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Theoretical dielectronic recombination rate coefficients for ground-state hydrogen-like neon
Volume 116, N u m b e r 4
PHYSICS LETTERS A
9 June 1986
T H E O R E T I C A L 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 RATE C O E F F I C I E N T S FOR G R O U N D - S T A T E H Y D R O G E N - L I K E N E O N K.R. K A R I M and C.P. B H A L L A Department of Physics, Kansas State University, Manhattan, KS 66506, USA Received 7 February 1986; revised manuscript received 7 April 1986; accepted for publication 11 April 1986
Dielectronic recombination rate coefficients are calculated and presented for ground-state hydrogen-like neon as a function of electron temperature, Doubly excited nl n'l' configurations of intermediate resonance states are restricted to n, n' = 2, 3, and 4 with all allowed values of l and l'. Computations have been performed in intermediate coupling with configuration interaction,
Neon has been used for diagnostic purposes in laser-induced fusion devices [1-3]. X-ray satellite spectra from doubly excited helium-like neon yield information about plasma density and the compressed core radius in neon-seeded glass microballoons [2,3]. At low electron densities Ne IX ions are formed by dielectronic recombination (DR) of free electrons with hydrogen-like neon ions. As the electron density increases, electron collisional excitation and three-body recombination processes become important. The states of doubly excited Ne IX ions are selectively populated by these competing processes; the relative intensities of the resonance and X-ray satellite lines can therefore be used as important probes of plasma state. Consider a Ne 9+ ion in its ground state G; a continuum electron with energy c collides with it and excites the ls electron to a higher orbital nl. In this process the continuum electron may loose enough energy and is captured to form an intermediate state I of doubly excited helium-like neon. Dielectronic recombination is achieved if this state makes a radiative transition to a state F which lies sufficiently low in energy and cannot autoionize. The D R rate coefficient ct(I --, F) for this process is given by [4,5]
172
a (I ~ F ) : ½(2~rh2/mkTe)3/2
F2(I
~ F)
× exp[ - ' t (I)/kT~ ] = 2.07 × 10-16[K × exp[ - q
(la)
cm2/T~]3/2F2(I --* F)
(I)/kT~],
(lb)
where F2(l -~ F) = ( 2 J 1 + 1 ) R ( I ) ~ ( I --* V)
/[(2Jo + a)r(i)].
(2)
J1 and JG are, respectively, the total angular m o m e n t u m of the intermediate resonance state I and initial ground state G; Te is the electron temperature in the plasma, c(I) is the Auger electron energy and R(I), ~0(I ~ F ) , and T(I) are, respectively, the nonradiative branching ratio, fluorescence yield and life time of the state I. It is easy to see that a(I -~ F) is maximum when ~(I) = 3kTc/2. The function F2(I-o F) is related to the ratio of intensities of resonance to satellite lines at a particular plasma temperature. Calculation of total D R rate coefficient is difficult because of the enormous number of intermediate resonance states over which summations have to be made. These states lie close in energy, and an accurate evaluation must incorporate the effects of configuration interaction and intermediate coupling [6,7]. Theoretical calculations of
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 116, Number 4
PHYSICS LETTERS A
D R rate coefficients with various approximations have appeared in the literature [8-18]. Measurements of D R rate coefficient and cross sections of selected ions have also been reported [19-22]. In this letter we report on our calculation of D R rate coefficients for ground state hydrogen-like neon. The calculations have been carried out in intermediate coupling with configuration interaction. D R rate coefficients for individual intermediate states I can be obtained from eq. (1). The H a r t r e e - F o c k - S l a t e r atomic model was employed to calculate radial one-electron orbitals. Configurations nlnl' of doubly excited helium-like neon ions were restricted to n, n' --- 2, 3 and 4 with all allowed values of l and 1". A multitude of inter125
! ~
! TOTAL R A T E / 6 A = R= C= O= E = F = G= H= I =
I00
%
2s2p 2s3d 2s4d 2s3p 2s4p 2s3s 2s4f 2s 2 2s4s
¢J
'o
75
n.-
ill:: E3
50
25
0 400
800
1200
1600
KTe (eV) Fig. 1. Dielectronic recombination (DR) rate coefficients versus kT~ where T~ is the electron temperature. The top curve represents the total DR rate coefficient divided by 6 while the bottom curves give partial rate coefficients of DR processes which proceed via intermediate resonance states 2snl with n = 2, 3 and 4. It is seen that for fixed values of 1, the D R state coefficient decreases as n increases.
