Yuri P. Raizer
`
`Gas Discharge Physics
`
`With 209 Figures
`
`Springer-Verlag
`Berlin Heidelberg New York London Paris
`Tokyo Hong Kong Barcelona Budapest
`
`Energetiq Ex. 2007, page 1 - IPR2015-01377
`
`

`
`Professor Dr. Yuri P. Raizer
`The Institute for Problems in Mechanics, USSR Academy of Sciences,
`Vernadsky Street 101, SU-117526 Moscow, USSR
`
`Editor:
`Dr. John E. Allen
`Department of Engineering Science, University of Oxford, Parks Road,
`Oxford OX13PJ, United Kingdom
`
`Translator:
`Dr. Vitaly I. Kisin
`24 Varga Street, Apt. 9, SU-117133 Moscow, USSR
`
`This edition is based on the original Russian edition: Fizika gazovogo razryada
`© Nauka, Moscow 1987
`
`ISBN 3-540-19462-2 Springer-Verlag Berlin Heidelberg New York
`ISBN 0-387-19462-2 Springer-Verlag New York Berlin Heidelberg
`
`Library of Congress Cataloging-in-Publication Data. Raizer, Y. P. (Yuri Petrovich) [Fizika gazovogo razryada.
`English] Gas discharge physics I Yuri P. Raizer. p. em. Abridged translation of: Fizika gazovogo razryada. Inclu(cid:173)
`des bibliographical references and index. ISBN 3-540-19462-2.- ISBN 0-387-19462-2 (U.S.) 1. Electric discharges
`through gases. I. Title. QC71l.R22713 1991 537.5'3-dc20 91-14976
`
`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, broadcasting, reproduction on
`microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only per(cid:173)
`mitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copy(cid:173)
`right fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
`
`©Springer-Verlag Berlin Heidelberg 1991
`Printed in Germany
`
`The use of 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 protective laws and regulations and therefore free for
`general use.
`
`Typesetting: Springer T EX inhouse system
`
`57/3140-543210- Printed on acid-free paper
`
`Energetiq Ex. 2007, page 2 - IPR2015-01377
`
`

`
`(2.50)
`
`(2.51)
`
`osed of the hydrodynamic flux of
`coefficient.3 The terms containing
`:/2 «: kTe, are neglected; accord(cid:173)
`ISSes in collisions with molecules
`, 17). The last term in (2.50) de-
`is the resultant creation rate that
`the ionization potential.
`that the equation for the rate of
`s is
`
`'le
`
`(2.52)
`
`;t to time, as in (2.49).
`1g the effects caused by spatial
`contribution of heat conduction.
`::m and passing thermal energy to
`uison with the term proportional
`
`~ve in (2.22, 50-52) in the form
`singled out of Fe. Ignoring the
`Lecount that f.i+ «: f.le, we obtain
`
`(2.53)
`
`l is, to the same accuracy,
`
`(2.54)
`
`red in Sect. 9.7.
`
`;ion (2.51) for F and >.e stem from the
`llian distribution functions and ~~rn(e:) =
`F 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(v / c)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 sin wt, 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 ::;p vm. We integrate the equation of collisionless motion,
`r = v,
`
`miJ = -eEosinwt,
`
`to give
`
`eEo
`.
`eEo
`v = - - cos wt + vo ,
`r = - -2 smwt+vot+ro.
`mw
`mw
`An electron oscillates at the frequency of the field; these oscillations are
`superimposed onto an arbitrary translation velocity v 0 • 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 /2. The limiting case of "collisionless" oscillations is approximately realized
`at optical frequencies, and also at microwave frequencies at low pressures, p ::::;
`10Torr.
`
`.
`
`(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. 2007, 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 = -eEosinwt- mvvm,
`
`r = v.
`
`(3.3)
`
`Vm
`r.p = arctan- ,
`w
`
`T =
`
`.
`
`The solution of (3.3), valid after several collisions, is
`eEo
`v = J
`cos(wt+r.p),
`m w2 +v~
`)
`. (
`eEo
`Sill wt + r.p
`mwJw2 +v~
`The amplitudes of displacement and velocity of the electron are less by a
`factor of )1 + v~/w2 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 1r /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 sinwt,
`and the other to its rate of change, E = wEo coswt:
`.
`Vm
`eEo
`eEo
`Sill wt + -
`cos wt ,
`m(w2 + v~)
`w m(w2 + v~)
`.
`weEo
`vmeEo
`2 ) cos wt -
`( 2
`2 ) Sill wt .
`( 2
`m w +vm
`m w +vm
`
`(3.4)
`
`(3.5)
`
`T =
`
`v =
`
`The ratio of the components is determined by the relative role of collisions
`and is unambiguously related to the phase shift r.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 = 21r j,
`which is greater than the frequency f by nearly an order of magnitude.1 In
`the limit v~ -«:: w2 , 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 A= cf /,but with A= A/27r: a/ A= eE0 jmw2 A= ufc.
`
`36
`
`Energetiq Ex. 2007, page 4 - IPR2015-01377
`
`

