Resonance enhancement of neutrinoless double electron capture
Abstract
The process of neutrinoless double electron (ECEC) capture is revisited for those cases where the two participating atoms are nearly degenerate in mass. The theoretical framework is the formalism of an oscillation of two atoms with different total lepton number (and parity), one of which can be in an excited state so that mass degeneracy is realized. In such a case and assuming light Majorana neutrinos, the two atoms will be in a mixed configuration with respect to the weak interaction. A resonant enhancement of transitions between such pairs of atoms will occur, which could be detected by the subsequent electromagnetic deexcitation of the excited state of the daughter atom and nucleus. Available data of atomic masses, as well as nuclear and atomic excitations are used to select the most likely candidates for the resonant transitions. Assuming an effective mass for the Majorana neutrino of 1 eV, some halflives are predicted to be as low as years in the unitary limit. It is argued that, in order to obtain more accurate predictions for the ECEC halflives, precision mass measurements of the atoms involved are necessary, which can readily be accomplished by today’s high precision Penning traps. Further advancements also require a better understanding of highlying excited states of the final nuclei (i.e. excitation energy, angular momentum and parity) and the calculation of the nuclear matrix elements.
keywords:
neutrino mass, neutrinoless double beta decay, double electron capture, nuclear matrix elements1 Introduction
The question as to whether massive neutrinos obey a Dirac or a Majorana symmetry, presently constitutes one of the most important unresolved problems of particle physics and astrophysics. If neutrinos are Dirac particles, i.e. if neutrino and antineutrino are fundamentally different, then total lepton number must be conserved. Contrary, if neutrinos are Majorana particles, i.e. if neutrino and antineutrino are identical particles, then lepton number conservation is not required anymore. Indeed, lepton number (LN) conservation is one of the most obscure appearances in the Standard Model of elementary particles, since there is no known fundamental principle or symmetry, which would require this.
Already in 1939 Furry FURR39 noticed that the exchange of neutrinos (later termed Majorana neutrinos) between two neutrons could lead to the production of two protons and two electrons in the reaction
(1.1) 
Today, such a reaction is termed the neutrinoless doublebeta () decay, and an observation of this reaction is still the only unambiguous way to identify the Majorana character of the neutrino schv . Over the years, considerable efforts from experimentalists and theorists alike have been devoted to this process (for reviews see Ref. dbdreviews ). Although of fundamental importance, this process is unfortunately characterized by an excessively low rate, which poses a significant challenge to any experiment.
Assuming that the light neutrino mixing mechanism provides the dominant contribution to the decay, the decay rate for a given isotope is simply given by the product of the effective Majorana neutrino mass squared , the known 3body phasespace factor, and the much less wellknown nuclear matrix element squared, which is particular to every nuclear transition under study. The phasespace factor contains a dependence on the nuclear charge (), the Qvalue of the reaction () and the Fermicoupling constant (). The main objective of every experimental decay search is the determination of the absolute value of the effective Majorana neutrino mass . However, a mere observation of the decay would already constitute a significant advancement in neutrino physics.
In the 3neutrino mixing scenario, the effective Majorana neutrino mass takes the form
(1.2) 
Here, () are the elements of the PontecorvoMakiNakagawaSakata (PMNS) neutrino mixing matrix, which mixes the mass eigenstates in weak interaction. It contains the usual 3neutrino mixing angles plus a CPviolating phase, which appears in oscillations, and two additional Majorana phases, .
The recent claim for an observation of the decay of Ge with years evidence implies eV by assuming the renormalized QRPA (RQRPA) nuclear matrix element and its uncertainty given in Ref. src . The goal of the upcoming GERDA experiment gerda is to put this claim to a test by improving the sensitivity limit of the detection by more than an order of magnitude. The next generation experiments, which will be using several other candidate nuclei, will eventually be able to achieve this goal as well avig .
