# How to estimate $d$-wave nuclear scattering cross section?

I would like to estimate the $$\ell=2$$ ($$d$$-wave) contribution to elastic scattering cross section (or scattering length) for collisions such as $$n$$-$$n$$ and $$n$$-$$p$$ at low energy (a few MeV or less). I found the following resource

https://doi.org/10.1016/j.nds.2006.11.001

which gives the total cross section and also quotes scattering lengths $$a_0$$ and $$a_1$$ for n-p scattering, but it does not (as far as I can tell) pull out a value for $$a_2$$. I am hoping for either a known value, or a rough number or even an order-of-magnitude estimate. For example, the above paper says $$a_0 \simeq -23.72\,$$fm and $$a_1 = 5.414\,$$fm so $$|a_1/a_0| \simeq 0.23$$ for n-p scattering. Would it be reasonable to estimate $$|a_2| \simeq 0.2 |a_1|$$? Or is there a resource which would give the answer more precisely?

If $$\sigma$$ is the total cross section (20 barn) then one finds $$\sigma - (\pi a_0^2 + \pi a_1^2) \simeq 1.4\,{\rm barn}$$ so this seems to suggest $$|a_2| \simeq 6.7\,$$fm, which is similar to (in fact a little larger than) $$|a_1|$$. So which is the better estimate, $$|a_2| \simeq |a_1| \qquad or \qquad |a_2| \simeq 0.2 |a_1| ?$$

• I think you mis-interpret these references. By a_1 they mean the scattering length in the spin-triplet s-wave (l=0) channel. The p-wave (and d-wave) contribution to the cross section is of course energy dependent. It vanishes at threshold, and becomes more important at higher energies Dec 21, 2022 at 4:36
• @Thomas yes I realised that but decided to leave the question unchanged since it got a good answer Dec 21, 2022 at 13:21

For unpolarized beam and target, I believe the contribution to the total n-p cross-section from the triplet state is $$3\pi a_1^2$$, not $$\pi a_1^2$$. (See Amsler Eq. 21.37 or Babenko & Petrov Eq. 3.)

The total cross-section $$\sigma_0$$ is only plotted in the ENDF nuclear reaction data library paper, but its value can be found in the source they cite (Dilg 1975). Using the values from these sources: $$\sigma_0=20.491(14)\,\textrm{barns}$$, $$a_0=-23.719(5)\,\textrm{fm}$$, and $$a_0=5.414(1)\,\textrm{fm}$$, we have: $$\sigma_0-\left(\pi a_0^2 + 3\pi a_1^2\right)=0.051(14)\,\textrm{barn}$$

So there isn't convincing evidence for any higher angular momenta contribution; it is certainly very small. This, of course, isn't surprising since low energy np scattering is dominated by the s-wave deuteron pole. (Dilg 1975 seems to have been careful, but there are quite a few corrections and so I use the HEP $$5\sigma$$ criteria for "convincing".)

• Thanks, that is v. helpful. I also found it puzzling that one often sees $\sigma = 4\pi a^2$ but it appears the ENDF paper is not using that convention, which is why I skipped the factor 4. But I was expecting the d-wave part to be larger (of order 1 barn) owing to spin-related effects so this answer is very informative for me. Dec 20, 2022 at 23:51
• I think you mis-interpret these references. By a_1 they mean the scattering length in the spin-triplet s-wave (l=0) channel. The p-wave (and d-wave) contribution to the cross section is of course energy dependent. It vanishes at threshold, and becomes more important at higher energies. Dec 21, 2022 at 4:35
• Thanks @Thomas. You are right, $a_0$ and $a_1$ are for S-wave spin-0 singlet and spin-1 triplet. I have corrected my answer. Dec 21, 2022 at 6:16

I am adding this answer which I found after following the helpful directions indicated by user David Bailey. I found my way to the following websites, both of which provide partial wave phase shift values for n-n and n-p scattering:

https://nn-online.org

https://gwdac.phys.gwu.edu/

The second only gives results for incident kinetic energies above 10 MeV. The first gives plots down to 0 energy but I suspect those plots may be questionable at very low energies. The outcome of using those sites is that I now think the d-wave cross section is extremely small compared to the s-wave cross section at energies below 1 MeV.