

energies a, in terms of the extracted renormalized Hartree-Fock energies 8, are likely

to be incorrect.

In the following calculations, we shall limit ourselves to the renormalization effects

due only to two major shells, the one containing the Fermi level I and the one nearest

to it. The obtained values for the Hartree-Fock energies E,, although indicative of the

goodness of the model, are expected to depend on the truncation of the configuration

space.

5.4. THE RENORMALIZED EFFECTIVE FORCE STRENGTH V,.

In the IGE analysis, the two-body force strength V, of eq. (17) is uniquely deter-

mined as the inverse of the highest (positive) eigenvalue of the pairing matrix [eq. (20)].

284 V. GILLET et U/.

INDEPENDENT QUAS[-PARTICLE MODEL 285

The results for all nuclei studied here are summarized in fig. 6, where V o is plotted as a function of the atomic number.The upper curves, denoted by (A), take the restricted configuration space nl and the lower curves (B) the extended configuration space n2. (See sect. 4 for the definition of nl and rt 2 spaces.) The results are given for three different values of the triplet-to-singlet strength ratio c~ = 0.0, - 0.4, - 1.0 as indicated on fig. 6.

5.4.1. Dependence of Vo on the size of the configuration space. It is apparent from figs. 4 and 6 that the extracted IGE values for V o are strongly dependent on the pairing interaction of particles occupying even far-away configurations. Comparison of curves A and B in fig. 6 shows that the solutions obtained with only the major shell containing the Fermi level 2 are quite different from those corresponding to the inclusion of the next major shell. In all previous pairing calculations only the space n t was considered.

5.4.2. Dependence of V o on the triplet-to-sin, qlet strength ratio. The only parameter of the IGE analysis is the triplet (S = 1) to singlet (S = 0) strength ratio. The dependence of Vo on the precise value of the strength ratio is weak as shown in fig. 6, where the ratio is varied (c~ = - 1, - 0.4, 0.0). The dependence becomes even weaker with the inclusion of more configurations (compare the set of curves A and B). The dependence is the strongest for the lighter nuclei involving smaller configuration spaces and the weakest for heavier nuclei where the shells are highly degenerate. As mentioned before, the pairing matrix elements are indeed insensitive to the triplet part of the force (involving only the relative angular momentum L = 1) as compared to the singlet part (which corresponds to L = 0, leading to large Slater integrals and geometrical coefficients).

It is clear from the discussion given above that as pairing is acceptable approximation for the interaction of nucleons only in the outside open shell, the obtained values of V o are strongly model-dependent. Furthermore the obtained values differ from one nuclear region to another since they depend on the values for the oscillator length parameter b which is somewhat uncertain. Accordingly it is not reasonable to ask for a V o constant throughout the table for a check of the validity of the quasi-particle picture. However, if the model holds at all, one should expect relatively constant values in a given nuclear region.

It is remarkable that in the case of the enlarged configuration space n2 (curves B) a strong regularization of Vo takes place. For the Ni and Sn isotopes, the N = 50 isotones, and 2°3pb, /°SPb and 2°TPb, the obtained values of V o vary by less than 1 MeV around their average value within each region. However, the large variations observed in the N = 28 isotones indicate that the model does not apply to these light nuclei as expected from the large energy gaps between shell-model states and the weak degeneracies. The variations in 197pb, t99pb and 2°lPb should not be taken to be too meaningful since the experimental data are mostly lacking and had to be supplemented by guesses. The discontinuity of V 0 in going from 2°7pb to 2°9pb results from shell closure. Here the quasi-particles go to the limit of holes and particles, respec-

286 v. GILLET e t al.

