# Atomic corrections for the unique first forbidden $\beta$ transition of
187Re

O. Niţescu1,2,3<EMAIL_ADDRESS>R. Dvornický1<EMAIL_ADDRESS>F. Šimkovic1,4<EMAIL_ADDRESS>1Faculty of Mathematics, Physics
and Informatics, Comenius University in Bratislava, 842 48 Bratislava,
Slovakia  2 International Centre for Advanced Training and Research in
Physics, P.O. Box MG12, 077125 Măgurele, Romania  3“Horia Hulubei” National
Institute of Physics and Nuclear Engineering, 30 Reactorului, POB MG-6,
RO-077125 Bucharest-Măgurele, Romania  4Institute of Experimental and Applied
Physics, Czech Technical University in Prague, 110 00 Prague, Czech Republic

###### Abstract

In this paper, we reexamine one of the most promising candidates for
determining the neutrino mass scale – the unique first forbidden $\beta$
transition from 187Re($5/2^{+}$) to 187Os($1/2^{-}$). With the lowest-known
ground-state to ground-state $Q$-value for a $\beta$ transition at $2.4709$
keV, rhenium’s $\beta$ decay can offer insights into the neutrino mass scale
puzzle. However, understanding its electron spectrum is a complex task.
Besides involving a mixture of $s_{1/2}$-state and $p_{3/2}$-state electrons,
the rhenium $\beta$ spectrum could be strongly influenced by various atomic
corrections. In addition to our previous paper, we have incorporated finite
nuclear size, diffuse nuclear surface, screening, and exchange corrections
into the rhenium $\beta$ decay model. We have accounted for the last two
effects within the framework of the Dirac-Hartree-Fock-Slater self-consistent
method. We have discovered that both screening and exchange effects
significantly alter the partial decay rates for the $s_{1/2}$\- and
$p_{3/2}$-state emission channels, while still maintaining the experimentally
confirmed dominance of the $p_{3/2}$-state emission. The ratio between the
respective decay rates has been found to be approximately $10^{4}$. When
compared to the other corrections, the exchange effect stands out due to the
modification it induces in the spectrum shape. We demonstrate that
calculations with and without the exchange effect lead to entirely different
shape factors for the decay spectrum. Finally, we illustrate that to preserve
the linearity of the Kurie plot, it is essential to include the exchange
correction in its definition. We conclude that atomic effects, especially the
exchange effect, should be taken into account in current and future
investigations of the neutrino mass scale from $\beta$ decays.

††preprint: APS/123-QED

##  I Introduction

An open question in particle physics is related to the absolute values of the
light neutrinos masses. The information gathered from solar neutrinos,
atmospheric neutrinos, nuclear power reactors, and experiments at accelerators
has established that at least two of the three neutrinos have a non-zero mass.
However, the result is based on neutrino oscillations, which depends on
neutrino masses differences, not on their absolute values, so the absolute
neutrino mass scale is uncertain. Besides the cosmological observations, which
recently yielded an upper limit for the sum of the neutrino masses of $\sim
0.12$ eV [1], other puzzle pieces can be obtained from nuclear $\beta$ decay
and neutrinoless double $\beta$ decay [2, 3, 4].

The distortion in the endpoint measurements of the spectrum of electrons
emitted in a $\beta$ decay offers a direct means of determining the values of
neutrino masses, denoted as $m_{k}$ ($k=1,2,3$). However, the number of events
emitted near the endpoint, within an interval $\Delta T_{e}$, is proportional
to $(\Delta T_{e}/Q)^{3}$ [5, 6]. Therefore, a low $Q$-value $\beta$
transition is desirable to enhance sensitivity. Consequently, some experiments
are based on the ground-state to ground-state $\beta$ decays of tritium (3H)
and rhenium (187Re) with $Q$-values of $18.6$ keV [7] and $2.4709$ keV [8],
respectively. Other experiments, such as HOLMES [9], NuMECs [10], and ECHo
[11], aim to utilize the lowest energy electron capture of 163Ho, which has a
ground-state to ground-state $Q$-value of $2.833$ keV [12]. Recently, there
has also been a growing interest in ultra-low $Q$-value (under $1$ keV)
ground-state-to-excited-state $\beta$ transitions [13, 14, 15], which
represent potential candidates for future neutrino mass scale determination
experiments.

The current best upper limit on effective neutrino mass,

$m_{\beta}=\sqrt{\sum_{k=1}^{3}\left|U_{ek}\right|^{2}m_{k}^{2}}$ (1)

was recently fixed from the tritium $\beta$ decay, measured by the KATRIN
experiment [16], $m_{\beta}\leqslant 0.8$ eV. This limit far exceeds the
previous investigations by Troitsk experiment, $m_{\beta}\leqslant 2.2$ eV
[17], and Mainz experiment, $m_{\beta}\leqslant 2.3$ eV [18], also based on
tritium $\beta$ decay. Even more recently, Project 8 has demonstrated that
cyclotron radiation emission spectroscopy (CRES) can constrain $m_{\beta}\leq
155$ eV from the tritium $\beta$ spectrum using only a cm3-scale physical
detection medium [19]. This suggests that CRES is an attractive technique for
next-generation direct neutrino mass experiments and for measuring the $\beta$
spectra in general.

While the number of events near the endpoint is favored by the $Q$-value of
rhenium decay, which is nearly one order of magnitude smaller than that of
tritium decay, understanding its spectrum shape is a complex task. Firstly,
the emission of low-energy electrons suggests that various atomic and
molecular effects could have a significant impact on the spectrum shape of the
rhenium ground-state to ground-state $\beta$ transition. An example of this is
the oscillations observed in the rhenium $\beta$ spectrum due to environmental
fine structure from the target crystal [20]. Secondly, in contrast to the
decay of tritium, which is an allowed transition, the transition
187Re($5/2^{+}$)$\rightarrow$187Os($1/2^{-}$) is classified as a unique first
forbidden transition. Consequently, its spectrum is a mixture of
$s_{1/2}$-state and $p_{3/2}$-state electrons. The exceptionally low $Q$-value
of rhenium decay results in a clear dominance of the $p_{3/2}$-state electron
emission channel over the $s_{1/2}$-state electron emission channel. This
dominance has been experimentally confirmed [21], and our previous
investigation calculated the ratio between the channels to be around $10^{4}$
[6].

The experimental efforts to achieve sub-eV sensitivity to the neutrino mass
from the rhenium $\beta$ spectrum culminated with the combination of the MANU
and MIBETA groups into the Microcalorimeter Arrays for a Rhenium Experiment
(MARE) [22]. The MARE project eventually transitioned to the holmium efforts
of ECHo and HOLMES [23]. One possible reason for closing the project is the
lack of theoretical knowledge about the rhenium $\beta$ spectrum. In contrast,
the theoretical description of tritium $\beta$ decay involves many more
corrections, including screening, exchange with atomic electrons, finite
nuclear size effects, radiative corrections, recoil effects, etc. [24, 25].

In this paper, we investigate the $\beta$ decay of 187Re, incorporating all
relevant corrections to its spectrum. We begin with the same relativistic wave
functions for the emitted electrons as used in our previous study [6]. Next,
we enhance the precision of the theoretical rhenium decay spectrum by
including finite nuclear size, diffuse nuclear surface, and screening
corrections. The latter is calculated using the self-consistent Dirac-Hartree-
Fock-Slater description of the atomic bound electrons surrounding the electron
emitted during rhenium decay. We have observed significant differences in the
decay rates for both emission channels compared to our previous paper, but
negligible modifications to the spectral shape resulting from the
aforementioned corrections.

