Plasma
`Physics
`and
`Engineering
`
`Alexander Fridman
`Lawrence A. Kennedy
`
`C\ Taylor & Francis
`~ Taylor & Francis Group
`
`NEWYORK • LONDON
`
`Energetiq Ex. 2022, page 1 - IPR2015-01277
`
`

`
`Denise T. Schanck, Vice President
`Robert L. Rogers, Senior Editor
`Summers Scholl, Editorial Assistant
`Savita Poornam, Marketing Manager
`Randy Harinandan, Marketing Assistant
`
`Susan Fox, Project Editor
`Shayna Murry, Cover Designer
`
`Published in 2004 by
`Taylor & Francis
`29 West 35th Street
`New York, NY 10001-2299
`
`Published in Great Britain by
`Taylor & Francis
`4 Park Square
`Milton Park
`Abingdon OX14 4RN
`
`www.taylorandfrancis.com
`
`Copyright © 2004 by Taylor & Francis Books, Inc.
`
`Printed in the United States of America on acid-free paper.
`
`All rights reserved. No part of this book may be reprinted or reproduced or utilized in any
`form or by any electronic, mechanical, or other means, now known or hereafter invented,
`including photocopying and recording, or in any information storage or retrieval system,
`without permission in writing from the publishers.
`
`10 9 8 7 6 5 4 3 2 1
`
`Library of Congress Cataloging-in-Publication Data
`
`Fridman, Alexander A., 1953-
`Plasma physics and engineering I by Alexander
`A. Fridman and Lawrence A. Kennedy
`p. em.
`ISBN 1-56032-848-7 (hardcover: alk. paper)
`1. Plasma (Ionized gases). 2. Plasma engineering. I. Kennedy, Lawrence A., 1937-
`II. Title.
`
`QC718.F77 2004
`530.4'4-dc22
`
`2003022820
`
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`

`
`404
`
`Chapter 6: Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`2
`2
`)
`co
`c
`J.LP
`c
`- - - = 1 + __ P_ = 1 + _o_"'-
`(co/ k )2
`co Bco Bi
`v!
`Bz
`
`(6.214)
`
`This introduces again the Alfven wave velocity vA (Equation 6.139); here
`p = neM is the mass density in completely ionized plasma.
`'
`Only a few principal types of electromagnetic waves in magnetized plasma have
`been discussed. For example, numerous possible modes are related to electromag(cid:173)
`netic wave propagation not along the magnetic field. Some of them are very impor(cid:173)
`tant in plasma diagnostics and in high-frequency plasma heating. More details on
`the subject can be found in Ginsburg215 and Ginsburg and Rukhadze. 216
`
`6.7 EMISSION AND ABSORPTION OF RADIATION IN PLASMA:
`CONTINUOUS SPECTRUM
`
`6.7.1 Classification of Radiation Transitions
`
`Because of its application in different lighting devices, radiation is probably the
`most commonly known plasma property. Radiation also plays an important role in
`plasma diagnostics, including plasma spectroscopy, in the propagation of some
`electric discharges, and sometimes even in plasma energy balance.
`From a quantum mechanical point of view, radiation occurs due to transitions
`between different energy levels of a quantum system: transition up conesponds to
`absorption of a quantum, E1 - Ei = nco; transition down the spectrum conesponds
`to emission Ei- E1 = nco (see Section 6.7 .2). From the point of classical electrody(cid:173)
`namics, radiation is related to the nonlinear change of dipole momentum, actually
`with the second derivative of dipole momentum (see Section 6.7.4).
`It should be mentioned that neither emission nor absorption of radiation is
`impossible for free electrons. As was shown in Section 6.6, electron collisions are
`nece~sary 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 interaction cannot.
`It is convenient to classify different types of radiation according to the different
`types of an 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 electron 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 positive ion or in interaction with neutrals.
`The emitted energy in this case is a continuum (usually infrared) called bremsstrahl·
`ung (direct translation: stopping radiation). The reverse process is the bremsstrahl(cid:173)
`ung absorption.
