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Waves in a plasma in a magnetic field
SECTION B
INTERACTION
OF MICROWAVES
WAVES IN A PLASMA
AND IONIZED
IN A MAGNETIC
AIR
FIELD
R. J. PAPA and W. P. ALLIS M~h~tts Institute of Technology Ah&met-The propagation of plane electromagnetic waves through an i&kite low temperature electron plasma in the presence of a constant magnetic field is to be considered. A dispersion equation is derived by solving for a self-consistent electric field using the Boltzmann equation and Maxwell’s equations. Since small deviations from the Maxwell-Boltdistribution are taken into account, the dispersion equation yields three double solutions for the index of refraction. In addition to the two waves whose phase velocity is of the order of light velocity, a plasma wave is present whose velocity is of the order of sound velocity. Normal wave surfaces for the three waves are plotted for various values of the parameters. The polarization of the waves is aIso considered. ZERO SEPTUM
PLASMA
In this section, the thermal motions of the electrons will be neglected, the collision frequency assumed zero, and the ions of infinite mass. The wave equation may be written:
field under the influence of an alternating electric field. The wave equation (2) may be written in matrix form as follows:
Vx(VxE)+y=O
t;/
Assume E to be proportional to exp ju (i-~nfc), where n is a vector normal to the wave front. II = 1II 1is the index of refraction of the medium. A constant magnetic field is taken along the Z-axis. The vector n, which determines the direction of propagation, lies in the X-Z plane at an angle 8 with the Z-axis. The wave equation then becomes: nx(nxE)+fiE=O
(21
where K, the dielectric tensor, is defined by : k’d+O
(3)
@so 0, the conductivity tensor, may be derived by considering the equations of motion of an electron in the presence of a constant magnetic
$if)
-n2~2+R,~
ai,= a12 =I& a13 =
$
=
= 0
a22 =
--a21
n2&l-f-K,,
[;I
= a31
-a2+K22
a23 =1(23=
a33 =
(4)
-a32
-n2r2+1Y33
(5) and f = sin 8, C = cos B are the direction cosines of Il. The components of the dielectric tensor for a cold plasma are:
&l
=&,=I---
K33
=
K,,
= --K21=
2 l-B2
l-c? - ja"j?
1_Bz
f6>
WAVES IN A PLASMA
taken T- 0 and v, = 0. It may be noted that n2 changes sign only by going through zero or infinity. When n* = co, this is termed a resonance, when na = 0, this is termed a cut-off. The family of solutions of equation (9) are conveniently represented on a plot of j3”against aa. The $ - aa plane is divided up into eight regions by two resonance lines and two cut-off lines. Within each region the normal wave surfaces remain topologically the same. A normal wave surface is a polar plot of phase velocity n = c/n, In Fig. 1, sample normal wave surfaces are drawn within each region. The radius of the dotted circle represents the velocity of light. lotion (9) may be rewritten so that B is expressed in terms of a, #3and n :
where
*&!3=-_
r
mco
Co
There is no non-zero solution of equation (4) unless the determinant of the coefficient matrix is set equal to zero. Thus, the dispersion equation is obtained from det [a] = 0, and may be written : An4 -Bn2+C=0
(8)
where A, B and C are functions of a, /3 and 8. The solutions of equation (8) may be written n1,2
=
4’jcW+4W41
101
(9)
where DB = Ba - 4AC is always positive. n is either real or purely imaginary because we have
tan20 =
-K,(?&K,)(+-K,) (2 - KJo&r - K,K,)
2
i
eB mw
1
u,-co
,,/-<-. 0 P
,’
‘.
\
XI
u,=lJ,=8
:
\..____d
0
j’
\
Fig. I.
H-PSS
Wave normal surfaces
In
a
plasma in a magnetic field.
(10)
R. J. PAPA and W. P. ALLIS
102
Where &,, Kr and RI are the components of the dielectric tensor in rotating co-ordinates. K T=K1l;
I-8
KR = 1-a’
s=eT a2 mc*
a2
a2
K,=l--;
where
The matrix form of the wave equation is still
KJ= l-1+8; WI
For propagation along magnetic field, B = 0, there are two waves n,* = K, which is right circularly polarized, n12= KJ which is left circularly polarized. For propagation across magnetic field : 8 = a/z
no* = Kp is the ordinary wave polar-
ized along B. n2_Kr
x-
-
K 1. K
LOW RERAN
For a equation K+E/ca = different.