tively. Although the IGE secular problem such as written here has nothing singular for the limit v~ ~ 1 or 0, it is doubtful whether the pairing model is of any use for inter- preting the dynamics of these two nuclei. Nonetheless there is a need to understand why particles and holes behave so differently in the analysis of the data. As seen in fig. 6, the value of V o obtained for 2°7pb is in reasonable agreement with the one obtained for the other Pb isotopes. On the other hand, the obtained Vo for 2 ° 9 p b is 10 to 15 MeV higher, a jump not observed in any of the other nuclei. We may understand this result in the following way: up to 2°vPb, all the levels beyond N = 82 shell are treated as states available for particle occupation. Therefore the continuity in Vo from 2°3Pb to 2°spb is not destroyed by the presence of the N -- 126 magic shell. As soon as the N = 126 shell is filled up such as in 2°gPb, the states below this shell are effectively screened off and therefore there is a sudden decrease of the size of the configuration space. Since, roughly speaking, V o is inversely proportional to the "effective" size of the configuration ~pace, the sudden jump in Vo is a consequence of the stability of the N = 126 magic shell.

5.5. T H E SELF-ENERGIES A N D O C C U P A T I O N A M P L I T U D E S

The energy gaps within a normalization constant ~ are the components, all positive, of the eigenvector associated with the positive highest eigenvalue - 1/Vo of the IGE matrix [eq. (20)]. The normalization constant ~ is determined by variation until the eq. (23) is fulfilled. This corresponds to the condition on the conservation of the average number of nucleons. The fulfillment of eq. (23) is in general possible and determines uniquely ~. In some cases there remains a residual error 5(N} = I N - ( N ) [ (see fig. 7) indicating in all likelihood that some input data may be too far from a pure quasi-particle image to yield a solution when analysed with the present method. The minimization of the quantity 6 ( N ) has been taken as one of the criteria for the choice of the sets of sign functions s~ [eq. (26)], when there were several possible alternatives for the filling order of the subshells.

The residual errors 6 ( N ) constitute also a test of the validity of the pure quasi- particle picture. They are displayed graphically in fig. 7 for the cases A and B, respectively. It can be seen at once that going from A to B one improves almost always the resolution of the particle-number equation either by diminishing the error 6 ( N ) left in A or in some cases, by completely cancelling it. The normalized gaps A v are listed in tables 2-6.

From the gaps A~, one derives straightforwardly the occupation probability amplitude vv of a level v from eq. (24) and its self-energy relative to the chemical potential 0~ = g~ - 2 from eq. (25). The numerical values obtained for these quantities are given in tables 2-6 and are displayed graphically in figs. 8-12. On these figures the g~ are given relative to the self-energy g~o of a reference level v o in order to eliminate the dependence of the ~ on the Fermi level energy 2 [eq. (27)]. The IGE method yields only the quantities 0~. The Fermi level energy 2 is not easily attainable from experimental

INDEP/~:NDE.~T Q U A S I - P A R T I C L E M O D E L 287

I

<~cn

I:

~c

#

(z)

~ d

. o o


~v - ' ~1 f7 /2

19 9/2

If 5/2 ~

2p 3/2 /

A

~'~ - '~ l f7/2

lg9/2 ' 7 i w J :

6

5 If 5 / 2 ~

z, 2p 3/2 /

B

0 i i i i L 0 i i , i ,

49Sc 51 v 53t4n 55Co 49Sc 51 v 531,4 n 55Co

, " O ' v

1

J J

/ J

0,75 / ' ~ 0,75 J

s

/ /

0,5 t 0,5 /

/ /

I /

z A , , If 7/2 1 f 7/2

0,25~-F1g 9/2 0,2 5~-i-1 g9/2 |lr2p1/2 ] l r2pl /2 /[h2p 3/2 012s [ ~ - ~ /11,2p3,2

/-, 9 S e 51 V 53Mn 55Co /'9Sc 51V

ld3/2 , 2 s l / 2 , ld5/2

s s

i J

.4 s

J

f /

/ /

B

1

,,m

5~Mn 5~o

Fig. 8. The self-energies ~v relative to ~afZ~ (upper part) and the occupation amplitudes v v ( lower part)

for the isotones N = 28. The calculation was performed with a Gaussian force o f range 1.7 fm and a triplet-to-singlet strength ratio 0¢ = - -0 .4 . Case A (left part) is a one-major shell calculation.

