Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers
A new exact integral form of the fully relativistic permittivity tensor for plasmas in a magnetic field is given. It is suitable for numerical applications for arbitrary wave numbers since all integrals in it are one-dimensional ones. This form is interesting for applications to study propagation...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2016
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nasplib_isofts_kiev_ua-123456789-1153252025-02-23T17:40:45Z Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers Точное вычисление релятивистского тензора диэлектрической проницаемости плазмы в магнитном поле для произвольных волновых чисел Точне обчислення релятивістського тензора діелектричної проникності плазми в магнітному полі для довільних хвильових чисел Pavlov, S.S. Basic plasma A new exact integral form of the fully relativistic permittivity tensor for plasmas in a magnetic field is given. It is suitable for numerical applications for arbitrary wave numbers since all integrals in it are one-dimensional ones. This form is interesting for applications to study propagation and absorption of electron Bernstein waves in the laboratory thermonuclear plasmas and of arbitrary electron and ion cyclotron waves in the hot astrophysical plasmas. Даётся новая интегральная форма полностью релятивистского тензора диэлектрической проницаемости плазмы в магнитном поле. Она годится для численных приложений при произвольных волновых числах, поскольку все интегралы в ней являются одномерными. Эта форма представляет интерес с точки зрения приложений для изучения бернштейновских электронных циклотронных волн в лабораторной термоядерной плазме и произвольных электронных и ионных циклотронных волн в горячей астрофизической плазме. Дається нова інтегральна форма повністю релятивістського тензора діелектричної проникності плазми в магнітному полі. Вона годиться для численних додатків при довільних хвильових числах, оскільки всі інтеграли в ній є одновимірними. Ця форма являє інтерес з точки зору додатків для вивчення бернштейнівських електронних циклотронних хвиль в лабораторній термоядерної плазмі і довільних електронних та іонних циклотронних хвиль в гарячій астрофізичної плазмі. 2016 Article Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers / S.S. Pavlov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 92-95. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.27.Ny https://nasplib.isofts.kiev.ua/handle/123456789/115325 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma Basic plasma |
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Basic plasma Basic plasma Pavlov, S.S. Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers Вопросы атомной науки и техники |
| description |
A new exact integral form of the fully relativistic permittivity tensor for plasmas in a magnetic field is given. It is
suitable for numerical applications for arbitrary wave numbers since all integrals in it are one-dimensional ones.
This form is interesting for applications to study propagation and absorption of electron Bernstein waves in the
laboratory thermonuclear plasmas and of arbitrary electron and ion cyclotron waves in the hot astrophysical
plasmas. |
| format |
Article |
| author |
Pavlov, S.S. |
| author_facet |
Pavlov, S.S. |
| author_sort |
Pavlov, S.S. |
| title |
Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers |
| title_short |
Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers |
| title_full |
Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers |
| title_fullStr |
Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers |
| title_full_unstemmed |
Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers |
| title_sort |
exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Basic plasma |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/115325 |
| citation_txt |
Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers / S.S. Pavlov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 92-95. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
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| first_indexed |
2025-11-24T04:17:18Z |
| last_indexed |
2025-11-24T04:17:18Z |
| _version_ |
1849643854907572224 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2016. №6(106)
92 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 92-95.
EXACT RELATIVISTIC MAXWELLIAN MAGNETIZED PLASMA
DIELECTRIC TENSOR EVALUATION FOR ARBITRARY WAVE
NUMBERS
S.S. Pavlov
Institute of Plasma Physics of the NSC KIPT, Kharkov, Ukraine
E-mail: pavlovss@ipp.kharkov.ua
A new exact integral form of the fully relativistic permittivity tensor for plasmas in a magnetic field is given. It is
suitable for numerical applications for arbitrary wave numbers since all integrals in it are one-dimensional ones.
This form is interesting for applications to study propagation and absorption of electron Bernstein waves in the
laboratory thermonuclear plasmas and of arbitrary electron and ion cyclotron waves in the hot astrophysical
plasmas.
