Shell-model calculations on the N = 82 ODD-proton nuclei (II)

Nuclear Plzysics Al67 (1971) 545-576; @ N~~~~-~e~~~~~ ~~~~~s~jng CO., A~~@~~u~

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

SHEL&MOHEL CALCULATIONS

ON THE N = 82 ~~~P~~T~N NUCLEI @I)

K. HEYDE and M. WAROQUIBR t

Received 2 November 1970

(Revised 26 February 1971)

Ahstraet: The q~~sj~a~i~~e T~-~a~~o~ approximation has been applied to a calctiation of the low-lying states in single-closed-shell nuclei with N = 82 neutrons. A central force of Gaussian shape with spin exchange is taken as the residual nuc~~o~-~uc~~n interaction, An extensive study is made of the resemblance with the realistic Elliott mteraction. The single- particle energies are determined by the iterative inverse-~a~-~ua~o~ method, The Ham& tonian matrix is diagonalized in a one- and ~r~-quasip~~~~e space after eS~~atio~ of spurious states. Transition rates, half-lifes, magnetic dipole morne~~ and electric quadrupole moments are cakulated, discussed in detail and compared with results obtained with the realistic Elliott interaction and the experimental data, The evaluated spectroscopic factors are studied and compared with the experimental one-nucleon transfer data w&b good quantitative agreement.

Ildmduction

In an earlier investigation of spherical nuclei with a single~el~sed-neutron-shell with N = 82, the odd-proton nuclei were studied with a rather schematic force, a surface delta intera~~on, in order to examine the main feat~es of the excitation spectra of the nuclei “3. Since then, there have been numerous experiments on these nuclei, both reaction studies “‘“> and p-decay studies, The variation with atomic mass number of the strongly excited states in the t3He, d) stripping reaction “> has establish~ the o~u~a~o~ probability varia~on and it is now pos~ble to look for a detailed comparison with theory. ~irnii~l~, the’ new experiments yieIding y-ray branching ratios, magnetic moments and q~~drupole moments were performed in order to compare results with the BCS theory of pairing.

Freed and Miles ‘* “) ma d e an analogous approach conferring the method but used a force directly deduced from the nucleon-nucleon smattering phase shifts, i.e. the Elliott reduced integrals. In order to use a realistic force for detaited nuclear structure c~cula~ons in a restricted mass region, 137 S; A r 145, specific renormalization processes from the excitation of the underlying 2 = 50, N = 82 core should be considered as discussed in the fundamental articles of Ruo and Brown ‘* “1. For this reason, we use a central interaction of Gaussian shape with a spin exchange

t As~iraut of the Nationaal Fonds voor Wetenschappe~ij~ Onderzoek,

545

546 K. HEYDE AND M. WAROQUIER

admixture, in which the spin triplet to spin singlet ratio is determined lo) as t = 0.2, for a good description of the low-lying excited states and the high nuclear level densities in the energy region of 1.2 MeV s E, 2 2.5 MeV. The contributions from 3p-lh and 2p intermediate states, which have been calculated in an exact way by the Tri&ste group ‘) in the Sn isotopes with the same valence shells as the N = 82 nuclei, can be analysed with the dominant multipole modes (pairing + quadrupole). From this exact renormalization process, matrix elements that are equivalent to Gauss- ian interaction matrix eIements with positive t-values are obtained. For this reason, the Freed and Miles calculations “) resemble a bare matrix element calculation with the known effect of too little binding of the ground state ‘) and a compressed level structure “). This fact is also found in the very high level densities for the low-lying 3QP states in the calculation of ref. “) compared to the experimental situation. The first objection is nicely studied in the comparison of the pairing matrix elements for the Elliott and Gaussian forces and the second effect is studied in the comparison of the level schemes with both forces.

Together with the results of the standard shell-model calculation of Wiidenthal 2), who used a SD1 as a residual interaction, we thought that at this stage a detailed comparison could be made of the force dependence as both schematic (SDI), effective (Gaussian with spin exchange) and realistic (Elliott matrix elements) forces have been used in the same model space for calculating the level schemes and other properties. Also the influence of enlarging the model space is studied as SD1 has been used both by Wildenthal “) (matrices of the order 300 x 300) and Waroquier and Heyde ‘) (38 x 38).

Some mathematical formulation of the electromagnetic transition probability formalism has been developed in sect. 2 with applications to the calculation of magnetic dipole moments, electric quadrupole moments and also transition rates between 3QP and 1QP states. In sect. 3 the residual interaction is discussed in view of the other forces used in similar approaches, and differences are pointed out. The single- particle energies are determined from the experimental level schemes, by using an iterative IGE procedure, coupled with TDA calculations, which account for interactions of the 1QP 1eveI with ail other 3QP states in an exact way. In sect. 5, the energy spectra are compared and in sects. 6-8, electromagnetic properties are calculated and discussed. Finally, in sect. 9, the spectroscopic factors for (3He, d) stripping reactions are compared with the experimental data and the extracted occupation probabilities are discussed.

2. Electromagnetic transition probabilities: Formalism

Since this work is a continuation of our first study ‘) on the N = 82 odd-proton nuclei, we refer to it for the theoretical formalism of the procedure followed for the construction of the Hamiltonian matrix with a central force of Gaussian shape with a spin exchange admixture as the residual nucleon-nucleon interaction, in com-

N = 82 ODD-PROTON NUCLEI 547

parison with an SD1 force used in ref. ‘). Also, the electromagnetic moments and transition rates are discussed in detail in this work.

2.1. THE ONE-BODY TENSOR OPERATOR IN SECOND QUANTIZATION

Let ol, be any single-particle tensor operator of rank I. This operator can always be written in terms of quasiparticle operators,

with

___ [zt,u,A+(ffblm)+(-I)‘-“u,u,A(abl-m)], (2.2)

(=j;g) = c <+Lll4lljb) -_ J2j, + 1

[tt&--( - l)‘c’& ~,]~O(dh%), (2.3) a,b

in which

Here sB = (- l)ibmmb and the factor c’ in eq. (2.3) come from the intrinsic properties of the single-particle operator 6, under the combination of time-reversal and hermitian conjugation. More details of it are given in an earlier work lo) on the doubly even N = 82 nuclei. The matrix elements of an electric multipole field

I &Yi = er”P, (2.6)

and of a magnetic multipole field

J1 = &21+l)rr-“I:~-, 0 P,]’ (2.7)

are expressed in ref. ’ “)_ For both operators, the factor c’ becomes + 1.

2.2. TRANSITIONS BETWEEN TWO (IQP+3QP) STATES

In order to calculate the reduced electromagnetic transition probabilities B(d; 1), magnetic dipole moments and electric quadrupole moments in the scheme of one and three quasiparticles, we consider the general expression of the matrix elements ( drJZJ6,, I$,JZ,> which contains four contributions.

As monopole transitions do not occur in the odd-A nuclei, we drop out the term (2.4) in the expression of the operator O,, in second quantization form. The matrix


elements, occurring in the four contributions, all contain the same Clebsch-

Gordan coefficient <$i&iZm]#r&?,>. The rest of their explicit evaluation is

given by the following expressions:

cl1 ~ <ffllsIllo@i> ______ (Uf ui - c’( - l)'vp Vi),

J2$,+ 1 (2.8)

t Cl3 z q-l)/‘-> 1/ 2$i+1 cr ~ 2/,+1 31.

