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Spatial Dispersion and Warm Plasma*

10
Spatial Dispersion and Warm Plasma *

In Chapter 9, we discussed wave propagation in cold isotropic plasma. The effects of dispersion, to be more precise, temporal dispersion, are emphasized. In literature, cold plasma is sometimes referred to as “temperate plasma” [1,2]. The permittivity of warm plasma, in addition to being the function of the frequency ω, is also a function of the wave number k. For this reason, the warm plasma is said to be spatially as well as temporally dispersive [1,2,3]. Metal at optical frequencies can be modeled as warm plasma [4], the Fermi velocity playing the role of the thermal velocity of the warm plasma. While the model based on the cold plasma is said to represent local optics, the warm plasma model represents nonlocal optics. Section 10.7 gives rudimentary explanation of the technical terms used in describing the plasma state and the references give a deeper exposure.

10.1Waves in a Compressible Gas

A compressible gas satisfies the equation of state

(10.1)

P = k B N T,

$P = k_{B} N T,$

where P is the pressure, N the number density, T the temperature, and k_B Boltzmann’s constant (1.38 × 10⁻²³ J/K).

Let

(10.2)

P = P 0 + p,

$P = P_{0} + p,$

(10.3)

N = N 0 + n .

$N = N_{0} + n .$

The values with subscript zero are the average values and the lower-case ones are a.c. values.

For an isothermal process (T constant), from Equation 10.1, Equation 10.2, Equation 10.3,

P 0 = k B N 0 T,

$P_{0} = k_{B} N_{0} T,$

p = k B n T .

$p = k_{B} n T .$

For acoustic waves, the adiabatic process where no heat transfer is taking place is more appropriate; P and N satisfy the relation

(10.4)

P N γ = P 0 N γ 0,

$\frac{P}{N^{γ}} = \frac{P_{0}}{N_{0}^{γ}},$

where γ is the ratio of specific heats.

The value of γ depends on the type of gas (monatomic, diatomic, or polyatomic). For a one-dimensional plasma problem, γ = 3 is appropriate.

From Equation 10.2, Equation 10.3, Equation 10.4, we have

(10.5)

p = γ k B n T .

$p = γ k_{B} n T .$

The hydrodynamic equation (Navier–Stokes equation) is

(10.6)

m d v d t = - 1 N \nabla p,

$m \frac{d v}{d t} = - \frac{1}{N} \nabla p,$

where m is the mass of the particle.

The continuity equation is

(10.7)

\nabla \cdot [N v] + \partial N \partial t = 0.

$\nabla \cdot [N v] + \frac{\partial N}{\partial t} = 0.$

The RHS of Equation 10.6 is nonzero in a compressible gas.

For small-signal a.c. values, after linearizing, Equations 10.6 and 10.7 are approximated as

(10.8)

m d v d t = - 1 N 0 \nabla p,

$m \frac{d v}{d t} = - \frac{1}{N_{0}} \nabla p,$

(10.9)

N 0 \nabla \cdot v = - \partial n \partial t .

$N_{0} \nabla \cdot v = - \frac{\partial n}{\partial t} .$

Taking the divergence of Equation 10.8, we obtain

(10.10)

\partial ( \nabla \cdot v ) \partial t = - \nabla 2 p N 0 m .

$\frac{\partial (\nabla \cdot v)}{\partial t} = - \frac{\nabla^{2} p}{N_{0} m} .$

From Equations 10.9 and 10.5, we obtain

(10.11)

\nabla \cdot v = - 1 N 0 γ k B T \partial p \partial t .

$\nabla \cdot v = - \frac{1}{N_{0} γ k_{B} T} \frac{\partial p}{\partial t} .$

Substituting Equation 10.11 into Equation 10.10, we obtain the wave equation for the pressure:

(10.12)

\nabla 2 p - 1 a 2 \partial 2 p \partial t 2 = 0,

$\nabla^{2} p - \frac{1}{a^{2}} \frac{\partial^{2} p}{\partial t^{2}} = 0,$

where the acoustic velocity a is given as

(10.13)

a = (γ k B T m) 1 / 2 .

