59
A~X~-AI,S OF PHYSICS: 30, 269-327 (1964) Statistical Analysis of Neutron Resonance Parameters* JOHN D. GARRISON? Brookhaven National Laboratory, I’pton, Sew l’ork The neutron resonance parameter data have been examined with emphasis on recent measurements. The results obtained here provide general support for the t,heory of nuclear reactions and the statistical theory of spectra. I. INTRODUCTION A number of analyses of neutron resonance parameters have been published in the past’;’ however, much uew data have accumulated and new theoretical developments” in the statistics of level spacings make this study rewarding. The attempt here has been to undertake the detemination of the various distribu- tions of resonance parameters and their associated resonance parameter correla- tions, and t’he deterniination of specific average resonance parameter quantities only where these analyses have not been performed in the past or where the accumulation of new improved data makes the analyses desirable. The analysis of the data and its interpretation is conveniently divided into two categories. One category concerns the distributions of the various resonance parameters or resonance parameter combinations about their mean values and the associated correlations between these parameters. The other category COII- terns the mean values of the resonance parameters or certain resonance parame- ter combinations. This paper treats primarily the first category, the distributions. Thus, for example, surh quantities as the neutron st.rength functions and average level spacing are not considered here, except for the case of nuclei with measured resonance spins. Although the reduced neutron width distribution has been t)reated estensively in the past, its general import,ance and the information it yields concerning the qualit’y of the neutron cross section data make it, desirable to analyze this dis- tribution in some detail, including the determination t,o t,he correlation between the reduced neutron widths of neighboring levels. This correlation is conveniently obtained from the distributions of the sums of successive neutron widths. * Research supported by the IJnited States Atomic Energy Commission. t On leave of absence from San Diego State College, San Diego, California, and General Atomic/General Dynamics, San Diego, California. 1 See, for example, refs. 1-5. 2 See, for example, refs. 6, 7. 269

Statistical analysis of neutron resonance parameters

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Page 1: Statistical analysis of neutron resonance parameters

A~X~-AI,S OF PHYSICS: 30, 269-327 (1964)

Statistical Analysis of Neutron Resonance Parameters*

JOHN D. GARRISON?

Brookhaven National Laboratory, I’pton, Sew l’ork

The neutron resonance parameter data have been examined with emphasis on recent measurements. The results obtained here provide general support for

the t,heory of nuclear reactions and the statistical theory of spectra.

I. INTRODUCTION

A number of analyses of neutron resonance parameters have been published in the past’;’ however, much uew data have accumulated and new theoretical developments” in the statistics of level spacings make this study rewarding. The attempt here has been to undertake the detemination of the various distribu- tions of resonance parameters and their associated resonance parameter correla- tions, and t’he deterniination of specific average resonance parameter quantities only where these analyses have not been performed in the past or where the accumulation of new improved data makes the analyses desirable.

The analysis of the data and its interpretation is conveniently divided into two categories. One category concerns the distributions of the various resonance parameters or resonance parameter combinations about their mean values and the associated correlations between these parameters. The other category COII-

terns the mean values of the resonance parameters or certain resonance parame- ter combinations. This paper treats primarily the first category, the distributions. Thus, for example, surh quantities as the neutron st.rength functions and average level spacing are not considered here, except for the case of nuclei with measured resonance spins.

Although the reduced neutron width distribution has been t)reated estensively in the past, its general import,ance and the information it yields concerning the qualit’y of the neutron cross section data make it, desirable to analyze this dis- tribution in some detail, including the determination t,o t,he correlation between the reduced neutron widths of neighboring levels. This correlation is conveniently obtained from the distributions of the sums of successive neutron widths.

* Research supported by the IJnited States Atomic Energy Commission. t On leave of absence from San Diego State College, San Diego, California, and General

Atomic/General Dynamics, San Diego, California.

1 See, for example, refs. 1-5.

2 See, for example, refs. 6, 7.

269

Page 2: Statistical analysis of neutron resonance parameters

GARRISOK

In an analogous fashion, the distribution of level spacings, the distributions of the sums of successive spacings and correlations of neighboring spacings have been obtained for selected even-even nuclei having resonances of only one spin state and for selected odd-A nuclei having two classes of resonances with spins I + ,14 and I - $$. I is the target nucleus spin. The neutron resonances are in a region of high nuclear excitation where the level spacing distribution can be predicted theoretically from very general statistical assumptions. No thorough comparison between the t,heoretical predictions and experiment has been made in the past, and it is important that such a comparison be made at this time. In making this comparison it has been found that the corrections which have been applied for missing and extra levels are generally important in providing improved agreement between theory and experiment.

There now exists a limited number of measurements of the compound nucleus total angular momentum quantum number, J, for particular resonances. The data have been tested for the dependence of the mean level spacing upon J and also for t’he dependence of the strength function on J.

No attempt has been made to deal statistically with the measurements of the total radiation widths at this t’ime. The accuracy of the measurements of the total radiat#ion width is t’oo poor to determine their distribution for individual nuclei, since this distribution is generally quite narrow.

Measurements of the partial radiation widths have been examined for channel- channel correlation. The distribution of partial radiation widths has been dis- cussed in a review article recently by Bartholomew (8) and more recently by Bollinger, Cot& Carpenter, and Marion (9). The partial radiation width dis- tribution is consistent with a chi-squared distribution with one or possibly two degrees of freedom, but the measurements still lack the accuracy and precision necessary to determine the distribution well. In this paper the analysis of this distribution has not been attempted.

The correlations between pairs of resonance parameters is important in the analysis of the resonance parameter data as well as in the theory of nuclear reac- tions (10). For this reason a number of correlation determinations have been attempted. In addition to the correlations mentioned above, the reduced neutron widths and neighboring spacings of selected isotopes and the fission widths and reduced neutron widths of fissile isotopes have been tested for correlation.

A considerable quantity of new data for the fissile isotopes are now available. This fact, coupled with the fact that the fission process is currently not well understood in detail, makes an analysis of the fission width distribution and fission width correlations desirable at this time.

Finally, the equivalence of positive and negative energy neutron resonances has been re-examined (3) in the light of improved data by determining the con- tribution of the positive and negative energy resonances to the thermal capture cross section of a large number of isotopes.

Page 3: Statistical analysis of neutron resonance parameters

SETJTROS RESONANCE PARAMETERS 271

The data considered here has been obtained solely by low energy (1 = 0) neutron measurements of the heavier isotopes.

Sections II t’hrough VIII which follow deal solely with the discussion and analysis of the dat’a and comparison of the results with the theory. All dis- cussion of t$e origin of the theoretical results is delayed for presentat,ion with the final discussion and summary in Section IX.

One basic assumption used throughout this paper should be noted: It has been assumed that the distribution of neutron widths and the distributions of level spacing for levels of the same spin are identical for all nuclei t!reated here. The experimental evidence appears to be in agreement with this assumption.

II. METHOD

For each isotope, in addition to the resonance parameters, the following data were provided as input for a simple IBM-7090 sorting and calculational code: - Z, the atomic number; A, the mass number; r not the average reduced neutron width; DOhs , the average observed level spacing; uzzoo , the thermal capture cross section; I?? , the average radiation width; I, the spin of the target nucleus; N, the total number of resonance levels; Em,, , the highest resonance energy t)o be considered in the “good data” group; g, the mean statistical weighting fact,or; and the class number. The statistical weighting factor is g = (2J + l)/ 2(21 + 1). g is one for even-even nuclei, and its mean value for both possible spin states of odd A and odd-odd nuclei is one-half.

Somewhat arbitrarily, the isotopes were divided initially into three classes: Class 1, the even-even nuclei; Class 2, the odd-A nuclei which were measured in a target containing essentially just, the one odd-A isotope; and Class 3, the odd- A and odd-odd nuclei measured in targets containing isotopic mixtures. The fission width data was calculat’ed later by hand.

Early in the development, of this paper it seemed desirable to group the data into classes according to qua1it.y. It later became apparent that only the best data is useful for careful analysis of distributions and that the remaining data is primarily of value only as an indication of the general quality of the resonance parameter data. Accordingly, a new Class A consisting of t,he high resolution measurements of P7, Tal*l, and Au197 made at Columbia University,3 and the recent very high resolution Harwell measurements of U2s8 (13) and Columbia University measurements of Thz3* (14), was formed.

The data used for input were obtained primarily from refs. 11 and 15; how- ever, the more recent, data which have been measured subsequent to these com- pilations have been included in the analyses. References to the newer data will be given where the data are discussed.

Table I presents the input data, other than the resonance parameters, used in t.his work. Only nuclei with mass number, A, greater than 7.5 have been used.

3 The results of these measurements may be found in refs. 11 and 12.

Page 4: Statistical analysis of neutron resonance parameters

272 GARRISON

TABLE I GENERAL INPUT DATA FOR THE STATISTICAL ANALYSES OF RESONANCE PARAMETERS

z .4 iv I

33 75 3

35 79 7

35 81 3

36 83 2

42 95 4

43 99 9

44 101 11

45 103 10

46 105 7

47 107 4

47 109 7

48 111 6

48 113 7

49 113 7

49 115 8

50 112 2

50 116 2

50 117 5

50 118 2

50 119 3

51 121 7

51 123 4

52 123 5

52 125 5

53 127 61

53 129 6

54 129 2

54 131 3

55 133 12

56 135 11

59 141 10

60 145 2

61 147 9

62 147 13

62 149 29

62 151 5

63 151 21

63 153 18

64 155 24 64 157 5

65 159 16

66 161 27

66 162 4

m0 (mV) (ii) 35 320

9 390

15 280 14.4 220 10.8 210

1.68 280 1.92 280 2.08 155 1.3 140 2.86 140 2.65 140 2.4 115 2.6 115 0.78 80

0.54 78

7.5 110 7 110 2.82 110

11.2 110 9.1 110 1.20 90 2.97 64

3.71 100 6.6 100

2.0 107 3.53 100

12.3 90 12.2 90

4.2 110 9.7 115

49 150 21.3 50

2.58 80

6.0 59 2.27 62 0.57 62 0.42 91 0.51 97 0.76 109 2.4 96 0.72 100 0.68 120

21.6 140

Dot,% (eV)

70

45

75

120 180

24 24

26

13 27

25 24 26

7.8

6.7

150 100

47

320 130

14

28

35 55

13.3 21

73 68

21 44

90

28 3

7 2.7

0.95 0.70 1.1 1.9 7.5

2.3 2

120

L,, (eV)

253

53.7 135.7

233 162

195 114

156

94 51.4 87.4

164.3

108.5 45.6

48.6

280 149

259

368 460

90

105 159 290

815 95.6

92 76

240 223

239 43.1

7

18.3 14.9 4.10

9.06 8.87

14.7 17.1

14.4 51.8 71.3

5.4

10.4 3.1

205

13.9 22

150

35.0

95.0

19,000

58 200

1.3

-

-

6.8 4.1

410 1.56

6.2 28.0

45 120

29

5.8 10.9

60 180

87

41,000 15,400

8,200 450

58,0C0

240,000 46

580 140

2

3

3 3

3 2

3 2

3

3 3

3 3

3 3

1 1

3

1 3

3

3 3 3

2 2

3 3

2

3 2

3 2

3

3 3 3 3 3

3 2 3 1

Page 5: Statistical analysis of neutron resonance parameters

Z A N

(iti 163 9

(ii 165 15

A8 167 4 69 169 10

71 175 l(i

51 176 21

72 177 12

72 179 23

73 181 77

74 182 5

74 183 13

74 186 3

75 185 Ii 75 187 10

77 191 14

77 193 15

78 192 2

78 195 9

79 197 56

80 198 5

80 199 8

80 301 7

83 209 18 90 232 49

92 233 10 92 234 15

92 235 29 92 236 9

92 238 55

93 237 74

94 239 19

94 240 11

94 241 ci

94 242 2

95 241 53

95 443 11

Total 1123

I’,,0 (mV) (2) Dotis (eV) Ji”K,X (eV)

3 .ti 103 9 55.8 2.32 06 5.8 87.2

1.44 70 4 9.41 2.04 70 6.8 Hi.8

1.19 00 3.3 41.3

0.82 00 2.3 33.6

1.08 61 3 33.2

1.08 GO 3 93.G

1 .(i5 59 4.35 329 15 47 75 253

6.3 77 15 66 .2

22 46 105 221

1.26 52 3 2.15(i

1.26 45 3 4.41

0.88 82 2.2 25.3 2.4 85 tj 20.2

0.0 100 30 53

7.6 125 21 189

5.38 140 lti.8 791

13 140 80 89.8 24 280 75 175 24 320 75 210

800 300 8,000 12,000 1.4 30 18.5 998

0.14 53 0.8 10.33 1.4 20 14 191

0.12 35 0.0 19.3 1.7 29 17 133

1 .9 24 19 997 0.13 34 0 0 12.63

0.05 37 2.7 52. (i

1.68 34 14 119

0.38 43 1.6 8. (iti

4.8 27 45 53.6

0.16 42 0.65 43.25 0.29 42 1.2 15.3

uzzw (barns) Class

120 ti5

12i

30

2,450 380

65

20 20 11

36 112

M

960

130 -

27

98.8

2,500

0.5 7.44

53.0 94

101 tj

2.72 109

280;

290 375

19

630

3 2

3 2

3

3 3

3 2

1

3 1

3 3

3 3

1

3 2

1 3

3

2 1

2 1

2 1

1 2

2

1 3 1

3 3

Using the input data listed in Table I, the sorting and calculational code de- temiued the various quantities needed for the distributions, obtained the dis- tributions for each isotope, and combined the distributions for each of the classes.

Two simple codes were programned for calculating correlation coefficients,

Page 6: Statistical analysis of neutron resonance parameters

274 GARRISON

and for obtaining near neighbor level spacing distributions and moments of these distribut’ions. The second code also proved useful in obtaining the moments and distributions of the sums of successive neutron widt’hs for the Class A nuclei.

In addition to the codes programmed for handling the data, certain codes were written t,o be used with the random matrix diagonalization results of C. E. Port’er.” These involved codes for the overlapping of eigenvalues from pairs of matrices to simulate the level spacing distributions of nuclei with two possible spin states, and codes removing and adding levels to provide corrections to the experimental dat’a. These will be discussed in somewhat more detail later in the paper.

Details of the calculations and the distributions will now be discussed for each case.

III. THE REDUCED NEUTRON WIDTH DISTRIBUTIONS

A. DISCUSSION

This section involves three different analyses which are interconnected: (1) the analysis of the distribution of individual neutron widths of selected high quality data to determine the mathematical form of the distribution; (2) the analysis of general classes of neutron width data to determine the quality of the neutron width measurements and to provide information on missing levels, some of this information to be used in lat,er sections of this report; and (3) the analysis of the distributions of the sums of k + 1 (0 5 lc 5 10) successive neutron widths as a means of determining the correlation between near neighbor neutron widths and higher powers of these neutron widths.

The mathematical form of the distribution of t’he individual neutron widths was well established by the work of Porter and Thomas (I), who also have given a plausibility argument providing a theoretical basis for their distribution. As is well known, their analysis indicated that the reduced neutron width dis- tribution is well represented by the chi-squared distribution with one degree of freedom, the “Porter-Thomas Distribution.” Much new data, measured with high resolution and good st’atistical accuracy, have accumulated since t’he analy- sis of Porter and Thomas. Because of the importance of this distribuCion, it is well to repeat this analysis with the better data which are now available.

