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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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related: physics.stackexchange.com/q/73899 –  Ben Crowell Aug 12 '13 at 4:18
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related: physics.stackexchange.com/q/73970 (which explains why Ted Bunn's answer is incorrect) –  Ben Crowell Aug 12 '13 at 16:07

3 Answers 3

up vote 12 down vote accepted

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
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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
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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
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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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Your calculation is exactly the same incorrect one that I initially did in answer this similar question: physics.stackexchange.com/q/73899 In fact it's far too crude to estimate the WKB tunneling probability by approximating the barrier as having a constant height equal to its maximum height. To get a halfway decent estimate, you have to use the Gamow formula, as in the revised version of my answer to the other question. –  Ben Crowell Aug 12 '13 at 4:18

If we're hoping for fusion to occur in trees, houses, and our own bodies, then by far the most likely scenario is one that results from natural radioactivity. For example, Brazil nuts contain quite a bit of radium, which is an alpha emitter. If you eat a Brazil nut and it undergoes alpha decay in your body, the most likely fate for the alpha particle is to dissipate all its energy by ionizing atoms, but there is also some smaller probability that it will fuse with one of the nuclei in your body. The latter probability is smaller simply because nuclei are small targets, but it's nonzero and probably does happen during your lifetime.

But the OP probably had in mind spontaneous fusion between stable nuclei, a la cold fusion. Our best fighting chance for getting this to happen is fusion of hydrogen nuclei, which have the lowest electrical repulsion, and we should make the reaction $^1\text{H}+^2\text{H}$, since then the fusion can proceed directly, without requiring the additional low-probability factor of a weak interaction to produce a stable product. (As we'll see below, $^2\text{H}+^2\text{H}$ is less probable for kinematic reasons.)

The tunneling probability $P$ is discussed in this answer. The result is

$\ln P=-\frac{\pi kq_1q_2}{\hbar}\sqrt{\frac{2m}{E}} $ .

At room temperature, the typical kinetic energy is $E\sim kT \approx 0.03$ eV. The reduced mass $m$ for $^1\text{H}+^2\text{H}$ is 2/3 the mass of a nucleon. (For $^2\text{H}+^2\text{H}$ the factor is 1 rather than 2/3, which is less favorable.) The result is $P\sim e^{-5000}$, which is so tiny that the process will never have happened at room temperature anywhere in the observable universe.

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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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