Plasma
`
`Physics
`and
`Engineering
`
`second edition
`
`Alexander Fridman
`
`Drexel University, Philadelphia, USA
`
`Lawrence A. Kennedy
`
`University of Illinois at Chicago, USA
`
`@ CRC Press
`
`Ta
`Boca
`
`& Francis Group
`on London NewYork
`
`ress is an imprint of the
`a or & Francis Group, an informa business
`
`Energetiq Ex. 2046, page 1 - |PR2015-01279
`
`Energetiq Ex. 2046, page 1 - IPR2015-01279
`
`

`
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`Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`409
`
`(cid:8).
`
`(6.6.42)
`
`1
`− ωBωBi
`
`ω2
`
`ω
`
`be derived taking into account the ion motion
`(cid:7)
`
`k2c2
`ω2
`
`= 1 − ω2
`
`p
`ω2
`
`1 ± ωB
`#
`If the electromagnetic wave frequency exceeds ion-cyclotron frequency
`(ω (cid:5) ωB), influence of the term ωBωBi
`ω2 in the denominator is negligible
`#
`and the dispersion equation 6.6.42 coincides with Equation 6.6.39. Conversely,
`if the wave frequency is low (ω (cid:10) ωB), then the term ωBωBi
`ω2 becomes dom-
`inant in the dispersion equation. In this case, the dispersion equation 6.6.42
`can be rewritten as
`
`#
`c2
`k)2
`(ω
`
`= 1 + ω2
`p
`ωBωBi
`
`= 1 + μ0ρ
`B2
`
`≈ c2
`v2
`A
`
`.
`
`(6.6.43)
`
`This introduces again the Alfven wave velocity vA (Equation 6.5.17); here,
`ρ = ne M is the mass density in completely ionized plasma (see Ginsburg,1960;
`Ginsburg and Rukhadze, 1970).
`
`6.7 Emission and Absorption of Radiation in Plasma,
`Continuous Spectrum
`6.7.1 Classification of Radiation Transitions
`Radiation occurs due to transitions between different energy levels of a quan-
`tum system: transition up corresponds to absorption of a quantum Ef − Ei =
`ω, transition down the spectrum corresponds to emission Ei − Ef = ω (see
`Section 6.7.2). From the point of classical electrodynamics, radiation is related
`to the nonlinear change of dipole momentum, actually with the second deriva-
`tive of dipole momentum. Neither emission nor absorption of radiation is
`possible for free electrons. Electron collisions are necessary in this case. It will
`be shown in Section 6.7.4 that electron interaction with a heavy particle, ion, or
`neutral is able to provide emission or absorption, but electron–electron inter-
`action cannot. It is convenient to classify different types of radiation according
`to the different types of electron transition from one state to another. Electron
`energy levels in the field of an ion as well as transitions between the energy
`levels are illustrated in Figure 6.32. The case when both initial and final elec-
`tron states are in continuum is called the free–free transition. A free electron
`in this transition loses part of its kinetic energy in the Coulomb field of a pos-
`itive ion or in interaction with neutrals. The emitted energy in this case is a
`continuum, usually infrared and called bremsstrahlung (direct translation—
`stopping radiation). The reverse process is the bremsstrahlung absorption.
`
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`
`410
`
`Plasma Physics and Engineering
`
`Free–free
`
`Bound–free
`
`Bound–bound
`
`FIGURE 6.32
`Energy levels and electron transitions induced by ion field.
`
`Electron transition between a free state in continuum and a bound state in
`atom (see Figure 6.32) is usually referred to as the free–bound transition. The
`free–bound transitions correspond to processes of the radiative electron–ion
`recombination (see Section 2.3.5) and the reverse one of photoionization (see
`Section 2.2.6). Such kinds of transitions also could take place in electron-
`neutral collisions. In this case, these are related to photo-attachment and
`photo-detachment processes of formation and destruction of negative ions
`(see Section 2.4). The free-bound transitions correspond to continuum radia-
`tion. Finally, the bound–bound transitions mean transition between discrete
`atomic levels (see Figure 6.32) and result in emission and absorption of spec-
`tral lines. Molecular spectra are obviously much more complex than that of
`single atoms because of possible transitions between different vibrational and
`rotational levels.
`
`6.7.2 Spontaneous and Stimulated Emission: Einstein Coefficients
`Consider transitions between two states (upper “u” and ground “0”) of an
`atom or molecule with emission and absorption of a photon ω, which is
`illustrated in Figure 6.33. The probability of a photon absorption by an atom
`per unit time (and, hence, atom transition “0” → “u”) can be expressed as
`P(“0”, nω → “u”, nω − 1) = Anω.