The analysis of Porter and Thomas provided corrections for the finite size of the samples, the missing of weak levels, and the uncertainties in the measured values for the reduced neutron widths. The method of analysis adopted here differs from that of Porter and Thomas. It was adopted primarily t’o avoid mak- ing the very imporOant but somewhat arbit,rary correction for the weak levels whose neutron widths are missing from the experimental distributions. How-

4 These results were generously made available on magnetic tape by C. E. Porter, and

are discussed in Brookhaven National Laboratory Report, BNL-6763, by C. E. Porter.

Page 7: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 275

ever, historical precedent also contributed, as the method of forming the dis- tributions tjo be used here was used earlier in the examination of the quality of fission product neutron width data (16). After the experimental neutron width distributions have been presented and the problems associated with them dis- cussed in some detail, the method of analysis of these distributions will be treated.

In Fig. 1 is shown the distribution of reduced neutron widths for the Class A nuclei. Aulg7 with the comparatively large number of 56 measured neutron widths has the smallest sample size of the five Class A nuclei which together provide a sample of 449 neutron widths. These data provide the best data avail- able for t,he study of neutron width distributions, and of level spacing distribu- tions to be treated in the next section. Also shown in the same figure is the neu- tron width distribution (dashed histogram) for the odd-A nuclei, Pz7, Tal*r, and Aulg7, alone.

In Figs. 2-5 are presented the distribution of reduced neutron widths for classes 1, 2, 3 and for classes 1, 2, and 3 combined. Shown in these figures are smooth curves representing the Porter-Thomas Distribution (solid line) and, in all but Figs. 1 and 5, the exponential distribution (dashed line) for comparison. The distribut’ions in the figures present the quantity Nz/g as a function of x, where N is the number of cases per one-tenth interval in II^, and R: is the ratio - grnO/gr?&“, rat.her than the desired ratio, PILO/PnO, since few resonances have measured spin. This method of presentation makes more evident the fact t,hat weak resonances, which were undetected in the measurements, are missing from the distributions. The lower bound of the first histogram interval is zero. Be- cause of t’he nature of the neutron-width distributions, the histogram intervals increase with Z, in order to include approximately equal numbers of cases in

FIG. 1. The reduced neutron width distribution of the Class A nuclei. The solid histogram

gives the distribution for Ilz7, Ta’*l, Aulg7, Thzz2, and W* combined. The dashed histogram

gives the distribution for the odd-A nuclei I 127, Tal**, and Aul97. The smooth curves are the

Porter-Thomas distribution.

Page 8: Statistical analysis of neutron resonance parameters

276 GARRISON

144 CASES CLASS I EVEN-EVEN NUCLE, 320 CASES CLASS 2

FIG. 2. The reduced neutron width distribution of the Class 1 nuclei. The smooth solid curve is the Porter-Thomas Distribution. The smooth dashed curve is the exponential distribution.

FIG. 3. The reduced neutron width distribution of the Class 2 nuclei. The smooth solid

curve is the Porter-Thomas distribution. The smooth dashed curve is the exponential

distribution. The gap between the Porter-Thomas distribution and the histogram for small reduced neutron widths corresponds to approximately 21 cases.

FIG. 4 FIG. 5

FIG. 4. The reduced neutron width distribution of the Class 3 nuclei. The smooth solid curve is the Porter-Thomas distribution. The smooth dashed curve is the exponential dis-

tribution. The gap between the Porter-Thomas distribution and the histogram for small

reduced neutron widths corresponds to approximately 125 cases. FIG. 5. The solid histogram is the reduced neutron width distribution for the Class 1, 2

and 3 nuclei combined. The dashed histogram is the reduced neutron width distribution for the resonances which lie above the upper bound, EMAX , of the good data used for Classes

1, 2 and 3. The smooth solid curve is the Porter-Thomas distribution appropriate for the

solid histogram. The smooth dashed curve is the Porter-Thomas distribution appropriate for the dashed histogram.

Page 9: Statistical analysis of neutron resonance parameters

SEUTHON RESONANCE PAI-LAMETEHS 277

each interval. Data from refs. 17 and 18 have also been included in the neut’ron widt,h distributions.

In obt,aining the distributions only isot,opes wit,h 5 or more well-measured resonances were used. For Classes A, 1 and 2 the large majorit,y of the cases are contributed by a very few isotopes with a large number of relatively well- measured resonances. The average reduced neutron width for each isotope was obtained by the following equation:

(1)

This procedure was used since the combined uncertainty in the strength func- - - tion, r,,O/D, and Dohs is usually smaller than the uncertainty in I’~O obtained by averaging the neutron widths of the measured resonances. This procedure tends to reduce the narrowing of the distribution which occurs for small samples when using the sample mean for rn”.

The above procedure neglects the difference in level spacing and rno/D for the two possible spin states of other than the even-even nuclei, as is necessary at this time, since generally resonance spins are unmeasured. Apparently this has only a snlall effect on the neutron width distribution since the experimental distribut’ion for the Class 1 even-even nuclei with only one possible resonance spin state is not different statistically from the distribution for the other classes containing two possible spin states except in the region of small grno/@‘?Lo where differing numbers of levels are missed. The effect of a different average neutron width for t,hc two spin states and evidence for it is discussed lat’er in this paper (see Fig. 8).

Since the average uncertainty in the measured values of rno is of the order of 30%, it is clear also from Fig. 8 that the neutron width distribution is far too broad to be sensitive Do the uncertaint,ies in neutron widt’hs. It is estimated from Fig. 8 that an average factor of 3 uncertainty in the neutron widths might lower t)he number of degrees of freedom of a v = 1.0 dist’ribution lo v - 0.9. A similar argument applies to errors in r,o when a number of nuclei are contribut- ing to the distribution, provided these values are nol systematically large or small. The errors in I‘,[) are expected to be generally smaller than t’hose of the individual r,,O. A correction fact’or, discussed in Appendix I, has been applied to the .r-values of each class where required t’o assure that, t#heir average is 1.00.

The strength furl&ions used in Eg. (1) were obtained from resonance parame- t)er measurements and from measurements in t,he kilovolt energy region, as described in ref. 16. The average level spacing was obtained from a “stairstep” plot of t)hc nmilber of resonances versus energy, also described in ref. 16.

The energy of the highest. energy resonance used in the “good data” distribu- tions, IL,, , is det’ermined by examining the “st,airstep” plots and the plotted

Page 10: Statistical analysis of neutron resonance parameters

278 GARRISON

cross section data to decide where the missing of levels becomes noticeable. The distribution for measured resonances above E,,,,, for all classes is also included in Fig. 5 to indicate the large number of resonances which are missed with the reduced instrumental resolution available at higher energies.

One of the difficulties of making measurements up to high energies with good resolution and counting statistics is that p-wave resonances with a different average reduced neutron width may be included in the s-wave distribution. The Columbia University data for U238 (11) has listed 12 levels which could be p- wave levels. It seems likely, however, that some of these are s-wave levels. For isotopes with mass number in the region of A = 100, the p-wave strength func- tion is much larger than the s-wave strength function and thus the stronger of the p-wave resonances may be detected even at relatively low energies. This leads to the possibility of including an appreciable number of p-wave resonances in the s-wave distribution. This is also true to a lesser degree for the very heavy elements. For this reason, a number of the nuclei with mass numbers in the vicinity of A = 100 have been omitted from these distributions and those treated have included resonances only up to moderate energies. Thus, no attempt is made here to estimate quantitatively the number of p-wave levels contributing to the neutron width distributions. Undoubtedly some p-wave resonances are included in the distributions. In addition, a few false resonances may be included in the distributions which arise from sample impurities or possibly even counting fluctuations. Therefore, it is to be expected that more weak s-wave resonances are missed than indicated by comparison of the histograms with the Porter- Thomas distribution. Because of the barrier penetrability factor (19), the p- wave resonances will be present in the distribution only as very weak resonances for the neutron energies at which these data have been obtained.

B. THE ANALYSIS OF THE SINGLE REDUCED NEUTRON WIDTH DISTRIBUTIONS

The reduced-neutron width distributions for Classes A, 1, 2, and 3 have been tested by the Maximum Likelihood Method’ to determine the parameter, v, the number of degrees of freedom in the analytical Chi-Squared Frequency Function family,

p(.& v) = (px)“-’ pe-pz r(P) ’

where p = v/2, which provides the best fit to the reduced neutron width dis- - tribution. I’(p) is the gamma function of p and x = gI’,O/gp,o as before.

Although the data are generally of much better quality than t,he data ex- amined by Porter and Thomas, there are still a number of weak resonances

5 See for example, ref. 80.

Page 11: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 279

missing from the distributions, so that the distributions are distorted for low values of x. Rather than try to make the rather uncertain correction to the dis- tribution for missing levels, only the cases with z > (Y have been considered, and the (Xi-Squared Distribution renormalized accordingly:

Q! is the lower cutoff point for x-values such that essentially no levels are missing from the distribution above (Y. The renormalization constant F(p, LY) was cal- culated as a function of p = v/2 for the values of (Y selected for the two classes.

The Maximum Likelihood Method was applied to the cases sorted in histo- gram intervals, with the x-values of cases in each group assumed to have the value of the average for the distribution over the group. In averaging, the dis- tribution for v = 1, the anticipated result, was used. However, the average ob- tained is not sensitive to the distribution, within the range of v-values observed experimentally.

Table II presents the values of v determined by the Maximum Likelihood Method for Class A. Also shown for comparison purposes are the values of v determined from classes 1, 2, and 3. Also indicated in the table are the values of a! used for each class and the size of the samples. Additional details of t’he cal- culations are presented in Appendix I. The error shown with each value of v

is the statistical error determined by Eq. (24) in Appendix I. The statistical error is the primary source of uncertainty in the determination of v. The results

TABLE II

THE NUMBER OF DEGREES OF FREEDOM, Y, ASSOCIATED ~VITH THE REDUCED NEUTRON WIDTH DISTRIBUTION

Class Number of cases

A (449

416

335

1 (144

137 117

2 317 0.01 1.03 f 0.11

258 0.1 1.03 f 0.16

3 234 0.38 1.00 f 0.13

0 1.04 f 0.06)

0.01 1.04 f 0.10

0.1 1.20 f 0.16

0 1.22 f 0.12)

0.01 1.27 f 0.20 0.10 1.62 f 0.40

a The data for Class A are not independent of Classes 1 and 2.

Page 12: Statistical analysis of neutron resonance parameters

280 GARRISOK

are consistent with a value of one for v. It should be pointed out that alt’hough the cutting off of the neutron width distribut’ion at x = CY avoids the correction for missed weak levels or extra weak levels which do not belong to the distribution, the statistical accuracy is considerably impaired, bot#h by a reduction of t,he number of cases considered and by removal of the region of the distribution most sensit’ive to V.

The Maximum Likelihood Method selects the parameter v of a family of dis- tributions which provides the best fit to the data. To complete the analysis, the data have been compared to the Porter-Thomas distribution using the Chi- Squared Goodness of Fit Test. As is obvious from Figs. l-5, this test indicates that all the neutron width distributions are consistent with the Port’er-Thomas dist’ribution.

By assuming the Porter-Thomas distribution to correctly represent’ the distribution of reduced neutron widths, the deviations of the experimentma dis- tributions from the Porter-Thomas distribution for weak resonances gives a measure of the number of weak resonances missed by the experiment)ers. For Classes 1, 2, and 3, the numbers of missed levels and t#heir proportion of the total number of levels in percent are: 5 & 3 (3 %), 21 f 5 (6 %), and 125 f 20 (30%), respectively. These numbers may be small since undoubtedly some p- wave resonances are included among the weak s-wave resonances in the experi- mental distributions.

The histogram of the poor data in Fig. 5, representing the measured resonances lying above the “cutoff” energy, E,,, , or cases where the number of good reso- nances is less than five have -40 % of the levels missing.

C. HIGHER DISTRIBUTIONS AND CORRELATIONS

The distributions, Pk , of the sums of successive widths of the 449 neighboring levels in Class A have been computed and are presented in Fig. 6. Also shown in the figure for a number of cases are Chi-Squared Distributions with k: + 1 de- grees of freedom which are expected to fit the data if there is no correlation be- tween any of the near neighbor neut,ron widths. Here Ic + 1 is the number of neighboring levels which have been summed to obtain the distribution Pk . Thus the distributions Pk are distributions of the quantities Yki = xf=o ai+j , where .L’~+~ is the ratio of reduced neutron width to averaged reduced neutron width for the (i + j)th resonance (see Appendix II).

The moments for the distributions, Pk , of the sums of widths of neighboring levels would normally be computed by:

where x = gPnO/#p,o, and iV is the total number of widths in the sample. Using Eq. (4), M,O, the first moment or mean of the single neutron width distribution

Page 13: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 281

- 0.05

0 5 IO 1520 0 IO 20 30 40

y=xgr,o/q c

FIG. 8. The distributions, Pk , of the sums of successive reduced neutron widths. The sub-

script k is one less than the number of the neutron widths involved in the sum. The smooth dashed curves are the chi-squared distribution with k + 1 degrees of freedom.

is exactly one; however, the first moment of the higher distributions (X: > 0) are generally not integral since not all the neutron widths contribute equally (an edge effect). For improved statistical precision in the higher moments, the fol- lowing equation which normalizes the first moments of all of t.he distributions, Pkr to t,he integral value k + 1 has been used:

Mnk’ = [

JL” I[ L1 + n(n - l)CX2

(MI”/k + 1)” 2(N - k) 1 . (5)

Here uZ2 = M20 - (~I4~0)~ is the variance of the Porter-Thomas (single neutron width) distribution. The denominator of the first term in brackets in Eq. (.5) assures that 41; = L + 1 while t,he second brackets factor removes the bias from the moments nil”,‘. It can be seen, as is discussed in Appendix II, that’ since the .K~+~ of Eq. (4) are the ratio of neutron widt,h to average neutron width, the result for Eq. (4) for Ic = 0 is of the same form as the first) brackets of Eq. (.5) (i.e., I’,O = MlO) and a bias correction is also required in this case. Therefore, Eq. (3) should be used for all Ic.

Page 14: Statistical analysis of neutron resonance parameters

282 GARRISON

TABLE III

MOMENTS OF NEUTRON WIDTH DISTRIBUTIONS FOR THE CLASS A NUCLEI

K Mlk Mzk Mak M4k

Expt Theory Expt Theory Expt Theory

0 1 2.79 f 0.27 1 2 7.5 f 0.6

2 3

3 4

4 5

5 6

6 7 61.8 f 1.9

7 8 78.3 f 2.2

8 9 96.9 f 2.4

9 10

10 11

14.2 f 0.8

23.1 f 1.1

34.0 f 1.4

46.9 f 1.6

117 f 3

140 f 3

3 .ooo

8.000

(7.58) 15.00

(14.3) 24.00

(23.2) 35.00

(34.0) 48.00

(46.7) 63.00

(61.5) 80.00

(78.3) 99.00

(97.1) 120.0

(118) 143.0

(141)

12.1 f 2.8

39 f 7

88 f 10

166 f 17

281 f 26

438 f 36

643 f 50

890 f 70

1200 f 90

1560 f 100

1950 f 200

15.00 68 f 24 48.00 248 f 40

(40.9)

105.0 651 f 100

(92.5) 192 1402 f 250

(173) 315 2700 f 400

(288) 480 .O 4680 f 700

(444) 693 .O 7630 f loo0

(647 ) 960.0 11400 f 1400

(902 1 1287 16600 f 2000

(1220) 1680 23200 f 3000

(160(J) 2145 31700 f 4000

(2040)

105.0 384

(279 1 945

(734) 1920

(1558)

3465

(2900 ) 5760

(4930) 9009

(7850)

13440 (11900)

19305 (17200)

26880 (24200)

36470 (33100)

Table III presents the experimental moments up to the fourth moment ob- tained by Eq. (5) of the distributions Pk shown in Fig. 6 for the Class A nuclei. The errors given with the experimental moments have been obtained from the deviations of the moments of each of the five Class A nuclei from their weighted mean given in Table III, and from calculations discussed in Appendix II. Also shown under ‘(theory” in the table are the moments for the Chi-Squared Family of distributions with k + 1 degrees of freedom and, in parentheses, the moments for distributions (k > 0) calculated from the experimental moments for k = 0 assuming no correlation.

