Dispersion. February 12, 2014

Transcription

1 Dispersion February 1, 014 In aterials, the dielectric constant and pereability are actually frequency dependent. This does not affect our results for single frequency odes, but when we have a superposition of frequencies it leads to dispersion. We begin with a siple odel for this behavior. The variation of the pereability is often quite weak, and we ay take µ = µ 0. Frequency dependence of the perittivity Recall that our earlier treatent of the dielectric constant began with electric dipole oent produced by a static electric field E in the presence of a linear restoring force, F = 0x, to produce a olecular polarization p ol = ex = ee 0 Then, if there are N olecules per unit volue with Z electrons per olecule, the dipole oent per unit volue is P = NZp ol χ e E so that Next, using χ e = NZe 0 D = E + P E = E + χ e E the dielectric constant is = 1 + χ e = 1 + NZe 0 This result changes when there is tie dependence to the electric field, with the dielectric constant showing frequency dependence. Suppose the aterial is sufficiently diffuse that the applied electric field is about equal to the electric field at each ato, and that the response of the atoic electrons ay be odeled as haronic. Again let x represent the displaceent of the charge fro equilibriu, and include daping, so that now we have [ ẍ + γẋ + 0x ] = ee x, t) 1

2 We assue everything here negligible agnetic effects, low-aplitude oscillations. enough to give iportant general features. Let the electric field vary haronically, E = E x) e it then the position of the electron with have the sae tie dependence, so and the electric dipole oent is [ iγ + 0] x = ee x) p = ex = e E 0 iγ) The odel is still Let there be N olecules per unit volue with Z electrons per olecule, with a fraction f i of the electrons having binding frequency i0 and daping γ i. This is reasonable since the different electrons in each olcule are bound differently to the nucleus. The total of all the f i should be the total nuber of electrons, i f i = Z. The dipole oent for each olecule is then p ol = i f i e i0 iγ i ) E Then, since the total dipole oent per unit volue is P = Np ol = χ e E, we have i Nf i e i0 iγ i ) E = χ e E Now, using D = E = E + P = E + χ e E, the dielectric constant is = 1 + χ e = 1 + N f i e i0 iγ i ) = 1 + Ne f i i0 iγ i This expression is accurate if f i, i0 and γ i are found quantu echanically. Anoolous dispersion and resonant absorption The frequency dependence of the dielectric constant has certain regular properties. iaginary parts, we have = + Ne = + Ne Re = + Ne I = Ne f i i0 iγ i ) f i i0 + iγ i i0 ) + γ i f i i0 ) i0 ) + γi f i γ i i0 ) + γ i Separating real and

3 and we note that the daping constant γ i is usually sall. At low frequencies, < i0 for all i, each ter in the real part of is positive and >. As the frequency increases, ore and ore of the ters becoe negative, until at high frequency, Re = Ne f i i0 i0 ) + γi and <. The real and iaginary parts of have peaks whenever is near one of the resonant frequencies, i0, of the olecule. At these frequencies, the corresponding ter of the denoinator becoes i0 ) + γ i γ i which is very sall but larger the higher the driving frequency). This produces a resonance peak in the iaginary part of, but since the ter is also changing sign in the real part, the peak is double positive for < i0 and negative once > i0. At these peaks, the aterial is absorbing energy fro the field the daping γ i plays an iportant role. The effect is easily seen fro the wave vector k. Suppose we have a plane wave travelling in the z direction, E = Ee ikz t), which passes near a resonant ato. Then k = µ Re + ii = µ 0 0 Re + ii Then, setting = c ε 0 k β + 1 iα we have β + 1 iα = c β + iαβ 1 4 α = c Re + ii ε 0 Re + ii ) so that and the electric field is given by β 1 4 α = c Re αβ = c I E = Ee ikz t) = Ee 1 αz e iβz t) The intensity of the wave, which varies as E then falls off as e αz 3

4 and α is called the attenuation constant. We see that the attenuation is larger near each resonant frequency, i0, where we expect the radiation to be exciting the electrons of the olecule. Relating this back to the resonance behavior, with α β, β β 1 4 α = c Re α = c β I I c Re Suppose, for siplicity, we have only a single resonance. Then with 0 so the attenuation constant is approxiately Re = 1 + Ne 0 0 ) + γ 1 I = Ne γ 0 ) + γ Ne γ α = Conductivity and low frequency behavior Ne c γ In a conductor there are free electrons. Since these have no restoring force, they ay be thought of as having a resonance frequency of zero. This akes the response of conductors at low frequency very different fro that of insulators. For insulators, the lowest resonant frequency is 10 > 0 and the equations above give a good approxiation to the actual response. For conductors, there is a very strong response at zero frequency. Consider the fraction of free electrons, f 0, separately, writing = + Ne = + Ne f i i0 iγ i f i i0 + Ne f 0 iγ i iγ 0 i 0 Ne f 0 = b + i γ 0 i) To see what is happening in ters of conduction, consider the Maxwell equation involving current, H D t = J We copare two different ways of handling this expression. First, let b be real so that if there are any other resonances they are far enough away in frequency to be negligible), and assue the ediu satisfies Oh s law, J = σe 4

5 Then, with the haronic applied field and a real dielectric constant, b, we would have H be) t = σe H + i b E = σe H = σ i b ) E = i b + iσ ) E Now copare this to the result without current but with the conduction electrons included in the dielectric constant. Then we have instead H D = 0 t H = ie Ne ) f 0 = i b + i E γ 0 i) Coparing the two expressions, we see that the effect of the zero-frequency resonance is the sae as a conductivity σ given by iσ Ne f 0 = i γ 0 i) σ = Ne f 0 γ 0 i) a result produced in 1900 by Drude. It requires substantial correction because the free electrons actually for a Feri gas. High frequency We expand our expression for the dielectric constant for i0 for all i. The denoinator is approxiately i0 iγ i and we have = + Ne Ne fi f i i0 iγ i Replacing the su over fractions by the atoic nuber, f i = Z, where = 1 NZe = 1 p p = NZe is called the plasa frequency. The plasa frequency depends only on NZ, the total nuber of electrons in the syste. Cobining with the wave nuber k, we have k = µε 5

6 k = 1 c = c 1 p ) so that = k c + p This provides a dispersion relation, k), for plasas. We will explore ore about dispersion relations soon. Exaple: Water Jackson has collected all available data to the date of first publication, anyway) on the index of refraction and attenuation coefficient of water as functions of frequency. The graphs on page 315 are fascinating! Take soe tie to digest what is going on. The window in the visible range is particularly interesting. 6