How were neutrons discovered?
In 1932, the physicist James Chadwick conducted an experiment in which he bombarded Beryllium with alpha particles from the natural radioactive decay of Polonium. The resulting radiation showed high penetration through a lead shield, which could not be explained via the particles known at that time.
With the postulate of an uncharged (neutral) particle, of about the same weight as a proton, however, Chadwick's interpretation problems disappeared quite naturally. Thus, his results could be explained within the known laws of nature, in particular with regard to energy and momentum conservation.
Experimental proof was only achieved later, in connection with the impressive discovery of nuclear fission by Meitner, Hahn and Strassmann during the Christmas of 1938.