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ELECTRON IMPACT EXCITATION OF FINE-STRUCTURE LEVELS OF NEON-LIKE TITANIUM (Ti XIII)
Atomic Data and Nuclear Data Tables 71, 103–115 (1999) Article ID adnd.1998.0802, available online at http://www.idealibrary.com on
ELECTRON IMPACT EXCITATION OF FINE-STRUCTURE LEVELS OF NEON-LIKE TITANIUM (Ti XIII) G. P. GUPTA,* N. C. DEB, and A. Z. MSEZANE Department of Physics and Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, Georgia 30314
We present results of a Breit–Pauli R-matrix calculation for the electron impact excitation of neon-like titanium, in which the 27 lowest fine-structure target levels arising out of the 4 lowest configurations 2s 2 2p 6 , 2s 2 2p 5 3s, 2s 2 2p 5 3p, and 2s 2 2p 5 3d are included. These target levels are represented by configuration interaction wave functions using the 1s, 2s, 2p, 3s, 3p, and 3d basic orbitals. The relativistic effects are included in the Breit–Pauli approximation via one-body mass correction, Darwin, and spin– orbit interaction terms in the scattering equations. For many transitions, complex resonance structures are found in the excitation cross sections. The excitation cross sections are integrated over a Maxwellian distribution of electron energies to give electron excitation rate coefficients over a wide temperature range from 150 to 600 eV. The relative populations for different electron densities and temperatures are also presented. © 1999 Academic Press
* Permanent address: Department of Physics, S.D. (Postgraduate) College, Muzaffarnagar, University of Meerut, India
0092-640X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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e 1 Ti XIII
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Theoretical Atomic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 EXPLANATION OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 TABLES I. Radial Function Parameters for Optimized Orbitals of Ti XIII . . . . II. Calculated and Experimental Energy Levels of Ti XIII . . . . . III. Comparison of Present Oscillator Strengths and Radiative Rates with Other Available Results . . . . . . . . . . . . . . . . . . . . . . . IV. Fine-Structure Collision Strengths for Ti XIII . . . . . . . . . . . . . V. Excitation Rates for Ti XIII Fine-Structure Levels at Electron Temperatures of 150, 400, and 600 eV . . . . . . . . . . . . . . VI. Relative Level Populations for Ti XIII at Various Electron Temperatures and Densities . . . . . . . . . . . . . . . . . . . . . . . .
108 109 110 113 114 115
INTRODUCTION In recent years, much attention has been focused on neon-like ions due to their wide application in astrophysical and laboratory plasma diagnostics. Some of these ions, such as Ti XIII, are also found as metallic impurities in fusion plasmas [1]. The possibility of laser amplification in transitions between the 2s 2 2p 5 3s and 2s 2 2p 5 3p configurations at VUV and soft x-ray wavelengths has further stimulated [2– 6] the need for a more comprehensive study of the atomic data for neon-like ions. Identification of 2p 5 3s–2p 5 3p and 2p 5 3p–2p 5 3d spectral lines of Ti XIII was reported by Jupe´n and Litzen [7]. Most theoretical investigations [8 –12] subsequent to this identification reported atomic data such as energy levels, wavelengths, oscillator strengths, and radiative rates. Only a few calculations [11, 12] reported collision strengths for some important transitions. Accurate atomic data such as energy levels, transition probabilities, and electron excitation rate coefficients are needed for the calculation of level populations and spectral line intensities. In the present paper we systematically built a reasonably good wave function set for the Ti XIII ion and checked its accuracy by comparing our energy levels, wavelengths, oscillator strengths, and transition probabilities against the most reliable results. Then, we used our wave functions to calculate collision strengths, excitation rate coefficients, and level populations in the Breit–Pauli R-matrix formalism for the 27 lowest fine-structure levels of Ti XIII belonging to the [1s 2 2s 2 ]2p 6 , 2p 5 3s, 2p 5 3p, and 2p 5 3d configurations. The relativistic effects are included in the Breit–Pauli ap-
proximation via one-body mass correction, Darwin, and spin– orbit interaction terms in the scattering equations [13]. Adequate care has been taken to ensure that the higher partial wave contributions, which may be particularly important for the electric-dipole-allowed transitions, are properly included. Theoretical Atomic Data In the present work, we used the six orthogonal oneelectron orbitals: 1s, 2s, 2p, 3s, 3p, and 3d. The 1s, 2s, and 2p radial functions are taken as the Hartree–Fock orbitals given by Clementi and Roetti [14] for the ground 1s 2 2s 2 2p 6 state. The parameters for the 3s, 3p, and 3d orbitals are optimized on the energies of the 2p 5 3s( 1 P o 1 3 o P ), 2p 5 3p( 3 D) and 2p 5 3d( 3 F o) states, respectively, using the CIV3 program of Hibbert [15]. The optimized parameters are displayed in Table I. We next constructed the J-dependent CI wavefunctions by using the expansions of the form [16].
