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The effect of interference on the average neutron cross sections of fissile nuclei
1
2.5
1
Nuclear Physics A159 (1970) 305 -- 323; @ Not fo be reproduced
THE EFFECT
by photoprint
SECTIONS JOHN
Publishing
or microfilm without written permission
OF INTERFERENCE CROSS
North-Holland
ON THE AVERAGE OF FISSILE
Co., Amsterdum from the publisher
NEUTRON
NUCLEI
D. GARRISON
GulfGeneral Atomic Incorporated Received
and San Diego State 13 April
College,
Californiu
1970
Abstract: The ratio of the average fission, capture and resonance scattering cross sections to the corresponding average cross sections obtained neglecting resonance-resonance interference have been calculated by a Monte Carlo method for a variety of interference conditions typical of fissile nuclei. It is shown in general that the average total cross section is unaffected by interference. The importance of interference increases with the number of open fission channels. The effective number of resonances with which a given resonance interferes is obtained for different numbers of open channels. Results for the cross-section ratios for both S- and P-wave neutrons have been obtained for the common tissile nuclei by fitting calculated cross sections to the average capture and fission cross sections over the neutron energy range from 0.3-100 keV. The ratios
for S-wave
R _,,, = 1.23+0.03, R sEa,l = 1.10+0.01;
R,..,, ‘s:Pu:
neutrons
at low energy
= l.45110.10;
(z
2;;U:
RIirl 1 0.99=0.005,
R,.,,
500 eV) are:
2s:U:
Rr,u 7 0.94+0.01, = 1.00~0.002,
The process of cross-section fitting also yields values for the average these nuclei. The effect of the (n,yf) process is presented.
Rrl,, = 0.95hO.01, R,.,,
.= 1.1130.01,
and R,.,,
= 1.02_CO.o05.
resonance
parameters
for
1. Introduction Resonance peaks are commonly observed in the collision cross sections of nuclear particles. Theories of nuclear reactions have beeen developed which successfully reproduce the energy dependence of these cross sections and provide a mechanism for understanding resonance phenomena rd4). When the widths of the resonance peaks are small compared to the resonance spacing, these cross sections exhibit the simple, Breit-Wigner energy dependence in the vicinity of each resonance, with a slowly varying energy dependence between resonances. When the resonance widths are of the order of the resonance spacing, or larger, the energy dependence of these cross sections is considerably more complicated; the cross section exhibits resonanceresonance interference. The interference between overlapping resonances, in addition to causing a more complicated energy dependence of the cross section, affects the average partial cross sections. It is found, that the areas under the resonances are altered by interference from what they would be for a simple Breit-Wigner energy dependence. These areas are changed in such a way that the average area and thus the average cross section is changed. It is the purpose of this paper to study the effect of interference on the average cross sections of fissile nuclei. There are two reasons for pursuing this study. One is to 305
306
J. D. GARRISON
obtain a better understanding of the phenomenon of resonance-resonance interference. The other is to provide information of interest in the design of nuclear reactors. To perform this calculation, the R-matrix theory of Wigner and Eisenbud ‘) and the distributions the theory of Kapur and Peierls “) have been used. In addition, for the R-matrix resonance parameters are required ‘). The R-matrix resonance parameters serve to determine the resonance areas and average cross sections which would have been obtained without resonance-resonance interference, while the KapurPeierls resonance parameters serve to determine the correct resonance areas and average cross sections including the effect of interference. The Kapur-Peierls resonance parameters are obtained from the corresponding R-matrix parameters using the mathematical connection between the R-matrix and Kapur-Peierls theories 6-8). Some evidence concerning the effect of interference on average cross sections was discussed in an earlier work which was limited to interference between just two resonances ‘). This paper extends the earlier work to include many interfering resonances and a broader treatment of the phenomenon, including both S- and P-wave neutrons (I = 0 and I = 1). Specific application of these results to ‘FLU, ‘i:lJ and ‘GiPu is included. In the next section the pertinent equations are presented. A more complete discussion of many of the equations has been given in the earlier work ‘). In the remaining sections the method of calculation and results are discussed.
2. Theory The neutron total, scattering, total fission compound nucleus of spin J are
and total capture
ai, = _?I1- ~;l”12~ k2
ai, = z J$ IUiC12,
cross sections
for a
(2)
r = f or y.
In eq. (3) the sum over channels is over all the fission channels (f) for the total fission cross section, and is over all the capture channels (r) for the total capture cross section. These and the R-matrix equations which follow are presented in standard is the collision matrix for entrance channel c’ (in this notation “). The quantity UiC paper c’ = n the neutron channel) and c is the exit channel. The quantity k is the magnitude of the propagation vector. In R-matrix theory, the collision matrix is given by
(4)
NEUTRON
CROSS
wl
SECTIONS
The quantity TLC is the partial level width for level % and channel c. The quantity is the potential scattering phase shift for channel c. For the neutron channel
The inverse of the level matrix
A,, is given by
(A-‘);., where the width matrix
$C
= (E,-Q%,--N-j.,,
(6)
rrlv = C r&CC. C
(7)
is given by
The quantitites E and EA are the c.m. energy and resonance energy for level 1. The shift matrix AA,,, normally found in eq. (6), has been set equal to zero for both S- and P-wave neutrons at all energies, although this procedure can be done exactly only for S-wave neutrons 6v‘). For the purpose of calculating the total fission and total capture cross sections, the sum in eq. (7) is separated into its contributions from fission, capture and scattering
(8) where (9)
(11) For all but the P-wave levels of intermediate spin, the sum in eq. (11) consists of a single term. In the calculations performed here, eq. (10) has been assumed to give the elements of a diagonal matrix
r.
AYY
The total radiation
=
rydAv.
width ry is taken to be a constant,
(14 the same for all levels.
For the purposes of obtaining resonance areas and average cross sections, it is desirable to use the Kapur-Peierls form of the cross-section equations. Since the joint probability distribution of Kapur-Peierls resonance parameters is not known direct9,10), the distribution of the R-matrix parameters and the mathematical connection lY between the R-matrix and Kapur-Peierls resonance parameters must be used. The connection between the R-matrix and Kapur-Peierls resonance parameters is through the complex orthogonal matrix transformation which diagonalizes the level matrix and its inverse &(T-‘),,(A-‘),,T,, = &,-E-W;)& (13)
J. D.
308
GARRISON
The quantities EL and rh are the Kapur-Peierls resonance energy and total width for level p. For a complex orthogonal transformation, the inverse matrix is the transpose matrix, i.e., (T- l)pl equals T,,.This fact is used to obtain eq. (14) below. Using the transformation matrix T,the collision matrix of eq. (4) can be expressed as (14)
’
where
The Kapur-Peierls formulation of the cross sections follows on combining eqs. (l)-(3), (14) and (15) [some of the equations differ in form from those of ref. ‘) to more nearly match the form of the single level eqs. (17)-(29) which follow].
(16) r;,
cos
II
24, rg - 2 sin 24, IA(E;- E)
d;=; T p Cr--ZIL(W] , (E;-Ey+(gg2
r = f or y,
(E;-E)'+(+r;)'
,
(17)
(18)
where r;, = i [Re fA, - Im tACtg 24,], c I;,
[Re Frc + Im f,, ctg 2&l,
= - i c
w;
r;=L$x
I, =c [
+ wiyr
&
(E:,-E:)PJ,,,-t(r;+r:)Q:,,
y
1 1’
+(E; - E:)c%,
In ” C (E;-q2+[+(r;+r:)]2
(E;-E:)2+[+(r;+r:)]2
Qk, = Im Jk, 9 ,
pi,, = Re Rk,,
(19) (20)
(21)
(22) (23) (24) (25) (26)
In the case of widely separated resonances, interference can be neglected, which means that the off-diagonal elements of the level matrix and its inverse can be taken to be zero. The cross-section equations for this case are given by a sum of single-level
NEUTRON
CROSS
SECTIONS
309
Breit-Wigner terms plus a potential scattering term, involving the R-matrix parameters directly. J OnT
x
(
=
k”(20 --OS
-2
2(1-cos
24,)+
F
c [L
vL)+
~0s 24, Lj.
1
(EL
-
- 2 sin 24, L(J% -4) El2
,
(27)
+w,)2
II (28)
rZ”(rllcos 24”- rlly - rAlf) - 2 sin24, r&~ - E) (E,-E)‘+(~LA)~
CT;, =
(r=for
VI*
It can be seen that eqs. (16)-(18) and (27)-(29) are similar in form. The difference between them is the effect of resonance-resonance interference. Foreqs. (27~(29), the only interference present is that between resonance and potential scattering. The areas under a single resonance I of eqs. (16)-( 18) are given by the usual singlelevel area formulas I1 ) (30) (31)
&J’An_ - &J,AT _&Jr-&J! If AY
*
(32)
Here, h and m are Planck’s constant and the reduced mass. The average resonance cross sections are obtained from the average areas by (4;)
= (CEY.!?,
DJ
r = y, f, n or T,
(33)
where the quantity DJ is the mean level spacing for resonances of spin J. Since we wish the average to be determined at a particular energy, all areas are evaluated at the same energy. The area equations and average resonance cross sections corresponding to eqs. (30)-(33) for the no interference case are &J
= AT
&i, = ‘&:, al;,
(cT;l,)=
bn cos2&
f
4m
El
’
r.Ur
rlAcos 24,
r = y or f, ’
= ‘GJ;T-L+djf)
9
)
r =
(34) (35) (36)
y,
f, n or T.