`
`e full range of displacement (3.2)
`1 starts oscillating anew after each
`.tive to the instantaneous direction
`;count, we add the rate of loss of
`>f motion of the "mean" electron.
`¥e have the equation for the mean
`
`)llisions, is
`
`Vm n- ' w
`
`(3.3)
`
`(3.4)
`
`city of the electron are less by a
`:illations. The higher the effective
`'m is determined by the velocity of
`targes than the oscillation velocity;
`hase relative to the field, the phase
`~ role of collisons vmfw increases
`
`(3.4) can always be resolved into
`:nitude of the field E = Eo sin wt,
`:::oswt:
`
`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~ )i = 1.6cm, 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~ ~ w2
`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 - --;:;:;T '
`
`(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 ~ 10 MHz, vm ~
`3 x 109 ps-I, 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 / p in a constant field.
`Therefore, at f ~ 10MHz and Eo/p ~ 10V /(em· Torr), we have A,....., 0.1 em
`regardless of pressure.
`
`oswt,
`
`,;t.
`
`(3.5)
`
`3.2 Electron Energy
`
`d by the relative role of collisions
`tift r.p. This form of presenting the
`1e subsequent sections.
`' collisions is characterized by the
`Lilar frequency of the field w = 271' J,
`tearly an order of magnitude.1 In
`lose to (3.1) for free oscillations.
`:xample of microwave radiation at
`x 1010 8-1. Let p ~ 1 Torr, then
`~hly corresponding to the threshold
`
`evaluated, the displacement amplitude must
`':!; = >.j2Tr: a/~= eEo/mw2 ~ = 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) =- eE~ (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 mv~/2, corresponding to the mean velocity v 0 = {v(t)),
`and that of oscillations. In the case of free oscillations, the latter energy is
`
`37
`
`Energetiq Ex. 2007, page 5 - IPR2015-01377
`
`

`
`(3.8)
`
`In the example given at the end of Sect. 3.1.2, f = 3 GHz, Eo= 500V 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.
`
`3.2.2 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)
`
`)
`
`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 j2 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
`L1c: E, equal to twice the mean kinetic energy of oscillations:
`jmv2
`e2E 2
`.
`L1EE = 2m(w2 ~ v~) = 2 \ -2- = 2cosc.
`Ihis result can be given the following interpretation. In the interval between
`two collisions, the electric field imparts to an electron an average kinetic energy
`Eosc· If the electron goes through a large number of oscillations in this period,
`then cosc 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 Eosc 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 b of its energy c;, then
`
`(3.11)
`
`38
`
`Energetiq Ex. 2007, page 6 - IPR2015-01377
`
`

