A group of theoretical scientists at JINR has applied a new approach to study the properties of hot systems, i.e., systems where the excitation energy is uniformly distributed over many degrees of freedom. Applying the superoperator method, the scientists have formulated a statistical approach to semileptonic weak reactions with hot thermally excited nuclei in the dense and hot stellar plasma. Such reactions play an important role in the late stage of massive star evolution, and their rates and cross sections are used as nuclear input in core-collapse supernova simulations. A series of 15 scientific papers on this topic was awarded the second JINR Prize in the category of theoretical physics research 2022.

The interest in supernovae is related to the key role they play in our understanding of many cosmic phenomena. It is supernova explosions that determine the chemical evolution of the universe, as they eject heavy elements synthesised in the star into their surroundings. Supernova blast waves create dense clumps of interstellar gas, in which new stars and planets are then born. Closely related to supernovae are such questions as the formation of neutron stars and black holes, the synthesis of superheavy elements, the origin of cosmic rays, the nature of cosmic gamma-ray bursts, and many others.

To date, there is no generally accepted model capable of quantitatively describing the explosion mechanism of core-collapse supernovae. It is important to emphasise that the results of computer simulations of various explosion mechanisms largely depend on the input data on the cross sections and rates of semileptonic nuclear reactions (electron and neutrino capture, neutrino inelastic scattering, etc.) used. The problem is complicated by the fact that deep in the stellar interior temperature is so high (T≈10⁹-10¹⁰ Kelvin) that nuclear excited states are thermally populated. Therefore, for an accurate estimation of the nuclear input for core-collapse models, semileptonic reactions with hot thermally excited nuclei should be considered. Since the experimental study of such reaction is not feasible, their rates and cross sections can be obtained only theoretically.

Databases on rates and cross sections of semileptonic nuclear reactions (electron and neutrino capture, neutrino inelastic scattering, etc.) presently used in astrophysical simulations are obtained either by large-scale shell model calculations (for iron-group nuclei) or with the help of the so-called “hybrid model” for heavier neutron-rich nuclei. In order to get around the problem of the rapid growth of the configuration space with increasing number of nucleons in the nucleus and/or its excitation energy, various approximations, such as the Brink-Axel hypothesis or the “method of back-resonances”, are used in these calculations. However, these assumptions make the calculations thermodynamically inconsistent, violating, for example, the principle of detailed balance, which relates the excitation and deexcitation probabilities of the hot nucleus in a weak reaction. The superoperator method developed by the JINR scientists allows us to increase the accuracy of the calculation of nuclear stellar weak-interaction rates and cross sections and thus contributes to the understanding of the core-collapse supernova explosion mechanism.

The superoperator method is a mathematical technique that allows us to consider the density matrix operator, which describes the mixed state of the hot nucleus, as a pure state (vector) in the Liouville space, which is a linear vector space of operators on the Hilbert space. The superoperators are operators acting on the elements of this Liouville space. Due to the transition to the Liouville space and the introduction of superoperators it becomes possible to generalise to the case of hot nuclei those nuclear models that are based on the wave function concept.

“The superoperator method is based on a statistical approach to the nuclear many-body problem at finite temperature. Within the statistical approach, the temperature-dependent nuclear response function to a weak-interaction multipole excitation is calculated first, and then the semileptonic reaction rates and cross sections are expressed through it. This approach has the advantage that it does not require calculations in a very large configuration space, and therefore can be applied at any temperature and has no restrictions on the mass of nuclei,” Head of the Nuclear Structure Sector of the Atomic Nucleus Theory Department of the Bogoliubov Laboratory of Theoretical Physics at JINR, Doctor of Physics and Mathematics Alan Dzhioev said. “In addition, it was shown in our studies that the superoperator method allows us to perform calculations without invoking the Brink-Axel hypothesis and without violating the principle of detailed balance. It is also possible to perform a successive refinement of the wave function of a hot nucleus by taking into account the coupling of simple and complex configurations. For cold nuclei, such a method was developed in BLTP by Professor V. G. Soloviev and his colleagues in the framework of the quasiparticle-phonon nuclear model. We managed to generalise this method to the case of hot nuclei.”

