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

Question

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, $$ |a_2| \simeq |a_1| \qquad or \qquad |a_2| \simeq 0.2 |a_1| ? $$

Answer 1

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".)

Answer 2

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.