O b f ~a L S JM !, K
C i ~ JM J ! 5
ij
j
j
j j
J
(1)
j51
where each of the K single-configuration functions f j is constructed from one-electron functions, and a j defines the coupling of the orbital L j and the spin S j angular momenta to give the total angular momentum J. The mixing coeffi104
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
e 1 Ti XIII
G. P. GUPTA, N. C. DEB, and A. Z. MSEZANE
TABLE A Comparison of Calculated and Observed Wavelengths (in Å) of 3s–3p lines in Ti XIII Observed a
Calculated
2p 5 3p 2p 5 3p 2p 5 3p 2p 5 3p 2p 5 3p
Transitions
3s, 3p, 3d
Large CI
3 3 3 3 3
263 297 457 469 530
281 320 458 471 554
1
So So 3 P2 1 D2 3 S1 1
a b
2p 5 3s 3 P 1 2p 5 3s 1 P 1 2p 5 3s 3 P 1 2p 5 3s 1 P 1 2p 5 3s 3 P 2
rates in the length approximation for all the optically allowed and intercombination transitions among the 27 lowest levels. In Table III we compare our weighted oscillator strengths ( gf L ) and radiative decay rates ( A L ) with those calculated by Hibbert et al. [8] and Bhatia et al. [12]. We note that for all the strong transitions, where the oscillator strengths are larger than 0.1, the general agreement among the three calculations is very good. Even for some relatively weaker transitions there is remarkably good agreement among the three calculations. The transitions 2–9, 3– 6, 3–14, 5–9, 5–10, 6 –21, 7–23, 7–24, 7–25, 8 –21, 8 –24, 9 –25, 10 –17, 10 –27, 12–18, 13–21, 14 –18, and 14 –23 are some of the examples of such agreement. We thus infer that the wave functions used in our calculation are fairly accurate for a scattering calculation. The scattering wavefunction for each total angular momentum J and parity p combination is expanded in the inner region (r # a) in the R-matrix basis as [17, 18]
282 b 322 b 459 472 552
Jupe´n and Litzen [7]. Values obtained from experimental level energies [7].
cients b ij are obtained by diagonalizing the Breit–Pauli Hamiltonian with respect to the basis C i . The Hamiltonian here consists of the nonrelativistic terms plus the Breit– Pauli corrections via one-body mass correction, Darwin, spin– orbit, spin– other– orbit, and spin–spin terms [16]. The wave functions given by Eq. (1) are then used to calculate the excitation energies of the 27 fine-structure levels of Ti XIII. These energies are displayed in Table II. In Table II our fine-structure energies relative to the ground state are compared with the experimental results of Jupe´n and Litzen [7] and the theoretical results of Hibbert et al. [8] and Qiu et al. [9]. It is clear from this Table that the ordering of our calculated excitation energies is the same as that of the experiment. Our energy values for all 26 excited levels agree within 0.5% of the measured values and are in close agreement with the other theoretical results. The results of Hibbert et al. [8] are in best agreement with the measured values because they used a large CI expansion and their ab initio energies were adjusted with the experimental energies. In this paper, our main aim is to perform a Breit–Pauli calculation for the collision strengths and other related parameters. These are rather impractical to obtain with a large CI expansion— hence, the need to work with a limited CI which gives reasonably good energies and oscillator strengths as well as allows us to perform the Breit– Pauli calculations for the scattering part without much difficulty. The wavelengths for some important transitions calculated from our energies, given in Table II, are compared with the observed wavelengths [7] in Table A. There is reasonably good agreement between theory and observations. As a further check we also calculated these wavelengths with a very large CI wave function expansion, including in addition the following configurations: (1s 2 2s 2 2p 5 ) 4s, 4p, and 4d; (1s 2 2s2p 6 ) 3s, 3p, 3d, and 4s; (1s 2 2s 2 2p 4 ) 3p 2 , 3p4p, and 4p 2 where the 4l orbitals are optimized on the 3l target terms. Excellent agreement is achieved in this case, as seen in Table A. We then used our limited CI wave function set to calculate absorption oscillator strengths and radiative decay
C k~ J p ! 5A
OC
ijk
F Ji ~ x 1 , x 2 , . . . , x N ; r N11 , s N11 !u ij ~r N11 !
