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Frequency Dispersion Characteristics of Dielectrics, Conductors, and Plasmas { 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 { Questions?