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Feynman Diagram Method for Atomic Collisions

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Pramfi.na, Vol. 6, No. 5, 1976, pp. 312-322. © Printed in India. Feynman diagram method for atomic collisions T PRADHAN* and AVINASH KHARE Institute of Physics, Bhubaneswar 751007 MS received 1 December 1975; after revision 5 January 1976 Abstract. Feynmandiagram method of treating atomic collision problems in per- turbation theory is presented and matrix elements are calculated for a number of pro- cesses. The result for the resonant charge transfer in hydrogen is identical to the well known OBK value. However, in processes like collisional ionisation, the results are different from those obtained by conventional methods. Keywords. Electron exchange; ion exchange; vertex function; collisional ionisation; photo-effect charge transfer. 1. Introduction Feynman diagram method has proved to be very useful in a variety of problems in relativistic as well as non-relativistic physics. Apart from its indispensability in relativistic particle physics (Akhiezer and Beretetskii 1965), it has been found to be a very successful tool in handling many-body problems in nuclear, solid state (Fetter and Walecka 1971)and atomic physics (Kelley 1974). However, no attempt seems to have been made to apply this method to atomic collision problems. While exploring such possibilities the authors have found that in problems such as charge exchange processes the diagrammatic method not only gives an elegant physical picture of the phenomena but at the same time makes the calculation simple. The lowest order diagram consists of an electron exchange quite similar to the photon exchange in elastic scattering of charged particles. For the resonant charge transfer in hydrogen proton collision one obtains the familiar OBK amplitude (Brinkman and Kramers 1930); there is no diagram corresponding to the JS amplitude (Jackson and Schiff 1953a) in the lowest order. Another charge transfer mechanism operates in exchange scattering of electrons by hydrogen which can lze represented by a Feynman diagram with a proton exchange. Tb.e success of the diagrammatic method in these processes has prompted us to apply it to a host of other atomic collision processes. Principal among these are the photo-electric effect and its inverse (the radiative recombination), ionisation of atoms by electron and ionic impact. Detailed calculation of the amplitude of these processes are presented in this paper. The theory of charge exchange in positive ion-atom collision relevant to Feynman diagram method is presented in section 2 where amplitudes of other typical rearrange- ment processes of experimental interest have also been evaluated. Section 3 is *On leave of absence from Saha Institute of Nuclear Physics, Calcutta 700009. 312 Feynman diagram method for atomic collisions 313 devoted to the calculation of photo-electric effect and its inverse process of radiative recombination where relativistic and Coulomb effects have been ignored for the sake of simplicity. In section 4, calculation of matrix element for ionisation of atoms by electron and iortic impact is presented. Section 5 is devoted to discussions. 2. Charge transfer collisions We consider the process A+ + B--~ A + B+ where A+ is a positive singly ionized atom which picks up art electron from the atom B after collision. In other words, exchange of an electron between B and A+ characterises the collision. This exchange is, "literally" expressed by the Feynman diagram, of figure 1. The momenta of various particles indicated in tb.e diagram refer to the centre of mass system. The rules for writing down the matrix elements being quite standard (Bhasin et al 1965) need not be repeated here. Using these rules, we obtain M = ? (X,, K, -- X,, to)/ F (X, -- X,, to, m) × I"(K,, to,; Kt--K,, to) (1) where $I, = St, -- (2rr) 4 8(8) (P~ -- PI) ~ ( E, -- Et) M, or = KI~/2M. +, ~'~ = K*2/2Ma +, to = 12~- to1 = f2t- tot, I2~ = K,2/2Ma ~ ~B, 01 -- KI~ 2M a ~'~' a,~, an being bittding energy of the electron in atoms ' A' and ' B' respectively. r' (p~, to1; Pz, to~) is the vertex function representing the interaction of particles 1 and 2 and the bound state composed of these particles and AF (P, ~o, m) is the propagator for the electron. The vertex function/~(Pz, wx; P2, w2)is related to the Fourier transform ( : m pq gxs ~, mx + ms / of the Schr6dinger wave-function of the bound state formed by particles 1 and 2 as follows (see Appendix): (s) (~,, ~-,) (B+) (.k'f, wf) ,) i (A~) (~,~) t (A) (~'f,_n.f) lllgure I. Feyaraan diagram for charge transfer process A+ + B.