Z. Phys. A - Atoms and Nuclei 303, 123-130 (1981) Zeitschdft A ~ ' f ~ r . ~ for Physik A i~, l&jr l I I ~

and Nuclei �9 Springer-Verlag 1981

The Structure of the Odd-A Nuclei with A 130 and the Problem of -Softness

D. Chlebowska

Institute of Nuclear Research, Warsaw, Poland

Ch. Droste and T. Rz~tca

Institute of Experimental Physics, Warsaw, Poland

Received June 29, 1981

The negative parity states in the 123"125'~31Cs and the 131La nuclei are described in the framework of the particle-core coupling model. In order to study the problem of gamma softness, the following two core models are used: (t) the rigid triaxial rotor model,~tnd (2) the v-unstable model with v-dependent inertial functions. The properties of the odd-A nuclei with rigid and soft cores are compared with the experimental data. The results do not allow to draw a definite conclusion about the softness. When seeking properties which could help to distinguish between soft and rigid nuclei it has been found that some spectro- scopic factors for the proton stripping reaction are sensitive to the gamma softness.

1. Introduction

It has been established that the particle-core coupling model can reproduce the properties of the transitional odd-A nuclei. This means that such quantities as the level energies are usually reproduced with an accuracy of several dozen kilo-electron-volts. Having such a model we can ask which nuclear properties can be studied by analysing the structure of the odd-A nuclei. It appears that such properties as the nuclear shape or softness can be investigated. Nowadays, the problem of nuclear softness is of special interest. The reasons for which this problem has arisen are the following. The transitional even-even nuclei (see e.g. [1]) and transitional odd-A nuclei [2, 3] are well described in the framework of the rigid triaxial rotor model (the Davydov-Filippov (D-F) model). The rigidity assumed in the D-F model is in disagreement with the results of the microscopic dynamic calcula- tions [4, 5]. In the latter works the Bohr collective Hamiltonian with the inertial functions and potential energy surface (PES) both calculated microscopically was used. The results of these microscopic dynamical calculations show that transitional nuclei (e.g. from the 50 < Z, N < 82 region) should be soft with respect to the gamma deformation. The 7-softness means that the nuclear wave function spreads in the v-direction

and the nuclear shape is not "frozen" as it is assumed for the rigid nuclei. It deserves mentioning that the PES for the 7-soft nucleus is flat in the v-direction. A satisfactory description of the even-even nuclei from the 50<Z, N__<82 region was obtained (see e.g. [6, 7]) in the framework of the semi-microscopic model of Dobaczewski et al. [8], which takes into account the results of microscopic calculations for inertial functions and the potential energy surface. This model calculations confirm the conclusion of the full micro- scopic calculations [5] that the considered nuclei should be soft. The odd-A nuclei in the 50 < Z, N < 82 region are also well described [9] when their even- even cores are assumed to be soft. The problem of V-softness has not been solved yet. The experimental data are equally well described when the rigid (fixed) or soft (fluctuating) nuclear shape is assumed. This is so, because many of the observables (e.g. the probabilities of the E2 transitions between g.s. band states in even-even nuclei) are weakly sensitive to the V-softness. For the odd-A transitional nuclei the problem of ?,-softness was studied in [9-121. To investigate this problem it is convenient to use the particle-core coupling model with a spherical single-particle basis

0340-2193/81/0303/0123/$01.60

124 D. Chlebowska et al.: Structure of Odd-A Nuclei with A ~ 130

(see e.g. [13]). This model allows us to calculate easily the properties of the odd-A nuclei with dif- ferent cores, since the only information needed about the core are its excitation energies and the reduced matrix elements (R]IE2]IR') between the core states IR) and [R'). The general idea of the present investigation is the same as in [9], and can be briefly summarized as follows. Experimental data (usually very scarce) about even-even nuclei are used to determine the best parameters of the core models. These parameters allow us to calculate a large number of core energies and matrix elements (R[IE2]IR') which are used as the input data in the odd-A nucleus model. The calculated properties of odd-A nuclei are compared with the experimental data. A core model is sought which gives the best description of the odd-A nuclei�9 In the present work (like in I-9]) two such core models were applied: (1) the model proposed by Dobaczewski et al. I-8] for V-soft cores and (2) the Davydov-Filippov model for rigid cores. In the present paper the structure of 123,125,13 ~ Cs and a3~La is studied in the framework of the above men- tioned models which are described in Sect. 2. The theoretical results are compared with experiment and the properties which could help to distinguish between soft and rigid nuclei are discussed in Sect. 3. Conclusions are given in Sect. 4. It is worth mentioning that the properties of the neighbouring nuclei (viz. 127 '129Cs

and 133La) were studied in [9].

