IMPACT OF NEW GAMOW–TELLER STRENGTHS ON EXPLOSIVE TYPE IA SUPERNOVA NUCLEOSYNTHESIS

The formation of SNe Ia and their progenitors have been an issue of debate (e.g., Nomoto et al. 1995; Hillebrandt & Niemeyer 2000; Maoz et al. 2014). In typical cases of accretion from a non-degenerate companion star, known as the single-degenerate model, the WD mass approaches the Chandrasekhar mass limit and SNe Ia are induced ("Chandra model"). In double-degenerate models, two WDs merge to produce an SN Ia (Iben & Tutukov 1984; Webbink 1984); in recent violent merger models (e.g., Pakmor et al. 2013; Sato et al. 2016), thermonuclear explosions are induced in sub-Chandrasekhar-mass WDs ("sub-Chandra model"). Important differences between the two models are the central densities (ρ_c) of the exploding WDs. In the Chandra model, ρ_c > 10⁹ g cm⁻³, while ρ_c ≲ 10⁸ g cm⁻³ in the sub-Chandra models (Wang & Han 2012; Nomoto et al. 2013).

In Chandra models, thermonuclear burning triggered in the central region of the WD propagates outward as a subsonic deflagration (flame) front (Nomoto et al. 1976, 1984). Rayleigh–Taylor instabilities at the flame front cause the development of turbulent eddies, which increase the flame surface area, enhancing the net burning rate and accelerating the flame (Müller & Arnett 1982; Arnett & Livne 1994; Khokhlov 1995). In some cases the deflagration may be strong enough to undergo a deflagration to detonation transition (DDT: Blinnikov & Khokhlov 1986; Khokhlov 1991; Iwamoto et al. 1999). The turbulent nature of the flame propagation, including the possible DDT and associated nucleosynthesis have been studied in full 3D simulations (e.g., Gamezo et al. 2005; Röpke et al. 2006; Seitenzahl et al. 2013).

The main observable characteristics of SNe Ia are their optical light curves and spectra. The light curves are powered primarily via the decays of ⁵⁶Ni and its daughter ⁵⁶Co (Arnett 1979). The early spectra are characterized by the presence of strong absorption lines of Si; the intermediate-mass elements such as Ca, S, Mg, and O; and the Fe-peak elements Fe, Ni, and Co (Branch et al. 1985; Parrent et al. 2014). The late-time spectra show emission lines of Fe-peak elements, which include those of stable Ni, i.e., neutron-rich ⁵⁸Ni. It is thus evident that the light curves and spectra are closely related to the nucleosynthesis, which is crucial to study the unresolved issues regarding the explosion models and the progenitors of SNe Ia.

Explosive nucleosynthesis calculations in both the Chandra and sub-Chandra models predict the production of reasonable amounts of Fe-peak elements and intermediate-mass elements (Ca, S, Si, Mg, and O). In the inner parts of the WD, temperatures behind the deflagration or detonation wave exceed ∼5 × 10⁹ K, so that the reactions are rapid enough to synthesize mainly ⁵⁶Ni. In the surrounding parts with lower densities and temperatures, explosive burning produces the intermediate-mass elements Si, S, Ar, and Ca. Both models are successful in producing the basic features of SN Ia light curves and spectra. This implies that it is difficult to distinguish these models.

However, the synthesized amounts of some neutron-rich species, such as ⁵⁸Ni, ⁵⁴Fe, and ⁵⁵Mn relative to ⁵⁶Fe differ between the Chandra and sub-Chandra models because of the difference in the central densities of the WDs (Yamaguchi et al. 2015).