$a = {(\frac{γ k_{B} T}{m})}^{1 / 2} .$

10.2Waves in Warm Plasma

Let us consider the case where the gas has charged particles. In Chapter 9, we considered free electron gas, called cold plasma, and the force equation was based on the Lorentz force:

(10.14)

m d v d t = - q E .

$m \frac{d v}{d t} = - q E .$

In the cold plasma case, we ignored the term on the RHS of Equation 10.6 assuming that it is negligible compared to the Lorentz force. The assumption is valid as long as the temperature of the gas is low and the gas is considered as incompressible. In the warm plasma case, we consider the contributions to the force due to pressure gradients and we modify Equation 10.6 for the warm plasma case as

(10.15)

m d v d t = - q E - 1 N \nabla p .

$m \frac{d v}{d t} = - q E - \frac{1}{N} \nabla p .$

Taking the divergence of Equation 10.15 (and approximating d/dt → ∂/∂t),

(10.16)

\partial ( \nabla \cdot v ) \partial t = - q m (\nabla \cdot E) - 1 N m \nabla 2 p .

$\frac{\partial (\nabla \cdot v)}{\partial t} = - \frac{q}{m} (\nabla \cdot E) - \frac{1}{N m} \nabla^{2} p .$

Using Equation 10.11 in Equation 10.16, we obtain the wave equation for the pressure:

(10.17)

\nabla 2 p - 1 a 2 \partial 2 p \partial t 2 + N 0 q \nabla \cdot E = 0.

$\nabla^{2} p - \frac{1}{a^{2}} \frac{\partial^{2} p}{\partial t^{2}} + N_{0} q \nabla \cdot E = 0.$

The variable E in Equation 10.17 can be eliminated by using the Maxwell equation

(10.18)

\nabla \times H = ε 0 \partial E \partial t - N 0 q v,

$\nabla \times H = ε_{0} \frac{\partial E}{\partial t} - N_{0} q v,$

(10.19)

00 \nabla \cdot E N 0 q \nabla \cdot E N 0 q \nabla \cdot E = = = = = \nabla \cdot (\nabla \times H) = ε 0 \partial ( \nabla \cdot E ) \partial t - N 0 q (\nabla \cdot v), ε 0 \partial ( \nabla \cdot E ) \partial t + q γ k B T \partial p \partial t, - q γ k B T ε 0 p, - N 0 q 2 γ k B T ε 0 p = - N 0 q 2 m ε 0 m γ k B T p, - ω 2 p a 2 p .

$\begin{matrix} 0 & = & \nabla \cdot (\nabla \times H) = ε_{0} \frac{\partial (\nabla \cdot E)}{\partial t} - N_{0} q (\nabla \cdot v), \\ 0 & = & ε_{0} \frac{\partial (\nabla \cdot E)}{\partial t} + \frac{q}{γ k_{B} T} \frac{\partial p}{\partial t}, \\ \nabla \cdot E & = & - \frac{q}{γ k_{B} T ε_{0}} p, \\ N_{0} q \nabla \cdot E & = & - \frac{N_{0} q^{2}}{γ k_{B} T ε_{0}} p = - \frac{N_{0} q^{2}}{m ε_{0}} \frac{m}{γ k_{B} T} p, \\ N_{0} q \nabla \cdot E & = & - \frac{ω_{p}^{2}}{a^{2}} p . \end{matrix}$

Thus, the wave equation for the pressure variable p for the warm plasma case is given by

(10.20)

\nabla 2 p - 1 a 2 \partial 2 p \partial t 2 - ω 2 p a 2 p = 0.

$\nabla^{2} p - \frac{1}{a^{2}} \frac{\partial^{2} p}{\partial t^{2}} - \frac{ω_{p}^{2}}{a^{2}} p = 0.$

To obtain the wave equation for the electric field, we start with the Maxwell equation

(10.21)

\nabla ¯ ¯ ¯ \times E = - μ 0 \partial H \partial t

$\bar{\nabla} \times E = - μ_{0} \frac{\partial H}{\partial t}$

and take the curl on both sides

\nabla \times \nabla \times E = - μ 0 \partial ( \nabla \times H ) \partial t

$\nabla \times \nabla \times E = - μ_{0} \frac{\partial (\nabla \times H)}{\partial t}$

and from Equation 10.18, we obtain

\nabla \times \nabla \times E = - μ 0 \partial \partial t (ε 0 \partial E \partial t - N 0 q v),