`Electron transition between a free state in continuum and a bound state in atom
`(see Figure 6.32) is usually refened to as the free-bound transition. These transi(cid:173)
`tions conespond to processes of the radiative electron-ion recombination (see Sec(cid:173)
`tion 2.3.5) and the reverse one of photoionization (see Section 2.2.6). These kinds
`
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`

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`Fluid Mechanics of Plasma
`
`Emission and Absorption of Radiation in Plasma: Continuous Spectrum
`
`405
`
`2
`
`-= 1+1-loP ""~
`vz
`B 2
`A
`
`Bi
`
`(6.214)
`
`·ave velocity vA (Equation 6.139); here,
`'I ionized plasma.
`tagnetic waves in magnetized plasma have
`Jossible modes are related to electromag(cid:173)
`netic field. Some of them are very impor(cid:173)
`~quency plasma heating. More details on
`ad Ginsburg and Rukhadze. 21 6
`
`N OF RADIATION IN PLASMA:
`
`ansitions
`ghting devices, radiation is probably the
`Radiation also plays an important role in
`•ectroscopy, in the propagation of some
`n plasma energy balance.
`' view, radiation occurs due to transitions
`tum system: transition up conesponds to
`ransition down the spectrum conesponds
`2). From the point of classical electrody(cid:173)
`~ar change of dipole momentum, actually
`entum (see Section 6.7.4).
`emission nor absorption of radiation is
`wn in Section 6.6, electron collisions are
`in Section 6.7.4 that electron interaction
`Jle to provide emission or absorption, but
`
`·pes of radiation according to the different
`;tate to another. Electron energy levels in
`~tween the energy levels are illustrated in
`md final electron states are in continuum
`electron in this transition loses part of its
`ositive ion or in interaction with neutrals.
`urn (usually infrared) called bremsstrahl·
`). The reverse process is the bremsstrahl-
`
`:e in continuum and a bound state in atom
`the free-bound transition. These transi(cid:173)
`tive electron-ion recombination (see Sec(cid:173)
`mization (see Section 2.2.6). These kinds
`
`bound-bound
`
`Figure 6.32 Energy levels and electron transitions induced by ion field
`
`------~------~1- upper level
`
`1iro
`
`j
`
`lower level
`
`Figure 6.33 Radiative transition between two energy levels
`
`of transitions can also take place in electron-neutral collisions. In such a case these
`are related to photoattachment and photodetachment processes of formation and
`destruction of negative ions (see Section 2.4 ). The free-bound transitions conespond
`to continuum radiation.
`Finally, the bound-bound transitions mean transition between discrete atomic
`levels (see Figure 6.32) and result in emission and absorption of spectral lines.
`Molecular spectra are obviously much more complex than those of single atoms
`because of possible transitions between different vibrational and rotational levels
`(see Section 3.2).
`
`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 nffi, which is illustrated in Figure
`6.33. The probability of a photon absorption by an atom per unit time (and thus
`atom transition "0" ~ "u") can be expressed as:
`
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`
`406
`
`Chapter 6: Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`P("O" ~ "u" n - 1) = An
`' co
`,nro
`
`ro
`
`(6215)
`
`Here, nO) is the number of photons and A is the Einstein coefficient, which depends
`on atomic parameters and does not depend on electromagnetic wave characteristics.
`Similarly, the probability of atomic transition with a photon emission is:
`
`P("u" n ~ "0" n + 1) = _! + Bn
`'t
`co
`co
`
`'
`
`(0
`
`'
`
`(6.216)
`
`Here, 111: is the frequency of spontaneous emission of an excited atom or molecule
`which takes place without direct relation with external fields; another Einstei~
`coefficient, B, characterizes emission induced by an external electromagnetic field,
`The factors B and 1: as well as A depend only on atomic parameters. Thus, if the
`first right-side term in Equation 6.216 corresponds to spontaneous emission, the
`second term is related to stimulated emission.
