Download Comparison higher order modified effective-range theory for elastic scattering angular differential cross-sections e-Ar

22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Comparison higher order modified effective-range theory for elastic scattering
angular differential cross-sections e-Ar
S. Hassanpour1, S. Nguyen-Kuok2, A. Ageev2 and A. Sanaat1
1
Faculty of Basic Sciences, Department of physics, Nour Branch, Islamic Azad University, Nour, 4817935861, Iran
2
Laboratory of Plasma Physics, National Research University MPEI, Krasnokazarmennya Str. 14, RU-111250
Moscow, Russia
Abstract: A six-parameter MERT (Modified Effective-Range Theory) fit gives a better
representation of recent e-Ar angular differential cross-sections experimental measurements
at energies less than 35.0 eV and the angular range 𝜃 = 20 − 100∘ . Differential
cross-sections for the elastic scattering of low- and intermediate-energy ranges electrons by
argon atoms could be calculated using MERT6 formulae.
Keywords: partial
cross-sections
wave,
phase-shifts,
1. Introduction
Calculations of average Collision Cross-section provide
a basis for the computation of important transport
coefficients (diffusion, thermal diffusion, thermal
conductivity, and viscosity) for plasma. A major part of
the evaluation of transport properties is successful
expression of quantum cross-sections and collision
integrals. One of the most effective factors in plasma is
calculation of differential elastic scattering cross section
of electron-argon atoms. For their calculations it's
necessary to investigate interaction between electron and
argon atom. The former are highly dependent on the
partial wave phase-shifts, whose calculation in turn relies
on exact knowledge of the potential.
There was
developed long ago a way of several steps (a) the solution
of the Schrodinger equation using an approximate
potential energy function in order to provide phase-shifts;
(b) use of the phase-shifts to provide collision cross
sections; (c) insertion of the cross-sections into the
collision integrals; and (d) use of the collision integrals in
formulas for the transport properties to be compared with
experimental values. Agreement, or lack of it, is then
used to improve the potential function in step (a), and the
cycle is repeated so that best agreement is found for all
experimental values for the transport coefficients [1].
The quantum expression has the nature of probability
amplitude and is given by the square modulus of the
scattering amplitude
𝜎(𝐸, 𝜃) = 𝑑𝜎⁄𝑑𝛺 = |𝑓(𝐸, 𝜃)𝑓 ∗ (𝐸, 𝜃)|
(1),
Here 𝜎(𝐸, 𝜃) has the dimensions of area and may be
viewed as a quantum differential cross section. Quantum
cross sections are expressed in terms of a quantity called
the asymptotic phase-shift 𝜂𝐿 which may be viewed as the
shift in phase of the asymptotic wave function of the
scattered particle as compared to the incoming plane wave
of the particle. Collision integrals are required to
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MERT
coefficients,
elastic
differential
calculate transport coefficients and account for the
interaction between colliding species i and j. The
collision integrals are calculated assuming that the species
are in their ground state and defined as follows [2-4]:
(𝑙,𝑠)
𝑄𝑖𝑖 (𝑇) =
2(𝑙+1)
(𝑠+1)![2𝑙+1−(−1)𝑙 ]
∞
𝐸
𝐸
× ∫0 𝑒 −𝐾𝐾 � �
𝐾𝐾
(2)
𝑠+1
𝐸
(𝑙)
𝑄𝑖𝑖 (𝐸)𝑑 � � .
𝐾𝐾
The indices(𝑙, 𝑠)are directly related to the order of
approximation used for the transport coefficients.
(𝑙)
(𝑙)
𝑄𝑐𝑐 (𝐸) 𝑜𝑜 𝑄𝑖𝑖 (𝐸) are classical integral cross-sections,
(𝑙)
the quantum mechanical form 𝑄𝑐𝑐 (𝐸)are produced by
replacement of 𝑏 𝑑b with
(𝑙)
𝜎(E, 𝜃) sin 𝜃 𝑑𝜃 ,𝑄𝑞𝑞 (𝐸) =
𝜋
2𝜋 ∫0 (1 − cos 𝑙 𝜃)𝜎(E, 𝜃) sin 𝜃 𝑑𝜃
(3)
2. Phase-shift theory
When the interaction between a projectile and a target is
central, it is convenient to use the method of partial waves
to obtain the elastic scattering amplitude, the
corresponding differential and integrated cross-sections.