$\nabla \times \nabla \times E = - μ_{0} \frac{\partial}{\partial t} (ε_{0} \frac{\partial E}{\partial t} - N_{0} q v),$

and from Equation 10.15

- \nabla \times \nabla \times E - μ 0 ε 0 \partial 2 E \partial t 2 + N 0 μ 0 q (- q m E - 1 N 0 m \nabla p) = 0, - \nabla \times \nabla \times E - 1 c 2 \partial 2 E \partial t 2 - μ 0 ε 0 ω 2 p E - μ 0 q m \nabla p = 0.

$\begin{matrix} - \nabla \times \nabla \times E - μ_{0} ε_{0} \frac{\partial^{2} E}{\partial t^{2}} + N_{0} μ_{0} q (- \frac{q}{m} E - \frac{1}{N_{0} m} \nabla p) = 0, \\ - \nabla \times \nabla \times E - \frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}} - μ_{0} ε_{0} ω_{p}^{2} E - \frac{μ_{0} q}{m} \nabla p = 0. \end{matrix}$

Using Equation 10.19 to eliminate p from the above,

- \nabla \times \nabla \times E - 1 c 2 \partial 2 E \partial t 2 - ω 2 p c 2 E + μ 0 q m \nabla ¯ ¯ ¯ (N 0 q a 2 ω 2 p \nabla \cdot E) = 0.

$- \nabla \times \nabla \times E - \frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}} - \frac{ω_{p}^{2}}{c^{2}} E + \frac{μ_{0} q}{m} \bar{\nabla} (\frac{N_{0} q a^{2}}{ω_{p}^{2}} \nabla \cdot E) = 0.$

By simplifying, we obtain the wave equation for E as

(10.22)

- \nabla \times \nabla \times E - 1 c 2 \partial 2 E \partial t 2 - ω 2 p c 2 E + μ 0 ε 0 a 2 \nabla (\nabla \cdot E) = 0.

$- \nabla \times \nabla \times E - \frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}} - \frac{ω_{p}^{2}}{c^{2}} E + μ_{0} ε_{0} a^{2} \nabla (\nabla \cdot E) = 0.$

We should be able to recover the wave equation for E for the cold plasma case by substituting a = 0 in Equation 10.22:

- \nabla \times \nabla \times E - 1 c 2 \partial 2 E \partial t 2 - ω 2 p c 2 E = 0, - \nabla (\nabla \cdot E) + \nabla 2 E - 1 c 2 \partial 2 E \partial t 2 - ω 2 p c 2 E = 0.

$\begin{matrix} - \nabla \times \nabla \times E - \frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}} - \frac{ω_{p}^{2}}{c^{2}} E = 0, \\ - \nabla (\nabla \cdot E) + \nabla^{2} E - \frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}} - \frac{ω_{p}^{2}}{c^{2}} E = 0. \end{matrix}$

For the cold homogeneous plasma case (incompressible gas), p = 0 and ∇ · E = 0, and hence we obtain

(10.23)

\nabla 2 E ¯ ¯ ¯ - 1 c 2 \partial 2 E ¯ ¯ ¯ \partial t 2 - ω 2 p c 2 E ¯ ¯ ¯ = 0.

$\nabla^{2} \bar{E} - \frac{1}{c^{2}} \frac{\partial^{2} \bar{E}}{\partial t^{2}} - \frac{ω_{p}^{2}}{c^{2}} \bar{E} = 0.$

Let us solve Equation 10.22 for a harmonic wave with a phase factor e^jωt e^−jk·r. Noting that ∇ can be replaced by (−jk), for such a phase factor, Equation 10.22 becomes

(10.24)

- (- j k) \times (- j k) \times E + ω 2 c 2 E - ω 2 p c 2 E + a 2 c 2 (- j k) (- j k \cdot E) = 0, k \times k \times E + ω 2 c 2 E - ω 2 p c 2 E - a 2 c 2 k (k \cdot E) = 0.

