E nv 86 85 solids local field at an atom is influenced

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Unformatted text preview: er of molecules / volume ⇒ ε = 1 + nvα, i.e., χ = nvα (8.6) Solids: local field at an atom is influenced by dipoles of neighbours. Local field at an atom: (local = (ext + (sample P = ε0nvα(local (8.7) (8.8) Calculation of (local 8-4 a) Field EI acting on atom at the center of an imaginary sphere (I depends on crystal structure. (I = 0 if atom sites have cubic symetry, e.g. for: sc, fcc, bcc, NaCl, and CsCl structure (see practical course for derivation) b) Imaginare sphere is cut out of the solid Choise of radius R of imaginary sphere: • 2R << λ of electromagnetic field ⇒ (I = 0 for cubic symmetry. • R large enough to describe polarization in center due to atoms outside sphere by quasicontinuous dipole distribution, i.e., macroscopic polarization P ⇒ Field due to P described by polarization charges ρP on surface (see Fig. 8.3). 8-5 ρp = -Pn (8.9) Pn: normal component of P Note: unit normal points inside hollow sphere ⇒ ″-″ sign. Cf.: sphere with homogeneous surface charge density ρ ⇒ ( = (n = ρ/ε0 (Rigorous proof of eq. (8.9): see, e.g. Kittel) Fig. 8.3 In a continuum model the field at the center of a hollow sphere due to the polarization of the atoms outside the sphere can be described in termens of polarization charges on the surface of the sphere. Charge in ring at angle θ: 14 42 3 dq = − P cosθ ⋅ (2πa sin θ ) adθ Pn (8.10) Contribution to field in center: d( = − 1 dq cosθ 4πε 0 a 2 <( ⊥ P> = 0, (8.11) ( || P =( cosθ ⇒ polarization field in center of hollow sphere (″Lorentz field″) π P 1 (L = cos 2 θ sin θdθ = P 2ε 0 ∫ 3ε 0 0 8-6 (8.12) c) Account for depolarization Fig. 8.4 The depolarization field is opposite to the polarization and is due to the net surface charges originating from the polarization of a body in an external field. The polarization is homogeneous for an elipsoid of revolution. In this case the depolarizing field (N can be described in terms of a geometry-dependent depolarization factor N. Depolarizing fi...
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