The primary focus of this paper lies in the inclusion of the so-called
exchange correction, which considers the possible interchange between emitted
electrons and the atomic bound electrons. It turns out that the exchange
correction not only alters the decay rates for the $p_{3/2}$\- and
$s_{1/2}$-state channels but also affects the shapes of the spectra for these
channels. While the most significant shape modification occurs in the low-
energy region of the total spectrum, the exchange correction remains
substantial near the endpoint, potentially impacting the analysis for neutrino
mass scale determination from rhenium $\beta$ decay. When studying the
deviation of the rhenium spectrum from an allowed one, we discover that the
exchange correction strongly transforms the shape factor, shifting it from an
increasing linear behavior to a decreasing quadratic one. We present the best-
fit parameters in both cases. To preserve the linearity of the Kurie plot in
the scenario of zero effective neutrino mass, it is necessary to revise its
definition to incorporate the exchange correction. We illustrate how varying
effective neutrino masses impact the Kurie plots near the endpoint of the
$\beta$ decay of 187Re. While this paper is primarily aimed on the rhenium
$\beta$ decay, we emphasize that the corrections introduced here could
potentially have a significant effect on any low and ultra-low $Q$-value
$\beta$ transition.

##  II Electron Wave functions

###  II.1 Continuum states

The relativistic wave function for an electron emitted from a $\beta$ decay in
a continuum state can be expanded in spherical waves,

$\displaystyle\psi(E_{e},\bm{r})=$
$\displaystyle\psi^{(s_{1/2})}(E_{e},\bm{r})+\psi^{(p_{1/2})}(E_{e},\bm{r})$
(2) $\displaystyle+\psi^{(p_{3/2})}(E_{e},\bm{r})+\dots,$

where $E_{e}$ is the total electron energy. The superscript in the spherical
waves represents the orbital and total angular momentum
$(\ell_{j}=0_{1/2},1_{1/2},1_{3/2},...)$ written in the spectroscopic notation
$(\ell_{j}=s_{1/2},p_{1/2},p_{3/2},...)$. The relevant spherical waves for the
first-unique forbidden $\beta$ transition of 187Re are as follows [6]:

$\displaystyle\psi^{(s_{1/2})}(E_{e},\bm{r})=\begin{pmatrix}g_{-1}(E_{e},r)\chi_{\mu}\\\
f_{+1}(E_{e},r)(\bm{\sigma}\cdot\hat{\bm{p}})\chi_{\mu}\end{pmatrix},\hskip
11.38092pt\psi^{(p_{3/2})}(E_{e},\bm{r})=i\begin{pmatrix}g_{-2}(E_{e},r)\left[3(\hat{\bm{r}}\cdot\hat{\bm{p}})-(\bm{\sigma}\cdot\hat{\bm{r}})(\bm{\sigma}\cdot\hat{\bm{p}})\right]\chi_{\mu}\\\
f_{+2}(E_{e},r)\left[(\hat{\bm{r}}\cdot\hat{\bm{p}})(\bm{\sigma}\cdot\hat{\bm{p}})-(\bm{\sigma}\cdot\hat{\bm{r}})\right]\chi_{\mu}\end{pmatrix},$
(3)

where $\hat{\bm{p}}=\bm{p_{e}}/p_{e}$ is defined by the momentum of the
electron, $p_{e}=\sqrt{E_{e}^{2}-m_{e}^{2}}$. Here, $\bm{r}$ stands for the
position vector of the electron, $r=\left|\bm{r}\right|$ and
$\hat{\bm{r}}=\bm{r}/r$. Throughout the paper, we are using the units
$\hbar=c=1$.

The functions $g_{\kappa}(E_{e},r)$ and $f_{\kappa}(E_{e},r)$ are the large-
and small-component radial wave functions, respectively, which satisfy the
following system of coupled differential equations [26]

$\displaystyle\left(\frac{d}{dr}+\frac{\kappa+1}{r}\right)g_{\kappa}-(E_{e}-V(r)+m_{e})f_{\kappa}=0,$
$\displaystyle\left(\frac{d}{dr}-\frac{\kappa-1}{r}\right)f_{\kappa}+(E_{e}-V(r)-m_{e})g_{\kappa}=0,$
(4)

where $V(r)$ is the atomic potential of the final system.

The relativistic quantum number, $\kappa$, takes positive and negative integer
values, and specifies both the total angular momentum, $j$, and the orbital
angular momentum, $\ell$, by

$\displaystyle j=\left|\kappa\right|-1/2,\hskip
5.69046pt\ell=\begin{cases}\kappa&\quad\text{if }\kappa>0,\\\
\left|\kappa\right|-1&\quad\text{if }\kappa<0.\\\ \end{cases}$ (5)

The large- and small-component radial functions satisfy, for large values of
$p_{e}r$, the following asymptotic conditions

$\displaystyle\begin{Bmatrix}g_{\kappa}(E_{e},r)\\\
f_{\kappa}(E_{e},r)\end{Bmatrix}$ (6)
$\displaystyle\sim\frac{1}{p_{e}r}\begin{Bmatrix}\sqrt{\frac{E_{e}+m_{e}}{2E_{e}}}\sin(p_{e}r-l\frac{\pi}{2}+\delta_{\kappa}-\eta\ln(2p_{e}r))\\\
\sqrt{\frac{E_{e}-m_{e}}{2E_{e}}}\cos(p_{e}r-l\frac{\pi}{2}+\delta_{\kappa}-\eta\ln(2p_{e}r))\end{Bmatrix}$

where $\delta_{\kappa}$ is the phase shift and $\eta=\alpha ZE_{e}/p_{e}$ is
the Sommerfeld parameter. Here, $Z$ is the atomic number of the final nucleus
and $\alpha$ is the fine-structure constant. The energy-dependent
normalization ensures the correct behavior of the Fermi functions (see Sec.
sec:RheniumDecay) when the electrostatic interaction is switched off, i.e.,
$V(r)=0$.

To obtain the electron continuum states in the potential of the final positive
ion, 187Os+, we employ the RADIAL subroutine package [27]. A comprehensive
manual of the package can be found in the Supplementary Material of [27]. In
what follows, we briefly present different approximation schemes for the
calculation of $g_{\kappa}(E_{e},r)$ and $f_{\kappa}(E_{e},r)$.

The approximation scheme A. We assume the final nucleus as a uniformly charged
sphere, generating the following potential,

$V(r)=\begin{cases}-\frac{\alpha Z}{r}&\quad\text{for }r\geq R,\\\
-\frac{\alpha Z}{2R}\left[3-\left(\frac{r}{R}\right)^{2}\right]&\quad\text{for
}r<R.\\\ \end{cases}$ (7)

Here, $R$ is the radius of the final nucleus, $R=r_{0}A^{1/3}$ with
$r_{0}=1.2$ fm. By keeping the lowest power of the expansion of $r$, the
radial wave functions for the $s_{1/2}$ wave and $p_{3/2}$ wave states are
given by [28]

$\begin{pmatrix}g_{-1}(E_{e},r)\\\
f_{+1}(E_{e},r)\end{pmatrix}=\begin{pmatrix}A_{-1}\\\ A_{+1}\end{pmatrix},$
(8)

and

$\begin{pmatrix}g_{-2}(E_{e},r)\\\
f_{+2}(E_{e},r)\end{pmatrix}=\frac{p_{e}r}{3}\begin{pmatrix}A_{-2}\\\
A_{+2}\end{pmatrix},$ (9)

respectively.

The normalization constant can be expressed in a good approximation as

$A_{\pm k}\simeq\sqrt{F_{k-1}(Z,E_{e})}\sqrt{\frac{E_{e}\mp m_{e}}{2E_{e}}}$
(10)

where $k=\left|\kappa\right|$ and the Fermi function $F_{k-1}(Z,E_{e})$ is
given by

$\displaystyle F_{k-1}(Z,E_{e})$ $\displaystyle=$
$\displaystyle\left[\frac{\Gamma(2k+1)}{\Gamma(k)\Gamma(2\gamma_{k}+1)}\right]^{2}(2pR)^{2(\gamma_{k}-k)}e^{\pi\eta}$
(11) $\displaystyle\times\left|\Gamma(\gamma_{k}+i\eta)\right|^{2}.$

The remaining quantity is given by

$\displaystyle\begin{aligned} \gamma_{k}&=\sqrt{k^{2}-(\alpha
Z)^{2}},\end{aligned}$ (12)

and $\Gamma(z)$ is the Gamma function. We mention that this approximation
scheme, was also used in our previous investigation [6].

