Annaleii der Physik. 7. Folge, Band 36, Heft 4, 1979, S. 341-352 J. A. Barth, Leipzig Electromagnetic Fields of Non-Equilibrium Plasmas By A. ISIHARA Statistical Physics Laboratory, Department of Physics, State University of New York at Buffalo (U.S.A.) A b s t r a c t. A statistical treatment of the electromagnetic fields of a fully ionized plasma shows t h a t there are two types of radiation. One of the radiation fields is determined b y the time change of the distribution function and is characteristic of such plasmas. When the time change is due to the many-body couplings of the charges in the plasma, the radiation can be related to the nonequilibrium stress tensor. In particular, for large distances the radiation is determined by the total internal pressure in a way similar t o the classical Larmor formula for a point charge. Elektromagnetische Felder in Nicht-Gleichgewichtsplasmen I n h a 1t s u b e r s i c h t. Die statistische Behandlung des elektromagnetischen Feldes eines vollstLndig ionisierten Plasmas zeiat daB zwei Arten von Strahlungen bestehen. Eines der Strahlungs’.. felder wird durch die zeitliche Anderung der Verteilungsfunktion bestimmt und ist typisch fur solche Plasmen. Wenn die zeitliche Anderung durch die Vielkorperkopplungen der Ladungen im Plasma verursacht wird, kann die Strahlung mit dem Spannungstensor des Nichtgleichgewichtzustandes in Beziehung gebracht werden. I m Spezialfall fiir groBe Entfernungen wird die Strahlung durch den inneren Gesamtdruck i n einer Weise bestimmt, die der klassischen Larmorformel fur eine Punktladung iihnlich ist. 1. Introduction The energy loss due t o radiation is an important factor in the production of high temperature plasmas, and, much work concerning such radiation has been reported. However, very little is known about the radiation characteristic of the plasma state which is caused by niany body interactions between plasma particles. Radiation which is essentially caused by interactions is bremsstrahlung. Bremsstrahlung induced by two colliding particles has been discussed by many investigators [l,21. However, it is clear that in plasmas we cannot neglect the many-particle interactions due to long-range Coulomb forces. Because of this, treatments of plasmas become very difficult, and various macroscopic hydrodynamics1 descriptions have been developed. However, such methods are not suitable for the discussion of bremsstrahlung where a treatment based on a particle model is required. The importance of such an individual particle response has been discussed by ROSENBLUTH and ROSTOKER [3] in their treatment of the kinetic equation for the distribution function. The correlation between plasma particles is known to yield the so-called collective oscillation. It has been shown that there exist two different types of organized plasma oscillation, transverse and longitudinal [41. The longitudinal type of oscillation does not radiate. However, if a plasma is not homogeneous either in density or in temperature, the longitudinal mot,ion will couple with the transverse motion, and the transverse oscil16 Ann. Physik. 7. Folge, Bd. 36 A. ~ I H A R A 242 lations may give rise to radiation. This coupling was discussed by VLASOVand more recently by FIELD and TIDDIAN[5] by a macroscopic method. We shall, instead, adopt a statistical method [6] to find a new type of radiation which is due to the collective couplings of the plasma. For this purpose, the electromagnetic fields due t o a charge distribution will be treated first in the next section. We shall show that both the electric and magnetic fields consist of three different component fields, one static and two radiation fields. Of the two radiation fields, one is simply a sum of individual particle radiation due to acceleration and the other is due to the time change of the particle distribution function. Before applying these results, we shall in Section 3 discuss exact conditions for the so-called Vlasov equation and show that this equation, being characterized by a n effective field in which the particles are smeared out, is not suitable t o our discussion of a particular radiation field due to particle correlations. We shall then apply the BBKGY equation to the radiation formulas given in Section 2 and derive a new