ij
1
O d f ~x , x , . . . , x jk
J j
1
2
N11
!,
(2)
j
where A is an antisymmetrization operator, and F iJ are channel functions representing the 27 fine-structure atomic levels coupled with the angular and spin functions of the scattered electron to form channel functions of J and p. The f jJ are (N 1 1)-electron configurations constituted from the atomic orbitals and are included to ensure completeness of the total wave function expansion and to allow for shortrange correlation, and the u ij are the orthogonal set of continuum basis functions (orbitals). The coefficients C ijk and d jk are obtained by diagonalizing the (N 1 1)-electron Breit–Pauli Hamiltonian in the inner region. We have chosen the boundary radius a 5 3.0 a.u. Twelve continuum orbitals for each angular momentum are included to obtain convergence within the energy range of interest. The maximum number of channels retained in our calculation is 109. The coupled equations are solved using a perturbation method developed by Seaton [19] to yield K matrices and then the collision strengths. The R-matrix method is used to calculate partial collision strengths from J 5 0.5 to J 5 9.5. These partial waves produced convergent collision strengths for the forbidden and intercombination transitions, but higher partial wave contributions are needed for the dipole-allowed transitions and are calculated using the Bethe approximation [20]. In Table IV we present our results for the total collision strengths at impact energies of 45, 100, and 150 Ry. Our R-matrix results at 45 Ry are compared with the corresponding results of the distorted-wave calculation of Bha105
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. P. GUPTA, N. C. DEB, and A. Z. MSEZANE
tia et al. [12]. The two sets of results are in good agreement except for the transition 2p 6 1 S 0 3 2p 5 3d 3 D 1 , where the distorted-wave result is lower by a factor of 10. However, the oscillator strength for this transition agrees very well in the two calculations. The excitation rate coefficients for a transition from a lower level i to an upper level f at a temperature T e (in k) is given by [21]
C~i 3 f ! 5
8.63 3 10 26 2 g i T 1/ e
E
C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106 (1985) 3. D. L. Matthews, P. L. Hagelstein, M. D. Rosen, M. J. Eckart, N. M. Ceglio, A. U. Hazi, H. Medecki, B. J. MacGowan, J. E. Trebes, B. L. Whitten, E. M. Campbell, C. W. Hatcher, A. M. Hawryluk, R. L. Kauffman, L. D. Pleasance, G. Rambach, J. H. Scofield, G. Stone, and T. A. Weaver, Phys. Rev. Lett. 54, 110 (1985) 4. R. A. London, M. D. Rosen, M. S. Maxon, D. C. Eder, and P. L. Hagelstein, J. Phys. B 22, 3363 (1989)
`
V E ~i 3 f !
DE if
3 exp(2E/~kT e !)d~E/~kT e !!,
5. J. A. Cogordan, S. Lunell, C. Jupe´n, and U. Litzen, Phys. Rev. A 32, 1885 (1985)
(3)
6. G. P. Gupta, K. A. Berrington, and A. E. Kingston, J. Phys. B 22, 3289 (1989)
where g i 5 (2J i 1 1) is the statistical weight of the lower level i; DE if 5 E f 2 E i is the excitation energy; V E(i 3 f ) is the collision strength; k is the Boltzmann constant; and E is the electron impact energy. In Table V we present our excitation rates at electron temperatures 150, 400, and 600 eV. Finally, the level populations N j are calculated by solving the 27 coupled rate equations [22] N j@
O A 1 n ~ O C 1 O C !# 5 n ~O N C 1 O N C ! 1 O N A , ji
i,j
d ji
e
7. C. Jupen and U. Litzen, Physica Scripta 30, 112 (1984) 8. A. Hibbert, M. LeDourneuf, and M. Mohan, ATOMIC DATA AND NUCLEAR DATA TABLES 53, 23 (1993) 9. Y.-H. Qiu, S.-C. Li, and Y.-S. Sun, ATOMIC DATA NUCLEAR DATA TABLES 55, 1 (1993)
e ji
i,j
i
i,j
e ij
i
i.j
d ij
i
ij
AND
10. J. A. Cogordan, S. Lunell, C. Jupen, and U. Litzen, Physica Scripta 31, 545 (1985)
i.j
e
e 1 Ti XIII
(4)
11. H. L. Zhang and D. H. Sampson, ATOMIC DATA NUCLEAR DATA TABLES 43, 1 (1989)
i.j
AND
12. A. K. Bhatia, U. Feldman, and J. F. Seely, ATOMIC DATA AND NUCLEAR DATA TABLES 32, 435 (1985)
where C ij are the electron excitation rate coefficients; the superscripts e and d refer to the excitation and deexcitation, respectively; and n e is the electron density. In this calculation, we have used our A values and electron collisional rates. We considered all transitions among the 27 lowest fine-structure levels. The relative populations are normalized so that the sum of the level populations is unity. In Table VI we have tabulated our results for relative populations for a range of electron densities and temperatures appropriate to the condition inside the laser-produced plasma.
