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Ionization of atoms by positive ion bombardment near threshold energy
Volume 29A, number 1
PHYSICS LETTERS
can be e x p l a i n e d in two ways: e i t h e r with a b o i l e r m o v i n g at 1.2 × 108 c m / s o r with d e u t e r o n s a c c e l e r a t e d in the a x i s d i r e c t i o n to about 75 keV. T he flux a n i s o t r 0 p y c o m e s f r o m both the CM m o t i o n and the angle dependence of the d i f f e r e n tial c r o s s s e c t i o n in the t a r g e t m o d e l but in the b o i l e r m o d e l it is due only to the p l a s m a motion. The t h e o r e t i c a l r e l a t i v e flux v a r i a t i o n s ~ ( 0 ) / ~ ( 9 0 °) r e f e r r e d to the 90 ° flux ~(90 °) is shown fo r the t a r g e t (curve a) and b o i l e r m o d e l s ( c u r v e b) on fig. 2, the CM v e l o c i t y being 1.2 × 108 c m / s . M e a s u r e m e n t s of the angle dependence n eu t r o n flux a r e made e i t h e r with t h r e e p h o t o m u l t i p l i e r s o r with t h r e e s i l v e r a c t i v a t e d c o u n t e r s . F o r the s a m e i n i ti a l conditions the a n i s o t r o p y v a r i e s f r o m shot to shot and the e x p e r i m e n t a l points a r e a v e r a g e s of s h o t s y i e l d i n g the s a m e n u m b e r of n e u t r o n s (the total y i e l d i s m e a s u r e d at 90o). The a n i s o t r o p y i n c r e a s e s with the n e u t r o n y i e l d ( c u r v e s c and d). When the i n i t i a l d e u t e r i u m p r e s s u r e is l e s s than 2 T o r t the d i s c h a r g e m e c h a n i s m s e e m s to be d i f f e r e n t and the a n i s o t r o p y is a l w a y s i m p o r t a n t w h a t e v e r the n e u t r o n y i e l d ( c u r v e e) [3]. The flux v a r i a t i o n along the p l a t e a u i s c o n s i s t e n t with the b o i l e r m o d e l if one d i s r e g a r d s the v a l u e s n e a r 0 ° and 180 o. The m e a s u r e m e n t s at 180 ° have b e e n
IONIZATION
OF
24 March 1969
done with a head f r e e of m a t t e r f r o m one end of the a x i s to the o t h e r with the e x c e p t i o n of the g l a s s v e s s e l absorption. In the m e a s u r e m e n t s at 160 ° the a b s o r p t i o n of the e l e c t r o d e has b e e n evaluated. N e i t h e r a s i m p l e m i x t u r e of the b o i l e r and the t a r g e t m o d e l s nor a non-fully t h e r m a l p l a s m a could take the v a r i a t i o n into account. The u n e x p e c t e d n eu t r o n y i el d on the a x i s might be ex p l ai n ed supposing a p e r t u r b a t i o n of neutron t r a j e c t o r i e s due to the i n t e r a c t i o n of the neutron m a g n e t i c m o m e n t and the high m a g n e t i c f i el d which confines the hot p l a s m a . In p a r t i c u l a r this would explain that the a n i s o t r o p y is g r e a t e r f o r high neutron y i e l d s p r o d u c e d by the h o t t e r t h e r m a l p l a s m a and g r e a t e r confining m a g n e t i c field. Such a d i s c r e p a n c y f r o m both m o d e l s c e r t a i n l y n eed s to be ex p l ai n ed by f u r t h e r studies.
References 1. N.V. Filippov and T. I. Filippova - L. A. -TR 65261965. 2. P.J. Bottoms, J. P. Carpentier, J.W. Mather, K.D. Ware and A.H.Williams, Novosibirsk, August 1968, CN 24-G5. 3. C. Patou, A. Simonnet and J. P. Watteau, J. Phys. to be published.
