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Collective oscillations of electronic shells of atoms
Volume 14, number 1
PHYSICS LETTERS
COLLECTIVE
OSCILLATIONS
OF
ELECTRONIC
1 January 1965
SHELLS
OF
ATOMS
M. Ya. AMUSIA
A. F. Joffe Physical-Technical Institute, Academy of Science, Leningrad Received 7 December 1964
R e c e n t e x p e r i m e n t s on i n e l a s t i c c o l l i s i o n s of a r gon ions [1] have shown that t h r e e c h a r a c t e r i s t i c m a x i m a e x i s t in [he c r o s s s e c t i o n as a function of e n e r g y l o s s . The p u r p o s e of this a r t i c l e is to give a p o s s i b l e i n t e r p r e t a t i o n of this phenomenon, as a c o l l e c t i v e e x c i t a t i o n of an atom. It i s w e l l - k n o w n that c o l l e c t i v e o s c i l l a t i o n s e x i s t in a s p a c e - i n f i n i t e and h o m o g e n e o u s m a n y e l e c t r o n s y s t e m . We m a y c o n s i d e r them as a bound s t at e of a p a r t i c l e - h o l e p a i r . In an a t o m between a p a i r of e l e c t r o n s a l s o a c t Coulomb f o r c e s and the p h y s i c a l r e a s o n f o r the e x i s t e n c e of a bound state, the a t t r a c t i o n of a p a r t i c l e - hole p a i r - is the s a m e as in the e l e c t r o n gas. Let us suppose that such c o l l e c t i v e o s c i l l a t i o n s do e x i s t t. We can think that in an a t o m as a r e s u l t of a good s p ace and e n e r g y s e p a r a t i o n of d i f f e r e n t s h e l l s e a c h s h e l l o s c i l l a t e s s e p a r a t e l y . F o r an e s t i m a t e we find the f r e q u e n c y f r o m the p l a s m a formula tt 1
~ n = (4~e2Pn)~ ,
(1)
w h e r e Pn i s the e l e c t r o n density of the n ' t h shell. We can e x c i t e the o s c i l l a t i o n by i n e l a s t i c s c a t t e r i n g of e l e c t r o n s on a t o m s . The c r o s s s e c t i o n of this p r o c e s s is of the o r d e r of 10 -17 c m 2. The o s c i l l a t i o n in spite of being longitudinal can be e x c i t e d by d i r e c t a b s o r p t i o n of a ~ - q u a n turn. B e c a u s e o f P n N Z3n elf , a n ~ Zn~2eff , w h e r e Z n e f f is an e f f e c t i v e c h a r g e of n u c l e i Z for the n'~h shell. The s o - c a l l e d o n e - p a r t i c l e l e v e l s of the a t o m i c s h e l l - m o d e l depend on Z as Z2 eft " This d i f f e r e n c e m a k e s it p o s s i b l e to d e m o n s t r a t e the e x i s t e n c e of c o l l e c t i v e o s c i l l a "~ The assumption of the existence of atomic c o l l e c t i v e o s c i l l a t i o n i s n o t a n e w one [ 2 , 3 ] . It h a s b e e n discussed in literature and originates from J e n s e n ' s
and Bloch's papers of the thirties [2]. But the picture proposed h e r e g i v e s another number and energy of oscillations in an atom (see [2]). t t We don't write here a multiplier which is connected with bound conditions of a finite system (about 1.1 - 1.4). Having in mind rough estimates we omit t h i s correction. 36
tions by s y s t e m a t i c studying of d i f f e r e n t atoms. T h e s e q u a l i t a t i v e c o n s i d e r a t i o n s m a k e it p o s si b l e to explain the r e s u l t s of r e c e n t e x p e r i m e n t s [1]. The c o l l i s i o n of an ion and an A atom w as studied at e n e r g i e s in the r a n g e f r o m 10 to 50 keV. A f t er the c o l l i s i o n both p a r t i c l e s w e r e fixed and t h e i r k i n et i c e n e r g i e s and c h a r g e s a r e d e t e r m i n e d . T h r e e m a x i m a w e r e o b s e r v e d on the c u r v e of dependence of c r o s s s e c t i o n on the ine l a s t i c i t y R of the p r o c e s s . The p o s i t i o n of t h ese m a x i m a is independent of the i m p a c t p a r a m e t e r and the e n e r g y of the i n c o m i n g ion. When the c h a r g e of a p a r t i c l e is known, it is p o s s i b l e to c a l c u l