Finally, in Table IV are shown the correlation coefficients Ck for widths of two levels which have k levels between them.

The correlation coefficients Ck are defined by the equation6: --

Ck = xixi+k+l - xixi+k+l

u*2

where uzz = 2 is the variance of the reduced neutron width distribution and the

6 See, for example, ref. 65, page 277.

Page 15: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 283

TABLE 11

CORRELATION COEFFICIENTS BETWEEN NEAR NEIGHBOR NEUTRON WIDTHS

K Ck K Ck

0 -0.03 6 -0.11

1 -0.01 7 +0.03 2 SO.06 8 -0.08 3 0.00 9 -0.08 4 0.00 Error f0.07 5 0.00

Mean -0.01 f 0.02

bars denote averages. These Ck have been obtained from Table III using the equation:

(k-1) M; = M,O + 2M3Vl-’ + Mi-’ + 2a,2 c Ci .

i=o

Very few levels are missing in the data presented in Fig. 6 and Tables III and IV (of the order of a few percent are missing) and these missing levels are partially compensated by the observation of p-wave levels in Th232 and U2”8. It is to be expected that many (but not all) of the weak levels missed are missed at random relative to the position and strength of the other levels, and many of the p-wave levels are observed at random relative to the position and strength of other levels. This last statement is particularly valid for the Th232 and U238 data where measurements were made with very high resolution and where levels are unlikely to be close together. Since the correlation between neutron widths is small, the data is expect’ed to show little bias from the loss of levels and the in- clusion of extra levels; and the effect of missing and extra levels is expected to be small relative to the statistical uncert’ainty associated with the number of widths observed. No correction for missing or extra levels has been made for the neutron width data.

It can be concluded from Fig. 6 and Tables III and IV that the distributions, Pk , areconsistent with a Chi-Squared distribution with k + 1 degrees of freedom, or equivalently, that the correlation between all neutron widths of near neighbor- ing levels and powers of these widths is consistent with zero.

Details concerning t’he moments calculations and equations are presented in Appendix II in a general discussion also suitable for the analysis of level spacings which are treated later in this paper.

D. SEARCH FOR DEVIATIONS FROM THE PORTER-THOMAS DISTRIBUTION

It is the purpose of this section to examine the data for possible deviations from the Porter-Thomas distribution. From time to time various experimenters

Page 16: Statistical analysis of neutron resonance parameters

284 GARRISON

have cited evidence of possible deviations from the Porter-Thomas distribution. No evidence of any appreciable deviation was indicated in Section III, B, where data from groups of nuclides were examined. Here the distributions of individual nuclei will be examined.

Probably the most convenient approach for this study is to examine the dis- tribution of second moments of the reduced neutron width distribution for a group of nuclei to see if t’his distribution is consistent with theoretical expecta- t’ions and also to see if there are any individual cases which appear anomalous. Corrected second moment’s have been obtained by multiplying the moments calculated from Eq. (5) (with k = 0 and 12 = 2) by the factor (NJN). The ratio (Nt/N) corrects these second moments for missing levels (see Appendix II). N is the number of neutron widths used in the calculation of the second moments using Eq. (5), and Nt is the estimated true number of levels including levels missed. Only 38 second moments of neutron width samples have been used. In some cases, two separate sets of neutron widths from a given nucleus have been used when the second moments from two sets have already been obtained for other purposes.

Table V presents the data used for the distribution of second moments. Since

TABLE V

DATA USED FOR THE DISTRIBUTION OF SECOND MOMENTS

Nucleus N Mz’ Nt/N M2’t Nucleus N Mz NtIN M2’1

1’21

CP3 Sml@

Eu”’ Eu’63

Gd’= Dy161

Hole5 Lu’76

Lu”6

Hf’77 Hf”9

Ta’81

31

30

10 14

11 9

12 14

13

10 8

8 11 10

10 12 11

39

38

2.88

2.18

2.28 1.78

2.46 2.51

2.42 1.99

1.70

2.37 3.54

2.43 1.87 1.65 2.38

2.13 4.01

3.35 2.30

1.0 2.72

1.0 1.50

1.3 2.96 1.3 2.17

1.3 3.20 1.3 3.24

1.3 3.14 1.3 2.48

1.3 2.08

1.3 3.07 1.3 4.42 1.3 3.13

1.3 2.39 1.3 2.13 1.3 3.09

1.3 2.73 1.3 5.30

1.0 3.61 1.0 1.54

Rel*s

1rl91

w93

Au’Q1

Th2z2

U233

U234 U235 U238

Am243

9 5.56 1.3 7.01

8 2.71 1.3 3.46

10 2.90 1.3 3.76 8 2.95 1.3 3.74

28 2.54 1.0 2.16 28 2.26 1.0 1.70

77 2.95 1.0 2.71

77 2.34 1.0 1.04

10 1.97 1.3 2.55

10 1.87 1.3 2.42

10 1.64 1.3 2.12

50 3.13 1.0 3.18 50 3.49 1.0 3.99

37 3.03 1.3 4.73 10 1.70 1.3 2.21

10 1.83 1.3 2.37

27 2.10 1.3 2.49

26 3.30 1.3 5.02

10 1.99 1.3 2.57

Page 17: Statistical analysis of neutron resonance parameters

KEUTIt0I.G RESOKANCE PARAMETERS 28.5

the corrected second moments have generally been obtained from differing numbers of cases, it has been necessary to multiply the deviation of each cor- rected second moment from t,heir mean by the factor (N/10)“’ and add this to 3.07 to obtain t’he adjusted, corrected second moments, A&, , given in Table V. The distribution of these M:, can then be compared to the resuhs of a Monte- Carlo calculation which used 10 neutron widths per sample for calculation of the second moments. The number 3.07 is the number required by the data so that’ the mean of the A/i, is 3.00 as is expected theoretically.

Figure 7 gives the distribution of t,he MB, of Table V. Included in the figure is a distribution of 200 second moments each obtained from a sample of 10 cases drawn at randonl from the Porter-Thomas distribut,ion. This Monte-Carlo cal- culation has been normalized to the 38 experimental cases.

It is seen that the experimental distribution is in good agreement with the dist,ribution of second moments obt)ained from the Port)er-Thomas distribution. There appear to be no anomalous moments. Two points should be noted: first’, because the Porter-Thomas distribution is a very broad distribution, one can expect rather large deviations of moments of individual samples from the mean value to occur. Secondly, the nuclei treated here are those cases for which a reasonable sample size is available. It, is just these nuclei with relatively small level spacings and correspondingly complex internal wave functions which one would expect, according to the theor.y, to yield a dist.ribution of reduced neutron widths following the Porter-Thomas-distribution.

12

DISTRIBUTION OF SECOND MOMENTS OF REDUCED NEUTRON WIDTH

IO - OlSTRliYJTlONS --1 _-_

- EXPERIMENTAL s-

- - - - MONTE CARLO -___

J k4 6-

H 4 --. 4-

z

0 -.J

0 I 2 3 4 5 6 7 8 SECOND MOMENT Mp+

Fro. 7. The distribution of second moments of neutron widt,h distributions containing

ten widths. The solid line histogram represents the 38 experimental cases listed in Table I-. The dashed histogram represents 200 cases drawn at random from the Porter-Thomas distribution and normalized to the 38 esperimental cases.

Page 18: Statistical analysis of neutron resonance parameters

286 GARRISON

One further comment should be made. The nuclei listed in Table V were grouped into two classes, those whose correction factor (Nt/N) is believed to be close to one and those whose correction factor is believed to be of the order of 1.3. Correction factors based on somewhat better estimates of missing resonances will not appreciably alter the distribution given here.

IV. RESONANCES WITH MEASURED SPIN

Data from 153 resonances of odd-A nuclei with measured spin have been ex- amined to test the dependence of the level spacing and average reduced neutron width or strength function on spin.

The dependence of the average level spacing, D, , on spin, J, is predicted by the theory of Bethe (21):

0;’ = DO’(2J + 1) exp [-r(J + x)2] (7)

where Do and y are constants independent of J, which, however, depend upon such parameters as the excitation energy, mass number, and shell structure. More refined versions of Bethe’s formula have been developed by Lang and LeCouteur (22), Cameron (2S), and others.

Table VI gives the ratio of the number of resonances with spin J = I + x to J = I - x (I, the target nucleus spin), with the number of resonances used in determining the ratio in parenthesis following the ratio. If the constant y in Eq. (7) is small, the expected ratio is (I + 1)/I. This ratio is given in the right hand column of the table. The errors given represent counting statistics alone. The data in Table VI and also in Tables VII and VIII to be discussed next have been obtained from refs. 11, 15, and 2447.

The reason for separating the measurements into three groups is to test for experimental bias. At higher energies, the measurements of level spin usually can be performed only for the stronger resonances. If the value of gm is greater for one spin state than for the other, one might expect a bias in the measure- ments which could be rather serious at higher energies where many resonance spins are unmeasured. As evidence for this bias, the spin of the resonance which

TABLE VI

LEVEL DENSITY RATIOS

I All measured resonances (cases)

Selection I (cases) Selection II (cases) (I + 1)/Z

w 2.1 f 0.5 (88) 2.8 f 1.0 (38) 1.9 f 0.8 (26) 3.00 w 1.7 f 0.9 (16) 2.0 f 1.4 (9) 1.5 f 1.4 (5) 1.67 % 1.1 f 0.4 (27) 1.3 f 0.7 (16) 0.7 f 0.5 (10) 1.40 35 1.6 f 0.8 (18) 1.6 f 0.9 (13) 1.7 f 1.3 (8) 1.29

Page 19: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 287

TABLE VII

RATIOS OF AVERAGE REDUCED NEUTRON WIDTHS FOR TARGET ISOTOPES OF

SPIN ONE-HALF

Target isotope

Sei7

YE9

&

Wl83

PV95

HglQQ

Pll939

F+/Ew

0.067 + 0.07

- 0.03 (0.1

0.8 + 0.9 - 0.5

0.3 + 0.4 - 0.2

0.3 + 0.3 - 0.2

0.8 + 0.7 - 0.4

0.3 + 0.6 - 0.2

TABLE VIII

MEAN RATIOS OF AVERAGE REDUCED NEUTRON WIDTHS

Target isotope Spin class

Average ratio I I/(1

All measw~lr,esonance Selection I (cases) Selection II (cases) + 1)

I = f4 0.38 f 0.13 (86)

z=gi 0.8 + 0.7 (16) - 0.8

I=pj 1.2 +1 (22) - 0.6

z = w 0.8 + 0.7 (18) - 0.4

Weighted average for all 0.56 f 0.14 (145)” spins (experimental)

Comparison ratios 0.45

0.3 + 0.2 (36) 0.4 + 0.3 (24) 0.33

- 0.1 - 0.2 1.5 + 2 (9) 0.7 + 1.0 (5) 0.60

-1 - 0.5 1.8 + 2 (15) 1.7 + 2 (9) 0.71

-1 - 1

0.6 + 0.7 (13) 0.7 + 0.9 (8) 0.78 - 0.4 - 0.5

0.6 f 0.2 (76)” 0.7 f 0.3 (48)a

0.51 0.49

8 Includes In*15 for cases for which Z = K.

lies in energy just below the first resonance of unmeasured spin has been ex- amined for each nucleus. The ratio of I + x resonances to I - x resonances is 23 : 5 (11: 1 for I = 34 nuclei) indicating that gm for I + $4 resonances is larger for these particular nuclei. However, looking at the resonances of measured spin above the lowest energy resonance of unmeasured spin, the same ratio is 44:29 (32: 18 for I = s), which gives little indication of any bias. In Table VI, Selec-

Page 20: Statistical analysis of neutron resonance parameters

288 GARRISON

tion I includes only resonances which lie in energy below the first resonance of unmeasured spin, while Selection II in addition omits the resonance just below the first resonance of unmeasured spin.

From the result’s of Table VI it may be concluded that the data are in agree- ment with Eq. (7) for a small value of y wit’hin the rather large experimental uncertainties.

We turn now to considerat’ion of the difference in the average value of rno for resonances of each spin state. This has been done by examining t,he ratio of the average reduced neutron width for J = I + $5, E+, to the average reduced neutron width for J = I - 35, I‘,O-. This ratio will be designated hereafter by the symbol 1’. Only a few target nuclei have a measured ratio that is at all meaning- ful. These t’arget nuclei are all of spin I = $5. The experimental results for this ratio are presented in Table VII.

If the strength functions for each of t’he two possible resonance spin states are equal and there is a 2J + 1 dependence for the level density, then one expects a ratio of 0.33 for 1’ (= I‘,O+/r,O-) for I = $5. It is apparent in at least two cases in Table VII, for Se” and Ygg measured at’ Saclay (S4), that the ratio of 1’ may be appreciably lower than 0.33. Since the level densities are apparently about normal, at least for Se7’, this implies that the strength functions for each of the two spin states are quite different. The effect of a finite (not very small) value for the constant y in Eq. (7) would be to yield an even larger difference between the t’wo strength functions than that obtained assuming a ratio of [(I + I)/11 = 3 for the level densities, The ratio for Se77 also has been measured recently at Argonne. As originally reported (%5), the mean reduced neutron widths for each spin state were essentially the same, however, this conclusion was based on an incorrect spin assignment for the large resonance near 1 keV (96). Their results are apparently now in reasonable agreement with the Saclay results. It is clear t#hat the accuracy of the ratio r is not. well determined if it can be changed radi- cally by changing the spin assignment of one resonance.

For Se”, the ratio of the s-wave st’rengt,h functions for t’he two spin states is of t’he order of five, as measured at Saclay. Se77 lies in the mass region where the s-wave strength function is decreasing rapidly’ with increasing mass number. If the shape and radius of t,he potential is the same for both spin states, a differ- ence in well dept,h -2 X2eV for the two spin states of Se77 could account for a ratio of -5 for (I’,O/D)+/(I’,O/D)-. Apparently the ratio of strength function for YE9 is also large as measured at Saclay, and could be attribut’ed to a similar difference in well depth. It has been shown by Vogt (38) that the strength func- tion increases wit’h increase in surface diffuseness. Therefore, the strength func- tion ratios are not necessarily directly interpretable in terms of differences in well depth, since the surface diffuseness may differ for the two spin states.8

7 See, for example, ref. 16, Figure 2. * This point is due to Y. C. Tang.

Page 21: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 289

Because of uncertainties in the data and lack of sufficient data, no definite con- clusions can be drawn concerning the neutron-spin-target-spin interaction other than that it is small. The ratios, T, for the remaining nuclei in Table VII, which generally have masses in regions where the strength function is less rapidly vary- ing than it is near Se7l and Y 8s, do not differ significantly from 0.33 (or somewhat above, as expected for y not too small). P. P. Singh has recently reported (,@) that the strength functions for the two spin states of Trn16’ are the same within experimental error.

