Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources
The formation of a non-equilibrium distribution function (DF) of particles with the power-law interaction potentials is studied. Consideration is based on the non-linear kinetic equation of a Landau - Fokker - Planck type in the presence of particle (energy) sources. We compared our results with exp...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2005 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
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| Цитувати: | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources / I.F. Potapenko, V.I. Karas // Вопросы атомной науки и техники. — 2005. — № 1. — С. 60-62. — Бібліогр.: 4 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859719550062297088 |
|---|---|
| author | Potapenko, I.F. Karas, V.I. |
| author_facet | Potapenko, I.F. Karas, V.I. |
| citation_txt | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources / I.F. Potapenko, V.I. Karas // Вопросы атомной науки и техники. — 2005. — № 1. — С. 60-62. — Бібліогр.: 4 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The formation of a non-equilibrium distribution function (DF) of particles with the power-law interaction potentials is studied. Consideration is based on the non-linear kinetic equation of a Landau - Fokker - Planck type in the presence of particle (energy) sources. We compared our results with experimental data.
Досліджено формування нерівноважних функцій розподілу частинок зі степеневим законом взаємодії між ними. Розгляд основано на одновимірному нелінійному кінетичному рівнянні типу Ландау-Фокера-Планка за наявністю джерел частинок (енергії). Наведено порівняння з експериментом.
Исследовано формирование неравновесных функций распределения частиц со степенным законом взаимодействия между ними. Рассмотрение основано на нелинейном кинетическом уравнении типа Ландау- Фоккера-Планка при наличии источников частиц (энергии). Проведено сравнение с экспериментом.
|
| first_indexed | 2025-12-01T09:08:37Z |
| format | Article |
| fulltext |
QUASI STEADY-STATE DISTRIBUTIONS FOR COLLISIONAL PLASMAS
IN THE PRESENCE OF ENERGY AND PARTICLE SOURCES
I.F. Potapenko1, V.I. Karas`2
1Keldysh Institute of Applied Mathematics , RAS, Moscow, Russian Federation,
E-mail: irina@KELDYSH.ru;
2NSC “Kharkov Institute of Physics & Technology”, NAS of the Ukraine, Kharkov, Ukraine,
E-mail: karas@ kipt.kharkov.ua
The formation of a non-equilibrium distribution function (DF) of particles with the power-law interaction
potentials is studied. Consideration is based on the non-linear kinetic equation of a Landau - Fokker - Planck type in the
presence of particle (energy) sources. We compared our results with experimental data.
PACS: 52.25.Dg, 52.65.Ff, 52.50. Gj
1. INTRODUCTION
Non-equilibrium states of various physical systems
and the analysis of the non-equilibrium (NE) distributions
attract increasing interest particularly in connection with
the development and wide use of high-power particle and
energy sources. The energy (particle) source and sink can
be provided by ion beams, high-power laser radiation,
emission currents, fluxes of charged particles produced in
fusion or fission reactions. A comparison of the
characteristic times of ionization and relaxation shows
that, in the case at hand, the steady-state electron DF
should be determined mainly by electron - electron
collisions [1,2]. The analytical consideration of the NE
DF formation in the case of nonlinear equation and non-
stationary, non-localized, mismatched on intensity
sources and sinks is very problematic, thus numerical
treatment must be used. The original completely
conservative difference schemes having a very high
accuracy and allowing to provide the numerical
calculation during hundreds collisional times without
error accumulation, except for machine errors, are used.
Hence, it can be obtained from the condition for the
Boltzmann collision integral (CI) (for a semiconductor
plasma, the Landau or Fokker - Planck (LFP) collision
integral) to be zero. It follows from the above analysis [1]
that, for a semiconductor plasma in the energy range E –
EF > EF, (where EF is the Fermi energy) a power-law DF
with a nonzero flux of energy or particles in momentum
space (MS) can be established. This DF is formed both
due to collisions with electrons whose energy is in the
range E – EF > EF and background (equilibrium) electrons.
A specific feature of systems of particles interacting via
the Coulomb potential is that the scattering cross section
increases without bound as the momentum transferred
tends to zero. For gaseous and semiconductor plasmas
with a large Coulomb logarithm ln Λ = 10 - 15), one can
restrict himself to the expansion of the integrand in the
collision integral in small momenta transferred (a
diffusion approximation) and to represent the CI in the
LFP form [2-4], which are model representations of the
Boltzmann CI.