Case B (right part) is a two-major-shell calculation.


data. In these figures again the cases A and B correspond to calculations performed

in the spaces n, and n2, respectively.

We review now the results for v, and Ey-ElyO displayed in figs. 8-12. The quantities

t r, -12p3/2

A 5_

A- w lg9/2

3_

2_

C 6%

-1 _

57N , 5gN, C’N, ‘33~, 65~,-

:I> lg9/2

3_

1 ? 2 /

%$p?--- N, e3N, G5N,

Fig. 9. The relative self-energies and the occupation amplitudes in the nickel isotopes. The parameters were the same as for fig. 8. Note the instability of the 2p+ level which competes with the 14 level

in crossing B1Ni.

290 v. GILLET etal.

are expected to behave smoothly from one nucleus to the neighbouring ones, if the

input experimental spectra do indeed correspond to pure quasi-particle ones. More

precisely the occupation probabilities I_I~ should increase as mass increases within a

region.

87Rb 8gY “Nb 93lc *7Rb “Y 91Nb 93TC

’ -4

87Rb “Y “Nb 93 Tc 87Rb *‘Y “Nb 93TC

Fig. 10. The relative self-energies and the occupation amplitudes for the isotones N = 50.

5.51. The N = 28 isofones (fig. 8). The large energy gap between the low-lying lf%

and the next empty subshell 2p+ yields an almost trivial solution. With the pairing

force all the occupation probabilities of the shells other than If+ remain small, and

the inclusion of the core (2s, Id) (corresponding to the case B) leaves the results prac-


tically unchanged from the case A. The small effective degeneracy of the relevant configuration space (essentially the lf~ subshell) makes it likely that the pairing model does not apply in this region. This is made more evident when discussing the extracted V 0 (see the previous section) and the Hartree-Fock energies ev.

5.5.2. The Z = 28 isotopes (fig. 9). The characteristic feature of the results is the abnormal filling of the 2p~ subshell. This filling first increases up to s 7Ni and is thereafter depleted. It must be noted that small variations in the number of particles in a state of low degeneracy result in large variations in the values vv. The probability amplitudes for states such as 2p+ and 2p~ are accordingly very sensitive to the details of the numerical treatment. Fortunately the large variation of v2p ~ affects only slightly the overall

results, since the contribution of this state in the gap equation is weighted down by a small statistical factor~ Though the values of v v for nuclei other than 61Ni have a rather smooth behaviour, there is an inversion of the order of the gv in going from STNi and 59Ni to 63Ni and 65Ni. This suggests that the experimental ½- and -~- states are in-

deed more complex than the independent quasi-particle picture might suggest. Let us also mention that the quasi-particle assignments to the levels in 61Ni and 63Ni are uncertain. This uncertainty explains the behaviour of the IGE solutions for these two nuclei.

5.5.3. The N = 50isotones(fig. 10). Thev~andg~followasmoothtrend. In thefour nuclei considered here the experimental spectra are fairly well known, and the chosen input data of table 4 are likely to be rather pure quasi-particle states. The regular increase of the Fermi level 2 is easily apparent. It is near 2p~ for 87Rb, 2p~ for 89y and 1 g, for both 91Nb and 93Tc" If we take the average values of the g~ in the four nuclei,

we obtain glr~ = 0.0, gZp~ ~--- 0.9, gzp~ = 2.8 a n d glg~_ = 3.8, as compared to the

choice of Kisslinger and Sorensen 1) of 0.0, 0.6, 1.8 and 3.4. Thus our results confirm the level ordering adopted by KS as opposed to the usual shell-model ordering 2p~, l f~ and 2p½.

