A Systematic Study of Two-Neutrino Double Electron Capture

Abstract

In this paper, we update the phase-space factors for all two-neutrino double electron capture processes. The Dirac–Hartree–Fock–Slater self-consistent method is employed to describe the bound states of captured electrons, enabling a more realistic treatment of atomic screening and more precise binding energies of the captured electrons compared to previous investigations. Additionally, we consider all s-wave electrons available for capture, expanding beyond the K and $L_1$ orbitals considered in prior studies. For light atoms, the increase associated with additional captures compensates for the decrease in decay rate caused by the more precise atomic screening. However, for medium and heavy atoms, an increase in the decay rate, up to 10% for the heaviest atoms, is observed due to the combination of these two effects. In the systematic analysis, we also include capture fractions for the first few dominant partial captures. Our precise model enables a close examination of low Q-value double electron capture in ¹⁵²Gd, ¹⁶⁴Er, and ²⁴²Cm, where partial $KK$ captures are energetically forbidden. Finally, with the updated phase-space values, we recalculate the effective nuclear matrix elements and compare their spread with those associated with $2\nu {\beta }^{-}{\beta }^{-}$ decay.

1. Introduction

Exploring second-order weak interaction nuclear transitions represents a crucial channel in addressing unresolved challenges within modern physics. The detection of neutrinoless double beta decay (DBD) holds particular significance, as it would confirm the Majorana nature of neutrinos [1] (meaning they are their own antiparticles), and potentially provide insights into the mechanism generating their mass, as well as into charge-parity (CP) violation in the lepton sector [2,3,4,5,6,7,8]. Moreover, it would mark the first instance of the detection of a lepton number-violating process in laboratory. These types of beyond the Standard Model (SM) processes might play a significant role in certain leptogenesis mechanisms, which may elucidate the prevalence of matter over antimatter in the Universe [9,10].

Currently, the most studied modes of DBD involve the emission of two electrons from neutron-rich candidate isotopes: the two-neutrino mode ($2\nu {\beta }^{-}{\beta }^{-}$) and the neutrinoless mode ($0\nu {\beta }^{-}{\beta }^{-}$). While the latter remains experimentally elusive [11,12,13,14,15,16,17,18], the former serves as a playground to test different hypotheses and the predictive strengths of the nuclear structure models [19,20,21], and to constrain various parameters associated with physics beyond the SM [22,23,24,25,26,27,28,29,30,31,32]. A comprehensive and recent review on the latter topic can be found in [33].

DBD of proton-rich isotopes is also possible and is currently in the early stages of experimental exploration [34]. There are three such DBD modes [35,36]: the double-positron emitting ($0\nu /2\nu {\beta }^{+}{\beta }^{+}$) mode, the single electron capture with coincident positron emission ($0\nu /2\nu \mathrm{EC}{\beta }^{+}$) mode, and the double electron capture ($0\nu /2\nu \mathrm{ECEC}$) mode. Experimental measurements of these modes have received less attention due to their longer half-lives compared to the $2\nu {\beta }^{-}{\beta }^{-}$ decay mode, mostly due to their lower Q-values. However, their distinct signatures might help in their detection, especially if coincidence trigger logic can be employed [34]. There are some positive indications of the $2\nu \mathrm{ECEC}$ mode for ¹³⁰Ba and ¹³²Ba from geochemical measurements [37,38,39] and for ⁷⁸Kr [40,41]. Recently, the XENON1T collaboration reported the direct observation of the $2\nu \mathrm{ECEC}$ mode in ¹²⁴Xe [42,43].

From the theoretical point of view, the pioneer estimates of the decay rates for $2\nu$ECEC have been obtained by Primakoff and Rosen [44,45], followed up by the ones made by Vergados [46] and by Kim and Kubodera [47]. In these early stages, the predictions were based on a non-relativistic treatment for the captured electrons. Doi and Kotani were the first to present a detailed theoretical formulation for the process and to include the relativistic effects [35]. Still, the calculations are based on the atomic structure of a point-like nucleus, for which the electron bound states are known analytically. At the same time, Boehm and Vogel presented the results for some selected $2\nu$ECEC cases, but not many details of the calculations are provided [48]. Recently, the treatment of the atomic screening was improved using a Thomas–Fermi model for an atomic cloud [49,50,51]. The diffuse nuclear surface effects have also been accounted for through a realistic charge distribution inside the nucleus [50,51].

Although $2\nu$ECEC, by its very nature, stands at the interface between nuclear and atomic physics, some aspects regarding the atomic structure calculations have been overlooked or treated simplistically in previous investigations. In this paper, we employ the Dirac–Hartree–Fock–Slater (DHFS) self-consistent framework, providing an enhanced description of atomic screening for the systems involved in the decay and a more rigorous estimation of the binding energies for the captured electrons. We mention that the DHFS atomic potential includes the finite nuclear size and diffuse nuclear surface effects. Furthermore, we extend our analysis to include all s-wave electrons available for capture, surpassing the K and $L_1$ orbitals considered in prior studies.

We demonstrate that, for light atoms undergoing the $2\nu$ECEC process, the decrease in the decay rate resulting from more realistic screening is balanced out by the increase associated with opening the capture from higher orbitals than K and $L_1$. However, this balance is not observed for medium and heavy atoms, where the latter improvement in our model leads to a considerable increase in the decay rate, up to 10% for the heaviest atoms. For all cases undergoing $2\nu$ECEC, we provide the capture fractions for the first few dominant channels. We specifically address the low Q-value $2\nu$ECEC transitions of ¹⁵²Gd, ¹⁶⁴Er, and ²⁴²Cm, for which the capture of both K shell electrons is energetically forbidden. Finally, with the updated phase-space values, we reexamine the effective nuclear matrix elements and compare their spread with those associated with $2\nu {\beta }^{-}{\beta }^{-}$.