`To find relations between the Einstein coefficients A and B and the spontaneous
`emission frequency l/1:, analyze the thermodynamic equilibrium of radiation with
`the atomic system under consideration. 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:
`
`n = g" n exp(- 1iOJ)
`"
`8o o
`T
`
`(6.217)
`
`where 1iOJ is energy difference between the two states and g" and g0 are their statistical
`weights. According to the Planck distribution (see Section 4.1.5 and Equation 4.19),
`the averaged number of photons nO) in one state can be determined as:
`
`(6.218)
`
`Then, taking into account the detailed balance of photon emission and absorp(cid:173)
`tion for the system illustrated in Figure 6.33,
`
`which can be rewritten based on Equation 6.215 and Equation 6.216 as:
`
`n0An0) =n~~(~+BnO))
`
`(6.219)
`
`(6.220)
`
`The relations between the Einstein coefficients A and B and the spontaneous
`emission frequency 1/1: can be expressed based on Equation 6.217, Equation 6.218,
`and Equation 6.220 as:
`
`(6.221)
`
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`

`
`408
`
`Chapter 6: Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`describes the density of the radiation energy flux. Then, the radiation energy frolll
`the interval dw dQ absorbed by electrons with velocities v e + v e + dv e in unit volume
`per unit time can be presented as:
`
`This is the definition of the absorption coefficient aro(v_), which is actually calcu(cid:173)
`lated with respect to one electron and one atom. In its definition (Equation 6.224),
`n0 is the density of heavy particles and f (ve) is the electron distribution function.
`Similar to Equation 6.224, the stimulated emission of quanta related to electron
`collisions with heavy particles can be expressed as:
`
`(6.224)
`
`(6.225)
`
`Here, the coefficient of stimulated emission b ro ( v;) is introduced, which is also
`calculated with respect to one electron and one atom. Note that in accordance with
`the energy conservation law for mutually reverse processes of absorption (Equation
`6.224) and stimulated emission (Equation 6.225), the electron velocities v and v'
`e
`e
`are related to each other as:
`
`m(v;)
`2
`
`2
`
`= mv~ + nffi
`2
`
`(6.226)
`
`Similar to Equation 6.221, the coefficients of absorption aro(v_) and stimulated
`emission b ro ( v;) of an electron during its collision with a heavy particle are related
`to each other through the Einstein formula:
`
`Here, E is the electron energy (as well as E + nffi ). The stimulated emission coefficient
`is related to the spontaneous bremsstrahlung emission cross section as:
`
`(6.227)
`
`(6.228)
`
`6.7.4 Bremsstrahlung Emission due to Electron Collisions with Plasma
`Ions and Neutrals
`
`To be correct, the Bremsstrahlung emission cross section daJdw as well as coeffi(cid:173)
`cients a 0, and bro should be derived in the frameworks of quantum mechanics.
`However, classical derivation of these factors is more interesting and clear.
`According to classical electrodynamics, emission of a syste~. of electric charges
`is determined by the second derivative of its dipole momentum d. In the case of an
`
`beca
`brem
`by ar
`by th
`
`'1
`yield
`emitt
`
`parti•
`Equz
`in g.
`
`(freq
`whic
`inter
`
`Obv
`low
`radi<
`"ultr
`
`neor
`bee
`
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`

`
`Fluid Mechanics of Plasma
`
`Emission and Absorption of Radiation in Plasma: Continuous Spectrum
`
`409
`
`rgy flux. Then, the radiation energy from
`with velocities ve + ve + dve in unit volume
`
`(6.224)
`
`oefficient aOJ(ve), which is actually calcu(cid:173)
`~ atom. In its definition (Equation 6.224),
`" ( v e) is the electron distribution function.
`d emission of quanta related to electron
`•ressed as:
`
`(6.225)
`
`ssion b OJ ( v;) is introduced, which is also
`d one atom. Note that in accordance with
`reverse processes of absorption (Equation
`1 6.225), the electron velocities ve and v;
`
`mv2
`_ _ e +tiro
`2
`
`(6.226)
`
`;ients of absorption aOJ(ve) and stimulated
`;ol!ision with a heavy particle are related
`la:
`
`)112
`£
`(
`- - a (v)
`\£+lim
`OJ • e
`
`(6.227)
`
`-lim). The stimulated emission coefficient
`ng emission cross section as:
`
`2v' dcr (v')
`w
`_e
`e
`2
`dro
`
`(6.228)
`
`to Electron Collisions with Plasma
`
`·n cross section dcrJdw as well as coeffi(cid:173)
`:he frameworks of quantum mechanics.
