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Dielectronic recombination of highly charged ions
Volume 37A, number 1
PHYSICS LETTERS
DIELECTRONIC
R E C O M B I N A T I O N OF
25 October 1971
H I G H L Y C H A R G E D IONS *
R. A. BAIN and J. N. BARDSLEY
Department of Physics, University of Pittsburg, Pittsburg, Penn., USA Received 2 June 1971 Revised manuscript received 10 September 1971
Cross sections for dielectronic recombination of C5+, 0 7+ and Ne9+ are calculated. The results should be useful in calibration of experiments using beams of highly charged ions produced from Van der Graaff accelerators.
B e a m s of highly c h a r g e d ions can be p r o duced by p a s s i n g a beam f r o m a Van d e r Graaff a c c e l e r a t o r through a thin foil. T h e a c c u m u l a tion of s c a t t e r i n g data f o r such ions is r e q u i r e d f or a n a l y s e s of high t e m p e r a t u r e p l a s m a s in a s t r o p h y s i c s or t h e r m o - n u c l e a r r e s e a r c h . M. Fink and C. F. M o o r e (private c o m m u n i c a t i o n ) have p r o p o s e d to m e r g e an ion beam with an e l e c t r o n beam in o r d e r to m e a s u r e the c r o s s s e c t i o n f o r d i e l e c t r o n i c r e c o m b i n a t i o n , e.g. e + C 5 + - ' C 4 + + hu D i e l e c t r o n i c r e c o m b i n a t i o n is an i m p o r t a n t m o d e of r e c o m b i n a t i o n in high t e m p e r a t u r e p l a s m a s . Its s i g n i f i c a n c e in a s t r o p h y s i c a l p r o b l e m s has b e e n d i s c u s s e d by s e v e r a l a u t h o r s [1-6], and its ef f ects h a v e been o b s e r v e d r e c e n t ly in l a b o r a t o r y p l a s m a s [7]. The t h e o r y of the r e a c t i o n has b e e n d e v e l o p e d by B u r g e s s [1,3,6], T r e f f t z [8], S h o r e [9] and o th e r s . T h e m e c h a n i s m is a r e s o n a n t s c a t t e r i n g p r o c e s s , involving the f o r m a t i o n of a doubly e x c i t e d a t o m i c s t a t e which is s t a b i l i z e d by e m i s s i o n of a photon. We will a s s u m e that the r e s o n a n t l e v e l s do not o v e r lap, and will n e g l e c t d i r e c t r a d i a t i v e r e c o m b i nation. We will a l s o a s s u m e that the d e n s i t i e s a r e so low that t h r e e body c o l l i s i o n s can be neglected. F o r s i m p l i c i t y , we will r e s t r i c t our attention to the r e c o m b i n a t i o n of ions with only one or two e l e c t r o n s . The c o n t r i b u t i o n of each r e s o n a n c e to the r e c o m b i n a t i o n c r o s s s e c t i o n can then be w r i t t e n , u s i n g a t o m i c units, Wn
r a r nr
T h e r e s o n a n t p o s i t i o n and width a r e E n and Fn, and the l a t t e r can be d e c o m p o s e d into c o m p o nents due to r a d i a t i v e decay r r and a u t o - i o n i z a -
