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Atomki Preprint Hu Issn 0231-2468 INIS-mf—8669 ATOMKI PREPRINT HU ISSN 0231-2468 ATOMKI Preprint B/5 (1983) MULTIPLE IONISATION EFFECTS DUE TO RECOIL IN ATOMIC COLLISIONS* L. Vegh Institute of Nuclear Research of the Hung. Acad. of Sci. H-4001, Debrecen P.O.Box 51, Hungary ABSTRACT A description of atomic recoil excitation caused by the projectile nucleus is given using the two-potential formula. The relation of the direct and recoil ionisation amplitudes are discussed demonstrating that their inter- ference in dipol approximation may be destructive only approximately. The recoil mechanism can produce strong multiple ionisation and the strength of the sate]lite lines follow strictly the binomial distribution. The recoil ionisation in neutron induced reactions, the life-time of the inner-shell vacancy in recoiled ions and the recoil effects in the electron conversion coeffizients are discussed. 1. Introduction In the last few years there has been an increasing interest in atomic ioni- sation phenomena influenced by the recoil of target nucleus. The recoil of the nucleus of the atom may "shake off" some of the electrons (Migdal 1939, Levinger 1953, Ciochetti and Molinari 1965) and it is known to be important at very small values of the impact parameter b where it can give large contribution to the ionisation. The recoil matrix element (Migdal 1939, Levinger 1953, Andersen et al 1976, Feagin et al 1979) obtained in sudden approximation has the following form: where <jp« and (p^ are the initial and final atomic wave functions, respective- ly, depending on the electron coordinates ^b'4- •• *T) fixed to the centre of the target nucleus with atomic number "2r.% is defined as t, - . a (2) where m and Krare the electron and target nucleus masses, respectively, and-4 is the momentum of the recoiled nucleus in laboratory system. The sudden approximation can be used if the duration of the momentum transfer to the target nucleus is much shorter than the electron periodic time on the investigated orbit. This condition is fulfilled in nuclear processes except for some resonance reactions (Feagin 1982) governed by the short range strong interaction. * Submitted to Journal of Physics B., May 6 1983. In the case of the Coulomb interaction the dominant part of the recoil momentum is transferred at nucleus-nucleus distances close to the perihelion because this is the part of the classical orbit where the dominant part of the* change in the direction of the nuclei momenta occurs. Therefore it is a good estimate for the duration of the recoil momentum transfer the time needed to take the twice the distance of the closest approach at the given impact parameter b. This time denoted by tN is equal to tN- 2a [<• 0 •*/<*)]/« ,. a.OWil*. LEj.MeV where Ztis the atomic number of the incident nucleus, E andtf are the CM energy and velocity of the nuclei, respectively. If tN is much less than the periodic time of the investigated electron orbit with principial quantum number n, then the application of formula (1) for recoil calcula- tions is justified. This condition can be expressed as CO The sudden approximation condition (4) is fullfilled if b is small enough and if the incident particle is not very slow. The common feature of the most recoil calculations (see for example Andersen et al 1976, Amundsen 1978, Kleber and Unterseer 1979, Jakubassa and Amundsen 1979, Rosel et al 1982, Anholt and Amundsen 1982, Blair and Ariholt 1982, Reading et al 1982, Ford and Reading 1983) that they are essentially based on formula (1) if we retain only the first two terms in the expansion of the exponent operators in (1), that is e.xp • Tf and we neglect the second order terms ~Kmf\.'ri (Feagin 1982). Since for dif- ferent initial and final states the matrix element of the identity operator gives zero, the 8,,-body^ operator exp(ifc£ T$) disintegrates to the sum of one-body operators iS fer^ and the description of the recoil effects can be easily incorporated Into different treatments of the collision induced ionisation. The neglection of the higher order terms in expansion of the exponent operator in (1) can be justified if their contribution is small enough and this requirement can be formulated roughly for electrons with principial quantum number «. as loft .feJG^*- « A (5) where a, is the Bohr-radius, an is the radius of the Bohr orbit with n. It is possible that the condition (5) is fulfilled for inner shell e- lectrons but it is not true for electrons m higher shell. At very large recoil of the target condition (3) is not fulfilled already for the K-shell electrons. So it is an interesting problem to discuss the recoil effect in terms of formula (1) without any approximation. The discussion of large recoil effects for one electron target atom using the impulse approximation has been recently published by Feagin (1982). Using some elements of his work we discuss the scattering amplitude of the ion-atom collisions involving the recoil for many-electron atoms. 