[orcid=0000-0001-8586-6134] [1] [orcid=0000-0003-0003-6022]

1]organization=Department of Physics, University of Jyväskylä,
addressline=P.O. Box 35, postcode=FI-40014, state=Jyväskylä, country=Finland
[orcid=0009-0009-9658-2386] [1] 2]organization=International Centre for
Advanced Training and Research in Physics (CIFRA), addressline=POB MG-12,
postcode=RO-077125, state=Bucharest-Măgurele,
country=Romania3]organization=Faculty of Physics, University of Bucharest,
addressline=405 Atomistilor, POB MG-11, postcode=RO-077125, state=Bucharest-
Măgurele, country=Romania4]organization="Horia Hulubei" National Institute of
Physics and Nuclear Engineering, addressline=30 Reactorului, POB MG-6,
postcode=RO-077125, state=Bucharest-Măgurele, country=Romania
5]organization=KU Leuven, addressline=Instituut voor Kern- en Stralingsfysica,
postcode=B-3001, state=Leuven, country=Belgium 6]organization=Université de
Bordeaux, addressline=CNRS/IN2P3, UMR 5797, postcode=F-33170, state=Gradignan,
country=France [orcid=0000-0002-9598-8415] [orcid=0000-0003-4632-7327]
[orcid=0000-0003-3514-6678] [orcid=0000-0002-9898-660X] [1]
[orcid=0000-0002-6103-2845] [orcid=0000-0003-1082-7602]
[orcid=0000-0002-1990-0848] [orcid=0000-0002-0053-1691]
[orcid=0000-0003-4448-7650] [orcid=0000-0002-8851-7826]
7]organization=Instituto de Fisica Corpuscular, addressline=CSIC-UV,
postcode=46980, state=Gradignan, country=Spain 8]organization=GANIL,
addressline=CEA/DSM-CNRS/IN2P3, Bd Henri Becquerel, postcode=14000,
state=Caen, country=France 9]organization=Université de Bordeaux, CNRS/IN2P3,
LP2I Bordeaux, addressline=UMR 5797, postcode=F-33170, state=Gradignan,
country=France [orcid=0000-0002-1215-2269] [orcid=0000-0002-4317-3628]
[orcid=0000-0002-5279-0820] [orcid=0000-0002-0451-125X]
[orcid=0000-0001-9798-0655] [orcid=0000-0003-3368-7307] [2]
10]organization=Extreme Light Infrastructure - Nuclear Physics,
addressline=Horia Hulubei National Institute for R&D in Physics and Nuclear
Engineering (IFIN-HH), postcode=077125, state=Bucharest-Magurele,
country=Romania

[orcid=0000-0001-9883-1256] 11]organization=Natural Resources Institute
Finland, addressline=Yliopistokatu 6B, postcode=FI-80100, state=Joensuu,
country=Finland [orcid=0000-0001-9207-5824] 12]organization=Finnish Institute
for Educational Research, University of Jyväskylä, addressline=P.O. Box 35,
postcode=FI-40014, state=Jyväskylä, country=Finland 13]organization=Center for
Theoretical Physics, Sloane Physics Laboratory,addressline=Yale University,
city=New Haven, postcode=Connecticut 06520-8120, state=Connecticut,
country=USA [orcid=0000-0003-0934-8727] [orcid=0000-0001-6515-2409]
[orcid=0000-0003-0276-6483]

[cor1]Principal corresponding authors [fn2]Present address: Facility for Rare
Isotope Beams, Michigan State University, 640 South Shaw Lane East Lansing, MI
48824, USA

# High-precision measurements of the atomic mass and electron-capture decay
$Q$ value of 95Tc

Zhuang Ge<EMAIL_ADDRESS>Tommi Eronen  [  Vasile Alin Sevestrean
<EMAIL_ADDRESS>[  [  [  [  [  Ovidiu Niţescu  Sabin Stoica
Marlom Ramalho  Jouni Suhonen<EMAIL_ADDRESS>Antoine de Roubin
Dmitrii Nesterenko  Anu Kankainen  Pauline Ascher  Samuel Ayet San Andres
Olga Beliuskina  Pierre Delahaye  [  [  [  Mathieu Flayol  Mathias Gerbaux
Stéphane Grévy  Marjut Hukkanen  Arthur Jaries  Ari Jokinen  Audric Husson
Daid Kahl  [  Joel Kostensalo  [  Jenni Kotila  [  [  Iain Moore  Stylianos
Nikas  Marek Stryjczyk  Ville Virtanen

###### Abstract

A direct measurement of the ground-state-to-ground-state electron-capture
decay $Q$ value of 95Tc has been performed utilizing the double Penning trap
mass spectrometer JYFLTRAP. The $Q$ value was determined to be 1695.92(13) keV
by taking advantage of the high resolving power of the phase-imaging ion-
cyclotron-resonance technique to resolve the low-lying isomeric state of 95Tc
(excitation energy of 38.910(40) keV) from the ground state. The mass excess
of 95Tc was measured to be $-$86015.95(18) keV/c2, exhibiting a precision of
about 28 times higher and in agreement with the value from the newest Atomic
Mass Evaluation (AME2020). Combined with the nuclear energy-level data for the
decay-daughter 95Mo, two potential ultra-low $Q$-value transitions are
identified for future long-term neutrino-mass determination experiments. The
atomic self-consistent many-electron Dirac–Hartree–Fock–Slater method and the
nuclear shell model have been used to predict the partial half-lives and
energy-release distributions for the two transitions. The dominant correction
terms related to those processes are considered, including the exchange and
overlap corrections, and the shake-up and shake-off effects. The normalized
distribution of the released energy in the electron-capture decay of 95Tc to
excited states of 95Mo is compared to that of 163Ho currently being used for
electron-neutrino-mass determination.

###### keywords:

Penning trapmass measurements ultra-low $Q$ valueelectron captureneutrino mass

Neutrino oscillations in atmospheric, solar, and reactor neutrinos have
confirmed that at least two neutrino mass eigenstates have non-zero rest mass.
However, these oscillations cannot assess the absolute mass scale, but only
the squared differences of the mass eigenstates [1, 2, 3]. Neutrinos are the
second most abundant particles in the universe, and play an important role on
cosmological scales [4]. Accurate measurements of the total neutrino mass
involve their imprint on the cosmic microwave background (CMB) as well as on
structure formation in the early universe.

