- Email: [email protected]
On the dependence upon energy of the fission product poisoning of U235
J. Nuclear Energy I, 1957, Vol. 4, pp. 319 to 325. Pergamon Press Ltd.. London
ON THE DEPENDENCE FISSION PRODUCT
UPON ENERGY OF THE POISONING OF V3j*
U. L. BUSINARO,~S. GALLONE,?and D. MORGAN: (First received 22 June
1956; injnalform
1 September 1956)
Abstract-A calculationis made of the energy dependenceof the averagecross-sectionof the fission products of Ps. Two methods are used. One is based on the average over many resonancesof cross-sectionsand uses low-energy data on average widths and spacings: the other applies the “continuum theory” formulae normalized on experimental cross-sections at 1 MeV. INTRODUCTION THE importance of an accurate estimate as a function of energy of the cross-section for neutron capture by fission products of U 235has been emphasized by WEINBERG (1955). From the preliminary estimate that he makes of the ratio of the fissionproduct capture cross-section to the fission cross-section, it emerges that for an intermediate reactor (average neutron energy 100 eV, say) the high value of the ratio imposes serious difficulties in reactor operation. For example, frequent chemical processing would be necessary. 1.
CALCULATION
The object is to estimate the average cross-section 17~~ of the fission products defined as: 3FP = c y&?i”’ (1) where yi is the yield of the ith fission-product nucleus. For simplicity, only the stable end-products of fission are considered. The estimate is to be made for a range of energies from IO2to IO6eV. There is, however, only a limited amount of data on the Ziny’s. We have therefore, following the suggestion in WEINBERG’report, S sought to employ theoretical extrapolations. Two ranges of experimental data are used, and from these, independent estimates are derived. Method (a) proceeds from the low-energy data on level-widths and spacings, using a formula for the cross-section averaged over resonances. Method (b) applies the “continuum theory” of nuclear reactions normalized to experiments at 1 MeV. 2. METHOD (a) A first attempt in evaluating the average radiation capture cross-section, is made using the average over many well separated resonances of the Breit-Wigner formula. The statistical distribution of the neutron widths is also taken into account, as well as * This work was begun during the Summer School of Physics at Ravenna was given at the Monte Faito symposium of the EAES. t Istituto di Fisica dell’Universita e Laboratori CISE, Milano. $ Pembroke College, Cambridge. 319
last year.
A short account
of it
320
U.
L. BUSINARO, S. GALLONE, and D. MORGAN
the contribution of neutrons of angular momentum greater than zero. One can thus write (WEINBERGand WIGNER,1955)
where 7, is the average neutron width, Q(x) the distribution
r,
function for I’, xx==, ( r7l1 k the neutron wave number, and E the neutron energy measured in eV. Various empirical forms have been proposed (HUGHESand HARVEY,1955) for the function Q(X), The exponential form Q(X) = e-” has been adopted here. In (2) the bars indicate averages over many resonances. The square-bracketed factor in (2) roughly represents the contribution of neutrons of high angular momenta. With the assumed distribution function, the integral in (2) is straightforward, and gives
where f is the ratio of average reduced neutron width to average spacing:
f&;
i?, = 2yn2k
In the above formulae D is the average spacing between resonances of equal spin and parity. The energy dependence of B may be assumed to be given by the well-known Weisskopf semiempirical formula :
D = b,, exp {-u* [( W + E)” - I@]).