The value of the effective Majorana mass, as it appears in Eq. (1.2), contains several dependencies on phases and masses. Because of experimental uncertainties, different mass scenarios, like the normal () or inverted () hierarchy scenario, or degenerate () or nondegenerate cases, can presently still be entertained, which allow a wide range of possible mass values for , even zero in the most extreme and unfortunate situation of the normal hierarchy scenario hierar . Though even in that case the will still be allowed due to a contribution from the mass term in the neutrino propagator pepa , which one usually neglects, its decay rate would be utterly unobservable.
Recently, there has been an increased theoretical and experimental interest to another LN violating process, which is the neutrinoless double electron capture (ECEC) sujwy ; frekers ; barab1 ; barab2 . In this reaction two bound electrons from the atomic shell are captured by two protons, thereby lowering the charge of the final nucleus by two units:
(1.3) 
Here, the two asterisks denote the possibility of leaving the system in an excited nuclear and/or atomic state, the latter being characterized by two vacancies in the electron shell of the otherwise neutral atom. The energy excess given by the Qvalue of the reaction must still be carried away by an extra photon, in order to conserve energy. This is unlike the 2neutrino case, where the neutrinos can provide the energy balance. Thus, the reaction in Eq. (1.3) could in principle be detected by monitoring the rays or Auger electrons emitted from excited electron shell of the atom, the electromagnetic decay of the excited nucleus (in case of a nongroundstate transition) and the extra photon, whose energy would be
(1.4) 
We note that ECEC was considered by Winter WINT55 already in 1955.
The signature of the ECEC process is, therefore, different from the signature of the decay and would also require rather different coincident detection techniques. On the other hand, the coupling to an extra photon and/or Xray clearly makes the halflife excessively long to the extent that this process has not been considered a valid experimental option altogether.
The situation changes, however, if the energy difference in Eq. (1.4) approaches zero and no extra photon is required. This has been discussed by Bernabéu, De Rujula, and Jarlskog DERU , who pointed to the possibility of a resonant enhancement of the ECEC decay in case of a mass degeneracy between the initial and final nucleus. Their best candidate case was Sn, where the ECEC double Kshell capture process would lead to an exited state at 1871 keV in the final nucleus Cd. This possibility was recently excluded by a new mass difference measurement performed at Jyväskylä ( keV) RAKH09 , where it was shown that the energy to be paid by the double Kshell vacancy would not leave enough energy available for the excitation of the 1871 keV state in Cd.
The ECEC decays became a subject to a detailed theoretical treatment by Sujkowski and Wycech sujwy , who used a perturbative approach. Their conclusion was that an exact energy degeneracy could make the ECEC reaction competitive to the decay. However, a case with an exact energy degeneracy could not be identified.
Recently, another case for a near massdegeneracy was found and discussed in Ref. frekers . Here, it was argued that the 1204 keV state in Ge would be nearly degenerate to the ground state of the atomic nucleus Se in case of a double Lcapture process and given the experimental errors on the masses. A new mass difference measurement performed by Kolhinen et al. kolhinen essentially confirmed the previous central mass difference value of keV, however, with much higher precision. These authors therefore excluded a complete mass degeneracy with the 1204 keV state in Ge, even in the case of a double Lcapture, where the atomic energy for a double Lvacancy would amount to an extra 2.9 keV on top of the 1204 keV frekers . Prior to this, two experiments had already been performed by Barabash et al. barab1 ; barab2 , which gave lower bounds for the halflives of and :
In this paper we present a new theoretical framework for the calculation of resonant ECEC transitions, namely the oscillation of atoms. An improved theoretical description of the process includes the determination of relevant matrix elements for the most favored cases of capture of the and electrons. The ECEC transitions without and with the spatial parity violation are considered. Further, we provide an updated list of the most likely resonant transitions taking new nuclear spectroscopic data into account and using recent accurate measurements of values for several nuclei kolhinen ; RAKH09 ; penning1 ; penning2 ; penning3 ; SCIE09 . The selection of transitions is also based on accurate treatment of spintensor structures that arise in a product of the nuclear matrix elements and the electron wave functions of atomic shells. The reverse reaction
(1.5) 
of a neutrinoless production of two bound electrons (EPEP) will also be discussed.
