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ssp_11y - 4 Dynamics of atoms in crystals Physical...

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4-1 4 Dynamics of atoms in crystals Physical properties determined by either electrons (see Chap. 5) or movement of atoms about their equilibrium positions e.g.: - sound velocity - thermal properties (specific heat, thermal expansion, heat conduction of nonmetals) Adiabatic approximation (Born and Oppenheimer 1927) Electrons follow movements of nuclei without inertia (m c << M core ). No interaction between lattice vibrations and electron system - Displacement of nuclei new electron distribution of higher total energy restoring forces from energy gradient - Electron system remains in ground state * (Not always valid, e.g. superconductivity) * This means that, if the initial positions of the nuclei are restored, than the energy expanded is recovered in full and there remains no excitation of the electron system into electronic states of higher energies.
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4-2 Interatomic potential Fig. 4.1. Sketch of the interatomic potential U ( r ) and the harmonic approximation (dashed line). ( ) ( ) ( ) ( ) U r U r U r r r U r r r r r = + + + 0 0 2 2 0 2 0 0 1 2 ... (4.1) =U =0 =s 0 ( ) U r U U r s r = + 0 2 2 2 1 2 0 harmonic approximation (4.2) F = - grad U F U r s r = − 2 2 0 linear restoring force (4.3) Harmonic oscillator: F = - fs (4.4) (4.3), (4.4) f U r r = 2 2 0 (4.5) Curvature of U(r) determines interatomic coupling constants ( spring constants ).
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4-3 Fig. 4.2. Illustration of a simple harmonic oscillator. Lattice with monoatomic basis Approximation: only NN coupling, harmonic behavior ( ) ( ) ms s s f s s f s n n n n n n , : = − + + 1 1 displacement (4.6) Ansatz: harmonic oscillations ( ) s t s e n n i t = ω (4.7) ( ) = − + m s f s s s n n n n ω 2 1 1 2 (4.8) Translational symmetry: r r ma m n m n + = = ± ± 0 1 2 , , ,... s s s s e n n m n m n iqma and can only differ by phase factor: + + = (4.9) ( ) [ ] ( ) m f e e f qa iqa iqa ω 2 2 2 1 = + = cos (4.10) ω = = = f m qa f m qa f m qa 2 1 2 1 2 4 2 cos cos sin (4.11) ( ) 1 2 1 2 2 = cos sin x x Fig. 4.3. Sketch of a monoatomic linear chain. Equation of motion for n th atom.
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4-4 Fig. 4.4. Dispersion-curve ω (q) for a linear monoatomic chain. Because of eqs. (4.12a) and (4.12b) it is sufficient to plot ω (q) in one half of the Brillouin zone. ω = 4 2 f m qa sin ( ) ( ) ω ω q q = ( ) ( ) 3D: ω ω q q = (4.12a) ( ) ω ω π q q m a = + 2 3 , : D ( ) ( ) ω ω q q G = + (4.12b) qa a << >> 1 λ cq q m fa = = 2 ω (4.13) = ω q c constant velocity of sound wave similar to elastic continuum!
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4-5 Linear diatomic chain (lattice + basis M 1 , M 2 ) Fig. 4.5. Sketch of a linear
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