Unblocking of stellar electron capture for neutron-rich N=50 nuclei. The capture rates for hot and cold nuclei are represented by red and blue lines, respectively. The capture rates corresponding to the experimental ground-state GT strength are demonstrated with experimental uncertainties. The capture rates are shown as a function of the matter density of the collapsing stellar core.

According to the scientist, the first superoperator calculations of stellar weak-interaction rates and cross sections were performed applying an effective nuclear Hamiltonian with parameters fitted individually for each nucleus. However, the total cross sections and rates of semileptonic processes in the stellar matter are composed of individual contributions from many tens and hundreds of nuclides, including those far away from stability. Therefore, a self-consistent approach based on the Skyrme energy density functional was developed for global calculations. The problem of the configuration space growth at T≠0 was solved using the procedure of the residual interaction separabelization.

“To date, we have performed calculations of stellar weak-interaction rates and cross sections for several representative nuclei that dominate the isotopic composition of the central part of the star at different stages of collapse. These are a few isotopes of iron and nickel, which dominate the composition at the onset of collapse, and several neutron-rich nuclei with neutron number close to 50, which dominate at a later stage of collapse. Using these nuclei as examples, we have shown that thermodynamically consistent consideration of thermal effects leads to a more significant increase in reaction rates and cross sections than previously assumed. But the isotopic composition of the collapsing stellar core consists of tens and hundreds of different nuclides. Therefore, to use our method in computer models of stellar collapse, we need to carry out calculations for a wide variety of nuclides and tabulate rates and cross sections of semileptonic processes at different medium parameters, i.e. at different temperatures and densities of stellar matter,” Alan Dzhioev emphasised.

As a result, a group of scientists obtained the following main results:

• A new definition of fermionic superoperators in the Liouville space has been formulated. This definition extends the applicability of the equation-of-motion method for thermodynamically consistent calculations of strength functions and spectral densities of charge-exchange and charge-neutral multipole operators in hot nuclei.
• For nuclear models with a separable residual interaction, the equations of the thermal quasiparticle random phase approximation (TQRPA) have been obtained for the first time. These equations allow us to describe the excitation and deexcitation processes of hot nuclei preserving the detailed balance principle. A method has been developed to describe the fragmentation of the strength function of one-phonon states in hot nuclei. It is shown that the implementation of the principle of detailed balance requires redistribution of the thermal phonon vacuum, taking into account the connection of thermal quasiparticles and thermal phonons.
• Based on the TQRPA and applying the Donnelly-Walecka method of the multipole expansion of the operators of the nucleonic weak current, a statistical approach has been developed to calculate the cross sections and rates of semileptonic reactions with hot nuclei under astrophysical conditions.
• Using several iron group nuclei and neutron-rich nuclei with neutron number N≈ 50 as an example, thermal effects on the cross sections and rates of various semileptonic reactions (e^— -capture, ν-capture, neutrino scattering, etc.), which play an important role in the late stage of the massive star evolution, were studied for the first time. Meanwhile, it was shown that:

a.The refusal to use the Brink-Axel hypothesis in hot nuclei as well as the detailed balance principle satisfaction noticeably increase the contribution of thermally excited nuclear states to stellar weak-interaction rates and cross sections in comparison with the shell model predictions. In particular, the enhanced role of exoenergetic processes accelerates the temperature-induced growth of low-energy cross sections. This, in turn, increases the neutronization rate of stellar matter and its opacity to neutrino radiation.

b.The temperature-induced weakening of pairing correlations drastically increases the temperature dependence of the energy and strength of unblocked Gamow-Teller p→n transitions in neutron-rich nuclei. It is found that thermal effects are the main unblocking mechanism of low-energy Gamow-Teller transitions. As a result, contrary to the previous calculations made by other authors within simpler approaches, it is shown that the stellar electron capture on nuclei is not stopped at semimagic neutron-rich nuclei with N=50.

c.The enhanced contribution of charge-neutral transitions from thermally excited nuclear states broadens the energy spectrum of neutrinos scattered on hot nuclei in comparison with the results of shell model calculations.

• The spectra of neutrino-antineutrino pairs emitted by hot nuclei in stellar matter were calculated and the energy loss rates were estimated. It is shown that low-energy neutrinos ε_ν<10 MeV are mainly produced at temperatures T=1-2 MeV, which makes possible the so-called “thermostat effect”, i.e., a rapid rise of the energy loss rates as the temperature increases.