5.5.4. The Z = 50 isotopes (fig. 11). A considerable amount of experimental work has been devoted to the tin isotopes. The information on their low-lying spectra and associated single-particle strengths is fairly complete. The extracted v~ and gv indeed show the expected smooth behaviour except for the state of low degeneracy 2d~. In addition to the reason quoted in the case of the Z = 28 isotopes (i.e. the sensitivity of the occupation amplitudes of low degeneracy states to the details of the numerical treatment), we may also attribute the abnormal filling of the 2d~ shell to the competition between states of high and low degeneracy. The greater pairing energy in the state lh~ yields a tendency to deplete the occupation probability in the 2d~ state. We may also understand from fig. 11 why ordinary quasi-particle calculations locate the excited states in nuclei from 119Sn on at too high energies. This arises from the use of constant average values for the input g~. Here it is seen that the self-energies vary greatly throughout the region of the tin isotopes. The large variation of the extracted gv is not at all inconsistent with the independent quasi-particle picture,


since as we shall see in the next section the pure Hartree-Fock energies ev turn out to be rather constant. To supplement these variations of the ~'v associated with the varia-

O.

L ~'V- "~'2 d 5/2 A

2d 3/2 ~ \

. . . . . . . . . . . . . . . "b--- ; , ' ~ , . 2,

1.

l I I I I i I ~ 0

113Sn 11SSn117Sn 119Sn 121Sn 123Sn125Sn

£v-E2d5/2 B

,,' \ \ 2d 3/2 \ \

...*....,_...~,.-.'-....-..,_ \ / " -','_ ,-"*", I" ' - . / V ".. / " ' .L-.

3 s 1/2 ..... •

~S I I I I I I ~ 11 115Sn 1175n 119Sn 12~n 123Sn125Sn n

0.5

0.25

0.125

" ~ 4

2 d 5/2 19 7/2

p-,.. I "~.-""

.J-..4.e------r'"l ~---..,,

""" J # /

/ ; / ;

• - i 3~1/2 •

1 h 2 2 . ~ /

A

11~n 11,~j n 11~jn 11~:j n 12~Sn 12~ n 1,2~ n

2 d 5 / 2 ~

t ~ -° "°"~"'"°°~"~lel - t . . . . . . "Q"

0.25

0.125 B

I 1 1 k n I I I ,3so ~ns~ 119so 121so 1~3~ 4s~.

Fig. 11. The relative self-energies and the occupat ion ampli tudes for the tin isotopes.


tions of the self-energy corrections, the coupling to three-quasi-particle states has to

be introduced to bring down the 3’ and 4’ states as shown by Kuo et al. “).

6

5

L

3

2

1

0

-1

FY -zli13/2 2a9/2 f ~~',-~li13/2 29912

A ---d 6.

5

3~112 L.

/

3

1.

lh 111:

0.75

0.5

0.25

V”

I I1 I I)

197pb ~~,Db2~lpb2O+b2O5& 2o+b 209pb t+b l$+b 20~p,,203p@.+&+.b 209pb

Fig. 12. The relative self-energies and the occupation amplitudes in the lead isotopes.

294 V. GILLET ‘?f d.

5.5.5. The Z = 82 isotopes (fig. 12). The quasi-particle calculations assuming con-

stant E”, have been quite successful in the lead region. Their success may be understood

from the very smooth behaviour and near constancy throughout this region of & -E”“,

extracted from the IGE analysis. The only marked departure from constancy happens

in going from “‘Pb to ‘09Pb, where the model obviously breaks down. The rather

uniform trend makes it possible to use a set of self-energies S, with some weak A-

dependence as did Arvieux et al. ‘). Again the filling of the shell of lowest degeneracy

(the 3p*), is in marked contrast with the others as it competes with the filling of the

highly degenerate states.

5.6. THE HARTREE-FOCK ENERGIES

The Hartree-Fock energies E, in eq. (3) may be defined relative to any “reference filled

shells.” Any change of the reference filled shells induces merely a shift of each a, by

a quantity 6, which is constant from one nucleus to the other. We are here interested

in the deformation of the spectrum of the a, as a function of the number of nucleons A

and the choice of the reference state (filled shells) is irrelevant for our purpose. The c,

relative to some reference state are extracted from the y”, according to the expression

~~-1~ = g,,- 2 u:(vAIVlvA)+ C u~(vajl+a), (28) A a

where the summations A and a extend over the single-particle states which are un-

occupied and occupied, respectively, in the chosen “reference” shell-model state.

