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The variation of α of 92U235 with energy in the intermediate-energy range
Nuclear Energy
I. 1957.Vol. 5.
pp.
16 to 32.
Pergamon
Press, Ltd., London
THE VARIATION OF a OF 92U235WITH ENERGY IN THE INTERMEDIATE-ENERGY RANGE* SOPHIE OLEKSA Brookhaven National Laboratory (Received 2 July 1956; in revised form 25 September 1956) Abstract-Recent experiments have made possible a new evaluation of resonance parameters for some of the low-energy levels of g2Ua95. The data permit a numerical calculation of the variation of Q of ,,Uass with energy in the range lOO-10,000 eV in accordance with a method proposed by WIGNER. We examined the effects of fitting the experimental data with several different distributions. The results are: 1. The variation of a (Uza5) with energy from 100 eV to 10,000 eV is a slow one, no matter which of several distributions is assumed to hold for the fission and neutron widths. 2. The values of a obtained by means of WIGNER’Stheory are in good agreement with results of the KAPL integral experiments. 3. The calculated values of 5&f) agree reasonably well with the experimental values. 4. The theory can be applied with confidence to fissionable materials other than Usa for which comparable information about low-energy resonances is available. INTRODUCTION
of the radiative capture cross-section of a fissionable material to the fission cross-section, a, and the variation of a with neutron energy are among the most important properties of a reactor. PALEVSKY et al. (1956) showed that for U235, CIis practically constant in the energy range 0.006 eV to 0.2 eV. The measurement of 0: in the resonance region is a more difficult problem, because the partial reaction widths can vary from resonance to resonance, and the partial widths are hard to measure. The recent developments in the crystal spectrometer and the fast chopper (SAILOR, 1955; SEIDL et al., 1954), however, have made possible the location and resolution of 47 resonances in U 235between 0.3 eV and 35 eV. The values of a obttined in the region vary widely and their uncertainties are large. At energies greater than a few hundred electron volts, “integral experiments” have been done in which average or “cut-off” values of a have been obtained for Pu23g and U235(KANNE et al., 1955). The value of a corresponds to an average over the energy range above a certain cut-off energy. These values have proved to be useful, but their precise meaning is far from clear. WIGNER (1949) showed that it is possible to calculate the value of ccas a function of energy in the range 100 eV to 10,000 eV, provided that the total and fission widths of a reasonable number of resonance levels are known at neutron energies less than 100 eV. Information about cross-sections at intermediate energies can thus be obtained from the properties of resonances at relatively low energies. THE ratio
* This is a modification of a Brookhaven National Laboratory report (December 1954) which appeared in the June 1955 issue of Reactor Science nnd Technology.
USAEC. 16
Research carried out under the auspices of the
The variation of a of ,.JJes5 with energy in the intermediate-energy DISCUSSION
OF
THE
EXPERIMENTAL
range
17
DATA
The experimental data (EGELSTAFF and HUGHES,1956) on which the present study is based are listed in Table 1. The capture width, I’,,, varies somewhat from resonance to resonance, but the variation is not large and I?,, is assumed to be constant. The value given, 0.030 f 0.006 eV, is an average of the capture widths obtained for the TABLE I.-FISSION AND NEUTRON
WIDTHS OF 9J- ,235
-
E. WI
i L
0.29 1.142 2.04 2.82 3.14 3.61 4.84 5.45 5.83 6.12 6.40 7.10 8.82 9.32 9.83 10.21 1064 11.11 11.72 12.4 13.4 13.8 14.11 14.6 15.5 16.1 16.8 18.1 18.7 19.3
rt 0.005 & 0.010 i 0.03 zt 0.05 f 0.02 i 0.02 & 0.02 i 0.10 * 0.10 zk 0.10 f 0.05 & 0.05 f 0.07 & 0.1 f 0.1 f 0.1 * 0.1 i O*l & 0.1 f 0.1 f 0.1 & 0.1 * 0.1 i 0.2 f 0.2 f 0.2 f 0.2 * 0.2 f 0.2 * 0.2
99 120 13 70 135 66 17
* f * f f 5 &
28 * 26% 70*
0.0031 0.0154 0.0066 0.0026 0.028 0.050 0.052 0.022 0.016 0.027 0.30 0.11 1.2 0.11 0.025 0.065 0.033 0.045 0.59 1.2 0.08 0.15 0.20 0.17 0.27 0.29 0.29 0.46 0.20 3.2
5 15 3 45 25 10 10
12 18 16
42 + 18 43 f21
90 + 26
f OWO2 f 0.0008 rt 0.0004 i 0.0007 i 0.002 f 0.003 i 0.005 * 0.004 i 0.003 f 0.004 f 0.03 * 0.02 & 0.2 & 0.02 f 0@04 f 0.008 f 0.007 & 0.007 & 0.06 f 0.2 rfi 0.02 * 0.03 * 0.03 5 0.02 f 0.03 =t 0.04 f 0.04 & 0.07 f 0.06 * 0.3
I
0.0070 0.0144 0.0046 0.0015 0.016 0.026 0.024 0039 0.007 0.011 0.12 0.041 040 0.036 0.008 0.020 0.010 0.014 0.17 0.34 0.020 0.040 0.053 0.044 0.069 0.07 0.07 0.11 0.050 0.73
f o+lOO3 & 0.0007 f 0.0003 f 0.0004 i 0~001 f 0.002 f 0.003 f 0.002 & O+lOl f 0.002 f 0.01 i 0.006 If 0.06 f 0007 i O+lOl * 0.002 f 0.002 * 0002 & 0.02 * 0.05 f 0.006 * 0.008 f 0.008 f oGO4 f 0.008 f 0.01 xk 0.01 i 0.02 f 0.015 * 0.07
I
I’y=30+6mV. g = l/2 for all practical purposes. r,,O = r,(~/&)“* where E = 1 eV. Observed D = 065 eV.
first few resonances. The fission width, I’,, varies from resonance to resonance with a factor of about 10 from the smallest to the largest width. The neutron widths, I?,, fluctuate much more strongly; the largest value is several hundred times as large as the smallest value. The errors assigned to these widths are often as large as 60-70 % of the widths themselves. Because r, varies very nearly directly as the square root of the energy, the reduced neutron widths, I’,O, are used in the analysis of the data. rlzO = l?n(E/E)1/2 with E = 1 eV. The average neutron width, r,, is then 2
SOPHIE OLEKSA
18
equal to the product of the average reduced neutron width, r,O, and the square root of the energy. The calculation of ccinvolves the differential spectra produced by the fluctuations of the widths, i.e. the probability that FE lies between I’, and I’, + dr, where P, is either I?, or F,. The widths available, particularly the number of fission widths, are not enough to allow an adequate fitting of the differential spectrum. Consequently, it is necessary to fit the less sensitive integral spectrum in which the ordinate is the probability that I?, is greater than, or equal to, a given value. Unfortunately, the integral spectrum is often not sensitive enough to permit clear distinctions to be made between the effects of different distributions. Other difficulties are the experimental limitations which can distort the distribution, the neutron width distribution in TABLE 2.-AVERAGE
RESONANCE PARAMETERS FOR THE VARIOUS DISTRIBUTIONS
Distribution
Fno(ev)
-
-
-
-7
F, W
ccasE+O
Constant ExDonential
r$.-cr, Direct averages
0.03 0.03 0.03 0.09
x x x x
10-s* 10-s* 10-s* 10-a
0.43 0.57 0.40 0.48
0.069 0.053 0.074 0.063
-
-
f + f +
0.12 0.16 0.11 0.13
0.60 0.94 0.56
-
‘* These values were obtained from the data in Table 1 by ignoring those reduced neutron widths which are larger than 0.11 X 10WSeV.
particular. These involve the loss of very small levels simply because of their size, as well as the loss of levels by failure to resolve them. Consequently, any statistics obtained by fitting distributions to the data in Table 1 must be cautiously evaluated. Because of these,difhculties, the emphasis in this paper has been shifted from finding the distribution of widths which best fit the present data to considering several differential distributions, the integral form of each of which fits the data reasonably well. The results of the theoretical treatment can then be examined to find useful particular results or trends. In addition, the theoretical results can be checked against the results of the KAPL integral experiments. The following trial distributions have been assumed for I’, and rn; fyr,)dr,= (2F,)-r dr, for
02 r,s 2r,
= 0 for r. > 2r,
(la>
p(rzjdrz= (r,)-b-r&ir,c
(lb)
p(r,)dr,= 4(r,)-2rze-~rz%r,
UC)
r, is the average value of P5. We have used the same distributions for the fissions widths and for the neutron widths, but this does not mean that the distributions for r, and Pn must be the same. By fitting these distributions to the data given in Table 1, estimates can be made of the parameters involved. These estimates are shown in Table 2. It is of interest here to note that the direct averages of the data in Table 1 give rr = O-063eV and rlEo = 0.09 x 1O-3eV. The integral form of these distributions is compared with the experimental
19
The variation of 0: of JJSs6 with energy in the intermediate-energy range
values of the fission widths in Fig. 1, with the experimental values of the reduced neutron widths in Fig. 2. In fitting the reduced neutron widths, those values above O-11 mV were ignored because it was difficult to get any reasonable fits if the tail were included. For each of the distributions in equation (1) this gives an average value of P, O= O-03 x 1O-3eV. The calculations will be done not only for this value but for rlao = O-09 x 1O-3eV and I\,O = O-15 x 10e3eV, to allow for the fact that we have ignored the tail. If the tail really exists and is not caused by the 24 X
-.-
-------
Experimental
N=lQe-lsrf N=13&27rf
points
1
[271; tl]
18 16 14 12
=0*074eV(------I
10
I
4 2
I 0
0.01
0.03
0.05
0.07
0.09
0.11
0.13
. 0.15
rr
I 0.17 eV
FIG. I.-Integral spectrum of the fission widths in snUess.