Inelastic neutrino scattering cross section on the ⁵⁶ Fe nucleus as a function of neutrino energy E_ν at different T temperatures of the collapsing stellar core. Solid lines – results of the TQRPA calculations based on the superoperatormethod. Dashed lines – shell model results.

The scientist also noted that the superoperator method is applicable not only to describe the properties of hot nuclei, but also to describe arbitrary equilibrium and non-equilibrium statistical quantum systems. “In particular, we applied it to describe electron transport through open correlated quantum systems. For this purpose, with the help of superoperators, we managed to introduce the concept of nonequilibrium quasiparticles and use them to construct a perturbation theory for nonequilibrium steady states, as well as analogues of the configuration interaction and coupled cluster methods,” Alan Dzhioev said.

List of publications

1. A.A. Dzhioev, A.I. Vdovin, V.Yu. Ponomarev, J. Wambach, Charge-exchange transitions in hot nuclei, Physics of Atomic Nuclei, 72 (2009) 1320.
2. A.A. Dzhioev, A.I. Vdovin. On the TFD treatment of collective vibrations in hot nuclei, Int. J. Mod. Phys. E, 18 (2009) 1535.
3. A.A. Dzhioev, A.I. Vdovin, V.Yu. Ponomarev, J. Wambach, K. Langanke, G. Martinez-Pinedo. Gamow-Teller strength distributions at finite temperatures and electron capture rates in stellar environment, Phys. Rev. C, 81 (2010) 015806.
4. A.A. Dzhioev, D.S. Kosov. Super-fermion representation of quantum kinetic equations for the electron transport problem, J. Chem. Phys., 134 (2011) 044121.
5. A.A. Dzhioev, A.I. Vdovin, V.Yu. Ponomarev, J. Wambach. Thermal effects on neutrino-nucleus inelastic scattering in stellar environments, Ядерная Физика, 74 (2011) 1193.
6. A.A. Dzhioev, D.S. Kosov. Nonequilibrium perturbation theory in Liouville-Fock space for inelastic electron transport, J. Phys.: Condens. Matter, 24 (2012) 225304.
7. A.A. Dzhioev, A.I. Vdovin, Neutrino–antineutrino pair emission from thermally excited nuclei in stellar collapse, Physics of Atomic Nuclei, 77 (2014) 1166.
8. A.A. Dzhioev, A.I. Vdovin, J. Wambach, V.Yu. Ponomarev. Inelastic neutrino scattering off hot nuclei in supernova environments, Phys. Rev. C, 89 (2014) 035805.
9. A.A. Dzhioev, A.I. Vdovin, J. Wambach. Neutrino absorption by hot nuclei in supernova environments, Phys. Rev. C, 92 (2015) 045804.
10. A.A. Dzhioev, A.I. Vdovin, G. Martinez-Pinedo, J. Wambach, Ch. Stoyanov. Thermal quasiparticle random-phase approximation with Skyrme interactions and supernova neutral-current neutrino-nucleus reactions, Phys. Rev. C, 94 (2016) 015805.
11. A.A. Dzhioev, A.I. Vdovin, Ch. Stoyanov. The Skyrme-TQRPA calculations of electron capture on hot nuclei in pre-supernova environment, Phys. At. Nucl., 79 (2016) 1019.
12. A.A. Dzhioev, A.I. Vdovin, Ch. Stoyanov. Thermal quasiparticle random-phase approximation calculations of stellar electron capture rates with the Skyrme effective interaction, Phys. Rev. C, 100 (2019) 025801.
13. A.A. Dzhioev, K. Langanke, G. Martinez-Pinedo, A.I. Vdovin, Ch. Stoyanov. Unblocking of stellar electron captures for neutron-rich N=50 nuclei at finite temperature, Phys. Rev. C, 101 (2020) 025805.
14. A.A. Dzhioev, A.I. Vdovin, Thermodynamically consistent description of one-phonon states fragmentation in hot nuclei, Physics of Particles and Nuclei Letters, 18 (2021) 513.
15. A.A. Dzhioev, A.I. Vdovin, Superoperator approach to the theory of hot nuclei and astrophysical applications: I—Spectral properties of hot nuclei, Physics of Particles and Nuclei, 53 (2022) 1007.