Compact form and evaluation of Trubnikov’s plasma dielectric tensor
Trubnikov’s plasma dielectric tensor by means of integral transform is reduced to more simple and transperant form. Due that the problem evaluating this tensor for arbitrary plasma temperature and the angle of wave propagation is resolved. Розв'язується проблема точного обчислення плазмового ді...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2005 |
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| Формат: | Стаття |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
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| Цитувати: | Compact form and evaluation of Trubnikov’s plasma dielectric tensor / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2005. — № 2. — С. 61-63. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859824648329363456 |
|---|---|
| author | Pavlov, S.S. Castejon, F. |
| author_facet | Pavlov, S.S. Castejon, F. |
| citation_txt | Compact form and evaluation of Trubnikov’s plasma dielectric tensor / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2005. — № 2. — С. 61-63. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Trubnikov’s plasma dielectric tensor by means of integral transform is reduced to more simple and transperant form. Due that the problem evaluating this tensor for arbitrary plasma temperature and the angle of wave propagation is resolved.
Розв'язується проблема точного обчислення плазмового діелектричного тензору Трубнікова для довільних значень температури плазми і кута розповсюдження в ній хвиль.
Решается проблема точного вычисления Трубниковского плазменного диэлектрического тензора для произвольных значений температуры плазмы и угла распространения в ней волн.
|
| first_indexed | 2025-12-07T15:27:57Z |
| format | Article |
| fulltext |
COMPACT FORM AND EVALUATION OF TRUBNIKOV’S
PLASMA DIELECTRIC TENSOR
S.S. Pavlov1 and F. Castejon2
1Institute of Plasma Physics, NSC KIPT, Akademicheskaya str.1, 61108,Kharkov, Ukraine;
2Asociación EURATOM-CIEMAT PARA Fusión, 28040, Madrid, Spain
Trubnikov’s plasma dielectric tensor by means of integral transform is reduced to more simple and transperant form.
Due that the problem evaluating this tensor for arbitrary plasma temperature and the angle of wave propagation is
resolved.
PACS: 52.27.Ny
1. INTRODUCTION
The study of EC waves even in respectively low
temperature laboratory plasma requires taking into
account relativistic effects, especially for the quasi-
perpendicular waves propagation. Two original equivalent
integral forms of the exact relativistic dielectric tensor for
Maxwellian plasma were given by Trubnikov [1] but were
singular and as a result proved difficult to evaluate
numerically.
For low temperature plasma, in which relativistic
effects are relatively weak, gradually was developed the
theory of weakly relativistic approximation that widely
used now in applications. A review of vast number of
works in that direction is presented in the recent book of
Brambilla [2].
For plasma with more high temperature the theory is
developed in less detail. One approach was suggested by
Tamor [3], who simplified the problem studding only
non-singular anti-Hermitian parts of dielectric tensor.
This way, using recursive properties of anti-Hermitian
parts, allows him numerically to estimate those parts for
plasma with temperature below 30 KeV. For more high
temperatures this way failed due numerical instability of
the recursion.
Later, at the same way Bortnatici and Ruffina gave
exact analytical expressions for anti-Hermitian parts of
dielectric tensor for harmonics with numbers and
estimate Hermitian parts [4]. However, their results for
anti-Hermitian parts were not quite general and
estimations for Hermitian parts were approximate. That
didn’t allow them to derive whole dielectric tensor in the
exact closed form.
0≥n
Recently, a new way to evaluate relativistic tensor was
given by Swanson [5]. It allows for the case of
perpendicular waves propagation writing the exact
relativistic tensor elements in terms of five single singular
integrals over hyper-geometric functions. Those integrals
are relatively easy evaluated numerically. However, for
oblique propagation on this way double singular integrals
are necessary to calculate, that requires some
approximations.
