Statistics and the average cross section

~

.~A.I-~.

Nuclear Physics A344 (1980) 185-195: ( ~ North-HollandPublishino Co., Amsterdam

2.D

Not to be reproduced by photoprint or microfilm without written permission from the publisher

[

STATISTICS AND THE AVERAGE CROSS SECTION * P. A. MOLDAUER

Ar~tonne National Laboratory, 9700 South Cass Avenue, Argonne~ 1L 60439, USA Received 18 April 1980 Abstract : Average fluctuation cross sections were computed numerically from a unitary analytic K-matrix with normally distributed pole residue amplitudes and pole spacings having the Wigner distribution. Unexpected results included an elastic enhancement factor of 2.12 in the many channel limit, regardless of transmission factors. The enhancement factors were found to be very sensitive to variations in the statistical assumptions regarding both pole amplitudes and pole spacings. The results are discussed in the light of various theories of the fluctuation cross section.

I, Introduction and summary

In recent years much effort has been devoted to the theory of the nuclear fluctuation cross section, that is to say, the part of the energy averaged cross section arising from the fluctuating part of the reaction amplitude, rather than the direct cross section which arises from the energy averaged part of the amplitude. The recent review article by Mahaux and Weidenmfiller will serve as a guide to this literature t). While earlier theories had concentrated on the weak absorption limit 2), most of the recent theoretical work has sought to describe the case of strong absorption 3- 5). Formulas that described also intermediate absorption strengths have depended upon an unknown parameter, the elastic enhancement factor W [ref. 6)] or the channel degree of freedom parameter v [refs. 7,~)]. While some numerical efforts to determine the values of these parameters have been made 6- 8), no really systematic study over a wide range of channel numbers and channel absorption strengths appears to have been reported. It is one purpose of this paper to present the results of such a systematic study based upon a K-matrix unitary model with pole spacings having the Wigner distribution and normally distributed pole residue amplitudes. The theory and method of calculation are described in sects. 2 and 3 and they lead to the following result for the channel degree of freedom parameter in the absence of direct reactions

v(T, ~ T) = 1.78+(T1Zt2-0.78)e -0"228~T,

(1)

where T is the channel transmission coefficient which measures the absorption strength and ~ T is the sum of transmission coefficients for all competing channels. * This work has been supported by the US Department of Energy. 185 August 1980

186

P. A. Moldauer / Average cross sections

The resulting elastic enhancement factor is W = 1 + 2/v.

(2)

These results differ substantially from the formula for the enhancement of Hofmann et al. 6). They are qualitatively similar to the results in fig. 4 ofref. 7). The parameters v are to be used in the Hauser-Feshbach formula with width fluctuation correction as described in eqs. (23) and (24) of ref. 7). In some conditions this formula can be approximated by the cross section formula of Hofmann et al. 6), using the enhancement factor as given in eqs. (1) and (2). However this formula does not appear to offer a great computational advantage and it does fail in some circumstances, as shown in ref. 9). Direct reaction effects which are not discussed here can be taken into account by means of the Engelbrecht-Weidenmtiller transformation 10) as discussed in ref. 8). The strong absorption theories z-s) all predict v-- W---2 when T = 1. The numerical result of eq. (1) agrees with this prediction fairly well only for the case of a single channel with T = 1 (v = 1.96, W = 2.02). For large numbers of channels the parameters tend toward v = 1.78, W = 2.12 regardless of the channel absorption strength T, according to eq. (1). Presumably this effect arises from a channel-channel correlation that is caused by unitarity and is not incorporated in the theories cited above. The various theories of the fluctuation cross section all require the introduction of certain reasonable though somewhat arbitrary statistical assumptions. With the exception of the work of Mello and Seligman 5), these assumptions are quite similar. They are, however introduced in different formalisms where they apply to quantities having somewhat different meanings. Because of this and because of the inconsistency between the theoretical expectations 2-5) and the numerical K-matrix results (1), it is of interest to study the sensitivity of these results to changes in the statistical assumptions. Such sensitivity studies are described in sect. 3. They show the expected variation of v with changes in both the pole residue amplitude and the pole spacing distribution in the weak absorption limit 2). Less expected is the fact that also in the strong absorption limit the fluctuation cross sections change substantially with changes in the statistical assumptions regarding both the amplitudes and the spacings. These results are discussed in the light of various theories in sect. 4. 2. Numerical method

