Radioactive decays at limits of nuclear stability

M. Pfützner, M. Karny, L. V. Grigorenko, and K. Riisager

Rev. Mod. Phys. 84, 567 – Published 30 April 2012

Abstract

The last decades brought impressive progress in synthesizing and studying properties of nuclides located very far from the beta stability line. Among the most fundamental properties of such exotic nuclides, the ones usually established first are the half-life, possible radioactive decay modes, and their relative probabilities. When approaching limits of nuclear stability, new decay modes set in. First, beta decays are accompanied by emission of nucleons from highly excited states of daughter nuclei. Second, when the nucleon separation energy becomes negative, nucleons start being emitted from the ground state. A review of the decay modes occurring close to the limits of stability is presented. The experimental methods used to produce, identify, and detect new species and their radiation are discussed. The current theoretical understanding of these decay processes is reviewed. The theoretical description of the most recently discovered and most complex radioactive process—the two-proton radioactivity—is discussed in more detail.

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Received 21 January 2011

DOI:https://doi.org/10.1103/RevModPhys.84.567

Authors & Affiliations

M. Pfützner^* and M. Karny

Faculty of Physics, University of Warsaw, Hoża 69, PL-00-681 Warszawa, Poland

L. V. Grigorenko

Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, RU-141980, Dubna, Russia

K. Riisager

Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark

^*pfutzner@fuw.edu.pl

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Vol. 84, Iss. 2 — April - June 2012

