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Theory of transverse waves in Vlasov-plasmas
Physica 29
Felderhof, B. U.
662-674
1963
THEORY OF TRANSVERSE WAVES IN VLASOV-PLASMAS III, REFLECTION OF ELECTROMAGNETIC WAVES BY A PLASMA HALF-SPACE
by B. U. FELDERHOF Instituut voor Theoretische Fyaica, Rijksuniversiteit, Utrecht
Synopsis In this paper we solve the problem of reflection of normally incident electromagnetic waves by a plasma half-space. The equilibrium plasma is supposed to be spatially homogeneous and isotropic in velocity space; the particles are assumed to be reflected specularly at the plane boundary. From the multitude of stationary solutions, compatible with the latter assumption, one is selected by imposing a causality condition. The exact reflection coefficient found from this solution is compared with the results predicted by the Fresnel formula with the aid of the refractive index; the present problem leads to a natural extension of the definition of the refractive index, given in I, to the range of frequencies smaller than the plasma frequency. The relation to the theory of the anomalous skin effect in metals is also discussed.
1. 1 ntroduction. In parts I and II of this paper 1) we solved the initial-value problem of Vlasov and Maxwell equations for transverse waves in a homogeneous, unbounded plasma. It is interesting to consider in what manner such waves are excited when vacuum electromagnetic waves arrive at a bounded plasma. We shall solve this problem of reflection and transmission for the case of waves, normally incident on a half-space filled with a homogeneous, isotropic Vlasov-plasma, in the absence of external fields. It will be assumed that the plasma particles are reflected specularly by the plane boundary. The situation we consider can be regarded as a good approximation of real physical situations. In reality, a plasma confined by a vacuum magnetic field cannot be homogeneous over the whole region of space it occupies. There must be a region where the plasma is inhomogeneous, separating the homogeneous plasma and the adjacent vacuum. The thickness of this boundary layer is of the order of the gyration radius corresponding to the magnetic field at the boundary. Particle trajectories at the boundary are in accordance with the assumption of specular reflection of particles (henceforth to be called the speculum ass~~mption). Our approximation. will be a good one, provided the boundary layer thickness is much smaller than other characteristic dimensions, such as the vacuum wave length or the skin depth, i.e. the depth of penetration of the fields. -
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TRANSVERSE WAVES IN VLASOV-PLASMAS
-------------------------------
663
It is necessary to impose a causality condition in order to select the correct stationary solution, corresponding to a given frequency, from the multitude of solutions compatible with the speculum assumption. This causality condition, made precise in section 4, leads to a restriction on the coefficients of the expansion in solutions of different wave number. Afterwards it is proved from the explicit solution for distribution and fields that the causality condition is satisfied (section 7). The present problem has recently also been considered by Sh ur e P). Using the method of Ca s e "), Shure obtains the same expression for the expansion coefficients, but his justification of the solution from causality arguments is not sufficient. Shure also studies several different situations. The solution leads to a natural extension of the definition of the refractive index to the frequency range Iwl < Wp (section 8). In section 9 we compare the reflection coefficient, as obtained from the Fresnel formula with the aid of the refractive index, with the exact one resulting from the solution of our problem. There is a close relation between the present theory and the theory of the anomalous skin effect in metals, given by Reuter and So nd he im er s]. In that case causality is introduced automatically by the presence of a collision term in the equation for the distribution function. Moreover, Reuter and Sondheimer use the artificial device of introducing an external current source at the plasma surface; we avoid this and believe that our solution leads to a clearer understanding. It will be shown (section 10) that our methods yield the same results as the theory of Reuter and Sondheimer.
2. Basic equations, The basic equations for the present problem are Vlasov's linearized equation 1(2.2) and Maxwell's equations 1(2.3) for the half-space filled with plasma, and Maxwell's equations with p and j identically zero for the vacuum half-space. Vacuum electromagnetic waves of definite polarization normally incident on the boundary plane will be partly reflected, but will also excite waves in the plasma with the same direction of propagation and the same polarization, owing to isotropy of the plasma. It suffices to consider waves of linear polarization and we choose coordinates such that the x-axis is in the direction of the electric vector and z = 0 is the boundary plane; z < 0 will be taken to be the vacuum half-space, Z > 0 the plasma half-space. The equations for the plasma half-space can then be reduced to equations 1(2.6). The general solution of the basic equations can be obtained by superposition of solutions with time dependence of all quantities given by a factor exp(-iwt), where w is real. The dependence on space and time of the vacuum. electric and magnetic fields is then given by factors exp(±iwzlc -
664 -
B. U.FELDERHOF
iwt). Equations 1(2.6) for the plasma can be written in the form of~. = 2wf~
Vz -
8z
noe + -E~F(vz), m
8E~ = OZ
iw BW c u:
8B~ 8z
io:
-- = -
c
E~
(2.1 ) 4ne c
- --
f
f~(z,
vz) dvz.
Equations (2.1) may be regarded as a boundary-value problem, to be solved for given values f~(O, ve), E~(O) and B~(O) at z = O. In fact, however, in the problem of reflection these values are not known and the solution has to be found from the speculum assumption and the causality condition. Of course, we must connect the vacuum fields to the fields in the plasma in order to find the appropriate linear combination of vacuum plane waves. 3. Periodic and exponential solutions of (2.1). In the present section we consider solutions of (2.1) with space dependence of all quantities given by a factor exp(ikz), without being concerned with conditions at the plane z = O. Such solutions exist for all real k, and are of the form
gW,U(vz) = q2P
u 3F(ve} U -
Ve
+ A(w, u) o(~t -
EW'"" = -(4niejw) u 2, BW.1' = -(4niejw) CU,
vz), (3.1)
where u = wjk and q = wpjw. The solution (3.1) is again normalized to
J gW,1'(V e) dvz = u 2 - c2 ,
(3.2)
so that A(w, u) is given by A(w, u) = u 2
-
c2 + q2P
f u3F(vz) Ve -
(3.3)
U
The solutions (3.1) are the same as 1(3.9), but in the following w will be regarded as fixed and k as a variable. One can also look for solutions of (2.1) with complex k. These solutions are possible only, if u is a (complex) root of the equation (3.4)
Investigating the analytic behaviour of the function on the left-hand side, one finds that for q2 > I, i.e. for [co] < Wp, this function has two zeros ± Uo on the imaginary axis; for q2 < 1 there are no complex zeros. Hence for
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
[eo]
<
wp,
665
apart from the solutions (3.1), there are two solutions of the form
r ±1t.(vz) =
q2 ±ugF(vz) , ±1-to -
Vz
=
-(4niejw) u5,
Bw,h. =
+(4niejw) cUo.
Em.
h
.