(4)
W is the binding energy of the neutron in the compound nucleus; a is an empirical factor given by (SEGRE, 1953) u = 3.4
x lOA [A - 401”(eV)-l,
(5)
and b,, is the level spacing for low-energy neutrons (VOGT, 1955a; VOGT, 1955b; CARTERet al., 1954; HARVEYet al., 1955). Only a few data are available forf, lYv,and D in the published literature, and there are some discrepancies among the various authors.* The data used here are given in Table 1 together with other published data for comparison; in the case of the nuclei for which no data are available, those of the closest “known” nucleus (columns (a), (b), and (c) of Table 1) have been assumed. The results of the calculations are given in Fig. 1. A comparison of the result of the computation outlined in this section with the experimental data on radiative * Recently an unpublished independent calculation along the same lines was performed at Knolls Atomic Power Laboratory and kindly shown to us by the courtesy of Dr. P. GREEBLER. The dataused are much more complete than those which were available to us when the present computation was performed. KAPL results are shown for comparison in Fig. 1; they are roughly one-third those obtained using VOGT'S data, as outlined in Section 2. The difference is due to the fact that our calculations did not take into consideration the large D values associated with even-Z-even-N nuclei and magic neutron numbers.
56
R@’ . . . . In11S
Rhlo3
31 33 8 24 3.3
Xelsl . . . .Bala7
Ba138 ....Nd’45
Nd’46 . , . . Sm14s
NdlSo . . ..Smls2
Euxs3 . . . .Gdls8
cs=s
Prlrl
Smrao
ELI’S1
EuxSs
-
-_
24
2.4
50
25
from CARTER el al. (1954)
-
-
2.2
2.2
42
370
(1955)
et al.
from HARVEY
-
0.41
0.41
0.41
0.30
0.25
0.20
0.08
0.08
0.07
0.18
(b)
Values used in the calculation
-
0.43
0.43
0.16
0.27
!-
et al.
0.52
0.52
0.23
0.091
(1955)
et al.
(1954)
from HARVEY
-
from CARTER
f’. 1Ol3 (cm)
(CU Values taken from VOCT (1955a). lb)Values taken from trial fit of experimental data summarized by VOGT (1955b) (slide 1-13). I’) LEVINand HUGHES(1956). These data were communicated to use before publication, by the courtesy of Dr. A. M. WEINBERG.
12
Cd’la . . ..Telra
-
50
Mo8@. . . . MO’@”
Ruea
1’21
63
YE@ . . ..Mo8’
220
(a)
Moe5
-
Values used in the calculation
As=
. . . .SrS8
Nuclei assumed to have the same Cross-section
Brnl
Nucleus
-
TABLE 1
_
0.09
0.09
0.08
0.15
0.12
0,09
0.17
0.20
0.21
0.18
(c)
Values used in the calculation _
T
rr (ev)
0.09
0.085
0.11
0.10
0.15
0.15
0.26
from HUGHES and HARVEY (1955b)
_
c01
a. A &
b
9.
cd
F? Izh !z. 8 ? & c .z
3 D g 8 B %
% B 8
$
B
g
U. L. BUSINARO,S. GALLONE,and D. MORGAN
322
i
ii\
j
Continuum
i
jiiii
theory
‘.
--( -all
age .over resonances( Iz$35
iii-i
!‘:
----vogt
.-.*‘K.A.RL
fission cross section ’ II I III lllj_/
I I11111
I III
III
4
6 6104
2
4
6 8105
2
4 6 6106
eV E FIG. l-Dependence of fission-product poisoning upon neutron energy. U 236fission cross-section is given for comparison. The agreement between KAPL result and ours (continuum theory-all l’s) would further be improved if I > 0 contributions were taken into more detailed account in KAPL computations (see footnote, page 320).
cross-section obtained using an undegraded spectrum of neutrons of 1 MeV effective energy (HUGHES et al., 1953) is unsatisfactory, the computed value exceeding the experimental one by a factor greater than 2.