Large values of the ratio of 1’ should be detectable as deviations of the combined reduced neutron width distribution from the Porter-Thomas distribution. The expected reduced neutron width distribution for values of 7, 3, and 1 for the rat’io, r, along with the histogram for Class 2 nuclei (identical to that of Fig. 3), are shown in Fig. 8. Also shown in Fig. 8 is the Chi-Squared distribut’ion for Y = 0.5 (dashed curve) which deviates from the Y = 1 (1. = 1) curve in somewhat the same manner as t,he 1’ = 3 and 7 curves. It is clear from Fig. 8 that the ratio r for the nuclei contributing to the Class 2 distribution (and also for the Class 3 nuclei in Fig. 4 and the odd mass number Class A nuclei in Fig. 1) is not large, but it is also clear that it would be difficult to detect an average ratio of three. The curves in Fig. 8 were computed assuming equal numbers of resonances in each spin state.

Finally, in an effort to gain the last bit of information which is available in the data for t’he resonances of measured spin, average values of the ratio r’ for resonances of each spin and a weighted average for all spins have been obtained and are presented in Table VIII in a manner similar to that of Table VI. The mean ratio for each spin class is obtained from the equation:

r = CiWoni’/r,O> Cdl%-/ iT)

FIG. 8. The reduced neutron width distribution of the Class 2 nuclei. The smooth solid curves are the distributions obtained by summing two Porter-Thomas distributions in

equal proportion, but with different ratios, r, of the mean reduced neutron width. The smooth dashed curve is the chi-squared distribution with 0.5 degrees of freedom.

Page 22: Statistical analysis of neutron resonance parameters

290 GARRISON

- where 1’,,0 = ~(I’,O/D)D~I,, as before (Eq. (1)). The weighted averages for all spins for each selection are obtained by weighting the mean r for each spin by the number of cases. The comparison ratios are the ratios expected if the strength functions are equal for resonances of each spin and the ratio of level densities for each spin is (I + 1)/1 for all nuclei treated here.

It remains to interpret the results of Table VIII. The easiest interpretation which is consistent with Table VIII is to state that one might reasonably expect, because of the giant resonance behavior of the strength functions, that part of - - the time (rnO/D)+ would be greater than @,0/D)- and part of the time less than - (r,o/D)- such that on averaging over nuclei covering a wide range of mass num- bers, the two strength functions would be close to equal, particularly with a weak neutron-spin-target-spin force. This would lead therefore to an average r for each spin close to I/(1 + 1) and weighted averages of r over all spins close to the Comparison Ratios given at the bottom of Table VIII. Under the above in- terpretation, a value of y in Eq. (7) which is not small for the nuclei considered would lead t.o mean values of r which are higher than I/(1 + 1) for each spin and to weighted averages of r for all spins which are higher than the Comparison Ratios.

V. THE LEVEL SPACING DISTRIBUTIONS

In theory, the distribution of level spacings can be derived from the same as- sumptions as the distribution of neutron widths (6). Experimentally, the two distributions are not connected, except in so far as the data are systematically affected by the missing of weak levels and small spacings. The experimental aspects of t.he spacing distribution will be presented in this section along with a comparison with theoretical spacing distributions obtained by random matrix diagonalization (3.9). More general discussions will be deferred until the end of the paper.

In Section III, which concerns the distribution of reduced neutron widths, the difficulties connected with the experimental data were considered. In par- ticular, the possibility of missing weak levels and of observing extra levels was discussed. The missing of weak levels and the observing of extra levels provide the principal and essentially the only source of systematic error in the level spacing distributions. The eigenvalues are quite well measured. The experimental reduced neutron width distributions treated in Section III were not corrected for missing or extra levels; rather, an attempt was made to determine the form of the distribution without considering that part of the distribution containing the weak levels; a truncated distribution was used. Here, a definite attempt will be made to determine the corrections to the experimental level spacing distribu- tions required by the missing and extra levels. In obtaining these corrections, t’he Porter-Thomas distribution will be assumed to represent correctly the distribu- tion of reduced neutron widths.

Page 23: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 291

Only spacing distributions for the Class A nuclei will be considered. The cor- rections for these nuclei are (except for P8) relatively small. Class A must be divided into two subclasses, the even-even nuclei (ThZ3” and U238) and the odd-A nuclei (IL2’, Ta181, and Aulg7), because of the difference in the spacing distrihu- tions for the two subclasses.

The distributions, Pk , of the sums of successive level spacings, where k + 1 is t,he number of spacings in the sum, are useful in obtaining a visual comparison between theory and experiment. The distributions, PI, , are distribut’ions in Yki = c&o zi+j , where in this case IZ’~+~ is the ratio of level spacing to average level spacing for the (i + j)th spacing. (See Appendix II.) Figures 9 and 10 show histograms of the dist’ributions, Pk , for Th232 and U238 combined, and for 1127, TalsL, and Au197 combined, respectively. Also shown in the same figures are the smooth dashed curves obt~ained from t)he diagonalization of lO,OOO-10 X 10 matrices (40) which had matrix elements drawn at random from real gaussian distributions of zero mean, the rotation invariant, t)ime reversal invariant case

0 p, :: 2 3 4 6 12 IS 24

0

0.4 ; : S

;‘, 1: I ps 0.4

02 :

I-:,1’Ii”‘. : ,, ; -0.2

0 ,: :,

,.2 4 6 8 0 / :

7 14 21 28

04

Pb p2

t

: : ;i % 0.4

-f I6 24

0 32

:;ijl : : lip iI.1

5 hJ 15 20 10 20 30 40 Y= s/o

FIG. 9. The distributions, Pk , of the sums of successive level spacings for the Class A-even-even nuclei, Thza2 and IF. The subscript k is one less than the number of successive spacings summed for the distributions. The smooth dashed curves are the smoothed random matrix diagonalization results of Porter, BNL-6763.

Page 24: Statistical analysis of neutron resonance parameters

292 GARRISON

: 0 3 6 9 12 8 16 24 32

4 8 12 16 9 24 36

0.4. I’, S P9 -0.4

0.2- -02

0 : 0 5 IO I5 20 IO 20 30 40

FIG. 10. The distributions, Pk , of the sums of successive level spacings for the Class

A-odd-A nuclei, 1127, Tal81, and Aul97, The subscript k is one less than the number of suc-

cessive spacings summed for the distributions. The smooth dashed curves are the smoothed random matrix diagonalization results of Porter, BNL-6763.

(6). The 10 X 10 gaussian results were chosen for presentation here since their good st,atistical precision leads to smooth curves for visual comparison with the experimenbal histograms. For the distributions of odd-A nuclei found in Fig. 10, the srnoot,h curves were obtained by overlapping eigenvalues from pairs of the 10 X 10 gaussian matrices to simulate the two spin states of odd-A nuclei. The mean spacing ratio for the pairs of overlapped matrices was 1.4. This is the spacing ratio expected for a target nucleus of spin x if the mean spacing has a (2J + 1)-l dependence. The spins of P7, Ta181, and Aulg7 are 56, 35, and ,“$, respect’ively.

The histograms in Figs. 9 and 10 are not corrected for missing and extra levels. They do not lend t,hemselves readily to such corrections. For the purpose of making corrections t,o the dat,a and quantitatively comparing experimental with theoretical distribut#ions, t’he moments of the distributions, Pk , lend themselves

Page 25: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 293

particularly well. The nth moment (about zero) of the kth experimental dis- tribution should be determined by:

111,“’ = ( 116,”

(Mlkl@ + 1))” >

n(n - 1) (9)

. l+ 2(N _ k)UrP ( [ 1+ Gk ,g2 cjw - k - 1 - j)])

where each M,” is determined by Eq. (4), with the xi given by the ratio of the ith spacing to the mean spacing (sample mean) rather than the ratio of reduced neutron width t’o mean reduced neutron width as used in Section III, C where Eq. (4) occurs. Equation (9) is a more general case of Eq. (5) used with the re- duced neutron width distribution for which the correlation coefficients ck are consistent with zero. The correlation coefficients here are defined as previously by the equation:

c’k = Xi%+k+l - XiXi+k+l

uz2

but here t,he xi and .ri+k+l are the ratio of level spacing to average level spacing and aI2 z 0.3 and 0.5 for level spacings of even-even nuclei and level spacings of nuclei with two level spins, respectively. Equation (9) is derived in Appendix II. Its components have the same interpretation as was given for Eq. (5). The finite sample correction, the second brackets factor in Eq. (9), which makes Mnk’ an unbiased estimator of the distribution moments, is apparently a smaller correction for the level spacing distributions t’han it is for the neutron widt,h distribution since, as shall be shown, the near neighbor spacings are anticor- related and the summation over the ck in Eq. (9) is negative. In practice, the summation containing t,he ck cannot be evaluated with good accuracy. The biased moment,s:

(10)

of t,he spacing distributions of each half of the eigenvalues of the Class A nuclei and other nuclei have been calculated. The average of these pairs of moments has been compared with the moments obt’ained using all the eigenvalues. The differ- ence, on the average, can be attributed to a finite sample correction factor whose correction is of the order of one-half the correction obtained if the ck are all zero. Accordingly, the expression in square brackets in Eq. (9) has been set equal to 0.5 f 0.5. This correction is generally small relative to other uncertainties for moments of the lower distributions. For higher k this factor is more import,ant.

After calculating the moments, Mmk’ , for each sample of experimentjal spacings

Page 26: Statistical analysis of neutron resonance parameters

294 GARRISON

using Eq. (9) with the square brackets set equal to 0.5, these moments are cor- rected for missing and extra levels. To do this, an estimate of the number of missing and extra levels first must be made.

For the even-even nuclei, the smallest detectable reduced neutron width (the mean value over the energy range covered by the data), and the number of weak levels required to bring the experimental neutron width distribution into agree- ment with the Porter-Thomas distribution are estimated. The number of levels missed below the smallest detectable width is obtained from the integral of the Porter-Thomas distribution from zero up to the smallest detectable width. The number of extra levels is taken to be the algebraic difference between the number of missed levels and the number of weak levels needed to be added to (+) or t,aken away (-) from the neutron width distribution to bring it into agreement with the Porter-Thomas distribution. Few levels are missed due to small spacings because of the level repulsion effect; however, a smallest observable spacing was estimated from the data and was used in determining the corrections.

For odd-A nuclei, the smallest detectable reduced neutron width, the number of weak levels required to bring the experimental neutron width distribution into agreement with the Porter-Thomas distribution, and the mean smallest spacing which is observable are estimated from the experimental data. After deciding the mean smallest spacing, the correction for this smallest spacing is obtained directly from the corrections tables which are presented in Appendix III. The number of levels missed by small spacings is determined by the matrix eigenvalue calculations used to obtain the corrections tables. The matrix eigen- values calculations are discussed next. The number of levels missed because the neutron width is below the detectable limit and the number of extra levels are determined as for the even-even nuclei. The numbers of missing and extra levels determined by the above procedures are rather uncertain statistically.

The corrections to the moments for missing and extra levels, as tabulated in Appendix III, were determined after overcoming a number of difficulties. Ini- tially, levels were removed and added to the experimental data for the even-even nuclei, however, the corrections for Th232 and U238 did not agree well. It was therefore decided to determine the corrections from the 100,000 eigenvalues which are available on magnetic tape from the diagonalization of the 10,000 10 X 10 gaussian matrices. Only part of the tape was needed to provide good statistical accuracy. A program using this tape has been written which removes one eigenvalue at random from each matrix, adds one eigenvalue to each matrix, and/or removes one of two eigenvalues where the spacing between them is less than a particular value. The program records the number of levels removed because of close spacings. The moments of the spacings distributions, Pk , which are determined with this program, are compared with the moments of the unalt- ered matrix spacing distributions to determine the change in moments caused by

Page 27: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 29.7

the addition or removal of eigenvalues. The changes are small, as are the cor- rections to the moments for the nuclei treated here, so that the corrections for the nuclei can be calculated from these changes assuming that these changes vary linearly with addition or removal of levels. It is also assumed that the changes from the various causes can be treated independently. Inherent in the approach to these corrections is the assumption that there is no correlation between reduced neutron width and the level spacings, and that the extra levels occur at random relative to the levels already present, except for small spacings. The lack of any width-spacing correlation is indicated in the next section.

The corrections obtained from the nonoverlapped 10 X 10 matrices give reasonable agreement with the earlier corrections obtained from Th232 and UZ38 except for the higher distributions (/c 2 3) where it is clear that, in some manner, the finite size of the matrix makes the corrections small. Because of this difficulty, it was decided to diagonalize 200 20 X 20 gaussian matrices and put these eigenvalues on tape for determining the corrections. Even these corrections show the effect of the finite size of the matrix for k 2 6. In some cases these corrections have been adjusted upward in magnitude for the higher distributions.

One additional refinement has been found to be desirable. The need for this refinement became apparent when the corrections for extra levels were examined. The eigenvalues of the matrices occur on the average symmetrically about zero with zero occurring essentially at random between the two eigenvalues closest to zero. Therefore, for simplicity in determining the extra level corrections, an extra level has been inserted among the matrix eigenvalues first at zero and then at some larger number not expected to place it beyond the outside eigenvalue of the matrix. The average of these changes has been used to determine the cor- rections; however, these two changes are quite different. This has led to the realization that the mean level spacings are smaller near the middle of the dis- tribution than at the outside edges. The eigenvalues of the nuclei used here are away from the edge of the distribution and lie in a region where the mean level spacing is uniform. It therefore seems more appropriate to compare the cor- rected moments of the spacing distributions for the nuclei with the moments of the spacing distribution for the central eigenvalues of the 20 X 20 gaussian matrices, the largest matrix for which a diagonalization program is available here at Brookhaven at this time. The machine program for determining the corrections and calculating the moments accordingly has been modified for the removal of the outer eigenvalues.

The moments of both the 10 X 10 and 20 X 20 gaussian matrices are lowered by the removal of the outer eigenvalues, as is expected, since the superposition of a number of distributions with different mean will tend to broaden the com- bined distribution and raise the moments. The change in the corrections between the 20 X 20 matrix values and the corrections obtained from 20 X 20 matrices

Page 28: Statistical analysis of neutron resonance parameters

296 GARRISON

with 4 outside eigenvalues removed [(20 X 20) (--8)] from each end has been tested. The change is small relative to other uncertainties, and the original 20 X 20 corrections are those presented in Appendix III with some modifications which are discussed there.

Table IX presents the corrected moments of the distributions, Pk , for the neighbor level spacings of Th232 and U238 separately and their weighted mean. Also shown for comparison are the moments of the spacing distributions of the 20 X 20 random matrix eigenvalues for the case where 4 outer eigenvalues have been removed from each end [(20 X 20) (-8)] before computing the moments. The statistical errors in the random matrix moments are negligible relative to the errors in the experimental moments. The errors in the experimental moments are obtained by combining the statistical error (see Appendix II) with the error in the corrections for missing and extra levels, and the error in the finite sample correction. The errors in the corrections arise principally from the uncertainty in the numbers of missing and extra levels which, for the cases treated here, are of the order of 50 %. All of the correction errors have been assumed to be f50 %. The statistical errors of the second moments are shown in parentheses after the second moments to indicate the type of precision possible without the corrections.

Although all 99 spacings for U238 have been used in Fig. 9, it was later decided that the data for U2a8 was rather poor at higher energies and only the first 50 eigenvalues have been used in Table IX. Even these 50 levels seem to include many p-wave levels, making the corrections relatively large and possibly non- linear. By the methods indicated earlier it was decided that 3% of missed and 19% of extra levels (corresponding to 8 f 3 levels above the Porter-Thomas distribution) are a best interpretation of the data. Similarly, for Th232, using the first 154 eigenvalues, 3 % missed and 7 % extra levels are indicated by the data. It is of interest that both the corrections for missed and extra levels lower the moments.

