Gross Theory Model for Neutrino-Nucleus Cross Section

The nuclear gross theory, originally formulated by Takahashi and Yamada for the $\beta$-decays, is applied for the electronic-neutrino nucleus reactions, employing a more realistic description to the energetics of the Gamow-Teller resonances. The model parameters are gauged from the most recent experimental data, both for $\beta^-$ decay and electron-capture, separately for even-even, even-odd, odd-odd, odd-even nuclei. The numerical estimates for neutrino-nucleus cross sections agree fairly well with previous evaluations done within the framework of microscopic models. The formalism presented here can be extended to the heavy nuclei mass region, where weak processes are quite relevant, which is of astrophysical interest because of its applications in supernova explosive nucleosynthesis.


Introduction
The nucleosynthesis of heavy elements is only understood if stellar reactions take place in regions of nuclear chart far away from the β-stability line, involving a large number of unstable or even exotic nuclear species for which the experimental data are very scarce. For instance, the steps of nucleosynthesis in the r-process occurs out to and just along the neutron drip line where many of principal nuclear properties are still unknown. Great theoretical and experimental efforts have been invested in the last decades in order to describe the nuclear properties of different species along the β-stability line, as well as those of exotic nuclei involved in explosive nucleosynthesis processes [1,2,3].
The theoretical models can be separated generically into: i) the macroscopic models which describe the global nuclear properties [4,5,6,7,8,9,10], and where special attention is paid to the gross theory of the β-decay (GTBD); and ii) the microscopic formalisms i.e., the shell model or RPA based calculations [10,11,12,13] where detailed nuclear structure of each species is considered.
The GTBD was first proposed by Takahashi and Yamada [4] nearly forty years ago to describe the global properties of allowed β-decay processes. It is essentially a parametric model, which attempted to combine the single-particle and statistical arguments in a phenomenological way. Afterwards, different versions of the 'gross theory' have been developed and used for practical applications very frequently [5,6,7,8,9]. This is due to: i) their simplicity when compared with the hard computational work involved in the implementation of the microscopic models, and ii) their capability to reproduce the available experimental data, and to be extrapolated later on to unknown nuclei far away from the β-stability line. In fact, as these theoretical approaches account systematically and fairly well for the properties of stable nuclei, they have been extensively applied to describe: 1) the β-decay half-lives and other nuclear observables participating in the r-process, and 2) the properties of a great number of exotic nuclei that are involved in the nucleosynthesis.
It also should be mentioned that the gross-theory approach has been also used by N. Itoh et al., in Refs. [14] for the calculation of the total capture of a neutrino by 37 Cl, 16 O, 20 Ne and 56 Fe nuclei, which are used in the detection of solar neutrinos.
The aim of the present work is twofold. First, motivated by the simplicity of the original GTBD, we use it to evaluate the half-lives of allowed weak-transitions (β-decay and electron-capture) in nuclei with A < 70, which are of major importance for the presupernova collapse processes. We also analyze the consequences of employing a more realistic estimate for the energetic of the Gamow-Teller resonance (GTR) than in the previous works. This will lead us to a new trend for the adjustable parameter related to the energy spread of the GTR caused by the spin dependent part of the nuclear force. Second, we use the same gross-theory approach to describe the nuclear neutrino capture over a great number of nuclei involved in presupernova structure with the purpose to extend in the future the calculation to r-process in neutrino rich environment. Since within the stellar conditions no experimental data exist, our results are confronted with those achieved in the framework of microscopic approaches.
The paper is organized as follows. In Sect. 2 we briefly sketch the conventional gross-theory for nuclear β-decay and electron capture rate. In Sect. 3 we introduce the gross theory for the evaluation of the neutrino-nucleus reaction cross-section. The single-particle strength functions are discussed in Sects. 4 and 5, together with the estimate of the GTR energy and the procedure used to derive the corresponding spread of the transition strength. In Sect. 6 we exhibit and discuss our results. Summarizing conclusions and future extension of the present work are drawn in Sect. 7.