formula for dipole radiation due to the collective couplings of the particles. The radiation is determined by the internal pressure of the plasma. Hence, we shall show that a coupling between the longitudinal collective mode and the transverse mode results in the characteristic radiation. A formal spectral analysis of the radiation field will be given in Section 5. Throughout this paper, we shall assume a fully ionized state. Therefore, spectral emission due t o neutral molecules which may be contained in a plasma will not be considered, although the line spectra due t o such neutral molecules, if any, may be strong [7]. We shall also not discuss that radiation which is in principle a n individual particle event. Therefore, cyclotron radiation will not be discussed. 2. Retarded Potential and Radiation I n this Section, a statistical description of the electromagnetic fields in a plasma will be presented. We shall assume a fully ionized state and consider the case where the ions are smeared out. The latter assumption is made only for the purpose of simplifying equations. Of course, in bremsstrahlung electron-ion collisions are as important as the electron-electron collisions, but the results in this paper can automatically be extended t o the case where ions are explicitly taken into consideration. If we denote the vector and scalar potentials b y Z ( 7 , t ) and @(r, t ) respectively, the electric and magnetic fields are determined by A & H = rot A , 1 a2 E=----grad@. A c at For the purpose of calculating the radiation fields, it is convenient to use the potential equations. These are given by - 1 a22 AA---= c2 at2 - 47ce -JGf(T’,G;t)dC, C Here, A and @ are related to each other by Lorentz’ condition: - 1 a@ divA+--=O. c at Electromagnetic Fields of Non-Equilibrium Plasmas 243 f ( F ' , %; t )is the distribution function of the particle at the position T' with the velocity v and satisfies the following normalization condition : $ f ( r ' , ' v ; t ) d r ' d v ^= n , (2.4) where n is the number-density of electrons. The solutions to eqs. (2.2) are given by the retarded potentials. Tor instance, ths vector potential is expressed as follows: e Gf(?',G;t- Rfc) dT'dG, A = - JC R - where R is the distance between the observer and the source: R = IT - 7'1. (2.6) Equations (2.5) and (2.1) determine the electromagnetic properties of our plasma. It is to he noted that the distribution function is at a retarded time, t - Rjc, which appears in the integrands of eq. (2.5) does not give the total number of charges when integrated over the phase space. This is due to the fact that the normalization of the distribution function is carried out at a fixed time as in eq. (2.4) and the distribution function in eq. (2.5) is a function of a retarded time which depends on the positions of the charges. Therefore, the integrations in eq. (2.5) are not defined in the ordinary phase space, and the volume element in ey. (2.5) is not the time invariant volume element in the phase space. For this reason, the apparently simple integrations shown in eq. (2.5) must be carried out carefully. We note that j(r',%; T - R/c)&'is associatedwith the particles having the velocity v = dr'/& in the position range T ' N 7' d?' at a time t = t - R/c. Therefore, it is convenient to introduce Dirac's &function of a new time variable: + A u=t-t+R/c, in terms of which eq. (2.5) may be rewritten as follows: (2.7) To calculate H , we differentiate eq. (2.8) with respect to r as shown in eq. (2.1), and integrate the results by parts using the relation: . - t + - )R= ( 1 - - ~ )T di . % '(s-t+R), C C where 6' is the derivative of -6 with respect to the argument. Thus, we find the following results after collecting terms in a suitable way: Iy'= Z8+ IFI f ITII, (2.10) where %X2 (1 (Rc - R w ) RC ~ f)j (7, ;i;;t - C (2.11) (2.12) e I? " -- c / R ( R c %XZ -R .W) (2.13) A. IsIHARa 244 Similarly, for the electric field we arrive a t z=z,i-z14-ZII, (2.14) with - E, s = ec2 ZC-Z.