The outline of the paper is as follows: First we discuss the mixing of atoms with different lepton charges. This effect leads to the oscillations of atoms. In Sect. 2, we discuss the relevant formalism of the oscillations. We will show that the oscillation of stable atoms produces a too small effect to be measured experimentally. However, oscillation between a stable and an excited atom can lead to a resonant enhancement of lepton number violating decays.
Sect. 3 presents the estimated halflives of the decays. In the calculations we use the data on the Auger and radiative widths of excited electron shells and the information on the Coulomb interaction energy of two electron holes. We consider nuclei with arbitrary spinparity and take into account the fact that the spinparity uniquely determines a combination of upper and lower components of the relativistic electron wave functions entering the matrix elements associated with the capture. In Appendix A the procedure of averaging the electron wave functions over the nucleus is discussed. The transition matrix elements are derived for the and states of the daughter nuclei in Appendix B. The problem of calculating matrix elements is very complicated, and the result depends sensitively on the particular transition. We identify the most promising nuclei in the search for ECEC decays. Such nuclei will continue to be analyzed in future. In this paper, the halflives are normalized to the nuclear matrix element , which is close to the maximum evaluated value of the matrix elements for mediumheavy nuclei. In Sect. 3, we also give a complete list of the most likely resonance transitions, in which the unitary limit of resonant enhancement gives halflives of less than years for eV. We argue that accurate measurements of the mass differences between initial and final states of the nuclei are necessary, if future experiments of ECEC decays with halflives below years were to become a possibility. Experimental signatures of ECEC decays are discussed in Sect. 4.
2 Lepton number violating transitions between ground state and excited atoms
If lepton number is not conserved, then the weak interaction mixing between a pair of neutral atoms and is a natural occurrence, which leads to an oscillation between these two manybody quantum systems. In the present description we focus on a system, in which one of the atoms (usually the daughter atom) is left in an excited atomic or nuclear state. In fact, for EC processes the daughter atomic system is always excited, as the capture process always leaves a vacancy in the electron shell. If the ECEC Qvalue is of the order of the excitation of the atomic shell with two electron vacancies, one may expect a resonantlike transition. A few examples do exist in the nuclear chart, which have this property. On the other hand, if the ECEC Qvalue is significantly larger than the atomic excitation, one may find a situation, where an excited nuclear state matches the available energy (i.e. ), allowing again a resonantlike transition to an excited nuclear state. The latter type of oscillations may even have a practical experimental signature: one or even several Xray photons or Auger electrons from the deexcitation of the atomic shell being coincident with a ray (or a cascade of rays) from the deexcitation of the nucleus. In fact, the detection of a coincident ray cascade, if existent, may already be sufficient to uniquely identify the transition. It may be worth reiterating that any such transition requires the neutrino to be of Majorana type, as there is no phase space available for the emission two extra neutrinos.
The present description of a resonant enhancement of the ECEC transition will be done in the context of oscillations. We wish to point out, that our results are consistent with the results of Bernabéu et al. DERU and Sujkowski and Wycech sujwy for the physically interesting case, where the frequency of oscillations is much smaller than the width of the excited atom. In the opposite limit, when the frequency of oscillations is high, the standard formulas of the time evolution of a twolevel system are retained.
2.1 Oscillations in arbitrary systems
Specific features of the oscillations in the system of two atoms were discussed earlier in Ref. SIM08 . Two coupled oscillators, one of which experiences friction, constitute the mechanical analogue of the system, which we are considering.
Lepton number violating interactions induce transitions . These transitions can be described phenomenologically by nonHermitian Hamiltonian matrix
(2.1) 
Here and are the masses of the initial and final atom. is the decay width of the excited daughter atom. The offdiagonal matrix elements of are complex conjugate. The transition potential can always be made real by changing the phase of one of the states, i.e. . The diagonal matrix elements of the Hamiltonian are determined by strong and electromagnetic interactions, which conserve lepton number. The offdiagonal elements provide the mixing of the neutral atoms, and thereby, violate lepton number by two units as a result of the weak interaction with massive Majorana neutrinos. Using the Pauli matrices, the Hamiltonian can be written as:
(2.2) 
where
(2.3) 
The evolution operator can be expanded over the Pauli matrices to give
(2.4) 
where .
