Q value of the 100Mo Double-Beta Decay

Penning trap measurements using mixed beams of 100Mo - 100Ru and 76Ge - 76Se have been utilized to determine the double-beta decay Q-values of 100Mo and 76Ge with uncertainties less than 200 eV. The value for 76Ge, 2039.04(16) keV is in agreement with the published SMILETRAP value. The new value for 100Mo, 3034.40(17) keV is 30 times more precise than the previous literature value, sufficient for the ongoing neutrinoless double-beta decay searches in 100Mo. Moreover, the precise Q-value is used to calculate the phase-space integrals and the experimental nuclear matrix element of double-beta decay.


Introduction
Neutrinos are one of the least understood fundamental particles. For half a century physicists thought that neutrinos, like photons, had no mass. But recent data from the neutrino oscillation experiments at SuperKamiokande [1], SNO [2], and KamLAND [3] overturned this view and confirmed that the neutrinos are massive particles. However, oscillation experiments can yield only the differences in the square of neutrino masses, therefore, no absolute mass values can be determined. In addition, another question remains concerning the fundamental character of neutrinos, whether being Dirac or Majorana particles. Neutrinoless double-beta decay (0νββ-decay) is a process which can address both issues raised above. This decay process is forbidden according to the Standard Model of Particle Physics since it violates the lepton-number conservation and is only allowed if neutrinos are massive Majorana particles.
The Heidelberg-Moscow Collaboration [4] has claimed the observation of the 0νββ-decay using high-sensitivity [5,6] 76 Ge semiconductor detectors [7]. 100 Mo is another suitable nucleus to study the 0νββ-decay for the following reasons. 100 Mo is rather easy to produce as enriched material and its 2νββ matrix element is known [8]. Furthermore, it has a high Q-value and the 0νββ-decay rate scales with Q 5 . The MOON [9] and NEMO-3 [10] experiments aim to use 100 Mo for the 0νββ-decay search. The signal for a 0νββ-decay in a detector measuring the total energy of both electrons would be a peak at the position of the Q-value of the involved transition. Thus it is vital for the ongoing search of the 0νββ-decay that the Q-value is known to a fraction of the expected detector resolution.
Additional motivation for the precise Q-value measurement is to calculate the precise phase-space integral G and the experimental nuclear matrix element of the two-neutrino double-deta decay [11]. The half-life T 1/2 of the double-beta decay process is expressed as for the 2νββ-decay and for the 0νββ-decay. Here M is the nuclear matrix element between the initial and final states of the decay, m ν is the effective neutrino mass [11] and m e is the electron rest mass. For the 2νββ-decay case T 2ν 1/2 can be derived from experiment and G 2ν calculated reliably. Hence using Eq. (1) the experimental M 2ν is estimated. In the case of the 0νββ-decay the experimental half-life is unknown and hence we give the half-life of 100 Mo as a function of the effective neutrino mass for a given set of matrix elements using the G 0ν value taken from this work.
In this paper we present the precise double-beta decay Q-value of 100 Mo and report on the development of an off-line ion source for producing stable (or long-lived) mixed-ion beams for fundamental physics studies. 2 Experimental method JYFLTRAP [12] is an ion trap experiment for high-precision mass measurements of radioactive ions [13,14,15,16,17]. JYFLTRAP has been used for direct Q EC -value measurements of the superallowed beta emitters [18,19,20] and recently for double-beta decay. Our Q-value measurement is based on a comparison of the ion cyclotron frequency of a stored ion in a Penning trap [21]. The cyclotron frequency ν c of an ion is given by where B is the magnetic field, m is the mass and q the charge of the ion. The ions of interest are produced at the Ion Guide Isotope Separator On-line (IGISOL) facility [22] which has an advantage that both mother and daughter nuclei can be produced in the same reaction. On the other hand, an off-line ion source was used to produce a mixed ion beam of long-lived (mother and daughter) nuclei. Thus it enabled the measurement of the cyclotron frequencies of the mother and daughter nucleus in a consecutive manner. Hence the Qvalue can be obtained by using the following formula: where m m and m d are the masses of the mother and daughter ions and ν d νm is their cyclotron frequency ratio. The daughter nucleus was used as a reference ion and its mass excess value was obtained from ref. [23].
In the case of the double-beta decay, 100 Mo (mother nucleus) and 100 Ru (daughter nucleus) ions were produced by using an off-line ion source shown in Fig. 1(A). The ion source consists of two electrodes and it is similar in size to the light ion guide used for indiced fusion reactions at the IGISOL facility [22]. One of the electrodes was designed to hold the metal plates or powder of the enriched element of need and the other one was the ground electrode. To create ions a continuous spark was ignited by applying 500 V between the two electrodes at a 15 mbar helium pressure. In general this off-line ion source can be used to produce any stable mixed-ion beam. In this work it was used at JYFLTRAP to create 76 Ge -76 Se and 100 Mo -100 Ru pairs of singly-charged ions.
Ions were extracted from the off-line source by helium flow and guided by the sextupole ion guide (SPIG) into a differential pumping stage where they were accelerated to 30 keV and mass-separated with a 55 0 dipole magnet with a mass resolving power M/∆M of ∼ 500. The ions were then transported to a radiofrequency quadrupole (RFQ) structure where they were cooled, accumulated and bunched [24]. Finally, they were injected into the double Penning trap system.
The JYFLTRAP Penning traps are placed inside the warm bore of a 7 T superconducting magnet and they are separated by a 2-mm channel. The first trap is called the purification Penning trap, where the mass-selective buffergas cooling technique is applied for further axial cooling and isobaric cleaning [25]. The mass resolving power of the purification trap was on the order of 10 5 . Figure 1(B) displays a mass spectrum of ions ejected from the purification trap after the buffer gas cooling. The purification trap was operated to center selectively the mass region A = 100. The one of the selected ion samples was transported to the second trap called the precision trap.
In addition, time-separated (Ramsey method) [26] dipolar excitation was applied in the precision trap to ensure a single ion species is selected [27]. This was performed in the following way: at first the mass-selective reduced cyclotron frequency ν + (removal frequency) was applied for one ion species as two time-separated fringes of 5 ms with a waiting time of 20 ms (5-20-5 ms). This increases the reduced cyclotron radius of the unwanted ions. At the end of the excitation the ions were sent back to the purification trap, thus the excited ions (i.e. the unwanted ions) will not pass the 2-mm channel between the traps. This method allows us to clean even isomeric states that are only 200 keV higher than the ground state (in the case of 54 Co) [28]. In the purification trap, the buffer-gas cooling technique was re-applied and finally a pure ion sample was transported to the precision trap for the cyclotron frequency (ν c ) measurement.
At first, the cyclotron frequency was determined by employing the time-offlight (TOF) technique [29] with an excitation time of 100 ms in a singlefringe quadrupolar excitation scheme (conventional excitation scheme). Once the cyclotron frequency was determined, a quadrupolar Ramsey excitation was applied in order to determine the cyclotron frequency with higher precision [30].
A typical TOF resonance is shown in Fig. 2 (top panel) for 100 Ru ions with a conventional excitation scheme and an excitation time of 100 ms. The middle and bottom panels in Fig. 2 show the Ramsey resonances for 100 Ru and 100 Mo ions, respectively. A Ramsey excitation with two time-separated fringes of 50 ms and a waiting time of 400 ms (50-400-50 ms) was applied for these particular cases.