9 June 1986
mediate resonance states I nlnl' L S J ) were generated from these configurations; eigenstates of the hamiltonian constructed from this basis were used to calculate the life time T(I) in eq. (1). The fluorescence yield w(I ~ F) and nonradiative branching ratio R(I) were calculated as Wx(I ~ F) = Fx(I ~ F ) / F ( I ) and R(I) = FA (I)/F(I), where F x, FA, and F(I) are, respectively, the X-ray, Auger, and total widths of the state I. Partial D R rate coefficients for each intermediate state I were summed to obtain the total rate coefficient. This is shown in fig. 1 as a function of k T e, where Te is the electron temperature. This may be compared with the rate coefficients of helium-like argon [5] and iron [18]. Auger transition energies in the present case are much smaller; the rate coefficient reaches maximum at lower electron temperature and falls off rapidly. The total rate coefficient, however, is greater than the rate coefficient of iron and is comparable to that of argon. Theoretical calculations on transition energies and rates of doubly excited helium-like systems have been performed for magnesium, calcium, titanium, and iron by different authors [23] Vainshtein and Safronova [24] have employed a 1 / Z expansion technique to calculate X-ray wavelengths and transition rates of 2121' configurations for elements with Z = 4 to 34. The wavelengths and radiative transition rates from our calculations are within 0.1% and 8% respectively of those reported by Vainshtein and Safronova. The Auger decay rates of Vainshtein and Safronova are systematically higher than our results by about 20%. Similar agreement for energy levels and radiative transition probabilities, and disagreement in autoionization rates with the calculations of Vainshtein and Safronova have been reported by Dubau et al. [23] for Mg 1°÷. For ions of low atomic numbers nonradiative branching ratio R(I) are usually much greater than fluorescence yield (I ~ F); the function /72(I F) may then be approximated as F2(I --+ F) = (2Jr + 1)W(I ~ F ) / ( 2 J o + 1), where W(I--+ F) is simply the decay rate for the radiative transition I--+ F. This approximation is applicable only to 2121" configuration in the pres173
Volume 116, Number 4
PHYSICS LETTERS A
e n t c a s e w i t h t h e e x c e p t i o n of 2p z 3P2,1.0 states w h i c h c a n n o t A u g e r d e c a y b e c a u s e o f p a r i t y select i o n rules. E l e c t r i c d i p o l e t r a n s i t i o n b e c o m e s inc r e a s i n g l y i m p o r t a n t for 2lnl' c o n f i g u r a t i o n s as n increases. To investigate the dependence of partial DR rate coefficients on the principal quantum numb e r s of i n t e r m e d i a t e r e s o n a n c e states w e p l o t in fig. 1 the D R r a t e c o e f f i c i e n t s for 2snl c o n f i g u r a t i o n s w i t h n = 2, 3 a n d 4. It c a n b e s e e n t h a t for a f i x e d v a l u e o f l t h e r a t e c o e f f i c i e n t d e c r e a s e s as n increases. T h i s w o r k was s u p p o r t e d b y the D i v i s i o n o f C h e m i c a l Sciences, O f f i c e of B a s i c E n e r g y Sciences, O f f i c e of E n e r g y R e s e a r c h , U . S . D e p a r t m e n t of Energy.