2. Models

In the particle-core coupling model (see e.g. [13, 14]) the odd-A nuclei states are described by a single nucleon motion which is coupled to a collective motion of the even-even core. The total Hamiltonian H is written as:

H = Hco n 4- Hs.p. 4- Hin t , (1)

where Hoon is the collective Hamiltonian of the core, Hs.p. denotes the single-particle Hamiltonian and Hin t describing the interaction between the odd nucleon and the core is taken in the form

Hin t = -- K L ( - - 1)u0(us'P')h (c~ , (2) #

where

0(~ s'p') = r2p. g2# (3)

is the quadrupole moment operator of the odd nucleon and

(r = (3 AR2/4 ~) fl {D 2 o [cos V + c~ cos 2 V] /1

+ 1 ~ [D, 2z + D2- 2] [sin V - a sin 27] } (4)

with

= (2/7) 5 ] ~ f l

denotes the collective quadrupole moment operator of the core. In (2) ~: is the strength of the particle-core interaction. Diagonalization of the total Hamiltonian H is performed in the basis of eigenstates of Hooll coupled to spherical shell model wave functions of the odd nucleon. Matrix elements of Hint in such a basis are expressed by the following formula:

(,jR'cI [Himlj'R'z'I) = ( - 1) R + j ' + I + 1 k(4rc/3AR~)

�9 1/(5/4~z)(2j' + 1)U'21/201jl/2) J

�9 (ez[10(c~ ' ) (5)

where R 0 = 1.2A 1/3 fm and j, R and I are the angular momenta of the odd nucleon, core and odd-A nucleus, respectively. The parameter z distinguishes the core states with the same R. The constant k=40 MeV is taken instead of the expression

tc (3 AN2~4 re) (N 1 j lr2p. [ N l'j')

(here IN l j ) denotes the radial wave function of the odd nucleon)�9 Diagonalization of the total Hamiltonian H (1) is done in two steps. Firstly, using a core model one has to calculate the excitation energies E R of the core and the reduced matrix elements (R~[IQ(C~ Se- condly, these values are inserted into the energy matrix of the odd-A nucleus. Then, this matrix is diagonalized. The eigenfunctions of the total Hamil- tonian have the form:

]IM)= ~ w,(j; R z) Ij R z; 1M), (6) jR~:

where wf is the probability that an odd-A nucleus consists of a particle in the state ~) coupled to a core in the state IR z). The model can be applied in the case when the odd- particle states are well above or below the Fermi level. This requirement is fulfilled in our case (see [9]). In order to study the problem of 7-softness two models were applied to describe the cores: (1) the triaxial rigid rotor model of Davydov-Filippov, and (2) the v-unstable model with ~-dependent inertial functions [8] which was used to describe ~-soft nuclei. The latter model is based on the Bohr collective Hamiltonian with the following simplifying assump-

, tions: (i) the potential energy surface (PES) is v-in- dependent, what reproduces fairly well the results of the microscopic calculations [15], (ii) the inertial func- tions Bpa = constant and Bar = O, (iii) the other inertial functions B~ (where e =x, y, z, VY) are taken in the form

B~(7) = fB~i~r (V) + b

D. Chlebowska et al.: Structure of Odd-A Nuclei with A ~ 130 125

\ 0.0 0.2 0.4 0.6

gl'F -

to

[ [ I I 0 ~ 20 ~ 40 ~ 50"

y ~

Fig. 1. Density of matter distribution fo r the ground state of the 122Xe nucleus. The presented results have been calculated in the framework of the ?-unstable model with ?-dependent inertial func- tions [8]

