*2.1. Electron Capture on Nuclei*

At the stellar conditions early in the collapse at which the core composition (described by nuclear statistical equilibrium) is dominated by nuclei from the iron-nickel mass range (*p f*-shell nuclei), electron capture is dominated by Gamow-Teller (GT+) transitions. The subscript refers to the isospin component in the GT operator such that in GT+ transitions a proton is changed into a neutron, in GT<sup>−</sup> transitions, which are relevant for *β*<sup>−</sup> decay of nuclei with neutron excess, a neutron is changed into a proton, and the GT0 strength, important for describing low-energy inelastic neutrino-nucleus scattering, refers to transitions between proton states and neutron states. It is now possible to derive converged low-energy spectra and transitions of *p f*-shell nuclei in the respective model space [2]. In fact, it turned out that, in addition to a constant renormalization of the Gamow–Teller operator [29,30], such highly correlated wave functions are required to describe the strong fragmentation and total value of GT+ strength [31], as experimentally determined by charge-exchange experiments [32,33].

The formalism for the calculation of electron capture rates has been introduced in ref. [34,35]. Note that the strong energy dependence of phase space as well as the fact that the electron Fermi energy is of the same order as the Q-value (the energy difference between initial and final nuclear states) of the abundant nuclei under presupernova conditions makes a detailed and accurate description of the GT+ distribution an important requirement for a reliable description of stellar electron capture during this phase of the collapse. That the diagonalization shell model is up to this task and indeed the method of choice to describe stellar weak-interaction rates during presupernova collapse has been demonstrated by Cole and collaborators [36]. In [36], the capture rates, derived from experimental GT+ data for all *p f*-shell nuclei for which data exist, are compared with rates calculated within the shell model using two different residual interactions. As shown in Figure 1, the agreement is quite satisfactory at the conditions at which these nuclei are abundant and relevant for the core dynamics. A tabulation of shell model capture rates for *p f*-shell nuclei has been made available based on large-scale studies using a variation of the Strasbourg–Madrid KB3 interaction [10]. More recent studies using an improved residual interaction basically confirmed prior calculations. These studies led to slight improvements for selected mid-*p f*shell nuclei; see Figure 1.

**Figure 1.** Comparison of electron captures rates, calculated from experimental GT+ data and distributions, derived from the large-scale shell model calculations with two different interactions (KB3G [37] and GXPF1 [38]) and from a Quasiparticle Random Phase Approximation (QRPA) approach [36]. See text for details. The conditions correspond to the early stage of the collapse where the capture rates are sensitive to details of the GT+ distribution. The shell model rates have been quenched with typical factor of (0.74)2, as derived in [30]. *ρYe* and *T* denote the electron density and temperature, respectively. KVI, UCSL and IUCF stay for the laboratories at which the experiments were performed. Taken from [39] with permission.

Stars in the mass range of 8–12*M* (*M* denotes the Sun mass) received a lot of attention recently as they fill the gap between low-mass stars, which end their lives as white dwarfs and massive stars which, as discussed above, run through the full circle of hydrostatic burning stages ending finally as core-collapse supernovae. The intermediate mass stars are not massive enough to ignite all advanced hydrostatic burning stages and instead degenerate ONe or ONeMg cores. Electron captures are crucial for the final fate of the stars, where the most abundant nuclei, 24Mg and 20Ne, are of key importance together with selected Urca pairs, which reduce the temperature of the core. Shell model rates for *sd*-shell nuclei exist since several years [8]. The important capture rate on 24Mg has been recently updated, mainly due to improved experimental data [11]. The capture rate on 20Ne has also been updated with, however, two remarkable highlights. First, it has been pointed out that the rate at the relevant astrophysical conditions could be decisively altered due to the influence of the second forbidden transition between the 20Ne and 20F ground states [11]. Such a situation is a novum, as the electron capture process is usually dominated by permitted transitions and (first) forbidden transitions are contributed only in high-temperature, high-density environments. The transition was very recently measured in a dedicated experiment [15], and it was indeed confirmed that it increases the capture rate in the astrophysically relevant range by orders of magnitude (Figure 2). The measured transition strength also agrees with the value calculated within the shell model [40]. Secondly, the electron capture rate on 20Ne at the astrophysical conditions, relevant for the core evolution of intermediate-mass stars, is now completely determined experimentally [15,40]. The improved 20Ne electron capture rate has interesting consequences for the final core evolution as the faster electron capture supports the ignition of oxygen burning at slightly smaller densities and off-center. Simulations, exploiting the larger rate, indicate that some intermediate-mass stars might explode as thermonuclear rather than electron capture supernovae [40]. Final conclusions can, however, only been drawn after multidimensional simulations of the core evolution with improved treatments of convection becoming available [41,42]. Other nuclei, for which the electron capture rates are dominated by second forbidden transitions under astrophysical conditions, are 24Na and 27Al [43]. The latter is expected to play a minor role on the evolution of ONeMg cores. The former may trigger convectional instabilities that again require multidimensional modeling.