The most direct method to measure the absolute mass scale of antineutrinos
involves studying the electron energy spectrum of $\beta^{-}$ decay. Though
the neutrinoless double $\beta^{-}$-decay experiments can be used to infer the
effective Majorana-neutrino mass from the measured lifetime, the exact
relation depends on the mediator model and relies on the calculation of the
involved transition matrix elements [5, 6, 7, 8]. The ongoing leading
experiment for the absolute neutrino mass scale determination is the Karlsruhe
Tritium Neutrino (KATRIN) $\beta^{-}$-decay experiment which is designed to
measure the electron-antineutrino mass, $m_{{\overline{\nu}_{e}}}$, with a
sensitivity of 0.2 eV/c2 at 90% C.L. Most recently, KATRIN has set a limit of
$m_{{\overline{\nu}_{e}}}$ < 0.8 eV/c2 (90% C.L.) [9, 10, 11]. Another
experiment, Project 8, takes advantage of the cyclotron radiation emission
spectroscopy (CRES) technique via measurements of the tritium end-point
spectrum. The new technique CRES will allow for an eventual sensitivity to
m${}_{{\overline{\nu}_{e}}}$ down to 0.04 eV/c2. The first frequency-based
neutrino mass limit of electron-weighted neutrino mass < 155 eV/c2 is
extracted from the background-free measurement of the continuous tritium
$\beta$ spectrum in a Bayesian (frequentist) analysis [12]. An alternative
method in the ECHo [13, 14, 15, 16] and HOLMES [17, 18] experiments, uses
electron capture (EC) on 163Ho, and has reached a current limit of 150 eV/c2
for the electron-neutrino mass [15].

A $Q$ value as small as possible is desired in these single decay experiments
for electron (anti)neutrino mass determination. The effective fraction of
decays in a given energy interval $\Delta{E}$ at the endpoint area will be
larger with a lower $Q$ value [19, 20]. Currently, only ground-state-to-
ground-state (gs-to-gs) decay cases 3H ($\beta$ decay) and 163Ho (electron
capture), are being used for direct neutrino-mass-determination experiments.
Ongoing intensive searches for isotopes undergoing $\beta$/EC decays from the
ground state to an excited state with a low $Q$ value are actively conducted
at JYFLTRAP, LEBIT, CPT, ISOTRAP and SHIPTRAP Penning traps [21, 22, 23, 24,
25, 26, 27, 28, 29, 28, 30, 31, 32, 33, 34]. Penning trap mass spectrometry
(PTMS) is the leading technique for accurate and precise mass and $Q$ value
determination, and it is hitherto the only direct method to measure the decay
$Q$ value to a sub-keV precision or better to verify whether a potential
candidate is an ultra-low (< 1 keV) $Q$-value transition or not.

In this article, we report on the first direct measurement of the gs-to-gs EC
$Q$ value of 95Tc with the JYFLTRAP PTMS. The precise $Q$ value obtained in
this study, in conjunction with nuclear energy level data for excited states
of 95Mo, is utilized to ascertain their ground-state-to-excited-state (gs-to-
es) $Q$ values. In the case of 95Tc, there are two potential low $Q$-value gs-
to-es EC transitions, that could be used for neutrino-mass detection. To
explore this potential, we have utilized two computational approaches, the
atomic self-consistent many-electron Dirac–Hartree–Fock–Slater method and the
nuclear shell model, to predict the partial half-lives and energy-release
distributions for the EC-decay transitions in question.

##  1 Experimental method

The experiment was performed at the Ion Guide Isotope Separator On-Line
facility (IGISOL) using the JYFLTRAP double Penning trap mass spectrometer
[35] at the University of Jyväskylä, Finland [36, 37].

To generate 95Tc ions, a natural Mo target foil was irradiated with a few
$\mu$A proton beam at 45 MeV from the K-130 cyclotron at the Accelerator
Laboratory of the University of Jyväskylä. A helium-filled small volume gas
cell was used to stop the recoils produced from the proton-induced fusion-
evaporation reaction, and the ions were extracted using gas flow and guided
through a sextupole ion guide [38] with a combination of DC and RF fields.
Subsequently, the ions were accelerated with a 30 kV electric potential,
followed by mass separation using a 55∘ dipole magnet with a typical mass
resolving power of $M/\Delta{M}$ $\approx$ 500\. After isobaric separation for
ions of $A/q=95$, including the reaction products 95Nb+, 95mTc+, 95Tc+ and
95Mo+, they were directed to a radiofrequency-quadrupole cooler-buncher (RFQ-
CB) [39], where they underwent accumulation, cooling, and bunching.

Decay-daughter ions of 95Mo+ were prepared using the upstairs offline glow-
discharge ion source. A 90∘ electrostatic bender selected ions either from the
online target station or the offline ion source for downstream transmission.

JYFLTRAP comprises two cylindrical Penning traps in a 7-T superconducting
solenoid. The first trap, functioning as a purification trap, is filled with
helium buffer gas and is used for isobaric purification through the sideband
buffer gas cooling technique [40]. This method achieves purification with a
mass resolving power of $\approx$ $10^{5}$. In the purification trap, all
cooled and centered ions (95Nb+, 95mTc+, 95Tc+, and 95Mo+) are initially
excited to a large magnetron motion orbit. This is accomplished by applying a
dipole excitation at the magnetron motion frequency $\nu_{-}$ for
approximately 11 ms. Subsequently, a quadrupole excitation is executed for
approximately 100 ms to center the ions of interest through collisions with
the buffer gas. The buffer gas cooling technique eliminated 95Mo+ but did not
have enough mass resolving power to remove the other aforementioned ions. To
prepare mono-isotopic samples of 95Tc+, the coupling of the dipolar excitation
with Ramsey’s method of time-separated oscillatory fields [41] and the phase-
imaging ion-cyclotron-resonance (PI-ICR) technique [42, 43] was utilized, as
described in details in [34]. A plot of the Ramsey-type dipole excitation
frequency scan with a 5 ms (On) - 17 ms (Off) - 5 ms (On) excitation pattern
in the second (precision) trap, filtered by the positional gates using the PI-
ICR identification with a 755 ms phase accumulation time, is shown in Fig. 1.

