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

Example of Sturm’s functions.

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

Brown et. alApJS (2002)

We found more lines below O13

We obtained many morelines than observed inEBIT experiment.

Open core transitions for Fe XVIII

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)

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