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A SEMI-EMPIRICAL, FORMULA FOR THE TOTAL K-SHELL IONIZATION CROSS SECTION BY
ELECTRON IMPACT
C. Jakoby, H. Genz, A. Richter
To cite this version:
C. Jakoby, H. Genz, A. Richter. A SEMI-EMPIRICAL, FORMULA FOR THE TOTAL K-SHELL
IONIZATION CROSS SECTION BY ELECTRON IMPACT. Journal de Physique Colloques, 1987,
48 (C9), pp.C9-487-C9-490. �10.1051/jphyscol:1987978�. �jpa-00227399�
C o l l o q u e s u p p l e m e n t a u n 0 1 2 , Tome 48, d b c e m b r e
A SEMI-EMPIRICAL FORMULA FOR THE TOTAL K-SHELL IONIZATION CROSS SECTION BY ELECTRON IMPACT"'
C. JAKOBY, H. GENZ a n d A. RICHTER
Institut fifr Kernphysik, Technische Hochschule Darmstadt, D-6100 Darmstadt. F.R.G.
A b s t r a c t :
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The e v a l u a t i o n o f some 6 0 0 m e a s u r e m e n t s o f e l e c t r o n i n d u c e d K - s h e l l i o n i z a t i o n c r o s s s e c t i o n s f o r e l e m e n t s w i t h 6i
Z5
9 2 a n d i m - p a c t e n e r g i e s o f I < E o5
5 GeV y i e l d e d a s e t o f r e l i a b l e d a t a , w h i c h w e r e u s e d f o r t h e 3 e ~ i v a t i o n o f a s e m i - e m p i r i c a l f o r m u l a , t h a t e n a b l e s t h e p r e d i c t i o n o f O K f o r a l l e l e m e n t s a n d e l e c t r o n b o m b a r d i n g e n e r g i e s . B a s e d o n t h e t w o - p a r a m e t e r B e t h e f o r m u l a a t h r e e - p a r a m e t e r f o r m u l a was was d e r i v e d , w h i c h d e p e n d s o n f3 a n d Z a n d t h e t h r e e Z - d e p e n d e n t c o n - s t a n t s . They a r e n e e d e d t o d e s c r i b e p r o p e r l y p o s i t i o n , s l o p e a n d h e i g h t of t h e c r o s s s e c t i o n maximum a s w e l l a s t h e e n e r g y b e h a v i o u r . They w e r e d e d u c e d by f i t t i n g t h e t h r e e - p a r a m e t e r f o r m u l a t o t h e e v a l u a t e d d a t a o f e l e v e n e l e m e n t s . S i n c e t h e c o n s t a n t s e x h i b i t a Z d e p e n d e n c e , a power l a w o r p o l y n o m i a l s w e r e f i t t e d t o t h e s e q u a n t i t i e s . The m l y d e r i v e d f o r m u l a p r e d i c t s t h e t h e e l e c t r o n i n d u c e d K - s h e l l i o n i z a t i o n c r o s s s e c t i o n f o r a l l Z a n d i m p a c t e n e r q i e s t o a b o u t ' 1 3 % .1 . I n t r o d u c t i o n
D e s p i t e t h e f a c t t h a t a v a s t b o d y o f e x p e r i m e n t a l d a t a h a s b e e n c o l - l e c t e d d u r i n g t h e l a s t y e a r s o n t h e m e a s u r e m e n t o f t h e e l e c t r o n i n d u c e d X - s h e l l i o n i z a t i o n c r o s s s e c t i o n oK
,
t h e r e a r e s t i l l n o d a t a f o r many e l e m e n t s a t a l l , o r f o r c e r t a i n e l e m e n t s o n l y i n a v e r y s m a l l e n e r g y i n - t e r v a l,
a n d s o m e t i m e s t h e y a r e a l s o v e r y c o n t r a d i c t o r y . S i n c e t h e know- l e d g e o f oK i s n o t o n l y o f t h e o r e t i c a l i n t e r e s t b u t a l s o o f t e n u s e f u l i n many b r a n c h e s o f b a s i c a n d a p p l i e d r e s e a r c h , i t seems t o b e d e s i r a b l e t o e v a l u a t e t h e d a t a a n d combine t h e r e l i a b l e v a l u e s i n o r d e r t o d e r i v euppott potted by the Deutsche ForschungsgemeinechaZt
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987978
C9-488 JOURNAL DE PHYSIQUE
a formula that describes aK for all elements and energies.
