Finite Nuclear Size Corrections Research Articles

Systematic trends of the oscillator strengths for n=3 and 2 electric dipole transitions in oxygenlike ions.

Energies and oscillator strengths for electric dipole transitions between states with principal quantum numbers 3 and 2 of singly excited oxygenlike ions have been calculated for 32 ions with atomic numbers 10\ensuremath{\le}Z\ensuremath{\le}79 using the multiconfiguration Dirac-Fock method. The calculations include finite nuclear size, Breit interaction, and quantum electrodynamic corrections. The Z dependence of the oscillator strengths throughout the sequence is analyzed for some selected transitions. Effects of both relativity and configuration interaction are seen to be very important for medium and highly charged ions. Numerous irregularities and some sharp discontinuities in the Z dependence of the oscillator strengths are found for many transitions due to the effect of level crossings. A rapid interpolation based on results obtained for a few ionization stages can therefore lead to serious errors.

Residual limitations of theoretical atomic-electron binding energies.

Relativistic calculations of atomic-electron binding energies have been refined by using the relativistic LS-average scheme to treat open outer shells, by including self-energy corrections for shells up to 3p as well as 4s, and accounting for energy shifts caused by interaction with Coster-Kronig continua. The contributions of ground-state correlation were estimated from the pair energy calculated through nonrelativistic many-body theory. The need for a relativistic theory of correlations is noted. As in our previous work, the calculations include relaxation, the effect of finite nuclear size, Breit interaction, and quantum-electrodynamic (QED) corrections. Results are compared with binding energies measured on free atoms and with solid-phase measurements on metals that have been corrected for solid-state shifts; these shifts were calculated under the assumption of complete screening with the core-ionized site treated as a neutralized metallic impurity atom in the original metallic host. Discrepancies between experimental energies and relativistic independent-particle calculations including relaxation, QED, and finite-nuclear-size corrections are traced to correlation corrections, uncertainties in the self-energy, and neglect of the effect of (super-)Coster-Kronig fluctuations.

Nuclear finite-size effects in light muonic atoms

The finite nuclear size corrections to the s-state energy levels of light muonic (or hydrogenic) atoms are calculated analytically through order ( Zα) 6. In addition to the usual expression of order ( Zα) 4, the ( Zα) 5, ( Zα) 6 and ( Zα) 6 log( Zα) contributions have been determined. These corrections have been separated into terms of nonrelativistic and relativistic orders. The results have been checked by solving perturbatively the exact eigenvalue equations of the Schroedinger and Dirac problems of a particle orbiting in the Coulomb field of a uniform charge distribution of fixed radius. Application is made to the case of the μ- 4He atom. Finite-size contributions to the hydrogen Lamb shift and the relativistic recoil effect are briefly discussed.

Analysis of the recoil ion spectra for the β-decays of 6He, 19Ne and 35Ar

The recoil energy spectrum is calculated for allowed β-decay with all form factors up to the weak magnetism and axial tensor terms included. Finite nuclear size and electromagnetic corrections including bremsstrahlung are also given. The results are equivalent to evaluation of the electron neutrino angular correlation but are directly related to the observed quantities and thus more easily interpreted in terms of the higher order form factors. The formulae were applied to the experimental + 6Li + spectrum of Johnson et al. for the decay 6 He → 6 Li + + e − + v and the best fit yields a tensor form factor consistent with zero, in agreement with the approximate analysis of Calaprice. The implications for a possible second-class current are discussed. The theoretical recoil spectra for the 19Ne and 35Ar mirror decays are also calculated and it is shown that measurements of these spectra, with accuracy equivalent to the 6He data, would yield conclusive evidence for a possible second-class tensor form factor.

Effect of field theoretical corrections on X-ray spectra

The wavelengths of\(K_{\alpha _1 } , K_{\alpha _2 } , K_{\beta _1 } , K_{\beta _2 } , K_{\beta _2 } I and K_{\beta _2 } II\) lines of heavy elements of atomic number 70–92 and the associated energy levels, K, LII, LIII, MII, MIII, NII and NIII have been calculated using a modified form of the relativistic energy formula of Sommerfeld, incorporating in it both the total and internal screening constants in a closed form. All the field theoretical corrections in the second and the fourth order as well as the finite nuclear size correction have been included.