a t e the total i o n l s a t i o n potential ~i U / - the e n e r g y which p r o d u c e d the ionlsation. Then the dependence was studied of c r o s s s e c t i o n s of p r o c e s s e s on R* = R - ~i U/. T h r e e n a r r o w m a x i m a w e r e also o b s e r v e d and t h e i r p o s i t i o n s w e r e independent of the c h a r g e s of ions a f t e r c o l l i s i o n . The widths of t h e s e l i n e s was of the o r d e r 10-20 eV, and the magnitude R*-53, 263, 475 ~ 15 eV). A c c o r d i n g to [1] it s e e m s they a r e not onep a r t i c l e . The f i x ed position of l e v e l s allows us to make an a s s u m p t i o n that they c h a r a c t e r i s e the e x c i t a t i o n s p e c t r u m of a single atom. The p r o b a b i l i t i e s of being e m i t t e d for one o r s e v e r a l e l e c t r o n s f r o m an e x c i t e d a t o m a r e of the s a m e o r d e r [1]. This is in a c c o r d a n c e with our h y p o t h e s i s of a c o l l e c t i v e n a t u r e of t h e s e l e v e l s . Indeed, if the o s c i l l a t i o n is e x c i t e d in a shell, all e l e c t r o n s of h i g h e r s h e l l s a r e in an " e x t e r n a l " f i el d (apart f r o m the s e l f - c o n s i s t e n t H a r t r e e Fock field}. We can show that ~2n i s much l a r g e r than the f r e q u e n c y of r o t a t i o n of e l e c t r o n s in ex t e r n a l s h e l l s . As a r e s u l t the " e x t e r n a l " f i el d suddenly a c t s upon t h e s e e l e c t r o n s . If the amount of e n e r g y is sufficient any n u m b e r of e l e c t r o n s m ay l e a v e the atom. Let us c o m p a r e the e x p e r i m e n t a l data obtained in [1] with e s t i m a t i o n s b a s e d on eq. (1). It is n e c e s s a r y to b e a r in mind the following. In the t h e o r y of the e l e c t r o n gas we c o n s i d e r it as an e l e c t r i cal l y n e u t r a l s y s t e m by including the s p r e a d - o u t e x t e r n a l p o s i t i v e c h a r g e . This is the r e a s o n why
Volume 14, number 1
PHYSICS LETTERS
the i o n i s a t i o n p o t e n t i a l is independent of the n u m b e r of e l e c t r o n s r e m o v e d . In an a t o m the e n e r g y which is n e c e s s a r y f o r r e m o v i n g , f o r e x a m p l e , two e l e c t r o n s is m u c h m o r e than twice the e n e r g y of r e m o v i n g one. It is a r e s u l t of the b r e a k i n g of the e l e c t r o - n e u t r a l i t y during ionisation. T h e r e f o r e , if we would like to u s e f o r m u l a s which a r e c o r r e c t f o r the e l e c t r o n gas we have to b r i n g the e x p e r i m e n t a l data in a c c o r d a n c e with the a s s u m p tion that the i o n i s a t i o n p o t e n ti a l is independent of the n u m b e r of p a r t i c l e s r e m o v e d . It is a c h i e v e d by p a s s i n g f r o m R to R*. F o r e s t i m a t e s the f r e q u e n c i e s of o s c i l l a t i o n s of L, M s h e l l s of A w e r e c a l c u l a t e d ¢ and they c a m e into s e m i - q u a n t i t a t i v e a c c o r d a n c e with exp e r i m e n t a l data 40 - 70 eV and 2 2 0 - 320 eV. The wide s p r e a d of e s t i m a t e s of ~2n is a r e s u l t of the i n a c c u r a t e definition of the " e l e c t r o n d e n s it y of a shell". In c o n c l u s i o n l et us c o n s i d e r , why the d a m p in g of the o s c i l l a t i o n is s m a l l . If all s h e l l s in an a t o m would be d i s c o n n e c t e d the d e c o m p o s i t i o n on p a r t i c l e s and h o l e s would be p o s s i b l e only if the ene r g y of o s c i l l a t i o n is m o r e than Un, the e n e r g y of the i o n i s a t i o n f r o m the o s c i l l a t i n g shell. If ~2n < Un the d e c o m p o s i t i o n would be p o s s i b l e only * The plasma formula is inapplicable for systems with only a few electrons. This is the reason why we don't calculate K-frequency.