Here f and b are phenomenological scaling parameters, micr B, (?) denotes microscopically calculated inertial

functions [16] taken at fi=0.2. That value of fl is close to fi . . . . (see Table 2b). Using this model the 122'124'13~ nuclei were described. For the tS~ nuclei we found that better results could be obtained when PES is y-dependent. The y-dependent part of PES assumed as proportional to cos 37 has produced a non-zero prolate-oblate difference V•o (Table 2a). The fundamental difference between both considered cores is connected with the localization of their wave functions. Calculations done in the framework of the y-unstable model for 122'124'13~ and 13~ show that the wave functions of these nuclei are quite well localized in the/3-deformation space and spread over a large region in the y-deformation space (see Fig. 1 and Table 2b). This is the reason why such nuclei are called y-soft. The same nuclei described by the Davydov-Filippov model have their wave functions localized in one point (/3o, ?o).

Table 1. Parameters for rigid cores

Parameter lZ2Xe 124Xe 13~ lS~

70 24~ 24~ 27~ 23~ /30 0.22 0.20 0.17 0.22 E(2/) l-keV] 302.3 317.6 453.2 324.3

Table 2a. Parameters for soft cores. The potential energy surface is characterized by equilibrium deformation/3o, the depth of poten- tial O = V( /3=0) - V(/3o ) and its stiffness C=(d 2 V/d/32)t~_r Other parameters are explained in the text

Parameter ~ 2 2 X e 124Xe 13OXe ~ 30Ba

/30 0.18 0.17 0.07 0.17 D [MeV] 1.35 0.97 0.06 0.98 a C [-MeV] 228.0 210.0 86.2 230.0 Vpo [MeV] b 0 0 0 1.07 f 7.35 7.35 5.0 4.0 b [hZ/MeV] -280 .4 -288 .5 - 8 1 . 4 -133 .3 B~t ~ [hZ/MeV] 43.0 43.2 130.0 52.0

a Depth of potential given for ? = 0 ~ b Vpo = min V(/3, 7 = 60 ~ - min V(/3, ? = 0 ~

Table 2b. Expected values and dispersions [8] of variables/3 and 7 for the ground state in the soft cores

Parameter 122Xe 124Xe 130Xe 130 Ba

? . . . . 27~ 28~ 29~ 25~ a(cos 3?) a 0.53 0.57 0.62 0.56 /3 . . . . 0.22 0.21 0.14 0.21 a(/3) 0.06 0.06 0.04 0.05

a(cos 37)for the ground state in the Wilets-Jean model is equal to 0.58

3. Calculations and Comparison with Experiment

The 122'124't3~ and 13~ nuclei are considered as the cores of 123"tzs'lalcs and 131La, respectively. The values of the parameters used to describe the rigid cores are given in Table 1. The model parameters /3o, ?o and E(2~-) (the latter quantity is used instead of the mass parameter B) were chosen in the following way. Parameters /3o were obtained from the value of B(E2, 2 + --, 0~-). The asymmetry parameters ?o and energies E(2 +) were fitted to reproduce the experi- mental level schemes and electromagnetic properties of the cores. The parameters used for the soft cores were obtained from the fit to all experimental data of even-even nuclei and are summarized in Table 2a. The resultant mean values of/3 and ? as well as their dispersions are presented in Table 2b.

To have an idea to what extent are the y-unstable and rigid triaxial rotor models able to describe the proper- ties of the even-even nuclei taken here as the cores, the results for l Z Z X e (a typical example) are presented in Fig. 2 and Table 3. The rigid rotor model does not reproduce the experimental level energies as well as the y-unstable model does. The predicted energy spectrum is too much stretched and some levels known from experiment are not present in the calculations. The electromagnetic properties of the 122Xe nuclei shown in Table 3 do not favour any of the models. All observables in Table 3, except for those in the fifth row, are well described by both models. The discrepancy which appears in the fifth row may be due to the fact that the spin assignment (6~-) of the 2,057 keV level is uncertain. The results (not presented here) and conclusions for the other even-even nuclei (124.13OXe ' 13~ are similar to that given above.

