Numerical values of momentum transfer cross sections sigma sub m for ion-dipole collisions are compared with the corresponding capture cross sections sigma sub c as a function of ion velocity and rotational temperature. For values of dipole moment mu from 1 to 4 Debyes the sigma sub m/sigma sub c ratio is in the range 1.2 to 2.0 (roughly). This is in contrast to the simple relation for Langevin collisions where sigma sub m/sigma sub c is equal to or approximately 1.10 independent of polarizability of the target atom. At low temperatures, the momentum transfer cross sections can be as large as 2000 A squared but they are only about 15 to 30 percent of the total scattering cross sections sigma sub S

Dugan, J. V., Jr.

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THE RELATION BETWEEN MOMENTUM TRANSFER AND CAPTURE AND
TOTAL SCATTERING CROSS SECTIONS FOR ION-DIPOLE COLLISIONS
by John V. Dugan, Jr.
Lewis Research Center
Cleveland, Ohio
TECHNICAL PAPER proposed for presentation at
Twenty-fifth Gaseous Electronic Conference
London, Ontario, October 17-20, 1972
https://ntrs.nasa.gov/search.jsp?R=19730001958 2020-03-23T09:11:18+00:00Z
THE RELATION BETWEEN MOMENTUM TRANSFER AND CAPTURE
AND TOTAL SCATTERING CROSS SECTIONS FOR
ION-DIPOLE COLLISIONS
by John V. Dugan, Jr.
Lewis Research Center
National Aeronautics and Space Administration
Cleveland, Ohio
ABSTRACT
Numerical values of momentum transfer cross sections a for ion-m
dipole collisions are compared with the corresponding capture cross sections
CT as a function of ion velocity and rotational temperature. For values of
\s
dipole moment M from 1 to 4 Debyes the (o"m/0"c) ratio is in the range 1,2
to 2.0 (roughly). This is in contrast to the simple relation for Langevin
collisions where (or /a ) = 1.10 independent of polarizability of the target
Lii ' \s
atom. At low temperatures, the momentum transfer cross sections can be
as large as 2000 A , but they are only about 15 to 30 percent of the total
scattering cross sections cr .
s
INTRODUCTION
Totalscattering cross sections for ion-dipole collisions have been
calculated from a scattering ratio (refs. 1 and 2) just as capture cross
sections for these encounters were calculated from a capture ratio ? r f l l " ;
(refs. 3-5). The total scattering cross section, a , is
rCTS = 7Jo C(b2)d(b2) (1)
where C is the fraction of trajectories for which scattering in the center-
S ' ' •
of-mass system is greater than a minimum angle 9 for a fixed impact
parameter b. Generally one hundred trajectories were studied per point
and results plotted as histograms of fractional scattering intensity I_(0)
S
versus the scattering angle 0 (in increments of 0.2 radians). This intensity
I (9) is already integrated over the iiicremental solid angle sin 9 d&
S ' "-' • '
so that
where I is the total number of trajectories studied at fixed b, i.e., the
incident intensity. Results of I ' (6) distributions are somewhat rough but
S ' ,
indicate that there is considerable small angle scattering at large impact
parameters, i.e., 25 « b s 50 A, so that total scattering cross sections
a_ range from 1000 to 8000 A2 (ref. 2).S
DETAILS OF THE CALCULATION
The momentum transfer cross section which is of interest for the
diffusion of ions through polar gases is effectively the total scattering cross
section weighted for large angle scattering, i. e.,
/
r®d(b2)/
... X:
(2k)
where C is a normalization constant. Thus we can simply calculate
this momentum transfer cross section from a modified scattering ratio
C via
r77/
4/0
C' (b2)d(b2) (2b)
O i
where the approximate scattering ratio is
ft-:-.-.
Cs(b) =G I ( 0 ) (1 -cos0) . (3)
C.,, the normalization constant, is the reciprocal of the incident intensity.
The summation in Eq. (3) is done for^qual 0.2 radian increments with
(1 - cos 0) in each case evaluated at the midpoint of the interval.
Total scattering cross sections calculated for symmetric top polar
targets were larger^ihan for linear molecules of identical dipole moment
(refs. 1 and 2). This effect was expected on the basis of the Stark Effect
(ref. 2). In the case of the momentum transfer cross section, however, the
(1 - cos 9) factor minimizes any significant differences in the amplitude of
small angle scattering which is dominant at large impact parameters. Thus,
momentum transfer cross sections for ions in polar gases are not expected
to be very sensitive to target geometry;
' 4Numerical calculations were done for ion velocities of 5X10 and
5 - 110 cm sec and for target rotational temperatures of 77 and 300 K; Most
of the data consists of 100 trajectory sets for fixed impact parameter b.
The exception to this were 200 trajectory collision sets which were done for
constant b to obtain a relatively smooth Ia(0j for T0 = 77 K, v = 5X10 cm
1 ! S ' *V ' . - . ( • " ' ' . '
sec" . Generally, the scattering distributions are somewhat rough* At
first it might appear that Ia(Q) could be assumed independent of scatterings . .