2. The scattering amplitude We discuss ion-atom processes where the incident ion is fully ionised. For the description of the collision in laboratory system we have the following coordinates. Let R» • • •R* be the coordinates of the N electron and ftrand" R#the coordinates of the target and incident nuclei, respectively. The notions of the corresponding momenta are ?„ ..-•*«, PT and P, . We define a new set of coordinates as *»•«>- K CM Mr * Mm » Mi j, where M» is the mass of the incident nucleus and ftA is the coordinate of the centre of mass of the target atom: (7) The conjugated mosnenta to the coordinates in (6) have the form: Nm This choose of the CM coordinates can be considered as generalization of the treatment of Feagin(1982)which has been constructed for a two nuclei- one electron three-body system. The Hamiltonian has the form: where TCM is the kinetic energy operator of the centre of mass and T** is that of the atom-incident nucleus relative motion. For our treatment of the ionisation process we split the Hamiltonian (9) into three part in the following way: H - Ho + V, • VH (13) with H. - TCM . T.. + H, ,^3 The inital and final asymptotic wave functions corresponding to the eigenfunctions of H# (14) have^the form: where u and Cf are the conjugated momenta (9). <f{ and (f>4 are eigenfunctions of Hn , <fk(^ ,•••&) is the initial ground state atomic wave function, <fcl*n ••• ,*«>) describes the final state of the N-electron-target nucleus system. The motion of the centre of mass of the whole system is separated off and not considered. For the sake of simplicity the spin indices are neglected. The exact transition matrix element has the form: where the T operator depends on the nucleus-nucleus, electron-incident nucleus two-body interactions. We have calculated the matrix element (18) using the well-known two- potential formula of Gell-Mann, Goldberger and Watson (Gell Mann and Goldberger 1953)» Tfl (19) where the first term is the \matrix element of the Vj interaction which is assumed in this approximation to be well-known. The second term is the ^JjWRA matrix ^element of the Vu interaction with the scattering functions Vt$ and Vj* which are the solutions of the Schrodinger equation with Hamiltonian The first term of (19) can be calculated easily and accurately because the v*t interaction (15) depends only on the distance l«a-f$T|. Taking into account that the variable R in integral (18) can be expressed as see (7), and applying (15), (18) and neglecting in the nuclear amplitude the off -energy shell effects due to the binding of atomic electrons (impulse approximation), we have where we have denoted the first term of (19) by T4t . cj,* ^ -^ H is the momentum transfer to the target nucleus, fc is defined by (2), t Cep is the nucleus-nucleus scattering amplitude, Rfk(fc) is given by (1). For the calculation of the second, DWRA term of (19) we need the know- ledge of the fu and V/^* scattering wave functions. They can be obtained from the "^-matrix given in (21). Applying the general relations and neglecting the influence of the atomic electron energies on the nuclear motion we obtain - N » where *^(Rw)is the nuclear scattering wave function and the (-) motion in (flir relates to the continuum electron wave functions. Writing these scattering wave functions into the second term of (19) which we denote by T** • in the further we can write In order to calculate the 77^ matrix element we insert <x completness relation into it: , w (23) with £ .iiij ^v /.#_ <-» I ^ i l£-> denotes nuclear scattering states and £ l^'X^lis the projection operator of the N electron states including sum" for the bound and integral for the scattering states, respectively. Taking into account that the Coulomb interactions in the second product can produce one-electron transi- tions only and for antisymmetrized N -electron wave function the N interac- tions in the sum give equal contributions, T^"' can be written as where <^(^ •?„) describes an atondc state differs from the initial ground state in one single-electron orbit and $ relates on the entity of -tie elec- tron states. Using (21) and (24), T+i has the form (25) Now we investigate the matrix-element where a K-electron goes into a positive energy state. In R,M(fc.) \n (25) we introduce the following approxima- tion * .. where pe is the momentum of the continuum electron and we can write with -*S(5> w ^ where q?"( jf4 ... ^denotes an N-< electron wave function which can be gener- ated by taking away one Is orbit in the initial state and the scattering orbit in the final state, respectively.
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