# Exchange correction for allowed $\beta$-decay

O. Niţescu1,2,3 S. Stoica3 F. Šimkovic1,4,5<EMAIL_ADDRESS>1Faculty of Mathematics, Physics and Informatics, Comenius University in
Bratislava, 842 48 Bratislava, Slovakia  2“Horia Hulubei” National Institute
of Physics and Nuclear Engineering, 30 Reactorului, POB MG-6, RO-077125
Bucharest-Măgurele, Romania  3 International Centre for Advanced Training and
Research in Physics, P.O. Box MG12, 077125 Măgurele, Romania  4Bogoliubov
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
141980 Dubna, Russia  5Institute of Experimental and Applied Physics, Czech
Technical University in Prague, 110 00 Prague, Czech Republic

###### Abstract

We investigate the exchange effect between the final atom’s bound electrons
and those emitted in the allowed $\beta$-decay of the initial nucleus. The
electron wave functions are obtained with the Dirac-Hartree-Fock-Slater self-
consistent method, and we ensure the orthogonality between the continuum and
bound electron states, in the potential of the final atom, by modifying the
last iteration of the self-consistent method. We show that orthogonality plays
an essential role in calculating the exchange correction. After imposing the
orthogonality, we found considerable differences in magnitude and energy
dependence compared to previous results. We argue that our findings can solve
the mismatch between the previous predictions and experimental measurements in
the low-energy region of the $\beta$ spectrum. First, we calculate the
exchange effect for the low-energy $\beta$ transitions in 14C, 45Ca, 63Ni, and
241Pu, recently investigated in the literature. Next, we compute the total
exchange correction for a large number of $\beta$ emitters, with $Z$ from $1$
to $102$. From the systematic study, we found that for ultra-low energy, i.e.,
$5$ eV, the $Z$ dependence of total exchange effect is affected by $s_{1/2}$
and $p_{1/2}$ orbitals closure. We also show that the contributions from
orbitals higher than $2s_{1/2}$ orbital are essential for correctly
calculating the total effect, especially for low energies and heavy $\beta$
emitters. Finally, we provide an analytical expression of the total exchange
correction for each atomic number for easy implementation in experimental
investigations.

††preprint: APS/123-QED

##  I Introduction

The continuous development of the experimental techniques for measuring the
nuclear $\beta$ decay spectra has recently increased the interest in
calculating the atomic corrections for the process. Precise investigations of
the $\beta$ spectra are currently used for getting information on some
fundamental, still unknown, issues. For example, one can answer the most
pressing question in particle physics: What is the absolute value of the
neutrino mass? The most sensitive experiments aiming to measure the neutrino
mass are based on $\beta$ transitions with low kinetic energy released in the
decay. The Karlsruhe tritium neutrino mass (KATRIN) experiment, using the
superallowed $\beta$ decay of tritium, recently reported the best upper bound,
$m_{\beta}\leqslant 0.8$ eV [1]. Precise measurements of the $\beta$ spectra
shape could also uncover the existence of exotic currents beyond the standard
electroweak model [2, 3, 4, 5, 6, 7]. Lorentz invariance violation can also be
investigated by searching for sideral variations in the $\beta$ decay spectra
[8]. Not only that the study of $\beta$-decay has solved fundamental physics
problems, but it can still solve some of them, mainly in neutrino physic.
Therefore, predictions of the $\beta$ spectra should include all relevant
corrections to match the experimental precision [7].

The primary atomic effects on the $\beta$ spectrum arise from the screened
$\beta$ particle wave function due to the atomic electron cloud, the exchange
between the emitted and bound electrons, and the sudden change of the nuclear
charge. The latter can induce internal ionization (shake-off) or atomic
excitations (shake-up), and its effect is not expected to exceed $0.1\%$.
Contrary, the screening, and the exchange corrections can significantly
influence the $\beta$ spectrum shape and the decay rate, especially for low
energetical transitions. The exchange correction arises from creating a
$\beta$ electron in a bound orbital of the final atom corresponding to one
occupied in the initial atom. Simultaneously, an atomic electron from the
bound orbital makes a transition to a continuum orbital of the final atom.

The study of the exchange correction started in 1962 in Bahcall’s paper [9],
which was only focused on the interchange of the emitted electrons from
different $\beta$ transitions with the bound electrons from only the $1s$
orbital. The approximation led to a lower emission probability at low energy.
Twenty years later, Haxton showed that the exchange correction for tritium
$\beta$ decay enhances the process at low energy [10]. The latter conclusion
is in agreement with the recent experimental measurements. The enhancement at
low energy was also confirmed by Harston and Pyper [11] in the case of
multiple nuclear $\beta$ decays, the allowed $\beta$ transitions of 14C, 35S,
106Ru and the nonunique first forbidden $\beta$ transition of 241Pu.

Recently, the exchange correction has been revisited for the nonunique first
forbidden $\beta$ decay of 241Pu [12]. Although the wave functions for the
emitted and bound electrons were the analytical solutions of the Dirac
equation in the hydrogenic approximation given by Rose [13], it was clear that
the electron spectrum, with the exchange correction, better fits the
experimental values. The results for 241Pu were refined with more
sophisticated wave functions in [14], where the allowed $\beta$ transition of
63Ni is also presented. In a review of the analytical corrections for allowed
$\beta$ decays [7], it can be found some further investigations on the
exchange corrections for the decay of 45Ca and 241Pu. Besides the matching
between the predicted and the experimental spectrum, the exchange effect is
also important when excluding background $\beta$ events from sensitive
experimental analysis, e.g., LUX-ZEPLIN [15], XENONnT [16], XENON1T [17]. In
this regard, the atomic exchange correction was investigated for the unique
first forbidden transition of 85Kr and the nonunique first forbidden
transitions of 212Pb and 214Pb [18].

Considering the increasing interest in the field and current experimental
capabilities of measuring the low-energy region of $\beta$ spectra, for
example, with metallic magnetic calorimeters (MMCs) [19, 20, 21], we reexamine
the exchange effect calculation. For the electron wave functions, we employ a
modified Dirac-Hartree-Fock-Slater self-consistent method, which ensures the
orthogonality between the continuum and bound electron states in the potential
of the final atom. We found that the final atom’s non-orthogonal continuum and
bound states lead to undesired errors in the overlaps between the initial
atom’s bound states and the final atom’s continuum states, which translates
into a downturn in the total exchange correction. We present the calculations
in detail for the $\beta$ decays of 14C, 45Ca, 63Ni and 241Pu. Then we perform
the exchange correction calculation for a large number of $\beta$ emitters,
with an atomic number in the range $Z=1-102$. Except for the low-energy
region, the total exchange correction progressively increases with the nuclear
charge. Still, even for $Z=100$, the total exchange effect is under $1\%$ at
$200$ keV kinetic energy of the emitted electron so that the correction can be
completely neglected after this region, depending on the aimed accuracy. We
also investigate how the partial contributions, coming from orbitals higher
than $2s_{1/2}$ orbital, influence the calculation of the total effect. Those
partial contributions are important, especially for heavier atoms, and their
impact decreases with the increasing energy of the emitted electron. Finally,
we provide an analytical expression of the total exchange correction for each
atomic number for easy implementation in experimental investigations.

