Adiabatic ion-sound waves
A low frequency potential electric field in plasma is considered. Solutions in the form of the ion-sound waves in
 supposition, that the considered process is an adiabatic one for rapid particles (electrons) and isothermal one for slow
 particles (ions), are obtained. Coming from the...
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| Date: | 2009 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2009
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| Cite this: | Adiabatic ion-sound waves / A.A. Stupka // Вопросы атомной науки и техники. — 2009. — № 1. — С. 83-85. — Бібліогр.: 5 назв. — англ. |
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| citation_txt | Adiabatic ion-sound waves / A.A. Stupka // Вопросы атомной науки и техники. — 2009. — № 1. — С. 83-85. — Бібліогр.: 5 назв. — англ. |
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| description | A low frequency potential electric field in plasma is considered. Solutions in the form of the ion-sound waves in
supposition, that the considered process is an adiabatic one for rapid particles (electrons) and isothermal one for slow
particles (ions), are obtained. Coming from the Boltsmann distribution adiabatic Debye radius is obtained and similarly
adiabatic Jeans wave-length for the gravitating system is obtained. Due to kinetic description dispersion equation for
adiabatic sound-waves is defined more precisely.
Розглянуто низькочастотне потенційне електричне поле у плазмі. Одержані розв’язки у вигляді іонно-
звукових хвиль за припущенням, що це адіабатичний процес для швидких частинок (електронів) й ізотермічний
для повільних (іонів). Виходячи з розподілу Больцмана отримано адіабатичний радіус Дебая і, аналогічно йому,
адіабатичну довжину хвилі Джинса для гравітуючої системи. Завдяки кінетичному опису уточнено дисперсійне
рівняння для адіабатичних іонно-звукових хвиль.
Рассмотрено низкочастотное потенциальное электрическое поле в плазме. Получены решения в виде ионно-
звуковых волн в предположении, что это адиабатический процесс для быстрых частиц (электронов) и
изотермический для медленных (ионов). Исходя из распределения Больцмана получен адиабатический радиус
Дебая и, аналогично ему, адиабатическая длина волны Джинса для гравитирующей системы. Благодаря
кинетическому описанию уточнено дисперсионное уравнение для адиабатических ионно-звуковых волн.
|
| first_indexed | 2025-12-07T18:25:19Z |
| format | Article |
| fulltext |
ADIABATIC ION-SOUND WAVES
A.A. Stupka
Dnipropetrovsk National University, Dnipropetrovsk, Ukraine
A low frequency potential electric field in plasma is considered. Solutions in the form of the ion-sound waves in
supposition, that the considered process is an adiabatic one for rapid particles (electrons) and isothermal one for slow
particles (ions), are obtained. Coming from the Boltsmann distribution adiabatic Debye radius is obtained and similarly
adiabatic Jeans wave-length for the gravitating system is obtained. Due to kinetic description dispersion equation for
adiabatic sound-waves is defined more precisely.
PACS: 52.35.Fp, Qz
1. ADIABATIC LINEARIZATION
OF THE BOLTSMANN DISTRIBUTION
Considering subsystems as ideal gases we obtain for
characteristic motion velocities of ions and electrons
e eu T /
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2009. № 1. 83
Series: Plasma Physics (15), p. 83-85.
m
e
e
such a relation toward the rate of the
examined movement (ionic sound)
iu u u . (1)
That is why electrons in a sound-wave have relaxed and
behave as equilibrium gas in the external field with the
Boltsmann distribution [1]
0 exp( )en n e Tϕ= / , (2)
where ϕ is scalar potential of the electric field of charged
particles of plasma with the density of charges of
every sign. As is generally known [1], in a linear theory it
is possible to separate the potential and vortical field, and
the last has high-frequency optical branches only, that is
why here it will not be studied.
0n
According to (1), in the first order one can
consider ions "cold" (and ignore influence of thermal
motion of ions on spreading waves), i.e. proceed from
motion equation
d e M
dt
ϕ= − ∇ / .iv (3)
Also we use a linearized continuity equation
0 0i
i
n
n div
t
δ∂
+ =
∂
v (4)
and the Poisson equation
4 ( )i ee n nδϕ π δ δ= − − . (5)
If one supposes electronic components to be isothermal,
then linearization of equation (2) gives
0en n e Teδ δϕ= / (6)
and we will get for the wave solution of the system (3)-(6)
the known [1] sound-waves eT
ST Mu = .