> (1 Rc(Rc - R . v ) ~ c (2.15) (2.16) (2.17) In these expressions, the acceleration v is assumed to be known as a function of the coordinates and the velocity and z = t - 17 - T‘l/c. These equations reduce to that 1 d(%- G1), whererlandZlaretheposition for one particle if we replace f by - d(7’ n rl) and velocity of the particle. At large distances the fields Ha and E, are both inversely proportional t o the square of the distance R . Therefore, they are simply the static field. The field composed of HI, and EIIvaries as 1/R a t large distances if df -, the time change in the charge distridz butions is local and is independent of R , and represents a radiation field, This also applies t o the field H I and EI. The distinguishing feature of these two radiation fields H I , E, and HI,, EII is clear. The former is induced by the acceleration and is the summation of the radiation of the individual particles. It was derived by MASON and WEAVER[8] by a different method. The latter field is induced by the time rate of change of the distribution function and is characteristic of non-steady state plasmas. The change in time of the distribution function is caused, for example, by a n injection of a beam of charged particles into the system or by an application of the time dependent external forces. The collisions between the plasma particles also cause the change -the so-called collisional change. We are particularly int<erestedin the radiation induced by collisions. The reason why (df/dz),-the collisional change of the distribution function-determines the field may be understood by considering a non-steady state plasma wherein the state of the particles is changing in time. In this case, all the particles in our system must rearrange their momenta from time t o time to meet new situations into which they are brought. However, since the particles are in the plasma state and are strongly coupled with each other such rearrangements are coupled with each other and would require an extra energy input. Radiation caused by such a mechanism may be characteristic of the plasma state and may have a certain relation t o the statistical behaviors of the system. Therefore, we may call this “the plasma radiation”. Of course, collisions take place irrespective of whether our system is in equilibrium or not. In equillbrium cases, however, the velocity distribution of particles does not change in time. Therefore, the radiation which we expect must be characterized by this constant distribution function. An ideal example of an equilibrium situation is Planck‘s black body radiation. In actual plasmas, the radiation equilibrium which is necessary for black body radiation may not be attained. Electromagnetic Fields of Non-Equilibrium Plasmas 246 3. Kinetic Equations I n order t o calculate the plasma radiation, it is necessary t o know the charge distribution. The differential equation which determines the distribution function is obtained by applying the methods developed by Born, Green and others to the Bystem composed of charged particles and of the radiation field [7, 91. We start with a Hamiltonian for this system of the following form [ l ] : where the vector potential A" is assumed to have the form: The first term corresponds to the external field and the second term is the Fourier expansion of the radiation field. - is the canonical coordinate of the wave of wave qrl ( 4 are Coulomb interaction potentials between the plasma particles. length 1. The The distribution function which describes the total system satisfies Liouvill's equation which is obtained by setting the Poisson bracket between the distribution function and the above Hamiltonian equal t o aflat. The differential equation satisfied by the charge distribution function is easily obtained by integrating the Liouville equation over the states of the radiation field and over the states of all but one particle. The result is given by the BBGKY equation: wheref2(7, V ;T 2 ,7j2;t ) is the two-body distribution function which gives the probability of finding two particles of velocities ?j and ?j2 a t the positions r and r2 respectively. f2 satisfies the following normalization condition. Jf2dC2 dG = ( N - l ) f ( T , % ; t ) , (3.4) where N is the total number of particles, @(I ? - T2I) is the Coulomb potential between these particles and F* is defined by E"*=Z*fc(%X2). 