superposition of the widths of several resonances, then a reasonable fit to the present data is the integrated form of the differential distribution [K/(b2 + I’,2)Jdl?,. BETHE (1956) prefers the distribution e-” dx, where x = 2/2I’2/r,O. HUGHES and HARVEY (1955) have considered the distribution of l?,“/r,O for 150 resonances of several isotopes, where pm0 is the average reduced neutron width of a particular isotope. They find that the exponential distribution agrees well with the experimental ditferential spectrum. R. G. THOMAS (1956) analysed the reduced neutron widths and the fission widths for fluctuations by means of the maximum likelihood method. He considered the special forms for the &i-squared distributions
(2) where l? = the gamma function, x = the width, 2 = the average width, v = the degree of freedom
20
SOPHIE OLEKSA
The best fit for the reduced neutron widths of the 150 resonances for the several The exponential isotopes is obtained for Y = 1, the Porter-Thomas distribution. distribution, equation (lb), is Y equal 2; the distribution of the form I’ze-crc, equation (lc), is v equal 4. For the fission widths of U235, THOMAS found originally that Y equal 4 gave the best fit. Later data changed this value to Y equal 2.5. The final distribution to consider is the one that WIGNER (1949) used in his original 26 r 24 22 20 18 16
10 8
6 4 2
0.01
I 0.02
0.03
I
0.04
I
0.05
\
0.06
I 0.07
r; FIG.
2.-Integral
I 0.08 mV
spectrum of the reduced neutron widths in 9zU235.
study. He assumed a distribution that is symmetrical and constant on the logarithmic scale, i.e. for F2 < x0/r* P (r,) dl’, = 0 = (2r, Z
In r,)-l dl?, for
0
for
x,/r,
<
r, < xorz
(3)
r, > xor,
where x,, = geometric centre of the distribution, r,” = the spread in the values of the widths,
For
by n.
the fission widths, x is replaced by
f; for the neutron widths x is replaced
The variation of a of paU*s5with energy in the intermediate-energy
range
21
THEORY
According to the usual theory of compound nucleus formation, the cross-section o(n, x) is given by r, o(n, x) = 0, i;’ (4) where oC= the cross-section for the formation of the compound nucleus, I’%= the partial width for the process X, and I? = the total width. According to resonance theory, dy,r 'c= (E - E~)Z+ ry4
(5)
so that (T(tz, x) =
n22gr,r, (6)
(E - E,)E+ ry4
Equation (6) is the one-level Breit-Wigner formula; R is the wavelength and E the energy of the incoming neutron; E,. is the energy of the neutron at resonance; g, a statistical weighting factor, is equal to (2J + 1)/(2s + I)(21 + I), where J is the spin of the compound nucleus, Z is the spin of the target nucleus, and s is the spin of the neutron. If it is assumed that R, l?,, and PO are constant over the energy spread dE of the neutron source, then the average of equation (6) over a single level gives
s
a(n,x)dE =
rr
2x2A2g j?f
barns eV
4.1 x loegr,rz: XL= r barns E
eV
(7)
where E now is the energy of the level. The energy and the widths should be expressed in electron-volts. For the energy range IOO-10,000 eV, only neutrons with I = 0 need be considered. Since Z = 7/2 for U235, g will have the values 7/16 and 9/16, but for all practical purposes g can be taken as l/2. The partial width, I?,, will be rr or I?,, depending on whether the radiative capture cross-section or the fission cross-section is wanted. The total width, I’, is equal to the fission width, I’,, plus the radiation width, I?:,, plus the neutron width, rn. It is believed that the average values of the radiation and fission widths do not vary appreciably with energy. This would seem to imply that a should not change with energy, since a = r,/I’, for an individual level. However, if I?, fluctuates from level to level, around some average value, then a will have different values for different levels. The data of Table 1 show that r, does fluctuate in this way. This variation of I?, from level to level, together with the weighting of each level with the probability that a reaction will occur, is the basis for the variation of the average value of a with energy. (This point is discussed in greater detail by WIGNER(1949).) At
22
SOPHIE
OLEKSA
low energies, l?,, which varies with the square root of the energy, is negligible compared with rY and l?,. The probability for a reaction, (l?, + I’,)/@?, + rY + J?,), is then practically equal to unity. At higher energies, rn approaches rr + r,. Since lYYis constant, the probability of the reaction will be affected mainly by the fluctuation of l?,. The effect of the fluctuations on Z(n, y) and $n,f) can be expressed as Z(n, y) = B’ and
= d
a,f)
(8)
r,r, Om om~m~(rt) ss
rri+ r,+ rr
dr, dr,
(9)
where c’ = (4.1 x lo6 g)/DE barns/eV, and the averages are over many levels. Equations (8) and (9) apply only for neutrons with angular momentum I equal zero. D, the average level spacing of the compound nucleus levels, is assumed to be larger than the widths; it appears in the expression because the averaging is taken over many levels. The capture-to-fission ratio is then given by
Oh y)
a=a(n,f> The integrations of equations (8) and (9) give the Z(n, y) and C(n,f) in terms of the average fission width and the average neutron width, and u is then obtained as a function of those parameters. The value of u obtained in this way depends on the energy because of the dependence of p, on energy. The formulae for IXbecome quite complicated for some of the distributions. Many of the main features of the results, however, can be seen much more easily in the limits of zero and very large energy. In the limit E-t co the neutron width becomes much larger than the fission and radiation widths and a approaches the same limit for all the distributions. This limit is:
ssmvmr,
dr, dr,
JS
drdr,
m
0
a--7
co
0
‘co
0
on
0
fvw(w,
I
rr
cc
w,)r,
0
r
-
dr,
-v
I‘,
(11)
In the limit E-t 0,the average neutron width approaches zero. For each of the distributions used here, cc, as E + 0,is given by
Ct=
s0 s0
or
Tyrfj &- dr, f
Y
f
Y
(12)
mwf&drf
s co
0
SC==
m,) *
l-00
I-
?0
dr, P
pm *
Y
n f+
Y
dr,
The variation of a of BzU*86 with energy in the intermediate-energyrange
23
The next step in. the analysis is the derivation of the actual expressions for a(n, y), Z(n,f), and a for the different distributions. This is done in the Appendix, where the expressions for the various quantities of interest are given in some detail to save other workers the trouble of deriving them. The work is simpltied by the occurrence of recursion formulas. RESULTS It is evident from the previous section that the calculations can be tedious, but that many of the general properties of the behaviour of a can be seen more easily from the behaviour of a in the limits E--f 0 and E -+ co. In particular, a good idea
distribution FIG. 3.-Variation of a with r,/F, for E < 0,E -+ Q).The Porter-Thomas the Bethe distribution do not apply to the fission widths. They are included here for comparison. Fig. 3 can be used for nuclei other than g2UBs6.
and
can be obtained of the possible range of variation of a, and the effects of using different fission width distributions can be estimated. The Bethe distribution and the PorterThomas distribution apply to the neutron widths and not to the fission widths. For purposes of comparison, however, we calculated a in the limit E + 0 with the fission widths also described by these distributions. In the limiting cases of very large and very small values of the neutron energy, a can be expressed as a function of the ratio I’$,, and the variation of a with this parameter is shown in Fig. 3. All of the distributions give the same result in the limit E-t 00, i.e. a = I’,jrp Except for the constant distribution (equation la) and the
24
SOPHIEOLEKSA
rfeecrf distribution (equation lc), each distribution gives a different curve in the limit E-t 0, and in each case the value of tc is greater in the lower limit than in the higher limit. For the distributions in equation (1) the values of tl in the limit E -s 0 and E -+ co are tabulated in Table 2 for the best values of the parameters obtained from the TABLE 3.-VARIATION
OF CCWITH
l?~,/ri
FOR
E+ 0
AND
E +'a
_
I-v/r,
WIGNER dist.*
dist.
0
0
0
0.10
0.13
0.16
0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.26
0.30
0.38
0.43
0.50
0.56
0.61
0.68
0.72
0.79
0.83
0.91 1.02 1.14 1.25
0.94 1.05 1.16
I
0 0.18 0.32 044 0.56 0.67 0.79 0.90 1.oo 1.11 1.22
’
0 0.46 0.68 0.87 1.04 1.20 1.35 1.50 1.64 1.77 1.90
0
0.25 0.43 0.58 0.72 0.86 0.99 1.11 1.24 1.36 1.48
0 0.67 0.97 1.22 144 1.64 1.83 2.01 2.18 2.35 2.51 I
-!-
0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1 .oo
-I
* Tf = 2.12 t PORTER-THOMAS
data in Table 1. It will be seen later that the constant and I’,+e-crf distribution for the fission widths give values of cc that agree with experiment. The exponential distribution in rr gives values of ct that do not agree with experiment. The same is true for the Porter-Thomas and the Bethe distributions in rr. The results for the variation of dl with ry/!Zf are listed in Table 3. TABLE 4.-VARIATION
OF a WITH
ri FOR E --f 0
AND
E --f co
(r,/h, = 0450) cr(E --f co)
a(E + 0)
1.50 2.00 2.50 2.72 3.00 3.50
0.60 0.61 0.62 0.62 0.63 0.64
I ) ’
0.58 0.55 0.52 0.51 0.49 0.47
(
Act
0.02 0.06 0.10 0.11 0.14 0.17
For the Wigner distribution, the value of GIdepends on the parameter rf, which is a measure of the spread in the fission widths. In the results presented in Fig. 3 and Table 3, the value of rf was taken as 2.72. This value was used by WIGNEK, and its use here permits comparison with his results. At the same time, the value rf = 2.72 fits the new experimental data quite well. The effect on the value of M of varying rf is shown in Table 4 for P,,/fo = O-60, and in the limits E -+ 0 and E--f a.
