Magnetic properties of odd-odd deformed nuclei

Nuclear Physics A286 (1977) 371-376 ; © North-Holland Publishing Co., A»raterdam

1 .E.3 : 1,E .4

Not to be reproduced by photoprlnt or microfilm without written peemlesion from the publisher

MAGNETIC PROPERTIFS OF ODD-0DD DEFORMED NUCLEI t JEAN KERN Physics Department, University, 1700 Fribourg, Switzerland and GORDON L. STRUBLE Lawrence Iivermore Laboratory, University of California, Z.ivernrore, California 94550 and Department of Physics, University of Munich, Munich, Germmry Received 1 April 1977 A6etract : Magnetic moments and Ml transition probabilities depend on the parameters gx, gn , and gao in odd-odd deformed nuclei . The parameter gx can be obtained from an intraband branding ratio and a magnetic moment . Available data are analyzed and compared with pr~ediçtions of the unified model.

In deformed odd-mass nuclei, the g-factor of a state ~ IIC) is often parameteriDed in terms of the quantities gn and gx . The parameter gn depends on the unpaired nucleon and can be compared with predictions of Nlsson-type models t). The parameter gx is the gyromagnetic ratio for colléctive motion but may be significantly influenced by the presence of the unpaired nucleon Z). The combination of experimental magnetic dipole moments and intraband y-ray branching ratios determines both gx and ga [refs. s-6)] " So far, little attention has been given to the corresponding problem in odd-odd deformed nuclei. Hook') derived a relation for the magnetic moment in the strong coupling limit and compared the available experimental results with theoretical values that he calculated using this relation, Nlsson wave functions 1), a value of gx = 0.4, and free-neutron and free-proton values for the gyromagnetic ratios of the valence particles. Similar approaches have been followed by Gailagher 8), and Hûbel et al. 9). In this paper we show that for an odd-odd nucleus, the value of gx and a linear combination of gnp and gna (referring to the protons and neutrons, respectively) t Work suppôrted in part by the Fonds National Suisse de la Recherche Scientifique and under the auspices of the US Energy Research and Development Administration under contract No . W-7405Eng_48. .

371 ~uguat l9n

37 2

l . KERN AND G . L. STRUBLE

can be obtained by combining the value of the magnetic moment u and an intraband y-ray branching ratio ~, (or E2/M1 mixing ratio). To calculate ~., one needs an expression forthe reducedprobabilities for E2 and M1 transitions. For odd-odd nuclei these expressions are B(E2, hK-~ IFK)

where

16n

(h2K0~IfK)Ze2Qô, .

(1)

B(M1 ; Ii K-~ IfK) = kN(h1~~IrK)2(GgK)Z+ 4~

and Qo is the intrinsic quadrupole moment of the nucleus: The signs of f1p and Lï in eq. (3) are taken as in the expression and

K = ~lp+il,

The expression for the magnetic moment is ~ = 9x1+

K g` I+1 G '

(S)

Eqs. (2) and (5) are formally the same for odd-mass nuclei . In that case, however, only one term in eq. (3) is retained . As in the case of odd-mass nuclei 3 ), the quantity (G~lQo) can be extracted from the branching ratio ~, of a cross-over (I -+ I- 2) to cascade (I --> 1-1) transition, or extracted from the mixing parameter S of an I -=" I-1 transition . Then, if Qo is known, G~ (with the sign chosen according to the predictions of the Nilsson model) can be obtained and introduced into eq . (5). From this, a value for gx is uniquely determined. This value may then be introduced into eq. (3), from which a linear combination of gnp and gna can be deduced. For K = 0 bands, Ëlo = -~v so that eq . (3) reduces to G`a = éiP(gnP-gna), which is independent of gx . Eq. (5) =educes to ~ = gxl. A determination of ~ for any level with angular momentum not equal to zero in a K = 0 band yields gx directly . , The quantities gnp and gn o are characteristic of the valence nucleons and should be the same in odd-mass and odd~dd nuclei if corrections due to the Coriolis force, changes in deformation, and the neutron-proton residual interaction are carefully taken into account. For very deformed nuclei, these corrections are often small, and so a comparison of the gnp and gnp values obtained from odd-odd nuclei with values from odd-mass nuclei or with values from calculations that use Nlsson-type wave functionsare a measure of the validity ofthe unified model when applied to odd-odd nuclei . In table 1 we present experimental values for parameters in even-even, odd-mass,