`
`(3.8)
`
`.2, f = 3 GHz, Eo = 500 V js, and
`s much less than the mean energy
`tain a discharge. The absence of
`rgy of the field and no deposition
`
`to the electrons, via the electric
`1it time that the field performs on
`
`(3.9)
`
`that oscillates in phase with the
`d in phase by 1r /2 does no work,
`notion is not associated with any
`1 electron gains the mean energy
`,f oscillations:
`
`(3.10)
`
`pretation. I11 the interval between
`lectron an average kinetic energy
`Jer of oscillations in this period,
`1 energy (3.8). An act of elastic
`mges the direction of motion but
`Then the field starts swinging the
`velocity, that is, imparts to it an
`.lt of energy gained from the field
`collision is thus transformed into
`roscopically, the field does work
`he electron. Everything proceeds
`:ole of the drift energy is played
`
`n the field and transferring it to
`ic losses. If an electron loses in
`
`Vm'
`
`(3.11)
`
`where the amplitude Eo is replaced with the mean-square field E defined by the
`equality E 2 = (E2(i)) = E6f2. If w2 ~~'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 (w2 ~ v~) behave indistinguishably from constant
`fields; the similarity l = f(E jp) holds. At high frequencies, w2 ~ v~, the mean
`electron energy is independent of vm, p, and the similarity in field frequency
`holds: l = ft(Ejw). If 8 = const, l e< (E/w)2 • In equivalent situations, E e< w.
`This is the reason why gas breakdown at optical frequencies (w ,.._, 1015 s- 1)
`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 i 1
`and have, immediately after SCattering, a velocity VI and energy ci = mvf j2.
`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 i 1• In the time i ~ it and until the next collision, the electron
`is driven by the force -eEo sinwi at a velocity v(i) = u(cos wi- cos wit)+ VI,
`u = eE0 fmw. Its energy c: = mv2 /2 at each moment of this period is
`
`c:(i) =; [vr + 2vt u(cos wi- cos wit)
`
`+ u2
`
`( cos2 wi - 2 cos wi cos wit + cos2 wit)]
`
`At the moment iz 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:(i2), which is also the energy of random motion.
`Therefore, between two collisions the random-motion energy changes by
`
`Llc:(it, iz) = c:(iz)- ct ~ mvt · u(coswiz- cos wit) .
`
`39
`
`Energetiq Ex. 2007, page 7 - IPR2015-01377
`
`

`
`This last (approximate) transition takes into account that the random-motion
`velocity v1 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 tz - t1 between two collisions, and the corre(cid:173)
`lation between the field phases at the moments of collisions wt1 and wtz vanishes
`owing to the random nature of collisions, i.e., the phases may be arbitrary. The
`value of L.lc: then varies in this interval from the maximum gain L.lc:+ = 2mv1 u
`and the maximum loss L.lc:_ = -2mv1u, the extremal values corresponding to
`parallel velocities v1 and u and certain phases, wt1, wtz. When averaged over
`many collisions, however, that is, over moments t1 and tz, an electron gains the
`energy
`Llc: E = (L.lc: ( t1, tz) )t 1 , t 2 = mu2 12 = 2c:rr.osc.
`which we found above by calculating the mean work done by the field. 2 Since
`u 1 v ~ 1, the actual changes that the electron energy experiences in collisions, be
`they positive or negative, are of first order of smallness in ulv, IL.lc:±llc:,...., ulv,
`while the resulting positive L.lc: E Is ,...., ( u I v )2 is of second order. This latter
`quantity is a small difference of two relatively large ones; in symbolic form,
`L.lc:E,...., (L.lc:+ -IL.lcl).
`
`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 elm ratio) and assume that
`electrons collide only with electrons. Sum up over all electrons the equation of
`motion mv = -eEo sinwt+Pco1, where Peal 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 2::: mv of the gas oscillates as cos wt, with a 1r 12
`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:
`
`! L m;z = 2:mv ·v2 =- (:Eosinwt) (2:mv)
`
`IX sin wt cos wt IX sin 2wt .
`
`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 t2 -
`t 1 is exp[ -Vm(t2 -
`tl)lvmdtz. Correlations add an additional factor to LleE of (3.10): w2 /(w2 + v;.) [3.1].
`
`40
`
`Energetiq Ex. 2007, page 8 - IPR2015-01377
`
`