The variation of a of 81Uas5with energy in the intermediate-energy
range
25
Although, for simplicity, the limits E -+ 0 and E +- co have been used, the calculation of a is valid only in the range 100 eV to 10,000 eV. Since the method is a statistical one, the energy interval at the energy E, over which the average is taken, should be large enough to include many levels yet small compared to E. For this reason, a lower limit of 100 eV is chosen. It will be seen from the graphs, however, that for the average neutron widths, r,, that have been used in these calculations, the value of u at 100 eV differs very little from the value obtained at
E FIG. 4.-Variation
of a of gzUaa5with energy for the I’se-cr*
eV distribution.
E = 0. Equations (8) and (9) apply only for neutrons with angular momentum 2 equal zero. At neutron energies greater than 10,000 eV, the effects of neutrons with angular momentum greater than zero must be taken into account. At 100,000 eV, reactions such as inelastic scattering must be considered. The variation of cc with energy for two of the distributions is shown in Fig. 4 and Fig. 5. In Fig. 4 the fission widths, I’,, and neutron widths, I’,, have distributions of the form r,e-cr*. In Fig. 5 we consider a constant distribution for the fission widths, r,, and for the neutron widths, I’%. The calculations have been done for four different values of I‘, to show how a change in the average neutron width affects the variation of c( with energy. We also include an earlier calculation for r,/r‘f = 0.33 and r, = 0*13E112x 1O-3eV. Once again it is obvious that there is little difference between the results of the constant distribution and the l?Z.Crz distribution. The theoretical results are also compared with the values obtained for a in the KAPL jntegral experiment; the latter values which are denoted by the circles are also listed in Table 5. In that table, the energies given are median energies; the median energy is defined so that half of the neutron flux is below the median energy. According to J. STEHN (private communication), the values of a obtained in the KAPL integral experiments seldom have uncertainties smaller than 20 per cent of the value of a itself. It is evident from Fig. 4 and Fig. 5 that theory and experiment
SOPHIE OLEKSA
26
agree quite well in the range 100 eV to 10,000 eV for the constant and I’,e-crz distributions, particularly if the errors in the average values are considered. Russian measurements of 7 (KURCHATOV, 1956) in the range 8-130 eV give an a value that is
0.8 0.7 0.6 8
0.5 0.4 o-3 0.2 0.1 0
1
10
102
105
104
Id
16
eV
E FIG. 5.-Variation
of cc of gzU23S with energy for the constant
distribution.
larger than the KAPL data, i.e. a = 0.64 & 0.05. In the keV region the Russian values agree with the KAPL results. These data have also been tabulated in Table 5. The data in Table 1 give I’Jrr = 0.57 + 0.16 for the exponential distribution. Even at the lower limit of 0.41, the exponential distribution in F, gives an a that is TABLE 5.-V.4RtATIoN 0~ a WITH (experimental values) KAPL Energy
(eV)
Russian
data*
I
3,OOO 15,000 215,000 300,oOO
datat
CI _
I 100 150 1,200
ENERGY
0.52 0.42 0.47 0.42 0.41 0.10 0.17
I
S-130 30,000 140,000 250,000 9OO,ooo
1
0.65. 0.32 0.17 0.12 0.10
i 0.05 f 0.03 i 0.06 i 0.08 f 0.08
* The energies given are median energies. The median energy is defined such that half the neutron flux is below this energy. J. R. STEHN states that the values of OLgiven here are seldom better than 20%. j’ The values of cx were calculated from the values of q. Up to 30 keV, Y = 2.45 i 0.02, after 80 keV, v = 2.47 i 0.03 was used, where v = average number of neutrons per fission.
higher than the experimental a. Not until F,/rf is approximately 0.3 do the theoretical values begin to agree with the experimental values. Thus far we have considered the same distributions for the fission widths and for the neutron widths. In Fig. 4 both were of the form I’sce-cr*; in Fig. 5 both were
The variation of a of pBU*36with energy in the intermediate-energy range
27
constant. A calculation has been made in which the fission widths have a different distribution from the neutron widths. The fission widths were described in terms of a rEeecr * distribution and the neutron widths were described by an exponential distribution. The parameters are the same as those used for the calculations in Fig. 4. Changing the neutron width distribution from the l?ze-cr= distribution to the exponential distribution did not affect the final results. These results are shown in Table 6. We also include in Table 6 the results obtained when both l?, and I’* have exponential distributions. TABLE 6.-COMPARISON OF c?FOR DIFFERENTDISTRIBUTIONS
i;,,!w*
x 106
9
IS
Energy (eV) 1
OP;6 0.56 0.55 0.55 0.55 0.54 053 0.50 0.48 0.47
16 64 100 , 1,600 6,400 10,000 160,000 640,000 1,000,000
9
15
- -
-
-
-
-
0.:6 0.56 0.55 0.55 0.54 0.53 053 0.50 0.48 0.47
0.256 0.55 0*55 0.55 0.54 0.52 0.52 0,48 0.47 0.46
Ok 0.94 0.93 0.91 0.90 0.88 0.87 0.78 0.73 0.71
Op94 0.94 0.93 0.92 0.88 0.85 0.84 0.70 0.62 0.60
-
Cols.1,2: P(l',), P(l',) both have form rre-“’ Cols. 3, 4: P(l?,) as above; P(r,J exponqtial; Cols. 5, 6: both exponential;
qr,), p(r,)
__-.
I
OP;6 055 055 0.55 0.54 053 0.52 0.48 0.47 0.46
15
9
e, r,/l; = @4. I?#, = 0.4.
ry/Ff = 057
At 100,000 eV, the experimental value is much smaller than the theoretical value. We have assumed that rf does not vary systematically with energy. If, however, one considers r, as slowly increasing with energy (BETHE, 1955), so that at 100,000 eV it has increased by a factor of approximately 3, then good agreement will exist distribution for fission, l?$‘, will between theory and experiment. In the r+Prf change from O-40 to O-13. It can be seen from Fig. 3 that for l?,/r, of O-13 a will be between O-2 and O-13, closer to the latter. This is well within the experimental errors. In addition to the possible increase of r,, at higher energies the effects of angular momenta 2 > 0 and inelastic scattering appear. For inelastic scattering, the position of the excited states must be known. WEISSKOPF (unpublished report) did a preliminary study of a at these higher energies, in which he considered these effects and got agreement with the experimental data. In addition to a, the formulae can be used for the calculation of Z(n, y) and a(n,f). The results for the case where the fission widths have the PF-crf distribution and the neutron widths the exponential distribution are shown in Table 7. Instead of S(n, y) and ii(n,f), we have tabulated DZ(n, r) and DZ(n,f). For U2s it is dilllcult to compare the calculated radiative capture cross-section with experiment, but we can compare the calculated fission cross-section with experiment if we consider the experimental results as average values. This is done in Table 7. Column 3 gives the theoretical DZ(n,f), column 4 gives the experimental Z(n,f),
SOPHIE OLEKSA
28
column 5 gives the D obtained from columns 3 and 4. From 100 eV to 10,000 eV, D does not vary by much; at IO6eV, however, D has changed by a factor of 10. According to the usual level density formula of D proportional to e- 2/aZ, where for 92U235a is approximately 48 MeV-l (BLATT and WEISSKOPF, 1952), D should decrease by a factor of about 3 to 4 from zero to 1 MeV (the excitation energy changes from approximately 6.39 MeV to 7.39 MeV). The fact that D varies by a factor of 10 TABLE 7.-COMPARISON
Energy (eV)
D&f 1
D&J) (barns
OF THE THEORETICAL AND
eV)
(barns
&f
eV)
EXPERIMENTAL
1 (exp)
(Z)
(barns)
100
6.43
24.2
0.48
0.50
1,600
1.50
2.77
6.5
0.43
0.50
6,400
0.695
1.30
4.4
0.30
040
10,000.
