The Dodd-Greider series for four-body collisions | 13

ABSTRACT

In order to solve (12.1), we shall adopt the distorted wave formalism. By way of introduction, we will first recall the salient features of this theory [260, 261]. In the distorted wave formalism, instead of directly solving the Schro¨dinger equation (12.1) with rigidly determined interactions, one customarily considers a related model problem in which the real channel interactions Vi and Vf are replaced by certain distorting potential operators Wi and Wf . The following Green operators are associated with these latter two interaction potentials

g+i = (E −Hi −Wi + iε)−1 (13.1)

g−f = (E −Hf −Wf − iε)−1 (13.2)

or equivalently

g+i = (1 + g + i Wi)G+i ≡ ω+G+i (13.3)

g−f = (1 + g − f Wf )G−f ≡ ω−G−f . (13.4)

Here, G+i and G−f are the Green functions defined by

G+i = (E −Hi + iε)−1 (13.5)

G−f = (E −Hf − iε)−1 (13.6)

where ω± are the Møller wave operators. Next, in lieu of the full wave functions Ψ±i,f = (1 +G

±V di,f )Φ ± i,f we introduce the related distorted waves

χ±i,f = (1 + g ± i,fWi,f )Φi,f = ω

±Φi,f . (13.7)

The distorted waves χ+i and χ − f satisfy the following equations in the limits

ε→ 0±

(E −Hi −Wi)χ+i = 0 (E −Hf −Wf )χ−f = 0. (13.8)

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The connection of the model problem (13.8) with the original Schro¨dinger equation (12.1) is provided through the requirement that χ±i,f and Ψ

the same asymptotic behaviors as ri,f →∞ χ+i −→ Ψ+i −→ Φ+i ri →∞ (13.9)

χ−f −→ Ψ−f −→ Φ−f rf →∞ . (13.10) The prior form of the transition amplitude which is defined by

T−if = 〈Φf |Ω−†(Vi−Wi)ω+ +Ω−†[1−(Vi−Wi)g+i ]Wi|Φi〉. (13.12) This relation can be readily verified by employing the definitions for g+i and ω+ or by an algebraic derivation

Vi = Vi(1 + g+i Wi)−Wi(1 + g+i Wi) + [1− (Vi −Wi)g+i ]Wi = (Vi −Wi)ω+ + [1− (Vi −Wi)g+i ]Wi. (13.13)

Using the well-known Chew-Goldberger operator identity 1/A − 1/B = (1/A)(B−A)(1/B) with 1/A = G+ and 1/B = g+i the second term in (13.12) becomes (E+iε−Hf )(ω+−1) = iε(ω+−1).We write the transition amplitude T−if as follows

T−if = 〈Φf |Ω−†(Vi −Wi)ω+|Φi〉+ T dif (13.14) T dif = lim

ε→0 iε〈Φf |ω+|Φi〉 = lim

ε→0 iε〈Φf |χ+i 〉. (13.15)

Here, the contribution from the additive term T dif will vanish in the limit ε→ 0. The corresponding condition

lim ε→0

iε〈Φf |χ+i 〉 = 0 (13.16)

is satisfied by choosing a distorting potential which leads only to elastic scattering in the considered channel and, as such, does not cause rearrangement. This can be achieved by choosing the distorting potential to depend only on the relative coordinate between the projectile and the target. By a simple transformation, the wave operator Ω− from (13.11), can be

rewritten via the matrix element

Ω− = [1 +G−(Vf −Wf )]ω−. (13.17) Then, by employing (13.16), it follows from (13.15)

We recall that the Hermitean conjugated relation to (13.17) is given by Ω−† = ω−†[1 + (Vf − W †f )G+]. Hence, the exact transition amplitude T−if in the distorted wave theory reads as

T−if = 〈χ−f |(Vi −Wi) + (Vf −W †f )G+(Vi −Wi)|χ+i 〉. (13.19) Similarly, we can obtain the exact post form of the transition amplitude in the distorted wave formalism via

T+if = 〈χ−f |(Vf −W †f ) + (Vf −W †f )G+(Vi −Wi)|χ+i 〉 (13.20) provided that

lim ε→0

iε〈Φf |ω−†|Φi〉 = 0. (13.21)

Using the expression

Ω− = [1 + Ω−G−f (Vf −Wf )]ω− (13.22)

the transition operator U−if introduced in (13.18) can be written as the following integral equation

U−if = ω −†(Vi −Wi)ω+ + ω−†(Vf −W †f )G+f U−if . (13.23)

This can alternatively be cast into the form

U−if (1−K) = ω−†(Vi −Wi)ω+ (13.24)

