Fluctuations of a nuclear cross section around an average value varying with energy

Volume 16, number 3

PHYSICS LETTERS

c r o s s s e c t i o n a r o u n d which the f l u c t u a t i o n s occur. E x p r e s s i o n (2), in fact, is only f o r m a l l y different f r o m the Hauser and F e s h b a c h f o r m u l a for the a v e r a g e c r o s s s e c t i o n in a s t a t i s t i c a l p r o c e s s [7]:

<%~,(E)> = ~x2 ~

g(J) sl

J s~,l TJOt's'l' Tolsl ' (3)

J

1 June 1965

a d e c r e a s i n ~ function of the energy b e c a u s e the p r o d u c t of x~(E) with a t e r m (N(E)) i n c r e a s i n g with the e n e r g y gives a (aaa,(E)) d e c r e a s i n g with the energy. Let us c o n s i d e r now the c a s e of low energy. We can see that:

ll'

~ TJ c

×~--~2~(e)

r e f e r s to the i n i t i a l channel; T J , s, l, to the final channel; TJc to all the p o s s i b l e decay channels. F r o m this e x p r e s s i o n one can see that the average c r o s s s e c t i o n v a r i e s with the e n e r g y either b e c a u s e i n c r e a s i n g the e n e r g y of incident p a r t i c l e s their t r a n s m i s s i o n function i n c r e a s e or b e c a u s e of an i n c r e a s i n g in the n u m b e r of all possible decay modes of the compound nucleus. ( a a o t, (E)> is, at f i r s t , an i n c r e a s i n g function of E, then r e a c h e s a m a x i m u m and, l a t e r on, dec r e a s e s with the energy. R e t u r n i n g to e x p r e s s i o n (2) we note: 1) Theory of fluctuations gives for N(E) the e x p r e s s i o n :

'

Jll' asl s'l' (T J ~2(TJ ~2"

Tasl

asl a's'l', TJ

(5)

ot's'l'

where

TJsl

~

J

J ss' ll'

(2J+ I) g(J) - (2i+ 1)(2I+ 1) "

N(E) =

~

(4)

Ir~2(2J+ I)

(6)

(We obtain this expression straightforward from the comparison of eqs. (2) and (3) by introducing the sharp-cut-off approximation over the transmission functions.) Using the McDonald e x p r e s s i o n for 1 / c~ T J [8] and a s s u m i n g that the exponential cut-off f a c t o r s of the level d e n s i t i e s which appear in this e x p r e s sion a r e about equal to 1 (approximation which should not be too bad at s m a l l e n e r g i e s of the incident particle) 1 / ~ TcJ ~ 1/(2J+ 1)H(E)), where H(E) is an i n c r e a s i n g function of the energy. Then, Kj(E) i s a p p r o x i m a t e l y independent f r o m J and

1

ss' Jll' " clsl" • ~ ' s ' l "

If we would derive from this expression the exact dependence of N(E) on the energy, we should know the exact expression of the transmission functions as a function of the energy; it is possible however to estimate the behaviour of N(E) with the energy by making the rough approximation of a sharp cutoff over the transmission functions [1], i.e., we can neglect in the sums which appear in eq. (4) the smaller terms and consider equal to 1 the main terms. This approximation is in many cases not too bad. In this approximation N(E) is equal to the number of terms appearing in the considered sums. This number is an increasing function of the energy owing to the increase of the ingoing angular momenta. For sufficiently high values of the energy (when the new ingoing angular momenta are bigger than the initial channel spins) it varies about as E~. Some numerical calculations have been done to test effectively the above predictions giving good agreement. 2) The comparison between eq~. (2) and (3) allows us to obtain the behaviour of X~(E) with the2energy For high energy it is easy to deduce that X6(E) is

1

Kj(E) - (2i+1)(2l+1) ~ T J c c

~X2

Xot~) ~ -2 (2i + 1)(2[+ I)HiE)

(7)

it is again a decreasing function of the energy. We observe that Xo2(E) is always a decreasing function of the energy also when the last approximation is not completely valid. In fact, starting from the eqs. (5) and (6) one can see that, for a fixed J value, Kj(E) is a decreasing function of the energy and Fj(E)

J TJ = ~ ~ Totsl ot's 'l ' ss' ll'

is a i n c r e a s i n g one, so, if the n u m b e r of the t e r m s in the p r e c e d i n g s u m would, be constant, we could e a s i l y deduce that ×~(E) is always a dec r e a s i n g function of the energy. We m u s t c o n s i d e r that the n u m b e r of J which e n t e r in the s u m is i n c r e a s i n g with the e n e r g y . Its i n c r e a s i n g is however slow (it v a r i e s about as E½) c o m p a r e d to the v a r i a t i o n of each Kj(E) with the energy, and Kj(E) i n c r e a s e s quite slowly with J so the new t e r m s which appear in the sum, at a given E, cannot influence a p p r e c i a b l y the t r e n d

