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Effect of atomic ionization on the K-X transition energy
Volume 35A, number 5
EFFECT
OF
PHYSICS LETTERS
ATOMIC
IONIZATION
ON T H E
28 June 1971
K-X
TRANSITION
ENERGY
V. M. M1KHAILOV and M. A. HANONKIND
Institute of Physics, Leningrad State University, Leningrad, USRR Received 18 May 1971
The change in the K-X-ray energy due to the ionization of the atom is calculated in perturbation theory for heavy elements (Z >~37). The strong dependence of the shift on the orbital angular momentum of the removed valence electron is explained.
T he K - X - r a y e n e r g y shift d E due to the i o n i z a t i o n of t h e atom is defined as a d i f f e r e n c e of e n e r g i e s of the c o r r e s p o n d i n g ions. It can b e d e s c r i b e d in t e r m s of p e r t u r b a t i o n s Aq~ and in n o n r e l a t i v i s t i c H a r t r e e - F o c k p o t e n t i a l of a n e u t r a l atom. Aq~ = ~01s - ~ is t h e p e r t u r b a t i o n c a u s e d by the " h o l e s " in t h e ' ~ n t e r n a l s u b s h e l l s and 4? = ~ k qnldk ~°nl~ k is t h e p e r t u r b a t i o n c a u s e d by the " h o l e s ~ In the v a l e n c e e u b s h e l l s . H e r e ~0nl is the p o t e n t i a l f o r m e d by t h e n / - e l e c t r o n , the exchange t e r m s included, qn~l b is the n u m b e r of the r e m o v e d v a l e n c e e l e c t r o h s : d E = d E ( l ) + AE(Z);'-" AE(1) is the shift of the X - r a y line in t h e f i r s t o r d e r of p e r t u r b a t i o n t h e o r y ; it r e p r e s e n t s th e d i f f e r e n c e of the i n t e r a c t i o n e n e r g i e s of the v a l e n c e e l e c t r o n with the I s - and r i p - e l e c t r o n s ( d E ( l ) > 0):
k &E (2) is the major contribution of the second order t e r m s to dE; i t is the correction to the
binding e n e r g i e s of t h e i n n e r l s - and n p - e l e c t r o n s r e s u l t i n g f r o m t h e change of the a t o m i c f i e l d (AE{2) < 0): AE(2) = -
~
2Nco)Iba(A~o)Iba(~)(~Eba)-1
a,b kk;b I5a(~O) - f R(a)R Co) ~ ' 2 d r ;
qk =-qnklk ; ~ k - ~n k l k '
where index ha" labels a l l possible states i n cluding excitations in the continuous spectrum, index "b" indicates the occupied bound states; N (b) is the number of the n/-electrons in the ion subshells, R(b) is the one-electron radial wave function for a neutral atom and AEab is the single p a r t i c l e energy difference. A more detailed theoretical analysis of eqs. (1), (2) is given in
r a . [2]. d E (table 1) is c a l c u l a t e d u s i n g the w a v e functions of the p o t e n t i a l [3]. T h e t o t al e r r o r in
Table 1 The shifts of the K-X-ray energies per one removed valence electron (in a.u. × 10 -3) Atom
nkl k
(AE) teor.
(AE) (el)xp.
(AE) teor.
KrYl~2 Rb
(AE) (:)p.
(AE) teor.
