Volume 8, number 4
PHYSICS
CALCULATION
He, He*,
OF ABSOLUTE
He +, Ne, Ne*,
LETTERS
15 F e b r u a r y 1964
IONISATION
CROSS
Ne +, Ar,
Ar*,
Hg
SECTIONS
and Hg*
OF
L. VRIENS
Physical Laboratory of the University, Utrecht, Netherlands
Received 27 January 1964
R e c e n t l y G r y z i n s k i d e v e l o p e d a t h e o r y 1,2) of
a t o m i c c o l l i s i o n s by w h i c h it i s p o s s i b l e to c a l c u l a t e
in good a p p r o x i m a t i o n a b s o l u t e c r o s s s e c t i o n s f o r
i o n i s a t i o n of a t o m s by e l e c t r o n s . T h e p r i n c i p a l i d e a
of t h i s t h e o r y i s t h a t i o n i s a t i o n of an a t o m can be
a p p r o x i m a t e d a s a t w o - p a r t i c l e c o l l i s i o n , in w h i c h
the i o n i z i n g e l e c t r o n " h i t s " one a t o m i c e l e c t r o n .
T h e a t o m i s i o n i s e d when the e n e r g y t r a n s f e r to t h e
a t o m i c e l e c t r o n i s g r e a t e r than i t s b i n d i n g e n e r g y .
This is valid for each atomic electron separately;
thus the t h e o r y c a n a l s o be a p p l i e d to i n n e r - s h e l l ionis a t i o n . F o r the c a l c u l a t i o n of an a b s o l u t e c r o s s s e c t i o n u s e i s m a d e of t h e e x p e r i m e n t a l l y known i o n i s a t i o n e n e r g y U and the n u m b e r of e l e c t r o n s in the
s h e l l . In t h i s t h e o r y no f r e e p a r a m e t e r i s i n t r o d u c e d . Some a p p r o x i m a t i o n t h a t a r e of no f u n d a m e n t a l i m p o r t a n c e a r e m a d e . We r e m a r k t h a t the
a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t b e c o m e s
b e t t e r b e c a u s e of t h e s e a p p r o x i m a t i o n s .
In c o n t r a d i c t i o n to G r y z i n s k i ' s f i r s t p a p e r 1) he
now 2) c h o o s e s an e n e r g y d i s t r i b u t i o n of the a t o m i c
electron:
(@2
1
f ( E 2) dE 2 : ~ _ . , exp -
dE 2 ,
(1)
w h e r e E 2 i s the k i n e t i c e n e r g y of the a t o m i c e l e c t r o n , and w h e r e he t a k e s E e q u a l to t h e a v e r a g e
k i n e t i c e n e r g y of t h a t e l e c t r o n . T h i s g i v e s the f o l lowing e x p r e s s i o n f o r the c r o s s s e c t i o n f o r d e t a c h m e n t of one e l e c t r o n f r o m the a t o m :
When the number of electrons in the shell k is Nk,
corresponding to a binding energy Uk, the cross
section becomes Nk Qik. In order to get the total
cross section for ionisation of an atom we have to
sum only over the different shells:
Qi tot : F , NkQik "
k
(4)
W i t h eq. (4) we c a l c u l a t e d 3) i o n i s a t i o n f u n c t i o n s
f o r H e , H e * , He + , N e , Ne*, Ne+, A r , A r * , Hg
and Hg*. We i n d i c a t e d t h e m e t a s t a b l e s t a t e s by an
a s t e r i s k . Our r e s u l t s a r e g i v e n in f i g s . l t o 7 . We
c o m p a r e o u r r e s u l t s with e x p e r i m e n t a l l y m e a s ured absolute cross sections, except for ionisation
f r o m t h e m e t a s t a b l e s t a t e s , a s f o r t h i s p r o c e s s the
i o n i s a t i o n f u n c t i o n s h a v e not y e t b e e n m e a s u r e d .
T h e r e a s o n why we c a l c u l a t e d t h e s e i o n i s a t i o n f u n c tions for metastable atoms is twofold: first because
t h e y a r e i m p o r t a n t f o r g a s e o u s d i s c h a r g e s and s e c o n d l y b e c a u s e we a r e b u i l d i n g a p p a r a t u s to m e a s ure them.
We s e e t h a t in m o s t c a s e s t h e a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t i s v e r y good, e s p e c i a l
ly f o r i o n i s a t i o n of A r and He +.
An e x c e p t i o n we m u s t m a k e f o r i o n i s a t i o n of Ne.
