Rotational energies and moments of inertia of non-axial nuclei

Rotational energies and moments of inertia of non-axial nuclei

Nuclear Physics 17 ( 1960 ) 169 -- 174; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written pe...

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Nuclear Physics 17 ( 1960 ) 169 -- 174; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher




A. S. D A V Y D O V , N. S. R A B O T N O V and A. A. C H A B A N

Moscow State University, Moscow Received 18 N o v e m b e r 1959

A b s t r a c t : I t is s h o w n t h a t the ratios of rotational energies of a non-axial nucleus depending on the ratios of energies of two rotational states w i t h spin 2 change b u t little w i t h the deviation of nuclear m o m e n t s of inertia f r o m their h y d r o d y n a m i c values.

1. I n t r o d u c t i o n In the papers of Davydov and Filippov 1) and of Davydov and Rostovsky 2) a theory of the rotational states was developed for nuclei which do not possess axial symmetry. It was shown that the ratio of the energy of all the rotational levels to the energy of the first excited state with spin 2 is uniquely determined if the experimental value of the corresponding ratio for the second spin 2 state is available. It was further shown that the relative probabilities for quadrupole electric transitions between rotational levels are also determined uniquely through the same ratio of energies. These simple results are obtained if one makes two simplifying assumptions: a) the internal nuclear state does not change with its rotation (adiabatic approximation), b) the principal nuclear angular momenta are expressed only through two parameters A and y with the help of the equalities I , = A sin* (r--~2~i),

i = 1, 2, 3.


Such dependence of angular momenta on ~ is, in particular, obtained in the hydrodynamic nuclear model and therefore we shall call this approximation a hydrodynamical one. It seems natural to inquire to what extent the results of refs. x *) are to be affected if one abandons both simplifying assumptions. MacDonald ? has used the expression for the moments of inertia in the form

z , = I , HL\/-~/ +p

]-' ,


where I , R are the moments of inertia of a solid body, I~H are the moments ot inertia coinciding with I, of formula (1) at A = 4Bfl*, and p is a new parameter which ? The a u t h o r s wish to t h a n k N. MacDonald for kindly sending the p r e p r i n t of his work, 169


A. S.









was taken to be equal to 0.1 and 0.2. The peculiar feature of expression (2) is that I i --> [Ri for p --> 0; for/9 :/: 0 and y --> 0, the moment of inertia 13 --> 0; for p :~ 0 and fl --> 0, all the values I x -> 0. Formula (2) can be considered as an empirical formula which takes into account the deviations of the nuclear moments of inertia from their hydrodynamical values. MacDonald has investigated only rotational levels of spin 2. We shall show below that in this case it is impossible to determine which of the corrections is more important: nonadiabatic conditions or deviation of the moments of inertia from their hydrodynamical values. The present work attempts to investigate in adiabatic approximation the rotational states of non-axial nuclei with three arbitrary principal moments of inertia. We shall show that in the general case the ratio of the rotational energies is expressed in terms of the following two parameters: ~, which is the energy ratio of two levels with spin 2, and ~], which is a parameter depending on the character of collective motions responsible for the rotation of the nucleus. Comparison of the results with experiment allows us to establish that the hydrodynamical approximation is sufficient for calculating energy ratios of the rotational levels. The reason for deviations of the theoretical results from experiment is to be sought in the interaction of rotation with the internal nuclear states.

2. C a l c u l a t i o n of the R o t a t i o n a l


Let us write down the operator of the rotational energy of a non-axial even nucleus in adiabatic approximation 3

H :

½~a,J}, i=l

where a i = h2/I, and J~ are the projections of the angular momentum on the principal directions of the nucleus; I, are the principal moments of inertia of the nucleus. It has been pointed out in ref. 2) that the rotational states of even nuclei belong to the total symmetry representation of group D 2. In the present article we shall consider only such states. It is easy to show that the energies of the rotational states of spin 2 is determined by the equation

E2--2(al +a~+a3)E + 3(ala~+aaaz+a2a3) = O. If we express the roots of this equation by El(2 ) and E~(2), and use the properties of the coefficients of a quadratic equation, we shall be able to write

~_. a~

ala2+alas+a2a3 _ 1~,




where E2(2) ~=-->1. El(2) The energies of all rotational states will be expressed in dimensionless units

# = E / E I ( 2 ). Then the energies of the rotational states with spins 3 and 5 will be directly expressed through the experimentally determinable ratio } with the help of the formulae o"(3) = 1+~,

8¢a(5) = 4 + ~ ,

~2(5) = 1+4~.

The energy of the rotational levels of other spin values depends not only on the parameter se but also on the other parameter =




Thus, for instance, the energy levels of spin 4 and 6 are defined by the respective equations #a--5(1+~)#2+4[~2+½-9-~+1]#--40[½~(1+~)+7~] ---- 0, #a

14(147 ~)o~3+49 (1 + 4 ~ + ~2)o¢ 2 - [36(~a-}- 1)+578~(1 + ~)+3888~]d ~ + [252~ (1 + ~2) + 889~2 + 13608 (l + ~)~] = 0.

