1. Jackson Section 7.5 A-C Emily Dvorak – SDSM&T
Frequency Dispersion Characteristics of Dielectrics, Conductors, and Plasmas
Jackson Section 7.5 A-C
Emily Dvorak – SDSM&T
Emily Dvorak - Jackson Section 7.5 (A-C)

2. Section Overview Introduction Simple Model for ε(ω)
Anomalous Dispersion and Resonant Absorption
Low-frequency Behavior, Electric Conductivity
Model of Drude (1900)
Section Overview
Emily Dvorak - Jackson Section 7.5 (A-C)

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

4. Simple Model for ε(ω)
Emily Dvorak - Jackson Section 7.5 (A-C)

5. Harmonic Oscillating Fields
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
Emily Dvorak - Jackson Section 7.5 (A-C)

6. 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 reveals
Finally solve for the exponential and plug into equation for x which when used in equation 4.72
Dipole Moment
Emily Dvorak - Jackson Section 7.5 (A-C)

7. 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
Dielectric Constants
Emily Dvorak - Jackson Section 7.5 (A-C)

8. Anomalous Dispersion and Resonant Absorption
Emily Dvorak - Jackson Section 7.5 (A-C)

9. Resonant Frequencies 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
Emily Dvorak - Jackson Section 7.5 (A-C)

10. Dispersion Types and Absorption
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
Emily Dvorak - Jackson Section 7.5 (A-C)

11. Constants 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
Emily Dvorak - Jackson Section 7.5 (A-C

12. Low-frequency Behavior, Electric Conductivity
Emily Dvorak - Jackson Section 7.5 (A-C)

13. Low Frequency Behavior
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
Emily Dvorak - Jackson Section 7.5 (A-C)

14. 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
Emily Dvorak - Jackson Section 7.5 (A-C)

15. 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
Model of Drude (1900)
Emily Dvorak - Jackson Section 7.5 (A-C)

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

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