Rick's Cosmology Tutorial: Appendix A3 Abstract

Derivation of the Reaction Rate for He3 formation from p + D

We have noted in Chapters 13 and 14 of the tutorial that the crucial slowness of the reaction which initiates stellar burning of hydrogen, pp --> D, is due in part to the need for the protons to penetrate the Coulomb barrier before they can react. This being the case, we have queried why the second reaction, i.e. p + D --> He3, should not also be slow, since a similar Coulomb barrier must be penetrated in this case also. Yet these two reactions differ in rate by about a factor of ~10^17. In this Appendix we derive the rate of the p + D --> He3 reaction from first principles. We shall see that it is indeed suppressed by the Coulomb barrier. This is the reason for this reaction rate reducing extremely steeply, effectively switching off, below temperatures of ~10^7 K. We conclude that the 17 orders of magnitude difference in the pp and pD reaction rates must actually be due entirely to the weakness of the weak nuclear force, contrary to the impression given by many popular physics books.

The method used to derive the reaction is rate is similar to that used in Appendix A2 for np capture. A Schrodinger representation of the relevant matrix elements is used. However, rather than deriving analytic expressions for the matrix elements, the Schrodinger equation is integrated numerically. This permits solutions with both the nuclear force and the Coulomb potential present. The integration routine was checked by reproducing the np matrix elements found analytically in Appendix A2.

The process starts by finding a effective potential well representation of the nuclear force which is appropriate for helium-3 by fitting the known binding energy. It is shown that the only relevant matrix element is for the electric dipole interaction between a bound S-wave state and a free P-wave state, both state having net spin 1/2. The Coulomb barrier is found to cause a steep reduction in the matrix elements at lower energies, the energy dependence in this regime being in line with the usual simple analytical approximation (exponential - this is derived).

In one illustrative case the wavefunctions of the bound and free states are plotted against radial distance, both with and without the Coulomb potential. This shows that the energy dependence of the matrix element derives from that of the free state in the region a < r < 20 fm, where a is the range of the nuclear force. The bound state is little affected by the Coulomb potential. Hence, the matrix element depends upon 'how much' of the free state manages to "penetrate the Coulomb barrier" from outside. Note, though, that this does not mean "how much of the free state exists in the interior region r < a". In fact, very little of the free wavefunction does penerate into the region of the nuclear force (r < a), at any energy, and indeed this is the basis of the zero-range approximation. It is actually the size of the free wavefunction within the region where the Coulomb potential is significant (a < r < rc) that matters. The region r < rc is where the free state has negative kinetic energy because V > E.

There is a band of energies (roughly 12 to 25 MeV) where the n-p and p-D matrix elements are of similar magnitude. For other energies the p-D matrix element tends to be much smaller. Its largest value is about half that for n-p. At sufficiently large and sufficiently small energies, the p-D matrix element is orders of magnitude smaller than that for n-p. At low energies, this is the result of the Coulomb barrier.

The cross-section of the p + D --> He3 reaction for a given reaction energy is found from the matrix elements derived. The reaction rate is thus found for that energy. To find the reaction rate at a given temperature, integration over the Maxwell distribution of energies is required (the usual Gamow peak integral). This is carried out and the derived numerical reaction rate against temperature are compared with solutions in the literature. Given the sensitivity to temperature, the overall agreement is very good.

The temperature dependence of the reaction rate includes contributions from several sources (the matrix element, the conversion of the matrix element to a cross section, and the Maxwell distribution). A simple analytic approximation of the Gamow peak integral confirms the numerically derived temperature dependence.

Read Appendix A3 (pdf): Derivation of the reaction rate for Proton-Deuteron Capture to form helium-3

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