The WKB calculation of the nuclear isotope shift

Volume 23, number 8

THE

WKB

PHYSICS LETTERS

CALCULATION

OF

THE

21 November 1966

NUCLEAR

ISOTOPE

SHIFT

V. P. K R A I N O V I. V.Kurchatov Atomic Energy Institute, Moscow and M. A. MIKULINSKY All-Union Scientific Institute of Physical-Technical and Radio-Technical Measurements, Moscow Received 10 October 1966

A semi-classical formula for calculations of the change of the mean-square nuclear radii caused by the addition of a neutron is suggested.

The isotope shift in heavy nuclei is caused mainly by the change of the nuclear electric charge distribution. The isotope shift was calculated by using the theory of finite Fermi systems [I, 2]. The change of the nuclear electric charge distribution due to the addition of the neutrons is caused by the nucleon interaction only and vanishes when it is absent. Therefore the problem is of a special interest. In this note the results of an analytical calculation (WKB approximation) of the isotope shift are demonstrated. This method may be applied to the consideration of some problems, for example, the nuclear quadrupole moments and the dipole transitions. It was shown in ref. 2 that the isotope shift c a l c u l a t i o n fits the e x p e r i m e n t only , when a nucleon i n t e r a c t i o n is u s e d which is diff e r e n t in sign and magnitude on the s u r f a c e and in the i n t e r i o r of the nucleus. H e r e this c o n c l u sion is a l s o c o n f i r m e d by c o m p a r i s o n with the e x p e r i m e n t a l isotopic shifts and f r o m the condition of r e a l i t y of the monopole t r a n s i t i o n f r e q u e n c i e s and by the data on the sign of the nucleon i n t e r a c tion f r o m ref. 3. The c o m p a r i s o n b e tw e e n the t h e o r e t i c a l and e x p e r i m e n t a l data on the i s o t o p e shifts did not m a k e it p o s s i b l e to find the i n t e r a c t i o n a m p l i t u d e s but it g i v e s the dependence between the i n t e r i o r and s u r f a c e i n t e r a c t i o n s constants. We obtained the following s i m p l e a n a l y t i c a l e x -

p r e s s i o n f o r the proton r . m . s , radius change due to addition of two neutrons taking into account the n u c l e a r shell s t r u c t u r e : g~(r2>

R2

l2

- a + b Tt

' 1

1

0.4 + 1.3A-~ 4

2.7

- 6.8

b-

1

+ 0.5A-~

2.7 - 6.8A-

"

Here Z, R and A are nuclear charge, radius and atomic number respectively; Io2 = 2~oR, where co is the fermi energy; I is the angular momentum of the shell model state occupied by the added neutrons. We have used this expression for the isotope shift calculation of the optical and the X-ray lines and for the isomeric shift calculation. Our results agree well with the experiments and the computer calculations [2, 4]. The detailed consideration will be published in ref. 5. 1. A. B. Migdal, The theory of finite fermi systems and the nuclear structure (Moscow, Nauka, 1965). 2. G.G.Bunatyan and M. A. Mikulinsky, J. Nuel. Phys. (Ud. SSR) I (1965) 38. 3. A. B. Migdal, A.A.Lushnikov and D. F. Zaretsky, Nuel. Phys. 66 (1965) 193. 4. V.A. Belyakov, Phys. Letters 16 (1965) 279. 5. V. P. Krainov and M. A. Mikulinsky, J. Nucl.Phys. (Ud. SSR) 4 (1966) II.

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