Yuri P. Raizer
`
`Gas Discharge Physics
`
`With 209 Figures
`
`Springer
`
`Energetiq Ex. 2011, page 1 - IPR2015-01377
`
`

`
`Professor Dr. Yuri P. Raizer
`The Institute for Problems in Mechanics, Russian Academy of Sciences,
`Vernadsky Street 101,117526 Moscow, Russia
`
`Editor:
`Dr. John E. Allen
`Department of Engineering Science, University of Oxford, Parks Road,
`Oxford OX1 3PJ, United Kingdom
`
`Translator:
`Dr. Vitaly I. Kisin
`24 Varga Street, Apt. 9, 117133 Moscow, Russia
`
`This edition is based on the original second Russian edition: Fizika gazovogo razryada
`© Nauka, Moscow 1987, 1992
`
`1st Edition 1991
`Corrected 2nd Printing 1997
`
`ISBN 3-540-19462-2 Springer-Verlag Berlin Heidelberg New York
`
`Library of Congress Cataloging-in-Publication Data.
`Raizer, IU. P. (IUril Petrovich) [Fizika gazovogo razrlada. English] Gas discharge physics I Yuri P.
`Raizer. p. em. "Corr. printing 1997"- t.p. verso. Includes bibliographical references and index.
`ISBN 3-540-19462-2 (hardcover: alk. paper) 1. Electric discharges through gases. I. Title.
`QC711.R22713 1997 537.5'3-dC21 96-53988
`
`This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
`concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad(cid:173)
`casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of
`this publication or parts thereof is permitted only under the provisions of the German Copyright
`Law of September 9, 1965, in its current version, and permission for use must always be obtained
`from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
`© Springer-Verlag Berlin Heidelberg 1991
`Printed in Germany
`The use of general descriptive names, registered names, trademarks, etc. in this publication does not
`imply, even in the absence of a specific statement, that such names are exempt from the relevant pro(cid:173)
`tective laws and regulations and therefore free for general use.
`Typesetting: Springer TEX inhouse system
`Cover design: design & production GmbH, Heidelberg
`SPIN 10565824
`54/3144- 54 3 2 1 o- Printed on acid-free paper
`
`Energetiq Ex. 2011, page 2 - IPR2015-01377
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`

`
`(2.50)
`
`(2.51)
`
`:l of the hydrodynamic flux of
`fficient. 3 The terms containing
`~ kTe, are neglected; accord(cid:173)
`s in collisions with molecules
`). The last term in (2.50) de(cid:173)
`the resultant creation rate that
`ionization potential.
`t the equation for the rate of
`
`(2.52)
`
`J time, as in (2.49).
`me effects caused by spatial
`ntribution of heat conduction.
`and passing thermal energy to
`on with the term proportional
`
`in (2.22, 5{}-52) in the form
`:gled out of r e· Ignoring the
`ount that f-l+ ~ f-le, we obtain
`
`(2.53)
`
`, to the same accuracy,
`
`(2.54)
`
`lin Sect. 9.7.
`
`1 (2.51) for F and >.e stem from the
`a distribution functions and vm(c) =
`' and >.e, ~' are set equal to 2; the
`
`3. Interaction of Electrons in an Ionized Gas
`with Oscillating Electric Field
`and Electromagnetic Waves
`
`3.1 The Motion of Electrons in Oscillating Fields
`
`Both the equations of electrodynamics and the equations of motion of electrons
`are linear with respect to the fields E, H and the velocity v of the electron. For
`this reason, the superposition principle holds. Any periodical field can be resolved
`into harmonic components, so that it is sufficient to consider only the sinusoidal
`field, all the more so because one normally deals with monochromatic fields and
`waves. In the case of nonrelativistic motion, the magnetic force of the wave,
`e(vjc)H, is much less than the electric force eE. Furthermore, the amplitude of
`electron oscillations in discharge processes is usually small in comparison with
`wavelength .\. We assume, therefore, that the electron is in a spatially uniform
`electric field E =Eo sinwt, Eo= const.
`
`3.1.1 Free Oscillations
`Assume that an electron moves without collisions, an assumption that is meaning(cid:173)
`ful if the electron performs a large number of oscillations in the interval between
`collisions, w ~ vm. We integrate the equation of collisionless motion,
`mv = -eEo sinwt,
`r = v,
`to give
`
`.