K = ω−†(Vf −W †f )G+f (13.25) where K represents the so-called kernel i.e. the homogeneous term of the integral equation. Since ω− is given by ω− = 1 + g−f Wf , it follows that the form of K is independent of the choice of the distortion in the entrance channel. Expanding U−if in powers of K i.e. in an infinite perturbation series, we obtain

U−if = I(1 + ∞∑ n=1

Kn) (13.26)

I = ω−†(Vi −Wi)ω+ (13.27) where I is an inhomogeneous term of the integral equation (13.23). However, this latter expansion diverges in the case of rearrangement collisions due to the existence of the so-called disconnected diagrams. These Feynman diagrams correspond to collisional paths describing three constituents interacting pairwise with each other in the presence of a fourth body as a freely

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propagating particle. In the impulse space, this physical situation is described by means of the Dirac δ-function in the kernel of the integral equation. The presence of this δ-function indicates the conservation of momentum. Such a kernel is said to contain the mentioned disconnected diagrams. The free motion is mediated via the free-particle Green resolvent G+0 which leads to the factor 1/(E − E0 + iε), where E0 = k2/2. Since in the T -matrix we have an integration over k in the whole space, it is clear that we may have a situation where E = E0, and this causes divergence of the energy-dependent term 1/(E−E0+ iε) in the limit ε→ 0. The typical kernel (Vf −W †f )G+0 (Vi−Wi) from the iterated transition T operator would not contain any disconnected diagrams if none of the two-body interactions in the perturbation Vf −W †f is repeated in Vi −Wi. This can be achieved with the introduction of a virtual intermediate channel “x” as originally suggested by Dodd and Greider [260, 261]. The Green operator associated with this virtual channel is

g±x = (E −H + Vx ± iε)−1 (13.28) where Vx is an appropriate channel potential. A conveniently chosen Vx can eliminate all disconnected diagrams. Using the relation

G+ = g+x (1 + VxG +) (13.29)

we obtain the integral equation

U−if (1−K1)=ω−†(Vi−Wi)ω++ω−†(Vf−W †f )g+x (Vi−Wi)]ω+ (13.30) where K1 is a new kernel

K1 = ω−†(Vf −W †f )g+x G+f . (13.31) By employing certain suitably chosen potentials Vx andWf , the kernel K1 can become free from any disconnected diagram. An example of such a situation occurs when the potential Vx does not appear in Vf −W †f , as in the CDW-4B method [84]. Inserting (13.29) into (13.19), we arrive at the expression

T−if = 〈χ−f |(Vi −Wi) + (Vf −W †f )g+x (1 + VxG+)(Vi −Wi)|χ+i 〉. (13.32) If we neglect the term with the Green operator G+, we obtain a first-order approximation (for simplicity also denoted by T−if ) for the prior form of the transition amplitude

T−if = 〈χ−f |{1 + g−x (Vf −Wf )}†(Vi −Wi)|χ+i 〉. (13.33) Introducing an auxiliary distorted wave ξ−f as follows

|ξ−f 〉 = {1 + g−x (Vf −Wf )}|χ−f 〉 (13.34) we have

T−if = 〈ξ−f |Vi −Wi|χ+i 〉 . (13.35)

13.1.1 The ZP − (ZT; e1, e2)i collisional system According to the requirement (13.9) and the correct asymptotic behavior of Ψ+ which is

Ψ+ −→ ϕi(x1,x2)eiki·ri+iνi ln(kiri−ki·ri) ≡ Φ+i (ri →∞) (13.36) the following factorized form for the function χ+i appears as optimal

χ+i = ϕi(x1,x2)ψ + i . (13.37)

Here, ψ+i is an unknown function to be determined according to a particular choice of the distorting potential. Inserting (13.37) into (13.8), we obtain

ϕi(E − Ei −H0 − Vi)ψ+i + 1 b

+ ψ+i (Ei − hi)ϕi = 0 (13.38)

Ui = Vi −Wi . (13.39) The term

(Ei − hi)ϕi ≡ Oi(x1,x2) ≡ Oi (13.40) appearing in (13.38) is equal to zero only for the exact eigen-solutions ϕi(x1,x2) ≡ ϕi and Ei of the target Hamiltonian hi. However, since these latter solutions are unavailable, the term Oi should, in principle, be kept throughout, as suggested by Belkic´ [87] within the CB1-4B method. The explicit computations for double charge exchange [87] and transfer ionization [127] have shown that the contribution from this term is ∼ (10− 15)%. This correction will presently be neglected, in which case (13.38) becomes

ϕi

( k2i 2µi

−H0 − ZPZT R

+ ZP s1

+ ZP s2

) ψ+i

+ 1 b

∇xkϕi ·∇xkψ+i + Uiϕiψ+i = 0. (13.41)