If the moments of inertia are determined by formula (1), then = rjH ----- ~2[18(1-[-~)J-x and the energies of all the rotational states will be functions of only one parameter ~, which in this case equals or exceeds 2, while in the general case .~j is a second parameter, whose possible values lie in a certain interval which can be determined through the value of ~. In order to determine the limits of this parameter variation depending on ~, we have to take into account that according to eqs. (3) and (4) the values a,/El(2 ) (i = 1, 2, 3) are the roots of the equation of the third degree =


In conformity with the requirement that the roots of this equation should be real and positive, we find that the value ~/ can lie in the following intervals, determinable through the value of ~: ~:2(3--~) ~ 547 g 32--1


1 < ~ ~ 3;

0 <__ 54,/ _< 3~--1




The ratios o~1(4) and d"2(4),as functions of ~ for different values of ~/satisfying the inequalities (5), are represented in fig. 1. The shaded area lies between the









curve corresponding to the hydrodynamical approximation (designated by ~]~) and the curve corresponding to the moments of inertia determined with the help of eq. (2) at fl = ~ = 0.2 (designated by ~]HD). Fig. 2 shows the possible values of e'1(6 ) ratios for different values of ~ and the values of ~ that satisfy the inequalities (5).



t2. ~(4)


9- -


,, - - } "


- f







Fig. 1. The possible theoretical values of the ratios 8, (4) and 8~ (4) for different values of ~ and ~/.


c166 ~




o ~{S6








Fig. 2. The possible theoretical values of 6',(6) for different values of ~ and r/.







One should certainly keep in mind that the values of ~ satisfying the inequalities (5) correspond to all possible relations between the principal angular momenta and in particular to some which obviously cannot be realized in a nucleus. Such is, for instance, the case of a solid-body rotation of a spherical body (I 1 = 12 = 13, ~ = 1) marked by a cross in figs. 1 and 2. Similar are the cases for which ~ is near to or equals zero. 3. C o m p a r i s o n



and Discussion

Recently it has become customary to consider that in nuclei the moments of inertia have values which are intermediate between those given by a hydrodynamical model and b y a solid-body rotation model. Therefore, the real relation between the principal moments of inertia of a nucleus apparently corresponds to those values of ~ for which the energy ratios ~ ( J ) are shifted from the hydrodynamical values in the direction of the shaded area. This being taken into account, the energy ratios 6~,(J) will only slightly change with the change of ~ values tolerated for atomic nuclei. This dependence becomes particularly slight for ~ ~ 4. Almost all the experimental ratios of 8 1 ( 4 ) and 5~1(6) available at the present moment (see table 1) lie lower than the possible theoretical TABLE


E x p e r i m e n t a l ratios Nucleus OS190

Mg 24 Fe~6 0S188

OslS~ SII~. is2

ErlSe Erl6S Dyl~0 GdlSS Wlfl2

El(2 ) (keV)








5.12 4.61 4.85 6.17 7.73 11.68 11.87 12.47 13.35 15.34 14.68



i 186.7 1368 845 155 137.2 122.3 80.7 79.9 87.0 89.0 100.9

2.99 3.09 3.49 4.09 5.60 8.92 9.76 10.29 ll.16 13.01 12.11

4.04 3.82 4.54


5.1o 6.63 10.14 10.67 ll.2~6 12.11 14.0 13.20

2.94 3.01 2.47 3.08 3.16 3.01 3.29 3.31 3.27 3.24 3.26


6.76 6.86 6.70 6.56

3) e) 4) i


4) 7) 7) 8) 3,4)


values calculated in adiabatic approximation (the shaded areas in figs. 1 and 2), while the experimental energy ratios @2(4) lie lower than the theoretical ones at any ~7 satisfying inequalities (5). This indicates that the agreement of the theory with experiment cannot be improved by passing from the hydrodynamical moments of inertia to values intermediate between these and the moments of inertia obtained in a solid-body model. Moreover, it follows from fig. 1 that if we formally choose those relations between the three principal moments of inertia which improve the agreement with the experimental ,#1(4) ratios, we shall at the same time make this agreement worse for the



6*3(4) ratios. Thus, it m a y be stated that the disagreement of the theory with experiment is occasioned by the adiabatic approximation used in the calculations. The results obtained in the present article permit to expect that a theory which will take into account the coupling between rotation and the internal excited states of a nucleus can be developed on the basis of the dependence of the angular momenta in the hydrodynamical model on the parameter 7 responsible for the deviation of the shape of the nucleus from axial symmetry. The dependence of the relative energies 6*1(4), 6*2(4) and o'1(6 ) on the deviation of the parameter ~/from its hydrodynamical value ~/n is less than the corrections due to the violation of the adiabatic conditions. References 1) 2) 3) 4) 5) 6) 7) 8)

A. S. Davydov and G. F. Filippov, J E T P 35 (1958) 440; Nuclear Physics 8 (1958) 287 A. S. Davydov and V. S. Kostovsky, J E T P 36 (1959) 1788; Nuclear Physics 12 (1959) 58 B. S. D~elepov and L. K. Peker, Decay Schemes of Radioactive Nuclei (Moscow, 1958) B. S. D~elepov and L. K. Peker, Excited States of Radioactive Nuclei, UINR, Dubna, p. 218 O. Nielson, N. Roy Poulson, R. Sheline and ]3. S. Jensen, Nuclear Physics I0 (1959) 475 S. Cook, Nuclear Physics 7 (1958) 480 I(. P. Jacob, J. W. Mihelich, R. Hantz and T. Handey, ]3ull. Am. Phys. Soc. 3 (1958) 558 ]3. Bitterlikh, E. P. Grigoriev, ]3. S. D~elepov, A. P. Zolotavin and B. Kratzik, Izv. Acad. Nauk SSSR, Ser. Fiz. 23 (1959) 868