Investigation on Rare Nuclear Processes in Hf Nuclides

Simple Summary

Nuclear instability is an interesting topic which plays an important role in nuclear models, electroweak interaction e conservation laws. Herein, investigations on rare nuclear processes in Hf isotopes, using HP-Ge spectrometry and Hf-based crystal scintillators, are described. This type of investigation can be crucial in developing models in nuclear and astroparticle physics. In addition to the already-observed alpha decay of ¹⁷⁴Hf, some other rare nuclear processes, such as the alpha decay of ¹⁷⁶Hf in ¹⁷²Yb and ¹⁷⁷Hf in ¹⁷³Yb are near the theoretical expectations, giving hope to their first observation in the near future. In addition, a short emphasis on several types of Hf-based crystal scintillators is reported.

Abstract

In this work, a review of recent studies concerning rare nuclear processes in Hf isotopes is presented. In particular, the investigations using HP-Ge spectrometry and Hf-based crystal scintillators are focused; the potentiality and the results of the “source = detector” approach are underlined. In addition, a short introduction concerning the impact of such kind of research in the context of astroparticle and nuclear physics is pointed out. In particular, the study of $\alpha$ decay and double beta decay of ${}^{174}$Hf, ${}^{176}$Hf, ${}^{177}$Hf, ${}^{178}$Hf, ${}^{179}$Hf, ${}^{180}$Hf isotopes either to the ground state or to the lower bounded levels have been discussed. The observation of $\alpha$ decay of ${}^{174}$Hf isotope to the ground state with a T${}_{1/2}=7.0\left(1.2\right)\times {10}^{16}$ y is reported and discussed. No decay was detected for $\alpha$ decay of ${}^{174}$Hf isotope at the first excited level of daughter and of ${}^{176}$Hf, ${}^{177}$Hf, ${}^{178}$Hf, ${}^{179}$Hf, ${}^{180}$Hf isotopes either to the ground state or to the lower bounded levels. The T${}_{1/2}$ lower limits for these decays are at the level of ${10}^{16}$–${10}^{20}$ y. Nevertheless, the T${}_{1/2}$ lower limits for the transitions of ${}^{176}$Hf$\to {}^{172}$Yb ($0^{+}\to 0^{+}$) and ${}^{177}$Hf$\to {}^{173}$Yb ($7/2^{-}\to 5/2^{-}$) are near to the theoretical predictions, giving hope to their observation in the near future. All the other experimental limits ($\sim {10}^{16}$–10${}^{20}$ y) are absolutely far from the theoretical expectations. The experiments investigating the $2ϵ$ and $ϵ{\beta }^{+}$ processes in ${}^{174}$Hf are also reported; the obtained half-life limits are set at the level of ${10}^{16}$–${10}^{18}$ y. Moreover, we estimate the T${}_{1/2}$ of $2\nu 2ϵ$ of ${}^{174}$Hf decay at the level of (0.3–6) × ${10}^{21}$ y (at now the related measured lower limit is $7.1\times {10}^{16}$ y).

1. Introduction

A fascinating topic of nuclear physics is the nuclear instability, which plays an important role in nuclear models, electroweak interaction, and conservation laws. Here, the current status of the experimental searches for rare $\alpha$ and double beta decay (DBD) in Hf-isotopes is reviewed.

The DBD is a significant nuclear decay for neutrino physics, to test, e.g., the calculations of different nuclear shapes and the decay modes that involve the vector and axial-vector weak effective coupling constants ($g_A$), etc. The DBD with neutrinos is a higher-order effect with respect to the $\beta$-decay: the expected half-lives are longer than those of the $\beta$-decay, and roughly of the order of $\simeq {10}^{20}$ y. Therefore, special experimental care has to be taken to study this process. The DBD without neutrinos emission is a process requiring lepton-number violation, in addition to a non-vanishing neutrino mass that requires a Majorana mass component. Therefore, it could be often regarded as the golden-standard process for probing the fundamental nature of neutrinos. Thus, the $0\nu$DBD is of great interest because it could open a new window beyond the Standard Model. The number of candidate isotopes to this study are 69, in particular, 35 via the emission of two $e^{-}$ and 34 in positive channels: either two positron emission (2${\beta }^{+}$) or a positron emission with an electron capture ($ϵ{\beta }^{+}$) or a double electron capture (2$ϵ$) [1,2]. The simultaneous study of positive and negative DBD can constrain the theoretical parameters with very high confidence, giving mutual information. Additionally, the nuclear matrix elements for the $2\nu$ transition and $0\nu$ transition can be connected through significant parameters: in the free nucleon interaction, the $g_A$ value is 1.2701, but, considering a nuclear decay, the phenomenological axial-vector coupling value seams reduced to $g_A<1$, more precisely: $g_A\sim 1.269A^{-0.12}$ or $g_A\sim 1.269A^{-0.18}$, depending on the adopted nuclear model [1]. Moreover, the “quenched” $g_A$ observed is shown to encode the emergence of chiral-scale symmetry hidden in QCD in the vacuum; the $g_A$ value could impact the in-medium modified nucleon weak and electromagnetic form factors on the neutrino mean free path in dense matter; the $g_A$ role in the values of the cross sections of neutrino scattering from astrophysical sources on nuclei; and many other issues. Thus, DBD investigation considering different nuclei would help to constrain these and other significant model-dependent parameters. In addition, in case of the $0\nu 2ϵ$ transition, it is possible to study also the so-called “resonant effect” [1,2].

