On a Special Zero of Semi-Classical Elastic S-Matrix Element

Abstract

A formula for semi-classical elastic s-matrix element has been derived by Brink and Takigawa (see [1]) for a potential having three turning points with a potential barrier. If Slj denotes the S-matrix element corresponding to angular momentum l and total angular momentum j, Slj is given, in the usual notation, by       + + = ( ) exp(2 ) 1 ( ) exp(2 ) exp(2 ) 32 32 1 N i iS N i iS S i lj ε ε δ , where N(z) is defined by ( ) ( ) ( ) e z z z N z exp ln 2 ( ) 2 1 Γ + = π and 21 S i π ε − = . If 2 2 h E k µ = is the wave number corresponding to a zero of semi-classical S-matrix element, it can be shown that exp(2 ) 0 ( ) 1 exp(2 ) 1 31 = + + iS N iε πε and one obtains         + = + + 1 exp(2 ) ( ) ln 2 1 2 2( )1 31 πε π N iε i S n , which is a necessary and sufficient condition for a zero of semi-classical S-matrix element. Now, = 0 lj S means the absence of an outgoing wave. Since the asymptotic wave boundary condition for the corresponding partial wave ) U (k,r lj is given by ( ) ~ ( , ) ( , ) ( ) ( ) U kr U k r S U k r lj l lj l − − − , where (−) Ul and (+) Ul stand for incoming and outgoing Coulomb wave functions respectively. A new phenomenon was discovered by M. Kawai and Y. Iseri (See [2]) in case of elastic scattering of nucleons on composite nuclei. They found that elastic S-matrix element becomes very small for special combinations of energy (E), orbital angular momentum (l), total angular momentum (j) and target nucleus as shown in the figure. It has been found that this phenomenon is universal (see [3]) for light ion elastic scattering.