A Modified Formal Theory on Semi-Relativistic Effects during Electron Emission from Metallic Surfaces upon the Impact of High Energy Particles

In this paper we introduce a formal theory on unveiling relativistic effects during electron emission from clean metallic surfaces upon high charged particle impact using a Jellium-type wave function including suitable spinors. In addition image charge final state electron surface interactions have been initiated in the relativistic region as well as the scattering of the projectile from the multi-center bulk potential. Finally, a semi-relativistic condition is considered in place of the ionization mechanism of scattering from an aluminium semi-infinite solid target by non-relativistic electrons to determine multiple differential cross-section.


Introduction
In the last three decades significant progress has been made in understanding the atomic physics of electronatom ionization both theoretically and experimentally for relativistic energies [1][2][3][4][5][6][7][8][9][10][11][12][13][14] as well as for non-relativistic energies [15][16][17][18][19][20][21][22][23][24][25][26].Capabilities of different theoretical models in reproducing various features of experimental results have been widely tested for different kinematical conditions.Study of energy spectrum of ejected electrons could be very interesting.The complexity of this reaction is already revealed in the simplest case of low-energy electron impact ionization of atomic hydrogen where final state interactions of escaping particles strongly modify the observed electron spectra [16,21].Hence a realistic approximate eigenfunction of the non-separable three body Hamiltonian is essential.Generally, a theoretical description of this process from solid targets has to deal with various aspects of the beam-solids interactions.A relativistic charge particle impinging on a metallic target, which is the study of this work, leads to charge-density fluctuation of the solid.Asymptotically, this causes an image-charge distribution of the incoming and outgoing particles [22].Moreover, the motion of the various electrons is periodically distributed by the interaction with the bulk potential [24,25].The aim of this paper is to investigate the energy spectrum of electron ejection from clean metalic surfaces upon higher charged particle impact for relativistic energies [10,12,23].The present study of the formal theory is the assumption that the degrees of freedom of the projectile can be decoupled from those of the target.This is justified, since we assumed the momenta of the incoming and outgoing electron to be considerably larger than the Fermi momentum.To obtain the analytical theory that can be analyzed, a Jellium wave function [23] with suitable spinors [12] of the initially bound electron is assumed.Asymptotic image-charge distribution of the Vacuum electron is taken into account.As well as the binary collision of the projectile with the bound electron, the scattering of the incident particle from the multicenter bulk potential is treated in the kinematic approximation [25,26].A screened Coulomb muffin-tin bulk potential is adopted.Recently using the multiple scattering theory [17] multiplied with some spinors has calculated the energy spectrum of scattered electrons in k-shell ionization of medium heavy atoms by relativistic electrons [10][11][12] which agree nicely with experiments.In the present study we developed the theory of relativistic   e, 2e scattering mechanism process using a Jellium wave function [23] of the undistributed surface in the initial state.In addition, image-charge final state electron surface interactions have been included.Moreover, the scattering of the projectile from the multicentre bulk potential is considered.The proposed theory will provide scattering cross-sections results of different kinematical conditions.Hence we introduced a modified theory on semi-relativistic effects during electron emission from aluminium surfaces upon the impact of high energy particles.

Theory
In this theory, we consider an atomic scattering system consisting of a projectile with nuclear change p Z and mass p m being inelastically scattered from a clean metallic semi-infinite solid ejecting one electron into the vacuum level of the solid.Then the total Hamiltonian of the projectile-solid system is where e H is the Hamiltonian of the simi-infinite solid in the absence of the projectile, W pe is the projectileejected electron interaction, and H p represents the projectile-crystal interaction and contains the plasmon and phonon modes and their interaction with electron.The initial and final state boundary conditions are specified by eigenstates , i f of asymptotically unperturbed initial and final channel Hamiltonian      0, where are the corresponding asymptotic eigen energies.The transition amplitude T for the scattering system from initial state or the post form where   and   are the eigensates of the total Hamiltonian H according to the state i and f respectively.From Equations (4) and (5) the perturbations potential and The eigenstates   of H can be written as where the MØller wave operator is given by where G  is the Green operator of the total Hamiltonian H. Combining Equations ( 9), ( 8) and ( 4), the T matrix element may be written as , where Here assuming i V to be a multiple-center potential, the first term of Equation (10), that is sin T describes the transition of the system from state i to f due to a single scattering from each individual scattering center.One center and multicenter multiple scattering is contained in the matrix element in Equations ( 10) and ( 12), since the Lippmann-Schwinger equation of the total Green operator leads to the expansion where 0 G  and V are the full free propagator and the total potential of the projectile-solid compound.It should be noted here that the labels sin and mu of the amplitudes and refer to single and multiple scattering specifically from the potential i .The potential in the initial channel can be written as The operator s p W vac p W stands for the particle-bulk interaction and its explicit functional form is specified below.The potential is less prominent.Since it is an asymptotic (imagine charge) perturbation.For the asymptotic final-channel Hamiltonian [23] is given by .
Here the kinetic-energy operator of the secondary electron is referred to by e , whereas amounts to the asymptotic final state interaction of this electron with semi-infinite solid.The choice Equation ( 15) leads to the final-channel distribution operator where s e W denotes the short-range interaction of the secondary electron with the surface.Upon substitution of Equation ( 14) into Equation (11) , .
The amplitudes, given by Equations ( 18) and ( 19), provide the first-order approximation of the matrix element T. The term in Equation ( 20) is less prominent since is an asymptotic (image charge) perturbation.Hence it has no significant contribution to the term.So, the amplitude, given by Equation ( 20) is neglected here.