The approximation scheme B. In this approximation scheme, we consider the
case, where the final nucleus generates a point-like potential, $V(r)=-\alpha
Z/r$. The radial wave functions can be expressed analytically as [29]

$\displaystyle\begin{aligned}
g_{\kappa}(E_{e},r)=\frac{\kappa}{k}\frac{1}{pr}\sqrt{\frac{E_{e}+m_{e}}{2E_{e}}}\frac{\left|\Gamma(1+\gamma_{k}+i\eta)\right|}{\Gamma(1+2\gamma_{k})}(2pr)^{\gamma_{k}}e^{\pi\eta/2}\\\
\times\imaginary\\{e^{i(pr+\zeta)}\hskip
0.28436pt_{1}F_{1}(\gamma_{k}-i\eta,1+2\gamma_{k},-2ipr)\\},\\\
f_{\kappa}(E_{e},r)=\frac{\kappa}{k}\frac{1}{pr}\sqrt{\frac{E_{e}-m_{e}}{2E_{e}}}\frac{\left|\Gamma(1+\gamma_{k}+i\eta)\right|}{\Gamma(1+2\gamma_{k})}(2pr)^{\gamma_{k}}e^{\pi\eta/2}\\\
\times\real\\{e^{i(pr+\zeta)}\hskip
0.28436pt_{1}F_{1}(\gamma_{k}-i\eta,1+2\gamma_{k},-2ipr)\\},\\\ \end{aligned}$
(13)

with

$e^{i\zeta}=\sqrt{\frac{\kappa-i\eta m_{e}/E_{e}}{\gamma_{k}-i\eta}}.$ (14)

Here, ${}_{1}F_{1}(a,b,z)$ is the confluent hypergeometric function. We
mention that the numerical solutions from the RADIAL package, with the input
$rV(r)=-\alpha Z$, are equivalent to the analytical solutions presented in Eq.
(eq:WFPointAnalytical), if we fix $r$ on the nuclear surface.

The approximation scheme C. We consider the final nucleus as a sphere filled
with protons following a Fermi distribution [30]

$\displaystyle\rho_{p}(r)=\frac{\rho_{0}}{1+e^{\left(r-c_{\text{rms}}\right)/a}},$
(15)

where we chose for the half-way radius $c_{\text{rms}}=1.07A^{1/3}=6.118$ fm
and for the surface thickness $a=0.546$ fm. $\rho_{0}$ is determined from the
normalization to $Z$.

Thus, the electrostatic interaction of an electron at $r$ with the final
nucleus is described by

$\displaystyle
V_{\text{nuc}}(r)=-\alpha\int\frac{\rho_{p}(r^{\prime})}{|\bm{r}-\bm{r^{\prime}}|}d\bm{r^{\prime}}.$
(16)

The approximation scheme D. In our final approximation scheme, we consider for
the final positive ion, 187Os+, the following electrostatic potential,

$\displaystyle
V(r)=V_{\text{nuc}}(r)+V_{\text{el}}(r)+V^{\text{Slater}}_{\text{ex}}(r),$
(17)

which is a sum of the nuclear, electronic and exchange potentials. The nuclear
potential, $V_{\text{nuc}}(r)$, is presented in the approximation scheme C,
i.e. Eq. (eq:nuclearPotential). The electronic potential, $V_{\text{el}}(r)$ ,
describes the interaction energy of an electron at $r$ with the atomic cloud,
and it is found from integrating over the volume of the atomic electron
density, $\rho(r)$,

$\displaystyle
V_{\text{el}}(r)=\alpha\int\frac{\rho(r^{\prime})}{|\bm{r}-\bm{r^{\prime}}|}d\bm{r^{\prime}}.$
(18)

Considering Slater’s approximation [31], we can write the exchange potential
in terms of the atomic electron density,

$\displaystyle
V_{\text{ex}}^{\text{Slater}}(r)=-\frac{3}{2}\alpha\left(\frac{3}{\pi}\right)^{1/3}\left[\rho(r)\right]^{1/3}.$
(19)

The atomic electronic density, $\rho(r)$, is discussed in the following
section, where we present the calculation for the electron bound states.

Our choice to include a local exchange component in the potential for the
continuum states can be arguable, but, as motivated later, it ensures the
orthogonality between the continuum states of the emitted electron and the
bound states of the atomic electrons of 187Os+. Orthogonal continuum and bound
electron wave functions for the final nucleus are crucial ingredients for
correctly calculating the exchange correction [32, 33, 34].

Figure 1: The electrostatic potential for 187Os+ as function of $r$, where the
emitted electron is located, in four different approximation schemes: (A) The
final nucleus as an uniform charged sphere. (B) A point-like final nucleus.
(C) The final nucleus as a charged sphere filled with protons following a
Fermi distribution. (D) The same as the preceding case but the atomic electron
screening is taken into account.

In Fig. fig:PotentialsFor187OsPositiveIon, we plot all quantities $rV(r)$ as
functions of $r$, for each approximation scheme considered above. In the
simplest case of a point-like nucleus (scheme B), we can see that
$rV(r)=-76\alpha$ is a straight line. In the most complex case (scheme D),
where we include the finite nuclear size, diffuse nuclear surface, and the
atomic screening corrections, for large values of $r$, $V(r)=-\alpha/r$. It is
an expected asymptotic result because the potential describes the interaction
of the emitted electron with a positive ion of charge $+1$. We can see that
when moving from a uniformly charged sphere (scheme A) to one filled with
protons following a Fermi distribution (scheme C), the difference between
potentials is slightly visible. Still, the electron wave functions A are
approximated by keeping the lowest power in the expansion in $r$. By solving
the radial Dirac equation numerically for the potentials A and C, we confirm
that there are no differences between the wave functions due to the diffuse
nuclear surface correction. Thus, any differences between the electron wave
functions A and C are associated with missing terms in the expansion.

Figure 2: The radial wave functions for an electron emitted from the $\beta$
decay of 187Re, evaluated on the nuclear surface of the final nucleus $R$, as
functions of the electron kinetic energy $E_{e}-m_{e}$. In the top panel, the
components of on electron emitted in $s_{1/2}$ state, $g_{-1}(E_{e},r)$ and
$f_{+1}(E_{e},r)$, and in the bottom panel, the components associated with the
emission in $p_{3/2}$ state, $g_{-2}(E_{e},r)$ and $f_{+2}(E_{e},r)$. We
consider four different approximation schemes in the evaluation of the radial
wave functions (see text for details).

In Fig. fig:WaveFunctions, we depict the radial wave functions of an electron
in the $s_{1/2}$ wave state, $g_{-1}(E_{e},r)$ and $f_{+1}(E_{e},r)$ (top
panel), and in the $p_{3/2}$ wave state, $g_{-2}(E_{e},r)$ and
$f_{+2}(E_{e},r)$ (bottom panel), evaluated on the nuclear surface of the
final nucleus, $r=R$, and as functions of the electron kinetic energy,
$E_{e}-m_{e}$. We present the results for the $\beta$ emitter 187Re using all
the approximation schemes. In the case of the large component radial wave
functions, $g_{-1}(E_{e},r)$ and $g_{-2}(E_{e},r)$, there are no shape
deviations between different approximation schemes. Their amplitudes
progressively decrease from scheme A to D. The missing terms in the expansion
in $r$ from scheme A strongly influence the small component radial wave
functions, $f_{+1}(E_{e},r)$ and $f_{+2}(E_{e},r)$. Instead of overlying the
wave functions C, they are almost zero and constants for any electron kinetic
energy. As a final remark, the effect of the atomic screening for an electron
emitted in the $p_{3/2}$ wave state is substantial, as can be seen in the wave
functions D. The emission of electrons in the $p_{3/2}$ wave state is dominant
for the $\beta$ decay of 187Re, so we expect a considerable deviation when the
screening correction is switched on.