PACS: 52.27.Ny
INTRODUCTION
Theoretical studying electromagnetic waves
propagation and absorption in magnetized plasma in the
electron cyclotron frequency range requires accurate
taking into account relativistic effects, associated with
increasing masses of fast enough electrons, especially in
the case of wave propagation almost perpendicular to
magnetic field lines and for high cyclotron harmonics
numbers under consideration [1]. In both these cases in
the ion cyclotron resonance frequency range similar
effects can also arise for hot enough plasmas [2].
The basis for studying linear electron cyclotron
waves in plasmas is an accurate evaluation of the
relativistic plasma dielectric tensor. Two original
equivalent exact integral forms of this tensor were given
on the ground of the relativistic form of Vlasov kinetic
equation [3], however, in general case of arbitrary
plasma and wave parameters their applicability has been
rather limited for numerical applications. Later, on the
base of the exact relativistic plasma dispersion functions
(PDF) and the rather compact form of this tensor, this
limitation was been essentially weakened [4].
Neglecting the ion dynamics, tensor elements in
this form are represented as double series: on the
electron cyclotron harmonic numbers and on the exact
relativistic PDFs multiplied with coefficients of
expansion of the functions )()(
nn IeA in the small
parameter . Here n is the number of the electron
cyclotron harmonic; 2)( k ,
k is transverse wave
number,
сTV /0 is Larmor radius of electrons,
00 / mTVT and )/( 0сmeBc are their thermal
velocity and fundamental cyclotron frequency, e is
the electron charge; )(nI are modified Bessel function
of the integer index. The tensor itself is presented in the
form
ij
p
ijij D
2
0
),,(
k , (1)
0
23
2
11
1
l
ln
ln
n
n
ZAnD II
II
,
0
232112
l
ln
ln
n
n
Z
d
Ad
niDD II
II
,
0
23
2
1122
2
l
ln
ln
n
n
Z
d
dA
DD II
II
,
0
25
3113
2
1
l n
lnln
n
n z
Z
AnDD
IIII
,
0
25
3223
2 l n
ln
ln
n
n z
Z
d
dAi
DD
II
II
,
n l n
ln
ln
n
ln
ln
n
z
ZA
ZAD
0
2
27
2
2533
2
II
II
II
II .
Above the next designations were used: is the
angular frequency of electromagnetic wave; k is the
wave vector; /Tсmμ 2
0 ,
0c,m ,T are the speed of light
in vacuum, the electron rest mass and temperature,
respectively; /2μNa 2
II , c/ωkN IIII is longitudinal
refractive index; nz )2/()( 0Tс Vkn II ;
0p is
plasma frequency of the electrons with rest mass,
3 2 3 2( , , )n l n l nZ Z a z I I I I
is the exact relativistic
PDF with the index 23 lnII . Superscript ln II of
brackets in expressions
ijD means the order of
expansion of the term in brackets in the parameter .
This designation significantly reduces recording. For
example, the expansion of functions )()(
nn IeA ,
mentioned above
00 2!)!()!2(
!)(2)1(
)(
l
ln
nl
ln
l
ln
l
n A
llnln
ln
A IIII
IIIIII
II
(2)
can be presented on this way in a much more short form
0
)(
l
ln
nn AA
II
. (3)
mailto:pavlovss@ipp.kharkov.ua
ISSN 1562-6016. ВАНТ. 2016. №6(106) 93
In applications of relativistic tensor (1) for
investigation of the fast electron cyclotron waves in
laboratory thermonuclear plasmas usually takes place
the condition 1 and, consequently, series in this
parameter (formally in the index l ) converges so rapidly
that its accurate calculation requires to summarize a few
terms only.