`(6.7.1)
`Here, nω is the number of photons; A is the Einstein coefficient, which
`depends on atomic parameters and does not depend on electromagnetic wave
`
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`Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`411
`
`hω
`
`Absorption
`
`Upper level
`
`Emission L
`
`ower level
`
`FIGURE 6.33
`Radiative transition between two energy levels.
`
`characteristics. Similarly, the probability of atomic transition with a photon
`emission is
`
`P(“u”, nω → “0”, nω + 1) = 1
`
`τ
`
`+ Bnω.
`
`(6.7.2)
`
`Here, 1/τ is the frequency of spontaneous emission, which takes place without
`direct relation to external fields; Einstein coefficient B characterizes emission
`induced by an external electromagnetic field. The factors B and τ as well
`as A depend only on atomic parameters. If the first right-hand side term
`in Equation 6.7.2 corresponds to spontaneous emission, the second term is
`related to the stimulated emission. To find relations between the Einstein
`coefficients A, B and the spontaneous emission frequency 1/τ, analyze the
`thermodynamic equilibrium of radiation with the atomic system. In this case,
`the densities of atoms in lower and upper states (Figure 6.33) are related in
`accordance with the Boltzmann law (Equation 4.1.9) as
`(cid:3)
`(cid:4)
`nu = gu
`g0
`
`n0 exp
`
`− ω
`T
`
`,
`
`(6.7.3)
`
`where ω is energy difference between the two states, gu, g0 are their statistical
`weights. According to the Planck distribution, the average number of photons
`¯nω in one state can be determined as
`3(cid:3)
`(cid:4)
`¯nω = 1
`
`exp
`
`ω
`T
`
`− 1
`
`.
`
`(6.7.4)
`
`Taking into account, the balance of photon emission and absorption (see
`Figure 6.33)
`n0P(“0”, nω → “u”, nω − 1) = nuP(“u”, nω → “0”, nω + 1),
`
`(6.7.5)
`
`(6.7.6)
`
`which can be rewritten based on Equations 6.7.1 and 6.7.2 as
`(cid:3)
`(cid:4)
`n0A ¯nω = nu
`+ B¯nω
`
`.
`
`1 τ
`
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`412
`
`Plasma Physics and Engineering
`
`The relations between the Einstein coefficients A, B, and the spontaneous
`emission frequency 1/τ can be expressed based on Equations 6.7.3, 6.7.4, and
`6.7.6 as
`, B = 1
`A = gu
`τ .
`g0
`
`(6.7.7)
`
`1 τ
`
`The emission probability can be rewritten from Equations 6.7.2 and 6.7.7 as
`+ nω
`P(“u”, nω → “0”, nω + 1) = 1
`
`(6.7.8)
`
`.
`
`1 τ
`
`τ
`
`The contribution of the stimulated emission nω times exceeds the spon-
`taneous one. Although the expressions (Equation 6.7.7) for the Einstein
`coefficients were derived from equilibrium thermodynamics, the final Equa-
`tion 6.7.8 can be applied for the nonequilibrium number of photons nω.
`
`6.7.3 General Approach to Bremsstrahlung Spontaneous Emission:
`Coefficients of Radiation Absorption and Stimulated Emission during
`Electron Collisions with Heavy Particles
`The bremsstrahlung emission is related to the free–free electron transitions,
`and does not depend on external radiation like the spontaneous emission.
`An electron slows down in a collision with a heavy particle, ion, or neu-
`tral and loses kinetic energy, which then partially goes to radiation. A free
`electron cannot absorb or emit a photon because of the requirements of
`momentum conservation. Describe the bremsstrahlung emission at a specific
`frequency ω in terms of the differential spontaneous emission cross section:
`dσω(ve) = dσω/dωdω, which is a function of electron velocity ve. The emis-
`sion probability per electron of quanta with energies ω/ω + d(ω) can be
`given by the conventional definition:
`
`dPω = 1ωdQω = ven0dσω(ve) = ven0
`dσω
`dω dω.