We have chosen as reference state the one corresponding to the filling of all the

shells up to, and including, the lowest subshell of the major shell containing the

chemical potential 1. This subshell is precisely the one adopted as a reference in the

presentation of the gV in the previous section. Namely the reference Hartree-Fock

state corresponds to the shells filled up to lf+ for the N = 28 isotones, up to 2p, for

the Z = 28 isotopes, up to If+ for the N = 50 isotones, up to 2d+ for the Z = 50

isotopes and up to 1 i F for the Z = 82 isotopes.

The extracted Hartree-Fock energies relative to the Fermi level J_ are presented in

fig. 13, and their values in the tables 2-6. They correspond to the case B above with the

inclusion of two major shells. A large part of the slope of the plotted quantities is due

to the variation of the relative spacing between i and the HF energies which is modi-

fied when filling up the shells. The quantity 1 is not easily attainable with accuracy

from the experiment. In order to avoid the /2 dependence one may choose, as in the

case of the &,, a reference HF single-particle state vo. In this case, however, the varia-

tions of & and &,, are entangled and difficult to separate. Thus it appears more appro-

priate, even if somewhat arbitrary, to choose as a reference energy the average value

(29)

In this way the relative HF energies are defined as

&,--8 = (&,-A)-(E-a). (30)

+“9

- A

(M

eV)

8. -

7.

_

-\

\

!$I2

5. _ '8.

2PU2

%

:<

If 512

-0. bA\i.

*PSI2

2. _

0. 4 \

lf7/2

-2.

-3.

-4. I

\ 19

9/z

I I

I II

1 ‘c

"3s"

11S5nl175n 1195n 1215" 1235n125sn

; i lh9/2

I

I

I

I

I

I

)

197pb 1'Bpb201pb203pb205Pb207Pb

Fig.

13

. T

he

true

H

artr

cc-F

ock

ener

gies

z,

re

lativ

e to

th

e Fe

rmi

leve

l I

insi

de

each

nu

cleu

s.

Aga

in,

the

inte

ract

ion

is a

Gau

ssia

n fo

rce

of

rang

e 1.

7 fm

and

with

a

trip

let-

to-s

ingl

et

stre

ngth

ra

tioa

: -0

.4.

-2~3

; 7

2~3/

2

If 51

2

-4 _5

/

if71

2

i I..

.) 1.

. .)

. .

. .*

. ,

. ,

49sc

5’

v 53

t.l”

55co

57

N1

59N

, 61

N,

63N

, 65

N,

87R

b 89

Y g’

Nb

g31c

11

3s,l1

5s,

117S

,119

S,12

1S,1

23S,

125S

,~

197p

b 19

9pb2

olpb

2~pb

~pb2

07pb

*

Fig.

14

. T

he

true

H

artr

ee-F

ock

ener

gies

E

, re

lativ

e to

th

eir

aver

age

valu

e E

.


T h e y are g iven in fig. 14.

All these resul ts a re o b t a i n e d wi th a t r ip le t - to -s ing le t s t reng th ra t io ~ = - 0 . 4 ,

wh ich is one m o s t o f ten a d o p t e d in she l l -mode l spec t roscop ic ca lcu la t ions . In con t r a s t

to the pa i r ing in t e r ac t ion , the field f o r m i n g m a t r i x e lements en te r ing in the H a r t r e e -

F o c k H a m i l t o n i a n [eq. (3)] are m o r e sensi t ive to the exchange c h a r a c t e r o f the force.

H o w e v e r this a - d e p e n d e n c e is n o t s t rong e n o u g h so tha t a change o f ~ w o u l d m o d i f y

dras t ica l ly the conc lu s ions we m a y d r a w f r o m figs. 13 and 14. Th is p o i n t is i l lus t ra ted

in tab le 7 where the ev o b t a i n e d in the t in r eg ion w i t h th ree d i f ferent va lues o f c~ are

c o m p a r e d . T h e changes o f t he e~ wi th ~ are m o d e r a t e .

