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Recently, I have learn't about the Gamows factor, and I have plotted it. Then I realized that the probability of Deuterium and tritium fusing at a temperature of 150 million kelvin when they collide, was about 0.00048. But, I know that the nuclear cross section for fusion is extremely small (about 1-1000 barns). But why is this so? I f the probability of fusion is 0.00048 when they collide, then why is the nuclear cross section so small? Is it because of the collisions? Are the collisions really that rare? Is the collision cross section really that small? Why is the nuclear cross section so small?

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  • Here the x-axis is the temperature and the y-axis is the probability of fusion when a collision occurs.
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    $\begingroup$ It is funny that you call a "barn" small, because the terminology came from the bulding called "barn", so large you cannot miss it. " "couldn't hit the broad side of a barn refers to someone whose aim is terrible. " en.wikipedia.org/wiki/Barn_(unit)#Etymology " A barn is approximately the cross-sectional area of a uranium nucleus" $\endgroup$
    – anna v
    Mar 12, 2017 at 9:49
  • $\begingroup$ Yeah! I know, but I mean, it still pretty small compared to interactions in cold gases. $\endgroup$
    – Chandrahas
    Mar 12, 2017 at 9:56
  • $\begingroup$ And... if the cross section were a little bit bigger, we would probably have nuclear fusion already. $\endgroup$
    – Chandrahas
    Mar 13, 2017 at 3:40

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A crucial thing to realize is that 'temperature' does not equal particle energy. Temperature is a macroscopic quantity (i.e. a property of a system) while particle energy is a microscopic quantity (i.e. a property of the constituents of the system). The Gamow factor holds for particles, not for the system.

Ions in a (fusion) plasma have a certain energy (velocity) distribution (such as the Maxwellian distribution if the ions are in thermal equilibrium), where the 'temperature' describes some sort of average of the particle energies (you can define temperature in subtly different ways). The crucial thing here is that there is a significant number of particles that have (much) higher energies than the energy corresponding to the temperature of the system ($k_BT$). It is mostly those ions in the high energy tail of the distribution that actually undergo fusion reactions.

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