Transition from direct to sequential two-proton decay in $s$-$d$ shell nuclei

Transitions among different mechanisms of two-proton decay are studied in general. The introduced improved direct-decay model generalizes the semi-analytical models used before and provides flawless phenomenological description of three-body correlations in $2p$ decays. This is demonstrated by examples of the low-lying $^{16}$Ne state decays. Different forms of transition dynamic are shown to be highly probable beyond the proton dripline for the $s$-$d$ shell nuclei. It is demonstrated that transition dynamic of $2p$ emitters can provide means for extraction of a width of the ground-state resonance of a core+$p$ subsystem of the core+$2p$ system. Practical applicability of the method is demonstrated by properties of the $^{14}$F ground state derived from the $^{15}\mbox{Ne}\rightarrow ^{\,13\!\!}\mbox{O}+2p$ decay data and of the $^{29}$Cl ground state derived from the $^{30}\mbox{Ar}\rightarrow ^{\,28\!\!}\mbox{S}+2p$ decay data.


Introduction
Two-proton (2p) radioactivity has attracted a large interest in theory and experiment in recent years. This peculiar radioactive decay mode was predicted for even-proton systems in the year 1960 [1] and experimentally discovered in 2002 [2,3]. Since then, important advances in the studies of two-proton (2p) radioactivity, and, more generally speaking, of true two-proton decays [4] were made. These studies comprise part of a general effort to extend our knowledge to the limits of existence of nuclei, up to the neutron and proton driplines and as far beyond them as possible. It is known that the true 2p decay mechanism, where both protons are emitted simultaneously, is realized under certain energetic conditions, in particular when the relation between one-and two-proton separation energies (S p and S 2p , respectively) makes sequential decay (two single-proton emissions separated in time) * Corresponding author. p > 0, where A denotes the mass number of the mother nucleus. However, in light proton-rich and in neutron-rich nuclei, where the potential barriers governing the decay are relatively small, the genuine three-body decay mechanism can materialize in the form of a democratic decay [5]: these democratic decays exhibit distinct correlations between the decay products and follow a different systematic of lifetimes compared to true 2p decays [4]. Together, and at variance to the sequential decay mechanism, which requires the population of intermediate narrow states, the true 2p and democratic decays are often characterized as direct decays. A schematic view of the various decay mechanisms and their energy balance is shown in Fig. 1 (a,b,d,e).
The transition from the true 2p mechanism to the democratic decay dynamic is typically governed by the relation of the two quantities, Ŵ r (the width of the resonance ground state) and E r (the ground state decay energy in the core + p subsystem of the three-body system core + p + p). The characteristic of this transi- systematically. This is the purpose of the present Letter, and the motivation will be further detailed in the following.
In general, the decay dynamics of 2p emission is fully characterized by three parameters: (i) the 2p decay energy E T =−S (A) 2p , (ii) the ground state (g.s.) decay energy in the core + p subsystem 2p , and (iii) the width Ŵ r of the intermediate state. As the width Ŵ r is a function of E r , the delimitation of different 2p decay mechanisms can be depicted by a kind of "phase diagram" in the {E T , E r } plane, see Fig. 1 (c).
Here, Fig. 1 provides an overall view of the transition dynamic in 2p decays. At first glance, one would expect that the direct 2p and the sequential decay are separated by the line where E r = E T .
However, it was shown in Refs. [6,7] that this transition happens at lower values E r = (0.75-0.85)E T . Similarly, it was found experimentally for decays of 6 Be and 16 Ne in Refs. [9,10] that the transition "Democratic 2p"↔"Sequential 2p" does not even happen for energies E T considerably higher than E T = 2E r ; this issue remains an open question. In the present work we investigate systematically the transition regime "True 2p"↔"Sequential 2p", and, while moving along this line, we obtain detailed quantitative results for the transition region "True 2p"↔"Democratic 2p" in Section 5. It is demonstrated that this transition region is often involved in ground-state decays of s-d shell nuclei beyond the proton dripline.
