ABSTRACT
In this chapter we consider a process called transfer ionization, as already abbreviated by TI, where according to (11.2) simultaneous electron capture and target ionization take place
ZP + (ZT; e1, e2)i −→ (ZP, e1)f + ZT + e2(κ ) (15.1) where κ is the momentum vector of the free electron in the target frame. Here, the analysis of the entrance channel is the same as for the corresponding double electron capture (11.1). Therefore, we only need to focus on the exit channel for reaction (15.1). To this end we first determine the distorted wave ξ−f which satisfies the following equation [127]
(E −H + Vx − iε)|ξ−f 〉 = −(iε− Vx)|χ−f 〉 (15.2) which is obtained from (13.34). Choosing the intermediate channel potential Vx in such a way that the constraint
Vx|χ−f 〉 = 0 (15.3) is automatically satisfied, we have in the limit ε→ 0+
(E −H + Vx)|ξ−f 〉 = 0. (15.4) Writing ξ−f in a factored form similar to the distorted wave χ
ξ−f = ϕfG−f (15.5) we arrive at
G−f (Ef −H0 − VP)ϕf + ϕf (E − Ef −H0 − Vf )G−f +
1 a1 ∇s1ϕf ·∇x1G−f + Vxξ−f = 0 (15.6)
where a1 =MP/(MP+1). It is important to realize that (15.6) can be solved without any further approximations [127] if the model potential Vx is
Vx = ZP
( 1 R − 1 s2
) − ( 1 x1 − 1 r12
) − 1 a1 ∇s1ϕf ·∇x1 ◦
1 ϕf
. (15.7)
ION-ATOM
Thus, using the mass limit MP,T À 1, it follows that (15.6) is reduced to[ E − Ef −H0 + ZT − 1
x1 + ZT x2
− ZP(ZT − 1) ri
] G−f = 0 (15.8)
where the independent variables are separated, and this permits the exact solution. The possible nodes of ϕf would render Vx singular in (15.7). To bypass this difficulty, we introduce the symbol ◦ in (15.7) to indicate that Vx acts only on those functions that contain ϕf in the factored form, as in (15.5). In other words, the symbol ◦ determines the domain of the definition of the operator Vx , which is allowed to act only on a subspace of the complete Hilbert space containing wave functions with a factored hydrogen-like bound state ϕf . This will be the case if we seek G−f in a factored form, such as
G−f = C−f ϕ−q1(x1)ϕ−q2(x2)ϕ−qi(ri ) (15.9) which permits the exact solution in the form of the C3 function
ϕ−q1(x1) = Γ(1 + iν ′ T)e
piν′T/2+iq1·x1 1F1(−iν′T, 1,−iq1x1 − iq1 · x1) (15.10)
ϕ−q2(x2) = Γ(1 + iζ ′)epiζ
′/2+iq2·x2 1F1(−iζ ′, 1,−iq2x2 − iq2 · x2) (15.11)
ϕ−qi(ri) = Γ(1− iν′′)e−piν ′′/2+iqi·ri
1F1(iν′′, 1,−iqiri − iqi · ri) (15.12)
where C−f is a constant, ν ′ T = (ZT − 1)a1/q1 , ζ ′ = ZTa2/q2 , ν′′ =
ZP(ZT − 1)µi/qi and a2 = a1 = MP/(MP + 1). The unknown vectors q1 , q2 and qi can be determined by imposing the required simultaneous constraints
E − Ef = q 2 1
2a1 +
+ q2i 2µi
(15.13)
q1 · x1 + q2 · x2 + qi · ri = −kf · rf + κ · x2. (15.14) Then, (15.13) and (15.14) together with the relation rf = −ari−b(x1+x2)/µf (a = MP/[MP + 2] and b = MT/[MT + 2]), as well as with the mass limit MP,T À 1 lead to
q1 = a
µf kf ' 1
µf kf ' v (15.15)
q2 = a
µf kf + κ ' 1
µf kf + κ ' p qi = akf ' kf (15.16)
p = κ+ v . (15.17)
In this way, the distorted wave ξ−f from (15.5) is obtained as
ξ−f = N −(ζ)N−(νT)N−(ν)φfϕf (s1) 1F1(−iζ, 1,−ipx2 − ip · x2)
× 1F1(−iνT, 1,−ivx1 − iv · x1) 1F1(iν, 1,−ikfri − ikf · ri) (15.18) where the function φf is defined by (11.78) and N−(ζ) = Γ(1 + iζ)epiζ/2 , N−(νT) = Γ(1+ iνT)epiνT/2 , ζ = ZT/p , νT = (ZT − 1)/v and ν = ZPZT/v. Using (13.43), (13.57) and (15.18), the expression for the prior form of the transition amplitude, which is defined by
T−if = 〈ξ−f |Ui|χ+i 〉 (15.19) becomes
T−if = N¯ ∫∫∫
ds1ds2dR eiα·s1+iβ·x1−iκ·x2Rν(ri, rf )ϕ∗f (s1)
× 1F1(iνT, 1, ivx1 + iv · x1) 1F1(iζ, 1, ipx2 + ip · x2) × [ZP(1/R− 1/s2) 1F1(iνP, 1, ivs1 + iv · s1)ϕi(x1,x2) − ∇x1ϕi(x1,x2) ·∇s1 1F1(iνP, 1, ivs1 + iv · s1) − 1F1(iνP, 1, ivs1 + iv · s1)Oi(x1,x2)] ≡ T−if ;ν(η ) (15.20)
where N¯ = (2pi)−3/2N+(νP)N−∗(νT)N−∗(ζ). Here Oi is from (13.40) and
Rν(ri, rf ) = N−∗(ν)N+(ν) × 1F1(−iν, 1, ikfri + ikf · ri) 1F1(−iν, 1, ikirf + iki ·rf ) (15.21)
with
ki · ri + kf · rf = α · s1 + β · x1 α = η −
( v
2 − Ei − Ef − Eκ
v
) vˆ
β = −η − ( v
2 + Ei − Ef − Eκ
v
) vˆ (15.22)
where Eκ = κ2/2. The difference Ei − (Ef + Eκ) ≡ Q˜ from (15.22) between the initial and final electronic energies represents the so-called inelasticity factor, or equivalently, the Q˜−factor1. This observable is of key importance for translational spectroscopy, which through measurement of the inelasticity factor determines the energy gain or loss of the scattered projectiles.
