Computer calculation of probability for binary collisions of electrons with ions and molecules
Computer calculation of rate coefficient for binary collision i < σᵢν > as a function of temperature is presented, and the Maxwell electron velocity distribution function is chosen. The finite elements of the fifth order made it possible to significantly speed up the process of calculation...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2021 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2021
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Computer calculation of probability for binary collisions of electrons with ions and molecules / Yu.M. Marchuk, Yu.S. Kulyk, V.Е. Moiseenko // Problems of Atomic Science and Technology. — 2021. — № 4. — С. 154-156. — Бібліогр.: 5 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860247892111917056 |
|---|---|
| author | Marchuk, Yu.M. Kulyk, Yu.S. Moiseenko, V.Е. |
| author_facet | Marchuk, Yu.M. Kulyk, Yu.S. Moiseenko, V.Е. |
| citation_txt | Computer calculation of probability for binary collisions of electrons with ions and molecules / Yu.M. Marchuk, Yu.S. Kulyk, V.Е. Moiseenko // Problems of Atomic Science and Technology. — 2021. — № 4. — С. 154-156. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Computer calculation of rate coefficient for binary collision i < σᵢν > as a function of temperature is presented, and the Maxwell electron velocity distribution function is chosen. The finite elements of the fifth order made it possible to significantly speed up the process of calculation < σᵢν >. The result of the approximation is a smooth function and the values of this function, its first and second derivatives, have no jumps at the mesh nodes and the accuracy of calculation is within the limits of statistical errors for the source data. These advantages and the results will be used in future tasks.
Запропоновано комп’ютерний розрахунок вірогідності парних зіткнень < σᵢν > як функції від температури. Вибрана максвелiвська функція розподілу електронів за швидкостями. Скінченні елементи п'ятого порядку дозволили значно прискорити процес обчислення < σᵢν >. Результатом апроксимації є гладка функція. Значення цієї функції, її першої та другої похідних не мають стрибків у вузлах сітки, а точність знаходиться в межах статистичних помилок вхідних даних. Переваги даного методу будуть використані в майбутніх задачах.
Предложен компьютерный расчет вероятности парных столкновений < σᵢν > как функции от температуры. Выбрана максвелловская функция распределения электронов по скоростям. Конечные элементы пятого порядка позволили значительно ускорить процесс расчета < σᵢν >. Результатом аппроксимации является гладкая функция. Значения этой функции, ее первой и второй производных не имеют скачков в узлах сетки, а точность расчета находится в пределах статистических ошибок исходных данных. Эти преимущества и результаты будут использованы в будущих задачах.
|
| first_indexed | 2025-12-07T18:39:07Z |
| format | Article |
| fulltext |
ISSN 1562-6016. ВАНТ. 2021. № 4(134) 154
https://doi.org/10.46813/2021-134-154
COMPUTER CALCULATION OF PROBABILITY FOR BINARY
COLLISIONS OF ELECTRONS WITH IONS AND MOLECULES
Yu.M. Marchuk, Yu.S. Kulyk, V.Е. Moiseenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
Computer calculation of rate coefficient for binary collision i as a function of temperature is presented,
and the Maxwell electron velocity distribution function is chosen. The finite elements of the fifth order made it pos-
sible to significantly speed up the process of calculation i . The result of the approximation is a smooth func-
tion and the values of this function, its first and second derivatives, have no jumps at the mesh nodes and the accu-
racy of calculation is within the limits of statistical errors for the source data. These advantages and the results will
be used in future tasks.
PACS: 29.17.+w; 41.75.Lx
INTRODUCTION
In the non-equilibrium plasma, elastic and inelastic
collisions determine the rates of many collision proc-
esses. Collisions of electrons with heavy particles (ion
or with atom) are of the certain interest for plasma pro-
duction and sustain. If you consider the case of elastic
collisions, scattering is not accompanied by an appre-
ciable loss of energy; in the case of inelastic interac-
tions, the energy losses are higher.
The probability of electron scattering through a cer-
tain angle is determined by the value of the interac-
tion potential of the scattering particle with the incident
electron. It is proportional to the differential scattering
cross section /d d d d and the angular
dependence of the scattering cross section /d d
can take as a simple form, which simplifies calculations
and can be quite complex. Total rate coefficient for
collisions with the relative velocity is defined as
/i d d d .
The frequency of collisions for monoenergetic elec-
trons is determined by the expression e i n
where n is the density of particles which electron col-
lides.