tion F n a. The multiplicities of the initial ionic state and the resonant state a r e ~ i and wn. For the recombination of an ion in a "S state, the maximum cross section is then ~max=a(En) = (~/ 2En) Wn r a n Frn / (Fn)2 If the spread of relative energies in the beams is greater than Fn, a more useful quantity is the integrated cross section Qn = f ~ (E)dE = (Tr2/4En) wn F a F nr/rn . For singly charged ions, F r is much smaller then Fna, and the cross sections a r e small. For large nuclear charge Z, F a n is independent of Z to lowest order, whereas F r varies as Z 4. For a specific resonance E n varies as Z 2. As Z is increased amax and Qn increase approximatelya as Z 2 until F r becomes of the same order as F n" Further increases in Z lead to a reduction in the cross section, ultimately as Z -2. The cross section for recombination of electrons with C5+, 0 7+ and Ne 9+ have been calculated. The important p a r a m e t e r s for the lowest five resonances a r e given in tables 1-3. The resonant energies and auto-ionization rates have been taken from the sources listed in table I. The radiative decay rates were calculated by using single configuration wave functions, composed of products of Slater orbitals. This simple method is adequate for our purposes, but should be improved in future calculations. Estimates of the cross sections for other members of the isoelectronic s e r i e s can be obtained by using the rules given above. * This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by Army Research Office-Durham under Contract No. DA-31-124-ARO-D440 75
PHYSICS
V o l u m e 37A. n u m b e r 1
25 O c t o b e r 1971
LETTERS
~fable 1 E l e c t r o n r e c o m b i n a t i o n with C ° + i o n s
St ate
En (cV)
a
Fn {eV)
Fr
n (eV)
qm ax (cm2)
Qn (cm2eV)
(2s) 2 1S (*)
264,3
0.18
2.1 × 10 -4
5.3 x 10 -21
1.5 × 10 -21
(2p) 2 1S (*)
281.6
0.009
7.3 × 10 .4
2.9 × 10 -19
4.5 x 10 -21
(2s) (2p) 1p [10]
273.7
0.09
3.3 x 10 .4
4.8 x 10 -20
7.9 x 10 -21
(2s) (2p) 3 p [101
265.8
0.009
3 3 x 10 . 4
1.3 x 10 -18
2.0 x 10 . 2 0
(2p) 2 1D (~)
273.0
0.17
8.8 × 10 .4
1.1 × 10 -19
3.0 x 10 -21
(*) D r a k e ( p r i v a t e c o m m u n i c a t i o n ) : (11;) By e x t r a p o l a t i o n f r o m P e r r o t t and S t e w a r t [11]. Table 2 E l e c t r o n r e c o m b i n a t i o n with O7+ions n (eV)
I "a n (eV)
(2s) 2 IS
461.4
0.19
(2p)2 1S
485.3
O.OlO
State
E
r Fn (eV)
(Ymax ( c m 2)
Qn (cm2eV)
7.0 × I0 -4
9.2 × i 0 -21
2.8 × 10-21
2.4 × 10-3
3.9 x 10-19
7.4 x 10-21
(2S) (2p) 1p
474.5
0.10
1.2 × 10 .3
9.0 × 10 -20
1.4 x 10 - 2 0
(2s) (2p) 3 p
463.5
0.009
1.2 ~ 10 -3
2.3 × 10 -18
3.9 x 10 . 2 0
(2p) 2 1D
473.6
0.19
2.9 x 10 -3
1.9 x 10 -19
5.7 x 10 -20
Table 3 E l e c t r o n r e c o m b i n a t i o n with Ne9+ions Crmax (cm 2)
(cm~ae V)
0.20
1.7 x 10 -3
1.4 × 10 . 2 0
4.5 x 10 -21
0.010
5.9 × 10 -3
3.8 x 10 -19
9.4 × 10 -21
729.8
0.11
3.2 × 10 .3
1.4 × 10 -19
2.4 x 10 -20
(2s) (2p) 3p
715.6
0.009
3.2 x 10 -3
2.8 × 10 -18
5.6 × 10 -20
(2p) 2 1D
728.5
0.20
7.3 x 10 -3
2.8 × 10 -19
7.2 x 10 -20
En (eV)
(2s) 2 1S
713.0
(2p) 2 1S
743.5
(2s) (2p) 1p
r a n (eV)
These results s h o w t h a t t h e 3 p a n d 1D r e sonances should be most easily observed. The m a g n i t u d e of t h e c r o s s s e c t i o n s i s n o t u n d u l y small, and the resonances are well spaced. This reaetion therefore provide an ideal test for the calibration of beam experiments using highly charged ions. Experimental measurements for recombination into higher states, and for more complex ions would then be most valuable, since t h e a c c u r a c y of t h e o r e t i c a l c a l c u l a t i o n s i s u n certain in these situations.