2. Formalism for the Two-Neutrino Double Electron Capture

2.1. Energetics

We investigate the two neutrino double electron capture ($2\nu \mathrm{ECEC}$) process,

$$2e^{-}+(A,Z+2)\to (A,Z)+2{\nu }_e,$$

(1)

in which the initial nucleus $(A,Z+2)$ captures two atomic electrons, changing its atomic number by two units and emitting two neutrinos. If the electrons are captured from x and y atomic orbitals, we denote the process as $2\nu \mathrm{xy}$. To keep the notation simple, we use the X-ray notation for the orbitals, i.e., $\mathrm{x}\mathrm{and}\mathrm{y}=K$, $L_1$, $M_1$, $N_1$, … Following the $2\nu \mathrm{xy}$ process, the final atomic system, while electrically neutral, remains in an excited atomic state, with vacancies in the orbitals x and y from which the electrons were initially captured. Consequently, the $2\nu \mathrm{ECEC}$ process is followed by atomic de-excitation, involving a sequence of X-ray emissions and Auger electrons released from the outer shells. Figure 1 provides a schematic illustration of two channels of the $2\nu$ECEC process in ${ }^{124}$Xe. The top panel shows that both electrons are captured from the K orbital ($2\nu KK$), while the bottom panel shows that they are captured from $L_1$ orbital ($2\nu L_1{L}_1$). It is worth noting that the atomic de-excitation results in different energy deposition in the detector depending on whether the capture is from the K or $L_1$ shells. This is due to the fact that the atomic relaxation following $L_1{L}_1$ capture emits less energy compared to the $KK$ capture.

Figure 1. A simplified schematic representation of the $2\nu$ECEC process in ${ }^{124}$Xe. Top panels: both electrons are captured from K shell. Bottom panels: both electrons are captured from L shell. Left panels: two bound electrons from K (top) or L (bottom) shell are captured by the initial nucleus ${ }^{124}$Xe. Right panels: the atomic relaxation of the final neutral atomic system ${ }^{124}$Te, with two holes in K (top) or L (bottom) shell. The de-excitation is performed via X-ray emissions and Auger electrons from outer shells. The figure is reproduced from [52] by adding the schematic representation of the $2\nu L_1{L}_1$ process in ${ }^{124}$Xe.

We adopt the following energy balance,

$${\mathcal{M}}_{\mathrm{gs}}(A,Z+2)-{\mathcal{M}}_{\mathrm{gs}}(A,Z)={\omega }_{\mathrm{x}}+{\omega }_{\mathrm{y}}+{\mathcal{M}}_{\mathrm{xy}}(A,Z)-{\mathcal{M}}_{\mathrm{gs}}(A,Z),$$

(2)

which indicates that the difference in the atomic masses of the initial and final systems in ground states, usually denoted as $Q={\mathcal{M}}_{\mathrm{gs}}(A,Z+2)-{\mathcal{M}}_{\mathrm{gs}}(A,Z)$, is shared between neutrinos with energies ${\omega }_{\mathrm{x}}$ and ${\omega }_{\mathrm{y}}$, and the atomic relaxation of the final atom. The mass ${\mathcal{M}}_{\mathrm{xy}}(A,Z)$ corresponds to an atomic excited state of the final neutral system, with holes in the shell $\mathrm{x}$ and $\mathrm{y}$ from where the electrons were captured in the initial atom, and it can be written as

$${\mathcal{M}}_{\mathrm{xy}}(A,Z)=E_f+Z{m}_e-B_{\mathrm{xy}}\left(Z\right)$$

(3)

where $E_f$ is the nuclear mass (energy) of the final nucleus, $Z{m}_e$ is the rest energy of the final atomic cloud ($m_e$ is the mass of the electron), and $B_{\mathrm{xy}}\left(Z\right)$ is the total electron binding energy of the final system with a configuration with two holes. We adopt the units in which $\hslash =c=1$. One can see that the atomic relaxation energy can be written as a difference in the total electron binding energies of the final atom with gs and xy configurations,

$$R_{\mathrm{xy}}=B_{\mathrm{gs}}\left(Z\right)-B_{\mathrm{xy}}\left(Z\right).$$

(4)

The total energy of the emitted neutrinos can be written as

$${\omega }_{\mathrm{x}}+{\omega }_{\mathrm{y}}=Q-R_{\mathrm{xy}}$$

(5)

and, making the usual approximation that $B_{\mathrm{xy}}\left(Z\right)-B_{\mathrm{gs}}(Z+2)=-\left|t_{\mathrm{x}}\right|-\left|t_{\mathrm{y}}\right|$, we finally obtain [35,49,51]

$$\begin{array}{ccc}\hfill {\omega }_{\mathrm{x}}+{\omega }_{\mathrm{y}}& =& E_i-E_f+2m_e-\left|t_{\mathrm{x}}\right|-\left|t_{\mathrm{y}}\right|\hfill \\ & =& E_i-E_f+e_{\mathrm{x}}+e_{\mathrm{y}},\hfill \end{array}$$

(6)

where $t_{\mathrm{x}}$ and $e_{\mathrm{x}}=m_e-\left|t_{\mathrm{x}}\right|$ are, respectively, the binding and the total energy for the electrons in shell $\mathrm{x}$.

2.2. Half-Life

Following the separation of the decay rate from [35,49], the inverse half-life of the $2\nu$ECEC process from $0^{+}$ ground state of the initial nucleus to the $0^{+}$ ground state of the final nucleus is given by

$${\left[T_{1/2}^{2\nu \mathrm{ECEC}}\right]}^{-1}=g_A^4{\left|M^{2\nu \mathrm{ECEC}}\right|}^2{G}_{K\to \mathrm{edge}}^{2\nu \mathrm{ECEC}},$$

(7)

where $g_A$ is the axial coupling constant, $G_{K\to \mathrm{edge}}^{2\nu \mathrm{ECEC}}$ is the phase-space factor (PSF) discussed later, and $M^{2\nu \mathrm{ECEC}}$, the nuclear matrix element (NME), is given by

$$M^{2\nu \mathrm{ECEC}}=-\frac{m_e}{\tilde{A}}\left[M_{\mathrm{GT}}^{2\nu \mathrm{ECEC}}-{\left(\frac{g_V}{g_A}\right)}^2{M}_{\mathrm{F}}^{2\nu \mathrm{ECEC}}\right].$$

(8)