1 (3.5) Here, we used the relation: A H = rot 2, A. where the vector potentials 1 and are that which are averaged over the radiation stat,es in terms of the distribution functions for the total system. E* is the external field plus the radiation field. The right hand side of eq. (3.3.) is the so-called collision term. If the ions are explicitly taken into consideration, we must add t o this right hand side the following term : A. I~IH~LRA 246 where fll is the probability distribution function of an electron-ion pair interacting with the potential ly -Ziand siare respectively the velocity and coordinates of an ion. We introduce a n average potential function W a r by the following equation: - f-sgpdi&dF2 = (3.9) Then, eq. (3.1) can be transformed into aj 1 a (3.10) which is equivalent to (3.11) where P is the Lorentz force defined by 1 = 7 (% X 2), ” A E‘ + (3.12) with the electric field (3.13) As is clear from eq. (3.9), the right hand side of eq. (3.11) is generally not equal to zero. However, if the dependence of the distribution function f2 on the velocity and the coordinates is separable or if f2 is the product of the one body distribution functions: v f2 = f (7, ; t ) f (72, v z ; t ) the right hand side of eq. (3.11) vanishes and we have the so-calledVlasov equation: Y (3.14) It is to be noted that the particles are now smeared out into the form of an effective field. Therefore, this equation may not be suitable to discuss bremsstrahlung. For our later reference, we give here two additional equations. One is the equation of continuity: an a at ar - + -. (nu) = 0 (3.15) with the number-density n and the mean velocity u defined by $fdi?=n, $zfd?i = n % . Another is the dynamical equation: (3.16) (3.17) where F* is the mean Lorentz force defined by - 1 F* = ;SF* f dv 1 = E* + - u x H , C (3.18) Electromagnetic Fields of Non-EquilibriumPlasmas 247 and alar. P is the divergence of the stress tensor: a af a@ ar ar ar - - P = nzJ(vv-uu)-dv +$n2-dr2 (3.19) where n2 is the pair distribution function : n2(r,r2;t ) = J f 2 dv dv,. These equations can easily be derived from eq. (3.3). It can also be shown that the pressure tensor takes the following form: a a P = P o - q - * u 1-2211-24, 0 ar ar ( ) (3.20) if our system is not in equilibrium and the mean velocity u depends on coordinates. Here ajar u is the divergence free tensor, Po is the equilibrium constant pressure, and qo and q are the coefficients which come from the deviation of the distribution functions f 2 and f from the equilibrium functions. 4. Dipole Radiation Following the discussions in Section 2, we shall calculate the plasma radiation induced by collisions. The physical significance of this radiation may be obtained in consideration of the dipole field. Let us use the collision term on the right hand side of eq. (3.3) as a source of radiation. Then, we have from eq. (2.13) (4.1) Passing to the limit of large distances and integrating over the velocity space, eq. (4.1) is reduced to ilr2 x S$ H n =- n2 (r', r2;t - (4.2) C Here, %(r', r2) is the pair distribution of two particles at r' and r2. We may express the pair distribution function n2 as a function 5, of the relative coordinates r12 = r2 - r' and the two particles [101 : 1 + The dependence of C2(r12, x) on the center of gravity coordinates x = r' r12/2x = r' r12/2may be considered to be small. Moreover, the main contribution to the integral of eq. (4.2) comes from the region where the two molecules are not very far apart, namely where r12 is small. Therefore, we may approximate G2(r12,2) as follows: + qf-12, + -21 r12 x; t)= %(r12, r'; t) a nz(r12, r'; z) - 248 A. ISIHARA Thus, we have the following relation: Jg n2(r*,1 2 ; z) dr, r = fi2(r12,r ’ ; z) @’ 2 dr,, + -i 7.a r J r12E2(r12,r ’ ;z) @’ * d r l , . r12 2 ar r12 Replacing r12 by -r12 in this expression and using the symmetry property Z2(r12,x) = n2(-r12, x), we obtain the following expression: Jg Z,(r’, rz;z) dr, = - J ii2(r12,r’ ;t)@’ rX d r , , i a +- ‘12 J r12n2(r,,, r ‘ ; z) @‘ -dr12. 