References [1] B. Yaakobi, D. Steel, E. Thorsos, A. Hauer and B. Berry, Phys. Rev. Lett. 39 (1977) 1526. [2] K.B. Mitchell, D.B. Vanttusteyn, G.H. McCall and P. Lee, Phys. Rev. Lett. 42 (1979) 232. [3] J.F. Seely, Phys. Rev. Lett. 42 (1979) 1606. [4] C.P. Bhalla and T.W. Tunnel, J. Quant. Spectrosc. Radiat. Transfer 32(1984) 141. [5] C.P. Bhalla and T.W. Tunnel, Phys. Lett. A 108 (1985) 22. [6] K.R. Karim, M.H. Chen and B. Crasemann, Phys. Rev. A 28 (1983) 3355. [7] D.H. Sampson, A.D. Parks and R.H. Clark, Phys. Rev. A 15 (1977) 1393. [8] A. Burgess, Astrophys. J. 139 (1964) 776; 141 (1965) 1589. [9] B.W. Shore, Astrophys. J. 158 (1969) 1205. [10] A.H. Gabriel, Mon. Not. R. Astron. Soc. 160 (1972) 99. [11] C.P. Bhalla, A.H. Gabriel and L.P. Presnyakov, Mon. Not. R. Astron. Soc, 172 (1975) 359. [12] J.N. Gau and Y. Hahn, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 121; J.N. Gau, Y. Hahn and J.A. Ritter, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 131, 147;
174
9 June 1986
Y. Hahn, J.N. Gan, R. Luddy and A. Retter, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 65; Y. Hahn, J.N. Gau, R. Luddy, M. Dube and N. Shkolnik, J. Quant. Spectrose. Radiat. Transfer 24 (505) 1980. [13] I. Nasser and Y. Hahn, J. Quant. Spectrosc. Radiat. Transfer 29 (1983) 1; K. LaGattuta and Y. Hahn, Phys. Rev. A 27 (1983) 1675; D.J. McLanghlin and Y. Hahn, Phys. Rev. A 27 (1983) 1389; D.J. McLaughlin, I. Nasser and Y. Hahn, Phys. Rev. A 31 (•985) 1926; M.D. Dube, R. Rasoanaivo and Y. Hahn, J. Quant. Spectrosc. Radiat. Transfer 33 (1985 13. [14] K.J. LaGattuta, Phys. Rev. A 30 (1984) 3072. [15] L.J. Roszman and A.W. Weiss, J. Quant. Spectrosc. Radiat. Transfer 30 (1983) 67. [16] A.K. Pradhan, Phys. Rev. A 30 (1984) 2141. [17] D.C. Griffin, M.S. Pindzola and C. Bottcher, Phys. Rev. A 31 (1985) 568. [18] F. Bely-Dubau, A.H. Gabriel and S. Volonte, Mon. Not. R. Astron. Soc. 189 (1979) 801; F. Bely-Dubau, J. Dubau, P. Faucher and A.H. Gabriel, Mon. Not. R. Astron. Soc. 198 (1982) 239; F. Bely-Dubau, J. Dubau, P, Faucher, A.H. Gabriel, M. Loulergue, L. Steelman-Clark, S. Volonte, E. Antonucci and C.G. Rapley, Mon. Not. R. Astron. Soc. 201 (1982) 1155. [19] F. Bely-Dubau, M. Bitter, J. Dubau, P. Faucher, A.H. Gabriel, K.W. Hill, S. von Goe)er, N. Sauthoff and S. Volonte, Phys. Lett. A 93 (1983) 189. [20] J.B.A. Mitchell, C.T. Ng, J.L. Forand, D.P. Levac, R.E. Mitchell, A. Sen, D.B. Miko and J.Wm. McGowan, Phys. Rev. Lett. 50 (1983) 335. [21] D.S. Belic, G.H. Dunn, T.J. Morgan, D.W. Muller and C. Timmer, Phys. Rev. Lett. 50 (1983) 339. [22] P.F. Dittner, S. Datz, P.D. Miller, C.D. Moak, P.H. Stelson, C. Bottcher, W.B. Dress, G.D. Alton and N. Neskovic, Phys. Rev. Lett. 51 (1983) 31. [23] J. Dubau, M. Loulergue and L. Steenman-Clark, Mon. Not. R. Astron. Soc. 190 (19800 125; L. Blanchet, M. Cornille, J. Dubau, P. Faucher, J. Lion and S. Volonte, Astron. Astrophys. 152 (1985) 417; Bitter et al., Phys. Rev. A 29 (1984) 661; J. Dubau, A. H. Gabriel, M. Loulergue, L. Steenman-Clark and S. Volonte, Mon. Not. R. Astron. Soc. 195 (1981) 705. [24] L.A. Vainshtein and U.I. Safronova, At. Data Nucl. Data Tables, 21 (1978) 49,
PHYSICS LETTERS A
9 June 1986
T H E O R E T I C A L 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 RATE C O E F F I C I E N T S FOR G R O U N D - S T A T E H Y D R O G E N - L I K E N E O N K.R. K A R I M and C.P. B H A L L A Department of Physics, Kansas State University, Manhattan, KS 66506, USA Received 7 February 1986; revised manuscript received 7 April 1986; accepted for publication 11 April 1986
Dielectronic recombination rate coefficients are calculated and presented for ground-state hydrogen-like neon as a function of electron temperature, Doubly excited nl n'l' configurations of intermediate resonance states are restricted to n, n' = 2, 3, and 4 with all allowed values of l and l'. Computations have been performed in intermediate coupling with configuration interaction,