13. N. S. Scott and P. G. Burke, J. Phys. B 13, 4299 (1980) 14. E. Clementi and C. Roetti, ATOMIC DATA DATA TABLES 14, 177 (1974)
AND
NUCLEAR
15. A. Hibbert, Comput. Phys. Commun. 9, 141 (1975) 16. R. Glass and A. Hibbert, Comput. Phys. Commun. 16, 19 (1978)
Acknowledgments
17. N. S. Scott and K. T. Taylor, Comput. Phys. Commun. 25, 347 (1982)
This research is supported by the Air Force Office of Scientific Research and Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, United States Department of Energy. One of us (G.P.G.) is supported by the National Science Foundation under Grant AST-9528945.
18. K. A. Berrington, P. G. Burke, K. Butler, M. J. Seaton, P. J. Storey, K. T. Taylor, and Y. Yu, J. Phys. B 20, 6379 (1987) 19. M. J. Seaton, private communication (1987) 20. A. Burgess and V. B. Sheorey, J. Phys. B 7, 2403 (1974)
References 1. R. K. Janev, Physica Scripta 37, 5 (1991)
21. W. B. Eissner and M. J. Seaton, J. Phys. B 7, 2533 (1974)
2. M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. J. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch,
22. U. Feldman, J. F. Seely, and A. K. Bhatia, J. Appl. Phys. 56, 2475 (1984) 106
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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e 1 Ti XIII
EXPLANATION OF TABLES TABLE I.
Radial Function Parameters for Optimized Orbitals of Ti XIII Orbital Coefficients Powers Exponents
TABLE II.
The The The The
one-electron orbitals in spectroscopic notation. expansion coefficients of the (unnormalized) Slater-type orbitals. powers of r in the Slater-type orbitals. exponents of the Slater-type orbitals.
Calculated and Experimental Energy Levels of Ti XIII (in a.u.) Index Level Present Experiment HDM QLS
A number assigned to each level in ascending order of energy. Energy levels with complete configurations. Present calculated values. Experimental values [7]. Calculated values of Hibbert et al. [8]. Calculated values of Qiu et al. [9]. 1 a.u. 5 219,474 cm 21
TABLE III.
Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results I, J gf L(I-J) A L(J-I) Present HDM BFS
TABLE IV.
Lower and upper levels as designated by the index numbers given in Table II. Weighted oscillator strength, where g is the statistical weight factor for the lower level and f L is the absorption oscillator strength. Radiative decay rates from an upper state J to a lower state I. Values in the present calculation. Calculated values of Hibbert et al. [8]. Calculated values of Bhatia et al. [12].
Fine-Structure Collision Strengths for Ti XIII BFS Present
Distorted-wave results of Bhatia et al. (Ref. [12]) at 45 Ry. Present results of Breit–Pauli R-matrix calculation at 45, 100, and 150 Ry.
TABLE V.
Excitation Rates (in cm 3 s 21) for Ti XIII Fine-Structure Levels at Electron Temperatures of 150, 400, and 600 eV
TABLE VI.
Relative Level Populations for Ti XIII at Various Electron Temperatures and Densities Te ne
Electron temperature. Electron density (in cm 23).
