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Calculation of average radiative-capture cross sections for 103–105 eV neutrons
Journal
of Nuclear
Energy
Parts A/B.
1966, Vol. 20, pp. 675 to 682.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
CALCULATION OF AVERAGE RADIATIVECAPTURE CROSS SECTIONS FOR 103-lo” eV NEUTRONS* A. G. DOVBENKO, S. M. ZAKHAROVA, V. E. KOLESOV and A. V. MALYSHEV (Received 31 January
1964)
Abstract-The average cross sections for radiative neutron capture have been calculated for 30 isotopes of Rb, Zr, MO, Sn and Sm on the basis of the statistical theory of nuclear reactions. Nuclear surface penetrations obtained from the optical model, and level densities corresponding to the Fermi gas model were used in the calculation. The results of the calculations are compared with the available experimental data at 25 keV. It is shown that there is a possibility of satisfactory quantitative evaluations of the average capture cross sections in isotopes for which there are no experimental data on the mean neutron resonance parameters. IT HAS been shown(1-4) that calculation of the energy dependence of the average radiative capture cross sections in the framework of the statistical theory, with nuclear surface penetrations corresponding to the optical model, agrees satisfactorily with experiment. It is also known that the absolute value of the cross section is strongly dependent on the mean distance between the levels of the compound nucleus and on the mean radiation width. These parameters are usually taken in accordance with experimental data in the low-energy region. If such data are not available, the mean parameters are selected on the basis of comparing the calculated average radiative capture cross sections with experiment. Many questions in the field of reactor construction can only be solved if one knows the average radiative capture cross sections as a function of energy. Experimental data available at present were obtained in the main for isotopes activated on the capture of a neutron. There are practically no experimental data for the majority of non-activating isotopes or for unstable isotopes. For this reason it is interesting to calculate cross sections for radiative capture on the basis of the systematics of the mean parameters. (5S6) The available experimental data at 25-30 keV can be used for the quantitative comparison of the cross sections.“-1s) In addition to this, by summing the calculated cross sections for individual isotopes it is possible to make a comparison with experimental cross sections for a natural mixture of isotopes, which in many cases are known in a wide energy range. Calculations were made for the following isotopes : *jRb, 90,91J’2.94.96zr, 92.94-9g,100M0
112.114&120.122,124~n
and
144,147-150,152,154Sme
The statistical theory equation normally employed tive capture cross sections was employed(1s2s15):
Here E is the kinetic energy of the incident incident and scattered neutron respectively; * Translated by J. J.
CORNISH
for calculating
radia-
neutron; 1, I’ the orbital moments of the J the total moment of the compound
from Atomnuya Energiya 18, 114 (1965). 675
average
676
A. G. DOVBENKO, S. M. ZAKHAROVA, V. E. KOLESOV and A. V. MALYSHEV
nucleus; j = I * 4, j, = Ik & + are the spins of the input and output channel respectively; Z the spin of the ground state of the target nucleus; Z, the spin of the k-th excited level of the target nucleus; ji and .~j”,l’,factors allowing for the number of open channels and equal respectively to the number of j and j, values satisfying conditions
??
IJ--11
(2)
level of the target nucleus at which the scattering takes place; the distribution of reduced neutron widths taken from LANE TZ’(E - Ek) the penetrations of the nuclear surface computed potential
1+ 2
v,
V(r) = -
(3) at a potential well depth V, = 45 MeV, diffuse layer thickness d = 0.5 f,E = O-1 and nuclear radius R = r,A1’3, where r,, = 1.25f. The sum over I’ only contains terms satisfying the conservation of parity law. In the concrete calculation the penetrations were selected on the condition of best agreement between experiment and calculated total cross sections and radiative capture cross sections of neighbouring monoisotopic elements. The ratio of the mean distance D(U + E,J) between levels with total moment J to the mean radiation width I’,(U + E) at excitation energy U + E has the form
D(U + E,J)
D(U,J) D(U + E,J)
I?,( U + E) = -*F,(U) Factor f(E)
was calculated p(U,J)I’,(u)
D(U,J)
r,(U)
on the simplifying
~
1
D(U,J)
= const.
-
‘r&U + E) assumptions
+f
‘/
(E).
