# A New Determination of the $(Z,A)$ Dependence of Coherent Muon-to-Electron
Conversion

Léo Borrel  David G. Hitlin  Sophie Middleton  California Institute of
Technology
Pasadena, CA 91125 USA

###### Abstract

Should muon-to-electron conversion in the field of a nucleus be found in the
current generation of experiments, the measurement of the atomic number
dependence of the process will become an important experimental goal. We
present a new treatment of the (Z,A) dependence of coherent muon-to-electron
conversion in 236 isotopes. Our approach differs from previous treatments in
several ways. Firstly, we include the effect of permanent quadrupole
deformation on the charged lepton flavor violating matrix elements, using the
method of Barrett moments. This method also enables the addition of muonic
X-ray nuclear size and shape determinations of the charge distribution to the
electron scattering results used previously. Secondly, we employ a Hartree-
Bogoliubov model to calculate neutron-related matrix elements for even-even
nuclei. This takes into account the quadrupole deformation of the neutron
distributions and the fact that neutrons are, in general, in different shell
model orbits than protons. The calculated conversion rates differ from
previous calculations, particularly in the region of large permanent
quadrupole deformation. Finally, we propose an alternative normalization of
the muon-to-electron conversion rated, which related more closely to what a
given experiment acturally measures, and better separate lepton physics from
nuclear physics effects.

††preprint: APS/123-QED

##  I Introduction

The most stringent experimental upper limit on charged lepton flavor violation
(CLFV) via muon-to-electron conversion ($\mu\rightarrow e$) in the field of a
nucleus ($R_{\mu e}\leq 7\times 10^{-13}$ at $90\%$ C.L.) was set by the
SINDRUM-II experiment using a gold target [1]. The Mu2e [2] and COMET [3]
experiments, both nearing data-taking, will search for $\mu\rightarrow e$
using an aluminum target. The Mu2e experiment at Fermilab aims to reach a
single event sensitivity (SES) of $\sim 3\times 10^{-17}$ for the conversion
rate relative to muon capture. The COMET experiment at J-PARC, has a planned
sensitivity of $2\times 10^{-15}$ for Phase I and a similar sensitivity to
Mu2e for Phase II [4]. CLFV limits derived from the decays $\mu^{+}\to
e^{+}\gamma$ and $\mu^{+}\to e^{+}e^{-}e^{+}$, mainly sensitive to dipole
coupling, provide comparable or better limits. The new generation of
experiments, MEG-II for $\mu^{+}\to e^{+}\gamma$ [5] and Mu3e for $\mu^{+}\to
e^{+}e^{-}e^{+}$ [6], are underway at PSI.

Should the decay or conversion experiments find evidence for CLFV decays or
$\mu\\!\rightarrow\\!e$ conversion in aluminum, interest will turn to
understanding the effective operators that mediate CLFV at low energies. This
can be done by studying the Dalitz plot distributions of polarized muon or tau
decays to three leptons [7, 8, 9] or by measurements of the conversion rate
with other nuclear targets, as the ($Z,A$) dependence of the conversion rate
can probe the Lorentz structure [10, 11, 12]. Future proposed experiments,
such as Mu2e-II [13] and a conversion search at the Advanced Muon Facility
[14], will provide opportunities to measure $\mu\rightarrow e$ conversion in
other target materials.

The $\mu\rightarrow e$ conversion process manifestly involves the nuclear
physics of the target elements. We discuss herein several aspects of
extracting the New Physics from nuclear physics. In particular, we present a
new systematic study of the atomic number dependence of the coherent
conversion rate. We utilize the method of Barrett moments [15] to add a
substantial amount of muonic X-ray data to previous studies that mainly employ
electron scattering data on nuclear charge distributions. This allows us to
take into account the fact that many nuclei have permanent quadrupole
deformations, which affects the evaluation of matrix elements of New Physics
operators. We also employ a new treatment of the neutron distribution and
present a new treatment of the normalization of experimental and theoretical
results.

The study of the ($Z,A$) dependence of conversion is most cleanly done with a
target consisting of a single stable isotope, as with ${}^{27}_{13}$Al, the
target in both the Mu2e and COMET experiments. Potential higher $Z$ targets
such as Ti have many stable isotopes and are typically not readily available
as separated isotopes in sufficient quantity to meet experimental conversion
sensitivity requirements. Therefore, in Sec. III we discuss conversion rates
for both targets composed of individual isotopes and targets with natural
elemental abundance. Section IV addresses the question of the historical
approach to the presentation of experimental limits on $\mu\rightarrow e$
conversion. The usual approach, in analogy to the concept of a decay branching
fraction, is to normalize the conversion rate to the rate of muon capture on a
particular nucleus, and therefore to present limits on this ratio, referred to
as $R_{\mu e}$:

$R_{\mu e}(A,Z)=\frac{\Gamma(\mu^{-}+N(A,Z)\rightarrow
e^{-}+N(A,Z))}{\Gamma(\mu^{-}+N(A,Z)\rightarrow\text{all captures})}.$ (1)

We introduce an alternative normalization, which hews more closely to what a
given experiment actually measures and better separates the lepton physics and
the nuclear physics, removing extraneous contributions to the $Z,A$ dependence
of conversion.

Section V is a discussion of the practicalities involved in choosing future
targets.

##  II Review of existing literature

Should CLFV be observed in $\mu\to e$ conversion in the upcoming round of
experiments, interest will turn to identifying the Lorentz structure of the
CLFV coupling. This can be accessed by a sufficiently precise measurement of
the atomic number dependence of the conversion rate [10, 11, 12, 16].

The most widely cited treatment of the atomic number dependence of coherent
$\mu\to e$ conversion is that of Kitano, et al.[10], extended by Cirigliano et
al.[11], which includes 55 isotopes whose charge distributions, measured using
elastic electron scattering, were compiled by De Vries et al.[17]. These
measurements are parameterized using a variety of models for the nuclear
charge distribution: harmonic oscillator, two and three-parameter spherical
Fermi, two and three-parameter Gaussian, as well as model-independent sum-of-
Gaussians, and Fourier-Bessel expansion treatments where available. The spin-
independent contribution to the coherent conversion rate has an $A^{2}$
enhancement and in most, but not all, models is dominant. Three types of
effective operators, scalar, vector and dipole, contribute to coherent
conversion. The rate of $\mu\to e$ conversion is calculated using the overlap
integrals between the Dirac wave functions and nucleon densities for these
three cases. While some direct measurements of neutron distributions are
explored in Refs.[10, 11], for the main results the neutron and proton
distributions are assumed to be identical and are scaled by $N/Z$.

The recent calculations of Haxton et al. [18] improve these analyses for a
selected number of nuclei by employing nuclear-level effective field theory to
determine the matrix elements. Effective field theory has also been exploited
to refine the interpretation of experimental CLFV limits in terms of New
Physics couplings [19].

The Kitano et al. and Cirigliano et al. studies present the $Z$-dependence of
the conversion rates for 28 and 55 isotopes respectively, that have had their
muon capture lifetime measured [20] in the form of a ratio: $R_{\mu
e}(A,Z)=R_{\mu e}^{Z}/R_{\mu e}^{Al}$. This ratio has a great deal of
structure, largely due to the influence of “magic numbers” associated with
closed shells in the nuclear shell model on the nuclear size and shape. The
influence of shell structure on the nuclear size, and thus the evaluation of
the New Physics matrix elements for the largely coherent conversion process is
unavoidable. The additional structure in the $Z$ dependence, introduced by the
division by the muon capture rate, a non-coherent Standard Model process, can,
however, be avoided. This formulation is traceable in part to the original
conversion normalization proposal of Weinberg and Feinberg [21]. The question
of normalization of the conversion rate will be discussed in Section IV.

###  II.1 Treatment of the neutron distribution

Kitano et al. [10] used measured neutron distributions when possible. These
measurements, derived from charged pion, proton or alpha particle scattering,
were available for only 16 nuclei. Their primary method of including neutrons
in determining the $(Z,A)$ dependence of conversion was therefore to scale the
proton distribution by a factor of $N/Z$. Particularly in heavy nuclei,
protons and neutrons populate levels with different quantum numbers and are
thus in different shells. This produces rather different shapes and sizes for
protons and neutrons. In particular quadrupole deformations, as measured by
B(E2) (the electric quadrupole transition probabilities) determined by Coulomb
excitation and by muonic X-ray hyperfine structure, show a different
dependence for protons and neutrons [22]. We have therefore resorted to a
model calculation to account for these details. To improve on simple $N/Z$
scaling, we account for the fact that neutron distributions are typically
larger than proton distributions by using a comprehensive set of calculations
employing a deformed relativistic Hartree-Bogoliubov model [23, 24] to
estimate the size and shape of the neutron distributions. This approach more
accurately accounts for the fact that in most nuclei, particularly those in
the region of large permanent quadrupole deformation ($Z=60-80$), neutrons are
in different shells than protons. Zhang et al. [25] have developed a deformed
relativistic Hartree-Bogoliubov model [23, 24] for even-even nuclei ranging
from $Z=8$ to $Z=120$ to determine individual neutron and proton
distributions, accounting for quadrupole deformation. This work is being
extended to other nuclei [26], but is not yet comprehensive beyond the even-
even-sector.

###  II.2 The effects of deformed nuclei

Many nuclei, especially those with protons or neutrons far from a closed
shell, are non-spherical, having permanent quadrupole or higher multipole
shapes. The question addressed herein is what is the effect of these
deformations on the calculation of $\mu\rightarrow e$ conversion rates? The
$\mu$ to $e$ conversion occurs primarily from the 1$S$ state, so one can
calculate conversion rates using only the radial Dirac equation with an
equivalent spherically symmetric charge (or neutron) distribution. However,
characterizing the average over non-spherical distributions with a derived
$rms$ radius does not account for the effective smearing at the outer edge of
a nucleus due to, for example, a quadrupole deformation.

The effective nuclear charge or neutron distribution skin thickness enters
into the evaluation of the CLFV matrix elements. The relative importance of
the skin thickness in determining the matrix element depends on the atomic
number (c.f. $N/Z$) and the quadrupole deformation. We calculate the CLFV
matrix elements using a Dirac equation with spherical symmetry, the angular
integration accounting for the effect of the quadrupole deformation, whether
oblate or prolate. This produces an effect largely concentrated at the nuclear
surface, increasing the effective skin thickness of the charge distribution.
The $N/Z$ ratio increases with atomic number; the excess neutrons producing a
“neutron skin”, a larger effective skin thickness for the neutrons, as well as
effects on the charge distribution, e.g. isotope shifts in electronic or
muonic atom spectra. These effects are further modulated by both proton and
neutron shell structure, particularly around magic number closed shells.

