7.5 Dispersion Characteristics of Dielectrics, Conductors and Plasmas.

7.5.1

a. In a dispersive medium, ε and μ are functions of the angular frequency ω. Hence n = \sqrt{εμ} is also a function of ω. For simplicity, we assume that μ = 1 to a good approximation and concentrate on ε(ω), but much of the discussion holds for μ(ω) and n(ω) as well.

b. The important results will be:

(i) ε(ω) is in general complex and is an analytic function of ω for Im (ω) > 0 at thermal equilibrium;

(ii) Im ε can be identified with 4πσ/ω, where σ is the real, frequency dependent conductivity, which is positive for a medium in thermal equilibrium. Im ε → 0 for ω → ∞.

(iii) Re ε ≥ 1 for ω = 0 and Re ε → 1 for ω → ∞, but in general Re ε is not necessarily positive. The conditions Re ε < 0 (for ω large enough), Im ε ≪ |Re ε| are characteristic of a plasma.

(iv) The fundamental solutions for the fields in a medium are proportional to exp(i(k · x−ωt)), same as in vacuum, but k is in general complex and its modulus is given by k = nω/c (or k^{2} = εω^{2}/c^{2} for μ = 1).

c. Two further generalizations are sometimes needed:

(i) If the medium is anisotropic, the relation between D and E is still linear, but D is not always in the same direction as E. The general relation is D_{p} = ∑_{q} ε_{pq}E_{q}, where ε_{pq} is the dielectric tensor. For an isotropic medium, ε_{pq} = εδ_{pq.}

(ii) If the field varies rapidly compared to some characteristic length of the medium, the relation between D and E can be non-local: for an isotropic medium,

D(r) = ∫ d^{3}r′ε(r, r′)E(r′).

When the relation is local, ε(r, r′) = εδ(r − r′). More generally, for a homogeneous medium, ε(r, r′) depends only on r − r′.

7.5.2

a. The polarizability of a molecule in a uniform E field of frequency ω is defined by e ⟨X_{p}⟩ = ∑_{q} χ_{pq}E_{p}, where X_{p} (for p = 1, 2, 3) is the sum of all coordinates x_{np} of the electrons in the molecule. The general quantum formula for χ_{pq} is

χ_{pq} = −\frac{e^{2}}{h̄} ∑_{ij} ρ_{i} [\frac{⟨i | X_{p} | j⟩ ⟨j | x_{q} | i⟩}{ω − ω_{ij} + iη/2} − \frac{⟨i | X_{q} | j⟩ ⟨j | x_{p} | i⟩}{ω + ω_{ij} + iη/2}] ,  (5.1)

where ρ_{i} is the occupation probability of the initial state |i⟩ , h̄ω_{ij} = E_{j} − E_{i} is a transition energy and η → 0^{+}.

b. If the initial distribution of atomic states |i⟩ is isotropic, as it usually is in the absence of static external fields, χ_{pq} = χδ_{pq} and (5.2) below holds for any component of x.

c. More generally, χ_{pq} may be reduced to a diagonal form by a change to new axes (see for instance 7.6.2b); if x is in the direction of one such axis, the corresponding diagonal element of the polarizability tensor is

χ = \frac{e^{2}}{m} ∑_{ij} ρ_{i} \frac{f_{ij}}{ω_{ij}^{2} − ω^{2} − iωη} ,  (5.2)

where f_{ij} is the oscillator strength: f_{ij} = ω_{ij} |⟨i | x | j⟩|^{2} 2m/h̄.

d. The dipole sum rule holds:

∑_{j} f_{ij} = Z,

where Z is the number of electrons in the molecule. In the optical range, the deep-lying electrons do not contribute appreciably to the polarizability; then one has an approximate partial sum rule, where Z is the number of “polarizable” electrons (typically, these are the “valence” electrons).