In the Chandra models, the central densities of the WDs are high enough that the Fermi energy of electrons tends to exceed the energy thresholds of the electron captures (ECs) involved. Electron captures reduce the electron mole fraction, Y_e, which is the number of electrons per baryon,

$$\begin{eqnarray}&&{Y}_{e}\equiv \displaystyle \sum _{i}\,{Z}_{i}{Y}_{i},\end{eqnarray} \tag{ 1 }$$

where the sum is over all nuclear species, and Y_i is the abundance for a given species i with Z_i protons. As a result of EC, the Chandra model synthesizes a significant amount of neutron-rich Fe-peak elements. The detailed abundance ratios with respect to ⁵⁶Ni (or ⁵⁶Fe) depend on the convective flame speed and the central densities (e.g., Benvenuto et al. 2015 for rotating WD models), which must be studied in multi-D hydrodynamical simulations.

On the other hand, the sub-Chandra models undergo little EC, so that the amount of stable Ni is much smaller. Such a difference in densities at nuclear burning can be tested with various observations. Specifically, the following observations are sensitive to the central density of the WD, and hence, whether the model is a Chandra or sub-Chandra model.

• 1.  
  The late-time spectra of some SNe Ia show features of stable Ni and Fe (e.g., Maeda et al. 2010; Nomoto et al. 2013).
• 2.  
  X-ray spectra of SN Ia remnants provide abundance ratios such as stable Ni and Mn with respect to Fe (Yamaguchi et al. 2015).
• 3.  
  Solar abundance patterns of Fe-peak isotopes would constrain the ratios of Ni/Fe, Mn/Fe, etc. (e.g., Nomoto et al. 2013), depending on the chemical evolution models of our Galaxy and the produced abundances in indiviual type Ia events (e.g., Nomoto et al. 2013; Seitenzahl et al. 2013).

To constrain the explosion conditions and thus the explosion models, it is important to accurately predict the EC rates relevant to nucleosynthesis in SNe Ia.

Most of the nucleosynthesis studies so far employ EC rates based on shell model estimations when available (Fuller et al. 1982a; Langanke & Martínez-Pinedo 2001), most of which are simpler than what is currently possible. Nuclear physics experiments and improved theoretical descriptions of weak rates (for example, Caurier et al. 1999) have constrained SN Ia nucleosynthesis in attempts to accurately predict the abundance ratios of Fe-peak nuclei. These "second generation" results have greatly improved upon earlier nucleosynthesis predictions (Langanke & Martínez-Pinedo 2003).

The current push in describing SN Ia nucleosynthesis is toward reducing the uncertainty in the yields while also attempting to replicate experimental results, which are now made possible by improved techniques (for example, Honma et al. 2004; Sasano et al. 2011). Work in this direction will provide not only convergence in nuclear models toward an accurate description of weak rates, but also a measure of the sensitivity of SN Ia models to the nuclear physics inputs.

Because of their relevance to SNe Ia, the GT strengths of pf-shell nuclei have undergone significant experimental investigations over the past several years. Various techniques have been employed to extract the GT strengths in the pf-shell nuclei. A compilation of GT measurements of many of the pf-shell nuclei with 45 ≤ Z ≤ 64 was performed by Cole et al. (2012). This study includes studies of β-decay results (Alford et al. 1991; Williams et al. 1995; Popescu et al. 2007; Anantaraman et al. 2008), charge-exchange results using (n, p) reactions (Vetterli et al. 1987, 1989; Anantaraman et al. 2008; Yako et al. 2009), charge-exchange results using (d, ²He) reactions (Alford et al. 1993; Bäumer et al. 2003, 2005; Cole et al. 2006; Hitt et al. 2009), charge-exchange results using (t, ³He) reactions (Hagemann et al. 2004, 2005; Grewe et al. 2008), and charge-exchange results using (p, n) reactions assuming isospin symmetry (Williams et al. 1995; Anantaraman et al. 2008).

In addition to experimental work, there has been theoretical interest in computing GT strengths. In nuclear reaction networks, the commonly used KBF model (Caurier et al. 1999) is often employed to generate EC rates. The KB3G and KBF rates are generally the standards used in nucleosynthesis calculations (Langanke & Martínez-Pinedo 2001) and have been for over a decade. However, more recent models have been proposed, including the GXP-type shell model family (Suzuki et al. 2011). These two models have been found to produce GT strength distributions, which are in disagreement, resulting in potentially different EC rates, and thus possible different nucleosynthetic yields in SNe Ia. Which model is correct is the subject of the aforementioned experimentation.