For $Q$-value measurements, the PI-ICR method is used to measure the cyclotron
frequency, $\nu_{c}={qB}/({{2\pi}m})$, where $B$ is the magnetic field
strength, $q$ is the charge and $m$ the mass of the stored ion. The PI-ICR
technique [43] provides around 40 times better resolving power than the
conventional time-of-flight ion-cyclotron-resonance (TOF-ICR) method [43, 44,
45]. Two timing patterns are needed for the determination of $\nu_{c}$. The
patterns differ only in their quadrupolar conversion pulse, separated in time
by the defined phase-accumulation time, $t_{acc}$. The phase images of these
two are projected onto a position-sensitive MCP detector after the trap.
Additionally a center point, measured without any excitations, is needed for
angle determination.

The angle between two phase images of the projected radial motions with
respect to the center spot is denoted as $\alpha_{c}=\alpha_{+}-\alpha_{-}$,
where $\alpha_{+}$ and $\alpha_{-}$ represent the polar angles of the
cyclotron and magnetron motion phases. The cyclotron frequency $\nu_{c}$ is
derived from: $\nu_{c}={(\alpha_{c}+2\pi n_{c})}/{2\pi{t_{acc}}}$, where
$n_{c}$ represents the full number of revolutions made by the measured ions
during the phase accumulation time $t_{acc}$. Different accumulation times for
95Tc+ were utilized to unambiguously assign $n_{c}$. An accumulation time of
574 ms was employed for the actual measurements to determine the final
$\nu_{c}$ for both 95Tc+ and 95Mo+ ions; the choice also ensures that any
leaked isobaric contaminant would not overlap with the ions of interest. The
positions of the phase spots for magnetron and cyclotron motion were carefully
selected to maintain an angle $\alpha_{c}$ within a few degrees. This choice
aimed to minimize the shift in the $\nu_{c}$ ratio of the 95Tc+-95Mo+ pair due
to the conversion of the cyclotron motion to magnetron motion and the possible
distortion of the ion-motion projection onto the detector to a level well
below 10-10 [44]. The excitation of the $\nu_{+}$ delay was systematically
scanned over one magnetron period, while the extraction delay varied over one
cyclotron period. This accounted for any residual magnetron and cyclotron
motion that might have shifted the different spots. The total data
accumulation time of interleaved measurements of $\nu_{c}$ for 95Tc+-95Mo+
ions was $\approx$ 4.9 hours, respectively.

The gs-to-gs electron-capture $Q$ value, $Q_{\rm EC}$, can be derived from the
mass difference of the decay pair:

$Q_{\rm EC}=(M_{p}-M_{d})c^{2}=(R-1)(M_{d}-qm_{e})c^{2}+(R\cdot B_{d}-B_{p}),$
(1)

where $M_{p}$ and $M_{d}$ represent the masses of the parent and daughter
atoms, respectively, and $R$ (=${\nu_{c,d}}/{\nu_{c,p}}$) denotes their
cyclotron frequency ratio for singly charged ions ($q=1$), with $m_{e}$ being
the mass of an electron. The electron binding energies of the parent and
daughter atoms, denoted as $B_{p}$ and $B_{d}$, are neglected due to their
small values (on the order of a few eVs [46]), and $R$ is close to 1. Since
both the parent and daughter ions have the same $A/q$ and their relative mass
difference $\Delta M/M<10^{-4}$, the mass-dependent error becomes negligible
compared to the statistical uncertainty achieved in the measurements. Also,
the contribution of uncertainty to the $Q$ value from the mass uncertainty of
the reference (daughter), which is 0.12 keV/c2 for 95Mo [47], can be
neglected.

Figure 1: (Color online). (a) Ramsey-type dipole excitation frequency scan with a 5 ms (On) - 17 ms (Off) - 5 ms (On) excitation pattern in the second trap filtered by the positional gates shown in (b) using the PI-ICR identification (755 ms phase accumulation time) plot. The used angular gates are highlighted. The vertical dashed blue line shows the chosen optimal frequency to transmit 95Tc ions while suppressing the others.  Table 1: Potential candidate transitions of initial state (ground state) of parent nuclei 95Tc (9/2+) to the final states (excited states) of daughter 95Mo with ultra-low $Q$ values. The first column lists the excited final state of 95Mo for the low $Q$ value transition. The decay type is provided in the second column. The third and fourth columns present the derived decay $Q_{\rm EC}$ values in keV, sourced from literature (Lit.) [47] and this work (T. W.), respectively. The fifth column displays the experimental excitation energy $E^{*}$ with its experimental error [48, 49, 50] in keV. The sixth column shows the confidence ($\sigma$) of the $Q^{*}_{\rm EC}$ being positive/negative. Columns seven to nine, denoted as $\Delta{x}$, represent the distance of $Q_{\mathrm{EC}}$ values to the computed atomic relaxation energy following the electron capture $\varepsilon_{x}$ in the daughter atoms [51]. FNU means forbidden non-unique. Spin-parity assignments and energy values enclosed in braces {} signify uncertain assignments or uncertainties in excitation energy, resulting in uncertainties in the decay type or decay energy. All the energies are in unit of keV.  Final state | Decay type |  $Q^{*}_{\rm EC}$ (Lit.) |  $Q^{*}_{\rm EC}$ (T. W.) | E∗ |  $Q/\delta Q$ (T. W.) |  $\Delta_{\mathrm{K}}$ (T. W.) |  $\Delta_{\mathrm{L1}}$ (T. W.) |  $\Delta_{\mathrm{L2}}$ (T. W.)
---|---|---|---|---|---|---|---|---
95Mo (9/2+) | allowed | 15.6(50) | 20.52(61) | 1675.40(60) | 33 | 0.47(61) | 17.64(61) | 17.89(61)
95Mo ({7/2+, 9/2+}) | {allowed} | 8.0(51) | 12.9(10) | 1683.0(10) | 13 |  | 10.0(10) | 10.3(10)
95Mo (1/2+) | 4th FNU | -1.0({51}) | 3.92({13}) | 1692({}) | {29} |  | 1.04({13}) | 1.29({13})