0.536
1.01
3.7
0.27
0.39
0.092
0.187
1.5
0.12
0.38
160,000
11.6
t+Z,f)
640,000
0.034
0.070
1.2
0.06
0.38
1,000,000
0.025
0.052
1.3
0.04
0.35
I r, = 0.09I?'2 x IO-3eV Energy (eV)
100
D&y) (barns
eV)
10.5
D&f 1 (barns
eV)
I
;;cn,f 1
(2)
(barns
19.2
24.2
0.79
1,600
2.37
4.43
6.5
0.68
0.82 0.80
6,400
1.06
2.02
4.4
0.46
0.62
10,000
0.807
1.56
3.7
0.42
0.60
160,000
0.125
0.260
1.5
0.17
0.54
640,000 1,ooo,ooo
0.043
0.094
1.2
0.08
0.53
0.030
0.067
1.3
0.05
0.44
IT;, = 0.15 E112 x 1O-3 eV
D* is the value obtained
when the cross-section
is corrected
for the higher angular momenta.
indicates that the higher angular momenta have begun to appear. We can show these effects if we assume with WICNER (1949a) that in first approximation the higher angular momenta can be accounted for by (1 + kR)2, where k is the wave number of the neutron and R is the radius of the nucleus, R = 1.45 Ali3 x lo-l3 cm. The D obtained from the cross-sections which were corrected for higher angular momenta is shown in column 6. It now varies at most by a factor of 2, which is in better agreement with the behaviour of D predicted by the level-density formula. This analysis, however,’ is a qualitative one. The experimental value of D is 0.65 eV. CONCLUSIONS
The theory and results treated in the earlier sections of this report can be summarized briefly in the statements which follow. 1. The variation of LXof 92U235with energy from 100 to 10,000 eV is a slow one,
The variation of c(of g$J235with energy in the intermediate-energyrange
29
no matter which of several distributions is assumed to hold for the fission and neutron widths. 2. The values of tl obtained by means of WIGNER’S theory are in good agreement with results of the KAPL integral experiments. 3. The calculated values of ii(n,f) agree reasonably well with the experimental values. 4. The theory can be applied with confidence to fissionable materials other than 92U235for which comparable information about low-energy resonances is available. Although the above conclusions are highly favourable, the available experimental data still leave some questions open. Thus, it would be highly desirable to have enough information for a particular isotope so that differential distributions could be obtained which fit fhe experimental fission and neutron widths well. For a more detailed understanding of the variation of CIwith energy, a better understanding of the partial widths is needed. Thus, I’? has been assumed to be constant; actually it fluctuates slightly from level to level, but the radiation process involves such a large number of final states that rY is already an average quantity. The variation of I’? with energy is not large, and rr increases at most by a factor of 1.3 over the energy range up to 1 MeV (WEISSKOPF, unpublished report). The assumption of constant I‘,, is, therefore, probably a good one, but again more detailed information would be helpful. Similarly, it has been assumed that there is no systematic variation of r, with energy and that the fluctuations in I’, from level to level are responsible for the variation of tc with energy. More information bearing on this assumption would be helpful, because an increase in r, by a factor of approximately 3 from 1 eV to 100,000 eV brings the calculated cc at the high-energy end into agreement with the experimental values. With regard to I?,, statistical theory predicts that this width should vary as the square root of the energy, and the experimental data seem to support this prediction. Finally, it would ‘be of value to know whether the widths, I’, and rr, and the spacing, D, depend upon the angular momentum 1 of the incident neutron; the assumption thus far has been that they do not. The available data point in the direction of different distributions for the fission and neutron widths. Calculations were done in which the same distributions, the. l?~e-crz distributions, were used for I’, and I’,. Other calculations were done in which the l?,e -cr= distribution was kept for the I’,, but an exponential distribution was used for I’,. The results were the same for both calculations. The effect of changing the distribution in r, can be seen in Fig. 3, where a change in the distribution of r, from the I?++‘~ distribution to the exponential distribution for the same r, can vary ccby a factor of 2. Finally, it is felt that the agreement between the calculated a(n,f) and the experimental values when the higher I values are considered are reasonably good. The results of the calculations, on the whole, are such as to increase confidence in the applications of this theory to the problem of getting values of a as a function of energy for use in reactor calculations. APPENDIX Expressions for Z(n, y), Z(n, f)and cc For the constant distribution defined by equation (la), t?(n, y) equation (8), a(n,_f) (equation 9), and o(in the limit E -+ 0 (equation 12), are not difficult to integrate. For those distributions which are related to the &i-squared distribution, i.e. the
SOPHIE OLEKSA
30
exponential distribution (equation lb) and the I’,e- crz distribution the work is simplified by the existence of a recursion formula. m 1
m
rfmr,r
ab _
mr -
ss 0
rf
0
+
&aI’d’r,)
r,
+
&‘,
(equation
dr,
lc),
(13)
r:,
can be obtained from
am+r I,, = (-I)“+’
(14)
-
aamabf IO0
where (15) dx
m
and E,(al?Y) is the exponential integral
5
e-” ;
al-./
with
(a, > 0)
El(Ul?,) = -E&$J
lo1= _-!(b
a)2
_
[earyE,(ar,,)
1 + (b -
2
a)4
2 + (b - a)3 c
Iz2 =
-2..(b _
a)5
-I’r,,ear~El(d’,,)
+ $
i
1
(16)
+ &
1
+ k
+ i -r,ebr~E,(br,)
[
-ryebr~El(br,)
r,2ebryE,(br,)
1 1
(17)
+ k
-
%+$I
-
i
(18)
[&JE&qJ - ebrYEr(brY)] 12 + (b -
2 + (b -
[
[ebrYE,(bI’Y)- ear~E,(ar,)]
-rrye”r,,E,(ar,)
-
ear~El(ar,)]
[ebr,JE,(bF,) -
1
---!L
ebr~E,(bI’,)l
r,ebr~El(bI’,)
+ (b -u)~
‘lz = (b _
-
a)
I,1 = (b _ a)8
,
a)” [
lY,e~r7E,(f2r,) a)” C
r,VrYEl(crY)
rv
- a
-
i + r,ebr~E,(bry)
i
+ a2 - l?,2ebr~El(br,)
1
‘+ 2
When the distribution of the fission widths and the distribution
-
b
1
(19)
of the neutron
The variation of a of ~8Usss with energy in the intermediate-energy
31
range
widths are both exponential, then
qn,y) = [I -
(20)
= o’abl,,
a(n,f) a+
o’ubr,l,l
aI’,ear~El(uI’,)]-l - 1 as
E--s- 0
with a = l/r,, b = l/?i,. When the distribution of the fission widths and the distribution widths are both of the form I’,eMcr~, then
of the neutron
z(~, Y) = uhwr,~12
@z,f) a+
(1 -
= o’a2b21,,
- dy,(dyj-l
dy
(21)
- 1 as
E+
0
with a = 2/r,, b = 2/p’,. When the distribution of the fission widths is of the form l?&crs and the distribution of the neutron widths is an exponential, then
tc+
(1 -
qn, r) =
fAz2br,-r,l
c&f) =
a’a2b12,
.
(22)
- I
ur,[l- dy,(a,)ly
as
E+
0
with a = 2/r,, b = l/l?,, and Izl = 112, with a and b interchanged. For the Bethe, Porter-Thomas, and Wigner distributions we are interested mainly in the limiting values of cc for very large and very small energies. WIGNER (1949) found that with his distribution a in the limit E-+ 0 is given by
(23) The Porter-Thomas
distribution gives er@‘f(l
a-t[l-XGr~~
- erf dr,/2rp
- 1
(24)
where erf is the error integral. The Bethe distribution (BETHE, 1955) gives dl-+ {I - 2/21?,/rf [sin 2/v, where
Cix=-
sz
cos
Ci 2/2r,pf
1/2=
* cos t 7 dt
(Si 2/2r,/r, - r/2)3)-r and
Si x =
1
(25)
‘sin t dt
s0
t
In the limit E-+ 00, the distributions all give (26)
32
SOPHIEOLEKSA
Acknowledgements-I
am particularly
with the calculations
and to JACK HARVEY for his advice, for many interesting dis-
cussions,
and for making
CHERNICK for suggesting
the experimental the problem
cussions and encouragement, for supplying
indebted
to NORMAN BAUMAN for his help
data available.
and for discussions,
I wish to thank JACK IRVING KAPLAN for dis-
ENNIS PILCHER for help with the data, and J. R. STEHN
the values of CIobtained
in the integral experiments.
I am grateful to
R. G. THOMAS for sending me preprints of his work.
REFERENCES BETHE H. A. (1955) Theoretical analysis of neutron resonances in fissile materials. Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, A/Conf. S/p/585. BLATT J. M. and WEISSKOPFV. F. (1952) Theoretical Nuclear Physics, p. 312. Wiley, New York. EGELSTAFFP. A. and HUGHES D. J. Resonance structure of U233, IJza5, and Puzs9. Physics and Mathematics, Progress in Nuclear Energy Vol. 1. pp. 55-89 McGraw-Hill, New York. HUGHESD. J. and HARVEYJ. A. (1955) Size distribution of neutron widths. Phys. Rev. 99, 3, 1032-1033. KANNE W. R., STEWARTH. B., and WHITE F. A. (1955) Capture-to-fission ratio of Pu239and Uz3j for intermediate energy neutrons. Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, A/Conf. 8/p/595. KURCHATOVI. V. (1956) Physics of the water-moderated reactor. Nuclear Engineering 1, 101-103. PALEVSKYH., HUGHES D. J., ZIMMERMAN R. L., and EISBERGR. M. Direct measurement of the energy variation of 7 for IJzs3, Uz3j and Puza9. J. Nucl. Energy (in press). PORTERC. E. and THOMASR. G. (1656) Fluctuations in nuclear reaction widths. Phys. Rev. 104, 483491. SAILOR V. L. (1955) The low energy cross section of Uz3j. Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, A/Conf. 8/p/586. SEIDL F. G. P., HUGHESD. J., PALEVSKYH., LEVINJ. S., KATO W. Y., and SJOSTRANDN. G. (1954) Fast chopper time of flight measurement of neutron resonances. Phys. Rev. 95,476-499. STEHNJ. (1954) private communication. WEI~SKOPFV. (1952) unpublished report. WIGNERE. P. (1949) On the variation of n with energy in the lOO-1OUOeV region. BNL-25. WIGNERE. P. (1949a) Amer. J. Phys. 17, 99.