The computations are based upon the K-matrix theory of nuclear reactions, A finite pole series representing the K-matrix is generated on the basis of statistical assumptions. From this the S-matrix elements and all cross sections are computed at one real energy at the center of the K-matrix pole distribution in the manner of Hofmann et al. 6). The process is repeated many times to obtain a distribution of cross section values which can then be averaged.

P. A. M o l d a u e r

/

A v e r a g e cross sections

187

The calculation starts with the specification of a number of channels N, a number of poles P and a complex average diagonal S-matrix element S.. for each of the N channels a (input average off-diagonal S-matrix elements are taken to be zero.) A real K-matrix is generated in the form

K.b = 6,,bKO + ~ 7,,yT,,b/E,,.

(3)

ii

Writing Ra = - i(1 - SJ / ( 1 + Sa.),

(4)

K ° = Re/~.,

(5)

we select

and we select the 7,. randomly and independently from a distribution f . whose mean is zero and whose standard deviation aa is given by ~r. = lm/~,/~.

(6)

The K-matrix pole positions E, are selected randomly from a distribution g which is approximately symmetric about zero, and which has a mean spacing between adjacent poles E, of unity. The selection of the distributions f, and g will be discussed is sect. 3. Next, the S-matrix is computed by matrix inversion S = (1 - i K ) - '(1 + iK),

(7)

and the cross sections between all channels are computed as

~.b = ]6ab-- S~b[2"

(8)

This procedure is repeated many times. Each time a separate independent sample is drawn from f~ and from g to generate the 7.. and E. parameters. The average S-matrix elements S.b and the average cross-sections #.b are identified with the sample averages of the many S.b and Cr.bvalues so calculated. The sample standard deviations of these quantities are also calculated. In all cases the calculated S was found to be in agreement, within statistics, with the input S. From the sample average S-matrix S the direct cross sections are calculated

d,r ff ab

= I,Lb- S~bl 2,

(9)

and from these the sample fluctuation cross sections are obtained fl dir ~r.b = ~ab - ~%.

(1 O)

The sample channel transmission coefficients T,~ = 1 - ~ ISa~l2, ¢

(11)

188

P. A. Moldauer / Average cross sections

are also calculated. Furthermore, the sample standard deviations for all of those quantities are computed. Theoretically the fluctuation cross sections are well represented by the HauserFeshbach formula with width fluctuation correction v) °awFcHF -- T"--~TbCab({ T c}, {re}),

(12)

c

where the correction factor Cab depends upon the transmission coefficients T~ and the fluctuation degree of freedom parameters vc associated with all channels c. The correction factors C are such that for any two statistically equivalent channels a and b T~= Tb, f ~ = J b W e h a v e v . = v b a n d .WFCHF ~a~ .WFCHF Oab

2 -

-

Wa =

1+

Ya

.

(13)

where Wa is the elastic enhancement factor. By including two or more equivalent channels in each calculation, we calculate the sample elastic enhancement factors as the ratios of the sample elastic fluctuation cross sections aaa,flaveraged over equivalent channels a, to the sample inelastic fluctuation cross section afar,likewise averaged over equivalent channels a and b. From these the sample degrees of freedom v, and their standard deviations are computed. The results are presented in the next section. 3. Results of calculations

The standard statistical assumptions used in these calculations were the following. The distributions f, are normal for all channels a. The distribution g is generated by drawing spacings between successive poles E, at random from the Wigner distribution of spacings D [ref. 11)] gw(D)dD

= ½~De-"D2/gdD.