Here, $\tilde{A}=1.12A^{1/2}$ in MeV and $g_V$ is the vector coupling constant. This choice of $\tilde{A}$ reproduces the average excitation energies in the intermediate nucleus for a wide range of isotopes [53]. Nevertheless, there are reasons supporting the use of a single value for both, as discussed in [53]. The double Gamow–Teller (GT) and Fermi (F) transition matrix elements, in closure approximation, are $M_{\mathrm{GT}}^{2\nu \mathrm{ECEC}}=⟨0_f^{+}\left|{\sum }_{j,k}{\tau }_j^{-}{\tau }_k^{-}{\sigma }_j{\sigma }_k\right|0_i^{+}⟩$ and $M_{\mathrm{F}}^{2\nu \mathrm{ECEC}}=⟨0_f^{+}\left|{\sum }_{j,k}{\tau }_j^{-}{\tau }_k^{-}\right|0_i^{+}⟩$[49]. Here, ${\tau }_{j,k}^{-}$ is the isospin-lowering operator, transforming a proton into a neutron and ${\sigma }_{j,k}$ is the nucleon spin operator.

It is important to highlight that the factorization of the half-life in Equation (7) assumes the closure approximation [35]. Consequently, the PSFs outlined below should be paired with NMEs computed within the same approximation. However, there have been various studies computing NMEs beyond the closure approximation and overlooking the constraint above [54,55,56]. Despite the lack of consistency, the predictions for half-lives are notably close to experimental values. In this study, we adhere to the closure approximation for comparison purposes. It is worth mentioning that an alternative factorization of the $2\nu$ECEC half-life, beyond the closure approximation, will be presented elsewhere.

In this paper, we consider that captures can occur from any occupied $s_{1/2}$ orbitals of the initial atom, i.e., $K,L_1,M_1,\dots$ The capture from other orbitals is suppressed, firstly because the higher orbital angular momentum of these orbitals reduces the capture probability, and secondly, smaller NMEs are associated with these captures [35]. Under these assumptions, we found that the PSF expression is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G_{K\to \mathrm{edge}}^{2\nu \mathrm{ECEC}}=& \frac{m_e{\left(G_F\left|V_{ud}\right|m_e^2\right)}^4}{16{\pi }^3}\frac{2\tilde{A}}{3m_e^2}\frac{1}{m_e^3}\sum _{\genfrac{}{}{0pt}{}{\mathrm{x},\mathrm{y}=K}{E_I-E_F+e_{\mathrm{x}}+e_{\mathrm{y}}>0}}^{\mathrm{edge}}{\mathcal{B}}_{\mathrm{x}}^2{\mathcal{B}}_{\mathrm{y}}^2\hfill \\ \hfill & \times {\int }_0^{E_I-E_F+e_{\mathrm{x}}+e_{\mathrm{y}}}\left[{⟨K_{n,\mathrm{xy}}⟩}^2+{⟨L_{n,\mathrm{xy}}⟩}^2+⟨K_{n,\mathrm{xy}}⟩⟨L_{n,\mathrm{xy}}⟩\right]{\omega }_{\mathrm{x}}^2{\omega }_{\mathrm{y}}^{2}d{\omega }_{\mathrm{x}}\hfill \\ \hfill =& \sum _{\genfrac{}{}{0pt}{}{\mathrm{x},\mathrm{y}=K}{E_I-E_F+e_{\mathrm{x}}+e_{\mathrm{y}}>0}}^{\mathrm{edge}}{G}^{2\nu \mathrm{xy}}.\hfill \end{array}\end{array}$$

(9)

where $G_F$ is the Fermi coupling constant and $V_{ud}$ the first element of the Cabibbo–Kobayashi–Maskawa (CKM) matrix. The sums are running over all occupied orbitals of the initial atomic system from where the electrons can be captured. The “edge” label refers to the last such orbital. The localization probability for an electron from shell $\mathrm{x}$ on the nuclear surface can be written as [35,49],

$${\mathcal{B}}_{\mathrm{x}}^2=\frac{1}{4\pi m_e^3}\left[g_{\mathrm{x}}^2\left(R\right)+f_{\mathrm{x}}^2\left(R\right)\right],$$

(10)

in terms of the large- and small-component radial wave functions describing the bound state, $g_{\mathrm{x}}\left(r\right)$ and $f_{\mathrm{x}}\left(r\right)$, respectively, evaluated on the nuclear surface $R=1.2A^{1/3}$. The factors $⟨K_{n,\mathrm{xy}}⟩$ and $⟨L_{n,\mathrm{xy}}⟩$ can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ⟨K_{n,\mathrm{xy}}⟩=& \frac{1}{-e_{\mathrm{x}}+{\omega }_{\mathrm{x}}+⟨E_n⟩-E_i}+\frac{1}{-e_{\mathrm{y}}+{\omega }_{\mathrm{y}}+⟨E_n⟩-E_i}\hfill \\ \hfill ⟨L_{n,\mathrm{xy}}⟩=& \frac{1}{-e_{\mathrm{x}}+{\omega }_{\mathrm{y}}+⟨E_n⟩-E_i}+\frac{1}{-e_{\mathrm{y}}+{\omega }_{\mathrm{x}}+⟨E_n⟩-E_i}\hfill \end{array}\end{array}$$

(11)

where $⟨E_n⟩$ is a suitably chosen excitation energy for the intermediate nucleus, $(A,Z+1)$. In the actual calculations, the energy difference $⟨E_n⟩-E_i=\tilde{A}-Q/2+m_e$.

For comparison with the results of the previous investigations, we define also the PSF for captures from K and $L_1$ orbitals only,

$$G_{K\to L_1}^{2\nu \mathrm{ECEC}}=G^{2\nu KK}+2G^{2\nu K{L}_1}+G^{2\nu L_1{L}_1}$$

(12)

which is a particularization of Equation (9). The factor of 2 stems from the equal contributions of $2\nu \mathrm{xy}$ and $2\nu \mathrm{yx}$ captures in the total decay rate for any $\mathrm{x}\ne \mathrm{y}$ orbitals.