`.ors is more interesting and clear.
`s, emission of a syste~. of electric charges
`[ts dipole momentum d. In the case of an
`
`t;~ectro.? (characterized by a radius vector r) scatteting by a heavy particle at rest,
`J == -er. In the case of electron-electron collisions,
`=
`··
`··
`e d
`d=-eT. -er =--·-(mv +mv )=0
`m dt
`
`I
`
`2
`
`I
`
`2
`
`because of momentum conservation. This relation explains the absence of
`bremsstrahlung emission in the electron-electron collisions. Total energy, emitted
`by an electron during the time of interaction with a heavy particle, can be expressed
`by the following relation: 217•217a
`
`..
`Using the Fourier expansion of electron acceleration r(t) in Equation 6.229
`yields the emission spectrum of the bremsstrahlung, which means the electron energy
`emitted per one collision in frequency interval ro + ro + dw:
`
`(6.229)
`
`+oo
`
`2
`
`dE = ~ J . ~(t) exp( -iwt)dt dw
`3rrt: 0c
`
`OJ
`
`(6.230).
`
`Taking into account that the time of effective electron interaction with a heavy
`particle is shorter than the electromagnetic field oscillation time, the integral in
`Equation 6.230 can be estimated as the electron velocity change ~ve during scatter(cid:173)
`ing. This allows simplification of Equation 6.230:
`
`(6.231)
`
`Averaging Equation 6.232 over all electron collisions with heavy particles
`(frequency v"', electron velocity ve) yields the formula for dQOJ (see Equation 6.223),
`which is the bremsstrahlung emission power per one electron in the frequency
`interval dro:
`
`(6.232)
`
`Obviously, this formula of classical electrodynamics can only be used at relatively
`low frequencies ro < mv; /2ti, when electron energy exceeds the emitted quantum of
`radiation. At higher frequencies ro > mv; /2ti, dQOJ = 0 must be assumed to avoid the
`"ultraviolet catastrophe."
`Based on Equation 6.232 and Equation 6.223, the cross section of the sponta(cid:173)
`neous bremsstrahlung emission of a quantum tiro by an electron with velocity ve can
`be expressed as:
`
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`

`
`410
`
`Chapter 6: Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`Figure 6.34 Absorption coefficients per one electron and per one Ar, He atom
`
`(6.233)
`
`Bremsstrahlung emission is related to an electron interaction with heavy parti(cid:173)
`cles; therefore, its cross section (Equation 6.233) is propmtional to the cross section
`am of electron-heavy particle collisions. Equation 6.227 and Equation 6.228 then
`permit the devivation of formulas for the quantum coefficients of stimulated emission
`bw and absorption aw. For example, the absorption coefficient aw(ve) can be presented
`in this way at co>> vm as:
`
`(6.234)
`
`where £ is electron energy before the absorption. Numerical examples of absorption
`and stimulated 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. 101
`Plasma radiation is more significant in thermal plasma, where the major con(cid:173)
`tribution in V111 and cr"' is provided by electron-ion collisions. Taking into account
`the coulomb nature of the collisions, the cross section of the bremsstrahlung emission
`(Equation 6.233) can be rewritten as:
`
`Z 2e 6
`16n
`dcr = -= - - - - - : : - ----::--=-:c--
`3 m 2c3v 2nco
`3-J3(4n£0
`w
`)
`Here, Z is the charge of an ion; in most of our systems under consideration, Z = l,
`but relation 6.235 in general can be applied to multicharged ions as well. Based on
`
`(6.235)
`
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`

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`'luid Mechanics of Plasma
`
`Emission and Absorption of Radiation in Plasma: Continuous Spectrum
`
`411
`
`o=1.78eV
`
`20
`
`E, eV
`
`1 and per one Ar, He atom
`
`dw,
`
`2
`,
`mv
`n(J)::; __ e
`2
`
`(6.233)
`
`an electron interaction with heavy parti-
`5.233) is proportional to the cross section
`~quation 6.227 and Equation 6.228 then
`antum coefficients of stimulated emission
`lrption coefficient a00(ve) can be presented
`
`(6.234)
`
`ption. Numerical examples of absorption
`ne electron and one atom of Ar and He
`tcy of a ruby laser are presented in Figure
`
`e1 thermal plasma, where the major con(cid:173)
`tron-ion collisions. Taking into account
`ss section of the bremsstrahlung emission
`
`(6.235)
`
`our systems under consideration, Z = 1,
`i to multicharged ions as well. Based on
`
`Equation 6.235, the total energy emitted in unit volume per unit time by the
`bremsstrahlung mechanism in the spectral interval dw can be expressed by the
`following integral:
`
`J!re/JISd(J) = f n(J)ninJ(ve)vedved(J())(vJ
`
`=
`
`(6.236)
`
`Bere, ni and ne are concentrations of ions and electrons; f(ve) is the electron velocity
`distribution function, which can be taken here as Maxwellian; and vmin = ,}2nwjm is
`the minimum electron velocity sufficient to emit a quantum tzw. After integration,
`the spectral density of bremsstrahlung emission (Z = 1) per unit volume (Equation
`6.237) finally gives:
`
`Obviously, this formula can only be applied to the quasi-equilibrium thermal plasmas
`that are able to be described by the one temperature T.