As is generally known [2], the ordinary
(isothermal) theory of ion-sound oscillations gives the
incorrect value of spread velocity. Using of adiabatic state
equation in a theory with hydrodynamic description of
electronic component [3] corrects the situation with the
value of spread velocity
e
S
T
u
M
γ
= , (7)
where γ is the adiabatic index, T is the temperature of
electronic gas,
e
M is the ion mass.
We will show how we obtained this result
proceeding from equilibrium and kinetic approaches to
description of electronic components of plasma. The state
equation in the case of adiabatic motion of electrons, as is
generally known, has the form
1TV constγ − = . (8)
Linearization of (8) gives
( 1) e
e
e
TT
n enδ γ δ= − . (9)
Linearization of equation (2) with the use of bond (9)
brings us over to expression
0
02
0
1 ( 1)e e
e
ee
n e T
n n
n TT
e
ϕ δϕ
δ γ
⎛ ⎞
+ − =⎜ ⎟
⎝ ⎠
. (10)
And also by the virial theorem average potential energy of
charge in the electric field (which has a coulomb form for
the main approximation of retired potentials) is expressed
in terms of kinetic energy in such a way [4]
2U K= − .
Obviously [1], transversal part of both velocity and
electromagnetic field must be cast aside, as not relating to
the longitudinal ion-sound waves in the main
approximation. That is why
2K T= /
and U T= − . And average potential energy of electrons is
U eϕ= − , (11)
therefore from (10) we have
0
e
e
e
n n e
T
δϕ
δ
γ
= . (12)
This equation differs from (6) only by the change of
temperature in γ times. It allows to express deviation of
scalar potential in terms of deviation of density. For the
long-wave sound oscillations in the Poisson equation it is
possible to neglect laplasian, that gives equality of density
deviations
in neδ δ= . (13)
Then we have instead of (3)
0
e
e
Td n
dt n M
γ
δ= − ∇ . (14) iv
that jointly with (4) after the calculations similar to the
isothermal case being conducted gives velocity of
adiabatic ion-sound waves, which coincides with (7).
In addition, by a standard method [1] in supposition of
mobility only of electronic components ( 0inδ = ) after
substitution of relation (12) in the Poisson equation (5)
the electronic radius of (adiabatic) screening turns out to
be
24
e
DSe
T
r
e
γ
π
= . (15)
Absolutely similarly it is possible to calculate a Jeans
wave-length at the system of gravitating particles of mass
and temperature T . Note that a result corresponds to
replacement in (15) , where is gravitation
constant, i.e.
m
2e Gm→
84
2 G
2 .
4DS
T
Gm
γλ
π
= (16)
2. KINETIC DESCRIPTION
An alternative to hydrodynamic consideration is
kinetic one. Thus we must solve the Vlasov equation
0
0 0a
a a
f f e E
t p
υ
∂ ∂
+ ∇ + =
∂ ∂
rr
r
0af . (17)
with the potential field
E ϕ= −∇
r
(18)
for the one-particle distribution functions of both
electronic and ionic components of plasma. It is easily to
verify, that the Maxwell-Boltsmann distribution is the
equilibrium solution [1]
0
( , ) ( )expa
F T V xf
T
ε⎛ −
= ⎜
⎝ ⎠
% ⎞
⎟ . (19)
We will consider small deviation from an equilibrium
using the Chepmen-Enskog method [5].
For the high-frequency field the temperature dependence
is not important (it is possible even to deal with «cold»
particles), a temperature does not have time to change -
isothermal state. Then small deviation from the
Boltsmann distribution will be
( , ) ( )expa
F T V x Pf V
T T
εδ δ
⎛ ⎞− ⎛ ⎞= −⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
%
% . (20)
Here VV
N
=% is specific volume per a particle. At the
constant amount of particles in the system we have for
deviation of density
2
Nn V
V
δ δ= − ( Nn
V
= ), (21)
i.e. after integration by velocities we have for ionic
component
3
i
N V
if d n n
V N
δδ υ ⎛ ⎞= − =⎜ ⎟
⎝ ⎠∫ δ . (22)
However, if the field frequencies are considerably less
than characteristic one for the subsystem of this sort of
particles, particles will have time to tune their
temperatures adiabatically. This for small deviation from
the Boltsmann distribution will give
2
( , ) ( )exp
( , ) ( )
a
F T V xf
T
F T V x S PT T V
T TT
εδ
ε δ δ δ
⎛ ⎞−
= ×⎜ ⎟
⎝ ⎠
⎛ ⎞−
× − − −⎜ ⎟
⎝ ⎠
%
%
%
. (23)
We remember that F TS E+ = . Integrating by velocity,
we have
3
2a
e Pf d n T V
TT
ϕδ υ δ δ⎛= − −⎜
⎝ ⎠∫ % ⎞⎟ . (24)
Equation of state (8) gives relationship between density
and temperature (9). I.e.