###  II.2 Bound states

For atomic bound states ($E_{e}<m_{e}$), each discrete energy level is
characterized by its quantum number $\kappa$, its principle quantum number
$n$, its binding energy $\epsilon_{e}^{n\kappa}$, and its total energy
$E_{e}^{n\kappa}=m_{e}-\left|\epsilon_{e}^{n\kappa}\right|$. The relativistic
wave function for a bound electron in a spherically symmetric atomic
potential, $V(r)$, can be written as [26]

$\displaystyle\psi_{n\kappa
m}(\bm{r})=\begin{pmatrix}g_{n,\kappa}(r)\Omega_{\kappa,m}(\hat{\bm{r}})\\\
if_{n,\kappa}(r)\Omega_{-\kappa,m}(\hat{\bm{r}})\end{pmatrix},$ (20)

Here, the spherical spinors, $\Omega_{\kappa,m}(\hat{\bm{r}})$, are defined by
[35, 36]

$\displaystyle\Omega_{\kappa,m}(\hat{\bm{r}})=\sum_{\mu=\pm
1/2}\innerproduct{\ell,\frac{1}{2},m-\mu,\mu}{j,m}Y_{\ell,m-\mu}(\hat{\bm{r}})\chi_{\mu}$
(21)

where $\innerproduct{j_{1},j_{2},m_{1},m_{2}}{j,m}$ are Clebsch-Gordan
coefficient, $Y_{\ell,m}(\hat{\bm{r}})$ are the spherical harmonics, and
$\chi_{\mu}$ are the usual Pauli spinors.

The large- and small-component radial functions for bound orbitals are
referred as $g_{n,\kappa}(r)$ and $f_{n,\kappa}(r)$, respectively, and are
obeying,

$\displaystyle\left(\frac{d}{dr}+\frac{\kappa+1}{r}\right)g_{n,\kappa}-(E^{n\kappa}_{e}-V(r)+m_{e})f_{n,\kappa}=0,$
$\displaystyle\left(\frac{d}{dr}-\frac{\kappa-1}{r}\right)f_{n,\kappa}+(E^{n\kappa}_{e}-V(r)-m_{e})g_{n,\kappa}=0,$
(22)

For the calculation of the bound orbitals, we employed the program DHFS.f,
included in the RADIAL subroutine package [27]. It solves the Dirac-Hartree-
Fock-Slater (DHFS) equations for the ground-state electron configuration of
neutral atoms or positive ions with $N_{e}$ bound electrons and $Z$ protons in
the nucleus. The procedure uses almost the same potential as described in Eq.
(eq:PotentialSchemeD), but with the difference that the Latter’s tail
correction [37] is imposed for the exchange potential. The electron density,
$\rho(r)$, entering the electronic and exchange potentials, is obtained self-
consistently [38, 39]. The procedure starts with an approximate electron
density obtained from the Molière parametrization of the Thomas-Fermi
potential [40]. Then, the electron density is renewed iteratively from the
obtained bound orbitals until neither the DHFS potential nor the binding
energies change. We call the procedure from the program DHFS.f a true DFHS
self-consistent method.

Table 1: Binding energies for neutral atom 187Re in eV. In the first column, we indicate all occupied shells using the spectroscopic notation [27, 41]. In the second and third columns, we present the true DHFS self-consistent method binding energies and the results obtained with the modified DHFS self-consistent method, respectively. In the last column, we present the experimental values taken from [42].  Orbital $(n\ell_{j})$ | $\epsilon_{e}^{n\kappa}$ (true) |  $\epsilon_{e}^{n\kappa}$ (modified) | $\epsilon_{e}^{n\kappa}\textrm{(exp)}$ [42]
---|---|---|---
$1s_{1/2}$ | -71857.5 | -71857.5 | $-71681\pm 2$
$2s_{1/2}$ | -12508.4 | -12508.4 | $-12532\pm 2$
$2p_{1/2}$ | -11993.7 | -11993.7 | $-11963\pm 2$
$2p_{3/2}$ | -10537.7 | -10537.7 | $-10540\pm 2$
$3s_{1/2}$ | -2911.9 | -2911.9 | $-2937\pm 2$
$3p_{1/2}$ | -2677.7 | -2677.7 | $-2686\pm 2$
$3p_{3/2}$ | -2360.0 | -2360.0 | $-2371\pm 2$
$3d_{3/2}$ | -1961.4 | -1961.4 | $-1953\pm 2$
$3d_{5/2}$ | -1891.9 | -1891.9 | $-1887\pm 2$
$4s_{1/2}$ | -615.7 | -615.7 | $-629\pm 2$
$4p_{1/2}$ | -516.7 | -516.7 | $-522\pm 2$
$4p_{3/2}$ | -442.2 | -442.2 | $-450\pm 2$
$4d_{3/2}$ | -277.8 | -277.8 | $-278\pm 2$
$4d_{5/2}$ | -263.9 | -263.9 | $-264\pm 2$
$5s_{1/2}$ | -91.2 | -91.2 | $-86\pm 2$
$5p_{1/2}$ | -60.6 | -60.6 | $-56\pm 2$
$4f_{5/2}$ | -55.0 | -55.0 | $-47\pm 2$
$4f_{7/2}$ | -52.3 | -52.3 | $-45\pm 2$
$5p_{3/2}$ | -49.0 | -49.0 | $-45\pm 2$
$5d_{3/2}$ | -9.28 | -9.24 | $-9.6\pm 1$
$5d_{5/2}$ | -8.24 | -8.20 | $-9.6\pm 1$
$6s_{1/2}$ | -7.98 | -7.67 | $-7.9\pm 1$

Because we want to ensure orthogonality between continuum and bound states, we
take the last iterated atomic electron density, $\rho(r)$, obtained from the
program DHFS.f, and construct the potential from the approximation scheme D,
i.e., Eq. (eq:PotentialSchemeD). When solving for bound states in this
potential, we call the procedure a modified DHFS self-consistent method. More
about the differences between the true and modified DHFS self-consistent
frameworks can be found in our previous paper about the exchange correction
for allowed $\beta$ decay [34].

In Table tab:BindingEnergies, we compare the binding energies obtained with
the true and modified DHFS methods with the experimental values for the
neutral atom 187Re. We can see that the deviation from the conventional (true)
DHFS method subtly influences the binding energies of the last three occupied
orbitals.

##  III First unique forbidden
$\beta$ decay of ${}^{187}\text{Re}$

The $\beta$ transition from the ground-state $5/2^{+}$ of 187Re to the ground-
state $1/2^{-}$ of 187Os is classified as a first unique forbidden $\beta$
transition. Due to the difference in the angular momentum and parity, $\Delta
J^{\pi}=2^{-}$, the electron and neutrino are emitted, respectively, in
$p_{3/2}$ and $s_{1/2}$ states or vice versa. An interesting feature of first
unique forbidden $\beta$ decay of 187Re is that due to its low transition
energy, $Q=2.4709$ keV [8], the emission of $p_{3/2}$ state electrons is
favored with four orders of magnitude then the emission of $s_{1/2}$ state
electrons [6]. The theoretical differential decay rate is a sum of two
contributions associated with emission of the $s_{1/2}$ and the $p_{3/2}$
state electrons,

