move a dislocation which would be expanded on by Frank Nabarro and called the Peierls–Nabarro force. In 1929, he studied solid-state physics in Zurich under the tutelage of Heisenberg and Pauli. His early work on quantum physics led to the theory of positive carriers to explain the thermal and electrical conductivity behaviours of semiconductors. He was a pioneer of the concept of "holes" in semiconductors. He established "zones" before Léon Brillouin, despite Brillouin's name being currently attached to the idea, and applied it to phonons. Doing this, he discovered the Boltzmann equations for phonons and the umklapp process. He submitted a paper on the subject for his habilitation, acquiring the right to teach at German universities. Physics Today noted that "His many papers on electrons in metals have now passed so deeply into the literature that it is hard to identify his contribution to conductivity in magnetic fields and to the concept of a hole in the theory of electrons in solids". Academic in exile In 1932, Peierls was awarded a Rockefeller Fellowship to study abroad, which he used to study in Rome under Enrico Fermi, and then at the Cavendish Laboratory at the University of Cambridge in England under Ralph H. Fowler. In Rome, Peierls completed two papers on electronic band structure, in which he introduced the Peierls substitution, and derived a general expression for diamagnetism in metals at low temperatures. This provided an explanation of the hitherto mysterious properties of bismuth, in which diamagnetic properties were more pronounced than in other metals. Due to Adolf Hitler's rise to power in Germany, he elected to not return home in 1933, but to remain in Britain. He declined an offer from Otto Stern of a position at the University of Hamburg. Granted leave to remain in Britain, he worked at the Victoria University of Manchester with funding from the Academic Assistance Council, which had been set up to help academic refugees from Germany and other fascist countries. Most of his immediate family also left Germany; his brother and his family settling in Britain, and his sister and her family, along with his father and stepmother, moved to the United States, where his uncle Siegfried lived. Peierls collaborated with Bethe on photodisintegration and the statistical mechanics of alloys when challenged by James Chadwick. Their results still serve as the basis for mean-field theories of structural phase changes in complete alloys. Although