Inhomogeneous relativistic plasma dielectric tensor
The paper is concerned with the method to derive the inhomogeneous relativistic plasma dielectric tensor on the
 base principle of analytic continuation for Cauchy-type integrals and the exact plasma dispersion functions concept. На основі принципу аналітичного продовження та поняття точних...
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| Date: | 2011 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2011
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| Cite this: | Inhomogeneous relativistic plasma dielectric tensor / S.S. Pavlov // Вопросы атомной науки и техники. — 2011. — № 1. — С. 53-55. — Бібліогр.: 4 назв. — англ. |
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| author_facet | Pavlov, S.S. |
| citation_txt | Inhomogeneous relativistic plasma dielectric tensor / S.S. Pavlov // Вопросы атомной науки и техники. — 2011. — № 1. — С. 53-55. — Бібліогр.: 4 назв. — англ. |
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| description | The paper is concerned with the method to derive the inhomogeneous relativistic plasma dielectric tensor on the
base principle of analytic continuation for Cauchy-type integrals and the exact plasma dispersion functions concept.
На основі принципу аналітичного продовження та поняття точних релятивістських плазмових дисперсійних
функцій представляється метод одержання тензору діелектричної проникності неоднорідної релятивістської
плазми.
На основе принципа аналитического продолжения и понятия точных релятивистских плазменных
дисперсионных функций представляется метод получения тензора диэлектрической проницаемости
неоднородной релятивистской плазмы.
|
| first_indexed | 2025-12-07T17:23:08Z |
| format | Article |
| fulltext |
INHOMOGENEOUS RELATIVISTIC PLASMA DIELECTRIC TENSOR
S.S. Pavlov
Institute of Plasma Physics, NSC “Kharkov Institute of Physics and Technology",
Kharkov, Ukraine
The paper is concerned with the method to derive the inhomogeneous relativistic plasma dielectric tensor on the
base principle of analytic continuation for Cauchy-type integrals and the exact plasma dispersion functions concept.
PACS: 52.27.Ny
1. INTRODUCTION
Deriving of the inhomogeneous fully relativistic
plasma dielectric tensor, taking into account the spatial
dispersion of plasma, is of current importance for the
theoretical study of electromagnetic waves in
thermonuclear plasma in the electron cyclotron (EC)
frequency range. An importance of the exact taking into
account relativistic effects follows from the fact that those
effects can even arise in laboratory plasmas with
moderate temperatures and especially in quasi-
perpendicular, in respect to magnetic field, propagation
regime. An importance of accounting the spatial
dispersion is defined by more and more deep development
of idea to use the electron Bernstein waves for plasma
heating and current drive.
The fully relativistic plasma dielectric tensor for
homogeneous plasma [1] in the form, suitable for
analytical and numerical applications, was derived on the
base of principle of analytic continuation and concept of
the exact plasma dispersion functions (PDFs), generating
the weakly relativistic PDFs to the case of arbitrary
plasma temperatures [2, 3].
The main purpose of the present work is an
extension of this method to the case of plasma,
inhomogeneous in transverse to magnetic field direction.
2. INHOMOGENEOUS RELATIVISTIC
PLASMA DIELECTRIC TENSOR
In the relativistic limit the Vlasov kinetic equation,
being the start point for deriving of plasma dielectric
tensor, has the form [1]
0, =
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
⎥⎦
⎤
⎢⎣
⎡ ++−
∂
∂
+
∂
∂
p
BBpE
r
p
0
f
mc
γef
mt
fγ , (1)
where ( )2/1 mcpγ += , ( )2/c-1m v/v=p and m
is momentum and the rest mass of the electron.