Table X presents the corrected moments of the distributions, Pk , for the neighbor spacings of Ilz7, Tal*l, and Au1g7, combined. The errors in the moments have been determined as for Th232 and U *38. The statistical errors of the second moments are again indicated in parentheses following the moments. The moments of the overlapped (20 X 20) (-8) matrices also are given in Table X for com- parison. The ratio of mean spacing for the two matrices is again 1.4. The experi- mental moments shown in Table X were corrected for the three nuclei jointly since they were measured under similar conditions and have approximately the same number of levels. It is assumed here that when levels are closer together than 0.05 Dabs one of the two levels is missed. It is also assumed that weak levels with a neutron width to mean neutron width ratio less than 0.01 were missed. With the Porter-Thomas distribution indicating that 7 f 4 levels are needed to raise the experimental distribution in the region of weak levels, one obtains 11 570

Page 29: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS

TABLE IS

297

MOMENTS OF THE SPACING C)ISTRIBI*TIONS FOR Th?a* ANL) 11*38

Distribution Expt M:

PII Th=” 1 1.29 f 0.05 (0.04 [-2sa 1.14 f 0.09 (0.07

PI :i::'~ ::t f 0.08 (O.OG [TYaa ~ 4.2 f 0.2 (0.1)

MeaIl~ 4.34 Th”= 9.45 zt 0.12 (0.07

1’238 9.23 f 0.28 (0.12,

Mean 9.40 Th’=’ 16.53 z!c 0.15 (0.08

I’?38 16.2 Mean ~16.45

f 0.4 (0.14,

Th=” 25.6 xt 0.2 (0.1) I?" ,25.3 It 0.4 (0.2)

Mean l”5.5 Id Thz3? ‘36.6 =t 0.2 (0.1)

I:?%a '36.3 f 0.5 (0.2) Mean 36.5

Tk13” 19.6 f 0.3 (0.1)

p?38 49.4 It 0.8 (0.2) Mean 49.6

Theory Expt Ma Theorq

1.28 1.9

4.42 10.6

9.50 31.6

16.0

1.96 + 0.15 1.37 f 0.25

1.81 10.4 f 0.5

9.0 * 0.9 10.1

31.0 * 1.2

29 f2

30.5 70 f2

G6 +4 B9

133 f 6 129 f 11 132

220 It 9

221 f 10 225

354 f 12

351 * 13 353

70.7

25.7 135

36.7

49.7

229

358 - -

3.1 f 0.5 1.0 f 1.0

2.6

25 + 3 18 f 5

23 104 f 15

89 f 17

100

290 f 20 269 f 35 289

702 f 32

654 f G5 690

1403 f GO 1356 f 110 1391

2562 f 90 2502 f 180 2540

3.29

27.3

110

311

724

452

618

TABLE S

~%)MENTS OF THE SPACING ~STRIBI-TION FOR Ilz7, Tal*l, AND Au197 COMBINEU

Dis- tri- bu- tion

Expt .Uz Theory Expt iW3 Theory Expt Ma Theory

P,, 1.49 i 0.08 (0.07) 1.48 2.8 + 0.4 2.73 0.0 f 1.4 5.84

PI 4.68 zk 0.13 (0.09) 4.71 12.3 + 1.0 12.6 35 + 5 37.0

P? 9.8 + 0.2 (0.10) 9.86 34.7 zt 1.7 35.0 131 f. 12 133

Pi 17.1 + 0.2 (0.12) 17.0 77 f3 75.9 371 zt 24 356

p4 26.3 f 0.3 (0.13) 26.1 144 f 4 141 817 f 45 797

PS 37.4 i 0.3 (0.14) 37.2 241 fG 237 1604 + 80 1556 P6 50.5 + 0.3 (0.15) 50.2 374 f 7 3G9 2841 f 110 2771

missing aud 7 $6 extra levels. Here the correction for small spacings raises the uioments while the other two corrections lower the nloments as before.

The experimental moments presented in Tables IX and X show reasonable agreement with the random matrix inonleuts giver1 in the same tables and also with other random matrix moments. It also should be apparent that the best

Page 30: Statistical analysis of neutron resonance parameters

298 GARRISON

data, Th2*l, requiring the smallest corrections and having the best statistical accuracy provides the best agreement with the random matrix moments, and that P8 requiring the largest correction and having the smallest sample size provides the poorest comparison with the random matrix moments.

The corrections for missing and extra levels leave much to be desired in the way of accuracy. However, if the random matrix moments are accepted as close representations of the true experimental moments, then the need for these cor- rections is clearly demonstrated. As mentioned earlier, the actual corrections to the moments for each case were determined from the correction tables in Ap- pendix III. It should be pointed out that the experimental results in Tables IX and X may appear more accurate than in fact they are, since a b-function dis- tribution would yield moments M,” = (k + I)“, so that perhaps better quanti- ties to consider in Tables IX and X would be the differences M,” -. (k + 1)“. It is apparent then that the moments Mnk are rather poorly determined.

The correlations between neighboring spacings are contained in the moments of Tables IX and X. The correlation coefficients, Ce , for pairs of spacings sepa- rated by k spacings have been calculated using Eq. (6) or Eq. (59) in Appendix II and the corrected experimental moments (before the numbers were rounded off) of Tables IX and X. These correlation coefficients are listed for Th232, UZ3* and for P, Tal*l, and Au1g7 combined in Table XI under the heading ‘Wo- merit.” The correlation coefficients have also been calculated from the spacings directly using the code mentioned in Section II. These latter coefficients are listed in Table XI under the heading “Direct. ” At the bottom of Table XI are given the approximate statistical errors of the correlation coefficients in each column. An estimation of the contribution of systematic error to the uncertainty in the correlation coefficients is more difficult and has not been attempted. In

TABLE XI

CORRELATION COEFFICIENTS FOR NEAR NEIGHBOR LEVEL SPACINGS

K Thzs2 Moment Direct

0 -0.33 -0.21

1 -0.07 -0.08 2 -0.02 -0.04 3 +0.09 +0.05 4 -0.17 -0.11

5 0.00 +0.04 Statistical Error fO.08

U*8 Moment Direct

-0.32 +0.07 (-0.45)”

-0.07 -0.10

-0.18 -0.19

+0.43 +0.10 -0.36 +0.10

+0.36 +0.18 f0.14

Theory

-0.25

1127, ‘&HI, AU197

Moment Direet

-0.31 -0.17

Theory

-0.26

-0.11

-0.03 +0.03

0.00

-0.13 +0.03

-0.03 -0.09 -0.08 t-o.10 +0.07 -0.05 -0.06 -0.08 f0.01 -0.10 -0.05 -0.03

0.00 -0.04 -0.01

f0.07 fO.03

a 12 weakest levels removed

Page 31: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 299

the “Direct” column for Uz3* in Table XI, the correlation coefficient Co has also been calculated with the 12 weakest levels (labeled “uncertain Z-value” by the workers at Columbia University (11)) omitted and is presented in parenthesis for comparison. This gives a measure of the effect of the p-wave contribution on the correlation coefficient calculations. The corrections applied to the U238 spacing distribution moments lead to a more reasonable value for Co , but also appear to cause an appreciable oscillation in the higher correlation coefficients. The cor- relation coefficients listed in Table XI under “theory” are obtained from the moments (before the numbers were rounded off) labeled “theory” of Tables IX and X, the nonoverlapped and overlapped (20 X 20) (-8) cases.

The experimental results for the correlation coefficients are in reasonable agreement with the theory. It can be concluded that the correlation coefficients Co , C1 , and probably Cz are negative in all cases. The correlation coefficients for higher values of k may be positive or negative; their sign is undetermined by these calculations. However, all the correlation coefficients except Co are probably small.

An immediate practical result of the negative correlation of close spacings is that the mean spacing of nuclei is more accurately determined by a few levels than would be the case for no correlation. For example, for even-even nuclei, the standard deviation for Dabs o btained from five successive levels is 4.19 Dabs rather than the -0.27 Dabs which would be obtained if there were no correlation between spacings. Similarly, for nuclei with levels of two spins, the standard deviation for Doks obtained from five successive levels is -0.25 Dabs while the standard deviation if there were no correlation would be -0.35 Dabs .

It is interesting to note that the negative correlation of near levels under “theory” in Table XI for both the overlapped and nonoverlapped cases are close to the same values, contrary to what one might intuitively expect from the super- position of two uncorrelated level sequences.

VI. CORRELATIONS

The correlation coefficients for neighboring neutron widths and for neighboring level spacings have been obtained in previous sections. It is desirable to consider possible correlations between other pairs of resonance parameters. (In particular, correlation coefficients associated with the fission widths should be obtained before considering the fission width distributions in the next section.)

The fissionable nuclei U233, U235, PUBIS, and Pu241 (11,41-46) have been examined for possible correlation between the total fission widths of nearest neighbor resonances (I’f:I’f) and for possible correlation between the total fission width and reduced neutron width of the same resonance (pf:grno). The mean correla- tion coefficients for these nuclei are shown in Table XII along with their errors. These numbers are thought to be the best representation of the data after con-

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300 GARRISON

TABLE XII

RESONANCE PAFL~METER CORRELATION COEFFICIENTS

Parameters Mean correlation coefficients

Fissionable nuclei I-,:rr -0.1 f 0.2 I-f : grn0 0.0 f 0.2

Even-even nuclei rmo:S 0.0 f 0.1 rno:rno 0.0 * 0.1

Partial radiation width nuclei: SeT7, WI*), PVQ6 r,:r, -0.1 f 0.2 ry:rno 0.0 f 0.2

sidering various combinations of the widths for the individual nuclei. The cor- relation coefficients for the individual nuclei tend to show somewhat more fluctuation than expected from the size of the samples considered if the true correlation coefficient is the same for each nucleus. This probably reflects the quality of the data rather than differences in the correlation coefficients among these nuclei. In fact, the correlation coefficients for the first 12 levels in U233 do not even agree very reasonably when calculated from the three sets of multilevel parameters which are now available (43, 45, 46).g The correlation coefficients associated with the total fission width are not simple channel-channel correlation coefficients since a number of fission channels contribute to each width. In addition, the facts that a number of levels in the fissionable nuclei have been missed and that two spin states are involved make the conclusions reached from these correlation coefficients less definite.

The even-even nuclei Th232, U234, U236, U23*, and Pu~~O (11, 13, 14) have been examined for possible correlation between the reduced neutron width and neigh- boring spacing (rnO: 8). The mean correlation coefficient along with its statistical error, obtained from these calculations, also is given in Table XII. The correla- tion coefficients for the individual nuclei agree with the mean value in the table within their statistical accuracy, except for U236, which contributes little to the mean correlation coefficient because of the small size of its sample. Since cor- relation coefficients obtained from the moments of the neutron width distributions in Section III involved some odd-A nuclei with two spin states, the (I’,O:rno) correlation coefficient for nearest neighbor neutron widths of even-even nuclei has been included in Table XII.

Finally, data on partial radiation widths and neutron widths for Ptlg5, WIB, and Se77 (31, 4’7) have been examined for possible correlation between radiation widths of the same resonance (I’,: I?,), and for possible correlation between partial radiation widths and reduced neutron width of the same resonance (I?, : r,O). The

9 Erich Vogt only has parameters for 10 levels listed.

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NEUTRON RESONANCE PARAMETERS 301

mean correlation coefficients for these two cases and their statistical errors are presented in the same Table XII. The @‘,:I’,) and (I’,:I’,~) correlation coefi- cients for the individual nuclei agree with the mean values given in Table XII within their statistical accuracy. Only the partial radiation widths for transitions to the ground and first excited states of Se78 and Wls4 have been used in the calculations for Table XII, while partial radiation widths for transitions to the lowest three states of Ptlg6 have been used.

The results presented in Table XII are consistent with zero correlation between all the pairs of resonance parameters tested. Combining the results of Table XII with the correlation coefficients obtained earlier in this paper, one can reasonably conclude (by induction, since not all conceivable tests have been made, nor all possible nuclei examined) that the correlation between the various partial widths of the same level (channel-channel correlation) and of different levels is zero or small. Also, there is zero or little correlation between the partial widths and the near neighbor spacings. The only correlation which has been observed is that found between near neighbor spacings, presented in the previous section.

Whether the conclusion of zero or small channel-channel correlation can be applied to the case of fission is not so clear. The many different channels cor- responding to the many possible pairs of fission products appear to have partial fission widths which are highly correlated: The fission mass yield distribution does not seem to vary greatly from resonance to resonance while the total fission width varies over a wide range of values.

VII. DISTRIBUTIONS OF FISSION WIDTHS

The resonance parameter data of fissionable nuclei and, in particular, the fission width data, are neither as well measured nor as numerous as the better data on neutron widths and level spacings. In addition, considerable confusion exists because there is not a sufficient number of resonance spin measurements for the fissionable nuclei. Spin measurements are more important in determining the fission width distributions than in determining the reduced neutron width distribution. The fact that the average fission width for the two possible spin states is generally rather different and the fact that the number of degrees of freedom associated with the fission width distributions is larger than that of the neutron width distribution accounts for the increased importance of spin measure- ments for the fission width distributions. Although no strong conclusions con- cerning the distribution of fission widths can be made, it is desirable and within the scope of this paper to examine the data in some detail at this time to deter- mine the present status of our information concerning the fission width distribu- tion.

The resonance parameters of U233, U235, Pu23g, and Pu24L are the only parameters of fissionable nuclei suitable for statistical analysis. Figure 11 compares the

Page 34: Statistical analysis of neutron resonance parameters

302 GARRISON

FIG. 11. The reduced neutron width distribution of the four fissionable nuclei U233, US”,

Pu288, and Put41. The smooth curve is the Porter-Thomas distribution. The gap between

the smooth curve and the histogram corresponds to approximately 15 cases.

combined reduced neutron width distribution of the 64 “good data” resonances of these nuclei with the Porter-Thomas distribution. These data have been taken from references (11, 41-46, 48). Approximately 20 % (-15 cases) of the levels have been missed. This is not very surprising, even though only the best data are used, since the mean level spacings are small and the widths of many of the resonances rather broad. The missing resonances place an additional limitation on the accuracy of the results to be discussed below.

The highest energy resonances whose fission and neutron widths are used here are at 20.76, 10.20, 52.6, and 10.20 eV for U233, U23s, Pu23g, and Puz41, re- spectively. Although fission widths and neutron widths for U235 are available up to 39 e‘CI’, the data treated here have been cut off at the 10.20 eV resonance since many resonances have unmeasured fission widths above 10.2 eV, and the mean fission width above 10.2 eV is appreciably smaller than it is below 10.2 eV.

Figure 12 presents the distributions of the ratios of fission width I’, to average fission width rf for U233, U235, Pu23g, and PUBIS. The average fission width used for each nucleus was obtained from the resonances at and below the cutoff energies given above. It should be noted that there is a general absence of levels with very small fission width. The absence of very small fission widths is believed to be real. There is no reason why levels of small pf would not be detected as is the case with small neutron widths. The absence of the very small fission widths for Pu23g is not so apparent in Fig. 12 since the histogram intervals are too broad. Com- parison of the four fission width distributions in Fig. 12 indicates that they all seem to exhibit the same general form, since each has a large group of resonances with I’~ less than rr and a smaller number of broad resonances extending with little or no peaking up to widths several times the average. Because of the simi- larity in form of these distributions, it seems desirable to combine them for the purpose of performing certain statistical tests, keeping in mind the limitations of this procedure. Figure 13 shows the combined distribution of fission widths.

Page 35: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 303

I-- FISSION WIDTH DISTRIBUTION

I--- 1

I5 CASES “235

EM,, 10.2 eV

0. m n rl I 0 1.0 2.0 3.0

RATIO OF FISSION WIDTH TO AVERAGE FISSION WIDTH r,/?,

FIG. 12. The fission width distributions of W3, V5, PtW, and PL@ presented separately.