2.3. Electron Bound States Description

Accurate computation of the PSFs for any $2\nu \mathrm{ECEC}$ process requires precise atomic structure calculations for the atomic systems involved in the process. For this purpose, we employed the Dirac–Hartree–Fock–Slater (DHFS) self-consistent framework. The nuclear, electronic, and exchange components of the DHFS potential, as well as the convergence of the self-consistent method, were described in detail in [57,58], based on the RADIAL subroutine package [59], which we also used in our calculations. In a relativistic framework and for a spherically symmetric potential of the atomic systems, the large- and small-component radial wave functions from Equation (10) obey the following system of coupled differential equations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left(\frac{d}{dr}+\frac{\kappa +1}{r}\right)g_{n\kappa }\left(r\right)-(e_{n\kappa }-V_{\mathrm{DHFS}}\left(r\right)+m_e)f_{n\kappa }\left(r\right)& =0,\hfill \\ \hfill \left(\frac{d}{dr}-\frac{\kappa -1}{r}\right)f_{n\kappa }\left(r\right)+(e_{n\kappa }-V_{\mathrm{DHFS}}\left(r\right)-m_e)g_{n\kappa }\left(r\right)& =0.\hfill \end{array}\end{array}$$

(13)

In previous sections, the shells referred to as $\mathrm{x}$ or $\mathrm{y}$ can be uniquely identified by the state $n\kappa$, where n represents the principal quantum number and $\kappa$ corresponds to the relativistic quantum number.

In Table 1, we present a comparison between experimental binding energies and values predicted by different models for the selected isotopes (Kr, Xe, and Ba) undergoing the $2\nu$ECEC process. The model proposed by [35], assuming a uniform charge distribution for the nucleus with no screening due to the atomic charge, fails to reproduce the experimental data. Models incorporating atomic screening and diffuse nuclear surface effects, as seen in [50,51], using a Thomas–Fermi screening function and a realistic proton density inside the nucleus, respectively, provide improved results but still yield binding energies significantly larger than the experimental ones. Notably, the DHFS self-consistent method employed in this work successfully reproduces experimental binding energies within a 1% margin.

Table 1. The binding energies, $t_{\mathrm{x}}$, in units of eV, for all occupied $s_{1/2}$ orbitals of the neutral atoms, Kr, Xe and Ba, undergoing $2\nu$ECEC process. The experimental data are sourced from [60], our values are obtained with the DHFS self-consistent method, and values from prior studies are taken from [35] and [50,51].

Shell$\left(\mathit{n}{\mathit{\ell }}_{\mathit{j}}\right)$	${\mathit{t}}_{\mathbf{x}}$ [35]	${\mathit{t}}_{\mathbf{x}}$ [50,51]	${\mathit{t}}_{\mathbf{x}}$ [This Work]	${\mathit{t}}_{\mathbf{x}}$ [Experiment] [60]
	(eV)	(eV)	(eV)	(eV)
Kr
$K\left(1s_{1/2}\right)$	$-17,936$	$-17,700$	$-14,280$	$-14,327\pm 2$
$L_1\left(2s_{1/2}\right)$	$-4507$	$-3100$	$-1902$	$-1927\pm 2$
$M_1\left(3s_{1/2}\right)$	–	–	$-278$	$-292\pm 2$
$N_1\left(4s_{1/2}\right)$	–	–	$-27.46$	$-27.51\pm 0.1$
Xe
$K\left(1s_{1/2}\right)$	$-41,340$	$-39,400$	$-34,556$	$-34,565\pm 2$
$L_1\left(2s_{1/2}\right)$	$-10,424$	$-7800$	$-5417$	$-5452\pm 2$
$M_1\left(3s_{1/2}\right)$	–	–	$-1122$	$-1149\pm 2$
$N_1\left(4s_{1/2}\right)$	–	–	$-208$	$-213\pm 2$
$O_1\left(5s_{1/2}\right)$	–	–	$-23.63$	$-23.40\pm 0.1$
Ba
$K\left(1s_{1/2}\right)$	$-44,610$	$-42,400$	$-37,450$	$-37,442\pm 2$
$L_1\left(2s_{1/2}\right)$	$-11,293$	$-8500$	$-5961$	$-5991\pm 2$
$M_1\left(3s_{1/2}\right)$	–	–	$-1274$	$-1293\pm 2$
$N_1\left(4s_{1/2}\right)$	–	–	$-256$	$-254\pm 2$
$O_1\left(5s_{1/2}\right)$	–	–	$-38$	$-31\pm 2$
$P_1\left(6s_{1/2}\right)$	–	–	$-4.64$	$-5.21\pm 0.1$

3. Results and Discussions

The PSFs for captures from K and $L_1$ shells only, $G_{K\to L_1}^{2\nu \mathrm{ECEC}}$, calculated for the $2\nu$ECEC cases previously investigated, are shown in Table 2. Here, we have assumed the same Q-values as in [49]. Our results are consistently lower by about 5% than the values reported in [49], with a couple of exceptions observed in ${ }^{92}$Mo and ${ }^{180}$W. Our results generally fall within 70% of the values reported in [50,51], with a noticeable dependency on A: for low mass numbers, our PSFs are lower than those in [50,51], but as the mass number increases, the factors in this work approach and even exceed the values reported in that reference, suggesting a discrepancy in the underlying models. In contrast, the $G_{K\to L_1}^{2\nu \mathrm{ECEC}}$ values derived in this work exhibit a consistent 10–20% reduction compared to the ones provided in [35], which could be attributed to the fact that, in [35], the screening effect is not accounted for. We chose not to display the $G_{K\to L_1}^{2\nu \mathrm{ECEC}}$ values from [48], since it does not employ the same definition of the PSF due to a different separation of the decay rate.

Table 2. Comparison of the $G_{K\to L_1}^{2\nu \mathrm{ECEC}}$ (in units of ${10}^{-24}$ yr${ }^{-1}$) values from [35,49,50,51], with values obtained through the method explained in this study. The Q-values cited in [49] were utilized. In cases where these values were not available, the Q was calculated using atomic masses provided in [61].