$\begin{matrix} - (- j k) \times (- j k) \times E + \frac{ω^{2}}{c^{2}} E - \frac{ω_{p}^{2}}{c^{2}} E + \frac{a^{2}}{c^{2}} (- j k) (- j k \cdot E) = 0, \\ k \times k \times E + \frac{ω^{2}}{c^{2}} E - \frac{ω_{p}^{2}}{c^{2}} E - \frac{a^{2}}{c^{2}} k (k \cdot E) = 0. \end{matrix}$

Since k × k × E = k(k · E) − k²E, we can write Equation 10.24 as

(10.25)

k (k \cdot E) (1 - a 2 c 2) - [k 2 - k 20 (1 - ω 2 p ω 2)] E = 0.

$k (k \cdot E) (1 - \frac{a^{2}}{c^{2}}) - [k^{2} - k_{0}^{2} (1 - \frac{ω_{p}^{2}}{ω^{2}})] E = 0.$

Equation 10.24 can be written separately for E_∥ and E_⊥, where E_∥ is a component parallel to k and E_⊥ is normal to k.

For the parallel component k ⋅ E = kE_∥, we have

(10.26)

k 2 E ∥ (1 - a 2 c 2) - [k 2 - k 20 (1 - ω 2 p ω 2)] E ∥ = 0, k 2 a 2 c 2 + k 20 (1 - ω 2 p ω 2) = 0, k 2 = k 20 ( 1 - ( ω 2 p / ω 2 ) ) a 2 / c 2 = ω 2 - ω 2 p a 2 .

$\begin{matrix} k^{2} E_{∥} (1 - \frac{a^{2}}{c^{2}}) - [k^{2} - k_{0}^{2} (1 - \frac{ω_{p}^{2}}{ω^{2}})] E_{∥} = 0, \\ k^{2} \frac{a^{2}}{c^{2}} + k_{0}^{2} (1 - \frac{ω_{p}^{2}}{ω^{2}}) = 0, \\ k^{2} = k_{0}^{2} \frac{(1 - (ω_{p}^{2} / ω^{2}))}{a^{2} / c^{2}} = \frac{ω^{2} - ω_{p}^{2}}{a^{2}} . \end{matrix}$

Let

(10.27)

a c = δ,

$\frac{a}{c} = δ,$

(10.28)

ω 2 p ω 2 = X,

$\frac{ω_{p}^{2}}{ω^{2}} = X,$

Then

(10.29)

k 2 = k 20 1 - X δ 2 .

$k^{2} = k_{0}^{2} \frac{1 - X}{δ^{2}} .$

From Equation 10.21, we obtain

(10.30)

H = - 1 μ 0 j ω k \times E,

$H = - \frac{1}{μ_{0} j ω} k \times E,$

and since k and E are parallel for this case

(10.31)

H = 0.

$H = 0.$

The parallel mode under discussion has an electric field component in the direction of propagation and the zero magnetic field. It is also called an electron plasma wave. It is an acoustic wave, like a sound wave, modified by the pressure of a charged compressible fluid. Its phase velocity is given by

(10.32)

V p = ω k = a 1 - X - - - - - \sqrt = c δ 1 - X - - - - - \sqrt .

$V_{p} = \frac{ω}{k} = \frac{a}{\sqrt{1 - X}} = c \frac{δ}{\sqrt{1 - X}} .$

The ω−k diagram for the electron plasma wave is shown in Figure 10.1.

FIGURE 10.1
ω−k diagram for the electron plasma wave.

Note that the phase velocity a becomes negligible for the cold plasma approximation and the electron plasma wave becomes a plasma oscillation at the plasma frequency ω_p. A simple way of interpreting the electron plasma wave is to say that the plasma oscillation becomes a longitudinal electron plasma wave when the plasma is considered as warm.

The electron plasma wave propagates for only ω > ω_p. For ω < ω_p, this wave is evanescent. Now let us examine the wave propagation when E is perpendicular to k.

From Equation 10.24,

(10.33)

[k 2 - k 20 (1 - ω 2 p ω 2)] E ⊥ = 0,

$[k^{2} - k_{0}^{2} (1 - \frac{ω_{p}^{2}}{ω^{2}})] E_{⊥} = 0,$

(10.34)

k 2 = k 20 (1 - ω 2 p ω 2) .