126 D. Chlebowska et al.: Structure of Odd-A Nuclei with A ~ 130

E ~eV]

~000

~000

I000

0 L

122 *,p

54xe

10 ~ ,2783 10" 3030 7 ~___.__~ 2727 8 ; } ~ 2646 5 ~ 2 6 2 7 6 ~ 2595

(7*) 2460 4 % - - ~ - , : 2492 8* ~ 2217 2' ~2375

(6 § 2057 8\ /20.83 - - ZU6U

4 ~ 1 9 5 6 5 + ~ 1774 6 , / _ . ~ - ~ 1 9 5 0

(2*)\ 1495 2 ~ ~1787 6 * " /1467 63\ /1432

1 3 9 2 4 ' " " 1403 3* ~ 1214 4 - - 1324

(0~/ "1149 0 ~ 1128

2 : ~ 843 4 ~ 846 - - 828 2 - - 791

1 0 - 3865

5 ~ 3433

7 3191

6 31t4 4 2864

8 ~ 2 6 5 0

5 ~ 1992

4 ~ 1686 6 ~ 1654

3 ~ 1085

4 ~ 875 2 783

2* 331 2 ~ 355 2 ~ 302

0 + 0 0 0 0 0 EXPERIMENT "SOFT . . . . RIGID"

Fig. 2. Positive parity states in the a22Xe nucleus. The experimental data [17, 18] are compared with the theoretical predictions of the ?-unstable (" soft ") and triaxial rigid rotor (" rigid") models, Energies are given in keV units

T a b l e 3. Experimental and calculated gamma transition probabilities between states of the ~22Xe nucleus

Experiment Calculation

soft core rigid c o r e

B(E2, 2[ ~O~)e2b 2 0.221(20) 0.196 0.210

B(E2, 4[ -~ 2[ )e2b 2 0.32(5) 0.300 0.294

B(E2, 64{ -~4[)e2b 2 0.31(6) 0.375 0.361

B(E2, 2~ -+Of) 0.035(8)" 0.049 0.038

B(E2, 2+ ~ 2~-)

B(E2, 6~- ,--* 4[ ) 0.004 (2) b 0.043 0.012

B(E2, 6~ - , 4 ~ )

B(E2, 6~ ~ 6i ~) <0.81 b 0.45 0.13 B(E2, 6~- -~ 4~-)

B(E2, 7i ~ --*6+) < 0.095 0.082 0.004

B(E2, 7, + ~ 5 ? )

Pure E2 transition between the states 2~ ~ ~ 2~ was assumed u The spin assignment of the 2,057 keV level (65) is uncertain

E I- 31 ............ 4404

31 3493

31- 3199 27 .... 3134 3000k

I 27

27" 2328

2000 ~i- 23 ........ 2064 21 1953

i 19 1811 23 21- 1573 21 23- 1528 19

(19-1 19 1143 1435

1000 L (17") 1004 17 1098 17 993 / / 19- ,843 19 986

15" 321 15 . . . . . 373 15 360

0 11" 0(159) II 0 11 0 EXPERIMENT "RIGID"CORE "RIGID"CORE

A B

2580

31 ...... 3479

27 2323

1738 21 1720 1633 19 1520 1472 23 1471

17 1121

t9 824

15 351

1t 0 'SOFT"CORE

A

Fig. 3. Negative parity states in 123Cs. The experimental data have been taken from [19-21]. Energies (in keV units) are given relative to the first 11/2- state. The spin values are expressed in h/2 units. The notat ion is explained in the text

E~

3000

2000

27 3229

277 2567 2 7 ~ 2 7 2 0

27 2484

2 3 ~ 2 t 1 7 21 2019 19 1888 23 1839 21 t79B

2E 1697 21 1721 23 1599 21" 1633 19 ~ 1 5 4 5 19 1570

(19) I486

19 1168 ( 1 7 " ) ~ 1 0 t 2 D ~ t 1 2 9 17 1033 t7 ......... 1149 t9- 937 19 I=== I=103 I 19 916

15- 365 15 378 15 371 15 , 400

11- 0(266) 11 0 EXPERIMENT "RIGID"CORE

A

11 0 11 0 "RiGID"CORE "SOFT"CORE

B A

Fig. 4. Same as in Fig. 3 but for IZSCs. The experimental data have been taken from [19, 20]

