- A L P H A - D E C A Y OF R d A c ON T H E C O L L E C T I V E M O D E L A N D T H E

SPIN OF T H E A c X N U C L E U S

S. G. R y z h a n o v

The most complete and exact energy level scheme for the AeX nucleus and th e a - d e c a y scheme for the RdAc nucleus to these levels is given in Reference [1]; the a -par t i c le yields in percent are also given (cf. Figure), According to the Bohr-Mottelson collect ive model of the nucleus [2] the ground state and excited levels of AeX with energies of 59, 286 and 332 key must be considered single-particle states because a - d e c a y to these levels is most probable (cf. Figure). Apparently the levels at 110, 178 and 938 key are rotational

satellites of the second level, forming a developed rotational band. Actually, according to [2] the energy of rotational excitations for an even-odd nucleus (AcX is R a ~ ) is given by the expression

E r = B[I(I + 1 ) - - K ( K + 1)], (1)

where I Is the rotational moment of the levelwhileK is the moment for the single-particle level; B is the n,

rotation eomtant B = - ~ (J Is the effective moment of inertia). In even-odd nuclei we have I = K. K + .1,

K + 2,etc. It is easy to show that the difference in energies between the levels 110 - 5 9 ; 1'13 -- 59; 238 -- 59 are tn good agreement with Equation (1) for K equal to 5/2, ' / /2 or 9 /2 with a rotational constant of 6.97, 5.55 and 4.62 key,respectively. The level at '/9 key is apparently of nomotational nature.

The probability for a - d e c a y to a given level of the daughter nucleus is given by the Ter-Martiroslan formula [3]

I L K

(2)

where C.~,~Kd,0 /s the Clebsch-Gordan coefficient, I is the orbital moment of the a -par t ic le , C / s a con-

stant for a given rotational band, a and 13 are semi-empir ica l constants the values of which are determined from empirical data. According to Equation (2) W K always falls off monotonically with increasing excitation

I energy. However, in RdAc the relative yield of a - p a n i c l e s to the 173-key level is twice as large as to the neighboring level. This departure may possibly be explained by assuming that single-particle levels of an odd nucleon in the field of a strongly-deformed nuclear core (cf.Ref. [4] ) with sp~n K are not very different in energy from rotational towels with spire K + 2 and .K + 3. Since the squares of the Clebsch-Gordan eceff ic ier~ fall off rather sharply with l~ereaslng I the probability of a - d e o z y to these rotational states becomes negl t~bly small and the observed yield~ ~f ~ -particles are actually determined only by the coatributio~ from the s in~e-parde le leveb with spin K, The factor C in front of the ex,eone~iZi is the same-for all ba~A~ being considered; henee, t l~

100

yield ratio for a-particles corresponding to the formation of levels with energies of 173 and 59 key (4%: 21%) is determined only by the exponential factor e" aEz which is due to the reduction in the penetrability of the potential barrier caused by the reduction in the e~ergy of the alpha particle owing to excitation of the daughter nucleus. The latter quantity is equal to the difference of energies of these levels. Thus. we find the factor a = 14.2. The above also applies for the energy levels at 238 and 173 key. Hence the yield ratio for a-particles

E

332 IS % $O7

t% Z85 :7%

,.8 (,Ts) (,,,,,3) w

t", x" e} ,73 0t,1 ,

.o (sf) 2%

59 t 0 ) - - �9 2 t %

Z9 " " 5 ~

0 ta% 17e x

Level scheme for AcX. E is the energy of the level with respect to the ground state in key; the figures in brackets are the energies of the rotational levels with respect to the level at 59 key. The figures at the right indicate the a-par t ic le yield (in percent) for a-,decay of RdAc with formation of the corresponding AcX level .

corresponding to the formation of these levels (2%: 4%), should be equal to the ratio of the exponential factors indicated above, i. e., e-'=-0-238/e--,t.0.t73 With a = 14.2, this ratio is close to the empirical value, L e., 1/2. To compute the constant 15 which appears in the exponential factor and which is determined by the height of the centrifugal barrier use can be made of the yield of a-particles corresponding to the formation of the level at 110 key (the first rotational satellite of the band).

We propose, that in accordance with conservation of parity, I can take on only even values of the natural series. The squares of the Clebsch-Gordan coefficients for K equal to 5/2, 7/2 and 9/2 and l = 2 are corres- pondingly 5/14. 56/165, 45/143 (the terms in Equation (2) with l > 2 can be neglected). The empirical ratio for the yields of a-particles corresponding to the form- ation of levels with energies of 110 and 59 key is 2~ 21%. while the energy exponent for these levels is 0.487. From this it follows that 15 = 0.365. The value of a is

1

clo~e to that of the exponent 170/#daeeay in the Bethe

formula [5] and is also close to the corresponding exponent for Am =41 [3] . The factor 15 is In exact agreement with that given for Am m [3]. This means that the nuclides AcX and Am ~r have the same elongation.* If ft is also assumed that decay is accompanied by a change of parity (odd l ) then 15 is found to be 0.166. This cannot be explained. Hence the authors of Reference [1] have appar- ently found it completely justifiable to assign the trans- Ition with an energy of 50,2 key to the levels with the energies of 286 and 238 key. However, the empirical multipolarity of the ),-lines found in Reference [1], do not find any direct reflection in the rotational scheme given in Reference [2].

Received May 31, 1957

L I T E R A T U R E C I T E D

[1] M. Fritley, S. Rosenblum, W. Waladares and G, Boussier, J. de Phys. et Rad. 15, 43 (1954); 16,378 (1955).

[2] A. Bo!-trandB. R. Mottelson, Problemy ,9~eremennoi Ftziki No. 9, 78 (1955); A. Bohr, and B, R, Mottel- son, Phys. Rev. 90, 717 (1953).

[2] L. L. Gol'din, L. K. Peker and G. I. NovDtova, Usp. Fiz. Nauk 59, 3,459 (1956).

[4] B, R. MotteL~on and S, Niel~en, P~oble~y Sovreme:nnoiF~zikINo. i , 186 (1956).

[5] H. Be =he, NuCle~rPh~ies (state Teeth P ~ , 1948) Chapter 2, p. 179,* *

�9 Hence ~he value K = 5/2 is tobe, ~r~fetred for ~ ,~:r band being considered. �9 * Russian translation~

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