(3.5)
If we write %0 = i [wi d, where d > 0, the spatial factor of the solution of (2.1) corresponding to +~to is exp(zld) for w > 0; for the solution corresponding to -1-to the spatial factor is exp( -z/d). For w < the situation is reversed. For convenience of notation we shall retain only the superscript ~t to denote the periodic solutions (3.1), whereas the exponential solutions (3.5) will be denoted by superscripts ±.
°
4. Specul-um assumption and causality condition. An arbitrary superposition of periodic and exponential solutions (3.1) and (3.5) will again be a solution of (2.1). With superposition coefficients C(u), C+ and C- such a solution can be written
vz) =JC(u) glt(v z) eiwz!u cht + :L: C±g±(vz) e±iwz!1to, ± E~(z) = -(4nielw) [J~t2C(~t) eiwz!" du +::L: C±:'i-t5 e±iWZ/UO],
f~(z,
(4.1)
±
B~(z) =
-(4nielw) c[JuC(u) eiroz!u du
±::L: C±uo e±iroZ!Uo], ±
where the sum of discrete terms is of course only present for Jw) < W p . It is possible to prove that an arbitrary solution of (2.1) can be written in the form (4.1), but for the following this proof is not needed. The assumption that particles are reflected specularly by the boundary surface implies (4.2) f~(O, v z) = f~(O, - v z). Using the fact that
gU(-v z) = g-u(v z),
g+(-vz)
=
g-(vz),
(4.3)
one finds from (4.1) and (4.2) that the speculum assumption imposes the following restriction on the expansion coefficients
J [C('i-t)-C( -%)J gu(vz) d#
+ (C+-C-) g+(v z) + (C--C+) g-(v z) = O.
(4.4)
Of course, equation (4.4) does not determine the coefficients, which means that for given amplitude of the incoming plane wave there are still many solutions satisfysing (4.2). We shall impose a condition of causality, which for every frequency wand given amplitude of the incoming wave selects a unique solution from the multitude of stationary solutions compatible
666
B. U. FELDERHOF
with (4.4). This unique solution must be such that by superposition of these solutions over different frequencies, it is possible to construct a solution of the basic equations which for z < 0 represents a wave packet of arbitrary shape travelling in the positive z-direction, while there are no waves travelling in the negative z-direction and while the plasma is at rest, ~tntil the wave packet arrives at the boundary plane. Hence, the causality condition excludes all those solutions in which the plasma is in a perturbed state before waves from the vacuum arrive at its boundary. It is intuitively clear that the restriction on the expansion coefficients corresponding to this causality condition is C(~t) =
0
c+
= 0
C-
= 0
u < 0, w>O, w < O.
for for for
(4.5)
That is, from the periodic solutions (3.1) and the exponential solutions (3.5), corresponding to a given frequency w, only the periodic solutions propagating into the plasma and the exponentially decreasing solution are admitted. It will be shown in the next section that the restrictions (4.4) and (4.5) uniquely determine the expansion coefficients, except for a constant factor. It is necessary to give a rigorous justification of the restriction (4.5), because a superposition of periodic solutions with negative phase velocity could be chosen such as to yield a vanishing contribution for large values of z. We shall justify (4.5) by showing that the explicit solution, which we shall obtain, satisfies the causality condition formulated above.
5. Solution of equations (4.4) and (4.5). Introducing the antisymmetric function A (u) and the discrete coefficient D by A(u) = C(u) - C(-u),
D = C+ - C-,
(5.1)
one can write for equation (4.4)
f A(u) gu(vz) du
+ D{g+(v z) -
g-(vz)} = 0;
(5.2)
or, using the explicit form of gu(vz) as given by (3.1),
{u2 - c2 + 2niq2u3F+(u)}A+(u) + {u2 - c2 - 2niq2u3F _(u)}A_(~t) = = -q2F(u) f [u2 + wu' + U'2J A(~t') d~t' - D{g+(u) - g-(u)}.
(5.3)
On separating the right-hand side of (5.3) into its positive- and negativefrequency parts and making use of the antisymmetry of A(u), one obtains
A(u) = a{l
+ 2niq2uF+ (u)} -
D{gt (u) -
.2"(u)
+
-a{l -
2niq 2uF_(u)}
t+ (u)}
+
- D{g:!:.(u) - g=(u)} .2"*(u)
(5.4)
TRANSVERSE WAVES IN VLASOV-l'LASMAS. III
667
where
a = (ij2n) J uA (~t) du,
(5.5)
and where .£Z'(u) is obtained from Z(~t)
= ~t2 - c 2
+ 2niq2~t3F+ (tt)
(5.6)
by shifting real zeros of Z(u) an infinitesimal distance below the real axis. We henceforth assume that F(v z) vanishes identically for Ivzl > c (d. section I, 6). In that case, for q2 < I, the function Z(u) has two real zeros ±~tt; also, for q2 < 1, equation (3.4) has no complex roots so that the terms in D are absent. For q2 > 1, Z(u) has no zeros on the real axis and in that case the coefficient D must be such that the numerators in (5.4) vanish at the zero Uo of f!l'(u) in the upper half plane and at the zero -Uo of f!l'*(u) in in the lower plane, respectively. This yields the following relation for D
D [u~
ac2
-- = -u~ 2ni
- c -q sf 2
2u
Uo
o
F(v z) ] dv z . (vz - uo)2
(5.7)
On summation of the two terms in (5.4)and explicit calculation of the factors in the numerators multiplying D, one finds that A(u) can also be written
.
A(~t) = (D~to - mal
2c2q2~tF(u)
1.£Z'(u)1 2
•
(5.8)
According to (4.5) and (5.1) the coefficients C(ttl. C+ and C- can easily be obtained from A(u) and D,
C(u) = A(u) C(u) = 0 C+ C+
= 0,
=D,
C-= -D C- = 0
for for for for
u u
> < >
0, 0, w 0, w
(5.9)
so that with the aid of (4.1) the factor Duo - nia in (5.8) can be expressed in terms of the magnetic field at z = O. When the relation obtained is also used in (5.7), one finds for A(u) and D
D =
A(u = iwc B(O) q2 uF (u) ) 2ne 1.£Z'(u)1 2 1 ieee
icoc _ . B(O) . 2ne
u~.£Z"(uo)
= -
2ne
'
-1
(5.10)
B(O) . --;;-~---:u~.2'*/(_~tO)
The discrete coefficient D of course only differs from zero for lcol < COp. The magnetic field B(O) appears as an undetermined proportionality factor.