The properties of the negative-parity states in 123,1250131Cs and ~ 31La were calculated using the soft and rigid cores described above. The lowest-lying core states with spins up to 16 h (43 and 29 states for the soft and rigid core, respectively) were taken into account. Two single-particle states ( lhtt /z and 2fw2) were assumed to be available for the odd proton. The values of all parameters used in the particle-core

model are the same as in [9]. The calculations are performed with the model matrix elements (R~[lQ(C~ ') while the core excitation energies are taken either from the model (case A) or from experiment (case B). The calculated negative-parity states of 123,125,13 l eg and 131La are compared with experimental ones in Figs. 3-6. Only theoretical states which have their

D. Chlebowska et al.: Structure of Odd-A Nuclei with A ~ 130 127

I ~oool

23

2000 (23") 2040

19 1559

19- 1197 15 1217 (15-) 1173

1000 13 830

(1 ~ 629 35--) 534 15 528

0 L 11- 0(775) 11 .... 0 EXPERIMENT "RIGID"CORE

A

2733

23 2159

23 1719

19 1338 18 1191 19 1069

15 1067 13 816

13 830 15 546 15 490

II 0 II 0 "R]GID"CORE "SOFT"CORE

B A

Fig. 5. Same as in Fig. 3 but for 13iCs. The experimental data have been taken from [19]

E keV:

3000

2000

I000

27"' 3364

27 2780

27- 2332 27 2312 23 2194

23 1871

23- 1540 23 1508

17 ,,.1221 17 1207 1195 17 1103 17- 1105 19

19 1043 19- 869 19 843

15- 336 15 387 15 375 15 326

11" 0(305) 11 0 11 0 11 0 EXPERIMENT "RIGID"CORE "RIGID"CORE "SOFT"CORE

A B A

Fig. 6. Same as in Fig. 3 but for 13lEa. The experimental data have been taken from [22-24]

experimental counterparts are shown there. It is seen that the calculations reproduce all observed negative-parity states. For further verification of the theory we should like to call attention to the i25Cs nuclei where an unknown parity band (19/2, 1,590 keV; 21/2, 1,771 keV; 23/2, 1,893 keV; 25/2, 2,139 keV) is built on the negative parity 17/2 state [19]. It would be interesting to determine the parity of these states.

Table 4. Root mean square deviations between the experimental and calculated excitation energies of the odd-A nuclei. The quantities are given in MeV. The values in parentheses correspond to the asymmetry parameter 7o = 20~ (see Sect. 3)

Nucleus Soft core Rigid core Rigid core A A B

123Cs 0.12 0.59 (0.45) 0.17 (0.15) 125Cs 0.10 0.37 0.10 iSiCs 0.16 0.36 0.12 aSiLa 0.05 0.57 0.26

2xl

[(MeV)l core / 21-

23-

1.0 15-

4 § 19- 2"

0.5 § -

2 15-

0.( -0 "~ 11-

; ' ~o ' 4 k(MeV)

Fig. 7. Energy of the negative-parity states versus strength of the particle-core interaction. The selected levels of the 123Cs nuclei with soft core are presented

Our calculations do not predict such states with negative parity. The root mean square deviation values (see Table 4), which have been obtained by comparing the calculated and experimental excitation energies indicate, that better results are obtained with the soft cores than with rigid ones described by the pure Davydov-Filippov model (in case A). Generally, in the rigid core calcula- tions the levels with increasing spins lie too high in energy. Description of the odd-A nuclei with the rigid core is improved in case B in which the experimental energies of the corresponding even-even nucleus are taken instead of the model ones. In this case the qualities of the calculations with the rigid and soft cores become similar. There are systematic disagreements in the description of the odd-A nuclei both when the soft and rigid cores are used. For the nuclei with the soft core the order of the states 19/22, 21/2i- and 23/2i- is incorrect and the spacing between the states 19/2i- and 17/2i- is too large. As it is demonstrated in Fig. 7 for the case of

128 D. Chlebowska et al. : Structure of Odd-A Nuclei with A ~ 130

--(MeV

2.0

1.5

1.0

0.5

experiment~ 23-

_ _ 2 1 - 23- 19-

1 7 - ~ 1 9 -

17 - 1 9 - ~ - 1 7 -

19-

~ 1 5 - - - 1 5 - 15-'-""'-"--

0.0 -- 11- 11 - II- �9 I I I I I

20 ~ 22 ~ 24 ~ -~,

Fig. 8. Energy of the negative-parity states versus the asymmetry parameter y. The selected levels of the 123Cs nuclei with rigid core are presented. The spin values are in h/2 units