(4)
:
= const. (25 + 1) exp [2(aU)1’2];
D( U, J)
where .sy is the energy of the y-quantum
1; &n”p(U -
emitted.
(5) E;,,J> ds,,
In this case
u aY3p(U -
E>,,J) de,,
.r L,+;
f(E)=
(6) e; p(U + E -
F:,,J) dEy
J‘0 In the energy region under study (E < U)f (E) FZ 1, so the assumptions (5) have little effect on the results of the calculation. In formulae (1) (4) (5) and (6) 6, + 6, U=B,-
6, 6 I On
for for for for
even-even, even-odd, odd-even, odd-odd compound
nuclei,
made above
Calculation
of average radiative-capture
671
cross sections for 103-lo5 eV neutrons
where I?, is the neutron binding energy in the nucleus; 6, and 6, are the pairing energies of two protons and two neutrons respectively taken from NEMIROVSKYand ADAMCHUK’~~). The mean distances D(U,J) between levels with the given total moment J and excitation energy U are computed from the Fermi gas model equations with parameters a given by MALYSHEVt5). The values of the mean radiation widths are taken on the basis of relation l?;,(A) from ZAKHAROVA and MALYSHEP). Calculations of onj, were performed on an electronic computer. The programme TABLE I.-PARAMETERS a (MeWI)
Isotope
5/2-
8.50
9.0 5;;; Of o+ Of 0+ 0’ 5/2+ o+ s/2+ Of o+ Of
s6Zr* ‘=Mo
g4M~ ‘=Mo neMo Q’M~ g”M~ lonMo l%n %Gn USn lqn* 117Sn
1:;+ I;;+ T I;;+ 0+ o+ Of Of l/ZO7/2Of 07 0’
ll”Sn 119sn* 12”Sn* lzzSn l%n ‘r?Sm* ‘4’Sm 1qm* 1.19Sm Is”Sm* Ia*Sm* 1WSm*
10.2 11.0 11.6 14.0 10.2 12.6 12.6 12.9 13.2 17.6 18.0 12.1 12.2 16.5 15.5 15.5 15.0 15.5 15.3 16.2 15.2 17.0 20.0 22.0 23.6 25.0 24.0 21.0
USED IN CALCULATION ii (eV) 750t
1.7 x 104 1 X 103 3.7 x 103 6.3 x lo3 2.6 x lo3 2.4 x lo3 1 x 103 220t 1 X 103 220t 270 430
U (MeV)
1‘:) (eV)
8.58 6.19
0.410 0.270 0.245 0.270 0.220 0,160 0.270 0.220 0.230 0.160 0.200 0.102 0.090 0.061 0.063 0,096 0.065 0,110 0.093 0.106 0.108 0.106 0,100 0.065 0.059 0.063 0.065 0,066 0.068 0.072
6.49 5.19 5.52 4.65 6.71 6.08 6.44
5.49 6.15 4.99 4.61
850
6.68 6.24 6.85 5.39 6.60 5.00 6.30 4.70 4.40 4.54 5.40 5.77 4.40 5.27 3.98 4.25 3.80
1.4 x 103
sot
600 1201_ 1 x 103 1807 1.7 X 103 1.7 >: 103
2 >. 103 325 14t 175 6t 115 87 740
* For these isotopes the experimental data for Dare unreliable or absent altogether. t For these nuclei D(U,J) was computed from 5 at 04 GYl/2.
was based on the method of solving the Schrijdinger of arbitrary form described by ERMAKOV et u/.@~) The mean
parameters
and characteristics
radial equation
of the ground
states
with a potential
of the isotopes
used
in the calculation are given in Table 1. As usual (15), for S-neutrons (I = 0). D, = mJ, where ij = 2D,,, at I + 0; D, = D/g,,where b = Dabsat I = 0; g, 2J+
1
= 2(2Z + I)’ When calculating on?, lggSn two other excited levels lying below 100 keV were taken into account in addition to the ground state of the target nucleus : 23.8 keV(3/2+) and 89.0 keV (1 l/2+). The results of the calculation are given in Figs. 1-3. For isotopes
A. G. DOVBENKO, S. M. ZAKHAROVA,V. E. KOLESOVand A. V. MALYSHEV
678 95Rb,
99.92.94.96Zr,
lOO&,
116-120Sn and
144*147-150*152Sm the calculated cross sections agree with the experimental cross sections? within the experimental errors. For isotopes lz2Sn 124Sn and glZr the deviation of the calculated cross sections from the experimental is G 50 per cent. .
