¹ In memoriam of Vladimir Lobashev and Jochen Bonn who significantly contributed to the development of the KATRIN neutrino mass experiment.

The Standard Model of particle physics assumes that all matter in the universe is composed of leptons and quarks. With the exception of neutrinos, the masses of these basic fermions are known [8]. Although not required by any law of physics, the mass of neutrinos used to be assumed to equal zero. However, neutrino oscillation experiments revealed that it is not the case. According to present knowledge [19] the neutrino v_α, created with flavor α in a charged-current weak interaction process, is a quantum superposition of the neutrino mass eigenstates v_i

(1)

Here, the index α = e, µ, τ marks three known kinds of the active³ flavor neutrinos ν_e, ν_µ and ν_τ. The index i denotes the individual neutrino mass eigenstates with mass m_i. U_{α i} are the elements of the Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix. Their absolute values are known from neutrino oscillation experiments with rather good accuracy [8]. Oscillation experiments also yield the differences of squares of the neutrino masses, Δ m_jk² = m_j²-m_k² or at least their absolute values. Since measured Δ m_jk² do not equal zero a fact of topical importance follows: at least two neutrino mass eigenvalues are different from zero. However, at the time of writing the masses m_i themselves remain unknown and the same holds for the mass ordering where all three options are likely: m₁ < m₂ < m₃ [the normal hierarchy (NH)], m₃ < m₁ < m₂ [the inverted hierarchy (IH)], or m₁m₂m₃[the quasi-degenerate mass spectrum]. With one exception [37] that has not yet been confirmed by any other experiment, almost 70 years of effort by experimentalists has resulted in knowing only the lower and upper limits of the neutrino masses m₁, m₂ and m₃ or their combinations. Oscillation experiments carried out with solar, atmospheric, accelerator and reactor neutrinos have proved to be an extremely powerful tool of neutrino physics [28] since they have demonstrated that neutrinos are massive particles. From current values of the squared mass differences Δm₂₁² = (7.53 ± 0.18) ∙10^-5 eV², |Δm₃₂²| = (2.44 ± 0.06) ∙10^-3 eV² for NH and |Δm₃₂²| = (2.52 ± 0.07) ∙10^-3 eV² for IH [8], it is possible to derive a lower limit on two of the mass eigenvalues. The limits (at the 90% CL) are m₂ > 8 meV, m₃ > 49 meV for NH and m₁ > 48 meV, m₂ > 49

meV for IH. Of course the real masses m_i can be much larger, i.e. it may hold that m₁m₂m₃. Possible values of the neutrino masses are depicted in Fig. (1)

Fig. (1). Neutrino masses mi as a function of the lightest mass m1 for the normal hierarchy NH (left panel) and m3 for the inverted hierarchy IH (right panel). We have calculated the curves for the mean values Δm212 = 7.53 ·10-5 eV2, |Δm312| ≈ |Δm322| = 2.44 ·10-3 eV2 for NH and 2.52 ·10-3 eV2 for IH, and for the admixtures | Ue1|2 = 0.6803, | Ue2|2= 0.2354 and | Ue3|2 = 0.0843 derived from [8]. Also shown is the effective neutrino mass mve, the β-spectroscopic quantity defined by Eq. (5), as well as the sum of neutrino masses derived from cosmological measurements. Also depicted is the upper limit of resulting from the latest measurements at Mainz [38] and Troitsk [39] and the expected limit from the upcoming KATRIN experiment [40].

If the neutrino is a Majorana particle (i.e. neutrino ν and antineutrino ͞ν are identical particles) the 0νββ decay, e.g. (A, Z) → (A, Z+2) + 2e^–, could develop into an important tool for determination of the effective Majorana neutrino mass, namely

(2)

Here are the yet unknown Majorana phases, 0 ≤ ≤ 2π. Assuming that the 0νββ decay proceeds dominantly via the emission and subsequent absorption of a virtual neutrino, its half-life can be expressed as

(3)

The kinematic factor G_0vis exactly calculable, in contrast to the value of the nuclear matrix element M_0ν, which depends on the applied nuclear model. Moreover, it is necessary to take into account all virtual states of the intermediate nucleus (A, Z+1), most of which are not known from experiment. A comparison of different evaluations indicates a spread by a factor of 2 to 3 in the calculated nuclear matrix elements [41].

The current state of the 0νββ decay theory was recently reviewed [42]. The authors argue that it would be possible to disentangle various mechanisms that may contribute to the neutrinoless double β-decay if this decay were examined in a sufficient number of isotopes. However, other investigators [43] recommend a more cautious interpretation of the experimental data of T_1/2^0vββ. According [44], the matrix elements from present calculations differ from one to another by up a factor of three. In addition to the possibility of experimental determination of the effective neutrino mass m_ββ according to Eq. (2), the 0νββ decay seems to be the only known way to examine a possible violation of the law of lepton number conservation. Long-standing searches for this extremely rare process are described [45, 46]. Requirements for current and future 0νββ experiments are discussed in [47, 48], while new expectations and uncertainties in 0νββ decay are reviewed [49] in the light of recent neutrino oscillation studies as well as cosmological observations. The latest results of the double beta decay searches along with the expected sensitivities of experiments in construction are reviewed [50-52].

With one exception, all the searches until now have found only a lower limit of , i.e. the upper limit of m_ββ . The only exception is the claim in [53] for an observation of the + 2e^- decay with a half-life Assuming dominance of the 0νββ-decay mode and choosing a particular value of the nuclear matrix element Moν , from Eq. (3) the authors derived⁴ the effective mass eV [37]. The GERDA experiment [56], aiming to check this claim, recently reported the results of its phase I: (⁷⁶Ge) > 2.1 · 10²⁵ y at 90% CL [57]. An example of the GERDA spectrum is shown in Fig. (2). Results of the recent searches for the 0νββ decay are summarized in Table 1.

Fig. (2). Spectrum of electrons and photons recorded in the GERDA experiment [56] with germanium detectors enriched in 76Ge (red spectrum with higher intensity) and non-enriched Ge detectors (blue spectrum with lower intensity). The continuous spectra correspond to the sum of energy of two electrons released in the 2νββ decay of 76Ge with the half-life of 1.5·1021 y. The low energy part of the spectrum is made mostly by β particles from the 39Ar decay and photon bremsstrahlung of these particles. Pure β emitters 39Ar (T½ = 269 y, Qβ = 565 keV) and 42Ar (T½ = 33 y, Qβ = 600 keV with daughter 42K) are produced via cosmogenic activation of atmospheric argon. The specific activity of 39Ar is 1 Bq kg-1. Spectral lines with higher energy are due to γ quanta emitted in a decay of natural radioisotopes. Vertical band at Qββ = 2039 keV with the bandwidth of ±20 keV denotes the region where the line from the 0νββ decay is expected. The background in this region is about 0.02 counts keV-1·kg-1· y-1.

Table 1. Lower limits of the half-lives determined at the 90% CL in recent experiments along with upper limits of the effective neutrino mass, m_ββ, derived for various values of the nuclear matrix elements.

Experiment	Isotope	T1/20vββ (y)	mββ (eV)
GERDA [57]	76Ge	> 2.1 · 1025	< (0.2 – 0.4)
NEMO-3 [58]	100Mo	> 1.1 · 1024	< (0.3 – 0.8)
CUORICINO [59]	130Te	> 2.8 · 1024	< (0.30 – 0.71)
EXO-200 [60]	136Xe	> 1.1 · 1025	< (0.19 – 0.45)
KamLAND-Zen [61]	136Xe	> 1.9 · 1025	< (0.12 – 0.25) *

* Combined with the former result of the EXO-200 experiment [62].

Important, though strongly model-dependent, information about the neutrino mass states can be gained from cosmological data. Of particular relevance are the precision measurements of temperature fluctuations of the cosmic microwave background (CMB) and investigation of large scale structures (LSS) of the universe [19]. The extensive sets of measured data enable the estimation not only of the usual six parameters of the standard cosmological model, but also of the sum of the neutrino masses and the effective number of the neutrino families N_eff. Details can be found, e.g. in reviews [63, 64] presented at the Neutrino 2014 conference. Theoretical aspects of interconnection between neutrino physics and dark matter issues have most recently been considered [65].

