Double electron capture

In order to complete the expression for the transition amplitude in the distorted wave theory for double electron capture, we look for the distorted waves in the exit channel. First, we determine the auxiliary distorted wave ξ−f defined by (13.34). Letting ε → 0+, it is seen that, according to (13.34) and (13.28), ξ−f is the solution of

(E −H + Vx)ξ−f = (E −H + Vf −Wf + Vx)χ−f . (14.1)

Since χ−f satisfies the relation (13.8), it follows that (14.1) can be reduced to

(E −H + Vx)ξ−f = Vxχ−f . (14.2) Under the assumption

V †xχ − f = 0 (14.3)

it follows that (14.2) becomes solvable analytically. In such a case, we write ξ−f in the factored form

ξ−f = ϕf (s1, s2)ψ − f (14.4)

where the unknown function ψ−f is the solution of the equation

ϕf (E − Ef −H0 − Vf )ψ−f + 1 a

∇skϕf ·∇skψ−f + Vxϕfψ−f = 0 . (14.5)

Choosing the potential Vx in the form used by Belkic´ and Mancˇev [84, 85]

Vx = −1 a

∇sk lnϕf ·∇sk (14.6)

we have

(E − Ef −H0 − Vf )ψ−f = 0 (14.7)

ION-ATOM

so that the function ψ−f finally reads as

ψ−f = µ 2iνT f N−(ν)[N−(νT)]2e−ikf ·rf 1F1(iν, 1,−ikfri − ikf · ri)

× 1F1(−iνT, 1,−ivx1−iv · x1) 1F1(−iνT, 1,−ivx2−iv · x2) . (14.8) Utilizing (13.42) for the distorting potential Ui and the corresponding distorted wave for the initial state χ+i = ϕiψ

+ i , where the function ψ

+ i is deter-

mined by (13.56), the transition amplitude for double electron capture in the CDW-4B method becomes [84]

T−if = −N2 ∫∫∫

dx1dx2dri eiki·ri+ikf ·rfL(ri, rf )ϕ∗f (s1, s2) × 1F1(iνT, 1, ivx1 + iv · x1) 1F1(iνT, 1, ivx2 + iv · x2) ×{ 1F1(iνP, 1, ivs2 + iv · s2)∇x1ϕi(x1,x2) ·∇s1 1F1(iνP, 1, ivs1 + iv · s1) + 1F1(iνP, 1, ivs1 + iv · s1)∇x2ϕi(x1,x2) ·∇s2 1F1(iνP, 1, ivs2 + iv · s2)}

(14.9)

where N = N+(νP)N+(νT) and

L(ri, rf ) = µ−2iνPi µ−2iνTf [N−(ν)]2 × 1F1(−iν, 1, ikirf+iki ·rf ) 1F1(−iν, 1, ikfri+ikf ·ri). (14.10)

A simplification of (14.10) follows from the eikonal approximation

[N−(ν)]2 1F1(−iν, 1, ikirf + iki · rf ) 1F1(−iν, 1, ikfri + ikf · ri) ' (kirf+ki ·rf )iν(kfri + kf · ri)iν'(µiµf )iν [(vR−v ·R)(vR+v ·R)]iν = (µiµf )iν [v2(R2 − Z2)]iν = (µiµf )iν(vρ)2iν ' (µvρ)2iν

∴ L(ri, rf ) ' µ−2i(νP+νT)(µvρ)2iν (14.11) where µ = MPMT/(MP +MT). Here, ρ is the projection of vector R to the XOY plane perpendicular to the Z-axis i.e. ρ = R − Z with ρ · Z = 0, where vector Z represents the projection of vector R to the Z-axis. The phase factor (µvρ)2iν , which stems directly from the inter-nuclear potential VPT = ZPZT/R, does not influence the total cross section, since

Q−if (a 2 0) =

1 (2piv)2

∫ dη ∣∣∣T−if (η )∣∣∣2 = ∫ dη

∣∣∣∣∣R − if (η ) 2piv

∣∣∣∣∣ 2

(14.12)

where

R−if (η ) = −N2 ∫∫∫

× 1F1(iνT, 1, ivx1 + iv · x1) 1F1(iνT, 1, ivx2 + iv · x2) ×{ 1F1(iνP, 1, ivs2 + iv · s2)∇x1ϕi(x1,x2) ·∇s1 1F1(iνP, 1, ivs1 + iv · s1) + 1F1(iνP, 1, ivs1 + iv · s1)∇x2ϕi(x1,x2) ·∇s2 1F1(iνP, 1, ivs2 + iv · s2)} .