(14)

In addition, in order to mock up the expected anticorrelation of neighboring spacings [ref. 12)], the spacings were drawn in pairs from the distribution (14) and within each pair the smaller spacing was used first, then the larger one. However, no statistically significant effects on the calculation could be detected when this anticorrelation feature was omitted. The number of poles P per sample were chosen so as to be large enough to eliminate the effect of the finite pole sample sizes 1a). Preliminary calculations showed that P should vary from about 20 for weak absorption to about 500 for strong absorption. It was also found that the fluctuation cross section depended only upon the transmission coefficients and not upon the phases or the relative phases of the average S-matrix elements. With these standard statistics, calculations were done for the channel numbers and transmission coefficients listed in table 1. The resulting channel degree of freedom

189

P. A. Moldauer / Average cross sections

TABLE 1A Calculated elastic enhancement factors W and degrees of freedom v for various numbers N of equivalent channels with transmission coefficients T T 0.19

0.36

0.64

0.84

0.99

2

2.69(7) 1.18(5)

2.51(8) 1.32(7)

2.25(4) 1.59(5)

2.12(2) 1.78(4)

2.02(3) 1.95(5)

5

2.57(4) 1.28(3)

2.33(4) 1.50(4)

2.15(3) 1.73(4)

2.08(2) 1.84(4)

2.06(3) 1.89(6)

10

2.42(5) 1.41(5)

2.25(4) 1.60(5)

2.13(4) 1.76(6)

2.12(3) 1.79(5)

2.10(4) 1.82(7)

15

2.39(5) 1.43(5)

2.26(5) 1.58(6)

2.18(5) 1.69(7)

20

2.31(4) 1.52(5)

2.20(5) 1.67(7)

2.15(5) 1.74(8)

2.14(6) 1.75(8)

2.11(6) 1.80(10)

30

2.23(5) 1.62(7)

2.22(6) 1.63(8)

2.20(8) 1.67(11)

2.13(8) 1.76(12)

2.10(9) 1.81(15)

2.11(6) 1.80(10)

parameters are plotted against the sum of channel transmission coefficients ~ T in fig. 1, where different symbols are used for different transmission coefficients. These results were least square fitted to the three-parameter formula

v(T, ~ T) = (1 +A)+(Tn-A)e -clxT~,

(15)

resulting in the formula of eq. (1). TABLE 1B Calculated W and v for mixed channel transmission coefficients and very weak absorption Total number of channels

Number of channels with T

15

5

0.19

5

0.64

5

0.99

I0

0.19

10

0.99

30

0.041

20

30

T

W v 2.19(8) 1.68(I I) 2.15(4) 1.74(12) 2.14(4) 1.76(14) 2.20(7) 1.67(10) 2.10(4) 1.82(13) 2.67( I 0) 1.20(7)

P. A. Moldauer ,' Average

190

section,s"

cross

2.0

1.8

; 1.6

i

i

"lJ T

1.4 ,

.19

•

.36 ]

•

1.2

.64

.84 .99

1.0

__

L

1 5

~

J 10

i

i 15

f

i 20

f

J

I 25

l_

30

2:T

Fig. 1. Channel degree of freedom parameters v versus the transmission coefficent sum ~T as calculated numerically with standard statistics for various channel transmission factors T. The solid lines give the results obtained from the formula of eq. ( 1).

Next we varied the statistical assumptions away from the standard statistics described above. First we changed the distribution of the 7,,. by flattening or peaking the normal distribution somewhat. The resulting amplitude distributions .ll, have a fourth central moment/~4 which is less or greater than the value of 3.0 which characterizes the normal distribution with unit standard deviation. To find the effect of the K-matrix pole spacing distribution upon the fluctuation cross section, we performed calculations with uniformly spaced poles (picket fence model) having a spacing distribution standard deviation al) = 0.0, and with random pole positions (exponential spacing distribution) with a D = 1.0, as well as with the Wigner distribution which has ao = 0.523. These changes in the statistics were first aplied to the calculation of the weak absorption case of three independent channels, each with T ~ 0.1. The resulting fluctuation cross sections were found to be in excellent agreement with the predictions of eqs. (80)-(82) of ref. 2), as shown in the top part of table 2. Similar calculations for five independent channels, each with T = ~ 0.19 already exhibit significant deviations from the formulas of ref. 2). However the strong dependence upon /~4 and aD remains. These results are shown in the bottom part of table 2. Finally, the variations of statistics were applied to the strong absorption cases of between 5 and 30 channels each with T = 0.99. The resulting values of v are plotted

