Table Of ContentEdited by
O. M. Belotserkovskii
Computational Center
Moscow, USSR
M. N. Kogan
Moscow Physicotecfmicallnstitute
Dolgoprudny, USSR
S. S. Kutateladze
and
A. K. Rebrov
Institute of Thermophysics
Novosibirsk, USSR
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library 01 Congress Cataloging in Publication Oata
International Symposium on Rarefied Gas Oynamics (l3th: 1982: Novosibirsk, R.S.F.S.R.)
Rarefied gas dynamics.
"Revised versions 01 selected papers lrom the Thirteenth International Symposium on
Rarefied Gas Oynamics, held July 1982, in Novosibirsk, USSR"-T.p. verso.
Bibliography: p.
Includes index.
1. Rarelied gas dynamics-Congresses. 1. Belotserkovskii, O. M. (Oleg Mikhailovich) II. Title.
QC168.155 1982 533/2 85-3719
ISBN 978-1-4612-9497-9 ISBN 978-1-4613-2467-6 (eBook)
DOI 10.1007/978-1-4613-2467-6
Revised versions 01 selected papers lrom the Thirteenth International Symposium on
Rarelied Gas Oynamics, held July 1982, in Novosibirsk, USSR
© 1985 Springer Science+ Business Media New York
Originally published by P1enum Press, New York in 1985
Softcover reprint of the hardcover 1s t edition 1985
AII rights reserved
No part 01 this book may be reproduced, stored in a retrieval system, or transmitted,
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record ing, or otherwise, without written permission lrom the Publisher
X. COLLISIONAL PROCESSES
ANALYTIC.AL :roRMULAE .R>R CROSS SECTIONS AND
RATE CONSTANTS OF ELEMENTARY PROCESSES IN GASES
G. V. Du.brovskiy, A. V. Bogdanov, Yu. Ji:. Gorbachev,
L. F. Vyunenko, V. A. Pavlov, and V. M. Strelchenya
Leningrad State University
Leningrad, 198904, USSR
In our review article presented here a quasiclassical kinetic
equation has been euggested witn a collision integral expressed
through the scattering T-matrix in the quasiclassical approximation.
Now we shall give some results on cross sections (CS) and rate
constants (RC) calculations carried out within a simplified version
of this expression (a generalized eieanal formula, GEF).
I. GEN.li.o'RALIZED EICONAL :roRMULAE FOR CROSS SECTIOUS
The GEF for a differential CS (DCS) can be written as1
rf
<5 <~ ,q) = ~~~~ ~dj exp(iA} ,11) f;n<f 2,
-~n<f). = ~ (:::)~ e;lCP(~ ~0) ( exp(it. S/b) - .S 'if). (I. I)
~ ~
Here ~= p- p' ~ the relative momentum transfer, q is the quantum
numbere change, \I ie a set of N angle variables of the internal
motion of collidin~ moleculea, '§ ~s their _telatille posi tion radius
vector in the point of maximal approach (j' = (f ,+)), t. S is the
claasical action increment
+-0s-0
.6.S(~,\i0) =- dt V(f+ ;(t)t,\..9 0 +~(t)t, t(t)),
where V is the interaction potential, v(t),~v (t) stand for relative
velocity and internal motion frequenciea vectora and are to be found
from the trajectory problern aolution. (Here we shall use an eieanal
approximation).
Thua to calculate CS (I.I) one haa to express both the molecu-
697
lar Rarnil t~ni~ and the ~ntera.ction potential in the action-angle
va.riablee I-~ (H ~ H( I)), to choose an adequate model for clae
sical trajectory, and to do the neceesa.r~ integrale. For dia.t~rnice
the rotating Moree oscillator (r10) model may be used for H( I ) with
the interaction potential V of a general type
(I.2)
Here r1 2 are the i~teratsmic distancee in the molecu1es, 2( I 2 are
the anelee between R and r1_2 , P (x) are Legendre polynomiai~. De
pending on the transition ubder cSnsideration (RT, RR, RV, VT, VV)
different terme in Fq.(I,2) ahould be reta.ined.
For polyatomics the rigid rotor - harmonic oecillator model is
the eimpliest one with the Hamiltonia.n H being
H(~) 1i.t)lk(~+I/2), it..(j,l,~).(I.3)
""Bj(j +I)+ (A- C)l2 +
k=I
A,B,C in Fq. (I. 3) a.re the molecule rotational oonata.nts (A))-B~O~,
j is the rotational quantum number corresponding to the molecule total
momentum, 1 ia the"quantum number" aasociated with the momentum pro
jection onto the symmetry axi~ for a symmetric to~ molecu1e, and being
calculat!i through the real rotationa.l spectrum 1 ~ (Bj(j+I) - E.k k)
~(A - C) for asymmetric tops. J - +
In many applications the inelaatic tra.naitions influence on the
ela.etic acattering can be neg1ected, that reaulta in the following
formu1ae for DCS and CS
00
G 2~ ~ I ~ 12
(q) "" db b '
Goh• Aoa. , 1- E -= 300 °1<. 6'o·r A"Z . 1-;=2.
2.- E "' 't50 °\( ~ 2.,-{=lf
3-E-=-618°1<. br::
30 30 3 - 6
1 - E -=-1 &8 °\( i
20 zo
z.
io iO
.' 3 E "K
~ 0 '
0 618 :fG8
a b
Fig. I. 'lhe Ar- N2 rotational excitation es va j' (a) and E (b).
698
r.
with (;" (®) being the elastic ecattering DCS, (q, j>(®)) the
"inelastlc ecattering profile" determined by the potintial anieotropy.
