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On the difference between “classical” and quantum mechanical collision theories
Volume 10, number 2
PHYSICS LETTERS
1June 1964
ON T H E D I F I ~ E R E N C E BETWEEN "CLASSICAL" AND QUANTUM MECHANICAL COLLISION THEORIES L. VRIENS
Physical Laboratory of the University, Utrecht, Netherlands Received 4 May 1964
When c a l c u l a t i n g c r o s s s e c t i o n s for i o n i s a t i o n (or excitation) of a t o m s by e l e c t r o n s , it is possible to make use of two types of collision theor i e s . F i r s t the so called " c l a s s i c a l " t h e o r i e s 15, in which the i o n i s a t i o n p r o c e s s is d e s c r i b e d as a c o l l i s i o n between two f r e e e l e c t r o n s , and second the well known q u a n t u m t h e o r i e s , for example the Born approximation. F o r high e n e r g i e s E 1 of the i m p i n g i n g e l e c t r o n t h e r e is an i m p o r t a n t d i s c r e p a n c y between these t h e o r i e s . Namely, the two p a r t i c l e collision theory p r e d i c t s 2) the i o n i s a t i o n and excitation c r o s s s e c t i o n s Q to be p r o p o r t i o n a l to E l - 1, w h e r e a s the quantum t h e o r i e s p r e d i c t 3) the Q's to be p r o p o r t i o n a l to E l - 1 In E 1. Seaton pointed out 4) that in quantum theory one obtains finite c o n t r i butions f r o m i m p a c t p a r a m e t e r s which a r e much l a r g e r than those which c o n t r i b u t e in " c l a s s i c a l " t h e o r i e s . How fully this i n d i c a t e s the cause of the difference in b e h a v i o u r of the c r o s s section formulae, is d i s c u s s e d in this l e t t e r in t e r m s of d i f f e r e n t i a l c r o s s sections. In the two p a r t i c l e c o l l i s i o n theory G r y z i n s k y d e r i v e d 5) a d i f f e r e n t i a l c r o s s section aAE, 0 d~E dO for an e n e r g y loss AE of the i m p i n g i n g e l e c t r o n to the atomic e l e c t r o n , and a s i m u l t a n e o u s s c a t t e r i n g of the i m p i n g i n g e l e c t r o n by an angle 0. We t r a n s f o r m e d this c r o s s s e c t i o n to a d i f f e r e n t i a l c r o s s s e c t i o n dependent on AE and K, where ~K is the magnitude of the m o m e n t u m change of the incident e l e c t r o n . The r e s u l t i s :
We r e s t r i c t ~ to positive values, c o r r e s p o n d i n g in our f o r m u l a e to an e n e r g y loss of the impinging e l e c t r o n . We a s s u m e that the atomic e l e c t r o n s have an isotropic v e l o c i t y - d i r e c t i o n distribution. F o r E 1 >> A E i s k1 >> K a n d eq. (15 s i m p l i f i e s to: 4 m 3 v e 4 dK
(TAE' K d K d A E ~ - d AE . ~6k2 k 12 /{4
Integration of eq. (3) over K will r e s u l t in the d i f f e r e n t i a l c r o s s section (~AE d ~E, which was given in a p r e v i o u s l e t t e r 2). F r o m eq. (2) it follows that K = Kmin = Kmax for E 2 = 0. Then only one m o m e n t u m change of the impinging e l e c t r o n is p o s s i b l e for one AE. Now we c o m p a r e eqs. (1), (2) and (3) with the corresponding quantum formulae. For simplicity we give the quantum f o r m u l a for excitation of an atom f r o m the ground state 0 to an excited state n, in the case that the t r a n s i t i o n is optically allowed. The c r o s s s e c t i o n s Q a r e known to be proportional to E 1-1 in E 1 for l a r g e E1 both for ioni s a t i o n and for an optically allowed excitation. The d i f f e r e n t i a l c r o s s section for m o m e n t u m change rdf in excitation of an atom to state n is 3):
Ion(K) dK - 8m2~e4 d K t
h4kl 2
Kmin
1
Kmin, max = ( ~ { ~ ± * / E 2 } 170
mAE
k-2kl
and
K m a x ~ 2 k1 .