ATOMS BY POSITIVE ION BOMBARDMENT NEAR THRESHOLD ENERGY
J. W. SHELDON School of Engineering Science, Florida State University, Tallahassee, Florida, USA Received 4 February 1969
An expression for the cross section for ionization of atoms by ion bombardment near threshold energy is obtained in terms of the photoionization cross section of the target atom and the elastic scattering cross section of the ion-atom collision partners.
R o s e n has f o r m u l a t e d a m e t h o d w h e r e b y low e n e r g y i n e l a s t i c c o l l i s i o n s can be a p p r o x i m a t e d by c o n s i d e r i n g the a t o m i c paths c l a s s i c a l l y ; then the p r o b a b i l i t y of one of the a t o m s u n d e r g o i n g an i n e l a s t i c t r a n s i t i o n is c a l c u l a t e d u s in g the sudden a p p r o x i m a t i o n . T h e m e t h o d r e q u i r e s that the t r a n s i t i o n e n e r g y be a c o n s i d e r a b l e f r a c t i o n of the i n i t i a l r e l a t i v e e n e r g y of the c o l l i s i o n p a r t n e r s . T h i s p r o c e d u r e w a s u s e d by R o s e n to c o m p u t e 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 h e l i u m -
h e l i u m and a r g o n - a r g o n [2] c o l l i s i o n s and has s i n c e been u s e d f o r hydrogen a t o m - a t o m [3] c o l l i sions. The r e s u l t s gave good a g r e e m e n t with the o l d e r e x p e r i m e n t a l w o r k of Rostagni [4] and i s a l s o in r e a s o n a b l e a c c o r d with the m o r e r e c e n t e x p e r i m e n t s of Hayden and A m m e [5]. In this p a p e r R o s e n ' s w o r k is u s e d a s the b a s i s f o r an a p p r o x i m a t e c a l c u l a t i o n of the c r o s s s e c t i o n f o r i o n i z a t i o n of a t o m s by ion b o m b a r d m e n t n e a r t h r e s h o l d energy. The r e s u l t is e x -
Volume29A, number 1
PHYSICS LETTERS
p r e s s e d in t e r m s of the photoionlzation c r o s s section, ~p and the i o n - a t o m elastic s c a t t e r ing c r o s s section, a e. R o s e n ' s method c o n s i s t s of a s s u m i n g the colliding p a r t i c l e s follow c l a s s i c a l paths dictated by their interaction potential, but at a time c o r responding to their distance of c l o s e s t approach, R a time-dependent perturbation is suddenly turned on. The p e r t u r b a t i o n m a t r i x element Vfi between the initial state i and final state f of the s y s t e m decays v e r y slowly b e c a u s e the incident p a r t i c l e has lost m o s t of i t s kinetic e n e r g y in the ionizing collision. The ionization section under these c i r c u m s t a n c e s is given [1] by c-1 (Ti(e) = 21r / F IVfi (R, E)[2bdbdE (1)
E=o b--O
(X, E)Z
w h e r e I is the ionozation e n e r g y of the t a r g e t atom, E is the kinetic e n e r g y of the ejected e l e c t r o n and e is the relative e n e r g y of collision. The perturbation m a t r i x element for the interaction can be written Vfi
(R, E) =J-r t~f (E) V(R) ~i
dT
(2)
where V(R) is the p e r t u r b a t i o n potential, ~i is the wave function for the active e l e c t r o n of atom B in its initial state and ~f(E) is the wave function for the s a m e e l e c t r o n in the continuum in e n e r g y normalization as used by Bates [6]. If it is a s s u m e d that the ion c o r e s do not take p a r t in the interaction, theh the p e r t u r b a t i o n potential is given by e2 e2 V(R) = -R- - ~/R2 + r 2 - 2 R • r (3) w h e r e r is the position v e c t o r of the active e l e c t r o n relative to the t a r g e t atom. The usual expansion of eq. (3) to dipole o r d e r leads to the perturbation e Vfi (R,E) ~-~--~ ~t (E)
(4)
in the rotating atom approximation [7] w h e r e (E) is the dipole m o m e n t for transitions of the
24 March 1969
e l e c t r o n f r o m the initial bound state of B to a state with e n e r g y E in the continuum. Inserting eq. (4) into eq. (1) yields
~i(e)
2~re4E~-lf_ f~ 1~2(E)bdbdE
(5)
=0 b R4(I+E)2 The e n e r g y integral can be evaluated in t e r m s of the photoiontzation c r o s s section, since the dipole m o m e n t is r e l a t e d to the c r o s s section for photeionization by [6] #2(E ) =
3~c 4~2(I+ E)e2 ~p(E).