Neutrino oscillation experiments yielded a lower limit ≥ 0.05 eV. The cosmological upper bounds of depend on the types of measurements included and details of their analysis. For example, the newest results of the

Planck collaboration range from < 0.4 eV up to < 1.7 eV at the 90% CL [66]. The often quoted estimate < 0.6 eV is considered to be a conservative one. On the other hand, Beutler et al. [67] derived from the current cosmological data the sum of the neutrino masses ∑ m_i = 0.36 ± 0.10 eV (68% CL). This estimate, together with the known elements of the neutrino mixing matrix [68], led to the values of the kinematically measured quantity m_ve (defined in Section 2.4) equal to 0.117 ± 0.031 eV for NH and 0.123 ± 0.032 eV for IH [67]. Most recently, Palanque-Delabrouille et al. analyzed the measured Lyα forest power spectra combined with the spectra of CMB and baryon acoustic oscillation data. The authors assumed validity of the ΛCDM cosmological model in the presence of massive neutrinos. Altogether, 7 cosmological, 6 astrophysical and 13 nuisance parameters were varied simultaneously. Both frequentist and Bayesian approaches were applied and the stability of the solution was examined. The analysis yielded the strongest cosmological constraint on the total neutrino mass, < 0.14 eV at the 95% CL [69]. Detailed discussion of neutrino constraints resulting from data on LSS and CMB can be found in [70]. Constraints on the expected from future observations of galaxy clustering and cosmic microwave background have recently been treated [71].

In principle, the kinematic methods could yield a m_i value for each of the neutrino mass states assuming that this state is involved in the corresponding weak-interaction state with a measurable intensity, and assuming that the applied instrument has sufficient energy resolution to separate individual mass states. However, the current oscillation experiments show that the splitting of the mass states is far too small to be resolved by current spectrometers. In case of β- decay, various effective neutrino masses are presented [72-74]. In accord with the Particle Data Group [8] we utilize the following definition of the effective neutrino mass m_vα

(4)

Ref. [8] gives the following limits for the electron-based, μ-based and τ-based neutrino masses⁵, respectively

(5)

(6)

(7)

The limit of m_ve in Eq. (5) originates from recent investigations of tritium β-spectra where the conservation of energy and momentum is applied [38, 39]. The limit in Eq. (6) is based on the precision measurement of the muon momentum in the two-body decay of pions at rest: π⁺ → μ⁺ + ν_μ or π^- → μ^- + ͞ν_μ [75]. The limit in Eq. (7) was derived from the kinematics of the τ^- decay [76]. The large differences among these limits reflect the absolute precision in momentum and energy determination achievable in the relevant experiments. Considering the relation among the total energy, momentum and the rest mass of a particle (in units c = ħ = 1):

(8)

One can hope for a high sensitivity for the neutrino mass determination only when the neutrino is almost nonrelativistic; otherwise the experimental precision is lost in the E_tot² – p² difference, as underlined in [36].

The mass of a particle can be determined via time-of-flight, which is also a kinematic method. However, for neutrinos with their small mass, extremely large distances between the source and the detector are necessary. Investigation of the neutrinos emitted by the supernova SN1987A yielded the upper limit m_ve< 5.7 eV at the 95% CL [77]. It should be noted that this result depends on assumptions of the supernova models that are not yet capable of predicting the time-spread of the neutrino emission with sufficient precision. An extensive analysis of the SN1987A data has also been given [78].

The relationship between the results of neutrino oscillation experiments, 0νββ-decay searches, cosmological observations and examination of β-spectrum shape has been explored by several investigators [41, 55, 79, 80]. The results of neutrino oscillation experiments enable not only the examination of the dependence of various neutrino quantities on the minimal neutrino mass m_min as shown in Fig. (1) but also the exploration the-relationship of these quantities. Newer graphical comparisons of m_ve, m_{β β} , and are depicted in Figs. (3 and 4).

Fig. (3). Regions of the three observables sensitive to the absolute neutrino masses that are allowed by the neutrino oscillation experiments: mβ≡ mve (Eq. 5), mβ β (Eq. 2) and for NH (blue dotted) and IH (red solid). Bands correspond to 3σ uncertainties of the oscillation data. Figure is reproduced from [81].

Fig. (4). Regions of the effective Majorana neutrino mass mβ β allowed by the neutrino oscillation experiments, depending on the assumed values of the smallest neutrino mass mmin = min (mi), mtot = and < mβ > = mve defined in Eq. 5. The blue horizontally-hatched band corresponds to NH, the slant red one to IH. The width of these bands reflects the uncertainty due to the unknown phases φ(k) in Eq. 2. Wider bands restricted by solid lines (in red for NH and in black for IH) include the experimental uncertainties of the oscillation parameters. The systematic uncertainties of the nuclear matrix elements are not included. This figure is reproduced from [41].

Fig. (3) shows that the strongest cosmological constraint on total neutrino mass, < 0.14 eV at the 95% CL [69], corresponds to the following limits on both m_ve and m_ββ of < 0.04 eV for NH and < 0.06 eV for IH. According to [41] and Fig. (4), the neutrino oscillation data indicate a lower limit of m_ββ for IH but no such limit for NH. The overlap of bands for NH and IH demonstrates that future 0νββ experiments could examine the neutrino mass ordering only if their sensitivity is better than 0.01 eV. In addition, the oscillation data allow the testing of whether the observed values of m_ββ and are in agreement with the assumptions of the 3-neutrino framework and the dominance of the light neutrino exchange in the 0νββ-decay.

The first successful theory of the nuclear β-decay was developed by Fermi [2] who demonstrated that the shape of the uppermost part of the β-spectrum (close to the maximum energy of β-particles and thus to the minimum energy of neutrinos) is sensitive to the neutrino mass squared. Assuming only one neutrino mass state and no excitation of the daughter ion, the β-spectrum of allowed β-transitions (ΔI_nucl = 0, 1 and Δπ_nucl = 0) takes its simplest form⁶ [31]:

(9)

Here A, E, p and m_e denote the spectrum amplitude and the electron kinetic energy, momentum and rest mass, respectively. ε = E₀ -E where E₀ is the endpoint energy, i.e. the maximal electron kinetic energy for m_v = 0. The Fermi function F (E, Z +1) takes into account the Coulomb interaction of the emitted β-particle with the daughter nucleus and its surrounding electrons. Θ is the Heaviside step function.

The Kurie plots in Fig. (5) show that the neutrino mass can be extracted from measured β-spectrum only if the spectrometer resolution ΔE_instr is comparable or better than m_v. In addition, the spectrometer must exhibit a high luminosity⁷ and low background in order not to miss a weak effect at the upper end of the β-spectrum. Simultaneous fulfillment of these requirements was not yet achieved to the required level, despite more than half a century’s effort. Thus all the measurements of the β-spectra up to now have yielded only an upper limit on m_v.

Fig. (5). Kurie plots for an allowed β-decay and two values of me demonstrating narrowness of the endpoint region sensitive to the neutrino mass [35]. The graphs are obtained from Eq. (9) expressing the parameter E - E0 in units of m equaled to assumed nonzero value of mv. Explicitly for mv = 0.

When the technique of β-ray spectroscopy was significantly improved by Bergkvist in 1972 [83], it became necessary to consider that a part of the β-decay energy may be spent for an excitation of the daughter ion. In this case the observed β-spectrum would be described as a superposition of partial β-spectra with endpoint energies E_0j = E₀ –m_v –V_j, where V_j is the energy of the j-th excitation state populated with the probability P_j:

(10)

In the case of β-decay of an isolated tritium atom, i.e. for a hydrogen-like ³He⁺ ion, the quantities V_j and P_j can be precisely calculated [84]. The β-decay of a gaseous tritium molecules (T₂) is more complicated. Nevertheless, reliable calculations for the molecular ion (³HeT)⁺ are available that take into account both electronic and rotational- vibrational molecular states, see Fig. (6). These predictions have not yet been experimentally verified but their precision and internal consistency indicates their applicability in the newest neutrino mass measurements. On the other hand, calculations V_j and P_j for β-decay of atoms in many-electron systems of complex molecules or for β-emitters imbedded into solids may also suffer from insufficient knowledge of the composition of the radioactive samples. Very recently, a detailed assessment of molecular effects on the determination of m_ve from tritium β-spectra has become available [85].

Fig. (6). Probability distribution for excitations of the (3HeT)+ ion populated the in β-decay of gaseous T2 [86]. The figure combines results from the calculations of excitation to the electronic bound states [87] and to the electronic continuum [88] of (3HeT)+. The electronic ground state is populated in 57% of the decays, 29% of the decays end in the excited electronic states and remaining 14% lead to the electronic continuum. The rotational-vibrational excitations of the molecular ion in its ground electronic state are shown in the insert. The mean energy of these excitations is 1.7 eV with the energy spread (FWHM) of about 0.4 eV. The electronic excitations of the (3HeT)+ ion starting at about 20 eV are shown in the lower part of the figure.

Since the discovery of neutrino oscillations, it is necessary also to consider the individual neutrino mass states in the analysis of the measured β-spectra using the formula

(11)

This is exemplified in Fig. (7) for tritium β-decay, assuming the normal neutrino mass hierarchy with m₁ = 200 meV. Since the best instrumental resolution is still much worse than the differences among the individual neutrino mass states, the measured spectra can still be analyzed with the help of Eq. (10), where the variable m_v should be interpreted as the effective electron neutrino mass m_ve defined in Eq. (5).