`eEo
`eEo
`r = - -2 smwt + vot + r 0 •
`v = - - cos wt + vo ,
`mw
`mw
`An electron oscillates at the frequency of the field; these oscillations are
`superimposed onto an arbitrary translation velocity vo. The displacement and
`oscillation velocities are
`eEo
`eEo
`a = -- , u = -
`mw2
`mw
`The displacement is in phase with the field, while the velocity is out of phase by
`1r j2. The limiting case of "collisionless" oscillations is approximately realized
`at optical frequencies, and also at microwave frequencies at low pressures, p ;S
`lOTorr.
`
`.
`
`(3.1)
`
`(3.2)
`
`3.1.2 Effect of Collisions
`Collisions "throw off' the phase, thereby disturbing the purely harmonic course
`of the electron's oscillations. A sharp change in the direction of motion after
`
`35
`
`Energetiq Ex. 2011, page 3 - IPR2015-01377
`
`

`
`scattering stops the electron from achieving the full range of displacement (3.2)
`that the applied force can produce; the electron starts oscillating anew after each
`collision, with a new phase and new angle relative to the instantaneous direction
`of velocity. In order to take this factor into account, we add the rate of loss of
`momentum due to collisions to the equation of motion of the "mean" electron.
`As in the case of constant fields (Sect. 2.1.1), we have the equation for the mean
`velocity:
`
`mv = -eE0 sinwt- mvvm,
`
`r = v.
`
`The solution of (3.3), valid after several collisions, is
`
`v =
`
`T =
`
`eEo
`cos(wt + <p) ,
`mJw2 +v~
`)
`. (
`eEo
`Sill wt + <p
`mwJw2 +v~
`
`.
`
`Vm
`<p = arctan- ,
`w
`
`(3.3)
`
`(3.4)
`
`The amplitudes of displacement and velocity of the electron are less by a
`factor of Jt + v~jw2 than those for free oscillations. The higher the effective
`collision frequency vm, the smaller they are (vm is determined by the velocity of
`random motion, which is much greater in discharges than the oscillation velocity;
`see Sect. 3.2). The displacement is shifted in phase relative to the field, the phase
`shift increasing from 0 to 11' /2 as the relative role of collisons vm/ w increases
`from 0 to oo.
`The oscillation displacement and velocity (3.4) can always be resolved into
`two components, one proportional to the magnitude of the field E = Eo sin wt,
`and the other to its rate of change, E = wEo cos wt:
`.
`eEo
`Vm
`eEo
`2 ) Sill wt + -
`( 2
`2 ) cos wt ,
`( 2
`m w + vm
`w m w + Vm
`.
`VmeEo
`weEo
`2 ) cos wt -
`( 2
`2 ) Sill wt .
`( 2
`m w + vm
`m w + vm
`
`(3.5)
`
`r =
`
`v =
`
`The ratio of the components is determined by the relative role of collisions
`and is unambiguously related to the phase shift <p. This form of presenting the
`solution adds visual clarity to the results of the subsequent sections.
`Expressions (3.4, 5) show that the role of collisions is characterized by the
`ratio of the effective frequency vm and the circular frequency of the field w = 211' j,
`which is greater than the frequency f by nearly an order of magnitude.1 In
`the limit v;, ~ w 2 , formulas (3.4, 5) are close to (3.1) for free oscillations.
`To illustrate numerical values, consider an example of microwave radiation at
`frequency f = 3GHz; ,\ = lOcm, w = 1.9 x 1010 s-1. Let p ~ 1 Torr, then
`vm ~ 3 x 109 s-1 ~ w; Eo= 500 V fern, roughly corresponding to the threshold
`
`1 When the degree of spatial uniformity of the field is evaluated, the displacement amplitude must
`be compared not with wavelength.\= cf J, but with:);= .\f27r: af A= eEo/mw2 A= ufc.
`
`36
`
`Energetiq Ex. 2011, page 4 - IPR2015-01377
`
`

`
`111 range of displacement (3.2)
`trts oscillating anew after each
`: to the instantaneous direction
`mt, we add the rate of loss of
`totion of the "mean" electron.
