Proton decays in ¹⁶Ne and ¹⁸Mg and isospin-symmetry breaking in carbon isotopes and isotones

GSM is a configuration interaction framework based on the Berggren basis Berggren (1968). The Berggren basis consists of one-body states generated by a finite-range potential, such as a Woods-Saxon potential. The Berggren basis is obtained from the real-energy Newton completeness relation Newton (2013), by deforming the contour of real energy states in the complex plane Berggren (1968). Narrow resonance states sufficiently close to the real axis must be included with the bound states and complex scattering states Berggren (1968). The bound, resonance, and complex-energy scattering states of the Berggren basis form a complete set of states:

where $\ket{u_{n}}$ is a bound or resonance one-body state and $\ket{u(k)}$ is a scattering state belonging to the $L^{+}$ contour of complex momenta (see Ref.Michel et al. (2009) for details). In order to use the Berggren basis in numerical applications, one discretizes the $L^{+}$ contour of Eq.(1) with the Gauss-Legendre quadrature Michel et al. (2009). One typically needs 30-50 discretized states in order to have converged results Michel (2011).

The discretized Berggren completeness relation can be formally identified to that generated by a set of harmonic oscillator (HO) states. Consequently, one can build configurations in GSM from the Berggren basis states of all partial waves, similar to the standard shell model (SM) Michel et al. (2009). The nuclear Hamiltonian is then represented by a complex symmetric matrix in GSM.

One of the fundamental difference from SM is the appearance of many-body scattering eigenstates in the Hamiltonian spectrum. Indeed, many-body resonance states are hidden among the many-body scattering eigenstates of the Hamiltonian. The scattering eigenstates apparently form the vast majority of the eigenspectrum, but they are not part of the eigenstates of the lowest energies, contrary to the eigenstates of interest in SM. Therefore, one had to develop the overlap method to solve the so-called identification problem Michel et al. (2003, 2009). For that matter, the Hamiltonian is firstly diagonalized using pole approximation, i.e. by suppressing all scattering configurations of the many-body Berggren basis, so that only resonant configurations, built from $S$-matrix poles, remain. The obtained eigenstate are zeroth-order approximations of the exact resonant eigenstates of the Hamiltonian. One diagonalizes the Hamiltonian in full space afterwards. The resonant eigenstate of interest is then that which bears the largest overlap with the eigenstate obtained at pole approximation level. This allows to uniquely determine the resonant eigenstates of the Hamiltonian, as the pole configurations are always dominant in practice compared to those associated to the non-resonant continuum.

The GSM Hamiltonian matrix is diagonalized using the Jacobi-Davidson method Sleijpen and van der Vorst (1996). The Jacobi-Davidson method targets the eigenstate closest to the used zeroth-order approximation, so that it converges quickly to the sought resonant state. In fact, the Jacobi-Davidson method in GSM replaces the Lanczos method of SM where only the eigenstates of lowest energies can converge quickly. The GSM code has also been recently parallelized using the efficient two-dimensional partitioning algorithm Michel et al. (2020b), so that it can be efficiently used on powerful parallel machines.

The Hamiltonian and model space used in the calculation of the many-body nuclear states of carbon isotopes and isotones are very close to those used in Ref.Michel et al. (2019) in the context of proton-rich oxygen isotones, so that we will only describe its overall features.

We work in the frame of the core + valence nucleon picture. The considered core is ¹⁴C for carbon isotopes, whereas it is the mirror ¹⁴O for carbon isotones.

A potential of Woods-Saxon (WS) type is used to mimic the core, which is fitted to the single-particle spectrum of ¹⁵C. The same WS potential is used for the ¹⁴O core, to which a Coulomb potential generated by the core charge density is added, which is taken of a Gaussian form for simplicity Michel (2011). The parameters of the WS potential are: $d=0.65$ fm for diffuseness, $R_{0}=2.98$ fm for radius, $V_{0}=$ 51.5 MeV ($\ell=0$) or 49.75 MeV ($\ell>0$) for central potential depth, and $V_{so}=$ 6.5 MeV for spin-orbit potential depth.

We take into account $spdf$ partial waves in the valence space, which are of proton (neutron) type for carbon isotones (isotopes). Partial waves of the $spd$ type are represented using the Berggren basis, whereas the HO states are used for the $f$ partial waves. This is justified by the large centrifugal barrier of the $f$ partial waves, whose effect on asymptotic many-body wave functions is negligible. The residual nucleon-nucleon interaction used is generated from EFT Machleidt and Entem (2011), to which A-dependence is added to simulate missing three-body interactions Michel et al. (2019). The Coulomb interaction is added when considering valence protons.