(6)
Combining eqs. (5) and (6) and using atomic units (length ao, e n e r g y Ry) one obtains 6 ~ I ~ p ( E ) d E f ~ bdb ~ i = ' ~ - - E = 0 (I+E)3 b~0 R 4 (7) w h e r e ct is the fine s t r u c t u r e constant. Photoionization c r o s s sections n e a r threshold can be obtained by fitting experimental data with the e m p i r i c a l e x p r e s s i o n
~p(E) = %(1 + Eh)-n
(8)
w h e r e ~uo is the photoionozation c r o s s section at threshold and n is an e m p i r i c a l constant. Using eel. (8) in the f i r s t integral of eq. (7) and m a k i n K . . ~ h a r d s p h e r e approximation (R=4~e/Trfor b<~andb=R for b> in the second integral finally yields 6~po ~i = ctCrel2(n +2)
[ 1_ (l/e)n+2 1
(9)
Rsfe~'ences 1. 2. 3. 4. 5.
P. Rosen, Phys. Rev. 109 (1958) 348. P. Rosen, Phys. Rev. 109 (1956) 351. J.W.Sheldon, N.A.S.A., TN I~3134 (1965). A. Rostagni, Nuovo Cimento 2 (1934) 621. H. C. Hayden and R. C. Amine, Phys. Rev. 141 (1966) 30. 6. Atomic and molecular processes, ed. D.R. Bates, (Academic Press, New York, 1962). 7. K. Katsuura, J.Chem. Phys. 42 (1965)3771.
PHYSICS LETTERS
can be e x p l a i n e d in two ways: e i t h e r with a b o i l e r m o v i n g at 1.2 × 108 c m / s o r with d e u t e r o n s a c c e l e r a t e d in the a x i s d i r e c t i o n to about 75 keV. T he flux a n i s o t r 0 p y c o m e s f r o m both the CM m o t i o n and the angle dependence of the d i f f e r e n tial c r o s s s e c t i o n in the t a r g e t m o d e l but in the b o i l e r m o d e l it is due only to the p l a s m a motion. The t h e o r e t i c a l r e l a t i v e flux v a r i a t i o n s ~ ( 0 ) / ~ ( 9 0 °) r e f e r r e d to the 90 ° flux ~(90 °) is shown fo r the t a r g e t (curve a) and b o i l e r m o d e l s ( c u r v e b) on fig. 2, the CM v e l o c i t y being 1.2 × 108 c m / s . M e a s u r e m e n t s of the angle dependence n eu t r o n flux a r e made e i t h e r with t h r e e p h o t o m u l t i p l i e r s o r with t h r e e s i l v e r a c t i v a t e d c o u n t e r s . F o r the s a m e i n i ti a l conditions the a n i s o t r o p y v a r i e s f r o m shot to shot and the e x p e r i m e n t a l points a r e a v e r a g e s of s h o t s y i e l d i n g the s a m e n u m b e r of n e u t r o n s (the total y i e l d i s m e a s u r e d at 90o). The a n i s o t r o p y i n c r e a s e s with the n e u t r o n y i e l d ( c u r v e s c and d). When the i n i t i a l d e u t e r i u m p r e s s u r e is l e s s than 2 T o r t the d i s c h a r g e m e c h a n i s m s e e m s to be d i f f e r e n t and the a n i s o t r o p y is a l w a y s i m p o r t a n t w h a t e v e r the n e u t r o n y i e l d ( c u r v e e) [3]. The flux v a r i a t i o n along the p l a t e a u i s c o n s i s t e n t with the b o i l e r m o d e l if one d i s r e g a r d s the v a l u e s n e a r 0 ° and 180 o. The m e a s u r e m e n t s at 180 ° have b e e n
IONIZATION
OF
24 March 1969