`
`account that the random-motion
`velocity u; hence, we can ignore
`i.
`llisions are rare: Vm ~ w. Then
`xn two collisions, and the corre(cid:173)
`•f collisions wit and wh vanishes
`he phases may be arbitrary. The
`1e maximum gain L.\c:+ = 2mvt u
`xtremal values corresponding to
`, wt1, wt2 . When averaged over
`s t 1 and t 2 , an electron gains the
`
`1 work done by the field. 2 Since
`ergy experiences in collisions, be
`tallness in ujv, JLlc:±J/c:"' ujv,
`is of second order. This latter
`r large ones; in symbolic form,
`
`d
`oms and molecules. In a strongly
`:ons with nearly equal frequency.
`taken into account in considering
`
`gas (an even more general case
`cal e/m ratio) and assume that
`ver all electrons the equation of
`te rate of change of an electron's
`entum of interacting particles is
`> oscillates as cos wt, with a 1r /2
`'articles is also conserved under
`: of the gas energy:
`
`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 + 4?r P
`and B = H + 4?r 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:
`
`(3.13)
`
`'
`
`1 &B
`curl E = - - -
`(3.14)
`c &t
`div B = 0 ,
`(3.15)
`div D = 4?re.
`(3.16)
`This system is not completely closed because the electric current j, polar-
`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 = uE, P = XeE,
`M = xH. Instead of the electric Xe and magnetic susceptibility x, one intro(cid:173)
`duces the permittivity c: = 1 + 4?rXe and magnetic permeability ft = 1 + 4?rx.
`Together with the equations
`j = uE, D = c:E, B = t-tH,
`(3.17)
`where the material constants c:, t-t and conductivity u are assumed to be known,
`system (3.13-17) is closed. In gases and plasmas, the approximation ft = 1 can
`be used with extremely high accuracy.
`
`3.3.2 Displacement, Polarization, Conduction, and Charge Currents
`
`:s is arbitrary because the moments of
`the interval t2 -
`t1 is exp[ -Vm(tz -
`of (3.10): w2 /(w2 +vi) [3.1].
`
`The right-hand side of (3.13) can be treated as a current density
`. &P
`1 &D
`1 &E
`.
`J +- -- = J + - +- -- '
`47r &t
`&t
`4?r &t
`
`(3.18)
`
`41
`
`Energetiq Ex. 2007, page 9 - IPR2015-01377
`
`

`
`times 4Jrjc. Maxwell called the term (1/47r)8Dj8t, 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 BPI 8t
`in the displacement current is indeed a current density: that of the polarization
`current. Together with the conduction current j, it forms the total charge current
`it· The term (1/47r)8E I 8t 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 O", 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 = Le;r;, Le; = 0,
`(3.19)
`where r; is the radius vector of the charge e;, v; = r; is its velocity, and
`summation is extended to absolutely all charges (free, bound, electrons, nuclei)
`within a unit volume.
`
`it= Le;v; = a:t,
`
`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 t1 are now determined less by the
`value of E(t1) than by the evolution of E(t) in the preceding period t < t1. 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 = eE) 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 Ew = Ew0 sinwt, then the total current is itw = itwO sin(wt + 'Pw), with itwO
`and 'Pw being the functions of Ew0 and w.
`
`42
`
`Energetiq Ex. 2007, page 10 - IPR2015-01377
`
`

