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Nuclear Physics A

Transmission through a real potential barrier treated by means of certain phase-integral approximations

Nanny Fröman, Per Olof Fröman

Published Jun 1, 1970• 81 citations
Publisher
DOI
10.1016/0375-9474(70)90604-4

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The transmission coefficient for a one-dimensional system of N potential barriers of general shapes (unequal or equal) is expressed exactly, by means of the phase-integral method of N. Fröman and P. O. Fröman, in terms of quantities characterizing the separate barriers and wells of the system. The exact formula, being of interest for further rigorous studies of transmission properties, can for evaluation readily be converted into a useful approximate formula by insertion of available phase-integral expressions (of an arbitrary order) for the characteristic quantities mentioned above. Error estimates are given. The sub- and superbarrier cases are treated in a unified way. Some key facts about the phase-integral method are given in Appendix A.
A smooth one-dimensional system of N potential barriers of arbitrary shapes (unequal or equal) is considered. A general necessary condition for complete transparency is obtained that can be understood as a constraint on the reflection coefficients pertaining to the single barriers of the system. A maximum transmission problem of a general kind is solved and the solution is used to give a physical interpretation of the necessary condition. The sub- and superbarrier cases are treated in a unified way. The exact final formulas can readily be converted into accurate approximate ones by insertion of available phase-integral expressions (of an arbitrary order) for certain characteristic quantities appearing in the formulas.