(25
(4)
(5)
F o r sufficiently s m a l l K, the m a t r i x e l e m e n t eon(/O can be expanded such that E0n(K5 ~ iKz0n , where zon is a constant for Ka o << 1 (a o = r a d i u s of f i r s t Bohr orbitS. Then eq. (45 s i m p l i f i e s to
Ion(K ) dK~, 8m27re4 d K •
2
ontK) I ,
where con(K) is given by M a s s e y 3). F o r high E 1 quantum theory (Born approximation) gives
(~AE, K dK dAE - ]~6k2k124 rn3~ e4 Iklkl- K / K ~dK
where ~k 1 and/~k 2 a r e the i n i t i a l m o m e n t a of the i m p i n g i n g and the atomic e l e c t r o n r e s p e c t i v e l y . Eq. (1) is valid if: k 2 is constant, K m i n < K < K m a x , and AE < E 1 - E 2. Here E 2 is the k i n e t i c energy of the atomic e l e c t r o n and
(3)
12
h4kl2 K IZ0n
"
(6)
Volttme 10, number2
PHYSICS LETTERS
To obtain the total c r o s s section for excitation we have to i n t e g r a t e eq. (4) f r o m Kmi n to Kma x. Since m o m e n t u m t r a n s f e r s with Kao > 1 a r e v e r y i m p r o b a b l e this i n t e g r a t i o n will r e s u l t in Q N - k l - 2 In Kmi n ~ E1-1 In E 1 . So the factor In E 1 is due to the fact that K mni is p r o p o r t i o n a l to E 1-~. But in the two p a r t i c l e c o l l i s i o n theory K m i n is independent of R 1 (see eq. 2). On substil:uting the " c l a s s i c a l " Kmi n in the i n t e g r a t i o n o w T K in eqs. (4) and (6) the r e s u l t would be: Q .... E l - 1 . This m e a n s that for l a r g e E 1 m o s t i o n i s i n g c o l l i s i o n s or c o l l i s i o n s r e s u l t i n g in excitation of the atom belong to v e r y s m a l l mome~atum changes ~K of the impinging electron, 1 1 n a m e l y such, that h-K((2rn AE + 2rnE2)~ - ( 2 m E 2)~. F r o m c o m p a r i s o n of eq. (3) with eq. (6) it follows that, for K a o << 1, the c o r r e s p o n d i n g diff e r e n t i a l c r o s s s e c t i o n s a r e v e r y different. As i n e l a s t i c c o l l i s i o n s with K a o < 1 a r e m o r e i m por~l:ant for l a r g e E 1 than for s m a l l E l , this m e a n s that (for this r e a s o n ) the application of the two p a r t i c l e c o l l i s i o n theory is the m o r e unj u s t i f i a b l e the higher the e n e r g y E 1 . However, for s m a l l E 1 (for example < 10U, where U i s the binding e n e r g y of the atomic e l e c tron) the Kmin, a c c o r d i n g to " c l a s s i c a l " and to quantum m e c h a n i c a l collision t h e o r i e s , a r e not v e r y different. Then application of eq. (6) is no longer p o s s i b l e b e c a u s e K m i n (eq. (5)) is too high, and quantum t h e o r i e s give r e s u l t s not yet very well in a g r e e m e n t with e x p e r i m e n t . F o r these r e a s o n s it is u n d e r s t a n d a b l e that the two p a r t i c l e c o l l i s i o n theory can be applied s u c c e s fully for these low E 1 . We conclude (see also ref. 2) that the p r i n c i p a l d i f f e r e n c e between " c l a s s i c a l and quantum m e c h a n i c a l collision t h e o r i e s is not due to the c l a s s i c a l t r e a t m e n t of the two p a r t i c l e c o l l i s i o n theory, but to the fact that the law of c o n s e r v a t i o n of m o m e n tum gives different values of K m i n for two p a r ticle ( e l e c t r o n - e l e c t r o n ) c o l l i s i o n s as c o m p a r e d with m o r e p a r t i c l e ( e l e c t r o n - e l e c t r o n ( s ) + nucleus) collision, and to the s p e c i a l f o r m of eq. (6). Akerib and Borowitz 6) applied t h e i r " i m p u l s e a p p r o x i m a t i o n " on the i o n i s a t i o n and excitation of the hydrogen atom. In this i m p u l s e a p p r o x i m a -
1June 1964