FINITE ELEMENT APPROXIMATION
Many problems in applied mathematics are reduced
to solving different equations of various types (linear
and non-linear). One of the powerful numerical meth-
ods is the finite element method, which is a universal
tool for solving problems of mathematical physics.
The finite element method is based on the fact that
any arbitrary and continuous function x in the
computational domain, for example, the temperature,
can be approximated by a set of piecewise continuous
functions defined on a finite number of subdomains
(finite elements) [1].
The region of integration is divided into elements in
such a way that on each of them the unknown function
could be approximated, for example, by polynomials.
Inside each element, an unknown function is repre-
sented by a linear combination of basis functions, un-
known coefficients, while the values of the function are
at the nodal points of the approximation. As coordinate
functions, functions are taken that are identically zero
everywhere, except for one finite element, inside which
they coincide with the basis functions. Thus, the prob-
lem of approximating a function is reduced to a system
of linear algebraic equations, the number of equations
of which coincides with the number of nodal points.
These elements have common anchor points and to-
gether they approximate the shape of the region.
The essence of the problem is that we need to ap-
proximate the known experimental data for electron
impact ionization average cross section. This method
meets our requirements, since excessive accuracy is not
needed and the time spent on the calculation is rather
small. Also, the advantages of the method are that the
first and second derivatives at the mesh nodes are
matching. For this reason, this method is suitable for
calculations and can be used in work [2].
FORMULATION OF THE PROBLEM
There are Lagrangian and Hermitian finite elements
and each of them has its own advantages. The main
disadvantage of Lagrangian finite elements is that they
include additional points on the segment and deriva-
tives at the main nodes are not matching. We use fifth
degree polynomials to approximate a function for
which the first and the second derivative are matching
at the mesh nodes. This is essential advantage of Her-
mitian finite elements. In accordance with the above
written we will be use the fifth-order Hermitian finite
element approximation for the rate coefficient. A pow-
erful tool for simulating processes in plasma is de-
scribed in the work [3].
Using the general formula for the rate coefficient for
probability of binary collisions:
3
i e iS f d , (1)
we assume that the velocity distribution function is the
normalized Maxwellian:
2
3/ 2 2 2
1 expe eM
Te Te
f f
, (2)
where is the velocity of electrons. Thermal velocity
2 /Te ekT m depends on temperature, where k is
the Boltzmann constant, T is the temperature of elec-
trons, em is the mass of the electrons.
ISSN 1562-6016. ВАНТ. 2021. № 4(134) 155
Next, we take the values for the ionization cross-
section of the hydrogen atom from review [4] and ap-
proximate with columns (piecewise constant elements)
which defined on finite number of subdomains (Fig. 1).
Fig. 1. Function approximation using piecewise
constant elements for electron impact ionization
cross-section [4]
The required function S depends on the electron
temperature. Next, we introduce a temperature data grid
with equidistant nodes i. We make the following
change of variable: 2 2/ Teu .
At each grid interval, we represent the as a con-
stant value and analytically calculate the integral. The
number of such integrals is equal to the number of in-
tervals; therefore, we can represent equation (1) as a
sum by the number of nodal points i .
1
1 1
1
1 exp 1 exp
N
i i i i
i
S A u u u u
, (3)
where N is the number of points, 2 /Te iA ,
iu and 1iu is the lower and upper integration.
It is also of interest to calculate the first and second
derivatives of the function S with respect to tempera-
ture:
1
2
1
/ exp 2 1
N
i i i
i
Ad S dT u u u
T
2
1 1 1exp 2 1i i i
A u u u
T , (4)
3 21
2 2
1
2 2 1
/ exp
2
N
i i i
i
i
u u uAd S dT u
T T
3 2
1 1 1
1
2 2 1exp
2
i i i
i
u u uA u
T T
. (5)
A program has been written that makes it possible
using equations (3) - (5) to calculate the S for any
value of temperature of electrons. S is the continuous
in temperature and all derivatives exist and this is the
advantage of this program. But the main disadvantage
is that the program calculates slowly, since (3) - (5)
contains summation over the number of intervals and it
takes time.
The task is to simplify the S calculation, we will
approximate the results using Hermitian finite elements
with a polynomial of the fifth degree. Next, we intro-
duce a new grid of temperature data.
M is the number of intervals, k is the discrete
variable which enumerates of mesh nodes and varies
from 1 to M .
