Model for Nuclear Reactions with Neutrons

H. Feshbach; C. E. Porter; V. F. Weisskopf

doi:10.1103/PhysRev.96.448

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Model for Nuclear Reactions with Neutrons

H. Feshbach, C. E. Porter^*, and V. F. Weisskopf

Department of Physics and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts

^*Now at Brookhaven National Laboratory, Upton, New York.

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Phys. Rev. 96, 448 – Published 15 October, 1954

DOI: https://doi.org/10.1103/PhysRev.96.448

Abstract

A simple model is proposed for the description of the scattering and the compound nucleus formation by nucleons impinging upon complex nuclei. It is shown that, by making appropriate averages over resonances, an average problem can be defined which is referred to as the "gross-structure" problem. Solution of this problem permits the calculation of the average total cross section, the cross section for the formation of the compound nucleus, and the part of the elastic-scattering cross section which does not involve formation of the compound nucleus. Unambiguous definitions are given for the latter cross sections.

The model describing these properties consists in replacing the nucleus by a one-body potential which acts upon the incident nucleon. This potential $V = V_{0} + i V_{1}$ is complex; the real part represents the average potential in the nucleus; the imaginary part causes an absorption which describes the formation of the compound nucleus. As a first approximation a potential is used whose real part $V_{0}$ is a rectangular potential well and whose imaginary part is a constant fraction of the real part $V_{1} = ζ V_{0}$ .

This model is used to reproduce the total cross sections for neutrons, the angular dependence of the elastic scattering, and the cross section for the formation of the compound nucleus. It is shown that the average properties of neutron resonances, in particular the ratio of the neutron width to the level spacing, are connected with the gross-structure problem and can be predicted by this model.

The observed neutron total cross sections can be very well reproduced in the energy region between zero and 3 Mev with a well depth of 42 Mev, a factor $ζ$ of 0.03, and a nuclear radius of $R = 1.45 \times 10^{- 13} A^{\frac{1}{3}}$ cm. The angular dependence of the scattering cross section at 1 Mev is fairly well reproduced by the same model. The theoretical and experimental values for the ratios of neutron width to level distance at low energies and the reaction cross sections at 1 Mev do not agree too well but they show a qualitative similarity.

References (55)