`Here, dQω is the emission power per electron in the frequency inter-
`#
`val dω; n0 is density of heavy particles. The differential cross section of
`dω(ve) characterizes the spontaneous emis-
`bremsstrahlung emission dσω
`sion of plasma electrons during their collisions with heavy particles in the
`same way as the frequency 1/τ characterizes spontaneous emission of an atom
`(Equation 6.7.2). Also, similar to the Einstein coefficients A and B for atomic
`systems (see relations (Equations 6.7.1 and 6.7.2)), introduce coefficients aω(ve)
`and bω(ve) for radiation absorption and stimulated emission during the elec-
`tron collision with heavy particles. To introduce the coefficients aω(ve) and
`bω(ve), first define the radiation intensity I(ω) as the emission power in
`spectral interval ω/(ω + dω) per unit area and within the unit solid angle
`
`(6.7.9)
`
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`Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`413
`
`(Ω, measured in steradians). The product Iωdω dΩ actually describes the den-
`sity of the radiation energy flux. Then the radiation energy from the interval
`dω dΩ absorbed by electrons with velocities ve/ve + dve in unit volume per
`unit time can be presented as
`(Iωdω dΩ)n0[ f (ve)dve] ×a ω(ve).
`This is the definition of the absorption coefficient aω(ve), which is actu-
`ally calculated with respect to one electron and one atom. In its definition in
`Equation 6.7.10, n0 is the density of heavy particles and f (ve) is the electron
`distribution function. Similar to Equation 6.7.10, the stimulated emission of
`quanta related to electron collisions with heavy particles can be expressed as
`(Iωdω dΩ)n0[f (v
`] ×b ω(v
`) is introduced with
`Here, the coefficient of stimulated emission bω(v
`respect to one electron and one atom. In accordance with the energy con-
`servation for absorption (Equation 6.7.10) and stimulated emission (Equa-
`tion 6.7.11), the electron velocities ve and v
`are related to each other as
`= mv2
`+ ω.
`2
`
`).
`
`(cid:2)e
`
`(cid:2)e
`
`(cid:2)e
`
`)dv
`
`(cid:2)e
`
`(cid:2)e
`
`e
`
`)2
`
`(cid:2)e
`
`m(v
`2
`
`(6.7.10)
`
`(6.7.11)
`
`(6.7.12)
`
`Similar to Equation 6.7.7, the coefficients of absorption aω(ve) and stimulated
`) of an electron during its collision with a heavy particle are
`emission bω(v
`related through the Einstein formula:
`(cid:3)
`(cid:4)
`aω(ve) =
`) = ve
`ε + ω
`v(cid:2)
`(6.7.13)
`aω(ve).
`bω(v
`Here, ε is the electron energy (as well as ε + ω). The stimulated emis-
`sion coefficient is related to the spontaneous bremsstrahlung emission cross
`section as
`
`ε
`
`1/2
`
`e
`
`(cid:2)e
`
`(cid:2)e
`
`(6.7.14)
`
`(cid:2)e
`
`)
`
`dσω(v
`dω .
`
`(cid:2)e
`
`) = π2c2v
`
`ω2
`
`(cid:2)e
`
`bω(v
`
`6.7.4 Bremsstrahlung Emission due to Electron Collisions
`with Plasma Ions and Neutrals
`According to classical electrodynamics, emission of a system of electric
`charges is determined by the second derivative of its dipole momentum
`
`¨(cid:20)d. In the case of an electron (characterized by a radius vector (cid:20)r) scattering
`¨(cid:20)d = −e¨(cid:20)r. In the case of electron–electron colli-
`¨(cid:20)d = −e¨(cid:20)r1 − e¨(cid:20)r2 = (−e/m) d/dt(m(cid:20)v1 + m(cid:20)v2) = 0 because of momentum
`
`by a heavy particle at rest:
`sions,
`
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`

`
`414
`
`Plasma Physics and Engineering
`
`conservation. This relation explains the absence of bremsstrahlung emission
`in the electron–electron collisions. Total energy, emitted by an electron during
`interaction with a heavy particle, can be expressed by (Landau, 1982):
`+∞(cid:2)
`
`dt.
`
`(6.7.15)
`
`(cid:17)(cid:17)(cid:17)2
`(cid:17)(cid:17)(cid:17)¨(cid:20)r(t)
`E = e2
`6πε0c3
`−∞
`The Fourier expansion of electron acceleration ¨(cid:20)r(t) in Equation 6.7.15 gives
`the emission spectrum of the bremsstrahlung, the electron energy emitted per
`one collision in the frequency interval ω/(ω + dω):
`+∞(cid:2)
`
`(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)
`
`−∞
`
`dEω = e2
`3πε0c3
`
`¨(cid:20)r(t) exp(−iωt)dt
`
`2
`
`(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)
`
`dω.
`
`(6.7.16)
`
`Taking into account that the time of effective electron interaction with heavy
`a particle is shorter than the electromagnetic field oscillation time, the integral
`in Equation 6.7.16 can be estimated as the electron velocity change Δ(cid:20)ve during
`scattering. This allows simplification of Equation 6.7.16:
`dEω =
`
`e2
`6π2ε0c3 (Δv2
`e) dω.