In order to compare a measured β-spectrum S_exp (E) with theoretical predictions for various neutrino masses S_th (E^', m_ve), it is necessary to know the spectrometer resolution function R(E,E’). This function is the response of the spectrometer to a monoenergetic signal of unit intensity

(12)

Analyses of the measured β-spectra are usually performed by the least-squares method with at least four fitted parameters: m_ve², E ₀, spectrum amplitude A and background b. When fitting only the uppermost part of the β-spectrum, the uncertainty in the absolute energy scale of the spectrometer does not introduce a recognizable systematic error in the determination of m_ve [89]. The use of a fixed value of E₀ , derived from precision measurement of the atomic masses of ³H and ³He, would greatly facilitate the task, since the m_ve² and E₀ variables are strongly correlated. However, in this case the precision of both E₀ and the absolute energy calibration of the β-ray spectrometer have to reach a level of a few-meV [90], far from the present state of the art. Still, the comparison of fitted and independently measured E₀ values proves to be an important check of the correctness of the evaluation of the β-spectrum. The newest measurement of the cyclotron frequency ratios of ³He⁺ to HD⁺ and T⁺ to HD⁺ in the Penning ion trap have yielded a Q value for tritium β-decay equal to 18 592.01 ± 0.07 eV [91], which will enable a sensitive test of the fitted value of E₀ .

Fig. (7). The uppermost part of the tritium β spectrum, wide only 7 meV, was calculated for the normal hierarchy of the neutrino mass states (m1 < m2 < m3) assuming the endpoint energy E0 = 18.575 keV and m1 = 200 meV. The same mass square differences and the admixtures |Uei|2 as in Fig. (1) were assumed. Under these assumptions m2 = 200.19 meV, m3 = 206.19 meV, and the relative intensities of partial spectra β1, β2 and β3 are |Ue1|2 = 0.6803, |Ue2|2 = 0.2354 and |Ue3|2 = 0.0843, respectively [8]. The experimentally observable β-spectrum is composed of the three components. Taken with sufficiently high instrumental resolution, the spectrum will exhibit kinks at energies E0 – mi (i = 2, 3). In our example, the average electron neutrino mass mve defined by Eq. (5) equals 200.57 meV. We have assumed for simplicity that all the 3H → 3He + e- + ͞νe decays proceed to the ground state of the daughter ion.

Fig. (8). Magnetic spectrometer [93] with an electron orbit of 40 cm. The iron pole pieces produced the radially inhomogeneous magnetic field that yielded better focus at 180° than a homogenous field. The set of beam-defining slits made of aluminum minimized electron scattering. The instrument was adjusted for the relative momentum resolution8 of 0.5% and solid angle of about 0.1% of 4π. The size of radioactive sources was (0.4 x 2.5) cm2. The energy scale was calibrated with photoelectrons ejected from the K-shell of a lead convertor by the 511 keV annihilation radiation.

The first upper limit on the neutrino mass, derived from a precision measurement of the shape of the β-spectrum, was obtained in 1948 by Cook et al. [92]. The authors investigated the β-spectrum of ³⁵S (Q_β = 167.18 keV, T_½ = 87.5 d) using a magnetic spectrometer shown schematically in Fig. (8). From Kurie plots depicted in Fig. (9) the authors concluded that m_ve < 5 keV, i.e. less than one percent of the electron rest mass.

Fig. (9). The Kurie plot of the β-spectrum of 35S measured by Cook et al. [92]. The plots shown correspond to the 35S sources of estimated thickness of 0.04, 0.14 and 0.17 mg cm-2 that were prepared on 0.5 mg cm-2 Nylon and 0.03 mg cm-2 Zapon backings. For electron energies above 90 keV, the Kurie plots are all straight lines as expected for the 35S(3/2+) → 35Cl(3/2+) β-transition. According to the authors the deviations for energies below 60 keV are influenced by source thickness and backing. The enlarged section of the Kurie plot close to the β-spectrum endpoint shows the theoretical Fermi curves for an allowed β-transition calculated for assumed neutrino masses of 0, 5 and 10 keV.

This upper limit of m_ve was already improved by 1949 owing to the development of proportional counters [94, 95]. These instruments enabled the measurement of the β-spectrum of gaseous tritium with extremely large geometrical transmission corresponding to the almost full solid angle of 4π. The low endpoint energy of tritium⁹, E₀ = 18.6 keV, was advantageous, since the relative intensity of β-particles in the interval ΔE close to the endpoint is proportional to (ΔE/E₀)³ The resolution ΔE_instr of the applied counter [95] was 1.9 keV for the 17.4 keV calibration K_α x-rays of molybdenum. The results were m_ve < 0.5 keV [96] and m_ve < 1 keV [97].

Langer and Moffat [98] investigated the β-spectrum of tritium using the same magnetic spectrometer as was used in the previous studies of ³⁵S [92]. The sources were prepared by thermal evaporation in vacuum of tritiated succinic acid (HOOC·CH₂CH₂·COOH) with specific activity of 33 MBq μg^-1 (i.e. about 10% of protium atoms were substituted by tritium ones). This compound was quite stable at room temperature and sublimated at 150°C thus avoiding any thermal damage of a 4 μg cm^-2 thick organic support during evaporation. Autoradiography revealed a complete uniformity of the invisible deposit with an estimated surface density of about 0.5 μg cm^-2. Two methods were applied to prevent electrostatic charging of the source. First, a 2 μg cm^-2 thick and 1 cm-wide copper band was evaporated onto the reverse side of the insulating organic support. The band resistivity of 10⁸ Ω was sufficient to insure good grounding. Second, an

oxide-coated cathode filament was installed just below the source where electrons emitted from the heated filament would neutralize any positive charge created on the source due to the emission of β-particles. Both methods applied separately or together yielded the same endpoint energy E₀, while the one using an ungrounded source was lower by 460 eV due to the positive charging of the source. The analyses of measured β-spectra yielded an upper limit m_ve<250 eV [98].

In order to avoid difficulties with the detection of low-energy electrons, Hamilton et al. [99] investigated the tritium β-spectrum with a spherical electrostatic analyzer [100]. Instead of measuring counting rates, the authors recorded the current of β-particles that overcame a variable retarding potential and reached a spherical collector. From the integral spectrum measured with a rather thick tritium source (100 μg cm^-2) the authors derived an upper limit m_ve < 500 eV at 80% CL. Their endpoint energy of E₀ = 17.6 ± 0.4 keV is 2.4 σ below its current value.

Salgo and Staub [101] examined the tritium β-spectrum with a retarding-potential spectrometer in a planar arrangement. The source was T₂O ice evaporated on a gold plated holder of 5.5 cm radius that was kept at the temperature of liquid nitrogen. The T₂O deposit of 39 μg mass and activity of 3.9 GBq was not homogenously distributed so that the surface density of about 20 μg cm^-2 estimated from interference fringes in the source center was 50 times higher than average density. Great attention was paid to secure a linear potential drop between the source and the planar retarding grid and to avoid high potential gradients near the source that could cause a field emission of electrons. Nevertheless, the background determined in the energy region above 18.75 keV with a statistical uncertainty of ± 0.056%, strongly dominated the effect. The results are m_ve < 200 eV at 80% CL and m_ve <320 eV at 90% CL.

Daris and St.-Pierre [102] investigated the tritium β-spectrum with an iron-core magnetic spectrometer adjusted for the resolution Δp/p = 0.25% (i.e.ΔE_instr = 90 eV at 18.6 keV) and fractional solid angle Ω/4π = 7.5 ·10^-4. Sources were made by absorption of tritium into an aluminum foil in a gaseous discharge of which the surface had been previously electropolished and de-oxidized. The sources, of dimensions 1mm × 54 mm, prepared at tritium pressure of 1.5 Torr and a discharge voltage of 600–700 V, had an activity of 0.7-1.2 MBq. Measurements of activity after successive electro peelings revealed that 90% of the tritium ions were contained in a thickness of less than 7.4 nm (2 μg cm^-2) of aluminum. The energy loss of 18 keV electrons within the Al foil was estimated to be 20 eV. The instrumental resolution function was derived from ThC calibration lines. The resulting endpoint energy of 18.570 ± 0.075 keV agrees with present determinations. From four combined β-spectra, the authors concluded that where the error is the standard deviation σ. This means m_ve < 120 eV at the 90% CL.

Bergkvist [103, 104] succeeded in increasing the luminosity of the π√2 magnetic spectrometer by two orders of magnitude without deterioration of the instrumental resolution. He developed an extended radioactive source, of which the parts were put on appropriate potentials in order to eliminate influence of the source width on the spectrometer resolution. The aberration defects were further reduced by an electrostatic corrector. For his study of ³H β-spectrum, Bergkvist [83] prepared the source, of (20 ×10) cm² area, by implantation of tritium ions of kinetic energy of 400–800 eV into an aluminium foil. The source also contained a set of small ¹⁷⁰Tm sources of monoenergetic conversion electrons. These electrons of 22.9 keV kinetic energy, emitted from the K shell of daughter ytterbium atoms with natural widths Γ =32 eV, enabled the determination of the instrumental resolution for the extended source to be ΔE_instr = 40 eV at 18.6 keV. In addition, the conversion electrons allowed the inspection of the stability of tritium source with respect to contamination over-layers that might develop during long runs.