Nucleus	Q	${\mathit{G}}_{\mathit{K}\to {\mathit{L}}_1}^{2\mathit{\nu }\mathbf{ECEC}}$ [35]	${\mathit{G}}_{\mathit{K}\to {\mathit{L}}_1}^{2\mathit{\nu }\mathbf{ECEC}}$ [49]	${\mathit{G}}_{\mathit{K}\to {\mathit{L}}_1}^{2\mathit{\nu }\mathbf{ECEC}}$ [50,51]	${\mathit{G}}_{\mathit{K}\to {\mathit{L}}_1}^{2\mathit{\nu }\mathbf{ECEC}}$ [This Work]
	(MeV)	$\left({\mathbf{10}}^{-\mathbf{24}}{\mathbf{yr}}^{-\mathbf{1}}\right)$	$\left({\mathbf{10}}^{-\mathbf{24}}{\mathbf{yr}}^{-\mathbf{1}}\right)$	$\left({\mathbf{10}}^{-\mathbf{24}}{\mathbf{yr}}^{-\mathbf{1}}\right)$	$\left({\mathbf{10}}^{-\mathbf{24}}{\mathbf{yr}}^{-\mathbf{1}}\right)$
${ }^{36}$Ar	0.43259	–	–	$2.900\times {10}^{-4}$	$4.168\times {10}^{-4}$
${ }^{40}$Ca	0.19351	–	$1.250\times {10}^{-5}$	$1.020\times {10}^{-5}$	$1.314\times {10}^{-5}$
${ }^{50}$Cr	1.1688	–	$4.220\times {10}^{-1}$	$2.380\times {10}^{-1}$	$4.161\times {10}^{-1}$
${ }^{54}$Fe	0.6798	–	$4.690\times {10}^{-2}$	$3.021\times {10}^{-2}$	$4.553\times {10}^{-2}$
${ }^{58}$Ni	1.9263	17.00	15.30	9.900	14.79
${ }^{64}$Zn	1.0948	–	1.410	1.030	1.364
${ }^{74}$Se	1.209169	–	5.656	3.410	5.454
${ }^{78}$Kr	2.8463	774.0	660.0	410.0	637.0
${ }^{84}$Sr	1.79	–	93.60	64.62	90.58
${ }^{92}$Mo	1.651	–	208.0	82.32	128.2
${ }^{96}$Ru	2.71451	$2.740\times {10}^3$	$2.400\times {10}^3$	$1.450\times {10}^3$	$2.328\times {10}^3$
${ }^{102}$Pd	1.1727	–	46.00	42.09	44.64
${ }^{106}$Cd	2.77539	$6.220\times {10}^3$	$5.410\times {10}^3$	$4.299\times {10}^3$	$5.269\times {10}^3$
${ }^{108}$Cd	0.27204	–	$2.070\times {10}^{-2}$	$6.820\times {10}^{-2}$	$1.975\times {10}^{-2}$
${ }^{112}$Sn	1.91982	–	$1.150\times {10}^3$	869.7	$1.120\times {10}^3$
${ }^{120}$Te	1.71481	–	888.0	840.3	866.3
${ }^{124}$Xe	2.8654	$2.020\times {10}^4$	$1.720\times {10}^4$	$1.510\times {10}^4$	$1.685\times {10}^4$
${ }^{126}$Xe	0.92	–	46.10	60.59	44.98
${ }^{130}$Ba	2.619	$1.630\times {10}^4$	$1.500\times {10}^4$	$1.477\times {10}^4$	$1.464\times {10}^4$
${ }^{132}$Ba	0.844	–	39.10	61.98	38.12
${ }^{136}$Ce	2.37853	$1.580\times {10}^4$	$1.250\times {10}^4$	$1.222\times {10}^4$	$1.224\times {10}^4$
${ }^{138}$Ce	0.698	–	18.40	34.47	17.92
${ }^{144}$Sm	1.78259	–	$5.150\times {10}^3$	$6.436\times {10}^3$	$5.055\times {10}^3$
${ }^{152}$Gd	0.0557	–	–	$1.120\times {10}^{-2}$	$5.989\times {10}^{-7}$
${ }^{156}$Dy	2.012	–	$1.760\times {10}^4$	$2.208\times {10}^4$	$1.734\times {10}^4$
${ }^{158}$Dy	0.284	–	$1.830\times {10}^{-1}$	3.191	$1.751\times {10}^{-1}$
${ }^{162}$Er	1.844	$1.810\times {10}^4$	$1.500\times {10}^4$	$2.008\times {10}^4$	$1.470\times {10}^4$
${ }^{164}$Er	0.02507	–	–	$8.300\times {10}^{-3}$	$7.392\times {10}^{-11}$
${ }^{168}$Yb	1.40927	–	$4.710\times {10}^3$	$7.872\times {10}^3$	$4.647\times {10}^3$
${ }^{174}$Hf	1.0988	–	$1.580\times {10}^3$	$3.432\times {10}^3$	$1.563\times {10}^3$
${ }^{180}$W	0.1432	–	$1.560\times {10}^{-3}$	1.478	$1.321\times {10}^{-3}$
${ }^{184}$Os	1.453	–	$1.290\times {10}^4$	$2.422\times {10}^4$	$1.275\times {10}^4$
${ }^{190}$Pt	1.384	–	$1.290\times {10}^4$	$2.815\times {10}^4$	$1.285\times {10}^4$
${ }^{196}$Hg	0.82	–	821.0	$3.587\times {10}^3$	815.8

In Table 3, we present the total PSFs $G_{K\to \mathrm{edge}}^{2\nu \mathrm{ECEC}}$, as well as the capture fractions for selected shell pairs, for all nuclei for which $2\nu$ECEC is energetically possible, and EC is energetically forbidden. When comparing with one of the most precise calculations available [49] (column four of Table 2), one can see the following trend. For light atoms, the more rigorous treatment of atomic screening balances out the effect of accounting for all $\kappa =-1$ shells. The only exception is ${ }^{40}$Ca, where the PSF value computed using all $\kappa =-1$ shells is higher by about 7.5% than the one in [49]. The difference is induced by the interplay between the binding energies and Q-value in the PSF integral (see Equation (9)). For medium and heavy atoms, there is an increase in the decay rate compared to results from [49], almost linear in Z, reaching about 10% for the heaviest atoms.

