Frequency Dispersion
Characteristics of
Dielectrics, Conductors,
and Plasmas
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Jackson Section 7.5 A-C
Emily Dvorak – SDSM&T
Introduction
 Simple Model for ε(ω)
 Anomalous Dispersion and
Resonant Absorption
 Low-frequency Behavior, Electric
Conductivity


Model of Drude (1900)
Section Overview

Previously no dispersion has been
evaluated




This can only be true when looking at limited
frequencies or in a vacuum
Earlier sections are true when looking at single
frequency
Interpret ε and μ for the individual frequency
Now we need to make simple model
dispersion for superposition of different
frequency waves
Introduction
Simple Model for ε(ω)







Extension of section 4.6
Valid for low values of density – equation 4.69 reveals deficiency
Electron bound by harmonic force acted on by electric field
Eqn 4.71
Eqn. 7.49
γ measures phenomenological damping forces
Magnetic damping force effects are neglected

Relative permeability is unity (μ->μo)
Harmonic Oscillating Fields


Approximation: Amplitude of oscillation is small enough to evaluate
the E field with the electrons average position
If E field varies harmonically in time we can write the dipole moment
-iwt

E =< x > e
iwt
Þ x = Ee
Solving for x, taking the derivative and plugging into eqn. 7.49 reveals
-eE = m[-w -iwg + w ]Ee
2

2
o
iwt
Finally solve for the exponential and plug into equation for x which
when used in equation 4.72
e
2
2
-1
p = -ex = (wo - w - iwg ) E
m
2
Dipole Moment





To determine the dielectric constant of the medium we need to combine
equations 4.28 and 4.36
P
1
Summing over the medium with
c
=
=
N
<
p
>
e
j
j
N molecules and Z electron per
e
E eo E j
o
molecule, all with dipole moment
pmol
1
fj electrons per molecule each
NZpmol
e =
with binding frequency ωj and
oE
damping constant γj
2
NZ e
Oscillation strength follows sum rule
2
2
-1
=
(
i
)
o
 Eqn.7.52
Sj fj = Z
o m
Quantum mechanical definitions of ωj γj fj give accurate description of
dielectric constant
å
c
e
e
w
w
wg
e (w )
Ne2
=1+
S j f j (w 2j - w 2 - iwg j )-1
eo
eo m
Dielectric Constants
Anomalous Dispersion
and Resonant Absorption

ε is approx. real for most
frequencies





γj is very small compared
to binding or resonant
frequencies (ωj)
The factor (ω2j-ω2)-1
negative or positive
At low ωj all terms in sum
contribute to positive ε
greater than unity
In the neighborhood of ωj
there is violent behavior
Denominator become
purely imaginary
Resonant
Frequencies

Normal dispersion



Anomalous dispersion



Increase in Re[ε(ω)] with ω
Occurs everywhere except near resonant
frequency
Decrease in Re[ε(ω)] with ω
Im part very appreciable
Resonant absorption


Large imaginary contribution
Positive Im[ε(ω)] part represents energy
dissipation from EM into medium
Dispersion Types
and Absorption

Wave number k, Im and Re part describe attenuation

α is attenuation constant or absorption coefficient
Connection between α and β comes from eqn 7.5

w
k




c
me
e
=c o o =c o
n
me
e
b2 -
α can be approximate when


=
α<<β
Absorption is very strong
Re[ε] is negative
b = Re[e / eo ]
a2
4
k = b +i
=
w2
c2
e
eo
Re[ ]
w2
w
e
ba = 2 Im[ ]
c
eo
Intensity drops as e-αz
c
Ratio of Im to Re is fractional decrease in intensity per wavelength
divided by 2π
Constants
Im[e (w )]
a»
b
Re[e (w )]
a
2
Low-frequency Behavior,
Electric Conductivity





As ω approaches zero the medium is qualitatively different
Insulators – lowest resonant frequency is non zero
When ω=0 the molecular polarizability is given by 4.73, see
7.51 lim as ω->0
This situation was discussed in section 4.6
Fo – fraction of free electrons in molecule




Free meaning ω0 = 0
Singular dielectric constant at ω = 0
Separately adding contribution from free electrons times εo
εb contribution of all dipoles
Ne fo
e (w ) = eb (w ) + i
mw (g o - iw )
2
Low Frequency Behavior

Use Maxwell – Ampere’s law to examine singular behavior
along with Ohm’s law
¶D
Ñ´H = J +
¶t




J =sE
Recall the field’s harmonic time dependence
“normal” dielectric constant εb
Plugging it all in we see
D = eb E µ eb e
s
Ñ ´ H = -iw (eb + i )E
w
-iwt
We can determine conductivity if we don’t explicitly use ohms
law but compare to dielectric constant ε(ω)
Conductivity
fo Ne
s=
m(g o - iw )
2





Electric Conductivity
f0N -> number of free electrons per unit volume of medium
γ0/f0 -> damping constant found empirically through experiment
Example – Copper
2
 N=8x1028 atoms/m3
o
At Normal Temp we achieve





σ = 5.9x107 (Ωm)-1
γo//fo = 4x1013 s-1
f Ne
s=
m(g o - iw )
Assuming f0~1 we see frequencies above the microwave range ω <
1011 s-1
Thus all metal conductivities are Real and independent of
frequency
At frequencies higher than infrared conductivity is complex and
evaluated through eqn. 7.58
Model of Drude (1900)



Conductivity is is quantum mechanical with a
heavy influence from Pauli principle
Dielectrics have free electrons or more commonly
the valence electrons
Damping comes from the valence electrons
colliding and transferring momentum


Usually from lattice structure, imperfections and
impurities
Basically dielectrics and conductors are no
different from each other when frequencies a lot
larger than zero
Quantum Connection
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