Table 5. Compar ison of the experimental and theoretical values of the reduced transition probabilities B(E2) (in eZb 2 units) and mixing ratios &(E2/M1). The mixing ratios were calculated using the experimental gamma transition energies

Nucleus Transition B(E2) B(E2) Experiment Calculation

soft rigid

131La 15/21 ---, 11/21 0.34(2) 0.36 0.33 19/21 ~ 15/21 0.33(3) 0.47 0.36

Nucleus Transition 5(E2/M1) &(E2/M1) Experiment Calculation

soft rigid

123Cs 21/21 ~ 19/21 -0 .27 - 0 . 2 4 -0 .36

(Y0 = 24~ -0 .37

(70 = 20~

125Cs (17/21) ~ 15/21 - 0 . 3 -0 .23 -0 .39 21/21 ~ 19/21 -0 .23 -0 .23 -0 .33

131Cs (13/20 ~ 11/21 - 0 . 3 -0 .36 -0 .39 (15/22) ~ (13/21) - 0 . 2 - 0 . 30 -0 .27

123Cs, this situation could be improved if one takes value of the particle-core coupling strength smaller than 40 MeV. Unfortunately, for decreasing k the position of the 15/2/- level becomes worse. For the odd-nuclei with rigid core the parameter 7 is taken (in our standard procedure) from the best fit of the Davydov-Filippov model to the adjacent even- even nuclei. Taking the y-parameter slightly different from its "s tandard" value one can improve the

description of the odd-A nuclei. The case of 123Cs can be used as an example. Figure 8 and Table 4 show that 7o=20 ~ (case A with /30=0.22 and E(2~-)=261 keV) gives better results than 7o=24 ~ ("standard" value of 7 determined from 122Xe). For Yo = 20~ it was found that the sequence of all the levels in 123Cs is correct and the level energies are reproduced better (Table 4) than for 7o = 24~ It follows from Fig. 8 that the posi- tions of the 19/2i- and the 17/% levels in 125Cs (Fig. 4) and 131La (Fig. 6) can be rectified by decreasing slightly the value of Y. Necessity of changing the 7- parameter may be connected with the core polariza- tion, what would suggest that the real core is not so stiff as assumed in the D-F model. As it was shown in [9], some theoretical E2 transition probabilities in 127Cs depend on the core properties. Unfortunately, in view of the lack of experimental information comparison with theoretical results was impossible for 12VCs. Experimental data for the E2 transition probabilities in the A~130 region exist only for 129'131La [24]. One of these nuclei (131La) is analysed in the present paper. The results of calcula- tions together with experimental values of B(E2, 15/2i- ~ 11/2[) and B(E2, 19/2i- ~ 15/2i-) are pre- sented in Table 5. One can see that the transition probability B(E2, 15/2 i- ~ 11/2i-) is well reproduced by both core models, but for the B(E2, 19/21 ~ 15/2i-) transition probability the calculation involving the use of the rigid core agrees better with experiment. The mixing ratios observed in 123'125'131Cs (see second part of Table 5) are equally well reproduced by the rigid and soft core calculations. Seeking other properties which could be helpful in investigating the softness we considered the spectro- scopic factors for the proton stripping reactions. The values of spectroscopic factor for the low-lying 11/2- states in ~23'125'1alCs and a31La were calculated using the soft and rigid cores (see Table 6). One can see that there are significant differences between the spectroscopic factors of the 11/2- states in the odd-A nuclei with soft and rigid cores. The influence of the particle-core interaction (k) on the values of spectroscopic factors was investigated. It was found that the small spectroscopic factors are sensitive to k. On the contrary, large spectroscopic factors (or their ratios) are relatively insensitive to k. The ratio of the spectroscopic factors for the 11/2j- state to the next 11/2- equally strong populated state in 123Cs with soft core can be taken as an example. This quantity is equal to 1.1 and 1.3 for k=40 and 30 MeV, respectively. The same ratios for 123Cs with a rigid core are equal to 2.1 and 2.3. This shows that the considered ratio depends on the core properties and does not depend strongly on k. Thus it could be used as a tool for studying the problem of y-softness.