`tave the equation for the mean
`
`:ons, is
`
`(3.3)
`
`(3.4)
`
`of the electron are less by a
`ons. The higher the effective
`determined by the velocity of
`; than the oscillation velocity;
`relative to the field, the phase
`of collisons vmfw increases
`
`can always be resolved into
`e of the field E = Eo sin wt,
`
`of microwave breakdown at such pressures. Formulas (3.2) show that a= 2.5 x
`w-3 em, u = 4.7 x 107 cmjs. We find that a ~ X = 1.6 em, that is, the field in
`the electromagnetic wave is "uniform".
`
`3.1.3 Drift Oscillations
`In the limit of very frequent collisions or relatively low frequencies, v;, ~ w 2
`the oscillation velocity drops to
`
`,
`
`.
`eE(t)
`eEo
`v ~ - - - smwt = - - - = -f.leE(t) = vd(t).
`mvm
`mvm
`
`(3.6)
`
`At each moment of time, the oscillation velocity coincides with the drift velocity
`that corresponds to the field vector at this moment. For brevity, we refer to such
`oscillations in the mobility regime as drift oscillations.
`An electron behaves as it would in a constant field, responding to relatively
`slow changes of the field. Its displacement,
`
`r ~ Acoswt, A= eEo = f.leEo
`mvmw
`w
`
`(3.7)
`
`has an amplitude A less than that of free oscillations in the same field by a factor
`Of Vm/W ~ 1.
`The oscillations of electrons in rf fields (and of course, at lower frequencies)
`are of drift type. For example, the collision frequency at f ~ lOMHz, vm ~
`3 x 109 ps- 1 , exceeds w"' 108 s-1 even at fairly low pressures of p"' 0.03 Torr.
`In order to maintain a low-pressure, weakly ionized plasma by an rf field, one
`usually needs the values of Eo j p of the same order as E j p in a constant field.
`Therefore, at f ~ lOMHz and Eofp ~ lOY /(em· Torr), we have A"' 0.1 em
`regardless of pressure.
`
`(3.5)
`
`3.2 Electron Energy
`
`he relative role of collisions
`This form of presenting the
`;equent sections.
`ions is characterized by the
`:quency of the field w = 2nf,
`an order of magnitude. 1 In
`(3.1) for free oscillations.
`~ of microwave radiation at
`J s-1. Let p ~ 1 Torr, then
`rresponding to the threshold
`
`• the displacement amplitude must
`·: aj A= eEofmw2 A= ujc.
`
`3.2.1 Collisionless Motion
`If collisions do not occur, the field does no work, on the average, on an electron;
`indeed, (3.1) implies that
`
`(-eE. v) =- eE5 (sinwtcoswt)- eEo · vo(sinwt) = 0,
`mw
`
`where angle brackets denote time averaging.
`The electric field pumps up the motion of the electron only once, when it is
`switched on; then the electron's energy mv2 /2 pulsates but remains unchanged
`on the average. The time averaged energy (mv2 /2) is made up of the energy
`of translational motion mv5J2, corresponding to the mean velocity vo = (v(t)),
`and that of oscillations. In the case of free oscillations, the latter energy is
`
`Energetiq Ex. 2011, page 5 - IPR2015-01377
`
`

`
`e2E~
`2
`2 = mu /4 .
`cfr.osc. = -
`4
`mw
`In the example given at the end of Sect. 3.1.2, f = 3 GHz, Eo = 500 V js, and
`the oscillation energy cfr.osc. = 0.31 eV, which is much less than the mean energy
`of random motion (1-lOeV) necessary to sustain a discharge. The absence of
`collisions thus means no dissipation of the energy of the field and no deposition
`of energy in the matter.
`
`(3.8)
`
`3.2.2 Gaining of Energy From the Field
`The collisions lead to a net transfer of energy to the electrons, via the electric
`field. According to (3.5), the mean work per unit time that the field performs on
`an electron,
`
`(3.9)
`
`)
`
`= 2c:0 sc.
