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Inelastic scattering of fission neutrons in uranium
J. Nuclear Energy I, 1957. Vol. 4. pp. 326 to 328. Pergamon Press Ltd.. London
RESEARCH
NOTE
Inelastic scattering of fission neutrons in uranium (Received 11 June1956; in revised form I September 1956) Abstract-Energy distribution of inelastically scattered neutrons in uranium, based on various models of a nucleus, has been considered. An approximate comparison with the neutron spectrum in the fast reactor, EBRI, shows that nuclear level density calculated on the assumption of a degenerate Fermi-Dirac gas gives the best fit.
IT is well known that the inelastic scattering of high-energy neutrons by the fuel atoms is the main cause of degradation of fission spectra in fast reactors. Theoretically the energy distribution of inelastically scattered neutrons has been studied for some of the nuclei like Zn, Sn, Yb, Th (FELD et al., 1951). From the point of view of reactor study it is of greater importance to consider the inelastic scattering of fast neutrons by uranium nuclei, and this problem is treated here in some detail. The energy distribution of inelastically scattered neutrons is given by the expression (BLATTand WEISSKOPF,1952). F(E) dl? = constant ,%,(E)p(E - B) dE
(1)
where E and ,??represent the energy of the incident and the scattered neutron respectively. p is the level density of the residual nucleus corresponding to the excited energy (E - E), and o,(_/?)is the cross-section for formation of compound nucleus at B. Since the available information for the level density is very meagre, various approaches, based on different models of the nucleus, have been made to get a suitable expression for the distribution function. (i) For a highly degenerate Fermi-Dirac gas (FELD et al., 1951), P(E - 6) = C, exp a,(E - I?)” where C, and a, are constants adjusted from experimental results. (2) in (l), F(8) dE’ = constant Eu,(B) exp a,(E - B)” d,??.
(2) Substituting (3)
The energy distribution of inelastically scattered neutrons as given by (3) for U235 is shown by curve (i). Here the energy of incident neutrons is 2 MeV (mean fission energy) and the values for a1 (= 7.32 MeV-l) and o,(E) for (R = 9 x lo-i3 cm) were taken from the graphs given by FELDet al. (1951). According to this distribution, the mean energy of the inelastically scattered neutrons comes out to be NO.55 MeV. Instead of considering the incident neutrons of energy 2 MeV, if we average the incident neutron energy over a fission spectrum we obtain an energy distribution 326
Research note
327
for the scattered neutrons as shown by curve (ii). This gives a mean energy 4.49 MeV. (ii) In deriving (3) it is assumed that the level-density formula holds right up to the ground state. However, if the level density is taken to be constant up to E, = 3 MeV (WEISSKOPF,1937; FELD et al., 1951), the distribution is of the form F(F(E)dl7 = constant &r,(E) dJ?
(4)
which is shown by curve (iii) for the case of U 235. This distribution gives a mean energy of scattered neutrons as -1.33 MeV. (iii) The expression for the level density as derived by BETHE(1937), considering the free-particle model, is p(E - 1) = C,(E - E)-” exp a,(E - I?‘)*
(5)
where C, = 5.88 A-2 a2 = O-488 A6
for R = 1.5 x lo-r3 AlI3 cm.