Gross theory of nuclear beta decay (GTBD)
The GTBD permits to evaluate the half-lives of β ± -decay and the rates for electron capture weak processes. As an example, we briefly sketch here the original GTBD [4] for the decay (Z, A) → (Z + 1, A) + e − +ν. The total rate for allowed transitions is written (in natural units) as where G = (3.034545 ± 0.00006) × 10 −12 is the Fermi weak coupling constant, g V = 1 and g A = −1 are, respectively, the vector and axial-vector coupling constants ‡. The argument of the matrix element (E) is the transition energy measured from the parent ground state. Note that the true β-decay transition energy is The usual integrated dimensionless Fermi function [15,16], f (E), is evaluated from the approximated formulas given in Ref. [4] that are correct up to ∼ 10% for standard decays. The Q β -value is the difference between neutral atomic masses of parent and daughter nuclei: with B(A, Z) and B(A, Z + 1) being the corresponding nuclear binding energies, and m(nH) = m n − m( 1 H) = m n − m p − m e = 0.782 MeV. The masses were obtained in the same way as in Ref. [7]. This means that, when available, they are taken from Wapstra-Audi-Hoekstra mass table [17] and, otherwise, they are determined from Tachibana-Uno-Yamada semi-empirical mass formula [18]. The squares of the Fermi (F) and Gamow-Teller (GT) matrix elements are determined as: Here, ǫ min is the lowest single-particle energy of the parent nucleus and ǫ max is the energy of the highest occupied state. The one-particle level density (proton or neutron), dn 1 /dǫ, is determined by Fermi gas model for the parent nucleus, and the weight function ‡ Finite nuclear size effects are incorporated via the dipole form factor g → g where k is the momentum transfer and Λ = 850 MeV the cutoff energy.
W (E, ǫ), constrained by 0 ≤ W (E, ǫ) ≤ 1, takes into account the Pauli blocking. Finally, D X (E, ǫ), normalized as +∞ −∞ D X (E, ǫ)dE = 1, is the probability that a nucleon with single-particle energy ǫ makes a β-transition. As in Ref. [4] we neglect the ǫ-dependence, i.e., it is assumed that all nucleons have the same decay probability, independently of their energies ǫ, D X (E, ǫ) ≡ D X (E). The GTBD characterizes this D X (E) through their energy weight moments (for example, in [14] these expressions were written explicitly).
The dependence on the odd-even proton and neutron numbers in the daughter nucleus is introduced through the values for the pairing gap ∆ and the single-particle level spacing d. In the present work we adopted those from Ref. [5]. More details on the probability function D X (E) are given in Sect. 4.
The original GTBD [4] has been gradually improved [6,7], and nowadays we have two new versions: the first is named the 2nd generation gross theory (GT2), and the second is the so called semi-gross theory (SGT) in which some parts of nuclear shell effects are considered. The most recent GT2 and SGT approaches use an updated mass formula, and they better account for the shell and pairing effects [7,9].

Gross theory of nuclear neutrino capture (GTNC)
In the most recent versions of r-processes nucleoshynthesis in supernova, one considers that these processes take place on the surface of a protoneutron star during the supernova collapse. The nuclei are exposed there to a thermal flux Φ ν (E ν ) of ν e with energy E ν , which causes the reaction ν e + (Z, A) → (Z + 1, A) + e − , with cross-section [12,10,19] where E th is the reaction energy threshold, which is equal to the Q β -value for stable nuclei and zero for unstable cases. For Φ ν (E ν ) we take a zero-chemical potential Fermi-Dirac distribution where T ν is the neutrino temperature, and N is the normalization constant of the spectrum [12]. The evaluation of the ν e -nucleus cross-section σ ν (E ν ), in a neutrino-rich environment, must be consistent with the procedure employed in calculating the βdecay rates. The allowed transition approximation (see [19,Eq.(2.19) can be applied for relatively small momentum transfer. The integration covers all possible nuclear states allowed by the selection rules, and the integration limits are determined from the energy conservation condition. When the energies are measured from the ground state of the parent nucleus (Z, A), this condition reads where E = E ν − E e > 0 is the excitation energy of daughter nucleus (Z + 1, A), and F (Z, E) is the usual scattering Fermi function which takes into account the Coulomb interaction between the electron and the nucleus.

