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Superoperator Approach to the Theory of Hot Nuclei and Astrophysical Applications: II—Electron Capture in Stars

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Abstract

Electron captures on nuclei play an essential role in the dynamics of the collapsing core of a massive star that leads to a supernova explosion. We propose a novel thermodynamically consistent approach to calculate capture rates and cross sections of \({{e}^{ - }}\) capture on hot nuclei in the stellar interior. The method is based on the quasi-particle random phase approximation extended to finite temperature using the tool of the superoperator formalism. By the example of \(^{{54,56}}\)Fe it is shown that thermodynamically consistent incorporation of thermal effects leads to a stronger temperature dependence of the \({{e}^{ - }}\)-capture rates and cross sections for iron group nuclei than predicted by the shell-model calculations. The combined action of thermal effects and pairing correlations on the unblocking of Gamow–Teller \(p \to n\)-transitions is considered for neutron-rich nuclei around N = 50. It is shown that it is thermal effects that lead to the unblocking of low-energy transitions. Due to this, as well as to the contribution of forbidden transitions, the electron capture does not stop at nuclei with N = 50.

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Notes

  1. First of all, we are talking about the colossal values of density, temperature, and magnetic field strength.

  2. A more detailed presentation of the physics of supernovae is given in review articles [2, 4, 19–23], as well as in the collection [6], which most fully reflects the current state of knowledge in this area.

  3. The value \({{Y}_{e}}\) is defined as a ratio of the density of number of electrons \({{n}_{e}}\) [cm–3] to the density of number of nucleons \({{n}_{N}}\) [cm–3]

    $${{Y}_{{\text{e}}}} = \frac{{{{n}_{{\text{e}}}}}}{{{{n}_{N}}}}.$$

    Since the substance density is \(\rho = {{n}_{N}}{{m}_{u}}\) [g/cm3], where \({{m}_{u}}\)is the atomic mass unit [g], and since 1 a. m. u., expressed in grams, is numerically almost equal to the inverse of Avogadro’s number \({{N}_{A}}\), then \({{Y}_{{\text{e}}}}\) is numerically equal to the number of moles of electrons per gram of the substance [mol/g],

    $${{Y}_{{\text{e}}}} = \frac{{{{n}_{{\text{e}}}}}}{{\rho {{N}_{{\text{A}}}}}}.$$

    The latter relation is often used as a definition of \({{Y}_{e}}\) (electron mole number) [21]. With this definition, the \(\rho {{Y}_{e}}\) value is numerically equal to the number of moles of electrons per unit volume [mol/cm3].

  4. To capture an electron in the ground state of \(^{{56}}\)Fe, a threshold energy \(Q = M{{(}^{{56}}}{\text{Fe}}) - M{{(}^{{56}}}{\text{Mn}}) = 4.2\) MeV is required. A temperature different from zero accelerates the capture of electrons.

  5. In stars with \(M \gtrsim 20{{M}_{ \odot }}\), due to the higher temperature, a collapse begins owing to the nuclear photodisintegration. With the higher mass, \(M \gtrsim 60{{M}_{ \odot }}\), the process of electron-positron pair production begins to contribute to the violation of stability [24].

  6. The low value of entropy per nucleon (\(s \sim 1\) ) corresponds to the fact that the fraction of free nucleons is small. Intuitively, this follows from the fact that a heavy nucleus has fewer degrees of freedom than many free nucleons or light nuclei.

  7. At low neutrino energies \({{\varepsilon }_{\nu }}\), for the cross section of coherent elastic scattering on nuclei, we have \(\sigma \sim {{N}^{2}}\varepsilon _{\nu }^{2}\).

  8. Homologous collapse means that the rate of matter falling on the center is proportional to the distance to the center.

  9. Afterwards, depending on the mass, the protoneutron star either evolves into a neutron star, or collapses into a black hole.

  10.  The inverse process of \({{\beta }^{ - }}\)-decay of nuclei is also possible, but it is rather quickly blocked due to the increasing Fermi energy of the electron gas [3, 60].