7.5.3

a. The imaginary part of χ arises when ω is resonant with some ω_{ij}. Explicitly

ω Im χ = \frac{e^{2}}{m} ∑_{ij} ρ_{i}f_{ij}π [δ(ω − ω_{ij}) − δ(ω + ω_{ij})] .  (5.3)

Note that Im χ(−ω) = − Im χ(ω). In fact, χ(−ω) = χ^{∗}(ω).

b. If the system is initially in the ground state, only the term i = 0 remains, with ρ_{0} = 1. In this case the oscillator strengths f_{0j} and the frequencies ω_{0j} are non-negative and ω Im χ(ω) ≥ 0.

c. It can be shown that ω Im χ ≥ 0 is also true if ρ_{i} is the thermal equilibrium distribution. However if the system has been “pumped” to an excited state, as in a laser, ω Im χ(ω) can be negative. Generally, ω Im χ < 0 means that the system is unstable with respect to the perturbation caused by an electric field of frequency ω.

d. If there are N non-interacting molecules per unit volume, the dielectric constant is given by (writing χ^{mol} instead of just χ):

ε = 1 + 4πNχ^{mol},  (5.4)

provided that local field effects can be neglected.

e. In truth, the interactions between molecules cause the states |j⟩ to have finite lifetimes. Even if these interactions are negligible, the excited states can decay back to |0⟩ or to some lower state by emitting a photon, and thus have a finite lifetime and an associated width (natural width or radiative width).

7.5.4

a. The treatment of finite lifetime effects is rather involved and depends on the system under consideration. A simple way out is suggested by the formal identity of (5.2) with the response of a set of harmonic oscillators (classical or quantum mechanical) of frequency ω_{ij} and infinitesimal damping η: finite lifetime effects can be included by giving each effective harmonic oscillator a finite damping constant γ_{ij} > 0.

b. Taking for simplicity a system in its ground state, so that ρ_{0} = 1 and all other ρ_{i}’s vanish, we have then

ε = 1 + 4πN \frac{e^{2}}{m} ∑_{j} \frac{f_{j}}{ω_{j}^{2} − ω^{2} − iωγ_{j}}  (5.5)

with f_{j} = f_{0j} , ω_{j} = ω_{0j} and γ_{j} = γ_{0j} . It is still true that ∑_{j} f_{j} = Z.

c. If the frequencies ω_{j} form a continuum, or if they are spread apart by less than γ_{j}, the precise value of γ_{j} does not matter as long as γ_{j} ≪ ω. For ω near an isolated frequency ω_{j} however, the behavior of Re ε and Im ε depends strongly on γ_{j} as shown in the figures, although the area under the peak of Im ε, or more precisely ∫ ω Im ε dω, does not depends on γ_{j}. (See end of this section for figures).

d. For interacting molecules, as in a metal or a covalent solid, χ^{mol} is meaningless. One writes then ε = 1 + 4πχ^{tot}, where the (total) electric susceptibility χ^{tot} is still given by formulae (5.1) and (5.2) of 7.5.2. Now however |i⟩ are eigenstates of a unit volume of the material. For an assembly of N molecules, χ^{tot} = Nχ^{mol} and (5.4) is recovered.

7.5.5

a. It is remarkable that the simple formula (5.5) describes also a system containing “free” electrons, as are found in an ionized plasma. It also describes “itinerant” electrons, as are found in the conduction band of a metal or semiconductor. (Itinerant electrons behave effectively as free electrons with an effective mass m^{∗}.) In these cases the lowest frequency ω_{j} in the sum is set equal to zero. It is also customary to write the corresponding product Nf/m in the form N_{c}/m^{∗}, where N_{c} is the number of conducting electrons per unit volume and m^{∗} is their effective mass; one also puts γ = 1/τ, where the relaxation time τ is related to the mean velocity v and to the mean free path l by v = l/τ. With these notations (5.5) becomes

ε(ω) = ε_{b}(ω) + i\frac{4πN_{c}e^{2}}{m^{∗}ω} \frac{τ}{1 − iωτ} ,  (5.6)

where ε_{b} is the contribution of the “localized” or “bound” electrons.

b. It is perhaps more natural to describe the conducting electrons by a “free current” term in the Maxwell-Ampère equation