Each partial wave corresponds to a definite angular
momentum of the system. For the fixed incident energy
of the projectile, the higher partial waves correspond to
larger impact parameters of the incoming projectile. If
the impact parameter is larger than the range of the
interaction, the contribution of the corresponding partial
wave to the scattering amplitude, for this particular
energy, is zero. Thus for short-range interactions between
projectile and target only a finite number of partial waves
make contributions to the scattering amplitude.
Moreover, the number of contributing partial waves
1
increases as the impact energy increases. Then we
considered the situation which applies more usually in
atomic physics when the potential has long-range terms
vanishing as the inverse fourth power of the distance.
Such terms arise from the polarization of the atom in the
field of the incident electron. We studied the scattering of
a particle by a potential field with particular reference to
elastic electron scattering by a neutral atom. This subject
is clearly of interest in its own right as a branch of
quantum mechanical scattering theory. We started our
discussion by considering scattering by the simplest type
of potential, one which is short-ranged, spherically
symmetric, spin independent, and local. We turn away
from successive generalizations in the form of the
potential and consider the simplification which results
when the energy is sufficiently low. The knowledge of
the phase-shifts enables one to obtain the scattering
amplitude. In this case we show that the phase-shift can
be expressed over an appreciable range of energy in terms
of at most two parameters: the scattering length and the
effective range. The scattering amplitude and the cross
sections are determined by the asymptotic behavior of the
stationary scattering wave function [5].
The radial Schrödinger equation for a particle moving
in the polarization potential 𝑉(𝑟) = −
�−
ħ2
�
𝜕2
2µ 𝜕𝑟 2
−
𝑙(𝑙+1)
𝑟2
�−
𝛼𝑒 2
2𝑟 4
𝛼𝑒 2
2𝑟 4
is:
− 𝐸� 𝜓𝑙 (𝑟) = 0
(4)
may be separated in spherical polar coordinates, and a
simple connection between the radial solutions and the
asymptotic form of the stationary scattering wave
function may be found. This procedure, which is called
the method of partial waves [6, 7]. The computation of
the phase-shifts, which play key role in the method of
partial waves, is the subject of next section the wave
function describes the colliding electron at a large
distance r from the scattering center, and is give:
exp(𝑖𝑖𝑖)
(+)
𝜓𝑘 (𝑘, 𝑟⃗) �⎯� 𝐴(𝑘) �exp�𝑖 𝑘�⃗. 𝑟⃗� + 𝑓(𝑘, 𝜃, 𝜑)
�
𝑟
𝑟→∞
(5)
here the z-axis is the direction of the incoming plane

wave, k the wave vector, and 𝑓(𝑘, 𝜃, 𝜙) the scattering
amplitude. Next, by matching the coefficients of the
(+)
outgoing spherical waves𝜓𝑘 (𝑘, 𝑟⃗), and by using
Legendre polynomials and spherical harmonics function
we find that the scattering amplitude is independent of 𝜙
and given by:
∞
𝑓(𝑘, 𝜃) =
1
�(2𝑙 + 1)(𝑒 2𝑖𝜂𝑙 − 1)𝑃𝑙 (cosθ)
2𝑖𝑘
𝑙=0
(6)
where 𝑃𝑙 (cosθ) is the lth Legendre polynomial. It is
apparent from the above formulae that the method of
partial waves is most useful when only a small number of
partial waves contribute to the scattering. This situation
2
arises at low incident energies. More precisely, if a is the
"range" of the potential and 𝑘 the wave number of the
particle, only those partial waves will be important for
which 𝑙 ≲ 𝑘𝑘. 𝑎 is the "range" of the potential. The
phase-shift according to the modified effective-range
theory (MERT4) is [8-11]:
tan 𝜂0
4𝛼 2
𝜋𝜋 2
𝑘 ln(𝑘𝑎0 )� −
𝑘 + 𝐷𝑘 3
3𝛼0
3𝛼0
+ 𝐹𝑘 4 (𝑙 = 0)
= −𝐴𝐴 �1 +
tan 𝜂1 =
𝜋𝜋 2
𝑘 − 𝐴1 𝑘 3 (𝑙 = 1)
15𝑎0
𝜋𝜋𝑘 2
tan 𝜂𝑙 = (2𝑙+3)(2𝑙+1)(2𝑙−1)𝑎 (𝑙 > 1)
(7)
(8)
(9)
0
Where a 0 is the Bohr radius, 𝛼 is the dipole
polarizability of the atom, 𝑘 is the wave number related to
the energy E (in ev) by E = 13.6057 (ka 0 )2, and
𝛼 = 10.899𝑎0 3 for argon.