$\displaystyle\frac{d{\Gamma}}{dE_{e}}$
$\displaystyle=\frac{d{\Gamma}^{p_{3/2}}}{dE_{e}}+\frac{d{\Gamma}^{s_{1/2}}}{dE_{e}}$
(23)
$\displaystyle=\sum_{k=1}^{3}|U_{ek}|^{2}~{}\frac{G_{F}^{2}V_{ud}^{2}}{2\pi^{3}}~{}p_{e}E_{e}(E_{0}-E_{e})$
$\displaystyle\times\sqrt{(E_{0}-E_{e})^{2}-m^{2}_{k}}~{}\theta(E_{0}-E_{e}-m_{k})$
$\displaystyle\times\left[B^{{p_{3/2}}}(E_{e},p_{\nu})+B^{s_{1/2}}(E_{e},p_{\nu})~{}\right]$

where

$\displaystyle B^{s_{1/2}}=\frac{1}{2}~{}g^{2}_{A}\times$ (24)
$\displaystyle~{}\left(|\langle{\rm
f}\|\sum_{n}\tau^{+}_{n}g_{-1}(E_{e},r)~{}j_{1}(p_{\nu}r)~{}\\{\bm{\sigma}_{n}^{~{}}\otimes\bm{\hat{r}}_{n}^{~{}}\\}_{2}\|{\rm
i}\rangle|^{2}.\right.$ $\displaystyle+~{}\left.|\langle{\rm
f}\|\sum_{n}\tau^{+}_{n}f_{+1}(E_{e},r)~{}j_{1}(p_{\nu}r)~{}\\{\bm{\sigma}_{n}^{~{}}\otimes\bm{\hat{r}}_{n}^{~{}}\\}_{2}\|{\rm
i}\rangle|^{2}.\right),$

and,

$\displaystyle B^{p_{3/2}}=\frac{1}{2}~{}g^{2}_{A}\times$ (25)
$\displaystyle~{}\left(|\langle{\rm
f}\|\sum_{n}\tau^{+}_{n}g_{-2}(E_{e},r)~{}j_{0}(p_{\nu}r)~{}\\{\bm{\sigma}_{n}^{~{}}\otimes\bm{\hat{r}}_{n}^{~{}}\\}_{2}\|{\rm
i}\rangle|^{2}.\right.$ $\displaystyle+~{}\left.|\langle{\rm
f}\|\sum_{n}\tau^{+}_{n}f_{+2}(E_{e},r)~{}j_{0}(p_{\nu}r)~{}\\{\bm{\sigma}_{n}^{~{}}\otimes\bm{\hat{r}}_{n}^{~{}}\\}_{2}\|{\rm
i}\rangle|^{2}.\right).$

Here, $G_{F}$ is the Fermi constant and $V_{ud}$ is the element of the
Cabbibo-Kobayashi-Maskawa (CKM) matrix. $E_{0}$ is maximal endpoint energy (in
the case of zero neutrino mass) of the electron.
$p_{\nu}=\sqrt{(E_{0}-E_{e})^{2}-m^{2}_{k}}$ is the neutrino momentum and
$\theta(x)$ is a theta (step) function. $g_{A}$ denotes an axial-vector
coupling constant. $|i\rangle$ ($|f\rangle$) is the initial (final) state of
187Re (187Os) with $J^{\pi}=5/2^{+}$ ($J^{\pi}=1/2^{-}$).
$\mathbf{r}_{n}^{~{}}$ ($r=|\mathbf{r}|$ and $\bm{\hat{r}}=\mathbf{r}/r$) is a
coordinate of the ${\rm n}$-th nucleon.

The differential decay rate in Eq. (eq.sp1) depends on four different squared
matrix elements incorporated in the $B^{s_{1/2}}$ and $B^{p_{1/2}}$ terms. For
the purpose of factorization of calculation of squared nuclear matrix element
and phase-space factor, the large- and small-component electron radial
functions are approximated as follows [6]:

$\displaystyle{g}_{-1}(E_{e},r)$ $\displaystyle\simeq$
$\displaystyle\frac{{g}_{-1}(E_{e},R)}{j_{0}(p_{e}R)}~{}j_{0}(p_{e}r)\simeq{g}_{-1}(E_{e},R),$
$\displaystyle{f}_{+1}(E_{e},r)$ $\displaystyle\simeq$
$\displaystyle\frac{{f}_{+1}(E_{e},R)}{j_{0}(p_{e}R)}~{}j_{0}(p_{e}r)\simeq{f}_{+1}(E_{e},R),$
$\displaystyle{g}_{-2}(E_{e},r)$ $\displaystyle\simeq$
$\displaystyle\frac{{g}_{-2}(E_{e},R)}{j_{1}(p_{e}R)}~{}j_{1}(p_{e}r)\simeq\frac{r}{R}{g}_{-2}(E_{e},R),$
$\displaystyle{f}_{+2}(E_{e},r)$ $\displaystyle\simeq$
$\displaystyle\frac{{f}_{+2}(E_{e},R)}{j_{1}(p_{e}R)}~{}j_{1}(p_{e}r)\simeq\frac{r}{R}{f}_{+2}(E_{e},R),$
(26)

where $R$ is a nuclear radius. For bound states, required in the atomic
exchange correction (see Sec. sec:ExchnageCorrection), the same approximation
holds leading to $g_{n,-1}(r)\simeq g_{n,-1}(R)$, $f_{n,1}(r)\simeq
f_{n,1}(R)$, $g_{n,-2}(r)\simeq(r/R)g_{n,-2}(R)$ and
$f_{n,2}(r)\simeq(r/R)f_{n,2}(R)$. For continuum states, only the leading
terms in the expansion of spherical Bessel functions $j_{0}(p_{e}r)$ and
$j_{1}(p_{e}r)$ were considered. In this way one ends up with the energy
distribution, which depends only on a single squared matrix element [6]:

$\displaystyle\frac{d{\Gamma}}{dE_{e}}$ $\displaystyle=$
$\displaystyle\frac{d{\Gamma}^{p_{3/2}}}{dE_{e}}+\frac{d{\Gamma}^{s_{1/2}}}{dE_{e}}$
(27) $\displaystyle=$
$\displaystyle\sum_{k=1}^{3}|U_{ek}|^{2}~{}\frac{G_{F}^{2}V_{ud}^{2}}{2\pi^{3}}~{}B~{}R^{2}~{}p_{e}~{}E_{e}~{}(E_{0}-E_{e})$
$\displaystyle\times$
$\displaystyle\frac{1}{3}\left[F_{1}(Z,E_{e})p_{e}^{2}+F_{0}(Z,E_{e})((E_{0}-E_{e})^{2}-m^{2}_{k})\right]$
$\displaystyle~{}~{}\times\sqrt{(E_{0}-E_{e})^{2}-m^{2}_{k}}~{}~{}\theta(E_{0}-E_{e}-m_{k}),$

with

$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{g_{A}^{2}}{6R^{2}}|\langle{\rm
f}\|\sum_{n}\tau^{+}_{n}\\{\bm{\sigma}_{n}^{~{}}\otimes\bm{r}_{n}^{~{}}\\}_{2}\|{\rm
i}\rangle|^{2},{~{}~{}~{}~{}}$ (28)

and

$\displaystyle F_{0}(Z,E_{e})$ $\displaystyle=$
$\displaystyle\frac{{g}_{-1}(E_{e}){g}_{-1}(E_{e})+{f}_{1}(E_{e}){f}_{1}(E_{e})}{j_{0}(p_{e}R)~{}j_{0}(p_{e}R)},$
$\displaystyle F_{1}(Z,E_{e})$ $\displaystyle=$
$\displaystyle\frac{{g}_{-2}(E_{e}){g}_{-2}(E_{e})+{f}_{2}(E_{e}){f}_{2}(E_{e})}{j_{1}(p_{e}R)~{}j_{1}(p_{e}R)}.~{}$
(29)

Here, $g_{\kappa}(E_{e})\equiv g_{\kappa}(E_{e},R)$ and
$f_{\kappa}(E_{e})\equiv f_{\kappa}(E_{e},R)$. In the limit the Coulomb
interaction is switched-off $F_{0}(Z,E_{e})=1$ and $F_{1}(Z,E_{e})=1$.
Different approximation schemes, for the electron wave functions, are
indicated in the Fermi functions as $F_{k-1}^{\rm I}(Z,E_{e})$, where I = A,
B, C or D.

##  IV Exchange correction

Besides the screening correction, another prominent atomic correction,
significant for low Q-value $\beta$ transitions, is the exchange effect. It
arises from creating a $\beta$ electron in a bound orbital of the final atom
corresponding to one occupied in the initial atom. An atomic electron from the
bound orbital simultaneously makes a transition to a continuum state of the
final atom [32, 33].