TABLE 7

Dependence of the Hartree-Fock energies in the tin isotopes on the triplet-to-singlet strength ratio c~

nO 1138n

0 --0.4 --1

i~5Sn 1178n 119Sn

0 --0.4 --1 0 --0.4 --1 0 0.4 --1

lg~ 0.61 0.48 0.31 0.70 0.65 0.57 0.39 0.33 0.24 0.47 0.44 0.41 2d~ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2d~ 2.52 2.36 2.36 3.29 3.26 3.21 2.72 2.71 2.68 1.68 1.66 1.66 3s~ 2.00 1.83 1.91 2.61 2.64 2.68 2.47 2.50 2.54 2.60 2 . 6 1 2.64 lh~t 3.09 3.04 3.11 3.74 3.76 3.79 3.25 3.29 3.35 3.03 3.12 3.24

nlj l~ISn l~aSn 125Sn

0.0 --0.4 --1.0 0.0 --0.4 --0.10 0.0 --0.4 --1.0

lg. I_ 0.34 0.20 0.16 --0.35 --0.69 --1.25 --0.29 --0.63 --1.17 2d} 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2d~ 1.85 1.90 1.77 1.44 1.23 0.91 1.20 1.06 0.79 3s~_ 2.89 2.01 2.63 2.07 2.20 2.40 1.94 2.07 2.26 lh~ 2.97 2.96 3.10 1.43 1.37 1.34 1.45 1.42 1.41

The HF energies are given relative to the 2dz~ level for the values ~ = 0, --0.4, --1.0. The force is purely central, of Gaussian shape with a range of 1.7 fm, The depth V0, in each case, is the one extracted from the experimental data with the IGE method (e.g. for ~ -- 0,4 the corresponding V0 values are listed in table 4).

W i t h o u r def in i t ion , the ev shou ld be c o n s t a n t t h r o u g h o u t each o f the nuc l ea r re-

g ions p r o v i d e d t h a t the e x p e r i m e n t a l i n p u t d a t a fo l l ow the p u r e one -quas i -pa r t i c l e

mode l . I t m u s t be s tressed, howeve r , t h a t the d r a w i n g o f such c o n c l u s i o n f r o m the

p re sen t t heo re t i ca l resul ts is s u b m i t t e d to the res t r ic t ions desc r ibed in subsec t ion 5.3.3.

above .

F o r m o s t o f the levels the va r i a t i ons o f the ev wi th A are less t h a n + 0 . 5 M e V a r o u n d

the i r m e a n values . T h e m a i n excep t ion in m o s t o f the nuc l ea r reg ions is the lowes t

degene racy s t a t e j = ½. I ts H a r t r e e - F o c k ene rgy exhib i t s r a ther m a r k e d var ia t ions .

2 9 8 v . GILLET et al.

6. Conclusion

The answer of the IGE method to the goodness of the pure independent quasi- particle model lies in the behaviour of the extracted two-body force strength and Hartree-Fock energies. In no case did they show the constancy required by their theoretical definition. Some deviations from constancy however had to be expected due to the fact that only a simple, central Gaussian force is used, and that the experimental data are never pure quasi-particle ones. As a matter of fact, the model does not yield any significant information in the N = 28 isotones. On the other hand, it is fairly obvious from the overall analysis that there is an underlying pairing picture in all the other regions studied. This holds with deviations of the force strength V 0 within _ 1 MeV and of the Hartree-Fock energies ev within +_0.5 MeV in a given nuclear region. We consider this as quite remarkable in view of the still considerable uncertainties in the distribution of the single-particle strengths. In that connection an important experimental program involving one-nucleon transfer process must still be carried out. With the collection of more complete data, the present analysis should give more definite information on the extent of the validity of the quasi-particle description of the s.c.s. nuclei.

Salient new results of the present work are the following: (i) The effective strength V o is strongly renormalized when including other major

shells than the one containing the chemical potential 2. (ii) The inclusion of a second major shell, while strongly affecting V o, modifies

very little the quasi-particle states near the Fermi level 2 and their related quantities (e~, A . . . . . ). This result justifies the use of limited configuration spaces and of renormalized forces.

