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1Emily Dvorak - Jackson Section 7.5 (A-C)

Frequency Dispersion Characteristics of Dielectrics, Conductors, and Plasmas

Jackson Section 7.5 A-CEmily Dvorak – SDSM&T

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Introduction Simple Model for ε(ω) Anomalous Dispersion and

Resonant Absorption Low-frequency Behavior,

Electric Conductivity Model of Drude (1900)

Section Overview

3Emily Dvorak - Jackson Section 7.5 (A-C)

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

4Emily Dvorak - Jackson Section 7.5 (A-C)

Simple Model for ε(ω)

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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

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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

7Emily Dvorak - Jackson Section 7.5 (A-C)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 pmol

fj electrons per molecule each with binding frequency ωj and damping constant γj

Oscillation strength follows sum rule Eqn.7.52

Quantum mechanical definitions of ωj γj fj give accurate description of dielectric constant

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Anomalous Dispersion and Resonant Absorption

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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

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Normal dispersion Increase in Re[ε(ω)] with ω Occurs everywhere except near resonant

frequency Anomalous dispersion

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

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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

α can be approximate when α<<β Absorption is very strong Re[ε] is negative

Intensity drops as e-αz Ratio of Im to Re is fractional decrease in intensity per

wavelength divided by 2π

Constants

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Low-frequency Behavior, Electric Conductivity

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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

Low Frequency Behavior

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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 εb

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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Model of Drude (1900)

Electric Conductivity f0N -> number of free electrons per unit volume of medium γ0/f0 -> damping constant found empirically through

experiment Example – Copper

N=8x1028 atoms/m3 At Normal Temp we achieve

σ = 5.9x107 (Ωm)-1

γo//fo = 4x1013 s-1

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

16Emily Dvorak - Jackson Section 7.5 (A-C)

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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17Emily Dvorak - Jackson Section 7.5 (A-C) Questions?