By C.G. & WHITFIELD, G.D. (eds) KUPER
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Extra info for Polarons and Excitons (Scottish Universities' Summer School)
Sample text
E. the group velocity of waves vanishes. 1) with the condition curl P(r) = 0. We shall Fourier analyse P(r) in terms of complex exponentials. For some aspects of the problem, for example in examining the localization condition, another set of basic functions might be more suitable. 1) a source term due to the electron. Now, since D(r) is the externally applied field and P(r) the effective induced dipolar field, the interaction energy density will be given by —D(r) . P(r). P(r). The Lagrangian L' is defined by = d3 r Y'(r).
P = d3 r V,. 2 1 — e313 (rel). D = N7 2 e = 42-te5(r— r e1). D 4np , since p is e6(r — r 1) for a point electron. 9) we can determine the electron equations of motion: 4ei = t el OH Opel = 1 in P ÔH — = — eV rop(rei), Pei &hi Pet= — m — V r et(1)(r We shall now recast the Hamiltonian into a more useful form. (1)(r). 12). It is mathematically convenient to consider a cubic region of the solid of volume V = L 3 , which is large compared to the phenomena under consideration, and to apply periodic boundary conditions.
E. the state with no phonons present. As a result of the interaction in the lowest order of perturbation theory, transitions will be induced between this state and states of the system in which one has 33 34 D. PINES present a single phonon of momentum hk and the electron possesses a momentum p—kh. Such a state is denoted by p _kha: I 0>. The matrix element for the transition is simply Vic+ . e. p hk2 occur if = co+ — . In these lectures we shall ignore such a 2m, mc complication by restricting our attention to slow electrons for which moi hk + — ; thus only virtual transitions will be considered.