Neon has been used for diagnostic purposes in laser-induced fusion devices [1-3]. X-ray satellite spectra from doubly excited helium-like neon yield information about plasma density and the compressed core radius in neon-seeded glass microballoons [2,3]. At low electron densities Ne IX ions are formed by dielectronic recombination (DR) of free electrons with hydrogen-like neon ions. As the electron density increases, electron collisional excitation and three-body recombination processes become important. The states of doubly excited Ne IX ions are selectively populated by these competing processes; the relative intensities of the resonance and X-ray satellite lines can therefore be used as important probes of plasma state. Consider a Ne 9+ ion in its ground state G; a continuum electron with energy c collides with it and excites the ls electron to a higher orbital nl. In this process the continuum electron may loose enough energy and is captured to form an intermediate state I of doubly excited helium-like neon. Dielectronic recombination is achieved if this state makes a radiative transition to a state F which lies sufficiently low in energy and cannot autoionize. The D R rate coefficient ct(I --, F) for this process is given by [4,5]
172
a (I ~ F ) : ½(2~rh2/mkTe)3/2
F2(I
~ F)
× exp[ - ' t (I)/kT~ ] = 2.07 × 10-16[K × exp[ - q
(la)
cm2/T~]3/2F2(I --* F)
(I)/kT~],
(lb)
where F2(l -~ F) = ( 2 J 1 + 1 ) R ( I ) ~ ( I --* V)
/[(2Jo + a)r(i)].
(2)
J1 and JG are, respectively, the total angular m o m e n t u m of the intermediate resonance state I and initial ground state G; Te is the electron temperature in the plasma, c(I) is the Auger electron energy and R(I), ~0(I ~ F ) , and T(I) are, respectively, the nonradiative branching ratio, fluorescence yield and life time of the state I. It is easy to see that a(I -~ F) is maximum when ~(I) = 3kTc/2. The function F2(I-o F) is related to the ratio of intensities of resonance to satellite lines at a particular plasma temperature. Calculation of total D R rate coefficient is difficult because of the enormous number of intermediate resonance states over which summations have to be made. These states lie close in energy, and an accurate evaluation must incorporate the effects of configuration interaction and intermediate coupling [6,7]. Theoretical calculations of
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 116, Number 4
PHYSICS LETTERS A
D R rate coefficients with various approximations have appeared in the literature [8-18]. Measurements of D R rate coefficient and cross sections of selected ions have also been reported [19-22]. In this letter we report on our calculation of D R rate coefficients for ground state hydrogen-like neon. The calculations have been carried out in intermediate coupling with configuration interaction. D R rate coefficients for individual intermediate states I can be obtained from eq. (1). The H a r t r e e - F o c k - S l a t e r atomic model was employed to calculate radial one-electron orbitals. Configurations nlnl' of doubly excited helium-like neon ions were restricted to n, n' --- 2, 3 and 4 with all allowed values of l and 1". A multitude of inter125
! ~
! TOTAL R A T E / 6 A = R= C= O= E = F = G= H= I =
I00
%
2s2p 2s3d 2s4d 2s3p 2s4p 2s3s 2s4f 2s 2 2s4s
¢J
'o
75
n.-
ill:: E3
50
25
0 400
800
1200
1600
KTe (eV) Fig. 1. Dielectronic recombination (DR) rate coefficients versus kT~ where T~ is the electron temperature. The top curve represents the total DR rate coefficient divided by 6 while the bottom curves give partial rate coefficients of DR processes which proceed via intermediate resonance states 2snl with n = 2, 3 and 4. It is seen that for fixed values of 1, the D R state coefficient decreases as n increases.
