116
6 Three-Body Approach to Structural Properties …
To write the equation for the scattering amplitude at energy E, we start from
Eq. (6.67) and introduce a function X(p, q; E) collecting the inhomogeneous terms
and rewrite the equation as:
T ( p, q; E) = X ( p, q; E) +
1
π
∫ dq
q 2
q 2 − k 2 − iε
Z 1 ( p, q ; E)τ −1 (q ; E)T (q , q; E)
+
2
π 2
q 2 dq Z 2 ( p, q ; E)χ −1 (q ; E)
dq
q 2
q 2 − k 2 − iε
Z 3 (q , q ; E)τ −1 (q ; E)T (q , q; E),
(6.75)
where
X ( p, q; E) ≡ Z 1 ( p, q; E) +
2
π
dq
q
2 Z 2 ( p, q
; E)χ
−1
(q
; E)Z 3 (q
, q; E)
(6.76)
Now, we subtract from Eq. (6.75) the equation corresponding to the amplitude
T ( p, q; ε c ) when E = ε c to get
T ( p, q; E) = T ( p, q; ε c ) + [X ( p, q; E) − X ( p, q; ε c )]
+
1
π
∫
dq
{
q
2
q 2 −k 2 −iε
Z 1 ( p, q
; E)τ
−1
(q
; E)T (q
, q; E)
−Z 1 ( p, q
; ε c )τ
−1
(q
; ε c )T (q
, q; ε c }
+
2
π 2
q
2 dq
Z 2 ( p, q
; E)χ
−1
(q
; E)
dq
q
2
q 2 − k 2 − iε
Z 3 (q
, q
; E)τ
−1
(q
; E)T (q
, q; E)
−
dq
q
2 Z 2 ( p, q
; ε c )χ
−1
(q
; ε c )
dq
Z 3 (q
, q
; ε c )τ
−1
(q
; ε c )T (q
, q; ε c )
(6.77)
To write this equation for half-off-shell scattering amplitude, we put q = k where
E = k
2
/2m d + ε c and express the scattering amplitude as T k ( p, E). Note that the
two amplitudes T k ( p, E) and T k ( p, ε c ) are different. However, for these functions
appearing within the integrals, we make a simplifying assumption, viz. T k (q
, E) =
T k (q
, ε c ) and T k (q
, E) = T k (q
, ε c ). This approximation is based on the fact that
below the breakup threshold, the values of k, for which we are interested here, are
going to be small as compared to ε c . With this simplification, Eq. (6.77) reduces to:
T k ( p, E) = T k ( p, ε c ) + [X k ( p, E) − X k ( p, ε c )]