frequency dispersion: dielectrics, conductors, and plasmas · frequency dispersion: dielectrics,...
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Frequency Dispersion: Dielectrics, Conductors, and Plasmas
Carlos Felipe Espinoza Hernandez
Professor: Jorge Alfaro
Instituto de FısicaPontificia Universidad Catolica de Chile
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Contents
1 Simple Model for ε(ω)
2 Anomolous Dispersion and Resonant Absorption
3 Low-Frequency Behavior, Electric Conductivity
4 High-Frequency Limit, Plasma Frequency
5 Example: Liquid Water
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Simple Model for ε(ω)
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Simple Model for ε(ω)
Electron of charge −e bound by a harmonic force to an atom, and acted onby an electric field
Equation of motion:
m[~x + γ~x + ω2
0~x]
= −e~E (~x , t) (1)
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
If the field varies harmonically in time with frequency ω.
~E ∼ ~Ee−iωt (2)
~x ∼ ~x0e−iωt (3)
Dipole moment contributed by one electron:
~p = −e~x =e2
m(ω20 − ω2 − iωγ)
~E =e2
m
(ω20 − ω2) + iωγ
(ω20 − ω2)2 + γ2ω2
~E (4)
Difference of phase between the electric field and the dipole moment:
φ = tan−1
(ωγ
ω20 − ω2
)(5)
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Atomic contribution to the dielectric constant for N molecules per unitvolume, fj electrons per molecule with binding frequency ωj and dampingconstan γj :
ε(ω)
ε0= 1 + χe = 1 + N
e2
ε0m
∑j
fj(ω2
j − ω2 − iωγj)(6)
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Anomalous Dispersion and ResonantAbsorption
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Anomalous Dispersion and Resonant Absorption
In a dispersive medium, the wave equation for the electric field reads
∇2~E = εµ0∂2~E
∂t2(7)
it admits plane wave solutions
~E (z , t) = ~E0ei(kz−ωt) (8)
with the complex wave number
k =√µ0εω (9)
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
We can write the wave number in terms of this real and imaginary parts
k = β + iα
2(10)
and the electric field becomes
~E (z , t) = ~E0e−α
2ze i(βz−ωt) (11)
Intensity proportional to E 2 (to e−αz), α is called the absorptioncoefficient.
Index of refraction: n =cβ
ω
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
ε(ω)
ε0= 1 + N
e2
ε0m
∑j
fj(ω2
j − ω2 − iωγj)
The damping constants γj aregenerally small compared withthe resonant frequencies ωj .
Normal dispersion region isassociated with the increase ofRe(ε) with ω.
Anomalous dispersion (resonantabsorption) is where theimaginary part of ε isappreciable, and that representsdissipation of energy.
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Low-Frequency Behavior, ElectricConductivity
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Low-Frequency Behavior, Electric Conductivity
ε(ω)
ε0= 1 + N
e2
ε0m
∑j
fj(ω2
j − ω2 − iωγj)
Fraction f0 of electrons per molecule free (ω0 = 0)
ε(ω) = εb(ω) +Ne2
m
f0−ω2 − iωγ0
= εb(ω) + iNe2
m
f0ω(γ0 − iω)
(12)
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Maxwell-Ampere equation: ~∇× ~H = ~J + ∂D∂t
Assuming medium obbeys Ohm’s law: ~J = σ~E , ~D = εb~E
~∇× ~H = −iω(εb + i
σ
ω
)~E (13)
Assuming all the properties due the medium: ~D = ε(ω)~E
~∇× ~H = −iωε(ω)~E (14)
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
By comparison we get the electric conductivity (model of Drude)
σ =f0Ne
2
m(γ0 − iω)(15)
The dispersive properties of the medium can be attributed as well to acomplez dielectric constant, as to a frequency-dependent conductivity and adielectric constant.
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
High-Frequency Limit, PlasmaFrequency
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
High-Frequency Limit, Plasma Frequency
ε(ω)
ε0= 1 + N
e2
ε0m
∑j
fj(ω2
j − ω2 − iωγj)
At frequencies far above the highest resonant frequency
ε(ω)
ε0≈ 1−
ω2p
ω2(16)
where ωp =NZe2
ε0mis called the plasma frequency of the medium.
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
The wave number is given by
ck =√ω2 − ω2
p (17)
In dielectric media, the approximation holds only for ω2 >> ω2p.
In plasmas, the electrons are free and the damping force is negligible, sothe approximation holds over a wide range of requencies, includingω < ωp.
In conductors, the approximation holds for frequencies ω >> γ0 and thebehavior of incident waves for ω << ωp is similar to the behavior forplasma
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Example: Liquid Water
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Bibliography
1 J. Jackson, Classical Electrodynamics. Chapter 7.5
2 D. Griffiths, Indtroduction to Electrodynamics. Chapter 9.4
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Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water
Thank You
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