Depth distribution of the tritium activity in the aluminum source backing was examined with the peeling technique. 93% of activity was found in the 3 μg cm^-2 thick natural oxide layer while only 2.5% of tritium ions penetrated deeper than 6.4 μg cm^-2. The overall experimental resolution, including energy losses within the tritium source, was estimated to be ΔE_exp = 55 eV at 18.6 keV. In spite of numerous slits in the spectrometer chamber, scattered β-particles produced about half of a high and energy dependent background above E₀ (see Fig. 10). The second half was an intrinsic background of the large size detector needed for the extended source. The uppermost part of the measured tritium β-spectrum is shown in Fig. (10).

Fig. (10). The endpoint region of the tritium β-spectrum measured with the extended source in the Bergkvist experiment [83]. The quantity on the x-coordinate is proportional to the β-particle momentum. The spectrum above E 0 was checked for slight curvature by recording the background in the range 480 – 510 Ω. This test allowed the use of a sloping straight line for background description, as shown.

Bergkvist was the first who took into account that only 70% of the tritium decays go to the 1s ground state of the ³He⁺ daughter ion, while the remaining decays populate higher states, mainly the 2s state (see also [84]). This feature was approximated by means of two lines of the same width, shifted by 43 eV to each other. The width of the overall resolution function, utilized in the analysis of measured tritium β-spectrum, thus increased to 70 eV. This function was assumed to incorporate all the atomic and experimental effects, including energy losses in the source. The analysis of the measured tritium β-spectrum yielded m_ve< 55 eV at the 90% CL [83] and E₀ = (18610 ±16) eV [105].

Röde and Daniel [106, 107] investigated the tritium β-spectrum with the (π/2)√13 iron-free magnetic spectrometer [108] shown in Fig. (11).The instrument was adjusted for Δp/p = 7.5 ·10^-4 (ΔE_instr = 28 eV at 18.6 keV) at a fractional solid angle of Ω/4π = 1.5 ·10^-2. A special anticoincidence proportional counter enabled the reduction of the background without the source inside the spectrometer to about 0.03 s^-1 for energies below 20 keV, regardless of the relatively large size of the detector.

Fig. (11). An iron-free magnetic β-ray spectrometer with radial focusing and axial defocusing at the curved exit slit [108]. This focusing principle offers better combination of resolution and transmission than conventional π √2 spectrometers at the price of a larger detector and focusing angle of (π/2)·√13 = 324.5°. The radius of a central orbit is 30 cm. Note the coils compensating external magnetic fields in three perpendicular directions.

In order to increase the luminosity of their instrument the authors used a technique of an extended source [103]. Instead of a set of narrow strips at different potentials they applied an electrostatic correction by a continuous potential drop. For this purpose the 10 μg cm^-2 nitrocellulose backing of the radioactive layer was covered by 6 μg cm^-2 silver layer with a resistance of 10⁹ Ω between the adjacent potential defining wires with 1.2 mm spacing. A thin tritium film, prepared by spreading out a solution of tritiated polystyrene in benzene on water, was deposited on the silver side of the backing. Four tritium sources with dimensions (22 x 23) mm² and surface density of 4 – 6 μg cm^-2 were prepared in this way. Measurements of monoenergetic electrons emitted from a similarly extended ²¹⁶Po source proved that the instrumental resolution achieved with these extended sources was the same as that for an equipotential source of (0.5 × 20) mm². The quality of one of the sources is demonstrated by a completely straight Kurie plot of the last 3 keV region of the β spectrum as shown in Fig. (12).The derived endpoint energy E ₀ = (18.649 ± 0.074) keV agrees within 1σ with its current value and the upper limit m_ve < 86 eV was determined at the 90% C. L. [106].

Fig. (12). The Kurie plot of the tritium β-spectrum measured by [106]. The 3 keV wide endpoint region is shown for a source with a thickness of the active layer of (4.5 ± 1.5) μg cm-2. Quality of the fit is demonstrated by a spectrum of residuals in the upper part of the figure.

Tretyakov [109] developed a toroidal iron-free magnetic spectrometer which proved extremely useful in searches for m_ve in tritium β-spectra¹⁰. A magnetic field of the instrument was produced by 72 rectangular coils placed symmetrically around the spectrometer axis, which contained a radioactive source and the detector itself. This arrangement enabled the analysis of electrons emitted in the angular intervals of ±7.5° and twice 120° with respect to the source normal. To increase the dispersion of the device, electrons were led through the analyzing field four times, turning their direction by 180° in each path. This magnetic deflection by 720° also almost eliminated background caused by scattered electrons. In measurements of tritium β-spectra, Tretyakov et al. [113] utilized an extended non-equipotential source of the Bergkvist type [103] situated in the spectrometer focal plane. The composed source consisted of 18 parts, each of the size (2.6 × 18) mm². The parts were arranged in 9 pairs emitting electrons in opposite directions, thus enabling more efficient use of the spectrometer azimuthal angle.

The luminosity L = S · Ω/4π = 0.07 cm² and resolution Δp/p = 1.2 ·10^-3 were measured for a similar composed source emitting calibration conversion electrons from the ¹⁶⁹Yb decay. The tritium sources were prepared by vacuum evaporation of tritiated valine, an amino acid (CH₃)₂CH·CH(NH₂)·COOH (where two of the protium atoms were substituted by tritium ones), at ~300°C on Al backing. The surface density of the source was estimated to be 2 μg cm^-2 of valine. Due to the small size of the low-pressure Geiger-Müller counter and the absence of scattered electrons in the detector region, the background above the tritium endpoint was 50 times smaller than that of the Bergkvist measurement shown in Fig. (10). In addition, the background did not depend on the electron energy and was the same regardless of the presence or absence of the tritium source in the spectrometer. Nevertheless, tritium itself slowly contaminated the spectrometer vacuum chamber, and the background after one month of measurements was four times higher than at the beginning.

The least-squares fits of four series of the measurement of the β-spectrum based only on the statistical uncertainties of the counting rates yielded a normalized χ² = 1.49 which was improbably large for 191 degrees of freedom. From the scattering of the counting rates in the overlapping regions of individual series, the authors estimated a total measurement uncertainty to be σ_tot = 1.2 σ_stat. Two additional uncertainties were taken into account: ±15% for the uncertainty of the population of the effective excited states of the ³He⁺ ion and 20% due to a possible inhomogeneity of the source thickness. The Kurie plot of the recorded part of the tritium β-spectrum is reproduced in Fig. (13). The final result was m_ve < 35 eV at the 90% CL [113]. The least-squares analysis of measured spectra yielded E₀ = (18574.8 ± 1.3) eV while caution consideration of systematic uncertainties led to E₀ = (18575 ± 13) eV. Both determinations agree perfectly with the current value of E ₀.

Fig. (13). Kurie plot of the endpoint region of the tritium β-spectrum measured with a toroidal magnetic spectrometer by Tretyakov et al [113]. Solid and dashed lines correspond to mve = 0 and 40 eV, respectively.

In order to reduce problems with electron scattering in radioactive sources and spectrometers as well as uncertainties connected with final state interactions, Simpson [114] investigated the β-spectrum of tritium implanted into a Si(Li) x-ray detector. The detector of 80 mm² area and 5 mm thickness had resolution of 215 eV at 5.9 keV. A diffuse beam of tritium ions of energy between 8 and 9.1 MeV penetrated through 25 μm thick Be window into 0.2 mm of Si(Li) crystal. This penetration depth far exceeded the 2.5 μm range of the 18.6 keV β-particles and 14 μm range of 1.7 keV Si x-rays. During implantation the detector was kept at liquid-nitrogen temperature. The detector resolution deteriorated at the end of implantation but returned to its original value after annealing. The atomic density of implanted tritium was estimated to be about 1.5·10¹³ cm^-3. A chopper emitting 8.05 and 17.48 keV K_α1-X rays of Cu and Mo respectively, was installed in front of the Si(Li) detector to enable energy calibration and stabilization during long tritium β-spectra runs. The chopper wheel with two slots allowing penetration of x-rays to the detector rotated with a period of a few seconds. In this way, two spectra were recorded with the same electronics setting almost simultaneously: a spectrum of β-rays plus x-rays and the other of β-rays only. The measured tritium β-spectra did not reveal any deviations from the theoretical shape expected for zero neutrino mass and the upper limit of m_ve < 65 eV was derived with 95% confidence.