2 ar r12 Thus, combining the above two equations and introducing the result into Eq. (4.2) we arrive a t the following equation : r12 e r 2mc2 r2 Hn=-- e C (4.3) r x (-JPi.nods). --mc r Here, the integration in the last expression of eq. (4.3) is extended over the surface of the plasma, nobeing the normal to the surface. Piis the internal pressure tensor defined by C (4.4) dr12 * The physical meaning of eq. (4.3) is interesting. We recall that a momentum change of an electron gives rise to radiation in accordance with the classical theory of Larmor. Since pressure is a momentum change per unit area eq. (4.3) agrees with Larmor’s theory in this respect. We get a similar expression for the electric field: E, e rxr = - -- x me2 13 JP,.n o d s . (4.5) Therefore, the power radiated is proportional t o the square of the surface integral. The difference between eq. (4.3) and Larnior’s formula lies in the fact that in the former the momentum change is associated with the entire system of particles. I n other words, the plasma radiation is due to many electron correlations. It is not produced by a single particle. Equation (4.3) is expressed only in terms of the electron pressure. This is simply because we have neglected the ions. It is clear that essentially the same type of equation is obtained even if the ions are taken int,o consideration. 5. Spectral Analysis of the Plasma Radiation We shall now go one step further for a non-steady state plasma, and calculate the spectrum of the plasma radiation and prove eq. (4.3) by a different method. For the purpose, it is convenient t o introduce the well known formula [ 111: 6 z - - t + -R) = - ~ e x p [ i n , ( r - - t + 1 dw. (5.1 ) ( c 212 ”)] c 249 Electromagnetic Fields of Non-Equilibrium Plasmas Then, ey. (2.13) may be rewritten as follows: H, = e -J 2nc V X T R(Rc - R V ) dt C dz dr' d% d o df2. (5.2) Next, the following equation w exp (ioRlc) = -J exp ( i o n . R/c)dQ (5.3) R 4nc may be substituted in eq. (5.2). Here n is a unit vector in a definite direction, and the integration is carried out over the angles between n and R . The result is OziXZ Hn=&JRe-R.v n'. R' df exp [iw (z - t + dt e)] dz dr' dv d o dQ. (5.4) Therefore, if we make a Fourier analysis of the radiation field given by eq. (5.4) in the following form H , = J H(Q, w ;t , r ) e-i'"t d w df2 + c.c., (5.5) we obtain ' x Z df exp (iw H(f2, w ;t , r ) = 8 z c a J R c- R v d t - [$G-+ z]) d t dr' d v . (5.6) Similarly, for the electric field we have E, = J E(f2, w ;t , r ) e-iwt dw df2 + c.c., E(f2, 0 ;t , T) = - - exp (iw r$-+ (5.8) z]) d t d7 ' dC . These expressions for the field take simpler forms when the distance between the source and the observer is larger. I n such a case, ii/R can be expanded about TJr, the latter being a constant under the integrations involved in these expressions. Then, we use the dynamical equation (3.17): and take the unit vector n normal to the right hand side of this equation: (5.10) Defining the following Fourier transform of the forces which appeared in eq. (5.10): n .F' d t dr' (5.11) x,, - = . P expaw t-- ) we finally arrive a t the following expression: H, = J Hw," exp [io (g- r - t)]dw dQ, (5.12) A. ~ I H A R A 250 It is to be noted that F* represents the mean Lorentz force due to externally imposed electric field plus the transverse component of the radiation field; the Coulomb interaction is contained in the pressure term. If an inhomogeneityin temperature or in density exists, pressure w ill not be diagonal but take such a form as eq. (3.20). Also, if an external electromagnetic field is applied to the plasma, n will lie in the direction of propagation and X,,%will be perpendicular to this direction. The corresponding expressions for the electric field are given by :( - 7 - z)] d o ds2, A En = $ E,,n exp [io (5.13) Equations (5.12) and (5.13) represent the dipole field, and, therefore the energy loss may be computed from the Poynting vector: C S = - E ~ x Hn. 4n The total energy loss is obtained by integrating eq. (5.14) over time: c o o 4n $ (EnXHn)dt = 2n$S,dw, (5.14) (5.15) where c S, = -%(E,xB,) (5.16) 2n and E, =JE, (5.17) e-i"t dcc, f c.c.; Hn = J ii, eimt dcc, + C.C. Therefore, by using eq. (5.12) and (5.13) the power density in the direction specified by n is given (5.18) Sqn = It is interesting to study these spectral resolutions in the limit where there are no external fields and where pressure tensor p is nearly independent of time. Then, we have = -2n -- d(cc,)d3". a pe-i,= c jp. (5.19) Therefore, the expression for the magnetic vector in the radiation field reduces to the following : Electromagnetic Fields of Non-Equilibrium Plasmas 2.51 This is simply eq. (4.3). We can obtain a similar expression for the electric field. A s we mentioned before, these limiting expressions which correspondto dipole radiation indicate that the momentum change of all the particles in the system causes the plasma radiation. I n this sense eqs. (5.16) and (5.17) may be taken to have the same meaning as eqs. (4.3) and (4.5). 