One can see that all components of the evolution operator behave like , with . Since the eigenfrequencies are complex, the norm of the states is not preserved in time.
A somewhat similar form of the Hamiltonian matrix is responsible for the oscillation of neutral kaons OKUN84 . The main difference between the oscillation of neutral atoms and that of kaons is the mixing, which is maximum for kaons and exceedingly small for atoms.
The Hamiltonian in Eq. (2.1) also describes the effect of neutronantineutron oscillation in nuclear matter DGR85 ; MIK96 . In this case, and would be the neutron and antineutron masses, the baryon number violating potential, and the antineutron width related to annihilation channels (in vacuum and ).
The formalism is also similar to the one used for describing oscillations and decays of unstable neutrinos Gonz08 .
2.2 Oscillations of two stable atoms
If the two atoms are stable, then and . The transition probability is determined by the offdiagonal matrix element of the evolution matrix (2.4):
(2.5) 
This is just the case of oscillations of a twolevel system described for instance in Ref. LLQM . If , the transition probability is determined by the potential only. However, the exposure time of atoms in double decay experiments (months and years) is greater than by many orders of magnitude. By taking the average over one period, we one arrives at
(2.6) 
In the transitions , the composition of valence electron shells changes and, thus, the chemical properties of the substance. This circumstance can in principal be used for registering the oscillations of atoms. However, the potential is at least 30 orders of magnitude smaller than the atomic mass difference. For a hypothetical mass difference of keV one finds . Since degenerate groundstate masses do not exist, this scenario is purely academic, and we turn to systems of a stable mother and an excited daughter atom.
2.3 Oscillations and resonant transitions between ground state and excited atoms
According to the arguments in Ref. SIM08 , we assume a potential strength of eV, a typical decay width of eV for a mediumheavy atom, and a typical mass difference of MeV. In the lowest order in , we obtain
(2.7)  
(2.8) 
where
(2.9)  
(2.10) 
Since the width is small, the imaginary parts of in Eqs. (2.7) and (2.8) are negative, therefore, the states decay. Equation (2.10) gives the decay rate of the initial atom in agreement with the BreitWigner formula.
The excited atom manifests itself as a resonance in the decay amplitude. The most favorable conditions for the detection of the violation of lepton number conservation occur in the transitions , where the masses of the initial and final states are equal (degenerate) and the decay width of the daughter atomic nucleus is small.
The amplitude of finding the initial atom seconds after its preparation in the same initial state is determined by the diagonal matrix element of the evolution operator:
(2.11) 
The second term oscillates with the frequency and decays with the rate . At a low frequency, the system is unable to return to the initial state and decays. In this case one can talk about a lepton number violating decay of the initial atom with a width (Eq. 2.10).
The decay width reaches the unitary limit
(2.12) 
in the case of a complete degeneracy between initial and final state. From an experimental standpoint, where one would search for such lepton number violating decays, this would be the case of highest interest.
3 Analysis of ECEC halflives throughout periodic table
The selection of atomic systems with the potentially shortest ECEC halflives is based on equation (2.10). The equation shows that the decay rate is determined by three quantities: the mass difference between the initial and final states, the decay width of the final state, and the transition potential.
The mass difference depends on the Qvalue of the ECEC decay and the energy of the two electron vacancies in the final atom. In atomic physics, the electron binding energies can usually be calculated with an accuracy of several eV. We borrowed the binding energies from Ref. LARK . Noticeable corrections will, however, arise from the Coulomb interaction of the two holes. These calculations are carried out on the basis of the Dirac equation taking into account the screening effects of the nuclear charge.