Table 3. Values of the $G_{K\to \mathrm{edge}}^{2\nu \mathrm{ECEC}}$ (in units of ${10}^{-24}$ yr${ }^{-1}$) and of the capture fractions for various shell pairs. The Q-values are computed using atomic masses provided in [61].

Nucleus	Q	${\mathit{G}}_{\mathit{K}\to \mathbf{edge}}^{2\mathit{\nu }\mathbf{ECEC}}$	$\mathit{KK}$	${\mathit{KL}}_1$	${\mathit{KM}}_1$	${\mathit{KN}}_1$	${\mathit{L}}_1{\mathit{L}}_1$
	(MeV)	$\left({\mathbf{10}}^{-\mathbf{24}}{\mathbf{yr}}^{-\mathbf{1}}\right)$	(%)	(%)	(%)	(%)	(%)
${ }^{36}$Ar	0.4326	$4.244\times {10}^{-4}$	83.40	14.23	1.62	–	0.61
${ }^{40}$Ca	0.1935	$1.347\times {10}^{-5}$	81.38	15.44	2.07	0.16	0.73
${ }^{50}$Cr	1.1705	$4.301\times {10}^{-1}$	81.35	15.39	2.18	0.12	0.73
${ }^{54}$Fe	0.68076	$4.717\times {10}^{-2}$	80.41	16.01	2.35	0.16	0.80
${ }^{58}$Ni	1.9264	15.22	80.45	15.97	2.37	0.15	0.79
${ }^{64}$Zn	1.09502	1.407	79.73	16.50	2.48	0.14	0.85
${ }^{74}$Se	1.20924	5.648	78.64	17.03	2.74	0.30	0.92
${ }^{78}$Kr	2.84767	661.9	78.64	16.92	2.81	0.35	0.91
${ }^{84}$Sr	1.78977	94.16	77.83	17.33	2.98	0.44	0.96
${ }^{92}$Mo	1.65044	133.8	76.69	17.89	3.25	0.56	1.04
${ }^{96}$Ru	2.7145	$2.437\times {10}^3$	76.64	17.85	3.31	0.60	1.04
${ }^{102}$Pd	1.20347	53.70	75.16	18.77	3.59	0.68	1.17
${ }^{106}$Cd	2.77539	$5.540\times {10}^3$	75.71	18.31	3.55	0.69	1.11
${ }^{108}$Cd	0.27179	$2.122\times {10}^{-2}$	65.39	24.87	5.12	1.01	2.26
${ }^{112}$Sn	1.91981	$1.181\times {10}^3$	74.78	18.82	3.72	0.76	1.18
${ }^{120}$Te	1.73558	975.8	74.06	19.20	3.87	0.83	1.24
${ }^{124}$Xe	2.85674	$1.756\times {10}^4$	74.22	19.00	3.88	0.86	1.22
${ }^{126}$Xe	0.91778	47.36	71.66	20.62	4.28	0.95	1.48
${ }^{130}$Ba	2.6237	$1.570\times {10}^4$	73.57	19.31	4.00	0.92	1.27
${ }^{132}$Ba	0.84407	40.87	70.45	21.26	4.50	1.03	1.59
${ }^{136}$Ce	2.37853	$1.303\times {10}^4$	72.88	19.67	4.14	0.97	1.33
${ }^{138}$Ce	0.69594	19.03	68.37	22.43	4.86	1.14	1.82
${ }^{144}$Sm	1.7824	$5.417\times {10}^3$	71.11	20.67	4.49	1.07	1.50
${ }^{150}$Gd	1.28728	$1.301\times {10}^3$	69.29	21.72	4.81	1.16	1.69
${ }^{152}$Gd	0.05567	$1.547\times {10}^{-6}$	–	–	$7.40\times {10}^{-4}$	$1.04\times {10}^{-3}$	38.56
${ }^{154}$Dy	3.31234	$2.490\times {10}^5$	71.29	20.48	4.51	1.09	1.47
${ }^{156}$Dy	2.00595	$1.837\times {10}^4$	70.19	21.14	4.69	1.14	1.59
${ }^{158}$Dy	0.28282	$2.028\times {10}^{-1}$	42.31	35.46	9.04	2.26	6.01
${ }^{162}$Er	1.84696	$1.601\times {10}^4$	69.29	21.64	4.87	1.19	1.68
${ }^{164}$Er	0.02508	$8.606\times {10}^{-9}$	–	–	–	–	0.87
${ }^{168}$Yb	1.40936	$5.052\times {10}^3$	67.42	22.71	5.20	1.27	1.90
${ }^{174}$Hf	1.09994	$1.724\times {10}^3$	64.94	24.06	5.62	1.39	2.20
${ }^{180}$W	0.14323	$2.522\times {10}^{-3}$	$5.94\times {10}^{-5}$	17.72	8.08	2.34	34.73
${ }^{184}$Os	1.45289	$1.399\times {10}^4$	65.02	23.90	5.64	1.43	2.17
${ }^{190}$Pt	1.40132	$1.519\times {10}^4$	63.83	24.50	5.84	1.50	2.32
${ }^{196}$Hg	0.81859	916.2	56.49	28.25	7.04	1.84	3.38
${ }^{212}$Rn	1.71019	$1.400\times {10}^5$	61.36	25.58	6.27	1.68	2.63
${ }^{214}$Rn	0.15031	$1.376\times {10}^{-2}$	–	1.07	1.38	0.52	39.52
${ }^{218}$Ra	1.42814	$6.758\times {10}^4$	58.58	26.92	6.72	1.83	3.03
${ }^{224}$Th	1.16975	$2.802\times {10}^4$	54.68	28.74	7.34	2.03	3.64
${ }^{230}$U	0.75251	$2.393\times {10}^3$	42.80	33.66	9.16	2.59	5.87
${ }^{236}$Pu	0.45693	95.16	16.51	39.25	12.33	3.64	13.66
${ }^{242}$Cm	0.08682	$1.243\times {10}^{-3}$	–	–	–	–	9.66
${ }^{252}$Fm	0.782	$8.121\times {10}^3$	34.11	36.07	10.34	3.05	7.96
${ }^{258}$No	1.051	$7.188\times {10}^4$	41.77	33.61	9.28	2.73	6.14

The low Q-value transitions of ${ }^{152}$Gd, ${ }^{164}$Er and ${ }^{242}$Cm exhibit an interesting behavior. In these cases, both the $KK$ and the $K{L}_1$ captures are energetically forbidden, and the highest contribution to the total PSF is given by the $L_1{L}_1$, $M_1{M}_1$ and $L_1{M}_1$ captures. The values of the PSFs and capture fractions for these low Q-value transitions are given in Table 4.