The primary purpose of the present work is giving the
way to evaluate the relativistic dielectric tensor of
Trubnikov for arbitrary plasma temperature and angle
waves propagation. It is made on the ground of
transformation of this tensor to more simple and
transparent form. On this way dielectric tensor is
presented in the manner of the non-relativistic
approximation as an expansion in the Larmor radius in
terms of specially introduced in the work [6] the exact
relativistic PDFs to reduce evaluation of tensor to
evaluation of these PDFs.
2. MUTUALLY CONSISTENT FORMS OF
NON-RELATIVISTIC AND RELATIVISTIC
DIELECTRIC TENSORS
Using analytical properties of modified Bessel
functions and non-relativistic PDF, which appear in non-
relativistic dielectric tensor, it is possible to write that
tensor in different analytical forms. The choice of the
concrete form is defined either by tendency to extremely
possible simplicity of final expression [7] or by somewhat
other considerations [2]. In particular, one can verify that
this tensor can be presented as well in the following form:
[ ]∑
∞
−∞=
′′+′+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+=
n
nnnnnn
p ZZSZZRZZPZ )()()(1
2
0 ω
ω
ε , (1)
where , )( nZZ )( nZZ ′ , )( nZZ ′′ is non-relativistic PDF of
the argument )2()( // Tcn VknZ Ω−= ω and its first and
second derivatives in this argument, is thermal
electron velocity and
TV
,000
02
0
2
2
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
′−′
′−
= nnn
nn
n AAnAin
AinAn
P λλ
λ
,022
200
200
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
′
′−=
λλ
λ
λ
nn
n
n
n
AinA
Ai
nA
R
,200
000
000
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
n
n
A
S
)()exp()( λλλ nn IA −= , λλλ ddAA nn )()( =′ .
The main peculiarity of such presentation is total
separation of perpendicular and longitudinal dispersion.
The perpendicular dispersion is described by tensors
, , and the longitudinal one is described by
plasma dispersion function, , and its first and
second derivatives in the parameter .
nP nR nS
)( nZZ
nZ
Let us show using the exact relativistic PDFs,
( )μ,,0
23 zaZq+ and indications from the work [6] that the
Trubnikov’s dielectric tensor can be presented in the
Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 61-63 61
form, which is similar to the form (1). First, we’ll prove
that functions, ( μ,,1
23 zaZq+
62
) , appearing in the elements of
dielectric tensor 13ε , ,31ε 23ε , and functions, 32ε
( μ,,2
23 zaZq+ ) , appearing in element 33ε can be expressed
in terms of the exact PDFs, ( μ,,0
23 zaZ q+ ) . Really, using
differentiation the expression (4) from the work [6] in
argument z one can prove the next identities:
( ) ( ) , (2) μμ azZdzazdZa qq ,2,2 1
21
0
23 ++ −=
( ) ( ) ( )μμμ azZazZzdazZda qqq ,,,2 2
21
0
21
20
23
2
−++ =+ , (3)
Note that the identities (2), (3) take place for both cases
and . Now let us change argument 10 // <≤ N 1// >N z in
relativistic PDFs into ( )azZ n 2= . Then it is obviously,
that )2()( // Tcn VknZ Ω−= ω . The advantage of such a
change consists of the fact that relativistic PDFs depend
now of the same argument that the non-relativistic PDF.
It is not difficult to verify now term by term that the
relativistic tensor (1) can be written in the form
[ ], (4) ∑
∞
−∞=
′′∗+′∗+∗⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
n
nnnnnn
p ZZSZZRZZP )()()(1
2
ω
ω
με
where ( )nZZ is an infinite vector containing all
relativistic PDFs with gradually increasing index starting
from 25=q :
( ) ( ) ( ) ( )[ ],...,,,,,,,, 0
29
0
27
0
25 μμμ aZZaZZaZZZZ nnnn = ,
here the sign denotes the scalar product of the infinite
vector
∗
( )nZZ with every left-side matrix element
considered as an infinite vector resulting from finite
Larmor radius (FLR)- expansion beginning from the term
of zeroth order. So, for instance
[ ] )()( 2
11 nnnn ZZAnZZP ∗=∗ λ ,
where [ ]...,,, 210 λλλλ nnnn AAAA = . Here the superscript
in
k
λk
nA denotes the order in FLR-expansion of λnA .