`
`is determined by that component of velocity that oscillates in phase with the
`field and is proportional to vm. The term shifted in phase by 1r /2 does no work,
`on the average, over one period. The random motion is not associated with any
`transfer of energy. In one effective collision an electron gains the mean energy
`Llc: E, equal to twice the mean kinetic energy of oscillations:
`e2E 2
`jmv2
`Llc:E = 2m(w2 ~ v~) = 2 \ -2-
`This result can be given the following interpretation. In the interval between
`two collisions, the electric field imparts to an electron an average kinetic energy
`C:osc· If the electron goes through a large number of oscillations in this period,
`then C:osc is of the order of the free oscillation energy (3.8). An act of elastic
`scattering of an electron by an atom sharply changes the direction of motion but
`leaves the absolute value of velocity unaltered. Then the field starts swinging the
`electron in a new direction with respect to its velocity, that is, imparts to it an
`energy of order C:osc as if anew. The mean amount of energy gained from the field
`in each scattering event following the preceding collision is thus transformed into
`the energy of translational random motion. Microscopically, the field does work
`on overcoming the friction due to collisions of the electron. Everything proceeds
`as in a constant field (see Sect. 2.3.2), but the role of the drift energy is played
`by that of oscillations.
`
`(3.10)
`
`3.2.3 Balancing of the Electron Energy
`
`The balance is made up of gaining energy from the field and transferring it to
`heavy particles as a result of elastic and inelastic losses. If an electron loses in
`each collision a fraction 8 of its energy c:, then
`
`(3.11)
`
`38
`
`Energetiq Ex. 2011, page 6 - IPR2015-01377
`
`

`
`(3.8)
`
`~=30Hz, Eo= 500V js, and
`1ch less than the mean energy
`a discharge. The absence of
`of the field and no deposition
`
`the electrons, via the electric
`me that the field performs on
`
`(3.9)
`
`oscillates in phase with the
`phase by 1r /2 does no work,
`on is not associated with any
`!Ctron gains the mean energy
`dilations:
`
`(3.10)
`
`ation. In the interval between
`·on an average kinetic energy
`::>f oscillations in this period,
`ergy (3.8). An act of elastic
`:s the direction of motion but
`n the field starts swinging the
`~eity, that is, imparts to it an
`f energy gained from the field
`lision is thus transformed into
`:opically, the field does work
`:lectron. Everything proceeds
`of the drift energy is played
`
`te field and transferring it to
`osses. If an electron loses in
`
`(3.11)
`
`where the amplitude Eo is replaced with the mean-square field E defined by the
`equality E 2 = (E2(i)) = E'J/2. If w2 ~ v;,, then (3.11) transforms into (2.12).
`In the w - t 0 limit, the R.M.S. of E plays the role of a constant field.
`
`3.2.4 Mean Equilibrium Energy
`The mean energy reached by electrons under stationary conditions, when they
`transfer the entire energy gained from the field, is
`
`(3.12)
`
`Low-frequency fields (w 2 ~ v~) behave indistinguishably from constant
`fields; the similarity l = f (E j p) holds. At high frequencies, w2 ~ v~, the mean
`electron energy is independent of vm, p, and the similarity in field frequency
`holds: l = fl(Ejw). If 8 = const, lex: (Ejw)2. In equivalent situations, E ex: w.
`This is the reason why gas breakdown at optical frequencies (w '"" 10t5 s-t)
`requires enormous fields (E '"" 107 V fern) in the light wave, realizable only
`when giant laser pulses are focused (Sect. 7.6). Indeed, electron avalanches can
`develop only if the electron energies are of the order of lOeV.
`
`3.2.5 Actual Change in Electron Energy in a Collision
`The situation in ac fields is also similar in this respect to that found in de fields
`(Sect. 2.3.3). An electron may either gain energy from the field or lose energy
`to it, in amounts that much exceed the mean change Llc: E averaged over a large
`number of collisions. The relative directions of the motion and the field and the
`phase of field oscillations at the moment of collision decide whether the electron
`is to gain or lose energy. This is a fact of fundamental importance, which contains
`a classical analogue of such purely quantum phenomena as the true absorption
`and stimulated emission of photons.
`To illustrate this, we calculate directly the change in the energy of an electron
`in a collision. Let an electron undergo its most recent collision at a moment it
`and have, immediately after scattering, a velocity Vt and energy ct = mvt /2.
