Particle radioactivity of exotic nuclei

The development of radioactive beam techniques has over the last few decades led to impressive progress in accessing new areas on the nuclidic chart. Over the last 20 years, more than 500 new nuclides have been identified [1], almost all of them located very far from the line of stability. Usually, the first information about the new nuclide is obtained by observation of its radioactive decay. The characteristic half-life, the decay modes, their branching ratios and the released energy are the first attributes of a new system providing insight into their structure. With the departure from the stability new phenomena set in. The β-decay energy (Q value) increases, enhancing the decay rates and opening new decay channels accompanied by emission of delayed particles. These processes of β-delayed particle emission are discussed in another contribution to this volume [2]. However, when the limits of nuclear stability (drip-lines) are crossed, the nucleon separation energy becomes negative and thus nucleons can be spontaneously emitted from the ground state. In this paper, such radioactive processes of particle(s) emission from exotic nuclei will be reviewed.

Of the two limiting frontiers of the nuclidic chart, the neutron-deficient side is much better explored. This is caused by the proton drip-line being located closer to stability than the neutron drip-line. In addition, the fusion-evaporation reaction is a natural and convenient tool to produce neutron-deficient nuclei. The result is that for almost all odd-Z elements up to bismuth the proton drip-line has been experimentally reached or crossed. The characteristic phenomenon for the odd-Z nucleus beyond the proton drip-line is the proton radioactivity. Over the last few decades, proton spectroscopy developed into a powerful tool yielding a wealth of information on the properties and structure of exotic nuclei. On the other hand, among the unbound even-Z nuclei, the two-proton (2p) radioactivity is expected. This type of decay was discovered most recently and is the least known. It was observed in a few cases but it is predicted to occur in many others. Its genuine three-body character represents a challenge to theoretical models and brings hopes of providing a novel spectroscopic tool to study nuclei beyond the proton drip-line. In most cases of proton-unbound and two-proton unbound nuclei, the particle emission must compete with β decay and can be observed only if the partial half-life for particle emission, determined by penetration through the potential barrier, is sufficiently short. However, in a few regions of the proton drip-line, above the closed shells Z = 50, N = 82 and Z = 82 it is the α emission which wins the competition with proton radioactivity and β decay. The α spectroscopy is a well-established technique and is successfully used to investigate nuclei at the proton drip-line. Moreover, the special and interesting case of the superallowed α transition is predicted in the region of the doubly magic ¹⁰⁰Sn. Discussion of the proton and two-proton radioactivity, as well as α emission of very neutron-deficient nuclei, constitutes the main part of the present paper. The basic features of these phenomena and the main concepts used to describe them will be presented. The selected cases will be used to illustrate the current status and to give a flavor of particle radioactivity studies at and beyond the proton drip-line.

On the neutron-rich side, the limits of stability are still far from experimental reach for most neutron numbers. This reflects the technical difficulties in synthesizing extremely neutron-rich nuclei. The neutron drip-line has been determined experimentally for even neutron number only up to N = 20 and for odd N up to N = 27. One of the main motivations behind the efforts to construct the next generation of radioactive beam facilities is to conquer large new areas on the neutron-rich side of the chart. The question of whether the radioactive emission of neutrons could be observed is still open. Because of the missing Coulomb barrier, the expected lifetime of a one-neutron or two-neutron unbound nucleus is so short that this qualifies it to the realm of resonances or nuclear reactions, rather than to the radioactivity regime. Nevertheless, for the very small decay energy and large angular momentum, the centrifugal barrier could in principle delay the emission of neutrons to a measurable half-life. At the end of the paper, such a possibility for radioactive emission of neutrons will be briefly considered.

Recently, nuclear structure studies at the proton drip-line by means of radioactive decays were discussed by Blank and Borge [3]. The two-proton radioactivity was reviewed in some detail by Blank and Płoszajczak [4]. The current knowledge about the limits of nuclear stability was summarized by Thoennessen [5]. A general review of radioactive decays at the drip-line was recently presented in [6].

The emission of α rays was one of the first radioactive phenomena which opened the way to a new and unexpected realm of subatomic physics at the end of the 19th century [7]. Soon they become important messengers from the interior of radioactive atoms. Those early studies yielded an intriguing observation, formulated by Geiger and Nuttal, that the half-lives of α radioactive atoms were extremely sensitive to the energy of emitted α particles [8]. This puzzle was solved in 1928 by Gamow who was the first to apply the newly formulated quantum mechanics to the atomic nucleus [9]. The essential concepts introduced by him remain valid until today and, in fact, are still used in the description of particle radioactivity.

Firstly, Gamow understood the crucial role of the potential barrier, created by the Coulomb repulsion between the emitted particle and the daughter nucleus. The tunneling through this barrier, so sensitive to the particle's energy, explained the Geiger–Nutall law. The second idea was to approximate the state of a particle in the potential of the daughter nucleus as a stationary one. Indeed, the radioactive decay is so slow that the width of the decaying state is extremely narrow compared to its energy. This, in turn, led to the concept of the complex energy. The wave function of the emitted particle is written as

$$\begin{equation} \Phi({\bf r},t) = \Psi({\bf r}) \, \mathrm{e}^{-\mathrm{i} E t/\hbar}, \end{equation} \tag{ 1 }$$

where

$$\begin{equation} E = E_0 - \mathrm{i} \, \Gamma/2. \end{equation} \tag{ 2 }$$

The imaginary part of the energy yields the width of the state Γ and its half-life T_1/2 = ln 2 ℏ/Γ. The function Ψ(r) is found by solving the Schrödinger equation with the outgoing wave boundary condition. Such a description of an unstable state is called the Gamow state. It remains to be a basic ingredient of many modern approaches to particle radioactivity [10], as will be illustrated in the following. In addition, the idea of the Gamow state is an important component of the recently developed Gamow shell model [11], which is one of the attempts to extend the nuclear shell model towards very weakly bound systems, strongly influenced by continuum.

A simple and rather general formula for the particle decay width, following the basic assumptions stated above and equivalent to the so-called Wentzel–Kramers–Brillouin (WKB) approximation, reads [12]

$$\begin{equation} \Gamma = S \, N \, \frac{\hbar^2}{4\mu}\exp \left[-2\int_{r_2}^{r_3}{k(r) \, \mathrm{d}r}\right] , \end{equation} \tag{ 3 }$$

where N is the normalization factor,

$$\begin{equation} N \, \int_{r_1}^{r_2}{\frac{\mathrm{d}r}{2 \, k(r)}} = 1, \end{equation} \tag{ 4 }$$

and $k(r) = \sqrt {2 \, \mu |Q - V(r)|}/\hbar$. Here Q denotes the decay energy and μ is the reduced mass of the particle and the daughter nucleus. The integration limits r_i are the classical turning points, defined by V (r_i) = Q, where V (r) is the radial part of the particle–nucleus potential. The region between r₂ and r₃ is classically forbidden and here the particle remains under the barrier. The potential V (r) is given as a sum of nuclear, Coulomb, centrifugal and spin–orbit terms [13]. The model based on equations (3) and (4) is commonly used to interpret the results of proton and α spectroscopy and provides a good description of experimental data as far as spherical nuclei are concerned. The factor S in equation (3) represents the influence of many-body nuclear structure on the probability of particle emission. In the case of proton radioactivity it is called the spectroscopic factor and gives the measure of the single-particle purity of the initial state. In the case of α decay, S has the meaning of the preformation factor, describing the probability that the α particle is formed inside the mother nucleus.