Besides these, various nuclear models are continuously extended or improved, inspired, for example, by studying long-lived or stable superheavy isotopes and the predictions of their half-lives. Thus, the study of rare alpha decay plays a crucial role in developing nuclear physics. It can offer details about the nuclear structure, its levels and properties. Furthermore, the phenomenon of $\alpha$ decay can provide information about the fusion-fission reactions since the $\alpha$ decay process involves sub-barrier penetration due to the interaction between the nucleus and the $\alpha$ particle [3]. Moreover, understanding the nuclear properties is essential also for nuclear and particle astrophysics studies, for example, $\alpha$–capture reactions (equivalent to the inverse $\alpha$–decay process) are important for nucleosynthesis and $\beta$–delayed fission, together with other fission modes, determine the so-called “fission recycling” in the r-process nucleosynthesis [4]. In addition, as a byproduct, such researches contribute to develop new detectors and perform news protocols to purify materials containing DBD or rare $\alpha$ emitters from radio-impurities [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. In this work, main aspects on status of the investigations on rare nuclear processes in Hf isotopes are reported.

2. Searching for Double Beta Decay in Hafnium Isotopes

The ${}^{174}$Hf isotope is a potentially DBD emitter via $ϵ{\beta }^{+}$ and $2ϵ$ modes with the energy decay $Q_{2\beta }=1100.0\left(23\right)$ keV [26] and an isotopic abundance $\delta =0.156\left(6\right)$% [27]. A simplified $ϵ{\beta }^{+}$ and $2ϵ$ decay scheme of ${}^{174}$Hf is shown in Figure 1. Theoretical half-life estimations of the $ϵ{\beta }^{+}$ and $2ϵ$ of ${}^{174}$Hf are not present in literature (to our knowledge) but the values of the phase space factor involved are (0.001–7.5)$\times {10}^{-30}$ y${}^{-1}$ and (0.003–1)$\times {10}^{-17}$ y${}^{-1}$ respectively for the $2\nu ϵ{\beta }^{+}$ and $0\nu ϵ{\beta }^{+}$ transitions [28] and $(1.6-3.5)\times {10}^{-21}$ y${}^{-1}$ for the $2\nu 2ϵ$ transition [28]. Considering that typically the magnitude of the nuclear matrix element of the $2\nu$DBD is of the order of 1–10, the half-life estimations are roughly ${10}^{28}$–${10}^{33}$ y and (0.3–6) $\times {10}^{21}$ y respectively for the $2\nu ϵ{\beta }^{+}$ and $2\nu 2ϵ$ transitions.

In 2020 and 2021 two independent experiments [29,30] (they are the first searches ever realized for $2ϵ$ and $ϵ{\beta }^{+}$ decay of ${}^{174}$Hf and for the work in Ref. [29] also by using a high-pure sample of hafnium), the first one performed at HADES laboratory of the Joint Research Centre of European Commission (Geel, Belgium) and the second one at the underground National Laboratory of Gran Sasso (LNGS) in Italy, searched for the $2ϵ$ and $ϵ{\beta }^{+}$ decay of ${}^{174}$Hf. In both cases, a passive approach (the source of the event searched for is outside the detector), using gamma-ray spectrometry technique, has been adopted. Both the experiments have adopted two different geometrical arrangements to increase the detection efficiency and to decrease as much as possible the energy threshold.

The experiment at HADES in Ref. [29] used a sample of metallic hafnium with sizes $\oslash 59.0$ mm × 5.0 mm (total mass of 179.8 g that contained ∼0.29 g of the isotope ${}^{174}$Hf). The experimental set-up was arranged using two different couples of HP-Ge detectors, and the Hf-sample was installed between the end-cap of two HP-Ge detectors arranged back-to-back (see Figure 2-left). The total exposure of the experiment was 42 g × d for the isotope ${}^{174}$Hf.

The experiment at LNGS on Ref. [30] used a foil of metallic hafnium of 55.379(1) g and the HP-Ge ($\oslash 70$ mm × 70 mm) was covered by this hafnium foil (see Figure 2-right). In particular, the foil thickness (0.25(1) mm) was optimized to minimize the self-absorption of low-energy $\gamma$ and X-rays within the sample itself and improve the overall detection efficiency; infact, the metallic foils were coupled to the Ge crystal (under its end-cap). The data were collected over 310 d.

The total exposures of those experiments were 26.9 kg × d and 16.5 kg × d for Ref. [29] and Ref. [30] respectively. In the

Table 1. The detection efficiency of some $2ϵ$ and $ϵ{\beta }^{+}$ processes in ${}^{174}$Hf as performed in Refs. [29,30] if comparable. E${}_{\gamma }$ is the $\gamma$ quanta used in both the experiments to study the half-life of the related transition.