`
`(6.237)
`
`6.7.5 Recombination Emission
`Recombination emission takes place during the radiative electron-ion recombination
`discussed in Section 2.3.5 as one of possible recombination mechanisms. According
`to the classification of different kinds of emission in plasma (Section 6.7.1), the
`recombination emission 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:
`nw =IE I+ _ e
`mv 2
`2
`
`(6.238)
`
`n
`
`Correct quantum mechanical derivation of the recombination emission cross
`section is complicated, but an approximate formula can be derived using the fol(cid:173)
`lowing quasi-classical approach. Equation 6.235 was derived for bremsstrahlung
`emission when the final electron is still free and has positive energy. To calculate
`the cross section aRE of the recombination emission, generalize the quasi-classic
`relation (Equation 6.235) for electron transitions into discrete bound states. Such
`generalization is acceptable, noting that the energy distance between high electronic
`levels is quite small and thus quasi classic (see Section 3.1).
`The electronic levels of bound atomic states are discrete, so here the spectral
`density of the recombination cross section should be redefined taking into account
`number of levels !'ill per small energy interval till= nllw (around E11
`
`):
`
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`
`412
`
`Chapter 6: Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`dcr"'
`dffi fl.(f) = (J REt,.n,
`
`dcr"' 1M
`(J RE = dffi fz t,.n
`
`(6.239)
`
`For simplicity, consider the hydrogen-like atoms (where an electron moves in
`the field of a charge Ze); then E11 = -JHZ2 jn 2
`. Here,
`
`is the hydrogen atom ionization potential and n is the principal quantum number.
`In this case, the ratio M/f::..n characterizing the density of energy levels can be taken
`as:
`
`Finally, based on Equation 6.236 and taking into account Equation 6.239 and
`Equation 6.240, 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:
`
`(6.240)
`
`Here, £ is the initial electron energy. Then, the total energy ]"'" dm, emitted as a
`result of photorecombination of electrons with velocities in interval ve + ve + dve
`and ions with Z = 1 and leading to formation of an excited atom with the principal
`quantum number n, per unit time and per unit volume, can be expressed as:
`
`(6.241)
`
`( IH
`-
`-
`nm)
`nine 2IH
`](() dm- nmn.ne(JREJ(ve)v dv - C-l/7 --3 exp --7 - - dm
`T - Tn
`Tn-
`T
`e
`e
`
`11
`
`'
`
`(6.242)
`
`In this formula, the parameter C = 1.08 ·10-45 W · cm 3
`• Kl/ 2 is the same factor as
`in Equation 6.237; f(ve) is the electron velocity distribution function, which was
`taken as Maxwellian; and the differential relation between electron velocity and
`radiation frequency from Equation 6.238 was written as mvedve = ndm.
