On the theory of primary specific ionization in helium

Abstract

It is the aim of the present paper to establish an improved formula for the non-relativistic primary specific ionization of electrons in helium, as a result of a carefully developed theory in which numerically computed helium wave functions are introduced instead of less accurate hydrogenic wave functions. Sect.1 presents the calculation of the differential cross section for inelastic scattering of an incident electron by a helium atom on the basis of a well-known method for treating rearrangement collisions which makes use of the Born approximation. In Sect.2, the resulting formula is applied to the specific problem under consideration and it is first shown that the terms due to spin-exchange effects can be neglected under the prevailing conditions of validity. The various parts of the remaining simplified formula for the differential cross section are then calculated explicitly and in Sect.3, the necessary summations and integrations which ultimately lead to the primary specific ionizationS(β), are carried out. One of these integrations is studied in particular detail since it necessitates the introduction of additional simplifying approximations. The new final result forS(β) is discussed and compared with a frequently adopted formula derived from the classical work of Bethe on primary specific ionization. A graphical representation also permits a comparison with some experimental data. The main conclusion is that the new formula is able to account for as much as 25% of the discrepancy between Bethe’s formula and the experimental points. The remaining deviation can be physically understood. Finally, Sect.4 is devoted to the velocity distributions of the secondary electrons ejected from the helium atoms during the process of primary ionization in various quantum states of angular momentum characterized by the non-negative integerl. The general behavior of the cross sections per unit ϰ-interval (ϰ being a convenient dimensionless velocity parameter) corresponding tol=0, 1 and 2 is discussed and physically interpreted. These cross sections are also compared with their respective analytic counterparts in which Coulomb wave functions are used to describe the final states of the ejected electrons approximately. It turns out that the agreement is excellent forl=2 as well as for all higherl-values.