I. 1957.Vol. 5.
pp.
16 to 32.
Pergamon
Press, Ltd., London
THE VARIATION OF a OF 92U235WITH ENERGY IN THE INTERMEDIATE-ENERGY RANGE* SOPHIE OLEKSA Brookhaven National Laboratory (Received 2 July 1956; in revised form 25 September 1956) Abstract-Recent experiments have made possible a new evaluation of resonance parameters for some of the low-energy levels of g2Ua95. The data permit a numerical calculation of the variation of Q of ,,Uass with energy in the range lOO-10,000 eV in accordance with a method proposed by WIGNER. We examined the effects of fitting the experimental data with several different distributions. The results are: 1. The variation of a (Uza5) with energy from 100 eV to 10,000 eV is a slow one, no matter which of several distributions is assumed to hold for the fission and neutron widths. 2. The values of a obtained by means of WIGNER’Stheory are in good agreement with results of the KAPL integral experiments. 3. The calculated values of 5&f) agree reasonably well with the experimental values. 4. The theory can be applied with confidence to fissionable materials other than Usa for which comparable information about low-energy resonances is available. INTRODUCTION
of the radiative capture cross-section of a fissionable material to the fission cross-section, a, and the variation of a with neutron energy are among the most important properties of a reactor. PALEVSKY et al. (1956) showed that for U235, CIis practically constant in the energy range 0.006 eV to 0.2 eV. The measurement of 0: in the resonance region is a more difficult problem, because the partial reaction widths can vary from resonance to resonance, and the partial widths are hard to measure. The recent developments in the crystal spectrometer and the fast chopper (SAILOR, 1955; SEIDL et al., 1954), however, have made possible the location and resolution of 47 resonances in U 235between 0.3 eV and 35 eV. The values of a obttined in the region vary widely and their uncertainties are large. At energies greater than a few hundred electron volts, “integral experiments” have been done in which average or “cut-off” values of a have been obtained for Pu23g and U235(KANNE et al., 1955). The value of a corresponds to an average over the energy range above a certain cut-off energy. These values have proved to be useful, but their precise meaning is far from clear. WIGNER (1949) showed that it is possible to calculate the value of ccas a function of energy in the range 100 eV to 10,000 eV, provided that the total and fission widths of a reasonable number of resonance levels are known at neutron energies less than 100 eV. Information about cross-sections at intermediate energies can thus be obtained from the properties of resonances at relatively low energies. THE ratio
* This is a modification of a Brookhaven National Laboratory report (December 1954) which appeared in the June 1955 issue of Reactor Science nnd Technology.
USAEC. 16
Research carried out under the auspices of the
The variation of a of ,.JJes5 with energy in the intermediate-energy DISCUSSION
OF
THE
EXPERIMENTAL
range
17
DATA
The experimental data (EGELSTAFF and HUGHES,1956) on which the present study is based are listed in Table 1. The capture width, I’,,, varies somewhat from resonance to resonance, but the variation is not large and I?,, is assumed to be constant. The value given, 0.030 f 0.006 eV, is an average of the capture widths obtained for the TABLE I.-FISSION AND NEUTRON
WIDTHS OF 9J- ,235
-
E. WI
i L
0.29 1.142 2.04 2.82 3.14 3.61 4.84 5.45 5.83 6.12 6.40 7.10 8.82 9.32 9.83 10.21 1064 11.11 11.72 12.4 13.4 13.8 14.11 14.6 15.5 16.1 16.8 18.1 18.7 19.3
rt 0.005 & 0.010 i 0.03 zt 0.05 f 0.02 i 0.02 & 0.02 i 0.10 * 0.10 zk 0.10 f 0.05 & 0.05 f 0.07 & 0.1 f 0.1 f 0.1 * 0.1 i O*l & 0.1 f 0.1 f 0.1 & 0.1 * 0.1 i 0.2 f 0.2 f 0.2 f 0.2 * 0.2 f 0.2 * 0.2
99 120 13 70 135 66 17
* f * f f 5 &
28 * 26% 70*
0.0031 0.0154 0.0066 0.0026 0.028 0.050 0.052 0.022 0.016 0.027 0.30 0.11 1.2 0.11 0.025 0.065 0.033 0.045 0.59 1.2 0.08 0.15 0.20 0.17 0.27 0.29 0.29 0.46 0.20 3.2
5 15 3 45 25 10 10
12 18 16
42 + 18 43 f21
90 + 26
f OWO2 f 0.0008 rt 0.0004 i 0.0007 i 0.002 f 0.003 i 0.005 * 0.004 i 0.003 f 0.004 f 0.03 * 0.02 & 0.2 & 0.02 f 0@04 f 0.008 f 0.007 & 0.007 & 0.06 f 0.2 rfi 0.02 * 0.03 * 0.03 5 0.02 f 0.03 =t 0.04 f 0.04 & 0.07 f 0.06 * 0.3
I
0.0070 0.0144 0.0046 0.0015 0.016 0.026 0.024 0039 0.007 0.011 0.12 0.041 040 0.036 0.008 0.020 0.010 0.014 0.17 0.34 0.020 0.040 0.053 0.044 0.069 0.07 0.07 0.11 0.050 0.73
f o+lOO3 & 0.0007 f 0.0003 f 0.0004 i 0~001 f 0.002 f 0.003 f 0.002 & O+lOl f 0.002 f 0.01 i 0.006 If 0.06 f 0007 i O+lOl * 0.002 f 0.002 * 0002 & 0.02 * 0.05 f 0.006 * 0.008 f 0.008 f oGO4 f 0.008 f 0.01 xk 0.01 i 0.02 f 0.015 * 0.07
I
I’y=30+6mV. g = l/2 for all practical purposes. r,,O = r,(~/&)“* where E = 1 eV. Observed D = 065 eV.
first few resonances. The fission width, I’,, varies from resonance to resonance with a factor of about 10 from the smallest to the largest width. The neutron widths, I?,, fluctuate much more strongly; the largest value is several hundred times as large as the smallest value. The errors assigned to these widths are often as large as 60-70 % of the widths themselves. Because r, varies very nearly directly as the square root of the energy, the reduced neutron widths, I’,O, are used in the analysis of the data. rlzO = l?n(E/E)1/2 with E = 1 eV. The average neutron width, r,, is then 2
SOPHIE OLEKSA
18
equal to the product of the average reduced neutron width, r,O, and the square root of the energy. The calculation of ccinvolves the differential spectra produced by the fluctuations of the widths, i.e. the probability that FE lies between I’, and I’, + dr, where P, is either I?, or F,. The widths available, particularly the number of fission widths, are not enough to allow an adequate fitting of the differential spectrum. Consequently, it is necessary to fit the less sensitive integral spectrum in which the ordinate is the probability that I?, is greater than, or equal to, a given value. Unfortunately, the integral spectrum is often not sensitive enough to permit clear distinctions to be made between the effects of different distributions. Other difficulties are the experimental limitations which can distort the distribution, the neutron width distribution in TABLE 2.-AVERAGE
RESONANCE PARAMETERS FOR THE VARIOUS DISTRIBUTIONS
Distribution
Fno(ev)
-
-
-
-7
F, W
ccasE+O
Constant ExDonential
r$.-cr, Direct averages
0.03 0.03 0.03 0.09
x x x x
10-s* 10-s* 10-s* 10-a
0.43 0.57 0.40 0.48
0.069 0.053 0.074 0.063
-
-
f + f +
0.12 0.16 0.11 0.13
0.60 0.94 0.56
-
‘* These values were obtained from the data in Table 1 by ignoring those reduced neutron widths which are larger than 0.11 X 10WSeV.
particular. These involve the loss of very small levels simply because of their size, as well as the loss of levels by failure to resolve them. Consequently, any statistics obtained by fitting distributions to the data in Table 1 must be cautiously evaluated. Because of these,difhculties, the emphasis in this paper has been shifted from finding the distribution of widths which best fit the present data to considering several differential distributions, the integral form of each of which fits the data reasonably well. The results of the theoretical treatment can then be examined to find useful particular results or trends. In addition, the theoretical results can be checked against the results of the KAPL integral experiments. The following trial distributions have been assumed for I’, and rn; fyr,)dr,= (2F,)-r dr, for
02 r,s 2r,
= 0 for r. > 2r,
(la>
p(rzjdrz= (r,)-b-r&ir,c
(lb)
p(r,)dr,= 4(r,)-2rze-~rz%r,
UC)
r, is the average value of P5. We have used the same distributions for the fissions widths and for the neutron widths, but this does not mean that the distributions for r, and Pn must be the same. By fitting these distributions to the data given in Table 1, estimates can be made of the parameters involved. These estimates are shown in Table 2. It is of interest here to note that the direct averages of the data in Table 1 give rr = O-063eV and rlEo = 0.09 x 1O-3eV. The integral form of these distributions is compared with the experimental
19
The variation of 0: of JJSs6 with energy in the intermediate-energy range
values of the fission widths in Fig. 1, with the experimental values of the reduced neutron widths in Fig. 2. In fitting the reduced neutron widths, those values above O-11 mV were ignored because it was difficult to get any reasonable fits if the tail were included. For each of the distributions in equation (1) this gives an average value of P, O= O-03 x 1O-3eV. The calculations will be done not only for this value but for rlao = O-09 x 1O-3eV and I\,O = O-15 x 10e3eV, to allow for the fact that we have ignored the tail. If the tail really exists and is not caused by the 24 X
-.-
-------
Experimental
N=lQe-lsrf N=13&27rf
points
1
[271; tl]
18 16 14 12
=0*074eV(------I
10
I
4 2
I 0
0.01
0.03
0.05
0.07
0.09
0.11
0.13
. 0.15
rr
I 0.17 eV
FIG. I.-Integral spectrum of the fission widths in snUess.