`It should also be taken into account that emission of quanta with the same
`energy nm can be provided by trapping electrons on different excitation levels with
`different principal quantum numbers n (obviously, the electron velocities should be
`relevant and determined by Equation 6.238). Thus, the total spectrum of recombi(cid:173)
`nation emission is the sum of similar but shifted terms:
`
`]recomb dm = '"' J
`£.,; ron
`ro
`n*(w)
`
`(6.243)
`
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`
`

`
`j Mechanics of Plasma
`
`Emission and Absorption of Radiation in Plasma: Continuous Spectrum
`
`413
`
`RE
`
`(6.239)
`
`ke atoms (where an electron moves in
`2
`. Here,
`
`- "'13.6eV
`
`td n is the principal quantum number.
`e density of energy levels can be taken
`
`(6.240)
`
`king into account Equation 6.239 and
`:tion of the recombination emission in
`an atom in an excited state with the
`
`0)
`
`Figure 6.35 Recombination emission spectrum
`
`Each of the individual terms in Equation 6.243, is related to the different principal
`quantum number n, but giving the same frequency ro, and each is proportional to
`exp(-tzro/T). The lowest possible principal quantum number n*(ro) in the sum for
`the fixed frequency can be defined from the condition IE".J < tzro < jE"._J The
`spectrum of recombination emission conesponding to the sum (Equation 6.243) is
`illustrated in Figure 6.35.
`
`(6.241)
`
`6.7.6 Total Emission in Continuous Spectrum
`
`the total energy J(fm dro, emitted as a
`ith velocities in· interval v + v + dv
`e
`e
`e
`of an excited atom with the princip"l
`t volume, can be expressed as:
`
`(6.242)
`
`• K 112 is the same factor as
`0-45 W · cm 3
`city distribution function, which was
`lation between electron velocity and
`written as mv edv e = tzdw.
`tt emission of quanta with the same
`)TIS on different excitation levels with
`usly, the electron velocities should be
`Thus, the total spectrum of recombi(cid:173)
`ted terms:
`.,
`• J
`ron
`.,J
`Ol)
`
`(6.243)
`
`The total spectral density of plasma emission in continuous spectrum consists of the
`bremsstrahlung and recombination components, Equation 6.237 and Equation 6.243.
`These two emission components can be combined in one approximate but reasonably
`accurate general formula for the total continuous emission:
`
`J rodw = c ;~~ '¥( n; )dw
`
`(6.244)
`
`The parameter C = 1.08 ·10-45 W · cm3
`· K 112 is the same factor as in Equation
`6.237 and Equation 6.243; '¥(x) is the dimensionless function, which can be approx(cid:173)
`imated as:
`
`'¥(x) =I,
`
`IEgl
`'f
`ft(f)
`1 x=-<x = -
`T
`T
`8
`
`(6.245a)
`
`(6.245b)
`
`I
`nro
`if
`(6.245c)
`x=->x =-
`T
`I T
`Here, l£81 is the energy of the first (lowest) excited state of atom calculated with
`respect to transition to continuum and I is the ionization potential.
`
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`414
`
`Chapter 6: Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`When the radiation quanta and thus frequencies are not very large (flro < /E
`j),
`8
`the contributions of free-free transitions (bremsstrahlung) and free-bound transitions
`(recombination) in the total continuous emission are related to each other as:
`
`Jrecomb/Jbrerns = exp flro -1
`T
`w
`w
`
`(6.246)
`
`According to Equation 6.246, emission flro < 0.7T is mostly due to the bremsstrahl(cid:173)
`ung mechanism, while emission of larger quanta flro > 0. 7T is mostly due to the
`recombination mechanism. At plasma temperatures of 10,000 K, this means that
`only infrared radiation (A,> 211m) is provided by bremsstrahlung; all other emission
`spectra are due to the electron-ion recombination.
