# How often does nuclear fusion occur within the human body?

I'm just curious. I figure atoms fuse occasionally just by chance, like quantum tunneling or rogue waves. Is this true? If so, any idea how often?

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I don't know numbers but I'd expect the answer to be something like "maybe once in the entire history of the universe" :-P So, pretty rare. I'm sure someone else can be more precise about it. – David Zaslavsky Apr 13 '11 at 6:22
You should ask the same question for a say, 50 kg mass of sun at sun core. – Georg Apr 13 '11 at 16:12

This'll be a very rough order of magnitude estimate, but as you'll see it's good enough.

Suppose that two hydrogen atoms bump into each other. In order to fuse, the nuclei have to tunnel to within about a nuclear distance of $10^{-15}$ m of each other. The tunneling probability is something like $e^{-(2mE)^{1/2}L/\hbar}$, where $E$ is the energy gap, $m$ is the particle mass, and $L$ is the distance. The distance is of order $10^{-10}$ m (a Bohr radius) and the energy is about an MeV (the electrical potential energy of two protons right next to each other. I work out the numbers to get a probability of about $e^{-20000}$.

You'd next have to multiply that by the number of "chances" (number of times two atoms collide with each other). That's a large number by ordinary standards, but it's not exponentially large in the same way that the probability is exponentially small. Say you've got $10^{29}$ atoms in you, and each one collides with something else $10^{10}$ times per second. Then the number of chances per second is a mere $10^{39}$. I made up that number $10^{10}$ out of nowhere, but whatever it is, it's not $10^{10^4}$, which is what it'd have to be for there to be any significant probability.

So it never happens.

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Wikipedia says the reaction rate "increases from virtually zero at room temperatures up to meaningful magnitudes at temperatures of 10–100 keV." Is that something different? And humans are not just bags of hydrogen. We have trace amounts of uranium and thorium in us decaying, producing high-energy particles, etc. – endolith Apr 13 '11 at 18:06
That temperature range is $10^8$ to $10^9$ K. If you want to heat yourself up that hot, maybe you'll manage to fuse things! Your second point is probably right: if fusion events do occur within your body, they'll be due to high-energy particles. I don't immediately know how to do a sensible calculation of that, but the fast-moving particles are not of the sort that would usually want to fuse: alpha particles are already the stablest kind of nucleus in their mass range, for instance. – Ted Bunn Apr 13 '11 at 18:50
keV is a unit of energy, not temperature. An individual particle can travel with that kinetic energy and fuse even if all the other particles nearby average out to room temperature, no? Like "rogue waves" where statistically the average is body temperature, but a single particle randomly gets raised to a high enough energy to fuse. Extremely unprobable? Can't high energy particles (neutrons from decay or cosmic rays) collide with other particles (hydrogen) and bring them up to high energy levels? – endolith Apr 13 '11 at 21:40
Physicists often quote temperatures in energy units. When someone says the temperature is 10 keV, they really mean $k_{\rm B}T=10$ kev, where $k_{\rm B}$ is Boltzmann's constant. – Ted Bunn Apr 13 '11 at 21:53
Sure, you should really integrate over the Maxwellian energy distribution, but it doesn't matter. By the time you get up into the temperature range where the tunneling probability rises significantly, the number of particles is down by something like $e^{-E/kT}\sim e^{-10^5}$. – Ted Bunn Apr 14 '11 at 13:12
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Do not hold your breath. The probability is infinitesimally small, though I do not think anybody has calculated it.

In simpler systems, like crystals, people have attempted tunneling like calculations in trying to propose cold fusion. Crystals have the possibility of coherent behavior and some models have been proposed, but imo hand waving. There was a discussion a while ago here.

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