107
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e 1 Ti XIII
TABLE I. Radial Function Parameters for Optimized Orbitals of Ti XIII See page 107 for Explanation of Tables
108
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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TABLE II. Calculated and Experimental Energy Levels of Ti XIII (in a.u.) See page 107 for Explanation of Tables
109
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TABLE III. Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results See page 107 for Explanation of Tables
110
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TABLE III. Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results See page 107 for Explanation of Tables
111
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TABLE III. Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results See page 107 for Explanation of Tables
112
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TABLE IV. Fine-Structure Collision Strengths for Ti XIII See page 107 for Explanation of Tables
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TABLE V. Excitation Rates (in cm 3 s 21) for Ti XIII Fine-Structure Levels at Electron Temperatures of 150, 400, and 600 eV See page 107 for Explanation of Tables
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TABLE VI. Relative Level Populations for Ti XIII at Various Electron Temperatures and Densities See page 107 for Explanation of Tables
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ELECTRON IMPACT EXCITATION OF FINE-STRUCTURE LEVELS OF NEON-LIKE TITANIUM (Ti XIII) G. P. GUPTA,* N. C. DEB, and A. Z. MSEZANE Department of Physics and Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, Georgia 30314
We present results of a Breit–Pauli R-matrix calculation for the electron impact excitation of neon-like titanium, in which the 27 lowest fine-structure target levels arising out of the 4 lowest configurations 2s 2 2p 6 , 2s 2 2p 5 3s, 2s 2 2p 5 3p, and 2s 2 2p 5 3d are included. These target levels are represented by configuration interaction wave functions using the 1s, 2s, 2p, 3s, 3p, and 3d basic orbitals. The relativistic effects are included in the Breit–Pauli approximation via one-body mass correction, Darwin, and spin– orbit interaction terms in the scattering equations. For many transitions, complex resonance structures are found in the excitation cross sections. The excitation cross sections are integrated over a Maxwellian distribution of electron energies to give electron excitation rate coefficients over a wide temperature range from 150 to 600 eV. The relative populations for different electron densities and temperatures are also presented. © 1999 Academic Press
* Permanent address: Department of Physics, S.D. (Postgraduate) College, Muzaffarnagar, University of Meerut, India
0092-640X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
103
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. P. GUPTA, N. C. DEB, and A. Z. MSEZANE
e 1 Ti XIII
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Theoretical Atomic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 EXPLANATION OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 TABLES I. Radial Function Parameters for Optimized Orbitals of Ti XIII . . . . II. Calculated and Experimental Energy Levels of Ti XIII . . . . . III. Comparison of Present Oscillator Strengths and Radiative Rates with Other Available Results . . . . . . . . . . . . . . . . . . . . . . . IV. Fine-Structure Collision Strengths for Ti XIII . . . . . . . . . . . . . V. Excitation Rates for Ti XIII Fine-Structure Levels at Electron Temperatures of 150, 400, and 600 eV . . . . . . . . . . . . . . VI. Relative Level Populations for Ti XIII at Various Electron Temperatures and Densities . . . . . . . . . . . . . . . . . . . . . . . .
108 109 110 113 114 115
INTRODUCTION In recent years, much attention has been focused on neon-like ions due to their wide application in astrophysical and laboratory plasma diagnostics. Some of these ions, such as Ti XIII, are also found as metallic impurities in fusion plasmas [1]. The possibility of laser amplification in transitions between the 2s 2 2p 5 3s and 2s 2 2p 5 3p configurations at VUV and soft x-ray wavelengths has further stimulated [2– 6] the need for a more comprehensive study of the atomic data for neon-like ions. Identification of 2p 5 3s–2p 5 3p and 2p 5 3p–2p 5 3d spectral lines of Ti XIII was reported by Jupe´n and Litzen [7]. Most theoretical investigations [8 –12] subsequent to this identification reported atomic data such as energy levels, wavelengths, oscillator strengths, and radiative rates. Only a few calculations [11, 12] reported collision strengths for some important transitions. Accurate atomic data such as energy levels, transition probabilities, and electron excitation rate coefficients are needed for the calculation of level populations and spectral line intensities. In the present paper we systematically built a reasonably good wave function set for the Ti XIII ion and checked its accuracy by comparing our energy levels, wavelengths, oscillator strengths, and transition probabilities against the most reliable results. Then, we used our wave functions to calculate collision strengths, excitation rate coefficients, and level populations in the Breit–Pauli R-matrix formalism for the 27 lowest fine-structure levels of Ti XIII belonging to the [1s 2 2s 2 ]2p 6 , 2p 5 3s, 2p 5 3p, and 2p 5 3d configurations. The relativistic effects are included in the Breit–Pauli ap-
proximation via one-body mass correction, Darwin, and spin– orbit interaction terms in the scattering equations [13]. Adequate care has been taken to ensure that the higher partial wave contributions, which may be particularly important for the electric-dipole-allowed transitions, are properly included. Theoretical Atomic Data In the present work, we used the six orthogonal oneelectron orbitals: 1s, 2s, 2p, 3s, 3p, and 3d. The 1s, 2s, and 2p radial functions are taken as the Hartree–Fock orbitals given by Clementi and Roetti [14] for the ground 1s 2 2s 2 2p 6 state. The parameters for the 3s, 3p, and 3d orbitals are optimized on the energies of the 2p 5 3s( 1 P o 1 3 o P ), 2p 5 3p( 3 D) and 2p 5 3d( 3 F o) states, respectively, using the CIV3 program of Hibbert [15]. The optimized parameters are displayed in Table I. We next constructed the J-dependent CI wavefunctions by using the expansions of the form [16].