(iii) Although the extracted Hartree-Fock energies are fairly constant, the self energies gv may vary strongly in a given nuclear region with a strong dependence on the quantum numbers involved. This explains the difficulties encountered in ordinary calculations, where the gv are taken as parameters either constant or varying smoothly (with weak A-dependence) from one nucleus to the other. The results obtained in the previous calculations with the gv as parameters to be varied independently from one nucleus to the other for a best fit do not have much meaning, since it is not clear whether the agreement comes from the model or from the phenomenology.

(iv) In practically all the nuclear regions studied here, the filling of the state of the lowest degeneracies is rather irregular as it has to compete with the filling of the high- degeneracy states. The effect persists in general in spite of reasonable modifications of the input data and of the parameters. No such effects has yet been reported experimentally, even in the tin isotope work of Cohen et al. 13).

(v) The triplet-to-singlet strength ratio plays only a very minor role in pairing model. Even the extracted Hartree-Fock energies are not much changed at the variation of this ratio.


As a p rog ram for fur ther work, the I G E method can be appl ied to the two fol lowing

prob lems:

(i) The analysis of the low-lying intr insic spect rum of deformed odd-mass nuclei.

There the exper imenta l quas i -par t ic le energies which are needed are much more

numerous than in the spherical case because of the removal of the m-degeneracy. Such

s tudy requires a long and difficult exper imental p rog ram for collecting the desired

data .

(ii) The calculat ion of the states of the doubly even s.c.s, nuclei in the f r amework of

the in terac t ing quasi -par t ic le model can be pe r fo rmed util izing as input da ta the

in te rpo la ted results of the I G E method for the neighbour ing odd-mass nuclei. Thus

one links direct ly the states of the doub ly even s.c.s, nuclei to the exper imental spec-

t ra of the odd-mass ones wi thout var iable parameters cor responding to the Har t ree-

Fock energies or the force strength. This will be the subject of a for thcoming paper .

The au thors t hank Professor C. Bloch for suggesting the possibi l i ty of invert ing the

gap equat ions and for helpful discussions. They are thankful to many experimental is ts

of the C E N Saclay for discussions and communica t ions of exper imenta l results pr ior

to publ ica t ion , in par t i cu la r to M. Bassani, M. Conjeaud, M. Delaunay , H. Faraggi ,

S. Har ra r , L. Pap ineau and J. Picard.

They are par t icu la r ly thankful to Mrs. N. Tichi t for carrying out the lengthy num-

er ical calculat ions.

References

1) L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 9 (1960) 2) R. Arvieu et al., Phys. Lett. 4 (1963) 119 and references quoted there;

M. Rho, Nuclear Physics 65 (1965) 497 3) T. T. S. Kuo, E. U. Baranger and M. Baranger, Nuclear Physics 79 (1966) 513 4) V. Gillet, Rendiconti della Scuola Internazionale di Fisica (Enrico Fermi), Session XXXVI,

(Academic Press, New York, 1966) 43 5) V. Gillet and M. Rho, Phys. Lett. 21 (1966) 82 6) L. S. Kisslinger, Nuclear Physics 78 (1966) 341 7) J. H. E. Mattauch, W. Thiele and A. H. Wapstra, Nuclear Physics 67 (1965) 1 8) D. D. Armstrong and A. G. Blair, Phys. Rev. 140 (1965) B1226 9) M. K. Brussel, D. E. Lundqvist and A. I. Yavin, Phys. Rev. 140 (1965) B838

10) E. R. Cosman, C. H. Paris, A. Sperduto and H. A. Enge, Phys. Rev. 142 (1966) 673 11) N. Auerbach, Nuclear Physics 76 (1966) 321 12) Nuclear Data Sheet (U.S. Government Printing Office, Washington, D.C., 1960) 13) B. Cohen, Prakash and Schneid, private communication;

B. Cohen and Price, Phys. Rev. 121 (1961) 1441 14) V. Gillet, E. Sanderson and A. M. Green, Nuclear Physics 88 (1966) 321

a phenomenological test of the independent quasi-particle model

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