P. A. Moldauer / Average cross sections

19 I

TABLE 2 Calculated and theoretical elastic enhancement factors for weak absorption and various statistics Number

of

Elastic enhancement factors

Poles

equivalent per channels sample

Calculated transmission coefficients

P4(7)

Spacing distribution calculated

weak absorption theory

ref. 2), eqs. (80)-(82) 3

20

0.0956(13) 0.0948(9) 0.0932(9) 0.0944(10) 0.0955(13) 0.0943(13) 0.0956(13)

3.00 2.77 3.00 3.26 2.77 3.00 3.26

picket fence exponential exponential exponential Wigner Wigner Wigner

2.70(14) 2.72(1 I) 2.96(12) 3.15(12) 2.56(14) 2.78(15) 2.98(16)

2.74 2.76 3.00 3.27 2.55 2.80 3.02

5

100

0.1916(11) O.1857(11) 0.1964(11) 0.1938(10) 0.1963(11)

3.00 3.00 2.77 3.00 3.26

picket fence exponential Wigner Wigner Wigner

2.44(8) 2.93(9) 2.33(8) 2.56(7) 2.77(9)

2.26 3.00 2.12 2.33 2.55

in fig. 2 against the transmission coefficient sum ~ T, together with the results for the standard statistics. Fig. 2 exhibits very strong effects due to variations in statistics also in the strong absorption limit. The degree of freedom parameter v decreases sharply with increases in both p, and aD.

4. Theoretical interpretations Starting from a variety of formalisms 2-4), including the K-matrix formalism of sect. 2, S-matrix elements can be written in the form

Sab =

3 . b S .o + i ~ 7..A~,,,?vb,

(16a)

pv

where A~,I = 6-. , , ( E - E , , ) - ~ ( S P1, + W . , , ) , •

(16b)

where the )'.c and the E., 5P.,,, F.,, are real and a phase factor exp [i((J a + bb)] has been ignored. Eq. (16) is assumed to exhibit explicitly the resonance energy dependence of S.b which is of interest here and it ignores the energy dependence due to thresholds. Ifeq. (16) is derived from the K-matrix formalism, then the 7.. and E. are the same quantities as those introduced in sect. 2 and both for this derivation and for others 2 - 4) the standard statistical assumptions of sect. 3 a.re thought to be reasonable. Using

192

P. A. Moldauer / Average cross sections 2,4

2,2

2.0

3"

O'O= .523 ]

I

~

.

.

.

.

1.8

z

,y

i¸

1.6

~_

i

=.= I,,14=3.26 ETO=.523

1.4

1.2

1.0 5

10

15

20

25

30

ET

Fig. 2. Strong absorption dependence of the channel degree of freedom parameters v upon K-matrix pole statistics as discussed in the text. All channels have transmission coefficients T = 0.99. The solid line is computed from eq. (1) with standard statistics. The dashed lines indicate the trend of the results with non-standard statistics.

this expression for S, the fluctuation cross section can be written fl * aab = < E YuaAuvTvbTKaA~,~];.~b)ave-I(E TuaAuv"dvb)avel2, #vK~

(17)

Ov

where the averages ( )ave a r e taken over energy. T h e shift and width factors 5f,~ and f , , , are themselves functions of the 7uc and in the isolated resonance limit they are diagonal. T h e n also A is d i a g o n a l and the fluctuation cross section in this isolated resonance limit b e c o m e s fl = ( ~

a.b

it

2 2 A

2

,

,

2

7u.Tubl .ul ) . r e - - 1 ( ~ 7.,,7.~Au~)a.,e[ •

(18)

