Analytical representations of the elastic scattering cross sections of electrons with energies of 0.01-1 keV in atmospheric gases of N2, O2, O are given. These representations are suitable for the Monte Carlo method

Ivanov, V. Y.

Shneyder, V. A.

Sipov, N. K.

English

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NASA 'CECHNICAL MEMORANDUM NASA
ANALYTICAL REPRESENTATION OP ELASTIC SCATTERING
CROSS SECTIONS OF LOW ENERGY ELECTRONS BY ATMOSPHERIC GASES
V.Ye. Ivanov, N.K. Osipov and V.A. Shneyder
Translation of "Anallticheskoye predstavleniye secheniy
uprugogo rasseyanlya elektronov malykh energiy atmosfernymi
gazami", Geomagnetism i Aeronomiyas^Vol. XVII, No. 2, 1977,
pp. 242 - 245* . - ' • - . : "
, NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D. C. 205^6 SEPTEMBER 1977
https://ntrs.nasa.gov/search.jsp?R=19780003865 2020-03-22T06:44:19+00:00Z
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1. Report No.
— MSA -TM-
2. Government Acceition No. 3>. Recipient'* Co'olog No.
4. Till, end Subtitlo ANALYTICAL REPRESENTATION OF'
ELASTIC SCATTERING CROSS SECTIONS OF LOW ENERGY
ELECTRONS BY ATMOSPHERIC GASES .
5. Report Dote
1977
6. Performing Organ! lotion Cod*
7. Avther(o) .
V.Ye. Ivanov, N.K. Osipov and V.A. Shneyder
8. Performing Organ) tollon Report No.
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vTranslation of "Analiticheskoye predstavleniye secheniy uprugogo
rasseyania 'elektronov malykh energiy attnosfernymi gazami", Geomagnetism
• i Aeronomiya, Vol. XVII?; No. 2, 1977, pp. 242 - 245.
Abltrocf
Analytical representations of the elastic•scattering cross sections,
of electrons with energies of OiOl-i-keV- in atmo.spnerTcT gases of
N9, 0-, 0 are given1. • These representations are suitable for the
• * ~ ' * ' • • • • : * . '
Monte Carlo method. ", 'V , • • : ' . . " •••.. : • ' •
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NASA-MO
ANALYTICAL REPRESENTATION OP ELASTIC SCATTERING
CROSS SECTIONS OP LOW ENERGY ELECTRONS BY ATMOSPHERIC GASES
\T.Ye. Ivanov, N.K. Osipov and V.A. Shneyder
The possibilities of using the kinetics of low energy
electrons in the atmosphere are greatly limited both in certain
methods [1-3] and in the Monte Carlo method [^] by the accuracy
and form of the differential cross sections of their elastic
scattering in atmospheric gases. Thus, when these cross sections
are given in a form corresponding to the results of theoretical
i
calculations and assuming their analytical integration over the
scattering angles in the Monte Carlo method there is a great
reduction in the time necessary for achieving the given statistical
accuracy. Taking this into account, we shall try to determine the
analytical representations of differential scattering cross sections
I(E,6) of electrons with energies of E = 0.01-1 keV in the form of
a polynomial in powers of the square of the change in the pulse of
the scattered electron, corrected for the screening effect of the
field of the electron shell nucleus. We shall limit ourselves to ;
three terms, each of which prevails in the scattering region at
small, average and^  large angles, respec^tiyely_
l-l
where (2E/R) (1-cos 9) is the square of the change in the pulse
of the scattered electrons; R — the constant equalling 13.6 eV;
* Numbers in margin indicate pagination in original foreign text
. 0 — scattering angle; n — screening parameter; Ai(E), cu
empirically selected coefficients depending only on energy.
f 1
fr,
V.
• When use is made of the VKB approximation, the expression for
the screening parameter has the form
T(T 1'13 + 3'76[l3^ r
where v — _ is the electron velocity; T — electron kinetic energy
[in atomic units; Z -*- nucleus charge. However, according to
>. calculations of [4] carried out for 1<E<20 keV and having a good
agreement with the experimental data on electron energy dissipation
[697]s the .value of the parameter Tirt for the elastic scattering
T C
'of low energy electrons (£l keV) may be assumed to be constant.
1
 Then
where B — is a constants E — energy in units of eV.
o to to . to . no iso'a!.
Figure 1.
to so no iso no o
Figure 2.
/2U3
Below we shall give the results of approximating the experi-
mental data on scattering by nitrogen molecules [8] for electrons
with E = 20 - 90 eV and the angular range of 3< 6 <160°, data of
[9] for E = 300,400j 500 eV and 2 £ 6* 40°, and the data of [10]