O b f ~a L S JM !, K
C i ~ JM J ! 5
ij
j
j
j j
J
(1)
j51
where each of the K single-configuration functions f j is constructed from one-electron functions, and a j defines the coupling of the orbital L j and the spin S j angular momenta to give the total angular momentum J. The mixing coeffi104
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
e 1 Ti XIII
G. P. GUPTA, N. C. DEB, and A. Z. MSEZANE
TABLE A Comparison of Calculated and Observed Wavelengths (in Å) of 3s–3p lines in Ti XIII Observed a
Calculated
2p 5 3p 2p 5 3p 2p 5 3p 2p 5 3p 2p 5 3p
Transitions
3s, 3p, 3d
Large CI
3 3 3 3 3
263 297 457 469 530
281 320 458 471 554
1
So So 3 P2 1 D2 3 S1 1
a b
2p 5 3s 3 P 1 2p 5 3s 1 P 1 2p 5 3s 3 P 1 2p 5 3s 1 P 1 2p 5 3s 3 P 2
rates in the length approximation for all the optically allowed and intercombination transitions among the 27 lowest levels. In Table III we compare our weighted oscillator strengths ( gf L ) and radiative decay rates ( A L ) with those calculated by Hibbert et al. [8] and Bhatia et al. [12]. We note that for all the strong transitions, where the oscillator strengths are larger than 0.1, the general agreement among the three calculations is very good. Even for some relatively weaker transitions there is remarkably good agreement among the three calculations. The transitions 2–9, 3– 6, 3–14, 5–9, 5–10, 6 –21, 7–23, 7–24, 7–25, 8 –21, 8 –24, 9 –25, 10 –17, 10 –27, 12–18, 13–21, 14 –18, and 14 –23 are some of the examples of such agreement. We thus infer that the wave functions used in our calculation are fairly accurate for a scattering calculation. The scattering wavefunction for each total angular momentum J and parity p combination is expanded in the inner region (r # a) in the R-matrix basis as [17, 18]
282 b 322 b 459 472 552
Jupe´n and Litzen [7]. Values obtained from experimental level energies [7].
cients b ij are obtained by diagonalizing the Breit–Pauli Hamiltonian with respect to the basis C i . The Hamiltonian here consists of the nonrelativistic terms plus the Breit– Pauli corrections via one-body mass correction, Darwin, spin– orbit, spin– other– orbit, and spin–spin terms [16]. The wave functions given by Eq. (1) are then used to calculate the excitation energies of the 27 fine-structure levels of Ti XIII. These energies are displayed in Table II. In Table II our fine-structure energies relative to the ground state are compared with the experimental results of Jupe´n and Litzen [7] and the theoretical results of Hibbert et al. [8] and Qiu et al. [9]. It is clear from this Table that the ordering of our calculated excitation energies is the same as that of the experiment. Our energy values for all 26 excited levels agree within 0.5% of the measured values and are in close agreement with the other theoretical results. The results of Hibbert et al. [8] are in best agreement with the measured values because they used a large CI expansion and their ab initio energies were adjusted with the experimental energies. In this paper, our main aim is to perform a Breit–Pauli calculation for the collision strengths and other related parameters. These are rather impractical to obtain with a large CI expansion— hence, the need to work with a limited CI which gives reasonably good energies and oscillator strengths as well as allows us to perform the Breit– Pauli calculations for the scattering part without much difficulty. The wavelengths for some important transitions calculated from our energies, given in Table II, are compared with the observed wavelengths [7] in Table A. There is reasonably good agreement between theory and observations. As a further check we also calculated these wavelengths with a very large CI wave function expansion, including in addition the following configurations: (1s 2 2s 2 2p 5 ) 4s, 4p, and 4d; (1s 2 2s2p 6 ) 3s, 3p, 3d, and 4s; (1s 2 2s 2 2p 4 ) 3p 2 , 3p4p, and 4p 2 where the 4l orbitals are optimized on the 3l target terms. Excellent agreement is achieved in this case, as seen in Table A. We then used our limited CI wave function set to calculate absorption oscillator strengths and radiative decay
C k~ J p ! 5A
OC
ijk
F Ji ~ x 1 , x 2 , . . . , x N ; r N11 , s N11 !u ij ~r N11 !