(37)
J. D. GARRISON
310
For a description of the effect of interference on the average resonance cross sections, the following ratios will be useful (38) It will now be shown that R, is unity under all conditions. That is, the average total cross section is unchanged by interference. From eqs. (30) (33) (34) and (37), RT is unity provided (r,,) = (f;,). Using eq. (19), we can write I-;* cos 24” = Re i f’le exp (2@,). c
(39)
The average of eq. (39) over N levels is cos 24, = f ; r;,, cos 24” = Re $ exp (2i#“) i T FA,.
(40)
From eq. (I 5) we have PA, = z: T,, TV,r&r&.
(41)
P.y
From the properties of the transformation
matrix T and its inverse we have
1 T,, TV, = 6,, .
(42)
i.
Inserting eqs. (41) and (42) into eq. (40) and summing over v and i., we have (r;,)
cos 24, = Re i exp (2i&) i T Trc = (r,,) c
cos 24,)
(43)
Q.E.D. The averages of the two neutron widths is the same for any N. This is meant in the mathematical sense where a set of N interfering resonances is selected. For actual nuclei we expect N % cc. The relationship between the average partial cross sections with interference and neglecting interference is much more complicated. Combining eqs. (21), (30) and (31), the capture and fission areas are given by
Eq. (44) indicates that the fission and capture areas and thus the ratios are independent of #, and thus independent of the channel radius a. They do depend on the R-matrix widths and resonance energies which go into the inverse level matrix. The scattering areas and ratios do depend upon 4,, and a. It is easy to show, but will not be shown here, that the interference effects do not depend upon the individual magnitudes of the average R-matrix partial level widths and spacings, but rather depend upon the magnitudes of the ratios of the average widths to spacings.
NEUTRON
CROSS
311
SECTIONS
For comparison with experiment, the partial cross sections for different values of J and I must be combined. This is done using the statistical weighting factor gJ = (25+ 1)/2(21+ l), where I is the spin of the target nucleus. We have
r = n, y, f or T.
= ,c, SJ<&
(45)
An identical relation holds for the primed cross sections. In both cases the potential scattering cross section 2n( 1 -cos 24,)/k’ must be added to the resonance scattering and total cross sections for comparison with experiment. 3. Method of computation A computer code GAMAC I’) has been written which computes multilevel average cross sections and the ratios, Rl(r = n, y, T or f) using a Monte Carlo approach. A set of R-matrix resonance parameters is drawn at random from their known distributions ‘) to form the inverse level matrix for N levels. The eigenvalues and eigenvectors of the inverse level matrix are then obtained. These lead to the Kapur-Peierls resonance parameters and resonance areas for these N levels, using the equations of the previous section. The set of R-matrix resonance parameters for these N levels is also used to obtain the resonance areas neglecting interference. Repetition of this process, until a desired statistical accuracy is obtained, leads to the average areas, the average cross sections and cross-section ratios which we seek. A correction for the effect of the (n, yf) process is made so that the part of the measured fission cross section arising from this process can be treated in the calculation as contributing to neutron capture while the calculated average cross sections include it as fission for comparison with experiment ’ 3, l”). The R-matrix resonance energies for each set of N levels are determined by diagonalization of a real symmetric Hamiltonian whose elements are drawn at random from a Gaussian distribution which has zero mean. Since the eigenvalues of this Hamiltonian follow a semicircular distribution, a simple transformation is used to convert this to a flat distribution. The mean spacing for spin .I is obtained from the following relation D, = ::‘
exp {[(25+
1)2-(2Z+
1)2]/8a2}.
Here Dabs is the observed mean S-wave level spacing including all spin states. The quantity a2 is of the order of 10 [ref. 1‘)I. The value 10 was used in the calculations discussed in this paper. The square roots of the R-matrix partial fission and neutron widths are drawn from a Gaussian distribution with zero mean, and with a variance in each case chosen to yield the assumed mean partial width. The mean R-matrix neutron width at energy E is determined by
= R@)S’D,,
(47)
312
J. D. OARRISON
where the penetrability factor is determined by
P1(E) = ($&o)2l[l +(ka)2,),
l=O 1 = 1,
(48)
and the channel radius by a=
5 v- 4n *
(49)
Here ap is the potential scattering cross section at low energy (ka < 1). The quantity S’ is the strength function for orbital angular momentum 1. For this calculation S’ is assumed independent of spin J. The mean R-matrix fission width for each channel is assumed to be determined by the relation 16a1’) (rt)
= 2 {l+exp [2x(&-E)/w]}-‘.
Here E,,, is the fission threshold energy for this channel. The quantity w is the fission barrier width. It is of the order of 0.5 MeV. The radiation width was taken to be the same for both S- and P-wave resonances. The program allows for different S- and P-wave radiation widths. However, there seems to be insufficient experimental or theoretical information indicating any need for making them different for the fissile nucleides treated here. The following quantities among those specified as input are important for determining the average cross sections Dab,, Tr, So, S ‘, and I$, and w (for each channel). The quantities Dabs and Tr were specified separately according to custom, although only their ratio affects the average cross sections and interference. The observed level spacing was obtained from the low-energy fission cross section where the instrumental energy resolution and Doppler widths were much smaller than the resonance spacings. A large correction for missing levels was applied, based on the work of Martin ‘*), who has calculated low-energy, Doppler broadened fission cross sections for a variety of cases for comparison with experimental cross sections. The value of TV was determined partly by fitting the cross sections and partly by the low-energy resonance parameter data I’). This will be discussed more fully in the next section. The potential scattering cross section at low energy ap was taken to be 10 b for all cases. This is approximately correct for the fissile nuclei as determined by the neighboring doubly even nuclei where ap is easily observed. The quantity a,, affects only the ratio R,. The S- and P-wave strength functions So and S’ were determined by fitting the capture and fission cross sections from w 0.3- x 100 keV. The fission thresholds were primarily taken from the work of Lynn 20), although some adjustment was made in fitting the cross sections. This also will be discussed in the next section. The target and compound nucleus spins are known lg). Prior to performing these calculations, the computer program was tested in a variety of ways. The point cross sections obtained using a set of Kapur-Peierls reso-
NEUTRON
CROSS SECTIONS
313
trance parameters obtained from R-matrix parameters for four resonances agreed with the same cross sections calculated using the same R-matrix parameters and inversion of the inverse level matrix. The average cross sections agreed closely with the program NEARREX “) for values of the parameter Q = 0.0 and Q = 1.O [ref. “)I at two different energies. In addition, all the cross-section ratios approach unity for weak interference as they should. The total cross-section ratio is always unity within the accuracy of the calculations. 4. Results 4.1. THE EXTRAPOLATION
PROCESS
In the calculation of the cross-section ratios it is desirable to represent the physical situation where each resonance is interfering with many (N x co) resonances on each
1.4 -
6
CHANNELS_--“-
3
CHANNELS
_____--
2 CHANNELS____--I
0
8
16 NUMBER
CHANNEL
24 OF
-------
1.40
I .22 I. I 5
I
.070
32
LEVELS
N
Fig. 1. Calculation of the capture ratio Ry as a function of the number of open fission channels and the number of levels N for (r,,>/D, = 0.009 and r,,/D, = 0.04. 0.85 2 CHANNELS
I
a
16 NUMBER
32 OF
LEVELS
N
Fig. 2. Calculation of the fission ratio Rf as a function of the number of open fission channels and the number of levels N for
314
I. D. GARRISON
3-
4 CHANNELS ____---
/_______________
ccc
3 CHANNELS
0
2@i
1 CHANNEL
2 a if z =: z 5 z?
6 CHANNELS 5 CHANNELS -1
-
-2
-
ii ar
-3 I-4
1
I
I
I
I
8
16
32
NUMBER
OF
LEVELS
N
Fig. 3. Calculation of the resonance scattering ratio as a function of the number of open fission channels and the number of levels N for
side of it. In practice, N cannot be too large as the computer time involved in matrix diagonalization increases approximately as the cube of N. To determine the extrapolation to infinite N, the cross-section ratios were obtained for a number of different values of N (8, 16 and 32) and a number of open fission channels (l-7 and 10). These results (for l-6 channels) are shown in figs. l-3 for R,,R,and R,.The values of the input mean resonance parameters are appropriate for ‘2:I-l and “G:IJ with a mean neutron width to spacing ratio of (T,)/D, x 0.009 eV (S-wave at 7000 eV). The changes necessary to apply these results to the other cases will be discussed later. Approximately 10 000 resonances provided the data for the smooth curves of figs. l-3. In figs. 1 and 2 it can be seen that even for N = 32 the interference has not reached the N = 00 value which appears as an extrapolation at the right. No extrapolation was performed for fig, 3. In order to obtain this extrapolation, it was assumed that each level interferes fully, on the average, with an effective number of resonances R, an equal number on each side of it, and no others. Since the resonances at each end of the set N must interfere with a lesser number of resonances, this leads to a reduction of the interference effect. Fig. 4 shows the results of the simple calculation of this effect as a function of N for different values of n. Unity on the ordinate axis means full interference. The value of N at which the curves of figs. 1 and 2 reach one-half their infinite N values is compred with fig. 4 to obtain the approximate number n for different numbers of open fission channels. These values of IZare given in table 1.