Besides the general interest, there is another motivation for the present study, which has practical impact. Any phenomena of the transition type are expected to be prone to abrupt changes of its properties in response to minor variation of the parameters. Here even small variations of the characteristic decay parameters can lead to strong modifications of the decay properties and ob- and sequential mechanisms are indicated by different colors. The transition region separating these regimes is shown in gray color. The dashed lines delineating the boarderline of the transition region correspond to the displayed ratios between the 2p-decay energies E T in a three-body system and the 1p-decay energies E r of the g.s. of the core + p subsystem.
served quantities (such as branching ratios, lifetimes, etc.). In turn, this sensitivity can be used for a precise derivation of the nuclear parameters from measurements or for imposing limitations on their relations. Among the relevant parameters {E T , E r , Ŵ r }, the most interesting one is the sensitivity to Ŵ r . While resonance energies can be determined with rather good precision from available experimental data, much better resolution and/or statistics is required to derive the width of a state. The latter conditions are presently often not attainable for exotic dripline systems. Moreover, there is a rather large experimental "blind spot" with respect to the g.s. resonance widths, which covers the interval from ∼ 1m e Vt o ∼ 10 keV (corresponding to resonance lifetimes ranging from ∼ 10 −19 st o ∼ 1p s ) : the invariant and missing mass methods using spectrometers or kinematical complete measurements can be applied when resonance widths exceed ∼ 10 keV (i.e.: resonance lifetimes 10 −19 s), while other techniques (such as the in-flight decay method, plunger, etc.) are applicable only for long-lived states with lifetimes exceeding a few picoseconds (i.e.: widths 1m e V ) . Therefore, we derive here an indirect method to obtain information on the widths Ŵ r of the proton subsystems in the transition regimes between different mechanisms of twoproton emission. The practical feasibility of this method is demonstrated with the examples of 14 F g.s. populated in the 2p decay of the 15 Ne g.s. [11] and of 29 Cl g.s. populated in the 2p decay of 30 Ar [12].

Dynamics of three-body systems near the proton dripline
As mentioned above, the relationship of S 2p is one of the decisive conditions for different forms of nuclear dynamics in the even-proton systems in the proximity of the proton dripline. To illustrate this behavior, the systematics of the p and 2p separation energies for Ne and Ar isotopes near the proton dripline is shown in Fig. 2. These nuclides represent the lower and the upper parts of the s-d shell, respectively. One can see that the evolution of S 2p values is different, and the corresponding curves intersect near the proton dripline, leading to rapid changes of the dynamical properties of nuclides in this region. Particle-stable nuclei that are closest to the dripline typically exhibit a so-called Borromean structure (this denotes the phenomenon that removal of one of the three bodies, either one of the two protons or the core, leads to disintegration of the whole three-body system) with the condition S 2p . Particle-unstable nuclei located just beyond the dripline often undergo direct decays, i.e. either 2p decay by the true (S p ∼ 0) mechanism; in addition, a transition situation is possi- ble. Nuclei located further beyond the dripline pass the transition area and arrive either at a domain of democratic 2p decays (very likely for the lightest nuclei) or at a domain of sequential 2p decays (rather expected for the heavier, higher-Z systems). Thus, the transition dynamics investigated here is a quite likely situation in the area between the direct (true or democratic) and sequential 2p-decay dominance.

Improved direct 2p-decay model
Simplified analytical models of 2p decay are useful theoretical tools, which can substitute complicated and time-consuming three-body calculations when "quick" estimates, systematic studies, or just a phenomenological comparison with experimental observations is needed. The derivation of the direct decay model for 2p emission was described in detail in Refs. [7,14]. The model is based on the utilization of simplified three-body Hamiltonians, which allow a factorization of the Green's function. This approach can be traced back to the ideas of Galitsky and Cheltsov [15]. The expressions originating from the resonant R-matrix-type approximation [14] look very similar to those arising from the quasiclassical approximation [16], later used for 2p emission in a number of works [17,18, and references therein]. However, the models have important differences, which are discussed for example in [4,14,19].
The direct decay models of Refs. [7,14] have some disadvantages, which are overcome in the present approach. Namely, the decay amplitudes must be factorized either in the "V" system [see Fig. 3 (a)], or in one of the "Y" systems [see Fig. 3 (b,c)]. In the "V" system, the anti-symmetrization between identical protons is straightforward. However, the assumption of an infinite core mass is required in this case. Also the interaction of the protons (including Coulomb repulsion) is neglected, which leads to some overestimation of the width. This type of approximation is called "no p-p Coulomb" (NPP), see Eq. (21) of Ref. [7]. Working in the "Y" systems, the p-p Coulomb interaction can be effectively considered by taking the Z i Z j term of the Coulomb interaction in the X dimension as Z c while in the Y coordinate as Z c +1. Unfortunately, a consistent anti-symmetrization is complicated in the "Y" system, and one of the core + p resonances is not treated appropriately.
