Collision Matrix for (<span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo>,</mo><mi> </mi><mi>d</mi></math></span>) and (<span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mo>,</mo><mi> </mi><mi>d</mi></math></span>) Reactions

R. G. Thomas

doi:10.1103/PhysRev.100.25

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Collision Matrix for ( $n, d$ ) and ( $p, d$ ) Reactions

R. G. Thomas

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico

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Phys. Rev. 100, 25 – Published 1 October, 1955

DOI: https://doi.org/10.1103/PhysRev.100.25

Abstract

The contributions to ( $n, d$ ), ( $p, d$ ) reactions and their inverses from the pickup and stripping mechanisms are considered as corrections to the compound-nucleus or $R$ -matrix theory of nuclear reactions. In an ( $n, d$ ) reaction, for example, the $R$ -matrix theory neglects the interaction of the incident neutron with the target-nucleus proton "tails" which extend beyond the nuclear radius. The pickup correction to the collision-matrix component, or reaction amplitude, appears as the matrix element of the neglected interaction involving an exact wave function and the approximate wave function of the compound-nucleus system not having the interaction; a distorted-wave Born approximation is used in which the former exact wave function is replaced by one of the latter type with the appropriate radiation condition. An explicit expression is given for the collision-matrix component which, together with the compound-nucleus contribution, can be substituted directly into the formulas of Blatt and Biedenharn for total reaction cross sections and angular distributions. In general, the angular distributions contain interference terms in addition to the straight pickup and compound-nucleus contributions. If the distorted neutron and deuteron spherical partial waves are assumed to depend only on the angular momenta, and not explicitly on the total spin and the channel spins, the formula of Tobocman is obtained for the pickup contribution, while Butler's formula is obtained if plane waves are used instead of distorted waves. There are discussions of the various approximations, the exchange terms, and the question of the nuclear radius.

References (25)