ij
1
O d f ~x , x , . . . , x jk
J j
1
2
N11
!,
(2)
j
where A is an antisymmetrization operator, and F iJ are channel functions representing the 27 fine-structure atomic levels coupled with the angular and spin functions of the scattered electron to form channel functions of J and p. The f jJ are (N 1 1)-electron configurations constituted from the atomic orbitals and are included to ensure completeness of the total wave function expansion and to allow for shortrange correlation, and the u ij are the orthogonal set of continuum basis functions (orbitals). The coefficients C ijk and d jk are obtained by diagonalizing the (N 1 1)-electron Breit–Pauli Hamiltonian in the inner region. We have chosen the boundary radius a 5 3.0 a.u. Twelve continuum orbitals for each angular momentum are included to obtain convergence within the energy range of interest. The maximum number of channels retained in our calculation is 109. The coupled equations are solved using a perturbation method developed by Seaton [19] to yield K matrices and then the collision strengths. The R-matrix method is used to calculate partial collision strengths from J 5 0.5 to J 5 9.5. These partial waves produced convergent collision strengths for the forbidden and intercombination transitions, but higher partial wave contributions are needed for the dipole-allowed transitions and are calculated using the Bethe approximation [20]. In Table IV we present our results for the total collision strengths at impact energies of 45, 100, and 150 Ry. Our R-matrix results at 45 Ry are compared with the corresponding results of the distorted-wave calculation of Bha105
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. P. GUPTA, N. C. DEB, and A. Z. MSEZANE
tia et al. [12]. The two sets of results are in good agreement except for the transition 2p 6 1 S 0 3 2p 5 3d 3 D 1 , where the distorted-wave result is lower by a factor of 10. However, the oscillator strength for this transition agrees very well in the two calculations. The excitation rate coefficients for a transition from a lower level i to an upper level f at a temperature T e (in k) is given by [21]
C~i 3 f ! 5
8.63 3 10 26 2 g i T 1/ e
E
C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106 (1985) 3. D. L. Matthews, P. L. Hagelstein, M. D. Rosen, M. J. Eckart, N. M. Ceglio, A. U. Hazi, H. Medecki, B. J. MacGowan, J. E. Trebes, B. L. Whitten, E. M. Campbell, C. W. Hatcher, A. M. Hawryluk, R. L. Kauffman, L. D. Pleasance, G. Rambach, J. H. Scofield, G. Stone, and T. A. Weaver, Phys. Rev. Lett. 54, 110 (1985) 4. R. A. London, M. D. Rosen, M. S. Maxon, D. C. Eder, and P. L. Hagelstein, J. Phys. B 22, 3363 (1989)
`
V E ~i 3 f !
DE if
3 exp(2E/~kT e !)d~E/~kT e !!,
5. J. A. Cogordan, S. Lunell, C. Jupe´n, and U. Litzen, Phys. Rev. A 32, 1885 (1985)
(3)
6. G. P. Gupta, K. A. Berrington, and A. E. Kingston, J. Phys. B 22, 3289 (1989)
where g i 5 (2J i 1 1) is the statistical weight of the lower level i; DE if 5 E f 2 E i is the excitation energy; V E(i 3 f ) is the collision strength; k is the Boltzmann constant; and E is the electron impact energy. In Table V we present our excitation rates at electron temperatures 150, 400, and 600 eV. Finally, the level populations N j are calculated by solving the 27 coupled rate equations [22] N j@
O A 1 n ~ O C 1 O C !# 5 n ~O N C 1 O N C ! 1 O N A , ji
i,j
d ji
e
7. C. Jupen and U. Litzen, Physica Scripta 30, 112 (1984) 8. A. Hibbert, M. LeDourneuf, and M. Mohan, ATOMIC DATA AND NUCLEAR DATA TABLES 53, 23 (1993) 9. Y.-H. Qiu, S.-C. Li, and Y.-S. Sun, ATOMIC DATA NUCLEAR DATA TABLES 55, 1 (1993)
e ji
i,j
i
i,j
e ij
i
i.j
d ij
i
ij
AND
10. J. A. Cogordan, S. Lunell, C. Jupen, and U. Litzen, Physica Scripta 31, 545 (1985)
i.j
e
e 1 Ti XIII
(4)
11. H. L. Zhang and D. H. Sampson, ATOMIC DATA NUCLEAR DATA TABLES 43, 1 (1989)
i.j
AND
12. A. K. Bhatia, U. Feldman, and J. F. Seely, ATOMIC DATA AND NUCLEAR DATA TABLES 32, 435 (1985)
where C ij are the electron excitation rate coefficients; the superscripts e and d refer to the excitation and deexcitation, respectively; and n e is the electron density. In this calculation, we have used our A values and electron collisional rates. We considered all transitions among the 27 lowest fine-structure levels. The relative populations are normalized so that the sum of the level populations is unity. In Table VI we have tabulated our results for relative populations for a range of electron densities and temperatures appropriate to the condition inside the laser-produced plasma.
