Nuclear astrophysical plasmas: ion distribution functions and fusion rates

Abstract

T his article illustrates how very small deviations from the Maxwellian exponential tail, while leaving unchanged bulk quantities, can yield dramatic effects on fusion reaction rates and discusses several mechanisms that can cause such deviations.Fusion reactions are the fundamental energy source of stars and play important roles in most astrophysical contexts. Since the beginning of quantum mechanics, basic questions were addressed such as how nuclear reactions occur in stellar plasmas at temperatures of few keV (1 keV ≈ 11.6 x 10 6 K) against Coulomb barriers of several MeV and what reactions or reaction networks dominate the energy production. It was soon realized that detailed answers to such questions involved not only good measurements or quantum mechanical understanding of the relevant fusion cross sections, but also the use of statistical physics to describe the energy and momentum distributions of the ions and their screening [1].Gamow understood that reacting nuclei penetrate Coulomb barriers by means of the quantum tunnel effect and Bethe successfully proposed the CNO and then the pp cycle as candidates for the stellar energy production: this description has been directly confirmed by several terrestrial experiments that have detected neutrinos produced by pp and CNO reactions in the solar core [2].In the past only a few authors (e.g., d'E. Atkinson, Kacharov, Clayton, Haubold) have examined critically the energy distribution and proposed that such a distribution could deviate from the Maxwellian form. In fact, it is commonly accepted that main-sequence stars like the Sun have a core, i.e. an electron nuclear plasma, where the ion velocity distribution is Maxwellian. In the following, we first discuss why even tiny deviations from the Maxwellian distribution can have important consequences and then what can be the origin of such deviations. Thermonuclear reaction in plasmas and distribution tailsIn a gas with n1 (n2) particles of type 1 (2) per cubic centimeter and relative velocity v, the reaction rate r (the number of reactions per unit volume and unit time) is given by (1)where σ = σ(v) is the nuclear cross section of the reaction. The reaction rate per particle pair is defined as the thermal average Therefore, the reaction rate per particle pair < σv > is determined by the specific cross section and by the velocity distribution function of the reacting particles. When no energy barrier is present and far from resonances, cross sections do not depend strongly on the energy. Most of the contribution to < σv > comes from particles with energy of the order of kT, and the dependence on the specific form of f (v) is weak. The same is true for bulk properties that receive comparable contributions from all particles: e.g., the equation of state.The (EG = 45.09 MeV, blue). Correspondingly, the peaks become (much) lower; note that the three curves have been multiplied times 10 5 , 10 17 , and 10 26 , respectively, to make them visible on the same scale.

Big Bang Nucleosynthesis With a Non-Maxwellian Distribution

“…Thus, only a few particles in the tail of the distributions contribute to the fusion rates. In fact, the possibility of a deviation of the Maxwellian distribution and implications of the modification of the Maxwellian distribution tail for nuclear burning in stars have already been explored in the past [21][22][23][24]. As we show in the next sections, a strong deviation from q = 1 is very unlikely for the BNN predictions, based on comparison with observations.…”

Modified Debye‐Huckel Electron Shielding and Penetration Factor

“…In stellar atmospheres, atomic processes such as radiative and dielectronic recombination exhibit rates that indicate deviations from a Maxwellian distribution of electrons (Maero et al 2006). In stellar cores, signals of possible deviations in the ion distributions are evident and, although small, should be considered because they are capable of meaningfully influencing nuclear fusion reaction rates (Ferro & Quarati 2005;Lissia & Quarati 2005).…”

Fusion reactions in plasmas as probe of the high-momentum tail of particle distributions

“…Fusion processes select high-momentum particles that are able to penetrate the Coulomb barrier and are, therefore, extremely sensitive probes of the distribution tail [2,3,4,5]. This broadening of the interacting particle energy-momentum dispersion relation has been proposed recently [6,7] as a possible explanation of the strong enhancement of the observed low-energy rate of the reaction d(d, p)t in deuterated metal target [8,9,10,11,12].…”