Perturbations in an Electron Gas at Non-zero Temperatures

Assuming the electrons to move in a common perturbing potential, which is to be determined self-consistently, a general theory is given for the evaluation of charge and potential distributions round perturbations in an originally uniform electron gas at elevated temperatures. The results reduce to those given recently by the writers for perturbations in metals in the limit as the temperature T → 0. In the non-degenerate case, and to first order in the perturbing potential, our results lead to a treatment which supersedes the Brooks-Herring theory of ionized impurity centres in semiconductors. However, a numerical application we have considered verifies that their treatment will often represent a very good approximate solution of the new equation. We have also examined the low-temperature problem in a high-density gas and results are presented which indicate how the long-range density oscillations round a perturbation are damped out by the temperature effect in this case. Some tentative conclusions about the effect of electron interactions on the long-range oscillations are thereby suggested. Finally it is pointed out that, when a slowly varying potential assumption is valid, the perturbation series may be summed, and the theory of Sakai and of Feynman, Metropolis and Teller is regained.