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The elements on either side of neptunium (uranium and plutonium) are, of course, widely used in both power plants and nuclear weapons.

So why isn't the highly fissile nuclide Np-237 used in reactors? Np-236 is also (somewhat) fissile.

Np-237 is often cited as a dangerous 'high-level' waste product of traditional fission reactors. Is there a reason it is not fissioned along with the U-235 and Pu-239 by the same thermal and/or fast neutrons flying around?

With all the recent research and investment into new types of nuclear power, there must be a reason.

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Neptunium-237 is fissile, with a critical mass of around 60 kg, but it has a very low cross-section for thermal neutrons. It needs fast neutrons, and so in a typical power reactor only a tiny percentage of the Np-237 waste will fission.

it has a low probability of fission on bombardment with thermal neutrons, which makes it unsuitable as a fuel for light water nuclear power plants (as opposed to fast reactor or accelerator-driven systems, for example).

It's not easy to separate Np-237 from the other elements in fission waste. Wikipedia says

As of 2009, the world production of neptunium-237 by commercial power reactors was over 1000 critical masses a year, but to extract the isotope from irradiated fuel elements would be a major industrial undertaking.

However, Wikipedia also links to this 2005 press release from Berkeley Lab: Getting the Neptunium out of Nuclear Waste about a Neptunium diamide complex

that could hold the key for improving the extractability of neptunium from other nuclear waste products.

A more recent article on a similar theme is Separation of neptunium from actinides by monoamides: A solvent extraction study.

Processing nuclear waste isn't easy. It requires highly skilled workers, and the radiation tends to cause structural damage to the equipment, and can even make that equipment itself radioactive, especially if the waste contains gamma or neutron emitters.

FWIW, Np-237 is used to make Pu-238 which is used in RTGs (radioisotope thermoelectric generators). Np-237 production has been increased fairly recently, but the quantity being extracted is still fairly small. According to this recent article

By 2024, the DOE plans to produce 1.5 kilograms (3.3 pounds) of Plutonium-238 dioxide per year, enough to fuel a full RTG every 3-4 years. 

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    $\begingroup$ From ENDF, the fission cross sectio of 237Np at thermal energies is about 1 barn, compared with $>10^{4}$ barns for 235U. Just to put some numbers to how low is low… $\endgroup$
    – Jon Custer
    Jul 9 at 15:02

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