on the variance of the fluctuating cross section

Volume 211, number 4 PHYSICS LETTERS B 8 September 1988

O N T H E VARIANCE O F T H E F L U C T U A T I N G C R O S S S E C T I O N

E.D. DAVIS a and D. BOOSI~ ~,b, 1 Max-Planck-lnstitutJ~r Kernphysik, D-6900 Heidelberg 1, Fed. Rep. Germany

b Cyclotron Institute and Physics Department, Texas A&M University, College Station, TX 77863, USA.

Received 22 March 1988; revised manuscript received 27 June 1988

We present an exact expression for the variance in the average cross section of a compound-nucleus reaction between two definite channels and explore its qualitative features. In particular, we discuss deviations from Ericson theory and their relevance to searches for time-reversal non-invariance in compound-nucleus reactions.

In this letter, we discuss the var iance in the energy-average o f the integrated cross section aab for compound- nucleus react ions (o f fixed spin J and par i ty n) between defini te entrance and exit channels a and b, i.e. we consider Vab = ( aab -- #ab ) 2 (where the ba r denotes the average ), or, in terms o f the relevant f luctuating S-matr ix e lements (S n = S - S ) , the 3- and 4-point funct ions I S~a 12S~a and I S~b [4, as

Vab [ S n ~ b [ 4 - ( l S a a b l 2 ) 2 - J a b ( 2 [ ( l -* n 2 n = - -Saa)[Saa [ S , a + C . C . ] - 2 l l - & a l 2 l S ~ a l 2 ) ,

where we have taken aab = I Sab-- 8~b I 2 and used the d iagonal i ty o f ~q for compound-nucleus processes. (We sup- press here, and in what follows, k inemat ica l factors and dependence on J, n and the scattering energy. )

Fol lowing the approach of ref. [ 1 ], we have been able to der ive exact and tractable expressions for I Sanb 14 and fl 2 fl I S ~ I Saa- In contras t to previous studies o f such higher moments [ 2 -4 ], they hold for all resonance regimes,

independen t o f the degree o f absorp t ion or the number o f open channels. Our invest igat ions o f their proper t ies shed new light on the appl icabi l i ty of Ericson theory, relevant to current studies of spacet ime symmet ry viola- t ions with the fluctating cross section.

Lack o f space prohibi ts an adequate descr ipt ion o f our der iva t ion here. However, we note that the start ing poin t for the calculat ion is p rov ided by the s imple observat ion that the energies of the fluctuating S-matr ix elements in these ensemble averages are all equal. Thus one can relate them to the two-point generating funct ion o f ref. [ 1 ], and so adopt several o f the methods and results therein %

We find u l t imate ly that both I Sanb I 4 and n 2 n [ Saa [ Saa can be writ ten in the form

0 0 0

where

# ( Z ) - (I-A)AIAI-221 [Al ( 1 -+-A, )A2( 1 + 2 2 ) ] 1 / 2 ( 2 - 1 - 2 1 ) 2 ( 2 + 2 2 ) 2 '

I - To2 ~(Z)--- l q cope. ( 1 + Tc~.l )1/2( 1 + Tc22)I/2'

t Permanent address: Groupe de Physique Nuclraire Th6orique, Centre de Recherches Nuclraires et Universit6 Louis Pasteur, F-67037 Strasbourg Cedex, France.

~' A detailed description of the essential technical issues in such applications of the 2-point generating function, including the general- isation of our present results to a form catering for direct reactions, is given elsewhere [ 5 ].

0370-2693 /88 /$ 03.50 © Elsevier Science Publ ishers B.V. ( Nor th -Hol l and Physics Publ ishing Div is ion )

379


in which the transmission coefficients T~ = 1 - I~-c 12, and the "pre-exponential" factor J is for (i) [g i l a ] 2 r " Saa •

J = - ~q~ (4 trg(#ol,~) + 2 t rg(#a) trg(Va) +ra trg(#~) ( t rg(# 2) + ½ [trg(#a) ]2}),

(ii) IS~b 14:

J = {trg(v~) + ½r~ [ t r g ( ~ ) ] 2}{trg(Vb) + ½rb[trg(#b) 12}

+6~b( [trg(Va) ] 2+4 t rg (v] ) + r~{ [trg(#~) ]2 trg(Va) + 8 t rg(#a) trg(#,Va) + 8 trg(#a2V~) }

+ r 2 t rg(# 2) {trg(# 2) + [ trg(#~)]2}) .