Analysis and Results
The statistical standard deviation σ(ν c ) of the cyclotron frequency ν c can be estimated as [31] where N = number of detected ions, T ex = excitation time and K is an empirical constant independent of ion number and excitation time. In this experiment the precision was maximized by collecting a large number of ions and by applying long excitation times. Also a higher ν c improves the relative precision achieved with a strong magnetic field (see Eq. (1)). In addition, the Ramsey excitation yields a narrower central peak compared to the conventional excitation scheme. Therefore, using the Ramsey excitation the precision is improved by a factor of about 2 or more [32]. All these factors together allow us to determine the Q-value of the double-beta decay of 100 Mo with high-precision. outer statistical uncertainties (δr) [35] of the weighted average frequency ratio are 1.6×10 −9 and 1.8 ×10 −9 , respectively. The ratio of these values (Birge ratio) is very close to 1 which confirms that the scattering of the data is statistical. However, the larger error (in this case outer error) was used to i M Fig. 3. Measured cyclotron frequency ratios r i between 100 Mo + and 100 Ru + relative to the weighted average valuer.
determine the final uncertainty.
In the determination of the cyclotron frequency ratio the following systematic uncertainties were taken into account: The number of ions present in the trap can cause a shift in the cyclotron frequency. This is taken into account by plotting the center value of the cyclotron resonance as a function of the detected number of ions [36]. The cyclotron frequency equivalent to one ion in the trap was determined from this plot via a linear extrapolation to the observed 0.6 ions (detector efficiency = 60%). This extrapolated cyclotron frequency was taken as a final value. Hence the uncertainty due to the count rate comes together with the statistical uncertainty.
The drift of the magnetic field was taken into account by an interpolation of the reference frequencies measured before and after the cyclotron frequency measurement of the ion of interest. This linear interpolation does not take into account the short term fluctuations of the magnetic field. This was taken into account by adding ∆B/B = 3.22(16)×10 −11 /min [16] multiplied by the time difference between the two consecutive reference measurements quadratically to the uncertainty of the frequency ratio. As the Q-value is determined by the cyclotron frequency ratio between the mother and daughter having same A q , mass-dependent and other systematic uncertainties (if it exists) should cancel.
The Q-value can be derived from the final weighted average frequency ratio using Eq. (4). The term inside the parenthesis in Eq. (4) is small (∼ 2 − 3 × 10 −5 ), thus the uncertainty contribution from m d to the Q-value is negligible. Figure 4 shows the weighted average Q-values of 100 Mo with their uncertainties for two different excitation times and these two sets of data agree each other. Table 1 Weighted average frequency ratios (r) and the Q -values of 76 Ge and 100 Mo measured at JYFLTRAP. The final uncertainty normalized with the χ 2 is given in the parenthesis. # and T ex represent the number of doublet measurements and excitation scheme, respectively. Final frequency ratios and Q -values are indicated in bold.
Mother  The final weighted average cyclotron frequency ratiosr and Q-values of 76 Ge and 100 Mo with their corresponding uncertainties in the parenthesis are given in Table 1.