This simple approximation can be applied to other decays, such as the emission of heavy clusters [14]. A simple argument based on barrier penetration factors was also used in the first estimate of the 2p emission [15, 16]. In the more general case, however, such as particle emission from deformed nuclei or for the detailed description of the 2p radioactivity, a more advanced and complex theoretical model has to be introduced. Examples of such more sophisticated formalisms will be mentioned in the following sections.

The observation of particle radioactivity is possible only when its partial half-life is of the same order or shorter than the partial half-life for any other decay channel in the given nucleus, in particular β decay. Away from stability, β lifetimes are short due to large beta Q values. They have, however, a lower limit of about 1 ms [6]. It follows that processes of particle radioactivity at the drip-lines are typically fast, with lifetimes of the order of milliseconds and shorter. This has implications for the experimental methods used for investigating them.

The most suitable method for the production and separation of short-lived particle emitters is based on the in-flight approach. Two categories of techniques can be distinguished depending on the energy scale of the ion beam used to synthesize the product of interest. At low energy, of the order of the Coulomb barrier, the fusion-evaporation reaction is the main mechanism to produce the extremely neutron-deficient nuclei and recoil separators are used to purify them from unwanted contaminants and to transport them to a detector station. Almost all ground state proton emitters and α-decaying nuclei at the proton drip-line were studied with help of this technique. In the second category, the projectile beam energy per nucleon is larger than the Fermi energy of nucleons. Then, the production occurs due to fragmentation reaction and/or fission and spallation in inverse kinematics while the filtering of desired reaction products is accomplished by means of a fragment separator. A crucial advantage of the high-energy method is the possibility to fully identify in flight the selected ions, which provides the single-atom sensitivity. That is why this approach is the method of choice for very rare processes and was instrumental in the discovery of 2p radioactivity.

The performance of the two categories of the in-flight technique can be illustrated by the following comparison. We consider two state-of-the-art attempts to synthesize an exotic nucleus for which the production cross section is expected (or estimated) to be of the order of 100 fb. First is the search for the superheavy element Z = 120 where the fusion-evaporation reaction of ⁵⁴Cr impinging on a ²⁴⁸Cm target of 0.5 mg cm⁻² thickness, and the velocity filter SHIP [17] is used. With the beam intensity of 750 pnA, the expected waiting time for one event is about 100 days of irradiation [18]. On the other hand, the production of the doubly magic ⁴⁸Ni was achieved using the fragmentation reaction of the ⁵⁸Ni beam on a natural nickel target of 580 mg cm⁻² thickness and the A1900 fragment separator [19]. Although the beam intensity was only 20 pnA, the observed rate of ⁴⁸Ni was 1 atom day⁻¹ [20].

A selection of the characteristic features of the two families of the in-flight technique is given in table 1. A more detailed description of the high-energy, in-flight technique with an overview of the present and planned facilities is given in [21].

The most widespread detection technique of the modern charged-particle spectroscopy is based on silicon detectors, which provide a good energy resolution and sufficiently good time characteristics. Particularly important in this context was the development of double-sided silicon-strip detectors (DSSSD) [28]. The pixilation achieved by two sets of perpendicular strip electrodes helps to establish the spatial correlation between an implanted ion and particles emitted in its decay. This increases detection sensitivity and allows us to accept high counting rates. The great majority of experiments devoted to proton and α radioactivity employ DSSSD detectors or arrays of them.

Detection of particle emission is particularly challenging when half-lives are very short, in the range of microseconds or shorter. A recently developed solution to this problem applies digital signal processing (DSP). In this approach, the signals delivered by preamplifiers are digitized and all further manipulations are performed on their digital representations. The functions, previously executed by analogue electronics modules, such as shaping, discriminating, establishing coincidences etc, are replaced by numerical algorithms, which in general offer much more complex and flexible operations. In addition, all information contained in the original signal shape can be preserved for the off-line analysis [29]. An example of the special feature, easily implemented in a DSP-type data acquisition, is the triggering mode allowing us to record only such events where a pileup of two signals occurs within a short time window. Such a mode, together with the storage of the full shape of the signal, was instrumental in the study of the fine structure in the proton emission [30] and in the search for the superallowed α decay [31, 32]. An example illustrating this method is shown in figure 1.

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Nuclei produced by the high-energy in-flight (fragmentation) technique have usually significant energy straggling and to accomplish an efficient study of their decays they are implanted into the material of the detector. Arrays of silicon detectors are frequently used for this purpose yielding the total decay energy with a good accuracy. However, in the case of low-energy multiparticle decays, such as 2p emission, silicon detectors have the drawback that the information on the momenta of the emitted particles and on the energy sharing between them is lost. To remedy this problem, gaseous detectors based on the principle of the time projection chamber were developed [33, 34]. Their main advantage is that they record individual tracks of all particles ejected within their active volume. This results in extreme sensitivity to rare, multiparticle decays, as may be illustrated by the discovery of β-delayed three-proton emission in ⁴⁵Fe [35] and in ⁴³Cr [36], based on a few events only. Evidence for one such event is shown in figure 2. Most of the current studies on the 2p radioactivity were carried out with the help of gaseous time projection chamber detectors, see section 6.

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When the lifetime of a particle emitter is extremely short, in the subnanosecond range, there is no time for the separation of the ion and its implantation into a detector setup. In such a case, decay occurs in flight, shortly after production, and it is identified by the tracking of all decay products. For the precise determination of the momenta of light particles, such as protons, silicon microstrip detectors are used [37]. With this technique the 2p decay of ¹⁹Mg with a half-life of 4.0(15) ps was investigated [38].

Before the proton drip-line was experimentally reached, the direct emission of protons was first observed from the isomeric state in ⁵³Co, which had an excitation energy (3.2 MeV) higher than the proton separation energy [39]. The first ground-state proton radioactivity was identified 30 years ago in ¹⁵¹Lu [40] and in ¹⁴⁷Tm [41]. Since then, many other cases were found and proton spectroscopy evolved to a mature research field providing a wealth of data on nuclear structure at the proton drip-line. At present, more than 50 proton lines have been identified, originating from both the ground and isomeric states of 34 nuclei. In the case of seven proton-emitting states the fine structure, i.e. transitions to more than one final state, has been established. The list of all known proton emitters with their basic properties and references to relevant experimental and theoretical works was recently compiled in [6].

A convenient classification of proton-emitting states uses the concept of the combined seniority, s, equal to the number of unpaired protons and neutrons. All 'classical' proton radioactive nuclei have s ⩽ 2. They can be further divided into two groups. In the first, s = 1, the initial state is composed of a odd, unpaired proton and the even–even core. In the second, s = 2, in addition to the unpaired proton, there is also an unpaired neutron, which is usually treated as a spectator member of the even–odd core. The class of s ⩽ 2 emitters comprises all the cases of the ground-state proton radioactivity and decays from low-lying isomeric states. All of them are located between ¹⁰⁹I and ¹⁸⁵Bi. It may be anticipated that these limits will be surpassed in the future. In the region of lighter nuclei, Z < 50, the expected half-lives are very short, which combined with low production cross-sections poses a challenge to experimental techniques.