Single-particle strength functions
A key element in the gross theory is the single-particle strength probability function D X (E). The successive improvements of the theory have used gaussian-, exponential-, and lorentzian-type functions [4,7]. The sec-hyperbolic functions have been employed in the GT2 [7]. Here we will mainly adopt the gaussian-like behavior for the transition strengths. To illustrate that the calculations are rather independent of the functional form adopted for D X (E), a comparison will be done between the results obtained with the gaussian-like distribution and those calculated with the lorentzian-type strength function Here E X is the resonance energy, σ X is the standard deviation, and the other quantities are defined as in Ref. [4]. When isospin is a good quantum number the total Fermi strength |M F (E)| dE = N − Z is carried entirely by the isobaric analog state (IAS) in the daughter nucleus. However, because of the Coulomb force, the isospin is not a good quantum number and this leads to the energy splitting of the Fermi resonance. We will use the estimates introduced by Takahashi and Yamada [4], namely for β ± decay, The total GT strength in the (ν e , e − ) channel is given by the Ikeda sum rule , but its distribution cannot be established by general arguments, and therefore must be either calculated or measured. Charge-exchange reactions (p, n) have demonstrated that most of the strength is accumulated in a broad resonance near the IAS [20]. In fact, even before these measurements have been performed, Takahashi and Yamada [4] have used the approximation while σ GT is expressed as with σ N being the energy spread caused by the spin dependent nuclear forces.
For the Fermi transitions we use the relation (10). Yet, for the GT resonance, instead of employing the approximation (11), we use the estimate obtained by Nakayama et al. [21] from the analytic fit of the (p, n) reaction data of nuclei near stability line [20], where δ is positive. For the standard deviation σ GT we preserve the expression (12), and σ N is treated as an adjustable parameter. Note that the two terms of δ in (13) have well defined physical interpretations. The first one is due to the SU(4) symmetry breaking imposed by the spin-orbit coupling, and it is of the same order of magnitude as the Bohr-Mottelson estimate for the spin-orbit splitting (∆ ls ∼ = 20A −1/3 MeV), obtained from the approximation l ∼ = A 1/3 [22]. The second term is responsible for the partial restoration of the SU(4) symmetry, having the same mass and charge dependence as the difference between the energy shifts produced by the GT and Fermi residual interactions. We remark that the Eq. (13) is frequently used in the study of r-process in neutron rich nuclei [10,23,24,25,26,27,28,29,30,31,32]. There δ < 0, and therefore the GTR falls below the IAS, as happens in the shell-model calculation [10]. S

Fitting Procedure
Another important aspect in implementing the GTBD is the choice of the χ 2minimization method that is used to derive the width parameter σ N . In the original work of Takahashi and Yamada [4] is minimized the quantity where N 0 is the number of experimental β-decay half-lives, τ exp 1/2 , fulfilling the conditions: 1) the branching ratio of the allowed transitions exceeds ∼ 50% of the total β-decay branching ratio, and 2) the ground state Q-value is ≥ 10A −1/3 MeV.
In the present work σ N is determined through the minimization of the function where ∆ log(τ exp and δτ exp 1/2 is the experimental error. Thus, the χ 2 B -function reinforces the contributions of data with small experimental errors. Moreover, we perform different fittings for eveneven, odd-odd, odd-even and even-odd nuclei. Needless to say that for τ exp 1/2 we use here the most recent data [33], instead of those that were available when the GTBD has been formulated [4]. The condition log f t ≤ 6 is imposed to include only the allowed β-decays.
S Occasionally is used the fit [7] which also reproduces satisfactorily the stable nuclei. The second term of δ ′ is interpreted in the same way as that of δ in (13), but the first term here does not have any direct physical significance.