1January 1965
by e m i t t i n g a r - q u a n t u m and the width of the l e v e l would be v e r y s m a l l . But b e c a u s e of the connection between d i f f e r e n t s h e l l s , the e n e r g y of o s c i l l a t i o n t r a n s f e r s to the o u t e r s h e l l and c a u s e s ionisation. The s m a l l n e s s of damping is a r e s u l t of the weak coupling b et w een the s h e l l s . But e s t i m a t e s show, that t h e r e a r e c a s e s when ~ n > Un" It s e e m s damping is s m a l l b e c a u s e it is n e c e s s a r y that the e n e r g y , angular m o m e n t u m , z - c o m p o n e n t of an g u l ar m o m e n t u m and p a r i t y of the c o l l e c t i v e o s c i l l a t i o n and p a r t i c l e - h o l e s y s tem, would be the s a m e . This r e q u i r e m e n t dec r e a s e s the n u m b e r of s t a t e s into which the o s c i l l a t i o n d e c o m p o s e s and g u a r a n t e e s the s m a l l n e s s of ~ c o m p a r e d with ¢o. It is a p l e a s u r e for me to e x p r e s s my g r a t i t u d e f o r d i s c u s s i o n s to V. V. A f r o s i m o f , Y.S. Gordeev, V. N. E f i m o v , M.N. Panov, L . A . Sliv, N.V. Fed o r en k o and O. B. F i r s o v .
References 1. V.V.Afrosimov, Y.S.Gordeev, M.N.Panov and N.V.Fedorenko, Zh.Tekhn. Fiz.34 (1964) 1613, 1624. 2. F. Bloch, Z. Phys. 81 (1933) 363. 3. W. Brandt and S. Lundqvist, Physics Letters 4 (1963) 47.
A 4 Z - STATE IN THE S P E C T R U M OF SnH
L. KLYNNING, B. LINDGREN and N. ASLUND Institute of Physics, University of Stockholm, Sweden Received 3 December 1964
The band s p e c t r u m of SnH was f i r s t r e p o r t e d by Watson and Simon [1]. They c a r r i e d out a r o tational a n a l y s i s of the blue bands and c l a s s i f i e d t h e m as belonging to a 2A_2~ t r a n s i t i o n . In addition a r e d s y s t e m wa s r e p o r t e d but although g r o u n d s t at e c o m b i n a t i o n d i f f e r e n c e s w e r e obtained f r o m s o m e of the b r a n c h e s no s a t i s f a c t o r y anal y s i s could be given. K l e m a n [2] in h i s t h e s i s pointed out that t h e s e bands p r o b a b l y w e r e due to a 4Z-27r t r a n s i t i o n . T h e r e a r e only a few known e x a m p l e s of t r a n s i tions i n v o l v i n g a 4 ~ s t a t e and in m o s t c a s e s the e x p e r i m e n t a l data have been f a r f r o m c o m p l e t e . E n e r g y e x p r e s s i o n s f o r the t e r m s a r e g i v e n by Bud6 [3] and l a t e l y by Hougen [4]. F o r m u l a e f o r
the i n t en si t y d i s t r i b u t i o n a r e given by Bud5 and Kov~tcs [5]. We have photographed the r e d SnH in a s e c o n d o r d e r of a 1 0 - m e t e r co n cav e g r a t i n g . Only twelve b r a n c h e s f r o m the 0-0 band and four f r o m the 1-1 band have been o b s e r v e d . B e c a u s e of the l a r g e d i f f e r e n c e in w a v e l e n g t h as well as in i n t e n s i t y between the two s u b s y s t e m s we b e l i e v e that the s t r u c t u r e of the band is b e s t s e e n f r o m a F o r t r a t
diagram (fig. I). The term values observed fit fairly well to Hougen's energy formula if this is modified to take account of the large splitting between 4 ~ and 4 ~½ and the transition from Hund's case b to case c connected therewith. 37
PHYSICS LETTERS
COLLECTIVE
OSCILLATIONS
OF
ELECTRONIC
1 January 1965
SHELLS
OF
ATOMS
M. Ya. AMUSIA
A. F. Joffe Physical-Technical Institute, Academy of Science, Leningrad Received 7 December 1964