As an example, recent measurements of the GT transition strength distributions for the ⁵⁶Ni $\to$ ⁵⁶Cu and the ⁵⁵Co $\to$ ⁵⁵Ni transitions (Sasano et al. 2011, 2012) are helpful in constraining nuclear shell models. Using the (p, n) charge-exchange reaction, the strengths were measured with a good degree of accuracy over a very wide range of excitation energies. Here, isospin symmetry-breaking effects are neglected, and the experimentally extracted GT strengths for ⁵⁶Ni $\to$ ⁵⁶Cu can be applied to EC and β⁺ decays of ⁵⁶Ni (Sasano et al. 2011). It was found that the transition strengths for these two transitions more closely match those computed with the GXPF1J shell-model interaction (Honma et al. 2004; Suzuki et al. 2011) than those computed with the KB3G or the KBF models (Caurier et al. 1999; Poves et al. 2001).

These experimental and theoretical results suggest that actual EC rates may differ from those predicted by the KBF-family shell model calculations. A reduction of the EC rates may be responsible for an overall enhancement in Y_e and a concurrent change in production of ⁵⁶Ni in SNe Ia. The effect may be magnified if all nuclei in the pf-shell region are considered. Here, strength calculations result in EC rates that are lower for many pf-shell nuclei (for ρY_e = 10⁷ g cm⁻³; Suzuki et al. 2011). However, depending on the temperature and ρY_e of the medium, rates may increase or decrease as different parts of the GT strength function are integrated over. While decreases in EC rates may change the production of ⁵⁶Ni in SNe Ia, increased rates cannot be ruled out from the data. This was verified by Cole et al. (2012), who found no systematic shift in rates if one shell model is used over another. Depending on the shell model, strengths may be higher or lower for different thermal conditions. This concept will be discussed later in this paper.

The effects of the GXP-type shell model on proton-rich pf-shell nuclei with 23 ≤ Z ≤ 30 are studied because they influence nucleosynthesis in SNe Ia. Here, we examine both stable and unstable nuclei in this region. In particular, the effect on Y_e as well as production ratios are evaluated. Mass fractions of nuclei produced in SNe Ia are compared using both GXP parametrizations and KBF models. Trajectories of mass shells in a WD are used as input into a nuclear reaction network to gauge the effects of variations in nuclear physics inputs, and the final nuclear mass fractions in individual shells are computed. Because of computational limitations, the explosion calculation is decoupled from the nuclear reaction network. However, the effects of the nuclear shell model used are evident in the resultant electron fractions and the final mass fractions. Comparisons to solar values indicate that the enhancement in Y_e, which arises from using the GXP-type model, reduces the overall ⁵⁸Ni/⁵⁶Ni and ⁵⁸Ni/⁵⁶Fe ratios, which has been an interesting problem addressed by prior evaluations (Iwamoto et al. 1999; Brachwitz et al. 2000).

A brief overview of the GXP shell model calculation is presented in Section 2 along with a comparison with KBF rates. Next, the nuclear reaction network calculations and the insertion of the GXP rates are presented. Results, including produced mass fractions are compared in Section 4; these results, using the GXP shell model, are compared to results using the KBF rates as well as solar mass fractions. Finally, conclusions, a discussion, and future prospects are presented.

As mentioned in Section 1, deflagration models, such as the W7 model (Nomoto et al. 1984; Thielemann et al. 1986), attempt to simulate the effects of increased nuclear burning because of an increase in the surface area of the flame front as it propagates to the stellar surface. The flame (deflagration) speed is prescribed by the mixing-length theory. It accelerates to 0.08c_s in t = 0.6 s and to 0.3c_s in t = 1.18 s, where c_s is the local sound speed (Nomoto et al. 1984).