(2.10)

The expression of C33 can be obtained in a straightforward way from the

appendix (A.8) of ref. 33) because of its very long structure.

These four expressions form the basis of the evaluation of the matrix elements

<$fdfIolml%idi)~ in the calculation of the reduced transition matrix elements

B(d, r), the magnetic dipole and electric quadrupole moments.

The results of the obtained transition probabilities are discussed in detail in

sect. 8. Everywhere, the agreement with known experimental values is very good,

except for the strongly retarded Z-forbidden Ml transition from the first excited state

(3’ or 3’) to the ground state.

As we shall see, we get a retardation of about 3.2 x IO5 s.p.u. against 400

experimentally in . 139La It means that the theoretically obtained value is about

800 times more retarded. We can give the following consideration to explain this

disagreement. As the Ml transition from the first +’ or 3’ state to the ground state

is I-forbidden, the C,, contribution is vanishing, so the dominant terms for B(Ml)

have to come from C, 3 and C3 1. Their expressions [eqs. (2.9) and (2.10)] become very

simplified by the factor 61,1b appearing in the matrix elements (n,Z,jJ]~i ]lnJ&,>.

Furthermore the D (cd& 1) coefficient rejects the solution a = b, so that only

<2d~ll~Il12d~) and (2d~ll~Il12d~~ result, giving a non-vanishing contribution

among the five available single-particle orbits: 3s+, 2d,, 2d,, lgl and Ih,.

The first term of eq. (2.9) always gives a zero contribution as the orthonormal space

of 3QP states with i?+ (cdJe; $+f,)O> is built according to the condition:

j, > j, > j, if the three angular momenta are all different and in the case of two

equal angular momenta j, = j,, 2 j,. With these constrains on the sum, we can sim-

t C’31 has the same form of Csl with the following permutation

i~f;cjc’;d~d’;J~J’;e~e’.


plify eq. (2.9) in the case of an I-forbidden Ml transition to

c31 = ,$L-a)( u+ u* - “+ u+)(2J + 1)%,*6,* (sCj‘ (

t B’i “) (1+( - l)J3Y,+). 9r 3 1

(2.11)

This expression depends practically neither on the considered two-body force nor

on the set of single-particle energies used. Only the factor with the occupation prob-

abilities varies smoothly according to the choice of the nucleon-nucleon interaction,

but this cannot explain the retardation which is too large for the I-forbidden Ml

transition rates from the first excited state to the ground state in the N = 82 nuclei.

The strong condition 6,$6$,s,, appearing in eq. (2.11) retains only a few

number of configurations from the total 3QP-1QP contribution C&‘, to give

It is clear that the component of the 3QP state B+(~JJ~; ff)16) plays an

important role in the magnitude of C:q(. Unfortunately, these configurations have

very small amplitudes in the wave functions describing the low-lying 1QP levels for

the odd-proton N = 82 nuclei. The amplitudes are also not stable against small

changes of the Hamiltonian matrix: They and any expression depending on them

have practically no physical significance. Since in the evaluation of the reduced

transition probability B(MI) for the considered I-forbidden Ml transition, we have

to take the square of the sum of these contributions, agreement with experiment

will only happen in a few accidental cases. We refer to sect. 8 for more quantitative

details. The structure of the lQP-3QP contribution C :“’ is the same as expressed in

eq. (2.12) after the permutation i ;A f.

2.3. MAGNETIC DIPOLE MOMENTS

The magnetic dipole moment operator is defined as

J- ^ p, = \/$r-rU, . (2.13)

The expression of &?r is mentioned in (2.7). The magnetic dipole moment of a state

8” in an odd-proton nucleus in defined as

p = J~(#.A = &[.&1,$.A = 2). (2.14)

The 3QP-1QP and IQP-3QP contributions are reduced to the same expression.

so that we can write p = p,r +2~~~+p~~.


2.4. ELECTRIC QUADRUPOLE MOMENTS

The electric quadrupole moment operator is given by

Q, = Jymr2~2. (2.15)

The electric quadrupole moment in a nuclear state in an odd-proton nucleus is de-

fined as

Q = <cc94 = dl&2old~ = 6)

and an application of the general expression gives the complete expression for Q,

where Q = Qll+2Q31+Q33.

3. Residual interaction

As a residual interaction, a central force with a Gaussian radial shape is considered

because of the large P,(cos (I,,) components, compared with a SDI, which has

been studied in ref. ‘). The expression is

V(r) = - V, e-8’2(P,+ tPr), (3.1)

where r is the relative distance between the two nucleons and Ps, PT the spin singlet

and triplet projection operators. The force strength V, is obtained from the IGE.

The other parameters p and t, the spin-triplet-to-spin-singlet ratio, have been obtained

from fitting procedures to the level schemes and transition rates for the doubly

even N = 82 nuclei lo) as /3 = 0.325 fmm2 and t = 0.2. The influence of t is quite

pronounced in the magnitude of the pairing matrix elements G(uu cc; J = 0).

In the work of Freed and Miles 5), the Elliott reduced integrals have been

(a) (b)

(C) (d) Fig. 1. Dominant renormalization processes occurring from the core polarization. I(a) shows the bare matrix elements, 1 (b) the bubble 3p-lh diagram, l(c) and l(d) the 4p-2h and 2p states,

respectively, as intermediate states.


used, but we can show that a properly renormalized realistic force will lead to matrix elements for J” = O+ and J” = 2+ quite similar to the Gaussian interaction matrix elements. The renormalization of the Elliott force, through core excitation, is shown in fig. 1. In fig. lb, the particle and hole are coupled to intermediate J”, T”. It has been shown by Brown and Kuo ‘I) that each J”, T” contribution to the renormalization process behaves as an irreducible tensor force (direct term) of the form

F(1, 2) = G,-, &ri, r2)[YJcg 0 YJf,]“‘. (3.2)

Actual calculations in the Ni region 11) show that the dominant contributions from fig. lb, come from J” = 2, T” = 0 and J” = 4, T” = 0. Although the radial dependence for a P2(cos fI12) force ( Z:Y: * r$) can differ from G,,.,., (r,, r2), the resemblance of the matrix elements of the bubble diagram and the quadrupole force is very good ‘l). The other two diagrams lc and Id are discussed in ref. I’), and are likely to contribute only significantly to the pairing matrix elements. In this way, a good approximation to a renormalized realistic interaction should be

V = vElliott+ vpairing + vquadrupole . (3.3)

From the contributions of the renormalization process, as calculated in an exact

t (MeV)

Fig. 2. The difference between the Elliott and Gaussian force (t = 0.2, p = 0.325 fm-‘) pairing matrix elements, as compared with a renormalization process, indicated by the pairing matrix

elements for the sum of a pairing and quadrupole force.


way by the Trieste group ‘) in the same nuclear region (the Sn isotopes) with the same valence shells i.e. Ig,, 2d,, 2d,, 3s, and lh,, the force strengths of Vpairing

and ~quadrupole can be obtained by fitting the interaction matrix elements of the force (3.3) to the exact contributions.