Physicists at the ITEP in Moscow continued to study tritium β-spectra using the same source material and the same toroidal magnetic spectrometer as in their previous study [113]. The application of a proportional counter decreased by a factor of 10 the background caused by a steadily growing contamination of the spectrometer with tritium. Later, a three-channel proportional chamber was installed that increased the rate of data accumulation. Analysis of the data collected during five years (including those from [113]) led to an astonishing claim that 14 ≤ m_ve ≤ 46 eV at the 99% CL [115]¹¹. This was the first claim for the non-zero neutrino mass. In the following seven years, the ITEP physicists

improved their apparatus as well as their methods of β-spectra measurement and evaluation. A proportional chamber with six channels was installed as a detector. A non-equipotential source with continuous correction potential was built improving the instrumental resolution to ΔE_instr = 20 eV at 18.6 keV. To achieve the continuous potential distribution the source backing was made of a weakly conducting glass with specific resistivity of 10⁷ Ω cm^-1. The β-spectrum was scanned by electrostatic field variation while the magnetic field was kept constant. In this way, an uncertainty in the dependency of the detector efficiency on the electron energy was eliminated. The magnetic field was adjusted to focus electrons of energy of 21.2 keV, exceeding the tritium endpoint energy. Nevertheless, the final result, published in 1987, remained substantially unchanged: eV [117]. In the same work, “the model independent mass interval”, 17 < m_ve < 40 eV was derived from the mass difference of the doublet ³H – ³He. However, the utilized mass difference [118] is in disagreement with more recent values [119, 91].

The ITEP claim motivated several groups to develop new β-spectrometers and new source-preparation techniques. Extensive discussion of these experiments, including the now disproved ITEP result, can be found in the reviews [7, 33-35, 120, 121]. In the following, we mention a few details of possible interest in future β-spectroscopic searches for massive neutrinos.

In order to account properly for the resolution function of their magnetic spectrometer as well as for the electron-energy losses and back-scattering effect in their tritium source, Kawakami et al. [122] utilized the same chemical compound for the ³H source as well as for the ¹⁰⁹Cd reference sources: an arachidic acid (C₂₀H₄₀O₂) in the form of its Cd salt. The former was labeled with tritium while the letter was labeled with ¹⁰⁹Cd emitting monoenergetic KL₂L₃ Auger electrons of (18511.7 ± 1.3) eV energy. The sources of dimensions (20 × 6) cm² were deposited in two monomolecular layers of total thickness of 5 nm (0.6 μg cm^-2) on high-resistivity Ru₂O₃-coated aluminum plates. Optical aberrations due to the source size were compensated by suitable voltages on the source plate. A proportional chamber with resistive wires had position resolution of 0.5 mm for 18 keV electrons. The endpoint region of the tritium β-spectrum and the KL₂L₃ Auger line were recorded with the overall resolution of 30 eV. The overall response function of the whole setup R_T shown in Fig. (14) was derived by deconvoluting a natural width (Γ) and the shake up/off effect (SUO) from measured Auger line:

(13)

Fig. (14). (a) The overall response function of the experiment [122] obtained from the measured KL2L3 Auger line by means of Eq. (13). The low-energy tail extends up to –1000 eV. The total intensity is normalized to one. (b) The calculated shake up/off spectrum convoluted with a Lorentzian corresponding to the natural line width Γ = 11.43 eV. The peak at the energy loss of 0 eV corresponds to the no-energy-loss component whose probability is 80%.

A major improvement in precision β-ray spectroscopy was achieved by Robertson et al. [110] at the Los Alamos National Laboratory, who applied a gaseous source with circulating molecular tritium. The main advantage was that the theoretical spectrum of final states of the (T³He)⁺ molecular ion is known more reliably than that of complex solid sources. Additionally, the energy losses of β-particles within tritium gas could be determined more precisely than in the case of solid sources. A solenoidal magnetic field guided to the spectrometer only those β-particles that did not backscatter on inner walls of the source vessel. Since no separation window was allowed between the source and the spectrometer, a steady flow of tritium into the source was needed, followed by an intensive differential pumping to avoid spectrometer contamination. No increase of background due to escaped tritium was observed and the background rate of ~0.005 s^-1 was caused mainly by cosmic rays. A toroidal magnetic spectrometer of the Tretyakov type [109] with focal length of 5 m, was adjusted to a fixed energy above the tritium endpoint and variable negative bias voltages were applied to the source tube to scan the spectra. Electrons were detected by an array of Si micro strip detectors, allowing the recording of 12 spectra simultaneously. Observed counting rates were renormalized for pressure variations in the source, the activity of which was monitored with another Si detector. Instrumental resolution was measured by circulating gaseous ^83mKr through the source and examining the shape of the K-conversion line with 17.8 keV. The natural width of this line, as well as shake-up and shake-off satellites, were carefully taken into account. The authors [110] also quantified 10 various contributions of uncertainties in the neutrino mass determination.

Fig. (15). Toroidal magnetic spectrometer [111] of the Tretyakov type [109] installed at the Zurich University and used in several β-decay experiments. Distance between source and detector is 2.65 m. Photo reproduced from [124].

Fig. (16). Resolution function of the toroidal magnetic spectrometer at Zurich University [123]. The solid line with the resolution of 17.0 eV (FWHM) was derived from the measured L1 conversion-electron line of the 30.898 keV transition in 195Au (Γ = 6.25 eV). Points were obtained from a Monte Carlo simulation.

Holzschuh et al. [123] investigated the tritium β-spectrum with an iron-free toroidal magnetic spectrometer [111] that is shown in Fig. (15). The cylindrical source consisting of 10 disks (at appropriate potentials to compensate for the axial source extension) had a total active surface of 157 cm². Spectra were taken by stepping the retarding potential on radial grid around the source, while the magnetic field was kept constant and adjusted to focus electrons of kinetic energy 2.2 keV. The measured and calculated resolution functions of the spectrometer with resolution of 17 eV (FWHM) are depicted in Fig. (16). The detector, a proportional counter with a resistive anode wire, had a position resolution of 1–2 mm. Although the source was cooled down to -50°C and a large liquid nitrogen trap was installed within the spectrometer, detector background continuously grew, and typically after three weeks a decontamination of the setup from tritium was necessary. Test measurements with the 7 keV conversion electrons from ⁵⁷Co decay did not reveal any deposition of a residual gas that would exceed 10 ng cm^-2 on a cooled source.

Stoeffl and Decman at the Lawrence Livermore National Laboratory also constructed a toroidal β-ray spectrometer with a gaseous tritium source [112]. The quality of the equipment, similar to that of Robertson et al. [110], was demonstrated by conversion electron spectra emitted by multiply charged ^83mKr in gaseous state. However, their tritium β-spectrum showed an anomalous structure near the endpoint yielding an unphysical result m_ve = –130 ± 20 eV.

Two groups at the Institute of Nuclear Research at Troitsk near Moscow and the Physics Institute of the University in Mainz independently developed a new type of an integrating electron spectrometer well suited for measurements of tritium β-spectra, since it can reach an energy resolution of units of eV while keeping an input solid angle of tens percent of 4π [125-127]. The instrument combines a retarding electrostatic field with a guiding inhomogeneous magnetic field (the so called MAC-E Filter). Contrary to common differential spectrometers, the resolution function of this device has no high-energy tail, a great advantage in the search for a signature of massive neutrino in the uppermost part of β-spectrum. Weinheimer et al. [128] employed the Mainz MAC-E-Filter to measure β-spectrum of molecular T₂ frozen on an aluminum substrate cooled down to 2.8 K. The spectrometer was set to the full width (FW) resolution of 4 eV near the tritium endpoint. The fresh source of 10⁸ Bq activity contained about 40 monolayers of T₂ as determined by ellipsometry. Mass spectrometry revealed about 30% of protium contamination, probably originating from previous storage of tritium in a stainless steel container. There was a substantial escape of tritium from the source with an effective half-life of one weak. Therefore the data were taken for about ten days per source. The electrons were detected with a segmented silicon detector with a resolution of 2.0 keV FWHM at 20 keV energy. Data were stored event-by-event, allowing the checking of the distribution of time differences between successive events. Observed sudden increases in the count rate, probably triggered by microsparks in the spectrometer, were rejected. The remaining data were in accord with the statistics. An observable β-spectrum already emerged clearly from the background at 20 eV below the endpoint but the deduced m_ve² shifted to increasingly more negative values with widening fit interval. Analyses of the last 137 eV of the tritium β-spectrum yielded an upper limit of m_ve < 7.2 eV at 95% confidence level. The final results of the Mainz and Troitsk experiments are discussed in Sect. 3.4.

The experimental results disproving the ITEP claim [117] are summarized in Table. 2. To facilitate the comparison we present the upper limits calculated according to the unified approach [130] with quantities constrained to be non-negative. Undoubtedly, the case of the 30 eV-neutrino considerably improved β-spectroscopic methods aiming at the neutrino mass determination. In the words of the Nobel prize-winner V. L. Ginsburg, “Sometimes science is better served by a wrong result if it is published than withheld” [124].

