In this paper general expressions for the width and shift of atomic lines produced by fast electron collisions are obtained. We use the recent quantum-mechanical theories developed to explain the pressure line shape in the impact approximation. Only isolated and Lorentzian lines are analyzed. The calculations are performed by expanding the scattering matrix up to the second Born approximation. The contributions of both elastic and inelastic virtual collisions are calculated. Degenerate and nondegenerate atomic states are considered.
In this paper general expressions for the width and shift of atomic lines produced by fast electron collisions are obtained. We use the recent quantum-mechanical theories developed to explain the pressure line shape in the impact approximation. Only isolated and Lorentzian lines are analyzed. The calculations are performed by expanding the scattering matrix up to the second Born approximation. The contributions of both elastic and inelastic virtual collisions are calculated. Degenerate and nondegenerate atomic states are considered.
The work of two previous papers is extended and a theory of pressure broadening is developed which treats the perturbers quantum mechanically and allows for inelastic collisions, degeneracy, and overlapping lines. The impact approximation is used. It consists in assuming that it takes, on the average, many collisions to produce an appreciable disturbance in the wave function of the atom, and it results in an isolated line having a Lorentz shape. Validity criteria are given. When the approximation is valid, it is allowable to replace the exact, fluctuating interaction of the perturbers with the atom by a constant effective interaction. The effective interaction is expressed in terms of the one-perturber quantum mechanical transition amplitudes on and near the energy shell and its close relationship to the scattering matrix is stressed. The calculation of the line shape in terms of the effective interaction is the same as when the perturbers move on classical paths. Results are written explicitly for isolated lines. If the interaction of the perturbers with the final state can be neglected, the shift and width are proportional to the real and imaginary part of the forward elastic scattering amplitude, respectively. By the optical theorem, the width can also be written in terms of the total cross section. When the interaction in the final state cannot be neglected, the shift and width are still given in terms of the elastic scattering amplitudes, in a slightly more complicated fashion. Finally, rules are given for taking into account rotational degeneracy of the radiating states.
Quantum mechanics is used to treat the motion of the perturbers broadening a line. Several simplifying assumptions are made, such as the Born-Oppenheimer approximation, and the assumption of no degeneracy. These assumptions can be removed. Also made is the "impact approximation," which is essential for the validity of the results and cannot be removed. With it, the line has a Lorentz shape, and its width and shift are expressed in terms of the two scattering amplitudes of the perturbers by the atom in its upper and lower state. The case where the perturbers do not interact appreciably with the lower state is particularly simple. Then, the width and shift are proportional to the imaginary part and the real part of the forward scattering amplitude, respectively. A quantity called the "collision volume" is defined. It is shown that the impact approximation is valid only if the collision volume is much smaller than the volume per perturber. There is a second validity condition, which has no classical analog.
Abstract An outstanding defect of the old quantum theory was its inability to treat the phenomena occurring in the collision of electrons with atoms, and Born’s theory showed how the new quantum mechanics was capable of removing this defect. The original theory of Born was a method of successive approximations to the solution of the wave equation representing the colliding systems, and could be applied to both elastic and inelastic collisions. In its actual applications, which could be worked out to the first approximation, the theory fell far short of expectations. The Ramsauer effect, the most interesting phenomenon exhibited in the elastic scattering of electrons, could not be explained, and in the field of inelastic collisions, it failed to provide a means of explaining the excitation of levels of a term system which does not combine with the ground state system, e. g., the excitation of the triplet lines of helium. These failures are, of course, failures of the theory in its first approximation. The first approximation in Born’s method of successive approximations neglected the disturbance of the electron waves by the field of the scatterer, and also the possibility of exchange of electrons between the atoms and the colliding beam. The neglect of both these effects becomes important for low velocity impacts, and it was clear that the Ramsauer effect, which is only observed for low velocity electrons, could only be explained by more accurate theory. The question was whether both effects, the distortion of the electron waves by the atomic field, and the exchange of electrons was important, or only one of them.
The theory of impact broadening of the isolated spectral lines of the Rydberg atomic series perturbed by neutral particles is developed within the framework of the impulse approximation. The analysis of the broadening process is based on the consideration of the inelastic nl to n' transitions between highly excited states of the Rydberg atom. The role of the potential and resonance scattering of the quasifree electron by a perturbing particle in the impact broadening of the Rydberg nl levels is investigated. It is shown that the general expressions for the broadening cross sections presented in this paper contain the results of asymptotic Alekseev and Sobel'man theory (1965) as special cases. Both elastic and inelastic scattering contributions of the perturber by the parent core of Rydberg atom to the broadening cross sections are also considered within the framework of the 'shake' model. The results obtained are used for a quantitative explanation of experimental data on the broadening of Rydberg atomic levels perturbed by alkali-metal atoms. The energies and widths of the 3 P resonances on the quasidiscrete levels of the negative K - and Rb - ions are determined.
The theory of Baranger is discussed, relating it to the approach taken by Anderson in the last chapter and that taken by Fano in the one to follow. Baranger is concerned to describe pressure broadening in a band of close, overlapping lines. His original concern was with line broadening by fast-moving electrons in a plasma, which allowed him to use the impact approximation, but not to assume that collisions may be associated with classical paths. For this reason, although matter here is in the form of neutral molecules, the use of Baranger’s theory in its most general form requires that collisions be treated in terms of quantum scattering theory. In forming the correlation function, Baranger sets the algebra in a product space where line vectors take the place of the energy states used by Anderson, and the optical cross-section that governs line broadening is replaced by the matrix of an operator in line space, with line widths and shifts on the diagonal and line coupling parameters for the other elements. In the case of isolated lines, Anderson’s theory may be regained, but the introduction of line space paved the way, later on, for a much more general viewpoint.