373

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374

J. KERN AND G. L. STRUBLE

and odd-odd nuclei . These values are required for malting . comparisons of linear combinations of gnP and gnn obtained from odd-odd nuclei with those obtained from odd-mass nuclei or from theoretical calculations . Such comparisons are presented in table 2, where we made the calculations using wave functions from ref. t~ and taking gaff = 0.60 gjt°° . The agreement among the last three columns oftable 2 is, in general, very good. In the case of i ssEu, there is noexperimental value of gn for the ~-- [505] neutron orbital from odd-mass nuclei. Our value is, therefore, the fast model-dependent experimental determination of gnn for this orbital, and it agrees well with the calculated value. A serious discrepancy occurs for t'°Tm. This can be resolved either by a larger value of gnn or a smaller absolute value of gnP Since the experimental value of 1.43 for gna is larger than the maximum predicted by Lamm t°), we suspect that the absolute value of gnP should be smaller. Using our value for G~ and the odd-A experimental one for gna, we find gnP = -1.14±0.07. This result can be reproduced in a model calculation using g~rf = 0.73g n°, which is quite reasonable it), The values of gx in table 2 are generally larger than those of the even~ven core (see table 1). Adding a valence nucleon to a core can change its gyromagnetic ratio by changing the deformation or pairing oorrelations (by blocking). The valence particle also has a small probability of being scattered to each of a large number of Nilsson orbitals because of the Coriolis force. This may affect the magnetic prop-

TAAL F

2

Comparison among parameters derived from experiment and/or predicted from models Nucleus

9a exp.

81e cale.

1saEu(3 - 3) ls° Tb(3 - 3)

0.33 f0.2 0.41 10.05

0.40

1°sHo(0 0) 1ssHo(7 -7)

0.88 10.7 `)

0.33

l'sLu(1 -0) l' 6Lu(7 -7) le~Ta(5 +5) lea~(5 - 3) l s°Re(1 - 1) lse~(1 - 1)

0.31910.003 0.55 10.05 °) 0.33 f0.07 0:37 10.03 0.57 10.14 0.65 10 .16

0.31 0.31 0.31 0.25 0.27 0.30

Parameter value

parameter tested

from odd-odd nucleus

from odd-mass nucleus

9n,+9ne 9nP i- EAa 9nP -9n,

-0.2310.04 b) 1 .2910.03 1.2410.02 1.2610.2

1.30 t0.a4 1.30 10.04 1.61 10.02

-0.209 1.36 1.36 1.04

BnP +BAa a~,-en~

0.2910.06 0.5110.14

-0.22 10.07 0.571 t0.012

0.13 0.46

2.9810.08 2.9210.14 2.8910.16

2.99 *0.09 2.76 t0.05 2.76 f0.05

eAe

3HAP +3HAe ~9nP -~a9^n,

~BnP -~An

.

from , talc. `)

2.94 ~ 2.93 ~ 2.93

') Taken from Lamm '~ or from calculations using the same parameters as in ref. ' q.

") Assuming the validity of the model and a value of 0.67110 .004 [ref. s)] for gnP .

~ Where no experimental branching ratios are known, we have used gn values from odd-mass nuclei to extract values of gle from experimental magnetic moments.

MAGNETIC PROPERTIES

375

erties of the system, and because it is not explicitly calculated, will renormalize gR . The same phenomena affect the moment of inertia, and to a good approximation t z) where .gyp is the contribution to the moment of inertia due to protons and .d (_ .lp+ .fin) is the total moment of inertia. Therefore, we can relate the change in gR between an even-even and odd-odd nucleus (SgR) to the change in the moment of inertia between the even~ven core and the odd-mass nuclei that are isotopic and isotonic with the odd-odd nucleus (8.>:p and b.~n , respectively) : S _ 8.dn gR SgR ~

+( 1 -.gR) .

.>fp

Using the experimental values of the moments of inertia quoted in table 1, we have calculated the values of gR listed in the third column of table 2 and compared them with the experimental values in the second column. Usually the agreement with experiment is excellent. We have no explanation for the large disparity for the K = 7 band in t' 6Lu. The disagreement for the Re isotopes is probably due to large changes in deformation for these transitional nuclei . F.q. (~ is valid only for small changes in parameters affecting the moment of inertia. Although the data set is still quite small, there is enough experimental data to demonstrate that the unified model is valid for predicting the magnetic properties of odd-odd deformed nuclei . Corrections and renormali7ations can be~extracted from odd-mass nuclei either through model calculations t~ or by use of the experimental data directly. The values of gR are closer to those for neighboring even-even nuclei than to the values of gR for odd=mass nuclei because valence neutrons and protons contribute corrections ofopposite signs. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

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376

J. KERN AND G. L. STRUBLE

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