S. T. Butler, Phys. Rev. 80, 1095 (1950) Proc. Roy. Soc. (London) A208, 559 (1951) Phys. Rev. 88, 685 (1952)
Bhatia, Huang, Huby, and Newms, Phil. Mag. 43, 485 (1952)
J. R. Oppenheimer and M. Phillips, Phys. Rev. 48, 500 (1935) H. A. Bethe, 53, 39 (1938) R. Serber, 72, 1008 (1947) D. C. Peaslee, 74, 1001 (1948) G. F. Chew and M. L. Goldberger, 77, 470 (1950)
P. B. Daitch and J. B. French, Phys. Rev. 87, 900 (1952) R. Huby, Proc. Roy. Soc. (London) A215, 385 (1952) N. Austern, Phys. Rev. 89, 318 (1953) E. Gerjuoy, 91, 645 (1953) F. L. Friedman and W. Tobocman, 92, 93 (1953) J. Horowitz and A. M. L. Messiah, 92, 1326 (1953) J. phys. et radium 14, 695 (1953) N. C. Francis and K. M. Watson, Phys. Rev. 93, 313 (1954) S. Yoshida, Progr. Theoret. Phys. (Japan) 10, 1, 370 (1953) Fujimoto, Hayakawa, and Nishijima, 10, 113 (1953) R. D. Dalitz, Proc. Phys. Soc. (London) A66, 28 (1953) M. Gell-Mann and M. L. Goldberger, Phys. Rev. 91, 398 (1953) E. Clementel, Nuovo cimento 11, 412 (1954) S. T. Butler and N. Austern, Phys. Rev. 93, 355 (1954) W. Tobocman, 94, 1655 (1954) J. Yocoz, Proc. Phys. Soc. (London) A67, 813 (1954) I. P. Grant, A67, 981 (1954) ibid.A68, 244 (1955) R. Huby, Progr. Nuclear Phys. 3, 177 (1953)
John M. Blatt and L. C. Biedenharn, Revs. Modern Phys. 24, 258 (1952)
Berthelot, Cohen, Cotton, Faraggi, Grjebine, Levêque, Naggiar, Roclawski-Conjeaud, and Szteinsznaider, Compt. rend. 238, 1312 (1954) Holmgren, Blair, Simmons, Stratton, and Stuart, Phys. Rev. 95, 1544 (1954) Jones, McEllistrem, Douglas, Herring, and Silverstein, 98, 241(A) (1954)
W. Tobocman and M. H. Kalos, Phys. Rev. 97, 132 (1955)
Austern, Butler, and McManus, Phys. Rev. 92, 350 (1953)
E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947) T. Teichmann and E. P. Wigner, 87, 123 (1952)
R. G. Thomas, Phys. Rev. 97, 224 (1955)
V. F. Weisskopf, Phys. Rev. 52, 295 (1937) V. F. Weisskopf and D. H. Ewing, 57, 472, 935 (1940) H. Feshbach and V. F. Weisskopf, 76, 1550 (1949) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, Inc., New York, 1952), Chaps. VIII and IX
N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Oxford University Press, London, 1949), Chap. VIII, Sec. 5
N. C. Francis and K. M. Watson, Phys. Rev. 92, 291 (1953) Feshbach, Porter, and Weisskopf, 96, 448 (1954) Lane, Thomas, and Wigner, 98, 693 (1955)
G. Breit, Phys. Rev. 58, 1068 (1940) E. P. Wigner, 70, 15 (1946)
K. M. Watson, Phys. Rev. 88, 1163 (1952) G. Breit and H. A. Bethe, 93, 888 (1954)
L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 25, 736 (1953) R. Huby, Proc. Phys. Soc. (London) A67, 1103 (1954)
B. H. Bransden, Proc. Phys. Soc. (London) A65, 738 (1952) W. N. Hess and B. J. Moyer, Phys. Rev. 96, 859(A) (1954)
R. D. Dalitz, [4] R. G. Thomas (unpublished)
G. Breit, Phys. Rev. 71, 215 (1947)
Wigner and Eisenbud, [9]
(Sec. II of [10]) [Sec. II-B of R. G. Thomas, Phys. Rev. 88, 1109 (1952)]
Biedenharn, Blatt, and Rose, Revs. Modern Phys. 24, 249 (1952)
A. Simon and T. A. Welton, Phys. Rev. 90, 1036 (1953) Albert Simon, 92, 1050 (1953) W. B. Cheston, 96, 1590 (1954) J. Horowitz and A. M. L. Messiah, J. phys. radium 14, 731 (1953) H. C. Newms, Proc. Phys. Soc. (London) A66, 477 (1953)
J. R. Holt and T. N. Marsham, Proc. Phys. Soc. (London) A66, 1032 (1953)
R. G. Thomas, Phys. Rev. 91, 453(A) (1953) Horowitz and Messiah, [4] Fujimoto, Kikuchi, and Yoshida, Progr. Theoret. Phys. (Japan) 11, 264 (1954) G. Abraham, Proc. Phys. Soc. (London) A67, 273 (1954) A. M. Lane and D. H. Wilkinson, Phys. Rev. 97, 1199 (1955)

[c1] S. T. Butler, Phys. Rev. 80, 1095 (1950) Proc. Roy. Soc. (London) A208, 559 (1951) Phys. Rev. 88, 685 (1952)

[c2] Bhatia, Huang, Huby, and Newms, Phil. Mag. 43, 485 (1952)

[c3] J. R. Oppenheimer and M. Phillips, Phys. Rev. 48, 500 (1935) H. A. Bethe, 53, 39 (1938) R. Serber, 72, 1008 (1947) D. C. Peaslee, 74, 1001 (1948) G. F. Chew and M. L. Goldberger, 77, 470 (1950)

[c4] P. B. Daitch and J. B. French, Phys. Rev. 87, 900 (1952) R. Huby, Proc. Roy. Soc. (London) A215, 385 (1952) N. Austern, Phys. Rev. 89, 318 (1953) E. Gerjuoy, 91, 645 (1953) F. L. Friedman and W. Tobocman, 92, 93 (1953) J. Horowitz and A. M. L. Messiah, 92, 1326 (1953) J. phys. et radium 14, 695 (1953) N. C. Francis and K. M. Watson, Phys. Rev. 93, 313 (1954) S. Yoshida, Progr. Theoret. Phys. (Japan) 10, 1, 370 (1953) Fujimoto, Hayakawa, and Nishijima, 10, 113 (1953) R. D. Dalitz, Proc. Phys. Soc. (London) A66, 28 (1953) M. Gell-Mann and M. L. Goldberger, Phys. Rev. 91, 398 (1953) E. Clementel, Nuovo cimento 11, 412 (1954) S. T. Butler and N. Austern, Phys. Rev. 93, 355 (1954) W. Tobocman, 94, 1655 (1954) J. Yocoz, Proc. Phys. Soc. (London) A67, 813 (1954) I. P. Grant, A67, 981 (1954) ibid.A68, 244 (1955) R. Huby, Progr. Nuclear Phys. 3, 177 (1953)

