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Gadolinium-157 has the highest thermal-neutron capture cross-section among any stable nuclide. Based on my layman's understanding of neutron capture – particularly the fact that (I assume) neutrons don't much care about electron shells – I would have guessed that neutron capture cross-section would be proportional to either:

  1. Size of the nucleus. I.e., strictly increasing with (N+Z).
  2. Density of the solid.

Gadolinium is middling with respect to both properties. So what is it about the phenomenon of neutron capture that results in the cross-sectional peak for promethium through gadolinium?

Neutron Cross Section of elements by atomic number

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Welcome to the world of nuclear physics, where the answer is "It's a little more complicated than that."

  1. Density of the solid

You can rule this out: cross sections are tabulated per target atom.

  1. Size of the nucleus, i.e., strictly increasing with (N+Z).

This is a good guess, but you miss an important feature of thermal neutron physics: the relevant size parameter isn't the diameter of the nucleus, but the size of the neutron's wavepacket --- whose scale parameter is something like the neutron's wavelength. Thermal neutrons have wavelengths of a few angstroms ($1\text{ Å} = 10^{-10}\,\rm m$), many orders of magnitude larger than the physical size of a nucleus.

The actual result has more to do with nuclear structure: in order for there to be a capture reaction, there has to be a final state available to receive the neutron with the correct energy and quantum numbers. If you look at a table of isotopes (see also), you'll find that gadolinium and its lanthanide neighbors are pretty far from any nuclear magic numbers. That means that they have a very high density of nuclear states and are easy to excite --- and it increases the probability that there's a resonance in the nucleus $\rm^{158}Gd^*$ whose energy and quantum numbers overlap with a ground-state $\rm^{157}Gd$ and a milli-eV neutron.

The nuclear structure data file for $\rm^{158}Gd$ cites this 1978 paper in a description of the structure of the resonance. That reference (which I can't access) apparently refers to a resonant state in $\rm^{157}Gd$ with an energy of about thirty milli-eV, which is approximately the energy of a room-temperature neutron. That statement doesn't make sense to me right away, but there is an inflection in the cross-section curve at a thermal-ish energy.

If you look at neutron capture cross sections on a table of isotopes (this link should work)

table of isotopes: neutron cross sections

you can see your promethium-to-gadolinium cluster of high-$\sigma$ isotopes just to the right of the $N=82$ magic number. Midway between the $N=50$ and $N=82$ magic numbers is another very strong absorber, cadmium. You can also see that the elements in the uranium-ish island of stability are also eager neutron absorbers.

There are also pairing effects happening in gadolinium. Nucleons don't like to be alone, so nuclei with odd $N$ or odd $Z$ (or both) are less stable than their even neighbors. Gadolinium, like many heavy even-$Z$ elements, has a whole pile of stable isotopes, but the even-$N$ isotopes are more tightly bound than the odd-$N$ isotopes. If you look at the neutron cross sections for all of the gadolinium isotopes, you can see how desperately the odd-$N$ species want to collect an extra neutron:

isotope   σ (barn)
-------   --------
Gd-152       735
Gd-153     22310
Gd-154        85
Gd-155     60740
Gd-156         1.8
Gd-157    253700
Gd-158         2.2
Gd-159    (unstable)
Gd-160         1.4
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