tion, the i n t e r a c t i o n s between the i m p i n g i n g e l e c t r o n and the n u c l e u s and between the atomic e l e c t r o n and the n u c l e u s a r e neglected during the collision. If this neglection m e a n s that the law of c o n s e r v a t i o n of m o m e n t u m i s valid for the coll i s i o n between the i m p i n g i n g and the atomic e l e c tron, then this i m p u l s e a p p r o x i m a t i o n will give e r r o n e o u s r e s u l t s (compared with the Born approximation) for high E 1. And if the law of cons e r v a t i o n of m o m e n t u m is not valid for the coll i s i o n between the e l e c t r o n s (which is the case in ref. 6), then the e l e c t r o n - e l e c t r o n collision i s not t r e a t e d c o r r e c t l y . F r o m eq. (32) of ref. 6, the d i f f e r e n t i a l c r o s s section c o r r e s p o n d i n g to eqs. (3) and (6), i.e. for l a r g e E 1 and s m a l l K, can be derived. This d i f f e r e n t i a l c r o s s s e c t i o n is of the s a m e type as eq. (3), which also shows that the continuous v e l o c i t y - d i s t r i b u t i o n of the atomic e l e c t r o n is not r e s p o n s i b l e for the logar i t h m i c a l factor in the c r o s s s e c t i o n s Q. We conclude: the i m p u l s e a p p r o x i m a t i o n is not r e l i able (compared with the Born approximation) for l a r g e E 1. The author wishes to e x p r e s s his gratitude to P r o f e s s o r Dr. B. R. A. N i j b o e r and P r o f e s s o r Dr. J. A. Smit for s t i m u l a t i n g d i s c u s s i o n s , and to them and Dr. J. M. Fluit for r e a d i n g the m a n u script. This investigation is p a r t of the r e s e a r c h prog r a m of the Foundation for F u n d a m e n t a l R e s e a r c h of Matter (F.O.M.), f i n a n c i a l l y supported by the N e t h e r l a n d s O r g a n i s a t i o n for P u r e Scientific Res e a r c h (Z.W.O.). References
1} M.Gryzinsky, Phys.Rev. 115 (1959) 374. 2) L.Vriens, Physics Letters 9 (1964} 295. 3) H.S.W. Massey, Handbuch der Physik (SpringerVerlag, Berlin, 1956) Vol. 36, p. 351-358. 4) M.J.Seaton, Atomic and Molecular Processes, ed. D.R. Bates (Academic Press, New York, 1962) p. 377. 5) M. Gryzinski, Reports No. 436 and 447/XV1TI (1963), Institut for Nuclear Research, Swierk k/Otwocka, Poland. 6) R.Akerib and S. Borowitz, Phys. Rev. 122 (1961) 1177.
171
PHYSICS LETTERS
1June 1964
ON T H E D I F I ~ E R E N C E BETWEEN "CLASSICAL" AND QUANTUM MECHANICAL COLLISION THEORIES L. VRIENS
Physical Laboratory of the University, Utrecht, Netherlands Received 4 May 1964
When c a l c u l a t i n g c r o s s s e c t i o n s for i o n i s a t i o n (or excitation) of a t o m s by e l e c t r o n s , it is possible to make use of two types of collision theor i e s . F i r s t the so called " c l a s s i c a l " t h e o r i e s 15, in which the i o n i s a t i o n p r o c e s s is d e s c r i b e d as a c o l l i s i o n between two f r e e e l e c t r o n s , and second the well known q u a n t u m t h e o r i e s , for example the Born approximation. F o r high e n e r g i e s E 1 of the i m p i n g i n g e l e c t r o n t h e r e is an i m p o r t a n t d i s c r e p a n c y between these t h e o r i e s . Namely, the two p a r t i c l e collision theory p r e d i c t s 2) the i o n i s a t i o n and excitation c r o s s s e c t i o n s Q to be p r o p o r t i o n a l to E l - 1, w h e r e a s the quantum t h e o r i e s p r e d i c t 3) the Q's to be p r o p o r t i o n a l to E l - 1 In E 1. Seaton pointed out 4) that in quantum theory one obtains finite c o n t r i butions f r o m i m p a c t p a r a m e t e r s which a r e much l a r g e r than those which c o n t r i b u t e in " c l a s s i c a l " t h e o r i e s . How fully this i n d i c a t e s the cause of the difference in b e h a v i o u r of the c r o s s section formulae, is d i s c u s s e d in this l e t t e r in t e r m s of d i f f e r e n t i a l c r o s s sections. In the two p a r t i c l e c o l l i s i o n theory G r y z i n s k y d e r i v e d 5) a d i f f e r e n t i a l c r o s s section aAE, 0 d~E dO for an e n e r g y loss AE of the i m p i n g i n g e l e c t r o n to the atomic e l e c t r o n , and a s i m u l t a n e o u s s c a t t e r i n g of the i m p i n g i n g e l e c t r o n by an angle 0. We t r a n s f o r m e d this c r o s s s e c t i o n to a d i f f e r e n t i a l c r o s s s e c t i o n dependent on AE and K, where ~K is the magnitude of the m o m e n t u m change of the incident e l e c t r o n . The r e s u l t i s :
