Rick's Cosmology Tutorial: Appendix A2 Abstract

Derivation of Neutron-Proton Capture Cross Sections

In this Appendix we derive the cross-section for neutron capture on a proton (or vice-versa) and the closely related cross-section for photodisintegration of a deuteron into a neutron and a proton. The derivations are carried out so as to demonstrate that the resulting formulae are equally applicable in an alternative universe in which the strong nuclear coupling constant, gs, has a different magnitude.

The method is based on evaluating explicitly the relevant Schrodinger wavefunctions, for both free and bound states. A feature of the bound state wavefunctions is that they continue to have substantial magnitude even well beyond the range of the nuclear potential (i.e. for r > a). This is because the Schrodinger equation does not allow the wavefunction to fall suddenly to zero.

Matrix elements of appropriate interaction Hamiltonians are calculated between these Schrodinger states. The analytical approach is tractable in the so-called 'zero range' limit, for which the range, a, of the nuclear potential is allowed to shrink to zero whilst adjusting V0 so that the deuteron binding energy is held constant. In this limit the nucleons lie (paradoxically) entirely outside the range of the potential. Thus, the nucleons are bound at a separation for which there would classically be no force.

Fermi's Golden Rule is used to translate the matrix element into a cross-section. Cross-sections are derived for both a magnetic dipole interaction, in which the magnetic field of a photon couples with the magnetic moment of a deuteron, and also for an electric dipole interaction, in which the electric field of the photon couples with the charge of the proton. The resulting cross-sections are compared with standard formulae from the literature. Numerical results are also given. The magnetic dipole cross-section is generally dominant.

This very detailed derivation of the np capture cross sections drives home some important messages. The first is that the magnetic dipole matrix element is proportional to the difference between the proton's and the neutron's gyromagnetic ratios. Thus, the magnetic dipole cross-section for np capture is non-zero only because the gyromagnetic ratio of the proton and the neutron are different. Secondly, the electric dipole matrix element between free and bound S-waves would be zero by orthogonality of states of different energy if the spin states were the same. A non-zero matrix element between free and bound S-waves occurs only if the states are of different spin and the nuclear potential is different for the two spin states. Hence, the electric dipole cross-section is non-zero only because the nuclear potential differs for singlet and triplet spin states.

These points are laboured because of the implications for proton-proton capture, considered in detail in Appendix A4. We shall see that both the magnetic dipole and the electric dipole pp capture cross-sections are zero, causing pp capture rates to be far slower than np capture rates due to reliance on quadrupole interactions.

Read Appendix A2 (pdf): Derivation of the Neutron-Proton Cross Section

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