In fig. 2, the renormalization needed to go from the Elliott matrix elements to the Gaussian interaction (with t = 0.2) for J” = O+ is shown and compared with the sum of the pairing and quadrupole contributions with

’ (3.4)

and as Vpairing, the usual pairing force expression. As strength parameter X2, we obtain the value 1.52, which is close to the Kisslinger and Sorensen estimate of 125,/A for the quadrupole force. The extra pairing contribution comes from a pairing force with strength parameter “) G = 0.08. This vaIue is somewhat smaller than the estimate 25/A since these pairing matrix elements have to correct for the too low pairing components in the realistic force. For large negative values of t = - 1 .O, the Elliott bare matrix elements are reproduced, so that the spin-triplet-to-spin-singlet ratio in the Gaussian force acts as a renormalization parameter.

When calculating, with the quadrupole force with X, = 1.52, the corrections to the G(u~c~, J = 2+) matrix elements, we reach the following conclusions:

(i) The extra quadrupole force correction makes the realistic JR = 2+ matrix elements still more attractive. In the case of the important matrix elements which determine the low-lying structure of 3QP states in the odd-proton nuclei such

as G(I$S$; 2’), G(%$@; 2+), G(s$3_5; 2+); h G t e aussian matrix elements are nicely reproduced.

(ii) In other cases, the renormalized Elliott force gives larger values for G(abcd; J = 2+) compared to the Gaussian force. This fact can result in a still larger depression of some high-lying 3QP states.

The results are shown in fig. 3. So, the Gaussian force with t = t-O.2 gives a good effective force, for nuclear structure calculations on the odd-proton N = 82 nuclei.

4. Single-particle energies and the odd-even mass difference

By inverting the gap equations, it is possible to obtain from experimental data for the N = 82 odd-proton nuclei, a set of single-particle energies ae-aE,,r where the lg;. singte-particle orbit is considered as the reference state ’ “). They show a smooth variation in the whole region of the I? = 82 nuclei. As in the case of I3 9La and 141Pr, the f” = +’ and $’ 1QP states become strongly mixed with the low-lying 3QP states (see sect. 9); it is necessary in order to obtain good 1QP energies, to use the IGE in a modified way which we call IGE -I- TDA.

In this procedure, after doing the IGE with a first set of initial 1QP energies


E lQp 9 9 the Hamiltonian matrix for each $” is constructed and diagonalized. In this way, the interaction of the 1QP state with all possible 3QP levels is considered in an exact way. The difference Eyp’ - w8, where Ok, the TDA eigenvalue corresponding to the level with largest 1QP amplitude, is added to EiQ” to obtain a corrected 1QP

(Me G tabcd,J=2+)

0.8

i

i

Ia

0

X O.?

Gaussian afhxtive force (t=+0.2)

Elliott force

Renormatited Elliott force with P2-force

Fig. 3. The Jn = 2’ matrix elements, as calculated with the Gaussian force (t = 0.2, /I = 0.325 fmd2), are compared with the Jn - 2+ matrix elements for the realistic Elliott reduced integrals, and the renormalized realistic force. The numbers 1,2,3,4,5 for a, b, c, d stand for 39 2d+ 2ds,

lg;, 1 h+, respectively.

energy and to start a new IGE procedure. This coupled set of IGE+TDA equations is solved in an iterative way until convergence is obtained (normally 2 to 3 iterations). In this way, we obtain a set of unperturbed 1QP energies, the single-particle energies, the force strength of the Gaussian interaction with t = +0.2 and /I = 0.325 fmW2, and the occupation probabilities.

The reversed order of the $” = 4’ and 3’ 1QP levels in 141Pr as obtained by


Freed “) is remarkable compared to our iterative IGE+TDA method. This can be

due to the too small pairing matrix elements of the Elliott force, which are input

parameters in the IGE method as performed by Freed ‘, “).

In fig. 4, we show the experimental odd-even mass differences P,,(A), which are

Fig. 4. Half the odd-even mass difference, from the experimental binding energies, as compared with the lowest 1QP energy (E#) and the lowest TDA eigenvalue WY.

used as an approximation to the 1QP energy of the lowest state in each odd-proton

nucleus. These quantities are defined as

P,(A) = 3[S,(A + 1) -S,(A)],

for the odd-A proton nuclei, where

(4-l)

S,(A) = E,(A) -E&4 - 1). (4.2)

Here, S,,(A) denotes the proton separation energy and EB(A) the total binding energy

of the nucleus with mass number A, as taken from Mattauch and Thiele 13). In the

BSC theory, one can show that

W,(A) = E,(A), (4.3)

where ,!ZY(_4) is the lowest 1QP energy in the nucleus A. Although sometimes large

errors are found in the experimental data, the agreement is very good (see fig. 4).

The agreement is still better if we take the lowest TDA eigenvalue o/(A ) as a measure

of the odd-even mass difference. Compared to other IQP calculations of Rho 14)

and Lombard 1 ‘), our values for the lowest EiQp are systematically smaller and

closer to the experiment.

Also, Freed relaxed the requirement of reproducing exactly the odd-even mass


differences l6 P ) ,( A), in order to obtain IGE eigenvalues with the Elliott matrix elements of ~3 z 1.0, which with exact reproduction give W z 0.75. This in fact can also be responsible for the difference in 141Pr, because in this case, the error on P,(A) is only 4 ‘A. So, a 10 % change as taken by Freed is too large in 141Pr.

EXPERIMENT (El H.WILDENlHAL et and

(G.HOLM ‘)

al?

s&2 fDA

(GAUSS wth t =0.2) IbtWARCdUER . K.HEYDE,

SHELL -MODEL (SOI)

Fig. 5. Comparison of the TDA with the Gaussian interaction and the Elliott interaction as the residual two-body force with the experimental data. Levels connected by dashed lines are well

established states from nucleon-transfer reactions. “) denotes ref. ‘) and “) denotes ref. I*).

5. Energy spectra

The Hamiltonian matrices for each 2 Z of total angular momentum are diagonalized exactly in the proposed space. From the eigenvalues nuclear level schemes are constructed and are shown in figs. 5-9. The results from a central interaction with a Gauss- ian shape as the residual two-body force are compared with those from a surface delta interaction ‘) and the realistic Elliott interaction ‘* “) and with results obtained from an approximate standard shell-model calculation ** “) with a modified SDI. We give a short discussion of each nucleus separately.

5.1. THE -‘Cs NUCLEUS

The experimental data are obtained from the 137Xe p-decay results of Holm r8) and the results of Wildenthal et al. “9 “) obtained from the 13’Ba(d, r) pick-up reaction. The experimental excited states are collected in fig. 5 with their proposed

0.0

% (MeV)

3c

j> smc

0x6 ~3199

‘mJl ~. ~~ m

Fig. 6. See caption to fig. 5.

“‘Pr (n,n’r) h’R.OAVE)

.‘%die d ?‘Pr fB.H WILDENTHALI

59pr*2 TDA -SHELL-MODEL

(GAm t =0.2) (SOI) IMWAROQUIER-KHEYDE) IBHWILDENTHALI

TDA (ELLIOTT 1

/N FREED f WMILESI

0145 ----A___ 0.110

0010 /1L; %+ I 00



Ex (Me’.‘)

3!

ii

o-

O-

3-

0 !