D. Chlebowska et al. : Structure of Odd-A Nuclei with A ~ 130

Table 6. Stripping spectroscopic factors calculated for the 11/2- states in the odd-A nuclei with soft and rigid cores. The quantities (2I+ 1) w~(hla/2; R =0i-) are presented. Definition of w 2 is given in (6)

Nucleus Soft core (case A) Rigid core (case A)

11/27 11/22 11/23 11/22 11/27 11/2~ ll/2g 11/22

123Cs 3.40 0.05 0.08 3.20 4.60 0.53 2.17 0.94 12SCs 3.74 0.15 0.50 2.89 4.93 0.61 2.25 0.95 13~Cs 5.24 1.69 2.71 0.07 5.73 1.98 1.85 0.40 131La 4.36 0.22 2.86 0.14 4.89 0.48 2.16 0.99

129

4. Conclusions

The odd-A ca lcula t ions using 7-soft and 7-rigid cores r ep roduce all obse rved negat ive pa r i ty states in 123'125'131Cs a n d l a l L a . The level pos i t ion is r ep ro-

duced be t te r for nuclei with soft cores. F o r the r igid core ca lcula t ions , the states with increasing spins lie too high in energy. This is due to the cons tan t m o m e n t s of iner t ia a s sumed by the D a v y d o v - F i l i p p o v model . Some k ind of softness i n t roduced by using experi- men ta l values of the core exci ta t ion energies (case B) gives a lower ing the h igh-spin states in the odd-A nuclei. In tha t case the qual i ty of the y-soft and v-rigid core ca lcula t ions is similar. E lec t romagne t i c p roper t i e s of the cons idered odd-A nuclei are r ep roduced by bo th the models with the same quali ty. The B(E2, 1 9 / 2 i - ~ 15/21-) p robab i l i t y in ~3~La is an except ion. Ca lcu la t ion involving the r igid core gives here a resul t closer to the exper imenta l value. The 19/2i- --+ 15/21- t rans i t ion between the yras t states of the o d d - A nucleus co r re sponds main ly to the 4 [ ~ 2~ t r ans i t ion in the even-even core. The 4~ ~ 2 [ t rans i t ion as well as the o ther E2 t rans i t ions inside the g.s. b a n d are no t sensit ive to the k ind of core mode l app l i ed (see also Tab le 3 for 122Xe and [-25]). On the con t ra ry , some E2 t rans i t ions in which non-yras t states of the even-even nuclei take pa r t depend upon the y-softness. The con t r i bu t ion of such non-yras t core states in the yras t s tate wave funct ion of the odd-A nucleus m a y p r o d u c e a no t iceab le dependence of B(E2) u p o n the softness. C o m p l e m e n t a r y in fo rmat ion con- cerning the t r ans i t ion p robab i l i t i e s in the odd-A nucleus is given in [-9]. It seems tha t the p robab i l i t i e s o r E 2 t rans i t ions between higher yras t states in the o d d - A might be used (if measu red with sufficient accuracy) to s tudy softness of nuclei. M o r e exper imenta l da t a on the t rans i t ion p robab i l i t i e s are needed since one case of ~31La which was ana lyzed in this pape r canno t be conclusive. Spec t roscopic factors of the 11/2- states can be an add i t i ona l tool for s tudying the softness. In the previous sect ion it was shown tha t these quant i t ies are sensitive to models . Unfor tuna te ly , such da t a do no t exist for

the A ~ 130 nuclei. F o r some Cs nuclei spec t roscopic factors could be measu red in the p r o t o n s t r ipping reac t ions (e.g. (e, t)) using separa ted Xe targets. N o w a d a y s , the avai lab le exper imenta l da t a on the o d d - A nuclei are yet too scarce to decide whether the t r ans i t iona l nuclei in the region A ~ 130 are soft or rigid.

We wish to thank Dr. J. Srebrny who participated in the early stages of this work. The critical reading of the manuscript by Prof. A. Sobiczewski and his numerous comments are deeply appreciated. We are also indebted to Drs. J. Dobaczewski and S.G. Rohoziriski for their permanent interest in the present research and many valuable discussions.

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D. Chlebowska Institute of Nuclear Research Ho• 69 PL-00-681 Warsaw Poland

Ch. Droste, T. Rz~ca Institute of Experimental Physics Ho2a 69 PL-00-681 Warsaw Poland