`Averaging Equation 6.7.18 over all electron collisions with heavy particles
`(frequency νm, electron velocity ve), yields the formula for dQω (see Equa-
`tion 6.7.9), which is the bremsstrahlung emission power per electron in the
`frequency interval dω:
`
`(6.7.17)
`
`e
`
`(6.7.18)
`
`dQω = e2v2
`νm
`3π2ε0c3 dω.
`Obviously, this formula of classical electrodynamics can only be used at rel-
`atively low frequencies ω < mv2
`e/2, when the electron energy exceeds the
`emitted quantum of radiation. At higher frequencies ω > mv2
`e/2, we should
`assume dQω = 0 to avoid the “ultra-violet catastrophe.” Based on Equations
`6.7.18 and 6.7.9, the cross section of the spontaneous bremsstrahlung emission
`of a quantum ω by an electron with velocity ve can be expressed as
`dσω = e2v2
`σm
`3π2ε0c3ω dω,
`Bremsstrahlung emission is related to an electron interaction with heavy
`particles; therefore its cross section (Equation 6.7.19) is proportional to the
`cross section σm of electron–heavy particle collisions. Equations 6.7.13 and
`
`e
`
`ω ≤ mv2
`2
`
`e
`
`.
`
`(6.7.19)
`
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`
`Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`415
`
`6.7.14 then permit the derivation of formulas for the quantum coefficients
`of stimulated emission bω and absorption aω. For example, the absorption
`coefficient aω(ve) can be presented in this way at ω (cid:5) νm as
`(ε + ω)2
`
`aω(ve) = 2
`3εo
`
`e2ve
`mcω2
`
`σm(ε + ω),
`
`εω
`
`(6.7.20)
`
`where ε is electron energy before the absorption. Absorption and stimu-
`lated emission coefficients per one electron and one atom of Ar and He as
`a function of electron energy at a frequency of a ruby laser are presented in
`Figure 6.34. Plasma radiation is more significant in thermal plasma, where
`the major contribution in νm and σm is provided by electron–ion collisions.
`Taking into account the Coulomb nature of the collisions, the cross section of
`the bremsstrahlung emission Equation 6.7.19 is
`
`dσω =
`
`√
`
`16π
`3(4πε0)3
`
`3
`
`Z2e6
`m2c3v2ω.
`
`(6.7.21)
`
`Here, Z is the charge of an ion; in most of our systems under consideration
`Z = 1, but relation (Equation 6.7.21) in general can be applied to multicharged
`ions as well. Based on Equation 6.7.21, the total energy emitted in unit volume
`
`aω, bω, 10–39cm5
`
`aω
`
`bω
`
`Ar
`
`hω=1.78 eV
`
`aω
`
`bω
`
`He
`
`10
`
`20
`
`ε, eV
`
`30
`
`20
`
`10
`
`0
`
`FIGURE 6.34
`Absorption coefficients per electron and per one Ar and one He atom.
`
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`
`416
`
`Plasma Physics and Engineering
`
`per unit time by the bremsstrahlung mechanism in the spectral interval dω is
`∞(cid:2)
`
`Jbrems
`
`ω
`
`dω =
`
`ωninef (ve)vedvedσω(ve).
`
`(6.7.22)
`
`vmin
`
`Here, ni and ne are concentrations of ions and electrons; f (ve) is the electron
`velocity distribution function, which can be taken here as Maxwellian; vmin =
`√
`2ω/m is the minimum electron velocity sufficient to emit a quantum ω.
`After integration the spectral density of bremsstrahlung emission (Z = 1) per
`unit volume, Equation 6.7.23 finally gives
`(cid:3)
`(cid:4)
`(cid:4)
`(cid:3)
`dω = 16
`e6neni
`2π
`m3/2c3(4πε0)3T1/2 exp
`(cid:4)
`3
`3
`= C
`− ω
`nine
`T1/2 exp
`T
`−45W cm3 K1/2 × ni(1/cm3)ne(1/cm3)
`= 1.08 × 10
`(T, K)1/2
`
`Jbrems
`
`ω
`
`1/2
`
`(cid:3)
`
`dω
`
`− ω
`T
`
`dω
`
`(cid:3)
`
`exp
`
`(cid:4)
`
`dω.
`
`(6.7.23)
`
`− ω
`T
`
`6.7.5 Recombination Emission
`This emission occurs during the radiative electron–ion recombination.