A separate category of proton emitter groups cases in which the odd proton is accompanied by one or more broken nucleon pairs (of protons or neutrons), s > 2. Such a situation corresponds to highly excited isomers of multiparticle character. At present, only three cases belong to this class: ^53mCo, ^54mNi and ^94mAg. They are not located beyond the drip-line; their proton unbound character results from the high excitation energy.

The value of proton radioactivity for nuclear structure studies lies in the fact that relatively simple measurements of the proton energy and partial half-life yield information about the nuclear wave function. This is because the decay probability is so sensitive to the details of the potential, and in particular to the orbital angular momentum l of the initial state via the centrifugal part. By comparing the measured observables with the simple model, equations (3) and (4), one can analyze the initial state in terms of spherical shell-model orbitals. We illustrate this with the case of ¹⁵¹Lu [40]. Assuming that the emitted proton originates from the πh_11/2 state (l = 5), formula (3) correctly reproduces the measured partial proton decay constant for the spectroscopic factor S_exp = 0.5. On the other hand, the expected value of the spectroscopic factor can be calculated in the BCS approximation by S_th = u²_j, where u² is the vacancy probability for the considered orbital in the daughter nucleus. For ¹⁵¹Lu S_th = 0.54 [42]. The good agreement between measured and calculated spectroscopic factors supports the validity of the assumptions made, including the assignment of the initial state.

In the general case, however, when the nuclear vibrational or rotational degrees of freedom cannot be neglected, a single spherical shell-model orbital does not offer an adequate description of the initial state. Such a problem appears, for example, in a number of proton emitters between cesium and thulium located in a region of strong quadrupole deformation. Therefore, a more advanced theoretical description is required. A general approach to particle emission from deformed nuclei—the coupled channels method—was pioneered already 50 years ago for α decay ([43] and references therein). Recently, this general framework was adapted to proton radioactivity and several variants of this approach were developed [10]. Here, we sketch only the general features of this model.

First, cluster approximation is made, i.e. the proton is considered to move in the potential of the core (the daughter nucleus). The Hamiltonian of the system is written as

$$\begin{equation} H = H_{\mathrm{d}} + H_{\mathrm{p}} + V_{\mathrm{d p}}, \end{equation} \tag{ 5 }$$

where H_d is the Hamiltonian of the daughter, H_p is that of the proton and V_dp represents the interaction between the proton and the daughter. Next, the wave function of the initial state (the parent nucleus) is assumed to have the form

$$\begin{equation} \Psi_{JM} = \frac{1}{r} \, \sum _{\alpha=\{\pi,\delta\}} u_\alpha (r) \, [ {\cal Y}_{\pi} \otimes \Phi_{\delta} ]_{JM}, \end{equation} \tag{ 6 }$$

where u_α(r) is the radial function describing the relative motion of the proton and the daughter, α labels the channel and is composed of quantum numbers of the proton (π) and of the daughter (δ), ${\cal Y}_{\pi }$ is the orbital-spin wave function of the proton, and Φ_δ represents the wave function of the daughter nucleus. The latter satisfies the equation

$$\begin{equation} H_{\mathrm{d}} \, \Phi_{\delta} = E_{\delta} \, \Phi_{\delta}. \end{equation} \tag{ 7 }$$

The spectrum of daughter states can be taken from the experiment or from the theoretical model. Different assumptions concerning the nature of the core states lead to different versions of the model. The considered options include rotational motion [44], quadrupole vibrations [45] or triaxial vibrations [46]. By inserting (6) into the Schrödinger equation and integrating over all variables except the radial one (r), one obtains the set of coupled differential equations for radial functions:

$$\begin{equation} \left[\frac{\hbar^2}{2 \, \mu} \, \frac{\mathrm{d}^2}{\mathrm{d}r^2} - \frac{\hbar^2 \, l_{\mathrm{p}}(l_{\mathrm{p}}+1)}{2 \, \mu \, r^2} + E_{\mathrm{p}} \right] \, u_{\alpha}(r) = \sum _{\alpha '} \hat{V}_{\alpha \alpha'}(r) \, u_{\alpha '}(r). \end{equation} \tag{ 8 }$$

The l_p and E_p denote the orbital angular momentum and the energy of the relative motion of the proton and the daughter, respectively. The matrix $\hat {V}$ is composed of the matrix elements of the potential V_dp integrated over angles. If it contains deformed terms, this matrix is non-diagonal. The set of equations (8) is solved with the conditions that the radial functions vanish at the origin and far from the nucleus represent a purely outgoing Coulomb wave:

$$\begin{equation} u_{\alpha} \rightarrow \, \sim \left[ G_{l_\mathrm {p}}(\eta,k r) + \mathrm{i} \, F_{l_\mathrm {p}}(\eta, k r) \right], \end{equation} \tag{ 9 }$$

where F and G are the Coulomb functions [47], regular and irregular at the origin, respectively, $k = \sqrt {2 \mu E_{\mathrm {p}}}/\hbar$ is the wave number, and η = μZe²/ℏ²k is the Sommerfeld parameter for the atomic number of the daughter nucleus Z. These conditions are satisfied for the complex values of E_p and the solution represents the Gamow state. In principle, the decay width is given by the imaginary part of the energy, see equation (2); however, due to the extremely small value of the Γ compared to the real part, this way of calculation is not possible for practical (numerical) reasons. Instead, the partial width Γ_α for each channel can be determined from the radial functions by computing the asymptotic outgoing flux [48]. Then, the total decay width is given simply by

$$\begin{equation} \Gamma = \sum _{\alpha} \Gamma _{\alpha}. \end{equation} \tag{ 10 }$$

In this formalism, the partial decay width corresponding to the transition to the excited daughter state (fine structure) can be calculated in a straightforward way:

$$\begin{equation} \Gamma _{\delta} = \sum _{\pi} \Gamma _{\{\pi, \delta \}}. \end{equation} \tag{ 11 }$$

It should be noted that to ensure the convergence of the solutions of (8), enough daughter states have to be taken into account in the expansion (6). In some cases this may even include states which are located outside the available energy window and thus not accessible in the decay.