The deflagration model W7 was explored using networks I and II in Table 1. Here, the thermodynamic trajectories of Nomoto et al. (1984) have been used as inputs to the nuclear reaction network. The evolution of the electron fraction Y_e is shown in Figure 5 for several trajectories in both models. The values of Y_e at the end of the nucleosynthesis calculations (t ≈ 3.5 s) are also shown in Table 2 for the same trajectories. In this figure, each line represents the evolution of a mass element for the W7 model assuming a specific shell model assumption. For all models, the electron fraction is lower near the core. It can be seen that the electron fraction changes little between the GXP and the KBF models indicating that, though the GXP shell model results in a significantly different B(GT), the net effects on the nucleosynthesis are potentially insignificant.

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Figure 6 shows the final mass fraction ratios for both calculations:

$$\begin{eqnarray}&&R\equiv \displaystyle \frac{X{(Z,N)}_{\mathrm{GXP}}}{X{(Z,N)}_{\mathrm{KBF}}}.\end{eqnarray} \tag{ 2 }$$

Ratios are shown for the central trajectory (at a mass–radius M_r = 0) and an off-center trajectory (at M_r = 0.1232 M_⊙). It can be seen that the GXP model results in a more proton-rich final abundance distribution, but the effect is small. While this is not surprising because the EC rates are lower in the pf-shell region for the GXP model, the overall increase in Y_e (as seen in Figure 5) is small.

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It is also noted that the final abundances for the GXP model are more proton-rich for nuclei outside of the region, where the rates differ. Of course, a reduction in the EC rates results in a global increase in Y_e, which will affect all EC rates. Thus, while the overall electron fraction may change only slightly, the final abundance distriubution for a particular species may change by a larger amount.

Using the final abundances for each trajectory in the W7 model, the final mass fraction profile has been determined. These profiles are shown in Figure 7 for the GXP model evaluated in this work. These mass fractions closely resemble those of prior work (Brachwitz et al. 2000).

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From the final abundance ratios in each trajectory, a total abundance weighted by the mass of each trajectory is computed for the isotopes in the network—the overproduction factor. This double ratio, explicitly defined as

$$\begin{eqnarray}&&\mathrm{DR}\equiv \displaystyle \frac{{Y}_{i}/{Y}_{\mathrm{Fe}}}{{Y}_{i,\odot }/{Y}_{\mathrm{Fe},\odot }}=\displaystyle \frac{{Y}_{i}/{Y}_{i,\odot }}{{Y}_{\mathrm{Fe}}/{Y}_{\mathrm{Fe},\odot }},\end{eqnarray} \tag{ 3 }$$

 provides a measure of the uniformity of the nucleosynthesis compared to solar observations. These double ratios are shown in Figure 8 for the GXP model. The ratios for the KBF model are nearly identical and comparable to the results of Brachwitz et al. (2000). Notable are the overabundances of ⁵⁸Ni compared to the production of lighter Z nuclei.

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The nuclei ⁵⁴Mn and ⁵⁴Cr can also be addressed in Figure 8. Although a small fraction of ⁵⁴Cr is created from ECs on ⁵⁴Mn, ⁵⁴Cr is produced in significant abundance in the inner 10⁻³ M_⊙ of the model, where the electron fraction Y_e < 0.46. Thus, while it is not seen in Figure 7 because the scaling of this figure is not fine enough, its production is still evident in Figure 8. It is not surprising that ⁵⁴Cr is produced in greatest abundance in the center mass shell as the Y_e of this shell reaches a value very close to that of ⁵⁴Cr (Y_e ∼ Z/A = 0.44) as seen from Figure 5 and Table 2. From a nuclear statistical equilibrium (NSE) argument, one would expect this to be the mass shell with the largest final abundance of ⁵⁴Cr. This is also the mass shell with the lowest final Y_e. It is possible that an overall increase in Y_e of the entire model by a small amount could result in dramatic changes in the ⁵⁴Cr production. This is because all of the mass shells above the central shell would have Y_e, which is even farther away from that of ⁵⁴Cr. If the Y_e of the central shell increases to above 0.46 (a shift of less than 1%), then the overall ⁵⁴Cr production would start to decrease. However, production of nuclei with a higher Y_e, such as ⁵⁶Ni, would not be expected to decrease significantly because it is produced in a large range of mass shells with a range of electron fractions above and below that of ⁵⁶Ni (Y_e ≈ Z/A = 0.5). In the case of ⁵⁶Ni, while a single shell closer to the surface may produce less ⁵⁶Ni, shells below it would have an increased production, thus compensating for the loss. The inner mass shell has no shells below it, and any loss suffered by an overall increase in Y_e could not be compensated for. Even more, a global shift of Y_e for all shells may also increase the overall production of α-cluster elements, such as ²⁸Si.