`According to the classification of emission in plasma, the recombination emis-
`sion is related to free-bound transitions because as a result of this process, a
`free plasma electron becomes trapped in a bound atomic state with negative
`discrete energy En. This recombination leads to emission of a quantum
`ω = |En| + mv2
`2
`Correct quantum mechanical derivation of the recombination emission
`cross section is complicated, but an approximate formula can be derived using
`the quasi-classical approach. Equation 6.7.21 was derived for bremsstrahlung
`emission when the final electron is still free and has positive energy. To cal-
`culate the cross section σRE of the recombination emission, generalize the
`quasi-classical relation (Equation 6.7.21) for electron transitions into discrete
`bound states. It can be done because the energy distance between high elec-
`tronic levels is quite small and hence quasi-classical. The electronic levels of
`bound atomic states are discrete; so here the spectral density of the recombi-
`nation cross section should be redefined taking into account number of levels
`Δn per small energy interval ΔE = Δω (around En):
`Δω = σREΔn,
`σRE = dσω
`dσω
`ΔE
`dω
`dω
`Δn
`
`(6.7.25)
`
`.
`
`1 
`
`e
`
`.
`
`(6.7.24)
`
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`

`
`Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`417
`
`Consider the hydrogen like atoms (where an electron moves in the field of a
`charge Ze), then En = −IHZ2/n2. Here, IH = me4/22(4πε0)2 ≈ 13.6 eV is the
`hydrogen atom ionization potential, n is the principal quantum number. The
`ratio ΔE/Δn characterizing the density of energy levels is
`(cid:17)(cid:17)(cid:17)(cid:17)dEn
`(cid:17)(cid:17)(cid:17)(cid:17) = 2IHZ2
`
`≈
`
`ΔE
`Δn
`
`dn
`
`.
`
`n3
`
`(6.7.26)
`
`Based on Equation 6.7.22 and taking into account Equations 6.7.25 and
`6.7.26, the formula for the cross section of the recombination emission in an
`electron–ion collision with formation of an atom in an excited state with the
`principal quantum number n is obtained:
`
`(6.7.27)
`
`= 2.1 × 10
`
`−22cm2 (IHZ2)2
`εωn3 .
`
`1 n
`
`3
`
`e10Z4
`(4πε0)5c34mv2
`e
`
`ω
`
`σRE = 4
`√
`3
`
`3
`
`Here, ε is the initial electron energy. The total energy Jωndω, emitted as a
`result of photo-recombination of electrons with velocities in interval ve/ve +
`dve and ions with Z = 1, leading to formation of an excited atom with the
`principal quantum number n, per unit time and per unit volume is
`(cid:3)
`(cid:4)
`
`Jrecomb
`
`ω
`
`dω =
`
`Jωn.
`
`(ω)
`
`(6.7.29)
`
`Each of the individual terms in Equation 6.7.29 is related to the different
`principal quantum number n but giving the same frequency ω, and each is
`proportional to exp(−ω/T). The lowest possible principal quantum number
`(ω) can be defined from the condition |En∗| < ω < |En∗−1|. The spec-
`∗
`n
`trum of recombination emission corresponding to the sum Equation 6.7.29
`is illustrated in Figure 6.35.
`
`Jωndω = ωnineσREf (ve)vedve = C
`
`dω.
`
`(6.7.28)
`
`− ω
`2IH
`nine
`IH
`Tn3 exp
`T1/2
`Tn2
`T
`The parameter C = 1.08 × 10
`−45 W cm3 K1/2 is the same factor as in Equa-
`tion 6.7.23; f (ve) is the electron velocity distribution, which was taken as
`Maxwellian; the relation between electron velocity and radiation frequency
`from Equation 6.7.24 was written as: mvedve = dω. Emission of quanta with
`the same energy ω can be provided by trapping electrons on different exci-
`tation levels with different principal quantum numbers n (obviously, the
`electron velocities should be relevant and determined by Equation 6.7.24).
`The total spectrum of recombination emission is the sum of similar but shifted
`terms:
`∞(cid:10)
`n∗
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`
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`
`Plasma Physics and Engineering
`
`Jωrec
`
`Jωn
`
`ω
`
`ωn
`
`ωn–1
`
`FIGURE 6.35
`Recombination emission spectrum.
`
`Ψ
`
`dω.