2. BASIC CONTENT
Below, we will consider potentials (U ∼ r -β, with
1 4β≤ ≤ , where r is the distance between the interac-
ting particles), for which a local NE particle DF can
form. Note that the dynamics of particles interacting via
the Coulomb repulsion potential ( β = 1) can be
considered using a kinetic equations in either the LFP
form (see [2,4]). The DF f(v, t) is bounded at v = 0 and
quite rapidly decreases as V → ∞ at t → ∞ . Below,
we use dimensionless variables: the velocity in units of
the thermal velocity VT and time in units of the electron -
electron relaxation time τ ee , which is
τ ee=
v
T3 m2
4πn pe4 ln Λ
in the case of Coulomb
interaction. The DF f(v, t) is normalized so that the par-
ticle density and the total energy and the constant Γ are
equal to unity. We consider the formation of a stea-dy-
state NE solution in the presence of particle or energy
flows in velocity space using the versions of the LFP
collision integrals. In this case, the right-hand side of the
kinetic equation is added with the terms accounting for
the presence of a particle (energy) source (sink):
∂ f
∂ t
= I FP , L[ f , f ]S−S− , where
S±~ I±exp {−α1 v−v±
2} , or
S±~ I±δ v−v± /v
2 or
S±~ I±
δ v−v±
v 2 f v− , t . (1)
The flow direction is determined by the positions of v-
and v+. Either a Maxwellian DF or a delta - function was
used as an initial DF. In [4], the formation of a NE DF
was numerically simulated for the kinetic equation with
either the LFP collision integral in the presence of energy
and particle flows in MS that were sustained by a source
and a sink. First, solutions were obtained for the case
where the positions of a source and a sink in MS were
matched with the direction of a flow sustained by
collisions. Note that analytic consideration of equations
for the case of a localized source and sink gives a correct
60 Problems of Atomic Science and Technology. 2005. № 1. Series: Plasma Physics (10). P. 60-62
mailto:irina@KELDYSH.ru
flow direction, a namely, from high to low velocities [1].
It was shown in [4] that, within the interval between the
source and sink, a stationary (S) NE DF (of the
Kolmogorov kind) of particles is established with time.
This DF corresponds to the presence of an energy flow in
MS, whereas beyond this interval, the DF is
thermodynamically equilibrium. As was noted above, the
positions of the source and sink and the direction of flow
in MS should be matched with one another. To make sure
once again that this requirement is important, we
performed calculations with the inter-changed positions
of the source and sink in energy space. It turned out that
variations in the flow intensity by several orders of
magnitude did not influence the equilibrium particle
distribution when the source and sink positions were not
matched with the flow direction.
It is shown the DF for different flow intensities. It is
found that, for low intensities of the source a universal
SNEDF is formed in the velocity range v ≈ v+. This is
due to (i) a decrease in the cross section for Coulomb
scattering with increasing velocity (~v-3) and (ii) the al-
ways present flow of energy and particles (due to
Coulomb diffusion) toward the region of the main
("back-ground") equilibrium DF. Consequently, as the
source (sink) intensity increases, a universal NE particle
distribution is formed that occupies a progressively larger
space between the source and sink. Such behavior is
related to a decrease in the fraction of the flow transfer-
red to the background plasma. It is worth noting that the
increase in the intensity is accompanied by an increase in
the magnitude of the NE DF in proportion to the flux
magnitude [1]. Let us examine the form of the DF for
power-law interaction potentials with the exponents 1
β≤ ≤ 4. Note that β = 1 corresponds to the Coulomb
interaction potential, β = 2 corresponds to dipole
interaction, and β = 4 describes the interaction of so-
cal-led Maxwellian molecules. It is shown that NE DFs
for the case of a steady-state energy flow with an
intensity of I = 0.01 and β = 1, 2, and 4, for all these β
values, the power indexes of the formed NE power-law
DF are close to one another, which agrees with the
analytical results [1]. The magnitude of the NE part of
distribution function decreases with increasing β . These
results are in qualitative agreement with the above
analytic predictions. It is well-known that a fast ion,
which velocity is commensurable or exceeds the speed of
an electron in the Bohr’s orbit, interacts mainly with
binding and free electrons. In doing so, the ion produces
the atom excitation and ionization by the direct electron
impact and from the excitation of wake plasma waves by
ion, as well. We evaluate the source and sink intensities,
their positioning in the MS, the characteristic times of
Coulomb relaxation and those of the sources and sink
action using parameters corresponded to the conditions of
the experiment [2]. Note that, the free path of ion H with
energy 1.25 MeV in GaAs is l = 3 10-6 m, the ion speed
corresponding to the energy of 1 MeV is equal to vH = 7.5
106 m/s. From this, we compute the time period while
which the proton loses the main part of its energy – ttr = 4
10-13 s. For a given test specimen of gallium arsenide the
electron density is ne = 5 1024 m-3 and the characteristic
velocity in such semiconductor plasma is vth = 6 105 m/s.