‘L3Sm-DECAY TDA 62 ( 61

TDA TDA 10 DEFRENNE *E JACo85) (GAUSS) (ELLIOTT) wJI)

+ STRIPRNG-REACTION ‘Wd I’He,d, “+m

(M.WAROQUiER+K.HEYDE) (N.FREEC+WMILES) (M.WAROWIER+K.HEYDE)

(B.HWILDENlHAL)

Vf vi

i

0273 0.306 ___~~ 0110 7/;

% I 00


63EU 62

STRIPPING-REACTION TDA SHELL-MODEL TDA

“‘Sm t3He,d) lL5Eu (GAUSS) cxll) (ELLIOTT)

,BH WILDENTHAL, ,MWAROO"IER+KHEYDE, ,E NEWMAN KSTOTH et a!) IN FREED*WMiLES,



spin assignments. We have calculated all possible low-lying states (E, < 3 MeV)

and have compared these results with an approximate standard shell-model calculation

of Wildenthal (see fig. 5). He restricted his model space, by considering only (lg,,

2d,)“, and (lg,, 2d,)“-‘. (2d,, 3~;)~ configurations, where 2’ is the number

of extra protons. As in I3 ‘Cs, there are only five extra protons; this restriction has to

give better agreement with experiment in this case, than in nuclei with a higher number

of extra nucleons (see 14’Eu). The resemblance of the two different shell-model cal-

culations is remarkable. The doublet of $+ and y’ at about E, = 1 MeV is repro-

duced in both cases, as well as the number of low-lying (E, < 2.5 MeV) g’ and y+

states, etc. The two theoretical level schemes are in excellent agreement with experi-

ment. The low-lying experimental levels at 0.849 MeV and 0.981 MeV probably

have 3’ and 3’ spin assignments respectively, as also suggested by Freed and Miles “).

We predict a 3QP state of spin +’ at 1.463 MeV, which corresponds closely to the +’

level seen at 1.490 MeV, in agreement with the +’ state at 1.370 MeV in ref. “) and

at 1.440 MeV in Wildenthal’s model.

5.2. THE ‘39La NUCLEUS

Many experiments 19-23) have been performed up to now leading to the con-

struction of the level scheme of the N = 82 nucleus l3 9La. In fig. 6 we have separated

the experimental results from the decay of 139Ba [ref. ‘“)I from those obtained

from the stripping reaction ‘38Ba(3He, d) [ref. ‘“)I. We have done this for clarity

in order to establish in a better way agreement of our theoretical predictions with the

reaction data of Wildenthal on the basis of the calculated and measured spectroscopic

factors (see sect. 9). The Gaussian residual interaction reproduces the level density

in the region 1.00 MeV-2.50 MeV in contrast to the SD1 where the lowest three-quasi-

particle levels are about 0.600 MeV higher. This finding is not unexpected since the

quadrupole matrix elements for the SD1 are usually much weaker than for the Gauss-

ian and the Elliott interaction “). This fact can also easily be seen in ref. lo) where

in the doubly even N = 82 nuclei the first excited Of and 2+ states are not lowered

enough for the SDI. Just as in 13’Cs, 2” = 4+, J$+, lTL* and J$’ states are

calculated for ‘39La. We mention the very low J$’ state at 1.646 MeV. No q+,

?$’ and-i5_+ states were calculated in ref. “). There exists some uncertainty about the

spin assignments of the 1.420 MeV level. In the /3- decay of ‘39Ba [refs. ‘OS “‘) I, a

logft value of 7.5 is found, which is considerable lower than values to the other states

above 1.200 MeV. It could be explained if the level had a negative parity (,-, $-,

-2,-). In the reaction work of Wildenthal, an y- assignment is assumed for the 1.420

MeV level, which, however, in the p- decay of 139Ba [refs. 20* “‘)I requires strong

y-feeding followed by E3 and M2 + E3 y-decay of that state. In fact this possibility

cannot be excluded since the theoretically obtained half-life of 3.7 ns for the JG--

level at 1.420 MeV in ‘39La agrees very well with the experimental 2ns proposed by Berzins et al. ‘l). Th eoreticahy we cannot expect low-lying states with

negative parity except for the Ih, single-particle level, because in the considered


restricted scheme of five singie-particle levels 3s+, 2d,, 2d,, lg, and lh+ the low- lying 3- state appearing in the doubly even N = 82 nuclei cannot be lowered enough, as discussed in detail in ref. ’ “). Extending the space with the negative single-particle levels of the N = 5 and N = 3 oscillator shell, can yield a solution for this problem.

5.3. THE 14’Pr NUCLEUS

The experimental data for 141Pr are taken chiefly from the 141Pr(n, n’ y) reaction work of Dave et al. 24), the /I’ decay of 141Nd of Beery et al. 25) and from the proton transfer reactions ’ 40Ce( 3He, d) and 142Nd(d, 3He) by Wildenthal et al. 2s “). Fig. 7 shows the three theoretical level schemes. The agreement of the TDA results and the standard shell-model results 26 ) is again remarkable. The level density in the calculation with the Elliott interaction is larger than in our study. The levels are all compressed in the region 1.0 MeV-2.0 MeV: This is caused by the small excitation energy of 0.010 MeV of the first excited 2” state, in comparison with 0.145 MeV experimentally and 0.146 MeV in our work. This small value is the consequence of the set of one-quasiparticle energies E, as used by Freed and Miles “) which produce the crossing of the 3’ and 3’ levels between 141Pr and ‘43Pm in contrast with the set used in this work. With a Gaussian two-body force and with a SDI as the residual interaction, the crossing occurs between ’ 3 gLa and 141Pr, which seems to agree with the study of 14*Ce in ref. 1 “). It is evident that this difference of about 0.135 MeV in the value of Elp3 influences the low-lying 3QP states by more than 0.200 MeV, which explains the high level density of the 3QP states in ref. “f.

5.4. THE 143Pm NUCLEUS

We have taken experimental information from a recent 143Sm p-decay work of De Frenne et al. “) and from a pick-up reaction study ‘44Sm(d, 3He)‘43Pm of Wiidenthal et al, ‘* 19). Fig. 8 shows the comparison of the results with different nucleon-nucleon interactions in the same model. We notice easily that the five one-quasiparticle levels are reproduced very well in the three cases, but that the low-lying 3QP state of spin 3’ at 1.060 MeV is best established in the calculation with an Elliott interaction of Freed and Miles. The SD1 force is unable to lower it sufficiently, the 3QP group is already separated from the IQP levels by an energy gap of about 600 keV in disagreement with the experiment and the other two similar calculations with other interactions. This fact shows once more the incapacity of the SD1 to lower 3QP states in a limited space. Extension of the space, as Wildenthal has done in his standard shell-model calculation 2), can revalue the surface delta interaction, as demonstrated in ref. lo). Once more the Elliott interaction, does not spread the energy levels su~ciently.