[c5] John M. Blatt and L. C. Biedenharn, Revs. Modern Phys. 24, 258 (1952)

[c6] Berthelot, Cohen, Cotton, Faraggi, Grjebine, Levêque, Naggiar, Roclawski-Conjeaud, and Szteinsznaider, Compt. rend. 238, 1312 (1954) Holmgren, Blair, Simmons, Stratton, and Stuart, Phys. Rev. 95, 1544 (1954) Jones, McEllistrem, Douglas, Herring, and Silverstein, 98, 241(A) (1954)

[c7] W. Tobocman and M. H. Kalos, Phys. Rev. 97, 132 (1955)

[c8] Austern, Butler, and McManus, Phys. Rev. 92, 350 (1953)

[c9] E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947) T. Teichmann and E. P. Wigner, 87, 123 (1952)

[c10] R. G. Thomas, Phys. Rev. 97, 224 (1955)

[c11] V. F. Weisskopf, Phys. Rev. 52, 295 (1937) V. F. Weisskopf and D. H. Ewing, 57, 472, 935 (1940) H. Feshbach and V. F. Weisskopf, 76, 1550 (1949) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, Inc., New York, 1952), Chaps. VIII and IX

[c12] N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Oxford University Press, London, 1949), Chap. VIII, Sec. 5

[c13] N. C. Francis and K. M. Watson, Phys. Rev. 92, 291 (1953) Feshbach, Porter, and Weisskopf, 96, 448 (1954) Lane, Thomas, and Wigner, 98, 693 (1955)

[c14] G. Breit, Phys. Rev. 58, 1068 (1940) E. P. Wigner, 70, 15 (1946)

[c15] K. M. Watson, Phys. Rev. 88, 1163 (1952) G. Breit and H. A. Bethe, 93, 888 (1954)

[c16] L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 25, 736 (1953) R. Huby, Proc. Phys. Soc. (London) A67, 1103 (1954)

[c17] B. H. Bransden, Proc. Phys. Soc. (London) A65, 738 (1952) W. N. Hess and B. J. Moyer, Phys. Rev. 96, 859(A) (1954)

[c18] R. D. Dalitz, [4] R. G. Thomas (unpublished)

[c19] G. Breit, Phys. Rev. 71, 215 (1947)

[c20] Wigner and Eisenbud, [9]

[c21] (Sec. II of [10]) [Sec. II-B of R. G. Thomas, Phys. Rev. 88, 1109 (1952)]

[c22] Biedenharn, Blatt, and Rose, Revs. Modern Phys. 24, 249 (1952)

[c23] A. Simon and T. A. Welton, Phys. Rev. 90, 1036 (1953) Albert Simon, 92, 1050 (1953) W. B. Cheston, 96, 1590 (1954) J. Horowitz and A. M. L. Messiah, J. phys. radium 14, 731 (1953) H. C. Newms, Proc. Phys. Soc. (London) A66, 477 (1953)

[c24] J. R. Holt and T. N. Marsham, Proc. Phys. Soc. (London) A66, 1032 (1953)

[c25] R. G. Thomas, Phys. Rev. 91, 453(A) (1953) Horowitz and Messiah, [4] Fujimoto, Kikuchi, and Yoshida, Progr. Theoret. Phys. (Japan) 11, 264 (1954) G. Abraham, Proc. Phys. Soc. (London) A67, 273 (1954) A. M. Lane and D. H. Wilkinson, Phys. Rev. 97, 1199 (1955)

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Collision Matrix for ( $n, d$ ) and ( $p, d$ ) Reactions

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Collision Matrix for (n, d) and (p, d) Reactions

Abstract

References (25)

Collision Matrix for ( $n, d$ ) and ( $p, d$ ) Reactions