The best model-independent upper limits of the effective neutrino mass m_ve, as defined by Eq. (5), originate from two measurements of the tritium β-spectrum using electrostatic spectrometers of the MAC-E-Filter type. Kraus et al. [38] improved the Mainz setup in the following way in comparison with its previous stage [128]. The shock-condensed, amorphous T₂ films were cooled to 1.8 K, which substantially slowed down the surface migration, leading to small crystals of size exceeding the average thickness of the T₂ layer. Unaccounted energy loss in these crystals formed at temperatures of 3–4 K was the main reason for previous negative values of m_ve² [128]. The source area and thickness were increased to 2 cm² and about 140 monolayers (~50 nm). Highly oriented pyrolytic graphite (HOPG) with low electron backscattering and atomic flat surface over wide terraces served as a source backing. A LHe-cooled chicane with inner carbon coating adsorbed traces of T₂ that escaped from the source. In this way, the source-dependent component of the detector background was eliminated. Electropolishing the spectrometer tank and its electrodes not only reduced outgassing and field emission but also removed tritium contamination from previous runs. The spectrometer was backed up to 330–420°C and conditioned up to ± 30 kV to further improve conditions of the experiment. Using this improved setup, 11 measurement series were carried out during five years. The following are examples of the effects considered in the β-spectra analysis: a continuously growing coverage of the source by 0.3 monolayers of H₂ per day; a T₂ loss of 0.17 monolayers/day obviously caused by the recoiling daughter molecules that each sputtered a few neighboring molecules from the source, and a self-charging of the T₂ film that reached about 2.5 V at the outer surface. Final states of the daughter molecule calculated by Saenz et al. [131] for gaseous T₂ were slightly modified for a solid T₂ and complemented with a prompt excitation of neighbors next to a decaying T₂ molecule. This effect (amounting to a surprisingly large value of ~5% of all decays) is due to local relaxation of the lattice following the sudden appearance of a (T³He)⁺ ion. The final result of the Mainz neutrino mass experiment is m_ve²= –0.6 ± 2.2_stat ± 2.1_syst eV², yielding m_ve< 2.3 eV at 95% CL [38].

Aseev et al. [39] re-analyzed all data acquired by the Troitsk experiment in which the MAC-E Filter with the resolution of 3.7 eV (FW) at 18.6 keV was utilized to analyze β-particles emitted from a windowless gaseous source of molecular tritium (WGTS), similar to that of [110]. Details of the apparatus are described in [132]. The authors succeeded in suppressing the partial pressure of tritium in the spectrometer (of which the decay would cause an undesirable background) to about 10^-18 mbar. Superconducting solenoids provided a maximum magnetic field of 5 T, while the minimum field in the spectrometer analyzing plane was 1 mT. Electrons were detected with a Si(Li) detector with sensitive area of 2.3 cm². Due to the point-by-point character of the measurement, the intensity of the WGTS was checked every 15 minutes at the monitor point of β-spectrum at 18.0 keV where the counting rate was high. The hydrogen-isotope concentration in the WGTS was measured every two hours but nevertheless the main systematic error arose from an uncertainty of the tritium column density in the WGTS. Only runs with reliable experimental conditions were included in the re-analysis of β-spectra acquired during the 1994-2004 period. The former anomaly – a step-like structure of 5–15 eV below the endpoint with roughly 6 months period and an amplitude of ~6·10^-11 of all β-decays – was no longer observed. The final result of the Troitsk neutrino mass experiment is m_ve²= –0.67 ± 1.89_stat ± 1.68_syst eV², i.e.m_ve < 2.05 eV at 95% CL [39].

From these two independent experiments it is possible to calculate their weighted average, –0.64 ± 1.95 eV², which according to [130] gives the current upper limit of m_ve < 1.6 eV at 90% CL.

Almost all experimental results in neutrino physics are consistent with the concept of three active neutrinos coupled to weakly interacting Z and W^± bosons. Former claims for an admixture of a 17 keV neutrino have been rejected, see Sect. 3.7.1. Nevertheless, several short-baseline neutrino oscillation experiments and some cosmological observations indicate (with about 2–3σ significance) a possible existence of one or even two additional light sterile neutrinos. These so far hypothetical neutrinos would not have any ordinary weak interactions except those induced by mixing. Those alternatives are often referred to as the 3+1 and 3+2 scenarios. Experimental and theoretical aspects of light sterile neutrinos are treated in detail in ref. [10, 79], and the current experimental status and proposed searches of sterile neutrinos are listed [13, 177, 178].

In order to illustrate the present situation we quote a few recent examples: Archidiacono et al. [179] analyzed all available cosmological data assuming the 3+1 scenario and derived m_sterille ≈ m₄ ≈ 0.44 eV, while the analysis of short-baseline oscillation experiments yielded m₄ = 1.27 eV (on 2σ level) [180]. On the other hand, the Planck collaboration that measures the temperature anisotropy of the CMB [66], analyzed their data in combination with other cosmological surveys and found no clear indications for sterile neutrinos and derived < 0.23 eV [181]. An et al. [182] searched for a light sterile neutrino using data of the Daya Bay experiment on oscillation of reactor neutrinos. The measured data are in accord with the concept of three active neutrinos and lead to upper limits for |U₁₄|² in the region of 10^-3 eV² ≤ |Δm²₄₁| ≤ 0.1 eV². Abe et al. analyzed the Super-Kamiokande atmospheric neutrino data in the 3+1 framework assuming |U_e4|² = 0. No evidence of sterile neutrinos was observed. The following limits, |U_μ4|² < 0.041 and |U_τ4|² < 0.18 (90% CL), were found for Δm² > 0.1 eV² [183]. Boyarsky et al. examined hypothetical keV sterile neutrinos as a possible component of the dark matter. He derived from astroparticle observations that an admixture of sterile neutrinos to the active ones should be less than 10^-7 [184]. Leistedt et al. found that the concept of sterile neutrinos did not improve the concordance among various cosmological observations [185]. In the following two sections, we summarize former β-spectroscopic searches for sterile neutrinos and outline their perspectives.

3.7.1. Previous Experiments

Already in 1980 Shrock examined a possibility for searching for additional neutrino mass states m_i (i > 3) in β-ray spectra also [6]. He emphasized that an admixture of each of such states would produce a discontinuity in the β-spectrum slope similar to those shown in Fig. (7). From the position and amplitude of the ith kink the values of m_i and |U_ei|² entering Eq. (5), could be determined. In principle, the signature of a specific mass state m_i should be observable in the spectrum of every β-emitter with the endpoint energy E₀ at kinetic energy equal to E₀ – m_i. Shrock analyzed the experimental Kurie plots of a large number of β-decays. Taking into account that Kurie plots often deviate from linearity due to energy losses of low energy β-particles, he found a possible admixture |U_ei|² ≤ 0.1 for 0.1 keV ≤ m_i ≤ 3 MeV.

Schreckenbach et al. [186] searched for an admixture of sterile neutrinos in the β-decay of ⁶⁴Cu (T_1/2 = 12.7 h) that proceeds via β⁺ and β^– branches with endpoint energies of E_0β+ = 653 keV and E_0β- = 579 keV, respectively. The ⁶⁴Cu activity of 3.3·10¹¹ Bq was produced by (n, γ) reaction in a thermal neutron flux of 3·10¹⁴ cm^-2 s^-1. A target of an area of (5 × 10) cm² and surface density of 500 μg cm^-2 was prepared by vacuum evaporation of 98% -enriched ⁶³Cu onto an Al backing. The β⁺ and β^– spectra in the energy range 150-680 keV and 75-600 keV respectively were measured with a double-focusing magnetic spectrometer [187] equipped with a 32-wire proportional counter. The overall resolution of the set-up, including effects of the target thickness, was determined to be 1 keV at 300 keV using internal conversion lines in the reaction ⁶³Cu (n, e^–). Analysis of the β⁺ and β^– spectra taken from the same source with the same instrument increased reliability of the analysis since the signature of a sterile neutrino with mass m₄ should appear in β⁺ spectrum as well as β^– spectrum at energies E_0β⁺ –m₄ and E_0β- –m₄, respectively. An upper limit for the admixture of a neutrino state with a mass between 110 and 450 keV was determined to be 0.8% at the 90% CL.

Van Elmbt et al. [188] investigated the β-ray spectrum of ²⁰F with Q_β = 7.0 MeV and T_1/2 = 11 s. The ²⁰F activity of 1.6 MBq was produced by the ¹⁹F (d, p) ²⁰F reaction and measurements were done with an iron-free magnetic spectrometer set for a resolution of 3 keV at 4 MeV with a transmission of 0.1% of 4π. The electron energy loss within the LiF target of 280 μg cm^-2 surface density was 0.7 keV at 4 MeV. Reanalysis of the spectra revealed no admixture of sterile neutrinos, with an upper limit ranging from 0.59 to 0.18% for 400 keV ≤ m₄ ≤ 2.8 MeV [189].

