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Helium Fusion Rates

The helium fusion processes divide into two sets: the primary processes, which create isotopes that are in composition multiples of He4, and the secondary processes, which convert carbon-13 and nitrogen-14 into heavier isotopes. Virtually all of the energy created during the burning of helium-4 is released through the primary processes. The secondary processes are responsible for creating isotopes that are not multiples of helium-4, either directly or through the release of neutrons that combine with the nuclei in the gas to create neutron-rich isotopes.

Primary Processes

The rate at which helium is converted into heavier elements is set by the triple-alpha process, which is the process that combines three helium-4 nuclei into a single carbon-12 nucleus. This rate is faster than any of the rates that convert C12 and its products into heavier nuclei, so the triple-alpha process destroys most of the He4 in a gas and generates most of the power generated through helium fusion.

The triple-alpha process is much more temperature dependent than the other primary helium fusion processes. This strong dependence arises because the intermediate state of the reaction, the creation of beryllium-8 through the fusion of two He4 nuclei, is endothermic, and because the Be8 rapidly decays back into helium-4, which makes the equilibrium density of beryllium-8 highly temperature dependent. The strong temperature dependence of the triple-alpha reaction sets a narrow range for the core temperature of a star undergoing helium fusion.

Once carbon-12 is created, it can combine with He4 to give oxygen-16. The oxygen in turn combines with He4 to give neon-20. This fusion chain continues to argon-36. But the rates for each of these processes is considerably lower than for the triple-alpha process when the temperature is above 100 million degrees Kelvin. This means that the C12 created in the triple-alpha process cannot be burned away before all of the He4 in the gas is exhausted. This implies that the end product of helium fusion is predominately C12 for temperatures above 100 million degrees Kelvin.

For temperatures below 100 million degrees, the rate of converting carbon-12 into oxygen-16 exceeds the triple-alpha process. Under this circumstance, all of the C12 created in the triple-alpha process is converted into O16, so that the end product of helium fusion is principally O16.

The reaction rates for heavier elements exceed the reaction rates for the triple-alpha process for temperatures above 400 million degrees Kelvin, but because the C12 + He4fusion rate remains below the triple-alpha rate, a bottle neck is created in the low rate of O16 production. In this temperature regime, the triple-alpha rate still governs the rate at which He4 is burned. The small amount of O16 that is created is rapidly converted into Ne20 and Mg24.

This figure shows the reaction rates for helium fusion. The triple-alpha reaction rate He4(2He4,γ)C12) is calculated for a helium-4 density of 105 gm cm-3. This rate is plotted on the plots for both the primary and the secondary processes. The reader can specify whether the units of temperature are in degrees Kelvin or in kilo-electron volts. The nuclear reaction notation is described at the bottom of the page. More information on how to control the applet is given by the Applet Control Guide.

Secondary Processes

The secondary processes convert C13 and N14, both products of the CNO hydrogen fusion process, into O16 and O18 respectively. While these reactions are unimportant from the standpoint of power generation, because the density of C13 and N14 is small in a star, they do have an effect on the isotopic composition of the gas in a star. In the case of the burning of C13, this isotope of carbon is lost and a neutron is released that can be absorbed by other nuclei to create heavier elements. The creation of O18 from N14 only changes the isotopic composition of the gas, but the O18 can combine with He4 to create Ne21 and Ne22. The creation of a Ne21 nucleus is accompanied by the release of a neutron.

The reaction rates for the destruction of C13 and N14 are both greater than the helium-4 reaction rate, so these elements are burned before the triple-alpha process can consume a significant amount of He4.

The creation of Ne21 is an endothermic reaction, so, as with the triple-alpha process, energy conservation gives this process a strong temperature dependence. Both it and the process that creates Ne22 are insignificant until the temperature rises above 200 million degrees Kelvin. By 300 million degrees Kelvin, both fusion processes burn all of the O18 before the helium-4 is exhausted. Of these processes, the process creating Ne21 is the more important.

A Note on Notation

In the figure a compact notation for nuclear reactions is used. The general form is A(b,c)D, which is equivalent to A + b gives c + D. So the reaction for creating Deuterium is written as H1(H1,γ)H2, which means H1 + H1 gives H2 plus a gamma-ray.

Also note that, depending on the computer fonts available on your computer, the symbol for a neutrino, ν, and the symbol for a gamma-ray, γ, may look very similar. The key for distinguishing them is that the neutrino is involved only in reactions that involve an electron (e-) or a positron (e+).

A Comment on Reaction Rates

The rates given in the figure are based on formulae given in Astrophysical Formulae by Lang.1

1 Lang, Kenneth R. Astrophysical Formulae: A Compendium for the Physicist and the Astrophysicist. 2nd edition. New York: Springer-Verlag, 1980.

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