W e a r e i n d e b t e d t o G. W. F . D r a k e f o r s e n d i n g to us his unpublished results on the autoionizat i o n w i d t h s f o r t h e 1S s t a t e s . 76
Q
r r n (eV)
State
References [1] A . B u r g e s s . A s t r o p h y s . J. 139 (1964) 776; 141 (1965) 1589. [2] L . G o l d b u r g . A u t o i o n i z a t i o n , ed. A . T e m k i n (Mono Book C o r p . . B a l t i m o r e , 1966) p.1. [3] A . B u r g e s s . ibid. p. 25. [4] W . H . T t i e k e r and R . J . Gould, A s t r o p h y s . J. 144 (1966) 266. [5] L . G o l d b u r g a n d A . D u p r e e , N a t u r e 215 (1967) 41. [6] A. B u r g e s s and H . P . S u m m e r s , A s t r o p h y s . J. 157 (1969) 1007. [7] A. H. G a b r i e l , C. J o r d a n and T. M. P a g e t , Sixth Int. Conf. on the P h y s i c s of e l e c t r o n i c and a t o m i c c o l l i s i o n s , A b s t r a c t s (MIT P r e s s . Cambridge, Mass., 1969)p.558. [8] E . T r e f f t z , Z . A s t r o p h y s . 65 (1969) 299. [9] B . W . S h o r e , A s t r o p h y s . J. 158 (1969) 1205. [10] G . W . F . D r a k e a n d A . D a l g a r n o . P r o c . Roy. Soc. A320 (1971) 549. [11] R . H . P e r r o t t a n d A . L . S t e w a r t . J. P h y s . B.1 (1968) 381.
PHYSICS LETTERS
DIELECTRONIC
R E C O M B I N A T I O N OF
25 October 1971
H I G H L Y C H A R G E D IONS *
R. A. BAIN and J. N. BARDSLEY
Department of Physics, University of Pittsburg, Pittsburg, Penn., USA Received 2 June 1971 Revised manuscript received 10 September 1971
Cross sections for dielectronic recombination of C5+, 0 7+ and Ne9+ are calculated. The results should be useful in calibration of experiments using beams of highly charged ions produced from Van der Graaff accelerators.
B e a m s of highly c h a r g e d ions can be p r o duced by p a s s i n g a beam f r o m a Van d e r Graaff a c c e l e r a t o r through a thin foil. T h e a c c u m u l a tion of s c a t t e r i n g data f o r such ions is r e q u i r e d f or a n a l y s e s of high t e m p e r a t u r e p l a s m a s in a s t r o p h y s i c s or t h e r m o - n u c l e a r r e s e a r c h . M. Fink and C. F. M o o r e (private c o m m u n i c a t i o n ) have p r o p o s e d to m e r g e an ion beam with an e l e c t r o n beam in o r d e r to m e a s u r e the c r o s s s e c t i o n f o r d i e l e c t r o n i c r e c o m b i n a t i o n , e.g. e + C 5 + - ' C 4 + + hu D i e l e c t r o n i c r e c o m b i n a t i o n is an i m p o r t a n t m o d e of r e c o m b i n a t i o n in high t e m p e r a t u r e p l a s m a s . Its s i g n i f i c a n c e in a s t r o p h y s i c a l p r o b l e m s has b e e n d i s c u s s e d by s e v e r a l a u t h o r s [1-6], and its ef f ects h a v e been o b s e r v e d r e c e n t ly in l a b o r a t o r y p l a s m a s [7]. The t h e o r y of the r e a c t i o n has b e e n d e v e l o p e d by B u r g e s s [1,3,6], T r e f f t z [8], S h o r e [9] and o th e r s . T h e m e c