The parameters of the used nuclear interaction and the method used to deal with the infinite-range of the Coulomb Hamiltonian are the same as those used in Ref.Michel et al. (2019). Therefore, we refer the reader to that paper for details.

The obtained energy spectrum of carbon isotones is depicted in Fig. 1. One can see that both energies and widths of experimentally known eigenstates are well reproduced.

The two first states of ¹⁵F have a single-particle character, i.e., the $1/2^{+}$ ground state and $5/2^{+}$ first excited state correspond to the proton $1s_{1/2}$ and $0d_{5/2}$ one-body states, respectively. Hence, as the ¹⁵C single-particle states have been fitted to their experimental data, one could expect their isobaric analog states to be close to experimental data as well.

The experimental energies of the $0^{+}$ and $2^{+}$ states of ¹⁶Ne are well reproduced, even though these states were not fitted. Consequently, theoretical widths are fully predictive. One can see that the calculated width of the ground state of ¹⁶Ne is about 10 keV, which is smaller than the experimental value of 122 keV of Refs.Woodward et al. (1983); KeKelis et al. (1978). However, our value is close to that obtained in the three-body model by Grigorenko et al. Grigorenko et al. (2002), where it is estimated to be in the interval of 0.15-3.1 keV. A width of a few keV at most is also supported by other experimental works where 80 keV is an upper bound Brown et al. (2014). Consequently, we also support a small value for the particle-emission width of ¹⁶Ne, of the order of 10 keV. Similarly, we obtain a small width for the $2^{+}$ first excited state of ¹⁶Ne, of about 40 keV, which is close to the values measured in the experimental studies of Refs.Wamers et al. (2014); Brown et al. (2015) and calculated in the theoretical analyses done in Refs.Fortune (2017); Grigorenko et al. (2002).

One has no definitive experimental data for the ground and excited states of ¹⁷Na ens . Indeed, even though ¹⁷Na has been synthesized, it might well be a combination of three different eigenstates ens . Consequently, we can only compare our results to those arising from other models. The energies and widths of the three first eigenstates of ¹⁷Na have been considered in an empirical potential model in Refs. Fortune and Sherr (2013, 2010). The energies and widths of the $1/2^{+}$ ground state and $5/2^{+}$ excited state of ¹⁷Na agree well with our results. In particular, the $1/2^{+}$ ground state width differs by only about a factor 2, as it is 845 kev in our model and 1.6 MeV in that of Ref.Fortune and Sherr (2010). However, the $3/2^{+}$ state is about 1 MeV higher in our calculations compared to those of Ref.Fortune and Sherr (2010), where it is only 170 keV above the $1/2^{+}$ ground state. Consequently, we also have different widths for the $3/2^{+}$ state, as it is about 300 keV in our model and 20 keV in that of Ref.Fortune and Sherr (2010).

As ¹⁸Mg has not been observed experimentally, only a comparison between our calculations and other theoretical estimates can be done. The binding energy of ¹⁸Mg has been calculated with an empirical potential model in Refs.Fortune and Sherr (2013); Fortune (2016). The obtained value is very close to ours as it differs by about 100 keV. However, those authors did not provide information of the width of ¹⁸Mg, except for a rough estimate of 9 keV. Added to that, the question of the identification of one-proton and two-proton widths has not yet been effected, as we concentrated on total widths for the moment. We will consider the partial emission widths associated to the one-proton and two-proton channels of ¹⁶Ne and ¹⁸Mg in Sec.(III.3).

The Hamiltonians described in Sec.(II.2) allow to describe both carbon isotopes or isotones. As they are mirroring nuclei, it is interesting to compare their energy spectra. Moreover, carbon isotopes of $A=$ 15-18 are well known experimentally ens , so that they can give insight to the structure of the proton-rich carbon isotonic states which cannot be measured experimentally (see also Ref.Jansen et al. (2014) for an ab initio calculation of carbon isotopes).

The GSM calculations of carbon isotopes and isotones of $A=$ 15-18 are illustrated in Fig. 2.

One can see that the spectra of carbon isotopes are well described, except for the $3/2^{+}$ excited state of ¹⁷C, which is too high by 800 keV. Consequently, its mirroring state in ¹⁷Na is presumably too high and too wide in energy and width as well. Conversely, all considered states of the spectra of ¹⁶C and ¹⁸C reproduce experimental data properly, as theoretical and experimental energies differ by a few hundreds of keV at most. Therefore, the calculated excited states of ¹⁶Ne and ¹⁸Mg should be close to experiment.