315
NEUTRON CROSS SECTIONS
Fig. 4 is then used to obtain
the extrapolated
values for infinite N shown for figs. 1 and
2. 4.2.
DEPENDENCE
OF INTERFERENCE
ON THE
NUMBER
OF FISSION
CHANNELS
In fig. 1 it can be seen that the effect of interference on the capture cross section increases approximately linearly with the number of channels. Comparing fig. 1 with fig. 2, it can be seen that interference is relatively more important in resonance capture
I
8
16
32
NUMBER
OF LEVELS
N
Fig. 4. Calculation of the interference effect as a function of the number of levels Nassuming different numbers n as the effective number of resonances with which each resonance interferes. Interference is defined as the ratio of the interference effect for a particular value of N to the interference effect for infinite N. TABLE Effective
number
Open fission channels rl 1 2 3 4 5 6
1
of interfering
levels
Interfering levels n 3 6 8 10 11 12
than in fission, under the conditions presented here. However, the absolute reduction in the fission cross section by interference is larger than the augmentation of the capture cross section by the amount by which resonance scattering is also increased. The fission cross section is generally much larger than capture. In fig. 2 it can be seen that the interference effect as exhibited by the ratio Rf decreases with increasing number of channels. This occurs because the magnitude of the fission cross section increases and the capture and resonance scattering decrease with increasing numbers of open fission channels. The absolute reduction in the fission cross section increases with increasing numbers of open fission channels. At higher energies there is an increase in the mag-
316
J. 5.
GARRISON
nitude of the reduction in the fission truss section because of the greater augmentation of the scattering cross section. In fig. 3 it can be seen that she scattering ratio l?, can be negative as well as positive. Calculations of this ratio have a rather farge statistical uncertainty for the number of open fission channels near four for the other input conditions as used for fig. 3. These features arise because the average resonance scattering cross sections (a,&> and (cr,,} are both quite small under these conditions and both change from positive to negative with increasing value of the average total width. The reason (a,,> changes from positive to negative with increasing (r,,) is because the quantity cos 24, is sufficiently less than unity that it makes the factor in parenthesis (FAAcos 24, - rllr - rllf) of eq. (28) go from positive to negative with increasing fiL. If cos 24, were unity, the factor in parenthesis wouId always be positive because of eq. 17) or (8). In a simifar fashion co;,> goes from positive to negative with increasing f; as shown by eqs. (17) and (25). ri and rli. are related by the fact that {r;) equals (r,,). This is true since the trace of the inverse Ievei matrix is invariant under the transformation T, eq. (I 3). Because of statistical uncertainties it is not clear exactly where (0,“) becomes zero. It occurs for the number of open fission channels q slightly larger than four here. The quantity (a&,), on the other hand, goes to zero for tl about 5-4. The reason that (a:,) does not go to zero for the same value of q as (tr”,,) is because interference makes the values of the Kapur-Peierls resonance parameters, their average values and distributions in eq. (25) different than the values and distributions of the corresponding R-matrix parameters found in the factor parenthesis in eq. (28). We wish now to indicate how the cross-section ratios vary with the value of the ratio of mean R-matrix neutron width to spacingjBJ and afso how they vary with the R-matrix radiation width to spacing ratio r,/Lr, so that these results can be extrapolated to other conditions of interference. According to eq. (47) <~,>/DJ depends on the strength function and the energy through the penetrability factor. The quantity r,/D, is essentially constant up to 100 keV for resonances of a given spin for each nucleus. It has values in the range RS0.03-0.04 for ‘3iI.I and “i$J. For “~~Pu its values are in the range z 0.005-0,02. The lower values of ‘~:Pu arise principally because of the larger mean level spacings found for this nucleus. Over the ranges of (F,>/DJ and r,/D$ covered in this report, the capture and fission ratios both vary rather slowly. The variation can be treated reasonably as Iinear. In addition the variations with /DJ and F,/D$ will be treated as inde~ndent and wifi be presented as changes from the values R,o and RfO given in figs. I and 2 where V*>/& = 0.009 and r,/DJ z 0.04.
(511 R, = ~,~+0.6(0.009-~r,)~~~)~~~~~0.3(0.04-r~~~~)fl, Rf = R~,+(0.009-(r,)~D,)-t-0.5(0.04-r,/~,). (52) The dependence of the variation in the fission crass-section ratio upon the number of open fission channels Q is less important than for the capture ratio and has been neglected in eq. (52). Eqs. (51) and (52) will allow extrapolation of the ratios given
NEUTRON
CROSS
317
SECTIONS
in figs. 1 and 2 to other values of the average input resonance parameters. Presentation of the dependence of the scattering ratio upon (r,)/DJ and Ty/DJis much more complicated. It has not been given here because of this and because the average resonance scattering is much smaller than the average capture and fission cross sections and the potential scattering cross section. Only around 100 keV does the average neutron width approach the radiation width in magnitude. Even at this energy, the average resonance scattering cross section is small compared to potential scattering (and is negative).
1
IIO0 m
B 2 0’ 5 ,”
J IO
I ENERGY
100
(keV)
Fig. 5. The average capture and fission cross sections of ‘~~lJ.Tl~e histograms are the measured cross sections. The smooth curves are the calculated cross sections. The average resonance parameters used for the calculated cross section are given in the figure. 100 (_
-. ---
--...
---_
235 92” D
0.5 obs ry = 0.035
ILLLLl.L_L-_LLl
, LLlu_L1-.
IO
I
ENERGY
cv ev
-LLLA 100
(keV)
Fig. 6. The average capture and fission cross sections of 235U. 92 The histograms are the measured cross sections. The smooth curves are the calculated cross sections. The average resonance parameters used for the calculated cross sections are given in the figure.
J. D. GARRISON
318
239 94
.I..
-I
=
1.0 x lo-4
Pu
=
0
obs
1.3
x
= 2.1
lo-4
ev
-1111111 IO
I ENERGY
100
(k&J)
Fig. 7. The average capture and fission cross sections of 239Pu. 94 The histograms are the measured cross sections. The smooth curves are the calculated cross sections. The average resonance parameters used for the calculated cross sections are given in the figure.
4.3. FImING
OF THE AVERAGE
CAPTURE
AND FISSION CROSS SECTIONS
In order to obtain the cross-section ratios found for the common fissile nuclei, the average cross sections of these nuclei have been fit by adjusting the average resonance parameters and fission thresholds which are used as input to the computer program. The average capture and fission cross sections of “i;U, “s:U and “SzPu are shown in figs. 5, 6 and 7. The histograms are the experimental average cross sections. For ‘~~IJ, the cross sections below 2 keV are those of Weston er al. 23). Above 30 keV, the ENDF/B file was used 24,25), as evaluated by Boroughs, Craven and Drake 26). Between 2 keV and 30 keV the data required re-evaluation in the light of the results of Weston and the results of our fitting “‘). The cross sections at these energies are a smooth interpolation between Weston et al., and Boroughs et al. For “GZU, the evaluated data of the ENDF/B file was used below 1 keV. Above 1 keV the evaluated data of Schmidt was used 2”). For 2~~Pu, the evaluation of Pitterle, Page and Yamomoto 29) was used. This consists of an adjustment of ENDF/B file data. The smooth curves are a line drawn through what is believed to be the best fit to cross sections. The fitting was done using eight for the number of resonances N. These results were extrapolated to infinite N. The final input variables are shown on the curves, except for the fission thresholds and widths. In fitting the cross sections, the strength functions are rather well determined by the magnitudes of the average capture and fission cross sections at the various energies. The value determined for the P-wave strength function may be slightly high because the small contribution from D-waves has not been included in the calculations.
NEUTRONCROSS
The ratio of the capture
SECTIONS
to fission cross section
319
is determined
by the ratio of the
radiation width to the mean total fission width. The magnitudes of the radiation widths and mean total fission width must be determined by other information. Information concerning the radiation widths comes from measurements of resonance parameters of individual low-energy resonances of these fissile nuclei and neighboring nuclei. Table 2 lists the mean measured radiation width for the low-energy resonances of 233U, 92 ‘z:IJ and ‘ZzPu and the mean of these i9). Included in the table TABLE2 Radiation widths Nucleus
Radiation width (eV) BNL 325-2D 19)
Cameron 30)
0.046
0.039
235~ 92
0.043
0.033
2gPll
0.042
0.032
2;p
mean
0.044 ___ -.
.--_
-..
-
0.035 -
‘;;Th
0.025
0.025
2;;LJ
0.024
0.021
mean
0.024
0.026
~_--
-
are measurements for ‘s;U and ‘z;Th and calculations using the formula of Cameron 30). Although there is expected to be considerable uncertainty in the calculated radiation widths (about a factor of two) they have been included because of the difficulties in obtaining good radiation widths, or other resonance parameters for that matter, for the fissile nuclei, because of interference and overlap of resonances. From the table, we can see that the radiation width for the odd-mass nuclei are expected to be larger than the doubly even nuclei by a factor of about 1.5. The radiation widths for the doubly even nuclei are rather well determined. We expect that the radiation widths to be used for the fissile nuclei then should be about 0.04 eV. Let us now consider the fission thresholds. Northrop et al. 31), Usachev et al. 32), and Eccleshall and Yates 33) have provided experimental information concerning the lowest energy fission thresholds. Wheeler 34) has speculated as to the order of the low-lying transition states of a fissioning doubly even nucleus. The locations of these states have been more firmly established by the work of Lynn 20). Table 3 gives the locations of these levels as suggested by Lynn. Table 4 gives the fission thresholds used for 233U cross sections to the measured 92 7 ‘z:IJ and ‘ZzPu for fitting the calculated cross sections as shown in figs. 5-7. The thresholds which are underlined for ‘;;IJ are somewhat different from what strict adherence to table 3 would yield. These can be considered adjustments to obtain a better fit to the measured cross sections. However, these adjustments should not be considered unique or even necessarily a best set.