We illustrate the application of coupled channels formalism with the example of the two most complex proton emitters studied to date. The first is the odd–even (s = 1) ¹⁴¹Ho. Its ground state and also an isomeric state at about 60 keV excitation energy were long known as proton emitters [23, 49]. Both levels were interpreted as deformed and were assigned Nilsson orbitals 7/2⁻[523] and 1/2⁺[411], respectively. Such an interpretation was supported by the observation of rotational bands built on both these states [50]. More recently, at the Holifield Radioactive Ion Beam Facility (HRIBF) at Oak Ridge National Laboratory (ORNL), the fine structure in proton emission from both states was observed [51]. The ground state (T_1/2 = 4.1(1) ms) was found to decay to the first excited 2⁺ state in ¹⁴⁰Dy at 202 keV with the probability of 0.9(2)%. For the isomeric state (T_1/2 = 7.4(3) μs) the transition to this 2⁺ state was found to proceed with the branching of 1.7(5)%. Detection of such a weak transition with the microsecond decay time is particularly challenging. It was achieved mainly due to the application of the DSP-type data acquisition system. The proton radioactivity of both the ground state and the isomeric state of ¹⁴¹Ho was calculated using the coupled channels method by taking into account the spectrum of rotational states in ¹⁴⁰Dy and using the version of the model described in [46]. Fairly good agreement of all the predicted properties with the measured values was achieved but only when a strong prolate deformation (β₂ = 0.35) of the ¹⁴⁰Dy nucleus was assumed, which is significantly larger than the value of β₂ = 0.23 suggested by the E(2⁺) systematics [52]. This increased deformation contributes to the shape coexistence effects predicted in this region by several mean-field models. The decay schemes of both ¹⁴¹Ho states together with the composition of their wave functions resulting from the theoretical calculation are presented in figure 3. The wave function of the ¹⁴¹Ho ground state is dominated by πh_11/2 orbital. Its lifetime, however, is governed by a small (≈1%) πf_7/2⊗0⁺ component. The isomeric state appears to be strongly mixed but its half-life is determined mainly by a πs_1/2⊗0⁺ component while the branching to the 2⁺ state depends on πd_2/2⊗2⁺ and πd_5/2⊗2⁺ components. This demonstrates the sensitivity of proton spectroscopy to various tiny details of the wave function, which in turn strongly depend on the relative position of single-particle levels and thus on the proper choice of the potential. The analysis of the proton emission in ¹⁴¹Ho triggered the theoretical investigation of single-proton states in holmium isotopes. The self-consistent HFB calculations indicated significant changes of proton shell structure when approaching the proton drip-line and the important role of two-body tensor interactions [51].

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The second case is the odd–odd, s = 2, ¹⁴⁶Tm, in which five proton transitions were observed, three from the 5⁻ ground state, T_1/2 = 68(3) ms, and two from the 10⁺ isomeric state at 182 keV, T_1/2 = 198(3) ms [53], see figure 4. The ¹⁴⁶Tm is expected to be in a region of change from prolate to oblate deformation and to have a moderate deformation β₂ = 0.18. Therefore, proton radioactivity in this case was analyzed by the coupled-channels formalism in the version where protons are coupled to the vibrations of the core [45] with an additional extension to include coupling of the odd neutron [53]. The deformation β₂ is considered here in a dynamical sense as a parameter describing vibrational phonons. The calculated wave function compositions are indicated in figure 4. The ground state is interpreted as dominated by the πh_11/2⊗νs_1/2 configuration coupled roughly in equal parts to the 0⁺ and 2⁺ states of the ¹⁴⁴Er core. The former part is responsible for the 68% proton transition to the ground state of ¹⁴⁵Er. The transitions to the excited states, interpreted as the 3/2⁺ of the νs_1/2⊗2⁺ structure and the 11/2⁻ of νh_11/2 character, are governed by small components of the wave function, 4% πf_7/2⊗νs_1/2⊗2⁺ and 2% πs_1/2⊗νh_11/2⊗0⁺, respectively. Note that the latter component is isospin symmetric to the dominant component of the wave function. The wave function of the isomeric state is dominated by the πh_11/2⊗νh_11/2 configuration coupled to the 0⁺ and 2⁺ core states. A small admixture of the πf_7/2⊗νh_11/2⊗2⁺ configuration is responsible for the 1.2% transition to the excited state, interpreted as 13/2⁻ of νh_11/2⊗2⁺ character. All observables, half-lives and branching ratios are well reproduced by the model. In this remarkable example, the proton radioactivity is effectively providing information on neutron states in a very exotic nucleus.

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α radioactivity is one of the oldest known forms of nuclear decay and its theoretical description, as discussed in sections 2 and 4, becomes a model for the general understanding of particle or cluster emission. The role of α decay in nuclear spectroscopy can be hardly overestimated. It yields information on binding energies and masses, on structural effects (single-particle levels, shell closures and deformation), provides a clean and efficient method for decay tagging and helps to identify rare reaction products, such as new superheavy elements. More general reviews of α spectroscopy were recently given by Roeckl [54] and Delion [55]. In this paper, we focus on recent results on α emission close to the proton drip-line.

Conditions favoring α emission against other decay modes, such as β decay or proton radioactivity, occur above shell closures, where due to the larger binding of magic nuclei the α-decay energies are increased. One such region is located above the doubly magic ¹⁰⁰Sn. Moreover, it is the only region of α radioactivity where valence neutrons and protons occupy orbitals having the same quantum numbers which may result in the enhancement of α-preformation factors leading to large α-decay widths. The predicted unusually fast α decays of nuclei above ¹⁰⁰Sn are referred to in the literature as superallowed α decays [56]. A particularly large enhancement is expected for the decay of ¹⁰⁴Te, which can be considered as composed of a doubly magic ¹⁰⁰Sn core and the two protons and two neutrons occupying identical orbitals, thus easily forming an α particle. If confirmed, the decay of ¹⁰⁴Te could represent the standard reference for the α emission. Note that at present the decay of ²¹²Po to the doubly magic ²⁰⁸Pb is used as the reference, although the two valence neutrons in ²¹²Po occupy orbitals of different angular momenta and even different parity than the two valence protons.

In the search for the superallowed α decay, the lightest tellurium isotope reached so far is ¹⁰⁵Te [31, 57]. In the work of Seweryniak et al, ¹⁰⁵Te was produced directly by the fusion–evaporation reaction ⁵⁰Cr(⁵⁸Ni, 3n)¹⁰⁵Te and selected using the Fragment Mass Analyzer (FMA) at the Argonne National Laboratory [23]. The application of fast recovery electronics resulted in the observation of 13 α-decay events, yielding a very short half-life of T_1/2 = 0.7^+0.25_−0.17 μs [57]. Liddick et al took a different approach by producing the longer-lived ¹⁰⁹Xe, which decays by α emission to ¹⁰⁵Te [31]. They used the fusion–evaporation reaction ⁵⁴Fe(⁵⁸Ni, 3n)¹⁰⁹Xe and the Recoil Mass Separator (RMS) of the HRIBF facility at Oak Ridge National Laboratory [24] to select the reaction products. The digital acquisition system was programmed to record only events corresponding to two α particles emitted shortly one after another, see figure 1. This method proved to be more efficient and about 100 α–α events were attributed to the ¹⁰⁹Xe → ¹⁰⁵Te → ¹⁰¹Sn decay chain, resulting in the more precise half-life value of T_1/2 = 0.62 ± 0.07 μs [31]. The extracted reduced α-decay width for ¹⁰⁵Te relative to ²¹²Po, W_α = δ²/δ²(²¹²Po) [58], was found to be 2.0 ± 0.3. This increase of the decay probability is even more pronounced if the reduced α-decay width is compared to the value for ²¹³Po, which has a similar valence composition (α + n). The ratio δ²(¹⁰⁵Te)/δ²(²¹³Po) amounts to 2.7 ± 0.7. The comparison of this ratio with other tellurium isotopes, plotted in figure 5, indicates a systematic increase towards ¹⁰⁴Te. The extrapolated lower limit for this ratio for ¹⁰⁴Te is about 3, which leads to the upper limit for the half-life of 100 ns [31]. Such a short lifetime represents the main challenge to future experimental attempts at identifying the presumably superallowed decay of ¹⁰⁴Te.