superposition of the widths of several resonances, then a reasonable fit to the present data is the integrated form of the differential distribution [K/(b2 + I’,2)Jdl?,. BETHE (1956) prefers the distribution e-” dx, where x = 2/2I’2/r,O. HUGHES and HARVEY (1955) have considered the distribution of l?,“/r,O for 150 resonances of several isotopes, where pm0 is the average reduced neutron width of a particular isotope. They find that the exponential distribution agrees well with the experimental ditferential spectrum. R. G. THOMAS (1956) analysed the reduced neutron widths and the fission widths for fluctuations by means of the maximum likelihood method. He considered the special forms for the &i-squared distributions
(2) where l? = the gamma function, x = the width, 2 = the average width, v = the degree of freedom
20
SOPHIE OLEKSA
The best fit for the reduced neutron widths of the 150 resonances for the several The exponential isotopes is obtained for Y = 1, the Porter-Thomas distribution. distribution, equation (lb), is Y equal 2; the distribution of the form I’ze-crc, equation (lc), is v equal 4. For the fission widths of U235, THOMAS found originally that Y equal 4 gave the best fit. Later data changed this value to Y equal 2.5. The final distribution to consider is the one that WIGNER (1949) used in his original 26 r 24 22 20 18 16
10 8
6 4 2
0.01
I 0.02
0.03
I
0.04
I
0.05
\
0.06
I 0.07
r; FIG.
2.-Integral
I 0.08 mV
spectrum of the reduced neutron widths in 9zU235.
study. He assumed a distribution that is symmetrical and constant on the logarithmic scale, i.e. for F2 < x0/r* P (r,) dl’, = 0 = (2r, Z
In r,)-l dl?, for
0
for
x,/r,
<
r, < xorz
(3)
r, > xor,
where x,, = geometric centre of the distribution, r,” = the spread in the values of the widths,
For
by n.
the fission widths, x is replaced by
f; for the neutron widths x is replaced
The variation of a of paU*s5with energy in the intermediate-energy
range
21
THEORY
According to the usual theory of compound nucleus formation, the cross-section o(n, x) is given by r, o(n, x) = 0, i;’ (4) where oC= the cross-section for the formation of the compound nucleus, I’%= the partial width for the process X, and I? = the total width. According to resonance theory, dy,r 'c= (E - E~)Z+ ry4
(5)
so that (T(tz, x) =
n22gr,r, (6)
(E - E,)E+ ry4
Equation (6) is the one-level Breit-Wigner formula; R is the wavelength and E the energy of the incoming neutron; E,. is the energy of the neutron at resonance; g, a statistical weighting factor, is equal to (2J + 1)/(2s + I)(21 + I), where J is the spin of the compound nucleus, Z is the spin of the target nucleus, and s is the spin of the neutron. If it is assumed that R, l?,, and PO are constant over the energy spread dE of the neutron source, then the average of equation (6) over a single level gives
s
a(n,x)dE =
rr
2x2A2g j?f
barns eV
4.1 x loegr,rz: XL= r barns E
eV
(7)
where E now is the energy of the level. The energy and the widths should be expressed in electron-volts. For the energy range IOO-10,000 eV, only neutrons with I = 0 need be considered. Since Z = 7/2 for U235, g will have the values 7/16 and 9/16, but for all practical purposes g can be taken as l/2. The partial width, I?,, will be rr or I?,, depending on whether the radiative capture cross-section or the fission cross-section is wanted. The total width, I’, is equal to the fission width, I’,, plus the radiation width, I?:,, plus the neutron width, rn. It is believed that the average values of the radiation and fission widths do not vary appreciably with energy. This would seem to imply that a should not change with energy, since a = r,/I’, for an individual level. However, if I?, fluctuates from level to level, around some average value, then a will have different values for different levels. The data of Table 1 show that r, does fluctuate in this way. This variation of I?, from level to level, together with the weighting of each level with the probability that a reaction will occur, is the basis for the variation of the average value of a with energy. (This point is discussed in greater detail by WIGNER(1949).) At
22
SOPHIE
OLEKSA
low energies, l?,, which varies with the square root of the energy, is negligible compared with rY and l?,. The probability for a reaction, (l?, + I’,)/@?, + rY + J?,), is then practically equal to unity. At higher energies, rn approaches rr + r,. Since lYYis constant, the probability of the reaction will be affected mainly by the fluctuation of l?,. The effect of the fluctuations on Z(n, y) and $n,f) can be expressed as Z(n, y) = B’ and
= d
a,f)
(8)
r,r, Om om~m~(rt) ss
rri+ r,+ rr
dr, dr,
(9)
where c’ = (4.1 x lo6 g)/DE barns/eV, and the averages are over many levels. Equations (8) and (9) apply only for neutrons with angular momentum I equal zero. D, the average level spacing of the compound nucleus levels, is assumed to be larger than the widths; it appears in the expression because the averaging is taken over many levels. The capture-to-fission ratio is then given by
Oh y)
a=a(n,f> The integrations of equations (8) and (9) give the Z(n, y) and C(n,f) in terms of the average fission width and the average neutron width, and u is then obtained as a function of those parameters. The value of u obtained in this way depends on the energy because of the dependence of p, on energy. The formulae for IXbecome quite complicated for some of the distributions. Many of the main features of the results, however, can be seen much more easily in the limits of zero and very large energy. In the limit E-t co the neutron width becomes much larger than the fission and radiation widths and a approaches the same limit for all the distributions. This limit is:
ssmvmr,
dr, dr,
JS
drdr,
m
0
a--7
co
0
‘co
0
on
0
fvw(w,
I
rr
cc
w,)r,
0
r
-
dr,
-v
I‘,
(11)
In the limit E-t 0,the average neutron width approaches zero. For each of the distributions used here, cc, as E + 0,is given by
Ct=
s0 s0
or
Tyrfj &- dr, f
Y
f
Y
(12)
mwf&drf
s co
0
SC==
m,) *
l-00
I-
?0
dr, P
pm *
Y
n f+
Y
dr,
The variation of a of BzU*86 with energy in the intermediate-energyrange
23
The next step in. the analysis is the derivation of the actual expressions for a(n, y), Z(n,f), and a for the different distributions. This is done in the Appendix, where the expressions for the various quantities of interest are given in some detail to save other workers the trouble of deriving them. The work is simpltied by the occurrence of recursion formulas. RESULTS It is evident from the previous section that the calculations can be tedious, but that many of the general properties of the behaviour of a can be seen more easily from the behaviour of a in the limits E--f 0 and E -+ co. In particular, a good idea
distribution FIG. 3.-Variation of a with r,/F, for E < 0,E -+ Q).The Porter-Thomas the Bethe distribution do not apply to the fission widths. They are included here for comparison. Fig. 3 can be used for nuclei other than g2UBs6.
and
can be obtained of the possible range of variation of a, and the effects of using different fission width distributions can be estimated. The Bethe distribution and the PorterThomas distribution apply to the neutron widths and not to the fission widths. For purposes of comparison, however, we calculated a in the limit E + 0 with the fission widths also described by these distributions. In the limiting cases of very large and very small values of the neutron energy, a can be expressed as a function of the ratio I’$,, and the variation of a with this parameter is shown in Fig. 3. All of the distributions give the same result in the limit E-t 00, i.e. a = I’,jrp Except for the constant distribution (equation la) and the
24
SOPHIEOLEKSA
rfeecrf distribution (equation lc), each distribution gives a different curve in the limit E-t 0, and in each case the value of tc is greater in the lower limit than in the higher limit. For the distributions in equation (1) the values of tl in the limit E -s 0 and E -+ co are tabulated in Table 2 for the best values of the parameters obtained from the TABLE 3.-VARIATION
OF CCWITH
l?~,/ri
FOR
E+ 0
AND
E +'a
_
I-v/r,
WIGNER dist.*
dist.
0
0
0
0.10
0.13
0.16
0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.26
0.30
0.38
0.43
0.50
0.56
0.61
0.68
0.72
0.79
0.83
0.91 1.02 1.14 1.25
0.94 1.05 1.16
I
0 0.18 0.32 044 0.56 0.67 0.79 0.90 1.oo 1.11 1.22
’
0 0.46 0.68 0.87 1.04 1.20 1.35 1.50 1.64 1.77 1.90
0
0.25 0.43 0.58 0.72 0.86 0.99 1.11 1.24 1.36 1.48
0 0.67 0.97 1.22 144 1.64 1.83 2.01 2.18 2.35 2.51 I
-!-
0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1 .oo
-I
* Tf = 2.12 t PORTER-THOMAS
data in Table 1. It will be seen later that the constant and I’,+e-crf distribution for the fission widths give values of cc that agree with experiment. The exponential distribution in rr gives values of ct that do not agree with experiment. The same is true for the Porter-Thomas and the Bethe distributions in rr. The results for the variation of dl with ry/!Zf are listed in Table 3. TABLE 4.-VARIATION
OF a WITH
ri FOR E --f 0
AND
E --f co
(r,/h, = 0450) cr(E --f co)
a(E + 0)
1.50 2.00 2.50 2.72 3.00 3.50
0.60 0.61 0.62 0.62 0.63 0.64
I ) ’
0.58 0.55 0.52 0.51 0.49 0.47
(
Act
0.02 0.06 0.10 0.11 0.14 0.17
For the Wigner distribution, the value of GIdepends on the parameter rf, which is a measure of the spread in the fission widths. In the results presented in Fig. 3 and Table 3, the value of rf was taken as 2.72. This value was used by WIGNEK, and its use here permits comparison with his results. At the same time, the value rf = 2.72 fits the new experimental data quite well. The effect on the value of M of varying rf is shown in Table 4 for P,,/fo = O-60, and in the limits E -+ 0 and E--f a.