~t

T h e e v a l u a t i o n of these averages was discussed in s o m e detail by the author in ref. 2),

P. A. Moldauer / Average cross sections

193

appendix B, where the result was shown to depend on the spacing distribution of the E.. Our numerical results confirm the importance of this dependence. The elastic enhancement factor arises from the fact that in the elastic channel ~.~ is averaged while in an inelastic channel ~..7.~ 2 ,2 is averaged where 7.. and Tub are statistically independent. The resulting elastic enhancement (7,a)/(Tu,) 4 2 2 equals the value of kt4 of the amplitude distribution and is 3 for normally distributed 7u, with zero mean. In this limit v = 2 / ( W - 1 ) is expected to be unity in agreement with our numerical results. To discuss the enhancement effect for overlapping levels when A is no longer diagonal, there exist two different procedures. The first of these is to diagonalize the A-matrix by means of a complex-orthogonal transformation 2) T = T - 1 (inverse transpose). Then eq. (16) becomes Sa b ~- (~abSa0 + i 20uaBt~Oub' Iz

(19)

where B = TAT-

~

(20)

and

o.°

= y

(21)

v

Because B is again a diagonal matrix the evaluation of the fluctuation cross section proceeds exactly as in the isolated resonance limit 2). The only difference is that the distributions have changed. Instead of the normally distributed real 7,c, the 0uc are linear combinations with complex coefficients of many 7,c. Therefore, we expect both the real and imaginary parts of the 0uc to be independently normally distributed with zero means. In the limit of strong absorption, where the number of terms with complex coefficients contributing to each 0,~ becomes very large we would expect the dispersions of the real and imaginary parts to become equal. In that limit the elastic enhancement factor (10,,~14)/(r0,~[2) 2 becomes equal to 2 and the corresponding value ofv is also 2. Our numerical results show that this value of the enhancement is approached with standard statistics only for the one or two channel case with T = 1.0. When the channel number increases, so does the enhancement, doubtless due to correlation effects produced by the unitarity condition. It should be noted that in this view the elastic enhancement is always caused by a statistical autocorrelation effect, namely the ratio of a fourth moment to the square of a second moment of a normally distributed resonance parameter. There is numerical evidence 13) that for overlapping resonances the pole spacing distribution of S approaches the exponential law of uncorrelated poles with a o = 1. It might be expected that this randomized pole distribution would be independent of the K-matrix pole distribution and therefore the fluctuation cross section should be independent of the latter in the strong absorption limit. Nevertheless, our numerical results do exhibit a very pronounced variation of 0 "fl with changes in. a o.

194

P . A . M o l d a u e r / A v e r a g e cross s e c t i o n s

A somewhat different way of deducing the enhancement factor for strong absorption cases has been adopted by Agassi, Weidenmtiller and Mantzouranis 4). They note that for independent normally distributed Y,c, and for a ¢ b, the second term in eq. (17) vanishes and the first term contributes to the average only when/~ = rc and v = 2, so that o'fl

a,b~a

=

(E

2 2 2 7U.Y,'blAu~l )ave"

(22)

t1"¢

For the elastic case the second term in eq. (17) contributes only if # = v and the first term has three possible contributions to the average. First/~ = ~, v = 2, secondly /~ = 2, v = ~c, and lastly ~ -- v, ~c = 2. This leads to the following expression for the fluctuation cross section: ), 7~.(21A,,~] + A , , , A ~ ) ) a v e _ l ( ~ o 2 '/..A~,,)avol 2.

a~. try

(23a)

#

Because of the assumed lack of correlation of the energy dependences of A.. and A.~ for/~ 4: v, and the negligible contribution from/~ = v, this expression is expected to equal fl 2 2 2 a.. = 2 ( ~ 7.aT~b[A.,,[ )ave"

(23b)