`
`(6.7.30)
`
`,
`
`(6.7.31a)
`
`6.7.6 Total Emission in Continuous Spectrum
`The total spectral density of plasma emission in continuous spectrum consists
`of the bremsstrahlung and recombination components (Equations 6.7.23 and
`6.7.29). These two emission components can be combined in one approximate
`but reasonably accurate general formula for the total continuous emission:
`(cid:4)
`(cid:3)
`Jωdω = C
`nine
`ω
`T1/2
`T
`The parameter C = 1.08 × 10
`−45W cm3 K1/2 is the same factor as in Equa-
`tions 6.7.23 and 6.7.29; Ψ(x) is the dimensionless function, which can be
`approximated as
`(cid:17)(cid:17)
`(cid:17)(cid:17)Eg
`Ψ(x) = 1,
`if x = ω
`< xg =
`T
`T
`Ψ(x) = exp[−(x − xg)],
`if xg < x < x1
`(6.7.31b)
`Ψ(x) = exp[−(x − xg)] +2x 1 exp[−(x − x1)],
`if x = ω
`> x1 = I
`.
`T
`T
`(6.7.31c)
`(cid:17)(cid:17)Eg
`(cid:17)(cid:17) is the energy of the first (lowest) excited state of atom calcu-
`Here,
`lated with respect to transition to continuum; I is the ionization potential.
`When the radiation quanta and hence frequencies are not very large (ω <
`|Eg|), the contributions of free-free transitions (bremsstrahlung) and free-
`bound transitions (recombination) in the total continuous emission are related
`to each other as
`− 1.
`= exp
`
`Jrecomb
`
`ω
`
`/Jbrems
`
`ω
`
`ω
`T
`
`(6.7.32)
`
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`
`419
`
`According to Equation 6.7.32, emission ω < 0.7T is mostly due to the
`bremsstrahlung mechanism, while emission of larger quanta ω > 0.7T
`is mostly due to recombination mechanism. At plasma temperatures of
`10,000 K, this means that only infrared radiation (λ > 2μm) is provided
`by bremsstrahlung, all other emission spectrum is due to the electron–ion
`recombination. To obtain the total radiation losses, integrate the spectral den-
`sity (Equation 6.7.30) over all the emission spectrum (contribution of quanta
`ω > I can be neglected because of their intensive reabsorption). These total
`plasma energy losses per unit time and unit volume can be expressed after
`integration by the following numerical formula:
`(cid:11)
`(cid:12)
`(cid:13)
`1 +
`
`J,
`
`kW
`cm3
`
`= 1.42 × 10
`
`−37
`
`T, K neni(cm
`
`−3)
`
`(cid:17)(cid:17)
`
`(cid:17)(cid:17)Eg
`
`T
`
`.
`
`(6.7.33)
`
`6.7.7 Plasma Absorption of Radiation in Continuous Spectrum:
`The Kramers and Unsold-Kramers Formulas
`The differential cross section of bremsstrahlung emission (Equation 6.7.21)
`with Equations 6.7.13 and 6.7.14 permits determining the coefficient of
`bremsstrahlung absorption of a quantum ω calculated per electron having
`velocity ve and one ion:
`
`aω(ve) = 16π3
`√
`3
`3
`
`Z2e6
`m2c(4πε0ω)3ve
`
`.
`
`(6.7.34)
`
`This is the so-called Kramers formula of the quasi-classical theory of
`plasma continuous radiation. Multiplying the Kramers formula by neni and
`integrating over the Maxwellian distribution f (ve), yields the coefficient of
`bremsstrahlung absorption in plasma (the reverse length of absorption):
`(cid:3)
`(cid:4)
`
`2
`3π
`
`1/2
`
`e6
`(4πε0)3m3/2c
`
`.
`
`(6.7.35)
`
`3
`
`κbrems
`ω
`
`= C1
`
`neni
`
`T1/2ν3 , C1 = 2
`
`To use Equation 6.7.35 for numerical calculations of the absorption
`−1), the coefficient C1can be taken as: C1 = 3.69 ×
`coefficient κbrems
`(cm
`108cm5/s3 K1/2; here the frequency ν = ω/2π, and Z = 1. Another mechanism
`ω
`of plasma absorption of radiation in continuous spectrum is photoion-
`ization. The photoionization cross section can be easily calculated, taking
`into account that it is a reverse process with respect to recombination
`emission. Based on a detailed balance of the photoionization and recom-
`bination emission, Saha equations 4.1.15 and 6.7.27, the expression for the
`photoionization cross section for a photon ω and an atom with principal
`
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`
`Plasma Physics and Engineering
`
`(cid:8)
`
`3
`
`ω
`
`neni
`
`quantum number n is
`(cid:7) ωn
`= 7.9 × 10
`σωn = 8π
`√
`−18cm2 n
`e10mZ4
`.
