Comparing numerically calculated capture cross sections with experimental reaction cross sections for NH3/+/ + NH

Dugan, J. V., Jr.

English

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NASA TM X-52903 
COMPARISON OF NUMERICAL CAPTURE CROSS SECTIONS WITH 
EXPERIMENTAL REACTION CROSS SECTIONS FOR NH; + NH3 
by John V. Dugan, Jr, 
Lewis Research Center 
Cleveland, Ohio 
TECNN'ICAL PAPER proposed for presentation at Twenty- third 
\ Gaseous Electronics Conference sponsored by the American 
I 
! Physical Society and the United Aircraft Research Laboratories 
1 Hartford, Connecticut, October 20-23, 19'70 
https://ntrs.nasa.gov/search.jsp?R=19700032585 2020-03-23T18:39:12+00:00Z
COMPARISON OF NmIC& C m L T E  C R O S S  SECTIONS WITH 
by John V. Dugan, Jr. 
Nat iona l  Aeronaut ics  and Space Administrat ion 
Lewis Research Center 
Cleveland, Ohio 
ABSTRACT 
Numerically c a l c u l a t e d  capture  c r o s s  s e c t i o n s  (i- a r e  compared 
C 
+- 
with  Ryan's experimental  r e a c t i o n  c ros s  s e c t i o n s  f o r  NH 3 + NH3 
NH; + NH The numerical r e a c t i o n  c r o s s  s e c t i o n  ii = 0.6 o i s  2 R c 
4- 
obta ined  assuming t h a t  v i b r a t i o n a l l y  e x c i t e d  NH do not  r e a c t ,  
3  
Theory and experiment agree  s a t i s f a c t o r i l y  from thermal energy 
t o  nea r ly  1 eV al though t h e i r  s lopes  a r e  s l i g h t l y  d i f f e r e n t  and both  
s t e e p e r  than  Langevin. 
INTRODUCTION 
Experimental c r o s s  s e c t i o n s  have been r epor t ed  by ~ ~ a n l  f o r  t h e  
NH -parent  i o n  r e a c t i o n  a t  low ion  ene rg i e s  where: 3 
The purpose of t h i s  no te  i s  t o  compare t h e  numerical ly  c a l c u l a t e d  
cap tu re  c ros s  s e c t i o n s  wi th  Ryan's experimental  r e s u l t s .  These numer- 
i c a l  cap tu re  c ros s  s e c t i o n s  f o r  ion-d ipole  c o l l i s i o n s  have been ob- 
t a i n e d  by so lv ing  many randomly genera ted  t r a j e c t o r i e s , 2 ' 3  These 
cap tu re  c ros s  s e c t i o n s  should s e t  upper limits t o  r e a c t i o n  c r o s s  
s e c t i o n s  i n  t h e  reg ion  where r o t a t i o n a l  quantum numbers a r e  l a r g e  so  
4 
t h a t  cLass i ca l  approximations a r e  v a l i d .  Chupka has shown t h a t  
NH; i n  h igher  v i b r a t i o n a l  s t a t e s  r e a c t  l e s s  e f f i c i e n t l y  t o  produce 
+ NH;. Ryan r e p o r t s  t h a t  t h e  v i b r a t i o n a l  d i s t r i b u t i o n  of NH s p e c i e s  3 .  
i n  h i s  experiment i s  such t h a t  t h e  observed r e a c t i o n  c r o s s  s e c t i o n  
should be  only 60% of t h e  capture  c r o s s  ~ e c t i o n , ~  (assuming t h a t  t h e  
r e a c t i o n  e f f i c i e n c y  f o r  unexci ted  i o n s  i s  u n i t y ) .  I n  t h e  r eg ion  of 
i o n  energy from thermal  t o  one eV t h e  permanent d ipo le  p l a y s  a 
dominant r o l e  i n  determining t h e  cap tu re  c ros s  s ec t ion .  293 Thus, 
one should not  expect  t h e  Langevin formula t o  p r e d i c t  a c c u r a t e  va lues  
o r  s lope  f o r  t h e  cap tu re  c r o s s  s e c t i o n .  
HePERIMENTAL REAClTION CROSS SECTIONS 
I n  Ryan's s tud ies1  an app l i ed  r e p e l l e r  vo l t age  s e t s  an upper l i m i t  
t o  t h e  i o n  energy a t  t h e  e x i t .  A t  e x i t  energ ies  E somewhat g r e a t e r  
M 
t han  thermal  ( 0 , 3  t o  4 e ~ )  t h e  r e a c t i o n  c ross  s e c t i o n  i s  obta ined  
from t h e  rileasured r a t e  c o e f f i c i e n t  k by t h e  r e l a t i o n  v = k/(v) 
1/2 - 
where (v)  = (2F1/ mr) ; i s  t h e  average ion  energy = E /4 and 