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The α decay of ¹⁰⁵Te brings information on excited states in ¹⁰¹Sn and thus on single-neutron levels outside the doubly magic ¹⁰⁰Sn core. Indeed, in a subsequent study of the ¹⁰⁹Xe → ¹⁰⁵Te → ¹⁰¹Sn α-decay chain at the HRIBF facility, the fine structure in the decay of ¹⁰⁴Te was observed [32]. The evidence for a transition to the first excited state in ¹⁰¹Sn was additionally supported by the observation of 172 keV γ-line in coincidence with α particles. This finding confirmed the previous evidence for this state obtained by the recoil-decay tagging technique and γ spectroscopy [59]. Surprisingly, however, the α transition to the excited state was found to be much stronger than the decay to the ground state of ¹⁰¹Sn, contrary to the pattern observed in the decay ¹⁰⁷Te → ¹⁰³Sn [60]. The detailed analysis led to the conclusion that the spins of the ground state of ¹⁰¹Sn (7/2⁺) and of the first excited state (5/2⁺) are reversed with respect to level ordering postulated for ¹⁰³Sn and heavier tin isotopes [32], see figure 6. This unexpected behavior is explained as a result of orbital-dependent pairing and a small energy difference between νg_7/2 and νd_5/2 orbitals. Due to this interpretation, the νg_7/2 state, determining the ground state of ¹⁰¹Sn, is lower in energy than the νd_5/2. However, the pairing energy in the (νg_7/2)²_J=0 configuration is stronger than in the (νd_5/2)²_J=0, which together with the Pauli blocking favors configurations resulting in the ground-state spin of 5/2⁺ for ¹⁰³Sn and heavier tin isotopes up to ¹⁰⁹Sn [32].

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Another interesting aspect of the island of α radioactivity above the ¹⁰⁰Sn is related to the termination of the astrophysical rp-process which proceeds as a sequence of proton captures and β decays near the proton drip-line during the explosive hydrogen burning in certain types of x-ray bursts [61]. According to models, in particular environments the rp-process could reach tellurium isotopes where it terminates due to their fast α decays in the so-called Sn–Sb–Te cycle [62]. At which point along the chain of tin isotopes this cycle can start depends critically on the proton-decay energies (Q_p values) of antimony isotopes. Relevant in this context is the long-standing controversy concerning the Q_p value of ¹⁰⁵Sn [63]. It was solved, paradoxically, by α spectroscopy. This value can be determined from the relation between decay energies: Q_p(¹⁰⁵Sb) = Q_p(¹⁰⁹I) + Q_α(¹⁰⁸Te) − Q_α(¹⁰⁹I). The last unknown on the right-hand side of this relation was the α-decay energy of ¹⁰⁹I, which decays predominantly by proton emission. However, a tiny branch of α decay from this nucleus, of the order of 10⁻⁴, was observed at the HRIBF facility [64]. Again, the sensitivity of the applied digital acquisition system [29] was a key factor in the measurement of this short-lived activity (T_1/2 ≈ 94 μs). We note that ¹⁰⁹I is so far the only nucleus in the region above ¹⁰⁰Sn in which both the proton and α radioactivity was identified. From about 16 counts in the α spectrum of ¹⁰⁹I, the value Q_p(¹⁰⁵Sb) = 356 ± 22 keV was determined [64]. Such a low value excludes the chance to observe proton emission from ¹⁰⁵Sb, ending the controversy concerning the decay of this nucleus. In addition, this value excludes the formation of a Sn–Sb–Te cycle at ¹⁰⁴Sn. Recently, an analogous approach was undertaken to determine the Q_p value for ¹⁰⁴Sn by searching for the α-decay branch of ¹¹²Cs [65]. From the proton spectrum of ¹¹²Cs, the improved half-life of T_1/2 = 506 ± 55 μs has been determined. For the α branch, only the upper limit of 0.26% could be concluded. This value, however, allows us to set a lower limit on the Q_p value for ¹⁰⁴Sb of 0.15 MeV [65], which safely excludes the creation of the Sn–Sb–Te cycle at ¹⁰³Sn. To make the story complete we should add that the recent direct measurement of masses in this region in the JYFLTRAP Penning trap facility yielded proton separation energy for ¹⁰⁶Sb, which is too small to allow a strong Sn–Sb–Te cycle starting at ¹⁰⁵Sn [66]. Hence, the role of this cycle in the termination of the rp-process is finally found to be much weaker than previously expected.

Decays of neutron-deficient nuclei above lead are also dominated by α radioactivity. As an illustration of α-decay studies in this region we mention one spectacular result—the evidence for shape coexistence in ¹⁸⁶Pb [67]. The experiment was carried out at the velocity filter SHIP at GSI Darmstadt [17] used to produce and select the nuclei of ¹⁹⁰Po. The α particles emitted by ¹⁹⁰Po were measured together with conversion electrons and x-rays. Three α lines were identified and interpreted as representing transitions to the ground state of ¹⁸⁶Pb and to its first two excited states. The spectroscopic data suggested that all these three states fed by α decay have spin and parity of 0⁺. The model analysis of α preformation factors led to the conclusion that these states have different shapes. While the ground state of ¹⁸⁶Pb is spherical, the first excited 0⁺ state at 532 keV has oblate deformation and the second excited 0⁺ state at 650 keV is prolate deformed [67].

When we move along the line of isotopes of an even-Z element towards the neutron-deficient side, the two-proton separation energy usually becomes negative first, before the single-proton separation energy. This fact, caused by the pairing interactions between protons, was noted already in 1960 by Goldansky, who predicted the phenomenon of two-proton (2p) radioactivity [15]. Following his insight and early analysis, we define the true or genuine 2p decay as the simultaneous emission of two protons when the emission of one proton, and thus also sequential emission of two protons, is energetically not possible or strongly suppressed. As it turned out, the crucial feature of the true 2p emission is its essentially three-body character, which means that its dynamics, in general, cannot be reduced to a sequence of two-body decays [6]. This feature together with the energy criterion clearly distinguishes the true 2p decay from the sequential emission of two protons.

Experimentally, the proton drip-line could be explored more easily for light elements, and this was the case of ⁶Be in which the three-body decay features were first observed [68]. Due to considerable decay width (≈100 keV) the case of ⁶Be belongs to the regime of resonant phenomena rather than to radioactive ones. For such a category of decays, when the width of the decaying state is comparable to the energies of both emitted protons and no strong correlations between outgoing fragments are present, a notion of democratic decays was introduced [69]. Nevertheless, democratic decays represent the limit of the true 2p emission and, as a recent detailed study of correlations in the decay of ⁶Be showed, share some essential features with the 2p radioactivity [70].