Log- f tables for beta decay

An extensive tabulation of log- f values is presented for use in the calculation of the comparative half-lives for beta transitions. Values of capture-to-positron ratios are also given. These log- f -and ε/β + -values incorporate screening and finite nuclear size corrections. They are tabulated for allowed and first-forbidden unique transitions for Z from 6 to 95 as a function of transition energy. The range of transition energies covered is 0.010 MeV to 10 MeV with energy increments chosen so that intermediate values can be obtained from the tables by means of linear extrapolation. The values for first-forbidden unique transitions are larger than those given in older tables by 1.079, the log of 12. [See Eqs. (19) and (21).]

Beta-Gamma Correlations and Matrix Elements for Some First-Forbidden Nonunique Beta Transitions

Results of $\ensuremath{\beta}\ensuremath{-}\ensuremath{\gamma}$ directional-correlation measurements in the nonunique first-forbidden $\ensuremath{\beta}$ decay of the ground states of several odd-odd nuclei (${\mathrm{Sb}}^{124}$, ${\mathrm{Eu}}^{152}$, ${\mathrm{Re}}^{186}$, ${\mathrm{Tm}}^{170}$, ${\mathrm{Rb}}^{84}$) to the first ${2}^{+}$ levels in the even-even daughter nuclei are presented. These measurements were made as a function of the $\ensuremath{\beta}$-ray energy using a small shaped-field magnetic spectrometer. The results, in combination with all other available experimental data on each $\ensuremath{\beta}$ transition, have been analyzed to obtain the nuclear matrix element parameters. The analysis is based on the theoretical expressions given by Morita and Morita using exact electron radial wave functions which include finite nuclear size corrections. In addition, we have analyzed all available experimental data for the nonunique first-forbidden $\ensuremath{\beta}$ decay of ${\mathrm{Re}}^{188}$ and ${\mathrm{Rb}}^{86}$. A unique set of parameters was found for ${\mathrm{Re}}^{186}$, ${\mathrm{Re}}^{188}$, and ${\mathrm{Tm}}^{170}$ although the set for ${\mathrm{Tm}}^{170}$ suffers from a lack of adequate experimental information. For ${\mathrm{Rb}}^{86}$, the matrix element sets depend on whether the shape of the $\ensuremath{\beta}$ spectrum is statistical or not. For ${\mathrm{Rb}}^{84}$, five distinct sets of parameters were found which fit all available data equally well. For both ${\mathrm{Sb}}^{124}$ and ${\mathrm{Eu}}^{152}$, two sets of parameters were found that fit the data equally well; however, it is shown that a measurement of the $\ensuremath{\beta}$-circularly-polarized $\ensuremath{\gamma}$ correlation as a function of energy can eliminate one set in each case. Appropriate discussions based on nuclear models are given.

Finite nuclear size and radiative corrections in the construction and assessment of kurie plots for allowed beta-decay

If only a small fraction of a beta-spectrum is available for Kurie plot analysis because of the existence of strong branches of lower energy, it may become important to take into account finite nuclear size and radiative effects in the construction and assessment of the Kurie plot. Failure to do this may result in errors of many percent in the extraction of the intensity of the high energy beta-branch. These effects are analyzed and numerical estimates are made of the possible errors.

Beta Decay to the Second 2+ Excited State of 122Te

Bata-gamma angular correlations and beta spectra of the first forbidden beta transitions of 122 Sb have been studied. The result of the angular correlation measurement of the (734 keV β, 1256 keV γ) cascade indicates deviation from the ξ approximation for the 734 keV β transition. The analysis using exact electron radial wave functions which include finite nuclear size corrections shows that cancellation exists among the matrix elements. Thus, the ξ approximation is not fulfilled. The nuclear matrix parameters fitted to the angular correlation data are x =0.34±0.44, u =2.03±0.96, y '=25.0±13.8, w =1.63±4.94 and v '=-18.4±53.5. These parameters describe well the measured spectrum of the 734 keV β transition which shows deviation from the allowed shape. A comparison between the experimental results and the theoretical prediction based on the CVC theory is given.