200 100 80 60 0 1 20 100 80 60
140
1 20
IO 6 6
\+
+ \
2u
40
+
1
I
\t+ I
103
2
I
I111!11
I
I
4
2
4
69104 E,
ILli
I
2o IO
68105
6’~
FIG. 1 .-Radiative capture cross sections of Zr and Sn isotopes as a function of incident neutron energy. Data from following Refs. * (11); 0 (13), @ ??0 (151, 0 (161, - (171, + Cl@, ??(21).
In some cases parameter a obtained from ii values measured with low accuracy The theoretical cross sections agree better with experiment falls out of the systematics. if the D values used in the calculation are computed with parameters a predicted by the systematics. s5Rb. The sole experimental result (B = 2000 eV) was obtained by NEWSON et ~1.‘~~) The average capture cross section calculated with this ij value and I’? = O-25 eW5) is approximately three times less than the experimental. Agreement with t It should be noted that for the majority of isotopes examined there is only experimental value cny.
Calculation of average radiative-capture
cross sections for 103-lo5 eV neutrons
679
- 1000 -800 -600 -400
-
I00 80 - 60 - 40
L
E.
U
FIG. 2.-Radiative capture cross sections of MO isotopes as a function of incident neutron energy. Data from following Refs: ??(7), x (9), z (lo), 0 (12), ‘I (14), A ? ?v A (15).
experiment is obtained if one takes b = 750 eV, computed with a = 8.50 and r,, = 0.41 eVt as obtained by ZAKHAROVA and MALYSHE@). lrgSn. Calculation with value Li = 300 eV obtained from only two resonances by HARVEY et aZ.(23)leads to a cross section which is approximately half the experimental value. Agreement within the experimental errors is obtained if one takes b = 180 eV, computed with a = 15.5. lS4Sm. The largest deviation between calculated and experimental cross section is seen in this case (Go,, exp/a,,,,, w 2.7). Agreement is improved if a slightly higher value a = 24 is used. However, the discrepancy still remains greater than the experimental error. In two cases there is a spread in the available experimental data. The cross section t Value I’? = 0.41 eV taken according to the systematics agrees well with value 0.44 i_ 0.15 eV for resonance E,, = 230 eV taken from KAPCHIGASHEV and PoPov~‘~.
680
A. G. DOVBENKO, S. M. ZAKHAROVA, V. E. KOLESOV and A. V. MALYSHEV I-
’ “““”
I I /1111,
’ 1000 800
600 400
1
1000 600 600
1000 600 600
FIG. 3.-Radiative capture cross sections of Rb and Sm isotopes as a function of incident neutron energy. Data from following Refs: ??(8), 0 0 (15).
of g8Mo at 25 keV according to data of VERVIER~~O), BOOTH et u/.(14)and MACKLIN et aZ.(17)is 415, 390 & 120 and 120 mbarns respectively. Calculation agrees with For looMo the results of early work(12*13) MACKLIN et aZ.(13),viz. 209 i 21 mbarns. differ substantially from results obtained later(7P11) which agree with the calculated cross section (see Fig. 2). Cross sections for natural mixtures of isotopes? obtained by summing the calculated cross sections of the isotopes agree well with experimental values for all elements examined except Sm. Agreement with experiment can be obtained if the cross section of 147Sm and 14sSm is increased by a factor of 1.5, which is unlikely. The calculations performed have shown that the average radiative capture cross sections of isotopes are very sensitive to b values and provide an opportunity of checking the parameters of the Fermi gas model. This is explained by the fact that the mean radiation widths of neighbouring nuclei with different parity of the number of t The cross section for R7Rb is not given since its contribution is small compared with that of sjRb.
to cross section of a natural mixture
Calculation of average radiative-capture cross sections for 10”lo5 eV neutrons
681
nucleons are approximately identical, whereas the mean distances between the levels of these nuclei can differ considerably. For all elements studied (Fig. 4) the cross
sections for even-odd isotopes are considerably higher than those for even-even isotopes. Therefore, in spite of the small percentage content they make a considerable contribution to the cross section of a natural mixture’**).