  11.  An analogy with a spring is appropriate here. The larger the spring, the more energy it can store when compressed.

  12.  Following [63], we included the rest mass \({{m}_{{\text{e}}}}{{c}^{2}}\) in the definition of \({{\mu }_{e}}\).

  13.  In [64] instead of \({{Y}_{e}}\), the so-called average electron molecular weight \({{\mu }_{{\text{e}}}} = {1 \mathord{\left/ {\vphantom {1 {{{Y}_{{\text{e}}}}}}} \right. \kern-0em} {{{Y}_{{\text{e}}}}}}\) is used, which is equal to the mass (in amu) per electron. Since 1 amu, expressed in grams, is numerically almost equal to the inverse of Avogadro’s number, then the value \({{\mu }_{e}}\), expressed in amu per electron, numerically coincides with the mass (in grams) per mole of electrons. On the left-hand side of (7), the quantity \(\rho {{Y}_{e}}\) has the dimensionality mol/cm3 (see footnote on page 941).

  14.  With \(T \ne 0\), a total number of electrons is composed of ionization electrons and electrons resulting from the production of electron-positron pairs (\(\gamma \to {{e}^{ - }} + {{e}^{ + }}\)) from \(\gamma \)-quanta with the energy \( \geqslant \,2{{m}_{{\text{e}}}}{{c}^{2}}\) (see [65, p. 854]). Since the chemical potential of \(\gamma \)-quanta is zero, then in conditions of thermodynamic equilibrium, \({{\mu }_{{\text{e}}}} + {{\mu }_{p}} = 0\) [27], i.e., positrons are described by distribution function (6) with the chemical potential \({{\mu }_{p}} = - {{\mu }_{{\text{e}}}}\).

  15.  In nuclei with \(N > Z\) \(p \to n\) Fermi transitions from the ground state are forbidden, since they do not satisfy the isospin selection rule (\(\Delta T = 0\)). However, these transitions are possible from excited states.

  16.  Prior to this, it was believed that at the stage of collapse, the capture of electrons is carried out mainly by free protons, which are formed during the collapse of nuclei. However, an increase in the rate of \({{e}^{ - }}\)-capture by nuclei due to the GT\(_{ + }\) resonance leads to a decrease in entropy, as a result of which the nuclei do not fall apart. Therefore, though free protons capture electrons faster than nuclei, their contribution to \({{e}^{ - }}\)-capture is insignificant due to the low concentration. However, an increase of the rate of \({{e}^{ - }}\)-capture by nuclei due to the GT\(_{ + }\) resonance leads to a decrease in entropy, as a result of which the nuclei do not fall apart. Therefore, though free protons capture electrons faster than nuclei, their contribution to \({{e}^{ - }}\)-capture is insignificant due to the low concentration.

  17.  It should be noted that expression (14) neglects the effect of blocking the phase space for the resulting neutrinos. The blocking of the neutrino phase space due to the action of the Pauli principle occurs at densities of the order of several \({{10}^{{11}}}\,\,{\text{g/c}}{{{\text{m}}}^{3}}\), after it becomes possible to block neutrinos and form a degenerate neutrino gas (thermalization) with a distribution function \({{f}_{\nu }}({{\varepsilon }_{\nu }})\). This effect can be considered by introducing a blocking factor \((1 - {{f}_{\nu }}({{\varepsilon }_{\nu }}))\) under the integral sign. At lower densities, neutrinos leave the star without hindrance, and therefore, when calculating the rates of electron capture by nuclei with \(A = 21{\kern 1pt} - {\kern 1pt} 60\), it can be set to \({{f}_{\nu }}({{\varepsilon }_{\nu }}) = 0\).

  18.  It is worth recalling once again that the product of the density of a substance \(\rho \) [g/cm3] and the electron component \({{Y}_{{\text{e}}}}\) is numerically equal to the density of the electron gas [mol/cm3] (see footnote 4 on page 5).

  19.  For simplicity, we neglected the correction for the Coulomb interaction, i.e., we put \(F(Z,{{\varepsilon }_{{\text{e}}}}) = 1\).

  20.  It is assumed that the Fermi transition strength is concentrated in the isobar-analog state. The reduced probability of transition to this state is given by the expression \({{B}_{{if}}}({\text{F}}) = T(T + 1) - {{T}_{{zi}}}{{T}_{{zf}}}\).

  21.  Most experimental values of matrix elements for Fermi and Gamow–Teller discrete transitions, as well as data on nuclear levels, were taken from tables [79]. The matrix elements of allowed transitions for which there were no measurements were determined as average values to which log(ft) = 5.0 corresponds.

  22.  That is, all possible configurations of nucleons inside the valence \(sd\)-shell are considered, while the \(sp\)-shell is considered as a core.