##  2 Results and discussion

The determination of $Q_{\rm EC}$ depends on the measured cyclotron frequency
ratio $R$ via Eq. 1. Two data sets for 95Tc+-95Mo+ were collected. A full
scanning measurement of the magnetron phase, cyclotron phase and center spot
in sequence (one cycle) was completed in less than 5 minutes for each ion
species. In the analysis, the position of each spot was fit with the maximum
likelihood method. A few cycles were summed to have reasonable statistics for
fitting. The phase angles were calculated accordingly to deduce the cyclotron
frequencies of each ion species. The cyclotron frequency $\nu_{c}$ of the
daughter 95Mo+ as a reference was linearly interpolated to the time of the
measurement of the parent 95Tc+ (ion of interest) to deduce the cyclotron
frequency ratio $R$. Only the bunches with less than five detected ions were
considered in the data analysis in order to reduce a possible cyclotron
frequency shift due to ion-ion interactions [52, 53]. The count-rate related
frequency shifts were not observed in the analysis. The temporal fluctuation
of the magnetic field $\delta_{B}(\nu_{c})/\nu_{c}=\Delta t\times
2.01(25)\times 10^{-12}$/min [42], where $\Delta t$ is the time interval
between two consecutive reference measurements, is considered in the final
results. Contribution of temporal fluctuations of the magnetic field to the
final frequency ratio uncertainty was less than 10-10 since the parent-
daughter measurements were interleaved with $\Delta t$ < 10 minutes. The
frequency shifts in the PI-ICR measurement due to ion image distortions were
well below the statistical uncertainty and thus ignored in the calculation of
the final uncertainty. Furthermore, decay pair ions 95Tc+-95Mo+, being mass
doublets, cancel many of the systematic uncertainties in the cyclotron
frequency ratio.

The weighted mean ratio $\overline{R}$ of all single ratios was calculated
along with the inner and outer errors to deduce the Birge ratio [54]. The
maximum of the inner and outer errors was taken as the weight to calculate
$\overline{R}$. In Fig. 2, results of the analysis including all data with
comparison to literature values are demonstrated. The final parent-to-daughter
frequency ratio $\overline{R}$ with their uncertainty is determined to be
1.000 019 183 9(15). The corresponding gs-to-gs $Q$ values is 1695.92(13) keV.

Figure 2: (Color online). The measured experimental results from this work
compared to the literature values [55, 47]. The deviations of the individually
measured cyclotron frequency ratios $R$ ($\nu_{c}$(95Mo+)/$\nu_{c}$(95Tc+))
from the measured value $\overline{R}$ (left axis) and $Q$ value (right axis)
in this work are compared to values adopted from AME2020. The red points with
uncertainties represent individual measurements using the PI-ICR method.
Vertical brown dashed lines separate measurements conducted at different time
slots. The weighted average value $\overline{R}$ is depicted by the solid red
line, and its 1$\sigma$ uncertainty band is shaded in red. The dashed blue
line illustrates the difference between our new value and the one referenced
in AME2020, with its 1$\sigma$ uncertainty area shaded in blue.

The gs-to-gs $Q_{\rm EC}$ value of 1695.92(13) keV from this work is $\approx$
37 times more precise than that deduced from the evaluated masses in AME2020
[55, 47]. The measured $Q_{\rm EC}$ value has a deviation of 4.9(50) keV from
the AME2020 value and is $\approx$1$\sigma$ larger. The $Q_{\rm EC}$ value in
AME2020 is derived primarily from two ${\beta^{+}}$-decay experiments
95Tc(${\beta^{+}}$)95Mo [56, 57]. Combined with the atomic mass of 95Mo (mass
excess: $-$87711.87(12) keV/c2) from AME2020 [47, 55], we deduce the mass
excess of its parent nucleus 95Tc (9/2+) to be $-$86015.95(18) keV/c2. The
mass of 95Tc in AME2020 is primarily evaluated from ${\beta^{+}}$-decay
experiments 95Tc(${\beta^{+}}$)95Mo and 95Ru(${\beta^{+}}$)95Tc with influence
of 97.4% and 2.6%, respectively [56, 58, 59].

The high-precision electron-capture energy from this work, together with the
nuclear energy level data from Ref. [48, 49, 50] of the excited states of 95Mo
as tabulated in Table 1, was used to determine the gs-to-es $Q$ value
($Q^{*}_{\rm EC}$) of three candidate states as shown in Table 1. The newly
determined $Q^{*}_{\rm EC}$ confirm that the decay of the ground state of 95Tc
to the three excited states of interest are energetically allowed.

In case of EC, the closer the $Q$ value of the decay to one of the ionization
energies of the captured electrons, the larger the resonance enhancement of
the rate near the end-point, where the effects of a non-vanishing neutrino
mass are relevant. The event-rate dependence on the $Q$ value near the end-
point for EC is steeper than that for $\beta^{-}$ decay. As tabulated in Table
1, $\Delta_{x}$ gives the distance of the $Q_{\mathrm{EC}}^{*}$ value to the
computed atomic relaxation energy $\varepsilon_{x}$ following the capture of
electrons in the allowed daughter atomic shells ($x=$ K, L1, L2, and other
electrons from s-levels and p1/2-levels from the third and higher shells). For
the state with the excitation energy of 1675.40(60) keV, the captures of
electrons occupying the K and higher shells for the transition 95Tc (9/2+)
$\rightarrow$ 95Mo∗ are energetically allowed, while for states with the
excitation energy of 1683.0(10) keV and 1692 keV, only electrons from s-levels
and p1/2-levels from the second (L) and higher shells can possibly be captured
due to angular momentum conservation and the finite overlap of their wave
function with the nucleus. The transition 95Tc (9/2-) $\rightarrow$ 95Mo∗
(1692 keV), giving the values of 1.04($\\{13\\}$) keV and 1.29($\\{13\\}$) keV
for the distance of $Q_{\mathrm{EC}}^{*}$ to the computed atomic relaxation
energy following the electron capture $\varepsilon_{L1}$ = 2.878 keV and
$\varepsilon_{L2}$ = 2.632 keV, is of the decay type of 4th FNU (forbidden
non-unique). It has a long half-life which will result in an extremely low
fraction of events landing near the endpoint. This transition is not of
interest for future neutrino mass determination due to the low branching
ratio. To confirm whether the emitted neutrino energy 1.04($\\{13\\}$) keV is
ultra-low, further high-precision measurements of the excitation energy of the
state are required. The parity of the 1683.0(10) keV state needs to be
determined to verify the decay type of the transition to this state. The gs-
to-gs $Q$ value of 95Tc is now well refined to sub-keV uncertainty, combined
with the energy level of 1675.40-keV state, an ultra-low distance (0.47 keV)
of $Q_{\mathrm{EC}}^{*}$ to the computed atomic relaxation energy following
the electron capture $\varepsilon_{k}$ = 20.054 keV [51] is observed.