We r e s t r i c t ~ to positive values, c o r r e s p o n d i n g in our f o r m u l a e to an e n e r g y loss of the impinging e l e c t r o n . We a s s u m e that the atomic e l e c t r o n s have an isotropic v e l o c i t y - d i r e c t i o n distribution. F o r E 1 >> A E i s k1 >> K a n d eq. (15 s i m p l i f i e s to: 4 m 3 v e 4 dK
(TAE' K d K d A E ~ - d AE . ~6k2 k 12 /{4
Integration of eq. (3) over K will r e s u l t in the d i f f e r e n t i a l c r o s s section (~AE d ~E, which was given in a p r e v i o u s l e t t e r 2). F r o m eq. (2) it follows that K = Kmin = Kmax for E 2 = 0. Then only one m o m e n t u m change of the impinging e l e c t r o n is p o s s i b l e for one AE. Now we c o m p a r e eqs. (1), (2) and (3) with the corresponding quantum formulae. For simplicity we give the quantum f o r m u l a for excitation of an atom f r o m the ground state 0 to an excited state n, in the case that the t r a n s i t i o n is optically allowed. The c r o s s s e c t i o n s Q a r e known to be proportional to E 1-1 in E 1 for l a r g e E1 both for ioni s a t i o n and for an optically allowed excitation. The d i f f e r e n t i a l c r o s s section for m o m e n t u m change rdf in excitation of an atom to state n is 3):
Ion(K) dK - 8m2~e4 d K t
h4kl 2
Kmin
1
Kmin, max = ( ~ { ~ ± * / E 2 } 170
mAE
k-2kl
and
K m a x ~ 2 k1 .
(25
(4)
(5)
F o r sufficiently s m a l l K, the m a t r i x e l e m e n t eon(/O can be expanded such that E0n(K5 ~ iKz0n , where zon is a constant for Ka o << 1 (a o = r a d i u s of f i r s t Bohr orbitS. Then eq. (45 s i m p l i f i e s to
Ion(K ) dK~, 8m27re4 d K •
2
ontK) I ,
where con(K) is given by M a s s e y 3). F o r high E 1 quantum theory (Born approximation) gives
(~AE, K dK dAE - ]~6k2k124 rn3~ e4 Iklkl- K / K ~dK
where ~k 1 and/~k 2 a r e the i n i t i a l m o m e n t a of the i m p i n g i n g and the atomic e l e c t r o n r e s p e c t i v e l y . Eq. (1) is valid if: k 2 is constant, K m i n < K < K m a x , and AE < E 1 - E 2. Here E 2 is the k i n e t i c energy of the atomic e l e c t r o n and
(3)
12
h4kl2 K IZ0n
"
(6)
Volttme 10, number2
PHYSICS LETTERS
To obtain the total c r o s s section for excitation we have to i n t e g r a t e eq. (4) f r o m Kmi n to Kma x. Since m o m e n t u m t r a n s f e r s with Kao > 1 a r e v e r y i m p r o b a b l e this i n t e g r a t i o n will r e s u l t in Q N - k l - 2 In Kmi n ~ E1-1 In E 1 . So the factor In E 1 is due to the fact that K mni is p r o p o r t i o n a l to E 1-~. But in the two p a r t i c l e c o l l i s i o n theory K m i n is independent of R 1 (see eq. 2). On substil:uting the " c l a s s i c a l " Kmi n in the i n t e g r a t i o n o w T K in eqs. (4) and (6) the r e s u l t would be: Q .... E l - 1 . This m e a n s that for l a r g e E 1 m o s t i o n i s i n g c o l l i s i o n s or c o l l i s i o n s r e s u l t i n g in excitation of the atom belong to v e r y s m a l l mome~atum changes ~K of the impinging electron, 1 1 n a m e l y such, that h-K((2rn AE + 2rnE2)~ - ( 2 m E 2)~. F r o m c o m p a r i s o n of eq. (3) with eq. (6) it follows that, for K a o << 1, the c o r r e s p o n d i n g diff e r e n t i a l c r o s s s e c t i o n s a r e v e r y different. As i n e l a s t i c c o l l i s i o n s with K a o < 1 a r e m o r e i m por~l:ant for l a r g e E 1 than for s m a l l E l , this m e a n s that (for this r e a s o n ) the application of the two p a r t i c l e c o l l i s i o n