The variation of a of 81Uas5with energy in the intermediate-energy
range
25
Although, for simplicity, the limits E -+ 0 and E +- co have been used, the calculation of a is valid only in the range 100 eV to 10,000 eV. Since the method is a statistical one, the energy interval at the energy E, over which the average is taken, should be large enough to include many levels yet small compared to E. For this reason, a lower limit of 100 eV is chosen. It will be seen from the graphs, however, that for the average neutron widths, r,, that have been used in these calculations, the value of u at 100 eV differs very little from the value obtained at
E FIG. 4.-Variation
of a of gzUaa5with energy for the I’se-cr*
eV distribution.
E = 0. Equations (8) and (9) apply only for neutrons with angular momentum 2 equal zero. At neutron energies greater than 10,000 eV, the effects of neutrons with angular momentum greater than zero must be taken into account. At 100,000 eV, reactions such as inelastic scattering must be considered. The variation of cc with energy for two of the distributions is shown in Fig. 4 and Fig. 5. In Fig. 4 the fission widths, I’,, and neutron widths, I’,, have distributions of the form r,e-cr*. In Fig. 5 we consider a constant distribution for the fission widths, r,, and for the neutron widths, I’%. The calculations have been done for four different values of I‘, to show how a change in the average neutron width affects the variation of c( with energy. We also include an earlier calculation for r,/r‘f = 0.33 and r, = 0*13E112x 1O-3eV. Once again it is obvious that there is little difference between the results of the constant distribution and the l?Z.Crz distribution. The theoretical results are also compared with the values obtained for a in the KAPL jntegral experiment; the latter values which are denoted by the circles are also listed in Table 5. In that table, the energies given are median energies; the median energy is defined so that half of the neutron flux is below the median energy. According to J. STEHN (private communication), the values of a obtained in the KAPL integral experiments seldom have uncertainties smaller than 20 per cent of the value of a itself. It is evident from Fig. 4 and Fig. 5 that theory and experiment
SOPHIE OLEKSA
26
agree quite well in the range 100 eV to 10,000 eV for the constant and I’,e-crz distributions, particularly if the errors in the average values are considered. Russian measurements of 7 (KURCHATOV, 1956) in the range 8-130 eV give an a value that is
0.8 0.7 0.6 8
0.5 0.4 o-3 0.2 0.1 0
1
10
102
105
104
Id
16
eV
E FIG. 5.-Variation
of cc of gzU23S with energy for the constant
distribution.
larger than the KAPL data, i.e. a = 0.64 & 0.05. In the keV region the Russian values agree with the KAPL results. These data have also been tabulated in Table 5. The data in Table 1 give I’Jrr = 0.57 + 0.16 for the exponential distribution. Even at the lower limit of 0.41, the exponential distribution in F, gives an a that is TABLE 5.-V.4RtATIoN 0~ a WITH (experimental values) KAPL Energy
(eV)
Russian
data*
I
3,OOO 15,000 215,000 300,oOO
datat
CI _
I 100 150 1,200
ENERGY
0.52 0.42 0.47 0.42 0.41 0.10 0.17
I
S-130 30,000 140,000 250,000 9OO,ooo
1
0.65. 0.32 0.17 0.12 0.10
i 0.05 f 0.03 i 0.06 i 0.08 f 0.08
* The energies given are median energies. The median energy is defined such that half the neutron flux is below this energy. J. R. STEHN states that the values of OLgiven here are seldom better than 20%. j’ The values of cx were calculated from the values of q. Up to 30 keV, Y = 2.45 i 0.02, after 80 keV, v = 2.47 i 0.03 was used, where v = average number of neutrons per fission.
higher than the experimental a. Not until F,/rf is approximately 0.3 do the theoretical values begin to agree with the experimental values. Thus far we have considered the same distributions for the fission widths and for the neutron widths. In Fig. 4 both were of the form I’sce-cr*; in Fig. 5 both were
The variation of a of pBU*36with energy in the intermediate-energy range
27
constant. A calculation has been made in which the fission widths have a different distribution from the neutron widths. The fission widths were described in terms of a rEeecr * distribution and the neutron widths were described by an exponential distribution. The parameters are the same as those used for the calculations in Fig. 4. Changing the neutron width distribution from the l?ze-cr= distribution to the exponential distribution did not affect the final results. These results are shown in Table 6. We also include in Table 6 the results obtained when both l?, and I’* have exponential distributions. TABLE 6.-COMPARISON OF c?FOR DIFFERENTDISTRIBUTIONS
i;,,!w*
x 106
9
IS
Energy (eV) 1
OP;6 0.56 0.55 0.55 0.55 0.54 053 0.50 0.48 0.47
16 64 100 , 1,600 6,400 10,000 160,000 640,000 1,000,000
9
15
- -
-
-
-
-
0.:6 0.56 0.55 0.55 0.54 0.53 053 0.50 0.48 0.47
0.256 0.55 0*55 0.55 0.54 0.52 0.52 0,48 0.47 0.46
Ok 0.94 0.93 0.91 0.90 0.88 0.87 0.78 0.73 0.71
Op94 0.94 0.93 0.92 0.88 0.85 0.84 0.70 0.62 0.60
-
Cols.1,2: P(l',), P(l',) both have form rre-“’ Cols. 3, 4: P(l?,) as above; P(r,J exponqtial; Cols. 5, 6: both exponential;
qr,), p(r,)
__-.
I
OP;6 055 055 0.55 0.54 053 0.52 0.48 0.47 0.46
15
9
e, r,/l; = @4. I?#, = 0.4.
ry/Ff = 057
At 100,000 eV, the experimental value is much smaller than the theoretical value. We have assumed that rf does not vary systematically with energy. If, however, one considers r, as slowly increasing with energy (BETHE, 1955), so that at 100,000 eV it has increased by a factor of approximately 3, then good agreement will exist distribution for fission, l?$‘, will between theory and experiment. In the r+Prf change from O-40 to O-13. It can be seen from Fig. 3 that for l?,/r, of O-13 a will be between O-2 and O-13, closer to the latter. This is well within the experimental errors. In addition to the possible increase of r,, at higher energies the effects of angular momenta 2 > 0 and inelastic scattering appear. For inelastic scattering, the position of the excited states must be known. WEISSKOPF (unpublished report) did a preliminary study of a at these higher energies, in which he considered these effects and got agreement with the experimental data. In addition to a, the formulae can be used for the calculation of Z(n, y) and a(n,f). The results for the case where the fission widths have the PF-crf distribution and the neutron widths the exponential distribution are shown in Table 7. Instead of S(n, y) and ii(n,f), we have tabulated DZ(n, r) and DZ(n,f). For U2s it is dilllcult to compare the calculated radiative capture cross-section with experiment, but we can compare the calculated fission cross-section with experiment if we consider the experimental results as average values. This is done in Table 7. Column 3 gives the theoretical DZ(n,f), column 4 gives the experimental Z(n,f),
SOPHIE OLEKSA
28
column 5 gives the D obtained from columns 3 and 4. From 100 eV to 10,000 eV, D does not vary by much; at IO6eV, however, D has changed by a factor of 10. According to the usual level density formula of D proportional to e- 2/aZ, where for 92U235a is approximately 48 MeV-l (BLATT and WEISSKOPF, 1952), D should decrease by a factor of about 3 to 4 from zero to 1 MeV (the excitation energy changes from approximately 6.39 MeV to 7.39 MeV). The fact that D varies by a factor of 10 TABLE 7.-COMPARISON
Energy (eV)
D&f 1
D&J) (barns
OF THE THEORETICAL AND
eV)
(barns
&f
eV)
EXPERIMENTAL
1 (exp)
(Z)
(barns)
100
6.43
24.2
0.48
0.50
1,600
1.50
2.77
6.5
0.43
0.50
6,400
0.695
1.30
4.4
0.30
040
10,000.