Channel of the Decay	Decay Mode	Level of Daughter Nucleus	E${}_{\mathit{\gamma }}$ (keV)	Detection Efficiency (%)
		J${}^{\mathit{\pi }}$, Energy (keV)		[29]	[30]
$2L$	$2\nu$	$2^{+}$, 76.5	76.5	$0.39$	$3.15$
$2K$	$0\nu$	g.s.	977.4	$4.53$	$7.59$
$KL$	$0\nu$	g.s.	1028.9	$4.46$	$7.32$
$2L$	$0\nu$	g.s.	1080.4	$4.39$	$7.09$
$2K$	$0\nu$	$2^{+}$, 76.5	900.9	$4.67$	$8.01$
$KL$	$0\nu$	$2^{+}$, 76.5	952.4	$4.59$	$7.72$
$2L$	$0\nu$	$2^{+}$, 76.5	1003.9	$4.51$	$7.45$
$K{\beta }^{+}$	$2\nu +0\nu$	g.s.	511	$10.6$	$11.8$
$L{\beta }^{+}$	$2\nu +0\nu$	g.s.	511	$10.7$	$11.8$

the detection efficiencies of both the experiments are compared (when possible); typically the experiment at Ref. [30] has higher values with respect to the other experiment.

Considering the $2ϵ$ transitions, in case of a $2K$ o $KL$ capture in ${}^{174}$Hf, a cascade of X-rays (and Auger electrons) of ${}^{174}$Yb atom with energies in the range of (50.8–61.3) keV is expected, while energies of the $2L$ capture X-ray quanta are ≃(7–10) keV. These latter were below the detectors’ energy thresholds of both the experiments. Thank to these X-rays, the decay modes $2K$ and $KL$ to the ground state (g.s.) of the ${}^{170}$Yb have been studied by both experiments, besides the transition to the first excited level of the daughter nucleus. In both the investigations, the energy spectrum in the region of interest for the 2$\nu$2K and 2$\nu$KL was dominated by the background due to the electron capture of ${}^{175}$Hf (T${}_{1/2}$ = 70(2) d; Q${}_{EC}=683.9\left(20\right)$ keV), but also from events related to U/Th chains, as clearly shown in the Figure 3a,b. In Refs. [29,30] no peculiarity have been observed concerning the DBD of ${}^{174}$Hf; thus, in

Table 2. The half-life limits of $2ϵ$ and $ϵ{\beta }^{+}$ processes in ${}^{174}$Hf as performed in Refs. [29,30].

Channel of the Decay	Decay Mode	Level of Daughter Nucleus	Experimental Limit of T${}_{1/2}$ (90% C.L. (y))
		J${}^{\mathit{\pi }}$, Energy (keV)	[29]	[30]
$2K$	$2\nu$	g.s.	≥$7.1\times {10}^{16}$	≥$1.4\times {10}^{16}$
$KL$	$2\nu$	g.s.	≥$4.2\times {10}^{16}$	≥$1.4\times {10}^{16}$
$2K$	$2\nu$	$2^{+}$, 76.5	≥$5.9\times {10}^{16}$	≥$7.9\times {10}^{16}$
$KL$	$2\nu$	$2^{+}$, 76.5	≥$3.5\times {10}^{16}$	≥$7.9\times {10}^{16}$
$2L$	$2\nu$	$2^{+}$, 76.5	≥$3.9\times {10}^{16}$	≥$7.9\times {10}^{16}$
$2K$	$0\nu$	g.s.	≥$5.8\times {10}^{17}$	≥$2.7\times {10}^{18}$
$KL$	$0\nu$	g.s.	≥$1.9\times {10}^{18}$	≥$4.2\times {10}^{17}$
$2L$	$0\nu$	g.s.	≥$7.8\times {10}^{17}$	≥$3.6\times {10}^{17}$
$2K$	$0\nu$	$2^{+}$, 76.5	≥$7.1\times {10}^{17}$	≥$2.4\times {10}^{18}$
$KL$	$0\nu$	$2^{+}$, 76.5	≥$6.2\times {10}^{17}$	≥$3.1\times {10}^{17}$
$2L$	$0\nu$	$2^{+}$, 76.5	≥$7.2\times {10}^{17}$	≥$9.4\times {10}^{17}$
$K{\beta }^{+}$	$2\nu +0\nu$	g.s.	≥$1.4\times {10}^{17}$	≥$5.6\times {10}^{16}$
$L{\beta }^{+}$	$2\nu +0\nu$	g.s.	≥$1.4\times {10}^{17}$	≥$5.6\times {10}^{16}$

, the half-life limits are reported.

3. Searching for Alpha Decays in Hafnium Isotopes

Natural hafnium consists of six isotopes: ${}^{174}$Hf, ${}^{176}$Hf, ${}^{177}$Hf, ${}^{178}$Hf, ${}^{179}$Hf, ${}^{180}$Hf; all of them are theoretically unstable concerning the $\alpha$ decay with a Q${}_{\alpha }$ value in the range of 1.3–2.5 MeV. The Hf isotopes with natural abundance and with Q${}_{\alpha }>0$ are listed in the Table 3. All of them can decay to g.s. or to excited levels of daughter nuclei. Figure 4 shows simplified decay schemes of the $\alpha$ decay of the naturally occurring Hf isotopes.