Here, r~ = 1 - T ~ and #~, z,~ are the matrices

#a=T~2o(I+T,~2o) -1, vaTZ2o( l+2o ) ( l+T ,~2o ) -2 ,

where 20 is the 4 × 4 diagonal matrix with entries on the diagonal of 21, 22, --2 and --2; " t rg" is the graded trace operation. For a 4 X 4 matrix M, trg (M) = ( m ~ ~ + m22 ) - ( m33 + m44 ), where m , is the ith entry on the diagonal.

A noteworthy feature of these results is their similarity to the final expression n n. for SabScd in ref. [ 1 ]: only the pre-exponential factors are different. (This property holds for any n-point function, n >t 2, derivable from the 2- point generating function.) Although the integrals in eq. ( 1 ) must, in general, be computed numerically, the nature of their dependence on the number of open channels [confined to the product ~(2) ] and the transmission coefficients is particularly convenient.

When the sum of transmission coefficients S~ = 3~copenTc >> l, the above expressions for n 2 n I Sa~lS~a and I S~b L 4 can be approximated by explicit (asymptot ic) expansions [ 5 ]. Such results are appropriate to strongly- overlapping resonances, for which it has been shown [2 ] that, to leading order in 1/S~ (i.e. order 1/$1 for lanai 2 fl Sa~ and order 1 /S 2 for L Sane I 4 ) , ] ari a [ 2Sflaa ~. 0 and I Sanb 14 = 21S~b 12 in apparent agreement with the assumptions of Ericson theory ~2. However, this conclusion overlooks the magnitude of the corrections.

These can be generated by following the method of section 4 in ref. [ 7], which exploits the behaviour of rt(2) as $1 becomes large. After the change of integration variables 2 ~ 2 2 , (i = 1, 2 ), 7t (2) is rewritten as

rt(2) = e x p ( - $1 p12) ( 1 + ½S2P2~ 2 +.. .) (2)

[in which Pk = (2 k +22 ~)/2 + ( - 1 )k+ 1 and Sk k = ~copen Tc ]. An expansion in inverse powers of $1 is thus indi- cated. In fact, we find it more convenient to expand in inverse powers of $1 + 1, because under the above change of variables

1 121-221 1 - 2 #('~)~2--3 ( 1 + 2 1 ) 2 ( 1 + 2 2 ) 2 ( 2 , 2 2 ) t / 2 ' # ~ , #~= [ ( 1 + 2 ~ 2 ) ( 1 + 2 2 2 ) ] 1 / 2 ,

and so #a.n(2) is naturally treated as a unit. Otherwise, we proceed as in ref. [ 7 ], and using the result for I Sflab 12 given there, we find that (to lowest non-

vanishing order)

2T]T~ iSaa In 2Saa= 8 f l ( - & ~ ) T a 3 ( S , + 1 ) 2 , iSnab14__2(iSflao12)2=(l+7tSab)[6__4(T~+Tb)+r21 (Sl + 1 ) ~ '

in which r2 ( Sz + l ) / ( Sl + l ). More generally [51, I Snab I 2 S a bfl = ~ab I S aafl 1 2 S aa.fl

~2 A "proof" that S n is necessarily gaussian distributed when S~ >> 1 has also been presented in ref. [ 6 ], but it implicitly ignores odd moments.

380


We note that deviations from gaussian behaviour are more significant for elastic processes that they are for inelastic (cf. section 4 in ref. [3] ). In particular, IS~a 2 n I Saa, far from being zero, is of the same order in 1/Sl (namely 1/S~ ) as I S~nh 14! Clearly, Ericson theory is inconsistent in that it retains terms like I S~b [4 but discards termslike I S~. 2 fl I Saa. At the same time, neglect of such terms should be of significance in applications outside the regime of strong absorption (each S'aa and Z~q~a << 1 ), notably in benchmark experimental determinations (with protons) of the elastic enhancement factor [ 8 ].

Another interesting feature of the result for n 2 n IS~ I Saa is the factor of - S ~ . It indicates that our result is a consequence of unitarity, which restricts Sab to the unit circle (in the complex plane): if ~q~b is non-zero, the distribution of S,nh should be correspondingly distorted in the direction opposite to ~qab by an amount propor- tional to its magnitude. In fact, one can also account for the S~ dependence of I S~a 12Sfla in the large Sl limit by appealing to unitarity [ 5 ].