The challenge of identifying 2p radioactivity was twofold: it is experimentally difficult to synthesize and observe heavier nuclei at the proton drip-line and one has to select the right candidates with the 2p-decay channel strong enough to compete with β decay. The in-flight fragmentation techniques helped us to overcome the former challenge, as discussed in section 3. The choice of the best candidates was narrowed by a series of theoretical studies focused on the precise prediction of masses of neutron-deficient nuclei based on the isobaric multiplet mass equation and experimentally measured masses of neutron-rich members of isospin multiplets [71–73]. The identification of the most promising cases, which included ⁴⁵Fe, ⁴⁸Ni and ⁵⁴Zn, motivated and guided the experimental search for 2p emission.

The first evidence for the 2p radioactivity was obtained for ⁴⁵Fe in the experiment using an FRS separator at GSI Darmstadt [74]. It was confirmed, almost simultaneously, by the results of a similar measurement made with a LISE separator at GANIL Caen [75]. In both cases the fragmentation reactions of the ⁵⁸Ni beam were used to produce ions of ⁴⁵Fe which after separation and identification in flight were implanted into a set of silicon detectors. The analysis of the total decay energy and lifetime sufficed to claim the discovery of a new decay mode. The technical advancement and difficulty of these first experiments is reflected by the number of collected events. The statistically sound conclusions were drawn on the basis of four counts at GSI and 12 counts obtained at GANIL. Later, with the same technique applied at GANIL 2p emission of ⁵⁴Zn was established [76] and one count was found in the spectrum for ⁴⁸Ni coinciding with the predicted 2p-decay energy [77]. For the next step of 2p emission studies, the special gaseous time projection chambers were developed [33, 34]. They provided the first direct observation of two protons ejected from ⁴⁵Fe [78] and from ⁵⁴Zn [79]. In the experiment at an A1900 separator at the NSCL/MSU laboratory about 100 decay events of ⁴⁵Fe were detected with the OTPC yielding the first information of proton–proton correlations in the 2p radioactivity [80]. These achievement represented a breakthrough in the 2p emission studies and will be discussed in the following. Recently, with the help of the OTPC detector, also the first direct evidence for the 2p emission from ⁴⁸Ni was obtained [20]. The example images of 2p-decay events of ⁴⁵Fe, ⁴⁸Ni and ⁵⁴Zn are shown in figure 7. With a different technique the 2p decay of ¹⁹Mg was identified at the FRS facility at GSI [38]. The ions of ¹⁹Mg were produced by a radioactive beam of ²⁰Mg impinging on a secondary beryllium target. The events corresponding to ¹⁹Mg were reconstructed from the measured tracks of the in-flight decay products. The data yielded the half-life for ¹⁹Mg of 4.0 ± 1.5 ps and the projected p–p momentum distributions [81]. The measured values of the decay energy and the partial half-life for the 2p radioactivity are collected in table 2.

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The single-particle radioactivity (proton, α, cluster) is characterized only by the transition energy and the partial half-life. In the case of the 2p emission there is additional information contained in the correlations between momenta of the particles in the final state. A description of the motion of three bodies is most conveniently done in the Jacobi coordinates [83]. For the two-proton plus core system there are two equivalent Jacobi coordinate systems, the T and the Y, see figure 8. They differ in the selection and coupling of the two-body subsystems, but each of them is sufficient to represent the full physical picture. The particular choice of the Jacobi system, however, may be preferred if approximations are introduced. If we consider a decay of the initial nucleus at rest when the released energy E_T is shared between three particles having masses m₁, m₂ and m₃ and momenta k₁, k₂ and k₃, respectively, the Jacobi energies and momenta of subsystems X and Y (see figure 8) are

$$\begin{eqnarray} &&E_{\mathrm{T}} = E_x + E_y = \frac{k^2_x}{2 M_x} + \frac{k^2_y}{2 M_y} , \nonumber \\ &&M_x = \frac{m_1 m_2}{m_1 + m_2} , \, \, M_y = \frac{(m_1 + m_2) m_3}{m_1 + m_2 + m_3} , \nonumber\\ &&{\bf k}_x = \frac{m_2 {\bf k}_1 - m_1 {\bf k}_2 }{m_1 + m_2} , \,\; {\bf k}_y = -{\bf k}_3 \, . \end{eqnarray} \tag{ 12 }$$

It can be shown that for the fixed value of E_T the complete correlation picture is determined by two parameters. The conventional choice is the energy fraction ε and the angle θ_k:

$$\begin{equation} \varepsilon = E_x/E_{\mathrm{T}} ,\quad \cos(\theta_k)=(\mathbf{k}_{x} \cdot \mathbf{k}_{y}) /(k_x\,k_y). \end{equation} \tag{ 13 }$$

In the Jacobi Y system, on which we will focus in the following, indexes 1 and 3 refer to both the emitted protons. Then

$$\begin{equation} E_x = \frac{k^2_x}{2 \mu}, \quad {\bf k}_x = {\bf k}_1 + \frac{\mu}{M} \, {\bf k}_3, \quad {\bf k}_y = -{\bf k}_3, \end{equation} \tag{ 14 }$$

where M is the mass of the core and μ is the reduced mass of the proton of mass m and the core. It can be seen that in the limit of the heavy core (M ≫ m) E_x is the kinetic energy of one proton and θ_k is the angle between momenta of both protons increased by π.

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First, we discuss briefly simplified models of 2p radioactivity. By approximations to the three-body Hamiltonian, the decay amplitude can be factorized into a product of two-body terms [84, 85]. In the T Jacobi system, such an approach leads to the diproton model in which one final state interaction (FSI), the one between protons, is taken into account exactly. The factorization in the Y Jacobi system results in the so-called direct decay model. In the limit of the very heavy core, two FSI interactions, between each proton and the core, are treated exactly and the p–p Coulomb interaction is either neglected or approximated effectively [84]. The direct model proved to be very useful, as it offers a simple analytical and reasonably good approximation to the 2p-decay widths. In addition, it was used as a tool to study the transition between simultaneous and sequential emission of two protons [6]. Consider the initial nucleus to be in the ground state with both protons occupying the same orbital. Further it is assumed that if one proton is removed, the remaining one is left in a resonant state of the core + p system, characterized by the energy E_p, relative to the three-body decay threshold, and by the orbital angular momentum l_p. Only one such state of lowest energy is taken into account. Then, the 2p-decay width in the direct model is given by

$$\begin{eqnarray} \Gamma_{\rm{dir}} & = & \frac{E_{\mathrm{T}}}{2\pi} \, (E_{\mathrm{T}} - 2 E_{\mathrm{p}})^2 \, \int_0^{1} \!\! \mathrm{d} \varepsilon \frac{ \Gamma_{x}(\varepsilon E_{\mathrm{T}})} {(\varepsilon E_{\mathrm{T}}-E_{\mathrm{p}})^{2}+\Gamma_{x}(\varepsilon E_{\mathrm{T}})^{2}/4} \nonumber \\ && \times \frac{\Gamma_{y}((1-\varepsilon)E_{\mathrm{T}})} {((1-\varepsilon)E_{\mathrm{T}}-E_{\mathrm{p}})^{2} + \Gamma_{y} ((1-\varepsilon)E_{\mathrm{T}})^{2}/4}. \end{eqnarray} \tag{ 15 }$$

