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sional-multichannel method and to demonstrate that it works well, it is of some relevance to further consider the comparison between the CPA and the space-time Magnus methods. The CPA starts to break down for nonresonant inelastic processes where the energy transfered is a large fraction of the kinetic energy in one of the channels. 1 Consequently, a comparison can be made with a simple model like the following: W 119 W 00 =<*> for x< 0, W 00 = 0 and W u = 1 for x> 0, and W 01 = V (a constant) for 0 < x^ A and W 01 = 0 for x>A.The full quantum method, the CPA method, and our method (for a single interaction interval) may be solved analytically for this model. We find that at high energies the space-time Magnus method and the CPA both agree with the exact quantum results, and at low energies both methods give good results for the average transition probability. However, at low energies the spacetime Magnus method very accurately reproduces the correct oscillations in the transition probability while the CPA is badly out of phase with the correct oscillations. (In the very-low-energy limit, the single-interaction-interval space-time Magnus method begins to develop some discrepancies.)The reason that the semiclassical method provides an improvement over the CPA can readily be understood. The only approximation inherent in the semiclassical method involves the choice of a finite interval size, as the Magnus decom-For many years the annihilation of positrons in solids has been used to investigate various characteristics of electrons in matter. It has usually been assumed that at the time of annihilation the positron, or sometimes the positronium atom, was in its lowest state in thermal equilibrium with its environment, although the experiments position and the semiclassical approximation to the space-time propagator both become exact as the interval size decreases. It is the use of heavy masses and slowly varying potentials that permits the use of rather large intervals, thereby making the method useful.We are grateful to Professor John Light for many helpful discussions and suggestions"Positrons in metals and positronium in quartz have been observed to thermalize down to nearly liquid-helium temperatures before annihilation. The achievement of such low temperatures by both positrons and positronium in approximately 10" 10 sec indicates that phonon scattering plays a major role in thermalization. These results have important implications for the ability to achieve high precision in Fermi-surface studies by the positron-annihilation technique. 852

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Positron motion has been observed in potassium, sodium, magnesium, and aluminum over a wide range of temperatures. The observations show that positrons are close to thermal equilibrium before annihilation even at liquid helium temperature. If a Gaussian momentum distribution is fitted to the data, it yields a positron apparent mass m*/m of 1.6 ± 0.1 in K, 1.55 ± 0.1 in Na, 1.2 ± 0.1 in Mg, and 1.3 ± 0.1 in Al. These new results are fairly close to the theoretical expectations, but show that the subject is not yet fully understood.

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