ni 
m i s  t h e  reduced mass. Because o f  t h e  u n c e r t a i n t i e s  about  t h e  ion  
r 
6 
energy d i s t r i b u t i o n  i n  t h e  thermal  reg ion  t h e  experimental  c r o s s  sec-  
t i o n  f o r  thernlal energy i s  h e r e i n  obta ined  both  from t h e  Maxwell 
averaged r e s u l t s  and by e x t r a p o l a t i o n  of t h e  h igher  energy r e s u l t s  t o  
t h e  thermal  reg ion .  
1\.IONOENERGETPC NUIE3RICA.L CROSS SECTIONS 
The nuraerical cap ture  c r o s s  s e c t i o n s  v a r e  obta ined  from t h e  
C 
cap tu re  r a t i o  C~ which i s  t h e  f r a c t i o n  of c o l l i s i o n s  i n  which t h e  
i o n  and molecule approach wi th in  a  s p e c i f i e d  sepa ra t ion  (22  was 
2  
used f o r  a l l  c a l c u l a t i o n s  i n  t h i s  paper ) .  The capture  r a t i o s  f o r  
4- 
NH + NH3 a r e  p l o t t e d  aga ins t  t h e  square o f  t h e  impact parameter i n  3 
F i g ,  1 f o r  3 d i f f e r e n t  i o n  v e l o c i t i e s .  F i f t y  t r a j e c t o r i e s  were 
ca l cu la t ed  f o r  each p o i n t .  The h igher  i o n  v e l o c i t i e s  correspond t o  
i on  energ ies  o f  0 .2 and 0 . 8  eV; t h e  lower v e l o c i t y  corresponds t o  
r e l a t i v e  thermal  energy. These CR values  a r e  obta ined  f o r  t h e  same 
random number s e t s ;  it has  been shown t h a t  t h e  v a r i a t i o n  of  c ros s  
s e c t i o n s  wi th  random number s e t s  i s  w i t h i n  20%. The simple s t e p  
func t ion  behavior: which would b e  expected f o r  Langevin capture  c o l l i -  
s i o n s  wi th  NH i s  shown f o r  comparison a t  b  = b The corresponding 3 L' 
monoenergetic c r o s s  s e c t i o n  u i s  simply n t imes  t h e  a r e a  under 
c  
2  t h e  curve o f  CR a g a i n s t  b  . Thi s  c r o s s  s ec t ion  i s  p l o t t e d  a g a i n s t  
r e l a t i v e  t r a n s l a t i o n a l  energy E i n  F ig .  2.  The experimental  c ros s  
s e c t i o n  U and Langevin c ros s  s e c t i o n  a (us ing  NfI p o l a r i z -  
exx' c  3 
a b i l i t y  = 2.26 i3) a r e  p l o t t e d  f o r  comparison. 
COMPARISON OF RESULTS 
The numerical cap ture  c ros s  s e c t i o n  o has a  s lope  E -0. Gg 
C 
whereas t h e  experimental  c ros s  s e c t i o n  has  a s t e e p e r  s lope  u~ -0.78 
- 1 0  Ryan obta ined  a  r e a c t i o n  r a t e  c o e f f i c i e n t  of  18.1~10 cm3 s e c - l  a t  
350 K which corresponds t o  a  c r o s s  s e c t i o n  a 02 = 181 A 
exp 
2  4 ( 1 . 8 1 ~ 1 0 ' ~ ~  cm ) f o r  an average v e l o c i t y  o f  ( ~ k T / r n n ) ~ / ~ =  9 . 5 ~ 1 0  cm s e c  -1 
( e  = 0,030 e ~ ) .  This  a va lue  i s  only s l i g h t l y  lower than  t h e  
exp 
O2 
ex t r apo la t ed  va lue  of  185 A . A t h e o r e t i c a l  va lue  of  t h e  thermal  
c o l l i s i o n  c o e f f i c i e n t  (o v) can be obta ined  by i n t e g r a t i n g  t h e  mono- 
C 
e n e r g e t i c  o  over a Maxbrellian d i s t r i b u t i o n  a,t T = 350 K, The 
c  
express ion  f o r  t h e  capture  c o l l i s i o n  c o e f f i c i e n t  i s  
where (oo )c  = 33 x2 (ev)"' 
n  = 1 - n '  and n '  de sc r ibes  t h e  power dependence of t h e  numerical 
-n' 
c ross  s e c t i o n ,  i . e . ,  o  = ( c * ) ~  t . The n' i s  0.69 f o r  o  
c  C 
and 0.78 f o r  c . The i n t e g r a l  o f  Eq. (1) i s  
exp 
where ( 1 3 )  i s  t h e  gamma funct ion  and kT = E = 0,030 eV. For 
4 thermal v e l o c i t y ,  ( v )  = 9.5X10 crn sec-', t h i s  correspond-s t o  
02 
a  c ros s  s e c t i o n  a x 347 A o r  a  t h e o r e t i c a l  r e a c t i o n  c ros s  s e c t i o n  