The level of PuzS9 for which p,/P, is 7.7 has been placed at I’,/P, equal 4 to keep this case

on the scale. The value of EMAX listed for each case is the energy of the highest energy resonance whose fission width has been used in the distribution.

FIG. 13. The solid histogram is the distribution of the fission widths of U*33, Ur35, Pu*39, and Puz41 combined into one distribution using the cases presented in Fig. 12. The dashed

histogram is the distribution of the fission widths of the 19 levels of smallest neutron width, drawn from the solid histogram distribution. The smooth curves are the chi-squared dis-

tributions for 1, 2 and 3 degrees of freedom.

In a simple, but not conclusive, test to see if the missing resonances leave the fission width distribution distorted, the distribution of I’f/rr was obtained for the 19 resonances of smallest m” (about one-third of the resonances from each nucleus). This distribution is also given in Fig. 13 as the dashed histogram. There seems to be no statistically significant difference between this distribution and the distribution for all the resonances. This result is expected provided there is no correlation between the fission width and neutron width of the same resonance, and also provided the weak missing levels are not mostly of large fission width.

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304 GARRISON

The correlation between fission and neutron width has already been shown to be close to zero for the resonances which have been measured. Although weak resonances with large fission widths are less easily detected than weak resonances with small fission widths, this difference in detection efficiency makes little difference when the weak resonance is overlapped by a larger resonance. Quite a few resonances are expected to be missing because of this overlapping for the fissionable nuclei which have broad resonances and small spacings. The moments of the spacing distributions for the fissionable nuclei generally tend to indicate that levels are missed both because they are below the threshold for detection and because they are too close to other levels. In particular, it appears that quite a few of the levels in Uz33 are missed which are close to other levels.

Following earlier analyses, the fission width distribution has been tested by the Maximum Likelihood Method to determine which of the Chi-Squared Family provides the best fit to the distribution. The number of degrees of freedom determined by this test is expected to be related in a complicated manner to the number of open fission channels which are associated with each spin state. The relation between the number of degrees of freedom and the number of open fission channels is complicated for a number of reasons: (1) This distribution is a combined distribution of four nuclei, (2) the resonances of two spin states are present which generally are expected to have different average fission widths, and (3) the mean fission widths for the channels contributing to resonances of the same spin are in general expected to be different. The differences in average fission width for the different fission channels and resonance spin states are thought to arise from differences in the barrier penetrability for the fission channels associated with each spin state. These differences in penetrability will lead to a fission width distribution which is not a member of the Chi-Squared Family. In spite of this, the spread of the fission width distribution will continue to be discussed in terms of the number of degrees of freedom which best describes the distributions. L. Wilets has discussed the effect of the barrier penetration factor on the statistical distribution of reaction widths (49).

The Maximum Likelihood Test of the combined fission width distribution yields v = 2.0 f 0.2 (exponential distribution) for the number of degrees of freedom. A Chi-Squared Goodness of Fit test comparing this distribution with the exponential distribution indicates that there is only ~5% chance that this distribution is drawn from an exponentially distributed population. For visual comparison, the curves for v = I,2 and 3 have been included in Fig. 13.

It is apparent from Fig. 13 that a rather easy division into resonances of large and small rr is possible. It has been suggested theoretically (50) (51) and has also been indicated by multilevel fitting of the fission cross sections (&-46), that the wide group of resonances should be mostly associated with one par- ticular spin state and the narrow group of resonances mostly associated with the

Page 37: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 305

other. Rather than try immediately to separate all the fission widths of Fig. 13 into a wide and narrow group, the evidence concerning the spins of these levels should be examined. This evidence comes from multilevel fitting, the ratio of symmetric to asymmetric fission, and from direct measurements of resonance spin. These three sources of information will now be considered briefly.

The multilevel fitting leads to two groups of levels, at least for U233 and PUBIS, a broad group and a narrow group, with no interference between levels belonging to different groups. Certain pairs of levels definitely show interference, and these pairs are assigned to the same group; however, for all the nuclei for which multi- level fitting has been attempted, there are a number of levels which show little or no interference with other levels. These levels could be assigned to either spin group; an ambiguity exists. Lack of interference between two near neighbor levels involving contributions from two or more channels does not necessariIy indicate that these two levels have different spin, though this is a likely prospect when few fission channels contribute to the cross section.

Theoretical considerations have indicated that the ratio of symmetric to asymmetric fission should depend upon the spin state of the compound nucleus (50). The measurements to date (52-57) generally indicate small differences in the ratio of symmetric to asymmetric fission for resonances of different spin. Two exceptions are the work of Nasuhoglu et aE. (54) and of Roeland et al. (55), where no differences are detected. In the experiments where differences in this ratio have been measured, the results lend support to the grouping of levels determined by multilevel fitting, although only a few low energy resonances have been tested.

Fraser and Schwartz (58) have measured the spins of eight resonances in Puz39 (I = >s+). Where comparison is possible, their spin assignments are in agreement with the multilevel fit of Vogt. (46). The average fission width for each spin state for the resonances measured by them is not appreciabIy different; however, they did not determine the spins of the levels with largest fission width. There is evidence from the distribution of fission widths and from the multilevel fitting of Pu 23s that the number of degrees of freedom associated with the fission width distributions of P~23~ is smaller than for the other fissile nuclei considered here. This implies a wider spread and probably more overlap of the fission width distributions for the two spin states in Puz39.

Only the resonances at low energy are fairly complete in having an assignment to one spin group or the other. Table XIII lists the low energy resonances of U233, Uz35, PUBIS, and Pu**l along with their assignment to the broad (b) or narrow (n) group of levels. The levels with a rather definite assignment have the symbols : (i), for interference; (f), for fission symmetry ratio; and/or (s), for spin, following the spin group assignment to denote the method by which the spin group assign- ment was determined. In making the assignments, resonances sufficiently close

Page 38: Statistical analysis of neutron resonance parameters

306 GARRISON

TABLE XIII

SPIN ASSIGNMENT OF THE LOWER ENERGY RESONANCES OF U233, UzS6, Pu939, AND Puz41

Nuclide Nuclide (cases) EO (eV) Group I-f/P, (cases) EO (eV) Group rJ/ff

0.188 n 0.21 1.61 b(i) 1.98

1.785 n(i) (0 0.69 2.307 n 6) (0 0.16

3.64 n(i) 0.51 U233 4.79 b 6) (f) 3.14

(12) 5.85 n 0.64

6.85 n(i) 0.54

7.57 n 0.30 9.30 n 0.64

10.41 n 0.78

0.273 1.14

U235 2.035 (7) 2.82

3.16

3.60 4.84

b(i) 1.70

b(i) 2.20

n(i) 0.21

n(i) (small)

b(i) 2.70 n 0.78 n 0.07

0.296

7.90 11.00

Put39 11.90 (9) 14.3

14.7

15.5 17.6

22.2

0.264 4.28

4.56 Pu241 5.91

(8) 6.94

8.60 9.56

10.26

n(i)(f) 0.56

b 6) (8) 0.42

b(i) (sf 1.48

b(s) 0.22

n(i) 0.61

n(i) (9) 9.33

b(i) 7.67

b(s) 0.46

n(s) 0.76

n 0.20

n(i) 0.12

b(i) 0.52

b(i) 3.71

n(i) 0.23

n(i) 0.19

n(i) 0.28

b(i) 2.75

(rf/Pf)t, = 2.23 (14 cases). (rrlff)n = 0.42 (21 cases).

enough to interfere, and which do not interfere, are assumed to belong to different spin groups. The remaining resonances are given the assignments given them in particular, nonunique, multilevel fits. These assignments usually are based on the level width rather than on interference, or lack of it. The broad interfering negative energy levels of U235 and Puz39 have not been included in the table. Included in the table for each resonance is the ratio l?,/rf which has been used in Fig. 13. The energy IeveIs above 10.41 eV in U233 which were fit by multilevel formula by Harvey and Pattenden have not been included since the spin assign- ments and widths of these levels are probably much less reliable.

Figure 14 gives separately the distribution of l?f/rf for the narrow and for the broad group of levels given in Table XIII. The small level at 2.82 eV in U235 has been omitted, since its fission width has not been measured. The Pu23g cases have been cross-hatched for identification since the fission width distribution of Pu23g, at least for the broad group, appears to have an appreciably greater spread than the distribution for the other three nuclei. The ratios for levels of uncertain spin assignment in Table XIII have distributions for the broad and for the nar- row group similar to those of the ratios for the levels with more definite spin assignment and have been included in the distributions of Fig. 14.

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NEUTRON RESONANCE PARAMETERS 307

FIG. 14. The distributions of the fission widths of the selected broad and narrow group of resonances of Uzra, UB6, Pu239, and Pu*(l found in Table X11. The Puts9 cases have been cross-hatched.

A Maximum Likelihood Test has been applied to the fission width distributions shown in Fig. 14 to determine the number of degrees of freedom of the Chi- Squared Family which provides the best fit for each distribution. For the broad group v - 3 (all cases) and v - 16 (PUBIS excluded), while for the narrow group v - 6 (all cases) and v - 3 (excluding PIY~~). These numbers are somewhat larger than the values v - 3 (U233) and v - l+ (PUBIS) quoted by Vogt (46) from multilevel fitting; however, the distributions of fission widths for the two groups are not sufficiently well established to consider this a real disagreement.

Since these two fission width distributions do tend to indicate larger values for v than the multilevel fits, it is well to ask the question: What limitations do the multilevel fits impose on v? For example, could v be as large as 6? The con- census of private communications from Harvey, Moore, and Simpson, and Vogt is that a value as high as 6 for v is either unlikeIy or impossibIe for either spin state of any of the four fissile nuclei considered here.

There remains one additional approach which can, in principle, yield informa- tion concerning the number of degrees of freedom associated with the fission width distributions. Bohr (50) has suggested that the number of fission channels should be associated with the number of excited states energetically accessible to the compound nucleus at the top of the fission barrier. These excited states are expected to be the low-lying states of the compound nucleus, since at the top of the fission barrier much of the excitation energy is converted into energy of deformation. The energy above fission threshold for neutrons of zero neutron energy has been measured by Northrup, Stokes, and Boyer (59) for U233, U235, and Pu239 using the (d, p) reaction. These energies are 1.47, 0.60, and 1.61 MeV for U2s3, U235, and Pu23g, respectively. These energies can, in principle, lead to the number of open fission channels by examining the level schemes of U234, U236, and Pu*~O. However, this approach has at least two difficulties: (1) The level schemes

Page 40: Statistical analysis of neutron resonance parameters

308 GARRISON

for these nuclei are not sufficiently complete; (2) under this model for fission, the average fission width for the Pu 23g levels of spin 0+ should be much larger than the fission widths for the l+ levels, while experimentally it appears that the l+ levels have the larger average fission width. Some modification of Bohr’s proposal seems required.

From the evidence examined here, it can be concluded that the fission width distribution is broad, though not as broad as the neutron width distribution, that it has a general absence of very small fission widths, and that the distribution is probably not simple in form; it probably does not belong to the Chi-Squared Family of distributions. Similar conclusions have been reported previously.

VIII. THE EQUIVALENCE OF POSITIVE AND NEGATIVE ENERGY RESONANCES

It is of interest in the theory of nuclear reactions and in the more applied field of nuclear reactor theory to study the relation of the resonance parameters of the resonances at positive neutron energy and the negative energy resonances, which lie at excitations below the neutron binding energy, to the thermal capture cross section. Such a study will yield information concerning the reliability and general usefulness of the cross section and resonance parameter data. The positive and negative energy resonances are called equivalent if they contribute on the average equally to the thermal cross section (3). The contribution of the negative energy resonances to the thermal capture cross section must be deduced in- directly, since their parameters are not measured.

Following in some respects the work of Egelstaff (3), the distribution in R, the fraction that the positive energy resonances contribute to the thermal capture cross section, has been determined for all isotopes with two or more measured, lowest energy resonances and a known thermal capture cross section :

R = u+/~2200 01) where :

+ u ZZ 4*13 g (

0

(E,i - o.o;~~';i (ri/2000)2 ) + D:,*,(;::x;;::) + $1 (12)

and :

ri = rri + P2 rii (E = 0.0253 eV).

The subscript i refers to the ith positive energy resonance. E,,, is the resonance energy of the highest energy resonance used in Eq. (12). All other symbols have been identified earlier in the paper. Bars again denote average values.

The term following the summation in Eq. (12) is an estimate for the contribu- tion of the unmeasured higher energy resonances. The numerical coefficients

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NEUTRON RESONANCE PARALETERS 309

are chosen so that all resonance energies, Dabs , E, and E,,, must be in electron volts, all partial and total resonance widths must be in millielectron volts, and all cross sections in barns.

Figure 15 gives the distribution in R for 71 cases. In addition to the 68 cases found in Table I, R-values for W18*, Pu*~I, and Pu238 have been included using data from refs. 12 and 60-62. Sketched on the same figure are approximate distributions calculated using the following models: (1) Assuming the thermal capture cross section arises from two resonances of average strength, separated by an average level spacing, and with the zero of energy located at random be- tween them, PI(R). (2) Same as (1) except including two additional fixed energy, average strength resonances with resonance energies at E. = AN Doks , P2(R).

The effect of using the distribution of neutron widths in the calculation of P(R) is to lower the center of the distribution slightly. No analytical expression including this effect has been obtained. It has been assumed here that the effect of interference between resonances on R will be very small because of the many channels contributing to the capture process.

The distribution in R is expected to fall in the range 0 $ R i 1 and to be symmetrical about 0.5, if the negative and positive energy resonances are equiva- lent. On the average the positive and negative energy resonances will then each contribute one-half to the thermal capture cross section. For the histogram of Fig. 15 there are 36 cases with R < 0.5 and 35 cases with R > 0.5. The cases with R > 1.0 represent a “spilling over” of the distribution primarily because of experimental uncertainties in I’,O, pr , and u2200 . The single case at R = 1.5 in

P, (RJ -

PJRI---

71 CASES

FIG. 15. The distribution in the ratio R for nuclei with two or more measured low positive energy resonances. R is the ratio of the contribution of all the positive energy resonances

to the thermal capture cross section b+ , to the total thermal capture cross section, q2200 .

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310 GARRISON

Fig. 15 is that of Dy162 where a+ = 214 b and Q,K,O = 140 f 40 b.l” The most likely explanation of this case is that the experimentally measured value of I?7 for the first resonance is higher than the true value, as it appears to be compared with rr for resonances of neighboring nuclei. The shape of the approximate, analytical expressions for the distribution in R, which treat the positive and negative energy resonances as equivalent, are in agreement with experiment within the accuracy of the data and calculations. The equivalence of positive and negative energy resonances is indicated.

The equivalence of positive and negative energy resonances has a sound theoretical basis and has long been accepted. It might be well to point out par- ticularly for application to nuclear reactor design that this equivalence implies that one can calculate closely the energy dependence of partial and total neutron cross sections of nuclei in the low positive energy region, if the parameters of the most important single negative energy resonance (or an effective single negative energy resonance which reasonably duplicates the cross section contribution of all the negative energy resonances) and the positive energy resonances are known. The parameters of the most important or an effective single negative energy resonance can often be derived from a very limited knowledge of the partial and total cross sections at low positive neutron energies. It has been assumed in the calculations of this section that the total radiation width is relatively constant from resonance to resonance so that when the total radiation width for a par- ticular resonance is unmeasured, the average total radiation width can be used with reasonable reliability. Experiment is in accord with this assumption for most nuclei. Experimental measurements also indicate that there is a fairly smooth variation of average total radiation width as a function of mass number.