`Effective collisions occurring at a frequency vm each time give the velocity a
`completely random direction; hence, ct is the energy of random motion at the
`moment of collision it. In the time i ;::: it and until the next collision, the electron
`is driven by the force -eEo sin wi at a velocity v(i) = u(cos wt- cos wit)+ Vt,
`u = eEo/mw. Its energy c: = mv2 /2 at each moment of this period is
`
`c:(i) =; [vr+2vtu(coswt -coswtt)
`+ u 2 ( cos2 wt - 2 cos wt cos wtt + cos2 wtt)]
`
`At the moment t2 of the next collision, the velocity is again directed in an
`arbitrary ,direction but remains virtually unchanged in magnitude. The electron
`resumes motion at an energy c:(h), which is also the energy of random motion.
`Therefore, between two collisions the random-motion energy changes by
`
`Llc:(tt, t2) = c:(t2)- ct ~ mvt · u(coswi2- coswtt) .
`
`39
`
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`
`

`
`This last (approximate) transition takes into account that the random-motion
`velocity vt is much greater than the oscillation velocity u; hence, we can ignore
`the specific form of the small term of order mu2 •
`For the sake of simplicity, assume that collisions are rare: vm «:: w. Then
`many oscillations occur in the time i 2 -
`t 1 between two collisions, and the corre(cid:173)
`lation between the field phases at the moments of collisions wit and wi2 vanishes
`owing to the random nature of collisions, i.e., the phases may be arbitrary. The
`value of Llc: then varies in this interval from the maximum gain Llc:+ = 2mvt u
`and the maximum loss Llc;_ = -2mv1u, the extremal values corresponding to
`parallel velocities Vt and u and certain phases, wit, wiz. When averaged over
`many collisions, however, that is, over moments it and iz, an electron gains the
`energy
`LlcE = (Llc:(it, tz))t,,t 2 = mu2/2 = 2c:rr.osc.
`
`which we found above by calculating the mean work done by the field.Z Since
`u/v «:: 1, the actual changes that the electron energy experiences in collisions, be
`they positive or negative, are of first order of smallness in u/v, jLlc:±i/c:"' u/v,
`while the resulting positive Llc:E/c "' (u/v? is of second order. This latter
`quantity is a small difference of two relatively large ones; in symbolic form,
`Llc:E"' (Llc:+ -IL1c:_j).
`
`3.2.6 Why Electron-Electron Collisions
`Do not Dissipate the Energy of the Field
`Electrons of a weakly ionized gas collide with atoms and molecules. In a strongly
`ionized gas they collide with ions and other electrons with nearly equal frequency.
`However, only electron-ion collisions need to be taken into account in considering
`the effects of the interaction with the field.
`To reveal the reason, consider an electron gas (an even more general case
`can be taken: a gas of particles with an identical efm ratio) and assume that
`electrons collide only with electrons. Sum up over all electrons the equation of
`motion mv = -eEo sin wt+Pcot> where Pcol is the rate of change of an electron's
`momentum due to collisions. As the total momentum of interacting particles is
`conserved, the total momentum L:mv of the gas oscillates as coswi, with a 1rj2
`phase shift. Recalling that the total energy of particles is also conserved under
`elastic collisions, we can find the rate of change of the gas energy:
`= Lmv · v2 =- ( : Eosinwt) (Lmv)
`ex sin wi cos wt ex sin 2wt .
`
`~ L m;2
`
`2 In the general case of Vm ~ w, only one of the phases is arbitrary because the moments of
`consecutive collisions are correlated. The probability of the interval f2 -
`t1 is exp[ -Vm(t2 -
`tJ)]vmdt2. Correlations add an additional factor to Llc:E of (3.10): w2 j(w2 + v;,) [3.1].
`
`40
`
`Energetiq Ex. 2011, page 8 - IPR2015-01377
`
`

`
`~aunt that the random-motion
`ocity u; hence, we can ignore
`
`ons are rare: vm ~ w. Then
`two collisions, and the corre(cid:173)
`)llisions wiJ and wtz vanishes
`phases may be arbitrary. The
`naximum gain L1c:+ = 2mv1 u
`~mal values corresponding to
`t1, wtz. When averaged over
`and iz, an electron gains the
`
`ark done by the field. 2 Since
`y experiences in collisions, be
`ness in ujv, jL1qj/c:"' ujv,
`of second order. This latter
`trge ones; in symbolic form,
`
`) and molecules. In a strongly
`; with nearly equal frequency.