13. N. S. Scott and P. G. Burke, J. Phys. B 13, 4299 (1980) 14. E. Clementi and C. Roetti, ATOMIC DATA DATA TABLES 14, 177 (1974)
AND
NUCLEAR
15. A. Hibbert, Comput. Phys. Commun. 9, 141 (1975) 16. R. Glass and A. Hibbert, Comput. Phys. Commun. 16, 19 (1978)
Acknowledgments
17. N. S. Scott and K. T. Taylor, Comput. Phys. Commun. 25, 347 (1982)
This research is supported by the Air Force Office of Scientific Research and Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, United States Department of Energy. One of us (G.P.G.) is supported by the National Science Foundation under Grant AST-9528945.
18. K. A. Berrington, P. G. Burke, K. Butler, M. J. Seaton, P. J. Storey, K. T. Taylor, and Y. Yu, J. Phys. B 20, 6379 (1987) 19. M. J. Seaton, private communication (1987) 20. A. Burgess and V. B. Sheorey, J. Phys. B 7, 2403 (1974)
References 1. R. K. Janev, Physica Scripta 37, 5 (1991)
21. W. B. Eissner and M. J. Seaton, J. Phys. B 7, 2533 (1974)
2. M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. J. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch,
22. U. Feldman, J. F. Seely, and A. K. Bhatia, J. Appl. Phys. 56, 2475 (1984) 106
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. P. GUPTA, N. C. DEB, and A. Z. MSEZANE
e 1 Ti XIII
EXPLANATION OF TABLES TABLE I.
Radial Function Parameters for Optimized Orbitals of Ti XIII Orbital Coefficients Powers Exponents
TABLE II.
The The The The
one-electron orbitals in spectroscopic notation. expansion coefficients of the (unnormalized) Slater-type orbitals. powers of r in the Slater-type orbitals. exponents of the Slater-type orbitals.
Calculated and Experimental Energy Levels of Ti XIII (in a.u.) Index Level Present Experiment HDM QLS
A number assigned to each level in ascending order of energy. Energy levels with complete configurations. Present calculated values. Experimental values [7]. Calculated values of Hibbert et al. [8]. Calculated values of Qiu et al. [9]. 1 a.u. 5 219,474 cm 21
TABLE III.
Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results I, J gf L(I-J) A L(J-I) Present HDM BFS
TABLE IV.
Lower and upper levels as designated by the index numbers given in Table II. Weighted oscillator strength, where g is the statistical weight factor for the lower level and f L is the absorption oscillator strength. Radiative decay rates from an upper state J to a lower state I. Values in the present calculation. Calculated values of Hibbert et al. [8]. Calculated values of Bhatia et al. [12].
Fine-Structure Collision Strengths for Ti XIII BFS Present
Distorted-wave results of Bhatia et al. (Ref. [12]) at 45 Ry. Present results of Breit–Pauli R-matrix calculation at 45, 100, and 150 Ry.
TABLE V.
Excitation Rates (in cm 3 s 21) for Ti XIII Fine-Structure Levels at Electron Temperatures of 150, 400, and 600 eV
TABLE VI.
Relative Level Populations for Ti XIII at Various Electron Temperatures and Densities Te ne
Electron temperature. Electron density (in cm 23).
107
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e 1 Ti XIII
TABLE I. Radial Function Parameters for Optimized Orbitals of Ti XIII See page 107 for Explanation of Tables
108
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e 1 Ti XIII
TABLE II. Calculated and Experimental Energy Levels of Ti XIII (in a.u.) See page 107 for Explanation of Tables
109
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e 1 Ti XIII
TABLE III. Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results See page 107 for Explanation of Tables
110
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e 1 Ti XIII
TABLE III. Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results See page 107 for Explanation of Tables
111
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e 1 Ti XIII
TABLE III. Comparison of Present Oscillator Strengths and Radiative Rates (in s 21) with Other Available Results See page 107 for Explanation of Tables
112
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e 1 Ti XIII
TABLE IV. Fine-Structure Collision Strengths for Ti XIII See page 107 for Explanation of Tables
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e 1 Ti XIII
TABLE V. Excitation Rates (in cm 3 s 21) for Ti XIII Fine-Structure Levels at Electron Temperatures of 150, 400, and 600 eV See page 107 for Explanation of Tables
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e 1 Ti XIII
TABLE VI. Relative Level Populations for Ti XIII at Various Electron Temperatures and Densities See page 107 for Explanation of Tables
115
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