Rick's Cosmology Tutorial: Chapter 21 Abstract

Breit-Wigner Reaction Rates, The Synthesis of Carbon and Oxygen
and The Hoyle Coincidence

In Chapter 13 we have studied in some detail how hydrogen undergoes nuclear fusion into helium-4 in the core of stars. In Chapter 17 we have described the sequence of events when the hydrogen in the core becomes exhausted. In this chapter we shall assume that conditions in the core, or in a shell around the core, have become conducive to helium fusion. This means temperatures in the region of 100 to 150 million K.

Helium is fused into carbon-12 via the triple alpha reaction. This proceeds via an unstable intermediate nucleus, beryllium-8. The triple alpha reaction works only because, (a)beryllium-8 has an unusually long half-life (albeit only 10^-16 seconds), and, (b)the capture of the third alpha particle by the unstable beryllium-8 nucleus is a resonant reaction. In this Chapter we show how the reaction rate of resonant reactions can be evaluated from the Breit-Wigner cross section. The reaction rate decreases exponentially with the resonance energy, defined from the reaction threshold. Our derived rates for the double alpha reaction to form beryllium-8, and the subsequent capture of a further alpha particle to form carbon-12, both agree well with published reaction rates.

We also show how to calculate the rate of reactions which, though not resonant, are nevertheless dominated by the tail of a Breit-Wigner cross section. The distinction is that resonant reactions are dominated by the contribution at the resonance energy, whereas the non-resonant Breit-Wigner reactions are dominated by the contribution at the Gamow peak energy. The latter is the peak in the product of the tails of the Maxwell distribution and the Coulomb barrier terms. Thus we are able to derive the rate at which oxygen-16 is formed by capture of an alpha particle on carbon-12. These are in good agreement with published rates. The same approach shows good agreement with published rates for the capture of a proton on lithium-7 to form two alpha particles (one of the pp sequence reactions).

This Chapter provides an introduction to the Hoyle coincidence. We show that the nearest resonances of beryllium-8, carbon-12 and oxygen-16 appear to be positioned just nicely to permit the formation of carbon whilst avoiding all the carbon being burnt into oxygen. These coincidences are illustrated via histograms of the resonance energies compared with the reaction thresholds. The actual rates of the reactions are used to demonstrate that crudely comparable quantities of carbon and oxygen will be produced in stars. This is important to conventional biochemistry, which requires both these elements. If the reaction rates are changed by a couple of orders of magnitude then it is shown that either very little carbon or very little oxygen would result, with serious consequences for the emergence of life as we know it. Whilst two orders of magnitude might seem a rather large change, recall that the reaction rates are exponentially sensitive to the resonance energies. Hence, really quite small changes in the resonance energies would appear to be fatal to life based on conventional biochemistry. The Hoyle coincidence is treated is greater detail in Chapter 8 of the "Critique of the Cosmic Coincidences".

In this Chapter we also consider the reaction rates involving lighter nuclei. These are crudely estimated by analogy with the p-n and p-D reaction rates derived in Appendices A2 and A3. The latter are based on Schrodinger wavefunctions corresponding to a square-well potential description of the strong force, in contrast to the above Breit-Wigner resonance approach. These are very rough and ready estimates, but at least rationalise the approximate order of magnitude of these light nucleus reactions. An important by-product of these investigations is obtained by applying this approach to the reactions which are actually Breit-Wigner dominated. This allows us to evaluate by how much the proximity of a resonance increases these reaction rates. Thus, in the case of alpha capture on carbon-12 to form oxygen-16, a non-resonant reaction, the true (Breit-Wigner dominated) reaction rate is about three orders of magnitude larger than the Schrodinger based estimate. In the case of alpha capture on beryllium-8 to form carbon-12, a resonant reaction, the true (Breit-Wigner dominated) reaction rate is about six orders of magnitude larger than the Schrodinger based estimate.

Read Chapter 21 (pdf): Breit-Wigner Reaction Rates: Introduction to the Hoyle Coincidence

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Dione, one of the larger moons of Saturn, 700 miles across. The image was taken in visible light with the Cassini spacecraft (NASA, European Space Agency and Italian Space Agency)