`~n into account in considering
`
`(an even more general case
`e / m. ratio) and assume that
`all electrons the equation of
`1te of change of an electron's
`Jm of interacting particles is
`.ciliates as cos wt, with a 1r /2
`des is also conserved under
`· the gas energy:
`
`arbitrary because the moments of
`interval t2 -
`t 1 is exp[ -11n(t2 -
`!.10): w2 /(w2 +vi) [3.1].
`
`The total energy of particles oscillates at double frequency, as in collisionless
`motion, and remains unchanged on average. No dissipation of the field energy
`occurs. Recall that electron-electron collisions in a de field do not contribute to
`resistance and Joule heat release (Sect. 2.2.3), though as in de fields, electron(cid:173)
`electron collisions may affect dissipation indirectly, by changing the electron
`energy distribution and Vm.
`
`3.3 Basic Equations of Electrodynamics of Continuous Media
`
`Sections 3.1 and 3.2 outlined what happens with electrons of an ionized gas
`placed in an ac electric field. Let us turn to a different aspect of the electron-field
`interaction: the effect of the ionized state on the behavior of ac fields and the
`propagation of electromagnetic waves.
`
`3.3.1 Maxwell's Equations
`The electromagnetic field and the state of the medium are described in terms
`of field strengths E, H and inductions D, B. By definition, D = E + 47r P
`and B = H + 4Ir M, where P and M are the electric and magnetic moments,
`respectively, per unit volume. The vectors E, H, D, B satisfy the system of
`Maxwell's equations:
`47r. 1 aD
`curl H = -3 + - -
`c at '
`c
`1 aB
`(3.14)
`cur1E= - - (cid:173)
`c &t
`(3.15)
`div B =0,
`(3.16)
`div D = 4Ire.
`This system is not completely closed because the electric current j, polar(cid:173)
`ization P, and magnetization M generated by the field, depend on material
`properties. Both experience and theory indicate that direct proportionality reigns
`in constant fields and fields that vary not too rapidly: j = (j E, P = xeE,
`M = xH. Instead of the electric Xe and magnetic susceptibility x, one intro(cid:173)
`duces the permittivity c: = 1 + 47rXe and magnetic permeability J.L = 1 + 4Irx.
`Together with the equations
`
`(3.13)
`
`l
`
`(3.17)
`j=(jE, D=c:E, B=J.LH,
`where the material constants c:, fl and conductivity (j are assumed to be known,
`system (3.13-17) is closed. In gases and plasmas, the approximation fl = 1 can
`be used with extremely high accuracy.
`
`3.3.2 Displacement, Polarization, Conduction, and Charge Currents
`The right-hand side of (3.13) can be treated as a current density
`. aP
`1 aD
`1 aE
`.
`)+---=)+-+---
`47r &t
`&t
`47r &t
`
`l
`
`(3.18)
`
`41
`
`Energetiq Ex. 2011, page 9 - IPR2015-01377
`
`

`
`times 47r I c. Maxwell called the term (1 I 47r )8D I &t, which he postulated should
`be added to the conduction current, the displacement current. Without the dis(cid:173)
`placement current, (3.13, 16) contradict the unassailable law of charge conserva(cid:173)
`tion, (2.39).
`A variable field changes the polarization of matter with time, namely, dis(cid:173)
`places negative charges relative to positive ones by applying the electric force. In
`fact, any displacement of charges in space is a current, so that the term 8P I at
`in the displacement current is indeed a current density: that of the polarization
`current. Together with the conduction current i, it forms the total charge current
`it· The term (1147r)8EI&t is in no way connected with the motion of charge and
`therefore, is not literally a current. (Its meaning will be discussed in Sect. 13.5.)
`The total charge current was divided into the conduction and polarization
`components only to facilitate the application of the equations to ideal dielectrics,
`where a, i = 0. This partition is by no means mandatory. The total polarization
`vector Pt can be defined for any electrically neutral medium; Pt is related to
`the total charge current it, so that it is sufficient to operate with just one of these
`quantities. Indeed, by definition
`
`Pt = L e;r; ' L e; = 0'
`
`it= L e;v; = a:t '
`
`(3.19)
`
`where r; is the radius vector of the charge e;, v; = i'; is its velocity, and
`summation is extended to absolutely all charges (free, bound, electrons, nuclei)
`within a unit volume.