capture
3. METHOD
(b)
Another attempt to compute the average capture cross-section may be made starting from experimental data on average cross-sections at high energy, instead of using statistical data over resonances at low energy. Using the compound-nucleus hypothesis, one may write for the radiative capture cross-section 4% Y) =,SO 4%)
r? ry + rc,l,
(6)
where ~~~cl)(n) is the cross-section for compound-nucleus formation by capture of an 1 angular momentum neutron. This cross-section may be written as follows : d” (21 + 1) P(n). e(n) = zk2
(7)
On the dependenceupon energy of the fissionproduct poisoningof U235
323
As was recently pointed out (FESHBACHet al., 1954), it is possible to give good theoretical predictions for the average cross-section of compound-nucleus formation. One is thus left with the problem of determining the branching ratios by comparison with experimental data at high energy. This evaluation reduces to the fitting of only one parameter, if the following assumptions are made: I‘, = const = r,(O)(E,) I’:) = 2@(E)&
’
(8)
vI being the penetration factor, a known function of I and E (BLATTand WEISSKOPF, 1952). The energy dependence of B is that given in (4). One thus obtains for the branching ratio : I?.. 1
Introducing this result in (6), the comparison with the experimental value of CT@,r) at, say, 1 MeV, gives the value of El. Such a calculation was first carried out for each nucleus from Krs3 to Gd158, considering only I= 0 neutrons and putting @E)/d(E,) = 1. The experimental data used are those at 1 MeV (HUGHESet al., 1953). As to the nuclei for which no experimental data are available, values for nonmagic nuclei are taken from the average curve published in the cited reference, and for magic nuclei, from known cross-sections of neighbouring magic nuclei. The data used are summarized in Table 2. For the evaluation of the compound-nucleus formation cross-section, the continuum-theory results have been used with some further slight approximations. Precisely, 4tk T$ z - v * (K = 1013cm-l) (10) K I’ which holds at relatively low neutron energy (BLATTand WEISSKOPF,1952). Unfortunately, at 1 MeV the contribution to the radiative capture cross-section of angular momenta higher than zero is quite important. For such energies one has to consider terms up to I= 5, thus adding troublesome length to the computation. Instead of repeating the computation for each nucleus, taking carefully into consideration the contribution of high-angular momenta together with the dependence of level spacing on energy, we have divided all nuclei, from KF3 to Gdr5s, into eight groups, and attributed to each group an average cross-section weighted over the fission yield. The relative error in the case of I = 0 neutrons only was computed, and does not exceed 11x, The results of the calculations are shown in Fig. 1. 4. CONCLUSION The comparison of the calculations outlined in Section 3, with more recent computations carried out at KAPL and based on statistical data over resonances (see footnote, page 320) are quite satisfactory. At high energy the agreement would perhaps be improved by taking into consideration-in the KAPL computation-
U. L. BUSINARO, S. GALLONE, and D. MORGAN
324
Nucleus
Fission yield (a) ____
AS76 GeT6 Se” Se’* Br’8 Sea0 Br81 Sesz Krs3 Kr** Rbs5 Krsa Rba’ Srss YES Zrso Zrol Zro2 Nbu3 Zrs4 Moe6 Zrss Moe7 MO’Rhea MO-” RUlOl Ru’Oa
TABLE 2
-
Fission
44 Y> at 1 MeV (b)
r\TucIeus
4% Y> at 1 MeV (b)
yield (a)
Fission yield
Nucleus
4n, 74 at 1 MeV (b)
_67)
-
0.001 (:0.004) 0.009 1 0.020 0.037 0.08 0.133 0.20 0.586 1 I .O9 1.0 2.09 (.2.8) 3.8 4.6 5.0 5.9 5-o (.6.5) 5.0 6.0 (6,5) 6.2 (:6.5) ( 5.4) ( 5.5) 4.9 4.2
0.0225 (0.023) (0.024) (0.025) 0.0425 (0.028) 0.017 (0.032) (0.032)
0.9 0.52 (0.2) 0.08 0.028 (0.02) 0.018 0.008:3 0.013 0.012 0.016:3 (0.0 1) 0.01 0.01 0.012 (0.013:1 0.014 (0.016)I 0.017 (0.020)I 0.023 0.1 0.2 (::A, (2.0)
-
-
0,109 0.03 1 (0.096) (0.098) 0.085 0.108 0.178 (0.100) (0.100) (0.100) (0.100) (0~100) (0.223) [O.loo] [0.015] [0.015] [0.015] >0.014 0.090 >0.012 (0~100) 0.019 (0.100) (0.100) 0.105 0.100 (0.100) (0.100)
-
Xelsl Xelaa c?P Xe13* Ba135 Xe136 Bata Bats8 La’3fl
3.0 4.4 6.6 5.7 (6.0) 614 6.2 6.4 6.3 6.17 5.7 5.7 5.4 5.3 3.62 4.0
Ce140 Pr’Q Cer4* Nd14S Nd’44 Nd’“” Nd’*B Smr4’ Nd14* SmrPg NdreO Eu’~’ Smlsa En’&?