IS the extraordinary wave polarized in the plane perpendicular to B.
finite temperature plasma, the wave has the same form (2) n x (n x E) + 0. Now, however, the dielectric tensor is It is derived from the conductivity
tensor,
K= 1 +a, as before, but in the jOsO calculation of CTone must take account of the finite size of the electron orbit compared to the wavelength. This has been done by Sitenko, Drummond, Bernstein, Mower* and others and their results have been used. An example of the change in the components of K is given by the x-x component: Kll=l3r2 + (1 -+)(l-4/32)
P&s.
012
a13
a21
a22
a23
a32
a33
)
dispersion relation, again obtained det[a] = 0, may be written:
The
-~ans+(A+ab)n4-(B+ec)na+C=O
from (14)
u, b and c characterize the temperature effects and are implicated functions of a, fi and 8. A, B and C are the zero-tempera~re coefficients of equation (91. For moderate temperatures the terms Eb and SCmay be neglected compared to the larger A and B, but the term in cane increases the order of the equation and must be kept. The dispersion equation can then be written --arP+An4-Bnz+C==O
w
For indices n of order one, the solutions rzr,* of the dispersion equation are the same as before since .xw+Jis much less than the other terms. These solutions may be called the fast wave solutions. There is now a third solution with large n. This solution is called a plasma wave, and denoted by n,. Jts phase velocity, u =c/n, is of the order of magnitude of the electron’s thermal velocity. If rr3” % nif2, the constant term C may be neglected in the dispersion equation. Then,
The first quadrant of the ag-/3*plane is now divided into thirteen regions, as shown in Fig. 2. There is a resonance at a2 = +. For regions 2, 3, 6,7 and 8 the full cubic, equation (1 S), was solved. All wave surfaces were calculated with E = 10m2. The wave surfaces are surfaces of rotation about the Z-axis and have a plane of symmetry neroendicular to B. Thev are shown in Fie. 3.
1 WI Rev. 116, I6
41
i a31
ELECTRON PLASMA
*For references see L. MOWER, (1959).
(13)
WAVES IN A PLASMA
103
II
Fig. 2 Topological regions for a warm plasma.
Starting below the /?” = 4 resonance line, as is right circularly polarCPincreases from region 1 to 2, the normal wave ized, surface for the plasma becomes larger, until in - a”>10 + 8) is left circularly polarregion 3 its phase velocity, u = c/n, is pretty close to the velocity of light. In regions 4 and 5 there ” = :: + &j/(1 + fi)3 ixed . is no plasma wave. The plasma wave surface The third wave for propagation along the has a butterily shape in regions 6 and 7. In magnetic fieId is the Bohm Gross plasma region 8, the plasma wave surface is like a oscillation. This wave is a purely longitudinal lemniscate. The wave surface labelled p-x is a wave. plasma wave near 0” and a fast wave near 180”. $ _ (1-d In regions 9 and 10 the plasma wave surface is 3E like a lemniscate. In region 11, the plasma wave For propagation across the magnetic field, surface is a four-leaved rose, which becomes a lemuiscate in regions 12 and 13. In region 11, the there are three waves in general. The ordinary wave is polarized with its E figure eight part of the plasma wave surface goes ofF to infinity as $ approaches 1. In regions 12 vector along the ma~etic field. and 13, the figure eight pattern can be entirely accounted for by the fast wave solutions. For propagation along the magnetic field, There are two extraordinary waves polarized there are in general three waves,
R. J. PAPA and W. P. ALLIS
104
with their E vector in the plane perpendicular to the magnetic field. One extraordinary wave is a fast wave and the other is the plasma wave. Their indices of refraction are given by equation (16) where
A = [(l-/P-a2)+e
12a2j?2
(l-/3Z)(1-4jP)
+ 4(1-a2-/32)(1-t-2j32) (l-jP)2(1-4/?32 B = [(I-a”)‘-fi”]
3E &lJ= 1_4Bz
__.----._ 0 3 REGION 6
REG Nil
P
,,,’
,/” -,-----. , .\
.\
;’ : \ *,..
; i
,/
~..__._./’
I’
P,:: 8 /II. .I_____/~~’
/’
REGION
2
REGION 12
REGION 9 ,__-----__-
,/’
+
\
‘x REGION 4 P
REGION 7
Fig. 3. Wave normal surfaces for a warm plasma.