Table 4. Values of the $G_{K\to \mathrm{edge}}^{2\nu \mathrm{ECEC}}$ (in units of ${10}^{-24}$ yr${ }^{-1}$) and of the capture fractions for the dominant shell pairs in case of low Q-value transitions. The Q-values are computed using atomic masses provided in [61].

Nucleus	Q	${\mathit{G}}_{\mathit{K}\to \mathbf{edge}}^{2\mathit{\nu }\mathbf{ECEC}}$	${\mathit{L}}_1{\mathit{L}}_1$	${\mathit{L}}_1{\mathit{M}}_1$	${\mathit{L}}_1{\mathit{N}}_1$	${\mathit{L}}_1{\mathit{O}}_1$	${\mathit{M}}_1{\mathit{M}}_1$	${\mathit{M}}_1{\mathit{N}}_1$	${\mathit{M}}_1{\mathit{O}}_1$	${\mathit{N}}_1{\mathit{N}}_1$	${\mathit{N}}_1{\mathit{O}}_1$
	(MeV)	$\left({\mathbf{10}}^{-\mathbf{24}}{\mathbf{yr}}^{-\mathbf{1}}\right)$	(%)	(%)	(%)	(%)	(%)	(%)	(%)	(%)	(%)
${ }^{152}$Gd	0.05567	$1.547\times {10}^{-6}$	38.56	35.94	10.12	1.84	7.56	4.17	0.75	0.57	0.21
${ }^{164}$Er	0.02508	$8.606\times {10}^{-9}$	0.87	26.59	12.06	2.33	28.11	20.49	3.79	3.62	1.33
${ }^{242}$Cm	0.08682	$1.243\times {10}^{-3}$	9.66	33.49	13.91	4.16	16.55	12.49	3.65	2.32	1.35

The inverse of the PSF is proportional to the half-life for each transition (see Equation (7)). This quantity is plotted in Figure 2 for all nuclei shown in Table 3 versus atomic number. One can see a decrease in the Q-value as the atomic number increases.

Figure 2. Theoretical half-life scaled by $g_A^4{\left|M^{2\nu \mathrm{ECEC}}\right|}^2$ as function of atomic number for nuclei presented in Table 3. The color scale indicates the Q-value for each transition.

Finally, we investigate the effective matrix elements for the $2\nu \mathrm{ECEC}$ process of ${ }^{78}$Kr, ${ }^{124}$Xe, ${ }^{130}$Ba and ${ }^{132}$Ba. These can be obtained as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|M_{\mathrm{eff}}^{2\nu \mathrm{ECEC}}\right|& =\frac{1}{\sqrt{T_{1/2}^{2\nu \mathrm{ECEC}}{G}_{K\to \mathrm{edge}}^{2\nu \mathrm{ECEC}}}}\hfill \\ \hfill & =\frac{m_e}{\tilde{A}}\left|g_A^2{M}_{\mathrm{GT}}^{2\nu \mathrm{ECEC}}-g_V^2{M}_{\mathrm{F}}^{2\nu \mathrm{ECEC}}\right|.\hfill \end{array}\end{array}$$

(14)

using the experimental half-lives and the PSFs from Table 3. The effective NME values are presented in Table 5. It should be noted that, in the case of ${ }^{132}$Ba, there is also an experimental half-life measurement [38]. The authors state that the obtained value is tentative, and indeed, we obtain an effective NME value that is more than one order of magnitude larger than the others. Hence, we employ the half-life limit from [37]. Future measurements might clarify the situation, but we note that, in general, geochemical measurements tend to underestimate the half-lives, leading to overestimated effective NMEs. This may be a consequence of the difficulty in identifying the relevant production channel of the final atom [34]. Another interesting hypothesis is the variation in the weak interaction constant with time [62,63,64].

Table 5. Values or limits of the experimental half-life $M_{\mathrm{eff}}^{2\nu \mathrm{ECEC}}$. The Q-values are computed using atomic masses provided in [61]. The bottom row corresponds to the $2\nu {\beta }^{-}{\beta }^{-}$ decay of ${ }^{134}$Xe. For this case, the value of the effective NME was obtained from the lower half-life limit, and the PSF computed following [65] ($0.225\times {10}^{-22}$ yr${ }^{-1}$ using $Q=0.8258$ MeV).

Nucleus	${\mathit{T}}_{1/2}^{2\mathit{\nu }\mathbf{ECEC}}$	$\left|{\mathit{M}}_{\mathbf{eff}}^{2\mathit{\nu }\mathbf{ECEC}}\right|$
	(yr)
${ }^{78}$Kr	$9.2_{-2.9}^{+5.7}\times {10}^{21}$ [40]	$0.{457}_{-0.098}^{+0.095}$
	$1.9_{-0.8}^{+1.3}\times {10}^{22}$ [41]	$0.{318}_{-0.073}^{+0.100}$
${ }^{124}$Xe	$(1.1\pm 0.2)\times {10}^{22}$ [43]	$0.{072}_{-0.006}^{+0.008}$
${ }^{130}$Ba	$(2.2\pm 0.5)\times {10}^{21}$ [38]	$0.{170}_{-0.017}^{+0.023}$
	$(6.0\pm 1.1)\times {10}^{20}$ [39]	$0.{326}_{-0.026}^{+0.035}$
${ }^{132}$Ba	≥$2.2\times {10}^{21}$ [37]	≤3.335
Nucleus	$T_{1/2}^{2\nu {\beta }^{-}{\beta }^{-}}$	$\left|M_{\mathrm{eff}}^{2\nu {\beta }^{-}{\beta }^{-}}\right|$
	(yr)
${ }^{134}$Xe	≥$2.8\times {10}^{22}$ [66]	≤0.398

In Figure 3, we compare the values from Table 5 with the corresponding $2\nu {\beta }^{-}{\beta }^{-}$ effective nuclear matrix elements. These are defined similarly to the ECEC effective NMEs, but replacing isospin-lowering with isospin-rising operators in the definition of the Gamow–Teller and Fermi matrix elements. We note that effective matrix elements from both processes span similar ranges.