The decay width of the final atom is determined by the dipole emission rate leading to the deexcitation of the electron shell. In the nonrelativistic approximation, the capture rates of two electrons from the higher shells, like L, M, or N shells, scale with the principal quantum numbers as , but we will see that in the unitarity limit of a resonant decay this strong reduction of the probability could possibly be compensated by smallness of the deexcitation rates. The total decay width of the system is given by the sum of the widths of the atomic and the nuclear state. In most cases the decay width of the excited nucleus is smaller than the one of the atomic state by at least an order of magnitude and can, therefore, be neglected. The process of Auger electron emission as the alternative deexcitation process of the atom is also taken into account following the results of Ref. CAMP . The Auger electron emission is faster than the electromagnetic decays for low atomic .
The transition potential contains the uncertainties of the transition matrix elements connected with the complicated structure of the nuclear excitation. In order to obtain numerical estimates, we factorize the ECEC matrix element on a product of the atomic physics factor and the nuclear matrix element. This simplification is justified due to weak radial dependence of the and electron wave functions inside nuclei. Further, we normalize all the ECEC nuclear matrix elements to the value of nuclear matrix element obtained for the ground state to ground state transition GdSm in Ref. erice11 . The contributions from the electron shell are determined by different combinations of the relativistic wave functions of electrons for the capture from different shells. We systematically examine transitions between all the states with the parities for the capture of two electrons with orbital angular momenta and the principal quantum numbers .
3.1 Coulomb interaction energy of electron holes
The binding energy of electrons in the inner atomic shells varies from eV in light nuclei up to keV in heavy nuclei. In the outer shells, the binding energy is a few eV, both in light and heavy nuclei. Since electrons are usually captured from the most inner shells, the electron binding energy gives sizeable contribution to the energy balance in the double electron capture. We use data of electron binding energies reported by Larkins LARK . Those are accurate to better than an eV for light nuclei and to a few tens of eV in heavy nuclei. The relevant binding energies in the context of ECEC are, of course, always those for the final daughter atom with .
In heavy elements, the electron hole interaction energies may reach values of a few keV (for a double Kshell vacancy), which is quite large for our problem in question. It is therefore essential to calculate the interaction energies of two electron holes and include them into the total energy equation.
We used an approach that takes screening of the Coulomb potential by electrons occupying other orbitals into account. The shielding effect can be estimated from the known energy of the bound electrons. In the nonrelativistic theory, the effective charge can be found from equation
(3.1) 
where is the electron mass, is the principal quantum number. In the nonrelativistic theory, the electron velocity increases with the nuclear charge, which requires a relativistic treatment for heavy nuclei.
The binding energies in the Coulomb field are known from the Dirac relativistic wave equation BERE . Given , the effective charge may be found from
(3.2) 
where , , , and . Near the limit of , the nonrelativistic formula is recovered.
The effective charge takes into account the screening of the Coulomb potential, as well as the finite nuclear size. Given that the electronshell wave functions are known, one can calculate the interaction energy of electron holes.
We consider transitions between nuclei with good quantum numbers , so the twoelectron wave function should have good total angular momentum , projection , and parity. This can be arranged by weighting the twoelectron wave function with the ClebschGordan coefficients
(3.3) 
Here, () is the relativistic wave function of the electron in the Coulomb field.
The wave function of two electrons can be written as follows:
(3.4) 
The interaction energy of two electron holes can be obtained from equation
(3.5) 
The case of two holes with identical quantum numbers and requires special attention. The states are symmetric over the permutations and do not exist. In the case the states are antisymmetric over the permutations, and the interaction energy should be divided by a factor of , since the superposition (3.3) changes the overall normalization of the twoelectron wave function, as it follows from .
To simplify notations, we label the final states by indices and the initial ones by . The labels take values to indicate first and second electrons. There are two possibilities for the final state, and , and two possibilities for the initial state, and .
Equation (3.5) can be written as
(3.6) 
where
(3.7) 
and
Further simplifications appear after the use of the expansion
(3.8) 
where , , and are unit vectors toward .
The angular integrals are calculated with the use of equation
(3.9) 
where are spherical spinors LLQM ,
(3.10) 
and .
The remaining twodimensional integral over the radial variables,
(3.11) 
can be calculated numerically or analytically using Maple. Here,
(3.12) 
and
(3.13) 
where , , and , and similarly for .