$k^{2} = k_{0}^{2} (1 - \frac{ω_{p}^{2}}{ω^{2}}) .$

Equation 10.34 is the same as the dispersion relation for the cold isotropic plasma, discussed in Chapter 9. One can obtain H_⊥ from Equation 10.30, which shows that

(10.35)

E ⊥ = η p H ⊥,

$E_{⊥} = η_{p} H_{⊥},$

where

(10.36)

η p = η 0 1 - X - - - - - \sqrt .

$η_{p} = \frac{η_{0}}{\sqrt{1 - X}} .$

While the electron plasma wave and TEM wave can exist independently in an unbounded warm plasma, the amplitudes of the two will be determined in a bounded warm plasma by the boundary conditions. Appendix 10A discusses such a problem. Appendices 10B and 10C deal with such problems for the cases of warm magnetoplasma media.

10.3Constitutive Relation for a Lossy Warm Plasma

For a lossy warm plasma, Equation 10.15 will be modified as

(10.37)

m d v d t + m ν v = - q E - 1 N \nabla p .

$m \frac{d v}{d t} + m ν v = - q E - \frac{1}{N} \nabla p .$

Since

(10.38)

J = - N 0 q v,

$J = - N_{0} q v,$

Equation 10.37 can be written as (multiply Equation 10.37 by −N₀q/m)

d J d t + ν J = ε 0 ω 2 p E + q m \nabla p, \nabla p = γ k B T \nabla n .

$\begin{matrix} \frac{d J}{d t} + ν J = ε_{0} ω_{p}^{2} E + \frac{q}{m} \nabla p, \\ \nabla p = γ k_{B} T \nabla n . \end{matrix}$

Since

ε 0 \nabla \cdot E \nabla n \nabla p q m \nabla p = = = = ρ ν = - q n, - ε 0 q \nabla (\nabla \cdot E), - γ k B T ε 0 q \nabla (\nabla \cdot E), - γ k B T ε 0 m \nabla (\nabla \cdot E) = - ε 0 a 2 \nabla (\nabla \cdot E) .

$\begin{matrix} ε_{0} \nabla \cdot E & = & ρ_{ν} = - q n, \\ \nabla n & = & - \frac{ε_{0}}{q} \nabla (\nabla \cdot E), \\ \nabla p & = & - γ k_{B} T \frac{ε_{0}}{q} \nabla (\nabla \cdot E), \\ \frac{q}{m} \nabla p & = & - \frac{γ k_{B} T ε_{0}}{m} \nabla (\nabla \cdot E) = - ε_{0} a^{2} \nabla (\nabla \cdot E) . \end{matrix}$

Thus, the constitutive relation is given by

(10.39)

d J d t + ν J = ε 0 ω 2 p E - ε 0 a 2 \nabla (\nabla \cdot E) .

$\frac{d J}{d t} + ν J = ε_{0} ω_{p}^{2} E - ε_{0} a^{2} \nabla (\nabla \cdot E) .$

One can model the warm plasma as a dielectric by using Equation 10.39 and the Maxwell equation

(10.40)

\nabla \times H = J + ε 0 \partial E \partial t .

$\nabla \times H = J + ε_{0} \frac{\partial E}{\partial t} .$

For harmonic variations in space and time e^{j(ωt − k·r)}, Equation 10.39 becomes

j (ω + v) J = ε 0 ω 2 p E - ε 0 a 2 (- j k) (- j k \cdot E) .

$j (ω + v) J = ε_{0} ω_{p}^{2} E - ε_{0} a^{2} (- j k) (- j k \cdot E) .$

Since

J = J ∥ + J ⊥ (with reference to k),

$J = J_{∥} + J_{⊥} (with reference to k),$

(10.41)

E = E ∥ + E ⊥, (j ω + ν) J ∥ = ε 0 ω 2 p E ∥ + ε 0 a 2 k 2 ∥ E ∥,

$\begin{matrix} E = E_{∥} + E_{⊥}, \\ (j ω + ν) J_{∥} = ε_{0} ω_{p}^{2} E_{∥} + ε_{0} a^{2} k_{∥}^{2} E_{∥}, \end{matrix}$

(10.42)

(j ω + ν) J ⊥ = ε 0 ω 2 p E ⊥ + 0.