The case of 27Al is of particular interest, as it is the target for the Mu2e
and COMET experiments. 27Al has a large quadrupole moment (0.146 b$e^{2}$)
[27]. Most shell model calculations conclude that this nucleus has a prolate
shape, with $\beta=0.39$, although Dehnhard [28] argues for an oblate ground
state deformation. The sign of the quadrupole deformation does not enter into
the calculation of the relevant integrals in the solution of the Dirac
equation.

###  II.3 The isotopes of Neodymium

Neodymium has five significant stable isotopes whose charge distributions have
been measured in both electron scattering [29] and muonic X-ray spectra[30].
This provides an opportunity to explore some of the subtleties encountered in
combining these different determinations of the nuclear charge distribution.
There are measurements on five even-even neodymium isotopes: $A$=142, 144,
146, 148, and 150, which range from spherical to deformed. This allows us to
explore the interplay of the effect of a quadrupole deformation on the
effective skin thickness, as well as the differences between the two
techniques, as shown in Fig. 1.

The values of the quadrupole deformation $\beta$ determined largely from
$B(E2)$ measurements [22], show a steady increase with neutron number,
culminating at $N=90$ (plotted in Figure 2). The value of $\beta$ in the
three-parameter fits to the charge distribution is held fixed for both the
muonic X-ray and electron scattering cases.

The calculations of Zhang et al. using a Hartree-Bogoliubov model show that
the rms charge radius of these Nd isotopes grows as $A^{1/3}$, while the
neutron radius grows more quickly, as the neutrons are in a higher shell and
further from a shell closure. This is our primary motivation for using the
Zhang et al. calculations in evaluating the $S(n)$ and $V(n)$ overlap
integrals, rather than scaling the neutron distributions by ${N/Z}$ as has
heretofore been the practice [10][11]. Related calculations by the Zhang et
al. nuclear theory group [26] also reproduce the behavior of the quadrupole
deformation in the Nd isotopes as a function of neutron number.

Figure 1: The square of the dipole CLFV matrix element for five even-even
isotopes of Nd as determined using muonic x-ray data [30] and electron
scattering data [29] and three treatments of the charge distribution. Figure
2: Quadrupole deformation of five even-even isotopes of Nd determined from
B(E2) measurements in Coulomb excitation [22] (with experimental uncertainty),
and calculated for the charge and neutron distributions in the model of Zhang
et al. [25].

###  II.4 Goal of this study

We present a new determination of the atomic number dependence of the coherent
conversion rate, using the method of Barrett moments [15] to add the many
muonic X-ray measurements of nuclear charge distributions to the electron
scattering data used in previous studies. The muonic X-ray data encompass
measurements of many strongly-deformed nuclei, using the resolved hyperfine
structure in the X-ray transitions to measure the nuclear quadrupole moment
shape, typically parameterized by a three-parameter Fermi distribution.

Our treatment of the $A$ and $Z$ dependence of conversion differs from
previous treatments in several ways:

  * •

While previous studies have mainly used electron-scattering-based nuclear size
determinations, we include the many measurements of nuclear charge
distributions determined using muonic X-rays. This substantially enlarges the
sample size, particularly in the regime above $Z=60$, where many nuclei have
substantial quadrupole deformations. We combine the elastic electron
scattering and muonic X-ray data using the method of Barrett moments, and
devise a procedure to incorporate the effect of permanent $Y_{20}$
deformations on the effective nuclear skin thickness. By this approach, we can
include the conversion rates for a total of 236 isotopes.

  * •

Rather than using a neutron distribution determined by scaling the proton
charge distributions by $N/Z$, the primary method used in previous treatments,
we use a deformed relativistic Hartree-Bogoliubov model [25] for the neutron
distributions and compare with other approaches. This model, bench-marked
against a wide variety of experimental determinations for even-even nuclei,
allows us to include a wider variety of nuclei and to explore isotopic effects
on the conversion rate.

  * •

We present the $\mu\rightarrow e$ conversion rates for the coherent New
Physics process normalized to the total muon lifetime, rather than to the muon
capture rate.

##  III Atomic number dependence of $\mu\rightarrow e$ conversion

###  III.1 Theoretical background

In the following study we concentrate on the dominant spin-independent
coherent conversion process in which the final and initial state of the
nucleus is identical and the coherent conversion rate is enhanced by $A^{2}$.

The lepton flavor-violating coherent conversion rate of a muon into an
electron is given by Kitano et al. [10] as:

$\begin{gathered}\omega_{conv}=2G_{f}^{2}\left|A^{*}_{R}D+\tilde{g}^{(p)}_{LS}S^{(p)}+\tilde{g}^{(n)}_{LS}S^{(n)}+\tilde{g}^{(p)}_{LV}V^{(p)}+\tilde{g}^{(n)}_{LV}S^{(n)}\right|^{2}\\\
\phantom{XXX}+2G_{f}^{2}\left|A^{*}_{L}D+\tilde{g}^{(p)}_{RS}S^{(p)}+\tilde{g}^{(n)}_{RS}S^{(n)}+\tilde{g}^{(p)}_{RV}V^{(p)}+\tilde{g}^{(n)}_{RV}S^{(n)}\right|^{2}\end{gathered}$
(2)

with $G_{F}$ the Fermi constant, $A^{*}_{R,L}$ and $\tilde{g}$ are
dimensionless coupling constants describing the strength of each component of
the Lagrangian and $D,S$ and $V$ represent overlap integrals for dipole,
scalar, and vector interactions respectively. The $L$ and $R$ subscripts
indicate left-handed and right-handed components and the ($n$) and ($p$)
superscripts denote the neutron and proton terms respectively.

The overlap integrals are defined as:

$\displaystyle D$
$\displaystyle=\frac{4}{\sqrt{2}}m_{\mu}\int_{0}^{\infty}\left(-E(r)\right)\left(g^{-}_{e}f^{-}_{\mu}+f^{-}_{e}g^{-}_{\mu}\right)r^{2}dr$
$\displaystyle S^{(p)}$
$\displaystyle=\frac{1}{2\sqrt{2}}\int_{0}^{\infty}Z\rho^{(p)}(r)\left(g^{-}_{e}g^{-}_{\mu}-f^{-}_{e}f^{-}_{\mu}\right)r^{2}dr$
$\displaystyle S^{(n)}$
$\displaystyle=\frac{1}{2\sqrt{2}}\int_{0}^{\infty}(A-Z)\rho^{(n)}(r)\left(g^{-}_{e}g^{-}_{\mu}-f^{-}_{e}f^{-}_{\mu}\right)r^{2}dr$
$\displaystyle V^{(p)}$
$\displaystyle=\frac{1}{2\sqrt{2}}\int_{0}^{\infty}Z\rho^{(p)}(r)\left(g^{-}_{e}g^{-}_{\mu}+f^{-}_{e}f^{-}_{\mu}\right)r^{2}dr$
$\displaystyle V^{(n)}$
$\displaystyle=\frac{1}{2\sqrt{2}}\int_{0}^{\infty}(A-Z)\rho^{(n)}(r)\left(g^{-}_{e}g^{-}_{\mu}+f^{-}_{e}f^{-}_{\mu}\right)r^{2}dr$

where $f$ and $g$ are the large and small components of the radial solution of
the Dirac equation describing a muon orbiting a nucleus and converting to an
electron. The functions $f_{\mu}$ and ($f_{e}$) are the muon (electron)
wavefunction [31]. The terms $\rho^{(n)}$ and $\rho^{(p)}$ describe the
neutron and proton densities in the nuclei respectively. The electric field,
$E(r)$, is fixed by Gauss’ Law:

$E(r)=\frac{Ze}{r^{2}}\int_{0}^{r}r^{\prime
2}\rho^{(p)}(r^{\prime})dr^{\prime}.$ (3)

The muon and electron wavefunctions in the vicinity of the 27Al nucleus are
shown in Figure 3. The conversion occurs overwhelmingly from the $1S$ state;
we therefore integrate over polar and azimuthal angles to account for the
effect of quadrupole deformations.

(a)

(b)

Figure 3: The two components of the wavefunctions of the muon (top) and the
electron (bottom) in the vicinity of the Al nucleus. Here
$u_{1}=rg^{-}_{e,\mu}$ and $u_{2}=rf^{-}_{e,\mu}$.

###  III.2 Methodology

####  III.2.1 Combining electron scattering and muonic X-ray measurements

In order to combine measurements of nuclear charge distribution obtained by
electron scattering and muonic X-rays we use the method developed by Barrett
[15]. Electron scattering experiments derive information on the nuclear charge
distribution from the energy and angular distribution of the scattered
electron, mapping the momentum transfer of the electron to the Fourier
transform of the charge distribution [17]. These treatments typically keep
only the first term in the $q^{2}$ expansion. Muonic X-ray experiments use the
energy of X-rays emitted in the atomic cascade of muonic atoms to determine
the size and shape of the nucleus. Due to the strong overlap of the muon
wavefunction with the nucleus, the energy of the $3D-2P$ and $2P-1S$
transitions is highly sensitive to nuclear size parameters. Analysis of the
hyperfine structure allows the model-dependent determination of permanent
quadrupole charge distributions.

While for light nuclei the $2P-1S$ transition energy of a muonic atom can be
matched to a single mean square radius $\left\langle r^{2}\right\rangle$, this
is not true for heavier nuclei, where nuclear distributions with the same RMS
radius can generate quite different transition energies. Barrett[15]
introduced a different moment, shared by all nuclear distributions yielding
the same transition energy. The Barrett moment is defined as:

$\left\langle r^{k}e^{-\alpha
r}\right\rangle=\frac{4\pi}{Ze}\int\rho(r)r^{k}e^{-\alpha r}r^{2}dr,$ (4)

where $k$ and $\alpha$, which are $Z$-dependent, are given in [15].

In both electron scattering and muonic X-ray analyses, the nuclear charge
distribution has often been described using a 2-parameter Fermi distribution
(2pF):

$\rho(r)=\frac{\rho_{0}}{1+\exp\left[\frac{r-c}{t}\right]}\ ,$ (5)

where $c$ is the radius at the half maximum and $t$ is the skin thickness,
defined as the region in which the charge density falls from 90% to 10% of the
central value. This model is central to our Barrett moment-centered treatment.

