Calculations of charge transfer cross section

Garcia, J. D.

Gerjuoy, E.

Welker, J. E.

English

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R -  3 $ -  o / l - - o 4 A -  
* 
Class ica l  Calculations of Charge Transfer Cross Sections 
J. D. Garcia and E. Gerjuoy 
Department of Physics 
University of Pi t tsburgh 
Pittsburgh , Pennsylvania 15213, U. S .A. 
and 
Jean E. Welker 
Greenbelt , Maryland , U . S .A .  iu NASA4oddard Space Flight Center a i 2  2 s  a *  
ce 
zz This paper describes r e s u l t s  obtained thus far i n  a recently 
1 i n i t i a t e d  c r i t i c a l  examination 
procedures2 fo r  estimating caarge transfer cross  sect ions.  
the s igpif icance of our work, it i s  necessary first t o  recognize t h a t  
Gryzinski's prescr ipt ion f o r  computing charge t r a n s f e r  cross sect ions re- 
quires 8. p r i o r i  knowledge of  each of the  following e s sen t i a l ly  d i s t i n c t  
items : 
6f the  u t i l i t y  of Gryzinski ' s  c l a s s i c a l  
TO grasp 
(i) The cross sect ion a;Eff(v,,v2) f o r  producing an energy t r a n s f e r  
AE i n  the  co l l i s ion  of two charged p a r t i c l e s  of  unequal masses m ( i n  
charge t r a n s f e r  the  masses respectively of t h e  incident ion and of the 
9 3  electron)  moving w i t h  arbitrary ve loc i t ies  vl ,v2 i n  the  laboratory system, 
averaged over all orientat ions of v1,v2 f o r  f ixed speeds v1,v2. 
3 3  
(ii) The range of energy t r ans fe r  -- from (AE)I1 t o  (AElU -- within 
GD WPOd A l l l 1 3 V d  which the re  can be s ign i f i can t  probabi l i ty  of e lectron capture , remembering 
t h a t  the  physically observed charge t r a n s f e r  cross section (as a function of 
incident ion speed v 
i ts  i n i t i a l l y  bound o r b i t )  i s  
for spec i f ied  speed v2 of t h e  captured electron i n  1 
https://ntrs.nasa.gov/search.jsp?R=19680018333 2020-03-12T10:02:43+00:00Z
2 
(iii) The d is t r ibu t ion  i n  speeds f(v2)dv2 of the i n i t i a l l y  bound 
to-be-captured e lec t rons ,  i n  each of  t he  o r b i t s  from which capture can 
take place,  
( i v )  
e f f ec t ive  number N 
which independently can co l l ide  with -- and be captured by -- the  incident 
I n  each of  t he  o r b i t s  from which capture can take place,  the  
of e lectrons ( e  . g o ,  presumably one i n  H ,  two i n  H e )  e 
ion.  
I n  terms of the  above items (i) - ( i v )  , our contribution can be 
described as follows : 
( 1) We have "improved" on Gryzinski ' s o r ig ina l  formulation by 
3 employing i n  Eq, (1) the  exact formula f o r  instead of the  approximate 
expression Gryzinski suggests. It is  not evident t h a t  use of the  exact 
ueff ac tua l ly  improves the  agreement between theory and experiment 
ever employment of the exact formula eliminates uncertaint ies  concerning 
the e f f ec t s  of approximating u;:, therewith permitt ing somewhat l e s s  
equivocal evaluation of t he  u t i l i t y  of Gryzinski's prescr ipt ion.  As a 
matter of  f a c t ,  wi th  t he  exact ueff Eq, (1) can be integrated i n  closed 
form. 
How- AE 
AE 
U 
For instance,  i f  f o r  a l l  0 < (AE) < AE 6 (AE) R 
V 
(2) 1 
V 2 (E1 - E2 - / E  - - E  - I )  AE 6 45%? 
(ml + m2) "1 "2 
(where E1,E2 a re  the  k ine t i c  energies of the p a r t i c l e s )  then f o r  an incident 
simply charged ion 
Simi la r  formulas f o r  ulo hold when t h e  a l t e rna t ive  inequal i t ies3  t o  Eq. ( 2 )  
are obeyed throughout t h e  allowed range of AE, or when t h e  inequa l i t i e s  
change as AE increases the  range from (AE)& t o  (AEIU, 
3 
(2 )  We have examined various p o s s i b i l i t i e s  f o r  (AE)R and (AE)u, 
i n  addition t o  the l i m i t s  spec i f ica l ly  proposed by Gryzinski,2 which 
were 
1 2  (AE$ = zmvl Q u2 - u1 4a) 
where U2 ,U1 respectively are the  ionization poten t ia l s  of the  t ransfer r ing  
electron i n  i t s  i n i t i a l  and f i n a l  bound o rb i t s .  Alternatives t o  ( h a )  are 
