Systematic trends in beta-delayed particle emitting nuclei: The case of beta-p-alpha emission from 21Mg

We have observed beta+-delayed alpha and p-alpha emission from the proton-rich nucleus 21Mg produced at the ISOLDE facility at CERN. The assignments were cross-checked with a time distribution analysis. This is the third identified case of beta-p-alpha emission. We discuss the systematic of beta-delayed particle emission decays, show that our observed decays fit naturally into the existing pattern, and argue that the patterns are to a large extent caused by odd-even effects.


Introduction
Beta-delayed particle emission is an important decay mode for exotic nuclei and allows many aspects of nuclear structure to be probed, see the two recent reviews [1,2] for a comprehensive overview. We report here the first observation of βα emission as well as the rare βpα emission from the nucleus 21 Mg. Based on these observations we have identified systematic patterns in the occurence of betadelayed particle decays in proton-rich nuclei. We shall present and discuss these as well.
A detailed description of beta-delayed particle emission must include consideration of local nuclear structure effects, but its occurence is in general dominated by the available energy, i.e. the difference between the Q β -value and the particle separation energy. As is well known, for an isobaric chain with mass number A the Q β values will increase and the proton and neutron separation energies decrease as one moves from the beta stability line towards the driplines (modulated for even A by the pairing term). The α particle separation energy tends for light nuclei to be minimal for N = Z nuclei, but the minimum moves towards more proton rich nuclei and reaches the proton dripline at A ∼ 50. This causes a clear pattern for beta-delayed multiparticle emission, with β2p and β3p taking place close to the proton dripline, β2n, β3n etc starting from about halfway to the neutron dripline, while β2α is seen from A = 8, 9, 12 nuclei close to stability. (To the extent that these decays are sequential one can of course regard them as βα decays to the unstable A = 5, 8 nuclei.) Similar patterns appear in beta-delayed single-particle emission although exceptions occur for the very light nuclei such as the large P n values for N = 10 nuclei and the βα emission from neutron-rich N-isotopes.
We focus first on the multi-particle βpα decay and return in the discussion to the general patterns of beta-delayed particle emission.

Experimental results
The 21 Mg activity was produced at the ISOLDE facility at CERN by a 1.4 GeV proton beam impinging upon a SiC target. The produced atoms were extracted, laser ionized, accelerated to 60 keV, led through a mass separator into the experimen- tal set-up, and implanted in the window of a gas-Si telescope opposed by a Si(DSSSD)-Si telescope. A full account of the experimental procedure is given in [3]. The collected source also contained a substantial amount of 21 Na. The data from the Gas-Si charged particle telescope are presented as a ∆E-E spectrum in Fig.  1. Rescaled stopping powers [4] for α particles and protons (evaluated for silicon, but representing the total energy loss in the collection foil, the gas detector and the Si dead layer) are drawn in the figure and match the data well, indicating the presence of βα decays on top of the previously established [5,6] βp. The β-particle component in the lower left corner of Fig. 1 overlaps with protons below 1150 keV and α-particles below 700 keV making particle identification difficult at low energy. The α-particles are stopped in the DSSSD and cannot be separated there from the more intense proton branches.
The α-particle spectrum extracted from the gastelescope by applying the gate drawn as a dashed black closed line in Fig. 1 is shown in Fig. 2. Apart from a remaining background component at low energy five α branches can be identified in the spectrum. The α 2 , α 3 , α 4 and α 5 lines naturally fit into the 21 Mg scheme put forward in [3] as βdelayed single α branches (see [3] for the full decay scheme). The α 1 line, with measured laboratory energy 714(12) keV, does not fit with a transition between known levels in 21 Na and 17 F. However, it does agree with a known α-particle transition from 20 Ne to 16 O observed in the decay of 20 Na [7] with a laboratory energy of 714(4) keV.
A conclusive particle identification for α 1 was not possible from the ∆E-E plot, but strong support for the above assignment comes from the observation of a coincident line in the DSSSD detector, assigned to be the preceding proton. This proton branch p 3 (the numbering is chosen to be consistent with the full data set discussed in [3]) is displayed as the inset in Fig. 2. From the measured energy we deduce E cm (p 3 ) = 919(18) keV which leads to the interpretation of α 1 and p 3 as being due to βpα decay of 21 Mg through the 5/2 + isobaric analogue state (IAS) at 8.975 MeV in 21 Na via proton emission to the 5.621 MeV 3 − level in 20 Ne and finally α emission to the ground state of 16 O. The total branching ratio of this decay branch is found to be 1.6(3) · 10 −4 . This proton branch from the IAS has not been observed earlier and α-emission from excited states of 21 Na have only been reported in one earlier experiment [8].