Table 3. Main potential $\alpha$ decay of Hf nuclides. Isotopes with natural abundance ($\delta$) greater than zero (i.e., naturally present in nature) and with Q${}_{\alpha }>0$ for transitions between g.s. or between g.s. and lowest bounded level (with spin/parity $J^{\pi }$) are listed. Theoretical predictions and experimental measurements (if exist) on the T${}_{1/2}$’s are reported in the latest columns.

Nuclide Transition	${\mathit{J}}^{\mathit{\pi }}$	$\mathit{\delta }$	Q${}_{\mathit{\alpha }}$	T${}_{1/2}$ (y)
	Parent→	(%)	(keV)			Theoretical
	Daughter Nuclei	[31]	[26]	Experimental
	and Its Level (keV)				[32]	[33]	[34]
${}^{174}$Hf$\to {}^{170}$Yb	$0^{+}\to 0^{+}$, g.s.	0.156(6) [27]	2494.5(2.3)	$7.0\left(1.2\right)\times {10}^{16}$ [27]	$3.5\times {10}^{16}$	$7.4\times {10}^{16}$	$3.5\times {10}^{16}$
	$0^{+}\to 2^{+}$, 84.2			⩾$2.8\times {10}^{16}$ [30]	$1.3\times {10}^{18}$	$3.0\times {10}^{18}$	$6.6\times {10}^{17}$
${}^{176}$Hf$\to {}^{172}$Yb	$0^{+}\to 0^{+}$, g.s.	5.26(70)	2254.2(1.5)	⩾$9.3\times {10}^{19}$ [27]	$2.5\times {10}^{20}$	$6.6\times {10}^{20}$	$2.0\times {10}^{20}$
	$0^{+}\to 2^{+}$, 78.7			⩾$3.0\times {10}^{17}$ [35]	$1.3\times {10}^{22}$	$3.5\times {10}^{22}$	$4.9\times {10}^{21}$
${}^{177}$Hf$\to {}^{173}$Yb	$7/2^{-}\to 5/2^{-}$, g.s.	18.60(16)	2245.7(1.4)	⩾$3.2\times {10}^{20}$ [27]	$4.5\times {10}^{20}$	$5.2\times {10}^{22}$	$4.4\times {10}^{22}$
	$7/2^{-}\to 7/2^{-}$, 78.6			⩾$1.3\times {10}^{18}$ [35]	$9.1\times {10}^{21}$	$1.2\times {10}^{24}$	$3.6\times {10}^{23}$
${}^{178}$Hf$\to {}^{174}$Yb	$0^{+}\to 0^{+}$, g.s.	27.28(28)	2084.4(1.4)	⩾$5.8\times {10}^{19}$ [27]	$3.4\times {10}^{23}$	$1.1\times {10}^{24}$	$2.2\times {10}^{23}$
	$0^{+}\to 2^{+}$, 76.5			⩾$1.3\times {10}^{18}$ [30]	$2.4\times {10}^{25}$	$8.1\times {10}^{25}$	$7.1\times {10}^{24}$
${}^{179}$Hf$\to {}^{175}$Yb	$9/2^{+}\to 7/2^{+}$, g.s.	13.62(11)	1807.7(1.4)	⩾$2.5\times {10}^{20}$ [27]	$4.5\times {10}^{29}$	$4.0\times {10}^{32}$	$4.7\times {10}^{31}$
	$9/2^{+}\to 9/2^{+}$, 104.5			⩾$2.7\times {10}^{18}$ [30]	$2.0\times {10}^{32}$	$2.5\times {10}^{35}$	$2.2\times {10}^{34}$
${}^{180}$Hf$\to {}^{176}$Yb	$0^{+}\to 0^{+}$, g.s.	35.08(33)	1287.1(1.4)	–	$6.4\times {10}^{45}$	$5.7\times {10}^{46}$	$9.2\times {10}^{44}$
	$0^{+}\to 2^{+}$, 82.1			⩾$1.0\times {10}^{18}$ [35]	$4.0\times {10}^{49}$	$4.1\times {10}^{50}$	$2.1\times {10}^{48}$

Table 3. Main potential $\alpha$ decay of Hf nuclides. Isotopes with natural abundance ($\delta$) greater than zero (i.e., naturally present in nature) and with Q${}_{\alpha }>0$ for transitions between g.s. or between g.s. and lowest bounded level (with spin/parity $J^{\pi }$) are listed. Theoretical predictions and experimental measurements (if exist) on the T${}_{1/2}$’s are reported in the latest columns.