Discussion
The precise 100 Mo Q-value is used to calculate the phase-space integrals G of the 2νββ-decay and 0νββ-decays. A detailed formula for these calculations can  be found in ref. [11]. Results of the calculations are summarized in Table 2. The uncertainty in G is estimated solely from our Q-value uncertainty. The other uncertainty (in addition to the uncertainty from the Q-value) in G comes from the Fermi function approximation for the exact solution of the Dirac equation for a homogeneously charged sphere. The relativistic Fermi approximation limits the accuracy of G to some three digits.
Using Eq. (1) the value of the nuclear matrix element M 2ν is estimated experimentally to be 0.122±0.004 where the precise phase-space integral value is taken from this work and the recommended half-life T 1/2 = (7.1 ± 0.4) × 10 18 years is taken from [37]. A comparison between the computed nuclear matrix elements and the experimental one is shown in Table 3. Theoretical values of M 2ν vary by a factor of five whereas the experimental value lies in between. The matrix elements 0.197 and 0.152 of the spherical QRPA and SU(3) theories are consistent with the range of the experimental matrix elements. On the other hand the matrix element 0.108 computed by using the deformed SU(3) theory falls out side this range. This would suggest that deformation is not very important for the ground-state to ground-state 2νββ decay of 100 Mo.
In the case of 0νββ-decay finding an estimate for the experimental value of the nuclear matrix element M 0ν is not so straight-forward since the decay halflife is unknown in Eq. (2). However the expected half-life can be plotted as a function of the effective neutrino mass as shown in Fig. 5. We have used our value for the phase-space integral and a set of nuclear matrix elements M 0ν calculated by using the pnQRPA [42,43] and RQRPA [43] models. In these models the nucleon-nucleon short-range correlations have been accounted for combined ( ) Fig. 5. Expected half-life band as a function of the effective neutrino mass for 100 Mo. The ranges of the nuclear matrix element M 0ν come from [42] in the upper panel and from [43] in the lower panel. The values of the phase-space integrals were used from this work.
by two different methods, the Unitary Correlation Operator Method (UCOM) and Jastrow. For both models, the pnQRPA and the RQRPA, the range of computed matrix elements stems from the uncertainty in the value of the axial-vector coupling coefficient, g A = 1.0 -1.25, and from the variation in the value of the proton-neutron particle-particle interaction strength g pp , used to fit the experimental range of the 2νββ half-life.

Conclusion
In this article an off-line ion source is demonstrated for creating stable (or long-lived) mixed beams of single-charged ions. A precise double-beta decay Q-value, 3034.40±0.17 keV for 100 Mo, is determined, a candidate for searching the evidence of 0νββ-decay. The Q-value precision of about 200 eV was obtained for mass 100 region. Experimental phase-space integral G is obtained and using this value an experimental M 2ν value is derived for 100 Mo. In the case of neutrinoless double-beta decay half-life is estimated as a function of the effective neutrino mass for a set of nuclear matrix elements.