h a n i s m is a r e s o n a n t s c a t t e r i n g p r o c e s s , involving the f o r m a t i o n of a doubly e x c i t e d a t o m i c s t a t e which is s t a b i l i z e d by e m i s s i o n of a photon. We will a s s u m e that the r e s o n a n t l e v e l s do not o v e r lap, and will n e g l e c t d i r e c t r a d i a t i v e r e c o m b i nation. We will a l s o a s s u m e that the d e n s i t i e s a r e so low that t h r e e body c o l l i s i o n s can be neglected. F o r s i m p l i c i t y , we will r e s t r i c t our attention to the r e c o m b i n a t i o n of ions with only one or two e l e c t r o n s . The c o n t r i b u t i o n of each r e s o n a n c e to the r e c o m b i n a t i o n c r o s s s e c t i o n can then be w r i t t e n , u s i n g a t o m i c units, Wn
r a r nr
T h e r e s o n a n t p o s i t i o n and width a r e E n and Fn, and the l a t t e r can be d e c o m p o s e d into c o m p o nents due to r a d i a t i v e decay r r and a u t o - i o n i z a -
tion F n a. The multiplicities of the initial ionic state and the resonant state a r e ~ i and wn. For the recombination of an ion in a "S state, the maximum cross section is then ~max=a(En) = (~/ 2En) Wn r a n Frn / (Fn)2 If the spread of relative energies in the beams is greater than Fn, a more useful quantity is the integrated cross section Qn = f ~ (E)dE = (Tr2/4En) wn F a F nr/rn . For singly charged ions, F r is much smaller then Fna, and the cross sections a r e small. For large nuclear charge Z, F a n is independent of Z to lowest order, whereas F r varies as Z 4. For a specific resonance E n varies as Z 2. As Z is increased amax and Qn increase approximatelya as Z 2 until F r becomes of the same order as F n" Further increases in Z lead to a reduction in the cross section, ultimately as Z -2. The cross section for recombination of electrons with C5+, 0 7+ and Ne 9+ have been calculated. The important p a r a m e t e r s for the lowest five resonances a r e given in tables 1-3. The resonant energies and auto-ionization rates have been taken from the sources listed in table I. The radiative decay rates were calculated by using single configuration wave functions, composed of products of Slater orbitals. This simple method is adequate for our purposes, but should be improved in future calculations. Estimates of the cross sections for other members of the isoelectronic s e r i e s can be obtained by using the rules given above. * This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by Army Research Office-Durham under Contract No. DA-31-124-ARO-D440 75
PHYSICS
V o l u m e 37A. n u m b e r 1
25 O c t o b e r 1971
LETTERS
~fable 1 E l e c t r o n r e c o m b i n a t i o n with C ° + i o n s
St ate
En (cV)
a
Fn {eV)
Fr
n (eV)
qm ax (cm2)
Qn (cm2eV)
(2s) 2 1S (*)
264,3
0.18
2.1 × 10 -4
5.3 x 10 -21
1.5 × 10 -21
(2p) 2 1S (*)
281.6
0.009
7.3 × 10 .4
2.9 × 10 -19
4.5 x 10 -21
(2s) (2p) 1p [10]
273.7