for the same energies and 4s9£ 150°. At B = 2.38 the coefficients
and cL(E) are approximated by the following expressions:
A,(E) -8,39 • 10-"[ 1+1,5
(2)
(3)
(4)
(5)
(6)
(7)
Figure 1 gives examples of comparing the experimental data
; (dark circles — data from [9], light circles — data from
; [10], crosses — data from [8]) with the results of calculations
; using Formula (1) with the icoefficientsl (2[) - (7) (solid curves).
' In a similar ways using data in [11, 12], we compiled the
cross sections 'for scattering by molecular oxygen which, however,
! due to the absence of experimental data for large scattering
angles >was taken into account the same way as for Np, i.e., the
j coefficients (6)s(7). The validity of this approximation was
, controlled by comparing the integral scattering cross sections
for Op, calculated according to (1) with the measured integral
cross sections. At B = 2.48 the coefficients Ai(E) and ai(E) of
, molecular oxygen have the following form: .
... „, ^
, . ..'-. (9)
(10)
(11)
r
*
1 IP ~.
w nU—i—i—i i i inii .8—i i .
Figure 3, Figure
i Figure 2 shows experimental data on the cross sections of
scattering by 02 (crosses — data from [11], circles — [12]);
!
 these data were compared with the results of calculations using ,
the approximations obtained.
There are no measurements of the differential cross sections
for atomic oxygen. Therefore, when compiling their analytical
expressions, we use the theoretical formulas obtained, introducing
corrections in the field of small scattering angles. For this pur
pose, we standardized the integral cross sections calculated in
the Bornov approximation to the measured Integral cross sections.
The initial material was the results of calculations in [13]
carried out using the Hartree-Fokker method. The approximating
.e!xpr els slo^i for the scattering cross sections in this case has the
form
[ 2E -i-* r 2E •*-*—(1-cos e+2t)) I -f At \ -—(1-cos e+2ri) I »7 1 J
 • . . « • « • • ; J (12)
where = 182, A2 = 5.28, B =0.7, o=-1.5.
The degree of agreement 'between expression (12) and the
. Mott and Massy theory is illustrated in Figure 3, where the
crosses designate the results of theoretical calculations. To
correct the cross section given by Formula (12), we introduced
the correction coefficient D(E)8 which decreases the scattering
cross section by small angles in the low energy region
I
tf ft
• The correction coefficient D(E) is found from the conditions for
standardizing the integral elastic scattering cross sections
calculated from (12) to the total scattering cross sections /2*J5
measured in
D(E)>
; where a (E) is the experimentally measured total scattering cross
section of electrons by atomic oxygen. The approximating expression
given below is obtained for the correction D(E) calculated
according to (-1*0:
< [~ Z?(£)Ml-exp(-0,02£);v
Let us compare the integral elastic scattering cross sections
calculated for Np and Op using (1) and for 0 using (13), with the
experimental measurements of the latter. Figure 4 compares the
Integral cross sections for ^  calculated by integrating Formula
(1) (solid line) with the total cross sections measured in [15]
(light circles), ;tl6]l (triangles), and with the integral cross
sections obtained by numerical integration of experimental data [8]
(solid circles) and [9] (crosses). Figure 4 performs a similar
comparison for molecular oxygen with measurements of [14] (light
circles), [17] (dark circles) and [11] (crosses). In addition,
'' - it also gives similar data for atomic oxygen ( calculated
according to (13) — solid lines, circles — [I1*], theoretical
calculations given in [13] — triangles; the dashed curve
corresponds to the integral cross section calculated according
to (12)). Figure 4 gives the results of comparing the experi-
mental data with the calculations. These data show that the
, approximations obtained represent reliable material which can be
; used in calculations of electron transfer.
The authors thank I.A. Zhulin, L.S. Yevlashin and
Yu,B, Belostotskiy for assisting in the study and G.V, Starkov
for useful discussions.
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Trajmar, S., D. C. Cartwright and W. Williams. Phys. Rev.
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Analytical representation of elastic scattering cross sections of low energy electrons by atmospheric gases

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