##  II Atomic Exchange Correction for allowed $\beta$ transitions

The electron spectrum for a nuclear $\beta$ decay is proportional with [22]

$\frac{d\Gamma}{dE_{e}}\propto
p_{e}E_{e}(E_{0}-E_{e})^{2}F_{0}(Z^{\prime},E_{e})C(E_{e}),$ (1)

where the term $p_{e}E_{e}(E_{0}-E_{e})^{2}$ is related to the statistical
phase space factor that reflects the momentum distribution between the
neutrino and the electron, $F_{0}(Z^{\prime},E_{e})$ is the so-called Fermi
function and the shape factor, $C(E_{e})$, contains the nuclear matrix
elements and all the possible remaining terms dependent on lepton energy. Here
$E_{e}$ is the total energy of the electron,
$p_{e}=\sqrt{E_{e}^{2}-m_{e}^{2}}$ is the electron momentum, $E_{0}=Q+m_{e}$
is the maximum energy of the electron and $Z^{\prime}$ is the atomic number of
the final nucleus.

The Fermi function, which encodes the electrostatic interaction between the
electron and the final atom, can be expressed in terms of the large- and
small-component radial functions, $g_{\kappa}(E_{e},r)$ and
$f_{\kappa}(E_{e},r)$ of the emitted electron,

$F_{0}(Z^{\prime},E_{e})={g}^{2}_{-1}(E_{e},R)+{f}^{2}_{+1}(E_{e},R),$ (2)

evaluated on the nuclear surface of the final nucleus. The radial components
of the continuum states and their normalization are discussed in the following
section.

A widely used approximation for the Fermi function is obtained by keeping the
lowest power of the expansion in $r$ of the radial wave functions
corresponding to a uniformly charged sphere potential [23],

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

with $k=1$ for allowed $\beta$ transitions. The remaining quantities are given
by

$\displaystyle\begin{aligned} \gamma_{k}&=\sqrt{k^{2}-(\alpha
Z^{\prime})^{2}},\\\ \eta&=\alpha Z^{\prime}\frac{E_{e}}{p_{e}},\end{aligned}$
(4)

and $\Gamma(z)$ is the Gamma function.

For allowed transitions, the shape factor is not energy dependent, so their
spectra are proportional with

$\frac{d\Gamma}{dE_{e}}\propto
p_{e}E_{e}(E_{0}-E_{e})^{2}F_{0}(Z^{\prime},E_{e}).$ (5)

The modification of the allowed spectrum due to the exchange correction is
given by the following transformation [11]

$\frac{d{\Gamma}}{dE_{e}}\Rightarrow\frac{d{\Gamma}}{dE_{e}}\times\left[1+\eta^{T}(E_{e})\right]$
(6)

where

$\displaystyle\eta^{T}(E_{e})$ $\displaystyle=$ $\displaystyle
f_{s}(2T_{s}+T_{s}^{2})+(1-f_{s})(2T_{\bar{p}}+T_{\bar{p}}^{2})$ (7)
$\displaystyle=$ $\displaystyle\eta_{s}(E_{e})+\eta_{\bar{p}}(E_{e})$

Here,

$\displaystyle f_{s}=\frac{g^{\prime 2}_{-1}(E_{e},R)}{g^{\prime
2}_{-1}(E_{e},R)+f^{\prime 2}_{+1}(E_{e},R)},$ (8)

and the quantities $T_{s}$ and $T_{\bar{p}}$ depend respectively on the
overlaps between the bound $s_{1/2}$ ($\kappa=-1$) and $\bar{p}\equiv p_{1/2}$
($\kappa=1$) orbitals wave functions in the initial state atom and the
continuum states wave functions in the final state atom,

$\displaystyle
T_{s}=\sum_{(ns)^{\prime}}T_{ns}=-\sum_{(ns)^{\prime}}\frac{\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi_{ns}}}{\innerproduct{\psi^{\prime}_{ns}}{\psi_{ns}}}\frac{g^{\prime}_{n,-1}(R)}{g^{\prime}_{-1}(E_{e},R)}$
(9)

and

$\displaystyle
T_{\bar{p}}=\sum_{(n\bar{p})^{\prime}}T_{n\bar{p}}=-\sum_{(n\bar{p})^{\prime}}\frac{\innerproduct{\psi^{\prime}_{E_{e}\bar{p}}}{\psi_{n\bar{p}}}}{\innerproduct{\psi^{\prime}_{n\bar{p}}}{\psi_{n\bar{p}}}}\frac{f^{\prime}_{n,+1}(R)}{f^{\prime}_{+1}(E_{e},R)}.$
(10)

The sums are running over all occupied orbitals of the final atom, which, in
the sudden approximation, correspond to the parent electronic configuration.
The radial components of the bound states, $g_{n,\kappa}(r)$ and
$f_{n,\kappa}(r)$, are discussed in the following section. All primed wave
functions refer to the final atom.

According to the definition of the quantities $T_{s}$ and $T_{\bar{p}}$, we
can write

$\eta_{s}(E_{e})=\sum_{n}\eta_{ns}+f_{s}\sum\limits_{\begin{subarray}{c}n,m\\\
n\neq m\end{subarray}}T_{ns}T_{ms}$ (11)

and

$\eta_{\bar{p}}(E_{e})=\sum_{n}\eta_{n\bar{p}}+(1-f_{s})\sum\limits_{\begin{subarray}{c}n,m\\\
n\neq m\end{subarray}}T_{n\bar{p}}T_{m\bar{p}}$ (12)

Here we defined the partial exchange correction of $n$th $s$ orbital and $n$th
$\bar{p}$ orbital as,

$\eta_{ns}=f_{s}(2T_{ns}+T^{2}_{ns})$ (13)

and

$\eta_{n\bar{p}}=(1-f_{s})(2T_{n\bar{p}}+T^{2}_{n\bar{p}}),$ (14)

respectively.

Although the contribution coming from the exchange with $p_{1/2}$ orbitals,
$\eta_{\bar{p}}(E_{e})$, is small, it should not be neglected if we aim for
high precision [7]. Depending on the specific application, the $p_{1/2}$
contribution was ignored [11, 12, 14] or included [7, 18]. Moreover, the mixed
sum in Eq. 11 was not taken into account in [11] for the reason that its
contribution is small in $\eta_{s}(E_{e})$. The approximation is motivated for
high energies, but its contribution can be about $1-3\%$ for kinetic energies
below $1$ keV [12].

The essential quantities in computing the exchange correction are the overlaps
between the bound orbitals electron wave functions in the initial atom and the
continuum state electron wave function, with energy $E_{e}$, in the final
atom, $\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi_{ns}}$. In numerical
calculation, we must overcome three difficulties. The first two are related to
the oscillatory nature of the continuum state wave function. First, good
knowledge of the wave function is required over a wide region of space, and
second, special care should be addressed for the integration method. The last
and the most important difficulty is to ensure that the final-state electron
continuum wave function is orthogonal to the wave functions of the final-state
bound orbitals, e.g.,
$\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi^{\prime}_{ns}}=0$. The overlap
should be zero because its wave functions are eigenfunctions of the same
Hamiltonian [11]. If the overlap integral
$\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi^{\prime}_{ns}}$ is not zero or at
least much smaller then $\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi_{ns}}$,
then the overlap $\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi_{ns}}$ may be
significantly in error [11]. The effect was also studied in different nucleon
removal reactions, $(\gamma,p)$ and $(e,e^{\prime}p)$, and significant
modifications were found in the polarization of the outgoing protons and
photon asymmetry for $(\gamma,p)$ reaction [24].