That is because the X coordinate is exactly the vector between core and p, but the Y coordinate is only approximately equal to the vector between the core and another p. The latter approximation becomes exact in the limit A c →∞ (in this limit "Y"→"V").
In the following, an improved direct decay model (IDDM) is developed which is based on simple R-matrix-type analytical approximations for amplitudes; it combines the positive features of both approximations, NPP and EFC. The general expression derived in Ref. [7] [see Eqs. (30)-(32) of this work], however with a modification, which accounts in a correct way for the angular momentum (Color online.) Three coordinate systems, "V"+"T", "Y 1 ", "Y 2 ", with their respective kinematical variables as used in this work for the description of the threebody system core + p + p, are shown in panels (a), (b), and (c), respectively. coupling of { j 1 , l 1 } and { j 2 , l 2 } of the first and the second core + p subsystems, is: where J is the total angular momentum, γ ={j 1 , l 1 , j 2 , l 2 }, ̹ = {ε, 1 , 2 }, and the angles i are related to the vectors r i (see where ϕ J γ is assumed to be the radial part of the three-body resonant wave function (WF) in jj coupling. The nature of the short-range potential V and its derivation and the underlying approximations are discussed in detail in Ref. [7]. The amplitude A JM J does not depend on the parameter R when it is chosen beyond the range of V . The integral in Eq. (2) can be evaluated by replacing the continuum WFs with the quasi-stationary WF in proximity of the two-body resonance energies: where k, v, E are the related momentum, velocity, and energy in the two-body channel. The pure radial function ψ jl (E r , r) is the quasi-bound WF: it has a resonant boundary condition at the twobody resonant energy E r and is normalized to unity in the internal domain aŝ where G l is irregular at the origin Coulomb WF with angular momentum l. In this definition, the value of parameter R should be chosen around the classical outer turning points for the core + p Coulomb barriers. The amplitude A jl of the resonance with the parameters E r and Ŵ r = Ŵ(E r ) is defined as The width as a function of energy is defined by the standard Rmatrix expression via the penetrability function P l , channel radius r cp , and reduced width θ 2 . In Eq. (4), the resonant term typically used in such calculations is augmented by taking into account the "potential scattering" contribution A jl . This allows additional tests of reliability of the obtained results. The potential scattering term can be approximated reasonably well by the scattering off a solid sphere with radius r sp : From Eq. (3) the amplitude is factorized into a momentumdependent term and a radial integral Because the spin degrees of freedom cannot be observed even in the most sophisticated modern radioactive-ion-beam experiments, the summation over the final state spin quantum number is performed. For systems with zero core spin there are two terms to be treated. The first type is with S = 0 where J = L, and the second is with S = 1 where L ={J − 1, J , J + 1}. After averaging over M J , it is obtained that The coefficient C JLS γ provides the jj to LS re-coupling for two protons in the orbitals j 1 and j 2 : mediate states in the subsystems with appropriate sets of quantum numbers and correctly couples them to the total spin of the system. However, there are important drawbacks in the approximation which do not allow the reproduction of realistic momentum distributions of fragments in 2p decays. Therefore, a phenomenological improvement of Eq. (9) is proposed by replacing the amplitude with a more realistic one. The latter uses terms composed in different Jacob systems instead of those in the "V" system. For the case without potential scattering contribution in Eq. (4), it reads The permutation operator O S provides the correct symmetry of the WFs with respect to proton permutations. As the isospin WFs of two protons are symmetric, it should be used aŝ The amplitude A where k T x and v T x are momentum and velocity associated with the p-p energy E T x . The meaning of such an approximation is to treat the protons as being emitted from a broad spatial region defined by the function φ(r). The source function φ(r) with a Gaussian type formfactor is used The radius parameter a pp in the above expression is equal to the root mean square radius of the function φ(r).