The parameters to be
determined by the fit between experiment and theory are
A, A1, D and F. The physically most interesting
parameter is the scattering length A which determines the
cross section at zero energy by the relation
𝜎(0) = 4𝜋𝐴2 . From the above phase-shifts, the crosssections are calculated in the usual way concerning the
contribution of the infinite sum of higher partial waves.
Consider the elastic scattering of a projectile of mass μ
and energy 𝐸 = ℏ2 𝑘 2 /2𝜇 by a central potential V(r)
which results in a change in momentum from ℏ𝐤 𝑖 to
ℏ𝐤𝑓 for the projectile�𝑘𝑖 = 𝑘𝑓 = 𝑘�.
The MERT5
expansion has the form [9, 10]:
tan 𝜂0
𝜋𝜋 2
4𝛼 2
𝑘 ln(𝑘𝑎0 )� −
𝑘 + 𝐷𝑘 3
3𝛼0
3𝛼0
+ 𝐹𝑘 4 (𝑙 = 0)
= −𝐴𝐴 �1 +
tan 𝜂1 = 𝑎1 𝛼𝑘 2 − 𝐴1 𝑘 3 + (𝑏1 𝛼 4 + с1 𝛼)𝑘 4
+ 𝐻𝑘 5 (𝑙 = 1)
tan 𝜂𝑙 = 𝑎𝑙 𝛼𝑘 2 + (𝑏𝑙 𝛼 4 + с𝑙 𝛼)𝑘 4 (𝑙 ≥ 2)
(10)
(11)
(12)
Where H is an additional fitting parameter, and the
coefficients 𝑎𝑙 , 𝑏𝑙 and 𝑐𝑙 are given by:
𝑎𝑙 =
𝑏𝑙
=
𝜋
(2𝑙 + 3)(2𝑙 + 1)(2𝑙 − 1)𝑎0
(13)
𝜋[15(2𝑙 + 1)4 − 140(2𝑙 + 1)2 + 128]
3
�(2𝑙 + 3)(2𝑙 + 1)(2𝑙 − 1)� (2𝑙 + 5)(2𝑙 − 3)𝑎0 2
с𝑙 =
3𝑎𝑙
𝑎 2
(2𝑙 + 3)(2𝑙 + 1)(2𝑙 − 1) 0
(14)
(15).
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Hincklemann and Spruch [19] have shown that there
should be a term, proportional to 𝐻 𝑙𝑙(𝑘𝑎0 )5 , in the
expansion of the p-wave phase-shift.
Simulations using published (theoretical) phase-shifts
indicate that extended versions of the standard effective
range theory with six adjustable parameters are required
to give an adequate description of the phase-shifts for
argon. The appropriate equation for the d-wave phaseshift then becomes:
tan 𝜂𝑙 = 𝑎𝑙 𝛼𝑘 2 + (𝑏𝑙 𝛼 4 + с𝑙 𝛼)𝑘 4 + S𝑘 5
(16).