H. Bethe, Phys. Rev. 57, 1125 (1940) Feshbach, Peaslee, and Weisskopf, 71, 145 (1947) H. Feshbach and V. F. Weisskopf, 76, 1550 (1949) E. P. Wigner, 73, 1002 (1948) Am. J. Phys. 17, 99 (1949)
H. Feshbach and V. F. Weisskopf, Phys. Rev. 76, 1550 (1949)
Phillips, Davis, and Graves, Phys. Rev. 88, 600 (1952)
B. T. Feld [Experimental Nuclear Physics, edited by E. Segrè (John Wiley and Sons, Inc., New York, 1953), Vol. 2]
Feshbach, Peaslee, and Weisskopf ([1]) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, Inc., New York, 1952)
H. Bethe, Phys. Rev. 57, 1125 (1940)
Fernbach, Serber, and Taylor, Phys. Rev. 75, 1352 (1949)
K. W. Ford and D. Bohm, Phys. Rev. 79, 745 (1950)
Morse, Lowan, Feshbach, and Lax, U. S. Navy Department of Research and Inventions Report No. 62.1R, 1945 (unpublished)
W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952)
L. Wolfenstein, Phys. Rev. 82, 690 (1951)
G. Racah, Phys. Rev. 61, 186 (1942) ibid.62, 438 (1942)
J. M. Blatt and L. C. Biedenharn, Revs. Modern Phys. 24, 258 (1952)
Biedenharn, Blatt, and Rose, Revs. Modern Phys. 24, 249 (1952)
L. C. Biedenharn, Oak Ridge National Laboratory Report ORNL-1501, May 28, 1953 (unpublished)
Omitted endnote
Omitted endnote
Omitted endnote
H. H. Barschall, Phys. Rev. 86, 431 (1952)
Miller, Adair, Bockelman, and Darden, Phys. Rev. 88, 83 (1952)
Walt, Becker, Okazaki, and Fields, Phys. Rev. 89, 1271 (1953)
Okazaki, Darden, and Walton, Phys. Rev. 93, 461 (1954)
N. Nereson and S. Darden, Phys. Rev. 89, 775 (1953) 94, 1678 (1954) unpublished data on Li and B (private communication)
C. F. Cook and T. W. Bonner, Phys. Rev. 94, 651 (1954) McCrary, Taylor, and Bonner, unpublished data on Li (private communication)
McCraryTaylorBonnerNeutron Cross Sections, U. S. Atomic Energy Commission Report AECU-2040 (Technical Information Division, Department of Commerce, Washington, D. C., 1952) three supplements (unpublished)
C. E. Porter, Bull. Am. Phys. Soc. 29, No. 5, 25 (1954)
Feshbach, Porter, and Weisskopf, Phys. Rev. 90, 166 (1953)
R. K. Adair, Phys. Rev. 94, 737 (1954)
M. H. Johnson and E. Teller, Phys. Rev. 93, 357 (1954)
M. Walt and H. H. Barschall, Phys. Rev. 93, 1062 (1954)
R. S. Carter et al., Phys. Rev. (to be published)
F. G. P. Seidl, Hughes, Palvesky, Levin, Kato, and Sjöstrand, Phys. Rev. 95, 476 (1954) private communication
R. S. Carter and J. A. Harvey, Phys. Rev. 95, 645(A) (1954)
Foote, Landon, and Sailor, Phys. Rev. 92, 656 (1953)
Sailor, Landon, and Foote, Phys. Rev. 93, 1292 (1954)
Pilcher, Carter, and Stolovy, Phys. Rev. 95, 645(A) (1954) private communication
Hughes, Kato, and Levin, Phys. Rev. 92, 1094 (1953) private communication
R. L. Christensen, Phys. Rev. 92, 1509 (1953)
Melkonian, Havens, and Rainwater, Phys. Rev. 92, 702 (1953)
L. Bollinger, unpublished data on Sb (private communication)
V. E. Pilcher and R. S. Carter (private communication)
A. Bohr and B. R. Mottelson, Kgl. Danske Videnskab. Selenskab Mat. fys. Medd. 27, 16 (1953)
Omitted endnote
Lonsjo, Taylor, and Bonner (private communication)
Morrison, Muirhead, and Rosser, Phil. Mag. 44, 1326 (1953)
M. Goldberger, Phys. Rev. 74, 1269 (1948)
V. F. Weisshopf, Science 113, 101 (1951)
P. C. Gugelot, Phys. Rev. 87, 525 (1952)
J. W. Burkig and B. T. Wright, Phys. Rev. 82, 451 (1951)
B. L. Cohen and R. V. Neidigh, Phys. Rev. 93, 282 (1954)
D. M. Chase and F. Rohrlich, Phys. Rev. 94, 81 (1954)
R. D. Woods and D. S. Saxon (private communication)
Darden, Field, and Adair, Phys. Rev. 93, 931 (1953)
Omitted endnote
E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947)

[c1] H. Bethe, Phys. Rev. 57, 1125 (1940) Feshbach, Peaslee, and Weisskopf, 71, 145 (1947) H. Feshbach and V. F. Weisskopf, 76, 1550 (1949) E. P. Wigner, 73, 1002 (1948) Am. J. Phys. 17, 99 (1949)

[c2] H. Feshbach and V. F. Weisskopf, Phys. Rev. 76, 1550 (1949)

[c3] Phillips, Davis, and Graves, Phys. Rev. 88, 600 (1952)

[c4] B. T. Feld [Experimental Nuclear Physics, edited by E. Segrè (John Wiley and Sons, Inc., New York, 1953), Vol. 2]

[c5] Feshbach, Peaslee, and Weisskopf ([1]) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, Inc., New York, 1952)

[c6] H. Bethe, Phys. Rev. 57, 1125 (1940)

[c7] Fernbach, Serber, and Taylor, Phys. Rev. 75, 1352 (1949)

[c8] K. W. Ford and D. Bohm, Phys. Rev. 79, 745 (1950)

[c9] Morse, Lowan, Feshbach, and Lax, U. S. Navy Department of Research and Inventions Report No. 62.1R, 1945 (unpublished)

[c10] W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952)

[c11] L. Wolfenstein, Phys. Rev. 82, 690 (1951)