We can see an example of the computation of the Barrett moment from two
different datasets for Neodymium in Table 1. The Barrett moments derived from
the two types of experiments are seen to be in good agreement, supporting our
approach to combining the electron scattering and muonic x-ray of measurements
in determining the $Z$ dependence of the conversion rate.

####  III.2.2 Quadrupole deformation

In order to compute the overlap integral, we need to solve the Dirac equation
to obtain the muon and electron wavefunctions (as described in Section III.1).
We used an existing code [32] to solve the Dirac equation for a muonic atom in
the potential of a spherically-symmetric 2-parameters Fermi distribution.

The method used to include the effect of nuclear quadrupole deformation on the
matrix elements is as follows:

  * •

employ a deformed 3-parameter Fermi distribution:

$\rho(r,\theta)=\frac{\rho_{0}}{1+\exp\left[\frac{r-c(1+\beta
Y_{20}(\theta))}{t}\right]}$ (6)

where we take parameters $c$, $t$ and $\beta$ from available literature
detailed in Section III.2.3.

  * •

We convert this to a spherically-symmetric 2-parameter Fermi distribution by
removing the quadrupole deformation $\beta$ and adjusting the skin thickness
$t$ to keep a constant Barrett moment (defined in equation 4). This equivalent
2-parameter Fermi distribution is then used to solve the Dirac equation.

Using the $t$ parameter to match the Barrett moment effectively smears out the
nuclear surface, accounting for the integration of the $Y_{20}$ deformation
over angles.

One could also modify the $c$ parameter rather than the $t$ parameter to match
the Barrett moment. This results in a negligible difference in the overlap
integrals.

####  III.2.3 Dataset

Our calculations employ a compilation of data from different sources, which we
list in the footnotes to Table A of the Appendix (in which BM stands for
Barrett Moment). When there are different sources for data on an isotope, we
use the data from the compilation listing the largest number of isotopes.

  * •

Nuclear charge distribution parameters:
The vast majority come from Fricke et al. [33] (Table III.A and III.C), a
compilation of muonic X-ray and electron scattering experimental data. We add
several muonic X-ray measurements [34, 35, 36, 37] that were not included in
Fricke et al.. When muonic X-ray measurements do not exist, we use the
compilation of electron scattering experimental data from de Vries et al.
[17]. When the only information available is the proton rms radius, we use
data from Angeli et al. [38].

For quadrupole deformations we use the model of Möller et al. [39] for most
isotopes. This employs a Finite-Range Drop Model (FRDM) to compute ground-
state quadrupole deformations, as well as many other nuclear properties. When
available, we also use the Zhang et al. [25] model that calculates separate
quadrupole deformations for proton and neutron distributions.

  * •

Neutron distribution:
We use the Zhang et al. [25] deformed relativistic Hartree-Bogoliubov model to
separately evaluate the proton and the neutron nuclear distributions. This
model only treats even-even nuclei, avoiding the additional single particle
effects on both charge and neutron distributions seen in even-odd and odd-even
nuclei.

####  III.2.4 Uncertainties

We consider two sources of systematic uncertainties: the quadrupole
deformation $\beta$ and the radius at half maximum $c$ in the Fermi
distribution.

We take the quadrupole deformation error from Stelson and Grodzins [22], where
uncertainty on $B(E2)$ is converted into uncertainty on $\beta$ assuming a
uniform ellipsoidal charge distribution. The uncertainty on $c$ is taken
directly from Fricke et al. [33].

The relative uncertainty on $c$ does not exceed $2\%$, while the relative
uncertainty on $\beta$ can be up to $20\%$. We therefore propagated only the
uncertainty in $\beta$ to the Barrett moment and ignored the $c$ uncertainty.

The resulting relative uncertainty on the Barrett moment on all the available
elements never exceeded $1\%$; we show the values for the Neodymium isotopes
in Table 1 and Fig. 2 as an example.

###  III.3 Results

The resulting overlap integrals calculated on the 236 stable isotopes with
natural abundance above $1\%$ are displayed in Figure 4. The plot compares the
results of Heeck et al. [12]; our new results are in general agreement. The
main difference appears in the deformed nuclei region ($60<Z<80$), where our
values are higher due to our explicit treatment of the effect of the
deformation on the matrix elements. This results in a smoother decrease of the
overlap integral value after $Z=60$. This illustrates our attempts to
deconvolute the New Physics interaction from the nuclear effects.

Figure 5 shows the overlap integrals weighted by natural abundance, in
comparison with work from Kitano et al. [10]. The latter does not use natural
abundance weighting but only displays one isotope per element. Again, the main
difference between the two datasets is in the deformed nuclei region.

From the overlap integrals, we can compute the conversion rate using Eq. 2 for
all the elements, and we obtain the sensitivity of different target materials
normalized to the conversion rate in 27Al. In Fig. 6 our result is compared to
that from Cirigliano et al. [11]. Four physics models are considered: dipole
($D$), scalar ($S$) and two vector type ($V^{(\gamma)}$ and $V^{(Z)}$).
$V^{(\gamma)}$ describes the scenarios where the transition charge radius
operator gives the dominant contribution to the CLFV Lagrangian, and $V^{(Z)}$
describes the case where the dominant operator is assumed to be an effective
$Z$-penguin [40]. Some candidate complementary target nuclei for future
conversion measurements are highlighted, further discussion is presented in
Sec. V.

Figure 7 removes the normalization relative to muon capture and shows the
relative conversion rates in each material relative to that in 27Al. The
objective is to separate the coherent muon conversion from the incoherent muon
capture our motivation for this presentation of our results is detailed in
Sec. IV.

Figure 4: The overlap integrals for dipole, vector and scalar couplings as a function of atomic number for isotopes having a natural abundance $>$ 90%, for the compilation of Heeck et al.[12] and this work. For clarity, we shift the result of Heeck et al. by $+\frac{1}{2}$ on the x-axis. Figure 5: The overlap integrals for dipole, vector and scalar couplings as a function of atomic number for each element, weighted by natural abundance, for the compilation of Kitano et al.[10, 11] and this work. Figure 6: Comparison with of our results on sensitivity as a function of $Z$ with those of Cirigliano et al.[11]. The conversion rates are normalized to the muon capture rate and are relative to the rate for aluminum. Figure 7: Our calculation of the experimental sensitivity of CLFV experiments as a function of atomic number. The conversion rate is not normalized to the muon capture rate. Table 1:  Comparison of muonic X-ray and electron scattering determinations of three-parameter Fermi distributions in neodymium isotopes. Isotope | c (fm) | t (fm) | $\beta$ |  rms radius (fm) | Barrett Moment
---|---|---|---|---|---
|  |  | Muonic X-ray[30] |  |
142Nd | $5.80\pm 0.03$ | $2.32\pm 0.08$ | $0.104\pm 0.02$ | $4.91\pm 0.08$ | $15.99\pm 0.03$
144Nd | $5.85\pm 0.03$ | $2.27\pm 0.08$ | $0.123\pm 0.02$ | $4.94\pm 0.08$ | $16.11\pm 0.04$
146Nd | $5.82\pm 0.03$ | $2.42\pm 0.08$ | $0.151\pm 0.02$ | $4.98\pm 0.08$ | $16.26\pm 0.06$
148Nd | $5.84\pm 0.03$ | $2.40\pm 0.08$ | $0.197\pm 0.04$ | $5.00\pm 0.08$ | $16.38\pm 0.13$
150Nd | $5.86\pm 0.03$ | $2.35\pm 0.08$ | $0.279\pm 0.005$ | $5.05\pm 0.08$ | $16.58\pm 0.01$
|  Electron Scattering [29]
142Nd | $5.7045\pm 0.03$ | $2.539\pm 0.013$ | $0.104\pm 0.02$ | $4.92\pm 0.08$ | $15.98\pm 0.03$
144Nd | $5.6634\pm 0.03$ | $2.696\pm 0.013$ | $0.123\pm 0.02$ | $4.96\pm 0.08$ | $16.11\pm 0.04$
146Nd | $5.6600\pm 0.03$ | $2.760\pm 0.013$ | $0.151\pm 0.02$ | $4.99\pm 0.08$ | $16.23\pm 0.06$
148Nd | $5.6871\pm 0.03$ | $2.798\pm 0.022$ | $0.197\pm 0.04$ | $5.04\pm 0.08$ | $16.46\pm 0.13$
150Nd | $5.7185\pm 0.03$ | $2.861\pm 0.031$ | $0.279\pm 0.005$ | $5.13\pm 0.08$ | $16.86\pm 0.01$

##  IV Normalization and $(Z,A)$ Dependence of the Conversion Rate

The conventional approach to the normalization of $\mu\to e$ conversion
experiments, quoting the conversion rate (experimental limit or theory
prediction) normalized to the measured rate of $\mu$ capture on a given
nucleus, has been in place for more than seventy years. We will discuss the
shortcomings of this convention and propose a revised presentation of $\mu\to
e$ conversion results that addresses our concerns.

The first limit on $\mu\to e$ conversion, by Lagarrigue and Peyrou [41], using
cosmic ray muons stopped in copper and tin targets in a Wilson cloud chamber,
employed previous measurements of the muon lifetime (dominated by muon capture
in this regime) in copper and antimony [42] as normalization. The first
accelerator experiment, using a copper target [43], also normalized to the
(directly measured) rate of muon capture.

The choice to normalize to the muon capture rate is not precisely analogous to
the idea of a branching fraction (the number of decays into a particular mode,
divided by all decays), which would be to divide the conversion rate in the
field of a particular nucleus by all possible fates of the muon
($\mu\rightarrow e$ conversion, decay in orbit or nuclear capture). This
normalization is essentially a historical convention initially codified by the
work of Weinberg and Feinberg in 1959 [21]. All results or predictions on
muon-to-electron conversion have henceforth been presented in the form
$R_{\mu\to e}$ (Eq. 1).

Compilations of the history of experimental limits on CLFV processes typically
present the 90% confidence level limits for decays and conversion on the same
plot. This analogy ignores the fact that decay and conversion experiments are
normalized differently. The decay limits are reported as true branching
fractions, while the conversion rate limits are on the fraction of muon
captures resulting in the production of a mono-energetic electron, which does
not account for all fates of a muon in a muonic atom. Indeed, the lifetime of
such a muon is determined in varying proportions by the conversion rate, a BSM
process, by the nuclear capture rate, an incoherent Standard Model process,
and by the lifetime of the decay-in-orbit muon, which is modified from the
free decay rate by the atomic binding energy, the so-called Huff factor (0.993
for aluminum, 0.981 for titanium and 0.850 for gold) [44].

