ABSTRACT
The transition amplitudes for single electron capture from a helium-like system (ZT; e1, e2)i by ZP i.e. for process (11.3), treated within the CDW-4B method [189] in the prior and post versions without the term (vρ)2iZP(ZT−1)/v
are given by
T (CDW)− if = N
+(νP)N−∗(νT) ∫∫∫
× 1F1(iνT, 1, ivx1 + iv · x1)[(∆VP2 +Oi)ϕi(x1,x2) × 1F1(iνP, 1, ivs1 + iv · s1) − ∇x1ϕi(x1,x2) ·∇s1 1F1(iνP, 1, ivs1 + iv · s1)] (17.1)
and
T (CDW)+ if = N
+(νP)N−∗(νT) ∫∫∫
dx1dx2dR eiα·s1+iβ·x1ϕi(x1,x2)ϕ∗f2(x2)
× 1F1(iνP, 1, ivs1 + iv · s1) { [∆VP2 +∆V12]ϕ∗f1(s1)
× 1F1(iνT, 1, ivx1 + iv · x1) − ∇s1ϕ∗f1(s1) ·∇x1 1F1(iνT, 1, ivx1 + iv · x1)
} (17.2)
with
∆VP2 = ZP
( 1 R − 1 s2
) ∆V12 =
( 1 r12
− 1 x1
) (17.3)
where N−(νT) = Γ(1 + iνT)epiνT/2 , N+(νP) = Γ(1− iνP)epiνP/2 , νP = ZP/v and νT = (ZT − 1)/v. The two momentum transfers α and β in (17.1) and (17.2) read as
α = η − ( v
2 − ∆E
v
) vˆ β = −η −
( v
2 + ∆E v
) vˆ
∆E = Ei − Ef Ef = Ef1 + Ef2 (17.4) where Ei and Ef1,2 are the initial (helium-like) and final (hydrogen-like) binding energies, respectively. As can be identified from (17.1) and (17.2), the
ION-ATOM
entrance and exit channel perturbations are
Vi = ∆VP2 +Oi −∇x1 lnϕi(x1,x2) ·∇s1 (17.5)
Vf = ∆VP2 +∆V12 −∇s1 lnϕ∗f1(s1) ·∇x1 . (17.6)
In a test computation using (17.1), we checked that the initial eigen-problem corrective perturbation Oi from (13.40) yields a small contribution relative to those from the other terms. Therefore, the term Oi will be ignored in all the illustrations that follow without incurring significant errors. In the term ∆VP2 = ZP(1/R−1/s2) from (17.5) and (17.6), the interaction
−ZP/s2 represents the Coulomb potential between the projectile ZP and the passive electron e2. The asymptotic tail of this potential is −ZP/R so that the difference ∆VP2 between these two latter interactions is of short range as R −→∞. The active electron e1 can be captured by the projectile without its direct interaction with ZP. This occurs when the passive electron e2 interacts directly with ZP and the subsequent transfer of e1 to the projectile is made possible through the e1 − e2 correlations in the bound state ϕi(x1,x2) of the target (ZT; e1, e2)i. This latter effect is known as static correlation, because it exists in helium-like systems as one of their spectroscopic features without any reference to collision. The interaction ∆VP2 yields a negligibly small contribution in the forward direction which predominantly determines the total cross sections. This potential is significant only at larger scattering angles in differential cross sections. In any case, in the CDW-4B method, the contribution from the potential ∆VP2 to both differential and total cross sections rapidly diminishes with increasing values of the impact energy E. The second term ∆V12 in (17.6) represents the dynamic inter-electronic
correlation through the corresponding interaction 1/x12, which is screened at large distances x1 À x2 by the potential V1 = 1/x1. The e1 − e2 potential V12 must be a constituent part of the interaction potential Vf , since ∆V12 emerges in the definition of the exit channel perturbation through the difference between the total interaction V = ZPZT/R − ZP/s1 − ZP/s2 − ZT/x1 − ZT/x2 + 1/x12 and the binding potentials in the non-interacting hydrogen-like atomic systems (ZP, e1)f1 and (ZT, e2)f2 . The mentioned residual potential 1/x1, as the limiting value of V12 at infinitely large x1 and finite x2, also enters the expression for Vf from (17.6). This is because at infinitely large x1, the active electron e1 from the (ZP, e1)f1 system cannot discern the individual constituents in the (ZT, e2)f2 system which is, therefore, conceived as the net point charge ZT − 1. In order to account for this correctly screened nuclear charge, the genuine potential −ZT/x1 is written as −ZT/x1 ≡ −(ZT − 1)/x1 − 1/x1. Here, the term −(ZT − 1)/x1 is used to yield the distortion N−(νT)1F1(iνT, 1, ivx1 + iv · x1) with νT = (ZT − 1)/v, whereas the potential 1/x1 is joined together with V12 to give the short-range potential ∆V12 in (17.3).