We have extended the exchange effect formalism for allowed $\beta$
transitions, presented in [32, 33], for the unique first forbidden $\beta$
transitions. We found that two components of the exchange correction
independently modify the spectra associated with the emission of electrons in
the $s_{1/2}$ and $p_{3/2}$ states,

$\displaystyle\frac{d{\mit\Gamma}^{s_{1/2}}}{dE_{e}}$
$\displaystyle\Rightarrow$
$\displaystyle\frac{d{\mit\Gamma}^{s_{1/2}}}{dE_{e}}\times(1+\eta^{\text{T}}_{1}(E_{e}))$
$\displaystyle\frac{d{\mit\Gamma}^{p_{3/2}}}{dE_{e}}$
$\displaystyle\Rightarrow$
$\displaystyle\frac{d{\mit\Gamma}^{p_{3/2}}}{dE_{e}}\times(1+\eta^{\text{T}}_{2}(E_{e}))$
(30)

The result is consistent with the one obtained in [43], for the unique first
forbidden $\beta$-decay of 85Kr. The total exchange correction for each
partial spectrum is given by,

$\displaystyle\eta_{k}^{T}(E_{e})$ $\displaystyle=$ $\displaystyle
f_{k}(2T_{-k}+T_{-k}^{2})+(1-f_{k})(2T_{+k}+T_{+k}^{2})$ (31) $\displaystyle=$
$\displaystyle\eta_{-k}(E_{e})+\eta_{+k}(E_{e})$

where $k=\left|\kappa\right|$ can take the values $1$ or $2$. Here,

$\displaystyle f_{k}=\frac{g^{\prime 2}_{-k}(E_{e},R)}{g^{\prime
2}_{-k}(E_{e},R)+f^{\prime 2}_{+k}(E_{e},R)},$ (32)

and the dimensionless quantities $T_{\kappa}$ depend on the overlaps between
the bound states of the initial atom and the continuum states of the final
atom with energy $E_{e}$,

$\displaystyle
T_{\kappa}=\sum_{(n\kappa)^{\prime}}T_{n\kappa}=-\sum_{(n\kappa)^{\prime}}\frac{\innerproduct{\psi^{\prime}_{E_{e}\kappa}}{\psi_{n\kappa}}}{\innerproduct{\psi^{\prime}_{n\kappa}}{\psi_{n\kappa}}}\frac{g^{\prime}_{n,\kappa}(R)}{g^{\prime}_{\kappa}(E_{e},R)},$
(33)

for electrons in $s_{1/2}$ ($\kappa=-1$) and $p_{3/2}$ ($\kappa=-2$) states,
and

$\displaystyle
T_{\kappa}=\sum_{(n\kappa)^{\prime}}T_{n\kappa}=-\sum_{(n\kappa)^{\prime}}\frac{\innerproduct{\psi^{\prime}_{E_{e}\kappa}}{\psi_{n\kappa}}}{\innerproduct{\psi^{\prime}_{n\kappa}}{\psi_{n\kappa}}}\frac{f^{\prime}_{n,\kappa}(R)}{f^{\prime}_{\kappa}(E_{e},R)},$
(34)

for electrons in $p_{1/2}$ ($\kappa=+1$) and $d_{3/2}$ ($\kappa=+2$) states.
All primed continuum and bound states refer to the final atom. The sums are
running over all occupied orbitals of the final atom, which, in the sudden
approximation, correspond to the parent electronic configuration.

Taking into account the sums inside the $T_{\kappa}$ quantities, we can write

$\eta_{\kappa}(E_{e})=\sum_{n}\eta_{n\kappa}+f_{\left|\kappa\right|}\sum\limits_{\begin{subarray}{c}n,m\\\
n\neq m\end{subarray}}T_{n\kappa}T_{m\kappa}$ (35)

for negative values of $\kappa$, and

$\eta_{\kappa}(E_{e})=\sum_{n}\eta_{n\kappa}+(1-f_{\kappa})\sum\limits_{\begin{subarray}{c}n,m\\\
n\neq m\end{subarray}}T_{n\kappa}T_{m\kappa}$ (36)

for positive values of $\kappa$. In this way, we can define the partial
exchange correction contributions, $\eta_{n\kappa}$, given by

$\eta_{n\kappa}=f_{\left|\kappa\right|}(2T_{n\kappa}+T^{2}_{n\kappa})$ (37)

for negative values of $\kappa$, and

$\eta_{n\kappa}=(1-f_{\kappa})(2T_{n\kappa}+T^{2}_{n\kappa}),$ (38)

for positive values of $\kappa$.

Figure 3: All partial exchange corrections for the decay of 187Re, i.e., Eqs.
(eq:PartialExchnageMinusK) and (eq:PartialExchnagePlusK). The results are
presented for atomic electrons in $s_{1/2}$ ($\kappa=-1$), $p_{3/2}$
($\kappa=-2$), $p_{1/2}$ ($\kappa=+1$) and $d_{3/2}$ ($\kappa=+2$) states.
Figure 4: The total exchange correction for electrons emitted in $s_{1/2}$
wave state, $\eta^{\text{T}}_{1}(E_{e})$ (solid black line), and for electrons
emitted in $p_{3/2}$ wave state, $\eta^{\text{T}}_{2}(E_{e})$ (dashed blue
line).

The essential quantities in computing the exchange correction are the overlaps
between the continuum state electron wave function, with energy $E_{e}$, in
the final atom and the bound orbitals electron wave functions in the initial
atom, i.e., $\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi_{ns}}$. This overlap
is given explicitly by

$\displaystyle\innerproduct{\psi^{\prime}_{E_{e}\kappa}}{\psi_{n\kappa}}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}r^{2}g^{\prime}_{\kappa}(E_{e},r)g_{n,\kappa}(r)dr$
(39) $\displaystyle+$
$\displaystyle\int_{0}^{\infty}r^{2}f^{\prime}_{\kappa}(E_{e},r)f_{n,\kappa}(r)dr.$

and its numerical calculation requires good knowledge of the continuum wave
function over a wide region of space, from the nuclear center to where the
bound wave function for the initial atom ends.

In the calculation of the exchange correction, the most important condition is
to ensure that the final-state electron continuum wave function is orthogonal
to the wave functions of the final-state bound orbitals, i.e.,
$\innerproduct{\psi^{\prime}_{E_{e}\kappa}}{\psi^{\prime}_{n\kappa}}=0$. They
are eigenfunctions of the same Hamiltonian, so they must be orthogonal. A non-
zero value of
$\innerproduct{\psi^{\prime}_{E_{e}\kappa}}{\psi^{\prime}_{n\kappa}}$ may lead
to significant error in
$\innerproduct{\psi^{\prime}_{E_{e}\kappa}}{\psi_{n\kappa}}$ [32]. The
implications in the exchange correction of non-orthogonal continuum and bound
final-states wave functions were discussed in detail in [34]. Our chose of a
modified DHFS self-consistent method for bound states and of the potential
presented in Eq. (eq:PotentialSchemeD) for continuum states, imposes
automatically the orthogonality condition,
$\innerproduct{\psi^{\prime}_{E_{e}\kappa}}{\psi^{\prime}_{n\kappa}}=0$,
without any further numerically expensive orthogonalization procedures.

In the case of 187Re, there is an abundance of possible switches between the
$\beta$ emitted electrons and the bound electrons from the atomic cloud. For
the emission of $s_{1/2}$-state electrons, ten orbitals contribute to
$\eta_{1}^{\rm T}$, and for the dominant emission in $p_{3/2}$-state, seven
orbitals contribute to $\eta_{2}^{\rm T}$. In Fig.
fig:PartialExchangeContributions, we present all partial exchange corrections
for the $\beta$ decay of 187Re, i.e., Eqs. (eq:PartialExchnageMinusK) and
(eq:PartialExchnagePlusK), as functions of the kinetic energy of the emitted
electrons, i.e., $E_{e}-m_{e}$. We note that the contributions from $p_{1/2}$
(top right panel) and $d_{3/2}$ (bottom right panel) states are much smaller
than the ones from $s_{1/2}$ (top left panel) and $p_{3/2}$ (bottom left
panel) states because they are associated with the small components of the
Fermi functions $f_{1}(E_{e})$ and $f_{2}(E_{e})$, respectively (see Eq.
(eq:FermiFunctions)). They can not be neglected for a precise calculation.

