Finite Nuclear Size Effects Research Articles

Direct pair production in heavy-ion—atom collisions

Direct pair production in approx.5-MeV/amu heavy-ion--atom collisions with uranium target atoms is calculated with the plane-wave Born approximation and the semiclassical approximation. Briggs's approximation is used to obtain the electron and positron wave functions. Since pair production involves high momentum transfer q from the moving projectile to the vacuum, use is made of a high-q approximation to greatly simplify the numerical computations. Coulomb deflection of the projectile, the effect of finite nuclear size on the elec- tronic wave functions, and the energy loss by the projectile exciting the pair are all taken into account in these calculations.

Virtual photon spectrum in second-order Born approximation

We have evaluated expressions for E1 and M1 virtual photon spectra for relativistic electron scattering in second order Born approximation and included the effects of finite nuclear size and charge. The scattering amplitude can be evaluated analytically provided the elastic and inelastic form factors contain only poles. We have observed that the virtual photon spectrum is insensitive to details of both form factors over a wide range of electron energy and depends only on the nuclear rms radius and transition radius. The integral over the physical momentum transfer is performed numerically. We have evaluated spectra for electric dipole and magnetic dipole radiation for electron energies in the range 10--200 MeV and compared with distorted wave calculations. Except for low energy and high Z, our results compare well. A range-of-validity plot is given for each multipole, distinguishing regions of electron energy and mass number where the second order calculation is adequate and also those where the first order term alone would be sufficient and where the conventional long wavelength limit is applicable. Comparison with data on the excitation of the 16.3 MeV isobaric analog state in /sub 40//sup 90/Zr yields excellent agreement with no adjustable parameters.

Finite-size effects on the relativistic electrons in a Coulomb field. I

The influence of the finite nuclear size and screening effects on the evaluation of the interaction matrix elements of the Dirac electron in the nuclear Coulomb field is manipulated into a closed analytic form. An application to the evaluation of the Bremsstrahlung cross section is discussed.

Double Beta Decay

The double β decay is investigated by the use of the relativistic Coulomb wave function including the finite nuclear size effect. A general form of the V±A interactions is considered. It is shown that the decay rate is enhanced considerably in comparison with the case of the plane wave multiplied by the non-relativistic Fermi factor. It is confirmed that the selection rule derived on the basis of the plane wave is still valid in the relativistic Coulomb wave case. The neutrino mass and the V+A current parameters are estimated from the analysis of experimental data.

Virtual photon intensity and finite nuclear size effects

Virtual photon intensities for electron-scattering reactions are calculated using the plane-wave Born approximation and the generalised Helm model to describe the nuclear transition charge and current densities.

Self-energy corrections to theK-electron binding in heavy and superheavy atoms

The self-energy corrections of order $\ensuremath{\alpha}$ to the $K$-electron binding energy in high-$Z$ atoms are studied numerically throughout the range $Z=70\ensuremath{-}160$. Nuclear finite-size effects are included in the electron wave functions and in the electron propagator to avoid the Coulomb singularity at $Z\ensuremath{\alpha}=1$. Denoting the self-energy correction by $\ensuremath{\Delta}E=\frac{\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{4}F(Z\ensuremath{\alpha})m{c}^{2}}{\ensuremath{\pi}}$, it is found that $F(Z\ensuremath{\alpha})$ increases smoothly from a minimum near $Z=90$ through $F(100\ensuremath{\alpha})=1.46\ifmmode\pm\else\textpm\fi{}0.01$ to $F(160\ensuremath{\alpha})=3.34\ifmmode\pm\else\textpm\fi{}0.16$. For comparison purposes, the self-energy corrections in a Coulomb field for $Z$ between 50 and 130 are evaluated, and the values of $F(Z\ensuremath{\alpha})$ thus obtained are in close agreement with previously published values; the Coulomb field values of $F(Z\ensuremath{\alpha})$ are very close to the finite-nucleus results for $Z<90$.

