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7 Usually the coefficient ζ is negative, ζ < 0, and the potential looks like the one 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.

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. 58) for ω2 . 48 Phonons in crystals υ Fig. 57) B(T ) B0 (1 − 6αγ T ) . e. 60), which will again give first-order melting. g. a transformation into another crystal structure. To check that we will indeed have melting, we have to show that the shear modulus (shear modes for q → 0) becomes unstable.

3 Bose and Fermi statistics This chapter is a short reminder and a collection of basic formulae on Fermi and Bose statistics. For noninteracting particles with the spectrum εk , the energy of the quantum state, in which there are nk particles, is Ek,nk = εk nk (we incorporated in the index k also spin indices, and other indices if they exist). 33), the thermodynamic potential for this quantum state is equal to k = −T ln e(μ−εk )/T nk . e. e. we necessarily have μ < εk (or μ < 0 if εk ∼ k 2 /2m), and after summation we obtain B k = T ln 1 − e(μ−εk )/T .

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