Analytical Calculation of the
The eigenfunction of f H of Equation ( 15) at a given asymptotic energy f [23] is readily deduced in semirelativistic form [1,12] as In addition the Jelliun wave function [23] can be expressed in terms of reflection and transmission coefficient may be written as in semi-relativistic form [1,12] as Here  , indicates the strength of this interaction.In case 0 a  , we end up with the final state being a product of the two free-particles states.The final state energy is given by and In the present case, the transition amplitude term sin pe T may be written as in semi-relativistic integral form [23] as where are the momenta and energies of the final two electrons and are the momentum and energy of the incident electron.Here .Then performing the integral over the projectile co-ordinates, Equation (29) simplifies to where , .
e e e k J t In Equation (33) we introduced the complex vector .Using the theory [23], we have where the vector is given by .By the Fourier transform Now after inserting the Equation (35) into a Equation (32), the remaining one dimensional integral can be algebraically transformed to the Beta function integral representation The functions i J are given by .
Making use Equation (36) and after calculation, the final expression

sin T p,s
In this case the transition amplitude term of the Equation ( 19), the screened Coulomb potential [26] for relativistic case is given by The effective parameter eff  account for the screaming of the pure ionic field due to the presence of the localized positive cores as well as delocalized electrons [26].For aluminum surface the numerical value of eff  is 0.886.Equation ( 19) can be written as in semi-relativistic , where is the ionic core potential at the state , N is the number of ions in the solid and W is periodic in each layer parallel to the x − y plane, but not in zdirections.The j-th ion in the layer has the coordinates , , Since W is periodic in the x and y directions, we can introduce [26] .
Also, the expression for may be written as The function , we have been defined as ; 0,1, Now the spin sums can be obtained So, we obtain from Equation (56),  are the solid angles of the emitted electron and scattered projectile respectively.This is the modified formal theory on semi-relativistic effects during electron emission from metallic surfaces upon impact of high energy particles to determine differential cross-section of different kinematic conditions.


A theoretical formulation of scattering cross-section of high energy change particles from clean metallic semiinfinite solid in the relativistic region has presented single scattering amplitude sin T term [23].In this theory, for multiply differential cross-sections, we used a semi-relativistic Jellium initial state wave function.A relativistic final state electron surface image charge interaction has been included in its asymptotic form.We will utilize another multiple scattering term [23]

H
respectively, i.e., Transition Amplitude Term sin pe T in Relativistic Effects In a position-space representation the transition operator occurring in the Equation (18) has the form 0 , p r and e are the position vectors of the projectile and ionized electron respectively.Also 0 r  is time structure constant and eff Z (effective nuclear charge) = Nuclearcharge   p z − 0.3.To avoid difficulties arising from the infinite range of coulomb interactions we introduce the cutoff potential[23] spinors of scattered and ejected electron.In the Equation (23) the term refection and transmission coefficient R and T are given by the spin co-ordinates in the initial and the final states of the electrons.
p and are the Dirac spinors of the atomic elec- is an asymptotic (image charge ) perturbation, so the termvac p W , sin p v Tis neglected here.Therefore the expression for the sin T -matrix element may be written as in semi-relativistic form , the matrix element two dimensional reciprocal vectors