Frequency dependence of the permittivity

Transcription

1 Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but when we have a superposton of frequences t leads to dsperson. We begn wth a smple model for ths behavor. The varaton of the permeablty s often qute weak, and we may take µ = µ 0. 1 Frequency dependence of the permttvty 1.1 Permttvty produced by a statc feld The electrostatc treatment of the delectrc constant begns wth the electrc dpole moment produced by an electron n a statc electrc feld E. The electron experences a lnear restorng force, F = mω 0x, ee = mω 0x where ω 0 characterzes the strength of the atom s restorng potental. The resultng dsplacement of charge, x = ee, produces a molecular polarzaton mω0 p mol = ex = ee mω 0 Then, f there are N molecules per unt volume wth Z electrons per molecule, the dpole moment per unt volume s P = NZp mol ɛ 0 χ e E so that Next, usng ɛ 0 χ e = NZe mω 0 D = ɛ 0 E + P ɛe = ɛ 0 E + ɛ 0 χ e E the relatve delectrc constant s ɛ = ɛ ɛ 0 = 1 + χ e = 1 + NZe mω 0 ɛ 0 Ths result changes when there s tme dependence to the electrc feld, wth the delectrc constant showng frequency dependence. 1

2 1. Permttvty n the presence of an oscllatng electrc feld Suppose the materal s suffcently dffuse that the appled electrc feld s about equal to the electrc feld at each atom, and that the response of the atomc electrons may be modeled as harmonc. Agan let x represent the dsplacement of the charge from equlbrum, and nclude dampng, so that now we have m [ ẍ + γẋ + ω 0x ] = ee (x, t) In addton to the lnear response of the atom, we assume neglgble magnetc effects and low-ampltude oscllatons. The model s stll enough to gve mportant general features. Let the electrc feld vary harmoncally, E = E (x) e ωt then the poston of the electron wll have the same tme dependence, x (t) = xe ωt, so and the electrc dpole moment s m [ ω ωγ + ω 0] x = ee (x) p = ex = e E m (ω0 ω ωγ) Let there be N molecules per unt volume wth Z electrons per molecule, wth a fracton f of the electrons havng bndng frequency ω 0 and dampng γ. Ths s reasonable snce the dfferent electrons n each molcule are bound dfferently to the nucleus. The total of all the f should be the total number of electrons, f = Z. The dpole moment for each molecule s then p mol = f e m (ω 0 ω ωγ ) E Then, snce the total dpole moment per unt volume s P = Np mol = ɛ 0 χ e E, we have Nf e m (ω 0 ω ωγ ) E = ɛ 0χ e E Now, usng D = ɛe = ɛ 0 E + P = ɛ 0 E + ɛ 0 χ e E, the relatve delectrc constant s ɛ = ɛ ɛ 0 = 1 + χ e = 1 + N f e ɛ 0 m (ω0 ω ωγ ) = 1 + Ne f ɛ 0 m ω0 ω ωγ Ths expresson s accurate f f, ω 0 and γ are found quantum mechancally. 1.3 Anomolous dsperson and resonant absorpton The frequency dependence of the delectrc constant has certan regular propertes. magnary parts, ɛ = 1 + Ne f ω0 ω ωγ ( ) = 1 + Ne f ω 0 ω + ωγ (ω0 ω ) + ω γ Separatng real and

3 we have Re ɛ = 1 + Ne ( f ω 0 ω ) (ω0 ω ) + ω γ Im ɛ = Ne ω f γ (ω0 ω ) + ω γ and we note that the dampng constant γ s usually small. At low frequences, ω < ω 0 for all, so each term n the real part of ɛ s postve and ɛ > 1. As the frequency ncreases, more and more of the terms become negatve, untl at hgh frequency, Re ɛ = 1 Ne f ω ω 0 (ω0 ω ) + ω γ and ɛ < 1. The real and magnary parts of ɛ have peaks whenever ω s near one of the resonant frequences, ω 0, of the molecule. At these frequences, the correspondng term of the denomnator becomes ( ω 0 ω ) + ω γ ω γ whch s very small (but larger the hgher the drvng frequency). Snce the resonant real part s changng sgn, the peak s double, spkng postve for ω < ω 0 and then spkng negatve once ω > ω 0. At the same tme, the magnary part of ɛ also has a peak, causng the materal to absorb energy from the feld. Clearly, the dampng γ plays an mportant role, governng the magntude of the resonances. These effects are easly seen from the wave vector k. Suppose we have a plane wave travellng n the z drecton, E = Ee (kz ωt), whch passes near a resonant atom. Then settng µ 1, k = ω µɛ We need the real and magnary parts of k, so set = ω Re ɛ + Im ɛ k β + 1 α The factor of 1 s helpful below. We have β + 1 α = ω Re ɛ + Im ɛ β + αβ 1 4 α = ω (Re ɛ + Im ɛ) so that β 1 4 α = ω Re ɛ αβ = ω Im ɛ and the electrc feld s gven by E = Ee (kz ωt) = Ee 1 αz e (βz ωt) 3