6. Velocity distribution and fields. Inserting the coefficients C(u), C+ and C-, as given by (5.9) and (5.10) into equation (4.1), one can evaluate the velocity distribution function and the electric and magnetic fields. The
668
B. U. FELDERHOF
results can be cast into rather simple forms by some manipulation of the occurring integrals. Only for the electric field we shall elaborate on the details of the calculation; the magnetic field is obtained from this by a simple differentiation. For w > 0, the electric field in the plasma is given by (6.1)
The integral in this expression equals
~f[ 2m'
__
o
1 _. -
.?t'(~6)
1
fZ'*(u)
]
whereas the second term is only present for w into an integral
f
eiwz!tt
clu
(6.2)
,
<
Wp
and can be transformed
00
e- i ",z/1to
-1
-.,.,----- = - -
a'*'(-uo)
2ni
eiwz!tt
fZ'*(~6)
-00
z > 0,
du,
0<
W
<
wp.
(6.3)
For co > Wp the integral in (6.3) amounts to zero. Hence one can write for allw>O 00
0
f
E~(z) = ~ B~(O) [J ~z::; d~6 + ~:~::t) d~t ] o
-00
00
For w> Wp, the term in brackets in the latter expression has real zeros at ±kt = co] ±Ut and it follows from the definition of .?t' and a'* that one should integrate below the zero on the positive k-axis and above the zero on the negative k-axis. A similar calculation for co < again leads to the second expression in (6.4), but with the prescription that, for co < -COp, one should integrate above the zero of the term in brackets on the positive k-axis and below the zero on the negative k-axis. Equations (4.1), (5.9) and (5.10) may also be used to calculate the reduced particle distribution function. The calculation is similar to, but slightly more complicated than that for the electric field. We shall only quote the final result, which is
°
669
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
we
!~(z, Vz) = - -B~(O). ;n;
noe F(v;,<) m
J[(w - k _w;wJw-kv F(v,~), dV~) (w-kvz+is)]-\ik2dk. 2 C2
2
(6.5)
z+1,s
This expression is valid for all frequencies w; for Iw) > Wp one should follow the same prescriptions for integrating across the poles as given for (6.4). We are now in a position to make the connection between the fields in the plasma and the vacuum electric and magnetic fields. By integration of (6.5) over Vz it is easily seen that the current density is finite everywhere; in particular there is no singularity at z = O. Hence it follows from Maxwell's equations that the electric and magnetic fields are continuous at z = 0. The vacuum electric and magnetic fields can be written e iW2/C R e-'iwz/c E~(z) = B~(O) - - - - - -
+
l-R
e iwz/C B~(z) =
B~(O)
_
z
R e- iWZ/ll
(6.6)
------
l-R
In the theory of metals it is customary to express the reflection coefficient R(w) in terms of the suriace impedance 5, defined by 4;n; Ea;(O) 5=----, c By(O)
(6.7)
R = 5 - 4;n;{c . 5 + 4;n;{c
(6.8)
so that
Making use of the continuity of the fields at z = 0, one finds from (6.4) that 5(w) is given by 00
5 = 4iw
J[
w2
-
k 2c2
-
w~ w
J
w -
F(vz) k Vz
J-
. dv z
+ 'LS
1
dk,
(6.9)
-00
with the same prescriptions for integrating across the poles as given for (6.4).
7. The causality condition, In this section we shall justify the restriction (4.5), imposed on the superposition coefficients, by showing that the unique stationary solutions, that we have obtained, satisfy the causality condition, formulated in section 4. First we remark that w 2 - k 2c 2 - w 2 W F(v z). dv z = k 2Z(k, u) for k> 0, p w - kv z + '/,8 (7.1 ) 2Z*(k, =k u) for k < 0,
f
670
B. U. FELDERHOF
where Z(k, tt) is the dispersion function of part I, corresponding to fixed k. (This function is to be distingnished from the function Z(w, u) of (5.6), which is for fixed co). Our prescriptions for integrating across the poles in (6.4) and (6.5) are tantamount to using fZ(k, u) instead of Z(k, u). It was shown in I, appendix B that Z(k, u) has no zeros in the upper half u-plane, and by using fZ(k, ~t) the zeros on the real axis are shifted to lie below the axis. Similarly fZ*(k, ~t) has no zeros in the lower half u-plane, including the real axis. Hence the integrands in (6.4) and (6.5) are holomorphic in the upper half of the complex w-plane, including the real axis. As the integrals over k in (6.4) and (6.5) converge uniformly for w in the upper half plane, these also are holomorphic functions in the upper half w~plane. The integrals diverge at w = 0, but the integrals multiplied by the factor co in front are finite on the real axis. It will be convenient to normalize the stationary solutions in such a way that the magnetic field at z = equals unity. We shall use an overhead bar to denote solutions normalized in this fashion. From (6.4) one finds that the asymptotic behaviour for large w of the normalized electric field is given by z > 0,
°
which vanishes for large w in the upper half plane. Similarly, J~(z, vz) can be seen to vanish as to approaches infinity in the upper half plane. Hence it follows that E~(z) for z > a and f~(z, v z ) are positive-frequency functions of co. It follows from (6.9) that Sew) approaches 4njc for large w. Moreover, it follows from (6.9) and the fact that F(v z) is nonnegative that the real part of SCw) is nonnegative for real w. That is, Re Sew) is nonnegative on a contour, consisting of the real axis and a large semi-circle surrounding the upper half plane. This implies that Re S(w) is nonnegative in the upper half plane, because 5(w) is holomorphic in this half plane. Hence S + 4njc does not vanish in the upper half co-plane, which implies that also R(w), given by (6.8), is holomorphic in the upper half plane. As moreover, according to (6.8), R(w) vanishes for large OJ, the function R(w) is a positive-frequency function of w. When we construct a superposition of the normalized stationary solutions with coefficients P(w), the magnetic field at z = 0 wiJl be given for all times by
By(O, t)
=
J P(w) e-';'wt dw,
(7.2)
and the electric field and the distribution function by Ex(z, t) =
tz(z, vz, t) =
J fJ(w) E~(z) e- iwt dco, f fJ(w) J~(z, vz) e- iwt dOJ.
(7.3)
671
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
In the vacuum the superposition with coefficients P(w) gives rise to an electric field eiwz/c + R e- iwz/ c Ex(z, t) = (3(w) e- iwt dw, (7.4) z < o.
f
l-R
Hence the superposition corresponds to an incoming wave packet
Ein(z, t)
=
f
eiw(z/c-t)
dw,
z
<
(7.5)
0,
where
(1 - R(w))
=
(7.6)
(3(w).
An incoming wave packet of arbitrary shape, which vanishes identically for zlc - t > 0, is represented by a positive-frequency function ct(w). It follows from (7.6) that this corresponds to a positive-frequency function (J(w), because R(w) is a positive-frequency function. As E~(z) for z > and f~(z, vz) are positive-frequency functions of co, it follows from (7.3) that for such a superposition the electric field in the plasma and the distribution function vanish identically for t < O. Of course, this implies that for t < the magnetic field vanishes for z > O. Hence we may conclude that the plasma is not excited before the arrival of the incoming wave packet at the boundary plane. The outgoing wave packet corresponding to (7.5) is given by
°
°
Eout(z, t)
=
J ct(w) R(w)
e-iw(z/c+t)
dw,
z
<
0.
