Nuclear Fusion – Why Else Does the Sun Shine?

There is another way to derive energy from atomic nuclei. Think about a gas of protons. The pressure exerted by the gas increases as the temperature of the gas increases. If you think about a container filled with gas atoms or molecules, the thermal pressure exerted on the walls of the container is due to all the atoms and molecules bouncing off the walls. Remember, temperature is just a measure of the mean kinetic energy of the gas particles. At low temperatures, the electrical repulsion between them prevents the close approach of any two protons. But as the temperature of the gas increases in the core of the Sun, the minimum approach distance decreases, and occasionally two protons approach close enough (10Symbol for minus one exponentSymbol for minus five exponent meters) to each other that the short-range but very strong nuclear attractive force will bind or fuse the protons together.

Three-part figure illustrating the fusion process. In part one, labeled 'cool gas', two blue spheres labeled p+ represent protons, each with a curved arrow protruding to represent their path. The paths curve to bring the two particles closer as they pass near to each other, but they still remain separated. In part two, labeled 'warm gas', the two protons pass nearer to each other but electrical repulsion still keeps them separated. In part three, labeled 'hot gas, enabling fusion', the two protons finally approach each other closely enough to interact. Both are still represented as blue spheres labeled p+, with arrows pointing toward each other and meeting in a bright yellow star shape (indicating that fusion occurs and energy is released). The final state of the two input protons is shown as a small purple sphere labeled neutrino, a blue sphere right next to a green sphere (representing a proton and a neutron), and a bright purple star shape labeled e+ and e- with an arrow with a sine wave as the body, labeled gamma for gamma-ray. Next to the purple star shape a label says 'Positron (e+) and an electron (e-) combine and annihilate, producing a gamma-ray. These are all shown with arrows pointing away from the yellow star to indicate that they are the final state of the process.
[NMSU, N. Vogt]

When two protons fuse, a chargeless, perhaps massless, particle called a neutrino (Symbol for the Greek letter nuSymbol for zero in the superscript) is spit out, and one of the protons emits an anti-matter particle called a positron (eSymbol for plus sign in the superscript) – like a positively charged electron. Having lost its electrical charge, the proton becomes a neutron. The shorthand for this reaction is:

Symbol for 1 in the superscriptHSymbol for 1 in the subscript + Symbol for 1 in the superscriptHSymbol for 1 in the subscript Symbol for right-pointing arrow to indicate 'implies' Symbol for 2 in the superscriptHSymbol for 1 in the subscript + eSymbol for plus sign in the superscript + Symbol for Greek letter nuSymbol for zero in the superscript

There is a short string of similar reactions that go on in any hot gas of hydrogen atoms. This is called the proton-proton cycle or the P-P cycle. Four protons are combined to form a helium atom, with by products of two positrons, two neutrinos, and released energy in the form of gamma rays.

Step I: Symbol for 1 in the superscriptHSymbol for 1 in the subscript + Symbol for 1 in the superscriptHSymbol for 1 in the subscript Symbol for right-pointing arrow to indicate 'implies' Symbol for 2 in the superscriptHSymbol for 1 in the subscript + eSymbol for plus sign in the superscript + Symbol for Greek letter nuSymbol for zero in the superscript
Step II: Symbol for 2 in the superscriptHSymbol for 1 in the subscript + Symbol for 1 in the superscriptHSymbol for 1 in the subscript Symbol for right-pointing arrow to indicate 'implies' Symbol for 3 in the superscriptHeSymbol for 2 in the subscript + Symbol for Greek letter gamma
Step III: Symbol for 3 in the superscriptHeSymbol for 2 in the subscript + Symbol for 3 in the superscriptHeSymbol for 2 in the subscript Symbol for right-pointing arrow to indicate 'implies' Symbol for 4 in the superscriptHeSymbol for 2 in the subscript + Symbol for 1 in the superscriptHSymbol for 1 in the subscript + Symbol for 1 in the superscriptHSymbol for 1 in the subscript + Symbol for Greek letter gamma

The whole process can be summarized as one net reaction:

4 Symbol for 1 in the superscriptHSymbol for 1 in the subscript Symbol for right-pointing arrow to indicate 'implies' Symbol for 4 in the superscriptHeSymbol for 2 in the subscript + 2 eSymbol for plus sign in the superscript + 2 nuSymbol for zero in the superscript + Energy (Symbol for Greek letter gamma)

Four blue spheres labeled p+ have arrows pointing towards a location between them all, marked with a bright yellow star to indicate that an energetic process occurs there when these input particles interact. The outputs are shown with outward pointing arrows: a tight cluster of two green spheres and two blue spheres labeled Helium^4_2; two tiny purple spheres labeled Neutrinos; two particles labeled e+; three arrows with sine wave bodies labeled gamma for gamma rays.
[NMSU, N. Vogt]

Here is the amazing thing about the fusion process:

Mass (4H) = 6.6943 × 10Symbol for minus two in the exponentSymbol for four in the exponent grams
Mass (Symbol for 4 in the superscriptHeSymbol for 2 in the subscript + 2 eSymbol for plus sign in the superscript + 2 nuSymbol for zero in the superscript) = 6.6466 × 10Symbol for minus two in the exponentSymbol for four in the exponent grams

The difference in masses means that 0.048 × 10Symbol for minus two in the exponentSymbol for four in the exponent grams of matter disappear in the P-P Chain. This mass has been converted into energy (gamma rays). For one reaction, the energy released is:

E = 0.048 × 10Symbol for minus two in the exponentSymbol for four in the exponent grams × c
= 4.8 × 10Symbol for minus two in the exponentSymbol for six in the exponent grams × cSymbol for two in the exponent
= 4.8 × 10Symbol for six in the exponent grams × (3 × 10Symbol for one in the exponentSymbol for zero in the exponent )Symbol for two in the exponent cmSymbol for two in the exponent / secSymbol for two in the exponent
= 4.3 × 10Symbol for minus five in the exponent grams cmSymbol for two in the exponent / sec
= 4.3 × 10Symbol for minus five in the exponent ergs

How much energy is contained within an erg?


For every 4 hydrogen atoms (6.7 × 10Symbol for minus two in the exponentSymbol for four in the exponent grams) we get this much energy out of the reaction, so we can calculate the efficiency of the P-P Chain, and compare it to the efficiency of coal power and gravitational potential energy:

6.4 × 10Symbol for one in the exponentSymbol for eight in the exponent ergs/gram from the P-P Chain
1 × 10Symbol for one in the exponentSymbol for five in the exponent ergs/gram from GPE conversion
4 × 10Symbol for one in the exponentSymbol for two in the exponent ergs/gram from chemical burning

Thanks to Mike Bolte (UC Santa Cruz) for the base contents of this slide.