However, for the study of the slow or plasma
electron cyclotron waves in plasmas of laboratory or
astrophysics magnetic traps the parameter can
significantly increase and reach of values of order 1~
and even ones of order 1 . In this case the series in
the index l with an increase of the parameter begin to
converge slower and slower, which can cause serious
difficulties in its summation even in the case of the
weakly relativistic plasmas [5]. Obviously, in the case
of fully relativistic plasmas these difficulties can be
proved even more significant. In these unfavorable
cases it makes sense trying to find some alternative
form to one (1), suitable for accurate numerical
applications as the form (1) for the case 1 .
The main goal of the present work is the further
progress in resolving the problem of exact evaluation of
fully relativistic Maxwellian plasma dielectric tensor for
arbitrary values of (or for arbitrary wave numbers).
For each harmonic number n this scope is achieved by
means of introducing the generating function for the
anti-Hermitian parts of tensor and of introducing the
generalized relativistic PDF with their evaluation on the
basis of Kramers-Kronig formulae.
FUNCTIONS GENERATING ANTI-
HERMITIAN PARTS OF PLASMA
DIELECTRIC TENSOR
In the case of unfavorable value of the parameter
for each element of the fully relativistic tensor (1) and
each cyclotron harmonic number n the summation of
the slow convergent series mentioned above can be
analytically reduced to a two-step procedure, suitable
for accurate numerical applications. The first step
introduces the function generating the anti-Hermitian
part of this series in the parameter , and then a
numerical calculation of the one-dimensional integral of
this function leads directly to the anti-Hermitian part of
the sum of the series. The second step calculating
numerically the principal value of the integral in the
sense of Cauchy of the anti-Hermitian part leads to the
Hemitin part of the sum of the series. Obviously, both
parts together give a value of the whole series for the
corresponding tensor element.
Let us begin the demonstration of this two-step
procedure with the element of plasma dielectric tensor
11 . From the first integral form of Trubnikov it follows
[6]
.
/)(2
1
2
2
2
02
2
2
2
0
11
n c
n
p
npN
pJ
n
e
pdppd
K
IIII
II
(4)
Here )(2 K is Macdonald function, )/(mcpp is
normalized momentum, 21 p ,
2 . Using the
last difinition, the expression (4) can be converted to the
form
.
/)(2
1
22
0
2
2
2
2
0
11
c
n
n
p
npN
pJe
pdppd
n
K
IIII
II
(5)
After a change of variable p in (5) with
transformation )1( 222
IIpp and its Jacobian
pdpd // , we obtain the expression
.
/
)]1([
)(2
1
222
1
2
2
2
2
0
11
c
n
n
p
npN
pJ
edpd
n
K
IIII
II
II (6)
After one more change in (6) x with transform
21 IIpx and Jacobian 1/ dxd and accounting
relation 212 IIpxxx 22 1 IIp , we obtain
.
/1
)12(
)(2
1
2
22
0
1
2
2
2
2
0
11
2
c
n
xp
n
p
npNpx
pxxJ
edxe
pd
n
K
IIII
II
II
II
II
(7)
The integral over x that appears in formula (7) with the
multiplier )1exp( 2
IIp is one of the Cauchy type
with real density, satisfying the Hölder condition of
continuity and tending to 0 when 0x and x at
the contour. It is known that the integral of the Cauchy-
type
0
1 ( )
( ) ,
2
d
F z
i z
(8)
with the density )( satisfying the former conditions at
the contour, is defined at the contour itself by the
formulas of Sokhotskii-Plemejj
).0(,
)(
2
1
2
)(
)(
0
z
z
d
P
i
z
zF
(9)
Here, the functions )(zF , )(zF are the boundary
values of integral (8) when the argument z tends to the
contour from the right or from the left-hand side with
respect to the integration direction, respectively. The
letter P before integral denotes its principal value in the
Cauchy sense. At the real axis out of the contour the
integral (8) is not singular and, consequently,
).0(,
)(
2
1
2
)(
)(
0
z
z
d
P
i
z
zF
(10)
Thus, for the case 0/1 2 cnpNp IIIIII
the anti-
Hermitian part of the integral over x in (7) with the
multiplier
21 IIp
e
can be obtained by substituting the
anti-Hermitian part )/1( 2 cnpNpi IIIIII
of
the second of the formulas (10), times i2 , which
corresponds to the Landau rule for passing the pole,
94 ISSN 1562-6016. ВАНТ. 2016. №6(106)
instead of the Cauchy integral in expression (7). In this
way
partHermitian
npNpx
pxxJ
edxe
c
n
xp
/1
12
2
22
0
1 2
IIIIII
II
II
2( / ) 2 2/ 1 .cN p n
n cie J N p n p
II II
II II II
(11)
In the expression (11) it is convenient to go to the
arguments introduced by Robinson in the weakly
relativistic approximation: 2/IIpx , )/1( cnz ,
/2μNa 2
II . Then (11) is converted to
2 2 2 2. 2 [ 2 ( 2 ) / (2 )] .ax z
nHerm part ie J z ax x z ax
(12)
From (12) the imaginary part of 11 is
.