[c12] G. Racah, Phys. Rev. 61, 186 (1942) ibid.62, 438 (1942)

[c13] J. M. Blatt and L. C. Biedenharn, Revs. Modern Phys. 24, 258 (1952)

[c14] Biedenharn, Blatt, and Rose, Revs. Modern Phys. 24, 249 (1952)

[c15] L. C. Biedenharn, Oak Ridge National Laboratory Report ORNL-1501, May 28, 1953 (unpublished)

[c16] Omitted endnote

[c17] Omitted endnote

[c18] Omitted endnote

[c19] H. H. Barschall, Phys. Rev. 86, 431 (1952)

[c20] Miller, Adair, Bockelman, and Darden, Phys. Rev. 88, 83 (1952)

[c21] Walt, Becker, Okazaki, and Fields, Phys. Rev. 89, 1271 (1953)

[c22] Okazaki, Darden, and Walton, Phys. Rev. 93, 461 (1954)

[c23] N. Nereson and S. Darden, Phys. Rev. 89, 775 (1953) 94, 1678 (1954) unpublished data on Li and B (private communication)

[c24] C. F. Cook and T. W. Bonner, Phys. Rev. 94, 651 (1954) McCrary, Taylor, and Bonner, unpublished data on Li (private communication)

[c25] McCraryTaylorBonnerNeutron Cross Sections, U. S. Atomic Energy Commission Report AECU-2040 (Technical Information Division, Department of Commerce, Washington, D. C., 1952) three supplements (unpublished)

[c26] C. E. Porter, Bull. Am. Phys. Soc. 29, No. 5, 25 (1954)

[c27] Feshbach, Porter, and Weisskopf, Phys. Rev. 90, 166 (1953)

[c28] R. K. Adair, Phys. Rev. 94, 737 (1954)

[c29] M. H. Johnson and E. Teller, Phys. Rev. 93, 357 (1954)

[c30] M. Walt and H. H. Barschall, Phys. Rev. 93, 1062 (1954)

[c31] R. S. Carter et al., Phys. Rev. (to be published)

[c32] F. G. P. Seidl, Hughes, Palvesky, Levin, Kato, and Sjöstrand, Phys. Rev. 95, 476 (1954) private communication

[c33] R. S. Carter and J. A. Harvey, Phys. Rev. 95, 645(A) (1954)

[c34] Foote, Landon, and Sailor, Phys. Rev. 92, 656 (1953)

[c35] Sailor, Landon, and Foote, Phys. Rev. 93, 1292 (1954)

[c36] Pilcher, Carter, and Stolovy, Phys. Rev. 95, 645(A) (1954) private communication

[c37] Hughes, Kato, and Levin, Phys. Rev. 92, 1094 (1953) private communication

[c38] R. L. Christensen, Phys. Rev. 92, 1509 (1953)

[c39] Melkonian, Havens, and Rainwater, Phys. Rev. 92, 702 (1953)

[c40] L. Bollinger, unpublished data on Sb (private communication)

[c41] V. E. Pilcher and R. S. Carter (private communication)

[c42] A. Bohr and B. R. Mottelson, Kgl. Danske Videnskab. Selenskab Mat. fys. Medd. 27, 16 (1953)

[c43] Omitted endnote

[c44] Lonsjo, Taylor, and Bonner (private communication)

[c45] Morrison, Muirhead, and Rosser, Phil. Mag. 44, 1326 (1953)

[c46] M. Goldberger, Phys. Rev. 74, 1269 (1948)

[c47] V. F. Weisshopf, Science 113, 101 (1951)

[c48] P. C. Gugelot, Phys. Rev. 87, 525 (1952)

[c49] J. W. Burkig and B. T. Wright, Phys. Rev. 82, 451 (1951)

[c50] B. L. Cohen and R. V. Neidigh, Phys. Rev. 93, 282 (1954)

[c51] D. M. Chase and F. Rohrlich, Phys. Rev. 94, 81 (1954)

[c52] R. D. Woods and D. S. Saxon (private communication)

[c53] Darden, Field, and Adair, Phys. Rev. 93, 931 (1953)

[c54] Omitted endnote

[c55] E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947)

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Model for Nuclear Reactions with Neutrons

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