(7.7)
As R(w) is a positive-frequency function, E out vanishes identically for zk: t < 0, if
+
°
8. The refractive index theory. In I, section 8 we have defined the refractive index for frequencies Iwl > wp by means of the real zeros of the dispersion function Z(k, %}. This definition is in agreement with the results of the present article, because, as follows from (7.1), in the frequency range Iwl > Wp the integrand in (6.4) has a pole corresponding to the zero Ut of Z(k, u), which gives rise to an undamped wave in the electric field of phase velocity +Ut travelling in the positive a-direction: the zero -Ut does not contribute. The remaining contribution of the integral is, beyond some value of z, of no significance compared to the contribution from the zero Ut. That is, at some distance from the boundary plane the fields are correctly described by the refractive index as defined in I (8.2). For frequencies smaller than the plasma frequency, the electric field tends to zero as z increases, instead of becoming a periodic function as for
672
D. U. FELDERHOF
[co] > Wp. The electric field will be a complicated function of z in general, but unless w is very small the contribution from the zero -Uo of .2"*(tt) in the first expression (6.4) will be dominant except for very small or very large values of z. In the region where the contribution from -Uo dominates, the electric field decreases exponentially, proportional to exp( -z/d), where d = luo/wl (in the corresponding expression for w < 0, the zero Uo of .2"(u) yields a dominant contribution). Hence, in the present theory it is natural to extend the definition of the refractive index to the frequency range Iwl < W p by means of the zeros ±uo. Accordingly (d. 1(8.2) and (3.4)) the refractive index n(w) is defined for all jrequencies by n = ck],», where k is to be solved in terms of w from w2 -
k 2c2
-
J
z) 2 F(v w1JwP -- - . dvz = O.
(8.1)
kv z
w -
The definition of n(w) suggests an approximate solution of the reflection problem. which we shall call the refractive index theory. In this theory the fields are assumed to be correctly described up to the boundary by the refractive index, with amplitudes found from continuity at the boundary. According to this simplifying theory the reflection coefficient is given by the Fresnel formula
R=
I-n I+n
.
(8.2)
For Iwl < Wp the refractive index is purely imaginary and hence in the refractive index theory IRI equals unity; in other words, in this frequency range there is total reflection. For Iwl > Wp the refractive index is real and incident waves are partly reflected, partly transmitted; there is conservation of electromagnetic energy flow for the reflected and transmitted waves. 9. Anomalous skin effect. Inasmuch as the predictions of the exact theory differ from those of the refractive index theory, one deals with an anomalous skin effect. As already remarked in the preceding section, for Iwl > OJp, the field profile approaches that of an undamped electromagnetic wave only for large values of z. According to (6.4) asymptotically the electric field becomes
E~(z) ~ -2OJcB~(O). [O(k
2Z(k,
ok
wlk))
I
J-1.eiWZ/U"
(9.1)
k=w/u,
for large z, from which one can define the transmission coefficient T by (d. 6.6)
2 T = -2wc [ o(k Z (Jl, OJlh)) 1- R ok where R is given by (6.8) and (6.9).
I k-wht.
J-
1 ,
(9.2)
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
673
The fraction of incident energy flow reflected is given by IRI2; the fraction of energy transmitted by n ITI2. In general, these fractions do not add to unity, which means that part of the incident electromagnetic energy is transformed into mechanical energy of the particles. For Iwl < w p there is no transmitted wave, but the reflection coefficient, as given by the exact expression (6.8), differs from unity in absolute value. This means that not all of the incident energy is reflected; part of it is absorbed and transformed into mechanical energy. Inasmuch as the exact behaviour of the electric field in the plasma deviates from purely exponential behaviour exp(ikz), with a suitable value of k(w), it cannot be described in terms of a refractive index. Under these conditions it is impossible to define a conductivity (or dielectric constant) depending only on frequency, because the relation between electric field and current density is essentially non-local. It would, of course, be possible to use this relation as obtained from the exact solution for defining a conductivity depending not only on frequency, but also on wave number. The resulting expression for a conductivity with spatial dispersion coincides with the usual one, obtained by calculating the response of the plasma to an external electric field, or, rather, to an external current. vVe shall not pursue this matter in detail. 10. Comparison with the theory of metals. Reuter and Sun dh eimer s) have provided a theory of the anomalous skin effect in metals. For frequencies smaller than the visible (where interbancl transitions come into play) the electromagnetic behaviour of metals is dominated by the conduction electrons. If these are treated in classical approximation, the basic equations are the same as we have used, except that a collision term, corresponding to collisions with impurities or phonons, must be written on the right-hand side of the Vlasov equation 1(2.2). Reuter and Sondheimer used a simple collision term of the form -/l/T, where T is a mean collision time. In the case of metals it is hard to assess how the particles are reflected back into the plasma by the surface, which may be of a complicated nature. Reuter and Sondheimer solved the problem for two typical cases, viz. specular and diffuse reflection. In the latter case the particles are assumed to leave the surface with a random velocity, with probability determined by the equilibrium distribution. For the case of specular reflection their results coincide with ours, except that 8 in (6.4), (6.5) and (6.9) is replaced by liT. For very low temperatures liT is small. Numerical calculations are necessary to find the reflection coefficient as a function of frequency. These calculations have been carried out for metals; in that case one uses the Fermi-Dirac distribution for the equilibrium distribution and an effective electron mass. The experimental results are in good agreement with the theory, when corrected for polarizability of the
674
TRANSVERSE WAVES IN VLASOV-l'LASMAS. III
ion cores; the experiments favour diffuse reflection of particles at the boundary surface 5). As remarked in the introduction, for magnetically confined plasmas the assumption of specular reflection of particles is reasonable. It would be of interest to evaluate the reflection coefficient given by (6.8) and (6.9) numerically for a suitable equilibrium distribution, e.g. the relativistic thermal equilibrium distribution (d. section I, 12). Comparison with experiment will, however, be more difficult than in the case of metals. Acknowledgement. The author is greatly indebted to Professor N. G. van Kampen for helpful discuss.ons and stimulating criticism. Received 5-3-62
REFERENCES I} 2) 3) 4) 5)
Felderhof, B. U., Physic a 20 (1963) 293 and 317. Shure, F. C., Dissertation, University of Michigan, 1962. Case, K. M., Annals of Physics 7 (1959) 349. Reuter, G. E. H. and Sondheimer, E. H., Proc. roy. Soc. limA (1949) 336. Parker Givens, 1'11., Solid State Physics, Volume 6, p. 313, New York, 1958.