Nuclide Transition	${\mathit{J}}^{\mathit{\pi }}$	$\mathit{\delta }$	Q${}_{\mathit{\alpha }}$	T${}_{1/2}$ (y)
	Parent→	(%)	(keV)			Theoretical
	Daughter Nuclei	[31]	[26]	Experimental
	and Its Level (keV)				[32]	[33]	[34]
${}^{174}$Hf$\to {}^{170}$Yb	$0^{+}\to 0^{+}$, g.s.	0.156(6) [27]	2494.5(2.3)	$7.0\left(1.2\right)\times {10}^{16}$ [27]	$3.5\times {10}^{16}$	$7.4\times {10}^{16}$	$3.5\times {10}^{16}$
	$0^{+}\to 2^{+}$, 84.2			⩾$2.8\times {10}^{16}$ [30]	$1.3\times {10}^{18}$	$3.0\times {10}^{18}$	$6.6\times {10}^{17}$
${}^{176}$Hf$\to {}^{172}$Yb	$0^{+}\to 0^{+}$, g.s.	5.26(70)	2254.2(1.5)	⩾$9.3\times {10}^{19}$ [27]	$2.5\times {10}^{20}$	$6.6\times {10}^{20}$	$2.0\times {10}^{20}$
	$0^{+}\to 2^{+}$, 78.7			⩾$3.0\times {10}^{17}$ [35]	$1.3\times {10}^{22}$	$3.5\times {10}^{22}$	$4.9\times {10}^{21}$
${}^{177}$Hf$\to {}^{173}$Yb	$7/2^{-}\to 5/2^{-}$, g.s.	18.60(16)	2245.7(1.4)	⩾$3.2\times {10}^{20}$ [27]	$4.5\times {10}^{20}$	$5.2\times {10}^{22}$	$4.4\times {10}^{22}$
	$7/2^{-}\to 7/2^{-}$, 78.6			⩾$1.3\times {10}^{18}$ [35]	$9.1\times {10}^{21}$	$1.2\times {10}^{24}$	$3.6\times {10}^{23}$
${}^{178}$Hf$\to {}^{174}$Yb	$0^{+}\to 0^{+}$, g.s.	27.28(28)	2084.4(1.4)	⩾$5.8\times {10}^{19}$ [27]	$3.4\times {10}^{23}$	$1.1\times {10}^{24}$	$2.2\times {10}^{23}$
	$0^{+}\to 2^{+}$, 76.5			⩾$1.3\times {10}^{18}$ [30]	$2.4\times {10}^{25}$	$8.1\times {10}^{25}$	$7.1\times {10}^{24}$
${}^{179}$Hf$\to {}^{175}$Yb	$9/2^{+}\to 7/2^{+}$, g.s.	13.62(11)	1807.7(1.4)	⩾$2.5\times {10}^{20}$ [27]	$4.5\times {10}^{29}$	$4.0\times {10}^{32}$	$4.7\times {10}^{31}$
	$9/2^{+}\to 9/2^{+}$, 104.5			⩾$2.7\times {10}^{18}$ [30]	$2.0\times {10}^{32}$	$2.5\times {10}^{35}$	$2.2\times {10}^{34}$
${}^{180}$Hf$\to {}^{176}$Yb	$0^{+}\to 0^{+}$, g.s.	35.08(33)	1287.1(1.4)	–	$6.4\times {10}^{45}$	$5.7\times {10}^{46}$	$9.2\times {10}^{44}$
	$0^{+}\to 2^{+}$, 82.1			⩾$1.0\times {10}^{18}$ [35]	$4.0\times {10}^{49}$	$4.1\times {10}^{50}$	$2.1\times {10}^{48}$

Figure 4. Simplified decay schemes of potential $\alpha$ decay of Hf isotopes considering the first two energy levels of the daughter nuclei. The corresponding gamma transitions and the probability related for a single energy level are also shown. The ${}^{175}$Yb isotope is unstable via ${\beta }^{-}$ decay with T${}_{1/2}=4.185\left(1\right)$ d [36], all the other Yb nuclei are stable.

Figure 4. Simplified decay schemes of potential $\alpha$ decay of Hf isotopes considering the first two energy levels of the daughter nuclei. The corresponding gamma transitions and the probability related for a single energy level are also shown. The ${}^{175}$Yb isotope is unstable via ${\beta }^{-}$ decay with T${}_{1/2}=4.185\left(1\right)$ d [36], all the other Yb nuclei are stable.

In the decays of such Hf isotopes to excited levels, $\gamma$ quanta are emitted. In this case, a search by using low-background $\gamma$ spectrometry can be implemented. Moreover, because the ${}^{175}$Yb is unstable via ${\beta }^{-}$ decay with T${}_{1/2}=4.185\left(1\right)$ d [36], the ${}^{179}$Hf to the g.s. of daughter ${}^{175}$Yb can be evaluated using the mentioned passive technique.

Besides, the nuclide ${}^{178m2}$Hf, the excited state of ${}^{178}$Hf with energy 2446.1 keV, has a long half-life T${}_{1/2}=31\left(1\right)$ y [37]. Ordinarily, its de-excitation happens through ${}^{178m2}$Hf ⟶${}^{178}$Hf transition. However, the $\alpha$ decay of ${}^{178m2}$Hf ⟶${}^{174}$Yb is energetically possible with Q${}_{\alpha }$ = 4530 keV. In addition, the ${\beta }^{-}$ decay to ${}^{178}$Ta (Q${}_{{\beta }^{-}}$ = 606 keV), the electron capture to ${}^{178}$Lu (Q${}_{ϵ}$ = 349 keV) and the spontaneous fission (Q${}_{SF}\simeq 100$ MeV) are also possible [38].