K~lfl3
5s
1.7
1.0±0.2
3.2
1.0~-0.4
Zr
4d 5s
-5.8 2.0
- 4.4±0.1 3.0±0.4
-3.2±0.2
S~
5s 5p
3.6 3.3
3.0±0.4 3.0±0.4
Tm
4f 5d 6s
-28 - 1.9 1.0
-17.0±0.9 - 4.4±0.1 2.6±0.4
- 7.0 2.3 1.4 3.3 -72 - 0.9 1.1
3.7±0.4 -60~2
(AE) " - - e(1) xD. Kfl2~4
13
4.2~0.8
15 -0.5 -5.0
8.0±0.2
-1.8 -15 -11 2.6
3.7±0.4 -15.8+9.5
313
Volume 35A, number 5
PHYSICS LETTERS
the value of AE is coImected with the e r r o r s of the c a l c u l a t i o n s and with neglecting of the hi~her o r d e r t e r m s and does not exceed 20% of RE (~). The m a j o r c o n t r i b u t i o n to the shift is p r o p o r t i o n a l to the n u m b e r of the r e m o v e d v a l e n c e e l e c t r o n s qnklk as can b e s e e n from eqs. (1),(2). The contributio~^0f q u a d r a t i c t e r m s to eq. (2) is about 10% of A ~ in the case of the r e m o v e d s - and p - v a l e n c e ~ l e c t r o n s and ~ 4% for d- and f-electrons. The s t r o n g dependence of RE K on the o r b i t a l a n g u l a r m o m e n t u m 1 k of the r e m o v e d v a l e n c e e l e c t r o n is d e t e r m i n e d by the p o s i t i o n of the m a i n m a x i m u m R(nkl k) , rnk l k" The a b s o l u t e value of RE(2) i n c r e a s e s with the d e c r e a s e of rnklk b e c a u s e in the n u c l e a r r e g i o n ~anklk i n c r e a s e s a s well a s the velocity of t h e p o t e n t i a l change (as a function of ' r ' ) does. RE (2) should be equal to zero, if ~nklk is constant. T h i s dependence of RE(2) on rn., k explains the a n o m a l o u s l y l a r g e negative s h ~ of the K - X - r a y e n e r g y with the r e m o v a l of a v a l e n c e f - e l e c t r o n . When the outer v a l e n c e e l e c t r o n s ( s - o r p-) a r e r e m o v e d , the shift is p o s i t i v e , RE(2) b e i n g s m a l l in this case. F o r example, r n k l k for T m in atomic u n i t s is r4f = 0.46 and r 6 s = 3.57, If we now c o n s i d e r the KB - X - r a y line, the " i n n e r p e r t u r b a t i o n n R~ i n c r e a s e s s i n c e the m a x i m u m of the n p - e l e c t r o n d i s t r i b u t i o n is the
314
28 June 1971
f a r t h e r from the I s - m a x i m u m , the l a r g e r the value of n. In the c a s e of the r e m o v a l of the f - v a l e n c e e l e c t r o n this effect leads for the KBI~3 X - r a y line the i n c r e a s e both of the second o r d e r t e r m s and the a b s o l u t e v a l u e of AE as soon as the f i r s t o r d e r is s t i l l s m a l l . But t h e r e is a q u a l i t a t i v e change in the shift of the Kfl2B4 X - r a y line: the m a x i m u m of the density d i s t r i bution of the 4 f - e l e c t r o n is in the r e g i o n of the m a i n 4 p - m a x i m u m , which leads to a s t r o n g i n c r e a s e of the f i r s t o r d e r t e r m , i.e. to the dec r e a s e of the total shift. If the 6 s - or 5 d - v a l e n c e e l e c t r o n is r e m o v e d the t r a n s i t i o n f r o m REX" to is very s i m i l a r to t h e case when the f-electro~i is r e m o v e d . But R E ( l ) i n c r e a s e s c o n s i d e r a b l y s l o w e r b e c a u s e the m a i n m a x i m a for the 6 s - or 5 d - e l e c t r o n s a r e located f a r from the m a i n m a x i m a for the 3pand 4 p - e l e c t r o n s .
References
[1] O.I.~nnbaev, Zh. Eksp. i Teor. Fiz. 57 (1969) 1716 and to be published, Zh. Eksp. i Teor. Fiz. Pis'ma 10 (1969)209. [2] V. M. Mikhailov and M. A. Hanonkind, Bull. Acad. Sci. USSR, Ser. Phys., 35 (1971) 86. [3] F.Herman and S. Skillman, Atomic structure calculations (Prentice Hall, New Jersey, 1963).