I n t e r e s t i n g i s t h a t the t h e o r y can be a p p l i e d v e r y
w e l l to i o n i s a t i o n of i o n s ; f r o m t h i s we c o n c l u d e
t h a t t h e e x t r a C o u l o m b a t t r a c t i o n in t h e s e c a s e s
h a s no g r e a t r e s u l t i n g e f f e c t on the i o n i s a t i o n c r o s s
sections.
2E+ U
Qi
=
%E ( E1 ~sI_~ + g2 ( 1 - ~ 1 )
~
\E+E1 /
1n(2.7+
w h e r e E 1 i s t h e k i n e t i c e n e r g y of t h e i m p i n g i n g e l e c t r o n and % = 6 . 5 6 × 1 0 -14 c m 2 eV 2. In f i r s t a p p r o x i m a t i o n G r y z i n s k i t a k e s E e q u a l to U w h i c h i s e x a c t ly t r u e f o r the h y d r o g e n a t o m . Then eq. (2) s i m p l i f i e s to:
%
Vi = ~22 g(x) , with x= E1/U and
(3)
1 x-]
7[
1
g(x) =x~+~ )
1 + 2 ( 1 - - ~ x ) In ( 2 . 7 + J ' x - 2 - ~ l .
260
)I(1-E~)
E+U
, (2)
T h e a u t h o r w i s h e s to e x p r e s s h i s g r a t i t u d e to
P r o f . D r . J. A. S m i t f o r s t i m u l a t i n g d i s c u s s i o n s and
to M r . A. J. L e e n h o u t s f o r c a r r y i n g out t h e n u m e r i cal calculations.
T h i s i n v e s t i g a t i o n i s p a r t of the r e s e a r c h p r o g r a m m e of t h e F o u n d a t i o n f o r F u n d a m e n t a l R e s e a r c h of M a t t e r (F. O. M. ), f i n a n c i a l l y s u p p o r t e d
by t h e N e t h e r l a n d s O r g a n i z a t i o n f o r P u r e S c i e n t i f i c R e s e a r c h (Z. W. O. ).
Volume 8, number 4
PHYSICS
15 F e b r u a r y 1964
LETTERS
®
®
F',.:,
'~' Ar
s3
~He 2'S(~
~2~S
A r s~ ".\:
Ne
tu
ss
"
E
Ne
"T
2'
S
He1'S
4o
so
12o
6
'
'
'4'o'
'
'sb'
'
'1~o
2
u
6.
;.°",,,
....... o
>::: .......
o
ii:I
'
'
' 4~'
'
' 80'
'
' 1~0
,
.................... o........ ,
0
40
04
80
120
0
40
electron energy(eV)
eo
120
0.3
02
(21
200
400
600
Figs. 1-7. The calculated ionisation functions a r e given by solid c u r v e s , the contributions
of the different shells by dashed c u r v e s . In fig. 2a and b c o r r e s p o n d to the 3s
r e s p e c t i v e l y 2p shell of Ne s 3 (c=a +b). In fig. 3a and b c o r r e s p o n d to the 4s
and 3p shell of A t s 3 (c=a+b). In fig. 4 we denote by a, b and c the 6s, 5d2J
and 5dl~ shell. In figs. 5 and 6 we denote by a, b, c and d the 6p, 6s, 5d2~ ~knd
5dl~ shall. The m e a s u r e d functions a r e f r o m Smith 4) (xxx), Blealmey 5, 6)~.
(ooo~} and Dolder et al. 7, 8) (+++).
1) M.Gryzinski, Phys.l~ev. 115 (1959) 374.
2) M.Gryzinski, Reports No. 436, 447 and 448/X'VIII, Institute for Nuclear Research, Swierk k/Otwocka, P o land (1963) *.
3) L.Vriens, Rapport F. O. M. 16847 (1963), unpublished.
4) P. T. Smith, Phys. Rev. 36 (1930) 1293.
5) W. Bleakney, Phys.Rev. 36 (1930) 1303.
6) W. Blealmey, Phys.Rev. 35 (1930) 139.
7) K. T. Dolder, M. F. A. H a r r i s o n and P. C. Thoneman,
P r o c . R o y . Soc. 264 (1961) 367.
8) K . T . D o l d e r , M. F . A . H a r r i s o n and P. C . T h o n e m a n ,
P r o e . R o y . Soc. 274 (1963) 546.
* T h e s e r e p o r t s a r e available f r o m the P o l i s h G o v e r n m e n t
C o m m i s s i o n e r for Use of N u c l e a r Energy, the office of
scientific, technical and e c o n o m i c a l information, P a l a c e
of Culture and Science, Warshaw, Poland.
261