The total exchange correction for $s_{1/2}$ ($p_{3/2}$) electron spectrum of
rhenium $\beta$ decay is depicted with a solid (dashed) line in Fig.
fig:TotalExchange as a function of the kinetic energy of the emitted electron.
We can see that the exchange corrections decrease with the increasing energy
of the emitted electron, starting from around $73\%$ for $s_{1/2}$ electrons
and $37\%$ for $p_{3/2}$ electrons, at $2$ eV kinetic energy. The exchange
correction remains significant even in the $Q$-value region, around $12\%$ for
$s_{1/2}$ electrons and $5.5\%$ for $p_{3/2}$ electrons. Considering the large
amplitude and the shape of the exchange correction, we expect that it will
induce considerable modifications in both the decay rate and spectrum shape of
the $\beta$-decay of rhenium. Those modifications are discussed in what
follows.

##  V Results and Discussions

In Table tab:DecayRates, we present the partial decay rates,
$\Gamma^{s_{1/2}}$ (second column) and $\Gamma^{p_{3/2}}$ (fourth column),
from which we excluded the square matrix element, $B$. The partial decay
rates, presented in MeV, have been obtained from the integration over the full
energy range of Eq. (eq.sp). In each integration, we have used different
approximation schemes (indicated in the first column) displayed in Section
sec:ApproximationSchemes. The last line corresponds to the approximation
scheme D, where we also included the atomic exchange correction, presented in
Section sec:ExchnageCorrection. In what follows, this combination is indicated
as $\rm D+ex$. We consider the approximation scheme A, used in our previous
investigation, as a reference for a percent deviation with the other
approximation schemes. The deviations are presented in the third (fifth)
column for the partial decay rate $\Gamma^{s_{1/2}}$ ($\Gamma^{p_{3/2}}$).
Using the experimental half-live, $4.33\times 10^{10}$ y [44], we also present
our prediction for the squared matrix elements in the last column of Table
tab:DecayRates.

Table 2:  The partial decay rates for 187Re, excluding the squared matrix element, associated with the emission of electrons in $s_{1/2}$ wave state (second colum) and $p_{3/2}$ wave state (forth column). The approximation scheme for the electron wave functions is indicated in the first column. In the last line, the addition of the exchange correction over the scheme D is indicated as $\rm D+ex$. In the third and fifth column, we present the decay rate percent deviation between the scheme A and the other schemes, $\delta^{s_{1/2}}=100(\Gamma_{\rm{A}}^{s_{1/2}}-\Gamma_{\rm{X}}^{s_{1/2}})/\Gamma_{\rm{X}}^{s_{1/2}}$, and $\delta^{p_{3/2}}=100(\Gamma_{\rm{A}}^{p_{3/2}}-\Gamma_{\rm{X}}^{p_{3/2}})/\Gamma_{\rm{X}}^{p_{3/2}}$, respectively, where X can be B, C, D or $\rm D+ex$. The last column presents the experimental squared matrix elements.  w. f. | $\frac{10^{41}}{B}\times\Gamma^{s_{1/2}}$ | $\delta^{s_{1/2}}$ | $\frac{10^{37}}{B}\times\Gamma^{p_{3/2}}$ | $\delta^{p_{3/2}}$ | $B\times 10^{4}$
---|---|---|---|---|---
| [MeV] | % | [MeV] | % |
A | 9.30 | - | 9.19 | - | 3.63
B | 8.33 | -10.41 | 8.92 | -2.95 | 3.74
C | 7.88 | -15.23 | 8.88 | -3.35 | 3.76
D | 7.58 | -18.48 | 6.98 | -24.02 | 4.78
D+ex | 9.46 | 1.75 | 7.92 | -13.84 | 4.22

As a general result, in all the approximation schemes, the ratios between
$p_{3/2}$-state electron emission channel and the $s_{1/2}$-state electron
emission one are always around $10^{4}$, so different corrections do not
change this particular feature of the rhenium decay. Still, in comparison with
scheme A, there are considerable differences in the decay rates due to the
screening and exchange corrections. There is a $24\%$ ($18.5\%$) decrease in
the partial decay rate $\Gamma^{p_{3/2}}$ ($\Gamma^{s_{1/2}}$) due to
screening correction alone (scheme D). If we also include the exchange
correction (scheme $\rm D+ex$), the decay rate $\Gamma^{s_{1/2}}$ returns
close to the one from scheme A (just a $1.7\%$ increase). It is not the case
of $\Gamma^{p_{3/2}}$, which in scheme $\rm D+ex$, is $13.8\%$ lower than the
value from scheme A.

Figure 5: The single electron differential decay rate normalized to the
particular decay rate ($\Gamma^{s_{1/2}}$ and $\Gamma^{p_{3/2}}$) for the
emission of $s_{1/2}$ and $p_{3/2}$ electrons as functions of the electron
kinetic energy $E_{e}-m_{e}$ for $\beta$-decay of 187Re. We indicate with D
the approximation scheme D and with ex the atomic exchange correction. The
lower portion of the figure gives the difference between the spectra with and
without the exchange correction for $s_{1/2}$ wave state electrons with a
dashed line and $p_{3/2}$ wave state electrons with a solid line. Spectra are
normalized over the full energy range.

The most striking modification due to the exchange correction is the change in
the spectrum shape of the rhenium decay. In Fig. fig:Spectra, we plot the
normalized single electron spectrum for both $s_{1/2}$\- and $p_{3/2}$-state
emissions, using the approximation schemes D and $\rm D+ex$. We mention that
there are no considerable shape differences between approximation schemes A,
B, C, and D. In the lower portion of Fig. fig:Spectra, we also present the
residuals between spectra. Although the dominant $p_{3/2}$ spectrum is not as
strongly influenced by the exchange correction as the $s_{1/2}$ spectrum, the
modification is big enough to change the shape of the total spectrum of the
decay.

We present the total spectrum of rhenium $\beta$ decay in Fig.
fig:TotalSpectra with the same convention as in Fig. fig:Spectra, but now the
electron kinetic energy takes values from $700$ keV up to the $Q$-value, and
the residuals are in percentages. Note that the spectra are normalized to
unity over the full range of the kinetic energy. One can see a different shape
of the total electron spectrum of the rhenium decay when the exchange
correction is switched on. The considerable spectrum shape modification
induced by the exchange correction indicates that this correction should be
included in the neutrino mass investigations from rhenium decay, particularly,
and other low $Q$-value $\beta$ transitions, in general.

Figure 6: The differential decay rate normalized to the total decay rate
($\Gamma$) as function of the electron kinetic energy $E_{e}-m_{e}$ for
$\beta$ decay of 187Re. We indicate with D the approximation scheme D and with
ex the atomic exchange correction. The lower portion of the figure gives the
percentage residuals between the spectrum with and without the exchange
correction. Spectra are normalized over the full energy range.