It is clear from these points that a revised approach is desirable. In
particular, one would like to avoid the spurious introduction of nuclear
effects into the presentation of a conversion result to the greatest extent
possible in order to facilitate a conceptually cleaner comparison with results
from decay experiments.

Normalizing an experimental conversion result requires determination of the
number of stopped muons within the sensitive time window of the experiment.
This involves muon decay as well as nuclear muon capture. In the upcoming Mu2e
experiment downstream detectors count $2P-1S$ muonic X-rays or other
transitions and delayed $\gamma$-rays from muon capture to infer the stopped
muon rate. This measured rate is then used to normalize the conversion rate
limit or observation.

Table 2:  Muon capture rates on 27Al leading to Mg, Na and Ne final states (from [45]). Reaction | Observed | Estimated | Missing | Total
---|---|---|---|---
|  $\gamma$-ray | ground-state | yields | yield
| yield | transition |  |
27Al$(\mu^{-}\\!,\nu)^{27}$Mg | 10(1) | 0 | 3 | 13
27Al$(\mu^{-}\\!,\nu n)^{26}$Mg | 53(5) | 4 | 4 | 61
27Al$(\mu^{-}\\!,\nu 2n)^{25}$Mg | 7(1) | 3 | 2 | 12
27Al$(\mu^{-}\\!,\nu 3n)^{24}$Mg | 2 | 3 | 1 | 6
27Al$(\mu^{-}\\!,\nu pxn)^{26-23}$Na | 2 | 2 | 1 | 5
27Al$(\mu^{-}\\!,\nu\alpha xn)^{23-21}$Ne | 1 | 2 | 0 | 3
Total | 75(5) | 14 | 11 | 100

Table 2 lists the muon capture reactions in 27Al, which produce states of Mg,
Na, and Ne [45]. Note that only 13% of captures result in the 27Mg ground
state. The majority of captures result in neutron emission, with the nucleus
left in either ground or excited states. The detailed calculation of the
relevant nuclear matrix elements is quite complex, which is the reason the
experimentally measured value of the total $\mu$ capture rate is used in
deriving a conversion experiment’s single event sensitivity. This is an
important part of the normalization: the number of stopped muons in the
sensitive time window of the experiment.

Figure 8: The fraction of muon decay in orbit (DIO) in a muonic atom compared
to the $\mu^{-}$ capture rate as a function of $Z$. Experimental values are
taken from the Measday compilation [20], a survey experiment [44] or a
specific previous experiment (Ref. [45] for 27Al).

The experimental measurement of the total muon lifetime, with its associated
uncertainties, thus unavoidably involves the calculation of the experimental
efficiency and therefore the calculation of the $\mu\to e$ conversion rate.
Since the overlap of the muon atomic wave function with the nuclear proton and
neutron distribution influences the effective lifetime, the New Physics and
the Standard Model nuclear physics are inextricably mixed: the measured rate
(or limit on the rate) manifestly depends on the muon capture lifetime. The
muon nuclear capture rate grosso modo follows Wheeler’s [46] $Z_{\rm eff}^{4}$
law, but in detail shows the effect of nuclear shell structure on nuclear
size. The convention of quoting a “capture fraction”, thereby exaggerates the
effect of nuclear shell model structure.

Figure 9: Chronology of 90% confidence limits on $\mu\rightarrow e\gamma$,
$\mu\rightarrow 3e$, $\mu$ to $e$ conversion, $\tau\to e\gamma$,
$\tau\to\mu\gamma$ and $\tau\to 3e$, including predicted limits for current
experiments. The $\mu$ to $e$ conversion results are normalized to the total
muon lifetime, not the heretofore conventional $\mu$ capture rate. Unfilled
symbols represent the goals of experiments currently in preparation. Further
improvement in search sensitivity is possible with new stopped muon beams
under study for decades hence.

An explicit example can be found in equation (3) in the report of the SINDRUM
II limit on the conversion rate in lead (Pb) [47], where the denominator in
the calculation of $B_{\mu e}$ is $f_{\rm capt}N_{\rm stop}\epsilon_{\rm
tot}$. Since $f_{\rm capt}$ for Pb is 0.95, the normalization to $\mu$ capture
is only slightly different from the normalization to muon stops. For lighter
nuclei, however, for example, aluminum, $f_{\rm capt}$ is 0.61, so the change
in normalization amounts to 64%. Further, since the $\mu$ capture rate as a
function of atomic number reflects nuclear shell structure as well the details
of the low-lying levels of individual nuclei, and is primarily an incoherent
process, its use introduces extraneous nuclear physics into consideration of
the $Z$ dependence of the coherent New Physics process of $\mu$ to $e$
conversion.

From a theoretical perspective, a model calculation of the rate of conversion
effectively yields an absolute rate (more specifically, a rate characterized
by $G_{F}^{2}$ and a mass-scale coupling factor). The conventional
normalization involves a hybrid ratio of the calculated rate of the coherent
conversion process divided by the experimental measurement of a partially
incoherent muon capture process. This has not mattered in any practical sense
up to this point, but makes conceptual comparisons with decay experiments
unnecessarily difficult. We should remove this inconsistency before we begin
to compare conversion rates for different nuclei to explore the Lorentz
structure of the conversion process.

Table 3:  Atomic number-dependent adjustment of $\mu\to e$ conversion limits to normalization to muon stops rather than captures. |  |  | 90% CL Limit | 90% CL Limit
---|---|---|---|---
Year | Reference | Nucleus | Normalized to | Normalized to
|  |  |  $\mu$ Capture |  $\mu$ Stops
1952  | [41] | Sn, Sb | $1.0\times 10^{-1}$ | $1.0\times 10^{-1}$
1955 | [43] | Cu | $5.0\times 10^{-4}$ | $5.4\times 10^{-4}$
1961 | [48] | Cu | $4.0\times 10^{-6}$ | $4.3\times 10^{-6}$
1961 | [49] | Cu | $5.9\times 10^{-6}$ | $6.4\times 10^{-6}$
1962 | [50] | Cu | $2.2\times 10^{-7}$ | $2.4\times 10^{-7}$
1964 | [51] | Cu | $2.2\times 10^{-7}$ | $2.4\times 10^{-7}$
1972 | [52] | Cu | $2.6\times 10^{-8}$ | $2.8\times 10^{-8}$
1977 | [53] | S | $4.0\times 10^{-10}$ | $5.3\times 10^{-10}$
1982 | [54] | S | $7.0\times 10^{-11}$ | $9.3\times 10^{-11}$
1988 | [55] | Ti | $4.6\times 10^{-12}$ | $5.4\times 10^{-12}$
1993 | [56] | Ti | $4.3\times 10^{-12}$ | $5.0\times 10^{-12}$
1995 | [57] | Ti | $6.5\times 10^{-13}$ | $7.6\times 10^{-13}$
1996 | [47] | Pb | $4.6\times 10^{-11}$ | $4.7\times 10^{-11}$
2006 | [1] | Au | $7.0\times 10^{-13}$ | $7.2\times 10^{-13}$

Crivellin et al. [58] write the conversion rate, not the conversion ratio, as

${\rm{\Gamma}}_{\mu\to
e}^{N}\,\,=\,\frac{{m_{\mu}^{5}}}{{4{{\rm{\Lambda}}^{4}}}}|eC_{L}^{D}{D_{N}}+4({G_{F}}{m_{\mu}}{m_{p}}\tilde{C}_{(p)}^{SL}S_{N}^{(p)}$
(7)

$\begin{gathered}+\ \tilde{C}_{(p)}^{VR}V_{N}^{(p)}+\,p\to
n)|^{2}+L\leftrightarrow R,\end{gathered}$ (8)

involving the proton and neutron fields, but otherwise similar to the muon
decay rates.

The historical normalization of this process to the muon capture rate
conflates the actual sensitivity of a conversion experiment with the nuclear
physics of muon capture. In the background-free case the experimental
sensitivity depends only on the number of conversion electrons in the signal
window over the sensitive time of the experiment, which is also true of a rare
decay experiment. Why then should we insert an extraneous muon capture factor
into the many theory comparisons of conversion vs. decay sensitivity against
particular models?

The limits normalized to muon stops and muon capture are listed in Table 3.
Fig. 6 presents our calculation of the $(Z,A)$ dependence of the CLFV matrix
elements without the muon capture normalization, demonstrating the much-
reduced extraneous structure heretofore introduced by normalization to the
muon capture rate.

Fig. 9 shows the existing $\mu\rightarrow e$ conversion and decay experimental
limits. Here we normalize the conversion results to muon stops, removing the
historical muon capture normalization, making the comparison with decay
experiments more intuitive.

##  V Spin-dependence and Target Design Considerations

Coherent muon-to-electron conversion in a nucleus results in the emission of a
mono-energetic electron with an energy $E_{\mu e}$ that is nuclear dependent
(104.97 MeV for aluminum (Al)). Radiative corrections to the conversion
electron spectrum have been calculated by Szafron [59]. The time distribution
of the conversion electron depends on the mean lifetime of the muonic atom and
is also nucleus-dependent.

The dominant intrinsic background in a conversion search is electrons from
decay-in-orbit (DIO). The momentum spectrum of these electrons differs from
that of electrons from free muon decay, which follows the Michel spectrum with
an endpoint of $\sim$52 MeV/c. When the muon decay occurs in the nuclear field
there is a recoil tail and there are significant radiative corrections [60,
61]. The resulting endpoint of the electron momentum spectrum is close to the
conversion electron signal. Consequently, to detect the conversion electron an
experiment must have detectors with a momentum resolution capable of
distinguishing these recoil tail decay electrons from the conversion signal
electrons. Energy straggling in the target material must be minimized to
maintain good signal-to-background discrimination. Radiative pion capture
(RPC) backgrounds are produced when pions in the muon beamline stop in the
target. The pion is captured into the orbit of the nucleus, resulting in
emission of a photon, $\pi^{-}+N(A,Z)\rightarrow\gamma^{(*)}+N(A,Z-1)$,
followed by an asymmetric ($\gamma\rightarrow e^{+}e^{-}$) conversion
producing electrons with energies nearly up to the charged pion mass ($139$
MeV/c2). The upcoming experiments Mu2e and COMET mitigate pion backgrounds
through the use of pulsed beams and by exploiting the short pion lifetime
($\sim$26 ns at rest) and ignore particles in their detectors arriving before
a few hundred ns. Cosmic ray-induced backgrounds also contribute significantly
if not vetoed, but have a small dependence on stopping target design [62].

