32 833 charges are sources of dielectric displacement

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Unformatted text preview: ω 2 − i − nv 1 τ τ 3 ε 0m (8.28) 1 e2 with ω = ω − 3 nv ε m 0 2 1 2 0 Separation: ε(ω) =ε1(ω) + iε2(ω) (8.29) nv e 2 ω12 − ω 2 ε1 (ω ) = 1 + ε 0 m (ω12 − ω 2 ) 2 + ω 2 / τ 2 (8.30) nv e 2 ω /τ ε 2 (ω ) = ε 0 m (ω12 − ω 2 ) 2 + ω 2 / τ 2 (8.31) 8-11 Maxwell equations for solid divB = 0 ⇔ ∫∫ Bd A = 0 divD = ρ ⇔ ∫∫ Dd A = Q (no magnetic monopoles) (8.32) (8.33) (charges are sources of dielectric displacement field) & curl ( = - B ⇔ & ∫ ( d r = − ∫∫ Bd A (8.34) (law of induction) & curl H = j + D ⇔ & ∫ H d r = I + ∫∫ Dd A (8.35) (current or charge in flux cause curled magnetic field) & (8.35) ⇒ curl H = j + D = σ ( + ε 0ε (& ⇒ & = − µ H = curl( ⇒ µ curl H = curl curl ( & & (8.34) ⇒ − B 0 0 − 1 curl curl ( = σ (& + ε 0ε (&& 23 µ 0 14 4 curl curl ( = ∇ × (∇ × ( ) = ∇ (∇( ) − ∇ ( 2 ∇( = 0 if ρ = 0 inside the sample ⇒ 2 ∇ ( = ε 0εµ 0(&& + µ 0σ (& Wave equation for nonmagnetic solid (µ ≈ 1) 8-12 (8.36) 2 ∇ ( = ε 0εµ 0 (&& + µ 0σ (& (8.36) Solution by damped wave ansatz: ε = ε 0e ~ i ( k ⋅ r −ω t ) ~ 2 εω 2 ω k = 2 + iσ c ε 0c 2 (8.37) in (8.37) ⇒ c = 1 / µ 0ε 0 , ⇒ (8.37) speed of light in vacuum ~ω σ ω~ k= =n ε +i c ε 0ω c ~ Complex refractive index n = n + iκ (8.38) (8.39) n: refractive index, κ: absorbtion coefficient ~ω ω k = n+i κ ⇒ c c (8.40) Plane wave propagating in z-direction: ω κω i n z −ωt z − (8.37) c c ⇒ ε = ε 0e e2 13 14 4 23 (8.40) damping in direction plane wave of propagation c in vacuum → c/n in solid 8-13 (8.41) Damping of amplitude by e Damping of intencity by e K= − − κω z c ⇒ 2κω z c 2κω : absorbtion constant c (8.42) Relation to ε(ω) =ε1(ω) + iε2(ω) ~ n 2 = ( n + iκ ) 2 = ε (ω ) = ε 1 (ω ) + iε 2 (ω ) (8.43) ε1(ω) = n2 - κ2 = ε (8.44) σ ε 0ω (8.49) ε 2 (ω ) = 2nκ = σ ~ n2 = ε +i ↑ (8.38) ⇒ ε 0ω 8-14 Fig. 8.5 General form of ε1(ω) and ε2(ω) for a dipole oscillator according to equations (8.30) and (8.31). ε2(ω) ≈ 0 outside resonance maximum Width of maximum: 1/τ ε2(ω) ≈ 0 ⇒ n ≈ ε 1 (ω ) e2 1 Resonance frequency: ω1 = ω 0 − 3 nv ε m < ω 0 0 ω0: resonance frequency of αel(ω) ω1 < ω0 due to effect of local field. Classical harmonic oscillator: ω 0 = 8-15 f /m Quantum mechnics: resonances are absorbtion frequencies due to electronic tansitions ⇒ always several resonance maxima in atom mainly in 1016 s-1 range (UV). Solid: also interband and intraband transitions Intraband transitions: between filled and empty states of conduction band in metals and semiconductors. ω < ω1 : ε1(ω) > 1 ⇒ n ≈ ε1 (ω ) > 1 lim ε 1 (ω ) = ε (0) ω →0 ε(0): statistic dielectric constant (already holds for visible range). ω > ω1 : ε1(ω) < 1 ⇒ n < 1 e.g. for X-rays lim ε1 (ω ) = 1 ω →∞ 8-16...
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