All the many-body wave functions of carbon isotopes and isotones have an exact isospin quantum number. This is the case because they only have valence neutrons or protons, respectively, i.e. they are isospin aligned (see Ref.Michel et al. (2010) where a similar situation had been encountered in $A=6$ systems). Moreover, one can consider that they are generated by the same Hamiltonian, even though one uses different cores for carbon isotopes and isotones. Indeed, it is equivalent to consider either two different Hamiltonians for carbon isotopes and isotones, which differ only by way of the Coulomb Hamiltonian, or a single Hamiltonian built from the same nuclear and Coulomb interactions as in Sec.(II.2), but defined with a ¹²C core, where nucleons in the $0p_{1/2}$ shells are only subject to the WS potential of Sec.(II.2). Due to the presence or absence of Coulomb Hamiltonian in carbon isotones and isotopes, respectively, energies of mirror states are different. Thus, this shows the presence of partial dynamical symmetry of isospin Leviatan (2011); Leviatan and Van Isacker (2002). Indeed, one has $[\hat{H},\hat{T}^{2}]\neq 0$ because of the presence of the Coulomb Hamiltonian, whereas $[\hat{H},\hat{T}^{2}]\ket{\Psi}=0$ for a many-body wave function $\ket{\Psi}$ of the considered carbon isotopes and isotones, because it is isospin aligned. Consequently, it is necessary to consider other operators instead of $\hat{T}^{2}$ to assess isospin-symmetry breaking. Therefore, we will concentrate in the following on the energy shifts between mirror nuclear states, and also on the Coulomb contribution, as the latter are maximal in carbon isotones and nonexistent in carbon isotopes.

The energy shift associated to isospin-symmetry breaking is the Thomas-Ehrman shift Ehrman (1951); Thomas (1952). It is particularly strong in the presence of many-body nuclear resonances. One can clearly see its effect in Fig. 2. Indeed, in the absence of the Coulomb interaction, spectra would be identical. The Thomas-Ehrman shift is generated by the different asymptotes of the many-body wave functions of the carbon isotopes and isotones. Consequently, it can be expected to be small when the considered states are bound or narrow, and, conversely, to be the largest in the presence of broad resonance states. Isotonic resonance states of smallest widths are the $0^{+}$ and $2^{+}$ states of ¹⁶Ne and ¹⁸Mg. As a consequence, the Thomas-Ehrman shift is very mild for these states, as it is around 150 keV. Conversely, the Thomas-Ehrman shift is the strongest in $A=15$ nuclei and in the highest excited states of $A=16$ nuclei, where widths are typically larger than 500 keV. In this case, the Thomas-Ehrman shift is typically of 500 keV to 1 MeV. Interestingly, the $A=17$ nuclei do not present a very large Thomas-Ehrman shift, even though their widths are of the same order of magnitude. Indeed, the largest energy difference therein is about 200 keV only. One then sees two effects competing in the generation of the Thomas-Ehrman shift. On the one hand, proton-rich nuclear states must be broad resonances for Thomas-Ehrman shifts to be the largest, as the asymptotes of the many-body wave functions of mirror nuclei are very different. On the other hand, continuum coupling strongly influences the Thomas-Ehrman shift, so that it can change in two different pairs of nuclear states, even though the nuclear excitation energies and widths of these two pairs of nuclei are similar (see Ref.Mao et al. (2020) for an analogous study with Li isotopes).

Another observable measuring isospin-symmetry breaking is the Coulomb contribution to the binding energy. Indeed, as shown in Ref.Michel et al. (2019) in the context of proton-rich oxygen isotones, it strongly varies with many-body wave function asymptotes. Consequently, the effect of the one-body and two-body contributions of the Coulomb Hamiltonian in the binding energies of carbon isotones has been considered (see Fig. 3).

For this, one firstly suppresses the two-body Coulomb energy of carbon isotone binding energies, to see the effect of the Coulomb correlations (GSM-2BC in Fig. 3). Then, the one-body part of Coulomb energy is removed from the previous value (GSM-1BC-2BC in Fig. 3). This also provides the expectation value of the nuclear strong interaction Hamiltonian (i.e., excluding the Coulomb interaction). Hence, if the many-body wave functions of carbon isotones and isotopes of the same isospin multiplet had the same configuration mixing, the obtained energies would be exactly that of carbon isotopes where the valence particles are only neutrons without the Coulomb interaction. The energy differences for the $A=15$ carbon mirror nuclei, which are represented by one-nucleon wave functions in the presence of a WS potential, are almost identical. Interestingly, the expectation value of the nuclear part of the Hamiltonian (i.e., excluding the Coulomb interaction) in ¹⁶Ne is slightly smaller in absolute value than that of ¹⁶C, while it is the opposite in $A=17,18$ carbon mirror nuclei (see Fig. 3). Added to that, the nuclear Hamiltonian expectation value increases in absolute value from $A=17$ to $A=18$. This behavior is, in fact, unexpected. Indeed, one would expect energy difference to vary monotonously with proton-emission width, as isospin-symmetry breaking is the largest in this case (see Fig. 3). On the contrary, one sees almost no difference in $A=15$ carbon mirror nuclei, comparable differences of opposite signs in $A=16,17$ cases, and the largest difference in $A=18$ (see Fig. 3).