Considering that approach and using HP-Ge detectors, two experimental set-ups were realized (see Refs. [30,35]). Such setups are the same as those described in Section 2; they are the first experiments performed to study the $\alpha$ decay of ${}^{176}$Hf, ${}^{177}$Hf, ${}^{178}$Hf, ${}^{179}$Hf and ${}^{180}$Hf, but considering only the transitions emitting gamma quanta. Thus, in Refs. [30,35], the decays, at the first excited level of the daughter nucleus, have been studied, as well as the ${}^{179}$Hf to the g.s. of the daughter ${}^{175}$Yb, thanks to its ${\beta }^{-}$ instability (see Figure 4). The $\alpha$ decay of ${}^{174}$Hf to the g.s. of ${}^{170}$Yb was studied for the first time in Ref. [39]. In that case, a large cylindrical ionization counter has been used for measurements of natural alpha radioactivity. The heavy element samples were deposited uniformly over the internal surface of the cylindrical chamber, directly on the copper or the steel of the chamber (total active surface of 1200 cm${}^2$). The gas mixture used was 94% argon, 5% ethylene, and 1% nitrogen. The measurements on the Hf sample were performed in two stages. In the first one a sample of HfO${}_2$ enriched in ${}^{174}$Hf was deposited with an average thickness of 0.10 mg × cm${}^{-2}$ and a weak ${}^{210}$Po standard was added. A small peak was observed at 2.50 MeV. In the second stage, the HfO${}_2$ sample was mixed with a small amount of Sm${}_2$O${}_3$, to try to improve the energy calibration of the data considering the peak position of ${}^{147}$Sm, claiming that the T${}_{1/2}=(4.0\pm 0.2)\times {10}^{15}$ y. However, the theoretical expectation (see Table 3) was in strong tension with the observation claimed.

Recently, an interesting experiment, using an active approach, was performed in Ref. [27] to study some potentially $\alpha$ decay of Hf isotopes to g.s. or exited levels of daughter nuclei and in particular the $\alpha$ decay of ${}^{174}$Hf to the g.s. of ${}^{170}$Yb. The best obtained results are reported in Table 3 with the theoretical estimations. In particular, Ref. [27] reports a direct study of the $\alpha$ decay of naturally occurring Hf isotopes by exploiting the “source = detector” approach with a Cs${}_2$HfCl${}_6$ (CHC) crystal scintillator. There, after 2848 h of data taking, the $\alpha$ decay of ${}^{174}$Hf was observed with T${}_{1/2}=7.0\left(1.2\right)\times {10}^{16}$ y. The experiment was carried out at the STELLA facility of the LNGS. The CHC crystal scintillator with mass 6.90(1) g was coupled with a 3-inch low radioactivity photomultiplier (PMT, Hamamatsu R6233MOD), and placed above the end-cap of the ultra-low background HP-Ge $\gamma$ spectrometer GeCris (465 cm${}^3$). A schematic cross-sectional view of the experimental set-up is shown in Ref. [27].

The crystal surface was wrapped by diffusive Polytetrafluoroethylene (PTFE) tape to enhance the light collection. The CHC and HP-Ge detectors were installed inside a passive shield, all in a plexiglas box continuously flushed by high purity nitrogen gas.

A CAEN DT5720B digitizer, as an event-by-event system, acquired the pulse profiles from the PMT and HP-Ge. In this way, it was possible to implement a coincidence logic between the events in the CHC and HP-Ge to study the $\alpha$ decay of Hf isotopes to the first excited level of the daughter nucleus when a $\gamma$–ray of such a decay is emitted and can reach the HP-Ge detector.

The pulse-shape discrimination (PSD) between $\beta /\gamma$ and $\alpha$ particles, the time–amplitude analysis of fast correlated $\alpha$ decays, and the so-called Bi–Po events analysis were adopted to estimate the radioactive contamination of the CHC crystal. In particular, to build the background model, the data of the radioactive contamination of the CHC crystal was considered.

The time shape of each event was utilised to determine its “mean time” ($\langle t\rangle$) according to:

$$\begin{array}{c}\hfill \langle t\rangle =\sum f\left(t_k\right)t_k/\sum f\left(t_k\right)\end{array}$$

(1)

where the sum is over the time channels, k, starting from the origin of the pulse up to 8 $\mathsf{\mu }$s. Moreover, $f\left(t\right)$ is the digitized amplitude (at the time t) of a given signal. Figure 5 shows the scatter plot of the $\langle t\rangle$ versus the energy for the acquired data. It confirms the good pulse-shape discrimination ability of the CHC detector [40]. The distribution of the $\langle t\rangle$ for the events with energies—using the $\gamma$ scale—in the interval (0.4–3.0) MeV is reported in the inset of Figure 5. In Figure 6, the spectrum of $\alpha$ events according to the PSD analysis is given.

The Q.F.’s of the $\alpha$ particles at the energies of ${}^{224}$Ra, ${}^{220}$Rn and ${}^{216}$Po $\alpha$ decays were measured to be 0.39(4), 0.40(3), 0.40(3), respectively, and shown in Figure 7 of Ref. [27] with the fit following the prescription of Ref. [41].