For this reason, ⁵⁴Cr could be a good indicator of the effectiveness of the nucleosynthesis in an SN Ia because it is produced predominately in a small region of the star. Small global shifts, which tend to be averaged into multiple regions of the star, have more significant effects for the central region. Because of this, it is worthwhile to explore nucleosynthesis and thermodynamic effects such as turbulence and mixing within the SN Ia. In this case, 3D modeling—or 1D and 2D approximations—would become important.

To compare the GXP and KBF models, we take a ratio of the abundances produced in each model weighted by the individual mass shells in the explosion. This ratio is shown in Figure 9. A small reduction of ≲5% is noted for the pf-shell nuclei with a slight increase in only a few cases. The production generally decreases in the GXP model for higher neutron number, consistent with a slight shift to higher Y_e in the GXP model.

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Despite the seemingly significant differences in B(GT) between the GXP and KBF models, why are the differences in the nucleosynthesis so small (and likely unobservable)? The answer lies in the intersection of nuclear physics and the hydrodynamics of the deflagration model.

The total EC rate for a transition from a parent state to a daughter state is

$$\begin{eqnarray}\lambda & = & \mathrm{ln}2\displaystyle \frac{{({g}_{A}/{g}_{V})}^{2}}{6143}\displaystyle \sum _{i}\displaystyle \frac{(2{J}_{i}+1){e}^{-{E}_{i}/{kT}}}{G(Z,A,T)}\\ & & \times \displaystyle \sum _{j}{B}_{{ij}}({GT}){f}_{{ij}}(T,\rho ,\mu )\end{eqnarray} \tag{ 4 }$$

where the sum is over transitions from parent to daughter states. The phase space factor, f_ij, accounts for the energetic feasibility of the reaction, including the population of electrons at a particular energy (Langanke & Martínez-Pinedo 2000):

$$\begin{eqnarray}{f}_{{ij}} & = & {\displaystyle \int }_{{\omega }_{l}}^{\infty }\omega p{({Q}_{{ij}}+\omega )}^{2}F(Z,\omega ){S}_{e}(T)\\ & & \times (1-{S}_{\nu }({Q}_{{ij}}+\omega ))d\omega \end{eqnarray} \tag{ 5 }$$

where Q_ij is the EC transition energy divided by the electron mass, determined from the nuclear masses, and transitions are summed in Equation (4) from initial states E_i to final states E_j. The distribution function is S_e,ν(ω); here, S_e is the usual Fermi–Dirac distribution, and S_ν = 0. The normalized total electron energy (kinetic plus rest mass) is ω = E_e/m_ec². Coulomb barrier penetration effects are contained within the function F(Z, ω). The integration limits are set by the reaction threshold ω_l, which is the threshold value for which EC is energetically feasible; ω_l = 1 if Q_ij > −1 and ${\omega }_{l}=| {Q}_{{ij}}|$ for Q_ij < −1. For Q_ij < −1, the electron must have enough kinetic energy to make the reaction possible, and ω_l > 1. An electron with zero kinetic energy will not exceed the reaction threshold.

We consider the case of ECs on ⁵⁴Fe. The ground-state to ground-state Q-value for ⁵⁴Fe EC (using nuclear masses) is −1.21 MeV, meaning that to integrate over the strength function shown in Figure 1, the electron must have a total energy of at least 1.21 MeV. This raises the integration threshold in Equation (5). In order for the Fermi distribution S_e(T) to have a population with the electron energy above the threshold ω > ω_l, either the density must be high enough to push the Fermi energy above the threshold, or the temperature must be hot enough so that the smearing of the Fermi surface creates a population of electrons above threshold, or both.

