Koonin & Nauenberg 1989: spontaneous fusion in a deuterium molecule - a starting point for low temperature D+D fusion rate discussions

Find this paper in our Zotero library under NUCLEAR > Nuclear physics basics

Like other papers in this series, this paper was also prompted by the announcement of high fusion rates at ambient temperatures and pressures (Fleischmann & Pons 1989). Koonin & Nauenberg 1989 is a short 2-page paper published in Nature that employs a textbook approach for estimating D+D fusion rates under basic assumptions. Maybe because of this, it has become an often cited starting point for discussions of enhanced D+D fusion rates. 

Koonin & Nauenberg start out by estimating the fusion rate in the Fleischmann & Pons experiment to be on the order of $10^{-12}/s$. This is in stark contrast to the previously estimated fusion rate of D+D at room temperature which the authors give as approximately $10^{-74}/s$.

The authors then go about calculating their own rate. They start with a brief discussion of interatomic potentials for hydrogen molecules (i.e. $H_2$ as well as $D_2$) that could be used to model the Coulomb barrier before they settle on a numerical potential given by Kolos & Wolniewicz 1968 (ex post one can say that a parameterized potential such as Frost & Musulin 1954 yields similar results). This results in an assumed distance between the two nuclei in the molecule of about 0.7 A (= 70 pm) . They then solve for the tunneling probability i.e. the probability that one of the deuterium nuclei of a $D_2$ molecule can be found in close proximity to its neighboring nucleus — which would then prompt a fusion reaction. 

Koonin & Nauenberg’s result is an expected $D_2$ fusion rate of $10^{-64}/s$* — which is 10 orders of magnitude higher than the previous estimate but still 52 orders of magnitude to low to explain Fleischmann & Pons’ claims. However, at the end of their paper, the authors also caution the reader that this calculation ought to be only considered a basic starting point. Specifically, they state: 

These calculations of the fusion of hydrogen nuclei embedded in a metal are only approximate. Screening by the electrons modifies the Born-Oppenheimer potential. Moreover, fluctuations present in many-body situations might significantly enhance fusion rates, although there are limits to the efficacy of this mechanism in equilibrium conditions.

Other authors have picked up this challenge and discussed the role of screening and fluctuations in more detail, and in some cases with quantitative estimates.

*in other words: if one had a deuterium gas bottle and one were to zoom in on a single deuterium molecule, one would need to wait on average $10^{64} s$ to observe a “spontaneous” fusion reaction.