Figure 3. Absolute values of the effective nuclear matrix elements for the measured nuclei undergoing $2\nu \mathrm{ECEC}$ process or $2\nu {\beta }^{-}{\beta }^{-}$ decay. The values for $2\nu \mathrm{ECEC}$ processes and $2\nu {\beta }^{-}{\beta }^{-}$ decay of ${ }^{134}$Xe are the ones from Table 5. The other $2\nu {\beta }^{-}{\beta }^{-}$ values are the ones reported in [20], computed using the PSFs from [65]. Vertical bars indicate uncertainties derived from the ones of the measured half-lives. Arrows indicate that the corresponding point is an upper limit for the effective nuclear matrix element.

4. Uncertainties and Further Improvements

We first perform a sensitivity analysis on the input parameters, namely the Q-value, the average energy of the excited $1^{+}$ states $⟨E_N⟩$ and the nuclear radius. The relative variations in the PSFs due to each parameter are summarized in Table 6, with the Q-value having the highest influence. However, we note that modern determinations of this quantity are highly accurate and the absolute values of the PSFs are practically insensitive to the choice of Q-value. The sensitivity of the PSFs with respect to $⟨E_N⟩$ and the nuclear radius are much smaller, as noted previously in [49].

Table 6. Summary of uncertainties in the PSFs due to input parameters.

Source	$\mathit{\delta }{\mathit{G}}_{\mathit{K}\to \mathbf{edge}}^{2\mathit{\nu }\mathbf{ECEC}}/{\mathit{G}}_{\mathit{K}\to \mathbf{edge}}^{2\mathit{\nu }\mathbf{ECEC}}$
Q-value	(5–7) $\times \delta Q/Q$
$⟨E_N⟩$	0.07 $\times \delta ⟨E_N⟩/⟨E_N⟩$
Nuclear radius	0.2 $\times \delta R/R$

Another source of uncertainties is related to the choice of the DHFS model for atomic structure computations. The accuracy of the binding energies obtained in the DHFS approach is around 1% relative to the experimental values [58]. The binding energies enter in the PSFs definition through the integration limit, through the energies of the emitted neutrinos, and through the $⟨K_N⟩$ and $⟨L_N⟩$ factors, always as additive quantities to the Q-value. Consequently, this uncertainty has a negligible effect as it translates to $\mathcal{O}\left({10}^{-4}\right)$ relative variation in the Q-value of the process. Low Q-value transitions are an exception, and the uncertainty in the binding energies is dominant. The values of the wave functions on the nuclear surface can also introduce some uncertainty in the PSFs. This effect was studied in [58] by comparing the Coulomb amplitudes (proportional to the wave function values on the nuclear surface) in the DHFS model with the ones obtained through more complex models. The study found that Coulomb amplitudes obtained in the DHFS framework agree within 0.25% with ones obtained through the more refined Dirac–Hartree–Fock (DHF) model for the $1s_{1/2}$ and $2s_{1/2}$ shells and for atomic numbers above 20. Considering this value as uncertainty within our model, we evaluate the relative error in the PSFs due to the wave functions alone to be 1%, since the wave functions enter the PSF definition to the fourth power. We note that this uncertainty is a systematic one, the Coulomb amplitudes in the DHFS model always overestimating the ones obtained in the DHF model.

Besides the uncertainties discussed above, one may expect variations in PSFs from improving the overall model of the $2\nu$ECEC process. Firstly, the summed neutrino energy determination has been obtained through a usual approximation (see Equation (6)), but it can be more precisely determined through Equation (5). Consequences are most important in low Q-value transitions. For example, the DHFS framework predicts $R_{K{L}_1}=55.05$ keV for ${ }^{152}$Sm. Consequently, the $2\nu K{L}_1$ process of ${ }^{152}$Gd is energetically allowed (in contrast to the results shown in Table 3), as it was also found in [67,68,69] when studying the resonant neutrinoless double electron capture. Secondly, the Pauli blocking of the decay of the innermost nucleon states is not accounted for in our model. This aspect can be improved by averaging the bound electron wave function, weighted with a realistic nuclear charge distribution [65,70]. From preliminary results, we expect an increase of a few % in the PSFs due to the Pauli blocking effect.

5. Conclusions

In this paper, we have conducted a systematic study of atoms undergoing the $2\nu$ECEC process. Despite the fact that $2\nu$ECEC, by its very nature, resides at the interface between nuclear and atomic physics, certain aspects related to atomic structure calculations have been either overlooked or treated simplistically in previous investigations. In our model, we introduced two enhancements to the atomic part calculations of double electron capture transitions. Firstly, by employing the DHFS self-consistent framework to describe the initial atomic system, we refined both the distortion of bound wave functions due to atomic clouds and the estimation of binding energies for captured electrons. Secondly, we extended our consideration beyond previous studies, where only K and $L_1$ orbitals were included, by also allowing captures from the outer orbitals of the initial atoms.

With our improved model, we have updated the phase-space values for all atoms undergoing the $2\nu$ECEC process. For light atoms, we observed almost no differences compared to a previous model, where a simplified atomic screening and only captures from K and $L_1$ orbitals were considered. This lack of difference is associated with a cancellation of the decrease in the decay rate from a more precise screening treatment with the increase associated with the possibility of captures occurring from higher orbitals. In contrast, for medium and heavy atoms, we observed an increase in the decay rate, almost linear in Z, reaching about 10% for the heaviest atoms. In the systematic study, we also provided capture fractions for the first few dominant partial channels and separately addressed the low Q-value $2\nu$ECEC transitions of ${ }^{152}$Gd, ${ }^{164}$Er, and ${ }^{242}$Cm, for which the $KK$ capture is energetically forbidden. Finally, we demonstrated that the effective nuclear matrix elements for both $2\nu$ECEC processes and $2\nu {\beta }^{-}{\beta }^{-}$ decays span similar ranges.