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Figure 1
The chart of nuclei. The stable nuclides are represented by black squares, while the radioactive ones, which were experimentally identified, are shown by the light shaded area. The nuclides predicted to have positive nucleon separation energy according to the FRDM mass model (337), but not yet observed, are shown by the dark shaded area. The lines indicate the position of magic numbers corresponding to the closed neutron and proton shells (the numbers smaller than 20 are not shown). The insets show the location on the chart of the decay products of the parent nucleus which is indicated by a dark square. The observed decay channels of the proton-rich and the neutron-rich nuclei are shown on the left and on the right inset, respectively.Reuse & Permissions
Figure 2
Characteristic time scales and decay widths of radioactive decays. Ranges of application of selected experimental techniques are sketched.Reuse & Permissions
Figure 3
The proton- and two-proton separation energies of iron and cobalt isotopes as predicted by the FRDM mass model (337).Reuse & Permissions
Figure 4
The neutron- and two-neutron separation energies for the $N = 26$ and $N = 27$ isotones, as predicted by the FRDM mass model (337).Reuse & Permissions
Figure 5
The difference between the neutron number of the lightest experimentally observed isotope for a given atomic number $Z$ and the corresponding prediction for the last isotope before the proton drip line according to the FRDM mass model (337) (line) and the HFB-17 model (202) (circles). The results for the even $Z$ and odd $Z$ are shown in the bottom and in the top, respectively. The experimental values were taken from 312 with corrections contained in 45.Reuse & Permissions
Figure 6
The difference between the proton number of the lightest observed isotone for a given neutron number $N$ and the prediction of the last stable isotone before the neutron drip line according to the two theoretical mass models. The results for the even $N$ and odd $N$ are shown in the bottom and in the top, respectively. The plot details are the same as in Fig. 5.Reuse & Permissions
Figure 7
Systematic overview on calculated isotopic production cross sections in different reactions. For clarity, only values above $100 μ b$ are shown. From 436.Reuse & Permissions
Figure 8
A general scheme of the two main methods used to extract and separate radioactive nuclei. The dashed lines indicate connections which are considered in planning future facilities.Reuse & Permissions
Figure 9
A schematic view of the optical time projection chamber (OTPC). For each recorded event, the data consist of a 2D image taken by a CCD camera in a given exposure time and the total light intensity detected by a photomultiplier (PMT) as a function of time, sampled by a digital oscilloscope. The gating electrode is used to block the charge induced by incoming ions.Reuse & Permissions
Figure 10
An example of a registered two-proton decay event of ${}^{45}{Fe}$ . Top: An image recorded by the CCD camera in a 25 ms exposure. A track of a ${}^{45}{Fe}$ ion entering the chamber from the left is seen. The two bright, short tracks are protons of approximately 0.6 MeV, emitted $535 μ s$ after the implantation. Bottom: A part of the time profile of the total light intensity measured by the PMT (histogram) showing in detail the $2 p$ emission. Lines show results of the reconstruction procedure yielding the emission angles $ϑ$ with respect to the axis normal to the image. From 332.Reuse & Permissions
Figure 11
Schematic drawings of the NERO detector. Left: Side view showing the beta counting station (BCS) chamber located inside of NERO with the DSSSD at the central position. Right: Backside showing the cylindrical cavity to house the BCS and the three concentric rings of gas-filled proportional counters. The labels A, B, C, and D designate the four quadrants. From 383.Reuse & Permissions
Figure 12
Part of the preamplifier signal waveforms recorded by the front and the back strip of the $65 − μ m$ DSSSD during the ${}^{145}{Tm}$ experiment. The recoil depositing about 14 MeV energy is followed after $0.55 μ s$ by the 1.73 MeV signal. From 269.Reuse & Permissions
Figure 13
Some of the possible energetically allowed beta-decay channels for a neutron-rich nucleus. The precursor ${}^{A}Z$ beta decays to the emitter ${}^{A}(Z + 1)$ that has particle unbound excited states. All energies are given relative to the ground state of the emitter.Reuse & Permissions
Figure 14
$Q_{EC}$ value (top panel) and beta-decay half-life (bottom panel) as a function of proton number $Z$ for the lightest nucleus for each element where beta decay remains the dominating decay mode. Dashed lines (for $73 \leq Z \leq 82$ ) are the values at the proton drip line. Experimental values from 22, 23, 3, and 152 are completed by estimates based on 337.Reuse & Permissions
Figure 15