3 1( 1)a
ef d n n V n
T n
ϕδ υ γ δ−⎛ ⎞= − − +⎜ ⎟
⎝ ⎠∫ %δ
e
, (25)
that is why from (11) and (25) we have for electrons
3
ef d nδ υ γδ=∫ . (26)
Here we used potentiality (longitudinal) of motion, that
can be interpreted as presence of only one degree of
freedom 1l = , i.e. on a standard formula for ideal gas
2 3P
V
c l
c l
γ +
= = = . (27)
On the other hand, carrying out standard
linearization of the Vlasov equation we have small
deviation of distribution function from the equilibrium
form due to the change of the field
0 0a a
a a
f ff e E
t
δ
υ δ
∂ ∂
p
+ ∇ + =
∂
rr
r
∂
,
(28)
whence after the Fourier transformations we will get
0a
k
aa k
fE
pf ie
k
ω
ωδ
ω υ
∂
∂
=
−
r
r
r
r
r r .
(29)
For frequencies of ionic sound, small for electronic and
large for ionic component of plasma, we will put in the
Poisson equation (5) expressions (22) for ionic and (26)
for electronic densities accordingly
3 14 ( )i ee f d f d 3δϕ π δ υ δ υ
γ
= − −∫ ∫ . (30)
We will define dielectric function ε by standard relation
appearance
0div Eε =
r
. (31)
We will write down dielectric function of plasma for
frequencies of ionic sound proceeding from (30)
( )
0 0
3 2 2
2
4 1, 1 (
i e
i e )
f fk k
p pk d e e
k k k
πε ω υ
γω υ ω
∂ ∂
∂ ∂
= + +
− −∫
r r
r rr
r rr rυ
. (32)
The wave solutions appear at 0ε = , then neglecting
decreasing, we have
2 2 2 2
2 2 2 2
3
1 1e i Ti
Se
k u
k u ω ω
⎛ ⎞Ω Ω
0,+ − + =⎜ ⎟
⎝ ⎠
(33)
where 2 e
Se
e
Tu
m
γ
= , that in main approximation gives
2 2 2
3 3
1 .i i e
Se
eSe e
T T
ku k
T Mk u
γ
ω
⎛ ⎞Ω +
≈ + ≈⎜ ⎟
+Ω ⎝ ⎠
iT (34)
This result coincides with the formal result of calculation
of velocity in a hydrodynamic theory [3], because 3γ = ,
however we did not suppose the adiabatic state equation
for heavy particles (ions) explained by nothing.
85
In addition, ignoring low temperature ionic
component, we can consider the Debye screening for
0ω → . From (32) we have
( )
0
3 2
2
4 1 1, 0 1 ( ) 1
e
e 2 2 .
DSe
fk
pk d e
k k
πε ω υ
γ υ
∂
∂
= = + = +
−∫
r
rr
r r k r
(35)
Whence the radius of screening of the potential field of
external charge coincides with (15).
3. CONCLUSIONS
Processes in plasma (low-frequency (adiabatic) ones
for rapid particles - electrons and high-frequency ones for
slow – ions) are studied . The solutions in the form of ion-
sound waves in supposition, that it is an adiabatic process
for rapid particles (electrons) and isothermal one for slow
particles (ions) are obtained. Proceeding from the
Boltsmann distribution the solutions in the form of
adiabatic ion-sound waves, the adiabatic radius of
screening of the electric field and, similarly, adiabatic
Jeans wave-length for the gravitating system are obtained.
Due to kinetic description dispersion equation for
adiabatic ion-sound waves is defined more precisely.
I am deeply grateful to Dr. Churilova M.S. for her
precious and careful linguistic consultations.
REFERENCES
1. Electrodynamika plasmy. /Ed. by A.I. Akhiezer.
Moscow: “Nauka”, 1974, (in Russian).
2. A.Y. Wong, N.D’ Angelo, R.W. Motley // Phys. Rev.
Lett. 1962, v. 9, p. 415.
3. B.A. Trubnikov. Theoriya plasmy. Moscow:
“Nauka”, 1986, (in Russian).
4. L.D. Landau, Е.М. Lifshyts. Teoreticheskaya fizika.
Moscow: “Nauka”, 1976, v. 5 (in Russian).