A
FIG. 4.-Radiative
capture cross section of Rb, Zr, MO, Sn and Sm isotopes as a function of mass number A at -30 keV. 0, A: radiative capture cross sections of Zr isotopes: A, n : even-odd nuclei; 0, 0 : eveweven nuclei; nuclei. x : oddwen
Thus, the present work has demonstrated the possibility of theoretical evaluations of average radiative capture cross sections in the 103-lo5 eV energy range to an accuracy not worse than 50 per cent, as out of 22 isotopes for which comparison was made with experiment, only for 154Smwas the deviation greater than 50 per cent. REFERENCES 1. MARGOLIS B. Phys. Rev. 88, 321 (1952). 2. LANE A. and LYNN J. E. Proc. phys. Sot. Lond. AIO, 551 (1957). Energ. 11, 56 (1961); KOLESOV 3. TOLSTIKOVV. A., KOLESOVV. E. and STAVINSKIIV. S. Atom. V. E. and STAVINSKIIV. S. Atornn. Energ. 13, 371 (1962). 4. NEMIROVSKIIP. and ELAGIN Yu. Nucl. Phys. 45, 156 (1963). 5. MALYSHEVA. V. Zh. eksp. tekh. Fiz. 45, 316 (1963). 6. ZAKHAROVAS. M. and MALYSHEVA. V. Paper at the International Congress on Low and Medium Energy Nuclear Reactions, Paris, July (1964). 7. KAPCHIGASHEVS. P. and POPOV Yu. P. Atomn. Energ. 15, 120 (1963). 8. POPOV Yu. P. and SHAPIROF. L. Zh. eksp. tekh. Fir. 42, 988 (1962). 9. WESTONL. et al. Ann. Phys. 10,417 (1960). 10. VERVIERJ. Nucl. Phys. 9, 569 (1959). 11. LEIPUNSKIIA. I. et al. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy Geneva, Vol. 15, p. 50. United Nations, N.Y. (1958). B. Phys. Rev. 82, 67 (1951). 12. HUMMELV. and HAMMERMESH 13. MACKLIN R., LASAR N. and LYON S. Phys. Rev. 107, 504 (1957). 14. BOOTH R., BALL W. and MACGREGORA. Phys. Rev. 112,226 (1958). 15. GORDEEVI. V., KARDASHEVD. A. and MALYSHEVA. V. Handbook of Nuclear Physical Constants, Gosatomizdat, Moscow (1963). 16. MACKLIN R., GIBBONS J. and INADA T. Nucl. Phys. 43, 353 (1963). 17. MACKLIN R., GIBBONSJ. and INADA T. Nature, Land. 197, 369 (1963); Bull. Am. phys. Sot. 8, 81 (1963). 5
682
A. G. DOVBENKO, S. M. ZAKHAROVA,V. E. KOLESOVand A. V. MALYSHEV
18. STAVISSKIIYu. YA. and SHAPAR’A. V. Atom. Energ. 15, 322 (1963). 19. NEMIROVSK~P. and ADAMCHUKYu. Nucl. Phys. 39,553 (1962). 20. ERMAKOVS. M., KOLESOVV. E. and MARCHUK G. I. Neutron Physics (Collection), Gosatomizdat, Moscow (1961). 21. MACKLIN R., INADAT. and GIBBONS3. Nature, Lond. 194, 1272 (1962). 22. NEWSON H. et al. Ann. Phys. 14,346 (1961). 23. HARVEY J. et al. Phys. Rev. 99, 10 (1955). 24. BELANOVAT. S. and KAZACHKOVSKII 0. D. Atomn. Energ. 14,185 (1963).
p. 314
of Nuclear
Energy
Parts A/B.