Felderhof, B. U.
662-674
1963
THEORY OF TRANSVERSE WAVES IN VLASOV-PLASMAS III, REFLECTION OF ELECTROMAGNETIC WAVES BY A PLASMA HALF-SPACE
by B. U. FELDERHOF Instituut voor Theoretische Fyaica, Rijksuniversiteit, Utrecht
Synopsis In this paper we solve the problem of reflection of normally incident electromagnetic waves by a plasma half-space. The equilibrium plasma is supposed to be spatially homogeneous and isotropic in velocity space; the particles are assumed to be reflected specularly at the plane boundary. From the multitude of stationary solutions, compatible with the latter assumption, one is selected by imposing a causality condition. The exact reflection coefficient found from this solution is compared with the results predicted by the Fresnel formula with the aid of the refractive index; the present problem leads to a natural extension of the definition of the refractive index, given in I, to the range of frequencies smaller than the plasma frequency. The relation to the theory of the anomalous skin effect in metals is also discussed.
1. 1 ntroduction. In parts I and II of this paper 1) we solved the initial-value problem of Vlasov and Maxwell equations for transverse waves in a homogeneous, unbounded plasma. It is interesting to consider in what manner such waves are excited when vacuum electromagnetic waves arrive at a bounded plasma. We shall solve this problem of reflection and transmission for the case of waves, normally incident on a half-space filled with a homogeneous, isotropic Vlasov-plasma, in the absence of external fields. It will be assumed that the plasma particles are reflected specularly by the plane boundary. The situation we consider can be regarded as a good approximation of real physical situations. In reality, a plasma confined by a vacuum magnetic field cannot be homogeneous over the whole region of space it occupies. There must be a region where the plasma is inhomogeneous, separating the homogeneous plasma and the adjacent vacuum. The thickness of this boundary layer is of the order of the gyration radius corresponding to the magnetic field at the boundary. Particle trajectories at the boundary are in accordance with the assumption of specular reflection of particles (henceforth to be called the speculum ass~~mption). Our approximation. will be a good one, provided the boundary layer thickness is much smaller than other characteristic dimensions, such as the vacuum wave length or the skin depth, i.e. the depth of penetration of the fields. -
662
TRANSVERSE WAVES IN VLASOV-PLASMAS
-------------------------------
663
It is necessary to impose a causality condition in order to select the correct stationary solution, corresponding to a given frequency, from the multitude of solutions compatible with the speculum assumption. This causality condition, made precise in section 4, leads to a restriction on the coefficients of the expansion in solutions of different wave number. Afterwards it is proved from the explicit solution for distribution and fields that the causality condition is satisfied (section 7). The present problem has recently also been considered by Sh ur e P). Using the method of Ca s e "), Shure obtains the same expression for the expansion coefficients, but his justification of the solution from causality arguments is not sufficient. Shure also studies several different situations. The solution leads to a natural extension of the definition of the refractive index to the frequency range Iwl < Wp (section 8). In section 9 we compare the reflection coefficient, as obtained from the Fresnel formula with the aid of the refractive index, with the exact one resulting from the solution of our problem. There is a close relation between the present theory and the theory of the anomalous skin effect in metals, given by Reuter and So nd he im er s]. In that case causality is introduced automatically by the presence of a collision term in the equation for the distribution function. Moreover, Reuter and Sondheimer use the artificial device of introducing an external current source at the plasma surface; we avoid this and believe that our solution leads to a clearer understanding. It will be shown (section 10) that our methods yield the same results as the theory of Reuter and Sondheimer.
2. Basic equations, The basic equations for the present problem are Vlasov's linearized equation 1(2.2) and Maxwell's equations 1(2.3) for the half-space filled with plasma, and Maxwell's equations with p and j identically zero for the vacuum half-space. Vacuum electromagnetic waves of definite polarization normally incident on the boundary plane will be partly reflected, but will also excite waves in the plasma with the same direction of propagation and the same polarization, owing to isotropy of the plasma. It suffices to consider waves of linear polarization and we choose coordinates such that the x-axis is in the direction of the electric vector and z = 0 is the boundary plane; z < 0 will be taken to be the vacuum half-space, Z > 0 the plasma half-space. The equations for the plasma half-space can then be reduced to equations 1(2.6). The general solution of the basic equations can be obtained by superposition of solutions with time dependence of all quantities given by a factor exp(-iwt), where w is real. The dependence on space and time of the vacuum. electric and magnetic fields is then given by factors exp(±iwzlc -
664 -
B. U.FELDERHOF
iwt). Equations 1(2.6) for the plasma can be written in the form of~. = 2wf~
Vz -
8z
noe + -E~F(vz), m
8E~ = OZ
iw BW c u:
8B~ 8z
io:
-- = -
c
E~
(2.1 ) 4ne c
- --
f
f~(z,
vz) dvz.
Equations (2.1) may be regarded as a boundary-value problem, to be solved for given values f~(O, ve), E~(O) and B~(O) at z = O. In fact, however, in the problem of reflection these values are not known and the solution has to be found from the speculum assumption and the causality condition. Of course, we must connect the vacuum fields to the fields in the plasma in order to find the appropriate linear combination of vacuum plane waves. 3. Periodic and exponential solutions of (2.1). In the present section we consider solutions of (2.1) with space dependence of all quantities given by a factor exp(ikz), without being concerned with conditions at the plane z = O. Such solutions exist for all real k, and are of the form
gW,U(vz) = q2P
u 3F(ve} U -
Ve
+ A(w, u) o(~t -
EW'"" = -(4niejw) u 2, BW.1' = -(4niejw) CU,
vz), (3.1)
where u = wjk and q = wpjw. The solution (3.1) is again normalized to
J gW,1'(V e) dvz = u 2 - c2 ,
(3.2)
so that A(w, u) is given by A(w, u) = u 2
-
c2 + q2P
f u3F(vz) Ve -
(3.3)
U
The solutions (3.1) are the same as 1(3.9), but in the following w will be regarded as fixed and k as a variable. One can also look for solutions of (2.1) with complex k. These solutions are possible only, if u is a (complex) root of the equation (3.4)
Investigating the analytic behaviour of the function on the left-hand side, one finds that for q2 > I, i.e. for [co] < Wp, this function has two zeros ± Uo on the imaginary axis; for q2 < 1 there are no complex zeros. Hence for
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
[eo]
<
wp,
665
apart from the solutions (3.1), there are two solutions of the form
r ±1t.(vz) =
q2 ±ugF(vz) , ±1-to -
Vz
=
-(4niejw) u5,
Bw,h. =
+(4niejw) cUo.
Em.
h
.