The order in FLR-expansion in tensor , that is
connected with elements of tensor , must be
considered after dividing these elements by
nR
32233113 ,,, εεεε
λ .
3. EVALUATING OF THE EXACT
RELATIVISTIC PDFs
From the form (4) of Trubnikov’s dielectric tensor and
the formulas for the 1st and 2nd PDFs derivatives (20)
from the work [6] it follows that the problem of
evaluation of this tensor is reduced to the problem of
evaluation of the exact PDFs, ( )μ,,0
23 zaZq+ .
Using the expression (4) for anti-Hermitian parts of
these PDFs from the work [6] one can verify the
identity
( ) ×⎥⎦
⎤
⎢⎣
⎡ −+−= ∫
∞
∞−
+ )2(2expIm),,(Im
2212210
21 μπμ xazxdxzaZq
( ) ⎥⎦
⎤
⎢⎣
⎡ −−−+ )2(22
221212 μxazxaxzFq ,
where is Dnestrovskii function with integer index
. It is easy to see that from theory of Cauchy type
integrals it follows the formula
)(zFq
q
( ) ×⎥⎦
⎤
⎢⎣
⎡ −+−= ∫
∞
∞−
+ )2(2exp),,(
2212210
21 μπμ xazxdxzaZq
× ( ) ⎥⎦
⎤
⎢⎣
⎡ −−−+ )2(22
221212 μxazxaxzFq . (5)
The integral form (5) provides the first way evaluating the
exact PDFs and is a generalization on the fully relativistic
case of the method evaluating the weakly relativistic
PDFs of Airoldi and Orefice [8].
The second way is based on the direct calculating of
singular integrals describing Hermitian parts of the exact
PDFs using for calculation of the principal value of these
integrals the exact non-singular formulas [9]
( ) ( ) ( ) ( )dt
bt
tbfdt
bt
tbftfdt
bt
tf b
∫∫∫
∞−
∞
−
−
−
−
−−
=
−
0
00
22
P , (6)
(for ) 10 // <≤ N
( ) ( ) ( )dt
bt
tbftfdt
bt
tf b
∫∫
∞−
∞
∞− −
−−
=
−
2
P , (7)
(for the case ). 1// >N
At the figures 1, 3 of Supplement there are given
graphics of the exact PDFs with indexes from =3/2 till
=9/2 for two values of longitudinal refractive index,
=0.3, =1.1, and two values of plasma
temperatures, = 20keV and =40keV, respectively.
For comparison, at the figures 2, 4 of Supplement there
are presented plots of the weakly relativistic and exact
PDFs for the same values of parameter and different
values of temperature.
q
q
//N //N
eT eT
//N
REFERENCES
1. B.A. Trubnikov// Plasma Physics and the problem of
Controlled Thermonuclear Reactions / M.A. Leontovich
Editor / v. 3, Pergamon Press, 1959, p.122.
2. M. Brambilla. Kinetic theory of plasma waves,
homogeneous plasmas. Oxford: Oxford University Press.
1998.
3. S. Tamor// Nuclear Fusion (18). 1978, p.229.
4. M.Bortnatici, U.Ruffina // IL Nuovo Cimento, 1985,
v.6D, N. 3, p.231.
5. D.G. Swanson // Plasma Phys. Control. Fusion. 2002,
v. 44, p.1329.
6. S.S. Pavlov and F.Castejon. Exact relativistic PDFs//
Book of abstr. 10th International Conference and School
on Plasma Physics and Controlled Fusion, Alushta,2004.
7. A.I. Akhiezer, I.A. Akhiezer, A.V.Polovin,
A.G.Sitenko, K.N. Stepanov. Plasma Electrodynamics.
Oxford: Pergamon Press. 1975.
8. A.C. Airoldi, A.Orefice // J. Plasma Phys. 1982, v. 27,
p.515.