In the calculations, a pp = 4f mi s used, for which |A The IDDM unifies the advantages of both the NPP and EFC models and also goes beyond, thus providing a reliable tool for quantitative estimates and a phenomenological description of 2p decays.

Application to 16 Ne decays
In the following, the IDDM is applied to the examples of the 16 Ne 0 + g.s. and the 2 + first excited state, both decaying to 14 O + p + p, and its properties and performance is illustrated.
In Refs. [10,20], the 16 Ne decay data have been compared with sophisticated full-scale three-body calculations. The comparison yields quantitative agreement, therefore the results of the detailed three-body calculations are used as a reference for the results obtained in the present work. This allows a direct and general comparison and avoids the use of Monte-Carlo (MC) simulations. The latter are needed for a realistic comparison of the semi-analytical model results with the data, which comprise the bias of the particular detector setup. In Fig. 4 16 Ne are taken as 67% [21] and 16% [10]. The different models yield the following results given in Table 1. The NPP model tends to overestimate the decay width [14]. The results of the IDDM lie in between the NPP and EFC width values. The NPP and EFC models also have different deficiencies in describing the momentum distributions of the 2p decay. The IDDM model reproduces all features of the momentum distributions qualitatively correct, see Fig. 4. Also a quantitative agreement with the full-scale three-body calculations is largely obtained. For example, one should note the correct positions and intensity-ratio of the two peaks in Fig. 4 (c). For the 2 + state, however, a complete agreement with full-scale three-body calculations is not expected, as several quantum configurations are needed for description of the data [10].
As an intermediate result, the example of 16 Ne decays shows that the proposed IDDM combines the advantages of the NPP and EFC models, and it provides results which are close to the sophisticated three-body calculations. This makes the proposed approach a reasonable substitute for complicated three-body calculations, at least for exploratory and systematics studies.

Transition patterns in the s-d shell nuclei
In this Section, the IDDM will be applied more generally to several s-d shell nuclei and the decay dynamics is studied. The first case is 30 Ar decaying to 28 S + p + p via the [s 2 1/2 ] 0 configuration, see Fig. 5. In the calculation, the {E T , ε} plane is scanned for a fixed proton resonance energy E r ( 29 Cl) = 1.8 MeV and several decay widths Ŵ r ( 29 Cl) = 60, 170, 330, 500 keV. In the considered "Y" Jacobi system ε = E(core-p)/E T is the relative energy content of the core + p subsystem.
Transitions between different decay mechanisms become visible and the following evolution pattern can be recognized in Fig. 5 (a). At low E T values, the true 2p decay mechanism prevails, which is characterized by a relatively narrow, bell-shaped distribution centered at ε ∼ 1/2 (i.e., where the energies of both protons are equal). For E T > E r , two new peaks, associated with the sequential emission of protons, appear on the rising and falling edge of the central bell-shaped distribution. At an energy of E T ∼ 1.2E r ∼ 2.2M e V , the sequential-emission peaks become higher than the central peak around ε ∼ 1/2, and this value marks the transition point to the sequential decay mechanism (this was already sketched in Fig. 1 using the label "True 2p"↔"Sequential 2p" transition). Overall, the triangular structure reflects the evolution of the double-peak sequential decay pattern as a function of E T . At the tip of the triangle, at E T = 2E r , the two peaks belonging to the sequential emission overlap; at still higher E T they split up again. Fig. 5 (b,c,d) shows the results for different, larger widths Ŵ r , which is achieved by variation of the reduced width θ 2 parameter in Eq. (5). At first glance, similar patterns are obtained as for case (a). In detail, one can see that with increasing Ŵ r the triangular sequential-decay pattern arises at higher energy E T . However, it gradually smears out, and finally vanishes with increase of Ŵ r . This qualitative difference reflects the transition to the democratic decay mechanism: the evolution observed in the decay patterns from (b) to (d) corresponds to the "Sequential 2p"↔"Democratic 2p" transition regime (which is also indicated in Fig. 1).