The $\alpha$ energy spectrum, above 4 MeV, was fitted (see Figure 6) using a model which includes the $\alpha$ peaks (see Ref. [27] for details) of ${}^{232}$Th, ${}^{238}$U and their daughters in order to study such contaminants (taking in account the measured Q.F.’s of the $\alpha$ particles). Considering all the $\alpha$ events, the total internal $\alpha$ activity in the CHC crystal is at the level of 7.8(3) mBq/kg. However, adopting the declared T${}_{1/2}$ of the ${}^{174}$Hf $\alpha$ decay in Ref. [39] (see before), the event numbers in 2848 h of data taking with this CHC crystal (see Ref. [27] for details) would be $\sim 1100$ counts. Nevertheless, all the measured $\alpha$ events was 553(23), even ascribing all of them to ${}^{174}$Hf $\alpha$ decay (although considering the information reported above); thus, the result given in Ref. [39] is safely refused; in fact, even in such an unlike hypothesis, the T${}_{1/2}$ value derived in Ref. [27] would be $4.01\left(17\right)\times {10}^{15}$ y, about 4.5$\sigma$ far from T${}_{1/2}$ = $2.0\left(4\right)\times {10}^{15}$ in Ref. [39]. Therefore, the half-life given in Ref. [39] was safely rejected. Let us now report the improved determination of the half-life of the ${}^{174}$Hf $\alpha$ decay thanks to the analysis in Ref. [27].

The background model in the range (1.1–3.9) MeV, where the ${}^{174}$Hf $\alpha$ decay is expected too, was made by an exponential function (to take into account the residual $\beta /\gamma$ events), some degraded alpha particles and suitable asymmetric Gaussian to shaping the $\alpha$ decay of ${}^{174}$Hf (Q${}_{\alpha }=2494.5\left(2.3\right)$ keV), ${}^{147}$Sm (Q${}_{\alpha }=2311.2\left(10\right)$ keV) and the data in the energy interval (3.0–3.9) MeV (see Ref. [27] for details). The fit, in the range (1.1–3.9) MeV, gives a ${\chi }^2/n.d.f.$ = 0.87 (${\chi }^2$ probability = 38.7%) and $31.7\left(5.6\right)$ events for the signal searched for (see Figure 7-left). The peak near 2.3 MeV is ascribed to the $\alpha$ decay of ${}^{147}$Sm and the estimated number of counts is 29.5(5.4) in very good agreement with that expected for ${}^{147}$Sm reported in Table 7 of Ref. [27]: 36(6) counts. The statistical impact of the other eligible radionuclides is negligible, as demonstrated in Ref. [27]. The Q${}_{\alpha }$ values of ${}^{147}$Sm and ${}^{174}$Hf determined by the fit procedure show a slight shift in the same direction of the mean value (∼5%) consistent with the uncertainty of the adopted Q.F. model discussed in Ref. [27].

To exclude that the data in the range (2–3) MeV could be due to one single peak, the previous fit was performed considering one single peak (see Figure 7-right). The fit result gives a p-value of 1.7%.

Thus, according to all the detailed analysis in Ref. [27] (2848 h of data taking with a CHC crystal of 6.09(1) g), the half-life for the ${}^{174}$Hf $\alpha$ decay, is:

$$\begin{array}{c}\hfill T_{1/2}=(7.0\pm 1.2)\times {10}^{16}\mathrm{y}.\end{array}$$

(2)

4. Perspectives and Conclusions

In this work, the experimental studies and few T${}_{1/2}$ predictions concerning the rare decay of naturally occurring isotopes have been briefly reported. In particular, in Table 2 the half-life limits of $2ϵ$ and $ϵ{\beta }^{+}$ processes in ${}^{174}$Hf are shown. The half-life limits are at the level of ${10}^{16}$–${10}^{18}$ y. These values have to be compared to the values of the phase space factor predicted for such transition considering a nuclear matrix element of the order of 1–10 as typical for that calculations (see Section 2). In this case, it is interest to notice, that the $2\nu 2ϵ$ of ${}^{174}$Hf decay has a T${}_{1/2}$ estimation of (0.3–6) × ${10}^{21}$ y (at now the related measured lower limit is $7.1\times {10}^{16}$ y). This encourages progress in such research, especially using the “detector = source” approach to profit a better detection efficiency with respect to the passive one.

In Table 3 the main potential $\alpha$ transition of Hf isotopes and the related experimental measurements and theoretical predictions of half-lives are reported. In particular, an experiment to investigate the $\alpha$ decay of naturally occurring hafnium isotopes to the first excited state and the ground state using a CHC crystal scintillator in coincidence with a HP-Ge detector was completed after 2848 h of live time (see Ref. [27]). The analysis performed ruled out the T${}_{1/2}$ value of $\alpha$ decay of ${}^{174}$Hf isotope given in Ref. [39]. Furthermore, Ref. [27] stated that the $\alpha$ decay of ${}^{174}$Hf isotope to the ground state was observed with a $T_{1/2}=7.0\left(1.2\right)\times {10}^{16}$ y. This value is in good agreement with the theoretical predictions reported in Table 3.