II
INTERACTION
OF MICROWAVES
WAVES IN A PLASMA
AND IONIZED
IN A MAGNETIC
AIR
FIELD
R. J. PAPA and W. P. ALLIS M~h~tts Institute of Technology Ah&met-The propagation of plane electromagnetic waves through an i&kite low temperature electron plasma in the presence of a constant magnetic field is to be considered. A dispersion equation is derived by solving for a self-consistent electric field using the Boltzmann equation and Maxwell’s equations. Since small deviations from the Maxwell-Boltdistribution are taken into account, the dispersion equation yields three double solutions for the index of refraction. In addition to the two waves whose phase velocity is of the order of light velocity, a plasma wave is present whose velocity is of the order of sound velocity. Normal wave surfaces for the three waves are plotted for various values of the parameters. The polarization of the waves is aIso considered. ZERO SEPTUM
PLASMA
In this section, the thermal motions of the electrons will be neglected, the collision frequency assumed zero, and the ions of infinite mass. The wave equation may be written:
field under the influence of an alternating electric field. The wave equation (2) may be written in matrix form as follows:
Vx(VxE)+y=O
t;/
Assume E to be proportional to exp ju (i-~nfc), where n is a vector normal to the wave front. II = 1II 1is the index of refraction of the medium. A constant magnetic field is taken along the Z-axis. The vector n, which determines the direction of propagation, lies in the X-Z plane at an angle 8 with the Z-axis. The wave equation then becomes: nx(nxE)+fiE=O
(21
where K, the dielectric tensor, is defined by : k’d+O
(3)
@so 0, the conductivity tensor, may be derived by considering the equations of motion of an electron in the presence of a constant magnetic
$if)
-n2~2+R,~
ai,= a12 =I& a13 =
$
=
= 0
a22 =
--a21
n2&l-f-K,,
[;I
= a31
-a2+K22
a23 =1(23=
a33 =
(4)
-a32
-n2r2+1Y33
(5) and f = sin 8, C = cos B are the direction cosines of Il. The components of the dielectric tensor for a cold plasma are:
&l
=&,=I---
K33
=
K,,
= --K21=
2 l-B2
l-c? - ja"j?
1_Bz
f6>
WAVES IN A PLASMA
taken T- 0 and v, = 0. It may be noted that n2 changes sign only by going through zero or infinity. When n* = co, this is termed a resonance, when na = 0, this is termed a cut-off. The family of solutions of equation (9) are conveniently represented on a plot of j3”against aa. The $ - aa plane is divided up into eight regions by two resonance lines and two cut-off lines. Within each region the normal wave surfaces remain topologically the same. A normal wave surface is a polar plot of phase velocity n = c/n, In Fig. 1, sample normal wave surfaces are drawn within each region. The radius of the dotted circle represents the velocity of light. lotion (9) may be rewritten so that B is expressed in terms of a, #3and n :
where
*&!3=-_
r
mco
Co
There is no non-zero solution of equation (4) unless the determinant of the coefficient matrix is set equal to zero. Thus, the dispersion equation is obtained from det [a] = 0, and may be written : An4 -Bn2+C=0
(8)
where A, B and C are functions of a, /3 and 8. The solutions of equation (8) may be written n1,2
=
4’jcW+4W41
101
(9)
where DB = Ba - 4AC is always positive. n is either real or purely imaginary because we have
tan20 =
-K,(?&K,)(+-K,) (2 - KJo&r - K,K,)
2
i
eB mw
1
u,-co
,,/-<-. 0 P
,’
‘.
\
XI
u,=lJ,=8
:
\..____d
0
j’
\
Fig. I.
H-PSS
Wave normal surfaces
In
a
plasma in a magnetic field.
(10)
R. J. PAPA and W. P. ALLIS
102
Where &,, Kr and RI are the components of the dielectric tensor in rotating co-ordinates. K T=K1l;
I-8
KR = 1-a’
s=eT a2 mc*
a2
a2
K,=l--;
where
The matrix form of the wave equation is still
KJ= l-1+8; WI
For propagation along magnetic field, B = 0, there are two waves n,* = K, which is right circularly polarized, n12= KJ which is left circularly polarized. For propagation across magnetic field : 8 = a/z
no* = Kp is the ordinary wave polar-
ized along B. n2_Kr
x-
-
K 1. K
LOW RERAN
For a equation K+E/ca = different.