The Γ_i is the width of the two-body subsystem given by the R-matrix formula:

$$\begin{equation} \Gamma_{i}(E) = 2 \gamma^{2}_i P_{l_{\mathrm{p}}}(E, R, Z_i), \end{equation} \tag{ 16 }$$

where P_lp is the penetrability expressed by the Coulomb functions

$$\begin{equation} P_{l_{\mathrm{p}}}(E, R, Z_i)=\frac{kR}{F_{l_{\mathrm{p}}}^{2} (\eta,kR)+G_{l_{\mathrm{p}}}^{2}(\eta,kR)}, \end{equation} \tag{ 17 }$$

and the γ²_i is the reduced width

$$\begin{equation} \gamma^{2}_i = \frac{\hbar^2}{2 \mu_i R^{2}}\theta^{2}_i \end{equation} \tag{ 18 }$$

containing the dimensionless spectroscopic factor θ². The channel radius R is usually taken as

$$\begin{equation} R = 1.4 \, (A_{\rm{core}}+1)^{1/3}\, \rm{fm}. \end{equation} \tag{ 19 }$$

The index i distinguishes two-body subsystems, X and Y . Thus, μ_x is the reduced mass of the proton and the core, while μ_y is the reduced mass of the proton and the core + p system. The p–p Coulomb interactions are taken into account effectively by using Z_x = Z_core and Z_y = Z_core + 1. For approximate calculations one can take θ_x = θ_y ≈ 1. Formula (15) can be further simplified by noting that the nominator of the integrand is sharply peaked at energy ε/2, while the denominator varies smoothly with energy. This leads to the approximate formula for the 2p-decay width [85]:

$$\begin{equation} \Gamma_{\rm{dir}} \approx \frac{8 E_{\mathrm{T}}} {\pi (E_{\mathrm{T}}-2E_{\mathrm{p}})^2} \int_0^{1} \!\! \mathrm{d} \varepsilon \, \Gamma_{x}(\varepsilon E_{\mathrm{T}}) \, \Gamma_{y}((1-\varepsilon)E_{\mathrm{T}}). \end{equation} \tag{ 20 }$$

The predictions of the direct model, based on equation (20), for the four cases of the 2p radioactivity are given in table 2. The value of l_p expected to be dominant from a simple shell-model consideration was taken into account. It can be seen that this simple model represents a lower limit of the 2p-decay half-life. The predicted values agree with the measured ones within about an order of magnitude.

Other theoretical approaches were also applied to describe the 2p emission process [4]. One is based on the R-matrix formalism and adopts the diproton approximation [87]. A different method makes use of the shell model embedded into continuum [88]. In the latter approach both diproton and direct approximations are considered. The predictive power of these different models is similar. All of them, although very useful in making estimates of the decay width, are based essentially on two-body approximation and thus cannot describe all the features of the 2p radioactivity. This becomes evident when correlations between emitted protons enter the stage, see figure 9. To overcome this limitation, the full three-body model has been developed by Grigorenko and Zhukov [89, 90]. It adopts the cluster approximation—the two protons are considered to move in the potential of the core. In the Jacobi coordinate system, the natural variables of the wave function are the hyper-radius ρ, defined as

$$\begin{equation} \rho^2=\frac{A_1 \, A_2 \, A_3}{A_1 + A_2 + A_3}\left( \frac{\mathbf{r}^2_{12}}{A_3} + \frac{\mathbf{r}^2_{23}}{A_1} + \frac{\mathbf{r}^2_{31}}{A_2} \right), \end{equation} \tag{ 21 }$$

where the A_i denote mass numbers of particles, the r_ij are the distances between particles, and the hyper solid angle Ω₅ = {θ_ρ,Ω_x,Ω_y}. The Ω_x and Ω_y are ordinary solid angles of the Jacobi vectors X and Y and $\tan (\theta _{\rho })=\sqrt {M_x/M_y} X/Y$. In analogy to equation (6), the wave function of the initial state is postulated in the form

$$\begin{equation} \Psi_{JM}(\rho,\Omega_5) = \frac{1}{\rho^{5/2}} \sum _{\alpha = \{K,...\}} \chi_{\alpha}(\rho)\,{\cal J}^{JM}_{\alpha}(\Omega_5) , \end{equation} \tag{ 22 }$$

where χ(ρ) are radial functions and ${\cal J}^{JM}_{\alpha }(\Omega _5)$ are the hyperspherical harmonics [83]. The index α denotes the complete set of quantum numbers for three clusters, including the hypermomentum quantum number K. This leads to the following set of coupled differential equations for radial functions:

$$\begin{equation} \left[\!\frac{\hbar^2}{2 \, \mu} \frac{\mathrm{d}^2}{\mathrm{d} \rho ^2} \!-\! \frac{\hbar^2 \, {\cal L}_K ({\cal L}_K + 1)}{2 \, \mu \, \rho^2} + E_{\mathrm{T}}\! \right] \chi_{\alpha}(\rho) \!=\! \sum _{\alpha '} \hat{V}_{\alpha \alpha'}(\rho) \chi_{\alpha '}(\rho) , \end{equation} \tag{ 23 }$$

where ${\cal L}_K = K + 3/2$. Note that equation (23) has exactly the same form as equation (8). Here the matrix $\hat {V}$ is non-diagonal due to the three-body potentials, in particular long-range Coulomb interactions. The solution with the complex energy E_T represents the Gamow state. In contrast to equation (8), however, the asymptotic form of the radial functions is not trivial. To find the solution with the proper outgoing boundary conditions, Grigorenko and Zhukov proposed the following method [89]. For large values of hyper-radius ρ only the Coulomb interaction remains significant. In the truncated basis one can diagonalize equation (23) and then the asymptotic solutions must have the form of equation (9). Thus, the asymptotic behavior of the final radial functions is given by

$$\begin{equation} \chi_{\alpha}^{(+)}(\rho) \rightarrow \, \sim \sum _{\alpha '} \skew6\hat{A}_{\alpha \alpha '} \left[ G_{{\cal L}_0}(\eta _{\alpha '},k \rho) + \mathrm{i} \, F_{{\cal L}_0}(\eta _{\alpha '}, k \rho) \right] , \end{equation} \tag{ 24 }$$

where the matrix $\skew6\hat {A}$ is the one which diagonalizes the matrix

$$\begin{equation} \hat{V}_{\alpha \alpha'}(\rho) + \frac{\hbar^2 \, {\cal L}_K ({\cal L}_K + 1)}{2 \, \mu \, \rho ^2} \, \delta _{\alpha \alpha'} , \end{equation} \tag{ 25 }$$

and the parameter ${\cal L}_0$ is fixed to the minimum possible value for all channels, usually in the range from 3/2 to 11/2. The results were found to be not sensitive to this choice. Having the functions χ⁽⁺⁾_α(ρ), and thus the full wave function Ψ_JM(ρ,Ω₅), the decay width can be computed from the total flux going through a hypersphere of large hyper-radius [90]. By varying components of the core-p interaction, the internal structure of the parent nucleus is approximated by two protons occupying the state of a given orbital angular momentum (l²_p configuration). The 2p-decay half-lives calculated in this way were found to be located between limits of the direct model for the same values of l_p, thus providing a narrower range of predicted lifetimes for a given decay energy [90]. Moreover, in all cases listed in table 2 the measured half-life was found to be within this range. A more detailed analysis was done for ⁴⁵Fe. Experimental values of the decay energy and the partial 2p half-life are reproduced by the three-body model assuming the initial state to be a mixture of p² and f² configurations with the weight of the p² component of 30 ± 10% [91].