IX. DISCUSSION

In the preceding several sections, a number of distributions and correlation coefficients have been presented and a number of statistical tests have been discussed. These have been compared with pertinent theoretical predictions. It is well now to discuss the results of these earlier sections jointly in more detail while attempting to provide some insight into the current theories and the relation of these theories to the experimental measurements.

In the R-matrix theory of nuclear reactions (64) the partial width of a resonance rhe , which is proportional to the probability of decay of the compound nucleus into channel c at energy Eh , is given by:

rxc = 2P& (13)

where P, is a penetrability factor for channel c and where the reduced width

10 The thermal cross section for the Dy isotopes were obtained from measurements made

by L. House and R. Frost (63).

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NEUTRON RESONANCE PARAMETERS 311

amplitude factor is given by the 3A - 1 dimensional surface integral:

yxc = (fi2/2M,ay2 / +,*x,.was. (14)

Here fi is Planck’s constant divided by 27r, M, is the reduced mass for channel c, a, is the channel radius, & is the surface wave function, and XuM is the in- ternal wave function at energy Ex , all in the usual notation. At nuclear excita- tions above the neutron binding energy the internal wave function is so complex that the yxC for the many resonances observed at these energies are expected to have a gaussian distribution about zero mean. The original argument Ieading to this distribution is as follows (1): The integral can be divided into many small cells of dimension 3A - 1, each cell being expected to provide for each resonance a contribution which is equally likely to be plus or minus. The central-limit theorem of statistics (65) predicts that the sum of the contributions of the many cells to obtain the yxC leads to a normal distribution for these yxC . A normal distribution about zero mean for the 7he leads to a chi-squared distribution with one degree of freedom for the phC , the Porter-Thomas distribution.

More recently a theory of level spacings for complex spectra, such as are observed with sufficient excitation in many electron atoms and at high nuclear excitation in all but the lightest nuclei, has been initiated by Wigner (66) and developed further by Wigner, Porter, Rosenzweig, Dyson, Mehta, and others.ll The model that appears to apply is that the Hamiltonian matrix of a complex many body system is a typical member of an ensemble of Hamiltonians. Time reversal invariance is expected to hold for these systems so that the Hamiltonian is a real symmetric matrix. The ensemble of Hamiltonians is characterized by the joint distribution of its matrix elements. Porter and Rosenzweig (6) have shown that the two assumptions of statistical independence of the Hamiltonian matrix elements and representation independence (rotation invariance of the matrix element distributions in the N-dimensional vector space associated with the N X N dimensional Hamiltonian) are sufficient to determine that the matrix elements have a normal distribution (about zero mean for the off-diagonial elements). This ensemble with elements distributed normally has been called the gaussian orthogonal ensemble. There is currently no justification for the assumption of statistical independence.

Since there are no matrix elements connecting states with different quantum numbers, levels with connecting matrix elements are treated by separate en- sembles and are statistically independent of all other levels. For nuclei, apparently J and r (parity) are the only good quantum numbers other than energy. For atoms L and S, the total orbital angular momentum and total spin are sometimes

11 A fairly complete set of references is contained in the papers by Porter and Rosenzweig

(6) and Dyson (7).

Page 44: Statistical analysis of neutron resonance parameters

312 GARRISON

good quantum numbers. Levels must be ordered in groups having the same quantum numbers to obtain the results predicted by these gaussian orthogonal ensembles (67).

The gaussian orthogonal ensemble, obtained from the two basic assumptions of Porter and Rosenzweig, leads to the joint distribution of eigenvalues (the Wishart distribution) and the distribution of eigenvector components. The distribution of eigenvector components is just the distribution obtained from N randomly oriented orthonormal vectors. This distribution leads, in the limit of large N, to the normal distribution about zero mean for the reduced width amplitude yxO .

The distribution of nearest neighbor and next-nearest neighbor level spacings has been obtained analytically from the joint distribution of eigenvalues for N = 2, 3 (matrix dimension), and for the nearest neighbor spacings in the limit of large N(6, 68-70). Mathematical difficulties in determining other near neighbor spacing distributions have led to Monte Carlo studies of the spacing distributions using computers to diagonalize matrices whose elements have been drawn from distributions at random.

It is clear from the preceding paragraphs that the form of the spacing distri- butions and the distribution of partial level widths are intimately connected, though the widths and spacings turn out to be statistically independent. We have seen support for this theoretical result of statistical independence in Section VI where the (I’,,O:S) correlation in Table XII is consistent with zero.

In Section III we have seen that the lack of sufficient statistical precision and the presence of missing and extra levels makes it impossible to make a good determination of distribution of IA0 for the case of the (single channel) reduced neutron widths, although the distribution is quite consistent with a chi-squared distribution with one degree of freedom. The other case where comparison is possible, the distribution of partial radiation widths, has not been treated in this paper. This distribution, discussed in the introduction, is consistent with a chi-squared distribution with one or two degrees of freedom. However, the data in this case are much less reliable and, further, it is not clear whether the same theory applies (64,SS).

We turn now to the level spacing distribution of even-even nuclei. The results obtained here are generally in agreement with distributions obtained from the gaussian orthogonal ensembles. Once again, however, the experimental data do not allow a good determination of the distributions. For example, it is not possible to differentiate between different gaussian ensembles such as the (20 X 20) or (10 X 10). It does appear that the moments of the spacing distributions of the inner spacings, (20 X 20) (-8) or (10 X 10) (-4), provide a slightly better fit to the experimental data than, for example, the (20 X 20) matrix results where the mean outside spacing is rather different from the mean inner spacings. This

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NEUTRON RESONANCE PARAMETERS 313

has not been investigated in detail. It should also be pointed out that the gaussian ensemble distributions are not essentially different from distributions obtained by diagonalizing matrices loaded with plus or minus ones at random, provided the probability of plus ones and minus ones is approximately equally likely. Porter (68) has shown that an energy gap in the eigenvalue distribution appears when the symmetry of the distribution of matrix elements about zero mean is broken. However, the onset of this energy gap is sufficiently gradual that a probability ratio of minus ones to plus ones of 45 : 55 in the random matrices, for example, leads to changes in the spacing distribution moments which would be undetectable if they existed in the experimental distribution moments.

The approach of Dyson (7) in the statistical study of spectra has been some- what different and more general. His work has led to three classes of circular en- sembles, the orthogonal, the unitary and the sympleetie. The circular orthogonal ensemble is equivalent to the gaussian ensemble in the spacing distributions it yields and is in accord with experiment.

In the case of the spacing distributions of the odd-A nuclei, P*‘, Tam, and Aulg7, the random matrix results have been obtained by overlapping the eigen- values of pairs of random matrices before obtaining the spacing distributions. The positions of the eigenvalues of the two overlapped matrices are uncorrelated. The spacing distributions of overlapped sets of eigenvalues have been obtained for mean spacing ratios for the two matrices of 3,1.4, and 1. As for the even-even nuclei, the experimental spacing distribution is not sufficiently well determined to distinguish between different classes or equal f 1 ensembles. All are in reason- able agreement with the experimental data. In addition, the distributions for the three spacing ratios are not sufficiently different to establish a definite preference.

We have seen that the neutron width distribution and level spacing distribu- tions provide some support for the statistical theory of spectra. We have also seen that difficulties with the data and the inherent statistical uncertainties indicate that it will probably be some time before the distributions are well determined: It is easy to differentiate between the exponential and the Porter-Thomas dis- tribution for the neutron widths, and between the unitary, the symplectic, and the orthogonal ensembles for the spacing distributions, but the more refined differentiations are impossible at this time.

One additional point should be made: The agreement between the spacing distributions of the gaussian orthogonal ensemble and experiment is less satis- factory without making the corrections for missing and extra levels, and these corrections are based on the validity of the Porter-Thomas distribution which itself is intimately connected with the Hamiltonian ensemble. It has only been shown then that the gaussian orthogonal ensemble leads to results which are consistent with experiment. The validity of the corrections gains some support from the general observation that the uncorrected moments of the experimental

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314 GARRISON

spacing distributions (and neutron width distributions) approach the moments of the gaussian orthogonal ensemble as the quality of the data improves, and the direction of the approach is the direction indicated by the moment corrections.

We turn now to a summary of the remaining parts of this paper. Krieger and Porter (10) have obtained a multivariate distribution for the reduced width amplitudes 7hc based only on the assumptions of representation independence and statistical independence of the -ykc from different levels. For this distribution only a positive or zero correlation is possible between the partial level widths I’hc of the same level. Experiment is not in disagreement with these results: the (rr:grnO), (r,:r,) and (rr:rno) correlations of Table XII are consistent with zero and the partial fission width correlation must be positive. It has been sug- gested (1) that the partial radiation widths for transitions to the low-lying rota- tional levels of the heavier nuclei should show correlation because of the similar character of these levels. No such correlation has been found (Table XII). On the basis of the correlation results obtained here, certain approaches in the theory of average cross sections can be excluded. As an example, the theory of Newton (71) assumes strong correlation between the yAc which apparently are not present.

It has been shown in Section IV that the mean level spacing and strength func- tions for each of the two classes of resonances accessible in the low energy neutron spectroscopy of other than even-even nuclei are consistent with the mean level spacing formula of Bethe (Eq. (7)) f or small values of the constant y and a weak neutron-spin-nuclear-spin force.

In Section VII, the fission width distribution has been examined in some detail, primarily to determine the present state of our knowledge concerning this distribution. Essentially no new conclusions have been reached.

In Section VIII the equivalence of positive and negative energy resonances was demonstrated. This demonstration also provides a simple minded test of the consistency of the neutron cross section measurements and nuclear reaction theory. In fact, the whole analysis of this paper provides an excellent measure of some aspects of the consistency and reliability of the neutron resonance param- eter measurements.

APPENDIX I

THE MAXIMUM LIKELIHOOD METHOD APPLIED TO THE TRUNCATED CHI-SQUARED FAMILY OF DISTRIBUTIONS

The Chi-Squared Frequency Function for Y degrees of freedom is given by:

P(z, p> = g (p>eP (0 5 z 5 00) (15)

where Y = 2p. The truncated Chi-Squared Frequency Function is given by:

Jqz, p, a) = Fb, a> (PW r(P)

(p)e-‘“. (a 5 x S w) (16)

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KEUTRON RESONANCE PARAMETERS 315

The normalizat,ion factor F(p, a) is determined by the condition:

s

m P(x, p, a) dx = 1.

L11 (17)

The AIaximum Likelihood method determines t.he value of Y for t.his truncated Chi-Squared Family of distributions which provides the best fit to a sample of N quantities zi(l 5 i 5 N) drawn from the populahion of ratios with each 5% 2 LY. The Likelihood Function is defined as :

UP, a> = 6 P(G, P, a>. (18)

The Likelihood Function is a maximum for the value of v providing the best fit, t.hus v is determined by the equat.ion:

(19)

or, more conveniently :

g [log UP, a)1 = 0. (20)

Using the expressions for L(p, (Y) and P(.r, p, CI) in Eq. (20):

- 1L(P) + (log P> + 1 + oog a> - xi = 0 (21)

where #(p) = (d/ap)[log I’(p)] (di-gamma function). Experimentally, N is sufficiently large for the data used here that, it, is con-

venient to group the zi into m histogram intervals, with effective value xj for each histogram interval and with Nj values of xi in the jt.h interval, Mroducing the above changes and rearranging Eq. (21):

1 aF -- F ap

- J/(P) + (log P> + 1 1 = - &El Ni[(log rj) - *rj]. (22)

The experimental sample of N ratios yields a number for t’he right hand side of the above equation; the equation is then solved for V.

The uncertainty in v is obtained, according to ref. 20, from the equat,ion:

and where (Var v) = 4 Var p. For the truncat’ed Chi-Squared Family of diskibutions:

1 a*(l0g F) (Var p>-’ = Nfi’( p) - - - ______

P w (24)

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316 GARRISON

where #l(p) = 8 In J?(p)/&* (tri-gamma function). Use has been made of the tables in ref. 72 in the evaluation of Eqs. (22) and (24) for determining Y and its variance. Since cx is generally a small quantity, approximate equations have been developed for F(p, a) and its logarithmic derivatives as follows:

I(p, a) = s,’ P(x, Y) dx = +I 6‘ (px)p-lpe-pzdx

F(P,~) = 1

1 - ~(P,(v)’

For (Y << 1 it is useful bo use the expansion:

(25)

(26)

(27)

Some manipulation yields the following two equations for the derivatives of F(p, a), when only the first two terms are retained in the expansion for I(p, a) of Eq. (27) for the first derivative, and only the first term for the second deriva- tive :

1 aF --= F ap

WA - +) - KPd’+‘l(P + 1)lM + l/(P + 1) - l/4/3)1 C2@ F(P) - D + CP~>~+~/(P + 1)

#(log F) B W(P) - DA

w

= D[A - dp)l A +

r--D 1 A - #(PI - r--D (29) B = ; + ; - $‘(P)]

and D = (l/p)(cup)‘. For p = 0.5, I’(p) = 1.77245 and #(p) = -1.96351, #l(p) = 4.93480, and log p = -0.69315.

The above procedure assumes that the mean value of x for each class is t,hat value which provides the best fit to the Chi-Squared Family. This may not be so; a systematic error may be present in the determination of the values of 5. The above analyses can be readily extended to a two parameter fit by substituting px for J: in Eq. (3), and determining the values of both p and p which maximize the Likelihood Function. In a manner similar to that used in the development of the equation used to determine the value of p providing the best fit, an equa- tion for optimum /3 can be obtained:

(30)

where Z = (l/N) X7& N+zzi . F or small values of (Y, the following expansion ob- tained in a manner similar to Eq. (28) is useful:

(31)

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NEUTRON RESONANCE PARAMETERS 317

For CY 5 0.1 and p N 1, p N 0.5, the first terms in the expansion are adequate:

1aF 1 --=- FW P

nYP)/bdYl - .

The simplified equation for p is then:

1

{

1

p = ii I+ [plyp)/(pa)p] - 1 1

(32)

(33)

where one value of ,6 entering in the equation has been set equal to one. For F = 1 the equation reduces t,o :

p = l/Z (F = 1) (31)

which shows t,hat the optimum p simply renormalizes the values of .I’ so that, their mean value will be the same as the mean of the Chi-Squared Family (.c = 1). This renormalization procedure has been performed where necessary for the four classes of nuclei whose neutron width distributions have been considered, though generally the adjustment required is small relative to the statistmica error.

Equation (22) for t’he determination of v now becomes:

[ k $ - #(p) + 1 1 = - & 2 NJln(pzj) - Prjl. (35)

APPENDIX II

THE ANALYSIS OF DISTRIBUTIONS BY THE USE OF MOMENTS-CORRECTIONS AND CORRELATIONS

A. INTRODUCTION

The discussion here applies to distributions obtained from the ratio of reduced neutron width to average reduced neutron width or from the ratio of level spacing to average level spacing. This ratio, in each case, for the ith resonance or for the ith spacing is called ICY . Properties of the distributions, PI, , of the quantities, Yki = xf=o ~i+j , which are the sums of k + 1 successive values of .r are obtained from a finite sample of N cases and used as estimators of the properties of the tot.al population.