`(6.7.36)
`Z2
`(4πε0)5c6ω3n5
`3
`3
`Here, ωn = |En| / is the minimum frequency sufficient for photoionization
`from the electronic energy level with energy En and principal quantum
`number n. As seen from Equation 6.7.36, the photoionization cross sec-
`tion decreases with frequency as 1/ω3 at ω > ωn. To calculate the total
`(cid:18)
`plasma absorption coefficient in a continuum it is necessary to add to the
`bremsstrahlung absorption Equation 6.7.34, the sum
`n0nσωn related to
`the photoionization of atoms in different states of excitation (n) with con-
`centrations n0n. Replacing the summation by integration, the total plasma
`absorption coefficient in continuum can be calculated for Z = 1 as
`−3)
`κω = C1
`T1/2ν3 exΨ(x) = 4.05 × 10
`−1 neni(cm
`−23cm
`exΨ(x)
`.
`(6.7.37)
`(T, K)7/2
`x3
`Here, factor C1 is the same as in Equation 6.7.35; parameter x = ω/T;
`and the function Ψ(x) is defined by Equation 6.7.31. Obviously, the replace-
`ment of summation over discrete levels by integration makes the resulting
`approximate dependence κω(ν) smoother than in reality; this is illustrated
`in Figure 6.36 (Biberman and Norman, 1967). The relative contribution of
`photoionization and bremsstrahlung mechanisms in total plasma absorption
`of radiation in continuum at the relatively low frequencies ω < |Egs| (|Eg| is
`energy of the lowest excited state with respect to continuum) can be charac-
`terized by the same ratio as in case of emission (Equation 6.7.32). The product
`neni in Equation 6.7.37 can be replaced at not very high temperatures by the
`gas density n0using the Saha equation 4.1.15. At the relatively low frequencies
`ω < |Eg| the total plasma absorption coefficient in continuum is
`(cid:4)
`(cid:3)
`ω − I
`κω = 16π
`√
`e6Tn0
`gi
`exp
`4cω3(4πε0)3
`ga
`T
`3
`3
`−3)
`−(x1−x)
`= 1.95 × 10
`−7cm
`−1 n0(cm
`(T, K)2
`x3
`
`e
`
`gi
`ga
`
`.
`
`(6.7.38)
`
`This relation for the absorption coefficient is usually referred to as the
`Unsold–Kramers formula, where ga, gi are statistical weights of an atom and
`an ion, x = ω/T, x1 = I/T and I is the ionization potential. As an example,
`ω of red light λ = 0.65μm in an atmospheric pres-
`the absorption length κ−1
`sure hydrogen plasma at 10,000 K can be calculated from Equation 6.7.38 as
`ω ≈ 180 m. Thus, this plasma is fairly transparent in continuum at such con-
`κ−1
`ditions. It is interesting to note that the Unsold–Kramers formula can be used
`only for quasi-equilibrium conditions, while Equation 6.7.37 can be applied
`for nonequilibrium plasma as well.
`
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`
`421
`
`xωϕ/N, cm2
`10–17
`
`Xe
`
`20,000 K
`
`10,000 K
`
`v, 103 cm–1
`
`10
`
`20
`
`30
`
`7 5 3
`
`2
`
`10–20
`
`7 5 3
`
`FIGURE 6.36
`Absorption coefficient per one Xe atom, related to photoionization. Dashed line—summation
`over actual energy levels. Solid line—replacement of the summation by integration.
`
`6.7.8 Radiation Transfer in Plasma
`The intensity of radiation Iω decreases along its path “s” due to absorption
`(scattering in plasma neglected) and increases because of spontaneous and
`stimulated emission. The radiation transfer equation in quasi-equilibrium
`plasma can be then given as
`(cid:3)
`(cid:5)
`(cid:4)(cid:6)
`
`ω = κω
`κ(cid:2)
`
`1 − exp
`
`− ω
`T
`
`ω(Iωe − Iω),
`= κ(cid:2)
`
`dIω
`ds
`
`.
`
`(6.7.39)
`
`Here, the factor Iωe is the quasi-equilibrium radiation intensity
`
`Iωe = ω3
`4π3c2
`
`1
`exp(ω/T) − 1
`
`,
`
`(6.7.40)
`
`which can be obtained from the Planck formula for the spectral density
`(Equation 4.1.21). Introduce the optical coordinate ξ, calculated from the
`
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`
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`
`Plasma Physics and Engineering
`
`plasma surface x = 0 (with positive x directed into the plasma):
`ξ = x(cid:2)
`
`ω(x) dx, dξ =κ(cid:2)ω(x) dx.