##  3 Theoretical predictions

In the following we employed two calculation methods in order to predict the
transition half-life and the distribution of energy released in the decay,
namely atomic many-electron Dirac–Hartree–Fock–Slater (DHFS) self-consistent
method and the nuclear shell model many-nucleon framework. The DHFS framework
has been proven adequate for this type of calculations in our previous work
[60].

Using the DHFS method we obtained the wave-functions and the energy levels of
the atomic electrons. The calculations were performed for both the initial
atom and the final atom. The initial atom was in its ground state. For the
final atom, we considered all possible states with the electron configuration
of the initial atom having a hole in each shell from which the electron could
be captured. We assumed a central field $V(r)$ for the atomic system. The
potential has three components as given in [61]:

$V_{\mathrm{DHFS}}(r)=V_{\mathrm{nuc}}(r)+V_{\mathrm{el}}(r)+V_{\mathrm{ex}}(r),$
(2)

where $V_{\mathrm{nuc}}(r)$, $V_{\mathrm{el}}(r)$, $V_{\mathrm{ex}}(r)$ are
the nuclear, electronic and exchange potential. The nuclear potential takes
into account a realistic Fermi charge distribution in the nucleus [62]. The
electronic potential is generated based on the charge distribution of the
entire electron cloud. The procedure is iterative: after solving for the wave
functions of the electrons, the charge distribution is recalculated and thus
the potential as well. Then the computation is restarted until convergence.
The exchange potential assures the correct asymptotic behaviour of the
potential at $r\rightarrow\infty$. For the atomic structure calculations we
made use of the RADIAL subroutine package [61], which also contains the DHFS.F
code.

We denote the electron states as $\ket{(n,\kappa)}$, where $n$ is the
principal quantum number and $\kappa$ is the relativistic quantum number. The
atomic relaxation energy following the capture of an electron from the
$x=(n,\kappa)$ shell is denoted as $\varepsilon_{x}$. The values for
$\varepsilon_{x}$ are calculated according to the refined energy conservation
in [60]:

$\varepsilon_{x}=|T_{\rm g.s.}|-|T_{x}|,$ (3)

where $T_{\rm g.s.}$ and $T_{x}$ are the total binding energy of the final
atom in the ground state and in the excited state with a hole in the $x$
shell.

For allowed transitions the energy distribution of an EC event is calculated
as a sum over all atomic shells with $\kappa=\pm 1$ as

$\rho(E)=\frac{G_{\beta}^{2}}{(2\pi)^{2}}C\sum_{x}n_{x}\beta_{x}^{2}B_{x}S_{x}p_{\nu}E_{\nu}\frac{\Gamma_{x}/(2\pi)}{\left(E-\varepsilon_{x}\right)^{2}+\Gamma_{x}^{2}/4},$
(4)

where $B_{x}$ and $S_{x}$ are the exchange and overlap corrections, and the
shake-up and shake-off corrections, respectively, as discussed in the
following subsections. Here we go beyond the formalism used in [27] by adding
the shake-up and shake-off corrections into the energy distribution $\rho(E)$.
Here $E$ is related to the energy of the neutrino $E_{\nu}$ and the Q value as
$E=Q^{*}_{\rm EC}-E_{\nu}$. The momentum of the neutrino is denoted as
$p_{\nu}=\sqrt{E_{\nu}^{2}-m_{\nu}^{2}}$. The Coulomb amplitude is represented
as $\beta_{x}$, while $n_{x}$ is the relative occupancy of the shell. The
intrinsic line-widths of Breit–Wigner resonances centered at $\varepsilon_{x}$
are denoted as $\Gamma_{x}$ and are taken from [63]. The Fermi constant
$G_{\rm F}$ and the Cabibbo angle $\theta_{\rm C}$ are combined in
$G_{\beta}=G_{\rm F}\cos\theta_{\rm C}$. The nuclear structure information is
contained in the shape factor $C$ in terms of the nuclear form factor
${}^{A}F_{101}^{(0)}$ [64]:

$C=\left[{}^{A}F_{101}^{(0)}\right]^{2}=\left[-\frac{g_{\rm
A}}{\sqrt{2J_{i}+1}}M_{\mathrm{GT}}\right]^{2},$ (5)

where $M_{\mathrm{GT}}$ is the Gamow–Teller nuclear matrix element [65]. The
angular momentum of the initial nucleus is denoted as $J_{i}$, while the
strength of the weak axial coupling is represented as $g_{\rm A}$. For this
calculation, we took values for $g_{\rm A}$ in the conservative interval 0.7
to 1 [66, 67, 68] and present the partial half-life corresponding to the mean
decay rate in Table 3. The mean rate corresponds to a $g_{\rm A}$ value equal
to $0.857$. The half-life values for all $g_{\rm A}$ are between $-33.3\%$ and
$+36.1\%$ of each mean value.

We define $\lambda(E)$ as the total decay probability in the interval $(0,E)$:

$\lambda(E)=\int_{0}^{E}\rho\left(E^{\prime}\right)dE^{\prime}$ (6)

and the total decay constant is calculated as
$\lambda\equiv\lambda\left(Q^{*}_{\rm EC}-m_{\nu}\right)$. After integration,
using the narrow-width approximation, the decay rate can be written as

$\lambda=\sum_{x}\lambda_{x},$ (7)

with the partial decay constant defined as

$\lambda_{x}=\frac{G_{\beta}^{2}}{(2\pi)^{2}}Cn_{x}\beta_{x}^{2}B_{x}S_{x}p_{\nu
x}\left(Q_{\mathrm{EC}}^{*}-\varepsilon_{x}\right),$ (8)

where $p_{\nu
x}=\sqrt{\left(Q_{\mathrm{EC}}^{*}-\varepsilon_{x}\right)^{2}-m_{\nu}^{2}}$.

While an electron is captured by the nucleus, the other electrons (the
spectator electrons) can undergo some processes which affect the decay rate.
In the following subsections we will present a couple of corrections related
to those processes.

###  3.1 Exchange and overlap corrections

In the lowest-order approximation of the EC-decay process an electron is
captured from an atomic shell $x$ by the nucleus and a hole in this atomic
shell is observed. However, the electrons are identical particles and one
cannot distinguish between them leading to the possibility of a higher-order
process where first an electron is captured from shell $y$, different from
$x$, and simultaneously an electron from shell $x$ is promoted to shell $y$
and, again, the hole is observed in shell $x$. In both situations the hole
appears in the same atomic shell, thus making those two paths for the EC decay
indistinguishable from one another. The related higher-order correction is
called the exchange correction.

