Frequency Dispersion: Dielectrics, Conductors, and Plasmas

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Albert Andreasen
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1 1/23 Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hernández Professor: Jorge Alfaro Instituto de Física Pontificia Universidad Católica de Chile

2 2/23 Contents 1 Simple Model for ɛ(ω) 2 Anomolous Dispersion and Resonant Absorption 3 Low-Frequency Behavior, Electric Conductivity 4 High-Frequency Limit, Plasma Frequency 5 Example: Liquid Water

3 3/23 Simple Model for ɛ(ω)

4 4/23 Simple Model for ɛ(ω) Electron of charge e bound by a harmonic force to an atom, and acted on by an electric field Equation of motion: ] m [ x + γ x + ω 2 0 x = ee( x, t) (1)

5 5/23 If the field varies harmonically in time with frequency ω. Dipole moment contributed by one electron: E Ee iωt (2) x x 0 e iωt (3) p = e x = e 2 m(ω0 2 E ω2 iωγ) = e2 (ω0 2 ω2 ) + iωγ m (ω0 2 ω2 ) 2 + γ 2 ω 2 E (4) Difference of phase between the electric field and the dipole moment: φ = tan 1 ( ) ωγ ω 2 0 ω2 (5)

6 6/23 Atomic contribution to the dielectric constant for N molecules per unit volume, f j electrons per molecule with binding frequency ω j and damping constan γ j : ɛ(ω) ɛ 0 = 1 + χ e = 1 + N e2 f j ɛ 0 m (ω 2 j j ω 2 iωγ j ) (6)

7 7/23 Anomalous Dispersion and Resonant Absorption

8 8/23 Anomalous Dispersion and Resonant Absorption In a dispersive medium, the wave equation for the electric field reads it admits plane wave solutions 2 E = ɛµ0 2 E t 2 (7) with the complex wave number E(z, t) = E 0 e i(kz ωt) (8) k = µ 0 ɛω (9)

9 9/23 We can write the wave number in terms of this real and imaginary parts k = β + i α 2 (10) and the electric field becomes E(z, t) = E 0 e α 2 z e i(βz ωt) (11) Intensity proportional to E 2 (to e αz ), α is called the absorption coefficient. Index of refraction: n = cβ ω

10 10/23 ɛ(ω) ɛ 0 The damping constants γ j are generally small compared with the resonant frequencies ω j. Normal dispersion region is associated with the increase of Re(ɛ) with ω. Anomalous dispersion (resonant absorption) is where the imaginary part of ɛ is appreciable, and that represents dissipation of energy. = 1 + N e2 f j ɛ 0 m (ω 2 j j ω 2 iωγ j )

11 11/23 Low-Frequency Behavior, Electric Conductivity

12 12/23 Low-Frequency Behavior, Electric Conductivity ɛ(ω) ɛ 0 = 1 + N e2 f j ɛ 0 m (ω 2 j j ω 2 iωγ j ) Fraction f 0 of electrons per molecule free (ω 0 = 0) ɛ(ω) = ɛ b (ω) + Ne2 m f 0 ω 2 = ɛ b (ω) + i Ne2 iωγ 0 m f 0 ω(γ 0 iω) (12)

13 13/23 Maxwell-Ampere equation: H = J + D t Assuming medium obbeys Ohm s law: J = σe, D = ɛ be H ( = iω ɛ b + i σ ) E (13) ω Assuming all the properties due the medium: D = ɛ(ω) E H = iωɛ(ω) E (14)

14 14/23 By comparison we get the electric conductivity (model of Drude) σ = f 0 Ne 2 m(γ 0 iω) (15) The dispersive properties of the medium can be attributed as well to a complez dielectric constant, as to a frequency-dependent conductivity and a dielectric constant.

15 15/23 High-Frequency Limit, Plasma Frequency

16 16/23 High-Frequency Limit, Plasma Frequency ɛ(ω) ɛ 0 = 1 + N e2 f j ɛ 0 m (ω 2 j j ω 2 iωγ j ) At frequencies far above the highest resonant frequency ɛ(ω) ɛ 0 1 ω2 p ω 2 (16) where ω p = NZe2 ɛ 0 m is called the plasma frequency of the medium.

17 17/23 The wave number is given by ck = ω 2 ω 2 p (17) In dielectric media, the approximation holds only for ω 2 >> ω 2 p. In plasmas, the electrons are free and the damping force is negligible, so the approximation holds over a wide range of requencies, including ω < ω p. In conductors, the approximation holds for frequencies ω >> γ 0 and the behavior of incident waves for ω << ω p is similar to the behavior for plasma

18 18/23 Example: Liquid Water

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21 21/23

22 22/23 Bibliography 1 J. Jackson, Classical Electrodynamics. Chapter D. Griffiths, Indtroduction to Electrodynamics. Chapter 9.4

23 23/23 Thank You

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