Where I can find a table with energy dependent cross sections for neutrons?
[Physics] Energy-dependent neutron cross section data
neutronsscattering-cross-section
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There are two different answers: the growth of the PDF's, or the Regge trajectory of the Pomeron.
Roughly, the reason why the proton-proton cross section grows with Mandelstam $s$ is because the parton density functions (PDF's) of a proton, in particular it's gluon PDF, grows faster than $s$, thus outcompeting the natural fall off rate of $\frac{1}{s}$ of a two particle cross section. The growth of the electron PDF's on the other hand do not outcompete the $\frac{1}{s}$ falloff rate.
A completely different answer comes from a different framework for QCD called Regge kinematics. The high energy cross section of QCD is controlled by a reggeon called the pomeron. Because the Regge trajectory has a positive slope, the cross section rises. In a sense, this is just repeating the statement that the cross section rises, but one can calculate the slope of the Regge trajectory of the Pomeron (see the BFKL equation) and find that it is indeed positive.
If you don't know what a parton density function is: The parton density function of a hadron is a measure of how many particles it contains. You might complain that a proton only has three particles, two up quarks and a down, but this description is only accurate at low energies and is mainly useful for hadron spectroscopy (classifying the hadrons). When calculating a cross section in QCD, the framework is such that you presume that there is some likelihood of finding any particle inside of a proton, for example $P_{\gamma}(E_{\gamma})$ may denote the likelihood of finding a photon of energy $E$ inside of the proton. Without going into too many details, the intuition is obvious, the more particles inside of a hadron, the greater the cross section.
Scattering cross sections can have other parameters besides angle. For example you commonly have cross sections vs. angle and final energy, $d\sigma/d\Omega\, dE_\text{final}$. This might reflect the fact that in a generic elastic scattering process forward-scattered particles tend to retain most of the beam energy, while backwards-scattered particles must deposit a lot of energy and momentum in the target.
If you're doing purely elastic scattering from a crystal, for example Bragg scattering, the cross section $d\sigma/d\Omega$ will essentially vanish except for some very particular angles. It's the values of those magic angles that are useful, and the relative strengths of scattering at one angle versus another. The fact that the entire incident beam has to go somewhere can sometimes be a useful check, but more often the unscattered / forward-scattered beam gets sent to some beam dump and ignored.
Best Answer
Neutron-matter interaction cross-sections vary somewhat depending on the material in question, and depends greatly on energy. In addition, the type of neutron-matter interaction that statistically dominates depends on the energy, with elastic collision being the sole contributor to material cross-sections below energies of 4 MeV:
If you're talking about elastic scattering, another thing to note is that protons are really good at thermalizing neutrons via elastic interaction, a fact which owes primarily to the fact that neutrons and protons have similar masses. In contrast, scattering by larger atomic nuclei results in less effective thermalization:
A good approximate parametrization of the neutron-hydrogen elastic cross-section in units of barns is given by $$\sigma(E)=\frac{3 \pi }{(0.13 E+0.4223)^2+1.206 E},$$ as shown here:
The slides are taken from Wolf-Udo Schroeder's PHY466 notes at University of Rochester.