(0.100) (0.100) (O*lOo) (0.100) (0.100) 0.001 (0.100) 00023 0.005 0.0054 0.011 0.0042 (0~100) (0~100) (0~100) owe 0.100 0.080 (0.100) (0.100) (0.100) (0.100) (0.100) (O*lOO) (0.100) (0.100) (0.100) (0.100)
(:::) (& 0.445 0.4 0.15 (0.09) 0.03 1 0.013 0.015 0.002
Sm16r Gd’SS GdlSB Gd’j? GdrS8
-
-
Ia)The figures in brackets are interpolated values; the other values are experimental and taken from FALLER et al. (1955). cb)The figures in round brackets are deduced using the average fit of cross-sections for nonmagic nuclei as given by HUGHES er al. (1953); the figures in square brackets are assumed magic-nuclei cross-sections. Other values are experimental, as reported by HUGHES et al. (1953). the 1 >
A
0contributions more
carefully than could be done using a formula such as (2).*
comparison of formula
As is well known
(2)with (6)can be made,
using the relation:
(BLATTand WEISSKOPF,1952), the two expressions essentially agree,
giving for the continuum theory
r=&
(12)
independent of the nucleus. * We are greatly indebted to Dr. A. M. WEINBERG for letting us have an unpublished article by E. P. WIGNER concerning the detailed calculations of I > 0 contributions.
On the dependence upon energy of the fission product poisoning of U235
325
A more refined model, such as the crystal-ball model, could be used for evaluating the compound-nucleus formation cross-section: but this does not seem worth while doing if the uncertainty in the available experimental data is considered. We are greatly indebted to Dr. A. M. WEINBERG for having suggested the problem, and for many discussions and helpful assistance. REFERENCES BLATT J. M. and WEISSKOPFF. V. (1952) TheoreticaI Nuclear Physics, p. 478 (J. Wiley & Sons, N.Y.). CARTER R. S., HARVEY J. A., HUGRESD. J., and PILCHERV. E. (1954) Phys. Reo. 96, 113. CHAPMANT. S., FALLER T. I., and WEST J. M. (1955) Report A.N.L. 4807. FESCHBACHH., PORTERC. E., and WEISSKOPFV. F. (1954) Phys. Rev. 96,448. HARVEY J. A., HUGHESD. J., CARTER R. S., and PILCHERV. E. (1955) Phys. Rev. 99, 10. HUGHESD. J., GARTH R. C., and LEVIN J. S. (1953) Phys. Reu. 91, 1423. HUGHESD. J. and HARVEY J. A. (1955a) Phys. Reu. 99, 1032. HUGHESD. J. and HARVEY J. A. (1955b) Report BNL-325 (on neutron cross-sections). LEVIN J. S. and HUGHESD. J. (1956) Phys. Reo. 101, 1338. SEGRI?E. (1953) Experimental Nuclear Physics, vol. 2, p. 45 (J. Wiley & Sons, N.Y.). VOGT E. (1955a) The widths and spacings of nuclear resonance lines, Report NDA-14. V~GT E. (1955b) in the Report BNL-331 of the Conference on Statistical Aspects of the Nucleus held in Brookhaven in Jan. 1955. WEINBERGA. M. (1955) Geneva Conference Report, Paper 862, volume 3. WEINBERGA. M. and WIGNER E. P. (1955) Forthcoming book on nuclear reactor theory kindly lent to us by the courtesy of Prof. A. M. WEINBERG. WIONER E. P. Unpublished paper on neutron absorption in the low-energy region.