Simpson continued in 1985 his investigation of the β-spectrum of tritium implanted into a Si(Li) x-ray detector [114] (see Sect. 3.2) and observed a distortion in the part of the spectrum below 1.5 keV that was consistent with an emission of a neutrino of mass about 17.1 keV and mixing probability of 3% [190]. Some of the subsequent measurements with semiconductor spectrometers, examining β-spectra of ³H, ¹⁴C, ³⁵S and ⁶³Ni, were consistent with about 1% admixture of the 17 keV neutrino, see e.g. Fig. (27). These claims stimulated several theorists to take such a mass state into account, see e.g. paper by Glashow [191].

Fig. (27). The shape factor of the β-ray spectrum of 35S measured with a cooled Si(Li) detector [192]. In the energy region of 17 keV below the endpoint of Qβ = 167 keV, the observed spectrum deviates by 8 standard deviations from the expected single component β-spectrum. Solid line includes the 17 keV neutrino mass component with an admixture of 0.0084 ± 0.0006stat ± 0.0005syst.

However, measurements with magnetic spectrometers were not consistent with the claim. In particular, an upper limit of the admixture of the 17 keV neutrino in the ⁶³Ni decay was put as low as 0.073% at 95% CL [193]. The hypothesis of the 1% admixture was rejected by 22 σ, see Fig. (28). An upper limit for an admixture of neutrino with mass between 10.5 and 25 keV was found to be |U_e4|² < 0.15% at the 95% CL. An example of their β-spectrum of ⁶³Ni is depicted in Fig. (28).

Fig. (28). The β-spectrum of 63Ni measured using a magnetic spectrometer compared with the best fit for |Ue4|2 free (a) and |Ue4|2 = 1% (b). The smooth curve in (a) illustrates the size of 1% effect of the 17 keV neutrino. The figure was reproduced from [193].

The newer measurements with semiconductor spectrometers also did not support the 17 keV neutrino claim. For example, Mortara et al. [194] examined a β-spectrum of ³⁵S (Q_β = 167.2 keV, T_1/2 = 87.5 d) with an apparatus consisting of a superconducting solenoid that collected β-particles from the source to a high-resolution Si(Li) spectrometer. This arrangement exhibited an input solid angle of 50% of 4π and thus weak sources (typically less than 4 kBq) of better spectroscopic quality could be employed. The mixing probability of the 17 keV neutrino was determined to be –0.0004 ± 0.0008_stat ± 0.0008_syst, in disagreement with several previous experiments using solid-state detectors. The authors proved sufficient sensitivity of their apparatus by introducing a known amount of ¹⁴C (Q_β = 156.5 keV, T_1/2 = 5730 y) into their ³⁵S source. Fitting the β-spectrum of the composite (³⁵S + ¹⁴C) source with a pure ³⁵S spectrum shape led to the reduced χ² value of 3.5 while fitting with a free combination of the ³⁵S and ¹⁴C β-spectra yielded reduced χ² value of 1.06. A fraction of decays of ¹⁴C derived from the β-spectrum fit, 1.4 ± 0.1% of the total decay rate, agreed with a value of 1.34% measured for ¹⁴C alone during the ³⁵S + ¹⁴C source preparation. The spectra of residuals of the composite β-spectrum fit are shown in Fig. (29). Further investigation of the ³⁵S β-spectrum [195] clarified that the 17 keV neutrino claim, reported in previous studies of this isotope [192, 196] originated mostly from an improper treatment of the electron energy losses in the source itself. Still another exploration of ³⁵S β-spectrum revealed that a thin metallic diaphragm in front a Si(Li) detector can generate a false 17 keV neutrino admixture of 0.3% [197].

Fig. (29). Residuals from fitting the 35S β-spectrum shape into a measured β-spectrum of the composite (35S + 14C) source [194]. The solid curve indicates residuals expected from the known admixture of 14C.

Although now disproved, the claim for the 17 keV neutrino (in a similar manner to the 30 eV claim discussed in Sect. 3.3) pushed forward the development of precision β-ray spectroscopy. The relevant experiments with both positive and negative results are listed [198, 199]. Their detailed account can be found [200, 201]. “The rise and fall of the 17 keV neutrino“ was also briefly described [202].

Nevertheless, searches for kinks in β-ray spectra in a wider region of m₄ continued. Hiddemann et al. [203] searched for a signature of the sterile neutrinos in their β-spectrum of tritium taken with a magnetic spectrometer but no evidence was found in the mass region 10 eV ≤ m₄ ≤ 4 keV. Holzschuh et al. [204] explored the β-spectrum of ⁶³Ni with a magnetic spectrometer [111] and found the m₄ mixing probability to be less than 1·10^–3 at the 95% CL for neutrino masses in the range from 4 to 30 keV. In their study of the β-spectrum of ³⁵S, the same upper limit of |U_e4|² was determined for 10 ≤ m₄ ≤ 90 keV [205]. Galeazzi et al. [206] searched for the kinks in the low-energy β-spectrum of ¹⁸⁷Re (Q_β = 2.44 keV, T_1/2 = 4.3·10¹⁰y) with cryogenic microcalorimeters, mentioned above in Sect. 3.5.2. A single crystal of superconducting rhenium with mass of 1.5 mg and activity of about 1.1 Bq served as a source as well as an absorber of ¹⁸⁷Re β-rays. No evidence for an additional neutrino mass state m₄ was observed. The upper limits of an |U_e4|² admixture between 0.9% and 4.4% for m₄ = 1000 and 200 eV, respectively, were established at the 95% CL.

An agreement between measured and theoretical β-spectra is often reached only after adding one or two shape correction parameters without clear physical meaning. Such a procedure not only deteriorates the sensitivity of the experiment (since it increases the number of fitted parameters) but it can also hide some spectrum details or create artificial ones²⁰. For example, Apalikov et al. [208] investigated the β-spectrum of ³⁵S with a magnetic spectrometer [109] and found limits for 5 ≤ m₄ ≤ 80 keV. An admixture of the 17.1 keV neutrino was set to be less than 0.17% at the 90% CL. However, in order to reproduce their measured β-spectrum shape, the authors had to introduce a phenomenological correction in the form

(18)

Thus the number of fitted parameters increased by two. The f_corr(E) varied from 1.012 to 1.29 for m₄ between 5 and 80 keV, respectively. For m₄ = 17.1 keV the spectrum correction amounted to 4.5%.

Fig. (30). The K-shell internal-conversion electron line of the 63.1 keV transition in Tm measured with an original and modified 169Yb source [209]. The original source was prepared by vacuum evaporation onto Al backing and was contaminated by a usual hydrocarbon over layer21 (described by a 3 nm thick carbon layer in Monte Carlo calculations). In the case of the modified source a supplementary Pu layer of average thickness of 0.4 nm was evaporated where electrons of the kinetic energy of 3.7 keV suffered additional elastic and inelastic scattering. The agreement of measured and calculated (smooth curves) shapes indicates correctness of the electron loss function in plutonium applied in calculations of the 241Pu β-spectrum shape. The natural width of this conversion line is 32.0 eV [212].

Within the search for an additional neutrino mass-state in the β-decay of ²⁴¹Pu (Q_β = 20.8 keV, T_1/2 = 14.4 y) [209], the electron scattering and energy losses within a vacuum-evaporated source were carefully examined. The measurements were carried out with two electrostatic spectrometers [210, 211]. The individual elastic and inelastic scattering events were simulated using the Monte Carlo method. The quality of the approximation was tested by measuring the shape of the 3.7 keV conversion electron line of the ¹⁶⁹Yb source, see Fig. (30). This approach allowed the description of the measured part of the ²⁴¹Pu spectrum (from 0.2 to 9.2 keV) down to 2 keV without any artificial fitting parameter.

Fig. (31) demonstrates why the β-spectra intended for sterile neutrino searches should be measured with sufficient instrumental resolution. Lines observed on both continuous β-spectra of ground-state-to-ground-state decays of ²⁴¹Pu

and ⁶³Ni at energies of about 260 and 510 eV are the KLL Auger lines of carbon and oxygen. The K-shell vacancies were created by the impact of β-particles from the sources on atoms in the contamination layer. The observed line intensity is in good agreement with the prediction based on the cross-section for electron-impact ionization amounting (10⁴ – 10⁵) •10^–24 cm² for the β-particles of given energy [213] and activity of the sources (6.5 MBq for ²⁴¹Pu and 1 MBq for ⁶³Ni). If the spectra were recorded at a worse instrumental resolution, the Auger lines would smear out and one could erroneously interpret the observed deviation as an anomaly in the β-spectrum shape.