To verify the orthogonality, we define as reference the following
dimensionless quantities,

$\displaystyle
T^{\textrm{ref}}_{ns}=-\frac{\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi^{\prime}_{ns}}}{\innerproduct{\psi^{\prime}_{ns}}{\psi_{ns}}}\frac{g^{\prime}_{n,-1}(R)}{g^{\prime}_{-1}(E_{e},R)},$
(15)

and

$\displaystyle
T^{\textrm{ref}}_{n\bar{p}}=-\frac{\innerproduct{\psi^{\prime}_{E_{e}\bar{p}}}{\psi^{\prime}_{n\bar{p}}}}{\innerproduct{\psi^{\prime}_{n\bar{p}}}{\psi_{n\bar{p}}}}\frac{f^{\prime}_{n,+1}(R)}{f^{\prime}_{+1}(E_{e},R)},$
(16)

which should be zero for any energy of the emitted electron.

##  III Electron Wave functions

The electron relativistic wave function for a spherically symmetric potential,
$V(r)$, is defined as [25]

$\displaystyle\psi(E_{e},\bm{r})=\sum_{\kappa
m}\begin{pmatrix}g_{\kappa}(E_{e},r)\Omega_{\kappa,m}(\hat{\bm{r}})\\\
if_{\kappa}(E_{e},r)\Omega_{-\kappa,m}(\hat{\bm{r}})\end{pmatrix},$ (17)

where $g_{\kappa}$ and $f_{\kappa}$ are the large- and small-component radial
functions, respectively, which satisfy the following system of coupled
differential equations

$\displaystyle\left(\frac{d}{dr}+\frac{\kappa+1}{r}\right)g_{\kappa}-(E_{e}-V(r)+m_{e})f_{\kappa}$
$\displaystyle=0,$
$\displaystyle\left(\frac{d}{dr}-\frac{\kappa-1}{r}\right)f_{\kappa}+(E_{e}-V(r)-m_{e})g_{\kappa}$
$\displaystyle=0,$ (18)

where $E_{e}$ is the total electron energy. Here, $\bm{r}$ stands for the
position vector of the electron, $r=\left|\bm{r}\right|$ and
$\hat{\bm{r}}=\bm{r}/r$. The relativistic angular momentum quantum number,
$\kappa$, takes positive and negative integer values, and specifies both the
total angular momentum, $j$, and the orbital angular momentum, $\ell$, by

$\displaystyle j=\left|\kappa\right|-1/2,\hskip
5.69046pt\ell=\begin{cases}\kappa&\quad\text{if }\kappa>0,\\\
\left|\kappa\right|-1&\quad\text{if }\kappa<0.\\\ \end{cases}$ (19)

The spherical spinors, $\Omega_{\kappa,m}(\hat{\bm{r}})$, are defined by [26,
27]

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

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.

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 bound
orbitals satisfy the orthonormality relation

$\displaystyle\innerproduct{\psi_{n\kappa
m}}{\psi_{n^{\prime}\kappa^{\prime}m^{\prime}}}=\delta_{nn^{\prime}}\delta_{\kappa\kappa^{\prime}}\delta_{mm^{\prime}},$
(21)

where $\innerproduct{\bm{r}}{\psi_{n\kappa m}}=\psi_{n\kappa m}(\bm{r})$. In
terms of the large- and small-component radial wave functions, the scalar
product can be written in the following explicit form:

$\displaystyle\innerproduct{\psi_{n\kappa
m}}{\psi_{n^{\prime}\kappa^{\prime}m}}=$
$\displaystyle\int_{0}^{\infty}r^{2}\left[g_{n,\kappa}(r)g_{n^{\prime},\kappa^{\prime}}(r)\right]dr$
(22)
$\displaystyle+\int_{0}^{\infty}r^{2}\left[f_{n,\kappa}(r)f_{n^{\prime},\kappa^{\prime}}(r)\right]dr.$

For the calculation of the bound orbitals, we employed the RADIAL subroutine
package [28]. A comprehensive manual of the package can be found in the
Supplementary Material of [28]. The program DHFS.f, included in the package,
solves the Dirac-Hartree-Fock-Slater (DHFS) equations for the ground-state
configuration of neutral atoms and positive ions with $N_{e}$ bound electrons
and $Z_{p}$ protons in the nucleus. In this section, $Z_{p}$ can be either the
atomic number for the final nucleus, $Z^{\prime}$, or the atomic number for
the initial nucleus, $Z$. Although the DHFS equations are obtained by
replacing the non-local exchange potential with a local exchange approximation
[29] the results are reliable, and the procedure is efficient. There is
evidence [30] that the local exchange approximation can lead to accurate
electron binding energies without the need for the extensive numerical
calculations of the non-local exchange potential entailed by the full Hartree-
Fock approach.

The system of differential equations Eq. (III) is solved in the DHFS
potential,

$\displaystyle
V_{\text{DHFS}}(r)=V_{\text{nuc}}(r)+V_{\text{el}}(r)+V_{\text{ex}}(r)$ (23)

which is a sum of the nuclear, electronic and exchange potentials.

For the nuclear potential, $V_{\text{nuc}}(r)$, it is considered the
electrostatic interaction of an electron at distance $r$ with a spherical
nucleus filled with protons following a Fermi distribution [31]

$\displaystyle\rho_{p}(r)=\frac{\rho_{0}}{1+e^{\left(r-R_{n}\right)/z}},$ (24)

where $R_{n}=1.07A^{1/3}$ fm, $z=0.546$ fm, and $\rho_{0}$ must be determined
from normalization. Thus, the nuclear potential is

$\displaystyle
V_{\text{nuc}}(r)=-\alpha\int\frac{\rho_{p}(r^{\prime})}{|\bm{r}-\bm{r^{\prime}}|}d\bm{r^{\prime}}.$
(25)

The electronic potential describes the interaction of an electron at distance
$r$ with the atomic cloud, and it is found from integrating over the volume of
the electron density, $\rho(r)$,

$\displaystyle
V_{\text{el}}(r)=\alpha\int\frac{\rho(r^{\prime})}{|\bm{r}-\bm{r^{\prime}}|}d\bm{r^{\prime}}.$
(26)

Due to Slater’s approximation [29], the exchange potential can be expressed in
terms of the electron density in the following way

$\displaystyle
V_{\text{ex}}^{\text{Slater}}(r)=-\frac{3}{2}\alpha\left(\frac{3}{\pi}\right)^{1/3}\left[\rho(r)\right]^{1/3}.$
(27)

Figure 1:  Modified self-consistent potentials (solid curves) for the initial
neutral atom, 45Ca (top), and the final positive ion, ${}^{45}\textrm{Sc}^{+}$
(bottom). See Eq.33 and text for details. In both cases, we indicate the
nuclear potential (dashed curve), the electronic potential (dot-dashed curve)
and the Slater’s exchange potential (dotted curve).

