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

[Laurence Marks]

Small suggestion -- you may want to make the Be RMT rather larger than that
for O. You can check in case.outputm for how much charge is leaking out of
the core for the states of interest, and adjust. (I suspect that this won't
make much difference.)

2010/4/19 Amlan Ray <amlan_ray2005 at yahoo.com>

> Dear Prof. Blaha,
> I am writing in response to your mail on this subject dated April 8, 2010.
> In the meantime, I have done the calculations you suggested.
>
> a) The electron capture rate by an electron-capturing  nucleus is
> proportional to the electron density averaged over the volume of the
> nucleus. The electron capture reaction of 7Be is 7Be+e- goes to
> 7Li+neutrino. In the case of 7Be decay, 10% of the time 7Li*(478 keV state)
> is produced and 90% of the time 7Li(g.s) is produced. So 10% of the time a
> 478 keV gamma ray photon is emitted as a result of the electron capture by
> 7Be. The intensity of this  478 keV gamma ray line is measured as a function
> of time to determine the half-life of 7Be (about 53 days). In this way, the
> electron capture rate of 7BeO was measured in normal condition and under 270
> kbar pressure (causing about 9% volume contraction of the lattice) and 0.6%
> increase of the electron capture rate was found under compression.
>
> Since the nuclear volume is very small, so the averaging over the nuclear
> volume should be a very small effect. So if we take electron density at a
> point within the nuclear volume and compare with the corresponding result
> under compression, then that should be approximately ok.
>
> b) As per your suggestion, I changed RT0 in case.struct file and performed
> both relativistic and non-relativistic calculations for BeO. WIEN2K is
> considering a point nucleus and so I took points very close and within the
> beryllium nuclear volume at R0=0.0001BU, 0.00005Bu and 0.00001BU. In the
> case of non-relativistic calculations, the absolute value of the electron
> density increased by about 0.15% from R0=0.0001BU to 0.00001BU. In the case
> of relativistic calculations, the corresponding increase is by 0.28% (as
> expected). The absolute value of the electron density is about 1% higher for
> the relativistic calculation compared to non-relativistic calculation.
> However the increase of the electron density at R0 due to the compression of
> the lattice volume of BeO  by 9% remains about 0.1% for both the
> relativistic and non-relativistic calculations.
>
> c) I performed calculations considering (for Be) 1s as core state and 2s as
> valence state. I also performed  calculations considering both 1s and 2s as
> valence states. I also performed calculations by increasing Muffintin-radius
> RMT for Be from 1.45 BU to 1.49 BU. If I try to increase RMT further, then
> I am getting nn-error. Similarly for the compressed case of BeO, I could
> increase RMT (for Be) from 1.41 BU to 1.45 BU only.
>
> I find that the electron density decreases by about 0.1% when both 1s and
> 2s states are treated as valence states (for Be) compared to when 1s state
> is treated as core and 2s as valence (for Be). However the increase of the
> electron density at R0 due to the compression of BeO lattice remains about
> the same (0.1%) irrespective of how 1s and 2s states are treated.
>
> Regarding the effect of increasing the Muffin-tin radius RMT, I always
> found that the electron density at R0 increases very slightly as RMT is
> increased from 1.45 to 1.49 for Be. It happens for both relativistic and
> non-relativistic cases. I also studied this effect by treating 1s as core
> and 2s as valence (for Be). I found that 1s electron density at R0 always
> increases as a result of increasing RMT of Be, but 2s electron density at R0
> usually decreases for increasing RMT of Be. The overall effect is dominated
> by the much larger increase of 1s electron density as a result of increasing
> RMT of Be. So the total electron density at R0 always increases if we
> increase RMT of Be (keeping all other things excatly the same). This
> observation agrees well with our final result from WIEN2K calculation that
> as a result of the compression of the BeO lattice 2s electron density at R0
> increases but 1s electron density at R0 decreases.
>
> d) The calculated result from WIEN2K (0.1% increase of the electron density
> at the nucleus as a result of 9% compression of BeO lattice) disagrees with
> the experimental result of 0.6% increase. If we assume that the 1s electron
> density at the nucleus would remain unchanged due to the compression and
> only consider the increase of 2s electron density as calculated by WIEN2K,
> then the total electron density at R0 increases by about 0.25% (which is
> about 40% of the experimental result and so a better result).
>
> e) I think the problem is with the calculation of 1s electron density of Be
> at the nucleus (at R0). As the RMT of Be is increased, 1s electron density
> at the nucleus also increases. In the case of valence 2s state electrons,
> you have the boundary condition that the atomic wave function should match
> with the interstitial wave function and so it is behaving properly unders
> compression. However in the case of 1s core state wave function, you are
> using a free atom model. As I understand 1s electrons are in scf crystal
> potential, but they are not constrained to stay within the RMT radius and
> probably there is no other boundary condition also. So when the lattice is
> compressed, the Hartree potential increases and more of 1s wave function
> leaks out of RMT radius, thus reducing the 1s electron density at the
> nucleus. But 1s electrons should be constrained within RMT radius or there
> hsould be some appropriate boundary condition on them.
>
> My suggestion is if it is possible to introduce the boundary condition that
> 1s state Be wave function should be constrained within the RMT radius i.e.
> 1s wave function must be zero outside the RMT radius. If we put this
> boundary condition on 1s wave function, then I think the 1s electron density
> at the nucleus would increase due to the compression of the lattice and
> the agreement with the electron capture data (under compression) would be
> much better.
>
> With best regards
>
> Amlan Ray
> Address
> Variable Energy Cyclotron Center
> 1/AF, Bidhan Nagar
> Kolkata - 700064
> India
>
>
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>


-- 
Laurence Marks
Department of Materials Science and Engineering
MSE Rm 2036 Cook Hall
2220 N Campus Drive
Northwestern University
Evanston, IL 60208, USA
Tel: (847) 491-3996 Fax: (847) 491-7820
email: L-marks at northwestern dot edu
Web: www.numis.northwestern.edu
Chair, Commission on Electron Crystallography of IUCR
www.numis.northwestern.edu/
Electron crystallography is the branch of science that uses electron
scattering and imaging to study the structure of matter.
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