$(j ω + ν) J_{⊥} = ε_{0} ω_{p}^{2} E_{⊥} + 0.$

(10.43)

(\nabla ¯ ¯ ¯ \times H ¯ ¯ ¯) ⊥ = J ⊥ + j ω ε 0 E ⊥ = (ε 0 ω 2 p j ω + ν + j ω ε 0) E ⊥, (\nabla ¯ ¯ ¯ \times H ¯ ¯ ¯) ⊥ = j ω ε p ⊥ E ⊥,

$\begin{matrix} {(\bar{\nabla} \times \bar{H})}_{⊥} = J_{⊥} + j ω ε_{0} E_{⊥} = (\frac{ε_{0} ω_{p}^{2}}{j ω + ν} + j ω ε_{0}) E_{⊥}, \\ {(\bar{\nabla} \times \bar{H})}_{⊥} = j ω ε_{p ⊥} E_{⊥}, \end{matrix}$

where

(10.44)

ε p ⊥ = 1 - ω 2 p ω ( ω - j ν ) .

$ε_{p ⊥} = 1 - \frac{ω_{p}^{2}}{ω (ω - j ν)} .$

Let us derive next an expression for ε_p∥.

From Equation 10.41,

(10.45)

J ∥ (\nabla ¯ ¯ ¯ \times H ¯ ¯ ¯) ∥ (\nabla ¯ ¯ ¯ \times H ¯ ¯ ¯) ∥ = = = = ε 0 ω 2 p j ω + ν E ∥ + ε 0 a 2 k 2 ∥ j ω + ν E ∥, J ∥ + j ω ε 0 E ∥ ε 0 ω 2 p j ω + ν E ∥ + ε 0 a 2 k 2 ∥ j ω + ν E ∥ + j ω ε 0 E ∥, j ω ε 0 ε p ∥ E ∥,

$\begin{matrix} J_{∥} & = & \frac{ε_{0} ω_{p}^{2}}{j ω + ν} E_{∥} + \frac{ε_{0} a^{2} k_{∥}^{2}}{j ω + ν} E_{∥}, \\ {(\bar{\nabla} \times \bar{H})}_{∥} & = & J_{∥} + j ω ε_{0} E_{∥} \\ = & \frac{ε_{0} ω_{p}^{2}}{j ω + ν} E_{∥} + \frac{ε_{0} a^{2} k_{∥}^{2}}{j ω + ν} E_{∥} + j ω ε_{0} E_{∥}, \\ {(\bar{\nabla} \times \bar{H})}_{∥} & = & j ω ε_{0} ε_{p ∥} E_{∥}, \end{matrix}$

where

(10.46)

$ε_{p ∥} = 1 - \frac{ω_{p}^{2}}{ω (ω - j ν)} - \frac{a^{2} k_{∥}^{2}}{ω (ω - j ν)} .$

If we neglect collisions, then

(10.47)

$ε_{p ∥} = 1 - \frac{ω_{p}^{2} + a^{2} k_{∥}^{2}}{ω^{2}} .$

Substituting Equation 10.26 for k_∥, we obtain

(10.48)

$ε_{p ∥} = 0.$

This will be true even if collisions are taken into account.

Equation 10.47 implies that E_∥ can be nonzero even if D_∥ = 0.

The dispersion relation Equation 10.26 and the ω−k diagram in Figure 10.1 can be obtained by imposing Equation 10.47.

10.4Dielectric Model of Warm Loss-Free Plasma

Maxwell’s equations and the constitutive relations for warm plasma, viewed as a dielectric medium, are given below:

(10.49)

$\nabla \times E = - \frac{\partial B}{\partial t},$

(10.50)

$\nabla \times H = - \frac{\partial D}{\partial t},$

(10.51)

$\nabla \cdot D = 0,$

(10.52)

$\nabla \cdot B = 0,$

(10.53)

$B = μ_{0} H,$

(10.54)

$D = η_{0} {\bar{ε}}_{p} \cdot E,$

(10.55)

${\bar{ε}}_{p} = [\begin{matrix} ε_{p ⊥} & 0 \\ 0 & ε_{p ∥} \end{matrix}],$

where

(10.56)

$ε_{p ⊥} = 1 - \frac{ω_{p}^{2}}{ω^{2}},$

(10.57)

$ε_{p ∥} = 1 - \frac{ω_{p}^{2}}{ω^{2} - a^{2} k_{∥}^{2}} .$

Note that Equations 10.47 and 10.57 are the same when ε_p∥ = 0.