Linearization of the equation (1) leads to equation for
small perturbations of electron distribution function
),,(~ tf pr with respect to the relativistic Maxwellian
distribution )( pfM
0
~~~
=⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
⋅−
∂
∂
+
∂
∂
+
∂
∂
p
E
r
p M
B
fγeff
mt
fγ
ϕ
ω , (2)
where mceBB /0=ω , ( ) ( )γATAfM με −=−= exp/exp ,
422 cmpc +=ε is the energy of the free electron,
( ) μμπ /)(4 2
31 KmcA =− , )(2 μK is the MacDonald
function, 2)/( Tс V=μ is the main relativistic parameter,
mTT /=V is thermal speed of electrons, ϕ is the
azimuth angle in the momentum space with the polar axis
along the base magnetic field . It’s worth note that the
equation (2) is true for the case of sharp changing of
magnetic field in comparison with changing of electron
density and temperature, what is typifying for plasma in
the depth of plasma column.
0B
Let us search a decision of the equation (2) for small
perturbations of electron distribution function and electric
field in the form ( ) ( )ω,~,,~
//kx,ftf =pr ( )[ ]tzkiexp ω−// ,
( ) ),,(, // ωkxt ErE = ( )[ ]tzkiexp ω−// by means of the
perturbation method in the finite electron Larmor radius.
In accordance with this technique a decision is looking for
in the form of series
( ) ...,,~ )2()1()0(
// +++= fffkxf ω , (3)
where the perturbation is defined from equations )(nf
( )
)(2
)0(
)0(//// pEf
mc
eff
m
kpi MB ⋅−=
∂
∂
−⎟
⎠
⎞
⎜
⎝
⎛ −
μ
ϕ
ωγω ,
x
ff
m
pff
m
kpi
n
M
x
n
B
n
∂
∂
=
∂
∂
−⎟
⎠
⎞
⎜
⎝
⎛ −
− )1()(
)(////
ϕ
ωγω , ( ). (4) 1≥n
Decisions for first three equations (4), using for shortness
symbols and , can be
written
)()()( nnn fff −+ += yx iEEE ±=±
( )
( ) ϕμ i
M eEDf
mc
iepf m
m ±
⊥
± = 12
)0(
2
,
1
////
1
−
⎟
⎠
⎞
⎜
⎝
⎛ ±−= Bm
kpD ωγωm
,
( )
( ) ( )0
2
212
2
)1(
4
DeDEDf
mcm
epf i
M +′= ±
⊥
±
ϕμ m
mm
,
(5)
( )
[ ( ) ( )++′′−= ±
⊥
±
ϕϕμ 3
312122
3
)2(
8
ii
M eDeDDEDf
mcm
iep
f m
m
m
mmm
( ) ( )( ) ].3
32102101
ϕϕϕμ iii
M eDDeDDDeDDEDf m
mm
m
mmm +++″ ±
±±
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2011. № 1. 53
Series: Plasma Physics (17), p. 53-55.
The plasma current additions ( ) can be
found on the ground of expressions (5) in line with
)(kj 2,1,0=k
pppj 3)()( )()( df
m
ren kk ∫=
γ
For example, in zero approximation for the perturbation
and for the component of plasma current one
has
)0(
−f
)0(
xj
.)exp(
)()(8
)(
//1
2
0 0
2
2
5
22
)0(
dpeEDp
dppd
Kmcm
rniej
i
x
x
ϕ
π
γ
μγ
ϕ
μπ
μ
−
∞+
∞−
⊥
+∞
⊥
∫
∫ ∫
−
×=
Here it is convenient to utilize the normalization
mcpp /=
.
)/(
)exp(
)(8
)(
////
0
3
//
2
22
)0(
ωωγ
γ
μγ
ωμ
μ
B
x
pN
E
pdppd
mK
rniej
−−
×
−
=
−
+∞
⊥⊥
+∞
∞−
∫∫
Now we make the change of variables of integration
⊥pp //,
into γ//,p with transform Jacobian
⊥p,γ .
( )[ ]
∫∫
+∞
+
−
+∞
∞− −−
+−−
=
2
//1 ////
2
//
2
//
2
22
)0(
)/(
1)exp(
)(8
)(
p B
x pN
Epdpd
mK
rniej
ωωγ
γμγγ
ωμ
μ .