ON THE DEPENDENCE FISSION PRODUCT
UPON ENERGY OF THE POISONING OF V3j*
U. L. BUSINARO,~S. GALLONE,?and D. MORGAN: (First received 22 June
1956; injnalform
1 September 1956)
Abstract-A calculationis made of the energy dependenceof the averagecross-sectionof the fission products of Ps. Two methods are used. One is based on the average over many resonancesof cross-sectionsand uses low-energy data on average widths and spacings: the other applies the “continuum theory” formulae normalized on experimental cross-sections at 1 MeV. INTRODUCTION THE importance of an accurate estimate as a function of energy of the cross-section for neutron capture by fission products of U 235has been emphasized by WEINBERG (1955). From the preliminary estimate that he makes of the ratio of the fissionproduct capture cross-section to the fission cross-section, it emerges that for an intermediate reactor (average neutron energy 100 eV, say) the high value of the ratio imposes serious difficulties in reactor operation. For example, frequent chemical processing would be necessary. 1.
CALCULATION
The object is to estimate the average cross-section 17~~ of the fission products defined as: 3FP = c y&?i”’ (1) where yi is the yield of the ith fission-product nucleus. For simplicity, only the stable end-products of fission are considered. The estimate is to be made for a range of energies from IO2to IO6eV. There is, however, only a limited amount of data on the Ziny’s. We have therefore, following the suggestion in WEINBERG’report, S sought to employ theoretical extrapolations. Two ranges of experimental data are used, and from these, independent estimates are derived. Method (a) proceeds from the low-energy data on level-widths and spacings, using a formula for the cross-section averaged over resonances. Method (b) applies the “continuum theory” of nuclear reactions normalized to experiments at 1 MeV. 2. METHOD (a) A first attempt in evaluating the average radiation capture cross-section, is made using the average over many well separated resonances of the Breit-Wigner formula. The statistical distribution of the neutron widths is also taken into account, as well as * This work was begun during the Summer School of Physics at Ravenna was given at the Monte Faito symposium of the EAES. t Istituto di Fisica dell’Universita e Laboratori CISE, Milano. $ Pembroke College, Cambridge. 319
last year.
A short account
of it
320
U.
L. BUSINARO, S. GALLONE, and D. MORGAN
the contribution of neutrons of angular momentum greater than zero. One can thus write (WEINBERGand WIGNER,1955)
where 7, is the average neutron width, Q(x) the distribution
r,
function for I’, xx==, ( r7l1 k the neutron wave number, and E the neutron energy measured in eV. Various empirical forms have been proposed (HUGHESand HARVEY,1955) for the function Q(X), The exponential form Q(X) = e-” has been adopted here. In (2) the bars indicate averages over many resonances. The square-bracketed factor in (2) roughly represents the contribution of neutrons of high angular momenta. With the assumed distribution function, the integral in (2) is straightforward, and gives
where f is the ratio of average reduced neutron width to average spacing:
f&;
i?, = 2yn2k
In the above formulae D is the average spacing between resonances of equal spin and parity. The energy dependence of B may be assumed to be given by the well-known Weisskopf semiempirical formula :
D = b,, exp {-u* [( W + E)” - I@]).
(4)
W is the binding energy of the neutron in the compound nucleus; a is an empirical factor given by (SEGRE, 1953) u = 3.4
x lOA [A - 401”(eV)-l,
(5)
and b,, is the level spacing for low-energy neutrons (VOGT, 1955a; VOGT, 1955b; CARTERet al., 1954; HARVEYet al., 1955). Only a few data are available forf, lYv,and D in the published literature, and there are some discrepancies among the various authors.* The data used here are given in Table 1 together with other published data for comparison; in the case of the nuclei for which no data are available, those of the closest “known” nucleus (columns (a), (b), and (c) of Table 1) have been assumed. The results of the calculations are given in Fig. 1. A comparison of the result of the computation outlined in this section with the experimental data on radiative * Recently an unpublished independent calculation along the same lines was performed at Knolls Atomic Power Laboratory and kindly shown to us by the courtesy of Dr. P. GREEBLER. The dataused are much more complete than those which were available to us when the present computation was performed. KAPL results are shown for comparison in Fig. 1; they are roughly one-third those obtained using VOGT'S data, as outlined in Section 2. The difference is due to the fact that our calculations did not take into consideration the large D values associated with even-Z-even-N nuclei and magic neutron numbers.