∇ × H = \frac{4π}{c} J+\frac{1}{c} \frac{∂D}{∂t}  (5.7)

with J related to E by Ohm’s law, J = σE, while the “bound” electrons are described by a polarization P = χ_{b}E, leading to D = (1 + 4πχ_{b})E = ε_{b}E. Thus

∇ × H = −\frac{iω}{c} (ε_{b} + i\frac{4πσ}{ω} ) E  (5.8)

c. Comparing (5.8) and (5.6) we find that the conductivity σ is given by Drude’s formula

σ = \frac{N_{c}e^{2}}{m^{∗}} \frac{τ}{1 − iωτ} .  (5.9)

There can be several groups of electrons (or holes), each with its m^{∗} and τ.

7.5.6

a. Drude’s formula is obtained more directly by considering the effect of E on N_{c} particles of charge e and mass m^{∗} in the presence of a frictional force: the velocity v of each particle satisfies m^{∗} [(dv/dt) + (v/τ)] = eE and the current is J = N_{c}ev. The most refined quantum statistical treatments yield a formula of the same type and in addition relate τ (which may depend, usually weakly, on ω) to the microscopic properties of the medium and to the temperature.

b. The separation between “free” and “bound” electrons is neither physically clearcut in general, nor mathematically convenient. Since a polarization P gives rise to a current J = dP/dt = −iωP, a single complex function, either χ^{tot}(ω) or σ(ω), describes the response of the system. Quite simply, σ = −iωχ. (From here on χ means χ^{tot}).

c. If the response of the system is purely resistive (current in phase with the field, leading to power dissipation) then σ is real and is the more natural function to use; if the response is purely reactive, χ is real and ε = 1 + 4πχ is the natural choice.

d. If the d.c. conductivity N_{c}e^{2}τ/m^{∗} (see(5.9)) is finite, χ and ε diverge as ω^{−1} as ω → 0 (see(5.6)). Clearly, σ is more convenient to use in this limit. It is also true that σ → 0 for ω → ∞, while ε → 1. These properties of σ make it easier to write dispersion relations (see 7.10). Note also that the stability condition ω Im χ > 0, or ω Im ε > 0, translates into Re σ > 0. The meaning of this is clear since the power loss is given by \frac{1}{2} Re σ |E|^{2} .

7.5.7

a. For ω larger than all the frequencies ω_{j} corresponding to significant oscillator strengths f_{j}, the dielectric constant (5.5) takes the form

ε = 1 − \begin{matrix}ω & 2 p \\ ω & 2 \end{matrix} ,  (5.10)

where ω_{p} is the plasma frequency: ω_{p}^{2} = 4πNZe^{2}/m.

b. In an ionized plasma, the formula

ε = 1 − \frac{4πN_{c}e^{2}}{mω} \frac{1}{ω − i/τ}  (5.11)

holds down to frequencies of order τ^{−1} . In the laboratory, N_{c} is 10^{12} − 10^{16} /cm^{3} and ω_{p} = 6 × 10^{10} − 6 × 10^{12} sec^{−1}, while τ^{−1} ≃ 10^{8} −10^{10} sec^{−1}

c. In a metal, N_{c} is about 10^{23} cm^{−3}, so that 4πN_{c}e^{2}/m^{∗} is about 10^{15} sec^{−1}, while τ^{−1} is typically 10^{12} sec^{−1} . Thus, over the optical range and the u. v.

ε(ω) ≃ ε_{b}(ω) − \frac{4πN_{c}e^{2}}{m^{∗}ω^{2}} .  (5.12)

This holds also for a doped semiconductor, where N_{c} is much lower (down to 10^{16} cm^{−3}).

d. The striking feature of (5.10) and (5.11) is that ε(ω) becomes negative below a cutoff frequency, which for (5.10) is ω_{p} and for (5.11) is [4πN_{c}e^{2}/m^{∗}ε_{b}]^{1/2} = ω_{p}^{∗}.

e. Negative ε implies that n = \sqrt{ε} is purely imaginary, and so is k = nω/c. Then waves do not propagate far in the medium, but decay exponentially, according to

E = E_{0} exp(−ω_{p}ξ/c − iωt), for ω ≪ ω_{p}.