With the exchange potential, $V_{\text{ex}}^{\text{Slater}}(r)$, the obtained
self-consistent potential does not respect the correct asymptotic behavior,

$\displaystyle\lim\limits_{r\rightarrow\infty}rV_{\text{DHFS}}(r)=-\alpha(Z_{p}-N_{e}+1),$
(28)

because the exchange term cannot cancel the self-interaction term from the
electronic potential. The drawback is solved with the introduction of the
Latter’s tail correction [32] for the exchange potential,

$\displaystyle
V_{\text{ex}}(r)=\begin{cases}V_{\text{ex}}^{\text{Slater}}(r)&r<r_{\text{Latter}},\\\
\\\ -\frac{\alpha(Z_{p}-N_{e}+1)}{r}-V_{\text{nuc}}(r)-V_{\text{el}}(r)&r\geq
r_{\text{Latter}}.\end{cases}$ (29)

The cutoff radius, $r_{\text{Latter}}$, is determined by solving the equation

$\displaystyle
V_{\text{nuc}}(r)+V_{\text{el}}(r)+V_{\text{ex}}^{\text{Slater}}(r)=-\frac{\alpha(Z_{p}-N_{e}+1)}{r}.$
(30)

The electron density, $\rho(r)$, is obtained self-consistently [33, 34]. The
procedure starts with an approximate electron density obtained from the
Molière parametrization of the Thomas-Fermi potential [35]. Then, the electron
density is renewed iteratively from the obtained bound orbitals, until neither
the DHFS potential nor the binding energies change in a specified tolerance.

Table 1: Binding energies for neutral atom 45Ca in eV. In the first column, containing the occupied shells, we indicate with $*$ the relevant shells for the calculation of the exchange correction. In the second and third column, we present the true DHFS self-consistent method binding energies and the results obtained with the modified potential, respectively. In the last column, the experimental values taken from [36], are presented.  Orbital $(n\ell_{j})$ | $\epsilon_{e}^{n\kappa}\textrm{(true)}$ | $\epsilon_{e}^{n\kappa}\textrm{(modified)}$ | $\epsilon_{e}^{n\kappa}\textrm{(exp)}$ [36]
---|---|---|---
$*1s_{1/2}$ | -4015.1 | -4015.1 | $-4041\pm 2$
$*2s_{1/2}$ | -434.1 | -434.1 | $-441\pm 2$
$*2p_{1/2}$ | -359.1 | -359.1 | $-353\pm 2$
$2p_{3/2}$ | -355.2 | -355.2 | $-349\pm 2$
$*3s_{1/2}$ | -53.2 | -53.2 | $-46\pm 2$
$*3p_{1/2}$ | -34.0 | -34.0 | $-28\pm 2$
$3p_{3/2}$ | -33.6 | -33.6 | $-28\pm 2$
$*4s_{1/2}$ | -5.45 | -5.08 | $-6.113\pm 0.01$

For continuum states ($E_{e}>m_{e}$), the wave functions are not square
integrable, so usually the normalization is done on the energy scale, i.e.

$\displaystyle\innerproduct{\psi_{E_{e}\kappa}}{\psi_{E_{e}^{\prime}\kappa}}=\delta(E_{e}-E_{e}^{\prime}).$
(31)

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

$\displaystyle\begin{Bmatrix}g_{\kappa}(E_{e},r)\\\
f_{\kappa}(E_{e},r)\end{Bmatrix}$ (32)
$\displaystyle\sim\frac{1}{p_{e}r}\begin{Bmatrix}\sqrt{\frac{E_{e}+m_{e}}{2E_{e}}}\sin(p_{e}r-l\frac{\pi}{2}+\delta_{\kappa}-\eta\ln(2p_{e}r))\\\
\sqrt{\frac{E_{e}-m_{e}}{2E_{e}}}\cos(p_{e}r-l\frac{\pi}{2}+\delta_{\kappa}-\eta\ln(2p_{e}r))\end{Bmatrix}$

where $\delta_{\kappa}$ is the phase shift. For calculating the wave functions
corresponding to continuum states, we have employed the program RADIAL.f
(included in [28]), which can offer results accurate to up to 13 or 14 decimal
figures.

The easiest way to ensure the orthogonality between continuum and bound
states, e.g., $\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi^{\prime}_{ns}}=0$,
is to use the same potential in the calculation of the continuum and bound
wave functions for the final positive ion atom. The procedure we used is that
when obtaining the bound states, in the last iteration of the DHFS self-
consistent method, we do not impose the Latter’s tail correction for the
exchange potential, Eq. (29). In this way the potential for both continuum and
bound states is written as,

$\displaystyle
V(r)=V_{\text{nuc}}(r)+V_{\text{el}}(r)+V_{\text{ex}}^{\text{Slater}}(r),$
(33)

where the electronic and exchange components are obtained from the last
iterated electron density. In this way, we respect the correct asymptotic
condition for a scattering potential,
$\lim\limits_{r\rightarrow\infty}rV(r)=-\alpha(Z_{p}-N_{e})$. In Fig. 1, we
present the modified potentials for the initial neutral atom 45Ca, and the
final positive ion ${}^{45}\textrm{Sc}^{+}$. In what follows, when we use the
potential from Eq. 33, for both continuum and bound states, we call the
procedure a modified DHFS self-consistent method. Contrary, when the Latter’s
tail correction is imposed on the exchange potential, we call the procedure a
true DHFS self-consistent method.

We indeed deviate from the true DHFS self-consistent method, but it turns out
that the modifications in the binding energies and the bound wave functions
are negligible. In Table 1, we present the binding energies of the occupied
states for the neutral atom 45Ca. The difference between the true self-
consistent DHFS potential and the modified potential leads to a negligible
difference just in the binding energy of the last occupied orbital,
$4s_{1/2}$.

##  IV Results and Discussion

###  IV.1 Exchange effect for 14C, 45Ca, 63Ni and 241Pu
$\beta$ decay

Table 2: The relevant nuclear data for the isotopes considered in this paper. For each ground state to ground state $\beta$ transition we present the endpoint (second column) and the initial and final spin-parity states, $J_{i}^{\pi}$ and $J_{f}^{\pi}$ (third column). The Q-values of each $\beta$-decay are from Ref. [37] Isotope |  $Q$-value (keV) | $J_{i}^{\pi},J_{f}^{\pi}$
---|---|---
10C | 156.476(4) | $0^{+},1^{+}$
45Ca | 259.7(7) | $7/2^{-},7/2^{-}$
63Ni | 66.977(15) | $1/2^{-},3/2^{-}$
241Pu | 20.78 (17) | $5/2^{+},5/2^{-}$
Figure 2: The dimensionless quantities $T_{ns}$ (solid black) and $T^{\rm
ref}_{ns}$ (dashed blue) defined in Eq. 9 and 15, respectively, necessary to
perform the exchange correction calculation for the $\beta$-decay of 45Ca. The
results are presented for $s_{1/2}$ orbitals with $n=1~{}(a)$, $n=2~{}(b)$,
$n=3~{}(c)$, and $n=4~{}(d)$. The bound electron wave functions for both
initial and final atoms are computed with the true DHFS method, and the
continuum states of the final nucleus just with the electric and nuclear
components of the potential.