After the new changing of variables γ,//p into
2
//// 1, pxp +−= γ one has
,
)/1(
)12())1(exp(
0 ////
2
//
2
//
2
//
//
)0( ∫∫
∞+
−
∞+
∞− −−++
++++−
=
ωω
μ
B
x
pNpx
Epxxpx
dxpdCj
ωμ
μ
)(8
)(
2
22
mK
rnieC = . (6)
Now on the base of the principle of analytic continuation,
one can only leave the anti-Hermitian part of (6) and then
one can evaluate the integral in x analytically.
( )[ ]( ),1/
))/(exp(
2
//
2
////
//////
)0(
//
//
yxB
B
p
p
x
iEEppN
pNpdCj
−−−+
×−−= ∫
+
−
ωω
ωωμ
where the integration limits ±
//p follow from the condition
of pole appearing in the expression (6). Hence from the
anti-Hermitian part, principle of analytic continuation and
definition of exact relativistic PDFs [2], finite expressions
for components of plasma dielectric tenor and
can be obtained:
)0(
xxε )0(
xyε
[ ]),,(),,(
2
11 1251252
2
)0( μμ
ω
ω
με −+−= zaZzaZp
xx
,
[ ]),,(),,(
2 1251252
2
)0( μμ
ω
ω
με −−= zaZzaZip
xy
,
where ),,(25 μzaZ is the exact relativistic PDF with index
25 which corresponds to the fundamental electron
cyclotron resonance, )2/()( //1 TB Vkz ωω ±=±
and
longitudinal wave number cNk ///// ω= . Integrating the
perturbation in the same manner as after more
bulky calculations one can obtain expressions for
components of plasma dielectric tensor после
( =1,2). Here we give the finite result:
)2(f )0(f
)2(
,kiε
ki,
⎟
⎠
⎞
⎜
⎝
⎛+= EE
dx
d
dx
d )2()0( εεε ,
where
)0()0(
xyyx εε −= , , )0()0(
xxyy εε =
))],,,(),,((
)),,(),,([(
12/712/7
22/722/7
2
)2(
μμ
μμ
ωω
ω
με
−
−
+
−+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
zaZzaZ
zaZzaZ
B
Tp
xx
v
))],,,(),,((2
)),,(),,([(
12/712/7
22/722/7
2
)2(
μμ
μμ
ωω
ω
με
−
−
+
−−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
zaZzaZ
zaZzaZi
B
Tp
xy
v
)2()2(
xyyx εε −= ,
)].,.(4)),,(),,((3
)),,(),,([(
02/712/712/7
22/722/7
2
)2(
μμμ
μμ
ωω
ω
με
zaZzaZzaZ
zaZzaZ
B
Tp
yy
++
−+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
−
−
v
It can also be demonstrated that present approach to
derive inhomogeneous plasma dielectric tensor leaves in
force and for the general case of taking also into account
in equation (2) the drift terms in background distribution,
i.e. without limitations for gradients of inhomogeneous
plasma parameters.
3. CONCLUSIONS
The next conclusions can be drawn from the previous
text.
1. The present technique, based on principle of analytical
continuation for Cauchy-type integrals and on the exact
plasma dispersion function concept, provides the exact
account of relativistic effects for plasma that is
inhomogeneous across direction of magnetic field.
2. In the case of relativistic plasma, inhomogeneous
across magnetic field direction likely to the case of
homogeneous plasma, instead nonrelativistic PDF ,
in the plasma dielectric tensor there appeared the exact
relativistic PDFs with coefficient
)(xW
μ2//N . Moreover an
index of those PDFs is defined by the order of Larmor
54
55
expansion in line with the rule of homogeneous case [2].
Hence a structure of differential operator for this tensor
corresponds exactly to non-relativistic case with the same
type of inhomogeneity.
3. Present results can be used as for development of EC
wave numerical models (for example, of the type [4]),
which take into account the spatial plasma dispersion,
including the strong one, so and of ICR wave models in
the case of hot relativistic plasma.
4. This way can be also used to clarify the exact limits of
the weakly and mild relativistic approximations in the
numerical wave calculations.
