Stopping of swift protons in matter and its implication for astrophysical fusion reactions

Abstract

The velocity dependence of the stopping power of swift protons and deuterons in low energy collisions is investigated. At low projectile energies the stopping is mainly due to nuclear stopping and charge exchange of the electron. The second mechanism dominates after E p ≥ 200 eV. A dynamical treatment of the charge exchange mechanism based on two-center electronic wavefunctions yields very transparent results for the exchange probability. We predict that the stopping cross sections vary approximately as v 1.35 p for projectile protons on hydrogen targets in the 1 keV energy region.Nuclear fusion reactions proceed in stars at extremely low energies, e.g., of the order of 10 keV in our sun [1,2]. At such low energies it is extremely difficult to measure the cross sections for charged particles at laboratory conditions due to the large Coulomb barrier. One often uses a theoretical model to extrapolate the experimental data to the low-energy region. Such extrapolations are sometimes far from reliable, due to unknown features of the low-energy region. E.g., there might exist unknown resonances along the extrapolation, or even some simple effect which one was not aware of before. One of these effects is the laboratory atomic screening of fusion reactions [3,4]. It is well known that the laboratory measurements of low energy fusion reactions are strongly influenced by the presence of the atomic electrons. This effect has to be corrected for in order to relate the fusion cross sections measured in the laboratory with those at the stellar environment. Another screening effect, arising from free electrons in the stellar plasma, will not be treated here. For about one decade, until 1996, one observed a large discrepancy between the experimental data and the best models available to treat the screening effect. The simplest (and perhaps the best of these models), the so-called adiabatic model, predicts that as the projectile nucleus penetrates the electronic cloud of the target the electrons become more bound and the projectile energy increases by energy conservation. Since the fusion cross sections increase strongly with the projectile's energy, this tiny amount of energy gain (of order of 10-100 eV) leads to a large effect on the measured cross sections. However, in order to explain the experimental data, it is necessary an extraamount of energy -about twice the value obtained by the adiabatic model. This is

Nuclear astrophysics with radioactive beams

“…This has been studied in Ref. [110] in the simplest situation; proton + hydrogen collisions. The calculated stopping power was added to the nuclear stopping power mechanism, i.e.…”

Influence of tunneling on electron screening in low energy nuclear reactions in laboratories

“…For 7 Li(p,α)α reaction, in addition to the direct measurement, an indirect measurement of the cross section using the Trojan horse method has recently been made [4]. The comparison between the two methods indicates again that the screening energy in the direct method exceeds the adiabatic limit by a large factor.One should, however, keep in mind that the stopping power is not well established, especially for gas target, at such low energies [5][6][7]3]. Also different values of the screening energy are obtained depending on the method of analysis […”

Tunneling of Atoms, Nuclei and Molecules

“…[38]. Because of the low bombarding energy, the electrons adjust themselves quickly and one can use the adiabatic approximation.…”

“…When the energy is very low, as seen in the lower panel, the electron jumps frenetically back and forth between the two atoms due to resonant tunneling. At very low energies the exchange probability as a function of impact parameter is given in terms of the simple formula [38] …”