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Electromagnetic Fields of Non-Equilibrium Plasmas.

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Annaleii der Physik. 7. Folge, Band 36, Heft 4, 1979, S. 341-352
J. A. Barth, Leipzig
Electromagnetic Fields of Non-Equilibrium Plasmas
Statistical Physics Laboratory, Department of Physics, State University of New York at Buffalo
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
[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
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
H = rot A ,
1 a2
For the purpose of calculating the radiation fields, it is convenient to use the potential
equations. These are given by
1 a22
Here, A and @ are related to each other by Lorentz’ condition:
1 a@
c at
Electromagnetic Fields of Non-Equilibrium Plasmas
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 ,
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)
A = - JC
where R is the distance between the observer and the source:
R = IT - 7'1.
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:
in terms of which eq. (2.5) may be rewritten as follows:
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 . %
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:
(1 (Rc - R w ) RC
f)j (7,
;i;;t -
" -- c / R ( R c
-R .W)
Similarly, for the electric field we arrive a t
= ec2
Rc(Rc - R . v ) ~
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
d(%- G1), whererlandZlaretheposition
for one particle if we replace f by - d(7’ n
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
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:
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 ) ,
where N is the total number of particles, @(I ? - T2I) is the Coulomb potential between
these particles and F* is defined by
Here, we used the relation:
H = rot 2,
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 :
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 =
Then, eq. (3.1) can be transformed into
which is equivalent to
where P is the Lorentz force defined by
7 (% X 2),
E‘ +
with the electric field
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:
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:
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
- + -.
(nu) = 0
with the number-density n and the mean velocity u defined by
$zfd?i = n % .
Another is the dynamical equation:
where F* is the mean Lorentz force defined by
- 1
F* = ;SF* f dv
= E* + - u x H ,
Electromagnetic Fields of Non-EquilibriumPlasmas
and alar. P is the divergence of the stress tensor:
- - P = nzJ(vv-uu)-dv +$n2-dr2
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:
P = P o - q - * u 1-2211-24,
0 ar
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)
Passing to the limit of large distances and integrating over the velocity space, eq. (4.1)
is reduced to
x S$
H n =-
n2 (r', r2;t -
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 :
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:
+ -21 r12
x; t)= %(r12, r'; t)
nz(r12, r'; z)
Thus, we have the following relation:
n2(r*,1 2 ; z) dr,
= fi2(r12,r ’ ; z) @’ 2 dr,,
+ -i 7.a
J r12E2(r12,r ’ ;z) @’ * d r l , .
2 ar
Z2(r12,x) =
n2(-r12, x), we obtain the following expression:
Z,(r’, rz;z) dr,
= - J ii2(r12,r’ ;t)@’
rX d r , ,
i a
J r12n2(r,,, r ‘ ; z) @‘ -dr12.
2 ar
Thus, combining the above two equations and introducing the result into Eq. (4.2)
we arrive a t the following equation :
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
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 rxr
= - -- x
JP,.n o d s .
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
(5.1 )
Electromagnetic Fields of Non-Equilibrium Plasmas
Then, ey. (2.13) may be rewritten as follows:
R(Rc - R V ) dt
dz dr' d% d o df2.
Next, the following equation
exp (ioRlc)
= -J exp ( i o n . R/c)dQ
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
n'. R'
exp [iw (z - t +
dz dr' dv d o dQ.
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.,
we obtain
x Z df exp (iw
H(f2, w ;t , r ) = 8 z c a J R c- R v d t
z]) d t dr' d v . (5.6)
Similarly, for the electric field we have
= J E(f2, w ;t ,
r ) e-iwt dw df2 + c.c.,
E(f2, 0 ;t , T) = -
- exp (iw
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:
Defining the following Fourier transform of the forces which appeared in eq. (5.10):
n .F'
d t dr'
- = . P expaw t--
we finally arrive a t the following expression:
J Hw," exp
[io (g- r -
A. ~ I H A R A
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,
En = $ E,,n exp [io
Equations (5.12) and (5.13) represent the dipole field, and, therefore the energy loss
may be computed from the Poynting vector:
S = - E ~ x Hn.
The total energy loss is obtained by integrating eq. (5.14) over time:
c o o
$ (EnXHn)dt
S, = -%(E,xB,)
e-i"t dcc, f c.c.;
Hn = J ii,
+ C.C.
Therefore, by using eq. (5.12) and (5.13) the power density in the direction specified by n
is given
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
Therefore, the expression for the magnetic vector in the radiation field reduces to the
following :
Electromagnetic Fields of Non-Equilibrium Plasmas
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
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.
The Quantum Theory of Radiation, Clarendon Press, Oxford 1954.
Atombau und Spectrallinien, Ungar, New York 1953, Vol. 2. L. LANDAU
Classical Theory of Fields, Addison-Wesley Publishing Co., Inc., Reading,
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
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).
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
Phys. Fluids 5, 1 (1960). A. ISIHARA,
Statistical Physics, Academic Press, NY 1971.
[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.
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.)
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