Electron Capture Ratios in Ho and the Influence of different final States Amand Faessler, Thanks to M. Krivoruchenko, F. Simkovic for their contribution.

Overlap and Echange Corrections J. N. Bahcall, Phys. Rev. Lett. 9 (1962); Phys. Rev. 132 (1963) 362 and Nucl. Phys. 71(1965) 267; small basis partially only 1s,2s; no holes in final nucleus; no p1/2 capture; non-relativistic in light atoms. A. Faessler, E. Huster, O. Kraft, F. Krahn, Z. Phys. 238 (1970) 352. First time modification of the final electron wave functions by electron hole in different states; non-relativistic; no p1/2 capture. M. J. Martin, P. H. Plichert-Toft, Nucl. Data 8 (1970) ; relativistic without holes. E. Vatai, Nucl. Phys., A402 (1983) 1; non-relativistic, no holes. T. Mukoyama, Bull. Inst. Chem Res., Kyoto, 65(1987) Ho  163Dy; Relativistic s1/2 and p1/2 and a general hole (not specified) in Dy.

Electron capture probability

163 Ho and 163 Dy needs relativistic treatment of electrons

Dirac-Fock Code of : A. l. Ankudinov, S. I. Zabinsky, J. J. Rehr, Comp.Phys. Com. 98 (1996) 359  r

Radial coordinate: k = ln(r/r 0 )/0.05= 1…251 r 0 = *10 -5 [au]

Present Ho s-wave functions at the nuclear radius R = 1.2*A 1/3 [fm] compared with results of J. B. Mann and J. T. Waber, Atomic Data 5 (1973) 201 at the origin r = 0.0. Mann + Waber: Slater approximation for exchange and uniform charge distribution of the nucleus. S-waves Ho1s2s3s4s5s6s large at radius small at radius Mann large  (0) Mann small  (0) 0.

P1/2 Ho2p1/23p1/24p1/2 „Large“ at radius „Small“ at radius „Large“ Mann at r = „Small“ Mann at r = Present Ho p1/2 wave functions at the nuclear radius R = 1.2*A1/3 [fm]compared with J. B. Mann and J. T. Waber, Atomic Data 5 (1973) 201 (table VII).

Electron Decay Spectrum measured by the Bolometer (Automatically summed over all final decays of the Electron hole)

Wave functions and overlap expressions:

Amplitudes F 0f with overlap corrections + exchange F 0f.

As an example electron capture in Ho to the M 1 (3s1/2‘) Dy hole.

Overlap of 143Holmium s-electrons in the ground state with 143 Dysprosium s-electrons with a hole in M 1 (3s1/2). |Ho> 1s2s3s4s5s6s <Dy|1s <Dy|2s <Dy|3s <Dy|4s <Dy|5s <Dy|6s

F 0f ) The amplitudes F 0 (ns 1/2 Ho  ns 1/2 Dy). F f ) The amplitudes F(ns 1/2 Dy) summed over Ho. B 0f ) The correction factors B 0 (ns 1/2 Dy) = {F 0 /  ns1/2 (0)} 2 B f ) The final correction factor B(ns1/2) = {F(ns1/2 Dy)/  ns1/2(0)} 2 F 0f FfFf B 0f BfBf M1M1 3s1/ N1N1 4s1/ O1O1 5s1/

F 0f ) The amplitudes F 0 (np 1/2 Ho  np 1/2 Dy). F f ) The amplitudes F(np 1/2 Dy) summed over Ho. B 0f ) The correction factors B 0 (np 1/2 Dy) = {F 0 /  np1/2 (0)} 2 B f ) The final correction factor B(np1/2) = {F(np1/2 Dy)/  np1/2(0)} 2 F 0f FfFf B 0f BfBf M2M2 3p1/ N2N2 4p1/ O2O2 5p1/