Nuclear matrix elements for the beta decay of 198Au and 122Sb

All presently available experimental data on the spectrum shape factor, β-gg directional correlations, β-circularly polarized γ-correlations, and angular distribution of γ-rays from oriented nuclei in the first-forbidden beta transitions of 198Au and 122Sb to the first excited states in 198Hg and 122Te, respectively, have been used to obtain the nuclear matrix elements parameters x, u, ξ′ y, w and ξ′ v. The analysis was based on the theoretical expressions given by Morita and Morita using exact electron radial wave functions which include finite nuclear size corrections. In the case of 198Au, a unique set of parameters was found, but for 122Sb, two sets of parameters were found that fit the data equally well. These parameters are: for 198Au, x = 0.047±0.006, u = −0.097±0.008, ξ′ y = −1.95±0.21; w = 0.20±0.09 and ξ′ v = −4.23±1.26; for 122Sb, Set I, x = −0.06±0.06, u = 0.08±0.10, ξ′ y = 1.15±0.57, w = 4.83±0.55 and ξ′ v = −54.7±5.8; Set II, x = 0.01±0.05, u = 0.10±0.11, ξ′ y = 0.06±0.6, w = 2.47±0.50 and ξ′ v = −29.3±5.8. However, it is shown that in the case of 122Sb, a determination of β-circularly polarized γ-correlation as a function of energy can eliminate one set. For both 198Au and 122Sb, the observed values of the matrix elements are not in agreement with single-particle predictions. However, in the case of 122Sb, they are in agreement with the predictions of Kisslinger and Wu.

Decay ofHf181

The decay of ${\mathrm{Hf}}^{181}$ to ${\mathrm{Ta}}^{181}$ was studied by angular correlation measurements ($133K\ensuremath{-}482\ensuremath{\gamma}$, $133\ensuremath{\gamma}\ensuremath{-}482\ensuremath{\gamma}$, and $133\ensuremath{\gamma}\ensuremath{-}482K$, energies being given in kev), conversion coefficient measurements [${\ensuremath{\alpha}}_{K}(133)=0.50\ifmmode\pm\else\textpm\fi{}0.06$, ${\ensuremath{\alpha}}_{K}(136)=1.2\ifmmode\pm\else\textpm\fi{}0.2$, ${\ensuremath{\alpha}}_{K}(346)=0.038\ifmmode\pm\else\textpm\fi{}0.007$, ${\ensuremath{\alpha}}_{K}(482)=0.023\ifmmode\pm\else\textpm\fi{}0.002$], $\frac{K}{L}$ ratios, and coincidence measurements. The weak 137-kev transition, first observed by Mihelich, was placed in the decay by delayed coincidence measurements. The spin assignments are in excellent agreement with the calculations of Nilsson on the strong-coupling model. Comparison of $K\ensuremath{-}\ensuremath{\gamma}$ and $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\gamma}$ correlation coefficients in the Hf${\mathrm{O}}_{2}$ sources fails to show any after effects of $K$-hole formation on the correlation coefficients. The discrepancy between the $\frac{E2}{M1}$ ratio obtained from $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\gamma}$ correlation measurements and conversion coefficient measurements in the 482-kev transition is discussed in terms of the improved finite nuclear size corrections to $M1$ conversion coefficients described by Church and Weneser.

The Scattering of Fast Positrons by Nuclei

It is shown that the effects of finite nuclear size and radiative correction cause considerable modifications to the Mott formula for the cross sections. An exact calculation of the phase shifts is carried through for a simple nuclear model and 20 MeV positrons scattered by gold nuclei. Two approximate phase shift calculations are made and the singularity occurring in the radial wave equation is investigated. The radiative correction for scattering from light nuclei is found to be the same as for electron scattering.