Frequency Dispersion Characteristics of Dielectrics, Conduc - PowerPoint Presentation

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Frequency Dispersion Characteristics of Dielectrics, Conductors, and Plasmas

Jackson Section 7.5 A-CEmily Dvorak – SDSM&T

Emily Dvorak - Jackson Section 7.5 (A-C)

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Introduction

Simple

Model for ε(

ω

)

Anomalous Dispersion and Resonant AbsorptionLow-frequency Behavior, Electric ConductivityModel of Drude (1900)

Section Overview

Emily Dvorak - Jackson Section 7.5 (A-C)

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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 frequencyNow we need to make simple model dispersion for superposition of different frequency waves

Introduction

Emily Dvorak - Jackson Section 7.5 (A-C)

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Simple Model for ε(

ω)Emily Dvorak - Jackson Section 7.5 (A-C)

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Extension of section 4.6

Valid for low values of density – equation 4.69 reveals deficiencyElectron bound by harmonic force acted on by electric field

Eqn 4.71

Eqn. 7.49

γ

measures phenomenological damping forcesMagnetic damping force effects are neglectedRelative permeability is unity (μ->μ

o)

Harmonic Oscillating Fields

Emily Dvorak - Jackson Section 7.5 (A-C)

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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

Solving for x, taking the derivative and plugging into eqn. 7.49 reveals

Finally solve for the exponential and plug into equation for x which when used in equation 4.72

Dipole Moment

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Emily Dvorak - Jackson Section 7.5 (A-C)Slide7

Dielectric Constants

To determine the dielectric constant of the medium we need to combine equations

4.28 and

4.36

Summing over the medium with N

molecules

and

Z

electron

per molecule, all with dipole moment

p

mol

f

j

electrons per molecule

each with

binding frequency

ω

j

and

damping

constant

γ

jOscillation strength follows sum ruleEqn.7.52

Quantum mechanical definitions of ωj γj f

j

give accurate description of dielectric constant

Emily Dvorak - Jackson Section 7.5 (A-C)

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Anomalous Dispersion and Resonant Absorption

Emily Dvorak - Jackson Section 7.5 (A-C)

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ε

is approx. real for most frequencies

γj

is very small compared to binding or resonant frequencies (

ω

j)The factor (ω2j-ω2)-1 negative or positive

At low ωj all terms in sum contribute to positive

ε greater than unityIn the neighborhood of

ω

j

there is violent behavior

Denominator become purely imaginary

Resonant

Frequencies

Emily Dvorak - Jackson Section 7.5 (A-C)

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Normal dispersion

Increase in Re[

ε(ω)] with

ω

Occurs everywhere except near resonant frequency

Anomalous dispersionDecrease in Re[ε(ω)] with ωIm part very appreciable

Resonant absorptionLarge imaginary contributionPositive Im

[ε(ω)] part represents energy dissipation from EM into medium

Dispersion Types

and

Absorption

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Emily Dvorak - Jackson Section 7.5 (A-C)Slide11

Wave number k,

Im

and Re part describe attenuationα

is attenuation constant or absorption coefficient

Connection

between α and β comes from eqn 7.5α can be approximate whenα

<<βAbsorption is very strongRe[ε

] is negativeIntensity drops as e-αz

Ratio

of

Im

to Re is fractional decrease in intensity per wavelength divided by 2π

Constants

Emily Dvorak - Jackson Section 7.5 (A-C

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Low-frequency Behavior, Electric Conductivity

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As

ω

approaches zero the medium is qualitatively differentInsulators – lowest resonant frequency is non zero

When

ω

=0 the molecular polarizability is given by 4.73, see 7.51 lim as ω->0This situation was discussed in section 4.6

Fo – fraction of free electrons in moleculeFree meaning ω

0 = 0 Singular dielectric constant at ω = 0

Separately adding contribution from free electrons times

ε

o

ε

b

contribution of all dipoles

Low Frequency Behavior

Emily Dvorak - Jackson Section 7.5 (A-C)

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Use Maxwell – Ampere’s law to examine singular

behavior along with Ohm’s law

Recall the field’s

harmonic time

dependence

“

normal” dielectric constant

ε

b

Plugging it all in we see

We can determine conductivity if we don

’

t explicitly use ohms law but compare to dielectric constant

ε

(

ω

)

Conductivity

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Emily Dvorak - Jackson Section 7.5 (A-C)Slide15

Model of

Drude

(1900)

Emily Dvorak - Jackson Section 7.5 (A-C)

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Electric Conductivity

f0N -> number of free electrons per unit volume of medium

γ0/f0 -> damping constant found empirically through experiment

Example – Copper

N=8x10

28

atoms/m

3

At Normal Temp we achieve

σ

= 5.9x10

7

(

Ωm

)

-1

γo//

fo = 4x1013

s-1Assuming f0~1 we see frequencies above the microwave range ω < 1011 s

-1Thus all metal conductivities are Real and independent of frequencyAt frequencies higher than infrared conductivity is complex and evaluated through eqn. 7.58Slide16

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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Questions?

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