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In physics textbooks with chapters about nuclear fission there is often a historical introduction about Otto Hahn. That he tried to generate transuranium but discovered the nuclear fission.

If you start with uran (atomic number 92) you need to increase the atomic number i.e. the number of protons in the nucleus. However if you just add (naively) a neutron you just and up with another isotope (but not with another element i.e. not transuranium).

So which facts were known in 1938 to Otto Hahn which lead him to try neutrons to generate transuranium? What was his reason to try it with neutrons?

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See the Wikipedia article on neutron capture. The aim to to generate an unstable nucleus with an extra neutron that decays by beta emission to the element with one higher atomic number. This is one of the ways heavy elements are made in stars.

With neutron capture there is no barrier due to electrostatic repulsion so it can occur at relatively low energies.

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