However, during the nucleosynthesis, the time at which the production of ⁵⁴Fe is reached does not occur until about 0.5 s in the mass shell, which has the largest mass fraction of Fe by the end of the nucleosynthesis, corresponding to a mass of 5.612 × 10⁻² M_⊙. By this time, the temperature has dropped significantly in the W7 model, and the Fermi energy is less than 1.5 MeV. The spread in the Fermi surface is roughly 1 MeV, so nearly all electrons have energies less than 2 MeV. This means that the sum over B(GT) in Figure 1 is only at excitation energies less than 2 MeV. One sees from Figure 1 that the differences between the KBF and GXP strength functions are small in this region. This is indicated in Figure 10, which shows the integrated strength functions versus the excitation energy of the daughter nucleus. For ECs on ⁵⁴Fe, only excitation energies less than 2 MeV are important, in a region where the KBF and GXP strengths are nearly equal.

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The case is similar for ECs on ⁵⁶Ni. Although the Q-value is positive for this reaction (1.622 MeV), the temperature and density by the time ⁵⁶Ni is reached in the nucleosynthesis are roughly the same as for ⁵⁴Fe. While a small population of electron energies exceeds 3 MeV, where there is some deviation between the GXP and KBF models, most of the electron population exists at energies less than 1 MeV, corresponding to excitation energies less than 2.6 MeV, where the EC rates are low and the difference between the GXP and KBF strength functions is negligible. This can be seen by examining the integrated strength functions in Figure 10 in which deviations between the GXP and KBF models do not occur until excitation energies greater than about 3 MeV for ⁵⁶Ni with less deviation for ⁵⁴Fe except at very high excitation energies.

This is shown schematically in Figure 11. In this figure, the GT strengths are shown for ⁵⁶Ni and ⁵⁴Fe for the KBF and GXPF1J shell models. Electron Fermi distributions for the mass shells that produce the largest amount of ⁵⁶Ni and ⁵⁴Fe are overlayed onto these figures for various times along the mass shell evolution. The GT strengths have been offset by the EC Q-value (ground state) in each case to provide an accurate picture of the actual electron energy necessary to reach a specific excited state in the daughter nucleus.

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It is seen from this figure that while the mass shell starts off hot enough to create a population of electrons energetic enough to populate parts of the GT strength where the differences are significant between the two shell models, one must keep in mind that the nucleosynthesis path for each mass shell does not reach ⁵⁶Ni and ⁵⁴Fe until later in the shell evolution, in both cases, about 0.5 s after the explosion starts. By this time, the mass shell has cooled to the point at which nearly the entire Fermi distribution lies in the energy region, where the difference between the GXP and KBF strengths is insignificant as shown in the figure. Thus, the difference in EC rates for these two nuclei is small for these explosion models.

From Figures 8 and 9, it can be seen that the difference between the abundances determined using the GXP model and the KBF model are very small—on the order of a few percent. Furthermore, it can be seen that the abundance differences only exist at Z > 19. By the time these nuclei are produced, the temperature of the environment is low and the nuclear abundances match those very close to an environment in NSE. Any heating from nucleosynthesis for Z > 19 is not expected to differ significantly from that of the model of Iwamoto et al. (1999). Thus, the decoupling of the reaction network from the explosion trajectories is not expected to produce significant uncertainties.

Nucleosynthesis calculations were also carried out for the delayed-detonation model (Iwamoto et al. 1999) using models III and IV in Table 1. In this model, the explosion transitions from a deflagration near its center to a detonation at low density (Khokhlov 1991). This particular model is parametrized by the transition density with the transition density parameter set to match observed light curves and nucleosynthesis.