320
I. D. GARRISON
They do, however, represent fairly accurately the correct effective number of open channels for the radiation width used, in the sense required by this paper. By this it is meant, for example, that whether we pick for ‘z;lJ four open channels for the 2+ spin and two open channels for the 3+ spin, or three for each, will not make much difference on the capture or the fission ratios obtained for “s$. It is to be noted that the radiation width chosen for “i:IJ as given in fig. 5 of 0.03 eV is somewhat smaller than suggested above. This is a compromise. The alternative is to use 0.04 for F,, and lower the fission thresholds to yield a larger effective number of open fission channels. This wouId raise the capture ratio somewhat and have little effect upon the fission ratio. The barrier width w was taken to be 0.5 MeV in all cases except for the I+ transition state of “$zPu where a value near 0.2 is apparently required to reproduce the variation of the average cross sections with energy. The results presented here are for cross sections averaged over energy intervals which are Iarge compared to the energy separation of the intermediate structure resonances which have been observed in these fissile nuclei 3 ‘*3“). 4.4. CROSS-SECTION
RATIOS
Table 5 gives the cross-section ratios which indicate the effect of the (n, yf) process and interference. Results are given for three different energies. Approximately 300 resonances contributed to the ratios obtained at each energy. The same starting random number was used at each energy to provide a smooth energy variation of the ratios. TABLE3 LyJm’s proposed order for transition states of a fissioning doubly even nucleus 20)
_
---
Energy above lowest fission threshold (MeV) 0.1
State spin and parity
Of, 2+, 4+, 6+, etc.
x 0.5
l-,
x 0.7
2+, 3+, 4+, etc.
x 0.9
l-, 2-, 3-, etc.
= 1.2
2-, 3-, 4-, etc.
X 1.4
I+, 2+, 3+, etc.
x 1.6
c I.7
Excitation energy by thermal neutron capture
3-, 5-, etc.
O+, 2+, etc. 4+. 5+, etc. l-, 2-, 3-, etc.
(
3-. 4-, 5-, etc.
*-
t
2%JU
94
=%I 92 234~ +- 92
TABLE 4 Fission thresholds Nucleus
2;;U
23SU 92
2gPu
Spin and parity
Thresholds (MeV)
2+
-1.5,
-0.8,
3+
-0.8,
-0.2,
-0.3, _-0.0 -0.2
1-
-1.0,
--0.6,
2-
-0.6,
-0.3
-
3-
-1.0,
-0.6,
-0.3
4-
-0.6,
-0.3
0+ 1+ 4+ 5+
-1.5) 0.0
I
(n,yf) only
_-1.5
-0.8
0.0 -
1
3-
-0.6,
-0.3,
4-
-0.3,
-0.1
2+
-1.1.
-0.4,
3f
-0.4,
t-o.3
4+
-1.0,
-0.4,
5+
-0.4,
+0.3
1256-
-0.6
+O.l +0.3 f0.3
\
0+ 1+ 01-
-1.0,
-0.6
2-
-0.6,
-0.3
2+ 3+
- 1.5 -0.7
((n, yf) only
TABLE 5 Cross-section Nucleus
ratios indicating the effect of the (n, yf) process and interference
Energy (kev)
Capture ratios
(n, yf)
R,
Fission ratios
combined
(my0
Rr
combined
0.5
0.86
1.23
1.06
1.02
0.95
0.97
7.0
0.84
1.21
1.02
1.02
0.95
0.97
100.0
0.76
1.15
0.87
1.06
0.92
0.98
0.5
0.87
1.11
0.97
1.04
0.94
0.98
7.0
0.88
1.09
0.96
1.04
0.94
0.98
100.0
0.92
1.09
1.00
1.03
0.92
0.95 1.14
0.5
0.89
1.00
0.89
1.15
0.99
7.0
0.88
1.01
0.89
1.10
0.97
1.07
100.0
0.81
1.09
0.88
1.04
0.94
0.97
322
J. D. GARRISON
The ratios labeled (n, yf) are the ratios of the multilevel average cross sections including the correction for the (n, rf) process to the corresponding cross sections without the (n, yf) correction. The (n, ;tf) calculation is not a Monte Carlo calculation I38l4 )* The ratios in table 5 labeled R, and Rf are the ratios of multilevel (Kapur-Peierls) average cross sections to single-level average cross sections using the same set of Rmatrix resonance parameters. The average cross sections used for these ratios do not include the correction for the (n, yf) process. The ratios in table 5 labeled combined are a product of the previous ratios and are the ratios of the multilevel average cross sections corrected for the (n, yf) process to the single-level average cross sections uncorrected for the (n, rf) process. As has been indicated by Lynn ‘O), the (n, yf) process and interference tend to cancel in their effect on the average cross sections. The interesting point here is that the two effects turn out to have close to the same magnitudes. In this work the level matrix was diagonalized to obtain the Kapur-Peierls parameters. This procedure is rather costly in computer time for large numbers of levels N. A program has been developed by de Saussure and Perez 37) which avoids the diagonalization procedure of obtaining the Kapur-Peierls resonance parameters. This program uses the direct partial fraction expansion of the Reich and Moore 38) expression of the collision matrix. Appreciation is hereby expressed to the members of the Methods Group of Gulf General Atomic Incorporated for their considerable support and encouragement, References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
E. P. Wigner and L. Eisenbud, Phys. Rev. 72 (1947) 29 P. L. Kapur and R. E. Peierls, Proc. Roy. Sot. A166 (1938) 277 J. Humblet and L. Roscnfcld, Nucl. Phys. 26 (1961) 529 H. Feshbach, Ann. of Phys. 5 (1958) 357 C. E. Porter, ed. Statistical theory of spectra: fluctuations (Academic Press, New York, 1965) A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30 (1958) 257 J. D. Garrison, Ann. of Phys. 50 (1968) 355 F. T. Adler and D. B. Adler, Trans. Am. Nucl. Sot. 5 (1962) 53; Proc. Int. Conf. nuclear structure with neutrons (Antwerp, Belgium, July 19-23, 1965) P. A. Moldauer, Phys. Rev. 136 (1964) B947; 171 (1968) 1164 R. A. Freeman and 3. 13. Garrison, Nucl. Phys. A124 (1969) 577 A. M. Weinberg and E. P. Wigner,The physical theory of neutron chain reactors (Univ. of Chicago Press, Chicago, 1958) p. 54 J. D. Garrison and G. L. Boroughs, Gulf General Atomic Incorporated Report GA-9353 (1969) V. Stavinsky and M. 0. Shaker, Nucl. Phys. 62 (1965) 667 J. D. Garrison, Gulf General Atomic Incorporated Report GA-9141 (1968) M. A. Preston, Physics of the nucleus (Addison-Wesley, Reading, Mass., 1962) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 J. E. Lynn, Nuclear data for reactors (Proc. Conf. Paris, 1966) vol. 2 (IAEA, Vienna, 1967) p. 89 W. J. Martin, Masters thesis, San Diego State (1970) J. Stehn ef al., Brookhaven National Lab. Report BNL 325, second edition, suppl. 2, vol. 3 (1965)
NEUTRON
CROSS
SECTIONS
323
20) J. E. Lynn, Theory of neutron resonance reactions (Clarendon Press, Oxford, i968) p. 396 21) P. A. Modauer, C. A. Engelbrecht and G. J. Duffy, Argonne National Laboratory Report ANL-6978 (1964) 22) P. A. Moldauer, Phys. Rev. 135 (1964) 8642 23) L. W. Weston et al., Oak Ridge National Laboratory Report ORNL-TM-2140 (1968) 24) H. C. Honeck, Brookhaven National Laboratory Report BNL-50066 (1966) 25) Status Report on the Evaluated Nuclear Data File (ENDF/B), Special Session, Trans. Am. Nucl. Sot. 11 (1968) 166 26) G. L. Boroughs, C. W. Craven, Jr. and M. K. Drake, Gulf General Atomic Incorporated Report GA-8854 27) J. D. Garrison and D. Mathews, Gulf General Atomic Report GAMD-9615 (1969) 28) J. J. Schmidt, Neutron Cross Sections for Fast Reactor Materials, Part I: Evaluation, KFK 120 (EANDC-E-35-U) (Feb. 1966) 29) T. A. Pitterle, E. M. Page and M. Yamomoto, Atomic Power Development Associates, Inc. Report APDA-216, Vol. 1 (1968) 30) A. G. W. Cameron, Can. J. Phys. 36 (1956) 666 31) J. A. Northrup, R. H. Stokes and K. Boyer, Phys. Rev. 99 (1955) 616 32) L. N. Usachev. V. A. Paulinchuk and N. S. Rabotnov, Atomn. Energ. 17 (1968) 479 33) Eccleschall and Yates, Proc. Symp. on physics and chemistry of fission, Salzburg (IAEA, Vienna, 1965) p. 77 34) J. A. Wheeler, in J. L. Fowler and J. B. Marion, eds., Fast neutron Physics, vol. 2 (Interscience, New York, 1963) p. 2051 35) B. H. Patrick and G. D. James, Phys. Lett. 288 (1968) 258 36) M. G. Cao, E. Migneco and J. P. Theobald, Phys. Lett. 27B (1968) 409 37) G. de Saussure and R. B. Perez, Oak Ridge National Laboratory Report, ORNL-TM-2599 (June 1969) 38) C. W. Reich and M. S. Moore, Phys. Rev. 111 (1958) 929
2.5
1
Nuclear Physics A159 (1970) 305 -- 323; @ Not fo be reproduced
THE EFFECT
by photoprint
SECTIONS JOHN
Publishing
or microfilm without written permission
OF INTERFERENCE CROSS
North-Holland
ON THE AVERAGE OF FISSILE
Co., Amsterdum from the publisher
NEUTRON
NUCLEI
D. GARRISON
GulfGeneral Atomic Incorporated Received
and San Diego State 13 April
College,
Californiu
1970
Abstract: The ratio of the average fission, capture and resonance scattering cross sections to the corresponding average cross sections obtained neglecting resonance-resonance interference have been calculated by a Monte Carlo method for a variety of interference conditions typical of fissile nuclei. It is shown in general that the average total cross section is unaffected by interference. The importance of interference increases with the number of open fission channels. The effective number of resonances with which a given resonance interferes is obtained for different numbers of open channels. Results for the cross-section ratios for both S- and P-wave neutrons have been obtained for the common tissile nuclei by fitting calculated cross sections to the average capture and fission cross sections over the neutron energy range from 0.3-100 keV. The ratios
for S-wave
R _,,, = 1.23+0.03, R sEa,l = 1.10+0.01;
R,..,, ‘s:Pu:
neutrons
at low energy
= l.45110.10;
(z
2;;U:
RIirl 1 0.99=0.005,
R,.,,
500 eV) are:
2s:U:
Rr,u 7 0.94+0.01, = 1.00~0.002,
The process of cross-section fitting also yields values for the average these nuclei. The effect of the (n,yf) process is presented.