EFFECT
OF
PHYSICS LETTERS
ATOMIC
IONIZATION
ON T H E
28 June 1971
K-X
TRANSITION
ENERGY
V. M. M1KHAILOV and M. A. HANONKIND
Institute of Physics, Leningrad State University, Leningrad, USRR Received 18 May 1971
The change in the K-X-ray energy due to the ionization of the atom is calculated in perturbation theory for heavy elements (Z >~37). The strong dependence of the shift on the orbital angular momentum of the removed valence electron is explained.
T he K - X - r a y e n e r g y shift d E due to the i o n i z a t i o n of t h e atom is defined as a d i f f e r e n c e of e n e r g i e s of the c o r r e s p o n d i n g ions. It can b e d e s c r i b e d in t e r m s of p e r t u r b a t i o n s Aq~ and in n o n r e l a t i v i s t i c H a r t r e e - F o c k p o t e n t i a l of a n e u t r a l atom. Aq~ = ~01s - ~ is t h e p e r t u r b a t i o n c a u s e d by the " h o l e s " in t h e ' ~ n t e r n a l s u b s h e l l s and 4? = ~ k qnldk ~°nl~ k is t h e p e r t u r b a t i o n c a u s e d by the " h o l e s ~ In the v a l e n c e e u b s h e l l s . H e r e ~0nl is the p o t e n t i a l f o r m e d by t h e n / - e l e c t r o n , the exchange t e r m s included, qn~l b is the n u m b e r of the r e m o v e d v a l e n c e e l e c t r o h s : d E = d E ( l ) + AE(Z);'-" AE(1) is the shift of the X - r a y line in t h e f i r s t o r d e r of p e r t u r b a t i o n t h e o r y ; it r e p r e s e n t s th e d i f f e r e n c e of the i n t e r a c t i o n e n e r g i e s of the v a l e n c e e l e c t r o n with the I s - and r i p - e l e c t r o n s ( d E ( l ) > 0):
k &E (2) is the major contribution of the second order t e r m s to dE; i t is the correction to the
binding e n e r g i e s of t h e i n n e r l s - and n p - e l e c t r o n s r e s u l t i n g f r o m t h e change of the a t o m i c f i e l d (AE{2) < 0): AE(2) = -
~
2Nco)Iba(A~o)Iba(~)(~Eba)-1
a,b kk;b I5a(~O) - f R(a)R Co) ~ ' 2 d r ;
qk =-qnklk ; ~ k - ~n k l k '
where index ha" labels a l l possible states i n cluding excitations in the continuous spectrum, index "b" indicates the occupied bound states; N (b) is the number of the n/-electrons in the ion subshells, R(b) is the one-electron radial wave function for a neutral atom and AEab is the single p a r t i c l e energy difference. A more detailed theoretical analysis of eqs. (1), (2) is given in
r a . [2]. d E (table 1) is c a l c u l a t e d u s i n g the w a v e functions of the p o t e n t i a l [3]. T h e t o t al e r r o r in
Table 1 The shifts of the K-X-ray energies per one removed valence electron (in a.u. × 10 -3) Atom
nkl k
(AE) teor.
(AE) (el)xp.
(AE) teor.
KrYl~2 Rb
(AE) (:)p.
(AE) teor.