The panels (a) and (b) in Fig. 5 correspond to the θ 2 values of 0.5 and 2, respectively. The first value should correspond to the structure of a nucleus with strong configuration mixing. The second value is the upper limit of the admissible range for θ 2 , which corresponds to a pure single-particle structure. The difference between the discussed E T evolution patterns is quite significant. For instance, the transition "True 2p"↔"Sequential 2p" in the calculation (b) takes place at an E T value, which is about 300 keV larger than that in the case (a). Unrealistically large θ 2 values of 4 and 6 are used in cases (c) and (d), respectively. They are considered for the study of the transition to the democratic 2p-decay mechanism. From these and analogous evaluations it is found that the "True 2p"↔"Democratic 2p" transition takes place when the width Ŵ r is described by the relation for this, see also Fig. 1.
It is obvious that the charge of 30 Ar is too large (hence the 29 Cl g.s. width is too small) to allow the democratic decay mechanism to materialize with the E r = 1.8M e Vv a l u e used. The evolution of 2p-decay mechanisms similar to the one shown in Fig. 5 (a)→(d) can be achieved by variation of a charge of the 2p precursor or/and energy E r . Fig. 6 illustrates this issue by presenting three examples of the calculated transition patterns ranging from light to heavy s-d shell 2p emitters. In the calculated cases of 15 Ne, 26 S, and 34 Ca g.s. decays, it can be seen that the observed ( 15 Ne) as well as the evaluated ( 26 S) separation energies provide the g.s. location near the transition points. This means that it can be expected that the latter cases exhibit a strong sensitivity of the decay correlation patterns to the parameters of the involved nuclear states. In the following section, these findings will be utilized to study the core + p subsystem of a core + p + p nucleus.

Decay of the 15 Ne ground state
The first illustration, where the 2p-decay transition dynamics is used to study the core + p subsystem, is provided by the 15 Ne g.s. decay. One can see in Fig. 6 (a), that the 2p-decay energy of 15 Ne g.s., E T (exp) = 2.52 MeV [11], is located in the region where the width of the ε-distribution strongly varies with E T . The shape of the ε-distribution by itself can be seen as a not very reliable, model-dependent indicator. However, the width of the 15 Ne g.s. is   [13], Ŵ r calculated assuming an s-wave decay with θ 2 = 2, which is the single-particle limit). The vertical dotted lines mark the E T values from the experiment for 15 Ne [11] as well as from the systematics [13] for the cases 26 S and 34 Ca. The thin dotted lines for 15 Ne indicate the E T region within experimental value of the g.s. width Ŵ = 0.59 MeV [11]. The distributions for each energy E T are normalized to unity maximum value. quite broad, Ŵ(exp) = 0.59 MeV, and one may in addition investigate the evolution of the ε distribution as function of E T depending on model parameters. In Fig. 7 the "Y" system ε-distributions are calculated for different decay energies of 15 Ne within the g.s.
As expected, the ε-distributions, calculated for the centroid (E T ) and for the limiting cases (E T ± Ŵ/2) of the 15 Ne g.s. resonance, differ from each other. In particular, the differences are quite pronounced with respect to the different values of the 14 F g.s. widths Ŵ r ( 14 F), which, in turn, allows to extract some information on the widths Ŵ r from the 15 Ne decay correlation data.
The data for 15 Ne g.s. measured in Ref. [11] are shown in Fig. 7 (a). A direct quantitative comparison is not appropriate, since the mea- The E r = 1.8M e Vv a l u e is fixed, while the curves represent different Ŵ r values. In panels (c) and (d), the experimental angular distribution between core and one of the protons measured for 30 Ar g.s. decay in Ref. [12] (histograms) is compared with those from the respective theoretical distributions shown in panels (a) and (b). The experimental bias is taken into account in the Monte-Carlo simulations. sured distribution is broadened by the experimental resolution and the counts is not sufficient to study the E T evolution in detail. However, we would like to mention that such an analysis is possible with better statistics and better experimental resolution, as can  (a,b,c) show the likelihood of agreement between theoretical and experimental angular distributions (see Fig. 8) of 2p-decays of the 30 Ar g.s.
The probability is shown as function of the {E r , Ŵ r } variables for fixed E T values as indicated in the panels. Panel (d) shows the band of {Ŵ r , Ŵ r } values correlated with E T .
The best-fit E T value and its 1σ -uncertainty are indicated by the vertical line and the hatched area, respectively. be seen in the similar example of the 16 Ne g.s. decay [20]. The precision of the latter experiment seems to be sufficient to distinguish the solid and dotted curves in Fig. 7 (a) or dashed and solid curves in Fig. 7 (b). So, the proposed analysis is in principle feasible.