Fig. (31). The KLL-Auger lines of carbon and oxygen superimposed on continuous β-spectra of 241Pu and 63Ni [214]. The spectra of the vacuum-evaporated sources were recorded with an electrostatic spectrometer [210, 215] by scanning the shown energy regions 10 000 and 18 000 times22 . The measurement time was 1 s per point in every scan.

Fig. (32). Best upper limits on the admixture |Ue4|2 of sterile neutrinos derived from measured β-ray spectra. All upper limits refer to the 95% CL except those for 64Cu [186] and 3H [219] that were published at the 90% CL. The numbers denote the following radionuclides and their investigators: 1 – 3H [218], 2 – 3H [219], 3 – 187Re [206], 4 – 3H [203], 5 – 3H [114], 6 – 63Ni [204], 7 – 63Ni [193], 8 – 35S [194], 9 – 64Cu [186], 10 – 20F [189].

Recently, Troitsk physicists extended the analysis of tritium β-spectra gathered in their search for the effective mass m_ve of light active neutrinos [39] and using the maximum-likelihood method they looked for any signatures of sterile neutrinos. A possible admixture of the neutrino mass state with 20 ≤ m₄ ≤ 100 eV was found to be less than 1% at the 95% CL [217, 218]. The surprising feature of their analysis is that this upper limit did not depend on systematic uncertainties of their experiment. This was not the case in their analysis of the same β-spectra for m_ve where statistical and systematic uncertainties played about the same role [39]. Even a new analysis of tritium β-spectra of the Mainz neutrino mass experiment [38], involving detailed examination of systematic uncertainties, did not reveal any admixture of the fourth neutrino mass state [219].

A broad energy range of β-spectroscopic searches for sterile neutrinos is illustrated in Fig. (32). Relevant data is summarized in the section “Kink search in nuclear β-decay” of the current issue of the Review of particle physics [8].

3.7.2. Future Experiments

Formaggio and Barret [220] examined whether the KATRIN experiment [40] with intended sensitivity of 200 meV for light active neutrinos could verify the claim of Mention et al. for the possible existence of a sterile neutrino with > 1.5 eV² and sin²(2θ₄) = 0.14 ± 0.08 at 95% CL [221]²³. The authors [220] estimated that KATRIN after three years of data-taking should be able to place the limits sin²(2θ₄) ≤ 0.08 and 0.20 for |Δm₄₁²| > 4 and 2 eV², respectively. The two values of sin²(2θ₄) correspond to |U_e4|² = 0.02 and 0.05. Similar results were obtained [224] with the conclusion that, for mixing angles sin²(2θ₄) ≤ 0.1, the KATRIN experiment could provide the most stringent limit on the active-sterile mass square difference |Δm₄₁²|.

Further possibilities of KATRIN-like setups were examined by Sejersen Riis and Hannestad [225] who considered the 3+2 scenario, i.e. the mixing of three almost massless active neutrinos with two additional massive sterile neutrinos coupled to ͞ν_e. Their Monte-Carlo-generated tritium β-spectra included the sterile mass states m₄ and m₅ up to 6.4 eV and mixings |U_e4|² and |U_e4|² in the interval from 5.5·10^-4 to 0.18. Projected parameters of the KATRIN setup, including an energy resolution of 0.93 eV, a background of 0.01s^-1 and systematic uncertainty of 0.017 eV² [142], were assumed. Possible future improvements such as a higher β-spectrum amplitude, better resolution and lower background were probed as well. The authors concluded that the KATRIN experiment should be sensitive enough to confirm or disprove sterile neutrinos with Δm₄₁² = 6.49 eV², |U_e4|² = 0.12 and |U_e5|² = 0.11, as suggested by some cosmologists. In the 2 × 2 mixing scheme (one active light neutrino with the effective mass m_ve according to Eq. 5 and one sterile neutrino), any of the mass states m₄ with |U_e4|² ≥ 0.055 should be detected with 3σ confidence. The same sensitivity is expected for any of the mixing angles for the mass state with m₄ ≥ 3.2 eV.

Mertens et al. [82] recently scrutinized the sensitivity of future tritium β-decay experiments towards hypothetical keV-scale sterile neutrinos that are one of the candidates for dark matter of the universe, see e.g. [10]. Experimental considerations of the authors were based on the statistics of 10¹⁸ β-decay events occurring in the KATRIN gaseous tritium source during the three-year exposure time. Three possible measurement modes were taken into account: (i) a point-by-point measurement, with variable retardation of the electrostatic spectrometer yielding the β-spectrum in its integral form, (ii) an application of the KATRIN high-luminosity tritium source together with a future semiconductor spectrometer consisting of up to 10⁶ independent pixels and capable of operating at 10⁵ events per second, (iii) the KATRIN tritium source and electrostatic spectrometer but operating in the time-of-flight regime [226] where the last two modes yield the β-spectra in their differential form. As can be seen in Fig. (33), the differential spectra exhibit an extremely high sensitivity to |U_e4|² even in the case of a moderate spectrometer resolution. Note, however, that the limits shown comprise only the statistical sensitivity; the systematic uncertainties are not included.

Fig. (33). The 90% exclusion limit for an admixture of keV sterile neutrinos obtained from simulated β-spectra [82] that correspond to differential measurements of the KATRIN-like tritium source with a future multi-pixel semiconductor spectrometer of indicated energy resolution.

The authors [227] estimated quantitatively the influence of eight different uncertainties of the shape of the theoretical β-spectrum, such as decay into excited final states, nuclear correction terms, etc, and concluded that these uncertainties decrease the sensitivity to keV neutrinos by about one order of magnitude. Ensuring the relative standard deviation of individual points of the overall β-spectrum on the level of 10^-6 during 3 measurement years is an extremely challenging task. Therefore the authors considered an alternative approach where the keV sterile neutrinos would be searched in a β-spectrum with the same tritium source as in the KATRIN experiment but with a high-resolution detector. Their simulations showed that the wavelet transform, i.e. the convolution of the β-spectrum with specific window functions, is largely insensitive to smooth spectral deformations that are caused e.g. by slow changes of the measurement conditions. A statistical sensitivity to sterile-to-active neutrino mixing |U_e4|² = 10^-6 is expected for the detector resolution of FWHM ≈ 100 eV. Nevertheless, the most reliable way to correct experimental results is a sufficiently precise knowledge of all systematic effects. Such approach may require a series of side measurements as exemplified by the Mainz neutrino mass experiment [38].

Filianin et al. [228] are developing methods of cryogenic microcalorimetry and high-precision Penning-trap mass spectrometry in order to probe the possible existence of sterile neutrinos with masses from 0.5 to 100 keV. The authors considered the electron-capture decay of eight isotopes including ¹⁶³Ho, ¹⁷⁹Ta, ¹⁹³Pt and ²³⁵Np. They also examined a rather complex contribution of a sterile neutrino within the shape of an electron-capture spectrum; see Sect. 3.5.2 and [150, 152].

Abdurashitov et al. [229] proposed a measurement of the tritium β-spectrum with proportional counters that has already proven long time stability in the SAGE experiment on solar neutrinos. The authors estimated that an analysis of a β-spectrum measured with relatively poor resolution of 10–15% and containing a total of 10¹² registered β-particles, could reach a sensitivity of 10^-3 to 10^-5 for an admixture of sterile neutrinos in the mass range from 1 to 8 keV. This estimate took into account the deterioration of the shape of the spectrum due to β-decays near the counter inner wall as well as a pile-up effect at counting rates up to 10⁶ s^-1. However, one should also include the β-decay of tritium adsorbed on the counter walls – an effect that has already been carefully examined and proved to be substantial [230] in connection with the 17 keV neutrino claim.

The Troitsk experimental set-up for an investigation of tritium β-spectra was recently equipped with a new MAC-E-filter of twice the diameter and ten times the volume of the older one [39, 132]. The field of the main magnet was increased by 20%. The energy resolution at 18.6 keV is now 1.5 eV (FW), more than twice better than previously. The new facility should enable searching for sterile neutrinos in a few keV mass range with a sensitivity for |U_e4|² up to ~2·10^–4 and a further improvement of an order of magnitude after future upgrades [231].

The role of sterile neutrino warm dark matter ²⁴ was examined using ¹⁸⁷Re and ³H β-decays [233]. Theoretical aspects of sterile neutrinos and right-handed currents were treated in the frame of the anticipated KATRIN experiment [234]. A possibility to record signatures of extra dimensional sterile neutrinos in this experiment was examined [235].

²⁴ According to ref. [232] these sterile neutrinos could not only be warm but also cold dark matter.

²⁵ It was achieved by means of a magnetic spectrometer equipped with a Geiger counter. Former recording of electron intensity with photographic plates enabled rather precise measurement of the energy of monoenergetic internal conversion electrons but due to nonlinearity of the plate blackening did not allow reliable investigation of β-rays.