5.5. THE l.-Eu NUCLEUS

Stripping studies ‘44Sm(3He, d)‘45Eu of Wildenthal I99 “) and fl’ decay of


‘*‘Gd of Newman et af. I’) give the experimental information. In order to compare the standard sheIl-model calculations for ’ 3 7Cs with onIy five extra protons with the analogous calculation for 145Eu with 13 extra protons, we have also mentioned their results in fig. 9. The difference is enormous: As the experimental agreement in 137Cs was excellent, it is poor in 145Eu. It is evident that the restriction of the space by taking only (Ig,, 2d,)” and (lg,, 2d,)“- ’ * (3s+ 2d,)’ configurations is insufficient. Two-particle excitations into these higher orbits 3d,, 2d, and lh+ have been excluded due to the too large dimensions of the matrices. Experimentally, the five single-particle levels are separated from the rest by an energy gap of about 500 keV, excellently reproduced by the TDA with a Gaussian interaction, This energy gap does not exist in the calculation of Freed and Miles using the non- renormalized Elliott interaction. Their 3QP states are lowered too much, which seems to be the general trend for the Elliott interaction as used in ref. “). From the description of the energy spectra of the five considered odd-A N = 82 nuclei, we can draw some genera1 conclusions:

(i) The restricted standard shell-model calculation of ~ildenthal with a modi~ed surface delta interaction as residual force, shows its success in the lighter N = 82 odd-proton nuclei with only about five extra protons, but fails in the higher region ( ldSEu) due to the restriction of the possible configurations to avoid too large dimensions of the energy matrix. In contrast to it, the QTDA proves its success in the higher region.

(ii) The influence of the choice of the nucleon-nucleon interaction on the energy spectra in the same configuration space is considerable. The SD1 does not lower sufliciently the low-lying 3QP states, due to the small quadrupole matrix elements. The comparison of the Gaussian and the Elliott interaction in the energy spectra is also interesting. Since the experimental levels are very well reproduced in both cases, a general trend for the Elliott interaction is the high level density of the low-lying 3QP states as well as the too low position of this group of 3QP levels. It can be explained by the fact that Freed and Miles have used the bare Elliott matrix elements, where the pairing matrix elements especially are too small (see sect. 3). They have to be renormalized for core-polarization effects, mainly by a P2 force which causes more interaction between the low-lying 3QP states and will decrease the high level density. Analogue effects have been studied by Gmitro et al. “) in the Sn region.

(iii) As demonstrated in the doubly even N = 82 nuclei lo), the influence of the configuration space on the energy spectra working with the same residual force, shows that the SDI gives a better trend for the low-lying states when extending the configuration space as expressed by the restricted standard shell-model calculation in comparison with the QTDA ‘).

(iv) In order to demonstrate the trend of the low-lying 3QP states through the whole N = 82 region, we refer to figs. IO, 11 and 12, where for each spin I” the position and main character of the low-lying levels is shown. The separation of the


1QP states from the 3QP levels with increasing A is very well demonstrated. We remark that in the lighter region the lowest 3QP levels have the (3 3 2+ 3) or ($5 2+ 3) configuration as the main component, which changes into (3 4 2+ 4) or (3 3 2+ 3) in th e h’ h ig er region. The (3 3 Of 4) and (3 5 Of 4) configurations with

05 ’ _ 'h IQP

0.5

Fig. 10. Low-lying one-quasiparticle and three-quasiparticle 4+, a+, 4+ and 3’ states throughout the whole N = 82 region. The main component of each level is mentioned. The number indicates the

spectroscopic factor for c3He, d) reactions to these levels.


intermediate J” = O+ coupling which become the main part of the lowest 3QP

2” = 2’ and 3’ states in 14rPr, ‘43Pmand 14’ Eu are striking, in contrast with ref. “)

where the 2+ intermediate spin remains predominantly. This is due to the too small

G(_5 3 3&O+) and G($s 2 s, O+) p airing matrix elements as used in ref. “) (see fig. 2).

E’ ( bw!

20-

15-

10-

OS-

oo-

cs La Pr Pm ELI - - - /- C’B-‘Yr 0’ y;,

#’ ’

__-- ------+yf)l;) ,’ ,;,- --==-~=~-___________c 7 ,

,_,‘,; ,” ,rh+fw)

q&$+&h ----_______.,_,,--j~~~~-~__:- ,________~;-_____* - =______ W?QJ 20

/ 1% F4*?;,,___ _- I I-,,’ ,i’ ‘1 s WP

/ ,________x ---\,$’

,/ (y;w$)

(7ii72+2’V~+h ,4 ,f’ ,__---

, , _- ____._’ *\

4’ ___-___,’

@rib@ \\

~A_<:‘__/ /’

__ _____m__________* /

/ 2I

1%. -.\ ‘\

‘\ /----------- __ ____ _ ___I op

‘\ ,’ 72’ -.

‘\ ,______4

IOP _________ -..__-__-- ,A h

. \__.-__ _-__ __-VIP I Yi

0.0

Fig. 12. See caption to fig. 10. The 4’ states are marked by solid lines, the 8’ states by dashed lines.

(v) The possibility of negative-parity levels in the odd-proton N = 82 nuclei is.

not excluded, as suggested by Berzins et al. 19) in ‘39La and by De Frenne et al. 27)

in ‘43Pm with regard to the observed log ft values, suggesting first-forbidden

P-decay. Theoretically it should be possible to reproduce these levels if the space is.

extended with the N =5 and N = 3 oscillator shells.

6. Magnetic dipole moments

Experimentally, magnetic dipole moments have been measured for the ground

state in the nuclei 137Cs, ‘j9La, 141Pr and 143Pm. As the ground state changes ,$”

value between 139La and ’ 41Pr, we have to use different gyromagnetic gS factors for

the nuclei 137Cs, ‘39La at one side and 141Pr, ‘43Pm and 145Eu at the other

side. The gS values have been obtained by a fitting procedure to all known magnetic

dipole moments. Therefore, the respective coefficients of gr and gS are calculated

separately and are shown in table 1. The considered values of gS, are equal to

2.6583 for the nuclei with the Ig+ 1QP state as ground state and 4.4913 for the second

group. These values give a very good fit to the experimental data. The influence of

the 3QP components on the value of&i and & is negligible in 13’Cs and 139La,

TABLET

Mag

netic

di

pole

m

omen

ts

(in

Boh

r m

agne

tons

) fo

r th

e gr

ound

st

ate

in

the

odd-

prot

on

N

= 82

nuc

lei

13’C

s 3.

821

-0.0

007

$0.0

126

3.83

3 -0

.382

3-

0.00

07

-0.0

015

-0.3

83

2.81

5 2.

113

2.83

8 “)

1.

69

_-

‘=‘L

a 3.

792

-0.0

025

+0.0

145

3.80

8 -0

.380

$0

.002

5 -0

.001

9 -0

.379

2.

801

2.10

6 2.

778

“)

1.70

t41P

r 1.

947

+ 0.

029

+0.0

20

1.99

5 +

0.48

7 -0

.029

+0

.006

+0

.463

3.

227

4.07

6 4.

240

“)

4.61

‘43P

m

1.94

0 fO

.066

+0

.014

2.

020

+0.4

85

-0.0

66

+0.0

06

+0.4

25

3.15

0 3.

929

3.75

0 b)

4.

57

-

L45

E~

1.94

1 +0

.112

+0

.009

2.

062

-to.

485

-0.1

12

+0.0

07

+0.3

t30

3.07

1 3.

767

4.50

“)

Ref

. 28

).

b,

Ref

. z9

).