6. Concluding Remarks The formulae (5.15), (5.16) and (5.17) in the previous section tell us that “the plasma radiation” is determined by Xa,a which comprises two terms, a force term and a pressure term. The contribution from the force term becomes very small if there are no time varying external forces. The pressure term contains the Coulomb interaction. Therefore, the pressure term indicates the coupling of the longitudinal Coulomb field with the plasma radiation. If there is no inhomogeneity in the system, the pressure tensor will be diagonal and constant. Therefore, its divergence vanishes. Thus, we do not expect radiation in this case. In this connection, it is interesting to note that only the off-diagonal terms of the pressure tensor have a direct relation to transport coefficients such as viscosity or heat conductivity associated with irreversible phenomena. The above conclusion concerning the coupling of the longitudinal field with the radiation field is in agreement with the conclusion reached by VLASOV,FIELDand TIDMAN [5] by entirely different methods. However, we must note tat we have treated only bremsstrahlung from non-equilibrium plasmas, for which not only spacial inhomogeneities but also the time rate of change of the pressure tensor is necessary. We have neglected the ions in our treatment. Nevertheless, it is clear that the results in this paper hold, with appropriate modifications of the equations, for the case where ions are explicitly taken into consideration. It is again stressed that the ions play important roles in collisions or in maintaining oscillations and radiation. Thus far only the general aspects of bremsstrahlung from non-equilibrium plasmas have been discussed. If we introduce a particular physical situation, we would obtain that radiation peculiar to the condition selected. According to eqs. (2.13), (2.17) and (5.15), we see the possibility of exciting plasma radiation by inducing a change of the distribution function in time. Injection of a beam of charged particles or application of an external alternating field would, therefore, induce radiation. The treatment of these special cases appears to be the next task. References [l] W. HEITLER, The Quantum Theory of Radiation, Clarendon Press, Oxford 1954. Atombau und Spectrallinien, Ungar, New York 1953, Vol. 2. L. LANDAU [2] A. J. F. SOHMERFELD, and E. LIFSCHITZ, Classical Theory of Fields, Addison-Wesley Publishing Co., Inc., Reading, 1959. [3] M. ROSENBLUTH and N. ROSTOKER, Phys. Fluids 3, 1 (1960). [4] L. LANDAU, J. Phys. (U.S.S.R.)10,25 (1946);D. Born and E. GROSS, Phys. Rev. 75,1851,1864 (1949). Many Particle Theory, State Publishing House, Moscow 1950; G. B. FIELD, [5] A. A. VLASOV, Astrophys. J. 124, 555 (1956); D. A. TIDMAN, Phys. Rev. 117, 366 (1960). [6] N. BOQOL~BOB, J. Phys. 10, 257 (1946); J. G. KIRKWOOD, J. Chem. Phys. 14, 180 (1946); M. BORNand H. S. GREEN,Proc. R. SOC.A 188,lO (1946); H. S. GREEN, The Molecular Theory Phys. Fluids 4,341 of Fluids, North Holland Publishing Co., Amsterdam 1952); H. S. GREEN, (1959); CEAN-MOUTCHEN,Phys. Rev. 114, 394 (1959); B. B. KADOMTSEV, Sov. Phys. JETP 6, 117 (1958); M. ROSENBLUTH and N. ROSMKER, Phys. Fluids 5, 1 (1960). A. ISIHARA, Statistical Physics, Academic Press, NY 1971. 252 A. ISIH~RA [7] F. H. CLAUSER, Symposium on Plasma Dynamics, Addison-Wesley Publishing Co. Inc. Reading, 1960, Chapter 11. The Electromagnetic Field, The University of Chicago Press, 1951. [8] M. MASONand W. WEAVER, [9] W. E. BRITTEN,Phys. Rev. 106, 843 (1957); E. G. HARRIS,NRL Report No. 4144 (1958); A. ISIHARA, PIB Report No. 628 (1959); J. E. KLEVONTOVICH, J. Exp. Theoret. Phys. (U.S.S.R.) 16, 1405 (1959). [lo] H. S. GREEN,LOC.Cit. [ll]J. SCBMTNGER, Phys. Rev. 76, 1912 (1949). Bei der Redaktion eingegangen am 9. Juni 1978. (Revidiertes Manuskript eingegangen am 7. September 1978.) Anschr. d. Verf.: Prof. Dr. A. ISIHARA Statistical Physics Laboratory Department of Physics State University of New York a t Buffalo N.Y. 14260 (U.S.A.)

1/--страниц