This differs from relation (2) in that here the coefficient multiplying the exponential is energy-dependent. Substituting (5) in (1), the distribution of neutrons inelastically scattered by U235has been calculated and is shown by curve (iv). (iv) The liquid-drop model (BETHE,1937) gives p(E: - E) = C,(E - J?)-5/7 exp a,@ - fl)4/7
(6)
where C, = 0.171 Al/‘; a3 = 0.71 A217. The distribution of neutrons, obtained by using (6), is shown by curve (v). The neutron spectrum in the centre of the core of EBRI (LICHTENBERGER et al., 1955) is shown by curve (vi), with mean energy No.69 MeV. Since the energy degradation of fission spectrum in a fast reactor is mainly due to the inelastic scattering process, one expects a rough similarity between the calculated curve and the neutron spectrum in EBRI. However, it is important to note that, in a detailed study of the neutron spectrum in a fast reactor, processes like elastic scattering, leakage, slowing down of neutrons in the coolant will have to be properly considered. A ten-group calculation on this problem has been done with fair amount of success by OKRENTet al. (1955). As we have not considered, either, further fission in U235 or elastic scattering processes, which contribute neutrons around 2 MeV, the theoretical curve would lie below the experimental curve at 2 MeV. It can be seen from the graph that the curve (ii) obtained from equation (3) when averaged over the fission spectrum shows a somewhat better resemblance with the experimental curve. Curve (ii) shows a maximum around O-3 MeV, which is close enough to O-15 MeV, the position for maximum of the neutron spectrum in EBRI. On the other hand, the curves (iv) and (v) for free-particle and liquid-drop model show rather flat maxima, lying at about 0.65 MeV and 0.85 MeV respectively. Further, the behaviour of these curves is in complete disagreement with the experimental curve. This shows that so far as statistical model of a nucleus is concerned the level density as given by relation (2) appears to give the best results. Using relation (2), we have also calculated the mean energy after a 2-MeV
328
Research note
neutron is inelastically scattered in U238. In a graphite moderator the age up to In resonance (144 eV) of fission neutrons (mean energy 2 MeV) which have been inelastically scattered once in U23*, is found to be 231 - 53 = 178 cm2 (KAPLAN and CHERNICK,1955). Using cs as given by HUGHESand HARVEY(1955), the mean 3 .-
A20 5 b
5
z 0
10 Gi ;;
0
0.5
1.0 Energy
1.5
2.0 MeV
Fig. I.-Energy distribution of inelastically scattered neutrons in uranium. Curves (i), (iii), (iv), and (v) are for different models. Curve (ii) corresponds to the energy distribution (i) when incident neutron energy is averaged over a fission spectrum. Curve (vi) gives the neutron spectrum in EBRI. All curves have been suitably normalized.
energy which would give the above age is ho.2 MeV. This may be compared with our calculated value, d-49 MeV, which is based on the assumption that the formula (2) holds right up to the ground state. However, BATCHELOR (1956) has observed in U238, which is an even-even nucleus, rotational levels below 1 MeV. These follow a level scheme quite different from that given by equation (2). If we consider these rotational levels below 1 MeV and neglect the conservation of momentum, as we have done so far, the mean energy of the inelastically scattered neutrons increases considerably. Acknowledgements-We
are thankful to Mr. L. S. KoTHARIfor suggesting the problem and to Dr. K. S. SINGWI for discussions. One of us (R. C. B.) is grateful to the Department of Atomic Energy for the award of a Junior Research Fellowship. R. C. BHANDARI R. D. JAIN
Atomic Energy Establishment, Apollo Pier Road Bombay 1, India
REFERENCES BATCHELOR R. (1956) Proc. Phys. Sot. 69A, 214. BETHEH. A. (1937) Rev. Mod. Physics 9,79-90. BLAT 3. M. and WEISSKOPF V. F. (1952) Theoretical Nuclear Physics John Wiley and Sons, New
York, page 367. CHERNICK J. and KAPLANI. (1955) Proceedings of the International Conference on the Peaceful Uses
qf Atomic Energy, AlConf. 8/p/606. FELDB. T. et al. (1951) Final Report of the Fast Neutron Project Data NYO-636. HUGHESD. J. and HARVEYJ. A. (1955) Neutron Cross Section, BNL 325. LICHTENBERGER H. V. et al. (1955) Proceedings of the International Conference on the Peaceful Uses
of Atomic Energy, AIConf. 8/p/813. OKRENTD. et al. (1955) Proceedings of the International Conference on the Peaceful Uses of Atomic
Energy, AlConf. 8/p/609. WEISSKOPF V. F. (1937) Phys. Rev. 52, 296.
