He obtained a 1851 Exhibition Scholarship from the Royal Commission for the Exhibition of 1851, and submitted his MSc thesis on the "Harmonic Analysis for Alternating Currents" in June 1922. Walker then suggested Cockcroft sit for a scholarship to St. John's College, Cambridge, Walker's alma mater. Cockcroft was successful, winning a £30 scholarship and a £20 bursary awarded to undergraduates of limited means. Metropolitan Vickers gave him £50 subject to his returning after completing his degree. Walker and an aunt made up the balance of the £316 fee. As a graduate of another university, he was allowed to skip the first year of the tripos. He sat the tripos exam in June 1924, achieved a B* as a wrangler, and was awarded his BA degree. Cockcroft married Elizabeth Crabtree on 26 August 1925, in a ceremony at the Bridge Street United Methodist Church in Todmorden. They had six children. The first, a boy known as Timothy, died in infancy. They subsequently had four daughters, Joan Dorothea (Thea), Jocelyn, Elisabeth Fielden and Catherine Helena; and another son, Christopher Hugh John. Ernest Rutherford accepted Cockcroft as a research student at the Cavendish Laboratory on the recommendation of Miles Walker and the director of research at Metropolitan Vickers. Cockcroft enrolled as a PhD student in 1924, with a Foundation Scholarship of St John's College, and a State Scholarship. Under Rutherford's supervision, he wrote his doctoral thesis "On phenomena occurring in the condensation of molecular streams on surfaces", which was published in the "Proceedings of the Royal Society". He was awarded his doctorate on 6 September 1925. During this time he was an assistant to the Russian physicist Peter Kapitza, who was working on the physics of magnetic fields in extremely low temperatures. Cockcroft helped with the design and construction of helium liquefiers. In 1919, Rutherford had succeeded in disintegrating nitrogen atoms with alpha particles emitted from decaying radium atoms. This and subsequent experiments hinted at the structure of atomic nuclei. To explore it further, he needed an artificial means of creating particles with a velocity high enough to overcome the charge of the nucleus. This opened a new line of research at the Cavendish Laboratory. He assigned Cockcroft, Thomas Allibone and Ernest Walton to the problem. They built what became known as a Cockcroft–Walton accelerator. Mark Oliphant designed a proton source for them.