After the EC decay the atomic number of the nucleus decreases by one unit and
a hole is left in the daughter nucleus. Those changes affect the spectator
electrons and thus their initial $\ket{(m,\kappa)}$ and final
$\ket{(m,\kappa)^{\prime}}$ wave functions do not perfectly overlap anymore,
i.e., $\bra{(m,\kappa)^{\prime}}\ket{(m,\kappa)}\neq 1$. We use here the
generalized formula from [69]:

$B_{n\kappa}=\left|\frac{b_{n\kappa}}{\beta_{n\kappa}}\right|^{2},$ (9)

where $\beta_{n\kappa}$ is the previously mentioned Coulomb amplitude, and

$\displaystyle b_{n\kappa}=$
$\displaystyle\left[\prod_{m,\mu}\bra{(m,\mu)^{\prime}}\ket{(m,\mu)}^{n_{m\mu}}\right]\bra{(n,\kappa)^{\prime}}\ket{(n,\kappa)}^{-\frac{1}{2|\kappa|}}$
(10) $\displaystyle\times\left[\beta_{n\kappa}-\sum_{\begin{subarray}{c}m\neq
n\end{subarray}}\beta_{m\kappa}\frac{\bra{(m,\kappa)^{\prime}}\ket{(n,\kappa)}}{\bra{(m,\kappa)^{\prime}}\ket{(m,\kappa)}}\right].$

This higher-order correction is called the overlap correction.

###  3.2 Shake-up and shake-off effects

During the EC decay, a spectator electron can be promoted to upper unoccupied
shells, and this process is called the shake-up effect. The spectator electron
can also be ejected to the continuum, and this process is called the shake-off
effect. We consider a maximum of one spectator electron undergoing this
transition, such that in the final state we would have at most two holes in
the atomic shell. We use the formalism developed in [69] to end up with the
following expression for the $S_{x}$ factor of Eq. (4)

$S_{n\kappa}=1+\sum_{m,\mu}P_{m\mu}.$ (11)

Here $P_{m\mu}$ represents the probability that a spectator electron from
$\ket{(m\mu)}$ undergoes the shake-up or shake-off process. A spectator
electron can remain in the initial shell, can swap shells with another
spectator electron or be promoted or ejected. Thus $P_{m\mu}$ can be computed
as unity minus the probability for the electron to remain in the initial shell
and the probability to swap shells, as

$\displaystyle P_{m\mu}=$ $\displaystyle
1-|\bra{(m,\mu)^{\prime}}\ket{(m,\mu)}|^{2n_{m\mu}}$ (12)
$\displaystyle-\sum_{l\neq
m}n_{l\mu}^{\prime}n_{m\mu}|\bra{(l,\mu)^{\prime}}\ket{(m,\mu)}|^{2},$

where $n_{x}$ represents the occupancy of the shell $x$.

We also computed the impact of the two corrections on our theoretical
calculations. Both the exchange and overlap corrections and the shake-up and
shake-off effects increase the total decay rate by 8.1% and 8.6%,
respectively. The impact of exchange and overlap corrections is 7.5% and 7.8%.
In comparison, the shake-up and shake-off effects account for 0.61% and 0.64%.
The first value in each pair of numbers refers to the decay corresponding to
$Q^{*}_{\rm EC}=20.52$ keV while the second one refers to $Q^{*}_{\rm
EC}=12.9$ keV. This increase in the decay rate is the result of the fact that
both corrections open new channels for the decay process.

Table 2: Comparison of the experimental excitation energies $E_{\rm exc}$ (measured in keV), electric quadrupole moments $Q$ (measured in $e$barns), and magnetic dipole moments $\mu$ (measured in nuclear magnetons, $\mu_{N}$), with those computed by the NSM using the jj45pna, jj45pnb, and glekpn Hamiltonians. The experimental evaluated energies are sourced from [50] and the nuclear moments from [70, 71, 72, 73]. The calculations employed effective charges of $e_{\rm eff}^{p}=1.5e$ for protons and $e_{\rm eff}^{n}=0.5e$ for neutrons, and the bare g-factors $g_{l}(p)=1$, $g_{l}(n)=0$, $g_{s}(p)=5.585$, and $g_{s}(n)=-3.826$ were used for determining the nuclear moments. Experimental Evaluation | jj45pna | jj45pnb | glekpn
---|---|---|---
Isotope $(J^{\pi})$ |  $E_{\rm exc}$ (keV) |  $Q$ ($e$barn) |  $\mu$ ($\mu_{N}$) |  $E_{\rm exc}$ (keV) |  $Q$ ($e$barn) |  $\mu$ ($\mu_{N}$) |  $E_{\rm exc}$ (keV) |  $Q$ ($e$barn) |  $\mu$ ($\mu_{N}$) |  $E_{\rm exc}$ (keV) |  $Q$ ($e$barn) |  $\mu$ ($\mu_{N}$)
95Tc -$(\frac{9}{2}^{+})$ | 0 | - | 5.94(6) | 11 | -0.1392 | 6.3286 | 0 | -0.1508 | 6.3342 | 0 | 0.0784 | 6.3127
95Mo $(\frac{5}{2}^{+})$ | 0 | -0.022(1) | -0.9132(3) | 0 | -0.0379 | -0.9418 | 0 | -0.0238 | -0.9768 | 574 | -0.3359 | -0.8376
95Mo $(\frac{3}{2}^{+})$ | 204 | - | -0.404(12) | 151 | 0.0672 | -0.0734 | 165 | 0.0656 | -0.1335 | 816 | -0.0782 | 0.6095