especial ly  needed when U1 3 U2, i n  which event Eq. (1) m a k e s  ulo i n f i n i t e  
1 2 2 at incident ion energies - a  v d U1 - U2. It i s  t r u e  tha t  Gryzinski 
proposes a geometrical upper bound t o  t h e  cross sect ion,  but it i s  not 
c lear  t h a t  the proposed bound i s  well-founded or  generally useful. In  any 
event,  t he  AE limits of Eq. ( 4 )  cause ul0 computed from Eq-. (1) t o  be in-  
consistent wi th  detailed balancing. 
be computed from Eq. (1) only when U2 3 U ; otherwise the desired c l a s s i ca l  
estimate of u10 f o r  electron capture by ion 1 from atom 2 i s  t o  be found 
by de ta i led  balancing from the c lass ica l ly  estimated -- using Eq. (1) -- 
a10 f o r  electron capture by ion 2 from atom 1. 
value of 
a, 
t r ans fe r ) .  
For t h i s  reason we suggest t h a t  dl0 
1 
With t h i s  procedure the  
actual ly  employed i n  Eq. (1) always i s  3 0, and the resu l tan t  
always is f i n i t e ,  except at zero energy when U = U1 (resonant charge 10 2 
( 3 )  Calculations of alQ have been performed and (wheneveq possible) 
compared w i t h  experiment for  protons incident on the  noble gases, and f o r  
protons incident on t h e  alkali atoms. The a&roximate as w e l l  as the exact 
expressions f o r  aeff have been employed, and a l te rna t ives  t o  Eqs. ( 4 )  have 
been examined. 
detailed balancing when U2 4 U1 also has been tested. 
AE 
The u t i l i t y  of the suggestion that u be estimated from 
The sequence of 
10 
of charge t r a n s f e r  co l l i s ions  from the  a l k a l i s  a r e  pa r t i cu la r ly  useful  
f o r  t h i s  purpose, This sequence has the  fu r the r  advantage t h a t  t he re  
no room f o r  argument concerning the  value of N 
sure ly  Ne = 1 f o r  t h e  a l k a l i s .  
defined i n  ( i v )  above; 
seemingly a re  equally 
e 
The values of N e 
wel l  known (they are  1 of course) f o r  the noble gas sequence. We a l so  
have examined the  behavior of u from Eq. (1) f o r  charge t r ans fe r  i n t o  
exci ted states of the impingingproton; i n  such react ions we must under- 
stand how t o  weigh -- within Gryzinski's c l a s s i c a l  framework -- the  f a c t  
tha t  f o r  capture i n t o  an 
able n independent quantum mechanical f i n a l  s t a t e s  e All calculat ions 
= U2, i . e . ,  t ha t  have assumed the  only s ign i f i can t  v s a t i s f i e s  -i-mv2 2 
f (v2)  of (iii> i s  a 6-function. 
10 
/ 
there  a re  avai l -  2 o rb i t  of energy U1 = - 13.6/n 
2 
1 2  
Results of t he  above calculat ions w i l l  be i l l u s t r a t e d  and dis- 
cussed. Because the  var ie ty  of possible charge t r a n s f e r  reactions i s  s o  
g rea t ,  we do not f e e l  t h a t  these l imi ted  invest igat ions should be the  
so l e  basis f o r  sweeping generalizations concerning the  u t i l i t y  of 
Gryzinski?s procedures f o r  estimating charge t r a n s f e r  cross sections.  
It does seem fair  t o  say, however, tha t  our experience suggests t hese .  
procedures a re  s ign i f i can t ly  l e s s  useful  f o r  charge t r a n s f e r  than f o r  
ionizat ion by e lec t rons ,  where Bauer and Bartky 
are  r e l i a b l e  t o  within a f ac to r  of about th ree .  
4 f e e l  Gryzinski ' s  methods 
R e f  e r ence s 
1. 3. E. Welker, H. T a i  and E, Gerjuoy, Bul le t in  h e r .  Phys. Soc. 11, 
167 (1966). 
2. 
3. 
4. 
M. Gryzinski , Phys. Rev. E@, A336 (1965). 
E. Gerjuoy, Phys. Rev. 148, 54 (1966). 
E. Bauer and C, D, Bartky, J. Chem. Phys. - 43, 2466 (1965). 
* 
This research was  sponsored i n  pa r t  by the National Aeronautics and Space 
Administration under Contract Nqber  - NGR-39-011-035, and i n  p a r t  by the  
Advanced Research Projects Agency through the  U.S. Army Research Office 
Durham. 


Classical calculations of charge transfer cross sections

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