Decay of the 30 Ar ground state
The observation and study of the new isotopes 30 Ar and 29 Cl were reported recently [12]. It was concluded that the 2p decay of the 30 Ar g.s. is located in the transition regime between true 2p and sequential 2p decays. The analysis was carried out in a simplified way with the decay energy as the only parameter. Here, the 30 Ar data are re-analyzed using IDDM calculations.
Several calculated correlations and the respective "experimental" distributions (simulated by taking into account the setup response function, along the lines described in [12]) are shown in Fig. 8 together with the experimental data. Despite strong effects of the experimental setup, the measured angular distributions of the 2p-decay products appear to be informative. One can see from the figure, that the specific patterns connected with the formation of sequential-decay correlations of the 30 Ar g.s. produce sizable changes in the angular distributions 28 S + p.
In order to find the best match with experimental data, the parameters {E T , E r , Ŵ r } were systematically varied; a Kolmogorov test provides the probability for agreement between simulation results and experimental histograms, see Fig. 9. In the previous study Ref. [12], the assigned parameters were E T = 2.25 +0.15 −0.10 MeV, E r = 1.8(1) MeV. On the basis of the present, more refined analysis, whose results are shown in Fig. 9, a more precise value E T = 2.45 +0.05 −0.10 MeV is inferred (in agreement with our previous findings), and also the width of 29 Cl g.s. is derived as Ŵ r = 85(30) keV.
A pleasant feature of this analysis is that uncertainties of the major parameters are correlated, which can be used for imposing further limits. For example, it is shown in Fig. 9 (d) that a smaller Ŵ r ∼ 15 keV can be established for a fixed E T value.
It should be emphasized that the analysis performed is in the "proof of concept" stage, and should be validated by direct measurements.

Sensitivity of IDDM results
The reliability of the proposed width determination of a core + p g.s. nuclear system from the energy distributions obtained in the "Y" coordinate system depends on the robustness of the distributions with respect to the model parameters used. Such a sensitivity was systematically investigated, and it was found that the only significant dependence results from the general decay parameters {E T , E r , Ŵ r }. For instance, the transition patterns strongly depend on the interference of two-body amplitudes at resonant and near-resonant energy. However, Fig. 10 shows that a modifi- cation of amplitudes outside the resonance peak does not lead to significant changes of the characteristic behavior.
The other important assumption is that only one quantum configuration provides a dominating contribution to the decay width of the state. However, this is a natural situation for the transition dynamics, which is connected to the strong variation of the partial width just in one selected channel. This may be estimated in a forthcoming theoretical study.

Summary
Nearly half of the neutron-deficient s-d shell nuclei located 1-2 neutron numbers beyond the proton dripline decay by 2p emission. This decay mode exhibits three different characteristic: true 2p, democratic and sequential decay mechanisms. Their occurrence depends mainly on the relationship of the three parameters E T , E r , and Ŵ r . The changeover from one decay mechanism to another, which is reflected by altered momentum distributions of the decay products, occurs within rather narrow transition areas, i.e. within small changes of the relevant 2p-decay parameters. The behavior of the momentum distributions of 2p-decay products in the transition regions can be described by the semi-analytical improved direct decay formalism, IDDM. At variance to previously used approximations, the IDDM depicts all qualitative features of 2p-decay distributions in the whole kinematical space; in particular it covers the mentioned 2p-decay transition regions. Even with moderate computing effort, this model yields reliable quantitative estimates and phenomenological evidence, ideally suited for large-scale systematic surveys. Further development and refinements of such a model are of importance for future studies. In the transition regions, the momentum distributions of the decay products exhibit a strong sensitivity with respect to the variation of the three general 2p-decay parameters. Therefore experimental data from the 2p correlations in transition regions can be used to extract resonance parameters in the core + p subsystem of the 2p-decay precursor. The practical feasibility of such an approach has been demonstrated here with the example of the 15 Ne g.s. 2p-decay. By relying on the transition dynamics, improved constraints for the parameters E T , E r , and Ŵ r for the 30 Ar and 29 Cl g.s. can be set in comparison with the values derived in Ref. [12]. Overall, this will contribute to a better understanding of structure and decay dynamics of nuclei and resonances at and beyond the driplines.