###  3.3 Nuclear Matrix Elements

For the nuclear-structure calculation we employed the NuShellX@MSU [74] code
using the shell-model formalism. We used interactions jj45pna [75], a two-
nucleon potential with a perturbative G-matrix approach with the single
particle energies adjusted in the Coulomb part to reproduce the recent results
in [76] and jj45pnb [77], both sharing the same jj45pn model space consisting
of the proton $0f_{5/2}$-$1p_{3/2}$-$1p_{1/2}$-$1g_{9/2}$ orbitals and the
neutron $0g_{7/2}$-$1d_{5/2}$-$1d_{3/2}$-$2s_{1/2}$-$0h_{11/2}$ orbitals with
no truncations. Additionally, we employed the glekpn [78] interaction with the
model space consisting of
$0f_{7/2}$-$0f_{5/2}$-$1p_{3/2}$-$1p_{1/2}$-$1g_{9/2}$ proton orbitals and
$0g_{9/2}$-$0g_{7/2}$-$1d_{5/2}$-$1d_{3/2}$-$2s_{1/2}$ neutron orbitals. For
this model space, we applied truncations by fixing the occupation of the
$0f_{7/2}$ proton orbital to 8 protons and the $0g_{9/2}$ neutron orbital to
10 neutrons, resulting in a shell-model closed core up to the magic numbers 28
for protons and 50 for neutrons. Lastly, for this interaction three sets of
single-particle energies are available, we utilized the one suitable for the
mass region $A=94-98$. To assess the validity of the applied interactions, we
have computed the energy-level schemes for both jj45pn model-space
interactions as can be seen in Figure LABEL:fig:Level-spectra. The glekpn
interaction did not reproduce the level schemes correctly so that the related
energy spectrum is not shown here. The results indicate that the jj45pnb
interaction provides better agreement with the evaluated data.

Further, we have compared the nuclear moments obtained using the three
interactions with available evaluated experimental data, as presented in Table
2. Both the jj45pna and jj45pnb interaction showed a higher level of agreement
with the observed values, showcasing their applicability and efficiency for
the current study, whereas the Hamiltonian glekpn performs quite poorly.
Nevertheless, engaging the glekpn interaction demonstrates the importance of
exploring different interactions to fully understand their performance and
limitations in this nuclear mass region.

###  3.4 Results of the calculations

We computed the energy levels of the daughter nucleus to identify the
theoretical states corresponding to the experimental states of interest
($1675.4$ keV, $1683$ keV). We compared the theoretical energies with the
experimental ones and concluded that the following are the theoretical states
closest to the experimental ones: for the experimental state with $J_{f}=9/2$
and $E^{*}=1675.4$ keV the best matches were $E^{*}_{\rm th}=1748$ keV for
jj45pna, $E^{*}_{\rm th}=1897$ keV for jj45pnb, and $E^{*}_{\rm th}=1703$ keV
for glekpn. All the mentioned theoretical states have $J_{f}=9/2$. For the
experimental state having the energy $E^{*}=1683$ keV and with the angular
momentum and parity uncertain {7/2+, 9/2+}, the closest correspondence is for
jj45pna the energy $E^{*}_{\rm th}=1584$ keV, for jj45pnb the energy
$E^{*}_{\rm th}=1642$ keV, and for glekpn the energy $E^{*}_{\rm th}=1707$
keV. The mentioned three theoretical states have the angular momentum and
parity $7/2^{+}$.

Table 3: Computed mean half-lives using $g_{\rm A}=0.857$ (see the main text) for the EC decay of 95Tc to the two excited states in 95Mo (with experimental energies $E^{*}=1675.4$ keV and $1683$ keV), using three shell-model interactions for the Gamow-Teller matrix element, with their experimental $Q$ values shown in column 1. The second column indicates the used interactions and the third column the Gamow–Teller nuclear matrix element [65]. The computed total half-life and partial half-lives are demonstrated in columns 4-12. The atomic subshells are denoted using the X-ray notation.  $Q^{*}_{\rm EC}$ | interaction | $M_{\rm GT}$ | Total half-life | $\mathrm{~{}K}$ | $\mathrm{~{}L}1$ | $\mathrm{~{}L}2$ | $\mathrm{M}1$ | $\mathrm{M}2$ | $\mathrm{N}1$ | $\mathrm{N}2$ | $\mathrm{O}1$
---|---|---|---|---|---|---|---|---|---|---|---
(keV) |  |  | ($10^{3}$yr) | ($10^{7}$yr) | ($10^{4}$yr) | ($10^{5}$yr) | ($10^{4}$yr) | ($10^{6}$yr) | ($10^{5}$yr) | ($10^{7}$yr) | ($10^{6}$yr)
$20.52$ | jj45pna | $-0.00696$ | $877$ | $20.0$ | $117$ | $708$ | $640$ | $267$ | $223$ | $146$ | $491$
| jj45pnb | -0.016533 | $155$ | $3.55$ | $20.8$ | $125$ | $81.6$ | $47.4$ | $39.4$ | $25.8$ | $87.1$
| glekpn | 0.0070667 | $850$ | $19.4$ | $114$ | $587$ | $446$ | $259$ | $216$ | $141$ | $477$
12.9 | jj45pna | $-0.02412$ | $173$ | - | $24.1$ | $143$ | $79.7$ | $46.0$ | $37.5$ | $24.5$ | $82.6$
| jj45pnb | -0.0198667 | $255$ | - | $35.6$ | $210$ | $117$ | $67.9$ | $55.3$ | $36.1$ | $122$
| glekpn | 0.2296 | $1.9$ | - | $0.266$ | $1.57$ | $0.880$ | $0.508$ | $0.414$ | $0.271$ | $0.911$
Figure 4:  Normalized distributions of released energy in the EC decay of 95Tc
in the transitions to the excited states of 95Mo, as functions of
$E-Q^{*}_{\rm EC}$. The experimental excitation energies are $E^{*}=1675.40$
keV and $E^{*}=1683.0$ keV, while the corresponding $Q$ values are $Q^{*}_{\rm
EC}=20.52$ keV (green) and $Q^{*}_{\rm EC}=12.9$ keV (orange). The K, L1, L2,
M1 and N1 notations indicate sub-shells from which the electron was captured.
The M2, N1 N2 and O1 sub-shells are harder to distinguish and are not labeled.
The inset indicates an enlarged endpoint region showing the effect of neutrino
masses of 0.8 eV and 0 eV. The dotted lines depict the spectra for a massless
neutrino, while the solid lines correspond to a neutrino mass of 0.8 eV.
Figure 5:  Normalized distribution of released energy in the EC decay of 95Tc
in the transition to the excited state $E^{*}=1675.40$ keV of 95Mo, with
$Q^{*}_{\rm EC}=20.06$ keV, as function of $E-Q^{*}_{\rm EC}$ in comparison to
that of 163Ho with gs-to-gs $Q_{\rm EC}=2.8632$ keV [79]. The red line
corresponds to the on-resonance EC decay using the experimental $Q^{*}_{EC}$
of 20.06 keV within 1$\sigma$ range of the central value 20.52 keV, while the
blue line corresponds to the $Q^{*}_{\rm EC}$ of 2.8632 keV. For the sub-shell
notation and the inset the reader is referred to the caption of Fig. 4