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The main advance achieved by the three-body model is the prediction of momentum correlations between emitted protons. The resulting momentum density distribution for ⁴⁵Fe in the Y Jacobi coordinates [70, 92] is shown in figure 9. The calculated correlation picture is seen to be in very good agreement with the result of the measurement [80]. It shows a characteristic pattern of two bumps, one larger for small angles between protons (cos(θ_k) ≈ −1) and the other smaller for large opening angles (cos(θ_k) ≈ 1). The latter bump is related to the contribution of the p² configuration. The presented calculations were performed for the weight of the p² component equal to 24%. It is noteworthy that a single assumption concerning the composition of the initial wave function reproduces very well both the decay width of ⁴⁵Fe and its p–p correlation picture. In addition, the observed pattern is significantly different from predictions of simplified models, such as the direct model (figure 9(a)) or the diproton model. This observation represents the strongest evidence that the 2p emission is a genuine three-body phenomenon. Moreover, the case of ⁴⁵Fe indicates that the correlations between emitted protons do depend on the details of the initial wave function. This finding opens a possible way to investigate the structure of even-Z nuclei beyond the proton drip-line by means of 2p radioactivity. The predictive power of the three-body model was confirmed by the case of ⁶Be. The democratic 2p emission from ⁶Be was measured and in particular the detailed p–p momentum distribution was determined with good statistics [93]. It was found to be in very good agreement with the three-body model [70]. It reproduces well also the p–p correlations observed in the 2p decay of ¹⁹Mg [81]. In this case, however, only momenta projected on the transverse plane were recorded. In the cases of ⁵⁴Zn [82] and ⁴⁸Ni [94] the recorded statistics was too small to allow a meaningful comparison with the model, but at least the results do not contradict it.

The phenomena of proton and two-proton radioactivity may have natural analogues in neutron radioactivity of nuclei located beyond the neutron drip-line. The significant difference, however, is that neutrons are not sensitive to the Coulomb interaction and the potential barrier is created only by the centrifugal term. An exploration of the possible radioactive emission of neutrons was undertaken recently by Grigorenko et al [95]. For the matter of rough estimate they selected some cases of light nuclei, experimentally accessible at the present facilities. A simple width estimation for the emission of one neutron was done with the R-matrix formula (16). The result is that for the neutron in the d orbital in ²⁵O the half-life longer than 1 ps requires the decay energy E_n < 1 keV. It seems very unlikely that such a fine-tuning of energy will be found in reality. More realistic chances of finding the case of single-neutron radioactivity may appear for f and the higher-l states and thus among much heavier drip-line nuclei, currently out of reach.

The situation is much more favorable for the emission of two neutrons. For the width estimate the direct decay model given by equation (20) can be applied. Analysis of ²⁶O showed that the limit of 1 ps for the partial 2n half-life is reached for decay energy E_2n of 200 and 600 keV for d² and f² configurations, respectively. Very recently, evidence for the ground state of ²⁶O which is bound by less than 200 keV and decays by emitting two low-energy neutrons was published [96]. In another work the upper limit for the lifetime of ²⁶O of 5.7 ns was given [97]. Before more precise information on this state is obtained ²⁶O is considered as a promising case for the observation of 2n radioactivity.

There exist a possibility that the true four-neutron emission may be found. Such a decay channel would require that both S_4n < 0 and {S_n,S_2n,S_3n} > 0. Two likely candidates to fulfill these conditions are ⁷H and ²⁸O [95]. The width estimate of such a process can be done by a straightforward extension of the direct model formula (15). For the case of ²⁸O and the configuration [s²   d²] the half-life longer than 1 ps will be reached for E_4n < 500 keV. For ⁷H with the configuration [s²   p²] this limit amounts to 150 keV. More detailed information on the masses of relevant nuclei is needed to figure out if this exotic decay channel indeed takes place.

Radioactive decays at the limits of nuclear stability provide important and usually very first information about properties of extremely exotic nuclei. This is particularly significant at the proton drip-line, which is experimentally accessible much more easily than the neutron drip-line and where emission of charged particles occurs which can be detected relatively easily and efficiently. Nuclear spectroscopy at the drip-lines is important for several reasons. It yields data helping to trace structural changes such as migration of shell gaps with increasing difference between the number of protons and neutrons. It allows to improve and verify nuclear models which are used to predict the properties of astrophysically relevant nuclei. Weakly bound and unbound nuclei belong to the realm of open quantum systems—they provide the testing ground for a unified description of nuclear structure and reactions.

Over the last 30 years, proton radioactivity has become a major tool of nuclear spectroscopy. More than 40 proton emitting states have been observed, in seven emitters the fine structure was established. An advanced analysis taking into account the rotational and vibrational degrees of freedom of the core nucleus yields the details of the wave function, including its small components responsible for proton emission. So far no ground-state emitters with Z < 50 have been seen. It may be hoped that this limit will be crossed in the near future mainly due to improvements of the detection systems involving DSP.

α spectroscopy is also a fruitful source of information on the masses and structure at the proton drip-line. It helped to clarify the question of proton separation energies of antimony isotopes. It gave insight into the ordering of single-particle levels close to the doubly magic ¹⁰⁰Sn. It shed light on shape coexistence in neutron-deficient lead isotopes. Perhaps most interesting is the prospect of observing the superallowed α decay of ¹⁰⁴Te, which may establish the new reference for α emission.

Two-proton radioactivity was discovered 10 years ago and the experimental information on this process as well as its theoretical understanding are growing fast. The detailed momentum correlations measured for ⁴⁵Fe unveiled the essential three-body nature of this phenomenon and, in addition, indicated the sensitivity to the composition of the initial wave function. All measured observables agreed well with predictions of the three-body model by Grigorenko and Zhukov. The universality of this model was demonstrated by the excellent agreement with the data obtained for the democratic decay of ⁶Be. While a good description of the three-body Coulomb asymptotics is a strong feature of this model, its drawback is the simplification of the initial state implicit in the cluster approximation. A proper connection between the many-body nuclear structure of the initial nucleus and the three-body final state is the goal of future theoretical work. The experimental search for new 2p emitters is continuing. Soon, the region above zinc comprising isotopes of germanium, selenium and krypton will be explored. Among the elements below iron the candidates include ²⁶S and ³⁰Ar. The latter cases are expected to decay with very short lifetimes, in the subnanosecond range, and experiments using the in-flight decay tracking technique are being planned.

Experimental progress at the neutron drip-line may bring exciting observations on neutron radioactivity. In particular, the process of two-neutron emission may proceed in some cases with a measurable half-life. An even more spectacular decay by the simultaneous emission of four neutrons—tetra-neutron radioactivity—cannot be excluded.

The ongoing developments in radioactive beam facilities allow us to expect further advances in particle radioactivity studies of exotic nuclei.

The author is grateful to colleagues who shared their knowledge, opinions, results, figures and many illuminating discussions. In particular, thanks go to L Grigorenko, M Karny, Ch Mazzocchi and K Rykaczewski.