56
R@’ . . . . In11S
Rhlo3
31 33 8 24 3.3
Xelsl . . . .Bala7
Ba138 ....Nd’45
Nd’46 . , . . Sm14s
NdlSo . . ..Smls2
Euxs3 . . . .Gdls8
cs=s
Prlrl
Smrao
ELI’S1
EuxSs
-
-_
24
2.4
50
25
from CARTER el al. (1954)
-
-
2.2
2.2
42
370
(1955)
et al.
from HARVEY
-
0.41
0.41
0.41
0.30
0.25
0.20
0.08
0.08
0.07
0.18
(b)
Values used in the calculation
-
0.43
0.43
0.16
0.27
!-
et al.
0.52
0.52
0.23
0.091
(1955)
et al.
(1954)
from HARVEY
-
from CARTER
f’. 1Ol3 (cm)
(CU Values taken from VOCT (1955a). lb)Values taken from trial fit of experimental data summarized by VOGT (1955b) (slide 1-13). I’) LEVINand HUGHES(1956). These data were communicated to use before publication, by the courtesy of Dr. A. M. WEINBERG.
12
Cd’la . . ..Telra
-
50
Mo8@. . . . MO’@”
Ruea
1’21
63
YE@ . . ..Mo8’
220
(a)
Moe5
-
Values used in the calculation
As=
. . . .SrS8
Nuclei assumed to have the same Cross-section
Brnl
Nucleus
-
TABLE 1
_
0.09
0.09
0.08
0.15
0.12
0,09
0.17
0.20
0.21
0.18
(c)
Values used in the calculation _
T
rr (ev)
0.09
0.085
0.11
0.10
0.15
0.15
0.26
from HUGHES and HARVEY (1955b)
_
c01
a. A &
b
9.
cd
F? Izh !z. 8 ? & c .z
3 D g 8 B %
% B 8
$
B
g
U. L. BUSINARO,S. GALLONE,and D. MORGAN
322
i
ii\
j
Continuum
i
jiiii
theory
‘.
--( -all
age .over resonances( Iz$35
iii-i
!‘:
----vogt
.-.*‘K.A.RL
fission cross section ’ II I III lllj_/
I I11111
I III
III
4
6 6104
2
4
6 8105
2
4 6 6106
eV E FIG. l-Dependence of fission-product poisoning upon neutron energy. U 236fission cross-section is given for comparison. The agreement between KAPL result and ours (continuum theory-all l’s) would further be improved if I > 0 contributions were taken into more detailed account in KAPL computations (see footnote, page 320).
cross-section obtained using an undegraded spectrum of neutrons of 1 MeV effective energy (HUGHES et al., 1953) is unsatisfactory, the computed value exceeding the experimental one by a factor greater than 2.