In Fig. 1 we show the Coulomb interaction energy of two K holes as a function of the nuclear charge. As expected, the interaction energy increases approximately linearly with .
More accurate estimates of the interaction energy of electron holes can be obtained on the basis of the Breit potential, which takes into account relativistic effects . The accuracy of the present estimate, therefore, can be evaluated to be of order . For the expected error in the interaction energy of two K electron holes is about 20%. The accuracy is, of course, better for highershell atomic excitations.
3.2 Natural widths of excited electron shells
Decays of excited atoms are dominated by electric dipole transitions with the emission of Xray photons and/or the emission of Auger electrons. Dipole decays are described in the literature in detail (see, e.g., BESA ). For K vacancies, characteristic Xray dipole emissions are dominant in atoms with . The dipole transition has a probability eV (see, e.g., BESA ). Since for a decay, and , the corresponding width decreases as for transitions from higher orbits. This effect is of interest, since the unitary limit of the width of the lepton number violating decay is inversely proportional to the width of the daughter atom (see Eq. (2.12)).
Auger transitions of excited atoms with one electron vacancy are well studied theoretically and experimentally (for a review see CAMP ). The width of a twohole state is represented by
(3.14) 
where is the deexcitation width of daughter nucleus. Numerical values of the onehole widths are taken from Ref. CAMP . These values cover the range and oneelectron vacancies from K to N7 shells (). Equation (3.14) neglects the contributions from twohole correlations.
3.3 Lepton number violating potential
We consider the ECEC process assuming the standard form of the decay Hamiltonian
(3.15) 
where and is the Cabibbo angle. The field operators of the electron and electron neutrino are denoted as and .
The lefthanded electron neutrino is a superposition of the lefthanded projections of Majorana neutrinos with diagonal masses :
(3.16) 
Here, is PMNS neutrino mixing matrix. Majorana neutrinos are truly neutral particles and obey , where is the charge conjugation matrix and is the phase factor.
The strangeness conserving charged hadron current has the form
(3.17) 
where and are the field operators of the neutron and the proton, and the vector and axialvector coupling constants are and .
The potential of the ECEC capture of two electrons with the total angular momentum and projection can be written as follows
Here, is the weak charged current in the Heisenberg representation. and are states of the initial and final nuclei. The sum is taken over all excitations of the intermediate nucleus . is a wave function of the bound electron with quantum numbers , projection of the total angular momentum , and energy . The factor takes statistics of the captured electrons into account: for the identical states and otherwise.
In the derivation of , we neglected the small neutrino masses ( eV) in the neutrino potential, since the average exchange momentum in the process is large, MeV/c. Further simplifications are as follows:

Nonrelativistic impulse approximation for the nuclear current:
(3.19) 
Closure approximation for the intermediate states: The excitation energies of the intermediate states are replaced by an average value MeV. In addition, we set . The sum entering Eq. (3.3) is then calculated using completeness condition .

We restrict the calculation of the Majorana neutrino exchange potentials to the most favorable cases of eveneven nuclei. Then the angular momentum of the initial nucleus is and the angular momentum of the final (possibly excited) nucleus must be balanced by the capture of the atomic electrons and the angular momentum of the atomic state.
The potential can finally be written as
(3.20) 
In the case of a capture of and electrons and of a favorable case for the nuclear transitions , the matrix elements are given in Appendix B.
The numerical analysis of the ECEC transition is performed by factorizing the electron shell structure and the nuclear matrix element:
(3.21) 
Here, are lepton factors averaged over the nuclear volume. For low transitions, the lepton factors are given in Table 1.
Transitions  

The nuclear matrix elements of transitions have the form
(3.22)  
(3.23)  
where
(3.24) 
We note that the nuclear matrix element also appears in the calculation of the decay process dbdreviews .
For the transition the lepton parts and the nuclear matrix elements are evaluated as discussed in Appendix A. We note that vanishes for transitions whenever the two electrons are captured from states with the same quantum numbers for . This is the reason, why e.g. the transition ErDy (1, keV) is excluded from the analysis. If electrons are captured from different states, e.g., two electrons from different shells () or from and states, the transition is allowed and is considered in our analysis.