Rrl,, = 0.95hO.01, R,.,,
.= 1.1130.01,
and R,.,,
= 1.02_CO.o05.
resonance
parameters
for
1. Introduction Resonance peaks are commonly observed in the collision cross sections of nuclear particles. Theories of nuclear reactions have beeen developed which successfully reproduce the energy dependence of these cross sections and provide a mechanism for understanding resonance phenomena rd4). When the widths of the resonance peaks are small compared to the resonance spacing, these cross sections exhibit the simple, Breit-Wigner energy dependence in the vicinity of each resonance, with a slowly varying energy dependence between resonances. When the resonance widths are of the order of the resonance spacing, or larger, the energy dependence of these cross sections is considerably more complicated; the cross section exhibits resonanceresonance interference. The interference between overlapping resonances, in addition to causing a more complicated energy dependence of the cross section, affects the average partial cross sections. It is found, that the areas under the resonances are altered by interference from what they would be for a simple Breit-Wigner energy dependence. These areas are changed in such a way that the average area and thus the average cross section is changed. It is the purpose of this paper to study the effect of interference on the average cross sections of fissile nuclei. There are two reasons for pursuing this study. One is to 305
306
J. D. GARRISON
obtain a better understanding of the phenomenon of resonance-resonance interference. The other is to provide information of interest in the design of nuclear reactors. To perform this calculation, the R-matrix theory of Wigner and Eisenbud ‘) and the distributions the theory of Kapur and Peierls “) have been used. In addition, for the R-matrix resonance parameters are required ‘). The R-matrix resonance parameters serve to determine the resonance areas and average cross sections which would have been obtained without resonance-resonance interference, while the KapurPeierls resonance parameters serve to determine the correct resonance areas and average cross sections including the effect of interference. The Kapur-Peierls resonance parameters are obtained from the corresponding R-matrix parameters using the mathematical connection between the R-matrix and Kapur-Peierls theories 6-8). Some evidence concerning the effect of interference on average cross sections was discussed in an earlier work which was limited to interference between just two resonances ‘). This paper extends the earlier work to include many interfering resonances and a broader treatment of the phenomenon, including both S- and P-wave neutrons (I = 0 and I = 1). Specific application of these results to ‘FLU, ‘i:lJ and ‘GiPu is included. In the next section the pertinent equations are presented. A more complete discussion of many of the equations has been given in the earlier work ‘). In the remaining sections the method of calculation and results are discussed.
2. Theory The neutron total, scattering, total fission compound nucleus of spin J are
and total capture
ai, = _?I1- ~;l”12~ k2
ai, = z J$ IUiC12,
cross sections
for a
(2)
r = f or y.
In eq. (3) the sum over channels is over all the fission channels (f) for the total fission cross section, and is over all the capture channels (r) for the total capture cross section. These and the R-matrix equations which follow are presented in standard is the collision matrix for entrance channel c’ (in this notation “). The quantity UiC paper c’ = n the neutron channel) and c is the exit channel. The quantity k is the magnitude of the propagation vector. In R-matrix theory, the collision matrix is given by
(4)
NEUTRON
CROSS
wl
SECTIONS
The quantity TLC is the partial level width for level % and channel c. The quantity is the potential scattering phase shift for channel c. For the neutron channel
The inverse of the level matrix
A,, is given by
(A-‘);., where the width matrix
$C
= (E,-Q%,--N-j.,,
(6)
rrlv = C r&CC. C
(7)
is given by
The quantitites E and EA are the c.m. energy and resonance energy for level 1. The shift matrix AA,,, normally found in eq. (6), has been set equal to zero for both S- and P-wave neutrons at all energies, although this procedure can be done exactly only for S-wave neutrons 6v‘). For the purpose of calculating the total fission and total capture cross sections, the sum in eq. (7) is separated into its contributions from fission, capture and scattering
(8) where (9)
(11) For all but the P-wave levels of intermediate spin, the sum in eq. (11) consists of a single term. In the calculations performed here, eq. (10) has been assumed to give the elements of a diagonal matrix
r.
AYY
The total radiation
=
rydAv.
width ry is taken to be a constant,
(14 the same for all levels.
For the purposes of obtaining resonance areas and average cross sections, it is desirable to use the Kapur-Peierls form of the cross-section equations. Since the joint probability distribution of Kapur-Peierls resonance parameters is not known direct9,10), the distribution of the R-matrix parameters and the mathematical connection lY between the R-matrix and Kapur-Peierls resonance parameters must be used. The connection between the R-matrix and Kapur-Peierls resonance parameters is through the complex orthogonal matrix transformation which diagonalizes the level matrix and its inverse &(T-‘),,(A-‘),,T,, = &,-E-W;)& (13)
J. D.
308
GARRISON
The quantities EL and rh are the Kapur-Peierls resonance energy and total width for level p. For a complex orthogonal transformation, the inverse matrix is the transpose matrix, i.e., (T- l)pl equals T,,.This fact is used to obtain eq. (14) below. Using the transformation matrix T,the collision matrix of eq. (4) can be expressed as (14)
’
where
The Kapur-Peierls formulation of the cross sections follows on combining eqs. (l)-(3), (14) and (15) [some of the equations differ in form from those of ref. ‘) to more nearly match the form of the single level eqs. (17)-(29) which follow].
(16) r;,
cos
II
24, rg - 2 sin 24, IA(E;- E)
d;=; T p Cr--ZIL(W] , (E;-Ey+(gg2
r = f or y,
(E;-E)'+(+r;)'
,
(17)
(18)
where r;, = i [Re fA, - Im tACtg 24,], c I;,
[Re Frc + Im f,, ctg 2&l,
= - i c
w;
r;=L$x
I, =c [
+ wiyr
&
(E:,-E:)PJ,,,-t(r;+r:)Q:,,
y
1 1’
+(E; - E:)c%,
In ” C (E;-q2+[+(r;+r:)]2
(E;-E:)2+[+(r;+r:)]2
Qk, = Im Jk, 9 ,
pi,, = Re Rk,,
(19) (20)
(21)
(22) (23) (24) (25) (26)
In the case of widely separated resonances, interference can be neglected, which means that the off-diagonal elements of the level matrix and its inverse can be taken to be zero. The cross-section equations for this case are given by a sum of single-level
NEUTRON
CROSS
SECTIONS
309
Breit-Wigner terms plus a potential scattering term, involving the R-matrix parameters directly. J OnT
x
(
=
k”(20 --OS
-2
2(1-cos
24,)+
F
c [L
vL)+
~0s 24, Lj.
1
(EL
-
- 2 sin 24, L(J% -4) El2
,
(27)
+w,)2
II (28)
rZ”(rllcos 24”- rlly - rAlf) - 2 sin24, r&~ - E) (E,-E)‘+(~LA)~
CT;, =
(r=for
VI*
It can be seen that eqs. (16)-(18) and (27)-(29) are similar in form. The difference between them is the effect of resonance-resonance interference. Foreqs. (27~(29), the only interference present is that between resonance and potential scattering. The areas under a single resonance I of eqs. (16)-( 18) are given by the usual singlelevel area formulas I1 ) (30) (31)
&J’An_ - &J,AT _&Jr-&J! If AY
*
(32)
Here, h and m are Planck’s constant and the reduced mass. The average resonance cross sections are obtained from the average areas by (4;)
= (CEY.!?,
DJ
r = y, f, n or T,
(33)
where the quantity DJ is the mean level spacing for resonances of spin J. Since we wish the average to be determined at a particular energy, all areas are evaluated at the same energy. The area equations and average resonance cross sections corresponding to eqs. (30)-(33) for the no interference case are &J
= AT
&i, = ‘&:, al;,
(cT;l,)=
bn cos2&
f
4m
El
’
r.Ur
rlAcos 24,
r = y or f, ’
= ‘GJ;T-L+djf)
9
)
r =
(34) (35) (36)
y,
f, n or T.
