9 Calculation of Electron Impact Ionization Cross Sections
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FRED
E.
STAFFORD
Department of Chemistry and the Materials Research Center, Northwestern University, Evanston, Ill.
The
Gryzinski
theory
for
electron
sections for atoms is examined the total ionization of the contribution shells,
efficiency
agrees well with
the
differences
those of Otvos
with
and
molecular
experimental data.
data. The
of the predictions Stevenson
varies
variation
examined.
cross
theory the from
Calculations
The ratio of the cal
to the
from 0.8 to 1.3.
electron
The
reasons for
are analyzed.
of
importance
of this theory
molecules.
cross section
cross sections are
curve, the relative
available
are made for some diatomic culated
ionization
to the total cross section of inner
and agreement
systematic
impact
with regard to the shape
sum The
of the
atomic
reasons for
this
J η this p a p e r t h e r e a c t i o n e~ + M = M
+
+ 2e~
(1)
w i l l b e c o n s i d e r e d . M m a y b e a n a t o m , or it m a y b e a m o l e c u l e , i n w h i c h case t h e p r o b l e m of f r a g m e n t i v e i o n i z a t i o n w i l l h a v e to b e c o n s i d e r e d ; e~ represents a n e l e c t r o n w i t h k i n e t i c e n e r g y greater t h a n the i o n i z a t i o n p o t e n t i a l of M . T h e rate of this r e a c t i o n is g i v e n b y J ( M ) =i(e-)n(M)la(E) +
(2)
w h e r e I a n d i are the respective currents, n ( M ) is the n u m b e r d e n s i t y of M i n the v o l u m e of l e n g t h I t r a v e r s e d b y the e l e c t r o n b e a m ; σ is a constant c h a r a c t e r i s t i c of M , the energy Ε of the i m p a c t i n g electrons, a n d the p a r t i c u l a r process b e i n g considered—i.e., i o n i z a t i o n to a p a r t i c u l a r state or t o t a l i o n i z a t i o n . T h e q u a n t i t y σ has the u n i t s of area a n d is c a l l e d a cross section. 115
Margrave; Mass Spectrometry in Inorganic Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
116
MASS
S P E C T R O M E T R Y
I N
INORGANIC
C H E M I S T R Y
*T"
3h
ο*
1A
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ÔO
OS/Exp
Dr
1
π
1—
Π—ΓΓ
IB 5#
· •
·
•
• •
•
Ζ
· • •
, 1„ . ,_
OS/Gry
R e a c t i o n 1 is of interest i n studies of stellar atmospheres, the solar c o r o n a , r a d i a t i o n d a m a g e , plasmas a n d n u c l e a r f u s i o n , a n d i n mass spec t r o m e t r y . T h e s p e c i a l interest of this p a p e r is the s t u d y of h i g h t e m p e r a t u r e systems w h e r e the p a r t i a l pressure F of species M is g i v e n b y P=
I(M )T/S +
ay
w h e r e Τ is the absolute t e m p e r a t u r e of the c r u c i b l e or reactor, S is a constant of the mass spectrometer w h i c h is a s s u m e d constant w i t h change of mass, a n d γ is the r e l a t i v e g a i n of a s e c o n d a r y e l e c t r o n m u l t i p l i e r , i f used. M a n y species of interest h a v e b e e n o b s e r v e d o n l y b y o p t i c a l a n d / o r mass s p e c t r o m e t r y ; it is not possible to o b t a i n a n i n d e p e n d e n t measure of t h e i r cross sections. E s t i m a t e s of the cross section, or of r e l a t i v e cross sections therefore f o r m a n i m p o r t a n t p a r t of the c a l c u l a t i o n of t h e r m o d y n a m i c properties u s i n g the absolute e n t r o p y ( T h i r d L a w ) m e t h o d . I n c e r t a i n cases, it m a y b e possible to c o m p a r e S e c o n d a n d T h i r d L a w results to o b t a i n cross sections. I n a d d i t i o n , p a r t i c u l a r l y w h e r e t h e
Margrave; Mass Spectrometry in Inorganic Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
9.