RESEARCH
NOTE
Inelastic scattering of fission neutrons in uranium (Received 11 June1956; in revised form I September 1956) Abstract-Energy distribution of inelastically scattered neutrons in uranium, based on various models of a nucleus, has been considered. An approximate comparison with the neutron spectrum in the fast reactor, EBRI, shows that nuclear level density calculated on the assumption of a degenerate Fermi-Dirac gas gives the best fit.
IT is well known that the inelastic scattering of high-energy neutrons by the fuel atoms is the main cause of degradation of fission spectra in fast reactors. Theoretically the energy distribution of inelastically scattered neutrons has been studied for some of the nuclei like Zn, Sn, Yb, Th (FELD et al., 1951). From the point of view of reactor study it is of greater importance to consider the inelastic scattering of fast neutrons by uranium nuclei, and this problem is treated here in some detail. The energy distribution of inelastically scattered neutrons is given by the expression (BLATTand WEISSKOPF,1952). F(E) dl? = constant ,%,(E)p(E - B) dE
(1)
where E and ,??represent the energy of the incident and the scattered neutron respectively. p is the level density of the residual nucleus corresponding to the excited energy (E - E), and o,(_/?)is the cross-section for formation of compound nucleus at B. Since the available information for the level density is very meagre, various approaches, based on different models of the nucleus, have been made to get a suitable expression for the distribution function. (i) For a highly degenerate Fermi-Dirac gas (FELD et al., 1951), P(E - 6) = C, exp a,(E - I?)” where C, and a, are constants adjusted from experimental results. (2) in (l), F(8) dE’ = constant Eu,(B) exp a,(E - B)” d,??.
(2) Substituting (3)
The energy distribution of inelastically scattered neutrons as given by (3) for U235 is shown by curve (i). Here the energy of incident neutrons is 2 MeV (mean fission energy) and the values for a1 (= 7.32 MeV-l) and o,(E) for (R = 9 x lo-i3 cm) were taken from the graphs given by FELDet al. (1951). According to this distribution, the mean energy of the inelastically scattered neutrons comes out to be NO.55 MeV. Instead of considering the incident neutrons of energy 2 MeV, if we average the incident neutron energy over a fission spectrum we obtain an energy distribution 326
Research note
327
for the scattered neutrons as shown by curve (ii). This gives a mean energy 4.49 MeV. (ii) In deriving (3) it is assumed that the level-density formula holds right up to the ground state. However, if the level density is taken to be constant up to E, = 3 MeV (WEISSKOPF,1937; FELD et al., 1951), the distribution is of the form F(F(E)dl7 = constant &r,(E) dJ?
(4)
which is shown by curve (iii) for the case of U 235. This distribution gives a mean energy of scattered neutrons as -1.33 MeV. (iii) The expression for the level density as derived by BETHE(1937), considering the free-particle model, is p(E - 1) = C,(E - E)-” exp a,(E - I?‘)*
(5)
where C, = 5.88 A-2 a2 = O-488 A6
for R = 1.5 x lo-r3 AlI3 cm.