10.5Conductor Model of Warm Lossy Plasma

Maxwell’s equations in the time domain and the constitutive relations for a warm lossy plasma, viewed as a conducting medium, are given below:

(10.58)

$\nabla \times E = - μ_{0} \frac{\partial H}{\partial t},$

(10.59)

$\nabla \times H = J + ε_{0} \frac{\partial E}{\partial t},$

(10.60)

$\nabla \cdot H = 0,$

(10.61)

$\frac{d J}{d t} + ν J = ε_{0} ω_{p}^{2} E - ε_{0} a^{2} \nabla (\nabla \cdot E) .$

Note that

(10.62)

$ε_{0} \nabla \cdot E = ρ,$

(10.63)

$ρ = - q n,$

(10.64)

$\nabla p = γ k T \nabla n .$

In Equations 10.62 and 10.63, ρ is the volume charge density.

For time-harmonic fields, Equation 10.61 can be written as

(10.65)

$J = \bar{σ} \cdot E,$

(10.66)

$\bar{σ} = [\begin{matrix} σ_{⊥} & 0 \\ 0 & σ_{∥} \end{matrix}],$

(10.67)

$σ_{⊥} = - j ε_{0} \frac{ω_{p}^{2}}{ω - j ν},$

(10.68)

$σ_{∥} = - j ω ε_{0} \frac{ω_{p}^{2}}{ω (ω - j ν) - a k_{∥}^{2}} .$

By noting

(10.69)

${\bar{ε}}_{p} = \bar{I} + \frac{\bar{σ}}{j ω ε_{0}},$

we can obtain the conductivity tensor $\bar{σ}$ from ${\bar{ε}}_{p}$ and vice versa.

10.6Spatial Dispersion and Nonlocal Metal Optics

The longitudinal wave in warm plasma has the dispersion relation

(10.70)

$ε_{p ∥} = 0 = 1 - \frac{ω_{p}^{2}}{ω (ω - j ν) - a^{2} k^{2}}$

and the ω−k diagram is sketched in Figure 10.1. In the sense that

(10.71)

$ε_{p ∥} = ε_{p ∥} (ω, k),$

the warm plasma is spatially as well as temporally dispersive.

A simple metal, such as sodium, is modeled as a lossy plasma in Section 8.3. Typical values for sodium are ω_p = 8.2 × 10¹⁵ rad/s and ν/ω_p = 3 × 10⁻³. The plasma frequency is in the optical range. Due to the high electron density in metals, it can be shown that a parameter called Fermi velocity v_f is also important in modeling metal as plasma [5]. A typical value of v_f for sodium is v_f = 9.8 × 10⁵ m/s. The Fermi velocity v_f plays the role of the acoustic velocity in warm plasma. Thus, the modeling of metal for studying the interaction of thin metal films with optical waves with frequencies near the plasma frequency require a warm plasma model, which takes into account the spatial dispersion due to the excitation of the electron plasma wave in the metal, when the source wave is a p wave. Spatial dispersion will have important consequences and such a study is labeled as the study of nonlocal optics in contrast to the local optics (Fresnel Optics). A comprehensive account of the nonlocal optical studies using the warm plasma model is given by Forstmann and Gerhardts [4].

10.7Technical Definition of Plasma State

We have earlier given a vague definition of plasma medium as an ionized medium with positive ions and electrons with an “overall” charge neutrality. We also used the terms cold plasma and warm plasma. While reading books on plasma science, we also come across terms such as collective behavior, temperate plasma, hot plasma, nonneutral plasma, Debye length and plasma sheath, and so on. This section is used to explain the various terms quantitatively in the context of the theory discussed in Sections 8.3 and 8.4, Chapter 9, and this chapter. The one-dimensional electron–proton plasma with one degree of freedom is assumed.