The center-of-mass energy that gives an $s$ -wave penetrability of $10^{- 2}$ (full lines), $10^{- 6}$ (dashed lines), or $10^{- 10}$ (dotted lines) for a beta-delayed proton or alpha particle are shown vs the charge $Z$ for the precursors shown in Fig. 14. Gamma emission can be expected to compete for penetrabilities below $10^{- 6}$ (cf. Figure 2). For illustration, the emitter $Q_{α}$ value (23) is shown for a few beta-decaying nuclei; see the text.Reuse & Permissions
Figure 16
The total branching ratio of beta decays to the IAS is shown as a function of relative proton excess for the light Ne and Ar isotopes. The dashed line gives the estimated branching ratio for Fermi beta decays to the IAS for the Ni isotopes. Only the Fermi part of the transition is included, the partial half-life is assumed to scale inversely with $Z - N$ and total half-lives are taken from 152.Reuse & Permissions
Figure 17
$Q_{β}$ value (top panel) and beta-decay half-life (bottom panel) as a function of neutron number $N$ for the lightest particle stable nucleus for a given $N$ . The experimental drip line position is used for $N$ up to 30, all other values are taken from 337 and 339. The dashed line gives the $Q$ values from an estimate based on the Weizsäcker mass formula.Reuse & Permissions
Figure 18
Nuclei with large beta-delayed neutron-emission probability are marked with an open square if the probability for emitting one or more neutrons is larger than 50% and with a filled square if the average number of emitted neutrons is larger than 1. The $P_{n}$ values are taken from experiment (81; 22; 512) for $N < 20$ and from 339 otherwise. The full lines indicate the line of beta stability and the two drip lines estimated from the Weizsäcker mass formula and the broken lines the corresponding estimates for where beta-delayed one-, two-, and three-neutron emission becomes energetically allowed.Reuse & Permissions
Figure 19
Beta-delayed multiparticle decays recorded with the optical time projection chamber described in Sec. 3c1. The left panel shows beta-delayed three-proton emission from ${}^{45}{Fe}$ (from 330) recorded so the incoming track is not visible, the right panel the track of a ${}^{8}{He}$ ion entering from the right that after beta decay breaks up into a triton (long weak track), an alpha particle and an invisible neutron.Reuse & Permissions
Figure 20
The branching ratio for beta-delayed alpha decay of ${}^{12}N$ (filled triangles) is shown as a function of the total energy (250). The solid line is a fit to the feeding to $0^{+}$ and $2^{+}$ states and does not include the contribution to the $1^{+}$ state at 12.7 MeV. The filled circles (open squares) give the contribution from decays that do (do not) proceed through the ${}^{8}{Be}$ ground state, the dashed lines are the corresponding fits. See the text for details.Reuse & Permissions
Figure 21
The beta-delayed alpha decay data for ${}^{12}N$ shown in Fig. 20 are displayed corrected for the beta-decay phase-space factor and alpha particle penetrability factors. The total fit is divided into contributions from $0^{+}$ and $2^{+}$ states in ${}^{12}C$ . Note the clear interference between $0^{+}$ states at low energy and the enhanced decay rate at high energy (from 249).Reuse & Permissions
Figure 22
Schematic view of the radial part of the nucleus-proton potential. The classical turning points $r_{i}$ for a particle with energy $E_{r}$ are marked. The nuclear contribution turns to zero around $r = r_{nuc}$ .Reuse & Permissions
Figure 23
The half-life for the proton emission as a function of nuclear decay energy $Q_{p}^{nuc}$ and the orbital angular momentum carried away by the proton. Calculations were done for the case of ${}^{151}{Lu}$ with $S_{p} = 0.54$ . The measured values of the decay energy and the half-life, indicated by the black square, suggest the transition with $L = 5$ .Reuse & Permissions
Figure 24
(a) Energy spectrum of the first ( ${}^{109}{Xe}$ ) and second ( ${}^{105}{Te}$ ) $α$ pulses obtained from the $α − α$ pile up traces (inset). The lines at 3910(10) and 4063(4) keV are assigned to the ${}^{109}{Xe} \to {}^{105}{Te}$ transitions, while the lines at 4711(3) and 4880(20) keV are assigned to the ${}^{105}{Te} \to {}^{101}{Sn}$ decay. (b) $γ$ spectrum in coincidence with the analyzed $α − α$ traces. From 130.Reuse & Permissions
Figure 25
Energy conditions for different modes of the two-proton emission: (a) typical situation for decays of excited states (both $1 p$ and $2 p$ decays are possible), (b) sequential decay via narrow intermediate resonance, and (c) true $2 p$ decay. The cases (d) and (e) represent “democratic” decays. The gray dotted arrows in (c) and (d) indicate the “decay path” through the states available only as virtual excitations.Reuse & Permissions
Figure 26
Coordinate systems for two-nucleon plus core problem. In (a) the Jacobi T system, the “two-proton subsystem,” and the core are explicitly in configurations with definite angular momenta $l_{x}$ and $l_{y}$ . For a heavy core (b) the Jacobi Y system is close to (c) the single-particle V system used typically in many-body approaches.Reuse & Permissions