of Xo2(E). Moreover, in the light of the p r e c e d i n g con-

289

Volume 16, number 3

PHYSICS LETTERS

s i d e r a t i o n s one can o b s e r v e that Xo2(E) at f i r s t d e c r e a s e s slowly with the e n e r g y , then s t r o n g l y ; it f i n a l l y b e c o m e s again a slow function of the e n ergy. E r i c s o n has shown that the a n a l y s i s of a fluctuating c r o s s s e c t i o n (when N(E) and ×2(E) can be c o n s i d e r e d c o n s t a n t with the energy) by m e a n s of its a u t o c o r r e l a t i o n function and its p r o b a b i l i t y d i s t r i b u t i o n make it p o s s i b l e to obtain F, the a v e r a g e width of the compound n u c l e u s and N(E). When the variation of N(E) and Xo2(E) with the energy is small into an energy interval z~E > 2-3F it is possible again to analyse the fluctuations of the cross sections by means of the autocorrelation function C(¢) suitably defined taking into account the local average cross section (a method for obtaining (a~ot, (E)> from the experimental results is given in ref. 9): (8) ((~(~,(E) / (~aa,(E+¢) -

\<~ctc~,(E + e)>

(The bar indicates an average over the whole available energy interval.) Its Lorentzian shape gives again the value of F and its value for ¢ = 0 is given by:

1 June 1965

C

,~, (o) = L N - ~ ] .

(9)

At the c o n t r a r y , it is h a r d to a t t r i b u t e a p r e cise m e a n i n g to the p r o b a b i l i t y d i s t r i b u t i o n to the c r o s s section. In p a r t i c u l a r one cannot, in p r i n ciple, deduce f r o m the p r o b a b i l i t y d i s t r i b u t i o n of the c r o s s s e c t i o n s a value of N that can be comp a r e d i n a s i m p l e way with that obtained f r o m the a u t o - c o r r e l a t i o n , e s p e c i a l l y for s m a l l v a l u e s of

N(E). References 1. T.Ericson, Ann. Phys. 23 (1963) 390. 2. T.Ericson, Physics Letters 4 (1963} 258. 3. D.M.Brink and R.O.Ste~hen, Physics Letters 5 (1963} 77. 4. R.O.Stephen, Clarendon Laboratory Report, Oxford 1963. 5. Reports of Villars Winter School of Nuclear Physics (1964). 6. H. Feshbach, Ann.Phys. 5 (1958) 357; 19 (1962) 287. 7. H. Feshbach, The Compound Nucleus, in Nuclear Spectroscopy part B, ed. F. Ajzenberg Selove (Academic Press). 8. McDonald, Nuclear Phys.33 (1962) llO. 9. G. Pappalardo, Physics Letters 13 (1964) 320.

*****

ON T H E

PROBABILITY

DISTRIBUTION

OF

TOTAL

CROSS

SECTIONS

J. BENNEWITZ

Institut p~r Theoretische Physik der Universitttt, Frankfurt (Main) Received 28 April 1965

P r o b a b i l i t y d i s t r i b u t i o n s of d i f f e r e n t i a l and i n t e g r a t e d c r o s s s e c t i o n s have r e c e n t l y been d i s c u s s e d for the case of m a n y o v e r l a p p i n g r e s o n a n c e s u n d e r the a s s u m p t i o n s of the s t a t i s t i c a l model [1-3]. Here, we d e r i v e a G a u s s i a n p r o b a b i l i t y d i s t r i b u t i o n for the total c r o s s s e c t i o n of p a r t i c l e s with a r b i t r a r y s p i n s in the r e g i o n of s t r o n g l y o v e r l a p p i n g r e s o n a n c e s . The v a r i a n c e of this d i s t r i b u t i o n will be given by the fluctuating e l a s t i c c r o s s s e c t i o n in f o r w a r d d i r e c t i o n . According to the optical t h e o r e m , the total c r o s s s e c t i o n ~.asq, tot of a p a r t i a l wave with given channel i n a e x channel spin s, and o r i e n tation q is found to be

~ s q , tot = 4 ~ Fasq, asq(O)

where 290

Ira F

sq, ~sq (0 = O) ,

is the e l a s t i c s c a t t e r i n g a m -

plitude. We take F . . . . . . (0) to be a r a n d o m v a r i a b l e . Its i m a g i n a ~ p a r t (as well as its r e a l part) has been shown to have a G a u s s i a n d i s t r i bution with a v a r i a n c e equal to half the m e a n fluctuating c r o s s s e c t i o n of e l a s t i c s c a t t e r i n g

n asq (o = 0)) (actsq,

[2,

3].

T h e r e f o r e we obtain the p r o b a b i l i t y d i s t r i b u tion of aasq, tot:

P(~asq,

1 tot) = ~ ] ~ ~,, x

x exp{-[aasq, with the v a r i a n c e

tot -

(~asq,

tot )]

2

2

/(2tasq)}