R e c e n t e x p e r i m e n t s on i n e l a s t i c c o l l i s i o n s of a r gon ions [1] have shown that t h r e e c h a r a c t e r i s t i c m a x i m a e x i s t in [he c r o s s s e c t i o n as a function of e n e r g y l o s s . The p u r p o s e of this a r t i c l e is to give a p o s s i b l e i n t e r p r e t a t i o n of this phenomenon, as a c o l l e c t i v e e x c i t a t i o n of an atom. It i s w e l l - k n o w n that c o l l e c t i v e o s c i l l a t i o n s e x i s t in a s p a c e - i n f i n i t e and h o m o g e n e o u s m a n y e l e c t r o n s y s t e m . We m a y c o n s i d e r them as a bound s t at e of a p a r t i c l e - h o l e p a i r . In an a t o m between a p a i r of e l e c t r o n s a l s o a c t Coulomb f o r c e s and the p h y s i c a l r e a s o n f o r the e x i s t e n c e of a bound state, the a t t r a c t i o n of a p a r t i c l e - hole p a i r - is the s a m e as in the e l e c t r o n gas. Let us suppose that such c o l l e c t i v e o s c i l l a t i o n s do e x i s t t. We can think that in an a t o m as a r e s u l t of a good s p ace and e n e r g y s e p a r a t i o n of d i f f e r e n t s h e l l s e a c h s h e l l o s c i l l a t e s s e p a r a t e l y . F o r an e s t i m a t e we find the f r e q u e n c y f r o m the p l a s m a formula tt 1
~ n = (4~e2Pn)~ ,
(1)
w h e r e Pn i s the e l e c t r o n density of the n ' t h shell. We can e x c i t e the o s c i l l a t i o n by i n e l a s t i c s c a t t e r i n g of e l e c t r o n s on a t o m s . The c r o s s s e c t i o n of this p r o c e s s is of the o r d e r of 10 -17 c m 2. The o s c i l l a t i o n in spite of being longitudinal can be e x c i t e d by d i r e c t a b s o r p t i o n of a ~ - q u a n turn. B e c a u s e o f P n N Z3n elf , a n ~ Zn~2eff , w h e r e Z n e f f is an e f f e c t i v e c h a r g e of n u c l e i Z for the n'~h shell. The s o - c a l l e d o n e - p a r t i c l e l e v e l s of the a t o m i c s h e l l - m o d e l depend on Z as Z2 eft " This d i f f e r e n c e m a k e s it p o s s i b l e to d e m o n s t r a t e the e x i s t e n c e of c o l l e c t i v e o s c i l l a "~ The assumption of the existence of atomic c o l l e c t i v e o s c i l l a t i o n i s n o t a n e w one [ 2 , 3 ] . It h a s b e e n discussed in literature and originates from J e n s e n ' s
and Bloch's papers of the thirties [2]. But the picture proposed h e r e g i v e s another number and energy of oscillations in an atom (see [2]). t t We don't write here a multiplier which is connected with bound conditions of a finite system (about 1.1 - 1.4). Having in mind rough estimates we omit t h i s correction. 36
tions by s y s t e m a t i c studying of d i f f e r e n t atoms. T h e s e q u a l i t a t i v e c o n s i d e r a t i o n s m a k e it p o s si b l e to explain the r e s u l t s of r e c e n t e x p e r i m e n t s [1]. The c o l l i s i o n of an ion and an A atom w as studied at e n e r g i e s in the r a n g e f r o m 10 to 50 keV. A f t er the c o l l i s i o n both p a r t i c l e s w e r e fixed and t h e i r k i n et i c e n e r g i e s and c h a r g e s a r e d e t e r m i n e d . T h r e e m a x i m a w e r e o b s e r v e d on the c u r v e of dependence of c r o s s s e c t i o n on the ine l a s t i c i t y R of the p r o c e s s . The p o s i t i o n of t h ese m a x i m a is independent of the i m p a c t p a r a m e t e r and the e n e r g y of the i n c o m i n g ion. When the c h a r g e of a p a r t i c l e is known, it is p o s s i b l e to c a l c u l a t e the total i o n l s a t i o n potential ~i U / - the e n e r g y which p r o d u c e d the ionlsation. Then the dependence was studied of c r o s s s e c t i o n s of p r o c e s s e s on R* = R - ~i U/. T h r e e n a r r o w m a x i m