7.5.8

a. In general, a complex ε implies a complex index of refraction n = \sqrt{ε}. Following Jackson, we will use n for the real part of \sqrt{ε} and introduce the attenuation constant α by

k = \sqrt{ε} \frac{ω}{c} = n\frac{ω}{c} + i\frac{α}{2},  (5.13)

or

n^{2} − (αc/2ω)^{2} = Re ε,  (5.14)
nα = (ω/c) Im ε.  (5.15)

b. Three limiting cases are of interest:

(i) ε nearly real and positive: then n ≃ \sqrt{Re ε} and α ≃ (ω/nc) Im ε; k is nearly real;

(ii) ε nearly real and negative: then α ≃ (2ω/c) \sqrt{− Re ε} and n = Im ε /2\sqrt{− Re ε}; k is nearly imaginary;

(iii) ε nearly imaginary: then n(ω/c) ≃ α/2 and k is of the form (α/2) (1 + i) . One often sets α/2 = δ^{−1}.

c. Case (i) corresponds to “ordinary” dielectrics and case (ii) corresponds to plasmas and metals for ω < ω_{p} and ωτ > 1. Case (iii) corresponds to good conductors at low frequencies (ωτ < 1) and will be taken up again in 7.7.

d. (Contrary to Jackson) it is not correct in general to say that α is an absorption coefficient. Absorption is proportional to Im ε, not to the imaginary part of \sqrt{ε}. In particular a negative ε (case (ii), plasmas) leads to a large α, but no absorption. The waves decay exponentially because the medium is unsuitable for their propagation, not because they are absorbed. Only in case (i) can α be called properly an absorption coefficient. See the figures at the end for plots of n and α in a simple case.

7.5.9

a. Jackson discusses in detail, as an example, the values of n and α for liquid water over the astounding frequency range 10^{2} − 10^{22} sec^{−1}. The most remarkable feature of the spectrum is a sharp transparency window in the visible region: below it are absorption lines due to vibrational and rotational transitions, above it are absorption lines due to electronic transitions, which are broadened into bands by intermolecular interactions and by admixture with rotational and vibrational excitations.

b. In the low frequency region the absorption is dominated by a phenomenon that we have neglected so far: ionic conductivity. It is described by a formula identical to (5.9), where N_{c} and m^{∗} should be replaced by the ion concentration N_{i} and the effective ion mass M. Typically N_{i} ≪ N_{c} and M is 10^{4} times m^{∗}, so that the ionic term in the conductivity (and in ε) is important only when all the electrons are effectively bound.

c. At wavelengths less than a few Ångstroms (frequencies greater than 10^{18} sec^{−1}) the concepts of dielectric constant and index of refraction do not make sense, because they are based on the assumption that the spatial variation of the field is negligible on the scale of molecular dimensions. It is possible to generalize the theory by introducing a non-local response function as hinted in 7.5.1 c(ii). Physically, one can think of the phenomenon of x-ray diffraction in crystals, where the wavelength (or k) is changed by interaction with the atoms of the lattice. One can still define a total absorption coefficient α by measuring the decrease in the total intensity along the path of a beam. For h̄ω > 1 keV, intraatomic transitions determine α.

Here are various plots for the simple one-resonance dielectric constant

ε = 1 + \frac{ω_{p}^{2}}{ω_{res}^{2} − ω^{2} − iωγ} = 1 + \frac{z^{2}}{1 − x^{2} − ixy}

For convenience, we measure frequencies in units of ω_{res} and put z = ω_{p}/ω_{res} and y = γ/ω_{res}. The solid line is Re ε and the dotted line is Im ε, for z = 1 and y = 0.1 :

Here are the corresponding plots of n = Re \sqrt{ε} and Im \sqrt{ε}

and the real and imaginary parts of kc/ω_{res} = (nω + iαc/2) /ω_{res} = x\sqrt{ε}