9 June 1986
mediate resonance states I nlnl' L S J ) were generated from these configurations; eigenstates of the hamiltonian constructed from this basis were used to calculate the life time T(I) in eq. (1). The fluorescence yield w(I ~ F) and nonradiative branching ratio R(I) were calculated as Wx(I ~ F) = Fx(I ~ F ) / F ( I ) and R(I) = FA (I)/F(I), where F x, FA, and F(I) are, respectively, the X-ray, Auger, and total widths of the state I. Partial D R rate coefficients for each intermediate state I were summed to obtain the total rate coefficient. This is shown in fig. 1 as a function of k T e, where Te is the electron temperature. This may be compared with the rate coefficients of helium-like argon [5] and iron [18]. Auger transition energies in the present case are much smaller; the rate coefficient reaches maximum at lower electron temperature and falls off rapidly. The total rate coefficient, however, is greater than the rate coefficient of iron and is comparable to that of argon. Theoretical calculations on transition energies and rates of doubly excited helium-like systems have been performed for magnesium, calcium, titanium, and iron by different authors [23] Vainshtein and Safronova [24] have employed a 1 / Z expansion technique to calculate X-ray wavelengths and transition rates of 2121' configurations for elements with Z = 4 to 34. The wavelengths and radiative transition rates from our calculations are within 0.1% and 8% respectively of those reported by Vainshtein and Safronova. The Auger decay rates of Vainshtein and Safronova are systematically higher than our results by about 20%. Similar agreement for energy levels and radiative transition probabilities, and disagreement in autoionization rates with the calculations of Vainshtein and Safronova have been reported by Dubau et al. [23] for Mg 1°÷. For ions of low atomic numbers nonradiative branching ratio R(I) are usually much greater than fluorescence yield (I ~ F); the function /72(I F) may then be approximated as F2(I --+ F) = (2Jr + 1)W(I ~ F ) / ( 2 J o + 1), where W(I--+ F) is simply the decay rate for the radiative transition I--+ F. This approximation is applicable only to 2121" configuration in the pres173
Volume 116, Number 4
PHYSICS LETTERS A
e n t c a s e w i t h t h e e x c e p t i o n of 2p z 3P2,1.0 states w h i c h c a n n o t A u g e r d e c a y b e c a u s e o f p a r i t y select i o n rules. E l e c t r i c d i p o l e t r a n s i t i o n b e c o m e s inc r e a s i n g l y i m p o r t a n t for 2lnl' c o n f i g u r a t i o n s as n increases. To investigate the dependence of partial DR rate coefficients on the principal quantum numb e r s of i n t e r m e d i a t e r e s o n a n c e states w e p l o t in fig. 1 the D R r a t e c o e f f i c i e n t s for 2snl c o n f i g u r a t i o n s w i t h n = 2, 3 a n d 4. It c a n b e s e e n t h a t for a f i x e d v a l u e o f l t h e r a t e c o e f f i c i e n t d e c r e a s e s as n increases. T h i s w o r k was s u p p o r t e d b y the D i v i s i o n o f C h e m i c a l Sciences, O f f i c e of B a s i c E n e r g y Sciences, O f f i c e of E n e r g y R e s e a r c h , U . S . D e p a r t m e n t of Energy.
