Osmiridium: Investigating Nuclear Structure with XRF Spectroscopy and SODFT

All physics majors at MIT are required to take a set of experimental physics classes with laboratory work. One of the experiments in these courses concerns x-ray fluorescence (XRF) spectroscopy. Using this experimental setup, one can easily confirm Moseley's Law - the \(E \propto Z^2\) relation between the energy of the XRF photons and the atomic number of the element being interrogated. The Law comes about by considering a simple hydrogenic model for all elements, but this treatment becomes inherently less accurate at large \(Z\) because relativistic effects become significant and the core orbitals contract. Some students made a splash a few years ago by publishing a paper about the observed deviation from Moseley's Law by using the Sommerfeld model to describe relativistic corrections to the core electron energies. They showed that the lab setup was sensitive enough to measure such deviations from the Moseley \(Z^2\) curve.

Photo of the X-Ray spectrometer in MIT Physics Dept. Junior Laboratory. The spectrometer contains a liquid nitrogen cooled Ge crystal drift detector and is connected to an MCA. Radioactive sources with known emission energies are used as calibration standards.

There is an additional correction to the XRF energy spectrum that holds interest for nuclear as well as atomic physicists: the isotope shift. The isotope shift refers to shifts in the energy levels of the atom that are due to variance in the mass and volume between the nuclei of different isotopes of the same element. For light nuclei, the mass shift between isotopes results in a shift of the orbital energies due to a change in the center of mass. For heavy nuclei, the mass shift is convolved with the change in nuclear volume from the light to the heavy elements, known as the volume of field shift. Heavy nuclei can no longer be treated as point particles with positive charge. Instead, the charge of the nucleus must be smeared over a small, but finite, volume. Since the 1s electrons are the closest to the nucleus, it is this level that is most affected by variations in the nuclear charge distribution. Higher energy levels, such as the 2p, 3p, and 4p orbital, are relatively unaffected. XRF spectroscopy can, in principle, be used to indirectly probe the charge distribution of the nucleus by calculating the energies of transitions between 2p, 3p, or 4p and 1s using a theory that with a point-like nuclear model and observing the deviation from those energies due to the finite nuclear volume in experiment. A combined theory and experiment approach would allow testing of theoretical models in atomic physics that properly account for energy changes in the atomic levels due to the isotope shift as well as provide a better understanding of nuclear structure without the need for MeV (or GeV) electron beams. Electrons or photons with energy less than 150 keV are sufficient to produce these characteristic XRF transitions in all naturally occurring elements.

I used the spin-orbit density functional theory module in NWChem to obtain the energies of the atomic orbitals for each of the six elements measured in the experiment. The PBE0 exchange-correlation functional and SARC-ZORA basis sets were used for all calculations. A first-order approach to calculating the \(K\alpha\) transition energies from the calculated SODFT energy levels is to simply take the difference between the 1s and 2p energies since it is this atomic transition that produces the majority of the energy of the fluorescent photon. For further accuracy, a true excited state calculation is required. Since XRF concerns the core-electrons, we can assume that the local chemical environment will have minimal effect on the observed energies.

NWChem provides a built-in RMS radius for nuclei composed of a Gaussian smearing of positive charge with \(r_{RMS} = 0.836A^{1/3}+0.57\) femtometers where \(A\) is the nuclear mass. For \(U^{238}\), this gives \(r_{RMS} = 5.751\) femtometers. The keyword "finite" or "point" must be provided in the "GEOMETRY" section header to switch between the two options. An example NWChem input file is provided below as a template.

start atom

TITLE "U_Finite"
CHARGE 0
GEOMETRY nucleus finite
   U 0.0 0.0 0.0
END

BASIS spherical
   * library SARC-ZORA
END

RELATIVISTIC
   zora on
   zora:cutoff 1d-30
END

DFT
   odft
   mult 5
   xc pbe0
   convergence energy 1e-8
   grid fine
   iterations 1000
   print "final vectors analysis"
END
task sodft

The experimental results indicate that the experimental setup is able to achieve a 100 eV spectral resolution for XRF spectroscopy measurements - pretty good as XRF spectroscopy goes and considering this was an undergraduate laboratory class. I measured \(K\alpha_1\) transition energies which conformed very well (within one standard deviation) to the NIST standard measurements. Both the current measurements and NIST values show good agreement with the level of theory used - better than Moseley's Law or Sommerfeld relativistic corrections - in the NWChem calculations, but they do continue to diverge at high \(Z\). This indicates that there is still some aspect of the physics that the NWChem model does not capture, which is not unexpected. One contributing factor might be that the formula NWChem uses to estimate the RMS radius of the nuclei underestimates the currently accepted experimental value of the nuclear radius of U (5.857 fm) by 0.1 femtometers.

Unfortunately, I did not get as far as testing whether the isotope shift could be measured experimentally, but the prospects for doing so on even this modest setup seem promising. With more work and better x-ray sources (I used radioactive materials), it seems reasonable that the spectral resolution might be pushed past 100 eV to make ordering isotopically pure samples of heavy elements worthwhile. The center of mass shift in light elements might also be measured. What I find exciting about this experiment is that it indicates that, in state-of-the-art experimental facilities, it might be possible to test theories in both atomic and nuclear physics simultaneously. Ab-initio electronic structure theories for large nuclei may be refined based on the experimental results and might provide new understanding of the relationship between nuclear and electronic structure.

The next logical step to take this line of research further is to find a more accurate theoretical treatment. I found a number of papers describing the multi-configuration Dirac-Fock approach for QED and relativistic corrections. Some of these references are below. I am not sure what software already exists for doing these calculations or if they include the volume of field corrections. This should resolve any remaining doubts caused by the known inadequacies of DFT. Experimentally, better models of the spectral background would greatly increase the resolution capabilities. Obtaining higher-quality samples of the heavy metals and modeling the effect of the chemical environment of XRF spectra is also important to understand systematic errors. It would also be advantageous to consider the less favorable transitions from higher-energy p-orbitals to the 1s shell since these orbitals interact even less with the nucleus than the 2p orbitals. Other transition schemes might be considered as well.

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Investigating Nuclear Structure with XRF Spectroscopy and SODFT

Further Reading