Time distribution analysis
As mentioned above our data are contaminated by 21 Na, other small contaminants could in principle also be present. The observed βα and βpα branches are quite weak, so a cross-check of the assignment is valuable. This is done by considering the time distribution of the events.
Several factors influence the time distribution of the recorded 21 Mg events, see [3] for an exhaustive discussion. We shall use as reference the experimental time distribution recorded for events within the proton gate in Figure 1 and with energy above 1150 keV. The energy gate ensures that the reference distribution only contains protons from the decay of 21 Mg. The halflives of 21 Mg and 21 Na, 122(2) ms and 22.49(4) s [9], differ greatly as do the corresponding time distributions. Other contaminants are also expected to differ from 21 Mg.
Some of the βα branches have quite low statistics and we therefore compare their time distribution directly to the reference distribution. This can be done very efficiently with the empirical distribution function (EDF) statistics [10] that give powerful goodness-of-fit tests. The basic principle is to compare the shape of the data sample to the reference shape by measuring the distance between the two cumulated distributions. For experimental and reference distributions with values EDF i and F i in bin i, the three most frequently used EDF statistics are [11] Kolmogorov-Smirnov

Cramer-Von Mises
and Anderson-Darling where N is the total number of counts and p i is the probability to be in bin i in the reference distribution. The second column of Table 1 gives the 95% confidence levels for the three EDF statistics obtained through Monte Carlo simulation, values below these levels indicate the time distribution for the different lines are consistent with the one of 21 Mg. More details on the confidence levels are given in [3]. The EDF test results in Table 1 show that all lines, except for α 1 , agree with the reference distribution. The agreement is particularly good for the strongest line, α 3 . As mentioned above there is a contamination of β-particles in α 1 that come from both 21 Na and 21 Mg. We would therefore expect the time distribution for α 1 to be mainly that of 21 Mg with a small component of 21 Na. The EDF tests are sufficiently sensitive to see the effect of

Other βpα cases
The βpα decay mode is very rare as described in [1] with only two previously established cases: 9 C and 17 Ne. For two further candidates, 13 O and 23 Si, the decay mode has not been seen so far. Most searches have concentrated on seeing particle emission from the IAS in the beta-daughter due to the large beta-strength to this state.
The case of 9 C is special in that all states populated in the beta-daughter 9 B break-up into two α-particles and a proton, see [12] and references therein. This could be presented as a 100% branching ratio for βpα or βαp decay to 4 He, but the decays of the A = 9 nuclei are special in several aspects [1,2] and are not typical for this decay mode.
Although βpα has not been observed so far for 13 O it must occur since β-decays to the IAS in 13 N have been observed [13] and close to half of the IAS decays are known from reaction experiments [14] to go via proton-emission to α unbound states in 12 C or α-emission to proton unbound states in 9 B. Actually, the final state in both cases will be a proton and three α-particles which makes the decay more challenging to observe. The total branching ratio for the decay mode can be estimated to be 0.9(3) · 10 −4 .
For 17 Ne both decay orderings, βpα and βαp, have been observed [15] with a total branching ratio for the decay mode of 1.6(4) · 10 −4 . All observed decays proceed through the IAS in 17 F and go to the final nucleus 12 C. Adding now our observation of 21 Mg(βpα) 16 O it is striking that all cases go through an α-conjugate nucleus, namely 8 Be, 12 C, 16 O and 20 Ne respectively. Before drawing any firm structure conclusions we shall consider the broader systematics of beta-delayed particle emission in Z > N nuclei.

