# Relationship between time, separation and neutron transfer probability?

As an ansatz, suppose we know that when a smaller nucleus is incident upon a larger one with 1 MeV of kinetic energy, there is a nontrivial probability that a neutron will tunnel from the smaller to the larger:

$$^{12}C(d,p)^{13}C$$

Even if the energy of the incident particle is less than needed to surmount the Coulomb barrier, the neutron can tunnel from one nucleus to the other with a probability $T,$ the tunneling probability. One way to understand this probability is as a function of separation of the nuclei and time that they spend in proximity to one another.

To simplify the ansatz, then, suppose that rather than having the smaller nucleus approach the larger one against Coulomb repulsion, instead it is simply held at a specific distance $d_1$ for a specific amount of time $t_1$ such that the neutron tunneling probability is $T_1,$ and then after $t_1$ expires the smaller nucleus is instantaneously moved far away.

1. With a knowledge of $t_1$ and $d_1,$ is there a straightforward way to obtain the time interval $t_2$ needed for the same interaction to occur with probability $T_1$ at a distance $d_2?$
2. If so, is this result a general one, or does it depend upon such details as the resonances of the interacting nuclei?

EDIT: It occurs to me that this thought experiment is quite relevant to muon-catalyzed fusion.

• Well, for thermal (low energy) neutrons the capture cross section decreases linearly with increasing velocity, that is it is proportional to the time spent traversing the nucleus. Not sure that is quite what you mean... – Jon Custer Sep 9 '15 at 2:49
• That's an interesting observation. But, as you say, it's a little different than what I was wondering about. – Eric Walker Sep 9 '15 at 4:06
• Nucleus-on-nucleus interactions will, unlike neutron-nucleus, have the small problem of the Coulomb barrier to deal with first. But, thinking about the simplest kind of reactions you might mean I come up with the D(d,p)T - deuterium-deuterium fusion producing a proton and a triton. Is that what you meant? That cross section peaks at about 3MeV, then drops off. Not much structure in the D-D cross section, unlike, say $^{12}$C(d,p)$^{13}$C which clearly shows the effects of nuclear energy levels. Given that, I'll say that there won't be an easy answer for nucleus-nucleus interactions. – Jon Custer Sep 9 '15 at 13:06
• I think $^{12}C(d,p)^{13}C$ is the kind of interaction I had in mind. Are you able to elaborate on why the problem is a difficult one? – Eric Walker Sep 9 '15 at 13:10
• How much do you know about nuclear energy level diagrams? One on-line source is TUNL, Triangle-area Universities Nuclear Laboratory, tunl.duke.edu/nucldata/index.shtml. These map out the measured energy levels, and a variety of other nuclear reaction data. – Jon Custer Sep 9 '15 at 13:19