,0
,
)(2Im
22
2
2
2
0
11
az
azJedx
K
e
n n
xa
x
x
z
n
p
(13)
Here were used )]2/()2(2[2 22 xazxxaz ,
)N11( 2
II a and the integration limits were obtained
from the condition that the pole must appear inside the
integral of expression (7), that is equivalent to
01//2/221 zxax . Consequently they are
)N1/())]2/(()/1([ 22
II zzazax . Thus, for
the cyclotron harmonic number n and for the imaginary
(anti-Hermitian) part of the tensor component 11 it was
obtained the expression that is an alternative to (1) and
in which the summation of a series in the index l is the
reduced to the numerical calculation of the one-
dimensional integral (13). After the change of variables:
tx in accordance with relation )/1( zaxt ,
leading to the symmetry of the new limits of integration
)2/(2 zzat about zero and one more
changing: ut in accordance with )2/(2 zzaut ,
leading to the normalization of integration limits to
1u , then from (13) it follows
2
0 2 2 (1 / )
2
1
11 2 2
1
2 ( )
Im
,
0, ,
z
p a z
n
a Ku
n
e
n Ke
K
z a
due J
z a
(14)
where for brevity were used the next designations
/)1(2 2uK and )2/(2 zzaK .
The integral (14) in the interval [-1, 1] is not
singular, and therefore for arbitrary values of the
parameter can be calculated without any problems.
The real (Hermitian) part of the component
11 can also
be calculated from an imaginary part (14) along with
one of Kramers-Kronig formulas, linking Hermitian and
anti-Hermitian parts of an integral of Cauchy type,
defined on the real axis [1]
.
),,(Im1
),,(Re 11
1111
zt
dtta
Pza
(15)
This formula corresponds to the Landau rule for passing
the pole in the expression (11). For passing to the
standard coordinates in (14) and (15) it is necessary to
make a change of variable:
nzaz 2 . Thus, during
calculation of plasma dielectric tensor element
11 for
given cyclotron harmonic number n an evaluation of
series converging slowly in the index l can be reduced
to one-dimensional numerical calculation of integral
(14) and subsequent calculation of the principal value of
an integral of Cauchy type (15).
In a similar way it can be demonstrated that
calculating the remaining components of relativistic
plasma dielectric tensor (1) can also be reduced to
computing the same kind one-dimensional integrals in
the same interval [-1,1]. A form of this tensor which is
alternative one to (1) and suitable for accurate numerical
applications for arbitrary is presented below:
12
2 (1 / ) 2 2
11
1
Im ,z a z a Ku
n
n
n
e Ke due J
(16)
nn
Kuazaz
n
JJeduKee
n
i 2
1
1
)/1(2
12Im
,
222
1
1
)/1(2
22
1
Im nn
Kuazaz
n
JJeduKee
,
22
1
1
)/1(22
13Im n
Kuazaz
n
JeudueKe
n
,
nn
Kuazaz
n
JJeudueKei 2
1
1
)/1(22
23
1
Im
,
22
1
1
2)/1(23
33 2Im n
Kuazaz
n
JeduueKe ,
2)(2
2
0
K
ep
, /)1(2 2uK ,
,
),,(Im1
),,(Re
zt
dtta
Pza
ij
ijij
where
ij is designation of Kronecker symbol. At last
transition to the standard coordinates usual in the non-
relativistic case can be made in (16) throw the change
nzaz 2 .