**Figure 2.** Electron capture (ec) rate for 20Ne as function of density and for a specific temperature (log *T*[K] = 8.6) relevant for the core evolution of intermediate-mass stars. The rate is broken down to the individual state-by-state contributions. In the density regime, particularly relevant for core evolution, the rate is dominated by the second-forbidden ground-state-to-ground-state transition. The rates labeled 'Takahara et al.' are derived from allowed transitions calculated in the shell model [44]. Taken from [11].

In the later stage of the collapse of massive stars, the nuclei present in the core composition become heavier and more neutron-rich. The appropriate model space to describe electron capture for such nuclei is too large (requiring two major shells) to allow for shell model diagonalization calculations. The calculations are then based on the SMMC variant of the shell model [7], which allows the determination of nuclear properties at finite temperatures and in large multi-shell model spaces taking the relevant nuclear correlations into account. Such correlations are particularly important for nuclei with proton number below and neutron number, *N*, above an oscillator shell closure (such as *N* = 40). In such states, GT+ transitions would be completely blocked by the Pauli principle in the Independent Particle Model (IPM) [45] suppressing electron capture on nuclei drastically. However, it has been shown in [17,46] that nuclear correlations induced by the residual interaction move nucleons across the shell gap, enabling GT+ transitions and making electron capture on nuclei the dominating weak interaction process during collapse [17,47]. Let us add two remarks. The unblocking of the GT+ strength across the *N* = 40 shell closure has been experimentally confirmed for 76Se (with 34 protons and 42 neutrons) [48], in agreement with shell model studies [49]. Furthermore, shell model studies certainly show that the description of cross-shell correlations is a rather slowly converging process that requires the consideration of multi-particle multi-hole configurations [49–51].

Recently, it has been pointed out that the *N* = 50 shell closure could serve as a severe obstacle for electron capture on the very neutron-rich nuclei encountered in the later stage of the collapse [52]. This finding was apparently confirmed by measurements of the GT<sup>+</sup> strength in the *N* = 50 nuclei 86Kr and 88Se, which showed basically the vanishing strength for the ground state [53,54]. However, the situation is decisively different at the high temperaures (about 1 MeV) present in the collapsing core when *N* = 50 nuclei are abundantly present. Here, thermal excitations mix orbitals across the shell gap and unblock the GT transitions in this way. This was confirmed in two independent calculations for neutron-rich *N* = 50 nuclei using a thermal Quasiparticle Random Phase Approximation (QRPA) approach [55,56], in agreement with the earlier results obtained within the SMMC studies [17,57].

Based on the diagonalization shell model and the SMMC results and assuming a nuclear statistical equilibrium distribution for the composition, electron capture rates have been tabulated for the range of astrophysical conditions encountered during collapse of massive stars [57]. These rates consider potential screening effects of the astrophysical surroundings. The rate tabulation of Ref. [57] is now incorporated in many of the leading supernova simulation codes. It turns out that the rates have significant impact on collapse simulations. In the presupernova phase (*ρ* < 1010 g/cm3), the captures proceed slower than assumed before, and for a short period during silicon burning, *β*-decays can compete [12,13]. As a consequence, the core is cooler, more massive and less neutron rich before the final collapse. However, for a long time simulations of this final collapse assumed that electron captures on nuclei are prohibited by the Pauli blocking mechanism, as mentioned above (see, e.g., [27]). However, based on the SMMC calculations, it has been shown in [17] that capture on nuclei dominates over capture on free protons. The changes compared to the previous simulations are significant [17–19]. Importantly, the shock is now created at a smaller radius with more infalling material to traverse, but the density, temperature and entropy profiles are also strongly modified [18].

Finally, let us note that the shell model electron capture rates [10,58], which are noticeably slower than the pioneering rates of Fuller et al. (FFN) for *p f*-shell nuclei [34], have important consequences in nucleosynthesis studies for thermonuclear (type Ia) supernovae assuming the single-degenerate scenario as they result in a smaller reduction in the electronto-nucleon ratio being the burning front [16]. As a consequence, very neutron-rich nuclei such as 50Ti and 54Cr are significantly suppressed compared to calculations, which use FFN rates [59]. In fact, in calculations using the shell-model rates, no nuclide is significantly overproduced compared to solar abundances [16].
