1. Accurate wavelenghts for X-ray spectroscopy and the NIST Hydrogen and Hydrogen-Like ion databases Svetlana Kotochigova Nancy Brickhouse, Kate Kirby, Peter Mohr, and Ilia Tupitsyn Collaborators: National Institute of Standards and Technology Temple University

2. Outline: Highly charged ions Multi-configuration Dirac-Fock-Sturm method (MCDFS) Efficient in describing correlations Dirac energy + other Relativistic effects + QED + nuclear size effects Interactive WEB database Hydrogen-like ions

3. Application of MCDFS: L-shell emission spectra of Fe XVIII to Fe XXII Improving database for X-ray diagnostics Relatively well studied Transition Probability database at NIST EBIT experiment by Brown et al, ApJS, 140 p589 (2002) Still far from complete Line blending Weak lines

4. MCDFS and MBPT2 Our theory is combination of Multi-configuration Dirac-Fock-Sturm and Many-body perturbation theory For this talk I focus on wavelengths of emission lines for 2p q 3s 2p q 3p ----------> 2p q+1 q=2, 3, 4, 5 2p q 3d However, other properties, such as oscillator strengths and photo- and auto-ionization x-sections, can be evaluated Initial and final states use different and thus a non-orthogonal basis sets

5. Brief overview of theory  = c  det           H  c  = E c     H D    We construct N-electron Slater determinants from one-electron four-component Dirac spinors and Sturm's orbitals. Total wave function: The c  are found from solving: with an iterative Davidson algorithm Second order perturbation theory is used to include higher-order correlation effects from highly excited states.

6. Dirac Fock and Sturmian orbitals: Valence electrons are calculated by solving Dirac-Fock equations. Virtual orbitals are included in the CI to improve description of the total wave function. 1s 2 2s 2 2p 3 3s, 3p, 3d (4s, 4p, 4d,..., 8s, 8p, 8d) Valence Virtual Example of valence and virtual orbitals in Fe XIX (O-like). Occupied Unoccupied

7. Continue: Usage DF-functions for virtual orbitals is ineffective; the radius of DF-orbitals grows fast with level of excitation, so their contribution to CI is small.

8. Our solution is to use Sturm's function for virtual orbitals Obtained by solving the Dirac-Fock-Sturm equation. Continue: [H DF -  j0 ]  j =  j W(r)  j Usual DF operator Fixed Energy equal to one of the valence energies Eigenvalue of operator Weight function

9. The Sturm's orbital has ~ the same radius and the same asymptotic behavior as the valence orbital. The mean radius increases slowly with n. It leads to efficient treatment of correlation effects. Sturm's wave functions create a complete and discrete set of functions. Weight function: In V. Fock, Principles of quantum mechanics (1976), R. Szmykowski, J Phys B 30, p825 (1997) W(r) ~ 1/r. For more complex systems we use W(r) = - 1-exp(-(  r) 2 ) (  r) 2 -1 for r goes to 0 1/r 2 for r goes to infinity

10. Example of Sturm’s functions.

11. More details about the method We use a Fermi-charge distribution for the nucleus CI includes single, double, and triple excitations. We include Breit magnetic and retardation corrections in the CI No QED corrections!

12. MCDFS + MBPT2 applied to Fe XVIII upto Fe XX We compare our calculation with EBIT experimental data and theoretical HULLAC calculations of Brown et al (2002). Scale of calculation determined by size of Hamiltonian matrix Number of relativistic orbitals for all three ions is 46 Number of electrons in 2p shell differs Not all orbitals are treated equally: 1s 2 2s 2 2p q 3s 3p 3d … 5d appear in CI Higher excited orbitals upto n=8 are treated perturbatively

13. Example of scale of Fe XIX calculation ~60000 determinants in CI ~10 7 determinants in perturbation theory Many zero matrix elements in Hamiltonian AMD PC, 2GHz clock speed, 2Gbyte memory 150 Gbyte hard drive A calculation of 2p 3 3s energy levels takes one day

14. Brown et. al ApJS (2002) We found more lines below O13

16. We obtained many more lines than observed in EBIT experiment.

18. Open core transitions for Fe XVIII

19. Conclusions of our Iron calculations We are reach an 10 -3 Angstrom agreement with experiment without QED corrections An estimate of QED corrections suggests corrections between 10 -3 - 10 -4 Angstrom We will include QED effects in the near future We will attack the problem of the unidentified lines and line blends in X-ray transitions of Iron ions Our wave lengths are always lower than HULLAC's. better treatment of the ground state which lowers its total energy

20. Energy levels and transition frequencies of Hydrogen-Like ions. NIST project, led by P. Mohr, to create an interactive database for H-like ions. The database will provide theoretical values of energy levels and transition frequencies for n = 1 to n = 20 and all allowed values of l and j Values based on current knowledge of relevant theoretical contributions including relativistic, QED, recoil, and nuclear size effects. Fundamental constants are taken from CODATA – LSA 2002. Uncertainties are carefully evaluated We now work on H-like ions from He + to Ne 9+ Web site will be published in the beginning of 2005. http://physics.nist.gov/PhysRefData/HLEL/index.html

22. Relativistic Recoil Self Energy Vacuum Polarization Two-photon Corrections Three-photon Corrections Finite Nuclear Size Radiative-Recoil Correction Nuclear Size Correction to Self Energy and Vacuum Pol. Nuclear Polarization Nuclear Self Energy Contributions to energy level The main contribution comes from the Dirac Energy Others include:

23. Comparison with experimental Lamb-shift of H and H-like ions. The error bars show theoretical uncertainty due to the uncertainty in the nuclear radius. The difference between theory and experimental data is small 0

24. Hydrogen energy levels Frequency interval(s) Reported value Theoretical value (kHz) (kHz) H(2S1/2 - 4S1/2) - 1/4 H(1S1/2 - 2S1/2) 4 797 338(10) 4 797 330(2) H(2S1/2 - 4D5/2) - 1/4 H(1S1/2 - 2S1/2) 6 490 144(24) 6 490 128(2) H(2S1/2 - 8S1/2) 770 649 350 012.1(8.6) 770 649 350 015(3) H(2S1/2 - 8D3/2) 770 649 504 450.0(8.3) 770 649 504 448(3) H(2S1/2 - 8D5/2) 770 649 561 584.2(6.4) 770 649 561 577(3) H(2S1/2 - 12D3/2) 799 191 710 472.7(9.4) 799 191 710 481(3) H(2S1/2 - 12D5/2) 799 191 727 403.7(7.0) 799 191 727 408(3) H(2S1/2 - 6S1/2) - 1/4 H(1S1/2 - 3S1/2) 4 197 604(21) 4 197 599(3) H(2S1/2 - 6D5/2) - 1/4 H(1S1/2 - 3S1/2) 4 699 099(10) 4 699 104(2) H(2S1/2 - 4P1/2) - 1/4 H(1S1/2 - 2S1/2) 4 664 269(15) 4 664 253(2) H(2S1/2 - 4P3/2) - 1/4 H(1S1/2 - 2S1/2) 6 035 373(10) 6 035 383(2) H(2S1/2 - 2P3/2) 9 911 200(12) 9 911 196(3) H(2P1/2 - 2S1/2) 1 057 845.0(9.0) 1 057 845(3) H(2P1/2 - 2S1/2) 1 057 862(20) 1 057 845(3)

28. The address is: http://physics.nist.gov/PhysRefData/HDEL