1
2
1 2 3 2|
k kT T T i
i i
dS d SS S
dT dT
2
4 1 5 6 2
1 1
i
i i
dS d SS
dT dT
. (6)
Below are the finite elements for the fifth degree
polynomial [5]:
3 4 5
1 1 10 15 6 ,
3 4 5
2 6 8 3k ,
2 2 3 4 5
3
1 3
2 k ,
3 4 5
4 1 10 1 15 1 6 1 ,
3 4 5
5 1 6 1 8 1 3 1k ,
2 3 4 52
6
1 1 3 1 1 1
2
,k
where
1k k kT T ,
1
k
k k
T T
T T
. (7)
NUMERICAL RESULTS OF MODELLING
The next step is to write a Fortran program. In order
to calculate S using formula (6) in Fortran, we need
to know the constants, namely iS , / idS dT , and
2 2/
i
d S dT which are calculated at each nodal
point. Since these constants are required, at the prepara-
tory stage, our program launches a program that calcu-
lates the equations (3) - (5) and fills with these values at
the nodes of its own mesh. When this stage is passed
our program is ready to work.
Fig. 2. S values which are calculated by direct
and approximation method at each temperature value
S is the deviation values of the direct method
from the approximation method
For numerical experiments, we chose a temperature
range from 0 to 1000 eV and chose 100000 temperature
ISSN 1562-6016. ВАНТ. 2021. № 4(134) 156
values in which we will calculate S . Next, we made
the calculation using the new program (we will call the
approximation method) and using the old program
(direct method) and compared the time of their calcula-
tion, the results are shown below:
0.109201apprt s is the approximation method,
3.08882cyclet s is the direct method.
Numerical simulation results are presented in Fig. 2.
The figure shows that the result for one function calcu-
lated in two different method is the same.
CONCLUSIONS
At the first stage, a method was developed for cal-
culating S for electronic collisions using tabular val-
ues of . The calculation of S as a function of tem-
perature is proposed. A distinctive feature is that S
function is quickly calculated, and also the result of the
approximation of S is a smooth function and the
values of this function have no jumps. The application
of polynomials in finite elements of the fifth order
made it possible to speed up the process of calculating
S by about 30 times 28.27cycle
appr
t
t
. These advantages
will be used in future tasks.
REFERENCES
1. O. Zenkevich, K. Morgan. Finite elements and ap-
proximation. New York: "John Wiley & Sons",
1983, 328 p.
2. Y.S. Kulyk, V.Е. Moiseenko, T. Wauters,
A.I. Lyssoivan. Modelling of radio-frequency wall
conditioning in short pulses in a stellarator. Prob-
lems of Atomic Science and Technology. Series
“Plasma Physics”. 2021, № 1(131), p. 9-14.
3. J.D. Lore et al. Implementation of the 3D edge
plasma code EMC3-EIRENE on NSTX // Nuclear
Fusion. 2012, v. 52, № 5, p. 054012.
4. R.K. Janev, D. Reiter, U. Samm. Collision proc-
esses in low-temperature hydrogen plasmas. Jülich:
Forschungszentrum, Zentralbibliothek, 2003,
t. 4105.
5. C.C. Ike, E.U. Ikwueze. Fifth degree Hermittian
polynomial shape functions for the finite element
analysis of clamped simply supported Euler-
Bernoulli beam //American Journal of Engineering
Research (AJER). 2018, v. 7, № 4, p. 97-105.
Article received 04.06.2021
КОМПЬЮТЕРНЫЙ РАСЧЕТ ВЕРОЯТНОСТИ ПАРНЫХ СТОЛКНОВЕНИЙ ЭЛЕКТРОНОВ
С ИОНАМИ И МОЛЕКУЛАМИ
Ю.М. Марчук, Ю.С. Кулик, В.Е. Моисеенко
Предложен компьютерный расчет вероятности парных столкновений i как функции от темпера-
туры. Выбрана максвелловская функция распределения электронов по скоростям. Конечные элементы пято-
го порядка позволили значительно ускорить процесс расчета i . Результатом аппроксимации является
гладкая функция. Значения этой функции, ее первой и второй производных не имеют скачков в узлах сетки,
а точность расчета находится в пределах статистических ошибок исходных данных. Эти преимущества и
результаты будут использованы в будущих задачах.