(37)
J. D. GARRISON
310
For a description of the effect of interference on the average resonance cross sections, the following ratios will be useful (38) It will now be shown that R, is unity under all conditions. That is, the average total cross section is unchanged by interference. From eqs. (30) (33) (34) and (37), RT is unity provided (r,,) = (f;,). Using eq. (19), we can write I-;* cos 24” = Re i f’le exp (2@,). c
(39)
The average of eq. (39) over N levels is
(40)
From eq. (I 5) we have PA, = z: T,, TV,r&r&.
(41)
P.y
From the properties of the transformation
matrix T and its inverse we have
1 T,, TV, = 6,, .
(42)
i.
Inserting eqs. (41) and (42) into eq. (40) and summing over v and i., we have (r;,)
cos 24, = Re i exp (2i&) i T Trc = (r,,) c
cos 24,)
(43)
Q.E.D. The averages of the two neutron widths is the same for any N. This is meant in the mathematical sense where a set of N interfering resonances is selected. For actual nuclei we expect N % cc. The relationship between the average partial cross sections with interference and neglecting interference is much more complicated. Combining eqs. (21), (30) and (31), the capture and fission areas are given by
Eq. (44) indicates that the fission and capture areas and thus the ratios are independent of #, and thus independent of the channel radius a. They do depend on the R-matrix widths and resonance energies which go into the inverse level matrix. The scattering areas and ratios do depend upon 4,, and a. It is easy to show, but will not be shown here, that the interference effects do not depend upon the individual magnitudes of the average R-matrix partial level widths and spacings, but rather depend upon the magnitudes of the ratios of the average widths to spacings.
NEUTRON
CROSS
311
SECTIONS
For comparison with experiment, the partial cross sections for different values of J and I must be combined. This is done using the statistical weighting factor gJ = (25+ 1)/2(21+ l), where I is the spin of the target nucleus. We have
r = n, y, f or T.
= ,c, SJ<&
(45)
An identical relation holds for the primed cross sections. In both cases the potential scattering cross section 2n( 1 -cos 24,)/k’ must be added to the resonance scattering and total cross sections for comparison with experiment. 3. Method of computation A computer code GAMAC I’) has been written which computes multilevel average cross sections and the ratios, Rl(r = n, y, T or f) using a Monte Carlo approach. A set of R-matrix resonance parameters is drawn at random from their known distributions ‘) to form the inverse level matrix for N levels. The eigenvalues and eigenvectors of the inverse level matrix are then obtained. These lead to the Kapur-Peierls resonance parameters and resonance areas for these N levels, using the equations of the previous section. The set of R-matrix resonance parameters for these N levels is also used to obtain the resonance areas neglecting interference. Repetition of this process, until a desired statistical accuracy is obtained, leads to the average areas, the average cross sections and cross-section ratios which we seek. A correction for the effect of the (n, yf) process is made so that the part of the measured fission cross section arising from this process can be treated in the calculation as contributing to neutron capture while the calculated average cross sections include it as fission for comparison with experiment ’ 3, l”). The R-matrix resonance energies for each set of N levels are determined by diagonalization of a real symmetric Hamiltonian whose elements are drawn at random from a Gaussian distribution which has zero mean. Since the eigenvalues of this Hamiltonian follow a semicircular distribution, a simple transformation is used to convert this to a flat distribution. The mean spacing for spin .I is obtained from the following relation D, = ::‘
exp {[(25+
1)2-(2Z+
1)2]/8a2}.
Here Dabs is the observed mean S-wave level spacing including all spin states. The quantity a2 is of the order of 10 [ref. 1‘)I. The value 10 was used in the calculations discussed in this paper. The square roots of the R-matrix partial fission and neutron widths are drawn from a Gaussian distribution with zero mean, and with a variance in each case chosen to yield the assumed mean partial width. The mean R-matrix neutron width at energy E is determined by
= R@)S’D,,
(47)
312
J. D. OARRISON
where the penetrability factor is determined by
P1(E) = ($&o)2l[l +(ka)2,),
l=O 1 = 1,
(48)
and the channel radius by a=
5 v- 4n *
(49)
Here ap is the potential scattering cross section at low energy (ka < 1). The quantity S’ is the strength function for orbital angular momentum 1. For this calculation S’ is assumed independent of spin J. The mean R-matrix fission width for each channel is assumed to be determined by the relation 16a1’) (rt)
= 2 {l+exp [2x(&-E)/w]}-‘.
Here E,,, is the fission threshold energy for this channel. The quantity w is the fission barrier width. It is of the order of 0.5 MeV. The radiation width was taken to be the same for both S- and P-wave resonances. The program allows for different S- and P-wave radiation widths. However, there seems to be insufficient experimental or theoretical information indicating any need for making them different for the fissile nucleides treated here. The following quantities among those specified as input are important for determining the average cross sections Dab,, Tr, So, S ‘, and I$, and w (for each channel). The quantities Dabs and Tr were specified separately according to custom, although only their ratio affects the average cross sections and interference. The observed level spacing was obtained from the low-energy fission cross section where the instrumental energy resolution and Doppler widths were much smaller than the resonance spacings. A large correction for missing levels was applied, based on the work of Martin ‘*), who has calculated low-energy, Doppler broadened fission cross sections for a variety of cases for comparison with experimental cross sections. The value of TV was determined partly by fitting the cross sections and partly by the low-energy resonance parameter data I’). This will be discussed more fully in the next section. The potential scattering cross section at low energy ap was taken to be 10 b for all cases. This is approximately correct for the fissile nuclei as determined by the neighboring doubly even nuclei where ap is easily observed. The quantity a,, affects only the ratio R,. The S- and P-wave strength functions So and S’ were determined by fitting the capture and fission cross sections from w 0.3- x 100 keV. The fission thresholds were primarily taken from the work of Lynn 20), although some adjustment was made in fitting the cross sections. This also will be discussed in the next section. The target and compound nucleus spins are known lg). Prior to performing these calculations, the computer program was tested in a variety of ways. The point cross sections obtained using a set of Kapur-Peierls reso-
NEUTRON
CROSS SECTIONS
313
trance parameters obtained from R-matrix parameters for four resonances agreed with the same cross sections calculated using the same R-matrix parameters and inversion of the inverse level matrix. The average cross sections agreed closely with the program NEARREX “) for values of the parameter Q = 0.0 and Q = 1.O [ref. “)I at two different energies. In addition, all the cross-section ratios approach unity for weak interference as they should. The total cross-section ratio is always unity within the accuracy of the calculations. 4. Results 4.1. THE EXTRAPOLATION
PROCESS
In the calculation of the cross-section ratios it is desirable to represent the physical situation where each resonance is interfering with many (N x co) resonances on each
1.4 -
6
CHANNELS_--“-
3
CHANNELS
_____--
2 CHANNELS____--I
0
8
16 NUMBER
CHANNEL
24 OF
-------
1.40
I .22 I. I 5
I
.070
32
LEVELS
N
Fig. 1. Calculation of the capture ratio Ry as a function of the number of open fission channels and the number of levels N for (r,,>/D, = 0.009 and r,,/D, = 0.04. 0.85 2 CHANNELS
I
a
16 NUMBER
32 OF
LEVELS
N
Fig. 2. Calculation of the fission ratio Rf as a function of the number of open fission channels and the number of levels N for
314
I. D. GARRISON
3-
4 CHANNELS ____---
/_______________
ccc
3 CHANNELS
0
2@i
1 CHANNEL
2 a if z =: z 5 z?
6 CHANNELS 5 CHANNELS -1
-
-2
-
ii ar
-3 I-4
1
I
I
I
I
8
16
32
NUMBER
OF
LEVELS
N
Fig. 3. Calculation of the resonance scattering ratio as a function of the number of open fission channels and the number of levels N for
side of it. In practice, N cannot be too large as the computer time involved in matrix diagonalization increases approximately as the cube of N. To determine the extrapolation to infinite N, the cross-section ratios were obtained for a number of different values of N (8, 16 and 32) and a number of open fission channels (l-7 and 10). These results (for l-6 channels) are shown in figs. l-3 for R,,R,and R,.The values of the input mean resonance parameters are appropriate for ‘2:I-l and “G:IJ with a mean neutron width to spacing ratio of (T,)/D, x 0.009 eV (S-wave at 7000 eV). The changes necessary to apply these results to the other cases will be discussed later. Approximately 10 000 resonances provided the data for the smooth curves of figs. l-3. In figs. 1 and 2 it can be seen that even for N = 32 the interference has not reached the N = 00 value which appears as an extrapolation at the right. No extrapolation was performed for fig, 3. In order to obtain this extrapolation, it was assumed that each level interferes fully, on the average, with an effective number of resonances R, an equal number on each side of it, and no others. Since the resonances at each end of the set N must interfere with a lesser number of resonances, this leads to a reduction of the interference effect. Fig. 4 shows the results of the simple calculation of this effect as a function of N for different values of n. Unity on the ordinate axis means full interference. The value of N at which the curves of figs. 1 and 2 reach one-half their infinite N values is compred with fig. 4 to obtain the approximate number n for different numbers of open fission channels. These values of IZare given in table 1.