Besides this result, no decay was detected for $\alpha$ and DBD of ${}^{174}$Hf, ${}^{176}$Hf, ${}^{177}$Hf, ${}^{178}$Hf, ${}^{179}$Hf, ${}^{180}$Hf isotopes either to the ground state or to the lower bounded levels considering the data in Refs. [27,30,35]. The determined T${}_{1/2}$ lower limits for these decays are reported in Table 3. In particular, the T${}_{1/2}$ lower limits for the transitions of ${}^{176}$Hf$\to {}^{172}$Yb ($0^{+}\to 0^{+}$) and ${}^{177}$Hf$\to {}^{173}$Yb ($7/2^{-}\to 5/2^{-}$) are near to the theoretical predictions (see Table 3); all the other limits ($\sim {10}^{16}$–${10}^{20}$ y) are absolutely far from the theoretical expectations.

In Ref. [27], it has also been evaluated the average quenching factor for alpha particles modelled according to the prescription of Ref. [41] and reported in Figure 7 of Ref. [27]; it varies in the range 0.3–0.4. In addition, the CHC crystal scintillator of Ref. [27] exhibits a powerful PSD between $\beta /\gamma$ and $\alpha$ events as proved in Figure 5.

For the future, the development and the use of scintillation detectors, containing the isotope of interest (as a “source= detector” approach), are a very reasonable way to improve the experimental sensitivity. One of the more promising scintillator is the mentioned Cs${}_2$HfCl${}_6$. In particular, such detector meets high detection efficiency, good particle discrimination capability, large amount of the isotope of interest, dedicated protocol of material selection and growing procedure, a low background, and a good energy resolution. Indeed, nowadays there is significant interest in the development of scintillating crystals from the metal hexachlorides Cs${}_2$MCl${}_6$ (M = Hf or other nuclei) family thanks to their outstanding scintillating properties—a high light yield (up to 50,000 photons/MeV), perfect linearity of the energy response, excellent energy resolution (<3.5% at 662 keV in the best configuration), a quenching factor for alpha particles around QF = 0.3–0.5, and excellent statistical pulse shape discrimination ability. For example, in Figure 8 we show the diagram T${}_{1/2}$ vs the inverse of the square root of alpha energy in MeV. The black symbols are the results in Ref. [27]. The blue band is the extrapolation of the predictions on T${}_{1/2}$ for all the Hf isotopes using the Geiger-Nuttall scaling law considering the data point observed in Ref. [27]. The red symbols represent the sensitivity that the measurement can reach using a CHC crystal scintillator with 43.83 kg × day of exposure. As evident, there is a good perspective to observe the $\alpha$ decay of ${}^{176}$Hf and ${}^{176}$Hf (see also Table 3). In addition, to further increase the sensitivity for the study of the rare $\alpha$ and 2$\beta$ decays of naturally occurred hafnium isotopes with the active source approach, it is also relevant to improve the radiopurity of Cs${}_2$HfCl${}_6$ crystal scintillators, in particular for what concerns the Sm component in traces.

Besides this, some potential crystal scintillator containing Hf isotopes are the transparent optical ceramics based on cerium doped alkaline earth hafnates: BaHfO${}_3$(Ce) and SrHfO${}_3$(Ce) [42]; but also crystal scintillators as CaHfO${}_3$[43], HfF${}_4$[44], HfO${}_2$[45], La${}_2$Hf${}_2$O${}_7$(Ti) [46], Tl${}_2$HfCl${}_6$ [47,48]. The main scintillation properties of such crystals are listed in

Table 4. Main properties of Hf-based crystal scintillators (See text).

Scintillator	Percentage of Hf in Weight (%)	Density (g/cm${}^3$)	L.Y. (phe/MeV)	Main Decay Time (ns)	Main Emission Peak (nm)
BaHfO${}_3$(Ce)	49	8.3	∼40,000	∼25	∼400
CaHfO${}_3$	67	6.9	∼10,000	∼33	∼439
Cs${}_2$HfCl${}_6$	31	3.8	∼50,000	∼10,000	∼400
HfF${}_4$	70	7.1	∼300	∼29	∼350
HfO${}_2$	85	9.7	∼30,000	∼9500	∼480
La${}_2$Hf${}_2$O${}_7$(Ti)	23	7.9	∼13,000	∼10,000	∼475
SrHfO${}_3$(Ce)	57	6.7	∼40,000	∼42	∼410
				∼36 (89%);
Tl${}_2$HfCl${}_6$	22	5.3	∼25,000	∼217 (6%);	∼380
				∼1500 (11%)

. In general, an other promising approach is also a crystal that can work as scintillating bolometers to study the decays to the ground state of daughter nuclei [49]. HP-Ge, instead, is a promising choice for the investigations of decay to excited levels of daughter nuclei thanks the excellent energy resolution. Finally, an important issue is to use a crystal scintillator with high percentage in weight of Hf isotopes (see Table 4) or the implementation of isotopically enriched materials in the isotope of interest. However, in the our knowledge, no suitable technology existed that could guarantee a high concentration of a specific hafnium isotope, and the expensive enrichment process is mainly achieved by using the electromagnetic separation method.