`To obtain the total radiation losses, integrate the spectral density (Equation
`6.244) over all the emission spectrum (contribution of quanta flro >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:
`
`(6.247)
`
`6.7.7 Plasma Absorption of Radiation in Continuous Spectrum:
`Kramers and Unsold-Kramers Formulas
`The differential cross section of bremsstrahlung emission (Equation 6.235) with
`Equation 6.227 and Equation 6.228 permits determining the coefficient of
`bremsstrahlung absorption of a quantum flro calculated per one electron having
`velocity ve and one ion:
`
`Z 2e6
`16n3
`a (v ) = - - -~---::---
`3-13 m 2c(4nc.oflro)3ve
`"' e
`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 function f(ve) yields the coefficient of bremsstrahlung
`absorption in plasma (which actually is the reverse length of absorption):
`
`(6.248)
`
`c - 2 ( 2 )
`brems - c neni
`1 y1'2v3, 1-3 3n
`K"'
`-
`
`112
`
`6
`e
`(4nc.o)3m3'2cfl
`
`(6.249)
`
`To use Equation 6.249 for numerical calculations of the absorption coefficient
`K~en!S (cm-1), the coefficient c1 can be taken as c1 = 3.69. 108 cm5/(sec3. K112); here,
`the frequency v ro/2n, and Z = 1.
`Another mechanism of plasma absorption of radiation in a continuous spectrurn
`is photoionization, which was discussed in Section 2.2.6 regarding the balance of
`
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`
`Fluid Mechanics of Plasma
`
`frequencies are not very large (nw < !£81),
`,remsstrahlung) and free-bound transitions
`mission are related to each other as:
`
`(6.246)
`
`nw
`=exp--1
`T
`w < 0.7T is mostly due to the bremsstrahl(cid:173)
`er quanta nw > 0.7T is mostly due to the
`mperatures of 10,000 K, this means that
`ided by bremsstrahlung; all other emission
`1bination.
`, integrate the spectral density (Equation
`1 (contribution of quanta nw >I can be
`;orption). These total plasma energy losses
`:pressed after integration by the following
`
`(6.247)
`
`tion in Continuous Spectrum:
`rmulas
`strahlung emission (Equation 6.235) with
`permits determining the coefficient of
`tum nw calculated per one electron having
`
`m 2 c( 41tcofl(j) )3 v e
`
`(6.248)
`
`Ia of the quasi-classical theory of plasma
`ramers formula by neni and integrating over
`v e) yields the coefficient of bremsstrahlung
`the reverse length of absorption):
`
`6
`
`e
`) 3m312cn
`
`(6.249)
`
`112
`
`2 ( 2 )
`=3 3n:
`
`(4n:c:0
`calculations of the absorption coefficient
`ken as C 1 = 3.69 · 108 cm5/(sec3 · K 112); here,
`
`)rption of radiation in a continuous spectrum
`~d in Section 2.2.6 regarding the balance of
`
`Emission and Absorption of Radiation in Plasma: Continuous Spectrum
`
`415
`
`charged particles. The photoionization cross section can be easily calculated, taking
`into account that it is a reverse process with respect to recombination emission.
`Then, based on a detailed balance of the photoionization and recombination emis(cid:173)
`sion, Saha Equation 4.15, and Equation 6.241, the expression for the photoionization
`cross section for a photon nw and an atom with principal quantum number n is:
`
`0
`
`(6.250)
`
`e!O:rzz4 - - =7.9·10-Iscm2-';-(wn)3
`cr = 8n:
`)' cn6WJn)
`z-
`3-J3 ( 4n:c:
`ffi
`wn
`Bere, ffi11 = IE11jl11 is the minimum frequency sufficient for photoionization from the
`electronic energy level with energy E11 and principal quantum number n. As seen
`from Equation 6.250, the photoionization cross section decreases with frequency as
`11w3 at w > ffi11 •
`To calculate the total plasma absorption coefficient in a continuum, it is
`necessary to add to the bremsstrahlung absorption (Equation 6.248), the sum
`~non crmn related to the photoionization of atoms in different states of excitation (n)
`with concentrations n011 • Replacing the summation by integration, the total plasma
`absorption coefficient in continuum can be calculated for Z = 1 as:
`
`K C neni
`= - - e r x =
`1 yu"v3
`w
`
`X\TJ(
`
`)
`
`n n.(cm-3
`-1 e 1
`4 05 10-23
`em
`·
`·
`(T,K)712
`
`) x'¥( )
`e
`X
`x3
`
`(6.251)
`
`Here, factor cl is the same as in Equation 6.249; parameter X = nw!T; and the
`function 'P(x) is defined by Equation 6.245. Obviously, the replacement of summa(cid:173)
`tion over discrete levels by integration makes the resulting approximate dependence
`Km(v) smoother than in reality; this is illustrated in Figure 6.36.218
`Note that the relative contribution of photoionization and bremsstrahlung mech(cid:173)
`anisms in total plasma absorption of radiation in continuum at the relatively low
`frequencies nw < j£8
`j (j£8
`1 is energy of the lowest excited state with respect to
`continuum) can be characterized by the same ratio as in case of emission (Equation
`6.246). The product ne n; in Equation 6.251 can be replaced at not very high tem(cid:173)
`peratures by the gas density n0 using the Saha Equation 4.15. At the relatively low
`frequencies nw < j£81, the total plasma absorption coefficient in continuum is obtained
`in the following form:
`
`(6.252)
`
`This relation for the absorption coefficient is usually referred to as the
`Unsold-Kramers formula, where ga, g; are statistical weights of an atom and an
`ion, X = nw!T, X! = liT, and I is the ionization potential.