The dominant combinations of upper and lower component of bispinors, which enter the lepton part of the matrix elements after the factorization, are listed in Table 1. The definition of functions and can be found in Appendix B. The decay rates of other transitions are estimated roughly with
(3.25) 
The parity nonconservation in the weak interactions allows for instance transitions accompanied by capture of two wave or two wave electrons.
3.4 Likely resonant ECEC transitions
We have considered all the nuclei and their excited states registered in the database of the Brookhaven National Laboratory BNL in August 2010, as well as all the combinatorial possibilities associated with the capture of two electrons. The selection criteria are as follows:

The excitation energies are usually known with precision much higher than the atomic groundstate masses. We selected those pairs, where degeneracy occurs within the bounds given by a three standard deviation error of the groundstate mass measurements.

The unitary limit for the normalized halflife is less than years.
Tables 2, 3, and 4 present the results of the selection over stable parent isotopes. We show the natural abundances NA, the spinparity of the final nucleus , the excitation energy of the final nucleus , and the total mass difference . The two errors indicate the errors of the groundstate mass measurements. Shown are as well the quantum numbers of the two hole states and in the electron shell, the energy of the holes (not including the electron rest mass), the Coulomb interaction energies of the holes , and the decay widths . The last two columns show the minimum and maximum normalized ECEC halflives.
Tables 2, 3, and 4 list all, but no more than 5 transitions with the lowest quantum numbers of electron holes for each pair of the elements. If the spin is not fully determined, we took its lowest suggested value.
Rigorous calculations of the nuclear matrix elements (NME) based on the structure of nuclear states have not yet been performed. The objective here is to first select promising pairs of nuclei on the basis of rough estimates of the matrix elements, as these won’t significantly change the global picture. The halflives are normalized to the nuclear matrix element of , which roughly corresponds to the maximum evaluated value of NMEs for mediumheavy nuclei src . Transitions to excited states are suppressed due to dissimilarity of the nuclear wave functions suhonen ; Kolh11 .
NA  Transition  
5.52%  RuMo  0  2742 1  24.1 7.9 1.9  310  410  0.50  0.06  0.02  
23.7 7.9 1.9  410  410  0.06  0.06  0.01  
1.25%  CdPd  2737 1  16.5 5.9 4.1  110  110  24.35  24.35  0.74  
4.8 5.9 4.1  110  210  24.35  3.60  0.23  
5.1 5.9 4.1  110  211  24.35  3.33  0.21  
7.9 5.9 4.1  110  310  24.35  0.67  0.07  
8.5 5.9 4.1  110  410  24.35  0.09  0.02  
0.095%  XeTe  []  2853.2 0.6  1.2 1.8 1.5  210  210  4.94  4.94  0.16  
1.6 1.8 1.5  210  211  4.94  4.61  0.16  
5.2 1.8 1.5  210  310  4.94  1.01  0.08  
5.4 1.8 1.5  210  311  4.94  0.87  0.06  
6.1 1.8 1.5  210  410  4.94  0.17  0.02  
0.185%  CeBa  0  2315.32 0.07  27.513.3 0.4  110  110  37.44  37.44  0.93  
0.185%  CeBa  [0]  2349.5 0.5  6.713.3 0.4  110  110  37.44  37.44  0.93  
25.413.3 0.4  110  210  37.44  5.99  0.30  
25.813.3 0.4  110  211  37.44  5.62  0.28  
30.313.3 0.4  110  310  37.44  1.29  0.09  
30.513.3 0.4  110  311  37.44  1.14  0.08  
0.185%  CeBa  2392.1 0.6  17.113.3 0.4  110  210  37.44  5.99  0.21  
16.813.3 0.4  110  211  37.44  5.62  0.29  
12.313.3 0.4  110  310  37.44  1.29  0.07  
12.113.3 0.4  110  311  37.44  1.14  0.08  
11.213.3 0.4  110  410  37.44  0.25  0.03 