We consider the phenomenological advantages of four proposed elements:
titanium, vanadium, lithium, and gold. Table 4 provides details of these
materials including their densities, mean muonic atom lifetimes, isotope
abundance, and the nuclear spins of these isotopes. All these features have an
impact on the New Physics we could observe in these targets. The experimental
values for capture and decay fractions are taken from the Measday compilation
[20], a survey experiment [44] or a specific previous experiment (Ref. [45]
for 27Al). Figure 8 shows the variation of the average capture and decay
fractions with atomic number. Heavier nuclei have smaller decay fractions and
larger capture fractions, which is beneficial, as it results in fewer decay
background electrons for the same number of stopped muons.

In reality, the conversion target choice is a compromise between maximizing
the number of stopped muons while minimizing the scattering and energy losses
experienced by the outgoing electrons. For example, the Mu2e experiment plans
to utilize a foil-based target consisting of 37 thin ($\sim$ 100 $\mu m$
thick) foils orientated along the beam axis, based on detailed simulations
[63].

Table 4: Elements and their characteristics (atomic number ($Z$), density ($\rho$), lifetime ($\tau_{mean}$), and relative average fraction of decay ($f_{dec.}$) to capture ($f_{capt.}$) along with their naturally occurring abundance ($Ab.$) for given isotopes ($I.$), and their nuclear spins ($s$)). Lifetimes taken from Ref. [44]. Element | $Z$ | $\rho$ | $\tau_{mean}$ | $f_{dec.}$ | $f_{capt.}$ | $I.$ | $Ab.$ | $s$
---|---|---|---|---|---|---|---|---
|  | [g/cm${}^{3}]$ | [ns] | [$\%$] | [$\%$] |  | [$\%$] |
Li | 3 | 0.534 | 2175 | 99 | 1 |  6Li | 7.6 | 1
|  |  |  |  |  |  7Li | 92.4 | 3/2
Al | 13 | 2.11 | 864 | 39 | 61 |  27Al | 100 | 5/2
Ti | 22 | 4.51 | 329 | 15 | 85 |  46Ti | 8.25 | 0
|  |  |  |  |  |  47Ti | 7.44 | 5/2
|  |  |  |  |  |  48Ti | 73.72 | 0
|  |  |  |  |  |  49Ti | 5.41 | 7/2
|  |  |  |  |  |  50Ti | 5.18 | 0
V | 23 | 6.11 | 284 | 13 | 87 |  50V | 0.25 | 6
|  |  |  |  |  |  51V | 99.75 | 7/2
Au | 79 | 19.3 | 73 | 3 | 97 |  197Au | 100 | 3/2

###  V.1 Practical Considerations

From Figs. 6 and 7 it is apparent that both titanium (Ti) and vanadium (V)
have the advantage that, if coherent conversion is measured in Al, they
provide good separation in the relative conversion rates produced by the
dipole, scalar, and vector interactions. This makes them useful for
determining the type of physics responsible for a coherent conversion. From a
practical point-of-view, Ti and V can be readily obtained in solid, foil form.
In addition, Table 4 shows that the mean lifetime of a muonic Ti or V atom is
much shorter than for Al, but is not so short that pion-induced background
would overwhelm the conversion signal. Both Ti and V have smaller average
decay fractions than Al, meaning lower decay backgrounds for the same number
of stopped muons. For all these reasons Ti and V are popular choices for the
next generation of conversion measurements, assuming a signal in Mu2e or
COMET.

Lithium (Li) is another potential alternative target material. The muonic
lifetime in Li is close to the free muon lifetime; as a result, the survival
fraction is just a few percent. The relative proportion of decays would be
much larger than in Al, meaning more decay background for the same number of
stopped muons. Due to the much smaller survival fraction, reaching the same
sensitivity as aluminum would require much longer running time. From a
practical point of view, a Li foil target is feasible. Lithium has been used
in foil form in a a transition radiation detector[64]. Since Li has a much
lower density, $\sim$ 4 times less than Al, stopping the same number of muons
would require a much larger target volume.

From Figs. 6 and 7 it is also clear that the conversion rates, relative to Al,
for dipole, scalar, and vector interactions show largest differences for high
$Z$ nuclei, making gold (Au) an obvious choice for establishing the nature of
the physics, assuming a signal in Al. Table 4 shows that the relative fraction
of decay to capture in Au is small, meaning on average 3 $\%$ of muons will
undergo decay, a significant reduction from 39 $\%$ in Al. The downside comes
from the fact that Au has a mean muonic atom lifetime of just 73 ns. It would
not be feasible to use a Au target in a Mu2e or COMET style design due to the
enforced time selection required to remove pion-induced backgrounds and the
beam structure at these current facilities. If pion contamination could be
eliminated using a Fixed Field Alternating-gradient machine such as in the
proposed Advanced Muon Facility [14], measuring conversion in Au could be
feasible.

###  V.2 Considering Incoherent Contributions

The mono-energetic electron experimental signature of $\mu\to e$ conversion
selects the spin-independent (SI) coherent interaction with the charge or mass
distribution, which is proportional to $A^{2}$. Spin-dependent (SD)
contributions do not have coherent enhancement and have been studied in Refs.
[16, 65]. Thus far we have considered only coherent contributions from a
dipole, scalar, and vector interaction these operators will contribute to the
SI rate, while axial, tensor, and pseudoscalar operators contribute to the SD
rate. The SD rate depends on the distribution of spin in the nucleus [66], and
therefore requires detailed modeling of the target nucleus. In most models, SI
conversion dominates due to the $A^{2}$ enhancement, but this is not true in
all New Physics models. Reference [67] describes how the SD interaction
induced by ALP$-\pi^{0}$ mixing is the leading contribution to $\mu\to e$
conversion in models where axion-like particles (ALPs) induce the conversion.
Consequently, in the event a conversion signal is observed, it is important to
consider of the spin of the target nuclei to understand the Lorentz structure
of the New Physics.

Natural titanium has several stable isotopes with different spins, however,
separated isotopes of Ti are not readily available in the quantities needed
for a target of $\sim 200$ g. Employing a natural Ti target complicates the
unraveling of SD and SI conversion rates. Vanadium is therefore an attractive
alternative to Ti; it provides the same physics benefits (increased survival
fraction, similar muonic lifetime, etc.) with the advantage that it has a
single stable isotope, 51V (spin $\frac{7}{2}$), with $>$ 99$\%$ abundance.
Vanadium is also readily available in thin foils that can be fabricated into a
target similar to the Al target used in Mu2e.

Comparing conversion in light targets with very different neutron-to-proton
ratios (for example lithium (Li) and Al) could allow us to distinguish
operators involving neutrons from those involving protons [65]. Additionally,
Ref. [16] suggests that a conversion measurement in lighter nuclei, such as
Li, would be of interest for detecting SD conversion, since the SD rate being
relatively suppressed by $1/A^{2}$ compared to the SI rate. As a result, the
ratio $\Gamma_{SD}/\Gamma_{SI}$ is larger in lighter nuclei. For these reasons
Au (which, in addition, has only a single stable isotope) is also an
attractive option.

When choosing the target material for the next generation of conversion
experiments it is important to consider more than the conversion rate relative
to Al. We must also take into account the relative decay fraction, muonic atom
lifetime, and availability of isotopes of a known spin; we have described four
possible target choices All have physics advantages; the specific target
chosen will depend on, and influence, the experimental design of any future
facility, which is beyond the scope of this paper.

##  VI Conclusions

To conclude, we have presented a new calculation of the $(Z,A)$ dependence of
coherent muon-to-electron conversion in 236 isotopes, significantly extending
the previous studies of Kitano et al. [10] and Cirigliano et al. [11]. Our
approach has several key improvements over these previous treatments:

  * •

We have included the effect of permanent quadrupole deformation on the CLFV
matrix elements, using the method of Barrett moments to add the substantial
catalog of muonic X-ray nuclear size and shape determinations of the charge
distribution to the electron scattering results used in previous calculations.

  * •

Rather than using neutron distributions scaled by $N/Z$ from charge
distributions, as in the previous work, we have employed a Hartree-Bogoliubov
model for even-even nuclei to calculate neutron-related matrix elements. This
takes into account the quadrupole deformation of the neutron distributions as
well as the fact that neutrons are in general in different shell model orbits
than protons.

  * •

We present the resulting CLFV overlap integrals for stable isotopes with
greater than 1% abundance as well as results weighted for natural abundance.

  * •

The resulting conversion rates differ from previous calculations, particularly
in the region of large permanent quadrupole deformation.

Finally, we propose a revised normalization for $\mu\to e$ conversion results,
quoting the measured conversion rate (or limit thereon) directly, instead of
presenting the conversion rate divided by the muon capture rate, which is
generally not measured in the same experiment.

Our sensitivity plots are presented in this form. Note the reduction in the
scatter of points, which is largely due to the removal of the additional shell
model structure observed in the incoherent muon capture process.

## Acknowledgements

This work was supported in part by DOE Grant DE‐SC0011925. It was initiated at
the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-2210452.

We wish to acknowledge helpful conversations with Wick Haxton, Julian Heeck,
Natalia Oreshkina, Robert Szafron and Petr Vogel, as well as the careful
reading of the text by Bertrand Echenard, James Miller, Ryan Plestid and Frank
Porter.