capture
3. METHOD
(b)
Another attempt to compute the average capture cross-section may be made starting from experimental data on average cross-sections at high energy, instead of using statistical data over resonances at low energy. Using the compound-nucleus hypothesis, one may write for the radiative capture cross-section 4% Y) =,SO 4%)
r? ry + rc,l,
(6)
where ~~~cl)(n) is the cross-section for compound-nucleus formation by capture of an 1 angular momentum neutron. This cross-section may be written as follows : d” (21 + 1) P(n). e(n) = zk2
(7)
On the dependenceupon energy of the fissionproduct poisoningof U235
323
As was recently pointed out (FESHBACHet al., 1954), it is possible to give good theoretical predictions for the average cross-section of compound-nucleus formation. One is thus left with the problem of determining the branching ratios by comparison with experimental data at high energy. This evaluation reduces to the fitting of only one parameter, if the following assumptions are made: I‘, = const = r,(O)(E,) I’:) = 2@(E)&
’
(8)
vI being the penetration factor, a known function of I and E (BLATTand WEISSKOPF, 1952). The energy dependence of B is that given in (4). One thus obtains for the branching ratio : I?.. 1
Introducing this result in (6), the comparison with the experimental value of CT@,r) at, say, 1 MeV, gives the value of El. Such a calculation was first carried out for each nucleus from Krs3 to Gd158, considering only I= 0 neutrons and putting @E)/d(E,) = 1. The experimental data used are those at 1 MeV (HUGHESet al., 1953). As to the nuclei for which no experimental data are available, values for nonmagic nuclei are taken from the average curve published in the cited reference, and for magic nuclei, from known cross-sections of neighbouring magic nuclei. The data used are summarized in Table 2. For the evaluation of the compound-nucleus formation cross-section, the continuum-theory results have been used with some further slight approximations. Precisely, 4tk T$ z - v * (K = 1013cm-l) (10) K I’ which holds at relatively low neutron energy (BLATTand WEISSKOPF,1952). Unfortunately, at 1 MeV the contribution to the radiative capture cross-section of angular momenta higher than zero is quite important. For such energies one has to consider terms up to I= 5, thus adding troublesome length to the computation. Instead of repeating the computation for each nucleus, taking carefully into consideration the contribution of high-angular momenta together with the dependence of level spacing on energy, we have divided all nuclei, from KF3 to Gdr5s, into eight groups, and attributed to each group an average cross-section weighted over the fission yield. The relative error in the case of I = 0 neutrons only was computed, and does not exceed 11x, The results of the calculations are shown in Fig. 1. 4. CONCLUSION The comparison of the calculations outlined in Section 3, with more recent computations carried out at KAPL and based on statistical data over resonances (see footnote, page 320) are quite satisfactory. At high energy the agreement would perhaps be improved by taking into consideration-in the KAPL computation-
U. L. BUSINARO, S. GALLONE, and D. MORGAN
324
Nucleus
Fission yield (a) ____
AS76 GeT6 Se” Se’* Br’8 Sea0 Br81 Sesz Krs3 Kr** Rbs5 Krsa Rba’ Srss YES Zrso Zrol Zro2 Nbu3 Zrs4 Moe6 Zrss Moe7 MO’Rhea MO-” RUlOl Ru’Oa
TABLE 2
-
Fission
44 Y> at 1 MeV (b)
r\TucIeus
4% Y> at 1 MeV (b)
yield (a)
Fission yield
Nucleus
4n, 74 at 1 MeV (b)
_67)
-
0.001 (:0.004) 0.009 1 0.020 0.037 0.08 0.133 0.20 0.586 1 I .O9 1.0 2.09 (.2.8) 3.8 4.6 5.0 5.9 5-o (.6.5) 5.0 6.0 (6,5) 6.2 (:6.5) ( 5.4) ( 5.5) 4.9 4.2
0.0225 (0.023) (0.024) (0.025) 0.0425 (0.028) 0.017 (0.032) (0.032)
0.9 0.52 (0.2) 0.08 0.028 (0.02) 0.018 0.008:3 0.013 0.012 0.016:3 (0.0 1) 0.01 0.01 0.012 (0.013:1 0.014 (0.016)I 0.017 (0.020)I 0.023 0.1 0.2 (::A, (2.0)
-
-
0,109 0.03 1 (0.096) (0.098) 0.085 0.108 0.178 (0.100) (0.100) (0.100) (0.100) (0~100) (0.223) [O.loo] [0.015] [0.015] [0.015] >0.014 0.090 >0.012 (0~100) 0.019 (0.100) (0.100) 0.105 0.100 (0.100) (0.100)
-
Xelsl Xelaa c?P Xe13* Ba135 Xe136 Bata Bats8 La’3fl
3.0 4.4 6.6 5.7 (6.0) 614 6.2 6.4 6.3 6.17 5.7 5.7 5.4 5.3 3.62 4.0
Ce140 Pr’Q Cer4* Nd14S Nd’44 Nd’“” Nd’*B Smr4’ Nd14* SmrPg NdreO Eu’~’ Smlsa En’&?