Systematics of beta-delayed decays
Similar patterns also appear in other betadelayed particle decays (see [1,9] for more data and for references to the original work). One closely related example is βα decays that occur for all bound A = 4n, T z = −1 nuclei up to A = 40: 8 B, 12 N, 20 Na, 24 Al, 28 P, 32 Cl, 36 K and 40 Sc. The βp decays are well established [1] to occur strongly in A = 4n + 1, T z = −3/2 nuclei. The β2p decays of 22 Al and 26 P and the β3p decay of 31 Ar [16] also all end up in an α-conjugate nucleus. The decays observed for the elements N to Si are shown in figure  3. Note that the βα and βp modes are not marked explicitly when the βpα or β2p modes also occur.
In the following we shall argue that the observed patterns are likely (except for the very lightest nuclei) to be related to odd-even effects rather than α-cluster structure. We start by considering the systematics of Q EC -values for nuclei with Z > N as illustrated in figure 4. Even though many effects contribute to the masses in this region, a liquid drop estimate reproduces the trend of Q EC for the odd-A nuclei with T z = −1/2 (dashed line) where only the Coulomb term enters, as well as for T z = −3/2 (dotted line) where the asymmetry term also contributes. Note that Q EC in the latter case varies little for A between 25 and 50.
The experimental data show that the Q EC -values are roughly the same for each "quartet" of four nuclei that, as illustrated in the left panel of figure 5, have proton and neutron numbers (Z,N), (Z,N+1), (Z-1,N), (Z-1,N+1) where both Z and N are even. This is pronounced for quartets where the even-even nucleus has T = 1, and holds to a lesser degree also for T = 2 for mass numbers up to 40. The reason for this is that the two odd-A nuclei are at the same distance from the beta-stability line and therefore have about the same Q-value, as also shown by the liquid drop estimate. Without a pairing term in the liquid drop formula the Q-value for the even-even nucleus would be larger and the odd-odd smaller, but the odd-even effects counteracts this and as can be seen from figure 4 the magnitudes are even reversed for most nuclei. For the quartet with T = 1 (and T z = −1) the odd-odd nucleus has N = Z and is therefore extra bound, this happens to result in Q-values that are almost the same for all four nuclei. The quartets are indicated in figure 3 by thicker lines.
The observed decay patterns now follow from the energetics and are illustrated in the right panel of figure 5. The βα decays should occur in oddodd nuclei, since they have slightly higher Q-values and the daughter alpha particle separation energies tend to be smallest here. The βp decays need low proton separation energies in the daughter nucleus and therefore are more prominent for even Z, starting (as one goes from stability towards the proton dripline) in an even-odd nucleus. The βpα decay should be favoured in even-odd nuclei, and β2p and β3p decays should occur in odd-odd and even-odd nuclei, respectively, by extending these arguments.
Experimentally, the βα, βp and βpα decays appear first in the quartets where the even-even nucleus has T = 2 and the odd-odd nucleus T = 1, but βp occurs also in 59 Zn, 65 Ge and heavier nuclei. The beta-delayed multi-proton decays appear in more exotic nuclei, but it is noteworthy that βα in these nuclei only has been observed in the oddodd 22 Al. Similar patterns can be expected for β − -delayed particle decays, although the grouping of Q-values is less pronounced here. The βα and βnα decay modes will in general occur further away from the beta-stability line.

Conclusion
Our study of the decay of 21 Mg has given the first evidence for the occurence of the βα and βpα decay modes in this nucleus. The assignment of these decay modes to 21 Mg has been verified through statistical tests of the time distribution of the events. The occurence of these decay modes in the β + decay of 21 Mg fits naturally into the systematics of previously observed β + -delayed decays. We presented a brief overview of this systematics and argued that it can be explained by the variation in decay energy due to odd-even effects and that there is no need to invoke specific structure effects such as alphaclustering in spite of α-conjugate nuclei occuring often as final state nuclei.
This interpretation can be tested when new instances of these exotic decays are discovered. The βpα decay mode may not occur in heavier T z = −3/2 nuclei than 21 Mg (the Q-value becomes more than 10 MeV in 61 Ge, but the Coulomb barrier for α-particle emission is substantial then), but may be found also in the T z = −5/2 nuclei 23 Si, 27 S, 31 Ar etc. If found in 20 Mg it may help to quantify the 15 O(α,γ) 19 Ne reaction rate [17]. A general overview of which energetically allowed decays have not yet been observed was given in [1].