W. G. Penney developed a solvable model of electrons in a periodic lattice with R. de L. Kronig. Their 1931 calculation replaced the complicated crystal environment by a repeating one-dimensional potential, exposing allowed and forbidden electron-energy intervals.[ref]
The Kronig-Penney model makes energy bands concrete: matching wave functions across repeated cells constrains the possible energies. Its value lies in isolating the effect of periodicity, not reproducing every interaction in a real solid.

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