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By Ruark A.E.

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59)), but the number of phonons with an already renormalized, new frequency ω, 1/(eω/T − 1). 60) (here we have simplified the mathematics by writing the equation not for ω2 , but for ω; this is sufficient for our qualitative treatment). 60) is a self-consistent equation for ω. 61) see Fig. 11. e. 2) ∼e+iωt is again diverging, which indeed signals an instability of the lattice, in our case melting. And in this theory melting would be a first-order phase transition, with a jump in ω, as it should be.

4 Anharmonic effects υ υ 3 45 3 2 2 1 1 Fig. 8 Fig. 9 at the lowest quantum level of the oscillator (state 1). At higher temperatures the average distance between atoms is given by averaging over all occupied higherlying quantum levels. In the harmonic case we have the situation shown in Fig. e. the average lattice parameter, is the same at each quantum level. Thus it does not change with temperature, and in the purely harmonic crystal we would not have any thermal expansion. On the other hand, in an anharmonic crystal the interaction potential qualitatively looks as shown in Fig.

E. it is steeper for negative relative distortion (when two atoms approach one another) and less steep for positive u1 − u2 , when the two atoms move further apart. 26). ) , where the momenta involved obey conservation laws q1 + q2 = q3 + q4 , k1 = k2 + k3 + k4 (the total momentum of created phonons is equal to the total momentum of annihilated ones). Anharmonic interactions have several consequences. One is that they lead to phonon–phonon interactions, schematically illustrated in Fig. 7. g. for thermal conductivity.

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