C 
02 
of 208 A (us ing  oR = 0.6 o ) . This  l a t t e r  va lue  i s  about 13% l a r g e r  
C 
than  t h e  ex t r apo la t ed  experimental  c ros s  s e c t i o n  which i s  s a t i s f a c t o r y  
agreement. 
The experimental r eac t fon  c ros s  s e c t i o n  of F ig ,  2 i s  0 .45 t o  
0.50 of  t h e  numerical values over t h e  range of r e l a t i v e  energy shown. 
The ion-d ipole  i n t e r a c t i o n  i s  t h e  ch i e f  p o t e n t i a l  term over t h i s  
range.  This  c ros s  s e c t i o n  r a t i o  i s  i n  s a t i s f a c t o r y  agreement wi th  
6 t h e  r e a c t i o n  e f f i c i e n c y  f a c t o r  of 0.6,  a l though t h e  s lope  of  a 
exp 
i s  a  l i t t l e  s t e e p e r  than  f o r  
ac, 
-t- The numerical ly  c a l c u l a t e d  capture  c ross  s e c t i o n s  f o r  NH on 3 
I\TH a r e  i n  good agreement w i th  experimental va lues  i n  t h e  low energy 3 
r eg ion  from thermal  t o  s e v e r a l  eV i f  a cons tan t  r e a c t i o n  e f f i c i e n c y  
i s  used. The dependence of t h e  c ros s  s e c t i o n  on r e l a t i v e  t r a n s -  
l a t i o n a l  ( o r  i on )  energy i s  a l s o  i n  s a t i s f a c t o r y  agreement w i t h  t h e  
exper imenta l ly  observed E -" 78 For tu i tous  agreement between 
Langevin theory  and experiment sugges ts  t h a t  t h e  r e a c t i o n  e f f i c i e n c y  
rnasks t h e  r o l e  which t h e  permanent d i p o l e  p l ays  i n  t h e  capture  mech- 
anism. B e t t e r  agreement could be obta ined  i f  t h e  r e a c t i o n  e f f i c i e n c y  
were known a s  a func t ion  of  r e l a t i v e  energy, E a r l i e r  experiments 
suggested t h a t  t h e  o va lues  were Langevin-like.'  Although t h e  
exp 
d i p o l e  term determines t h e  capture  c ros s  s e c t i o n  t h e  abso lu t e  va lue  
o f  t h i s  c r o s s  s ec t ion  i s  cons iderably  l e s s  than  t h e  maximum capture  
c ros s  s e c t i o n  and coinci .dental ly  near  t h e  Langecin capture  c ros s  
s e c t i o n .  Th i s  behavior  has  been d iscussed  f o r  o the r  p o l a r  t a r g e t s  i n  
mass spectrometry.  2,3 
NUMERlCAL CAPTURE C. S. oc 
NUMERICAL REACTION a~ = 0.6 ac 
D(PTIL REACTlON C. S. aexp 
--- LANGNIN  CAPTURE C. S. CJL 
MAXWELL AVERAGED 
(oc) = (ucv)/(v) for i = 356 K 
uexp 1181 i2) 
lo-' 10-1 
RELATIVE TRANSLATIONAL ENERGY, 6 ,  eV 
Figure 2. - Variation of numerical, experimental and Langevin capture cross sections 
w i th  relative (ion-molecule) translational energy for NH; + NH3 capture and reac- 
tive collisions. 
Figure 1. -Var iat ion of capture ratio w i th  impact parameter for NH; + NH3 
capture coll ision at 3 ion velocities. Target rotators are  chosen from a 
heat bath at 350 K. 
1. K. R. Ryan, paper presented at 18th Annual Conference on Mass 
Spectrometry and Allies Topics, San Francisco, June 1970 
(to be published in J, Chem. Phys.). 
2. J. V. Dugan, Jr, and J. L. Magee, J. Chem. Phys. - 47, 3 103 (1967). 
- .. 3. J. V. Dugan, Jr., J. H. Rice, and J. L. Magee, NASA TM X-1586 
(1968). 
4. W. R. Chupka, J. Chem. Phys. -J 48 1527 (1968). 
5. K. R. Ryan, C.S.I.R.O., Camden, Australia, private communication. 
6. E. W. McDaniel, in Methods of Experimental Physics, Vol. 7, 
Atomic and Electron Physics, Part A, Atomic Interactions 
(~cademic Press, New York, 1968), Ch. 4. 
7. B. M. Hughes and T. 0. Tiernan, Aerospace Research Lab. WPAFB, 
Dayton, Ohio, private communication. 


Comparison of numerical capture cross sections with experimental reaction cross sections for NH3/+/ + NH3

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