This differs from relation (2) in that here the coefficient multiplying the exponential is energy-dependent. Substituting (5) in (1), the distribution of neutrons inelastically scattered by U235has been calculated and is shown by curve (iv). (iv) The liquid-drop model (BETHE,1937) gives p(E: - E) = C,(E - J?)-5/7 exp a,@ - fl)4/7
(6)
where C, = 0.171 Al/‘; a3 = 0.71 A217. The distribution of neutrons, obtained by using (6), is shown by curve (v). The neutron spectrum in the centre of the core of EBRI (LICHTENBERGER et al., 1955) is shown by curve (vi), with mean energy No.69 MeV. Since the energy degradation of fission spectrum in a fast reactor is mainly due to the inelastic scattering process, one expects a rough similarity between the calculated curve and the neutron spectrum in EBRI. However, it is important to note that, in a detailed study of the neutron spectrum in a fast reactor, processes like elastic scattering, leakage, slowing down of neutrons in the coolant will have to be properly considered. A ten-group calculation on this problem has been done with fair amount of success by OKRENTet al. (1955). As we have not considered, either, further fission in U235 or elastic scattering processes, which contribute neutrons around 2 MeV, the theoretical curve would lie below the experimental curve at 2 MeV. It can be seen from the graph that the curve (ii) obtained from equation (3) when averaged over the fission spectrum shows a somewhat better resemblance with the experimental curve. Curve (ii) shows a maximum around O-3 MeV, which is close enough to O-15 MeV, the position for maximum of the neutron spectrum in EBRI. On the other hand, the curves (iv) and (v) for free-particle and liquid-drop model show rather flat maxima, lying at about 0.65 MeV and 0.85 MeV respectively. Further, the behaviour of these curves is in complete disagreement with the experimental curve. This shows that so far as statistical model of a nucleus is concerned the level density as given by relation (2) appears to give the best results. Using relation (2), we have also calculated the mean energy after a 2-MeV
328
Research note
neutron is inelastically scattered in U238. In a graphite moderator the age up to In resonance (144 eV) of fission neutrons (mean energy 2 MeV) which have been inelastically scattered once in U23*, is found to be 231 - 53 = 178 cm2 (KAPLAN and CHERNICK,1955). Using cs as given by HUGHESand HARVEY(1955), the mean 3 .-
A20 5 b
5
z 0
10 Gi ;;
0
0.5
1.0 Energy
1.5
2.0 MeV
Fig. I.-Energy distribution of inelastically scattered neutrons in uranium. Curves (i), (iii), (iv), and (v) are for different models. Curve (ii) corresponds to the energy distribution (i) when incident neutron energy is averaged over a fission spectrum. Curve (vi) gives the neutron spectrum in EBRI. All curves have been suitably normalized.
energy which would give the above age is ho.2 MeV. This may be compared with our calculated value, d-49 MeV, which is based on the assumption that the formula (2) holds right up to the ground state. However, BATCHELOR (1956) has observed in U238, which is an even-even nucleus, rotational levels below 1 MeV. These follow a level scheme quite different from that given by equation (2). If we consider these rotational levels below 1 MeV and neglect the conservation of momentum, as we have done so far, the mean energy of the inelastically scattered neutrons increases considerably. Acknowledgements-We
are thankful to Mr. L. S. KoTHARIfor suggesting the problem and to Dr. K. S. SINGWI for discussions. One of us (R. C. B.) is grateful to the Department of Atomic Energy for the award of a Junior Research Fellowship. R. C. BHANDARI R. D. JAIN
Atomic Energy Establishment, Apollo Pier Road Bombay 1, India
REFERENCES BATCHELOR R. (1956) Proc. Phys. Sot. 69A, 214. BETHEH. A. (1937) Rev. Mod. Physics 9,79-90. BLAT 3. M. and WEISSKOPF V. F. (1952) Theoretical Nuclear Physics John Wiley and Sons, New
York, page 367. CHERNICK J. and KAPLANI. (1955) Proceedings of the International Conference on the Peaceful Uses
qf Atomic Energy, AlConf. 8/p/606. FELDB. T. et al. (1951) Final Report of the Fast Neutron Project Data NYO-636. HUGHESD. J. and HARVEYJ. A. (1955) Neutron Cross Section, BNL 325. LICHTENBERGER H. V. et al. (1955) Proceedings of the International Conference on the Peaceful Uses
of Atomic Energy, AIConf. 8/p/813. OKRENTD. et al. (1955) Proceedings of the International Conference on the Peaceful Uses of Atomic
Energy, AlConf. 8/p/609. WEISSKOPF V. F. (1937) Phys. Rev. 52, 296.