Numerical results and discussion
6.1. β − decay and electron-capture half-lives For the single-particle strength probability function D X (E) we adopt the gaussian-like behavior (8) in most of the calculations. The corresponding values of the adjustable parameters at the minimal value of the χ 2 -function, χ 2 min , are listed in Table I for the four different parity families of nuclei. They are labeled as σ * N and σ N , when for E GT are used, respectively, the Eqs. (11) and (13). One sees that σ N is always larger than σ * N , which means that the effect of using more realistic energies E GT is reflected in the increase of the standard deviations. The values of σ * N derived in Ref. [4] are exhibited parenthetically in Table I. It is important to point out that the difference between the old and new values for σ * N does not comes from the fitting procedure itself, but from the different samples of nuclei employed here for each parity family. Figure 1 shows the dependence of χ 2 /χ 2 min on both: i) the energy of the GTR (left panels for (11), and right panels for (13)), and ii) the type of the minimization function (upper panels for (14), and lower panels for (15)). We note that the χ 2 B -functions present rather pronounced minima when compared with those of the χ 2 A -functions. More, in most of the cases the χ 2 B minima are located at smaller values of the standard deviations than the χ 2 A ones. This is a direct consequence of including the experimental errors in the minimization procedure of the χ 2 B -function. In order to estimate the average deviation of our results, we have computed the mismatch factor η defined as [4] showing their values for each σ N in Table 1, and similarly the values of η * corresponding to each σ * N . It can be observed that the χ 2 B minimization procedure considerably reduces the mismatch factor, in particular for odd-odd family of nuclei. Thus, we can say that the use of χ 2 B -function modifies σ N and leads to a better statistical agreement between the theoretical results and the experimental data. Figure 2 compares the experimental β − -decay half-lives within the Mn isotopic chain with our results obtained for the σ N values from Table 1. One can see that the GTBD overestimates the data. However, it should be pointed out that this disagreement is not characteristic of the GTBD, since other microscopic and global models lead to similar results. For instance, this is the case of: a) the extended Thomas-Fermi plus Strutinsky integral method combined with the continuum quasiparticle random phase approximation (ETFSI+CQRPA) [12], and b) the extended Thomas Fermi method combined with the semi-gross theory (ETFSI+GT2) [7]. Figure 3 shows the distribution of log(τ cal 1/2 /τ exp 1/2 ), as a function of Q β A −1/3 , for β −decay. We observe that the results obtained with Eqs. (11) and (13) are quite similar to each other for same parity families, the first one being somewhat larger. We can also see that for the odd-odd family a very good agreement between theoretical and experimental results is obtained for Q β A 1/3 ≥ 45 MeV, while for the other three families this happens already for Q β A 1/3 ≥ 40 MeV. Thus, as frequently mentioned in the literature [4,7,9], the best GTBD results are obtained for heavy nuclei.
In the evaluation of the allowed electron-capture and β + -decay rates for nuclei of A < 70 we have re-adjusted the parameter σ N , imposing again the constraint log f t < 6. The resulting values of σ N and η for the two χ 2 -function, with E GT calculated from Eq. (13), are presented in Table 2. Figure 4 shows the values of log(τ calc 1/2 /τ exp 1/2 ) as a function of Q β A 1/3 for the electron-capture rates calculated with the underlined σ N values listed in Table 2. Similar general features to those remarked in the β − -decay case are obtained.
Also, we briefly discuss the dependence of the χ 2 procedure on the functional form of the employed strength distribution. Thus, we repeat the calculations for β − -decay and electron-capture rates using now the lorentzian distribution D X , given by the Eq. (9), together with the Eq. (13) for the GT energy. The resulting Γ N energies are shown in Table 3, and the corresponding log(τ calc 1/2 /τ exp 1/2 ) values for the β − emitter nuclei with A < 70 exhibit similar Q β A 1/3 dependence to that shown in Figure 3. Figure 5 shows the results for the electron-capture rates along the Ni isotopic chain. The calculations with the gaussian and lorentzian strength functions turn out to be quite similar to each other and both show a reasonable agreement with the data.

Neutrino-nucleus cross section
The reduced thermal cross section σ ν /A of the four β − emitter families was evaluated for the A < 70 nuclei with two sets of parameters, σ * N and σ N . The results, confronted in Figure 6, indicate that the Eq. (13) always yields smaller values for this quantity than those obtained with the Eq. (11), the difference being more pronounced for A > 30. However, for some isolated light nuclei, the use of a more realistic GTR energy increases the cross section. This is the case of 12 B, for which the product σ(E ν )Φ(E ν ) is shown in the left panel of Figure 7. The increase of σ(E ν ) arises from the contribution of the 1 + states with energies below the GTR (see Refs. [19,34]). As another example, in the right panel of Figure 7 are shown the results for the Ni isotopes ( 67 Ni, 68 Ni and 69 Ni). One notes that for the three nuclei, the product σ(E ν )Φ(E ν ) decreases when the energy of the GTR is moved up. Also, because of the pairing effect, the cross-section in 68 Ni presents the lowest value for both GT energies.
On the other hand, from Figure 8 it can be seen that our results for the reduced thermal cross-section in Ni nuclei emphasizes the odd-even effect when compared with the microscopic ETFSI+CQRPA calculation [12], where this effect seems to be washed out. This leads a different trend of the ν e -nucleus cross section with respect to A.
For completeness, in Figure 9 we present the results for σ ν /A obtained with the GTNC, both for the β − decaying nuclei (with σ N from Table 1), and for the nuclei where take place electron-capture (with σ N from Table 2).
It is worth noticing that the gaussian and lorenzian strength functions given, respectively, by Eqs. (8) and (9) yield almost the same results for the reduced thermal cross-sections.
At this point it is important to clarify the meaning of the thermal neutrino flux presented in Eq. (5), which we have used for the calculation of the thermal neutrinonucleus cross section σ ν . This neutrino energy flux is given by a Fermi distribution, i.e., Eq. (5) depending explicitly on the temperature parameter T ν . In order to compare our results with those of Borzov and Goriely [12] we have used here a constant temperature T ν = 4 MeV. However, this situation could not be a realistic one for the supernova neutrino wind. Neutrinos (and antineutrinos) with different energies and flavors decouple at different points of the supernova core and the neutrino spectrum, in fact could be non thermal. This is due to the non-thermalization of neutrinos through their transport along hydrodynamics medium evolution [35,36]. Thus, it could be interesting to determine the consequences of employing a different neutrino flux such as a power law flux of the form The parameters ǫ ν and α are not fully determined and here we take ǫ ν ≈ 3.1514 T ν = 12.6056 MeV, and α ≈ 2.3014, which reproduces better the Fermi-Dirac neutrino distribution function in Eq. (5) using T ν = 4 MeV. These parameter values were obtained in Ref. [35]. The normalization constant N P L ensures unitary flux between 0 and 102 MeV. We have found that, for all practical purposes, the flux (18) yields the same results as the thermal flux (5). This is an expected result, since these two fluxes tend to behave differently only in the tail zone, far away of the integration interval used to obtain the σ ν (E) for astrophysical applications. Some possible deviations in the tail of these fluxes are important for the rate of nuclear reactions in studies of astrophysics plasmas [37].