315
NEUTRON CROSS SECTIONS
Fig. 4 is then used to obtain
the extrapolated
values for infinite N shown for figs. 1 and
2. 4.2.
DEPENDENCE
OF INTERFERENCE
ON THE
NUMBER
OF FISSION
CHANNELS
In fig. 1 it can be seen that the effect of interference on the capture cross section increases approximately linearly with the number of channels. Comparing fig. 1 with fig. 2, it can be seen that interference is relatively more important in resonance capture
I
8
16
32
NUMBER
OF LEVELS
N
Fig. 4. Calculation of the interference effect as a function of the number of levels Nassuming different numbers n as the effective number of resonances with which each resonance interferes. Interference is defined as the ratio of the interference effect for a particular value of N to the interference effect for infinite N. TABLE Effective
number
Open fission channels rl 1 2 3 4 5 6
1
of interfering
levels
Interfering levels n 3 6 8 10 11 12
than in fission, under the conditions presented here. However, the absolute reduction in the fission cross section by interference is larger than the augmentation of the capture cross section by the amount by which resonance scattering is also increased. The fission cross section is generally much larger than capture. In fig. 2 it can be seen that the interference effect as exhibited by the ratio Rf decreases with increasing number of channels. This occurs because the magnitude of the fission cross section increases and the capture and resonance scattering decrease with increasing numbers of open fission channels. The absolute reduction in the fission cross section increases with increasing numbers of open fission channels. At higher energies there is an increase in the mag-
316
J. 5.
GARRISON
nitude of the reduction in the fission truss section because of the greater augmentation of the scattering cross section. In fig. 3 it can be seen that she scattering ratio l?, can be negative as well as positive. Calculations of this ratio have a rather farge statistical uncertainty for the number of open fission channels near four for the other input conditions as used for fig. 3. These features arise because the average resonance scattering cross sections (a,&> and (cr,,} are both quite small under these conditions and both change from positive to negative with increasing value of the average total width. The reason (a,,> changes from positive to negative with increasing (r,,) is because the quantity cos 24, is sufficiently less than unity that it makes the factor in parenthesis (FAAcos 24, - rllr - rllf) of eq. (28) go from positive to negative with increasing fiL. If cos 24, were unity, the factor in parenthesis wouId always be positive because of eq. 17) or (8). In a simifar fashion co;,> goes from positive to negative with increasing f; as shown by eqs. (17) and (25). ri and rli. are related by the fact that {r;) equals (r,,). This is true since the trace of the inverse Ievei matrix is invariant under the transformation T, eq. (I 3). Because of statistical uncertainties it is not clear exactly where (0,“) becomes zero. It occurs for the number of open fission channels q slightly larger than four here. The quantity (a&,), on the other hand, goes to zero for tl about 5-4. The reason that (a:,) does not go to zero for the same value of q as (tr”,,) is because interference makes the values of the Kapur-Peierls resonance parameters, their average values and distributions in eq. (25) different than the values and distributions of the corresponding R-matrix parameters found in the factor parenthesis in eq. (28). We wish now to indicate how the cross-section ratios vary with the value of the ratio of mean R-matrix neutron width to spacing
(511 R, = ~,~+0.6(0.009-~r,)~~~)~~~~~0.3(0.04-r~~~~)fl, Rf = R~,+(0.009-(r,)~D,)-t-0.5(0.04-r,/~,). (52) The dependence of the variation in the fission crass-section ratio upon the number of open fission channels Q is less important than for the capture ratio and has been neglected in eq. (52). Eqs. (51) and (52) will allow extrapolation of the ratios given
NEUTRON
CROSS
317
SECTIONS
in figs. 1 and 2 to other values of the average input resonance parameters. Presentation of the dependence of the scattering ratio upon (r,)/DJ and Ty/DJis much more complicated. It has not been given here because of this and because the average resonance scattering is much smaller than the average capture and fission cross sections and the potential scattering cross section. Only around 100 keV does the average neutron width approach the radiation width in magnitude. Even at this energy, the average resonance scattering cross section is small compared to potential scattering (and is negative).
1
IIO0 m
B 2 0’ 5 ,”
J IO
I ENERGY
100
(keV)
Fig. 5. The average capture and fission cross sections of ‘~~lJ.Tl~e histograms are the measured cross sections. The smooth curves are the calculated cross sections. The average resonance parameters used for the calculated cross section are given in the figure. 100 (_
-. ---
--...
---_
235 92” D
0.5 obs ry = 0.035
ILLLLl.L_L-_LLl
, LLlu_L1-.
IO
I
ENERGY
cv ev
-LLLA 100
(keV)
Fig. 6. The average capture and fission cross sections of 235U. 92 The histograms are the measured cross sections. The smooth curves are the calculated cross sections. The average resonance parameters used for the calculated cross sections are given in the figure.
J. D. GARRISON
318
239 94
.I..
-I
=
1.0 x lo-4
Pu
=
0
obs
1.3
x
= 2.1
lo-4
ev
-1111111 IO
I ENERGY
100
(k&J)
Fig. 7. The average capture and fission cross sections of 239Pu. 94 The histograms are the measured cross sections. The smooth curves are the calculated cross sections. The average resonance parameters used for the calculated cross sections are given in the figure.
4.3. FImING
OF THE AVERAGE
CAPTURE
AND FISSION CROSS SECTIONS
In order to obtain the cross-section ratios found for the common fissile nuclei, the average cross sections of these nuclei have been fit by adjusting the average resonance parameters and fission thresholds which are used as input to the computer program. The average capture and fission cross sections of “i;U, “s:U and “SzPu are shown in figs. 5, 6 and 7. The histograms are the experimental average cross sections. For ‘~~IJ, the cross sections below 2 keV are those of Weston er al. 23). Above 30 keV, the ENDF/B file was used 24,25), as evaluated by Boroughs, Craven and Drake 26). Between 2 keV and 30 keV the data required re-evaluation in the light of the results of Weston and the results of our fitting “‘). The cross sections at these energies are a smooth interpolation between Weston et al., and Boroughs et al. For “GZU, the evaluated data of the ENDF/B file was used below 1 keV. Above 1 keV the evaluated data of Schmidt was used 2”). For 2~~Pu, the evaluation of Pitterle, Page and Yamomoto 29) was used. This consists of an adjustment of ENDF/B file data. The smooth curves are a line drawn through what is believed to be the best fit to cross sections. The fitting was done using eight for the number of resonances N. These results were extrapolated to infinite N. The final input variables are shown on the curves, except for the fission thresholds and widths. In fitting the cross sections, the strength functions are rather well determined by the magnitudes of the average capture and fission cross sections at the various energies. The value determined for the P-wave strength function may be slightly high because the small contribution from D-waves has not been included in the calculations.
NEUTRONCROSS
The ratio of the capture
SECTIONS
to fission cross section
319
is determined
by the ratio of the
radiation width to the mean total fission width. The magnitudes of the radiation widths and mean total fission width must be determined by other information. Information concerning the radiation widths comes from measurements of resonance parameters of individual low-energy resonances of these fissile nuclei and neighboring nuclei. Table 2 lists the mean measured radiation width for the low-energy resonances of 233U, 92 ‘z:IJ and ‘ZzPu and the mean of these i9). Included in the table TABLE2 Radiation widths Nucleus
Radiation width (eV) BNL 325-2D 19)
Cameron 30)
0.046
0.039
235~ 92
0.043
0.033
2gPll
0.042
0.032
2;p
mean
0.044 ___ -.
.--_
-..
-
0.035 -
‘;;Th
0.025
0.025
2;;LJ
0.024
0.021
mean
0.024
0.026
~_--
-
are measurements for ‘s;U and ‘z;Th and calculations using the formula of Cameron 30). Although there is expected to be considerable uncertainty in the calculated radiation widths (about a factor of two) they have been included because of the difficulties in obtaining good radiation widths, or other resonance parameters for that matter, for the fissile nuclei, because of interference and overlap of resonances. From the table, we can see that the radiation width for the odd-mass nuclei are expected to be larger than the doubly even nuclei by a factor of about 1.5. The radiation widths for the doubly even nuclei are rather well determined. We expect that the radiation widths to be used for the fissile nuclei then should be about 0.04 eV. Let us now consider the fission thresholds. Northrop et al. 31), Usachev et al. 32), and Eccleshall and Yates 33) have provided experimental information concerning the lowest energy fission thresholds. Wheeler 34) has speculated as to the order of the low-lying transition states of a fissioning doubly even nucleus. The locations of these states have been more firmly established by the work of Lynn 20). Table 3 gives the locations of these levels as suggested by Lynn. Table 4 gives the fission thresholds used for 233U cross sections to the measured 92 7 ‘z:IJ and ‘ZzPu for fitting the calculated cross sections as shown in figs. 5-7. The thresholds which are underlined for ‘;;IJ are somewhat different from what strict adherence to table 3 would yield. These can be considered adjustments to obtain a better fit to the measured cross sections. However, these adjustments should not be considered unique or even necessarily a best set.