‘)

Ref

. “)

. T

he

pll,

,ujl

and

~33

expr

ess

the

lQP-

lQP,

3Q

P-1Q

P an

d 3Q

P-3Q

P co

mpo

nent

s re

spec

tivel

y fo

r th

e or

bita

l (r

j cont

ribu

tion

or

for

the

spin

(s

) co

ntri

butio

n.


but is considerable in the case of 143Pm and 14’Eu, as demonstrated in table 1.

The (2~~~ +,u~~) term gives a contribution which adds up to the single-particle

estimate ,~ir in order to reproduce the experimental value of the magnetic dipole

moment. We mention that we shall use these values of gS for the calculation of Ml

and M2 transitions, discussed in sect. 8.

7. Electric quadrupole moments

The electric quadrupole moments of the ground state of the odd-proton N = 82

nuclei are mentioned in table 2. The theoretical values obtained with a Gaussian

TABLE 2

Electric quadrupole moments in e . b for the ground state in the odd-proton N = 82 nuclei (eerr = 2e)

-- Q 11

Q(e ’ b) with eCrr = 2e

2Q31 Q33 Q GtlUSS

tot

Qexp(e . b) (e,rr = e) 9 Q ElliOlt(e . b)

137CS 10.046 fO.039 -0.005 +0.080 +0.05 “) +0.002 r39La +0.173 +0.078 -0.004 +0.248 +0.23 “) - 0.006 141Pr -0.136 -0.046 -0.006 -0.188 - 0.07 “) +0.059 143Pm +0.033 +0.017 -0.006 $0.044 -0.021 145E~ +0.208 +0.076 -0.004 +0.280 -0.103

“) Ref. 2*). “) Ref. “).

interaction and an Elliott force are compared with the experimental data. The agree-

ment with the results of a Gaussian force is excellent, in contrast with the

moments obtained with the Elliott matrix elements “). Both the sign and magnitude

of Q, r are extremely sensitive to the factor (u$ -v:), so that the agreement of our

results with the experiment proves the goodness of the set of occupation prob-

abilities used. We have not neglected the Q33 contribution which in the case

of 143Pm exceeds 18 % of the Q,, value, in contrast with ref. “). The contribution

2Q3, is in all cases additive to the one-quasiparticle term Q,,, which for r3’La

increases the Q,, term of 0.173 e . b to 0.251 e. b, in comparison with 0.24 e * b

experimentally.

8. Transition rates and half-lifes

In order to test the wave functions obtained after a TDA calculation with a

Gaussian interaction as the residual effective interaction, we have considered

various electromagnetic transitions occurring in the series of odd-proton N = 82

nuclei. The branching ratios for the different transitions can be compared with

experimental intensity ratios. The agreement is qualitatively very good. We mention

that in all cases the C31, C, 3 and C, 3 contributions have been considered

in the calculation of the transition rates.


The +(:op, f 3& transition. The results are shown in table. 3. The E2 and Ml

transitions from the first excited state to the ground state have been measured for

13’La, 141Pr and ‘43Pm Table 3 shows an extensive comparison between the two .

theoretical calculations with experiment. The very large retardations are well reproduc-

ed theoretically, but those obtained from a Gaussian interaction are approximately

thousand times too large in the case of 13’La. The reason for this diagreement is

already given in subsect. 2.2. As the M 1 transition is of the Z-forbidden type, the M, 1 and M1 3 contributions have to determine the value of B(M1). Only spin-orbit part-

ners contribute to M,, and M,, as expressed in the simplified formulae (2.12) for

M3** Its magnitude is mostly dependent on the value of the 3QP components

\1/(3f) (BisJ$; j’r) f or 1 d’ff erent intermediate J” values. In 13’La, the components

II/‘,” (3$2+$;$) and I# ($+J4+ 4;;) take th e value of 0.00598 and 0.00199,

respectively, which causes the very small M,, of -3.71 . 10e4 (see also table 3).

In agreement with ref. 6), all three terms in the B(M1) add up coherently.

The y- decay. The results are shown in table 4. The relative pure lh, 1QP level,

experimentally found in one-nucleon transfer reaction work in all the odd-proton

N = 82 nuclei, decays by E3 of M2fE3 transitions to the first excited state and the

ground state. They are all retarded transitions, with regard to the single-particle

estimate by a factor of 10 to 20. The experimental estimate of about 2 ns for the

half-life of the #n = J$- state in I3 ‘La proposed by Berzins et al. “) agrees very

well with the theoretical predictions of 3.7 ns. The M2+ E3 transitions of the

,$” = J$- state to the $” = 3’ ground state or first excited state in 137Cs, r3’La

and 141Pr seems to be predominant, but changes to an E3 transition -‘s’_- -+ 5’ in

145Eu.

The 5&r, decay. The results are shown in table 5. It is also interesting to study

the decay modes of the lowest 3QP state which has some collective features. Although

the transition probabilities cannot be verified by experiment, comparison of the

branching ratios can yield some information concerning the goodness of the

corresponding wave functions. From table 5, it can easily be seen that the reduced

transition probabilities are predominantly determined by the E,, contribution and

in this way depend on the initial wave function. The B(E2; $&opj -+ $or,) in

137Cs and 13’La is strongly retarded with regard to the large enhancements in

141Pr, ‘43Pm and 145Eu. This sudden switching of the microscopic structure of

the lowest 3QP $” = 3’ state is also e x p ressed in fig. 11, where the ($3 2+ 3) and

the (3 4 2+ $) configurations cross between 13’La and i41Pr. It leads to a pre-

dominant +c30pj to ground state decay mode in 141Pr, ‘43Pm and 145Eu in

agreement with the experiment. We like to remark on the very good reproduction

of the experimental 949 keV decay to the 1QP f ” = 3’ state in 145Eu as seen

in the branching ratios.

The 3&opj decay. The results are shown in table 6. The changing of the main

component of the lowest 3QP 2” = $’ state throughout the N = 82 region is


TABLE Transition probabilities of the transition of the first excited

MICE2 %+w----- +%+<I,

Or t-

Gauss

ra7cs i3gLa

Elliott “) exp Gauss Eliiott “) exp ?

4 WV)

BOW Eli contributions &I

E I3

E33

B(E2) tee2 * 10-52cmZ)

B(Ml) MI1 contributions MB1

MI3

M33

% E2 96 0.24

PO% 1.22x1os 9.11x105

(set-1) hindrance factor 8.5 1140

B(ML) 5.60 x lo6 3.83 x iO*

(set - ‘) hindrance factor 5.0 x 105 8000

B@ot), 1.28 x lo8 3.83 x lo8

(see-‘) hindrance factor 2.4 x lo4 8000

r* of Pot or %‘t1, 5.43 x IO+ 1.81 x lO-g

0.462 0.453 0.456

0.202

0.075

-0.100

0.009 0

5.06 0.038

0.0 0.0

-7.50x10-5

-4.83 x LO- 5

-1.46x10-3

3.36 x IO.+ 2.30x 1O-4

0.166 0.179 0.166

0.069

0.079

-0.113

0.008 0

0.27 0.10 4.8

0.0 0.0

-3.71 x10-4

-3.08 x lo-&

-1.38~10-~

5.70 x lO+ 4.19 x 1O-4 4.65 x 1O-3

8.4

4.18 x lo4

162

4.57 x 105

3.2~10”

4.99 x 105

3.0 x 105

1.39 x 10-S

0.04 0.2

1.58 x lo4 7.48 x 10s

428 9

3.68 x 10’ 3.74 x lo8

4000 400

3.68 x 10’ 3.74 x lo*

4000 400

1.88x10-a 1.85~10-~

“) Ref. 6). b, Ref. 30). ‘) Ref. 31).

