Dependence of the angular distribution of fission fragments on the target nuclear spin

Nuclear Physics 27 (1901) 348-351 ; +@ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

E

ENCE F AG

F TH N S

ANGULAR DISTRIBUTION O JCLEA S N T TARGET

FISSION

V. M. STRUTINSKI I. V. Kurchatov Atomic Energy Institute, Academy of Scier-s, Moscow, USSR Received 20 February 1901 Abstract: Angular distribution of fission fragments is calculated for small anisotropy . The analytical expression obtained takes into account the dependence of the fission and neutron widths on nuclear angular momentum . The theoretical data are compared with experiment .

In ref. 1) a method of calculating the angular distribution of fission fragments was set down and analytical expressions were obtained for the angular distribution when the target nucleus spin is finite. In case of a Gaussian-type probability distrib ation in K (the projection of the angular momentum of the compound nucleus J on the direction of the fission) given by a(K) oc exp ( -K2/2KO1z),

(1)

the angular distribution of the fragments weakly depends on the initial spin: the spin-dependent terms in the angular distribution are of the order 12 s2 / (2Ko2 ) z , where s is the initial spin and 1 the neutron orbital momentum, while the principal angle-dependent term is of the order 12 /2Ku2 . The dependence of the total decay probability of the compound nucleus rt (J) on the nuclear angular momentum also leads to second order corrections in the angular distribution . This effect also leads to an additional weak depenuence of anisotropy on the initial spin. When the anisotropy of the angular distribution is small and dependence of rt on J is weak, one can obtain an analytical expression for the angular distribution taking account of the J dependence of rt. The total width of a compound-nucleus state is the sum of radiative, neutron and fission widths. The first of the three is usually negligibly small while the dependence of rf and rn on the momentum can be determined öy the statistical theory as F,, (J) ^ F,, (0) exp [ -(Otf-al)J(J+1)], rf( .J)

^'(2.J+1)-1rf( ®) exp

C(ai -oc1 *)J(J+1)]

K- -J

exp [ -K2 /2KO2] .

(3)

In (2) and (3) we use a = A2 /2.fT, where f is the nuclear moment of inertia and T is the temperature of the nucleus . The subscript i designates the quantities 348

DEPENDENCE OF THE ANGULAR DISTRIBUTION

349

for the initial compound nucleus, the subscript f those for the compound nucleus state after emission of a neutron and the asterisks the quanti.:ies for the deformed "transitional" nucleus . The constant al* = A2/2 .fl * T*, where ,f1 is the moment of inertia of the transitional nucleus with respect to the axis normal to the symmetry axis. In the other cases the moment of inertia is equal to that of a spherical nucleus Jo = 2 AmR2. Expanding rn(J) and we have rt (J) into a series where

yt U) = rfU)Irt(J) = vt(o)[1+qjU+1)+ . . .],

q=

yn(o)(at-ul *) +yf(o)/2Ko2, yr(o) = rt(o)lrt(o) .

A1o1 = rn(o)/rt(o),

This expression for yf(J) should be substituted into the general expression for the angular distribution of the fragments, which, with the fixed values of l, the channel spin S = Jo ± â and J, is the form 1) WisJ (v) = where

2j+1_ 2 (2Jo + 1 ) aj (K)

a

,,l

(C8Ii.)2 aj(K) I Y,.(v) I 2 ,

= yf U)

oc(K) .

The total angular distribution of the fragments is given by the expression W (v)

=

1SJ

~gW csj(v)l

1SJ

(21 + 1)~a,

where ~a is the neutron absorption coefficient assumed to be independent of J and S. Let us substitute eqs. (5) and (4) into (6) and perform the summation, for which we shall use the following formulas of summation of Clebsch-Gordan coefficients a~

I+S J = p-s~

J~ il-Si ...~ 1

(CSxEm)2 =

CJx 2 ( Sp am)

p

for IK -mi < S, 0 for IK-ml > S, 1

1(1+1)+S(S+1)+2mu for 0 for

I,uI Iyj

The last equation is taken from ref. 1) t . t Let us also note the following useful relations') : E`J ( CSP 1M)$j(J+1) = W.(j +1+S)ll(1+1)+S(S+1)~ where the summation is restricted to the even or the odd values of J.,

< S, > S,

ImI ç l, Iml

> 1.

SSO

v.