2. DIA'IDMIC MOLECULES
RT-proceeees. With the interaction potential for Ar - B2 system
taken from Ref. ,, lin can be obtained as followa
lfYint
2 = ((2j'+I)/(2j+I) (E-E*)/E)I/2 J 212(F/h), (2.1)
qj
where F~~.= -2~J"'~d..-2sh-\2Sivfd. (2E/.}t)1/ 2), q. = j- j','ll~ (j +
j'- I~, E • E (I- (b/j') ), E*= Er(j') - Er(jJ~ Er(j) = B JtJ +I),
E = E .- EJ2, 8J'C- is Ar - N2 reduoea ma.ss, B is the B mole8ule
JP
rltational oonetant, ie the maximal appr8ach dist;gce whioh ~ould
be2ex:preseed through the impact parameter b from the equation b =
5'
(I- W (f )/E ), where W (R) = C exp(-..I.R) is the elaetic interac
tion poteßtial. !n the furthgr calculations the foll~wing values or
parameters gave been taken: el. = 2.9,6; C = I.853"10 ; ~ = 3•10 4;
=
B 9. 2 • IO- ( a. u.). ihe ,,resul te of the CS calcula tions are pre
slnted in Fig. I. Eqs.(I.4) and (2.!) have been ueed to calculate
the rotational excitation RC, ite dependence ve quantum number and
temperature being given in Fig. 2.
V~processes. The deeactivation RC for a process N2(m + I) +
li {0) •N2(m) + N (o) has been calculated as an example of the rota
tfonally averaged2VT traneition RC. Under the adiabatic conditions
Eq. (!.4) gives the following analytical expression
The conven tional no tat ion is mainely ueed here, and V is the
0
699
co111eions frequency, n is the mo1ecular number density,
e~tio
~era!~!ep!~:n~~:fficient of (r-r8) containing term in the in-
The results of the calcu1ations of the Re according to Eq.
(2.2) (Tab1~ 1) show a good agreement with the ..t rajectory calcula
tions data •
VV'-:processes. Let us consider CO(O)+N2(11oo(l)+N2(o) process
RC as an example of a vv.• exchange reaction. 'l'w'o types of analyti
cal approximations for this RC have been proposed.
An analytical fit to the numerical calculations of CS (1.4)
has been considered. The rotationally averaged es of this process
äe a function of energy in log-log ecale has been approximated by
a strait line to yield an expression
"~J 01•10-o. 61 4•. 10- 5 B_1I ,644( cm- 1) a02 •
With this es parametrization the Re under consideration has been
calcu1ated in an analytical form
~~(T) = k*(T/T )'t, (2.3)
0
where k*=l.5·10-14 cm3/s, ~ =1.144, T = 350K.
0
'lhe secong. approximation can be obtained by integration in
Eq. ( 1.1) over f in adiaba tic limi t ( 'I/d .. v>>1) • In this case as
suming the interaction potential to be
V(R,r1,r2) a W0(R) ( l+avvr1r 2),
with avv being a conetant and W (R) being a ~...orse potential, the
RC of VV' exchange in CO-N2 sye~em becomes (n is CO molecule vib
rational quantum number, m is that of N2 one)
ro~(T) ~)(m+ i)(~ ~(2.fD)l/2)l
= 't2({)1/2(nt- (fdß-
( 1+ 2ftE*H(T -T))(stdr (~1/2l/3exp(- ~- ~~(T0-T)),
0
~ n• ~ en( 1 - 2x e (n + 1/2))/2, p1a (<iirß d Jf 42kfr)1/3, (2.4)
f»f,
, ... , 1 , T>T0 and T<T0, ~j •IVnj -Ymjl'
To • ('i{kd(.r)l/2)-1· (2E*)3/2/(3' ~/3_ ,~/33)3/2.
700
Table 1. The comparison of the RC of the process N2(1) +
N2(o) ~ 2~2(0) (upper line) with trajectory cal
culations (bottom line)
---------· ---
---------------------------------------
----------------------------------
TK 1000 2000 3000 4000
In Table 2 the results obtained using Eqs. (2.3) and (2.~
are compared with the other theoretical and experimental data 7.
The cornparison shows that both formulae obtained are in good
agreement with both the trajectory caloulations and experirnent. But
while Eq. (2.3) has been deduced for one specific process only, bq.
(2.4) may be applied to an arbitrary desactivation VV'-reaction of
t~e. kind AB(n1) + CD(n2) ~ AB(n1+ 1) + CD(n2- 1) in adiabatic con
dJ.tJ.one.
VV-proceseee. AB an example the following process hae been
considered N2(m) + N (n) + N (~1) ~ N2(n-l) using the previous
eectione techniques. ~e first approach again gives Eq. (2.3) but
with k* and l depending on m and n and still not depending on the
Table 2. The RC of the CO~~) +3N2(1) ~CO(l) + N2(o) process
multip1ied by 10 orn /e ( a- the reeuits obtained
with the help of' Eq. (2.3), b- the eame f'or Eq.
(2.4), c- tr~jectory calculations data 5, d- e~pe
rimental data , e - collinear model estimations ).
------
TK 100 150 200 250 300- 3-50- -500- -10-00- -20-00- -
-a- 0.35 0.56 0.7-9- 1-.02- 1.26 1.5 -2.4-5- 4-.9-8 --n.o
- -
b 0.87 1.2 ----- 2-37 ----
0 0.53 0.64 0.78 0.93 1.1 1.3 2.0 4.0 6.5
d 0.41 o.59 o.a6 1.2 1-.4 -1.-5 -2.0 6.0
-
e 0.04 -0.1-5 -0.-32 0.52 0.73 0.95
701