This study has then shown the subtle effects induced by isospin-symmetry breaking in mirror nuclei of bound and unbound nature. Indeed, considerations based on separation energy and width are not sufficient to assess the Coulomb contribution on proton-rich nuclei. The inter-dependence of nucleon-nucleon correlations and continuum coupling generates complex configuration mixing and thus sometimes counter-intuitive behavior of observables.

The ¹⁶Ne and ¹⁸Mg isotopes are proton-rich unbound nuclei ens . Added to that, both one-proton and two-proton separation energies are negative, so that two different particle-emission channels are open therein. Consequently, it is necessary to calculate separately one-proton and two-proton decay widths in order to delineate the detailed structure of the ground states of ¹⁶Ne and ¹⁸Mg. However, as particle-emission width is obtained from the imaginary part of the eigenenergy in GSM, one has solely access, in principle, to the total emission width.

Nevertheless, due to the special structure of ¹⁶Ne and ¹⁸Mg many-body wave functions, we have been able to indirectly determine both one-proton and two-proton decay widths therein. For this, we use the fact that one has only two open channels, on the one hand, and that their particle-emission thresholds vary differently as a function of the Hamiltonian parameters, on the other hand.

Thus, in order to evaluate one-proton and two-proton decay widths, we changed the central potential depth $V_{0}$ of the WS core potential, in order for the one-proton separation energy to become positive or very negative. The Woods-Saxon central potential depth fitted by experimental data (see Sec.(II.2)) will be denoted as $V_{0}^{\text{(fit)}}$ in this section, as it is used only for comparison with other calculations. The results are shown in Fig. 4.

As ¹⁶Ne and ¹⁸Mg possess one additional valence proton compared to ¹⁵F and ¹⁷Ne, the binding energy of the former nuclei increases faster with central potential depth than that of the latter. Consequently, it is possible to find a central potential depth for which only the two-proton decay channel is open, so that the obtained width is that of two-proton emission.

The increase of width above one-proton emission threshold is of one-proton type only, which can be noticed from its exponential increase as a function of one-proton separation energy. The width of $A=16,18$ carbon mirror nuclei can also be compared to the widths of $A=15,17$ cases in Fig. 4. As $A=15,17$ carbon mirror nuclei are one-proton resonances, their width increases steadily with the Hamiltonian central potential depth. On the contrary, one can see that the widths of $A=16,18$ carbon mirrors increase abruptly when the one-proton channel opens, which points out the different asymptotes of $A=16,18$ ground-state wave functions before and after one-proton emission threshold.

The two-proton decay width is almost constant with respect to the central potential depth below one-proton emission threshold and also about 500 keV to 1 MeV above (see Fig. 4). It is thus reasonable to assume that two-proton decay width is almost independent of energy. Therefore, the value obtained in Fig. 4, where only the two-proton channel is open, can be extrapolated to the physical case, i.e. when $V_{0}=V_{0}^{\text{(fit)}}$ (indicated by an arrow in Fig. 4). This two-proton decay width is about 10-15 keV for both ¹⁶Ne and ¹⁸Mg nuclei. As one only has two emission channels, the one-proton width is the difference between total width and the two-proton emission width of 10-15 keV. One-proton emission is negligible for ¹⁶Ne (see upper panel of Fig. 4). Conversely, that of ¹⁸Mg is then estimated to be about 85-90 keV (see lower panel of of Fig. 4 and Fig. 1). The obtained values are in agreement with current experimental data Woodward et al. (1983); KeKelis et al. (1978); Brown et al. (2014); ens .

As a consequence, despite the fact that GSM can only provide total emission widths, where all partial widths are summed, we have been able to determine both one-proton and two-proton widths of ¹⁶Ne and ¹⁸Mg. For this, one has only assumed that two-proton width can be considered as independent of energy. This is justified from two grounds. On the one hand, two-proton width is almost constant below or close to one-proton emission threshold. On the other hand, total width minus two-proton width, with the latter considered as constant, behaves as a one-proton width for separation energies above one-proton emission threshold.