`
`3.3.3 Expansion into Harmonics
`Equations (3.17) fail in rapidly varying fields. Owing to the inertia of the pro(cid:173)
`cesses that produce polarization and current, they cease to track the variations
`of the field. The polarization and current at it are now determined less by the
`value of E(tt) than by the evolution of E(t) in the preceding period t <it. For
`instance, if the field E pointed for a long time in one direction and then was
`suddenly reversed, the current would flow for some time in the former direction,
`against the new field, until the charges are brought to rest.
`Obtaining the material equations (like D = c:E) that take into account re(cid:173)
`tardation effects is greatly facilitated because the motion of charges in matter is
`described by equations linear in E, r, v. Since Maxwell's equations are also lin(cid:173)
`ear, all time-dependent quantities can be expanded into Fourier series or integrals;
`in view of the superposition principle, one can operate only with harmonic com(cid:173)
`ponents, as we have already done in Sects. 3.1, 2. Three parameters completely
`determine the evolution of harmonic quantities: amplitude, frequency, and phase,
`so that the entire retardation effect is contained in the relation between these pa(cid:173)
`rameters for material characteristics and the field. If the field for some harmonic
`is E w = E w0 sin wt, then the total current is i tw = i twO sin( wt + cp w), with j twO
`and Cf'w being the functions of Ew0 and w.
`
`42
`
`Energetiq Ex. 2011, page 10 - IPR2015-01377
`
`

`
`'&t, which he postulated should
`·ment current. Without the dis(cid:173)
`ailable law of charge conserva-
`
`matter with time, namely, dis(cid:173)
`y applying the electric force. In
`JITent, so that the term a PI fJt
`ensity: that of the polarization
`t forms the total charge current
`l with the motion of charge and
`vill be discussed in Sect. 13.5.)
`e conduction and polarization
`e equations to ideal dielectrics,
`ndatory. The total polarization
`1tral medium; Pt is related to
`1 operate with just one of these
`
`(3.19)
`V; = r; is its velocity, and
`free, bound, electrons, nuclei)
`
`ving to the inertia of the pro-
`cease to track the variations
`~ now determined less by the
`; preceding period t < t 1 . For
`1 one direction and then was
`~ time in the former direction,
`t to rest.
`E) that take into account re(cid:173)
`notion of charges in matter is
`~~:well's equations are also lin(cid:173)
`nto Fourier series or integrals;
`~ate only with harmonic com(cid:173)
`Three parameters completely
`Jlitude, frequency, and phase,
`he relation between these pa(cid:173)
`f the field for some harmonic
`· itwO sin(wt + 'Pw), with itwO
`
`3.3.4 Material Equations for Harmonic Components
`These can be given a convenient and lucid form if we retain the concepts of
`conductivity and dielectric permittivity, which are familiar from working with
`constant fields. The total current itw is given by a linear combination of sin wt and
`cos wt, which corresponds to a linear combination of E and a E I fJt. Returning to
`the original concepts of conduction current CJ E and polarization current a PI at,
`which is equal to [(c -1)14?r]OE I fJt if the field varies slowly, and equipping the
`new coefficients CJ and c with the subscript w (because now they are frequency(cid:173)
`dependent), we rewrite the material equation in the form
`
`Cw- 1 aEw
`.
`3tw = C7wEw + ~ T' Ew = Ewo sinwt.
`
`(3.20)
`
`The quantities CJ w and cw are called the high-frequency conductivity and di(cid:173)
`electric permittivity of the medium. It is these characteristics of the medium that
`affect the behavior of variable fields in it.
`
`3.3.5 Energy Equation
`We now form the scalar products of (3.13) with E and of (3.14) with H,
`and subtract the resulting equations from each other. In view of the fact that
`H curl E - E curl H = div[E x H], assuming the relations between D and E,
`B and H are linear, we obtain the relation
`a E·D+H·B
`. c
`.
`+div-[E x H] = -3 ·E.
`-
`fJt
`87r
`47r
`This formula expresses the law of conservation of energy of the electromag(cid:173)
`netic field. The quantity E · D l81r is the electric energy density, and H · B l81r
`is that of magnetic energy;
`c
`s = 4?r[E X H]
`
`(3.21)
`
`(3.22)
`
`is the electromagnetic energy flux density (Poynting vector); and j · E is the
`energy released per second in 1 cm3 of the medium, equal to the decrease in
`electromagnetic energy. As a time average, the dissipation of energy in harmonic
`fields in a plasma is caused only by the conduction current. The pola.-i.zation
`current does not result in dissipation because it is shifted by 1r 12 with respect to
`the field, and (sin wt cos wt) = 0 (cf. the results of Sects. 3.2.1, 2).