0.536
1.01
3.7
0.27
0.39
0.092
0.187
1.5
0.12
0.38
160,000
11.6
t+Z,f)
640,000
0.034
0.070
1.2
0.06
0.38
1,000,000
0.025
0.052
1.3
0.04
0.35
I r, = 0.09I?'2 x IO-3eV Energy (eV)
100
D&y) (barns
eV)
10.5
D&f 1 (barns
eV)
I
;;cn,f 1
(2)
(barns
19.2
24.2
0.79
1,600
2.37
4.43
6.5
0.68
0.82 0.80
6,400
1.06
2.02
4.4
0.46
0.62
10,000
0.807
1.56
3.7
0.42
0.60
160,000
0.125
0.260
1.5
0.17
0.54
640,000 1,ooo,ooo
0.043
0.094
1.2
0.08
0.53
0.030
0.067
1.3
0.05
0.44
IT;, = 0.15 E112 x 1O-3 eV
D* is the value obtained
when the cross-section
is corrected
for the higher angular momenta.
indicates that the higher angular momenta have begun to appear. We can show these effects if we assume with WICNER (1949a) that in first approximation the higher angular momenta can be accounted for by (1 + kR)2, where k is the wave number of the neutron and R is the radius of the nucleus, R = 1.45 Ali3 x lo-l3 cm. The D obtained from the cross-sections which were corrected for higher angular momenta is shown in column 6. It now varies at most by a factor of 2, which is in better agreement with the behaviour of D predicted by the level-density formula. This analysis, however,’ is a qualitative one. The experimental value of D is 0.65 eV. CONCLUSIONS
The theory and results treated in the earlier sections of this report can be summarized briefly in the statements which follow. 1. The variation of LXof 92U235with energy from 100 to 10,000 eV is a slow one,
The variation of c(of g$J235with energy in the intermediate-energyrange
29
no matter which of several distributions is assumed to hold for the fission and neutron widths. 2. The values of tl obtained by means of WIGNER’S theory are in good agreement with results of the KAPL integral experiments. 3. The calculated values of ii(n,f) agree reasonably well with the experimental values. 4. The theory can be applied with confidence to fissionable materials other than 92U235for which comparable information about low-energy resonances is available. Although the above conclusions are highly favourable, the available experimental data still leave some questions open. Thus, it would be highly desirable to have enough information for a particular isotope so that differential distributions could be obtained which fit fhe experimental fission and neutron widths well. For a more detailed understanding of the variation of CIwith energy, a better understanding of the partial widths is needed. Thus, I’? has been assumed to be constant; actually it fluctuates slightly from level to level, but the radiation process involves such a large number of final states that rY is already an average quantity. The variation of I’? with energy is not large, and rr increases at most by a factor of 1.3 over the energy range up to 1 MeV (WEISSKOPF, unpublished report). The assumption of constant I‘,, is, therefore, probably a good one, but again more detailed information would be helpful. Similarly, it has been assumed that there is no systematic variation of r, with energy and that the fluctuations in I’, from level to level are responsible for the variation of tc with energy. More information bearing on this assumption would be helpful, because an increase in r, by a factor of approximately 3 from 1 eV to 100,000 eV brings the calculated cc at the high-energy end into agreement with the experimental values. With regard to I?,, statistical theory predicts that this width should vary as the square root of the energy, and the experimental data seem to support this prediction. Finally, it would ‘be of value to know whether the widths, I’, and rr, and the spacing, D, depend upon the angular momentum 1 of the incident neutron; the assumption thus far has been that they do not. The available data point in the direction of different distributions for the fission and neutron widths. Calculations were done in which the same distributions, the. l?~e-crz distributions, were used for I’, and I’,. Other calculations were done in which the l?,e -cr= distribution was kept for the I’,, but an exponential distribution was used for I’,. The results were the same for both calculations. The effect of changing the distribution in r, can be seen in Fig. 3, where a change in the distribution of r, from the I?++‘~ distribution to the exponential distribution for the same r, can vary ccby a factor of 2. Finally, it is felt that the agreement between the calculated a(n,f) and the experimental values when the higher I values are considered are reasonably good. The results of the calculations, on the whole, are such as to increase confidence in the applications of this theory to the problem of getting values of a as a function of energy for use in reactor calculations. APPENDIX Expressions for Z(n, y), Z(n, f)and cc For the constant distribution defined by equation (la), t?(n, y) equation (8), a(n,_f) (equation 9), and o(in the limit E -+ 0 (equation 12), are not difficult to integrate. For those distributions which are related to the &i-squared distribution, i.e. the
SOPHIE OLEKSA
30
exponential distribution (equation lb) and the I’,e- crz distribution the work is simplified by the existence of a recursion formula. m 1
m
rfmr,r
ab _
mr -
ss 0
rf
0
+
&aI’d’r,)
r,
+
&‘,
(equation
dr,
lc),
(13)
r:,
can be obtained from
am+r I,, = (-I)“+’
(14)
-
aamabf IO0
where (15) dx
m
and E,(al?Y) is the exponential integral
5
e-” ;
al-./
with
(a, > 0)
El(Ul?,) = -E&$J
lo1= _-!(b
a)2
_
[earyE,(ar,,)
1 + (b -
2
a)4
2 + (b - a)3 c
Iz2 =
-2..(b _
a)5
-I’r,,ear~El(d’,,)
+ $
i
1
(16)
+ &
1
+ k
+ i -r,ebr~E,(br,)
[
-ryebr~El(br,)
r,2ebryE,(br,)
1 1
(17)
+ k
-
%+$I
-
i
(18)
[&JE&qJ - ebrYEr(brY)] 12 + (b -
2 + (b -
[
[ebrYE,(bI’Y)- ear~E,(ar,)]
-rrye”r,,E,(ar,)
-
ear~El(ar,)]
[ebr,JE,(bF,) -
1
---!L
ebr~E,(bI’,)l
r,ebr~El(bI’,)
+ (b -u)~
‘lz = (b _
-
a)
I,1 = (b _ a)8
,
a)” [
lY,e~r7E,(f2r,) a)” C
r,VrYEl(crY)
rv
- a
-
i + r,ebr~E,(bry)
i
+ a2 - l?,2ebr~El(br,)
1
‘+ 2
When the distribution of the fission widths and the distribution
-
b
1
(19)
of the neutron
The variation of a of ~8Usss with energy in the intermediate-energy
31
range
widths are both exponential, then
qn,y) = [I -
(20)
= o’abl,,
a(n,f) a+
o’ubr,l,l
aI’,ear~El(uI’,)]-l - 1 as
E--s- 0
with a = l/r,, b = l/?i,. When the distribution of the fission widths and the distribution widths are both of the form I’,eMcr~, then
of the neutron
z(~, Y) = uhwr,~12
@z,f) a+
(1 -
= o’a2b21,,
- dy,(dyj-l
dy
(21)
- 1 as
E+
0
with a = 2/r,, b = 2/p’,. When the distribution of the fission widths is of the form l?&crs and the distribution of the neutron widths is an exponential, then
tc+
(1 -
qn, r) =
fAz2br,-r,l
c&f) =
a’a2b12,
.
(22)
- I
ur,[l- dy,(a,)ly
as
E+
0
with a = 2/r,, b = l/l?,, and Izl = 112, with a and b interchanged. For the Bethe, Porter-Thomas, and Wigner distributions we are interested mainly in the limiting values of cc for very large and very small energies. WIGNER (1949) found that with his distribution a in the limit E-+ 0 is given by
(23) The Porter-Thomas
distribution gives er@‘f(l
a-t[l-XGr~~
- erf dr,/2rp
- 1
(24)
where erf is the error integral. The Bethe distribution (BETHE, 1955) gives dl-+ {I - 2/21?,/rf [sin 2/v, where
Cix=-
sz
cos
Ci 2/2r,pf
1/2=
* cos t 7 dt
(Si 2/2r,/r, - r/2)3)-r and
Si x =
1
(25)
‘sin t dt
s0
t
In the limit E-+ 00, the distributions all give (26)
32
SOPHIEOLEKSA
Acknowledgements-I
am particularly
with the calculations
and to JACK HARVEY for his advice, for many interesting dis-
cussions,
and for making
CHERNICK for suggesting
the experimental the problem
cussions and encouragement, for supplying
indebted
to NORMAN BAUMAN for his help
data available.
and for discussions,
I wish to thank JACK IRVING KAPLAN for dis-
ENNIS PILCHER for help with the data, and J. R. STEHN
the values of CIobtained
in the integral experiments.
I am grateful to
R. G. THOMAS for sending me preprints of his work.
REFERENCES BETHE H. A. (1955) Theoretical analysis of neutron resonances in fissile materials. Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, A/Conf. S/p/585. BLATT J. M. and WEISSKOPFV. F. (1952) Theoretical Nuclear Physics, p. 312. Wiley, New York. EGELSTAFFP. A. and HUGHES D. J. Resonance structure of U233, IJza5, and Puzs9. Physics and Mathematics, Progress in Nuclear Energy Vol. 1. pp. 55-89 McGraw-Hill, New York. HUGHESD. J. and HARVEYJ. A. (1955) Size distribution of neutron widths. Phys. Rev. 99, 3, 1032-1033. KANNE W. R., STEWARTH. B., and WHITE F. A. (1955) Capture-to-fission ratio of Pu239and Uz3j for intermediate energy neutrons. Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, A/Conf. 8/p/595. KURCHATOVI. V. (1956) Physics of the water-moderated reactor. Nuclear Engineering 1, 101-103. PALEVSKYH., HUGHES D. J., ZIMMERMAN R. L., and EISBERGR. M. Direct measurement of the energy variation of 7 for IJzs3, Uz3j and Puza9. J. Nucl. Energy (in press). PORTERC. E. and THOMASR. G. (1656) Fluctuations in nuclear reaction widths. Phys. Rev. 104, 483491. SAILOR V. L. (1955) The low energy cross section of Uz3j. Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, A/Conf. 8/p/586. SEIDL F. G. P., HUGHESD. J., PALEVSKYH., LEVINJ. S., KATO W. Y., and SJOSTRANDN. G. (1954) Fast chopper time of flight measurement of neutron resonances. Phys. Rev. 95,476-499. STEHNJ. (1954) private communication. WEI~SKOPFV. (1952) unpublished report. WIGNERE. P. (1949) On the variation of n with energy in the lOO-1OUOeV region. BNL-25. WIGNERE. P. (1949a) Amer. J. Phys. 17, 99.