[Wien] electron density at the nucleus (Electron capture nuclear decay rate work)

[Amlan Ray]

Dear Stefaan,
Thank you for your detailed message suggesting to check several things. I have now done those calculations and let me discuss the results and my thoughts.
 
Regarding the question whether the 1s electron density at the nucleus should increase because of the compression of the beryllium atom, I still think it should increase (although WIEN2K is predicting that it should decrease) and the presence of the 2s electrons should not reverse this result and cause a decrease. 1s and 2s states are orthogonal and the total electron density at the nucleus is simply the sum of the two as WIEN2K code is also showing. However 2s electrons would certainly screen 1s electrons and as a result of the compression of 2s orbitals and the corresponding increase of 2s electron density at the nucleus, this screening effect would also increase. But I think this would be a higher order effect and cannot reverse the increase of 1s electron density at the nucleus due to the compression effect. For example, J. Bahcall (Phys. Rev. 128, 1207 (1962)) and Hartree and Hartree (Proc. R. Soc. London ser. A 150, 9 (1935)) found from their
 calculations that even if both the 2s electrons are removed from a beryllium atom (making 2s electron density at the nucleus =0), then also 1s electron density at the nucleus changes by only about a few tenth of a percent. This is because when 1s electron is very close the nucleus, it sees only the bare nucleus. In our case, WIEN2K is predicting only 10% increase of 2s electron density at the nucleus and so its effect on 1s electron density at the nucleus would be much smaller. On the other hand, the compression of 1s orbital should definitely increase the electron density at the nucleus. So the prediction of WIEN2K (decrease of 1s electron density at the nucleus due to compression)  is still puzzling to me. 
 
Regarding the calculations I have done
1) R0 =0.0001BU = 5.29 Fermi in my calculation. So it is nuclear distance. I tried to make it smaller, but the code didi not take any smaller value and made it again = 0.0001 BU. 
2) I performed calculations making both 1s and 2s valence states (-8 Ry input). For 9% volume compression of BeO lattice, it predicts 0.033% increase of total electron density at the nucleus. When I kept 1s as a core state, then the corresponding increase of total electron density at the nucleus was 0.09%. However if I compress BeO lattice volume by 15%, and treat both the 1s and 2s states as valence states, the corresponding increase of total electron density at the nucleus = 0.18%. On the other hand, if 1s is treated as a core state, then for 15% volume compression, the increase of total electron density at the nucleus= 0.15%.The experimental result regarding the increase of electron density at the nucleus due to 9% compression of BeO lattice is about 0.6% -0.8% (W..K Henseley et al., Science, 181, 1164 (1973) and L.g. Liu et al., Earth. Planet. Sci. Lett. 180, 163 (2000)). (However Liu's result of 0.8% is for amorphous beryllium hydroxide.) 
 
Mossbauer isomer shift is proportional to the difference of contact densities, but the electron capture rate is directly proportional to the electron density at the nucleus. I do not know if anyone has studied Mossbauer isomer shift under the effect of compression of the atom.
 
I would like to know how WIEN2K is doing 1s state wavefunction calculation under compression. What is the relevant subroutine to look at and any reference about the wavefunction calculation? Is it solving Schrodinger equation under both Coulomb and Hartree potential? N. Aquino et al. have performed (Phys. Lett A307, 326 (2003)) density functional calculations of a single compressed He atom placed in a spherical box. They have also recently completed such density functional calculations for compressed Li atom placed in a spherical box. It is found from their calculations that the 1s state electrons of a compressed He or Li atom very quickly start looking like a Thomas-Fermi atom where the electrons are in a box of radius equal to the mean radius of 1s electrons and the electrons can be treated as free particles. The kinetic energy of 1s electrons increases as the inverse square of the radius of mean distance of 1s electrons from the nucleus
 (Thomas-Fermi atom result). 
 
I find if I take the increase of 1s electron energy (due to the compression) from WIEN2K or TB-LMTO calculations (both give similar results) and then apply simple Thomas-Fermi model of atom assuming that the increase of the energy of 1s electrons (mostly kinetic energy increase) is due to the reduction of 1s orbital volume, then I can get a number for the increase of electron density in the box. If this is interpreted as the increase of electron density at the nucleus, then I get reasonable agreement with the experimental numbers of Henseley et al. (regarding BeO) and also with our experimental results regarding the increase of electron density at the nucleus of 109In and 110Sn implanted in small Au lattice versus large Pb lattice (A. Ray et al., Phys. Lett B679, 106 (2009)). We have discussed this method of calculation in our paper, but it is very empirical without much theoretical grounding. 
 
TB-LMTO treats atomic core states  as frozen. I am interested to know about the core state wave function calculation of WIEN2K  and whether the calculation is agreeing with the density functional calculations of compressed atoms that have been done so far. 
 
With best regards
                                                                                                 Amlan Ray
Address:
Variable Energy Cyclotron Center
1/AF, Bidhan Nagar
Kolkata - 700064
India


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