`
`3.4 High-Frequency Conductivity
`and Dielectric Permittivity of Plasma
`
`The results of Sect. 3.1 permit the immediate calculation of these quantities, but
`two qualifications will be made first. We assume that ions do not move and
`make a negligible contribution to conduction and polarization currents. Next, we
`
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`
`

`
`single out the term eM - 1 in ew - 1 due to the electrons bound in molecules
`and ions. This term is of the same order of magnitude as in nonionized gases
`(unless the polarizability of the excited molecules in the plasma is greater than
`that of nonexcited ones). Under normal conditions, eM- 1 = 5.28 x 10-4 in air,
`2.65 X 10-4 in Hz, and 0.67 X 10-4 in He; it is even smaller at low pressure
`because eM - 1 is proportional to density. These figures refer to the visible part
`of the spectrum and to lower frequencies. The contribution of the molecular pirrt
`to the bulk polarizability of plasma is very small for any appreciable ionization.
`
`3.4.1 Calculation of u"' and ew
`First we substitute the general expression (3.5) of the mean electron velocity v
`into (3.19) for the charge current j 1• Replacing the summation over all electrons
`with the multiplication by ne (the term with random velocity vanishes upon
`averaging) and comparing the obtained expression with (3.20), we find:
`
`(3.23)
`
`(3.24)
`
`These formulas are of fundamental importance for the physics of interac(cid:173)
`tion between plasma and electromagnetic fields. The ratio of the amplitudes of
`conduction and polarization currents,
`~cond,O =
`4IrCTw = Vm/W '
`)polar,O wlew- 11
`is determined by the ratio of collision and field frequencies.
`
`(3.25)
`
`3.4.2 The High-Frequency Limit (Collisionless Plasma)
`This regime is reached when w2 ~ v~, that is, at not particularly high frequen(cid:173)
`cies, i.e., the microwave or very far IR range, even at atmospheric pressure. In
`fact, the molecular polarizability of most dielectrics and nonionized gases retains
`the value typical for de fields up to optical frequencies. In the high-frequency
`limit,
`
`e = 1 _ 4Ire2ne
`
`rnw2
`
`w
`
`'
`
`(3.26)
`
`that is, conductivity is proportional to collision frequency, while dielectric per(cid:173)
`mittivity is independent of w. According to (3.25), the conduction current is small
`compared with the polarization current. This limit corresponds to the collisionless
`plasma model (Sects. 3.1, 3.2.1).
`
`3.4.3 Static Limit
`If w 2 <t:: v~, then
`
`2
`
`ne
`"' = 1 _ 4Jre
`2
`mvm
`
`'-'W
`
`•
`
`(3.27)
`
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`
`

`
`~lectrons bound in molecules
`titude as in nonionized gases
`in the plasma is greater than
`eM- 1 = 5.28 X 10-4 in air,
`even smaller at low pressure
`gures refer to the visible part
`ribution of the molecular piut
`Jr any appreciable ionization.
`
`the mean electron velocity v
`summation over all electrons
`dom velocity vanishes upon
`with (3.20), we find:
`
`(3.23)
`
`(3.24)
`
`e for the physics of interac(cid:173)
`he ratio of the amplitudes of
`
`(3.25)
`
`quencies.
`
`'lasma)
`10t particularly high frequen(cid:173)
`n at atmospheric pressure. In
`. and nonionized gases retains
`~ncies. In the high-frequency
`
`(3.26)
`
`:quency, while dielectric per(cid:173)
`le conduction current is small
`Jrresponds to the collisionless
`
`The conductivity is indistinguishable from the ordinary de conductivity of
`the ionized gas. The dielectric permittivity also reaches a value independent of
`frequency. The polarization current is small in comparison with· the conduction
`current, and vanishes completely in the limit w ....... 0.
`3.4.4 Why Dielectrics Usually Have e > 1, and Plasma e < 1
`Electrons in atoms and molecules are bound, while in plasmas (and metals, where
`also