Hyperfine structure of light exotic atoms

Recoil corrections to the hyperfine structure of light exotic atoms, including the effect of the anomalous magnetic moment are calculated for light muonic and baryonic atoms. In the case of muonic atoms, the effects of recoil and finite nuclear size are significant in view of the presently attainable experimental precision.

Vacuum polarization in strong, realistic electric fields

Electron-positron vacuum polarization potentials, accurate to all orders $n$ in ${(Z\ensuremath{\alpha})}^{n}$, are calculated using realistic nuclear charge distributions for selected nuclei with $26\ensuremath{\le}Z\ensuremath{\le}114$, and for the quasimolecule U-U. We give further details of the numerical methods used in calculations previously reported, and of the improvements which have led to the present increased accuracy. The vacuum polarization potentials are employed to determine muonic energy-level shifts [order ${(Z \ensuremath{\alpha})}^{n\ensuremath{\ge}1}\ensuremath{\alpha}$] for most relevant states with $l\ensuremath{\le}6$. For the shifts of order ${(Z \ensuremath{\alpha})}^{n\ensuremath{\ge}3}\ensuremath{\alpha}$, interpolation procedures for all values of $Z<137$ and for other exotic particles are noted. Although the effect of finite nuclear size is small for high-lying states (\ensuremath{\sim}10% reduction of the ${(Z \ensuremath{\alpha})}^{n\ensuremath{\ge}3}\ensuremath{\alpha}$ terms in heavy atoms), it yields a reduction by a factor \ensuremath{\gtrsim} 2 for low-lying states. In the case of the quasimolecule U-U, we find that the total vacuum polarization is a minor correction to the dominant Coulomb interaction, with the first-order (Uehling) contribution remaining much larger than the higher-order terms. We also present further details of a previous calculation of order ${(Z \ensuremath{\alpha})}^{2}{\ensuremath{\alpha}}^{2}$ in muonic lead.

Muon decay in orbit

The decay spectrum of a muon bound in the 1s 1 2 state of an atom is calculated using V-A theory. Accurate muon and electron wave functions including finite nuclear size effects and vacuum polarization are used for the evaluation of the electron emission spectra of various elements. The formalism to perform the angular integration of the matrix elements is developed for the V-A weak interaction.

Nuclear-Size Effects on Vacuum Polarization in Muonic Pb

The effect of finite nuclear size on the vacuum-polarization charge density is studied. The results to third order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{3}$, and to all orders $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{n}$, $ng~3$, are presented with special attention focused on the $5{g}_{\frac{9}{2}}\ensuremath{-}4{f}_{\frac{7}{2}}$ transition in muonic Pb. In addition, the accuracy of analytic calculations exploiting the smallness of the electron mass and of the nuclear radius is discussed.

Cuolomb corrections to the induced pseudoscalar term in nuclear muon capture

Cuolomb corrections to the induced pseudoscalar term, which appears in impulse approximation treatment of nuclear muon capture, are calculated. We find that it is important to the effect of finite nuclear size in order to obtain a reliable result. These corrections are found to be too small to account for the difference between the PCAC value for this term and the values necessary to understand recent muon capture experiments in 16O and 40Ca.