M . STRUTYNSM

After the summation over J, we obtain Was (1') =

Wrsj(V) = C

I

m+s

Y, - m-5

lYamtv'~

t2

{1+q[l(1+1)+S(S+l)+2m(k--m) ] )a(K),

where the constant C does not depend on Z and S. Let us express a (K) as a(K)

1--K2 ~2K02 +~rIK4 1(2Ka2 ) 2+

. . ..

(8)

The arbitrary coefficient n was introduced in eq. (8) in order to take account of a possible deviation of the K-distribution from the Gaussia.n distribution for large values of K. The term with K4 leads to the corrections of second order. The correction of the angular distribution due to the J dependence of yt is as small as q{2K02, i.e ., is also a small magnitude of second order (q and ~Koa being of the same order) . Taking this into consideration we obtain from (6), (7) and (8) after a simple e-,7aluation A (v)

where

= Car(v)-ar(90') .II cri(90 °)

(j-2/4K02){1+5-2/6K02--q[(14/12-12+ S2) COS 2 Y - .4 (lî/2K02) C 13 (14' (1--sin4 v) +i2/S2 +-1. )coS2 v]}, a

is the average square of the momentum transferred to the nucleus by the neutron, 14 and 14' are similar averages of 12(1+1)2 and 1(12 --1) (1-~ 2), while . In (9) the terms of third order are discarded. To the first S2 - W 0+-1)2+Jorder with aspect to 12/2K02 , the angular distribution does not depend on the target nucleus spin. The change is the angular distribution anisotropy as S 2 changes by AS2 equals As A (0') =

12

4Ko2 {

_ AS 2. ( 1-- 3)/6Ko2'+- ¢ q}

(10)

According to the experimental data of ref . 2 ) there is a small systematic difference in the magnitude of anisotropy for the nuclei U23s , U233 and Pu289 ; one has A (0°) (U23b) -A (0°) (U233) ti A (()O) (U233) -A (00) (Pu239) s:w It is natural to interpret this as the initial spin effect . For the three abovementioned nuclei the values of AS2 equal 7 and 8 units, respectively. For a theoretical estimate of As A (0'), we shall put Tt = T* = T in (4) . The magnitude of = fo (1 .2z+5.6z2), where .00 is the moment of inertia for a spherical nucleus and z = 1-(Z2/A)/(Z2 IA ),,,,It (ref. 3)) . At T .= C,3 MeV, ya(°) .^., yf (O) .. 0.5 and the solid-oody value of fo, we obtain q se 0.01 . The

D2PENDENCE OF THE ANGULAR DISTRIBUTION

351

parameter /4Ko2 is determined directly by the experimental value of the anisotropy, since this parameter simply coincidences with the latter to a first approximation . For neutrons of energy from 3 to 5 MeV (l2j4KQ2) ~-, 0 .12, K02 = 12--15 (refs. 2 4 )) . From (10) we find that at n = 1 (i.e., at the Gaussian a (K)) q four or five times exceeding the thermodynamic value would correspond to the experimental value of the difference A (0°), which appears improbable. Another possible effect consists in cutting off the distribution a(K) as compared with the Gaussian at large K, which can well be expected since the dimensions of the nucleus are finite (see also ref. 1)) . The coefficient q in this case must be less than unity of even negative . The value 71 ~-, 0 would correspond to the experimental values of AA (0° ) if the above-mentioned thermodynamic value of q was used . A rough estimate of the maximum value Km. of K can be obtain : J from the conditions of equality of the nuclear rotation energy Emt

A2

K2 _

K2

2K02

and the excitat.aon energy of the "transitional" nucleus E* . Thus we obtain Km. sw

(E*IT*)2Ko2.

At E a 2®3 MeV, K.,,,, ms nKO , where the factor n is equal to several units. This value K... agrees with -1 s::w 0 since the distribution a(K) goes to zero at K sw 1/2Koa, n = 0. The author expresses his gratitude to Dr. J . Griffin for valuable discussions . References 1) 2) 3) 4)

V. L. G. E.

M . Strutinski, J?`TP 39 (1960) ?81 Blumberg and R. B. Leachman, Phys. Rev. 116 (1959) 102 A. Pik-Pichak, JETP 36 (1959) 962 Simmons and R. L. Henkel, Phys. Rev. 120 (1960) 198