(3.5)
If we write %0 = i [wi d, where d > 0, the spatial factor of the solution of (2.1) corresponding to +~to is exp(zld) for w > 0; for the solution corresponding to -1-to the spatial factor is exp( -z/d). For w < the situation is reversed. For convenience of notation we shall retain only the superscript ~t to denote the periodic solutions (3.1), whereas the exponential solutions (3.5) will be denoted by superscripts ±.
°
4. Specul-um assumption and causality condition. An arbitrary superposition of periodic and exponential solutions (3.1) and (3.5) will again be a solution of (2.1). With superposition coefficients C(u), C+ and C- such a solution can be written
vz) =JC(u) glt(v z) eiwz!u cht + :L: C±g±(vz) e±iwz!1to, ± E~(z) = -(4nielw) [J~t2C(~t) eiwz!" du +::L: C±:'i-t5 e±iWZ/UO],
f~(z,
(4.1)
±
B~(z) =
-(4nielw) c[JuC(u) eiroz!u du
±::L: C±uo e±iroZ!Uo], ±
where the sum of discrete terms is of course only present for Jw) < W p . It is possible to prove that an arbitrary solution of (2.1) can be written in the form (4.1), but for the following this proof is not needed. The assumption that particles are reflected specularly by the boundary surface implies (4.2) f~(O, v z) = f~(O, - v z). Using the fact that
gU(-v z) = g-u(v z),
g+(-vz)
=
g-(vz),
(4.3)
one finds from (4.1) and (4.2) that the speculum assumption imposes the following restriction on the expansion coefficients
J [C('i-t)-C( -%)J gu(vz) d#
+ (C+-C-) g+(v z) + (C--C+) g-(v z) = O.
(4.4)
Of course, equation (4.4) does not determine the coefficients, which means that for given amplitude of the incoming plane wave there are still many solutions satisfysing (4.2). We shall impose a condition of causality, which for every frequency wand given amplitude of the incoming wave selects a unique solution from the multitude of stationary solutions compatible
666
B. U. FELDERHOF
with (4.4). This unique solution must be such that by superposition of these solutions over different frequencies, it is possible to construct a solution of the basic equations which for z < 0 represents a wave packet of arbitrary shape travelling in the positive z-direction, while there are no waves travelling in the negative z-direction and while the plasma is at rest, ~tntil the wave packet arrives at the boundary plane. Hence, the causality condition excludes all those solutions in which the plasma is in a perturbed state before waves from the vacuum arrive at its boundary. It is intuitively clear that the restriction on the expansion coefficients corresponding to this causality condition is C(~t) =
0
c+
= 0
C-
= 0
u < 0, w>O, w < O.
for for for
(4.5)
That is, from the periodic solutions (3.1) and the exponential solutions (3.5), corresponding to a given frequency w, only the periodic solutions propagating into the plasma and the exponentially decreasing solution are admitted. It will be shown in the next section that the restrictions (4.4) and (4.5) uniquely determine the expansion coefficients, except for a constant factor. It is necessary to give a rigorous justification of the restriction (4.5), because a superposition of periodic solutions with negative phase velocity could be chosen such as to yield a vanishing contribution for large values of z. We shall justify (4.5) by showing that the explicit solution, which we shall obtain, satisfies the causality condition formulated above.
5. Solution of equations (4.4) and (4.5). Introducing the antisymmetric function A (u) and the discrete coefficient D by A(u) = C(u) - C(-u),
D = C+ - C-,
(5.1)
one can write for equation (4.4)
f A(u) gu(vz) du
+ D{g+(v z) -
g-(vz)} = 0;
(5.2)
or, using the explicit form of gu(vz) as given by (3.1),
{u2 - c2 + 2niq2u3F+(u)}A+(u) + {u2 - c2 - 2niq2u3F _(u)}A_(~t) = = -q2F(u) f [u2 + wu' + U'2J A(~t') d~t' - D{g+(u) - g-(u)}.
(5.3)
On separating the right-hand side of (5.3) into its positive- and negativefrequency parts and making use of the antisymmetry of A(u), one obtains
A(u) = a{l
+ 2niq2uF+ (u)} -
D{gt (u) -
.2"(u)
+
-a{l -
2niq 2uF_(u)}
t+ (u)}
+
- D{g:!:.(u) - g=(u)} .2"*(u)
(5.4)
TRANSVERSE WAVES IN VLASOV-l'LASMAS. III
667
where
a = (ij2n) J uA (~t) du,
(5.5)
and where .£Z'(u) is obtained from Z(~t)
= ~t2 - c 2
+ 2niq2~t3F+ (tt)
(5.6)
by shifting real zeros of Z(u) an infinitesimal distance below the real axis. We henceforth assume that F(v z) vanishes identically for Ivzl > c (d. section I, 6). In that case, for q2 < I, the function Z(u) has two real zeros ±~tt; also, for q2 < 1, equation (3.4) has no complex roots so that the terms in D are absent. For q2 > 1, Z(u) has no zeros on the real axis and in that case the coefficient D must be such that the numerators in (5.4) vanish at the zero Uo of f!l'(u) in the upper half plane and at the zero -Uo of f!l'*(u) in in the lower plane, respectively. This yields the following relation for D
D [u~
ac2
-- = -u~ 2ni
- c -q sf 2
2u
Uo
o
F(v z) ] dv z . (vz - uo)2
(5.7)
On summation of the two terms in (5.4)and explicit calculation of the factors in the numerators multiplying D, one finds that A(u) can also be written
.
A(~t) = (D~to - mal
2c2q2~tF(u)
1.£Z'(u)1 2
•
(5.8)
According to (4.5) and (5.1) the coefficients C(ttl. C+ and C- can easily be obtained from A(u) and D,
C(u) = A(u) C(u) = 0 C+ C+
= 0,
=D,
C-= -D C- = 0
for for for for
u u
> < >
0, 0, w 0, w
(5.9)
so that with the aid of (4.1) the factor Duo - nia in (5.8) can be expressed in terms of the magnetic field at z = O. When the relation obtained is also used in (5.7), one finds for A(u) and D
D =
A(u = iwc B(O) q2 uF (u) ) 2ne 1.£Z'(u)1 2 1 ieee
icoc _ . B(O) . 2ne
u~.£Z"(uo)
= -
2ne
'
-1
(5.10)
B(O) . --;;-~---:u~.2'*/(_~tO)
The discrete coefficient D of course only differs from zero for lcol < COp. The magnetic field B(O) appears as an undetermined proportionality factor.