REFERENCES
1. B.A. Trubnikov. Electromagnetic waves in a relativistic
plasma in a magnetic field // Plasma Physics and the
Problem of Controlled Thermonuclear Reactions / Ed.
Leontovich M. A., Pergamon Press, New York. 1959,
v 3.
2. F. Castejon, and S.S. Pavlov. Relativistic plasma
dielectric tensor based on the exact plasma dispersion
functions concept // Phys. Plasmas. (2006), v. 13,
p. 072105. Erratum // Phys. Plasmas. 2007, v. 14,
p. 019902.
3. F. Castejon, and S.S. Pavlov. The exact plasma
dispersion functions in complex region // Nuclear
Fusion. 2008, v. 48, p. 054003.
4. V.L. Vdovin. Electron Cyclotron Heating in tokamaks
with 3D full wave code // 33rd EPS Conference on
Plasma Physics. Roma, Italy, June 19-23, 2066 ECA,
2006.
Article received 28.09.10
ТЕНЗОР ДИЭЛЕКТРИЧЕСКОЙ ПРОНИЦАЕМОСТИ
НЕОДНОРОДНОЙ РЕЛЯТИВИСТСКОЙ ПЛАЗМЫ
C.C. Павлов
На основе принципа аналитического продолжения и понятия точных релятивистских плазменных
дисперсионных функций представляется метод получения тензора диэлектрической проницаемости
неоднородной релятивистской плазмы.
ТЕНЗОР ДІЕЛЕКТРІЧНОЇ ПРОНИКНОСТІ
НЕОДНОРІДНОЇ РЕЛЯТИВІСТСЬКОЇ ПЛАЗМИ
C.C. Павлов
На основі принципу аналітичного продовження та поняття точних релятивістських плазмових дисперсійних
функцій представляється метод одержання тензору діелектричної проникності неоднорідної релятивістської
плазми.
|
| id | nasplib_isofts_kiev_ua-123456789-90640 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:23:08Z |
| publishDate | 2011 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Pavlov, S.S. 2015-12-29T15:44:35Z 2015-12-29T15:44:35Z 2011 Inhomogeneous relativistic plasma dielectric tensor / S.S. Pavlov // Вопросы атомной науки и техники. — 2011. — № 1. — С. 53-55. — Бібліогр.: 4 назв. — англ. 1562-6016 1562-6016 PACS: 52.27.Ny https://nasplib.isofts.kiev.ua/handle/123456789/90640 The paper is concerned with the method to derive the inhomogeneous relativistic plasma dielectric tensor on the
 base principle of analytic continuation for Cauchy-type integrals and the exact plasma dispersion functions concept. На основі принципу аналітичного продовження та поняття точних релятивістських плазмових дисперсійних
 функцій представляється метод одержання тензору діелектричної проникності неоднорідної релятивістської
 плазми. На основе принципа аналитического продолжения и понятия точных релятивистских плазменных
 дисперсионных функций представляется метод получения тензора диэлектрической проницаемости
 неоднородной релятивистской плазмы. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы Inhomogeneous relativistic plasma dielectric tensor Тензор діелектрічної проникності неоднорідної релятивістської плазми Тензор диэлектрической проницаемости неоднородной релятивистской плазмы Article published earlier |
| spellingShingle | Inhomogeneous relativistic plasma dielectric tensor Pavlov, S.S. Фундаментальная физика плазмы |
| title | Inhomogeneous relativistic plasma dielectric tensor |
| title_alt | Тензор діелектрічної проникності неоднорідної релятивістської плазми Тензор диэлектрической проницаемости неоднородной релятивистской плазмы |
| title_full | Inhomogeneous relativistic plasma dielectric tensor |
| title_fullStr | Inhomogeneous relativistic plasma dielectric tensor |
| title_full_unstemmed | Inhomogeneous relativistic plasma dielectric tensor |
| title_short | Inhomogeneous relativistic plasma dielectric tensor |
| title_sort | inhomogeneous relativistic plasma dielectric tensor |
| topic | Фундаментальная физика плазмы |
| topic_facet | Фундаментальная физика плазмы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/90640 |
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