## Appendix

###  VI.1 Table of Overlap Integrals

Isotope | $Z$ | $N$ | BM | $D$ | $S(p)$ | $V(p)$ | $S(n)$ | $V(n)$
---|---|---|---|---|---|---|---|---
6Li111If the available literature does not provide a 2-parameter Fermi distribution - some electron scattering measurements use a Fourier-Bessel expansion -, we use a least square method to fit the distribution to a 2pF,222This is not to be confused with the spherical three-parameter Fermi distribution often used in electrons scattering from heavy nuclei. | 3 | 3 | 6.19 | 0.0013 | 0.00055 | 0.00055 | 0.00055 | 0.00055
7Li111Assume $\beta=0$,222Landolt-Börnstein (only rms radius) [68] | 3 | 4 | 5.36 | 0.0014 | 0.00058 | 0.00058 | 0.00077 | 0.00078
9Be111Assume $\beta=0$ | 4 | 5 | 5.42 | 0.0028 | 0.00117 | 0.00118 | 0.00146 | 0.00147
10B111Assume $\beta=0$ | 5 | 5 | 5.68 | 0.0047 | 0.00198 | 0.00200 | 0.00198 | 0.00200
11B111Assume $\beta=0$,222Landolt-Börnstein (only rms radius) [68] | 5 | 6 | 5.46 | 0.0047 | 0.00201 | 0.00203 | 0.00241 | 0.00244
12C111Assume $\beta=0$ | 6 | 6 | 5.83 | 0.0072 | 0.00306 | 0.00310 | 0.00306 | 0.00310
13C111Assume $\beta=0$ | 6 | 7 | 5.82 | 0.0072 | 0.00306 | 0.0031 | 0.00306 | 0.0031
14N111Assume $\beta=0$ | 7 | 7 | 5.87 | 0.0102 | 0.00434 | 0.00441 | 0.00434 | 0.00441
16O | 8 | 8 | 6.76 | 0.0134 | 0.00571 | 0.00582 | 0.00571 | 0.00582
19F | 9 | 10 | 7.90 | 0.0163 | 0.00694 | 0.00711 | 0.00771 | 0.00790
20Ne | 10 | 10 | 8.57 | 0.0193 | 0.00819 | 0.00842 | 0.00821 | 0.00844
22Ne | 10 | 12 | 8.31 | 0.0199 | 0.00850 | 0.00874 | 0.01015 | 0.0104
23Na | 11 | 12 | 8.50 | 0.0246 | 0.0105 | 0.0108 | 0.0115 | 0.0118
24Mg | 12 | 12 | 8.84 | 0.0287 | 0.0122 | 0.0127 | 0.0122 | 0.0127
25Mg | 12 | 13 | 8.59 | 0.0297 | 0.0127 | 0.0131 | 0.0138 | 0.0142
26Mg | 12 | 14 | 8.59 | 0.0298 | 0.0127 | 0.0131 | 0.0149 | 0.0154
27Al | 13 | 14 | 8.76 | 0.0350 | 0.0150 | 0.0155 | 0.0161 | 0.0167
28Si | 14 | 14 | 8.99 | 0.0406 | 0.0174 | 0.0181 | 0.0174 | 0.0181
29Si | 14 | 15 | 8.94 | 0.0405 | 0.0173 | 0.0180 | 0.0186 | 0.0193
30Si | 14 | 16 | 8.88 | 0.0410 | 0.0176 | 0.0183 | 0.0201 | 0.0209
31P | 15 | 16 | 9.11 | 0.0462 | 0.0198 | 0.0207 | 0.0211 | 0.0220
32S | 16 | 16 | 9.49 | 0.0519 | 0.0222 | 0.0233 | 0.0222 | 0.0233
34S | 16 | 18 | 9.62 | 0.0513 | 0.0220 | 0.0231 | 0.0247 | 0.0259
35Cl333deVries et al.[17] | 17 | 18 | 10.31 | 0.0548 | 0.0234 | 0.0247 | 0.0248 | 0.0261
37Cl333deVries et al.[17] | 17 | 20 | 10.31 | 0.0547 | 0.0234 | 0.0246 | 0.0275 | 0.0290
40Ar | 18 | 22 | 10.10 | 0.0617 | 0.0263 | 0.0279 | 0.0322 | 0.0341
39K | 19 | 20 | 10.10 | 0.0688 | 0.0294 | 0.0313 | 0.0309 | 0.0329
41K | 19 | 22 | 10.20 | 0.0682 | 0.0291 | 0.0310 | 0.0337 | 0.0359
40Ca | 20 | 20 | 10.30 | 0.0750 | 0.0320 | 0.0341 | 0.0320 | 0.0341
44Ca | 20 | 24 | 10.52 | 0.0734 | 0.0313 | 0.0334 | 0.0375 | 0.0401
45Sc | 21 | 24 | 10.63 | 0.0799 | 0.0340 | 0.0365 | 0.0389 | 0.0417
46Ti | 22 | 24 | 10.92 | 0.0840 | 0.0357 | 0.0385 | 0.0388 | 0.0419
47Ti | 22 | 25 | 10.87 | 0.0855 | 0.0364 | 0.0392 | 0.0413 | 0.0445
48Ti | 22 | 26 | 10.84 | 0.0852 | 0.0362 | 0.0391 | 0.0428 | 0.0461
49Ti | 22 | 27 | 10.75 | 0.0865 | 0.0368 | 0.0397 | 0.0452 | 0.0487
50Ti | 22 | 28 | 10.72 | 0.0868 | 0.0369 | 0.0398 | 0.0470 | 0.0506
51V | 23 | 28 | 10.82 | 0.0934 | 0.0397 | 0.0430 | 0.0484 | 0.0523
50Cr | 24 | 26 | 11.20 | 0.0962 | 0.0407 | 0.0443 | 0.0442 | 0.0481
52Cr | 24 | 28 | 10.96 | 0.0994 | 0.0422 | 0.0459 | 0.0492 | 0.0535
53Cr | 24 | 29 | 11.06 | 0.0983 | 0.0417 | 0.0454 | 0.0504 | 0.0548
54Cr | 24 | 30 | 11.30 | 0.0952 | 0.0403 | 0.0439 | 0.0504 | 0.0548
55Mn | 25 | 30 | 11.38 | 0.103 | 0.0435 | 0.0475 | 0.0522 | 0.0571
54Fe | 26 | 28 | 11.14 | 0.113 | 0.0478 | 0.0524 | 0.0515 | 0.0564
56Fe | 26 | 30 | 11.43 | 0.108 | 0.0457 | 0.0502 | 0.0527 | 0.0579
57Fe | 26 | 31 | 11.56 | 0.108 | 0.0456 | 0.0501 | 0.0544 | 0.0597
Isotope | $Z$ | $N$ | BM | $D$ | $S(p)$ | $V(p)$ | $S(n)$ | $V(n)$
59Co | 27 | 32 | 11.70 | 0.114 | 0.0482 | 0.0532 | 0.0572 | 0.0630
58Ni | 28 | 30 | 11.44 | 0.124 | 0.0524 | 0.0580 | 0.0562 | 0.0621
60Ni | 28 | 32 | 11.64 | 0.121 | 0.0508 | 0.0563 | 0.0580 | 0.0642
61Ni | 28 | 33 | 11.74 | 0.120 | 0.0506 | 0.0560 | 0.0596 | 0.0660
62Ni | 28 | 34 | 11.84 | 0.117 | 0.0494 | 0.0549 | 0.0598 | 0.0663
63Cu | 29 | 34 | 12.05 | 0.123 | 0.0517 | 0.0576 | 0.0606 | 0.0675
65Cu | 29 | 36 | 12.13 | 0.122 | 0.0511 | 0.0570 | 0.0635 | 0.0708
64Zn | 30 | 34 | 12.29 | 0.126 | 0.0527 | 0.0591 | 0.0598 | 0.0670
66Zn | 30 | 36 | 12.38 | 0.125 | 0.0524 | 0.0588 | 0.0629 | 0.0705
67Zn444Angeli et al.[38] | 30 | 37 | 12.28 | 0.127 | 0.0531 | 0.0595 | 0.0655 | 0.0734
68Zn | 30 | 38 | 12.43 | 0.125 | 0.0522 | 0.0585 | 0.0661 | 0.0742
69Ga | 31 | 38 | 12.59 | 0.129 | 0.0538 | 0.0607 | 0.0659 | 0.0744
71Ga | 31 | 40 | 12.72 | 0.127 | 0.0530 | 0.0598 | 0.0684 | 0.0772
70Ge | 32 | 38 | 12.87 | 0.133 | 0.0552 | 0.0626 | 0.0656 | 0.0745
72Ge | 32 | 40 | 12.96 | 0.133 | 0.0553 | 0.0628 | 0.0692 | 0.0785
73Ge | 32 | 41 | 13.02 | 0.130 | 0.0538 | 0.0612 | 0.0690 | 0.0784
74Ge | 32 | 42 | 13.09 | 0.132 | 0.0546 | 0.0620 | 0.0717 | 0.0814
76Ge | 32 | 44 | 13.02 | 0.130 | 0.0535 | 0.0608 | 0.0738 | 0.0839
75As | 33 | 42 | 13.16 | 0.134 | 0.0554 | 0.0633 | 0.0705 | 0.0805
76Se | 34 | 42 | 13.36 | 0.138 | 0.0566 | 0.0651 | 0.0701 | 0.0806
77Se | 34 | 43 | 13.38 | 0.137 | 0.0563 | 0.0647 | 0.0712 | 0.0818
78Se | 34 | 44 | 13.25 | 0.139 | 0.0571 | 0.0656 | 0.0741 | 0.0851
80Se | 34 | 46 | 13.25 | 0.138 | 0.0569 | 0.0653 | 0.0772 | 0.0886
82Se | 34 | 48 | 13.22 | 0.139 | 0.0569 | 0.0654 | 0.0807 | 0.0926
79Br | 35 | 44 | 13.47 | 0.141 | 0.0579 | 0.0668 | 0.0728 | 0.0840
81Br | 35 | 46 | 13.23 | 0.144 | 0.0594 | 0.0685 | 0.0781 | 0.0901
80Kr | 36 | 44 | 13.63 | 0.149 | 0.0609 | 0.0706 | 0.0745 | 0.0864
82Kr | 36 | 46 | 13.39 | 0.149 | 0.0609 | 0.0706 | 0.0778 | 0.0903
83Kr | 36 | 47 | 13.35 | 0.149 | 0.0612 | 0.0710 | 0.0799 | 0.0927
