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The quantum theory has provided a means of calculating the interaction energies of two atoms by a perturbation method. It appears that the short range interaction forces are due mainly to electron exchange phenomena between the two atoms, while the van der Waals forces arise from mutual polarization effects.* The theory gives the first of these forces in the first approximation, while the van der Waals forces appear only in the second approximation. At large distances, where the interaction is small, it is somewhat surprising that the first approximation is not sufficient, and one is led to doubt the accuracy of the method when applied at distances at which the first and second approximations give comparable results. At these distances the mutual potential energy is comparable with the mean kinetic energy of a gas atom at ordinary temperatures, and it is therefore clear that a study of gas-kinetic collision phenomena should provide a satisfactory test of the validity of the perturbation method in this region. It is the object of this paper to carry out a number of calculations with this aim in viewr.In a previous paperf the quantum theory of collisions was applied to gaskinetic collisions, and it was shown that, although the classical theory can be used with accuracy to determine the law of force from viscosity and diffusion phenomena associated with heavy gases, it cannot be applied with safety to hydrogen and helium. The method to be used in such cases was given, and it was also shown that the existence of a definite total collision area-a feature of the quantum theory of scattering bv a centre of force, the potential of which falls off more rapidly than r2 at large distances-provides a further means of determining the law of force. As this collision area can now be directly measured with accuracy by molecular ray experiments, the range of applic * Vide

functions of the various helium lines for electron energies up to, and in certain cases greater than 400 volts. In this paper we have extended the calculations in this way and have considered also ionizing collisions. Since the probability of ionization by electrons can be measured with considerable accuracy we can apply a very satisfactory test of the theory in this direction. In addition to the probabilities of ionization of hydrogen and helium the velocity and angular distributions of ejected and scattered electrons are also computed. The comparison of calculated and observed results is discussed in detail and it is found that Born's approximation is valid for electrons with energies greater than 200 volts. Section I. ExcitationHelium. § 1. Method of Calculation. -Employing throughout the sam that used in paper A we see that the differential cross-section l n ( §)d,T 0, are the wave functions for the initial and final states respectively, k j2 n being the wave number of the electron incident in the direction of the unit vector n 0 and k' /2tz th at of the electron scattered in the direction of the unit vectorWe may use for the wave functions V F0, T*n the same forms as those given in paper A for the P and D states. If we write (Z |y) for the wave function of a single electron in the nlm state in the field of a charge Ze, the approximation used in paper A corresponds to a wave function for the nlm state of helium of the formf

SummaryThe extension of the WKB method to a complex potential, as used in the optical "model of the nucleus, is discussed. The formula for the complex phase shifts is formally -deduced, and its accuracy tested against exact calculations for a square potential well and a well with sloping sides. At low energies there occur large discrepancies; the WKB phases vary regularly with energy, whereas the exact values oscillate violently .about the WKB values in a characteristic way and marked resonances occur. The factors affecting the accuracy of the method are discussed.At higher energies the fluctuation of the phases about the WKB value is less marked, and its effect largely cancels out as the result of a larger number of phases being involved in the scattering.1. INTRODUCTION The WKB method for the determination of phase shifts in collision problems has been remarkably useful, notably in the scattering of electrons by atoms (Massey and Burhop 1952), and its range of validity is well understood. With the success of the optical model of the nucleus, in which a potential with an imaginary component is used, there arises the question of the correct method of application of the WKB method to a complex potential and its range of validity.The method has been found valid for nucleons incident on nuclei with energies large compared with the nuclear potential (Mohr and Robson 1956), in which case one may use an" obvious approximation to the WKB formula (Massey and Mohr 1934). In this approximation the ratio of the imaginary to the real component of the phase is equal to the ratio of the imaginary to the real component of the well depth.At lower energies, in a calculation of IX-particle scattering with a comparatively large imaginary well depth, B. A. Robson (unpublished data) in this laboratory recently found that a tentative adaption of the WKB formula without approximation gave an imaginary component of the phase many times the value given by numerical integration of the wave equation.

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