22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Quantum calculation of the differential and average collision cross-sections for
elastic low energy e-Ar scattering
S. Nguyen-Kuok1, Yu. Malakhov1, I. Korotkikh1 and S. Hassanpour2
1
Laboratory of Plasma Physics, National Research University MPEI, Krasnokazarmennya Str. 14, 111250 Moscow,
Russia
2
Department of science, Noor Branch, Islamic Azad University, Noor, Iran
Abstract: This paper presents the calculation results of the differential cross-sections of
electron-atom in the argon with the low energies of electrons to ε = 10 eV and comparison
with experimental and theoretical data from other sources. In contrast to the works of other
authors we used the approximation of higher order (up to 4) to calculate the phase shift ηl
and a large amount of packet waves (up to a million). Calculation was performed for the
scattering angles (θ = 10° - 100°). We used also higher orders L = 1…6, S = 1…11 to
calculate averaged cross-sections of the electron-atom in argon. The results of our
calculations have a good agreement with the experimental data.
Keywords: calculation results, differential cross-sections, averaged cross-sections,
approximation of higher order, e-Ar scattering
Electron scattering on the atoms for inert gases, With the use of the corresponding boundary conditions it
including argon, with the low energies is characterized by is possible to obtain expression for the function of
wave properties and is described by quantum-mechanical scattering amplitude in the following form:
model. The quantum-mechanical description of the theory
1 ∞
1 ∞
of particle scattering is based on the solution of
f (θ ) =∑ ( 2l + 1) ( exp ( 2iηl ) − 1) ( Pl (cos θ ) ) =
∑ (2l + 1)exp(iηl )sinηl Pl (cosθ )
2
ik
kl 0
=
l 0=
Schrödinger wave equation taking into account
potential
(3)
interaction energy of the particles φ ( r ) . The solution of
Schrödinger wave equation is produced with the method where P (cos θ ) – Legendre's polynomials.
l
of separation of variables. For this the wave function is
A
phase
shift ηl can be used for determining the
determined in the form two functions, one of which
depends on coordinates, and the second – from the differential cross section in the following form:
scattering angle θ . Quantum-mechanical description
2
makes it possible to determine the probability of
ds
1 ∞
(2l + 1) exp(iηl )sin ηl Pl (cos θ )
scattering (i.e., differential cross sections) the particles to=
d Ω k 2 l =0
the specific angle only θ . This probability is expressed as
(4)
∑
the phase shift ηl of the radial wave function ψ ( r ) .
Phase shift ηl depends on the wave number k and the
function U ( r ) , which are determined as follows:
k=
2p p
= =
λ 
2mφ ( r )
2mE
, U (r ) =

2
The results of our calculation with large amount of packet
waves (up to a million) have a good agreement with the
experimental data. A comparison with experimental and
theoretical data from other sources presents on Fig. 1.
(1)
where p, m, E – pulse, reduced mass and the total energy
of system respectively. The Schrödinger equation can be
written down in the spherical coordinates taking into
account the symmetry as:
1 ∂  2 ∂ψ
r
r 2 ∂r  ∂r
P-I-2-48
∂ 
∂ψ
1

+ 2
 sin θ
∂θ
 r sin θ ∂θ 

2
0
 + ( k − U )ψ =

(2)
1
Fig. 1. The differential elastic cross section e-Ar.
∞
A phase shift ηl determined with the Modified
2( L + 1)
=
Q ( L , S ) (T )
exp(− x) x S +1Q ( L ) (kTx)dx
( S + 1)![2 L + 1 − (−1) L ] ∫0
Effective Range Theory (MERT) of the fourth
(5)
approximation degree [7]. We used also higher orders L =
2
1…6, S = 1…11 to calculate averaged cross sections of
mg
E
2
– the initial relative speed of
x g=
=
the electron-atom in argon (Fig. 2). The averaged where =
2kT kT
effective cross sections are determined as follows:
the being encountered particles.
Fig. 2. The averaged effective interaction cross sections e-Ar.
2
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References
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and Clusters, 7, 333 (1988)
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(1985)
[6] G.N. Haddad and T.F. O’Malley, Aust. J. Phys., 35,
35 (1982)
[7] S. Nguyen-Kuok, The Basics of Mathematical
Modeling of Low Temperature Plasma. (Moscow:
Publishing House MEI) (2013)
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