Summarizing conclusions
We have briefly revived the original version of the gross theory for the β-decay. The main improvement introduced is a more realistic estimate for the location of the GTR energy peak, E GT . After fixing the free parameter of our model (σ N or Γ N , depending on the parametrization adopted for the strength function) we have calculated the β −decay and electron-capture rates. A careful selection of input data for A < 70 nuclei, with small error bars in the measured half-lives, has been done in order to fix the model parameters in the fitting procedure. The model can be extended to the A > 70 nuclei, as well as to the transuranic nuclei, which are of interest for the study of the r-process in supernova. The first-and second-forbidden weak processes could play an important role in the exotic nuclei within this nuclear mass region. But, these transitions can be easily included in the gross theory framework, as has been done already by Nakata et al. [9] within the semi-gross theory .
A relation analogous to (13) was also derived for the first forbidden charge-exchange resonances [38], which is quite different to the one used in Ref. [9]. Thus it might be more appropriate to employ [38, The present results are encouraging, in the sense that the gross theory could be able to describe in a systematic way, not only the nuclear properties along the β-stability line, but also for exotic nuclei involved in presupernova composition. In particular, the results for the reduced thermal cross section σ ν /A in the region A < 70 are in fair agreement with previous calculation performed within more refined microscopic models, i.e., the ETFSI+CQRPA model [12]. The difference between the two descriptions could be attributed to the use of the Fermi gas model which contains more degrees of freedom that the EFTSI+CQRPA. Consequently, in general, σ ν (E ν ) calculated with the Fermi gas model leads to values higher than those obtained with microscopical nuclear models [39,40,41], particularly for light or intermediated nuclei (see, for instance, the results for the ν − 12 C reaction shown in [9, Fig. 2)] and [41, Fig. 32]).
The important aspect of the recent r-process calculations is that they take into account the neutrino-rich environment in supernova explosion, where the ν e -nucleus reaction are in competition with the β-decay processes [42]. To address this type of calculation we have evaluated the cross-section σ ν (E ν ) within the GTNC model, folded with a temperature dependent neutrino flux.
Finally, we want to remark once more the simplicity of the present model, which we are planing to extended in the near future to the r-process nuclei region, as well as to evaluate the isotopic abundances in presupernova scenario. Table 1. Standard deviations σ N (in units of MeV) and mismatch factors η for β −decay. Gaussian single-particle strength probability function D X (E) was adopted. σ N and η (σ * N and η * ) indicate the results obtained with E GT approximated from Eq. (13) (Eq. (11)). Parenthetically are shown the values obtained by Takahashi et al [4] for a different data set of nuclei. The electronic neutrino cross-section are evaluated with the underlined values of σ N .  Table 2. Standard deviations σ N (in units of MeV) and mismatch factors η for β + decay and electron-capture. Gaussian single-particle strength function D GT was used.
The energy E GT has been evaluated from Eq. (13). The remaining notation is the same as in Table 1. No minimum has been found for the χ 2 A -function in the case of even-even parent nuclei.  Table 3. Standard deviations σ N (in units of MeV) and mismatch factors η for β − -decay and electron-capture, obtained from the minimization of the χ 2 B -function. Lorentzian single-particle strength probability function was used. The energy E GT has been evaluated from Eq. (13).     I  P  I  Q  I  R  I  S  I  T  I  U  I