`
`) 1 fJt, which he postulated should
`!Cement current. Without the dis(cid:173)
`.ssailable law of charge conserva-
`
`1f matter with time, namely, dis(cid:173)
`i by applying the electric force. In
`current, so that the term oP I fJt
`: density: that of the polarization
`, it forms the total charge current
`ted with the motion of charge and
`5 will be discussed in Sect. 13.5.)
`the conduction and polarization
`the equations to ideal dielectrics,
`nandatory. The total polarization
`neutral medium; P 1 is related to
`t to operate with just one of these
`
`, . = oPt
`(3.19)
`'
`'
`fJt
`e;, v; = r; is its velocity, and
`:s (free, bound, electrons, nuclei)
`
`Owing to the inertia of the pro(cid:173)
`hey cease to track the variations
`are now determined less by the
`the preceding period t < t1. For
`e in one direction and then was
`ome time in the fonner direction,
`1ght to rest.
`= c:E) that take into account re(cid:173)
`le motion of charges in matter is
`Maxwell's equations are also lin(cid:173)
`ed into Fourier series or integrals;
`:~perate only with harmonic com(cid:173)
`, 2. Three parameters completely
`amplitude, frequency, and phase,
`in the relation between these pa(cid:173)
`d. If the field for some harmonic
`tw = 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 oE I fJt. Returning to
`the original concepts of conduction current a E and polarization current a PI fJt,
`which is equal to [(c: -1)14n]OE I fJt if the field varies slowly, and equipping the
`new coefficients a and c: with the subscript w (because now they are frequency(cid:173)
`dependent), we rewrite the material equation in the form
`
`C:w -1 OEw
`.
`Jtw = awEw + ~ T' Ew = Ewo sinwt.
`
`(3.20)
`
`The quantities aw and C:w 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,
`Band Hare linear, we obtain the relation
`
`.
`
`(3.21)
`
`s
`
`(3.22)
`
`8 E · D + H · B
`di c [E H] _
`. E
`+ v-
`-
`x
`--J ·
`fJt
`8n
`4n
`This formula expresses the law of conservation of energy of the electromag(cid:173)
`netic field. The quantity E · D l8n is the electric energy density, and H · B l8n
`is that of magnetic energy;
`c
`S=--[ExH]
`4n
`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 polarization
`current does not result in dissipation because it is shifted by n 12 with respect to
`the field, and (sinwtcoswt) =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
`
`43
`
`Energetiq Ex. 2007, page 11 - IPR2015-01377
`
`

`
`single out the term eM - 1 in c:w - 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 w-4 in H2, and 0.67 X w-4 in He; it is even smaller at low pressure
`because c:M - 1 is proportional to density. These figures refer to the visible part
`of the spectrum and to lower frequencies. The contribution of the molecular piut
`to the bulk polarizability of plasma is very small for any appreciable ionization.
`
`3.4.1 Calculation of u w and ew
`First we substitute the general expression (3.5) of the mean electron velocity v
`into (3.19) for the charge current iv 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)
`
`C:w = 1- 4tre2ne/m (w2 + v;)
`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,
`
`(3.24)
`
`icond,O =
`41Taw
`'
`jpolar,O wlc:w- 11
`is determined by the ratio of collision and field frequencies.
`
`= Vm/w
`
`(3.25)
`
`3.4.2 The High-Frequency Limit (Collisionless Plasma)
`This regime is reached when w 2 ~ 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,
`
`c: = 1 _ 47re2ne
`
`mw2
`
`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
`
`" = 1 _ 47re2
`ne
`2
`mvm
`
`"-W
`
`•
`
`(3.27)
`
`44
`
`Energetiq Ex. 2007, page 12 - IPR2015-01377
`
`

`
`. electrons bound in molecules
`511itude as in nonionized gases
`.s in the plasma is greater than
`. S, eM- 1 = 5.28 X 10-4 in air,
`s even smaller at low pressure
`figures refer to the visible part
`ntribution of the molecular part
`for any appreciable ionization.
`
`)f the mean electron velocity v
`he summation over all electrons
`-andom velocity vanishes upon
`)n with (3.20), we find:
`
`(3.23)
`
`(3.24)
`
`mce for the physics of interac(cid:173)
`. The ratio of the amplitudes of
`
`(3.25)
`
`frequencies.
`
`;s Plasma)
`at not particularly high frequen(cid:173)
`even at atmospheri