In order to investigate the steps of the exchange correction calculation, we
selected four low Q-value $\beta$ transitions that were recently investigated,
i.e., 14C [38, 39], 45Ca [7], 63Ni [14] and 241Pu [12, 14]. The relevant
nuclear data for the considered isotopes are presented in Table 2. From the
difference in the nuclear spin-parity states, the first three $\beta$
transitions are classified as allowed transitions and the last one as
nonunique first forbidden, but it can be treated as allowed in the $\xi$
approximation [40]. We perform the calculation of the exchange effect from
$200$ eV up to the Q-value kinetic energy of the continuum electron for each
transition (second column of Table 2).

To investigate the orthogonality effect on the exchange correction, we
consider two approaches for computing the electron wave functions. In the
first one, the bound wave functions for both the initial neutral atom and the
final positive ion are obtained from a true DHFS self-consistent method. The
continuum states for the emitted electron are obtained by solving Eq. III with
the nuclear and electronic potential of the final positive ion, i.e.,
$V(r)=V_{\text{nuc}}(r)+V_{\text{el}}(r)$. In the second approach, we consider
the modified DHFS self-consistent method discussed in the previous section.
For the bound and continuum electron states of the final positive ion, we use
the same potential presented in Eq. 33 and depicted in the bottom part of Fig.
1.

Figure 3: The dimensionless quantities $T_{ns}$ (solid black) and $T^{\rm
ref}_{ns}$ (dashed blue) defined in Eq. 9 and 15, respectively, necessary to
perform the exchange correction calculation for the $\beta$-decay of 45Ca. The
results are presented for $s_{1/2}$ orbitals with $n=1~{}(a)$, $n=2~{}(b)$,
$n=3~{}(c)$, and $n=4~{}(d)$. The bound and continuum electron wave functions
are computed with the modified DHFS self-consistent method. The zero value of
the quantity $T^{\rm ref}_{ns}$ indicates a perfect orthogonality between the
continuum states of the emitted electron and the bound states of the atomic
electrons of the final nucleus.  Figure 4: The total exchange correction and
the partial contributions from all occupied $s_{1/2}$ orbitals as functions of
the kinetic energy of the electron emitted in the $\beta$-decay of 45Ca. The
top figure is obtained with non-orthogonal continuum and bound states of the
final atom (see text). In the bottom part the orthogonality is ensured by the
modified DHFS self-consistent method.

The non-orthogonal bound and continuum states from the first approach lead to
non-zero values of the dimensionless quantities $T^{\rm ref}_{ns}$, as can be
seen in Fig. 2, where we present all the contributions from the occupied
$s_{1/2}$ orbitals in the case of 45Ca $\beta$-decay. In the second approach,
where the orthogonality is imposed as descried above, one can see in Fig. 3,
that $T^{\rm ref}_{ns}$ are constant and zero for any energy of the continuum
state. Thus, the dimensionless quantities $T_{ns}$, entering the total
exchange effect calculation, are strongly influenced by whether the
orthogonality is imposed or not. We can say that as long as the overlaps
$\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi^{\prime}_{ns}}$ are not zero then
the overlaps $\innerproduct{\psi^{\prime}_{E_{e}s}}{\psi_{ns}}$, and
implicitly the quantities $T_{ns}$, are in error.

For the $\beta$-decay of 45Ca, we display in Fig. 4 the total exchange effect,
$\eta^{T}$, and the partial contributions, $\eta_{ns}$. The result with non-
orthogonal states is presented in the top panel while the one with orthogonal
states in the bottom panel. Not only that the exchange effect presents a
completely different energy dependence, but the orthogonality of the wave
functions strongly influences its magnitude. We mention that the differences
in magnitude and energy dependence were pointed out also in [41], but a
concrete explanation was not provided. In the first approach, the exchange
effect has a maximum of $8\%$ at around $1$ keV, decreasing rapidly for lower
energies of the continuum states. A similar exchange correction was obtained
in [7] (see Fig.6, pg. 30), for $\beta$-decay of 45Ca. For orthogonal states,
the exchange effect increases rapidly with decreasing energy of the continuum
state and is about $35\%$ at $200$ eV. The downturn in the case of non-
orthogonal states is associated with the behavior of the quantities $T_{ns}$,
which are in error. From our investigation, the downturn can even change the
sign of the exchange effect if the energy of the continuum state is low
enough, which translates into a suppression of events in the low energy region
of the $\beta$ spectrum. Therefore, the calculation of the exchange correction
for ultra-low Q-value $\beta$ transitions should address the orthogonalization
between the continuum and bound wave functions for the final atom. We mention
that in the calculation of total exchange effect for 45Ca, we also included
the contributions from the exchange with $p_{1/2}$ orbitals. Still, they are
three orders of magnitude smaller than the $s_{1/2}$ contributions and can not
be plotted in Fig. 4. In the case of non-orthogonal states, the $p_{1/2}$
contributions also present a downturn, and they contribute to the downturn of
the total exchange effect.

Figure 5: The total exchange correction, $\eta^{T}$, and each partial
contributions, $\eta_{ns}$, coming from the exchange with $s_{1/2}$ occupied
orbitals as functions of the kinetic energy of the emitted electron,
$E_{e}-m_{e}$. The results are presented for the $\beta$ decay of 14C, 45Ca,
63Ni and 241Pu. The total exchange correction also includes the partial
contributions coming from the exchange with the $p_{1/2}$ occupied orbitals,
which are too small to be included in the plot.  Figure 6: The normalized
electron spectra for the $\beta$ transitions of 14C and 45Ca, with the
exchange effect included (solid black line) and without the exchange
correction (dashed blue line). All spectra are normalized to unity over the
full energy range. In the bottom panel of each $\beta$ emitter, we present the
ratio between normalized spectrum with exchange correction and the normalized
spectrum without the exchange correction.  Figure 7: Same as Fig. 6, but for
the $\beta$ transitions of 63Ni and 241Pu.  Figure 8: The normalized
antineutrino spectra corresponding to the $\beta$ transitions of 14C, 45Ca,
63Ni and 241Pu, with the exchange effect included (solid black line) and
without the exchange correction (dashed blue line).