1966, Vol. 20, pp. 675 to 682.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
CALCULATION OF AVERAGE RADIATIVECAPTURE CROSS SECTIONS FOR 103-lo” eV NEUTRONS* A. G. DOVBENKO, S. M. ZAKHAROVA, V. E. KOLESOV and A. V. MALYSHEV (Received 31 January
1964)
Abstract-The average cross sections for radiative neutron capture have been calculated for 30 isotopes of Rb, Zr, MO, Sn and Sm on the basis of the statistical theory of nuclear reactions. Nuclear surface penetrations obtained from the optical model, and level densities corresponding to the Fermi gas model were used in the calculation. The results of the calculations are compared with the available experimental data at 25 keV. It is shown that there is a possibility of satisfactory quantitative evaluations of the average capture cross sections in isotopes for which there are no experimental data on the mean neutron resonance parameters. IT HAS been shown(1-4) that calculation of the energy dependence of the average radiative capture cross sections in the framework of the statistical theory, with nuclear surface penetrations corresponding to the optical model, agrees satisfactorily with experiment. It is also known that the absolute value of the cross section is strongly dependent on the mean distance between the levels of the compound nucleus and on the mean radiation width. These parameters are usually taken in accordance with experimental data in the low-energy region. If such data are not available, the mean parameters are selected on the basis of comparing the calculated average radiative capture cross sections with experiment. Many questions in the field of reactor construction can only be solved if one knows the average radiative capture cross sections as a function of energy. Experimental data available at present were obtained in the main for isotopes activated on the capture of a neutron. There are practically no experimental data for the majority of non-activating isotopes or for unstable isotopes. For this reason it is interesting to calculate cross sections for radiative capture on the basis of the systematics of the mean parameters. (5S6) The available experimental data at 25-30 keV can be used for the quantitative comparison of the cross sections.“-1s) In addition to this, by summing the calculated cross sections for individual isotopes it is possible to make a comparison with experimental cross sections for a natural mixture of isotopes, which in many cases are known in a wide energy range. Calculations were made for the following isotopes : *jRb, 90,91J’2.94.96zr, 92.94-9g,100M0
112.114&120.122,124~n
and
144,147-150,152,154Sme
The statistical theory equation normally employed tive capture cross sections was employed(1s2s15):
Here E is the kinetic energy of the incident incident and scattered neutron respectively; * Translated by J. J.
CORNISH
for calculating
radia-
neutron; 1, I’ the orbital moments of the J the total moment of the compound
from Atomnuya Energiya 18, 114 (1965). 675
average
676
A. G. DOVBENKO, S. M. ZAKHAROVA, V. E. KOLESOV and A. V. MALYSHEV
nucleus; j = I * 4, j, = Ik & + are the spins of the input and output channel respectively; Z the spin of the ground state of the target nucleus; Z, the spin of the k-th excited level of the target nucleus; ji and .~j”,l’,factors allowing for the number of open channels and equal respectively to the number of j and j, values satisfying conditions
??
IJ--11
(2)
level of the target nucleus at which the scattering takes place; the distribution of reduced neutron widths taken from LANE TZ’(E - Ek) the penetrations of the nuclear surface computed potential
1+ 2
v,
V(r) = -
(3) at a potential well depth V, = 45 MeV, diffuse layer thickness d = 0.5 f,E = O-1 and nuclear radius R = r,A1’3, where r,, = 1.25f. The sum over I’ only contains terms satisfying the conservation of parity law. In the concrete calculation the penetrations were selected on the condition of best agreement between experiment and calculated total cross sections and radiative capture cross sections of neighbouring monoisotopic elements. The ratio of the mean distance D(U + E,J) between levels with total moment J to the mean radiation width I’,(U + E) at excitation energy U + E has the form
D(U + E,J)
D(U,J) D(U + E,J)
I?,( U + E) = -*F,(U) Factor f(E)
was calculated p(U,J)I’,(u)
D(U,J)
r,(U)
on the simplifying
~
1
D(U,J)
= const.
-
‘r&U + E) assumptions
+f
‘/
(E).
(4)
:
= const. (25 + 1) exp [2(aU)1’2];
D( U, J)
where .sy is the energy of the y-quantum
1; &n”p(U -
emitted.