In general, the presence of the coupling term ∇xkϕi · ∇xkψ+i precludes a separation of the independent variables in (13.41). However, at the same time, there is a flexibility provided by the perturbation potential operator

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Ui = Vi −Wi, which permits cancellation of the coupling term. An adequate choice has been made by Belkic´ and Mancˇev [84, 85] as

Uiχ + i = −

1 b

∇xkϕi ·∇xkψ+i . (13.42)

Alternatively, the following choices for Ui can be implemented [189, 190]

Uiχ + i = ZP

( 1 R − 1 s2

) χ+i −

1 b

∇xkϕi ·∇xkψ+i (13.43)

Uiχ + i =

[ ZP

( 1 R − 1 s2

) + ( 1 R − 1 s1

)] χ+i −

1 b

∇xkϕi ·∇xkψ+i . (13.44)

Although other choices are possible, in every single case the requirement that the function χ+i has the correct asymptotic behavior must be fulfilled. It is seen that the distorting potentials (13.42)–(13.44), contain the term −(1/b)∑2k=1∇xkϕi · ∇xkψ+i , which together with the eikonal approximation R ' −rf (MP,T À 1), provide the exact solution ψ+i of the differential equation (13.41). In this case, a separation of the independent variables s1, s2 and rf is possible i.e. we can write

where C+i is a constant to be determined. We shall first find the distorted wave χ+i for the distorting potential (13.42) following Belkic´ and Mancˇev [84, 85]. Inserting (13.45) into (13.41), we obtain(

1 2a ∇2sk +

ZP sk

+ p2k 2a

) F+k (sk) = 0 (k = 1, 2) (13.46)

( 1 2µf

∇2rf − ZPZT rf

+ p2f 2µf

) F+(rf ) = 0. (13.47)

The exact solutions of these equations are

F+k (sk) = N+(νPk)eipk·sk 1F1(iνPk , 1, ipksk − ipk · sk) (13.48)

F+(rf ) = N+(νPT)eipf ·rf 1F1(−iνPT, 1, ipfrf − ipf · rf ) (13.49) where N+(νPT)=e−piνPT/2Γ(1+ iνPT) , νPk=aZP/pk and νPT=ZPZTµf/pf . The auxiliary vectors p1, p2 and pf are determined from the conditions

E − Ei = p 2 1

2a + p22 2a

+ p2f 2µf

(13.50)

p1 · s1 + p2 · s2 + pf · rf = ki · ri (13.51)

C+i exp[iνPT ln (pfrf − pf · rf )− i 2∑

νPk ln(pksk − pk · sk)] −→

−→ exp[iνi ln(kiri − ki · ri)] ri →∞ . (13.52)

Relation (13.50) is introduced in order to obtain three separable equations from (13.41) for the independent variables s1, s2 and rf . Expressions (13.51) and (13.52) originate from the requirement that χ+i must satisfy the correct boundary conditions (13.9). It is easily shown that

pk ' −v (MP À 1) pf ' −ki (MT À 1). (13.53)

With these values of vectors p1, p2 and pf , the energy conservation law is satisfied within the eikonal mass limit, so that E−Ei = k2i /(2µi) and, moreover, the constant C+i is identified as

C+i = µ −2iνP i . (13.54)

This result for the constant C+i , which is needed in (13.52), follows from the usual asymptotic limit

1 µi

kiri − ki · ri vsk + v · sk −→ 1 ri →∞ (k = 1, 2). (13.55)

Hence, the solution for the function ψ+i becomes

ψ+i = µ −2iνP i N+(ν)[N+(νP)]2eiki·ri 1F1(−iν, 1, ikirf + iki · rf )

× 1F1(iνP, 1, ivs1 + iv · s1) 1F1(iνP, 1, ivs2 + iv · s2) (13.56)

where νK = ZK/v (K = P,T) and ν = ZPZT/v. Now, it is readily checked that the distorted wave χ+i = ϕiψ

+ i has the correct asymptotic behavior

(13.9). It should be noted that the proof of the correctness of the boundary conditions of the continuum distorted wave methodologies is consistent with the concept of the strong limit from the formal scattering theory [7, 41]. This has been demonstrated within the three-body distorted wave formalism for single electron capture [262]. Such a demonstration is not hampered by the presence of the kinetic energy perturbation from the three-body symmetric eikonal (SE-3B) approximation, or by the Coulombic behavior of the perturbative potentials from the corresponding CDW-3B method [262]. The same conclusions can be verified to hold also true for the scattering vector χ+i from (13.56), as encountered in double electron capture treated in the four-body distorted wave formalism. The role and the physical meaning of the strong limit is studied in several chapters on the formal scattering theory in the first part of the present book as well as in Ref. [7].