References [1] B. Yaakobi, D. Steel, E. Thorsos, A. Hauer and B. Berry, Phys. Rev. Lett. 39 (1977) 1526. [2] K.B. Mitchell, D.B. Vanttusteyn, G.H. McCall and P. Lee, Phys. Rev. Lett. 42 (1979) 232. [3] J.F. Seely, Phys. Rev. Lett. 42 (1979) 1606. [4] C.P. Bhalla and T.W. Tunnel, J. Quant. Spectrosc. Radiat. Transfer 32(1984) 141. [5] C.P. Bhalla and T.W. Tunnel, Phys. Lett. A 108 (1985) 22. [6] K.R. Karim, M.H. Chen and B. Crasemann, Phys. Rev. A 28 (1983) 3355. [7] D.H. Sampson, A.D. Parks and R.H. Clark, Phys. Rev. A 15 (1977) 1393. [8] A. Burgess, Astrophys. J. 139 (1964) 776; 141 (1965) 1589. [9] B.W. Shore, Astrophys. J. 158 (1969) 1205. [10] A.H. Gabriel, Mon. Not. R. Astron. Soc. 160 (1972) 99. [11] C.P. Bhalla, A.H. Gabriel and L.P. Presnyakov, Mon. Not. R. Astron. Soc, 172 (1975) 359. [12] J.N. Gau and Y. Hahn, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 121; J.N. Gau, Y. Hahn and J.A. Ritter, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 131, 147;
174
9 June 1986
Y. Hahn, J.N. Gan, R. Luddy and A. Retter, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 65; Y. Hahn, J.N. Gau, R. Luddy, M. Dube and N. Shkolnik, J. Quant. Spectrose. Radiat. Transfer 24 (505) 1980. [13] I. Nasser and Y. Hahn, J. Quant. Spectrosc. Radiat. Transfer 29 (1983) 1; K. LaGattuta and Y. Hahn, Phys. Rev. A 27 (1983) 1675; D.J. McLanghlin and Y. Hahn, Phys. Rev. A 27 (1983) 1389; D.J. McLaughlin, I. Nasser and Y. Hahn, Phys. Rev. A 31 (•985) 1926; M.D. Dube, R. Rasoanaivo and Y. Hahn, J. Quant. Spectrosc. Radiat. Transfer 33 (1985 13. [14] K.J. LaGattuta, Phys. Rev. A 30 (1984) 3072. [15] L.J. Roszman and A.W. Weiss, J. Quant. Spectrosc. Radiat. Transfer 30 (1983) 67. [16] A.K. Pradhan, Phys. Rev. A 30 (1984) 2141. [17] D.C. Griffin, M.S. Pindzola and C. Bottcher, Phys. Rev. A 31 (1985) 568. [18] F. Bely-Dubau, A.H. Gabriel and S. Volonte, Mon. Not. R. Astron. Soc. 189 (1979) 801; F. Bely-Dubau, J. Dubau, P. Faucher and A.H. Gabriel, Mon. Not. R. Astron. Soc. 198 (1982) 239; F. Bely-Dubau, J. Dubau, P, Faucher, A.H. Gabriel, M. Loulergue, L. Steelman-Clark, S. Volonte, E. Antonucci and C.G. Rapley, Mon. Not. R. Astron. Soc. 201 (1982) 1155. [19] F. Bely-Dubau, M. Bitter, J. Dubau, P. Faucher, A.H. Gabriel, K.W. Hill, S. von Goe)er, N. Sauthoff and S. Volonte, Phys. Lett. A 93 (1983) 189. [20] J.B.A. Mitchell, C.T. Ng, J.L. Forand, D.P. Levac, R.E. Mitchell, A. Sen, D.B. Miko and J.Wm. McGowan, Phys. Rev. Lett. 50 (1983) 335. [21] D.S. Belic, G.H. Dunn, T.J. Morgan, D.W. Muller and C. Timmer, Phys. Rev. Lett. 50 (1983) 339. [22] P.F. Dittner, S. Datz, P.D. Miller, C.D. Moak, P.H. Stelson, C. Bottcher, W.B. Dress, G.D. Alton and N. Neskovic, Phys. Rev. Lett. 51 (1983) 31. [23] J. Dubau, M. Loulergue and L. Steenman-Clark, Mon. Not. R. Astron. Soc. 190 (19800 125; L. Blanchet, M. Cornille, J. Dubau, P. Faucher, J. Lion and S. Volonte, Astron. Astrophys. 152 (1985) 417; Bitter et al., Phys. Rev. A 29 (1984) 661; J. Dubau, A. H. Gabriel, M. Loulergue, L. Steenman-Clark and S. Volonte, Mon. Not. R. Astron. Soc. 195 (1981) 705. [24] L.A. Vainshtein and U.I. Safronova, At. Data Nucl. Data Tables, 21 (1978) 49,