`
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`416
`
`Chapter 6: Electrostatics, Electrodynamics, and Fluid Mechanics of Plasma
`
`Xw<PIN, cm2
`10-17,-.--.---.---.--,---,---,
`Xe
`
`1Q-20
`
`7
`
`5
`
`3
`
`~--_L--L-~--J_ __ L_~
`30
`20
`10
`v, 1 os cm-1
`
`Figure 6.36 Absorption coefficient per one Xe atom, related to photoionization: dashed line= summation
`over actual energy levels; solid line = replacement of summation by integration218
`
`As an example, the absorption length K~l of red light "A = 0.65!lm in an atmo(cid:173)
`spheric pressure hydrogen plasma at 10,000 K can be calculated from Equation
`6.252 as K~l "" 180m. Thus, this plasma is fairly transparent in continuum at such
`conditions. It is interesting to note that the Unsold-Kramers formula can be used
`only for quasi-equilibrium conditions, while Equation 6.251 can be applied for
`nonequilibrium plasma as well.
`
`6.7 .8 Radiation Transfer in Plasma
`The intensity of radiation Iro (defined in Section 6.7 .3) 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 a quasi-equilibrium
`plasma can then be given as:
`
`Here, the factor lw is the quasi-equilibrium radiation intensity:
`
`1im3
`1
`I =
`4n:V exp( 1im/T) -1
`we
`
`(6.253)
`
`(6.254)
`
`which can be easily obtained from the Planck formula for the spectral density of
`radiation (Equation 4.21). Introduce the optical coordinate<;, calculated from the
`plasma surface x = 0 (with positive x directed into the plasma body):
`
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`
`

`
`, 'I
`
`,f
`j
`
`anics of Plasma
`
`Emission and Absorption of Radiation in Plasma: Continuous Spectrum
`
`417
`
`X
`
`~ = J K~(x)dx, d~ =K~(x)dx
`
`0
`
`(6.255)
`
`Then, the radiation transfer equation can be rewritten in terms of the optical coor(cid:173)
`dinate as:
`
`diOJ(~) -1 (~)=-!
`Ole
`d~
`OJ ~
`
`(6.256)
`
`Assuming that in the direction of a fixed ray, the plasma thickness is d; x = 0
`conesponds to the plasma surface; and there is no source of radiation at x > d, then
`the radiation intensity on the plasma surface Iwo can be expressed by solution of the
`radiation transfer equation as:
`
`30
`v,103 cm-1
`
`o photoionization: dashed line = summation
`1tion by integration218
`
`·red light A, =·0.65!-Lm in an atmo-
`can be calculated from Equation
`, transparent in continuum at such
`old-Kramers formula can be used
`quation 6.251 can be applied for
`
`i. 7 .3) decreases along its paths due
`d increases because of spontaneous
`~r equation in a quasi-equilibrium
`
`1tion intensity:
`
`/T)-1
`
`(6.253)
`
`(6.254)
`
`'ro
`
`IOJ 0 = f !Ole[T(~)]exp(-~)d~, 1: = JK' dx
`
`d
`
`OJ
`
`OJ
`
`(6.257)
`
`0
`
`0
`
`The quasi-equilibrium radiation intensity JOle[T(~)] is shown here as function of
`temperature and therefore