5. Е.М. Lifshyts, L.P. Pitaevskiy. Fizicheskaya kinetika.
Moscow: “Nauka”, 1979 (in Russian).
Article received 22.09.08
АДИАБАТИЧЕСКИЕ ИОННО-ЗВУКОВЫЕ ВОЛНЫ
А.А. Ступка
Рассмотрено низкочастотное потенциальное электрическое поле в плазме. Получены решения в виде ионно-
звуковых волн в предположении, что это адиабатический процесс для быстрых частиц (электронов) и
изотермический для медленных (ионов). Исходя из распределения Больцмана получен адиабатический радиус
Дебая и, аналогично ему, адиабатическая длина волны Джинса для гравитирующей системы. Благодаря
кинетическому описанию уточнено дисперсионное уравнение для адиабатических ионно-звуковых волн.
АДІАБАТИЧНІ ІОННО-ЗВУКОВІ ХВИЛІ
А.А. Ступка
Розглянуто низькочастотне потенційне електричне поле у плазмі. Одержані розв’язки у вигляді іонно-
звукових хвиль за припущенням, що це адіабатичний процес для швидких частинок (електронів) й ізотермічний
для повільних (іонів). Виходячи з розподілу Больцмана отримано адіабатичний радіус Дебая і, аналогічно йому,
адіабатичну довжину хвилі Джинса для гравітуючої системи. Завдяки кінетичному опису уточнено дисперсійне
рівняння для адіабатичних іонно-звукових хвиль.
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-88231 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:25:19Z |
| publishDate | 2009 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Stupka, A.A. 2015-11-10T21:08:46Z 2015-11-10T21:08:46Z 2009 Adiabatic ion-sound waves / A.A. Stupka // Вопросы атомной науки и техники. — 2009. — № 1. — С. 83-85. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.35.Fp, Qz https://nasplib.isofts.kiev.ua/handle/123456789/88231 A low frequency potential electric field in plasma is considered. Solutions in the form of the ion-sound waves in
 supposition, that the considered process is an adiabatic one for rapid particles (electrons) and isothermal one for slow
 particles (ions), are obtained. Coming from the Boltsmann distribution adiabatic Debye radius is obtained and similarly
 adiabatic Jeans wave-length for the gravitating system is obtained. Due to kinetic description dispersion equation for
 adiabatic sound-waves is defined more precisely. Розглянуто низькочастотне потенційне електричне поле у плазмі. Одержані розв’язки у вигляді іонно-
 звукових хвиль за припущенням, що це адіабатичний процес для швидких частинок (електронів) й ізотермічний
 для повільних (іонів). Виходячи з розподілу Больцмана отримано адіабатичний радіус Дебая і, аналогічно йому,
 адіабатичну довжину хвилі Джинса для гравітуючої системи. Завдяки кінетичному опису уточнено дисперсійне
 рівняння для адіабатичних іонно-звукових хвиль. Рассмотрено низкочастотное потенциальное электрическое поле в плазме. Получены решения в виде ионно-
 звуковых волн в предположении, что это адиабатический процесс для быстрых частиц (электронов) и
 изотермический для медленных (ионов). Исходя из распределения Больцмана получен адиабатический радиус
 Дебая и, аналогично ему, адиабатическая длина волны Джинса для гравитирующей системы. Благодаря
 кинетическому описанию уточнено дисперсионное уравнение для адиабатических ионно-звуковых волн. I am deeply grateful to Dr. Churilova M.S. for her precious and careful linguistic consultations. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы Adiabatic ion-sound waves Адіабатичні іонно-звукові хвилі Адиабатические ионно-звуковые волны Article published earlier |
| spellingShingle | Adiabatic ion-sound waves Stupka, A.A. Фундаментальная физика плазмы |
| title | Adiabatic ion-sound waves |
| title_alt | Адіабатичні іонно-звукові хвилі Адиабатические ионно-звуковые волны |
| title_full | Adiabatic ion-sound waves |
| title_fullStr | Adiabatic ion-sound waves |
| title_full_unstemmed | Adiabatic ion-sound waves |
| title_short | Adiabatic ion-sound waves |
| title_sort | adiabatic ion-sound waves |
| topic | Фундаментальная физика плазмы |
| topic_facet | Фундаментальная физика плазмы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/88231 |
| work_keys_str_mv | AT stupkaaa adiabaticionsoundwaves AT stupkaaa adíabatičnííonnozvukovíhvilí AT stupkaaa adiabatičeskieionnozvukovyevolny |