0.09
3.3 x 10 .4
4.8 x 10 -20
7.9 x 10 -21
(2s) (2p) 3 p [101
265.8
0.009
3 3 x 10 . 4
1.3 x 10 -18
2.0 x 10 . 2 0
(2p) 2 1D (~)
273.0
0.17
8.8 × 10 .4
1.1 × 10 -19
3.0 x 10 -21
(*) D r a k e ( p r i v a t e c o m m u n i c a t i o n ) : (11;) By e x t r a p o l a t i o n f r o m P e r r o t t and S t e w a r t [11]. Table 2 E l e c t r o n r e c o m b i n a t i o n with O7+ions n (eV)
I "a n (eV)
(2s) 2 IS
461.4
0.19
(2p)2 1S
485.3
O.OlO
State
E
r Fn (eV)
(Ymax ( c m 2)
Qn (cm2eV)
7.0 × I0 -4
9.2 × i 0 -21
2.8 × 10-21
2.4 × 10-3
3.9 x 10-19
7.4 x 10-21
(2S) (2p) 1p
474.5
0.10
1.2 × 10 .3
9.0 × 10 -20
1.4 x 10 - 2 0
(2s) (2p) 3 p
463.5
0.009
1.2 ~ 10 -3
2.3 × 10 -18
3.9 x 10 . 2 0
(2p) 2 1D
473.6
0.19
2.9 x 10 -3
1.9 x 10 -19
5.7 x 10 -20
Table 3 E l e c t r o n r e c o m b i n a t i o n with Ne9+ions Crmax (cm 2)
(cm~ae V)
0.20
1.7 x 10 -3
1.4 × 10 . 2 0
4.5 x 10 -21
0.010
5.9 × 10 -3
3.8 x 10 -19
9.4 × 10 -21
729.8
0.11
3.2 × 10 .3
1.4 × 10 -19
2.4 x 10 -20
(2s) (2p) 3p
715.6
0.009
3.2 x 10 -3
2.8 × 10 -18
5.6 × 10 -20
(2p) 2 1D
728.5
0.20
7.3 x 10 -3
2.8 × 10 -19
7.2 x 10 -20
En (eV)
(2s) 2 1S
713.0
(2p) 2 1S
743.5
(2s) (2p) 1p
r a n (eV)
These results s h o w t h a t t h e 3 p a n d 1D r e sonances should be most easily observed. The m a g n i t u d e of t h e c r o s s s e c t i o n s i s n o t u n d u l y small, and the resonances are well spaced. This reaetion therefore provide an ideal test for the calibration of beam experiments using highly charged ions. Experimental measurements for recombination into higher states, and for more complex ions would then be most valuable, since t h e a c c u r a c y of t h e o r e t i c a l c a l c u l a t i o n s i s u n certain in these situations.
W e a r e i n d e b t e d t o G. W. F . D r a k e f o r s e n d i n g to us his unpublished results on the autoionizat i o n w i d t h s f o r t h e 1S s t a t e s . 76
Q
r r n (eV)
State
References [1] A . B u r g e s s . A s t r o p h y s . J. 139 (1964) 776; 141 (1965) 1589. [2] L . G o l d b u r g . A u t o i o n i z a t i o n , ed. A . T e m k i n (Mono Book C o r p . . B a l t i m o r e , 1966) p.1. [3] A . B u r g e s s . ibid. p. 25. [4] W . H . T t i e k e r and R . J . Gould, A s t r o p h y s . J. 144 (1966) 266. [5] L . G o l d b u r g a n d A . D u p r e e , N a t u r e 215 (1967) 41. [6] A. B u r g e s s and H . P . S u m m e r s , A s t r o p h y s . J. 157 (1969) 1007. [7] A. H. G a b r i e l , C. J o r d a n and T. M. P a g e t , Sixth Int. Conf. on the P h y s i c s of e l e c t r o n i c and a t o m i c c o l l i s i o n s , A b s t r a c t s (MIT P r e s s . Cambridge, Mass., 1969)p.558. [8] E . T r e f f t z , Z . A s t r o p h y s . 65 (1969) 299. [9] B . W . S h o r e , A s t r o p h y s . J. 158 (1969) 1205. [10] G . W . F . D r a k e a n d A . D a l g a r n o . P r o c . Roy. Soc. A320 (1971) 549. [11] R . H . P e r r o t t a n d A . L . S t e w a r t . J. P h y s . B.1 (1968) 381.