КОМП'ЮТЕРНИЙ РОЗРАХУНОК ЙМОВІРНОСТІ ПАРНИХ ЗІТКНЕНЬ ЕЛЕКТРОНІВ
З ІОНАМИ ТА МОЛЕКУЛАМИ
Ю.М. Марчук, Ю.С. Кулик, В.Є. Моісеєнко
Запропоновано комп’ютерний розрахунок вірогідності парних зіткнень i як функції від темпера-
тури. Вибрана максвелiвська функція розподілу електронів за швидкостями. Скінченні елементи п'ятого
порядку дозволили значно прискорити процес обчислення i . Результатом апроксимації є гладка фун-
кція. Значення цієї функції, її першої та другої похідних не мають стрибків у вузлах сітки, а точність знахо-
диться в межах статистичних помилок вхідних даних. Переваги даного методу будуть використані в майбу-
тніх задачах.
|
| id | nasplib_isofts_kiev_ua-123456789-195421 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:39:07Z |
| publishDate | 2021 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Marchuk, Yu.M. Kulyk, Yu.S. Moiseenko, V.Е. 2023-12-05T10:06:25Z 2023-12-05T10:06:25Z 2021 Computer calculation of probability for binary collisions of electrons with ions and molecules / Yu.M. Marchuk, Yu.S. Kulyk, V.Е. Moiseenko // Problems of Atomic Science and Technology. — 2021. — № 4. — С. 154-156. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 29.17.+w; 41.75.Lx DOI: https://doi.org/10.46813/2021-134-154 https://nasplib.isofts.kiev.ua/handle/123456789/195421 Computer calculation of rate coefficient for binary collision i < σᵢν > as a function of temperature is presented, and the Maxwell electron velocity distribution function is chosen. The finite elements of the fifth order made it possible to significantly speed up the process of calculation < σᵢν >. The result of the approximation is a smooth function and the values of this function, its first and second derivatives, have no jumps at the mesh nodes and the accuracy of calculation is within the limits of statistical errors for the source data. These advantages and the results will be used in future tasks. Запропоновано комп’ютерний розрахунок вірогідності парних зіткнень < σᵢν > як функції від температури. Вибрана максвелiвська функція розподілу електронів за швидкостями. Скінченні елементи п'ятого порядку дозволили значно прискорити процес обчислення < σᵢν >. Результатом апроксимації є гладка функція. Значення цієї функції, її першої та другої похідних не мають стрибків у вузлах сітки, а точність знаходиться в межах статистичних помилок вхідних даних. Переваги даного методу будуть використані в майбутніх задачах. Предложен компьютерный расчет вероятности парных столкновений < σᵢν > как функции от температуры. Выбрана максвелловская функция распределения электронов по скоростям. Конечные элементы пятого порядка позволили значительно ускорить процесс расчета < σᵢν >. Результатом аппроксимации является гладкая функция. Значения этой функции, ее первой и второй производных не имеют скачков в узлах сетки, а точность расчета находится в пределах статистических ошибок исходных данных. Эти преимущества и результаты будут использованы в будущих задачах. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Gas discharge, plasma-beam discharge, and their applications Computer calculation of probability for binary collisions of electrons with ions and molecules Комп'ютерний розрахунок ймовірності парних зіткнень електронів з іонами та молекулами Компьютерный расчёт вероятности парных столкновений электронов с ионами и молекулами Article published earlier |
| spellingShingle | Computer calculation of probability for binary collisions of electrons with ions and molecules Marchuk, Yu.M. Kulyk, Yu.S. Moiseenko, V.Е. Gas discharge, plasma-beam discharge, and their applications |
| title | Computer calculation of probability for binary collisions of electrons with ions and molecules |
| title_alt | Комп'ютерний розрахунок ймовірності парних зіткнень електронів з іонами та молекулами Компьютерный расчёт вероятности парных столкновений электронов с ионами и молекулами |
| title_full | Computer calculation of probability for binary collisions of electrons with ions and molecules |
| title_fullStr | Computer calculation of probability for binary collisions of electrons with ions and molecules |
| title_full_unstemmed | Computer calculation of probability for binary collisions of electrons with ions and molecules |
| title_short | Computer calculation of probability for binary collisions of electrons with ions and molecules |
| title_sort | computer calculation of probability for binary collisions of electrons with ions and molecules |
| topic | Gas discharge, plasma-beam discharge, and their applications |
| topic_facet | Gas discharge, plasma-beam discharge, and their applications |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/195421 |
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