(5) E;,,J> ds,,
In this case
u aY3p(U -
E>,,J) de,,
.r L,+;
f(E)=
(6) e; p(U + E -
F:,,J) dEy
J‘0 In the energy region under study (E < U)f (E) FZ 1, so the assumptions (5) have little effect on the results of the calculation. In formulae (1) (4) (5) and (6) 6, + 6, U=B,-
6, 6 I On
for for for for
even-even, even-odd, odd-even, odd-odd compound
nuclei,
made above
Calculation
of average radiative-capture
671
cross sections for 103-lo5 eV neutrons
where I?, is the neutron binding energy in the nucleus; 6, and 6, are the pairing energies of two protons and two neutrons respectively taken from NEMIROVSKYand ADAMCHUK’~~). The mean distances D(U,J) between levels with the given total moment J and excitation energy U are computed from the Fermi gas model equations with parameters a given by MALYSHEVt5). The values of the mean radiation widths are taken on the basis of relation l?;,(A) from ZAKHAROVA and MALYSHEP). Calculations of onj, were performed on an electronic computer. The programme TABLE I.-PARAMETERS a (MeWI)
Isotope
5/2-
8.50
9.0 5;;; Of o+ Of 0+ 0’ 5/2+ o+ s/2+ Of o+ Of
s6Zr* ‘=Mo
g4M~ ‘=Mo neMo Q’M~ g”M~ lonMo l%n %Gn USn lqn* 117Sn
1:;+ I;;+ T I;;+ 0+ o+ Of Of l/ZO7/2Of 07 0’
ll”Sn 119sn* 12”Sn* lzzSn l%n ‘r?Sm* ‘4’Sm 1qm* 1.19Sm Is”Sm* Ia*Sm* 1WSm*
10.2 11.0 11.6 14.0 10.2 12.6 12.6 12.9 13.2 17.6 18.0 12.1 12.2 16.5 15.5 15.5 15.0 15.5 15.3 16.2 15.2 17.0 20.0 22.0 23.6 25.0 24.0 21.0
USED IN CALCULATION ii (eV) 750t
1.7 x 104 1 X 103 3.7 x 103 6.3 x lo3 2.6 x lo3 2.4 x lo3 1 x 103 220t 1 X 103 220t 270 430
U (MeV)
1‘:) (eV)
8.58 6.19
0.410 0.270 0.245 0.270 0.220 0,160 0.270 0.220 0.230 0.160 0.200 0.102 0.090 0.061 0.063 0,096 0.065 0,110 0.093 0.106 0.108 0.106 0,100 0.065 0.059 0.063 0.065 0,066 0.068 0.072
6.49 5.19 5.52 4.65 6.71 6.08 6.44
5.49 6.15 4.99 4.61
850
6.68 6.24 6.85 5.39 6.60 5.00 6.30 4.70 4.40 4.54 5.40 5.77 4.40 5.27 3.98 4.25 3.80
1.4 x 103
sot
600 1201_ 1 x 103 1807 1.7 X 103 1.7 >: 103
2 >. 103 325 14t 175 6t 115 87 740
* For these isotopes the experimental data for Dare unreliable or absent altogether. t For these nuclei D(U,J) was computed from 5 at 04 GYl/2.
was based on the method of solving the Schrijdinger of arbitrary form described by ERMAKOV et u/.@~) The mean
parameters
and characteristics
radial equation
of the ground
states
with a potential
of the isotopes
used
in the calculation are given in Table 1. As usual (15), for S-neutrons (I = 0). D, = mJ, where ij = 2D,,, at I + 0; D, = D/g,,where b = Dabsat I = 0; g, 2J+
1
= 2(2Z + I)’ When calculating on?, lggSn two other excited levels lying below 100 keV were taken into account in addition to the ground state of the target nucleus : 23.8 keV(3/2+) and 89.0 keV (1 l/2+). The results of the calculation are given in Figs. 1-3. For isotopes
A. G. DOVBENKO, S. M. ZAKHAROVA,V. E. KOLESOVand A. V. MALYSHEV
678 95Rb,
99.92.94.96Zr,
lOO&,
116-120Sn and
144*147-150*152Sm the calculated cross sections agree with the experimental cross sections? within the experimental errors. For isotopes lz2Sn 124Sn and glZr the deviation of the calculated cross sections from the experimental is G 50 per cent. .