The factor of S~a is relevant to previous Monte Carlo studies of the distribution o f S n within the microscopic model we employ:

( 1 ) The "asymmetry" in the distributions of Re (S~ ) and Im (S~na) is measured by

Re (S~na) 3 {3Re(lSn~a[2Sn~a) ~Re(Saa) Im(S~,)3 = ] i m ( [ S n a l e S n ) pc ( Im(Saa)"

Accordingly, the "mysterious" difference in asymmetries of these distributions seen in refs. [ 3,9 ], is a consequence of the fact that S'aa was chosen to be real.

(2) Gaussian behaviour of Re (Sn, a) and Im (S,n,) has been observed under conditions of strong absorption. This is not inconsistent with our result for I S,n, 12S~,, since, under these circumstances, the factor of ~q,a ensures it is negligible.

In studying the approach to the regime of strongly overlapping resonances, we have considered the particular combination C,h= (aab)2/(#ab) 2- 1, taking (for simplicity) a~b: Cab then involves the ratio of IS,nb [4 to ( ]Sflab [ 2 )2 , and is a function only of transmission coefficients. The expression for I S,nb 12 has been taken from ref. [ 1 ] and the numerical integrations involved have been performed with a programme developed in the study

fl fl* of S,bSca (and documented in section 5 of ref. [ 7 ] ). Some feeling for the generic behaviour of Cab is given by the plots in fig. 1 - refer to the caption for the choices

of transmission coefficients, etc. We note: (a) C,b would appear to be monotonic function of S~. In fact, this is not always so for 0<S~ < 1. There is an

obvious explanation for the trend and its onset in the presence of the product zr0.). Provided S, > 1, the factor exp (-S~p~2) will be dominant [cf. eq. (2) ]. Thus monotonic behaviour is to be expected, and, whether Cab increases or decreases, is determined by the sign of the next-to-leading order term in its asymptotic expansion [cf. eq. (3)] .

(b) Despite the way in which C,b has been defined, outside the regime of strongly overlapping resonances, its magnitude depends strongly on the values of Ta and Tb - cf. the A's and m's. Keeping S~ fixed, Cab increases with decreasing Ta" Tb or, roughly speaking, decreasing #ab -- in words, the smaller the average cross section, the larger the relative fluctuations. There seems to be no simple explanation of this phenomenon in terms of properties of the distribution o f S n within the unit circle.

(c) The first two terms in the asymptotic expansion corresponding to Cab, namely

1 l + 2 1 6 - 4 ( T , + T b ) + r 2 ] Sl +1 ' (3)

yield a reasonable estimate for S~ as little as 10, the expression becoming almost exact for S~ >i 20. The inclusion of higher-order terms gives rise to conflicting results (i.e. in some cases, improved agreement, in other cases, not), presumably a reflection of the asymptotic character of the expansion. These conclusions should apply to other moments o f S n. (Surprisingly, such simple matters do not appear to have been addressed previously. )

(d) Apropos the rate of convergence of C,b to its Ericson value of unity, it is apparent that this is, in general,

381


3.0 I I I

2.5 j

Cab ,

20 ,y • / /

1.5 / .*Y ~ * ~ J

1.0 ~ - . ~ _ . ~ . ~ _ . . . . •

0.5 I I 0.00 0.05 0.10 0.15 0.2,

1/(S 1 ..,1)

Fig. 1. C,~,, exact (symbols) and approximate [eq. (3) - lines], versus 1/(S~ + 1 ), S~ being the sum of transmission coefficients. The exact values have been generated by taking n multiples of the following sets of transmission coefficients: (0.1, 0.11, 0.12, 0 .13 ,0 .14)- e (T,=0.1, T~,=0.14); (0.88,0.89,0.9,0.91,0.92) - • (T,=0.9, T~,=0.92); (0.1, 0.5, 0.9) - • (Ta=0.5, T~=0.9) and • (T,=0.1, Th=0.9). For example, n=3 , 4, 6, 8, 10, 12, 14 for the • ' s and • ' s .

2.0

1.8 Cab

1.6

1.4

1.2 !

1.0

i I I i I

~1=3

• 6

• 12 i i • •

, , , . • * 24

I J I ~ I ~ I

0.50 0.60 0.70 0.80 r2

Fig. 2. Exact values of Cab for various T,. distributions such that St and the number of open channels are fixed ( Ta = Tb = 0.5 ); the abscissa r 2 = ($2 + 1 ) / ( Sl + 1 ).

rather slow. In particular, Cab can be significantly larger than unity in the interval 10 < S 1 < 20, where one would naively expect Ericson theory to be a reasonable approximation. This may account for the observation [ 10 ] of an incidence of large values of the fluctuating cross section in excess of the number expected on the basis of Ericson theory.