84Kr | 36 | 48 | 13.32 | 0.149 | 0.0611 | 0.0709 | 0.0816 | 0.0946
86Kr | 36 | 50 | 13.27 | 0.151 | 0.0618 | 0.0717 | 0.0859 | 0.0995
85Rb | 37 | 48 | 13.31 | 0.156 | 0.0637 | 0.0742 | 0.0826 | 0.0962
87Rb | 37 | 50 | 13.27 | 0.156 | 0.0641 | 0.0746 | 0.0866 | 0.1008
86Sr | 38 | 48 | 13.43 | 0.160 | 0.0654 | 0.0765 | 0.0826 | 0.0967
87Sr | 38 | 49 | 13.41 | 0.161 | 0.0657 | 0.0768 | 0.0847 | 0.0991
88Sr | 38 | 50 | 13.40 | 0.161 | 0.0657 | 0.0769 | 0.0865 | 0.101
89Y333deVries et al.[17] | 39 | 50 | 13.43 | 0.168 | 0.0686 | 0.0804 | 0.0879 | 0.103
90Zr | 40 | 50 | 13.53 | 0.170 | 0.0693 | 0.0819 | 0.0866 | 0.102
91Zr | 40 | 51 | 13.64 | 0.168 | 0.0682 | 0.0807 | 0.0870 | 0.103
92Zr | 40 | 52 | 13.74 | 0.166 | 0.0671 | 0.0795 | 0.0872 | 0.103
94Zr | 40 | 54 | 13.98 | 0.163 | 0.0657 | 0.0780 | 0.0887 | 0.105
96Zr | 40 | 56 | 14.25 | 0.158 | 0.0637 | 0.0757 | 0.0890 | 0.106
93Nb | 41 | 52 | 13.75 | 0.170 | 0.0688 | 0.0819 | 0.0873 | 0.104
92Mo | 42 | 50 | 13.69 | 0.178 | 0.0720 | 0.0860 | 0.0857 | 0.102
94Mo | 42 | 52 | 13.88 | 0.174 | 0.0699 | 0.0837 | 0.0866 | 0.104
95Mo | 42 | 53 | 13.93 | 0.172 | 0.0693 | 0.0831 | 0.0875 | 0.105
96Mo | 42 | 54 | 14.15 | 0.167 | 0.0673 | 0.0808 | 0.0863 | 0.104
97Mo | 42 | 55 | 14.20 | 0.167 | 0.0669 | 0.0803 | 0.0876 | 0.105
98Mo | 42 | 56 | 14.37 | 0.162 | 0.0649 | 0.0781 | 0.0862 | 0.104
Isotope | $Z$ | $N$ | BM | $D$ | $S(p)$ | $V(p)$ | $S(n)$ | $V(n)$
100Mo | 42 | 58 | 14.66 | 0.159 | 0.0634 | 0.0764 | 0.0873 | 0.105
96Ru | 44 | 50 | 14.07 | 0.182 | 0.0728 | 0.0881 | 0.0828 | 0.100
98Ru | 44 | 54 | 14.32 | 0.175 | 0.0700 | 0.0849 | 0.0857 | 0.104
99Ru | 44 | 55 | 14.41 | 0.174 | 0.0693 | 0.0841 | 0.0866 | 0.105
100Ru | 44 | 56 | 14.55 | 0.170 | 0.0676 | 0.0822 | 0.0857 | 0.104
101Ru | 44 | 57 | 14.61 | 0.169 | 0.0672 | 0.0818 | 0.0871 | 0.106
102Ru | 44 | 58 | 14.74 | 0.166 | 0.0659 | 0.0803 | 0.0866 | 0.105
104Ru | 44 | 60 | 15.02 | 0.162 | 0.0643 | 0.0785 | 0.0874 | 0.107
103Rh444Angeli et al.[38] | 45 | 58 | 14.49 | 0.176 | 0.0699 | 0.0854 | 0.0901 | 0.110
102Pd | 46 | 56 | 14.79 | 0.179 | 0.0708 | 0.0870 | 0.0859 | 0.105
104Pd | 46 | 58 | 14.94 | 0.175 | 0.0693 | 0.0852 | 0.0870 | 0.107
105Pd | 46 | 59 | 14.98 | 0.175 | 0.0689 | 0.0849 | 0.0884 | 0.109
106Pd | 46 | 60 | 15.09 | 0.172 | 0.0679 | 0.0837 | 0.0882 | 0.109
108Pd | 46 | 62 | 15.29 | 0.169 | 0.0666 | 0.0822 | 0.0895 | 0.111
110Pd | 46 | 64 | 15.46 | 0.165 | 0.0647 | 0.0801 | 0.0898 | 0.111
107Ag | 47 | 60 | 14.88 | 0.177 | 0.0695 | 0.0861 | 0.0887 | 0.110
109Ag | 47 | 62 | 15.01 | 0.174 | 0.0680 | 0.0844 | 0.0897 | 0.111
106Cd | 48 | 58 | 14.69 | 0.184 | 0.0726 | 0.0902 | 0.0873 | 0.109
110Cd | 48 | 62 | 14.90 | 0.179 | 0.0701 | 0.0874 | 0.0903 | 0.113
111Cd | 48 | 63 | 14.94 | 0.178 | 0.0697 | 0.0869 | 0.0914 | 0.114
112Cd | 48 | 64 | 15.04 | 0.177 | 0.0690 | 0.0862 | 0.0918 | 0.115
113Cd | 48 | 65 | 15.08 | 0.175 | 0.0681 | 0.0851 | 0.0922 | 0.115
114Cd | 48 | 66 | 15.17 | 0.175 | 0.0681 | 0.0851 | 0.0935 | 0.117
116Cd | 48 | 68 | 15.31 | 0.174 | 0.0678 | 0.0849 | 0.0960 | 0.120
113In | 49 | 64 | 14.93 | 0.183 | 0.0714 | 0.0895 | 0.0932 | 0.117
115In | 49 | 66 | 15.04 | 0.180 | 0.0701 | 0.0880 | 0.0944 | 0.119
116Sn | 50 | 66 | 14.95 | 0.186 | 0.0720 | 0.0909 | 0.0951 | 0.120
117Sn | 50 | 67 | 15.00 | 0.184 | 0.0715 | 0.0903 | 0.0958 | 0.121
118Sn | 50 | 68 | 15.07 | 0.183 | 0.0710 | 0.0897 | 0.0966 | 0.122
119Sn | 50 | 69 | 15.10 | 0.182 | 0.0704 | 0.0890 | 0.0971 | 0.123
120Sn | 50 | 70 | 15.13 | 0.181 | 0.0701 | 0.0887 | 0.0981 | 0.124
122Sn | 50 | 72 | 15.16 | 0.179 | 0.0693 | 0.0878 | 0.100 | 0.126
124Sn | 50 | 74 | 15.22 | 0.178 | 0.0686 | 0.0870 | 0.101 | 0.129
121Sb | 51 | 70 | 15.28 | 0.181 | 0.0697 | 0.0888 | 0.0957 | 0.122
123Sb | 51 | 72 | 15.31 | 0.180 | 0.0693 | 0.0883 | 0.0978 | 0.125
122Te | 52 | 70 | 15.37 | 0.182 | 0.0698 | 0.0895 | 0.0937 | 0.120
124Te | 52 | 72 | 15.37 | 0.181 | 0.0693 | 0.0889 | 0.0958 | 0.123
125Te | 52 | 73 | 15.38 | 0.181 | 0.0690 | 0.0887 | 0.0969 | 0.124
126Te | 52 | 74 | 15.39 | 0.180 | 0.0689 | 0.0885 | 0.0980 | 0.126
128Te | 52 | 76 | 15.42 | 0.179 | 0.0681 | 0.0876 | 0.100 | 0.128
130Te | 52 | 78 | 15.41 | 0.179 | 0.0682 | 0.0878 | 0.102 | 0.132
127I | 53 | 74 | 15.52 | 0.181 | 0.0690 | 0.0893 | 0.0963 | 0.125
128Xe | 54 | 74 | 15.70 | 0.181 | 0.0684 | 0.0891 | 0.0938 | 0.122
129Xe | 54 | 75 | 15.68 | 0.181 | 0.0684 | 0.0892 | 0.0950 | 0.124
130Xe | 54 | 76 | 15.66 | 0.181 | 0.0682 | 0.0889 | 0.0962 | 0.125
131Xe | 54 | 77 | 15.65 | 0.182 | 0.0686 | 0.0895 | 0.0979 | 0.128
132Xe | 54 | 78 | 15.68 | 0.181 | 0.0683 | 0.0891 | 0.0989 | 0.129
134Xe | 54 | 80 | 15.63 | 0.181 | 0.0684 | 0.0892 | 0.101 | 0.132
136Xe | 54 | 82 | 15.67 | 0.180 | 0.0680 | 0.0888 | 0.103 | 0.135
Isotope | $Z$ | $N$ | BM | $D$ | $S(p)$ | $V(p)$ | $S(n)$ | $V(n)$
133Cs | 55 | 78 | 15.65 | 0.183 | 0.0688 | 0.0903 | 0.0975 | 0.128
134Ba | 56 | 78 | 15.75 | 0.184 | 0.0687 | 0.0907 | 0.0959 | 0.127
135Ba | 56 | 79 | 15.74 | 0.184 | 0.0689 | 0.0911 | 0.0972 | 0.128
136Ba | 56 | 80 | 15.70 | 0.184 | 0.0689 | 0.0911 | 0.0986 | 0.130
137Ba | 56 | 81 | 15.70 | 0.185 | 0.0691 | 0.0913 | 0.100 | 0.132
138Ba | 56 | 82 | 15.72 | 0.184 | 0.0687 | 0.0909 | 0.101 | 0.133
139La | 57 | 82 | 15.77 | 0.186 | 0.0693 | 0.0922 | 0.100 | 0.133
140Ce | 58 | 82 | 15.85 | 0.187 | 0.0694 | 0.0930 | 0.0981 | 0.131
142Ce | 58 | 84 | 16.00 | 0.182 | 0.0672 | 0.0905 | 0.0974 | 0.131
141Pr | 59 | 82 | 15.92 | 0.189 | 0.0699 | 0.0943 | 0.0972 | 0.131
142Nd | 60 | 82 | 15.99 | 0.191 | 0.0699 | 0.0950 | 0.0956 | 0.130
143Nd | 60 | 83 | 16.05 | 0.189 | 0.0691 | 0.0940 | 0.0956 | 0.130
144Nd | 60 | 84 | 16.13 | 0.186 | 0.0679 | 0.0925 | 0.0950 | 0.130
145Nd | 60 | 85 | 16.25 | 0.182 | 0.0665 | 0.0908 | 0.0942 | 0.129
146Nd | 60 | 86 | 16.38 | 0.179 | 0.0649 | 0.0888 | 0.0931 | 0.127
148Nd | 60 | 88 | 16.60 | 0.171 | 0.0618 | 0.0850 | 0.0909 | 0.125
