# 13 ext 1 n p 0 p 0 0 ext 1 n 813a n

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Unformatted text preview: eld for homogeneous polarization (only for ellipsoid of revolution) (N =-NP / ε0 (8.13) Field inside the sample (″macroscopic field″): ( = ( ext − 1 N P, ε0 P = ε 0 χ( = N: depolarization factor 8-7 χ ε 0 ( ext 1 + Nχ (8.13a) Depolarization factor for special geometries: Sphere: N = 1/3, hollow sphere: N = -1/3 (see eq. 8.12) Long rod || ( : N = 0 ⇒ for (I = 0 : (local = (ext + (N + (L (8.14) Spherical sample: (N = -(L ⇒ (local = (ext (8.15) 1 1 2 Thin slap ⊥ ( : ( local = ( ext − ε P + 3ε P = ( ext − 3ε P (8.16) 0 0 0 1 ( local = ( ext + P Long rod || ( : (N = 0 ⇒ 3ε 0 8-8 (8.17) Clausius-Mosotti equation Solid with atom sites of cubic symetry ⇒ (I = 0 (local = ( + (L , ( = (ext + (N , (8.18) macroscopic field inside sample P = ε0nv α(local (8.8) 1 ⇒ P = ε0nv α(( + 3ε P ) 0 nvα ( 1 1 − nvα ⇒ 3 P = ε0 χ( P = ε0 ⇒ χ= (8.3) nvα 1 1 − nvα 3 ε = 1+ χ = 1+ ⇒ (8.19) (8.20) n vα 1 1 − n vα 3 ε −1 1 n vα = 3 ε +2 (8.21) Clausius-Mosotti equation (8.22) Measurement of ε ⇒ molecular polarizability α. 8-9 Lorentz’s oscillator model for the electronic polarizability Small displacement x of an electron in a lattice atom from it’s equilibrium position. ⇒ - linear restoring force - dipole moment -ex Applied electromagnetic field ⇒ ( = ( local e − iωt ⇒ forced oscillation. 0 Oscillating dipole emits adiation ⇒ damping ∝dx/dt Equation of motion: d 2x dx 2 0 m 2 + mβ + mω 0 x = −e(local e iωt dt dt (8.23) β: damping constant, ω0 = f / m : frequency of undamped vibration, Restoring force: F = -fx Analogous to forced mechanical oscillation ⇒ stationary state approached with relaxation time τ = 1 /β (8.24) 8-10 Complex amplitude: x=− 1 e ( iω local m ω2 −ω2 − 0 τ p = -ex = αelε0(local Dipole moment: e2 1 α el (ω ) = Electrical polarizability: ε 0 m ω 2 − ω 2 − iω 0 τ (8.25) (8.26) (8.27) (8.27) in (8.21) ⇒ nv e 2 ε (ω ) = 1 + ε 0m 1 1 nv e 2 = 1+ 2 ω1 e ε 0m ω 2 − ω 2 − i ω ω 02 −...
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