CONCLUSIONS
1. It was shown that a fully relativistic dielectric tensor
of plasma in magnetic field can be presented in the
alternative to (1) the one dimensional integral form (16),
ISSN 1562-6016. ВАНТ. 2016. №6(106) 95
suitable for numerical applications for an arbitrary value
of the parameter .
2. The form of the relativistic tensor (16) is rather of
interest from the point of view applications to studying
propagation and absorption of Bernstein electron
cyclotron waves in the laboratory thermonuclear
plasmas and arbitrary electron cyclotron waves in the
hot astrophysical plasmas.
3. On the same way can be obtained the form of the
exact fully relativistic dielectric tensor of magnetized
plasma for ion plasma components suitable for
applications with arbitrary wave numbers. This tensor
can be used for the studying propagation and absorption
of the arbitrary ion cyclotron waves in extremely hot
astrophysical plasmas, for example in such ones as in
the conditions of supernova explosions of stars.
REFERENCES
1. I. Fidone, G. Granata, and R.L. Meyer // Phys. Fluids.
1982, v. 25, p. 2249.
2. F. Castejon, S.S. Pavlov, and D.G. Swanson // Phys.
Plasmas. 2002, v. 9, p. 111.
3. B.A. Trubnikov. Plasma Physics and the Problem of
Controlled Thermonuclear Reactions / Edited by
M.A. Leontovich. Oxford: “Pergamon”. 1959, v. III,
p. 122.
4. F. Castejon, S.S. Pavlov // Physics of Plasmas. 2006,
v. 13, p. 072105.
5. Francesco Volpe // Physics of Plasmas. 2007, v. 14,
p. 122105.
6. M. Brambilla. Kinetic theory of plasma waves,
homogeneous plasmas. Oxford: “Clarendon”, 1998.
Article received 28.09.2016
ТОЧНОЕ ВЫЧИСЛЕНИЕ РЕЛЯТИВИСТСКОГО ТЕНЗОРА ДИЭЛЕКТРИЧЕСКОЙ
ПРОНИЦАЕМОСТИ ПЛАЗМЫ В МАГНИТНОМ ПОЛЕ ДЛЯ ПРОИЗВОЛЬНЫХ ВОЛНОВЫХ
ЧИСЕЛ
C.C. Павлов
Даётся новая интегральная форма полностью релятивистского тензора диэлектрической проницаемости
плазмы в магнитном поле. Она годится для численных приложений при произвольных волновых числах,
поскольку все интегралы в ней являются одномерными. Эта форма представляет интерес с точки зрения
приложений для изучения бернштейновских электронных циклотронных волн в лабораторной термоядерной
плазме и произвольных электронных и ионных циклотронных волн в горячей астрофизической плазме.
ТОЧНЕ ОБЧИСЛЕННЯ РЕЛЯТИВІСТСЬКОГО ТЕНЗОРА ДІЕЛЕКТРИЧНОЇ ПРОНИКНОСТІ
ПЛАЗМИ В МАГНІТНОМУ ПОЛІ ДЛЯ ДОВІЛЬНИХ ХВИЛЬОВИХ ЧИСЕЛ
C.C. Павлов
Дається нова інтегральна форма повністю релятивістського тензора діелектричної проникності плазми в
магнітному полі. Вона годиться для численних додатків при довільних хвильових числах, оскільки всі
інтеграли в ній є одновимірними. Ця форма являє інтерес з точки зору додатків для вивчення
бернштейнівських електронних циклотронних хвиль в лабораторній термоядерної плазмі і довільних
електронних та іонних циклотронних хвиль в гарячій астрофізичної плазмі.
|