Transcript 7.5(A-C)
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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