done with a head f r e e of m a t t e r f r o m one end of the a x i s to the o t h e r with the e x c e p t i o n of the g l a s s v e s s e l absorption. In the m e a s u r e m e n t s at 160 ° the a b s o r p t i o n of the e l e c t r o d e has b e e n evaluated. N e i t h e r a s i m p l e m i x t u r e of the b o i l e r and the t a r g e t m o d e l s nor a non-fully t h e r m a l p l a s m a could take the v a r i a t i o n into account. The u n e x p e c t e d n eu t r o n y i el d on the a x i s might be ex p l ai n ed supposing a p e r t u r b a t i o n of neutron t r a j e c t o r i e s due to the i n t e r a c t i o n of the neutron m a g n e t i c m o m e n t and the high m a g n e t i c f i el d which confines the hot p l a s m a . In p a r t i c u l a r this would explain that the a n i s o t r o p y is g r e a t e r f o r high neutron y i e l d s p r o d u c e d by the h o t t e r t h e r m a l p l a s m a and g r e a t e r confining m a g n e t i c field. Such a d i s c r e p a n c y f r o m both m o d e l s c e r t a i n l y n eed s to be ex p l ai n ed by f u r t h e r studies.
References 1. N.V. Filippov and T. I. Filippova - L. A. -TR 65261965. 2. P.J. Bottoms, J. P. Carpentier, J.W. Mather, K.D. Ware and A.H.Williams, Novosibirsk, August 1968, CN 24-G5. 3. C. Patou, A. Simonnet and J. P. Watteau, J. Phys. to be published.
ATOMS BY POSITIVE ION BOMBARDMENT NEAR THRESHOLD ENERGY
J. W. SHELDON School of Engineering Science, Florida State University, Tallahassee, Florida, USA Received 4 February 1969
An expression for the cross section for ionization of atoms by ion bombardment near threshold energy is obtained in terms of the photoionization cross section of the target atom and the elastic scattering cross section of the ion-atom collision partners.
R o s e n has f o r m u l a t e d a m e t h o d w h e r e b y low e n e r g y i n e l a s t i c c o l l i s i o n s can be a p p r o x i m a t e d by c o n s i d e r i n g the a t o m i c paths c l a s s i c a l l y ; then the p r o b a b i l i t y of one of the a t o m s u n d e r g o i n g an i n e l a s t i c t r a n s i t i o n is c a l c u l a t e d u s in g the sudden a p p r o x i m a t i o n . T h e m e t h o d r e q u i r e s that the t r a n s i t i o n e n e r g y be a c o n s i d e r a b l e f r a c t i o n of the i n i t i a l r e l a t i v e e n e r g y of the c o l l i s i o n p a r t n e r s . T h i s p r o c e d u r e w a s u s e d by R o s e n to c o m p u t e 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 h e l i u m -
h e l i u m and a r g o n - a r g o n [2] c o l l i s i o n s and has s i n c e been u s e d f o r hydrogen a t o m - a t o m [3] c o l l i sions. The r e s u l t s gave good a g r e e m e n t with the o l d e r e x p e r i m e n t a l w o r k of Rostagni [4] and i s a l s o in r e a s o n a b l e a c c o r d with the m o r e r e c e n t e x p e r i m e n t s of Hayden and A m m e [5]. In this p a p e r R o s e n ' s w o r k is u s e d a s the b a s i s f o r an a p p r o x i m a t e c a l c u l a t i o n of the c r o s s s e c t i o n f o r i o n i z a t i o n of a t o m s by ion b o m b a r d m e n t n e a r t h r e s h o l d energy. The r e s u l t is e x -
Volume29A, number 1
PHYSICS LETTERS
