PHYSICAL REVIEW VOLUME 181, NUMBER 1 5 MAY 1969 Elastic and Inelastic Electron Scattering by He and Ne Atoms in Their Ground States*t E. M. A. Peixoto, C. F. Bunge, and R. A. Bonham DePartment of Chemistry, Indiana University, BloornAgton, Indiana 47401 {Received 2 December 1968) A study of elastic and inelastic scattering of electrons by He and Ne atoms in their ground states is made within the framework of the first Born approximation. For the elastic case a partial-wave analysis is also carried out. Properly correlated configura- tion interaction wave functions, and also Hartree-Fock wave functions, are used through- out. It is shown that for an accurate description of both elastic and inelastic scattering, a correlated wave function is essential. A table of elastic and inelastic electron scattering factors for the elements studied is given. I. INIODUCTION Calculations of differential cross sections for atomic electron scattering and x-ray scattering have been the subject of many works. '-" The problem can be divided into separate con- siderations of elastic and inelastic scattering. For the elastic part the failure of the first Born approxi- mation has already been discussed, '~" especially at low energies and for atoms of large atomic num- ber. In the latter case an improvement can be made by means of the partial-wave method. How- ever, for the inelastic part of the scattered ampli- tude we may expect that the first Born approxima- tion should provide a more suitable description. ' Besides improving the method for treating the scattering process, a refinement can also be made by using very accurate wave functions to describe the scatterer. '-" This is especially true for the inelastic part where the first Born approximation contains a term-dependent on the reduced second- order density matrix, this one being sensitive to electronic correlation. For He, Bartell and Gavin' ~' and more recently Kohl and Bonham' and also Kim and Inokuti" have illustrated this fact: There is a 6/~ discrepancy between the in- elastic scattering factors obtained by means of Hartree-Fock (HF) wave functions and those com- puted through the more exact treatment. The in- elastic scattering factors are lower than those cal- culated from HF wave functions. The influence of electronic correlation on elastic scattering factors is small; however, properly correlated wave func- tions are to be employed when a careful compari- son between theory and experiment is sought. Current developments of experimental techniques in electron scattering'4 have made it possible to measure differential cross sections with an uncer- tainty of less than 0. 1%. This error is far smaller than the magnitude of the corrections introduced by electronic correlation. Thus it is of interest to perform more accurate calculations in order to determine (a) the accuracy of the scattering theory, (b) agreement between various atomic fields and experiment, and (c) details of the electronic charge cloud such as the electron-pair cor'relation func- tion +( ~ ) 12 15 16 22 ' One of the ways to take into account electronic correlation into the wave function is by the method of configuration interaction (CI). " Recently, a, properly correlated CI wave function for Ne has become available" and it has been used in this work together with a CI wave function for He. In the next section the theory is presented for the calculations of both elastic and inelastic parts of the scattered amplitude. This is followed by a brief commentary on the wave functions employed. Finally, the results obtained by means of CI wave functions are compared with those derived from HF wave functions. II. THEORY A. First Born Approximation IT(s) = If (s)l +(4P0/a0 s )S(s), B 2 2 4 where If (s)l =(4P0/a0 s )[Z — F (s)] (2) N N S(s)=Z — E'(s)+ ZQ (O'Ie ij I4'), (3) 2~j=1 Z (s)= 2 (@I e" 'f le), 2=1 s = (4m/X) sin(e/2), p, =l (v/c)' and ao=0. 529167 A, In the framework of the first Born approximation, the total electron intensity scattered by atoms can be written as "~ "