 0 • .
angle at smaller impact parameters (<20 A). However, scattering in multi-
ple reflection encounters is not always isotropic in the center-of -mass sys-
tem at the smaller impact parameters, and the single reflection captures
•' • |, • ' ' " ' - , • ;
result in mainly small-angle scattering, i.e. , 0 < 0. 4 radians (ref; 2).
RESULTS AND DISCUSSION
Results of scattering intensity I_(0) for symmetric top C Ho CN targets
S ' ' ! y
are shown versus scattering angle -6 for Tp = 77 K at three different
impact parameters in Figs. l(a) to l(c); Two hundred trajectories were
solved for b = 10 A, 15 A, and 20 A. As was mentioned, since nearly half
the collisions are multiple reflections (corresponding to ion-molecule complex
formation) at 10 and 15 A it is expected that the I0(0) might be nearly iso-s
tropic. However, this is not the case at the lower impact parameter where
I (0) resembles a monotonically decreasing distribution mainly because of
O ' • . .-,'•;•
I_(0) for single reflection capture collisions (refs. 1 and 2); The results'for
S
 o . • • . • .b =. 15 A are much more nearly isotropic,, In all cases the collisions with
greater than 15 captures were assumed to have an isotropic distribution in 8.
The variation in I (0) results suggests that scattering in ion-dipole
S • ' * i " " ' . . - ' .
captures is considerably different than the Langevin case where all scattering
is isotropic for b values less than the critical impact parameter (ref; 6).
A more systematic study of the variation of Ia(0) with various collisionS •
parameters was not undertaken because the computer time necessary was
prohibitive; However, comparison of the aforementioned results (for a
dipole moment of 3^92 Debyes) with 1(0) for HC1 (M = Iv08 Debyes) (to be
, ... • S
discussed later) suggest a strong dependence of the form of the scattering
intensity on dipole moment. Thus, the relation between a' and a is
III L/
not simple as in the Langevin case but rather a function of relative energy
and molecular constants, particularly dipole moment.
Representative variations of C! with b2 for linear''CHQCN" mole-
: S
 f : . - , . , .'••• O
cules (dipole moment and polarizability of CH«CN) selected from heat baths
at TR = 77 and 300 K is shown in Fig. 2(a) for v = 5X104 cm sec"1 (ref. 2).
The open triangles are for an additional 100 cases and are an indication of
the C* dependence on random number sets; Comparison between results for
O .
100 case and 200 case trajectory sets at the same impact parameter suggest
that C* can be estimated within 15 percent using only 100 trajectories at each
" • ' 2 'impact parameters The corresponding C'(b )- plots are shown in Figs. 2(b)s • ..
and 2(c). Since there is considerable scattering intensity at small
. •< 9 < 0.2 rad) C* is considerably less than C0 at b = 25 A. Also C'O S S n S
is not a simple monotonically decreasing function of b as are CQ ando
capture ratio Cw. The capture cross section a is'simply defined in terms
XX O • •.
of the capture ratio
°
 S{
At smaller b values, as mentioned, the rough I_(0) cannot be approxi-s
mated by a uniform scattering intensity so that C' is never as large as CL.s • s
The momentum transfer crosssection~for ~-T = 77 K is ir times the
integral under the curve in Fig. 2(b), i.e., =2100 A . The momentum
transfer cross section is considerably less,for the higher rotational tempera-
ture, TR = 300 K, am s 1100 A2 suggesting a rough dependence am ~ (TR)~°* 5
based on comparison for these two temperatures.
The variation of momentum transfer d and capture cross sections
cr with ion velocity is shown for TT> = 77 K in Fig. 3. These a depend-
• n 7ft •• o 4*\
ences on relative energy, e , and rotational temperature, T0 are
-0 • 5 -0 '43 ' •steeper than the e" ' and (kTjJ~ * dependences for a , the total
scattering cross section (refs. 1 and 2). The corresponding dependences
for a are se-°-76 and kT0~°*33 for CH,CN.C rt . o-
Finally,, the variation of momentum transfer cross section with dipole
moment value is shown in Fig. 4 where C^ results for CHgCN (/i =
3.92 D. U.) are compared with results for HC1 (ju = 1.08D.U.) at v = 5X10
cm sec"1, T0 = 77 K. The a value for HC1 s 485 A2 is almost twice
y iC ' III
the corresponding capture.cross section. The results indicate that a
values are nearly proportional to dipole moment just as in.the case of the
capture a and total scattering cross sections a_.
C . , 5s
Comparison of am values with a values is done for rotational
temperatures T,-. of 77 and 300 K at a velocity of v = 5x10 cm sec" in
r t : ' . • • • . . . • - • - .