In the following, we present results for the ground state to ground state
$\beta$ transitions of 14C, 45Ca, 63Ni and 241Pu, using the modified DHFS
self-consistent method and the orthogonality between the continuum and bound
states. In Fig. 5, the total exchange correction is presented for each
transition, along with the contributions from the exchange with all occupied
$s_{1/2}$ orbitals. The $p_{1/2}$ contributions are also included in the
calculation, and they have similar behavior to the $s_{1/2}$ contributions,
but are three orders of magnitude smaller than those ones. At the lowest
energy chosen in this subsection for the continuum state, i.e. $200$ eV, the
exchange corrections are about $37\%$, $35\%$, $32\%$ and $27\%$ for the
$\beta$ decay of 14C, 45Ca, 63Ni and 241Pu, respectively. At first glance, the
magnitude of the exchange effect decreases with the increasing nuclear charge,
at least at $200$ eV. We will see in the following subsection that the
magnitude of the exchange correction has a more complicated dependence on the
atomic number as a result of a complex combined effect of the spatial
extension of the atomic potential, the spatial extension of bound wave
functions for different $n$ and the closure of $s_{1/2}$ and $p_{1/2}$
orbitals. From this study of one light, two light-medium, and one heavy
nucleus, we can also see that the heavier the nucleus is, the higher energies
the exchange correction affects. We will come back to this in the following
subsection.

We present in Fig. 6 and Fig. 7 the normalized electron spectra corresponding
to the decay of the four isotopes considered. For each transition, we display
the spectrum with and without exchange correction with a solid and dashed
line, respectively. Because of our choice of atomic potential, we mention that
the spectra without the exchange effect already include the finite nuclear
size, diffuse nuclear surface, and atomic screening corrections. The continuum
wave functions for the emitted electrons, entering the Fermi functions from
Eq. 2, encode those corrections. We did not include in the calculations the
radiative corrections, which are non-static Coulomb corrections. They are
negligible for low $Q$-value $\beta$ transitions [14, 42]. The exchange effect
dramatically increases the number of events in the low-energy region of the
spectra and modifies its overall shape. In the case of non-orthogonal bound
and continuum states for the final atom, the spectrum with the exchange effect
presents a downturn, leading to a mismatch between the predicted spectrum and
the measured one [14, 39, 21]. A comprehensive investigation of the residuals
between the measured spectra and our result will be detailed elsewhere.

For completeness, we also show the corresponding antineutrino spectra of each
transition in Fig. 8. The antineutrino spectrum is obtained by the
replacement, $E_{e}\rightarrow E_{0}-E_{e}$, in the electron spectrum from Eq.
1. The inversion is symmetric, and due to the exchange effect, we obtain the
same spectrum shape modifications for antineutrino as for electrons. In the
case of larger Q-value $\beta$ transitions, for which the radiative correction
can not be neglected, the inversion is no longer symmetric because of the
different expressions of the radiative corrections for electrons and
antineutrinos. Important to notice is the fact that the exchange correction
affects the endpoint regions of the antineutrino spectra. We leave the
discussion open on how the exchange effect alters the cumulative antineutrino
spectrum of a nuclear reactor. Further investigations are in progress in this
direction.

###  IV.2 Analytical parametrization

In order to provide an estimation of the exchange effects on a $\beta$-decay
spectrum in general, we have performed the exchange correction calculations
for a wide range of atomic numbers for the initial nucleus, from 1 to 102. The
electrons emitted in the $\beta$ decay process were considered with kinetic
energy between $50$ eV and $200$ keV.

Figure 9: The energy dependence of the total exchange correction for $\beta$
decay of different initial nuclei with atomic numbers $Z=20,30,50,70,100$. The
quantities are computed using Eq. 7, which includes exchange with all occupied
$s_{1/2}$ and $p_{1/2}$ orbitals.

We depict in Fig. 9 the energy dependence of the total exchange correction for
the $\beta$ transitions of different initial isotopes with
$Z=20,30,50,70,100$. We included the contributions coming from the exchange
with all occupied $s_{1/2}$ and $p_{1/2}$ orbitals. The decrease of the
exchange effect with the increasing energy of the emitted electron is not a
new or surprising result [11]. Still, in comparison with the unscreened
hydrogenic approximation used in the previous investigation [11], the DHFS
self-consistent method employed here increases the overall magnitude of the
exchange correction in the very low energy region. Another contribution to
this difference is the fact that in [11], the calculations are performed
including just the exchange with occupied $s_{1/2}$ orbitals up to and
including the $3s_{1/2}$ orbital and completely neglecting the $p_{1/2}$
orbitals. Except for the low-energy region, the total exchange correction
progressively increases with the nuclear charge. Still, even for $Z=100$, the
total exchange effect is under $1\%$ at $200$ keV kinetic energy of the
emitted electron. Its calculation can be neglected after this threshold,
depending on the aimed accuracy.

We present in Fig. 10, the exchange correction as a function of the atomic
number of the $\beta$ emitter, from $Z=1$ to $Z=102$. The dependence is
presented at four different kinetic energies of the $\beta$ electron, i.e.,
$50$ eV, $3$ keV, $10.5$ keV and $62$ keV. We depict the total exchange
correction, $\eta^{T}$, with filled black circles and the partial
contributions, $\eta_{1s}$ and $\eta_{2s}$, with filled orange squares and
filled blue triangles, respectively. The empty black circles represent the sum
of the partial contributions coming from the exchange with $1s_{1/2}$ and
$2s_{1/2}$ orbitals, i.e., $\eta_{1s}$+$\eta_{2s}$. In this way, we can
understand the importance of the contributions from the remaining occupied
$s_{1/2}$ and $p_{1/2}$ orbitals.

At $5$ eV energy, the total exchange effect presents discontinuities. We can
understand this behavior from the expression of $\eta_{s}$ in Eq. 11. With the
increasing atomic number, new $s_{1/2}$ orbitals are filled with electrons,
and new partial contributions enter in-game. Those contributions added after a
shell closure are not zero in the low-energy region, which can explain the
discontinuities in the total exchange effect. We indicate with thin dashed
lines the atomic numbers where $s_{1/2}$ orbitals are filled with bound
electrons, and we can see a correlation between the shell closure and the
jumps in the total exchange correction. The $p_{1/2}$ orbitals closure also
contributes to this behavior but subtly. With increasing energy, the jumps
transform into a smooth dependence. Even for $3$ keV, the partial
contributions start from zero, and the total exchange effect is a smooth
function of the atomic number. It remains smooth with the increasing energy of
the emitted electron.

Regarding the partial contributions, $\eta_{1s}$ and $\eta_{2s}$, we point out
a completely different $Z$ dependence than the one presented in [7] at $3$
keV. We associate the differences in Z dependence and the magnitude with the
fact that the bound and continuum states of the final atom are orthogonal in
our study. In our case, at $3$ keV, $\eta_{2s}$ presents a smooth dependence
on the atomic number, but in [7] it jumps even on negative values for certain
nuclear charges, leading to the downturn of the total exchange effect. From
the sum of $\eta_{1s}$ and $\eta_{2s}$, we can see that, for really low
energies, the higher partial contributions are essential for the calculation
of the total exchange correction. Their importance decreases with the energy
increasing for the continuum state.

