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In a fascinating 30 June 2013 article in Nature Chemistry, researchers from the University of Leeds found that when molecules of hydroxyl (OH, a fairly stable radical) and methanol (CH$_3$OH) are cold enough to adhere to each other in deep space instead of bouncing apart, they combine to form methoxy molecules (CH$_3$O) and water at rates hundreds of times higher than at room temperature.

The reason for this unexpected increase in reactivity is both elegant and simple: By being cold enough to cling together loosely for long periods, the integral of very low quantum tunneling rates becomes large enough to enable appreciable rates of combination of the two molecules, despite the reaction having a high classical energy barrier.

Now what I find interesting is that there is a simple equivalent of this "let's stay close for a while" situation in nuclear chemistry. Specifically, it is the formation of an ordinary chemical molecule of two nuclei that have a high energy potential for fusion. The chemical bond becomes the analog of the gentle "let's say close" binding of cold hydroxyl and methanol molecules, and the nuclear fusion reaction becomes the analog of the exothermic recombination of the two molecules, albeit with an enormously higher energy barrier.

This analogy leads to only two or three interesting candidates: TH, tritium hydride; TD, tritium deuteride; and T$_2$, a molecule of two tritons.

TH would seem by far the most interesting, since the probabilities for protons to tunnel over atomic distances are orders of magnitude higher than those of deuterons or tritons. The details will be complicated by the larger fusion cross sections of D-T and T-T in comparison to H-T, however.

So my question: Has anyone ever specifically looked for this effect -- that is, for higher rates of spontaneous fusion in TH that can be attributed to increased tunneling probabilities due to the proton being kept at an atomic distance from the triton for an indefinitely long time?

I did a cursory online-only look for relevant articles, without much success. Nonetheless, my best guess is that spontaneous TH fusion rates were likely calculated out to the $n^{th}$ degree decades ago, and likely experimentally verified to the same degree. After all, everything in nuclear chemistry is about tunneling probabilities.

Nonetheless, the close analogy of long-term atomic-distance binding of nucleons to this more recent finding on increased tunneling between cold-bound molecules is intriguing. I was curious whether anyone out there is familiar with the topic and can talk about it in this forum.

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[Rewrote the answer because I found out that my initial approximation was too crude.]

In the WKB approximation, the tunneling probability is $\exp[-\int_a^b dx \sqrt{(2m/\hbar^2)(V-E)}]$, where the integral is over the classically forbidden region from $a\sim10^{-15}$ m to $b\sim 10^{-10}$ m. The first obvious thing to try is approximating the integrand as a constant, the width of the classically forbidden region as $b$, $m$ as the mass of a proton, and $V-E$ as the nuclear energy scale of ~1 MeV. This gives a tunneling probability of $\exp(-10^4)$.

But this is a little too crude, because V is strongly peaked within a small range of $x$. One way to tell that this isn't a good enough estimate is that the answer didn't depend on $E$ because $E\ll V$, whereas in reality the sun really does shine, because the particles in the sun's core have enough energy.

It turns out that the WKB integral can be evaluated for a Coulomb barrier, and the result, known as the Gamow factor, is $\exp[-\pi\sqrt{2m/\hbar^2}(V_{max}a/\sqrt{E})]$, where $V_{max}$ is the maximum height of the barrier. In our case this comes out to be about $e^{-1000}$, which although a lot bigger than $\exp(-10^4)$, is still ridiculously small. Note that this result does depend strongly on $E$, even for $E\ll V$.

In a tritium hydride molecule you do get the benefit of repeated barrier assaults at a high frequency. The frequency of barrier assaults might be $10^{14}$ Hz or something (i.e., molecular vibrations have frequencies that lie roughly in the infrared range). But clearly when you multiply the two factors, you get something that will never have happened in the observable universe in the time since the big bang.

Essentially the same estimate explains why cold fusion doesn't work.

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    $\begingroup$ 1nm is a 10X overestimate of how far apart the nuclei are. 1 angstrom is closer. Googling suggests 0.74 angstroms (0.074nm) as the H-H separation in H$_2$. $\endgroup$ – Steve Byrnes Aug 11 '13 at 21:01
  • $\begingroup$ @SteveB: Thanks for pointing out my mistake. When I wrote 1 nm it was a mistake; I had actually calculated the estimate using 0.1 nm. The estimate of $\exp(-10^4)$ was the correct estimate using 0.1 nm. $\endgroup$ – Ben Crowell Aug 11 '13 at 21:23
  • $\begingroup$ Steve, thanks, that's a really nice example "fair use" of ballpark approximations! Here's an example of a 1958 B.E. Conway paper that analyzes phenomena in which protons seem to tunnel over atomic distances: nrcresearchpress.com/doi/pdf/10.1139/v59-025. Am I correct that the key difference between these phenomena is that long-distance proton tunneling phenomena are enabled by the very low chemical energy barriers that must be crossed, versus the astronomically larger ones that must be crossed (albeit over shorter distances) for nuclear fusion within a single bound pair of nucleons? $\endgroup$ – Terry Bollinger Aug 12 '13 at 2:12
  • $\begingroup$ It's fascinating that even though acid chemistry requires protons to tunnel easily over several ångströms, a proton cannot tunnel over the much shorter distance to the triton in a TH molecule. That's of course because the tunneling probability is not just a function of XYZ distance, but also of energy barriers. The proton-proton repulsion barrier for TH tunneling is astronomical, while in chemistry it is XYZ distance that dominates the probability. If the proton-proton repulsion could be "translated" into an XYZ-like distance, I suspect the triton would appear to be meters away. $\endgroup$ – Terry Bollinger Aug 13 '13 at 20:05
  • $\begingroup$ Also, just to mention it: Proton tunnling keeps you alive. If you don't believe it, try giving some tiny critter heavy water for a few days and see what happens. Deuterons are just below the threshold of easy tunneling over molecular distances, which changes their chemistry (versus protons) rather dramatically. Consequently, deuterium is toxic to living organisms as a direct result of quantum effects. Viva la proton tunneling! $\endgroup$ – Terry Bollinger Aug 13 '13 at 20:09

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