6. Velocity distribution and fields. Inserting the coefficients C(u), C+ and C-, as given by (5.9) and (5.10) into equation (4.1), one can evaluate the velocity distribution function and the electric and magnetic fields. The
668
B. U. FELDERHOF
results can be cast into rather simple forms by some manipulation of the occurring integrals. Only for the electric field we shall elaborate on the details of the calculation; the magnetic field is obtained from this by a simple differentiation. For w > 0, the electric field in the plasma is given by (6.1)
The integral in this expression equals
~f[ 2m'
__
o
1 _. -
.?t'(~6)
1
fZ'*(u)
]
whereas the second term is only present for w into an integral
f
eiwz!tt
clu
(6.2)
,
<
Wp
and can be transformed
00
e- i ",z/1to
-1
-.,.,----- = - -
a'*'(-uo)
2ni
eiwz!tt
fZ'*(~6)
-00
z > 0,
du,
0<
W
<
wp.
(6.3)
For co > Wp the integral in (6.3) amounts to zero. Hence one can write for allw>O 00
0
f
E~(z) = ~ B~(O) [J ~z::; d~6 + ~:~::t) d~t ] o
-00
00
For w> Wp, the term in brackets in the latter expression has real zeros at ±kt = co] ±Ut and it follows from the definition of .?t' and a'* that one should integrate below the zero on the positive k-axis and above the zero on the negative k-axis. A similar calculation for co < again leads to the second expression in (6.4), but with the prescription that, for co < -COp, one should integrate above the zero of the term in brackets on the positive k-axis and below the zero on the negative k-axis. Equations (4.1), (5.9) and (5.10) may also be used to calculate the reduced particle distribution function. The calculation is similar to, but slightly more complicated than that for the electric field. We shall only quote the final result, which is
°
669
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
we
!~(z, Vz) = - -B~(O). ;n;
noe F(v;,<) m
J[(w - k _w;wJw-kv F(v,~), dV~) (w-kvz+is)]-\ik2dk. 2 C2
2
(6.5)
z+1,s
This expression is valid for all frequencies w; for Iw) > Wp one should follow the same prescriptions for integrating across the poles as given for (6.4). We are now in a position to make the connection between the fields in the plasma and the vacuum electric and magnetic fields. By integration of (6.5) over Vz it is easily seen that the current density is finite everywhere; in particular there is no singularity at z = O. Hence it follows from Maxwell's equations that the electric and magnetic fields are continuous at z = 0. The vacuum electric and magnetic fields can be written e iW2/C R e-'iwz/c E~(z) = B~(O) - - - - - -
+
l-R
e iwz/C B~(z) =
B~(O)
_
z
R e- iWZ/ll
(6.6)
------
l-R
In the theory of metals it is customary to express the reflection coefficient R(w) in terms of the suriace impedance 5, defined by 4;n; Ea;(O) 5=----, c By(O)
(6.7)
R = 5 - 4;n;{c . 5 + 4;n;{c
(6.8)
so that
Making use of the continuity of the fields at z = 0, one finds from (6.4) that 5(w) is given by 00
5 = 4iw
J[
w2
-
k 2c2
-
w~ w
J
w -
F(vz) k Vz
J-
. dv z
+ 'LS
1
dk,
(6.9)
-00
with the same prescriptions for integrating across the poles as given for (6.4).
7. The causality condition, In this section we shall justify the restriction (4.5), imposed on the superposition coefficients, by showing that the unique stationary solutions, that we have obtained, satisfy the causality condition, formulated in section 4. First we remark that w 2 - k 2c 2 - w 2 W F(v z). dv z = k 2Z(k, u) for k> 0, p w - kv z + '/,8 (7.1 ) 2Z*(k, =k u) for k < 0,
f
670
B. U. FELDERHOF
where Z(k, tt) is the dispersion function of part I, corresponding to fixed k. (This function is to be distingnished from the function Z(w, u) of (5.6), which is for fixed co). Our prescriptions for integrating across the poles in (6.4) and (6.5) are tantamount to using fZ(k, u) instead of Z(k, u). It was shown in I, appendix B that Z(k, u) has no zeros in the upper half u-plane, and by using fZ(k, ~t) the zeros on the real axis are shifted to lie below the axis. Similarly fZ*(k, ~t) has no zeros in the lower half u-plane, including the real axis. Hence the integrands in (6.4) and (6.5) are holomorphic in the upper half of the complex w-plane, including the real axis. As the integrals over k in (6.4) and (6.5) converge uniformly for w in the upper half plane, these also are holomorphic functions in the upper half w~plane. The integrals diverge at w = 0, but the integrals multiplied by the factor co in front are finite on the real axis. It will be convenient to normalize the stationary solutions in such a way that the magnetic field at z = equals unity. We shall use an overhead bar to denote solutions normalized in this fashion. From (6.4) one finds that the asymptotic behaviour for large w of the normalized electric field is given by z > 0,
°
which vanishes for large w in the upper half plane. Similarly, J~(z, vz) can be seen to vanish as to approaches infinity in the upper half plane. Hence it follows that E~(z) for z > a and f~(z, v z ) are positive-frequency functions of co. It follows from (6.9) that Sew) approaches 4njc for large w. Moreover, it follows from (6.9) and the fact that F(v z) is nonnegative that the real part of SCw) is nonnegative for real w. That is, Re Sew) is nonnegative on a contour, consisting of the real axis and a large semi-circle surrounding the upper half plane. This implies that Re S(w) is nonnegative in the upper half plane, because 5(w) is holomorphic in this half plane. Hence S + 4njc does not vanish in the upper half co-plane, which implies that also R(w), given by (6.8), is holomorphic in the upper half plane. As moreover, according to (6.8), R(w) vanishes for large OJ, the function R(w) is a positive-frequency function of w. When we construct a superposition of the normalized stationary solutions with coefficients P(w), the magnetic field at z = 0 wiJl be given for all times by
By(O, t)
=
J P(w) e-';'wt dw,
(7.2)
and the electric field and the distribution function by Ex(z, t) =
tz(z, vz, t) =
J fJ(w) E~(z) e- iwt dco, f fJ(w) J~(z, vz) e- iwt dOJ.
(7.3)
671
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
In the vacuum the superposition with coefficients P(w) gives rise to an electric field eiwz/c + R e- iwz/ c Ex(z, t) = (3(w) e- iwt dw, (7.4) z < o.
f
l-R
Hence the superposition corresponds to an incoming wave packet
Ein(z, t)
=
f
eiw(z/c-t)
dw,
z
<
(7.5)
0,
where
(1 - R(w))
=
(7.6)
(3(w).
An incoming wave packet of arbitrary shape, which vanishes identically for zlc - t > 0, is represented by a positive-frequency function ct(w). It follows from (7.6) that this corresponds to a positive-frequency function (J(w), because R(w) is a positive-frequency function. As E~(z) for z > and f~(z, vz) are positive-frequency functions of co, it follows from (7.3) that for such a superposition the electric field in the plasma and the distribution function vanish identically for t < O. Of course, this implies that for t < the magnetic field vanishes for z > O. Hence we may conclude that the plasma is not excited before the arrival of the incoming wave packet at the boundary plane. The outgoing wave packet corresponding to (7.5) is given by
°
°
Eout(z, t)
=
J ct(w) R(w)
e-iw(z/c+t)
dw,
z
<
0.