The MERT6 expansion has the form [10]:
tan 𝜂0
𝜋𝜋 2
4𝛼 2
𝑘 ln(𝑘𝑎0 )� −
𝑘 + 𝐷𝑘 3
3𝛼0
3𝛼0
+ 𝐹𝑘 4 (𝑙 = 0)
= −𝐴𝐴 �1 +
tan 𝜂1 = 𝑎1 𝛼𝑘 2 − 𝐴1 𝑘 3 + (𝑏1 𝛼 4 + с1 𝛼)𝑘 4
+ 𝐻𝑙𝑙(𝑘𝑎0 )5 (𝑙 = 1)
2
4
4
tan 𝜂𝑙 = 𝑎𝑙 𝛼𝑘 + (𝑏𝑙 𝛼 + с𝑙 𝛼)𝑘 + S 𝑘
5 (𝑙
(17)
(18)
≥ 2) (19).
It was found that an additional short-range term
(proportional to 𝑘 5 ), and additional higher-order terms
(proportional to 𝑘 4 ) due to the polarization potential,
when added to the p-wave expansion(MERT6) greatly
improve the quality of the fit, resulting in agreement with
the calculated phase-shifts.
3. Analysis and discussion
Argon elastic differential cross-sections have been
measured extensively since the 1930s, typically for
impact energies between a few and 100 eV.
Measurements of elastic differential cross sections are
usually limited to a scattering angle range between about
20° and 130°. Only recently, with the development of the
magnetic angle-changing technique, has the full angular
range of backward scattering become available. But in
this present work is used that differential cross-sections
were measured between about 20° and 100°. We finally
note that the present analysis does not take into account
any spin effects which may play a role for the heavier rare
gases. In performing the MERT fits, different data sets
have been used. The calculations are given in the
following table with other MERT coefficients used in
author's coefficients for comparison [12, 13].
The final MERT6 fits have been performed using the
full data sets for all angles measured (θ = 20-100°). The
provided figures 1(a, b, c, d, e, f) show that the MERT6
coefficients adopted in this work have given poor
agreement at energies 4.5-8 eV for all angles, which is
also a problem in previous works [12, 13]. However, in
others range of energy and all angles especially 90o is
good agreement with the experimental data than other
works. It's necessary to mention that our results have
given better agreement at energies less than 2 eV for all
angles. But the resulting differential cross-sections are in
substantially better agreement with experimental
measurements than previous calculations.
4. Conclusion
The experimental data have been compared to MERT4,
5, 6 calculations. The MERT parameters obtained from
these different fits were all consistent with each other and
the experimental data were generally very well fitted by
the MERT6 formulae, thus including waves considered,
depending on the energy of the electron; higher energies
require more terms.
We showed that the recently developed MERT
expansion gives very good agreement with experimental
data for electrons that the fit performed in the low energy
range especially (below 2 eV). The agreement with the
experiment can be improved for some angles using six
parameters in MERT expansion. It is appeared that the
MERT4, 5 expansions are not sufficiently suitable to
describe differential cross-sections especially when
applying them to experimental data at intermediate
energies.
Table 1. Comparison of current work with other jobs.
Author
Weyhreter M.,
al.(MERT4)
-1.609
Buckman S.
J.,
et.al.(MERT4)
-1.495
-1.548
-1.498
-1.497
-1.551
8.094
8.1467
7.852
8.567
8.498
8.51
60.52
64.811
61.55
64.495
64.215
62.15
-59.47
-86.141
-71.45
-83.68
-83.0891
-72.91
16.859
14.099
-18.50
Parameter
A
A
D
F
H
et
Hassanpour S.
et.al.(MERT4)
Buckman S. J.,
et.al.(MERT5)
Buckman S. J.,
et.al.(MERT5)
Present work
(MERT6)
a0
a03
a03
a04
-
-
a05
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3
S
-
-
-
-
-0.9954
a05
Here we provide selective plots of the calculated results (lines) and experimental data (dots).
(b)
(a)
(d)
(c)
(e)
(f)
Fig. 1. Energy dependence of the elastic differential cross-sections 𝑑𝜎⁄𝑑𝛺 (in units of 𝑚2 𝑠𝑠 −1) for electron-argon
atoms scattering at different scattering angles (shown in the figures 1(a,b,c,d,e,f).
5. Acknowledgments
This research was supported completely by a grant
No.L-55-27.8.1393 from Islamic Azad University, Nour
Branch.
This support is gratefully acknowledged
Research Council of University.
4
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