In the delayed detonation, a slow deflagration phase is calculated by Nomoto et al. (1984) with an assumed constant flame speed of 0.015c_s. This is a typical laminar deflagration speed without convection and flame instabilities. The subsequent detonation (shock) phase is calculated by Iwamoto et al. (1999). It is assumed that DDT happens when the density at the flame front (upstream density) decreases to 2.2 × 10⁷ g cm⁻³.

Detailed nucleosynthesis calculations of delayed-detonation models find that the problem of overproduction of neutron-rich isotopes may be remedied by constraining the density at which DDT occurs to ∼10⁷ g cm⁻³ (Iwamoto et al. 1999; Brachwitz et al. 2000), which agrees with more physically based estimates (Bychkov & Liberman 1995; Niemeyer & Woosley 1997; Woosley 2007). This is consistent with the results of multi-dimensional simulations of delayed-detonation models (Seitenzahl et al. 2013).

As with the deflagration models (I and II), calculations were done assuming both a GXP and KBF shell model. The resulting mass fractions, using the WDD2 hydrodynamic trajectories, are shown in Figure 12. These mass fractions compare to those found using the W7 model of Iwamoto et al. (1999).

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Nuclear overproduction factors for nucleosynthesis using the GXP model (model IV) are compared to solar abundances in Figure 13. Compared to the W7 model (model II), the production of Cr, Mn, Fe, Ni, Cu, and Zn isotopes seems to match the solar distributions more closely across the isotopic chains studied, though the underproduction of Cu and Zn still exists.

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As with the deflagration model, the GXP calculation is compared to a calculation done using KBF rates for the delayed-detonation model. The final abundance ratios are shown in Figure 14. The ratios are slightly closer to unity in this model than in the deflagration model, while the shapes are similar. This is an indicator that the delayed-detonation model is not as sensitive to rate changes as the deflagration model. The explanation for these small differences between the GXP and KBF rates is the same as that given in the comparison of the deflagration model. That is, transitions in SNe Ia are only to low excitation energies as the Fermi energy of the electrons is ∼1–2 MeV. In this region, the differences between the GT strength functions are small.

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A comparison of the total produced Fe and Ni mass is shown in Table 3 for all four models studied. One sees a slight shift to more ⁵⁶Fe production using the GXP shell model, corresponding to a very slight increase in Y_e as indicated in Table 2.

Given the temperatures of the environments of SNe Ia, it is conceivable that nuclei in their excited states may result in reaction rates that vary from the ground-state rates. For this reason, it is worth mentioning the contribution to the EC rates for a few cases of nuclei in their excited states.

In Figure 1, the GT strength functions are shown in the GXP shell model for the first excited states of ⁵⁶Ni and ⁵⁴Fe, and the integrated strengths are shown in Figure 10. In both cases, the first excited state of the parent nucleus is the 2⁺ state. In each case, the strength function shifts to higher excitation energy. The shift of the strength from the excited 2⁺ states can be understood as a manifestation of the Brink hypothesis, which is primarily applicable to giant dipole resonances and can be applied to GT strength distributions. This hypothesis states that the giant dipole resonances (as well as the GT strengths) reside at the same energies relative to excited states as in the ground state (Axel 1962; Misch et al. 2014; Guttormsen et al. 2016). While not exact, it can be used to understand the shift in the GT strengths in Figure 1. Details concerning the applicability of the Brink hypothesis have been discussed in Langanke & Martínez-Pinedo (2000). Detailed studies of the applicability of the Brink hypothesis for M1 strength functions are presented by Loens et al. (2012), Dzhioev et al. (2014), and Dzhioev et al. (2010). Thus, the shift of the strength function is commensurate with the excitation energy of the parent nucleus. In the case of SNe Ia nucleosynthesis, this shift would likely not change the resultant nucleosynthesis. This is because the excited states of the parent nuclei, the 2⁺ state in both ⁵⁶Ni and ⁵⁴Fe, are at 2.7 MeV and 1.4 MeV respectively. At the temperatures of the SN Ia environment, these states are not expected to have a significant population, and can be dismissed as possible contributors to the strength distributions.