Despite the sensitivity of Cab to Ta and Tb, our discussion under (a) suggests that it may otherwise depend essentially on only SI (and not on any other features of the distribution of transmission coefficients). Concrete evidence for the validity of this is given in fig. 2. The points in this diagram correspond to randomly chosen T,. distributions, subject to only two constraints: the sum SI and the number of open channels are fixed. (The choice of r2 as abscissa is arbitrary - it quantifies the differences between the T~ distributions, without taking on too wide a range of values. ) The same pattern is seen when the number of channels is varied, keeping & fixed.

This remarkable property combined with the monotonicity of Cab implies that once we know Ta, Tt, it is (in principle) possible to extract from a measurement of Cab the corresponding value of S~, thereby providing a consistency check of the phenomenological optical model calculation needed to yield the values of Ta, Tb in the first place. In practice, the value of S~ extracted should not be too large (or, equivalently, the measured value of Cat, should not be too close to unity), because, as &--,oe, Cab becomes increasingly insensitive to its precise value.

We turn finally to the relevance of our findings for studies of spacetime symmetry violations, considering, by way of illustration, recent theoretical analyses [ 11,12 ] which extract a competitive upper bound on the time- reversal non-invariant part of the nucleon-nucleon interaction from the detailed balance measurement of ref. [ 13 ] in the domain of strongly overlapping resonances.

Because of their physically distinct origin, the symmetric and anti-symmetric parts of the fluctuating S-matrix were taken to be independent stochastic variables. The assumption of independence, however, violates unitarity, and the example of n 2 n I Saa I Saa illustrates that correlations implied by unitarity can still be "felt" in the Ericson regime, particularly for reactions involving charged projectiles (like those considered in ref. [ 13 ] ).

The inclusion of such correlations could have a dramatic effect. Consider, for example, the observable studied in ref. [ 11 ], namely

( a ot, - a a~ ) ( a ot, - a t,a ) F~ = [ (Oat, - - a a b ) 2 l 1/2[ (O'ba --aba) 2 11/2'

382


which has the advantage that a comple te theoret ical analysis is possible without assumpt ions about the distr i- but ion of S n. To leading order in the t ime-reversal non- invar ian t interact ion, it is given by

1 - - 2 ( ~ +5~z -{ -C .C. ) / (0" - - ~ ) 2 •

Here, a = (G,b+O'ba)/2, ~ = ISSl218SI '' and ~ = (S~*) 2(~S) 2, with SS= (Sn~b+Sna)/2--~ab and k S = (Snb - Sna)/2. The independence hypothesis impl ies

~ = l a s l 2 " l S S I 2, and ~ 2 = 0 .

Whi le the factor isat ion in ~ may be inaccurate, the neglect o f ~ could be more serious. Uni ta r i ty implies that, when jus t two channels are open, F,:,b = 1 even i f 6 S ~ 0 [ 14 ]: with an explici t ly uni tary parametr i sa t ion , one sees that the posi t ive cont r ibut ion o f ~1 is exactly cancelled by a negative cont r ibut ion from ~ . Geometr ica l argu- ments, paral lel ing those appealed in the case of I Sana 12S~a, suggest that ~ remains negative and is not necessari ly negligible in compar i son to ~ for strongly over lapping resonances. A reinvest igat ion of Fab, entai l ing the (for- midab le ) calculat ion o f 6-point functions, is warranted.

We thank F.H. Fr6hner , H.L. Harney and H.A. Weidenmti l ler for the helpful discussions, and L. van der Merwe for assistance in typing. D.B. acknowledges f inancial suppor t from the Center for Theoret ical Physics, Texas A&M Universi ty.

References

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[ l 0 ] H.L. Harney, private communication. [ 11 ] D. Boos6, H.L. Harney and H.A. Weidenmiiller, Phys. Rev. Lett. 56 (1986) 2012. [ 12] J.B. French, A. Pandey and J. Smith, in: Tests of time reversal invariance in neutron physics, eds. N.R. Roberson et al. (World

Scientific, Singapore, 1987) p. 80. [13] E. Blanke et al., Phys. Rev. Lett. 51 (1983 ) 355. [ 14] E.M. Henley and B.A. Jacobsohn, Phys. Rev. 113 (1959) 225.

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on the variance of the fluctuating cross section

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