  23.  In the SMMC method, the strength function \({{S}_{\Omega }}(E)\) of the operator \(\Omega \) is found by numerical inversion of the relation

    $${{R}_{\Omega }}(\tau ) = \int\limits_{ - \infty }^\infty {{{e}^{{ - E\tau }}}} {{S}_{\Omega }}(E)dE,$$
    (18)

    which relates it to the response function \({{R}_{\Omega }}(\tau )\) [97]. Uncertainties arise both in the calculation of the response function and in the numerical inversion of the Laplace transform (18).

  24.  Blocking is a consequence of the faster increase in the Fermi energy of the electron gas (\({{\mu }_{e}} \sim {{\rho }^{{1/3}}}\)) compared to the average value \(\langle Q\rangle \) in the \({{\beta }^{ - }}\) decay.

  25.  For this, a formula was used that includes the binding energies of four neighboring nuclei [125].

  26.  It is known [88, 89] that the experimentally found strength of charge-exchange GT transitions in the resonance region does not satisfy the sum rule S– – S+ = 3(N – Z) and is approximately half of the required value. In essence, three possible mechanisms for suppressing the Gamow–Teller strength are discussed [149, 82]: (i) Configurational mixing of particle–hole (1p1h) configurations with more complex ones (2p2h, 3p3h, and so on) [150] leads to fragmentation of the Gamow–Teller strength in a wide energy range, including the region above the GT resonance, which makes it difficult to detect experimentally. (ii) The interaction of 1p1h configurations with configurations involving the \(\Delta (1232)\)-isobar excitation can push a part of the strength into the Δ-isobar excitation energy region (\( \approx \,300\) MeV) [151]. (iii) Renormalization of the axial-vector constant gA inside the nucleus due to meson-exchange currents [152]. Experimental studies of (p, n) [153] and (n, p) [154] reactions on 90Zr testify that configurational mixing is the most important mechanism for quenching the strength of GT transitions (see also review [82]). However, the results of theoretical calculations are contradictory. On the one hand, calculations within the so-called second RPA [155], which considers a coupling of 1p1h and 2p2h configurations, confirm this conclusion. On the other hand, calculations in the context of QPM [156] show that consideration of the interaction of quasi-particles with phonons cannot push the GT strength into the region of excitation energies above 30 MeV, at least in the case of central separable strengths. Self-consistent RPA calculations with Skyrme forces, including tensor interaction, demonstrate that the coupling of GT states to spin-quadrupole 1+ states can also contribute to the quenching of the Gamow–Teller strength (see review [157] and references therein). However, the relative value of this contribution depends on the parameters of the tensor interaction [158], which can vary over a wide range [159]. Thus, the question of the role of various mechanisms in the quenching of the GT strength remains open [149].

  27.  Table 1 in [86] gives the values S+ i n 54,56Fe obtained in LSSM calculations after multiplication by (0.74)2 (3.6 and 2.7, respectively). These values are close to the QPM–QRPA results after multiplying the latter by (0.74)2.

  28.  The position of the centroid of the GT\(_{ + }\) distribution (6.81 MeV) was obtained from the data given in Table 1 of work [86], by adding the difference between the masses of the daughter and parent nuclei \(M{{(}^{{56}}}{\text{Mn}}) - M{{(}^{{56}}}{\text{Fe}}) = 4.21\) MeV.

  29.  In the secular equation (345, Appendix C, Part I), the particle–hole matrix elements contain the factor \(1 - y_{{{{j}_{p}}}}^{2} - y_{{{{j}_{n}}}}^{2}\).

  30.  Recall that transitions from excited states of the nucleus are considered using two-thermal-quasi-particle configurations, including a tilde thermal quasi-particle: \(\beta _{{{{j}_{p}}}}^{\dag }\widetilde \beta _{{{{{\bar {j}}}_{n}}}}^{\dag }\), \(\widetilde \beta _{{{{{\bar {j}}}_{p}}}}^{\dag }\beta _{{{{j}_{n}}}}^{\dag }\) and \(\widetilde \beta _{{{{{\bar {j}}}_{p}}}}^{\dag }\widetilde \beta _{{{{{\bar {j}}}_{n}}}}^{\dag }\).

  31.  When calculating \({{\sigma }_{{{\text{ex}}}}}({{\varepsilon }_{e}},T)\), in the formula (23) only \( \downarrow \)-transitions were included.

  32.  Due to the absence of an \(E < 0\) component in the strength function of GT+ transitions, the threshold for electron capture by hot nuclei is also predicted by SMMC calculations [100].