Figure 27
Complete correlation picture for ${}^{6}{Be}$ g.s. decay, presented in (a), (c) T and (b), (d) Y Jacobi systems. (a), (b) is theory, and (c), (d) is experimental data. Qualitative illustration of the meaning of different kinematical regions is provided above the panels. Data and calculations are from 213.Reuse & Permissions
Figure 28
Lifetime of ${}^{45}{Fe}$ vs decay energy calculated in different models. (a) Simplified models of $2 p$ decay. All spectroscopic factors are taken as unity. (b) Three-body model (211; 215) and continuum shell-model results (420). The experimental points demonstrate the rapid improvement of the data: circle (387), square (195), triangle (153), and diamond (332).Reuse & Permissions
Figure 29
Momentum density distribution on the kinematical plane ${ε, \cos (θ_{k})}$ for ${}^{45}{Fe}$ . (a), (b) Diproton model, (c), (d) direct decay model, (e), (f) three-body model, and (g), (h) experimental distribution. Correlation patterns are provided in (a), (c), (e), (g) the T and (c), (d), (f), (h) Y Jacobi systems. The calculations (e), (f) are from 207 and data (g), (h) are from 332.Reuse & Permissions
Figure 30
An example of a two-proton decay event of ${}^{48}{Ni}$ recorded with the optical time projection chamber described in Sec. 3c1. Top: the image recorded by the CCD camera showing a long track of the ${}^{48}{Ni}$ ion entering the chamber from below and the two bright, short tracks of protons emitted 1.576 ms after the implantation. Bottom: a part of the time profile of the total light intensity measured by the PMT showing in detail the $2 p$ emission. From 393.Reuse & Permissions
Figure 31
Complete correlation picture for ${}^{19}{Mg}$ g.s. decay, presented in (a) T and (b) Y Jacobi systems. Comparison of two inclusive distributions projected on a plane is given in the lower row (346). The angles $θ_{p − p}^{'}$ and $θ_{k}^{'}$ are between (c) projected momenta of protons and (d) projected Jacobi momenta in the T system. Solid, dashed, and dotted curves correspond to three-body model, diproton model, and phase-space simulations. Data are from 346 and calculations are from 207.Reuse & Permissions
Figure 32
Energy distributions in the Y Jacobi system (core- $p$ channel) calculated in the direct decay model of Eq. (29). All distributions are normalized to unity at the maximum.Reuse & Permissions
Figure 33
(a) Classical one-proton flight path $R_{f}$ for ${}^{6}{Be}$ , ${}^{12}O$ , and ${}^{19}{Mg}$ connected with the time delay in the $core + p$ subsystem calculated by Eq. (39). (b) Probability of different $R_{f}$ values for the decay of ${}^{12}O$ estimated by Eq. (29).Reuse & Permissions
Figure 34
Transition from the true three-body to the sequential decay for ${}^{6}{Be}$ , ${}^{12}O$ , and ${}^{19}{Mg}$ in the direct decay approximation Eq. (29). (a) The width vs decay energy $E_{T}$ , (b) evolution of energy distributions in the Y Jacobi system (core- $p$ channel) for the case of democratic decay, and (c) for radioactive decay. The curves in (b), (c) are normalized to unity for the maximum of the bell profile at $ε \approx 0.5$ .Reuse & Permissions
Figure 35
Lifetime vs decay energy systematics for several known and prospective true $2 p$ emitters calculated in the three-body model. Hatching indicates lifetime ranges accessible to different experimental techniques. Experimental results are shown by diamonds. Gray circles show specific predictions, where available.Reuse & Permissions
Figure 36
Spatial correlations in the ${}^{45}{Fe}$ WF in the T system for $W (p^{2}) = 2 %$ . (a), (b) Different radial ranges. Logarithmic scale: two (a) and one (b) contours per order of the magnitude.Reuse & Permissions
Figure 37
(a) The energy distributions between two protons (T Jacobi system) in different $s − d$ shell nuclei and (b) with different assumptions about the internal structure of ${}^{45}{Fe}$ . $W (l^{2})$ is the weight of the [ $l^{2}$ ] configuration in the nuclear interior. All distributions are normalized to unity maximum value. From 214.Reuse & Permissions
Figure 38
Contour maps of the momentum density distribution on the kinematical plane ${ε, \cos (θ_{k})}$ for ${}^{45}{Fe}$ in the T Jacobi coordinate system without (a) and with (b) classical extrapolation. (c) Comparison with experimental data from 332. Experimental and theoretical events are convoluted with the experimental resolution. From 207.Reuse & Permissions
Figure 39
(a) Estimated widths for the neutron and (b) the two-neutron emission compared with widths for the proton and the two-proton emission. (c) Estimates for the four-neutron emission. Hatched areas indicate lifetime ranges accessible by different experimental techniques; see also Fig. 35.Reuse & Permissions

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