Then the corresponding characteristic electron - electron
relaxation time is in the order of eeτ = 3 10-14 s. We see
that sufficient quantity of collisions takes place in a time
of the electron free path and under this condition the non-
equilibrium distribution function is developed due to the
flux presence in the velocity space. Let us estimate the
intensity of the sources arisen due to the electron
ionization by the direct electron impact and owing to the
plasma wave excitation. Along the full free path ion
creates about 104 electrons, in this case the characteristic
volume, where ionization will be produced, equals 3 10-21
m3. The density of new electrons being produced by ion
in a time unit is about
I+ = 3.3 1037 m-3 s-1. In the CI the normalization units are ne
and eeτ , then the normalized intensity is about 0.01 –
0.1 and normalized time of the source action is of the
order of 10. Two different source positions in the velocity
space have correlation with the plasmon ionization at the
characteristic speed v+1 = 3.5 and with the ionization by
the electron impact at v+2 = 7. In the considering
experimental situation, the principal losses are the ion -
electron emission from the film surface. Note that for the
case under discussion, the sink is distributed over the MS
in such a way that it is equal to zero down to the energy
5.65 eV , which corresponds to the work function. The
sink intensity is assumed to be proportional to the
developing DF. As have been mentioned above, the DF in
the presence of rather intensive sources and sinks cannot
be a thermodynamically equilibrium one and has to be
find out from the solution of the nonlinear kinetic
equation. Thus, we examine the evolution of the DF
formation solving the equation for the above parameters.
We consider the source intensities’ range 0.01- 0.1, which
variation can be connected both with the different power
of the bombarding ion beams and with the difference in
electron density between samples. The sink intensity 0.5
is appropriate in the case of emerging of half of electrons
having sufficient energy v2 > 4 from the sample. It is
shown that a DF formation for the sources 1 and 2 with
the intensities 0.05. Both sources and losses are acting till
the time t = 100. The quasi stationary non equilibrium
distribution is formed in a time about 10 - 15 and has a
character which is substantially distinguished from the
power-like functional dependence obtained in [1]. The
explanation can be attributed to the fact that sources and
sinks being distributed over the MS do not provide the
constant energy flux. In accordance with the results of
previous section, the NE DF establishing is independent
of the intensity value. The sources are turned off at t=12.
The total energy in the system is changed approximately
to 20%, but the particle density varies from 2 to 5
depending on the source and sink intensities. The source
(sink) intensity increasing leads to the corresponding
function increasing within the inertial interval and to the
function decreasing in the cold region. If the sink acts
perpetually and the sources act during a while about 15 -
30 then further developed function takes the Maxwellian-
type distribution with the temperature so far lower as the
source intensity was higher. Let us compute the emission
61
current dependence on the retarding potential U, which is
need to analyze in the experiment the energetic spectrum
of the electron emission. For the experimental conditions
have been studying, the electron energetic spectrum of
ion-electron emission would be quasi stationary for the
ion beam currents exceeded 1 - 10mA. In this case, the
emission current is certainly defined by the different ion
tracks dispersed in a space, but the distribution function
of emission electrons is practically quasi stationary that
leads to the quasi stationary emission current. From the
comparison of variants, we see how the difference in the
emission electron current dependence is strong for the
different endurance of the source action. In the
experimental conditions [2], the current of the ion beam
does not exceed 10 µ A. In this case, the emission
current will fully reflect the non-stationary character of
the sources. On each ion track, the distribution function
has time enough to go through all stages of its formation.
That is why, the emission current dependence on the
retarding potential observed in the experiment is a
superposition of the currents existing in different time
stages. Obviously, the electron energetic spectrum differs
from one that formed under the stationary source action.
From the emission current dependence on the retarding
potential it can be seen, that the emission current is non-
stationary because of the substantial non-stationary
sources. The comparison of the simulation and
experimental results shows that the taking into account of
the non-stationary source character may be the
determining factor.
3. SUMMARY
NE quasi steady-state local DF exist inside the
momentum interval between the energy (particle) source
and the bulk (or sink) of the particle distribution and has
the form of gradually decreasing functions. Numerical
simulation is in good agreement with the analytical results
and with the results obtained in experiments on irradiation
of a thin GaAs film by a fast ion beam.