p r e s s e d in t e r m s of the photoionlzation c r o s s section, ~p and the i o n - a t o m elastic s c a t t e r ing c r o s s section, a e. R o s e n ' s method c o n s i s t s of a s s u m i n g the colliding p a r t i c l e s follow c l a s s i c a l paths dictated by their interaction potential, but at a time c o r responding to their distance of c l o s e s t approach, R a time-dependent perturbation is suddenly turned on. The p e r t u r b a t i o n m a t r i x element Vfi between the initial state i and final state f of the s y s t e m decays v e r y slowly b e c a u s e the incident p a r t i c l e has lost m o s t of i t s kinetic e n e r g y in the ionizing collision. The ionization section under these c i r c u m s t a n c e s is given [1] by c-1 (Ti(e) = 21r / F IVfi (R, E)[2bdbdE (1)
E=o b--O
(X, E)Z
w h e r e I is the ionozation e n e r g y of the t a r g e t atom, E is the kinetic e n e r g y of the ejected e l e c t r o n and e is the relative e n e r g y of collision. The perturbation m a t r i x element for the interaction can be written Vfi
(R, E) =J-r t~f (E) V(R) ~i
dT
(2)
where V(R) is the p e r t u r b a t i o n potential, ~i is the wave function for the active e l e c t r o n of atom B in its initial state and ~f(E) is the wave function for the s a m e e l e c t r o n in the continuum in e n e r g y normalization as used by Bates [6]. If it is a s s u m e d that the ion c o r e s do not take p a r t in the interaction, theh the p e r t u r b a t i o n potential is given by e2 e2 V(R) = -R- - ~/R2 + r 2 - 2 R • r (3) w h e r e r is the position v e c t o r of the active e l e c t r o n relative to the t a r g e t atom. The usual expansion of eq. (3) to dipole o r d e r leads to the perturbation e Vfi (R,E) ~-~--~ ~t (E)
(4)
in the rotating atom approximation [7] w h e r e (E) is the dipole m o m e n t for transitions of the
24 March 1969
e l e c t r o n f r o m the initial bound state of B to a state with e n e r g y E in the continuum. Inserting eq. (4) into eq. (1) yields
~i(e)
2~re4E~-lf_ f~ 1~2(E)bdbdE
(5)
=0 b R4(I+E)2 The e n e r g y integral can be evaluated in t e r m s of the photoiontzation c r o s s section, since the dipole m o m e n t is r e l a t e d to the c r o s s section for photeionization by [6] #2(E ) =
3~c 4~2(I+ E)e2 ~p(E).
(6)
Combining eqs. (5) and (6) and using atomic units (length ao, e n e r g y Ry) one obtains 6 ~ I ~ p ( E ) d E f ~ bdb ~ i = ' ~ - - E = 0 (I+E)3 b~0 R 4 (7) w h e r e ct is the fine s t r u c t u r e constant. Photoionization c r o s s sections n e a r threshold can be obtained by fitting experimental data with the e m p i r i c a l e x p r e s s i o n
~p(E) = %(1 + Eh)-n
(8)
w h e r e ~uo is the photoionozation c r o s s section at threshold and n is an e m p i r i c a l constant. Using eel. (8) in the f i r s t integral of eq. (7) and m a k i n K . . ~ h a r d s p h e r e approximation (R=4~e/Trfor b<~andb=R for b> in the second integral finally yields 6~po ~i = ctCrel2(n +2)
[ 1_ (l/e)n+2 1
(9)
Rsfe~'ences 1. 2. 3. 4. 5.
P. Rosen, Phys. Rev. 109 (1958) 348. P. Rosen, Phys. Rev. 109 (1956) 351. J.W.Sheldon, N.A.S.A., TN I~3134 (1965). A. Rostagni, Nuovo Cimento 2 (1934) 621. H. C. Hayden and R. C. Amine, Phys. Rev. 141 (1966) 30. 6. Atomic and molecular processes, ed. D.R. Bates, (Academic Press, New York, 1962). 7. K. Katsuura, J.Chem. Phys. 42 (1965)3771.