IS the extraordinary wave polarized in the plane perpendicular to B.
finite temperature plasma, the wave has the same form (2) n x (n x E) + 0. Now, however, the dielectric tensor is It is derived from the conductivity
tensor,
K= 1 +a, as before, but in the jOsO calculation of CTone must take account of the finite size of the electron orbit compared to the wavelength. This has been done by Sitenko, Drummond, Bernstein, Mower* and others and their results have been used. An example of the change in the components of K is given by the x-x component: Kll=l3r2 + (1 -+)(l-4/32)
P&s.
012
a13
a21
a22
a23
a32
a33
)
dispersion relation, again obtained det[a] = 0, may be written:
The
-~ans+(A+ab)n4-(B+ec)na+C=O
from (14)
u, b and c characterize the temperature effects and are implicated functions of a, fi and 8. A, B and C are the zero-tempera~re coefficients of equation (91. For moderate temperatures the terms Eb and SCmay be neglected compared to the larger A and B, but the term in cane increases the order of the equation and must be kept. The dispersion equation can then be written --arP+An4-Bnz+C==O
w
For indices n of order one, the solutions rzr,* of the dispersion equation are the same as before since .xw+Jis much less than the other terms. These solutions may be called the fast wave solutions. There is now a third solution with large n. This solution is called a plasma wave, and denoted by n,. Jts phase velocity, u =c/n, is of the order of magnitude of the electron’s thermal velocity. If rr3” % nif2, the constant term C may be neglected in the dispersion equation. Then,
The first quadrant of the ag-/3*plane is now divided into thirteen regions, as shown in Fig. 2. There is a resonance at a2 = +. For regions 2, 3, 6,7 and 8 the full cubic, equation (1 S), was solved. All wave surfaces were calculated with E = 10m2. The wave surfaces are surfaces of rotation about the Z-axis and have a plane of symmetry neroendicular to B. Thev are shown in Fie. 3.
1 WI Rev. 116, I6
41
i a31
ELECTRON PLASMA
*For references see L. MOWER, (1959).
(13)
WAVES IN A PLASMA
103
II
Fig. 2 Topological regions for a warm plasma.
Starting below the /?” = 4 resonance line, as is right circularly polarCPincreases from region 1 to 2, the normal wave ized, surface for the plasma becomes larger, until in - a”>10 + 8) is left circularly polarregion 3 its phase velocity, u = c/n, is pretty close to the velocity of light. In regions 4 and 5 there ” = :: + &j/(1 + fi)3 ixed . is no plasma wave. The plasma wave surface The third wave for propagation along the has a butterily shape in regions 6 and 7. In magnetic fieId is the Bohm Gross plasma region 8, the plasma wave surface is like a oscillation. This wave is a purely longitudinal lemniscate. The wave surface labelled p-x is a wave. plasma wave near 0” and a fast wave near 180”. $ _ (1-d In regions 9 and 10 the plasma wave surface is 3E like a lemniscate. In region 11, the plasma wave For propagation across the magnetic field, surface is a four-leaved rose, which becomes a lemuiscate in regions 12 and 13. In region 11, the there are three waves in general. The ordinary wave is polarized with its E figure eight part of the plasma wave surface goes ofF to infinity as $ approaches 1. In regions 12 vector along the ma~etic field. and 13, the figure eight pattern can be entirely accounted for by the fast wave solutions. For propagation along the magnetic field, There are two extraordinary waves polarized there are in general three waves,
R. J. PAPA and W. P. ALLIS
104
with their E vector in the plane perpendicular to the magnetic field. One extraordinary wave is a fast wave and the other is the plasma wave. Their indices of refraction are given by equation (16) where
A = [(l-/P-a2)+e
12a2j?2
(l-/3Z)(1-4jP)
+ 4(1-a2-/32)(1-t-2j32) (l-jP)2(1-4/?32 B = [(I-a”)‘-fi”]
3E &lJ= 1_4Bz
__.----._ 0 3 REGION 6
REG Nil
P
,,,’
,/” -,-----. , .\
.\
;’ : \ *,..
; i
,/
~..__._./’
I’
P,:: 8 /II. .I_____/~~’
/’
REGION
2
REGION 12
REGION 9 ,__-----__-
,/’
+
\
‘x REGION 4 P
REGION 7
Fig. 3. Wave normal surfaces for a warm plasma.
II