Table I. All of the differences in cross sections are outside the numerical
uncertainty due to different random number sets feo the quantity or AT is
; - - - ' HI t/
meaningful.
CONCLUDING REMARKS
Momentum transfer cross sections, a , for ion-dipole collisions are
much larger than Langevin cross sections oy for non-polar molecules of
moderate polarizability at rotational temperatures TR of 300 K and 77 K.
Also the a values are generally about L2 to 2.0 times the corresponding
capture cross section whereas the Langevin a /a is always 1.105. The
111 L/
a values for ion-dipole systems are only a small fraction of total scatter-
ing cross sections ag, Thus the momentum transfer cross section is
generally larger than the capture cross section but only 20 to 30 percent of
the total scattering cross section. Diffusion coefficients for ions in polar
gases should be smaller than for non-rpolar molecules. A more systematic
study of the variation of am with all collision parameters is desirable. It
is also desirable to use larger random number sets in order to obtain
smoother distributions in scattering intensity.
REFERENCES
1. J. V. Dugan, Jr. and J. L. Magee, Paper presented at 24th Annual
Gaseous Electronics Conference, Gainesville, Florida, October 1971.
2. J. V. Dugan, Jr.. and J. L. Magge, Submitted to J. Chem. Phys.,
(1972).
3. J. V. Dugan, Jr. and J. L. Magee, J. Chem. Phys., £7, 3103,
(1967).
4. J. V. Dugan, Jr., J, H. Rice and J. L. Magee, Chem. Phys. Letters,
2, 323 (1969).
5. J. V. Dugan, Jr. and R. B. -Canright, Jr., J. Chem. Phys., 56,
3623 (1972);
6. E. W. McDaniel, Collision Phenomena in Ionized Gases, (John Wiley
and Sons, Inc., New York, 1964), Ch. 9, pp. 427-487.
C H 3 C N , v = 5xl04 C M - S E C " 1
• T R .
K
77
300
3r
A2
2000-2100
1000-1100
p.,
°VA2
1310 ;
750 V;
. I '
Table I.
Ic(9)
40
30
20
10
0
20
10
(alb-10 A.
(b) b " 15 A.
30
20
10
.2 3 -v.
0 , R A D ' : • ' • ' . • • '
' (c) b • 20 A.
Figure 1. - Variations of scattering intensity with center-of-
mass scattering angle for ion collisions with CfoCN targets^
cfiosenlfrom a bath afrotational temperature of TR - 77 K
for impact parameters of 10, 15, and 20 A.
"^j 100 CASE SETS
77 - 200 CASE SETS
SCATTERING 6
FRACTION,
.2 -
0 600 1200 i 1800 2400 . 3000 3600
(a) VARIATION OF TOTAL SCATTERING CROSS SECTION WITH
SQUARE OR IMPACT PARAMETER FOR ION COLLISIONS AT
VELOCITY OF 5x10* CM SEC"1 WITH CH3CN SYMMETRIC
TOP TARGETS.
M.T.
SCATTERING
RATIO,
1.00,-
.75
.50
.25
bjj, ~ 650-670 A2
i 2000-2100 A2
'm
0 200 400 600 800 1000 1200 1400 1600
(b) VARIATION OF MOMENTUM TRANSFER SCATTERING RATIO
WITH SQUARE OF IMPACT PARAMETER FOR ION COLLISIONS
AT VELOCITY OF SxlO4 CM SEC"1 WITH CH3CN TARGETS
CHOSEN FROM A BATH AT TD - 77 K.
( M.T.
SCATTERING
'\ RATIO,
1.00
.75
.50
.25
b2,-330-345 A2
om ~ 1000-1100 A2
I I
100 200 300__4QO_ 500 600 700
(c) VARIATION OF MOMENTUM TRANSFER SCATTERING RATIO
WITH SQUARE OF IMPACT PARAMETER FOR ION COLLISIONS
AT VELOCITY OF SxlO4 CM SEC'1 WITH CH3CN TARGETS
CHOSEN FROM A BATH AT T - 300 K. (ALL 100 CASE SETS. )
Figure 2.
6X103,-
CH3CN TR - 77 K
CROSS SECTION,
o,
°9A2
103
102
10r2 ID-1
RELATIVE ENERGY, e, eV
1.0
Figure 3. - Variation of momentum transfer and
capture cross sections with relative transla-
tional energy for ion collisions with CH3CN
targets chosen fromja heat bath at rotational
temperature of 77 K.
M.T.
SCATTERING
RATIO,
C'
A HCI
o CH3CN
0 100 200 300
 0
(IMPACT PARAMETER)2 b2 ,A2
Figure 4. -Comparison of variations of momentum transfer scattering
ratios with square of impact parameter for ion collisions at velocity
of 5xl(r cm sec'1 with targets having dipole moments of CH3CN
~ and HCI chosen from bath at TR • 77 K.


The relation between momentum transfer and capture and total scattering cross sections for ion-dipole collisions

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