By Daniel I. Khomskii
Provides the most suggestions and crucial theoretical equipment of the trendy quantum conception of solids for graduate scholars and researchers.
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Additional resources for Basic Aspects of the Quantum Theory of Solids : Order and Elementary Excitations
5. The integration in d 3 q is carried out not over 40 Phonons in crystals D(ω) D(ω) ~ ω2 ω (a) ωD (b) Typical ‘real’ density of states for phonons ω Debye model (with the approximation ω = sq) Fig. 5 the real Brillouin zone, but over a sphere, with the volume equal to the volume of the Brillouin zone. 36) N= πq (2π )3 3 0 where q0 – the Debye wavevector – is the maximum wavevector of the equivalent sphere in q-space. The maximum frequency ωD = sq0 is called the Debye frequency (s is the sound velocity); D = − hωD is the Debye temperature.
Of the phonon–phonon interaction, this is no longer true. But one can still use this expression approximately in the so-called quasiharmonic approximation, accounting for anharmonicity in the following way. The phonon frequencies ωq in the anharmonic crystal in general depend on the specific volume, ωq (V ) (see Fig. 4) determines phonon frequencies, in the anharmonic case depends on x). 49) ω dV d ln V which is called the Gr¨uneisen approximation; γ is the Gr¨uneisen constant (usually, in ordinary crystals, γ ∼ 1–2).
Thus we would have in the free energy the terms G(∇η)2 + E(∇ 2 η)2 . 28) As a function of ∇η, has a form similar to the one shown in Fig. 2, see Fig. 11. It is convenient to go to the momentum representation: (∇η)2 → q 2 η2 . Then 18 General theory of phase transitions T disordered phase homogeneous phase Lifshits point * inhomogeneous phase certain parameter Fig. 12 would contain the terms Gq 2 η2 + Eq 4 η2 . 31) 2 Q2 = qmin =− (G < 0) . That is, in this case the structure with the wavevector Q, or with the period l = 2π/Q, will be formed.
Basic Aspects of the Quantum Theory of Solids : Order and Elementary Excitations by Daniel I. Khomskii