The Gaussian interaction has the parameters /? = 0.325 fmm2 and t = +0.2. The Z-forbidden Ml transition

shown in fig. 12. The result of this changing structure on the reduced matrix elements is shown in table 6. All transitions are largely hindered. Branching ratios are well reproduced, as well as the 458 keV transition to the lowest 3QP f’” = 4’ state in “43Pm observed by De Frenne et al. 27).

The $$qpj decay. Theory and experiment give nearly a pure +Aor, + 5 + z(tQP) decay mode. The transition probabilities are of the same order as the single-particle estimates.


3

state to the ground state in the odd-proton N = 82 nuclei

Gauss

‘4’PY

Elliott “) ew Y GaLKV

14sPm

Elliott “) exl, ‘)

‘4%U --

Gauss Elliott “) exP

0.146 0.010 0.145 0.308

0.082 +0.228

0.115 +0.091

-0.076 -0.031

-0.004 0 -0.004

1.16 0.03 13.3 6.79

0.0 0.0 0.0

2.66 x 10-s 7.91 x10-J

3.93 x 10-s 9.69 x 1O-J

3.41 x10-J 4.19X 10-S

0.170 0.273 0.329 0.297 0.329

0.383

0.061

0.021

0 -0.003

0.09 18.11 0.97

0.0 0.0 0.0

1.55 X 10-Z

1.81 x IO-*

4.72x 1O-3

6.71x10-s 2.07~10-~ 4.85~10-~ 3.56~10-~ 15.82x10-~ 1.10x10-~ 2.0 x10-3

2.5

9.10x 104

38

3.59 x 106

2.7 x IO4

3.68 x IO6

2.6X 104

1.88X 10-T

0.02 0.4 9.3 0.07 11 0.36

2.46~10s 1.04~10~ 1.26X10’ 1.76x 10s 8.55~ 10’ 4.56~ IO6

1420 3.5 6.7 480 2.5 48

1.11 X 10’ 2.60x IO* 1.23x lOa 2.36x 10s 6.90~ lo* 1.25 x lo9

8.8 x lo3 370 5300 2700 1600 900

1.11 x10’ 2.61 X 10s 1.36~10” 2.36~ lo* 6.54x 10~ 7.76x lo* 1.25x lo9

8800 370 4800 2700 975 1500 911

6.24~10-~ 2.65~10-~ 5.10x10-” 2.93~10-~ 1.06~10-~ 0.89x10-’ 0.55x10-’

has a 9, value of 2.6583 for 13’Cs and “9La, and y, = 4.4913 for 141Pr, ‘43Pm and 13”Eu.

TADLE 4

Decay study and half-life estimate for the +- single-particle level

f - decay 13’Cs lJsLa

‘h--g+ BW3) 5.44 4.15 BW2) 4.95 1.89

Y - -+ t’ B(E3) 140.69 134.83

(;;;I<;;:) thee 17 4

hindrance factor 8.4 19

T+ofY- theory 4.20 X lo-lo 3.69X1O-9

cxP u 2x10-97

2.98 2.68 1.99 2.26 0.72 0.17

105.82 92.80 51.32

2 0.38 0.07

9 Y 1 ‘) 13.5 17.5 18.2

1.73 x 10-s 1.24~10-~ 2.29~10-~

“) Ref. 2’). “) Ref. 24). ‘) Ref. =). See table caption 3 for 0‘ values.

K. HEYDE AND M. WAROQUIER

m ”

569

m

d

b X



9. Spectroscopic factors and occupation probabilities

Spectroscopic factors in one-nucleon transfer reactions yield very good data for testing the splitting of the 1QP strength and the occupation probabilities of the proton single-particle states in the “gdsh” shells between Z = 50 and Z = 82. The supposition that the 2 = 50 shell remains closed is verified as no stripping to the negative parity levels 16, 2p+ 2p+ is observed ‘). Practically all nuclei from 13’2

TABLE 8a

Spectroscopic factors Sy(O, 2) for the (3He, d) stripping reactions are compared with TDA results

Mexp

WW

g; ~_1_1_1 s_do, $1

(Elliott) exp Gauss Elliott “) SDI b,

=‘cs

t+

t+

if

P

4-

‘39La

t+

f”

%”

1.490 2.155

0.981

2.070 2.070**

0.456

0.000

1.870 1.780

1.210 1.600 1.780 1.987 2.310 2.416

I.382 1.295 1.560 I .406 1.780 1.730 1.850 2.035 1.960 2.069 2.240 2.236

0.166 1.220

0.000 0.000 0.270 0.486 0.43 1.420 1.394 0.980 0.962 0.84

1.463 2.135 2.390

0.926 1.506 1.727 1.964 2.013 2.103 2.08iF 2.320 0.462

0.000

0.199 1.093 1.346

0.994 0.988

0.990 0.979

0.944 0.856

0.440 0.533

0.990 0.974

0.986 0.978

0.981 0.963

0.864 0.626

0.07 0.86*

0.86 CO.1 %

0.79* 0.79

1.02

0.60

1.01

0.09 0.65 0.13* 0.13

0.06 0.73 0.26 0.16 o.og* 0.08 0.94

0.08 0.17* 0.10* 0.87

< 0.1 % 0.04 0.11 0.001* 0.20* 0.5s* 0.78 0.02 0.92 0.02 0.43 0.01 0.94

0.38 0.54 0.002* 0.07 0.002 0.02 0.49 0.26 0.15 0.002* 0.06 0.84 0.001* 0.007s 0.02 0.26 0.94 0.04

0.04 0.47* 0.43* 0.90 0.00

om+ 0.81

0.82

0.52

0.89

0.03 0.35 0.88 0.57

0.001* 0.03

0.04 0.59 0.48 0.14 0.22

0.15 om2* 0.04

0.61 0.74 0.002* 0.002* 0.02

0.48 0.40 0.89 0.92

0.03

TABLE 8b

-

P” Edexp) (MeV)

~_- ‘4’Pr

*+ 1.300 1.660 1.464**

a* ‘z 1.130 1.600 1.440 2.230

~(Gauss) TDA

(MN __--

1.547 2.033 1.590** 1.233 1.600 1.788 2.068

z++ 0.000 0.000 1.290 1.338 1.580 1.566

g+ 0.145 1.450

J&- 1.110

143Pm

++ 1.173 1.753

1.046 1.530

1.110

1.226 2.223

s+ H 1.060 1.403

1.444 1.433

&+ 0.000 0.000 1.515 1.650

g+ 0.273

+- 0.962

0.30%

1.002

-

5 2

(&*t)

SfCO9 8)