B. ESTIMATORS OF THE RiOMENTS OF THE DISTRIBUTIONS P,

Assume that a sample of N quant’ities, ~l’i , is drawn from a population in order to determine the moments of the distributions, Pk , associated with the popula- tion. The mean value of the Yki or first moment M1k for t*he total population is assumed to be I: + 1. The nth sample moment, for the distribution, Pk , is given by:

(36)

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318 GARRISON

These moments provide an unbiased estimate of the population moments; how- ever, the accuracy of the estimate may be improved by introducing the new quantities M”,” :

k

Mk, = (j&k;+ 1)“’ (37)

To demonstrate the improved accuracy, let Aki = YK - (k + 1) be the deviation of each quantity Yki from the populat,ion first moment (n = 1) of the distribution, Pk , and expand Eq. (37) in terms of these A~ki :

N-k

(N - k;;k + 1) i=l c &i

n(n - 1) + 2(N - k)(k + 1)2 i=l

y & + . . . ]

N-k

(iv - B:ik + 1) i=l c &i

n(n - 1) + 2(N - k)2(k + 1)” cg Akiy + -1

(38)

The second terms in the numerator and denominator are identical. These terms provide one of the contributions to the error of the numerator and denominat,or so that the error in the determination of Mk, is reduced over that of Mnk for moments which do not differ greatly from (k + 1)“. A physical explanat,ion of the above is the following: When the quantities Yki of a sample are larger than their population mean, then the mean of their higher powers will tend to be larger also ; the errors are correlated.

By the procedure of Eq. (37) and Eq. (51) below, the first moments of each distribution Pk have integral values corresponding to the theory. Comparisons between theory and experiment are made with the higher mon1ent.s.

The quantities M”, form a biased estimate of the population moments for finite samples. The correction for this bias may be obtained as follows:

Draw I samples of N quantities, xi , each from the total population, each sample denoted by the running index CY. Let the deviation of the M:, moment from the population mean k + 1 for t’he sample cx be

8, = M:, - (k + 1). (39)

Then k k

Mk” -

na - (M;$; 1))” = (1 + v:;i+ 1))” (40)

and

Mk,, = M:,z(l + V& + 1)“. (41)

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NEUTRON RESONANCE PARAMETERS 319

Averaging Eq. (41) over the I samples and let’ting 1 -+ cc, (populat’ion averaging for a finite sample of N cases each) :

ilrS,,= x + bilk” R(I * + h&L [

$-+l&q + . . . , (42)

where each bar denotes t’he averaging:

The fluctuations in illk”a arise from fluctuations in the spread of the Yi and are expected to have little correlation with the fluctuations from t,he mean, V, ,

so that the averages of hfy, and the powers of t.he V, can be taken independently. The mean 0, = 0. For reasonable sample size V, is small so that higher order terms may be neglect’ed. One then obtains to second order:

[

- nf:,E: 1+ n(n - 1)va2

2(lc + 1)2 1 . (44)

The t,erm in brackets is the correction t,erm t,o be applied to t’he hdkna, to obtain the unbiased est.imate, hI”,‘, present.ed in the body of t,he paper. It remains to - relate t,he Vn? t,o the standard deviat,ion of the distribution PO and the correlation between neighboring Z; . For N >> k, M:, can be approximated by:

(N >> 1~)

since each .rirr enters (k + 1) times except for edge effects. Let &, = .ria - 1, be the deviation of each :ria from population mean for distribution PO , then:

M:, w (k +1> +

5 = [hf!, - (k + l)]’ w

Sow for (; + j) 5 (N - 1;)

(4’3)

(47)

(48)

Putting Eq. (48) in terms of the variance of the population distribution I-‘, , rr?, and the correlation coefficients, ck , giving the correlation bet’ween each .ri and ~i+b+l we obtain:

2 5%

N-k-2

(N - k)~,’ + ACT,’ 2 Cj(N - 1, - 1 - j). (49)

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320

Therefore

GARRISON

and

M”,h- = ML I[ n(n - 1)

(ML/k + 1)” ’ + 2(N - k) uz

r 9 N-k-2 \l

(50)

(51)

. lf t ek go Cj(N - k - 1 - ql.

2 m 0.3, 0.5 and 2.0 for the level spacing distributions of even-even nuclei, yir the level spacing distributions ot nuclei with two level spins, and for the reduced neutron width distributions, respectively. The correlation coefficients Ck have been given in the body of the paper for the first few values of Ck .

C.RELATIONS BETWEEN R/IOMENTS OF DIFFERENT DISTRIBUTIONS PI, AND

CORRELATION

Moments of the distribution Pk can be related to the moments of t.he dis- tribution Pk+ and the distribution PO in the following fashion:

Mnk = j&ii)

Yi = Xi + k Xi-+-j = Xi + Vi j=l

M,” = &k[~~Xi..+n~X71Vi+ *.-I

-__ n(n- l>---- __ M,k=M,o+n~lMZ-‘+ 2 MO,-2 M$l + . . .

+ nCxn-lV ~~~-1 gV + n(n 2- ‘) CZn--2u2 ozn--l ~~2 $

where the quantities C are correlat,ion coefficienk:

uzn-l = 2/(,n-1 - -)2

- uv = d(, - 012, etc.

. . .

(52)

(53)

(54)

(55)

(56)

(57)

(33)

Page 53: Statistical analysis of neutron resonance parameters

NEUTRON RESOXANCE PARAMETERS 321

In the special case of second moments, by successive application of Eq. (5.3) from k = 1 up, one obtains for t’he second moment’ of distribution PO :

k-l T jlJ2k = (Ii + l)J1* + k(k + 1) + 20x2 c C*(k - 9

i=o

where, as before, t,he Ci give the correlation between two individual r-values which have i r-values bet,ween them. Similar equations can be developed for higher moments. For the case of zero correlation between all 5; and their powers, the moments of all higher distributions can be obt#ained from the moments of the lowest order distribution, PO .

D. ERRORS

The stat.istical error in the second moment, of t.he kt,h dist,ribut.ion is well represented by (65)

cr$ - (/.12k)2

Nk (60)

where pnk is the nth central moment of the kth distribution, and Nk is the sample size for the kth distribution. The values of the quantities l/pak - (p2k)2 are given in Table XIV for the central moments of the Chi-Squared Family of distribu- tions and for the nonoverlapped and overlapped (20 X 20) ( - 8) matrix spacings distributions.

Originally, the errors in the moments of the neutron width and level spacing distributions were obtained experimentally from the differences between the moments obtained for the first, half and the second half of the data. Later these original error estimates served only to indicate the relation between the errors in the second moments and t,hose in the third and fourth moments. The errors in the second moments given in this paper have been obt.ained by combining t.he

TABLE XI\-

TABI:LATION OF THE QUANTITIES dph' - (+z~)~

k Chi-squared (20 X 20) (-8)

Not overlapped Overlapped (1.4: l)-

0 8 0.5 0.9 1 11 0.7 1.2

2 15 0.9 1.4 3 18 1.0 1.6 4 21 1.1 1.8 5 24 1.2 1.9 6 27 1.3 2.0

Page 54: Statistical analysis of neutron resonance parameters

322 GARRISON

statistical error of Eq. (60) with the error in the corrections to the moments. The errors in the third and fourth moments were altered from the original errors in proportion to the changes required for the second moments. The errors given for the moments of the neutron width distribut,ions are the statistical errors alone.

The statistical errors for the correlation coefficients have been calculated from the expression (65) :

where Nk is again the size of the sample used to determine the Ck . This was done irrespective of whether the C’k were determined directly or from t,he moment’s.

(Note: Subsequent to t,he submission of t’his paper for publication, a Monte- Carlo calculation (which was also used to obtain the dashed histogram of Fig. 7) has been performed which indicates that Eq. (60) is pessimistic in its estimate of the standard deviation of the second moment,s. The standard deviation of the second, third, and fourth moments is well represented by:

M;, - (MnkY Nk

(instead of Eq. (60)), w h ere the constant C is approximately 0.35, 0.55, and 0.75 for second, third, and fourth moments. Thus, the statistical part of the errors given in the moments tables are large).

E. DETERMINATION OF THE MISSING LEVEL CORRECTION FOR THE SECOND MOMENTS OF THE NEUTRON WIDTH DISTRIBUTION

Let MIt and Mzt be the sample first and second moments (for k = 0) for the case where no levels are missing, obtained using Nt neutron widths. Let Ml and Mz be the corresponding moments for the case where levels are missing, obtained from N(< N2) neutron widths. Equation (5) for the statistically more precise AI:, with no missing levels can be expressed in t,erms of the corresponding M, with levels missing as follows :

M1, = [(l/Nt) Et& dl [(l/Nt) c;2’4, xi]’

Rearranging terms :

M, = (N/Nt)[(l/N) c:=‘=l xi2 + (l/N) c%+l %I 2t (N/N$[(l/N) cfzl xi + (l/N) ~?I+N d

Since the values of xi for missing levels are very small, they can be neglected:

[(l/N) C!‘=I xi? [(l/N) cf’zl xi]”

Page 55: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAIIETERS 323

Since the bias correction fact,or [l + (az2/Nt)] e [l + (az2/N)], M:, can be ap- proximat)ed by :

APPENDIX III

RANDOM MATRIX MOMENTS CALCULATIONS AND CORRECTIONS FOR MISSED AND EXTRA LEVELS

A. ,~/IOI\IENTS CORRECTIONS

Table XV lists approximate corrections to be added to the moments MZk, Mak, and JfJk of the near neighbor spacing distributions of even-even nuclei for the three special cases where: (1) levels are not observed closer than 0.15 D01?B , (2) 3 % of the levels are missed at random (too weak to be observed), and (3) 7 % extra levels are present (but not closer to other levels than 0.15 Do, J. The correc- tions applied to the spacing distribution moments of Thz3? and W’38 can be ob- tained from this table assuming that the corrections vary linearly with the IWI~-

ber of missed and extra levels and that each type of correction is independent of the other. The corrections for small spacings were taken to be the 0.15 Dabs values in Table XV for both Th232 and U238 since the correction is small and most of the IeveIs removed because of smal1 spacing are the (weak) extra levels. It is impossible to remove close levels using a different smallest observable spacing for the original levels than for the extra levels with the machine codes developed for this work, though in principle the necessary information for corrections using two different values of smallest spacing can be extracted from a series of code calculations.

Table XVI lists the approximate corrections to be added to the moments

TABLE XC

MOMENT CORRECTIONS FOR EYEN-EVEN NUCLEI

k AM*k

Corrections

M,fzk AM4k

0.15 hs M7zed 7v0 Extra 0.15 Dabs 3% Missed 7% Extra 0.15 Dabs 3%

Missed 77, Extra

+0.03 -0.01 -0.03 +0.1 -0.03 -0.2 +0.4 -0.1 -0.7

$0.03 -0.02 -0.09 +0.2 -0.1 -0.7 +1.5 -0.7 -4

+0.03 -0.04 -0.15 +0.3 -0.4 -1.4 f3 -2.6 -11

+0.02 -0.04 -0.2 f0.3 -0.6 -2.6 +3 -6 -24

+0.02 -0.05 -0.3 +0.2 -0.8 -4 +4 -9 -45

$0.01 -0.06 -0.3 +0.1 -1.2 -6 +1 -15 -76

-0.02 -0.07 -0.4 -0.5 -1.4 -9 -5 -19 -130

Page 56: Statistical analysis of neutron resonance parameters

324 GARRISON

MZ’, Mzk, and M,” of the near neighbor spacing distributions of odd-A and odd- odd nuclei for the three special cases where: (1) levels are not observed closer than 0.1 Dabs , (2) 8 % of the levels are missed at random (too weak to be ob- served), and (3) 10 % extra levels are present (but not closer than 0.1 D,,t,J. The corrections for the spacing distribution moments of P, TalQ1, and Au197 com- bined can be obtained from this table.

The corrections in Tables XV and XVI are primarily from the 20 X 20 random matrix calculations; however, the corrections from the 20 X 20 random matrix calculations have been increased somewhat for the higher values of k to adjust for the effect of the finite size of the matrices. Further, the corrections for levels

TABLE XVI MOMENT CORRECTIONS FOR ODD-A OR ODD-ODD NUCLEI

Corrections

k AM2 AM+ aM4k

0.1 Dabs % 10% 0.1 Dabs M;&g 10% 0.1 Dabs % 10% Missing Extra Extra Mlssmg Extra

0 +0.08 -0.01 -0.03 +0.3 -0.07 -0.2 +1 -0.3 -1

1 +0.09 -0.05 -0.13 +0.6 -0.4 -1.0 $3 -2 -6 2 +0.11 -0.06 -0.26 +l.O -0.7 -2.7 $7 -6 -20 3 $0.12 -0.08 -0.29 $1.4 -1.2 -3.8 +14 -11 -35 4 +0.13 -0.10 -0.38 +1.8 -1.7 -6.3 +20 -20 -70 5 $0.14 -0.14 -0.43 $2.2 -2.8 -8.1 $30 -37 -103 6 +0.15 -0.17 -0.46 +2.6 -3.8 -9.6 +42 -65 - 140

TABLE XVII SECOND MOMENTS OF ADDITIONAL RANDOM MATRIX DISTRIBUTIONS

Second moments kit”

K 2x2 10 x 10

Gaussian Gaussian

X 20 x 20 20 20

Overlapped (20 20)(-8) x

Gaussian Random f 1 Spacing ratio Spacing ratio

1.27 1.32 1.35 1.32 1.52 1.42 4.48 4.57 4.47 4.75 4.68 9.58 9.73 9.58 9.93 9.84

16.7 16.9 16.6 17.1 17.0 25.7 26.0 25.7 26.2 26.0 36.8 37.1 36.7 37.3 37.1 49.8 50.1 49.8 50.4 50.2 64.9 65.2 64.8 65.5 65.3 82.1 82.2 81.8 82.6 82.3

Page 57: Statistical analysis of neutron resonance parameters

NEUTRON RESONANCE PARAMETERS 325

missed at random were found to depend apparently OR the magnitude of the spacing distribution moments. Since the moments obtained from the 20 X 20 matrices with no removal of edge eigenvalues are large, these corrections have also generally been modified in line with the corrections anticipated from other random matrix calculations and an approximate statistical formula derived to calculate these corrections.

The corrections presented here are adequate considering the statistical uncer- tainties in the numbers of missing and extra levels. With improved experimental accuracy, these corrections should be redetermined.

B. ADDITIONAL RANDOIVI ~\IATHIX ~IOMENTS CALCULATIONS

Because there may be some interest in knowing how sensitive the moments of the spacing distributions are to the type of matrix, a number of second moments of spacing distributions obtained by the random matrix diagonalization code of Porter (40) are included here. Table XVII gives these results. RECEIVED: January 22, 1964

ACKNOWLEDGMENTS

Special appreciation is due Dr. L. Nordheim for encouraging the initiation of this project, and Dr. C. E. Porter for contributing many ideas to all parts of this work. Material aid in

the form of random matrix diagonalization codes, random matrix eigenvalues stored on

magnetic tape, and eigenvalue spacing distribution moments have also been generously provided for use in this work by Dr. C. E. Porter. The hospitality at the Sigma Center of

Brookhaven National Laboratory and help of the staff of Sigma Center during my year’s leave from San Diego is gratefully acknowledged.

Virginia Pr’ather programmed the sorting code used early in this work. Programs for

calculating distribution moments, sorting, and calculating correlation coefficients were

developed by Frances Pope, who also performed many helpful hand calculations. The cor- rections for missing and extra levels using the taped output of diagonalized random matrices

were determined using codes programmed by Lourdes Zavitsas who, in addition, pro- grammed the overlapping of pairs of matrices and handled the diagonalization of the ran-

dom matrices by the computer.

The section on the fission width distribution has profited from information contained in let’ters from M. Moore, 0. D. Simpson, E. Vogt, and J. A. Harvey.

Finally, appreciation is due all the experimenters providing data for this work, especially the group at Columbia University who generously provided the superior data on Th232 be-

fore these data were presented in the published literature. The early work leading to this paper was performed at General Atomic/General

Dynamics.

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