9. S. S. Pavlov, F. Castejón, H. Maasberg, and I.I. Koba.
Plasma dispersion functions for complex frequencies //
10th European Fusion Theory Conference, Helsinki,
Finland, 2003.
SUPPLEMENT
Fig. 1 PDFs for the case , keVTe 20= 3.0// =N
63
Fig. 2 Weakly relativistic (solid-line) and exact (dash-
line) PDFs for and for temperatures
and
2/7=q 3.0// =N
keVTe 20= keVTe 80=
Fig. 3 Exact PDFs for the case and keVTe 40= 1.1// =N
Te=80 keV
Te=160 keV
Te=20 keV
Te=40 keV
Te=40 keV
Te=20 keV
Te=160 keVTe=80 keV
Fig. 4 Weakly relativistic (solid-line) and exact (dash-
line) PDFs for 2/7=q and for temperatures 1.1// =N
keVTe 40= and keVTe 160=
КОМПАКТНАЯ ФОРМА И ВЫЧИСЛЕНИЕ ПЛАЗМЕННОГО ДИЭЛЕКТРИЧЕСКОГО ТЕНЗОРА
ТРУБНИКОВА
С.С. Павлов, Ф. Кастехон
Решается проблема точного вычисления Трубниковского плазменного диэлектрического тензора для
произвольных значений температуры плазмы и угла распространения в ней волн.
КОМПАКТНА ФОРМА І ОБЧИСЛЕННЯ ПЛАЗМОВОГО ДІЕЛЕКТРИЧНОГО ТЕНЗОРУ
ТРУБНІКОВА
С.С. Павлов, Ф. Кастехон
Розв'язується проблема точного обчислення плазмового діелектричного тензору Трубнікова для довільних
значень температури плазми і кута розповсюдження в ній хвиль.
COMPACT FORM AND EVALUATION OF TRUBNIKOV’S
PLASMA DIELECTRIC TENSOR
|
| id | nasplib_isofts_kiev_ua-123456789-79344 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:27:57Z |
| publishDate | 2005 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Pavlov, S.S. Castejon, F. 2015-03-31T13:51:47Z 2015-03-31T13:51:47Z 2005 Compact form and evaluation of Trubnikov’s plasma dielectric tensor / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2005. — № 2. — С. 61-63. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 52.27.Ny https://nasplib.isofts.kiev.ua/handle/123456789/79344 Trubnikov’s plasma dielectric tensor by means of integral transform is reduced to more simple and transperant form. Due that the problem evaluating this tensor for arbitrary plasma temperature and the angle of wave propagation is resolved. Розв'язується проблема точного обчислення плазмового діелектричного тензору Трубнікова для довільних значень температури плазми і кута розповсюдження в ній хвиль. Решается проблема точного вычисления Трубниковского плазменного диэлектрического тензора для произвольных значений температуры плазмы и угла распространения в ней волн. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Compact form and evaluation of Trubnikov’s plasma dielectric tensor Компактна форма і обчислення плазмового діелектричного тензору трубнікова Компактная форма и вычисление плазменного диэлектрического тензора трубникова Article published earlier |
| spellingShingle | Compact form and evaluation of Trubnikov’s plasma dielectric tensor Pavlov, S.S. Castejon, F. Basic plasma physics |
| title | Compact form and evaluation of Trubnikov’s plasma dielectric tensor |
| title_alt | Компактна форма і обчислення плазмового діелектричного тензору трубнікова Компактная форма и вычисление плазменного диэлектрического тензора трубникова |
| title_full | Compact form and evaluation of Trubnikov’s plasma dielectric tensor |
| title_fullStr | Compact form and evaluation of Trubnikov’s plasma dielectric tensor |
| title_full_unstemmed | Compact form and evaluation of Trubnikov’s plasma dielectric tensor |
| title_short | Compact form and evaluation of Trubnikov’s plasma dielectric tensor |
| title_sort | compact form and evaluation of trubnikov’s plasma dielectric tensor |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79344 |
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