a w e r e also o b s e r v e d and t h e i r p o s i t i o n s w e r e independent of the c h a r g e s of ions a f t e r c o l l i s i o n . The widths of t h e s e l i n e s was of the o r d e r 10-20 eV, and the magnitude R*-53, 263, 475 ~ 15 eV). A c c o r d i n g to [1] it s e e m s they a r e not onep a r t i c l e . The f i x ed position of l e v e l s allows us to make an a s s u m p t i o n that they c h a r a c t e r i s e the e x c i t a t i o n s p e c t r u m of a single atom. The p r o b a b i l i t i e s of being e m i t t e d for one o r s e v e r a l e l e c t r o n s f r o m an e x c i t e d a t o m a r e of the s a m e o r d e r [1]. This is in a c c o r d a n c e with our h y p o t h e s i s of a c o l l e c t i v e n a t u r e of t h e s e l e v e l s . Indeed, if the o s c i l l a t i o n is e x c i t e d in a shell, all e l e c t r o n s of h i g h e r s h e l l s a r e in an " e x t e r n a l " f i el d (apart f r o m the s e l f - c o n s i s t e n t H a r t r e e Fock field}. We can show that ~2n i s much l a r g e r than the f r e q u e n c y of r o t a t i o n of e l e c t r o n s in ex t e r n a l s h e l l s . As a r e s u l t the " e x t e r n a l " f i el d suddenly a c t s upon t h e s e e l e c t r o n s . If the amount of e n e r g y is sufficient any n u m b e r of e l e c t r o n s m ay l e a v e the atom. Let us c o m p a r e the e x p e r i m e n t a l data obtained in [1] with e s t i m a t i o n s b a s e d on eq. (1). It is n e c e s s a r y to b e a r in mind the following. In the t h e o r y of the e l e c t r o n gas we c o n s i d e r it as an e l e c t r i cal l y n e u t r a l s y s t e m by including the s p r e a d - o u t e x t e r n a l p o s i t i v e c h a r g e . This is the r e a s o n why
Volume 14, number 1
PHYSICS LETTERS
the i o n i s a t i o n p o t e n t i a l is independent of the n u m b e r of e l e c t r o n s r e m o v e d . In an a t o m the e n e r g y which is n e c e s s a r y f o r r e m o v i n g , f o r e x a m p l e , two e l e c t r o n s is m u c h m o r e than twice the e n e r g y of r e m o v i n g one. It is a r e s u l t of the b r e a k i n g of the e l e c t r o - n e u t r a l i t y during ionisation. T h e r e f o r e , if we would like to u s e f o r m u l a s which a r e c o r r e c t f o r the e l e c t r o n gas we have to b r i n g the e x p e r i m e n t a l data in a c c o r d a n c e with the a s s u m p tion that the i o n i s a t i o n p o t e n ti a l is independent of the n u m b e r of p a r t i c l e s r e m o v e d . It is a c h i e v e d by p a s s i n g f r o m R to R*. F o r e s t i m a t e s the f r e q u e n c i e s of o s c i l l a t i o n s of L, M s h e l l s of A w e r e c a l c u l a t e d ¢ and they c a m e into s e m i - q u a n t i t a t i v e a c c o r d a n c e with exp e r i m e n t a l data 40 - 70 eV and 2 2 0 - 320 eV. The wide s p r e a d of e s t i m a t e s of ~2n is a r e s u l t of the i n a c c u r a t e definition of the " e l e c t r o n d e n s it y of a shell". In c o n c l u s i o n l et us c o n s i d e r , why the d a m p in g of the o s c i l l a t i o n is s m a l l . If all s h e l l s in an a t o m would be d i s c o n n e c t e d the d e c o m p o s i t i o n on p a r t i c l e s and h o l e s would be p o s s i b l e only if the ene r g y of o s c i l l a t i o n is m o r e than Un, the e n e r g y of the i o n i s a t i o n f r o m the o s c i l l a t i n g shell. If ~2n < Un the d e c o m p o s i t i o n would be p o s s i b l e only * The plasma formula is inapplicable for systems with only a few electrons. This is the reason why we don't calculate K-frequency.