(7.7)
As R(w) is a positive-frequency function, E out vanishes identically for zk: t < 0, if
+
°
8. The refractive index theory. In I, section 8 we have defined the refractive index for frequencies Iwl > wp by means of the real zeros of the dispersion function Z(k, %}. This definition is in agreement with the results of the present article, because, as follows from (7.1), in the frequency range Iwl > Wp the integrand in (6.4) has a pole corresponding to the zero Ut of Z(k, u), which gives rise to an undamped wave in the electric field of phase velocity +Ut travelling in the positive a-direction: the zero -Ut does not contribute. The remaining contribution of the integral is, beyond some value of z, of no significance compared to the contribution from the zero Ut. That is, at some distance from the boundary plane the fields are correctly described by the refractive index as defined in I (8.2). For frequencies smaller than the plasma frequency, the electric field tends to zero as z increases, instead of becoming a periodic function as for
672
D. U. FELDERHOF
[co] > Wp. The electric field will be a complicated function of z in general, but unless w is very small the contribution from the zero -Uo of .2"*(tt) in the first expression (6.4) will be dominant except for very small or very large values of z. In the region where the contribution from -Uo dominates, the electric field decreases exponentially, proportional to exp( -z/d), where d = luo/wl (in the corresponding expression for w < 0, the zero Uo of .2"(u) yields a dominant contribution). Hence, in the present theory it is natural to extend the definition of the refractive index to the frequency range Iwl < W p by means of the zeros ±uo. Accordingly (d. 1(8.2) and (3.4)) the refractive index n(w) is defined for all jrequencies by n = ck],», where k is to be solved in terms of w from w2 -
k 2c2
-
J
z) 2 F(v w1JwP -- - . dvz = O.
(8.1)
kv z
w -
The definition of n(w) suggests an approximate solution of the reflection problem. which we shall call the refractive index theory. In this theory the fields are assumed to be correctly described up to the boundary by the refractive index, with amplitudes found from continuity at the boundary. According to this simplifying theory the reflection coefficient is given by the Fresnel formula
R=
I-n I+n
.
(8.2)
For Iwl < Wp the refractive index is purely imaginary and hence in the refractive index theory IRI equals unity; in other words, in this frequency range there is total reflection. For Iwl > Wp the refractive index is real and incident waves are partly reflected, partly transmitted; there is conservation of electromagnetic energy flow for the reflected and transmitted waves. 9. Anomalous skin effect. Inasmuch as the predictions of the exact theory differ from those of the refractive index theory, one deals with an anomalous skin effect. As already remarked in the preceding section, for Iwl > OJp, the field profile approaches that of an undamped electromagnetic wave only for large values of z. According to (6.4) asymptotically the electric field becomes
E~(z) ~ -2OJcB~(O). [O(k
2Z(k,
ok
wlk))
I
J-1.eiWZ/U"
(9.1)
k=w/u,
for large z, from which one can define the transmission coefficient T by (d. 6.6)
2 T = -2wc [ o(k Z (Jl, OJlh)) 1- R ok where R is given by (6.8) and (6.9).
I k-wht.
J-
1 ,
(9.2)
TRANSVERSE WAVES IN VLASOV-PLASMAS. III
673
The fraction of incident energy flow reflected is given by IRI2; the fraction of energy transmitted by n ITI2. In general, these fractions do not add to unity, which means that part of the incident electromagnetic energy is transformed into mechanical energy of the particles. For Iwl < w p there is no transmitted wave, but the reflection coefficient, as given by the exact expression (6.8), differs from unity in absolute value. This means that not all of the incident energy is reflected; part of it is absorbed and transformed into mechanical energy. Inasmuch as the exact behaviour of the electric field in the plasma deviates from purely exponential behaviour exp(ikz), with a suitable value of k(w), it cannot be described in terms of a refractive index. Under these conditions it is impossible to define a conductivity (or dielectric constant) depending only on frequency, because the relation between electric field and current density is essentially non-local. It would, of course, be possible to use this relation as obtained from the exact solution for defining a conductivity depending not only on frequency, but also on wave number. The resulting expression for a conductivity with spatial dispersion coincides with the usual one, obtained by calculating the response of the plasma to an external electric field, or, rather, to an external current. vVe shall not pursue this matter in detail. 10. Comparison with the theory of metals. Reuter and Sun dh eimer s) have provided a theory of the anomalous skin effect in metals. For frequencies smaller than the visible (where interbancl transitions come into play) the electromagnetic behaviour of metals is dominated by the conduction electrons. If these are treated in classical approximation, the basic equations are the same as we have used, except that a collision term, corresponding to collisions with impurities or phonons, must be written on the right-hand side of the Vlasov equation 1(2.2). Reuter and Sondheimer used a simple collision term of the form -/l/T, where T is a mean collision time. In the case of metals it is hard to assess how the particles are reflected back into the plasma by the surface, which may be of a complicated nature. Reuter and Sondheimer solved the problem for two typical cases, viz. specular and diffuse reflection. In the latter case the particles are assumed to leave the surface with a random velocity, with probability determined by the equilibrium distribution. For the case of specular reflection their results coincide with ours, except that 8 in (6.4), (6.5) and (6.9) is replaced by liT. For very low temperatures liT is small. Numerical calculations are necessary to find the reflection coefficient as a function of frequency. These calculations have been carried out for metals; in that case one uses the Fermi-Dirac distribution for the equilibrium distribution and an effective electron mass. The experimental results are in good agreement with the theory, when corrected for polarizability of the
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ion cores; the experiments favour diffuse reflection of particles at the boundary surface 5). As remarked in the introduction, for magnetically confined plasmas the assumption of specular reflection of particles is reasonable. It would be of interest to evaluate the reflection coefficient given by (6.8) and (6.9) numerically for a suitable equilibrium distribution, e.g. the relativistic thermal equilibrium distribution (d. section I, 12). Comparison with experiment will, however, be more difficult than in the case of metals. Acknowledgement. The author is greatly indebted to Professor N. G. van Kampen for helpful discuss.ons and stimulating criticism. Received 5-3-62
REFERENCES I} 2) 3) 4) 5)
Felderhof, B. U., Physic a 20 (1963) 293 and 317. Shure, F. C., Dissertation, University of Michigan, 1962. Case, K. M., Annals of Physics 7 (1959) 349. Reuter, G. E. H. and Sondheimer, E. H., Proc. roy. Soc. limA (1949) 336. Parker Givens, 1'11., Solid State Physics, Volume 6, p. 313, New York, 1958.