For this purpose we have evaluated the available measurements con- cerning their consistency with regard to the fluorescence yields and a unique error assignment and selected for various elements (Z=6,7,8,10, 18,22,28,29,47,50 and 79) spread - although not for all elements - over an impact energy range of I„ < E0 < 2 GeV , where I„ denotes the K-shell binding energy.
2. The Formula
To the evaluated set of data, obtained as pointed out above, we fit- ted a three-parameter formula, which was derived from the two-parameter relativistic Bethe expression [1], Besides the general E and Z depen- dence it had to obey in addition several conditions:(i) for non relati- vistic energies the formula should agree with the predictions of the exact expression of the Bethe formula [2]/(2) the cross section should vanish for Eo = I„ and (iii) the slope of the cross section has to be zero at its maximum, which occurs at about E„ j 4I„ . The fitting pro- cedure will be explained in detail in a forthcoming paper [ 3 ] . We thus obtained the semi-empirical formula in barn
aR = aFi(Z,g)[F2(B) + b F3( Z , 6 ) + F„(Z)F5(Z,6)c] (1)
with
F!(Z,g) = 254.9/(IK3s) (2)
F
2(6) = ln[eV(-|-B
2)] - B
2(3)
F3( z , e ) = 1 - ( e0 2/ e2) (4)F„(Z) = ln(1/Bo2) (5)
F5(Z,B) = B o V g2 (6)
32 = 1 - (1 + E0/ m c2) ~2 (7)
Bo2 = 1 - (1 + IK/ m c2) -2, (8)
where E0 denotes the electron impact energy and I the K-shell binding energy. The three constants a, b and c were found to be Z dependent. We thus fitted a power law to a and polynomials to b and c and thus re- ceived the expressions
a(Z) = 5 . 1 4 Z- 0 , 4 8 (9)
b(Z) = 5.76 - 0.04Z (10)
c(Z) = 0.72 + 0.039Z - 0.0006Z2 . (11)
- K - S H E L L I O N I Z A T I O N CROSS S E C T I O N - -
I N C I D E N T K I N E T I C ENERGY (keV)
Fig.1
-
Evaluated experimental data of the electron induced K-shell ionization cross section [ 3 ] . The solid line repre- sents the prediction of the semi-empirical formula according to Eq. (1).Applying Eqs. (1)
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(11) one obtains aK for all elements Z and within the considered impact energy range to an accuracy of +13%. A typical example is given for 4tAg as shown in Fig.1. It was chosen for pre- sentation since it is one of those targets where measurements exist from threshold to the GeV-region. The evaluated data points are the ex- perimental results obtained by twelve different groups. The solid line represents the calculations according to Eq.(l). The agreement between the semi-empirical formula and experimental data are of the same qua- lity for all elements.3. Conclusion
Using the evaluated set of data points it has become possible to de- rive a three-parameter semi-empirical formula that describes the elec- tron induced K-shell ionization cross section to within +13%. A com- parison with existing theoretical approaches shows a good agreement with the calculations by Scofield [ 4 ] and Eschwey and Manakos [ 5 1 . This will be subject of a more comprehensive forthcoming paper
C31.
C9-490 JOURNAL DE PHYSIQUE
R e f e r e n c e s [ I ] B e t h e , H . : Ann. P h y s . :
5
( 1 9 3 0 ) 325[ 2 ] P o w e l l , C.J.: Rev. Mod. P h y s .
48
( 1 9 7 6 ) 33[ 3 ] J a k o b y , C . , Genz, H . , R i c h t e r , A.: t o b e p u b l i s h e d [ 4 ] S c o f i e l d , J . H . : Phys. Rev. @ ( 1 9 7 8 ) 963
[ 5 ] Eschwey, P . , Manakos, P.: 2 . P h y s .
