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Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character

The collision of slow electrons with atoms. I.—General theory and elastic collisions

H. S. W. Massey, C. B. O. Mohr

Published May 2, 1932• 70 citations
Publisher

Abstract

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.

DOI
10.1098/rspa.1932.0082

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Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Feb 1, 1929•25 citations
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 Ramsauer effect does not correspond to the decrease in optical scattering by a particle when the wave-length of the light is increased. For one may compute the cross-section for elastic collision of an electron with an atom, without making the approximations used by Born. If one does this, and neglects the resonance of the colliding electron with the atomic electrons one finds that, for atoms with no dipole moment in the normal state, the cross-section approaches a finite limit when the electronic velocity is decreased. The only general result of this calculation is that the distribution of scattered electrons tends, in this limit, to become uniform over all angles.
The Bohr formulation—the original one—is now largely a matter of history. The usual career of a theory was its fate. It explained much, but it failed at the first signs of complication, even for the simplest molecules as hydrogen and helium. Some results were either only approximate in a loose or incomplete fashion, e.g. in the application of the correspondence principle to intensities, the only reliable predictions being only for an absence of certain lines, or they quite disagreed with experiment, e.g. in the case of the hydrogen molecule and the hydrogen molecule ion, the normal and excited states of helium, the anomalous Zeeman effect, the Ramsauer effect (the phenomenon of the deflection of electrons at low speeds being less than at higher speeds in atomic collisions), and in the treatment of problems where more than one electron was involved. In the last mentioned cases the perturbation method, so successful in solving “many-body” problems in celestial mechanics, failed completely because systems of several electrons, possessing frequencies of the order of light waves, cannot interact classically. A bit of much forced pragmatism was practiced in treating the many electron atom as a variation of the hydrogen atom. Besides, there were technical oddities, like ½ quantum numbers, and impenetrated anologies between visible and X-ray spectra.
Abstract A large number of investigations have been carried out on the motion of charged particles through gases, and the subject has received added impetus from the introduction of quantum mechanics, as this theory is capable of dealing with collision phenomena in a manner that was beyond the reach of Bohr’s quantum theory. In the early days of the wave theory, many experiments were devised with a view to testing the validity of the theory; but now the foundations of quantum mechanics have been so firmly laid that we may use the theory to clarify the more complicated phenomena observed in the passage of charged particles through gases. Both from the experimental and theoretical point of view the most convenient particles to study are electrons; and, as a consequence, the phenomena observed in the collisions of electrons with gas molecules are well known and to a large extent understood. Well-marked diffraction phenomena have been observed and the relative probabilities of elastic and inelastic collisions measured for a number of gases. Although detailed theoretical explanations have not always been given, it is possible in nearly all cases to give a general description of the main processes contributing to the observed effects. For example, the Ramsauer effect is now completely explained.
Abstract One of the outstanding differences between classical theory and quantum mechanics is the importance which the latter attaches to the identity of particles. Although perfectly well realised before the introduction of modern quantum theory, the fact that two electrons are experimentally indistinguish­able did not introduce any novel features into the dynamics of many electron systems on Bohr’s theory. However, Heisenberg showed that these identity relations are of fundamental importance in many body problems, and was able to explain the formerly very puzzling characteristic of the spectrum of neutral helium, its division into two non-combining term systems. Owing to the identity of the particles, the wave function representing such a system must have certain symmetry properties; for the particles known in Nature this symmetry characteristic is that the function be either symmetric or anti-symmetric in the co-ordinates of the particles. For particles with a spin such as the electron and proton, only antisymmetry is allowed (Pauli’s principle), while for α-particles the wave function must be symmetrical. Since Heisenberg’s explanation of the helium spectrum the theory of anti-symmetric electron wave functions has been extended to describe the general features of the spectra of all atoms, while it appears from the work of Heitler and London that the theory is of fundamental importance in the explanation of chemical combination between atoms. Again, in furnishing a new statistics it is an essential feature of the theory of metals. As the effect of the identity relations depends on the fact that exchanging the electrons does not alter the system in any way, these effects are known as exchange phenomena.
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.
Abstract The work about to be described is a continuation of an investigation begun by one of us at Princeton, New Jersey, on the scattering of electrons by gas atoms. The conditions there were so chosen that only single collisions were taken into account, and it was possible to investigate not only the energy loss on collision, as in the Franck and Hertz experiments, but also the direction of the path of the electron after collision. By employing a magnetic analysis of the velocity of the electron after impact it was found possible to obtain information as to the probability of excitation of various stationary states of the atom considered, on collision with electrons of much higher velocities than could previously be used. The curves relating scattered intensity with angle of scattering were found to have sharp maxima, which were strongly suggestive of diffraction maxima, as might be expected if the motion of electrons is governed by the wave mechanics. But whereas the wave mechanics has been eminently successful in explaining the scattering results of Davisson and Germer with single metal crystals, and of Thomson and Rupp with thin metal foils the results of Dymond have been inexplicable on this basis. It may be said at once that we have been unable to find any trace of the sharp maxima found by him in I, which were apparently due to a secondary cause. Indeed, as has recently been pointed out by Elsasser, no interference effects can be expected, when the electron loses energy on collision, the case in which Dymond found maxima, for the frequencies of the electron before and after interaction with the atom are different and there can consequently be no coherence of the scattered waves.