theory is the m o r e unj u s t i f i a b l e the higher the e n e r g y E 1 . However, for s m a l l E 1 (for example < 10U, where U i s the binding e n e r g y of the atomic e l e c tron) the Kmin, a c c o r d i n g to " c l a s s i c a l " and to quantum m e c h a n i c a l collision t h e o r i e s , a r e not v e r y different. Then application of eq. (6) is no longer p o s s i b l e b e c a u s e K m i n (eq. (5)) is too high, and quantum t h e o r i e s give r e s u l t s not yet very well in a g r e e m e n t with e x p e r i m e n t . F o r these r e a s o n s it is u n d e r s t a n d a b l e that the two p a r t i c l e c o l l i s i o n theory can be applied s u c c e s fully for these low E 1 . We conclude (see also ref. 2) that the p r i n c i p a l d i f f e r e n c e between " c l a s s i c a l and quantum m e c h a n i c a l collision t h e o r i e s is not due to the c l a s s i c a l t r e a t m e n t of the two p a r t i c l e c o l l i s i o n theory, but to the fact that the law of c o n s e r v a t i o n of m o m e n tum gives different values of K m i n for two p a r ticle ( e l e c t r o n - e l e c t r o n ) c o l l i s i o n s as c o m p a r e d with m o r e p a r t i c l e ( e l e c t r o n - e l e c t r o n ( s ) + nucleus) collision, and to the s p e c i a l f o r m of eq. (6). Akerib and Borowitz 6) applied t h e i r " i m p u l s e a p p r o x i m a t i o n " on the i o n i s a t i o n and excitation of the hydrogen atom. In this i m p u l s e a p p r o x i m a -
1June 1964
tion, the i n t e r a c t i o n s between the i m p i n g i n g e l e c t r o n and the n u c l e u s and between the atomic e l e c t r o n and the n u c l e u s a r e neglected during the collision. If this neglection m e a n s that the law of c o n s e r v a t i o n of m o m e n t u m i s valid for the coll i s i o n between the i m p i n g i n g and the atomic e l e c tron, then this i m p u l s e a p p r o x i m a t i o n will give e r r o n e o u s r e s u l t s (compared with the Born approximation) for high E 1. And if the law of cons e r v a t i o n of m o m e n t u m is not valid for the coll i s i o n between the e l e c t r o n s (which is the case in ref. 6), then the e l e c t r o n - e l e c t r o n collision i s not t r e a t e d c o r r e c t l y . F r o m eq. (32) of ref. 6, the d i f f e r e n t i a l c r o s s section c o r r e s p o n d i n g to eqs. (3) and (6), i.e. for l a r g e E 1 and s m a l l K, can be derived. This d i f f e r e n t i a l c r o s s s e c t i o n is of the s a m e type as eq. (3), which also shows that the continuous v e l o c i t y - d i s t r i b u t i o n of the atomic e l e c t r o n is not r e s p o n s i b l e for the logar i t h m i c a l factor in the c r o s s s e c t i o n s Q. We conclude: the i m p u l s e a p p r o x i m a t i o n is not r e l i able (compared with the Born approximation) for l a r g e E 1. The author wishes to e x p r e s s his gratitude to P r o f e s s o r Dr. B. R. A. N i j b o e r and P r o f e s s o r Dr. J. A. Smit for s t i m u l a t i n g d i s c u s s i o n s , and to them and Dr. J. M. Fluit for r e a d i n g the m a n u script. This investigation is p a r t of the r e s e a r c h prog r a m of the Foundation for F u n d a m e n t a l R e s e a r c h of Matter (F.O.M.), f i n a n c i a l l y supported by the N e t h e r l a n d s O r g a n i s a t i o n for P u r e Scientific Res e a r c h (Z.W.O.). References
1} M.Gryzinsky, Phys.Rev. 115 (1959) 374. 2) L.Vriens, Physics Letters 9 (1964} 295. 3) H.S.W. Massey, Handbuch der Physik (SpringerVerlag, Berlin, 1956) Vol. 36, p. 351-358. 4) M.J.Seaton, Atomic and Molecular Processes, ed. D.R. Bates (Academic Press, New York, 1962) p. 377. 5) M. Gryzinski, Reports No. 436 and 447/XV1TI (1963), Institut for Nuclear Research, Swierk k/Otwocka, Poland. 6) R.Akerib and S. Borowitz, Phys. Rev. 122 (1961) 1177.
171