Abstract:
A formula for semi-classical elastic S-matrix element has been derived by Brink and
Takigawa for a potential having three turning points with a potential barrier (see [1] ) . If
S, 1 denotes the S-matrix element corresponding to angular momentum l and total
angular momentum j, S1 J is given, in the usual notation, by
. s: {1 + N(iE;)exp(2iS32 )}
Su =exp(2zu1) N(z'c) exp( 2z.8 ) , 32
(l)
where N(z) is defined by N(z) = f rxp(zln(~)) and E = -i S21 . r 1 +z n
If k = ~2:~£ is the wave number corresponding to a zero of semi-classical S-matrix
element, it can be shown that
1 + 1 + exp(2nc) exp (2 1· s 31 ) -_ 0
N(ic)
and one obtains
s31 = (2n +I) H + __!__ ln(--N_(_ic_)_J
2 2i I + exp(2nc)
(2)
which is a necessary and sufficient condition for the semi-classical S-matrix element to
be zero.
Now, S U = 0 means the absence of an outgoing wave. Since the asymptotic wave
boundary condition for the corresponding partial wave U U (k,r) is given by
U lJ (k,r) ~ U1H (k,r)- SuUt) (k,r),
(3)
where U 1(- l and U j + l stand for the incoming and outgoing Coulomb wave functions
respectively.
A new phenomenon was discovered by M. Kawai and Y. Iresi (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
(!), total angular momentum (j) and target nucleus. It has been found that this
phenomenon is universal for light ion elastic scattering (see[3 ]).
To the zero S-matrix element corresponding to this phenomenon, we have found that 2_ ln __!!ii~) __ ~ 0 both in case of deuterons scattering on nuclei and
2i 1 + exp(27r£)
4 He scattering on 40 Ni ,which means Su = (2n + 1) 7l'. It can be shown [1] that th~ S-
2
2i.~ 2iS1
matrix element can be put into the form Su :::::: _e_ + -=---z = 178 + 171 assuming that
N N
I e ZiS32 I :-:; I N 12 , where 17 B and 171 stand for the amplitude of the reflected wave at the
external turning point and the amplitude of the reflected wave at the innermost turning
point, respectively. Then it is clear that Su = 0 is due to the fact that the destructive
interference of these waves in the asymptotic region.