The complex combined effect of the spatial extension of the atomic potential,
the spatial extension of bound wave functions for different $n$ and the
$s_{1/2}$ and $p_{1/2}$ shell closure makes the analytical parametrization of
the total exchange correction difficult over the full $E_{e}-m_{e}$ and $Z$
range. We propose the following analytical fit for total exchange correction,

$\eta^{T}(x)=\left(a+bx^{c}\right)\exp(-dx^{e}),$ (34)

as a function of the kinetic energy of the emitted electron, i.e.,
$x=E_{e}-m_{e}$ in keV. The jumps in the $Z$ dependence for low energies
forced us to tabulate the required fit parameters for each $Z$ individually.
The five fit parameters are given in Table LABEL:tab:fitParameters
(Supplemental Material). Because of the orthogonality imposed in our study,
the fit is more relaxed than the one from [7], which involves nine fit
parameters. The discrepancy between the analytical fit models arises because
the downturn in the exchange correction from [7] can not be approximated with
a simple model. In our case, the difference between the fitted and calculated
exchange effect never exceeded $10^{-3}$ over the full energy range tested,
from $5$ eV to $200$ keV. Over the full $E_{e}-m_{e}$ and $Z$ range, the
average of residuals is smaller than $10^{-4}$, and the fit is comparable in
accuracy with the one from [7]. We mention that our fit is designed for
extrapolation above $200$ keV kinetic energy of the emitted electron.

Table 3: (Supplemental Material) The fit parameters for the total exchange correction are tabulated individually for each atomic number of the initial nucleus. The dimension of each parameter is found from the fit equation, Eq.34, with $x=E_{e}-m_{e}$ in keV.  $Z$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---
1 | 40.878 | 41.395 | 1.0627 | 9.6525 | 0.2310
2 | 11.440 | 13.268 | 1.2686 | 7.3986 | 0.2962
3 | 5.5124 | 0.1003 | 0.2022 | 5.2566 | 0.3183
4 | 2.8551 | 0.4427 | 1.7697 | 4.1501 | 0.3819
5 | 2.4633 | 0.1412 | 1.7544 | 3.5635 | 0.3688
6 | 2.1398 | 0.0317 | 1.9398 | 3.1399 | 0.3583
7 | 1.9035 | 0.0043 | 2.2619 | 2.8400 | 0.3482
8 | 1.8035 | -0.1029 | 0.6761 | 2.6039 | 0.3244
9 | 4.4452 | 29.653 | 0.6901 | 5.5130 | 0.2581
10 | 5.1785 | 39.584 | 0.7405 | 5.7240 | 0.2470
11 | 4.3112 | 19.394 | 0.6981 | 5.0458 | 0.2539
12 | 3.8227 | 14.001 | 0.6797 | 4.7287 | 0.2549
13 | 3.8809 | 13.420 | 0.6866 | 4.6739 | 0.2506
14 | 3.4108 | 9.6263 | 0.6827 | 4.3715 | 0.2550
15 | 2.8742 | 6.3302 | 0.6779 | 4.0086 | 0.2626
16 | 2.4395 | 4.2232 | 0.6750 | 3.6740 | 0.2707
17 | 2.1242 | 2.9843 | 0.6725 | 3.3994 | 0.2773
18 | 1.8963 | 2.2428 | 0.6691 | 3.1818 | 0.2821
19 | 2.5709 | 4.0673 | 0.7067 | 3.6467 | 0.2615
20 | 2.1091 | 2.4885 | 0.7326 | 3.2774 | 0.2764
21 | 1.8382 | 1.8290 | 0.7168 | 3.0469 | 0.2815
22 | 1.6569 | 1.4564 | 0.7006 | 2.8800 | 0.2842
23 | 1.5267 | 1.2281 | 0.6830 | 2.7557 | 0.2850
24 | 1.2780 | 0.8165 | 0.6507 | 2.4783 | 0.2927
25 | 1.3602 | 1.0057 | 0.6422 | 2.6026 | 0.2823
26 | 1.3086 | 0.9690 | 0.6203 | 2.5654 | 0.2792
27 | 1.2748 | 0.9753 | 0.5974 | 2.5548 | 0.2748
28 | 1.2568 | 1.0258 | 0.5753 | 2.5714 | 0.2693
29 | 1.1136 | 0.8685 | 0.5282 | 2.4317 | 0.2695
30 | 1.2723 | 1.2967 | 0.5380 | 2.6960 | 0.2552
31 | 1.5426 | 2.0748 | 0.5547 | 3.0430 | 0.2393
32 | 1.7034 | 2.5015 | 0.5692 | 3.1979 | 0.2324
33 | 1.7962 | 2.6806 | 0.5814 | 3.2649 | 0.2292
34 | 1.7945 | 2.5440 | 0.5913 | 3.2375 | 0.2295
35 | 1.7239 | 2.2223 | 0.5996 | 3.1466 | 0.2323
36 | 1.6218 | 1.8588 | 0.6071 | 3.0246 | 0.2364
37 | 3.2275 | 6.6088 | 0.6445 | 4.0640 | 0.2018
38 | 2.5377 | 3.9319 | 0.6563 | 3.6498 | 0.2150
39 | 2.0575 | 2.5016 | 0.6589 | 3.3026 | 0.2263
40 | 1.7775 | 1.7946 | 0.6630 | 3.0608 | 0.2347
41 | 1.5791 | 1.3588 | 0.6678 | 2.8673 | 0.2420
42 | 1.4287 | 1.0662 | 0.6727 | 2.7055 | 0.2482
43 | 1.2091 | 0.7074 | 0.6834 | 2.4424 | 0.2605
44 | 1.1287 | 0.5908 | 0.6884 | 2.3352 | 0.2651
45 | 1.0633 | 0.5044 | 0.6915 | 2.2439 | 0.2689
46 | 1.0102 | 0.4394 | 0.6921 | 2.1667 | 0.2717
47 | 0.9644 | 0.3887 | 0.6921 | 2.0988 | 0.2741
48 | 0.9834 | 0.4124 | 0.6813 | 2.1319 | 0.2699
49 | 1.0483 | 0.4948 | 0.6587 | 2.2341 | 0.2607
50 | 1.0811 | 0.5367 | 0.6463 | 2.2829 | 0.2555
51 | 1.1207 | 0.5906 | 0.6343 | 2.3406 | 0.2502
52 | 1.1568 | 0.6395 | 0.6250 | 2.3902 | 0.2457
53 | 1.1788 | 0.6679 | 0.6207 | 2.4189 | 0.2429
54 | 1.1859 | 0.6723 | 0.6192 | 2.4260 | 0.2415
55 | 2.6675 | 3.8795 | 0.5667 | 3.6830 | 0.1882
56 | 2.4590 | 3.1784 | 0.5810 | 3.5353 | 0.1935
57 | 2.0825 | 2.2149 | 0.5909 | 3.2648 | 0.2026
58 | 1.9646 | 2.0665 | 0.5747 | 3.2011 | 0.2025
59 | 1.8991 | 2.0272 | 0.5593 | 3.1752 | 0.2013
60 | 5.7069 | -4.4807 | 0.0445 | 2.0158 | 0.0958
61 | 4.7160 | -3.6434 | 0.0472 | 1.8830 | 0.1018
62 | 1.9100 | 2.4129 | 0.5209 | 3.2733 | 0.1927
63 | 1.9756 | 2.7329 | 0.5116 | 3.3596 | 0.1886
64 | 2.0675 | 3.1615 | 0.5048 | 3.4651 | 0.1843
65 | 2.1927 | 3.7411 | 0.5007 | 3.5924 | 0.1797
66 | 2.3524 | 4.4939 | 0.4989 | 3.7361 | 0.1751
67 | 2.5481 | 5.4568 | 0.4998 | 3.8932 | 0.1706
68 | 2.7706 | 6.6084 | 0.5026 | 4.0523 | 0.1665
69 | 3.0281 | 8.0113 | 0.5070 | 4.2161 | 0.1626
70 | 3.5968 | 12.385 | 0.5171 | 4.5873 | 0.1551
71 | 3.3291 | 9.9229 | 0.5153 | 4.4002 | 0.1584
72 | 2.9497 | 7.5192 | 0.5117 | 4.1643 | 0.1631
73 | 2.5073 | 5.3125 | 0.5071 | 3.8723 | 0.1695
74 | 2.1592 | 3.8373 | 0.5040 | 3.6063 | 0.1760
75 | 1.9041 | 2.8882 | 0.5026 | 3.3813 | 0.1819
76 | 2.1691 | 3.8474 | 0.5010 | 3.6090 | 0.1746
77 | 0.0297 | 0.2031 | -0.2002 | 0.3543 | 0.3930
78 | 1.2728 | 1.1235 | 0.5013 | 2.6830 | 0.2041
79 | -0.1392 | 0.3445 | -0.1388 | 0.2235 | 0.3614
80 | -0.1217 | 0.3229 | -0.1476 | 0.2014 | 0.3780
81 | -0.0768 | 0.2711 | -0.1728 | 0.1647 | 0.4341
82 | -0.0763 | 0.2694 | -0.1750 | 0.1562 | 0.4366
83 | -0.0702 | 0.2624 | -0.1798 | 0.1488 | 0.4456
84 | -0.0712 | 0.2625 | -0.1809 | 0.1414 | 0.4469
85 | -0.0732 | 0.2638 | -0.1811 | 0.1359 | 0.4454
86 | 1.4297 | 1.2203 | 0.5313 | 2.7644 | 0.2004
87 | 52.470 | -39.955 | 0.0481 | 4.3172 | 0.0384
88 | 3.4173 | 5.9999 | 0.5508 | 4.0237 | 0.1635
89 | 0.1538 | 0.1231 | -0.3077 | 0.4941 | 0.3504
90 | 0.0872 | 0.1567 | -0.2640 | 0.3621 | 0.3864
91 | 0.0184 | 0.1954 | -0.2298 | 0.2262 | 0.4372
92 | -0.0480 | 0.2417 | -0.2002 | 0.1214 | 0.4769
93 | -0.0719 | 0.2662 | -0.1847 | 0.1198 | 0.4457
94 | 5.5119 | -4.1559 | 0.0482 | 2.0596 | 0.0783
95 | 4.6253 | -3.5213 | 0.0463 | 1.8487 | 0.0863
96 | -0.1264 | 0.3340 | -0.1500 | 0.1698 | 0.3550
97 | 3.3772 | -2.4923 | 0.0509 | 1.6164 | 0.0989
98 | 3.0700 | -2.2689 | 0.0504 | 1.5107 | 0.1049
99 | 2.7323 | -1.9692 | 0.0543 | 1.4570 | 0.1093
100 | 1.2245 | 0.8824 | 0.4761 | 2.4648 | 0.2000
101 | 1.2193 | 0.8939 | 0.4675 | 2.4613 | 0.1988
102 | 1.2219 | 0.9252 | 0.4585 | 2.4717 | 0.1971
Figure 10: The exchange correction as function of the atomic number of the
$\beta$ emitter, from $Z=1$ to $Z=102$. The dependence is presented at four
different kinetic energies of the $\beta$ electron. We depict the total
exchange correction, $\eta^{T}$, with filled black circles and the partial
contributions, $\eta_{1s}$ and $\eta_{2s}$, with filled orange squares and
filled blue triangles, respectively. The empty black circles represent the sum
of the partial contributions coming from the exchange with $1s_{1/2}$ and
$2s_{1/2}$, i.e., $\eta_{1s}$+$\eta_{2s}$. We also indicate with thin dashed
lines the atomic numbers where the $s_{1/2}$ orbitals are fully filled with
bound electrons.