150Nd | 60 | 90 | 16.47 | 0.168 | 0.0609 | 0.0833 | 0.0920 | 0.126
144Sm | 62 | 82 | 16.06 | 0.195 | 0.0711 | 0.0977 | 0.0940 | 0.129
147Sm | 62 | 85 | 16.31 | 0.185 | 0.0669 | 0.0927 | 0.0917 | 0.127
148Sm | 62 | 86 | 16.43 | 0.181 | 0.0646 | 0.0902 | 0.0898 | 0.125
149Sm | 62 | 87 | 16.46 | 0.180 | 0.0648 | 0.0902 | 0.0909 | 0.127
150Sm | 62 | 88 | 16.45 | 0.185 | 0.0671 | 0.0922 | 0.0957 | 0.132
152Sm | 62 | 90 | 16.55 | 0.179 | 0.0645 | 0.0888 | 0.0944 | 0.130
154Sm | 62 | 92 | 16.64 | 0.172 | 0.0614 | 0.0854 | 0.0917 | 0.127
151Eu | 63 | 88 | 16.71 | 0.176 | 0.0625 | 0.0879 | 0.0872 | 0.123
153Eu | 63 | 90 | 16.61 | 0.182 | 0.0656 | 0.0913 | 0.0938 | 0.130
154Gd | 64 | 90 | 16.64 | 0.180 | 0.0641 | 0.0898 | 0.0908 | 0.127
155Gd | 64 | 91 | 16.65 | 0.180 | 0.0642 | 0.0905 | 0.0912 | 0.129
156Gd | 64 | 92 | 16.74 | 0.173 | 0.0613 | 0.0864 | 0.0888 | 0.125
157Gd | 64 | 93 | 16.72 | 0.179 | 0.0638 | 0.0899 | 0.0927 | 0.131
158Gd | 64 | 94 | 16.84 | 0.171 | 0.0604 | 0.0853 | 0.0894 | 0.126
160Gd | 64 | 96 | 16.89 | 0.168 | 0.0590 | 0.0837 | 0.0890 | 0.126
159Tb555deWit et al. [34] | 65 | 94 | 16.62 | 0.176 | 0.0610 | 0.0884 | 0.0882 | 0.128
160Dy444Angeli et al.[38] | 66 | 94 | 17.06 | 0.174 | 0.0609 | 0.0875 | 0.0868 | 0.125
161Dy | 66 | 95 | 16.87 | 0.181 | 0.0639 | 0.0910 | 0.0919 | 0.131
162Dy | 66 | 96 | 16.91 | 0.175 | 0.0615 | 0.0877 | 0.0899 | 0.128
163Dy | 66 | 97 | 16.92 | 0.179 | 0.0632 | 0.0902 | 0.0929 | 0.133
164Dy | 66 | 98 | 17.01 | 0.172 | 0.0603 | 0.0863 | 0.0899 | 0.129
165Ho666Landolt-Börnstein (electron scattering) [68] | 67 | 98 | 17.42 | 0.168 | 0.0582 | 0.0843 | 0.0852 | 0.123
164Er444Angeli et al.[38] | 68 | 96 | 17.28 | 0.175 | 0.0605 | 0.0881 | 0.0854 | 0.124
166Er | 68 | 98 | 17.14 | 0.175 | 0.0609 | 0.0882 | 0.0880 | 0.127
167Er | 68 | 99 | 17.21 | 0.178 | 0.0619 | 0.0898 | 0.0901 | 0.131
168Er | 68 | 100 | 17.25 | 0.172 | 0.0595 | 0.0865 | 0.0876 | 0.127
170Er | 68 | 102 | 17.34 | 0.169 | 0.0583 | 0.0850 | 0.0876 | 0.128
169Tm666Landolt-Börnstein (electron scattering) [68] | 69 | 100 | 17.15 | 0.181 | 0.0628 | 0.0916 | 0.0910 | 0.133
170Yb | 70 | 100 | 17.29 | 0.170 | 0.0580 | 0.0861 | 0.0831 | 0.123
171Yb | 70 | 101 | 17.33 | 0.175 | 0.0596 | 0.0885 | 0.0860 | 0.128
172Yb | 70 | 102 | 17.38 | 0.169 | 0.0575 | 0.0855 | 0.0839 | 0.125
173Yb | 70 | 103 | 17.41 | 0.172 | 0.0585 | 0.0871 | 0.0860 | 0.128
Isotope | $Z$ | $N$ | BM | $D$ | $S(p)$ | $V(p)$ | $S(n)$ | $V(n)$
174Yb | 70 | 104 | 17.40 | 0.171 | 0.0581 | 0.0862 | 0.0866 | 0.128
176Yb | 70 | 106 | 17.50 | 0.165 | 0.0557 | 0.0834 | 0.0847 | 0.127
175Lu444Angeli et al.[38] | 71 | 104 | 17.75 | 0.163 | 0.0545 | 0.0826 | 0.0799 | 0.121
176Lu444Angeli et al.[38] | 71 | 105 | 17.77 | 0.162 | 0.0539 | 0.0819 | 0.0798 | 0.121
176Hf | 72 | 104 | 17.57 | 0.169 | 0.0567 | 0.0859 | 0.0817 | 0.124
177Hf | 72 | 105 | 17.59 | 0.170 | 0.0568 | 0.0862 | 0.0829 | 0.126
178Hf | 72 | 106 | 17.64 | 0.166 | 0.0555 | 0.0845 | 0.0816 | 0.124
179Hf | 72 | 107 | 17.64 | 0.166 | 0.0551 | 0.0843 | 0.0819 | 0.125
180Hf | 72 | 108 | 17.59 | 0.167 | 0.0558 | 0.0850 | 0.0837 | 0.127
181Ta | 73 | 108 | 17.61 | 0.168 | 0.0554 | 0.0855 | 0.0820 | 0.126
182W | 74 | 108 | 17.51 | 0.171 | 0.0567 | 0.0873 | 0.0826 | 0.127
183W444Angeli et al.[38] | 74 | 109 | 17.58 | 0.171 | 0.0567 | 0.0875 | 0.0835 | 0.129
184W | 74 | 110 | 17.60 | 0.168 | 0.0556 | 0.0859 | 0.0824 | 0.127
186W | 74 | 112 | 17.66 | 0.166 | 0.0544 | 0.0845 | 0.0822 | 0.128
185Re444Angeli et al.[38] | 75 | 110 | 17.54 | 0.174 | 0.0571 | 0.0887 | 0.0838 | 0.130
187Re444Angeli et al.[38] | 75 | 112 | 17.59 | 0.172 | 0.0564 | 0.0878 | 0.0842 | 0.131
186Os888Hoehn et al. [35] | 76 | 110 | 17.65 | 0.170 | 0.0554 | 0.0872 | 0.0801 | 0.126
187Os444Angeli et al.[38] | 76 | 111 | 17.66 | 0.170 | 0.0552 | 0.0871 | 0.0807 | 0.127
188Os | 76 | 112 | 17.70 | 0.166 | 0.0539 | 0.0852 | 0.0793 | 0.125
189Os444Angeli et al.[38] | 76 | 113 | 17.68 | 0.168 | 0.0545 | 0.0862 | 0.0810 | 0.128
190Os | 76 | 114 | 17.74 | 0.166 | 0.0534 | 0.0849 | 0.0800 | 0.127
192Os | 76 | 116 | 17.74 | 0.164 | 0.0526 | 0.0841 | 0.0803 | 0.128
Isotope | $Z$ | $N$ | BM | $D$ | $S(p)$ | $V(p)$ | $S(n)$ | $V(n)$
191Ir444Angeli et al.[38] | 77 | 114 | 17.65 | 0.170 | 0.0548 | 0.0874 | 0.0812 | 0.129
193Ir444Angeli et al.[38] | 77 | 116 | 17.68 | 0.168 | 0.0539 | 0.0864 | 0.0812 | 0.130
194Pt | 78 | 116 | 17.81 | 0.163 | 0.0513 | 0.0836 | 0.0761 | 0.124
195Pt | 78 | 117 | 17.83 | 0.162 | 0.0511 | 0.0835 | 0.0767 | 0.125
196Pt | 78 | 118 | 17.83 | 0.162 | 0.0508 | 0.0831 | 0.0768 | 0.126
198Pt | 78 | 120 | 17.87 | 0.161 | 0.0503 | 0.0825 | 0.0774 | 0.127
197Au | 79 | 118 | 17.82 | 0.164 | 0.0512 | 0.0843 | 0.0765 | 0.126
198Hg | 80 | 118 | 17.78 | 0.165 | 0.0513 | 0.0851 | 0.0755 | 0.125
199Hg | 80 | 119 | 17.79 | 0.165 | 0.0512 | 0.0849 | 0.0761 | 0.126
200Hg | 80 | 120 | 17.80 | 0.163 | 0.0507 | 0.0843 | 0.0759 | 0.126
201Hg444Angeli et al.[38] | 80 | 121 | 17.80 | 0.164 | 0.0507 | 0.0844 | 0.0767 | 0.128
202Hg | 80 | 122 | 17.83 | 0.162 | 0.0501 | 0.0836 | 0.0763 | 0.127
204Hg | 80 | 124 | 17.87 | 0.160 | 0.0493 | 0.0826 | 0.0764 | 0.128
203Tl | 81 | 122 | 17.80 | 0.164 | 0.0505 | 0.0849 | 0.0761 | 0.128
205Tl | 81 | 124 | 17.83 | 0.162 | 0.0498 | 0.0840 | 0.0762 | 0.129
204Pb | 82 | 122 | 17.79 | 0.165 | 0.0505 | 0.0856 | 0.0752 | 0.127
206Pb | 82 | 124 | 17.83 | 0.163 | 0.0497 | 0.0845 | 0.0752 | 0.128
207Pb | 82 | 125 | 17.85 | 0.162 | 0.0494 | 0.0842 | 0.0754 | 0.128
208Pb | 82 | 126 | 17.85 | 0.162 | 0.0494 | 0.0841 | 0.0760 | 0.129
209Bi | 83 | 126 | 17.94 | 0.158 | 0.0474 | 0.0821 | 0.0719 | 0.125
232Th999Zumbro et al. (1986) [37] | 90 | 142 | 18.70 | 0.137 | 0.0369 | 0.0720 | 0.0583 | 0.114
238U101010Zumbro et al. (1984) [36] | 92 | 146 | 18.93 | 0.132 | 0.0343 | 0.0695 | 0.0544 | 0.110

77footnotetext: Landolt-Börnstein (muonic X-ray) [68]

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