`κ(cid:2)
`
`
`
`(6.7.41)
`
`0
`
`Then the radiation transfer equation can be rewritten in terms of the optical
`coordinate as
`− Iω(ξ) = −Iωe.
`dIω(ξ)
`(6.7.42)
`dξ
`Assuming that in direction of a fixed ray, the plasma thickness is “d”; x = 0
`corresponds to the plasma surface; and there is no source of radiation at
`x > d, then the radiation intensity on the plasma surface Iω0 can be expressed
`by solution of the radiation transfer equation as
`τω(cid:2)
`
`Iω0 =
`
`Iωe[T(ξ)] exp(−ξ) dξ,
`
`κ(cid:2)
`ωdx.
`
`(6.7.43)
`
`τω = d(cid:2)
`The quasi-equilibrium radiation intensity Iωe[T(ξ)] is shown here as a
`function of temperature and therefore an indirect function of the optical coor-
`dinate. In Equation 6.7.43, another radiation transfer parameter is introduced:
`τω is the optical thickness of plasma.
`
`0
`
`0
`
`6.7.9 Optically Thin Plasmas and Optically Thick Systems: Blackbody
`Radiation
`First assume that optical thickness is small τω (cid:10) 1; this is usually referred to
`as transparent or optically thin plasma. In this case, the radiation intensity on
`the plasma surface Iω0 (Equation 6.7.43) is
`τω(cid:2)
`τω(cid:2)
`
`Iω0 =
`
`Iωe[T(ξ)] dξ =
`
`ωdx = d(cid:2)
`Iωeκ(cid:2)
`
`jωdx.
`
`(6.7.44)
`
`0
`
`0
`
`0
`
`Here, the emissivity term jω = Iωeκ(cid:2)
`ω corresponds to spontaneous emission.
`Equation 6.7.44 shows that radiation of optically thin plasma is the result of
`summation of independent emission from different intervals dx along the
`ray. In this case, all radiation generated in plasma volume is able to leave it.
`If plasma parameters are uniform, then the radiation intensity on the plasma
`surface can be expressed as
`
`
`Iω0 = jωd = Iωeκ(cid:2)ωd = Iωeτω (cid:10) Iωe.
`
`(6.7.45)
`
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`
`423
`
`From Equation 6.7.45, the radiation intensity of the optically thin plasma is
`seen to be much less (τω (cid:10) 1) than equilibrium value (Equation 6.7.40) cor-
`responding to the Planck formula. The opposite case takes place when the
`optical thickness is high τω (cid:5) 1. This is usually referred to as the nontrans-
`parent or optically thick systems. If the quasi-equilibrium temperature can be
`considered constant in the emitting body, then for the optically thick systems
`(Equation 6.7.43) gives: Iω0 = Iωe(T). The emission density of the entire sur-
`face in all directions is the same and is equal to the equilibrium Planck value
`(Equation 6.7.40). This is the case of the quasi-equilibrium blackbody emis-
`sion, discussed in Section 4.1.5. Unfortunately, it cannot be directly applied
`to continuous plasma radiation (where the plasma is usually optically thin).
`The total black body emission per unit surface and unit time can be found
`by integration of the quasi-equilibrium radiation intensity (Equation 6.7.40)
`over all frequencies and all directions of hemispherical solid angle (2π)
`(cid:2)
`(cid:2)
`(cid:2)
`Je =
`Iωe cos ϑdΩ =
`πIωe(ω, T)dω = σT4.
`
`(6.7.46)
`
`dω
`
`2π
`
`This is the Stefan–Boltzmann law of blackbody emission. σ is the Stephan–
`Boltzmann coefficient (Equation 4.1.25); ϑ is the angle between the ray (Ω)
`and normal vector to the emitting surface.
`
`6.7.10 Reabsorption of Radiation, Emission of Plasma as Gray
`Body: The Total Emissivity Coefficient
`Plasmas are usually optically thin for radiation in the continuous spectrum,
`and the Stephan–Boltzmann law cannot be applied without special correc-
`tions. However, plasma is not absolutely transparent; the total emission can be
`affected by reabsorption of radiation. This can be illustrated by Equation 6.7.43
`for the radiation intensity Iω0 assuming fixed values of the quasi-equilibrium
`temperature and absorption coefficient (Equation 6.7.37)
`
`Iω0 = Iωe[1 − exp(−τω)] =I ωe[1 − exp(−κ(cid:2)ωd)] = Iωeε.
`Here, ε is the total emissivity coefficient, which characterizes the plasma