STAFFORD
Ionization
Cross
117
Sections -ι
'
1
π
1—'—Γ
50
Xe
75 Hg Gry/Exp
ic
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ÔO
He
0 Δ o
Ne
Ar
30
Kr
40
Ζ
Journal of Chemical
Physics
Figure 1. a. Logarithm [(Otvos-Stevenson cross section)/(experimental cross section for single ionization X 3)] vs. atomic number Z.; fr. Logarithm [(OS cross section)/classical cross section Table I)] vs. Z.; and c. Logarithm [(classical cross section)/(experimental cross section)] vs. Z. Uncertainties in the measured cross sections are not indicated on the figure, but often are hrge; see especially Kieffer and Dunn ( 14 )
r a t i o of intensities varies g r e a t l y w i t h t e m p e r a t u r e , it is possible to c o m b i n e mass s p e c t r o m e t r i c measurements w i t h mass flow measurements 2, 3, 4, 6, 7, 8, 20, 24)
(I,
to d e d u c e r e l a t i v e cross sections. T h e s e m e a s u r e
ments h a v e p r o v e d v a l u a b l e , b u t often are l i m i t e d i n that t h e y i n v o l v e t a k i n g s m a l l differences b e t w e e n large, u n c e r t a i n q u a n t i t i e s . T h e w o r k of O t v o s a n d Stevenson (22)
has p r o v e d u s e f u l i n p r o
v i d i n g cross sections for atoms. T h e s e authors suggest also that m o l e c u l a r cross sections b e t a k e n e q u a l to the cross sections of the constituent atoms ( a d d i t i v i t y r u l e ) . A s o u r s o p h i s t i c a t i o n i n mass s p e c t r o m e t r y has g r o w n , a n d as w e d e m a n d m o r e accurate answers of the t e c h n i q u e , the O t v o s - S t e v e n s o n c a l c u l a t i o n s h a v e b e e n subjected to i n c r e a s i n g l y m o r e c a r e f u l e x p e r i m e n t a l v e r i f i c a t i o n as is c r i t i c a l l y r e v i e w e d elsewhere 27).
(14,
D e v i a t i o n s f r o m e x p e r i m e n t h a v e b e e n f o u n d . T h e results are s h o w n
i n F i g u r e 1 A (27),
w h e r e the l o g a r i t h m of O S to e x p e r i m e n t a l cross
section ( 60 e. v. ) is p l o t t e d for the atoms as a f u n c t i o n of a t o m i c n u m b e r , Z.
F o r the most r e l i a b l e d a t a a v a i l a b l e — h y d r o g e n , the rare gases, the
a l k a l i metals, a n d the a l k a l i n e earth a t o m s — t h e r e is a significant scatter.
Margrave; Mass Spectrometry in Inorganic Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
118
MASS
S P E C T R O M E T R Y
O f some interest is the p o i n t for S n ( Z =
I N
INORGANIC
C H E M I S T R Y
5 0 ) , for w h i c h the c a l c u l a t e d
cross section is s m a l l . T h e same is t r u e w h e n the c a l c u l a t e d cross section for G e ( Ζ =
32 ) is c o m p a r e d w i t h m o r e recent results. I n b o t h of these
cases a n a s s u m e d secondary electron m u l t i p l i e r g a i n has b e e n u s e d ; this m a y b e a significant source of error. Recently, a theory developed b y G r y z i n s k i (10)
on classical p r i n
c i p l e s has b e e n u s e d to g i v e d e t a i l e d p r e d i c t i o n s of cross sections as a f u n c t i o n of i m p a c t i n g e l e c t r o n energy.
A s u m m a r y of the l i t e r a t u r e as
w e l l as n e w c a l c u l a t i o n s for m a n y of the atoms are g i v e n i n the p r e c e d i n g Downloaded by UNIV OF NORTH CAROLINA on July 14, 2016 | http://pubs.acs.org Publication Date: June 1, 1968 | doi: 10.1021/ba-1968-0072.ch009
p a p e r of this series ( 2 7 ) .
T h e theory states that the cross section