1January 1965
by e m i t t i n g a r - q u a n t u m and the width of the l e v e l would be v e r y s m a l l . But b e c a u s e of the connection between d i f f e r e n t s h e l l s , the e n e r g y of o s c i l l a t i o n t r a n s f e r s to the o u t e r s h e l l and c a u s e s ionisation. The s m a l l n e s s of damping is a r e s u l t of the weak coupling b et w een the s h e l l s . But e s t i m a t e s show, that t h e r e a r e c a s e s when ~ n > Un" It s e e m s damping is s m a l l b e c a u s e it is n e c e s s a r y that the e n e r g y , angular m o m e n t u m , z - c o m p o n e n t of an g u l ar m o m e n t u m and p a r i t y of the c o l l e c t i v e o s c i l l a t i o n and p a r t i c l e - h o l e s y s tem, would be the s a m e . This r e q u i r e m e n t dec r e a s e s the n u m b e r of s t a t e s into which the o s c i l l a t i o n d e c o m p o s e s and g u a r a n t e e s the s m a l l n e s s of ~ c o m p a r e d with ¢o. It is a p l e a s u r e for me to e x p r e s s my g r a t i t u d e f o r d i s c u s s i o n s to V. V. A f r o s i m o f , Y.S. Gordeev, V. N. E f i m o v , M.N. Panov, L . A . Sliv, N.V. Fed o r en k o and O. B. F i r s o v .
References 1. V.V.Afrosimov, Y.S.Gordeev, M.N.Panov and N.V.Fedorenko, Zh.Tekhn. Fiz.34 (1964) 1613, 1624. 2. F. Bloch, Z. Phys. 81 (1933) 363. 3. W. Brandt and S. Lundqvist, Physics Letters 4 (1963) 47.
A 4 Z - STATE IN THE S P E C T R U M OF SnH
L. KLYNNING, B. LINDGREN and N. ASLUND Institute of Physics, University of Stockholm, Sweden Received 3 December 1964
The band s p e c t r u m of SnH was f i r s t r e p o r t e d by Watson and Simon [1]. They c a r r i e d out a r o tational a n a l y s i s of the blue bands and c l a s s i f i e d t h e m as belonging to a 2A_2~ t r a n s i t i o n . In addition a r e d s y s t e m wa s r e p o r t e d but although g r o u n d s t at e c o m b i n a t i o n d i f f e r e n c e s w e r e obtained f r o m s o m e of the b r a n c h e s no s a t i s f a c t o r y anal y s i s could be given. K l e m a n [2] in h i s t h e s i s pointed out that t h e s e bands p r o b a b l y w e r e due to a 4Z-27r t r a n s i t i o n . T h e r e a r e only a few known e x a m p l e s of t r a n s i tions i n v o l v i n g a 4 ~ s t a t e and in m o s t c a s e s the e x p e r i m e n t a l data have been f a r f r o m c o m p l e t e . E n e r g y e x p r e s s i o n s f o r the t e r m s a r e g i v e n by Bud6 [3] and l a t e l y by Hougen [4]. F o r m u l a e f o r
the i n t en si t y d i s t r i b u t i o n a r e given by Bud5 and Kov~tcs [5]. We have photographed the r e d SnH in a s e c o n d o r d e r of a 1 0 - m e t e r co n cav e g r a t i n g . Only twelve b r a n c h e s f r o m the 0-0 band and four f r o m the 1-1 band have been o b s e r v e d . B e c a u s e of the l a r g e d i f f e r e n c e in w a v e l e n g t h as well as in i n t e n s i t y between the two s u b s y s t e m s we b e l i e v e that the s t r u c t u r e of the band is b e s t s e e n f r o m a F o r t r a t
diagram (fig. I). The term values observed fit fairly well to Hougen's energy formula if this is modified to take account of the large splitting between 4 ~ and 4 ~½ and the transition from Hund's case b to case c connected therewith. 37