320
I. D. GARRISON
They do, however, represent fairly accurately the correct effective number of open channels for the radiation width used, in the sense required by this paper. By this it is meant, for example, that whether we pick for ‘z;lJ four open channels for the 2+ spin and two open channels for the 3+ spin, or three for each, will not make much difference on the capture or the fission ratios obtained for “s$. It is to be noted that the radiation width chosen for “i:IJ as given in fig. 5 of 0.03 eV is somewhat smaller than suggested above. This is a compromise. The alternative is to use 0.04 for F,, and lower the fission thresholds to yield a larger effective number of open fission channels. This wouId raise the capture ratio somewhat and have little effect upon the fission ratio. The barrier width w was taken to be 0.5 MeV in all cases except for the I+ transition state of “$zPu where a value near 0.2 is apparently required to reproduce the variation of the average cross sections with energy. The results presented here are for cross sections averaged over energy intervals which are Iarge compared to the energy separation of the intermediate structure resonances which have been observed in these fissile nuclei 3 ‘*3“). 4.4. CROSS-SECTION
RATIOS
Table 5 gives the cross-section ratios which indicate the effect of the (n, yf) process and interference. Results are given for three different energies. Approximately 300 resonances contributed to the ratios obtained at each energy. The same starting random number was used at each energy to provide a smooth energy variation of the ratios. TABLE3 LyJm’s proposed order for transition states of a fissioning doubly even nucleus 20)
_
---
Energy above lowest fission threshold (MeV) 0.1
State spin and parity
Of, 2+, 4+, 6+, etc.
x 0.5
l-,
x 0.7
2+, 3+, 4+, etc.
x 0.9
l-, 2-, 3-, etc.
= 1.2
2-, 3-, 4-, etc.
X 1.4
I+, 2+, 3+, etc.
x 1.6
c I.7
Excitation energy by thermal neutron capture
3-, 5-, etc.
O+, 2+, etc. 4+. 5+, etc. l-, 2-, 3-, etc.
(
3-. 4-, 5-, etc.
*-
t
2%JU
94
=%I 92 234~ +- 92
TABLE 4 Fission thresholds Nucleus
2;;U
23SU 92
2gPu
Spin and parity
Thresholds (MeV)
2+
-1.5,
-0.8,
3+
-0.8,
-0.2,
-0.3, _-0.0 -0.2
1-
-1.0,
--0.6,
2-
-0.6,
-0.3
-
3-
-1.0,
-0.6,
-0.3
4-
-0.6,
-0.3
0+ 1+ 4+ 5+
-1.5) 0.0
I
(n,yf) only
_-1.5
-0.8
0.0 -
1
3-
-0.6,
-0.3,
4-
-0.3,
-0.1
2+
-1.1.
-0.4,
3f
-0.4,
t-o.3
4+
-1.0,
-0.4,
5+
-0.4,
+0.3
1256-
-0.6
+O.l +0.3 f0.3
\
0+ 1+ 01-
-1.0,
-0.6
2-
-0.6,
-0.3
2+ 3+
- 1.5 -0.7
((n, yf) only
TABLE 5 Cross-section Nucleus
ratios indicating the effect of the (n, yf) process and interference
Energy (kev)
Capture ratios
(n, yf)
R,
Fission ratios
combined
(my0
Rr
combined
0.5
0.86
1.23
1.06
1.02
0.95
0.97
7.0
0.84
1.21
1.02
1.02
0.95
0.97
100.0
0.76
1.15
0.87
1.06
0.92
0.98
0.5
0.87
1.11
0.97
1.04
0.94
0.98
7.0
0.88
1.09
0.96
1.04
0.94
0.98
100.0
0.92
1.09
1.00
1.03
0.92
0.95 1.14
0.5
0.89
1.00
0.89
1.15
0.99
7.0
0.88
1.01
0.89
1.10
0.97
1.07
100.0
0.81
1.09
0.88
1.04
0.94
0.97
322
J. D. GARRISON
The ratios labeled (n, yf) are the ratios of the multilevel average cross sections including the correction for the (n, rf) process to the corresponding cross sections without the (n, yf) correction. The (n, ;tf) calculation is not a Monte Carlo calculation I38l4 )* The ratios in table 5 labeled R, and Rf are the ratios of multilevel (Kapur-Peierls) average cross sections to single-level average cross sections using the same set of Rmatrix resonance parameters. The average cross sections used for these ratios do not include the correction for the (n, yf) process. The ratios in table 5 labeled combined are a product of the previous ratios and are the ratios of the multilevel average cross sections corrected for the (n, yf) process to the single-level average cross sections uncorrected for the (n, rf) process. As has been indicated by Lynn ‘O), the (n, yf) process and interference tend to cancel in their effect on the average cross sections. The interesting point here is that the two effects turn out to have close to the same magnitudes. In this work the level matrix was diagonalized to obtain the Kapur-Peierls parameters. This procedure is rather costly in computer time for large numbers of levels N. A program has been developed by de Saussure and Perez 37) which avoids the diagonalization procedure of obtaining the Kapur-Peierls resonance parameters. This program uses the direct partial fraction expansion of the Reich and Moore 38) expression of the collision matrix. Appreciation is hereby expressed to the members of the Methods Group of Gulf General Atomic Incorporated for their considerable support and encouragement, References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
E. P. Wigner and L. Eisenbud, Phys. Rev. 72 (1947) 29 P. L. Kapur and R. E. Peierls, Proc. Roy. Sot. A166 (1938) 277 J. Humblet and L. Roscnfcld, Nucl. Phys. 26 (1961) 529 H. Feshbach, Ann. of Phys. 5 (1958) 357 C. E. Porter, ed. Statistical theory of spectra: fluctuations (Academic Press, New York, 1965) A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30 (1958) 257 J. D. Garrison, Ann. of Phys. 50 (1968) 355 F. T. Adler and D. B. Adler, Trans. Am. Nucl. Sot. 5 (1962) 53; Proc. Int. Conf. nuclear structure with neutrons (Antwerp, Belgium, July 19-23, 1965) P. A. Moldauer, Phys. Rev. 136 (1964) B947; 171 (1968) 1164 R. A. Freeman and 3. 13. Garrison, Nucl. Phys. A124 (1969) 577 A. M. Weinberg and E. P. Wigner,The physical theory of neutron chain reactors (Univ. of Chicago Press, Chicago, 1958) p. 54 J. D. Garrison and G. L. Boroughs, Gulf General Atomic Incorporated Report GA-9353 (1969) V. Stavinsky and M. 0. Shaker, Nucl. Phys. 62 (1965) 667 J. D. Garrison, Gulf General Atomic Incorporated Report GA-9141 (1968) M. A. Preston, Physics of the nucleus (Addison-Wesley, Reading, Mass., 1962) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 J. E. Lynn, Nuclear data for reactors (Proc. Conf. Paris, 1966) vol. 2 (IAEA, Vienna, 1967) p. 89 W. J. Martin, Masters thesis, San Diego State (1970) J. Stehn ef al., Brookhaven National Lab. Report BNL 325, second edition, suppl. 2, vol. 3 (1965)
NEUTRON
CROSS
SECTIONS
323
20) J. E. Lynn, Theory of neutron resonance reactions (Clarendon Press, Oxford, i968) p. 396 21) P. A. Modauer, C. A. Engelbrecht and G. J. Duffy, Argonne National Laboratory Report ANL-6978 (1964) 22) P. A. Moldauer, Phys. Rev. 135 (1964) 8642 23) L. W. Weston et al., Oak Ridge National Laboratory Report ORNL-TM-2140 (1968) 24) H. C. Honeck, Brookhaven National Laboratory Report BNL-50066 (1966) 25) Status Report on the Evaluated Nuclear Data File (ENDF/B), Special Session, Trans. Am. Nucl. Sot. 11 (1968) 166 26) G. L. Boroughs, C. W. Craven, Jr. and M. K. Drake, Gulf General Atomic Incorporated Report GA-8854 27) J. D. Garrison and D. Mathews, Gulf General Atomic Report GAMD-9615 (1969) 28) J. J. Schmidt, Neutron Cross Sections for Fast Reactor Materials, Part I: Evaluation, KFK 120 (EANDC-E-35-U) (Feb. 1966) 29) T. A. Pitterle, E. M. Page and M. Yamomoto, Atomic Power Development Associates, Inc. Report APDA-216, Vol. 1 (1968) 30) A. G. W. Cameron, Can. J. Phys. 36 (1956) 666 31) J. A. Northrup, R. H. Stokes and K. Boyer, Phys. Rev. 99 (1955) 616 32) L. N. Usachev. V. A. Paulinchuk and N. S. Rabotnov, Atomn. Energ. 17 (1968) 479 33) Eccleschall and Yates, Proc. Symp. on physics and chemistry of fission, Salzburg (IAEA, Vienna, 1965) p. 77 34) J. A. Wheeler, in J. L. Fowler and J. B. Marion, eds., Fast neutron Physics, vol. 2 (Interscience, New York, 1963) p. 2051 35) B. H. Patrick and G. D. James, Phys. Lett. 288 (1968) 258 36) M. G. Cao, E. Migneco and J. P. Theobald, Phys. Lett. 27B (1968) 409 37) G. de Saussure and R. B. Perez, Oak Ridge National Laboratory Report, ORNL-TM-2599 (June 1969) 38) C. W. Reich and M. S. Moore, Phys. Rev. 111 (1958) 929