Pair Production by Photons: Exact Calculation for Unscreened Atomic Field

Exact relativistic Coulomb wave functions, in the form of partial-wave expansions, are used to obtain analytic formulas for the differential cross sections and energy spectrum for electron-positron pair production by photons. Radiative corrections, which are believed to be small (of the order of 1%), are neglected. The errors introduced by the finite nuclear size and recoil effects are quite negligible for the energy region considered here. Apart from these approximations, the present formulas give the exact cross sections for the unscreened atomic field. Numerical results are obtained for positron energy spectra for a large number of photon energies and atomic numbers in the ranges $2{m}_{e}{c}^{2}<k\ensuremath{\le}10{m}_{e}{c}^{2}$ and $1\ensuremath{\le}Z\ensuremath{\le}100$. Total cross sections are obtained by numerical integration of the spectra. A representative choice of spectra is given. The approximately 500 total cross sections calculated are given in tables and diagrams. By one- or two-way interpolation in these data the total cross section for any photon energy and nuclear charge within the given limits may be obtained with an accuracy of 1% or better. For heavy elements large Coulomb corrections to the Bethe-Heitler Born-approximation results are found. The positron energy spectra obtained are asymmetric, favoring positrons of high energy. The total cross section $\ensuremath{\sigma}$ is smaller than the Bethe-Heitler total cross section ${\ensuremath{\sigma}}_{B}$ for photon energies very close to the threshold (owing to the repulsion of the positron), while $\ensuremath{\sigma}$ is generally larger than ${\ensuremath{\sigma}}_{B}$ in the energy region $2.2{m}_{e}{c}^{2}\ensuremath{\lesssim}k\ensuremath{\lesssim}10{m}_{e}{c}^{2}$.

Finite nuclear size effects in allowed β-decay

A formal definition is put forward for the phase-space factor for allowed β-decay taking into account finite nuclear size effects. A simple and accurate parameterization valid for Z⩽40, of the Behrens-Jänecke electron wavefunctions is given and is used to evaluate the finite size effects analytically in a simple formula valid for high energy transitions where the corrections have practical significance.

Empirical values of higher-order matrix elements in five hindered-allowed β-decays: 22Na, 182Ta, 56Mn, 58Co, 32P

The available experimental data on the spectrum shape, β-γ directional correlation, β-γ (circularly polarized) correlation, and γ-ray longitudinal polarization for the hindered-allowed β-decays of 22Na, 182Ta, 56Mn, 58Co and 32P have been utilized to deduce three higher-order matrix elements. Exact, electron radial functions which take into account finite nuclear size and de Broglie wavelength effects have been used in the present analysis. The values of matrix element ratios M( iγ 5 r)/ M( σ), M(( σ · r) r)/ M( σ) and − C v M( α × r)/ C A M( σ) have been obtained.

Experimental test of conversion coefficient ratios for M-, N- and O-shell electrons

Abstract Relative conversion-line intensities have been measured for five E2 transitions of energies from 81 to 412 keV in nuclei with Z ranging from 55 to 80 using the iron-free 1 2 π√ 13 β-ray spectrometer. The results for M-, N- and O-shells were found to agree with the calculations where screening is taken into account. The atomic screening and finite nuclear size effects in internal conversion of the N-subshell electrons were investigated. Two previously measured conversion electron spectra were analysed with four different computer codes and the consistency of the results was examined.

K shell electron wavefunctions in complex atoms

A simple procedure for obtaining 1s electron wavefunctions in complex atoms is presented which takes into account many body, relativistic and finite nuclear size effects.

Finite nuclear size effects in Fermi β-decay

Abstract Corrections due to the finite extension of the nucleus are calculated for the Ferm function tabulated in the new Landolt-Bornstein series where the Fermi function is evaluated for a finite charge distribution, but under the assumption of a point isovector charge distribution. We treat the corresponding correction in a power series up to and including terms of order Z2α2, and arrive at ⨍t values which contain the integrated effect of the extended nuclear size. Finally, we discuss effective ⨍t values which include the modelindependent part of the radiative corrections of order a for some Fermi transitions.

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.

Magnetic dipole hyperfine structure in muonic atoms

The nuclear finite-size effect in the M1 hyperfine structure of muonic atoms of 93Nb and 139La has been studied experimentally. Systematic differences of the finite-size effect between nuclei in which the spin and the orbital angular momentum of the odd proton are parallel or anti-parallel are discussed.

The evaluation of finite nuclear size effects in allowed β-decay

Analytical parameterizations are given that reproduce with good accuracy the exact value of [f12 + g−12]r = R for negaton and positon wave functions generated in the field of a uniform spherical nuclear charge distribution. Formulae are presented for the corrections to the conventional ƒ-function for allowed β-decay due to the finite extension of the nuclear charge distribution, to the finite de Broglie wave length and to the variation of the electron and neutrino wave functions inside the nucleus.