Comprehensive Table of Calculated Huff Factors

TL;DR

This work provides a comprehensive, mechanism-based table of Huff factors $Q$ for nuclei with $6 \le Z \le 94$ by marrying a self-consistent HF+BCS nuclear structure framework (including pairing and deformation) with a microscopic treatment of the Coulomb field and distorted-electron Dirac wavefunctions. The Huff factors are shown to decrease monotonically with $Z$, while isotope dependence is tiny (typically $<0.1\%$), enabling practical use of effective factors for natural targets. The authors validate their approach against prior benchmark calculations and deliver a unified dataset that can refine the extraction of the nuclear muon capture rate $\Lambda_{\text{cap}}$ and support updates to muon-nuclear data libraries. The work also documents convergence properties and distortion effects, providing guidance for precise, reproducible muon-capture analyses.

Abstract

We present a systematic calculation of the Huff factor for nuclei with atomic numbers ($ Z $) in the range of $ 6 \leq Z \leq 94 $. The Huff factor quantifies the increase in the partial lifetime of the decay-in-orbit (DIO) of the muonic atom and serves as an essential correction factor for extracting the nuclear muon capture rate from the measured lifetimes of the muonic atom. However, previous calculations typically provided only the atomic number dependence and neglected isotope dependence -- an assumption whose reliability had not been examined, despite its importance for a comprehensive understanding of the nuclear muon capture rate. In this work, we calculate the Huff factor using nuclear charge distributions obtained from a fully self-consistent microscopic nuclear structure model that incorporates pairing and deformation effects. The resulting Huff factors exhibit a monotonic decrease with increasing $ Z $, while the isotope dependence is found to be small. Our results also show good agreement with previous calculations, supporting the reliability of the present framework. The comprehensive set of Huff factors presented here constitutes the first unified values currently available and will serve as a basis for future evaluations of muon nuclear data.

Figures (4)

• Figure 1: The calculated charge densities of ${}^{40}_{}\mathrm{Ca}$, ${}^{48}_{}\mathrm{Ca}$, ${}^{152}_{}\mathrm{Sm}$, and ${}^{208}_{}\mathrm{Pb}$ compared with the experimental data obtained by the electron scattering DeVries1987At.DataNucl.DataTables36_495.
• Figure 2: The calculated Huff factors for each element obtained in the present study given in Table \ref{['tab:average_huff_factors']} (red circles) and those in Ref. Suzuki1987-aq (blue squares).
• Figure B.3: Contribution of different $\kappa$ for the Huff factor. The top and bottom panels are for ${}^{12}_{}\mathrm{C}$ and ${}^{208}_{}\mathrm{Pb}$, respectively. The red circles are the $\left| \kappa \right|$ contribution to the Huff factor, $Q_{\kappa} + Q_{-\kappa}$, and the blue squares are its cumulative sum $\sum_{\kappa' = 1}^{\kappa} \left( Q_{\kappa'} + Q_{-\kappa'} \right)$. Lines are guide for the eye.
• Figure C.4: Comparison of the Huff factor calculated among different treatments of the lepton wave functions. The red circles and blue squares, respectively, represent the Huff factors with and without the Coulomb distortion of the emitted electron. The horizontal axis shows the atomic number $Z$ of the nucleus and isotopes with the same $Z$ are plotted together. The light-blue dashed curve is given by the analytic formula in Eq. \ref{['eq:analytic_Huff_factor_nonrela_muon']}, where the electron wave function is approximated by a plane wave and the bound-muon wave function is obtained from the Schrödinger equation with a point-charge Coulomb potential.