200 100 80 60 0 1 20 100 80 60
140
1 20
IO 6 6
\+
+ \
2u
40
+
1
I
\t+ I
103
2
I
I111!11
I
I
4
2
4
69104 E,
ILli
I
2o IO
68105
6’~
FIG. 1 .-Radiative capture cross sections of Zr and Sn isotopes as a function of incident neutron energy. Data from following Refs. * (11); 0 (13), @ ??0 (151, 0 (161, - (171, + Cl@, ??(21).
In some cases parameter a obtained from ii values measured with low accuracy The theoretical cross sections agree better with experiment falls out of the systematics. if the D values used in the calculation are computed with parameters a predicted by the systematics. s5Rb. The sole experimental result (B = 2000 eV) was obtained by NEWSON et ~1.‘~~) The average capture cross section calculated with this ij value and I’? = O-25 eW5) is approximately three times less than the experimental. Agreement with t It should be noted that for the majority of isotopes examined there is only experimental value cny.
Calculation of average radiative-capture
cross sections for 103-lo5 eV neutrons
679
- 1000 -800 -600 -400
-
I00 80 - 60 - 40
L
E.
U
FIG. 2.-Radiative capture cross sections of MO isotopes as a function of incident neutron energy. Data from following Refs: ??(7), x (9), z (lo), 0 (12), ‘I (14), A ? ?v A (15).
experiment is obtained if one takes b = 750 eV, computed with a = 8.50 and r,, = 0.41 eVt as obtained by ZAKHAROVA and MALYSHE@). lrgSn. Calculation with value Li = 300 eV obtained from only two resonances by HARVEY et aZ.(23)leads to a cross section which is approximately half the experimental value. Agreement within the experimental errors is obtained if one takes b = 180 eV, computed with a = 15.5. lS4Sm. The largest deviation between calculated and experimental cross section is seen in this case (Go,, exp/a,,,,, w 2.7). Agreement is improved if a slightly higher value a = 24 is used. However, the discrepancy still remains greater than the experimental error. In two cases there is a spread in the available experimental data. The cross section t Value I’? = 0.41 eV taken according to the systematics agrees well with value 0.44 i_ 0.15 eV for resonance E,, = 230 eV taken from KAPCHIGASHEV and PoPov~‘~.
680
A. G. DOVBENKO, S. M. ZAKHAROVA, V. E. KOLESOV and A. V. MALYSHEV I-
’ “““”
I I /1111,
’ 1000 800
600 400
1
1000 600 600
1000 600 600
FIG. 3.-Radiative capture cross sections of Rb and Sm isotopes as a function of incident neutron energy. Data from following Refs: ??(8), 0 0 (15).
of g8Mo at 25 keV according to data of VERVIER~~O), BOOTH et u/.(14)and MACKLIN et aZ.(17)is 415, 390 & 120 and 120 mbarns respectively. Calculation agrees with For looMo the results of early work(12*13) MACKLIN et aZ.(13),viz. 209 i 21 mbarns. differ substantially from results obtained later(7P11) which agree with the calculated cross section (see Fig. 2). Cross sections for natural mixtures of isotopes? obtained by summing the calculated cross sections of the isotopes agree well with experimental values for all elements examined except Sm. Agreement with experiment can be obtained if the cross section of 147Sm and 14sSm is increased by a factor of 1.5, which is unlikely. The calculations performed have shown that the average radiative capture cross sections of isotopes are very sensitive to b values and provide an opportunity of checking the parameters of the Fermi gas model. This is explained by the fact that the mean radiation widths of neighbouring nuclei with different parity of the number of t The cross section for R7Rb is not given since its contribution is small compared with that of sjRb.
to cross section of a natural mixture
Calculation of average radiative-capture cross sections for 10”lo5 eV neutrons
681
nucleons are approximately identical, whereas the mean distances between the levels of these nuclei can differ considerably. For all elements studied (Fig. 4) the cross
sections for even-odd isotopes are considerably higher than those for even-even isotopes. Therefore, in spite of the small percentage content they make a considerable contribution to the cross section of a natural mixture’**).
A
FIG. 4.-Radiative
capture cross section of Rb, Zr, MO, Sn and Sm isotopes as a function of mass number A at -30 keV. 0, A: radiative capture cross sections of Zr isotopes: A, n : even-odd nuclei; 0, 0 : eveweven nuclei; nuclei. x : oddwen
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