To evaluate the deviation from an allowed spectrum, we write the $\beta$
spectrum of 187Re as,

$\displaystyle\frac{d{\Gamma}}{dE_{e}}=\frac{G_{F}^{2}V_{ud}^{2}}{2\pi^{3}}Bp_{e}E_{e}F^{\rm
I}_{0}(Z,E_{e})(E_{0}-E_{e})^{2}A_{F}^{\rm I}$ (40)

where, for the moment, the neutrino masses are neglected and I = A, B, C, or
D. The shape factor $A_{F}^{\rm I}=1$ for the allowed transitions, but for
unique first forbidden,

$\displaystyle A_{F}^{\rm I}=\frac{R^{2}}{3}\left[p_{e}\frac{F^{\rm
I}_{1}(Z,E_{e})}{F^{\rm I}_{0}(Z,E_{e})}+(E_{0}-E_{e})^{2}\right],$ (41)

where we did not include the exchange with bound electrons. If we want to take
into account this effect, the shape factor becomes,

$\displaystyle A_{F}^{\rm{D+ex}}=\frac{R^{2}}{3}\biggl{\\{}$ $\displaystyle
p_{e}$ $\displaystyle\frac{F^{\rm D}_{1}(Z,E_{e})}{F^{\rm
D}_{0}(Z,E_{e})}(1+\eta^{\text{T}}_{2}(E_{e}))$ (42) $\displaystyle+$
$\displaystyle(E_{0}-E_{e})^{2}(1+\eta^{\text{T}}_{1}(E_{e}))\biggl{\\}}.$

We can see from Fig. fig:ShapeFactors, that all $A_{F}^{\rm I}$ are linear
increase with energy in the experimentally available range, from $700$ eV to
$Q$-value. We reproduce perfectly $A_{F}^{\rm I}$ if we consider the following
model fit,

$\displaystyle A_{F}^{\rm I}=a^{\rm
I}\left(1+b_{1}T_{e}+b_{2}T_{e}^{2}\right)$ (43)

with $T_{e}=E_{e}-m_{e}$ in eV. The best fit parameters are $b_{1}=1.50\times
10^{-5}$ eV-1 and $b_{2}=4.82\times 10^{-11}$ eV-2 The parameters $a^{\rm I}$
simply scale through different approximations of the Fermi functions (for
example, $a^{\rm A}=1.44\times 10^{-5}$), but they do not change the linear
increasing behavior of $A_{F}^{\rm I}$.

Figure 7: The shape factor $A_{F}^{\rm I}$, i.e. Eq. (eq:ShapeFactorABCD), for
different approximation schemes, I = A, B, C and D, and the shape factor
$A_{F}^{\rm D+ex}$, i.e. Eq. (eq:ShapeFactorExchange).

In contrast, $A_{F}^{\rm{D+ex}}$ decreases with energy, and its shape is more
complex due to the exchange correction. The best fit in the range from $700$
eV to the $Q$-value is obtained with the following assumption,

$\displaystyle
A_{F}^{\rm{D+ex}}=a^{\rm{D+ex}}\left(\frac{b_{-1}}{T_{e}}+1+b_{1}T_{e}+b_{2}T_{e}^{2}\right),$
(44)

where we found the best fit parameters $b_{-1}=19.95$ eV, $b_{1}=-6.80\times
10^{-6}$ eV-1 and $b_{2}=3.05\times 10^{-9}$ eV-2 with
$a^{\rm{D+ex}}=1.46\times 10^{-5}$. Important to notice is the sign change of
$b_{1}$ when the exchange correction is included. The choice of a fit model
with terms proportional to $T_{e}^{-1}$ is based on the experimental shape
factors that can also involve such terms [45].

The best upper limit on effective neutrino mass, $m_{\beta}\leqslant 0.8$ eV,
recently reported by the KATRIN experiment [16], holds in the degenerate
neutrino mass region, i.e., $m_{1}\simeq m_{2}\simeq m_{3}\simeq
m_{\beta}=\sum_{k=1}^{3}\left|U_{ek}\right|^{2}m_{k}$. So, we replace all the
neutrino masses $m_{k}$ ($k=1,2,3$) with the effective neutrino mass
$m_{\beta}$. For the following discussion we consider just the approximation
scheme D for the electron wave functions and we also include the atomic
exchange correction.

Figure 8: The Kurie plots in arbitrary units (a.u.) for the $\beta$ decay of
187Re with different values of the effective neutrino mass: $m_{\beta}=0.0$,
$0.2$, $0.4$, $0.6$, and $0.8$ eV. The Q-value considered is $2470.9$ eV [8].

The Kurie functions for the unique first forbidden transitions is given by

$\displaystyle K(E_{e},m_{\beta})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{d\Gamma/dE_{e}}{p_{e}E_{e}(p_{e}R)^{2}F_{1}(Z,E_{e})(1+\eta^{\text{T}}_{2}(E_{e}))}}$
(45) $\displaystyle=$ $\displaystyle
G_{F}V_{ud}\sqrt{\frac{B}{6\pi^{3}}}(E_{0}-E_{e})\sqrt[4]{1-\frac{m^{2}_{\beta}}{(E_{0}-E_{e})^{2}}}$
$\displaystyle\times$
$\displaystyle\left[1+\frac{p_{\nu}}{p_{e}}\frac{F_{0}(Z,E_{e})}{F_{1}(Z,E_{e})}\frac{(1+\eta^{\text{T}}_{1}(E_{e}))}{(1+\eta^{\text{T}}_{2}(E_{e}))}\right]^{1/2}.$

In the case of $m_{\beta}=0$, the Kurie plot, which is a plot of
$K(E_{e},m_{\beta})$ versus $E_{e}$, is ensured to be linear by keeping the
exchange correction for $p_{3/2}$ wave state electrons in the denominator of
its definition. The term that can change the linear behavior is the last term
in squared brackets, but its deviation from unity is very small. This
deviation is below $6\times 10^{-5}$ after $1000$ eV kinetic energy, and in
the region of interest from $2300$ eV to the $Q$-value, the deviation is even
lower under $10^{-6}$.

##  VI Conclusions

The distortion in the endpoint measurements of low $Q$-value $\beta$ decay
spectra is a direct tool to measure the values of neutrino masses. Current
experiments are based on the ground-state to ground-state $\beta$ transitions,
and for next-generation investigations, some suitable candidates could be the
ultra-low $Q$ value (under $1$ keV) ground-state-to-excited-state $\beta$
transitions. With the increasing interest in measuring neutrino masses from
$\beta$ decays, accurate theoretical spectra predictions should be provided.
However, their description can be challenging considering the $\beta$
transition classification and the plurality of atomic effects in the low-
energy region.

We investigate the ground-state to ground-state unique first forbidden $\beta$
transition from 187Re($5/2^{+}$) to 187Os($1/2^{-}$), considering all relevant
corrections to its spectrum and decay rate. In addition to our previous paper,
we incorporate corrections for finite nuclear size, diffuse nuclear surface,
screening, and exchange in the rhenium $\beta$ decay model. The last two
effects are calculated using the self-consistent Dirac-Hartree-Fock-Slater
description for the atomic bound electrons of the final atom. As the rhenium
$\beta$ emission involves a mixture of $s_{1/2}$-state and $p_{3/2}$-state
electrons, our exchange correction calculation accounts for all possible
contributions from exchanges with $s_{1/2}$, $p_{3/2}$, $p_{1/2}$, and
$d_{3/2}$ bound orbitals. Our findings reveal significant alterations in the
partial decay rates for both $s_{1/2}$\- and $p_{3/2}$-state emission channels
due to the screening and exchange effects while maintaining the experimentally
confirmed dominance of $p_{3/2}$-state emission.

The key feature of this paper is that, in addition to changing the partial
decay rates, the exchange correction modifies the shape of the total electron
spectrum for rhenium $\beta$ decay. By studying the deviations from an allowed
spectrum, we demonstrate that calculations with and without the exchange
effect lead to entirely different shape factors, transitioning from an
increasing linear behavior to a decreasing quadratic one. We present the best-
fit parameters for both cases. The extent of this shape modification is so
pronounced that we need to include the exchange correction in the definition
of the Kurie plot to maintain its linearity in the case of zero effective
neutrino mass. We also demonstrate how different effective neutrino masses
affect the Kurie plots near the endpoint of the $\beta$ decay of 187Re. Our
conclusion underscores the importance of considering atomic effects,
especially the exchange effect, in current and future neutrino mass scale
investigations from $\beta$ decays.

## ACKNOWLEDGMENTS

F.Š. acknowledges support by the Slovak Research and Development Agency under
Contract No. APVV-22-0413, and by the Ministry of Education, Youth and Sports
of the Czechia under the INAFYM Grant No. CZ.02.1.01/0.0/0.0/16_019/0000766.
O.N. acknowledges support support of the Romanian Ministry of Research,
Innovation, and Digitalization through Project CNCS-UEFISCDI No. 99/2021
within PN-III-P4-ID-PCE-2020-237 and the project CNCS-UEFISCDI No. TE12/2021
within PN-III-P1-1.1-TE-2021-0343.

The figures for this article have been created using the SciDraw scientific
figure preparation system [46].

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