In Table 3, we present the predicted half-lives for the decay of 95Tc to the
two excited states of 95Mo with the experimental energies $E^{*}=1675.4$ keV
and $E^{*}=1683$ keV for all relevant atomic shells and in Fig. 4, the
normalized distribution of the released energy in the EC decay of 95Tc to
excited states of 95Mo is demonstrated. The transition spectrum of 95Tc (9/2+)
$\rightarrow$ 95Mo∗ (1675.4 keV) is indicated in green, with
$Q_{\mathrm{EC}}^{*}$ of 20.52 keV, situated 0.47 keV relative to the computed
atomic relaxation energy following the electron capture in the allowed K
shell. In contrast, the transition 95Tc (9/2+) $\rightarrow$ 95Mo∗ (1683.0
keV), shown in orange, with a $Q_{\mathrm{EC}}^{*}$ value of 12.9 keV, is
relatively farther from the computed atomic relaxation energy following the
electron capture of the allowed L1 shell (giving a value of 10 keV for the
distance). As illustrated in the inset of Fig. 4, a more pronounced resonance
enhancement in the last 0.8 eV region near the endpoint for the former
transition is observed, suggesting a preference for choosing this transition
as a candidate for determining a non-vanishing neutrino mass. This phenomenon
guides us to search for cases that have the smallest distance of
$Q_{\mathrm{EC}}^{*}$ to the highest ionization energy of the captured
electron of all allowed shells for neutrino-mass determination experiments.

The decay rate close to the endpoint is highly sensitive to small variations
of the $Q$ value as demonstrated by the $1s$ level, K, manifesting as a
resonance itself, and thus can radically increase the number of recorded
events near the endpoint. The current accuracy of measurement of the
$Q_{\mathrm{EC}}^{*}$ value does not allow to make an unambiguous conclusion
about the position of the $1s$ level, relative to the endpoint. Assuming
$Q_{\mathrm{EC}^{*}}=20.06$ keV, which is consistent with the 1 $\sigma$ range
of experimental error and is shifted by -0.46 keV relative to the central
value, the resonance at the endpoint provides the highest EC event rate in the
neutrino-mass sensitive region. Figure 5 shows the normalized EC energy
spectrum as a function of the energy, $E-Q_{\mathrm{EC}}^{*}$, deposited in a
calorimeter through the de-excitation of atomic shells for
$Q_{\mathrm{EC}}^{*}=20.06$ keV, in comparison to the EC energy spectra in
163Ho atom. The $s1$ level, K, has a significant EC counting-rate enhancement
for the transition 95Tc (9/2+) $\rightarrow$ 95Mo∗ (1675.40 keV) in the
neutrino-mass sensitive region, as shown on an enlarged scale in the inset of
Fig. 5. Assuming a $Q$ value of $Q_{\mathrm{EC}}^{*}=20.06$ keV, technetium is
about three orders of magnitude more effective than holmium. Based on these
findings, it can be conjectured that we have found a potentially strong
transition for direct electron-neutrino mass determination. However, the short
half-life of about 1 day for the 95Tc can prove to be a challenge
experimentally.

##  4 Conclusion and outlook

A direct high-precision gs-to-gs EC-decay $Q$-value measurement of 95Tc (9/2+)
$\rightarrow$ 95Mo (5/2+) was performed using the PI-ICR technique at the
JYFLTRAP Penning trap mass spectrometer. A $Q$ value of 1695.92(13) keV was
obtained and the precision was improved by a factor of around 37 compared to
literature. The measurement also improved the mass excess of 95Tc by a factor
of 28 compared to previous experiments. Three candidate transitions of 95Tc
(9/2+) $\rightarrow$ 95Mo∗ were validated to be energetically allowed. The
refined sub-keV precision of the gs-to-gs $Q$ value of 95Tc allows us to find
an ultra-low energy difference (0.47 keV) between $Q_{\mathrm{EC}}^{*}$ and
the atomic relaxation energy $\varepsilon_{k}$ = 20.054 keV in the K capture
for the allowed gs-to-es transition to the 1675.40-keV state. The spin of the
1683.0-keV state needs to be determined along with its energy with higher
precision in order to see if the related transition is allowed and of low $Q$
value.

The atomic self-consistent many-electron Dirac–Hartree–Fock–Slater method and
three nuclear shell-model interactions were utilized to predict the partial
decay half-lives and energy distributions of gs-to-es EC transitions in 95Tc
with low $Q$ values. We computed the energy levels of the parent and daughter
nuclei and a few electromagnetic moments to assess the validity of the three
shell-model interactions (jj45pna, jj45pnb, glekpn). Multiple corrections,
such as exchange, overlap, shake-up, and shake-off effects, were accounted for
in these predictions. From the calculations, the ultra-low distance to the
atomic line K, level 1s, for 95Tc (9/2+) $\rightarrow$ 95Mo∗ (1675.40 keV)
results in a significant increase in the number of EC events in the energy
region sensitive to the electron neutrino mass. These findings confirm a
potentially powerful transition for direct electron-neutrino mass
determination.

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Declaration of competing interest

The authors declare that there are no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

Acknowledgements

We acknowledge the staff of the Accelerator Laboratory of University of
Jyväskylä (JYFL-ACCLAB) for providing stable online beam. We thank the support
by the Academy of Finland under the Finnish Centre of Excellence Programme
2012–2017 (Nuclear and Accelerator Based Physics Research at JYFL) and
projects No. 306980, 312544, 275389, 284516, 295207, 314733, 315179, 327629,
320062, 354589, 345869 and 354968. The support by the EU Horizon 2020 research
and innovation program under grant No. 771036 (ERC CoG MAIDEN) is
acknowledged. This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No.
861198–LISA–H2020-MSCA-ITN-2019. V.A.S., O.N., S.S., J.S., and J.K.
acknowledge support from project PNRR-I8/C9-CF264, Contract No.
760100/23.05.2023 of the Romanian Ministry of Research, Innovation and
Digitization. The work leading to this publication was supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - AY
155/2-1. The paper was supported by the DAAD Grant No. 57610603.