(0.100) (0.100) (O*lOo) (0.100) (0.100) 0.001 (0.100) 00023 0.005 0.0054 0.011 0.0042 (0~100) (0~100) (0~100) owe 0.100 0.080 (0.100) (0.100) (0.100) (0.100) (0.100) (O*lOO) (0.100) (0.100) (0.100) (0.100)
(:::) (& 0.445 0.4 0.15 (0.09) 0.03 1 0.013 0.015 0.002
Sm16r Gd’SS GdlSB Gd’j? GdrS8
-
-
Ia)The figures in brackets are interpolated values; the other values are experimental and taken from FALLER et al. (1955). cb)The figures in round brackets are deduced using the average fit of cross-sections for nonmagic nuclei as given by HUGHES er al. (1953); the figures in square brackets are assumed magic-nuclei cross-sections. Other values are experimental, as reported by HUGHES et al. (1953). the 1 >
A
0contributions more
carefully than could be done using a formula such as (2).*
comparison of formula
As is well known
(2)with (6)can be made,
using the relation:
(BLATTand WEISSKOPF,1952), the two expressions essentially agree,
giving for the continuum theory
r=&
(12)
independent of the nucleus. * We are greatly indebted to Dr. A. M. WEINBERG for letting us have an unpublished article by E. P. WIGNER concerning the detailed calculations of I > 0 contributions.
On the dependence upon energy of the fission product poisoning of U235
325
A more refined model, such as the crystal-ball model, could be used for evaluating the compound-nucleus formation cross-section: but this does not seem worth while doing if the uncertainty in the available experimental data is considered. We are greatly indebted to Dr. A. M. WEINBERG for having suggested the problem, and for many discussions and helpful assistance. REFERENCES BLATT J. M. and WEISSKOPFF. V. (1952) TheoreticaI Nuclear Physics, p. 478 (J. Wiley & Sons, N.Y.). CARTER R. S., HARVEY J. A., HUGRESD. J., and PILCHERV. E. (1954) Phys. Reo. 96, 113. CHAPMANT. S., FALLER T. I., and WEST J. M. (1955) Report A.N.L. 4807. FESCHBACHH., PORTERC. E., and WEISSKOPFV. F. (1954) Phys. Rev. 96,448. HARVEY J. A., HUGHESD. J., CARTER R. S., and PILCHERV. E. (1955) Phys. Rev. 99, 10. HUGHESD. J., GARTH R. C., and LEVIN J. S. (1953) Phys. Reu. 91, 1423. HUGHESD. J. and HARVEY J. A. (1955a) Phys. Reu. 99, 1032. HUGHESD. J. and HARVEY J. A. (1955b) Report BNL-325 (on neutron cross-sections). LEVIN J. S. and HUGHESD. J. (1956) Phys. Reo. 101, 1338. SEGRI?E. (1953) Experimental Nuclear Physics, vol. 2, p. 45 (J. Wiley & Sons, N.Y.). VOGT E. (1955a) The widths and spacings of nuclear resonance lines, Report NDA-14. V~GT E. (1955b) in the Report BNL-331 of the Conference on Statistical Aspects of the Nucleus held in Brookhaven in Jan. 1955. WEINBERGA. M. (1955) Geneva Conference Report, Paper 862, volume 3. WEINBERGA. M. and WIGNER E. P. (1955) Forthcoming book on nuclear reactor theory kindly lent to us by the courtesy of Prof. A. M. WEINBERG. WIONER E. P. Unpublished paper on neutron absorption in the low-energy region.