4 The ntensty of the wave, whch vares as E then falls off as e αz and α s called the attenuaton constant. We see that the attenuaton s larger near each resonant frequency, ω 0, where we expect the radaton to be exctng the electrons of the molecule. Relatng ths back to the resonance behavor, wth α β, β β 1 4 α = ω Re ɛ α = ω β Im ɛ ω Im ɛ Re ɛ Suppose, for smplcty, we have only a sngle resonance. Then wth ω ω 0 so the attenuaton constant s approxmately Re ɛ = 1 + Ne ω0 ω (ω0 ω ) + ω γ 1 Im ɛ = Ne ω γ (ω0 ω ) + ω γ Ne ωγ Low frequences: conductvty α = Ne γ In a conductor there are free electrons. Snce these have no restorng force, they may be thought of as havng a resonance frequency of zero. Ths makes the response of conductors at low frequency very dfferent from that of nsulators. For nsulators, the lowest resonant frequency s ω 10 > 0 and the equatons above gve a good approxmaton to the actual response. For conductors, there s a very strong response at zero frequency. By ncludng a zero frequency mode n the delectrc constant, we can derve Ohm s law, J = σe, as an effect of the resultng magnary part of the delectrc constant. Separate out the fracton of free electrons, f 0, n our expresson for the relatve delectrc constant, ɛ = 1 + Ne n =0 f ω 0 ω ωγ = 1 + Ne f 0 ω ωγ 0 + Ne Ne f 0 = ɛ b + ω (γ 0 ω) n =1 f ω 0 ω ωγ where we defne the background delectrc constant, ɛ b 1+ Ne n f =1 ω0 ω ωγ. To see what s happenng n terms of conducton, consder the Maxwell equaton nvolvng current, H D t = J 4

5 Applyng ths for harmonc felds and a real delectrc constant, ɛ b, and no current gves Now nclude the zero mode of the delectrc constant, or, defnng the conductvty as we have ( k H + ω 1 H (ɛ be) ɛ 0 t = 0 k H + ωɛ b E = 0 ɛ b + σ 1 H (ɛe) = 0 ɛ 0 t Ne ) f 0 E = 0 ω (γ 0 ω) Ne f 0 (γ 0 ω) k H + ωɛ b E = σe Comparng ths to the full harmonc form of Ampère s law, we have derved the form of Ohm s law, k H + ωɛ b E = J J = σe where σ s no longer a phenomenologcal constant, but arses from zero modes of the materal. Our expresson for the conductvty, σ = Ne f 0 m (γ 0 ω) was frst produced by Drude n It requres substantal correcton because the free electrons actually form a Ferm gas. 3 Hgh frequences: The plasma frequency Now we expand our expresson for the delectrc constant for ω ω 0 for all. The denomnator s approxmately ω 0 ω ωγ ω for all modes and we have ɛ = 1 + Ne 1 Ne mω ɛ 0 f f ω 0 ω ωγ Replacng the sum over fractons by the atomc number, f = Z, ɛ = 1 NZe mω ɛ 0 = 1 ω p ω 5

6 where ω p NZe s called the plasma frequency. The plasma frequency depends only on NZ, the total number of electrons n the system. Combnng wth the wave number k, we have so that k = µɛ ω c k = 1 c ω ɛ ( ) = ω c 1 ω p ω ω = k c + ω p Ths provdes a dsperson relaton, ω (k), for plasmas. We wll explore more about dsperson relatons soon. 3.1 Example: Water Jackson has collected all avalable data (to the date of frst publcaton, anyway) on the ndex of refracton and attenuaton coeffcent of water as functons of frequency. The graphs on page 315 are fascnatng! Take some tme to dgest what s gong on. The wndow n the vsble range s partcularly nterestng. 6