Coulomb penetration factor

For slow neutron-induced reactions that do not involve resonances, we know (Chapter 10) that ct ( ) °c 1 /vn so that (ctv) is a constant. For charged particle reactions, one must overcome the repulsive Coulomb force between the positively charged nuclei. For the simplest reaction, p + p, the Coulomb barrier is 550 keV. But, in a typical star such as the sun, kT is 1.3 keV, that is, the nuclear reactions that occur are subbarrier, and the resulting reactions are the result of barrier penetration. (At a proton-proton center-of-mass energy of 1 keV, the barrier penetration probability is 2 x 10-10). At these extreme subbarrier energies, the barrier penetration factor can be approximated as ... 

The matrix elements in angle brackets contain nuclear factors and (in the case of charged particles) the Coulomb barrier penetration probabilities or Gamow factors, originally calculated in the theory of a-decay, which can be roughly estimated as follows (Fig. 2.7). 

We discuss briefly the factors that determine the intensity of the scattered ions. During collision, a low energy ion does not penetrate the target atom as deeply as in RBS. As a consequence, the ion feels the attenuated repulsion by the positive nucleus of the target atom, because the electrons screen it. In fact, in a head-on collision with Cu, a He+ ion would need to have about 100 keV energy to penetrate within the inner electron shell (the K or Is shell). An approximately correct potential for the interaction is the following modified Coulomb potential [lj ... 

Here, w r) is the attractive (or repulsive) tail of the potential, a is the diameter of spheres, and we have assumed that the fluid is uniform, therefore translational invariance is implied. The first equality in the above equation embodies the physical requirement that the center of a sphere can not penetrate the excluded volume of other spheres. The second equality is just obtained from (1.25) by linearizing the entire exponential factor. Actually, it is the asymptote of the direct correlation function at the infinite separation. The approximation is known to be superior for describing the critical phenomena. The radial distribution function, however, shows an ill-behavior for a Coulombic system, similar to those from the PY closure. 

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