-~ A + B+. ]P--7 314 T Pradhan and A vinash Khare I" (p 1 col; (m~pl- : m'P~ F ( p'2 -- ws -- w, Pz, w2)=gl~\ mi-}-ms /Lk2m,. ) c\2ml )3 f2) The non-relativistic propagator cart be written as 1 AF (P, % m) -- pS/2 m _ co -- b7 (3) Substituting eqs. (2) and (3) in eq. (1) and taking into account the fact that particles A+ and B+ are on the mass-shell i.e., pS/2m = oJ we get M = g,~+,(aK, -- Kt)g,~+,(bKt--K,)[(Kt~mm K')2 o~ ] , (4) where a = MB+/M~ b = M,~+/M,~, The general formula eq. (4) can now be applied to the particular cases of (a) H++ H(ls) -+ H(ls) 4- H -~, (b) e+ + (e+ e-)l. -+ (e+ e-)i, + e+, (c) H + + (tl+ e-)i, -+ H (ls) + t,+. For the process (a) M = g~n+. M K,- K t ) gall% (Mm + M 1(1 l- K? 1 X L(K -- K,)2/2m--Kill2 (M 4- m) 4- ~-M 4- e J, (5) where ~ is the binding energy of the hydrogen atom ground state and M is the mass of the proton. Using the fact that K~2 = Kt 2 -- (K s) which follows from over-all energy-momentum conservation and II 8 "/'/'1/2 S O/`' g,~+. (p) = ~<+-j~y, s = a: i , ao being Bokr radius for ground state, Eq. (5) simplifies to M = .--32rrs5 1 (6) m [#' + K 2 (/9 ~ 4- 4 sin s 0/2)] 3, with fi = m/M and cos 0 = K I • KdK 2 which is readily seen to be the OBK anapli- tude. It is worth pointing out that figure 1 being the only possible Feynman diagram for the charge transfer process in the lowest order, OBK amplitude (Brinkman and Kramers 1930) constitutes the entire matrix element. There being no other diagram we conclude that JS amplitude (Jackson and Schiff 1953 a) is not obtainable from diagrammatic formalism. It seems therefore that Wick's contention (Jackson and Schiff 1953 b) that proton-proton potential should not contribute to the capture amplitude is borne out by Feynman diagram formalism. In view of the absence of the JS term, one may say that diagram method suffers from the drawback that in the lowest order there ks no agreement with experiment. This is mainly due to ignoring Feynman diagram method for atomic collisions 315 the long-range Coulomb interaction between the bound electron and the colliding proton, which may be regarded as an irtitial state (or final state) interaction. The successful method of taking this effect into account is the Impulse Approximation initiated by one of us (TP) (Pradhan 1957) and refined by several others (Mcdowell and Coleman 1970). For the process (b) the matrix element is simpler on account of equality of masses of all the particles involved. We find IB im M = g.+,- (½ K, -- Kt) g°+,'- (½ K, -- K,) 1 × ~-~ Is, 2 + ¼K2 (1 + 8 sin s 0/2)], (7) where sp is the inverse Bohr radius of positronium. It is to be noted that we are using non-relativistic dynamics and as such ignoring spins of electron and positron. Using the formula x° 8rr112 s hlz g'+'-@) - + and the relation K, z = Kt z = K ~ in eq. (7) we obtain M = 32rrs~ 1 K 2 m [sp2 + ~-(1 + 8 sin e 0/2)] z (8) which is identical to the OBK amplitude for the process under consideration. For the process (C), the matrix element works out to 32~rss 1 M -- -- (9) m [ s" + K2(m 2~ + 4 sirt 2 0/2) )3j where K ~= K~2@ Kt z and cos0=K t . K~/ I Kt I K~ [ As in the two previous cases, this is also identical to tb.e corresponding OBK ampli. tude. The mechanism for all the charge transfer processes considered so far is electron exchange. It may be worthwhile to consider a process which occurs through ionic exchange. In electron scattering by atoms, apart from the direct scattering of the electron the process where the incident electron is captured and the atomic electron ejected must be considered.* This latter process e+A--X+e can be represented by the Feynman diagra m given in figure-2. It is clear from the diagram that the mechanism for this process is through the exchange of the iort A +. The matrix element for tb.is diagram is * This latter process is the only mechanism for inelastic electron scattering. e+He(l'S)~He(23P)+e by Helium. The direct and exchange amplitudes can be measured individually by spin analysis of the scattered beam in experiments using polarised electrons on atoms. 316 T Pradhan and Avinash Khare (A) (~i, .t~.,) (e')( K-'f, ~Of) < II ................. > ('e) ('ffi,~i) CA) CK~ ,~f) Figure 2. Feynman diagram for electron-atom scattering e + A ~ A + e through ion exchange. M = P (/(1, cot; Kt -- Ks, co) Av (KI -- K~) co, M,t÷) r (K, -- X,, ,o; K,, co,) fl0) For the particular case of electron scattering by hydrogen where the exchanged particle is a proton this works out to M = 32rrs5 1 -m- [s 2 -+- K 2 (1 + 4/3 sin z 0/2)] 3, ~ = m/M (I 1) which is somewhat different from the corresponding expression obtained by using the Born-Oppenheimer approximation (Mott and Massey 1965).
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