10.7.1Temperate Plasma

In cold plasma, we neglect the effect of the temperature. The thermal velocity V_th is related to temperature T by

(10.72)

$\frac{1}{2} m V_{th}^{2} = \frac{1}{2} k_{B} T .$

The RHS of Equation 10.72 is one-half of an electron volt (eV) and substituting for k_B, the Boltzmann constant, we obtain the equivalence

(10.73)

$1 eV = 11000 ° K .$

In linearizing the Maxwell equations and the force equation, we assume that the electron velocity v is small compared to V_th. The cold plasma approximation neglects the spatial dispersion, which will be a valid approximation if V_th is less than the phase velocity V_ph = ω/k of the wave. So the cold plasma approximation is valid, if [1]

(10.74)

$v ≪ V_{th} ≪ V_{ph} .$

Thus, v ≪ V_th is implied by calling cold plasma as temperate plasma.

A plasma is called a warm plasma if the temperature is such that V_th is approximately equals V_ph of the electron plasma wave.

10.7.2Debye Length, Collective Behavior, and Overall Charge Neutrality

A plasma medium shields out any excessive fields in the medium. A characteristic distance, called “Debye length” λ_D, is the length-scale parameter, which ensures the shielding of the excessive electric fields in plasma [6]. The Debye length depends on the temperature and the electron density and is given by

(10.75)

$λ_{D} = {(\frac{ε_{0} k_{B} T}{N q^{2}})}^{1 / 2} = \frac{a}{\sqrt{γ} ω_{p}},$

(10.76)

$λ_{D} = 69 {(\frac{T}{N})}^{1 / 2} .$

It can be shown [6] that any excess potential Φ in the plasma will decay as

(10.77)

$Φ = Φ_{0} e^{- (|z| / λ_{D})} .$

For |z| > 5λ_D, Φ = Φ₀e⁻⁵ ≈ 0, and hence the excess potential is screened out in about five times λ_D.

If the plasma system dimension L is much larger than λ_D, the bulk of the plasma has a charge neutrality, that is, the net charge in a macroscopic volume is zero:

(10.78)

$N_{e} \approx N_{i} \approx N .$

In Equation 10.78, N_i is the ion density and N is often referred to as the plasma density. The qualifying word “overall” before the word neutrality is used to indicate that the plasma is not so neutral to blot out all interesting electromagnetic forces. This shielding becomes possible, because the electric Coulomb force is inversely proportional to the square of the distance. This force is considered long range compared to the other short-range forces. If we assume that the electron is a point, then the charges in a volume $(4 π / 3) λ_{D}^{3}$ collectively shield the effect of this electron. It is this collective behavior, in contrast to the collision between single particles, that control the motion of the charged particles. The collective behavior can occur only when the particle density is such that there are enough particles in a Debye volume.

10.7.3Unneutralized Plasma

The phenomenon of Debye shielding in a modified form can occur in an electron stream in klystron, and a proton beam in a cyclotron. These are not strictly plasmas but because they use the concepts of collective behavior, and shielding and use the mathematical tools of plasma physics, they are referred to as unneutralized plasmas [7].

References

1.Allis, W.P., Buchsbaum, S. J., and Bens, A., Waves in Anisotropic Plasmas, MIT Press, Cambridge, MA, 1963.
2.Heald, M.A. and Wharton, C. B., Plasma Diagnostics with Microwaves, Wiley, New York, 1965.
3.Akira, I., Electromagnetic Wave Propagation, Radiation, and Scattering, Prentice Hall, Englewood Cliffs, NJ, 1991.
4.Forstmann, F. and Gerhardts, R.R., Metal Optics Near the Plasma Frequency, Springer, New York, 1986.
5.Kittel, C., Introduction to Solid State Physics, Wiley, New York, 2005.
6.Chen, F.F., Introduction to Plasma Physics and Controlled Fusion, Vol.1: Plasma Physics, Springer, New York, 1983.
7.Davidson, R.C., Physics of Nonneutral Plasmas, Addison-Wesley, New York, 1990.

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Table of Contents for 10. Spatial Dispersion and Warm Plasma

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Table of Contents for
10. Spatial Dispersion and Warm Plasma