  33.  This conclusion follows directly from Eq. (4) of work [115] and its discussion.

  34.  With \(\rho {{Y}_{{\text{e}}}} = {{10}^{{10}}}{\text{ g/c}}{{{\text{m}}}^{3}}\), the electron Fermi energy \({{\mu }_{{\text{e}}}} \approx 11\) MeV (see Fig. 1), which exceeds the GT\(_{ + }\) resonance energy in 56,56Fe.

  35.  For \(({{T}_{9}},\mathop {\log }\nolimits_{10} (\rho {{Y}_{{\text{e}}}})) = (10,7)\) a value of the chemical potential \({{\mu }_{{\text{e}}}}\) is much less than the energy of resonant transitions (see Fig. 1). However, the presence of high-energy electrons at the tail of the Fermi–Dirac distribution makes the resonance contribution decisive. The contribution of transitions with negative energy is quite significant due to the increased phase space of the outgoing neutrinos.

  36.  A similar method was used by Fuller et al. [64, 72, 71] to reproduce the contribution of low-energy GT\(_{ + }\) transitions to electron capture.

  37.  At the same time, both processes produce neutrinos, which then leave the star. Hence, an increase of the rates of \({{e}^{ - }}\)-capture and \({{\beta }^{ - }}\) decay relative to the LSSM results accelerates the loss of energy and reduces the entropy of matter.

  38.  For the rate of \({{e}^{ - }}\)-capture by nuclei of the iron group, Bruenn used a parametrization based on an estimate by Fuller et al. of the strength and energy of a single-particle GT\(_{ + }\) transition \(1{{f}_{{7/2}}} \to 1{{f}_{{5/2}}}\) [71].

  39.  The diagram of the single-particle levels of the 82Ge nucleus is given in Fig. 1 in [206] and is generally close to the scheme in Fig. 24 for 76Ge. In the independent particle model, protons occupy levels \(1{{f}_{{7/2}}}\) and \(2{{p}_{{3/2}}}\), while neutrons populate all levels up to \(1{{g}_{{9/2}}}\), inclusively. The unblocking of GT\(_{ + }\) transitions occurs both due to the transition of protons to higher levels, and due to the formation of neutron vacancies in the \(pf\)-shell.

  40. \(\sum B(G{{T}_{ + }}) = 0.7 \pm 0.2\) below the excitation energy of 5 MeV of the daughter nucleus.

  41.  \(U = {{A{{T}^{2}}} \mathord{\left/ {\vphantom {{A{{T}^{2}}} 8}} \right. \kern-0em} 8} \approx 10\) MeV at \(T \approx 1\) MeV and \(A \approx 80\).

  42.  The RPA method with incomplete filling of single-particle levels is described in [210].

  43.  Nuclei with \(N \approx 82\) make a significant contribution to \({{e}^{ - }}\)-capture at high densities before bounce [221].

  44.  With this simplified consideration, the unperturbed wave function of the ground state of the nucleus has the form of the Slater determinant, instead of having a BCS vacuum structure.

  45. Note that the sequence of single-particle levels in Fig. 24 is close to that used in [206] to calculate the rates of electron capture by the 82Ge nucleus.

  46.  In [225], similar effects were considered in detail using the example of 80Ge.

  47.  Due to the smallness of the matrix elements of the residual particle–hole interaction between particle–particle and hole–hole states, considering TQRPA correlations does not lead to noticeable mixing of two-thermal-quasi-particle configurations.

  48.  According to our BCS calculations, in 82Ge and 86Kr there are 0.2 and 0.4 protons per level \(1g_{{9/2}}^{p}\), respectively.

  49.  According to Table 1 in [229], for a star with a mass \(15{{M}_{ \odot }}\), the core temperature is in the range \(1.0 \times {{10}^{{10}}}{\kern 1pt} - {\kern 1pt} 1.4 \times {{10}^{{10}}}\) K at densities \(\rho {{Y}_{e}} = 2.4 \times {{10}^{{10}}}{\kern 1pt} - {\kern 1pt} 2.3 \times {{10}^{{11}}}\,\,{\text{g/c}}{{{\text{m}}}^{3}}\).

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Funding

The work was supported by the RF Ministry of Science and Higher Education, grant no. 075-10-2020-117.

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    A. A. Dzhioev & A. I. Vdovin

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Dzhioev, A.A., Vdovin, A.I. Superoperator Approach to the Theory of Hot Nuclei and Astrophysical Applications: II—Electron Capture in Stars. Phys. Part. Nuclei 53, 939–999 (2022). https://doi.org/10.1134/S1063779622050045

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