We are grateful to STCU (project # 1862) for partial
financial support of this work.
REFERENCES
1. V.I.Karas`, S.S.Moiseev, and V.E.Novikov.// JETP
Lett. (21). 1975, p.525-528.
2. S.I. Kononenko, V.M. Balebanov, V.P. Zhurenko,
O.V.Kalantar`yan, V.I. Karas`, V.T. Kolesnik, V.I.
Mu-ratov, V.E. Novikov, I.F. Potapenko, R.Z.
Sagdeev.// Plasma Phys. Rep. (30). 2004, p.671-686.
3. I.F. Potapenko, A.V. Bobylev, C.A. de Azevedo, and
A.S. de Assis.// Phys. Rev. E.(56). 1997, p. 7159.
4. V.I. Karas`, I.F. Potapenko.// Plasma Phys. Rep.
(28). 2002, p. 837-846.
КВАЗИСТАЦИОНАРНЫЕ РАСПРЕДЕЛЕНИЯ ДЛЯ СТОЛКНОВИТЕЛЬНОЙ ПЛАЗМЫ ПРИ
НАЛИЧИИ ИСТОЧНИКОВ ЭНЕРГИИ И ЧАСТИЦ
И.Ф. Потапенко, В.И. Карась
Исследовано формирование неравновесных функций распределения частиц со степенным законом
взаимодействия между ними. Рассмотрение основано на нелинейном кинетическом уравнении типа Ландау-
Фоккера-Планка при наличии источников частиц (энергии). Проведено сравнение с экспериментом.
КВАЗІСТАЦІОНАРНІ РОЗПОДІЛИ ДЛЯ ЗІШТОВХУВАЛЬНОЇ ПЛАЗМИ ЗА НАЯВНІСТЮ ДЖЕРЕЛ
ЕНЕРГІЇ ТА ЧАСТИНОК
І.Ф. Потапенко, В.І. Карась
Досліджено формування нерівноважних функцій розподілу частинок зі степеневим законом взаємодії між
ними. Розгляд основано на одновимірному нелінійному кінетичному рівнянні типу Ландау-Фокера-Планка за
наявністю джерел частинок (енергії). Наведено порівняння з експериментом.
62
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| id | nasplib_isofts_kiev_ua-123456789-78653 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T09:08:37Z |
| publishDate | 2005 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Potapenko, I.F. Karas, V.I. 2015-03-19T17:40:29Z 2015-03-19T17:40:29Z 2005 Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources / I.F. Potapenko, V.I. Karas // Вопросы атомной науки и техники. — 2005. — № 1. — С. 60-62. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.25.Dg, 52.65.Ff, 52.50. Gj https://nasplib.isofts.kiev.ua/handle/123456789/78653 The formation of a non-equilibrium distribution function (DF) of particles with the power-law interaction potentials is studied. Consideration is based on the non-linear kinetic equation of a Landau - Fokker - Planck type in the presence of particle (energy) sources. We compared our results with experimental data. Досліджено формування нерівноважних функцій розподілу частинок зі степеневим законом взаємодії між ними. Розгляд основано на одновимірному нелінійному кінетичному рівнянні типу Ландау-Фокера-Планка за наявністю джерел частинок (енергії). Наведено порівняння з експериментом. Исследовано формирование неравновесных функций распределения частиц со степенным законом взаимодействия между ними. Рассмотрение основано на нелинейном кинетическом уравнении типа Ландау- Фоккера-Планка при наличии источников частиц (энергии). Проведено сравнение с экспериментом. We are grateful to STCU (project # 1862) for partial financial support of this work. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources Квазістаціонарні розподіли для зіштовхувальної плазми за наявністю джерел енергії та частинок Квазистационарные распределения для столкновительной плазмы при наличии источников энергии и частиц Article published earlier |
| spellingShingle | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources Potapenko, I.F. Karas, V.I. Basic plasma physics |
| title | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources |
| title_alt | Квазістаціонарні розподіли для зіштовхувальної плазми за наявністю джерел енергії та частинок Квазистационарные распределения для столкновительной плазмы при наличии источников энергии и частиц |
| title_full | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources |
| title_fullStr | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources |
| title_full_unstemmed | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources |
| title_short | Quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources |
| title_sort | quasi steady-state distributions for collisional plasmas in the presence of energy and particle sources |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78653 |
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