11;,

Comparing the expressions (22) and (23) we obtain W = v = 2. In this approach the enhancement appears to arise not from the statistics, but rather from a symmetry in the formalism (not the symmetry of the S-matrix, since also an asymmetric Smatrix would yield the same result.) However, the sensitivity of our numerical results to variations in statistical assumptions does demonstrate the central role that statistics plays in determining the elastic enhancement and the deviations from the value of 2 for the case of many strongly absorbed channels again testifies to the power of correlations to modify the expected results. Though they are plausible and also amply supported by experiment in the weak absorption limit, there is no guarantee that the standard K-matrix statistics also holds true in the strong absorption limit. F r o m fig. 2 it is clear that one can produce functions /~4(T, S T ) and an(T, Z T ) which will yield W = v = 2.0 or almost any other result for T ~ 1.0. On the other hand, if we interpret the K-matrix as an Rmatrix, then the poles and residue amplitudes are the eigenvalues and eigenfunctions of the nuclear hamiltonian with boundary conditions 2), and it seems reasonable that the statistics of these quantities should not depend upon the channel energies and hence absorption strengths. Alternate statistics have been proposed by a number of authors. Of these the statistics of Agassi et al. 4) are most closely related to those of the K-matrix parameters. There also are poles and pole amplitudes. But while the statistical distribution of the amplitudes plays a role in that theory, the pole distribution does not enter

P. A. Moldauer / Average cross sections

195

into the calculations. In view of our numerical results this must be regarded as disturbing. Mello and Seligman 5) use an entirely different approach. They propose a distribution law for the strong absorption S-matrix elements themselves, based upon thermodynamic - information theoretic considerations. To do this they must abandon the analyticity condition on the S-matrix which expresses the physical principle of causality 8). The effect of this neglect remains to be investigated. In their theory, Kawai et al. 3) abandon unitarity or flux conservation on the grounds of the supposed influence of thresholds upon average cross sections t4). Traditionally one aims to calculate average cross sections in energy intervals that exclude thresholds but approach them arbitrarily closely. Admittedly this procedure may be in need of more detailed justification. But that is 3urely even more true of the Kawai et al. procedure of completely abandoning I N ( N + 1) constraints on the S-matrix elements at every energy on the basis of ignorance regarding threshold effects.

5. Conclusions

If one uses the unitary analytic K-matrix model with parameter statistics that are experimentally well supported for few channels and weak absorption, then one obtains in the many-channel limit fluctuation cross sections that are given by the Hauser-Feshbach formula with width fluctuation correction, where the channel degree of freedom parameters are v ~ 1.78, or the elastic enhancement factors are W ~ 2.12 for all channels regardless of absorption strength. This result is, however, very sensitive to the statistical assumptions, both as regards the pole residue amplitudes and the pole spacings. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ! 1) 12) 13) 14)

C. Mahaux and H. A. Weidenmfiller, Ann. Rev. Nucl. Part. Sci. 29 (1979) 1 P. A. Moldauer, Phys. Rev. 135 (1964) B642 M. Kawai, A. K. Kerman and K. W. McVoy, Ann. of Phys. 75 (1973) 156 D. Agassi, H. A. Weidenmiiller and G. Mantzouranis, Phys. Lett. C22 (1975) 145 P. A. Mello, Phys. Lett. B81 (1979) 103; P. A. Mello and T. H. Seligman, to be published H. M. Hofmann, J. Richert, J. W. Tepel, H. A. Weidenmiiller, Ann. of Phys. 90 (1975) 403 P. A. Moldauer, Phys. Rev. C l l 0975) 426 P. A. Moldauer, Phys. Rev. C12 (1975) 744 P. A. Moldauer, Phys. Rev. C14 (1976) 764 C. A. Engelbrecht and H. A. Weidenmiiller, Phys. Rev. C8 (1973) 859 E. P. Wigner, in Fourth Canadian Mathematical Congress (University of Toronto Press, Toronto, 1957) p. 174 F. J. Dyson, J. Math. Phys. 3 (1962) 166 P. A. Moldauer, Phys. Rev. 171 (1968) 1164 A. K. Kerman and A. Sevgen, Ann. of Phys. 102 (1976) 570