(Gauss) exp Gauss Elliott “) SD1 b,

0.977 0.975

0.971 0.966

0.712 0.659

0.155 0.208

0.971 0.965

0.958 0.967

0.955 0.953

0.449 0.416

0.132 0.160

0.960 0.957

TABLE 8c

0.61* 0.83” 0.51* 0.08* 1.12 0.91

0.006 1.04 0.72

0.11 0.07 0.07

0.64 0.69 0.006*

< 0.1 “/* o 0.02

0.28 0.16 0.002* 0.004

0.96 0.94 0.03

1.08 0.90 0.002* 0.06

0.05 0.11 1.13 0.78

0.07 0.54 0.44

0.001 0.01

0.25 0.13 0.01

0.82 0.94 0.02

0.61* 0.31* 0.92

0.62

0.64

0.20

0.93

0.89

0.84

0.40

0.16

0.93

0.89*

0.02* 0.91 0.09 0.55 0.01 0.25 0.06 0.56

< 0.1 %” < 0.1 o%* <O.l %

0.32 o.ooi* 0.01 0.92 0.03

0.91 0.002* 0.02

< 0.1 % 0.86 0.08 0.45 0.001 0.01 0.22

f 0.1 % 0.89 0.02

$” &(exP) &Gauss) l-DA 4 4

S$C% /)

(MeW WeW (Gauss) (Elliott) exp Gauss Elliott “) SD1 b,

14$Eu

B’ 0.808 2.494**

f’ 1.040 1.750

8’ omo 1.840

?I+ 0.329 0.329

-e- 0.716 0.714

0.800 3.500** 1.036 1.587

0.000 1.717

0.957 0.971 0.98 0.90 0.90 0.89 0.02 0.02 0.04

0.957 0.962 1.01 0.90 0.89 0.89 0.02 0.02 < 0.1 % 0.08 0.04 0.05

0.180 0.168 0.33 0.17 0.16 0.22 (o.lo)* < 0.1 %* < 0.1 %*

(0.10) 0.01 <O.l % 0.076 0.066 0.17 0.08 0.06 0.13

0.02 < 0.1 % < 0.1 % 0.964 0.973 0.83 0.94 0.95 0.90

“) Ref. 6). b, Ref. I). All levels with the same 8” and marked by * are considered in the mentioned summed strength

x+,(0, $); ** means that the centre-of-gravity energy is taken.


up to 145E~ have been studied by either (3He, d) stripping or (d, 3He) pick-up or both reactions leading to excited states in the odd-proton nuclei 2* r7). As the formulae have been given in ref. ‘) we only give the spectroscopic factor for a (3He, d) stripping reaction on doubly even target nuclei, as

S,(O> $) = +fC’(d)l”.

In table 8, we give the detailed comparison for the (3He, d) spectroscopic factors for the five odd-proton N = 82 nuclei, denoted as S,(O, $), in which comparisons with earlier SD1 calculations ‘) and with the recent realistic force calculations of Freed and Miles “) are made. A general point is the good agreement within z 15 % for the lowest states with 2” = 3+, 3’ and y-. However, the value of Sly(O, J*)

SHELL-MODEL ESTIMATE

EXPERIMENTAL DATA

THEORETICAL (GAUSSIAN FORCE)

0.25 -

Fig. 13. The experimentally determined occupation probabilities for the lgS and 2d+ shells are compared with the TDA calculations for the odd-proton and doubly even N = 82 nuclei, as well

as with the pure shell-model estimates.


seems to be systematically too small, a fact which is due to the occupation probabilities

and discussed later. This 15 % is interesting in view of the uncertainties in the

relative spectroscopic factors, which is given as 15 % by Wildenthal ‘). Concerning

the f” = f’ and 3’ levels for 143Pm, 145Eu and 137Cs, agreement within 15 y0 is

again obtained for the strongest single-particle strength in the (3He, d) reactions.

Even the fragmentation for the 8” = $’ case in ’ 3 9La in five levels is reproduced

very well. In many cases, however, (d, 3He) pick-up reactions have been per-

formed and can be used as a cross check for obtaining occupation probabilities for

the various proton single-particle orbits. A reliable set of values us has been

obtained from (3He, d) and (d, 3He) reactions by Wildenthal 2), and is shown in

fig. 13, together with the theoretical calculated occupation probabilities, in the odd-

proton as well as in the doubly even nuclei. One can find a continuous rise of VT,)

up to 14’Ce with a saturation value at about 0.75. The theoretical calculated values

in the region of 140 < A < 145 are systematically 15 % too high. The value of 2 vzdt is better reproduced in the mass dependence. The too low vacancy in lg, is

reproduced in the too low spectroscopic factors for the experimental and theoretical

values, in this order.

In the case of v&~ for r4’Ce, a bump is observed in the experimental and theoretical

value. The bump in the theoretical calculated curve is mainly due to the fact that in the

case of 14’Ce, with eight extra protons, the IGE cannot be applied in an exact way ’ “).

From the experimental values for v&, v:,, and &+, in the case of ‘44Sm [ref. “)I,

we find that the number of particles outside the (lg,, 2d,) shells is 2.30?::;. This

means that, if a standard shell-model calculation is performed for the nuclei with A 2

144, excitations of two to three protons from the (lg+ 2d,)Z- 5o config-

uration into the next 2d,, 3.~ and lh, proton shells must be considered.

10. Conclusion

We can state that a one- and three-quasiparticle calculation as performed with

an effective interaction of central Gaussian type with spin exchange dependence,

describes the experimental level schemes, electromagnetic properties and one-nucleon

transfer reactions with good overall agreement.

We think that a properly renormalized realistic force should give important pairing

and quadrupole matrix elements very close to the Gaussian effective force

matrix elements, as discussed in sect. 3, and thus give the same energy spectra and

electromagnetic properties.

Together with the earlier calculations for the odd-proton N = 82 nuclei with

a more schematic force ‘) and the theoretical study of the doubly even N = 82

nuclei lo) in a quasiparticle description, a survey of the wealth of experimental data,

reaction data as well as p-decay studies have been given in these works.

We also examined the force dependence of the calculation of nucleon level schemes


since both schematic, effective and realistic forces have been used, in the same model space and also compared with a pure, but restricted standard shell-model calculation of Wildenthal in an enlarged configuration space.

However, the problem of low-lying (< 1.5 MeV) negative-parity states with low angular momentum in the odd-proton nuclei, as observed in some decay studies 2’S ‘“) remains. These levels cannot be obtained from coupling (~~~~(~)lh~; I-), but should come from (~~~~(3-)~=; I-). A s in the adjacent doubly even nuclei, the 3- octupole state is lowered in excitation energy as the atomic mass number A increases. Since in the 2QP calculation, these low-lying strong collective octupole states were not de- scribed well, the low-spin states with negative parity will also not be obtained properly in the 3QP calculation. Further experiments should be performed in these cases to obtain unique parity assignments.

The authors would like to thank Prof. J. L. Verhaeghe for his interest in this work. They are very grateful to B. H. Wildenthal and N. Freed for communicating the recent experimental and theoretical developments on the odd-proton N = 82 nuclei. They also would like to thank D. De Frenne and E. Jacobs for communicating their experimental data on 143Pm, prior to publication aud are grateful to Prof. C. C. Grosjean and Dr. W. Bossaert for extending computing facilities on the IBM 360 ordinator of the Centraal Rekenlaboratorium R.U.G.

References

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Documents

Shell-model calculations on the N = 82 ODD-proton nuclei (II)