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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
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
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
MCDFS and MBPT2
Our theory is combination of Multi-configuration Dirac-Fock-Sturmand Many-body perturbation theory
For this talk I focus on wavelengths of emission lines for 2pq 3s 2pq 3p ----------> 2p
q+1 q=2, 3, 4, 5
2pq 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
Brief overview of theory
= c det
H c = E cHD
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.
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.
1s2 2s2 2p3 3s, 3p, 3d (4s, 4p, 4d, ..., 8s, 8p, 8d)
Valence Virtual
Example of valence and virtual orbitals in Fe XIX (O-like).
Occupied Unoccupied
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.
Our solution is to use Sturm's function for virtual orbitalsObtained by solving the Dirac-Fock-Sturm equation.
Continue:
[HDF - j0] j = j W(r) j
Usual DF operator
FixedEnergy equalto one of the
valence energies
Eigenvalueof operator
Weightfunction
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/r2 for r goes to infinity
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 correctionsin the CI
No QED corrections!
MCDFS + MBPT2 applied to Fe XVIII upto Fe XX
We compare our calculation with EBIT experimentaldata and theoretical HULLAC calculations ofBrown et al (2002).
Scale of calculation determined by size of Hamiltonianmatrix
Number of relativistic orbitals for all three ions is 46 Number of electrons in 2p shell differs Not all orbitals are treated equally:
1s2 2s2 2pq 3s 3p 3d … 5d appear in CI Higher excited orbitals upto n=8 are treated perturbatively
Example of scale of Fe XIX calculation
~60000 determinants in CI
~107 determinants in perturbation theory
Many zero matrix elements in Hamiltonian
AMD PC, 2GHz clock speed, 2Gbyte memory 150 Gbyte hard drive
A calculation of 2p33s energy levels takesone day
Conclusions of our Iron calculations
We are reach an 10-3 Angstrom agreement withexperiment without QED corrections
An estimate of QED corrections suggests correctionsbetween 10-3 - 10-4 Angstrom
We will include QED effects in the near future
We will attack the problem of the unidentified linesand 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
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 Ne9+
Web site will be published in the beginning of 2005. http://physics.nist.gov/PhysRefData/HLEL/index.html
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:
Comparison with experimental Lamb-shift of H and H-like ions.
The error bars show theoretical uncertainty due to the uncertaintyin the nuclear radius.
The difference between theoryand experimental data is small
0
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)