K~lfl3
5s
1.7
1.0±0.2
3.2
1.0~-0.4
Zr
4d 5s
-5.8 2.0
- 4.4±0.1 3.0±0.4
-3.2±0.2
S~
5s 5p
3.6 3.3
3.0±0.4 3.0±0.4
Tm
4f 5d 6s
-28 - 1.9 1.0
-17.0±0.9 - 4.4±0.1 2.6±0.4
- 7.0 2.3 1.4 3.3 -72 - 0.9 1.1
3.7±0.4 -60~2
(AE) " - - e(1) xD. Kfl2~4
13
4.2~0.8
15 -0.5 -5.0
8.0±0.2
-1.8 -15 -11 2.6
3.7±0.4 -15.8+9.5
313
Volume 35A, number 5
PHYSICS LETTERS
the value of AE is coImected with the e r r o r s of the c a l c u l a t i o n s and with neglecting of the hi~her o r d e r t e r m s and does not exceed 20% of RE (~). The m a j o r c o n t r i b u t i o n to the shift is p r o p o r t i o n a l to the n u m b e r of the r e m o v e d v a l e n c e e l e c t r o n s qnklk as can b e s e e n from eqs. (1),(2). The contributio~^0f q u a d r a t i c t e r m s to eq. (2) is about 10% of A ~ in the case of the r e m o v e d s - and p - v a l e n c e ~ l e c t r o n s and ~ 4% for d- and f-electrons. The s t r o n g dependence of RE K on the o r b i t a l a n g u l a r m o m e n t u m 1 k of the r e m o v e d v a l e n c e e l e c t r o n is d e t e r m i n e d by the p o s i t i o n of the m a i n m a x i m u m R(nkl k) , rnk l k" The a b s o l u t e value of RE(2) i n c r e a s e s with the d e c r e a s e of rnklk b e c a u s e in the n u c l e a r r e g i o n ~anklk i n c r e a s e s a s well a s the velocity of t h e p o t e n t i a l change (as a function of ' r ' ) does. RE (2) should be equal to zero, if ~nklk is constant. T h i s dependence of RE(2) on rn., k explains the a n o m a l o u s l y l a r g e negative s h ~ of the K - X - r a y e n e r g y with the r e m o v a l of a v a l e n c e f - e l e c t r o n . When the outer v a l e n c e e l e c t r o n s ( s - o r p-) a r e r e m o v e d , the shift is p o s i t i v e , RE(2) b e i n g s m a l l in this case. F o r example, r n k l k for T m in atomic u n i t s is r4f = 0.46 and r 6 s = 3.57, If we now c o n s i d e r the KB - X - r a y line, the " i n n e r p e r t u r b a t i o n n R~ i n c r e a s e s s i n c e the m a x i m u m of the n p - e l e c t r o n d i s t r i b u t i o n is the
314
28 June 1971
f a r t h e r from the I s - m a x i m u m , the l a r g e r the value of n. In the c a s e of the r e m o v a l of the f - v a l e n c e e l e c t r o n this effect leads for the KBI~3 X - r a y line the i n c r e a s e both of the second o r d e r t e r m s and the a b s o l u t e v a l u e of AE as soon as the f i r s t o r d e r is s t i l l s m a l l . But t h e r e is a q u a l i t a t i v e change in the shift of the Kfl2B4 X - r a y line: the m a x i m u m of the density d i s t r i bution of the 4 f - e l e c t r o n is in the r e g i o n of the m a i n 4 p - m a x i m u m , which leads to a s t r o n g i n c r e a s e of the f i r s t o r d e r t e r m , i.e. to the dec r e a s e of the total shift. If the 6 s - or 5 d - v a l e n c e e l e c t r o n is r e m o v e d the t r a n s i t i o n f r o m REX" to is very s i m i l a r to t h e case when the f-electro~i is r e m o v e d . But R E ( l ) i n c r e a s e s c o n s i d e r a b l y s l o w e r b e c a u s e the m a i n m a x i m a for the 6 s - or 5 d - e l e c t r o n s a r e located f a r from the m a i n m a x i m a for the 3pand 4 p - e l e c t r o n s .
References
[1] O.I.~nnbaev, Zh. Eksp. i Teor. Fiz. 57 (1969) 1716 and to be published, Zh. Eksp. i Teor. Fiz. Pis'ma 10 (1969)209. [2] V. M. Mikhailov and M. A. Hanonkind, Bull. Acad. Sci. USSR, Ser. Phys., 35 (1971) 86. [3] F.Herman and S. Skillman, Atomic structure calculations (Prentice Hall, New Jersey, 1963).