##  V Conclusions

The $\beta$-spectrum shape investigation is a powerful tool to answer and
further understand open questions related to physics beyond the Standard Model
or neutrino physics. With the increasing experimental capabilities of
measuring the low-energy region, i.e., the first few keV, and the increasing
statistics, an accurate theoretical prediction for the $\beta$-spectrum should
be provided. Still, its description is a challenging task considering the
multitude of effects that are important in the low-energy region. For allowed
transitions, the primary atomic effects arise from the screened wave function
of the emitted electron due to the atomic electron cloud and the exchange
between continuum and bound electrons. The latter is the so-called exchange
correction.

We investigate the exchange effect for the allowed $\beta$ transitions
including all contributions from the occupied $s_{1/2}$ and $p_{1/2}$
orbitals. For the electron wave functions, we employ the Dirac-Hartree-Fock-
Slater self-consistent method. We modified the last iteration of the self-
consistent method to ensure the orthogonality between the continuum and bound
electron states in the potential of the final atom. We found that
orthogonality is an essential ingredient in calculating the exchange
correction. Non-orthogonal continuum and bound states of the final atom lead
to undesired errors in the overlaps between the initial atom’s bound states
and the final atom’s continuum states which translates into a downturn in the
total exchange correction. After including the orthogonality in the
calculation of the exchange effect, we found considerable differences in
magnitude and energy dependence when compared with previous investigations. We
argue that our findings can solve the mismatch between the previous
predictions and experimental measurements in the low-energy region.

Then, motivated to obtain an analytical parametrization for the exchange
correction, we extended the calculation for a wide range of $\beta$ emitters,
with Z ranging from $1$ to $102$. Except for the low-energy region, the total
exchange correction progressively increases with the nuclear charge. We found
that for ultra-low energy, i.e., $5$ eV, the $Z$ dependence of total exchange
effect is affected by the closure of $s_{1/2}$ and $p_{1/2}$ orbitals. For
higher energies, the exchange effect presents a smooth dependence on the
nuclear charge, which is an entirely different result than previous
investigations. We associate this behavior with the orthogonality between the
continuum and the bound states of the final atom. We also show that the
contributions from orbitals higher than the $2s_{1/2}$ orbital are essential
for correctly calculating the total effect, especially for low energies.
Finally, we provide an analytical expression of the total exchange correction
for each atomic number for easy implementation in experimental investigations.

## ACKNOWLEDGMENTS

F. Š. acknowledges support by the VEGA Grant Agency of the Slovak Republic
under Contract No. 1/0607/20 and by the Ministry of Education, Youth and
Sports of the Czech Republic under the INAFYM Grant No.
CZ.02.1.01/0.0/0.0/16_019/0000766.

S.S acknowledges support by a grant of the Romanian Ministry of Research,
Innovation and Digitalization through the project PN19-030102-INCDFM.

The figures for this article have been created using the SciDraw scientific
figure preparation system [43].

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