John Van Vleck: Quantum Theory and Magnetism

By David L. Huber

At the APS March Meeting on Tuesday, March 22, 2011, the FHP honored the contributions of John Hasbrouck Van Vleck to quantum theory and magnetism. The session was chaired by Chun Lin, from the University of Wisconsin-Madison and its invited speakers were Michel Janssen, University of Minnesota; David Huber, University of Wisconsin; Nicolaas Bloembergen, University of Arizona; Charles P. Slichter, University of Illinois, Urbana-Champaign; and Horst Meyer, Duke University. Among the attendees were Dr. and Mrs. John P. Comstock. Dr. Comstock, the son of Abigail Van Vleck’s sister, inherited a collection of Van Vleck memorabilia from his aunt and has recently donated it to the University of Wisconsin.

The session began with Chun Lin presenting a brief outline of Van Vleck’s life and career. He was born in Middletown, Connecticut in 1899 and grew up in Madison, Wisconsin where his father, Edward B. Van Vleck, was a professor of mathematics at the University of Wisconsin. John Van Vleck received his bachelor’s degree in physics from the university in 1920. He then went to graduate school at Harvard University, where he received his PhD in 1922. After finishing his degree, he accepted a faculty position in physics at the University of Minnesota, where he remained until 1928 when he took a similar position at the University of Wisconsin. Van Vleck was at Wisconsin from 1928 until 1934. In 1934, he left to become a professor at Harvard University and was a member of the Harvard faculty from 1934 to 1969. He was President of the American Physical Society in 1952, received the National Medal of Science in 1966, and in 1977 shared the Nobel Prize in Physics with his former student Philip Anderson and Sir Nevill Mott.

The first invited talk was Michel Janssen’s lecture on “Van Vleck from Spectroscopy to Susceptibilities: Kuhn Losses Regained.” He pointed out that the old quantum theory, which Van Vleck had applied to the interpretation of optical spectra, did not give results for electric and magnetic susceptibilities that were compatible with confirmed results obtained with the classical theory – a situation referred to in the history and philosophy of science literature as “Kuhn Losses”. While at the University of Minnesota, Van Vleck pointed out that this situation was remedied when the susceptibilities were calculated in the new quantum theory using matrix mechanics. At Minnesota, he also developed a general theory for the magnetic susceptibilities of atoms and ions that showed that in addition to the standard Curie term, there was a temperature-independent term arising from matrix elements of the Zeeman interaction between the ground manifold and excited manifolds.

David Huber’s talk focused on Van Vleck’s investigation of magnetic susceptibilities while he was at Wisconsin. The talk began with a discussion of the work of Van Vleck and his graduate student Amelia Frank, who applied the general theory of magnetic susceptibilities developed at Minnesota to trivalent rare earth ions. He and Frank showed that the temperature-independent term was the dominant term at low temperatures for Eu3+ where the total angular momentum of the ground manifold was zero. They also showed that the temperature-independent term, often referred to as Van Vleck paramagnetism, made a significant contribution to the susceptibility of Sm3+. The calculations of Van Vleck and Frank were done for isolated atoms and ions. In 1932 Van Vleck and his post-doctoral students Bill Penney and Robert Schlap published a series of three papers on the influence of the electrostatic fields coming from neighboring ions on the magnetic susceptibilities of rare earth and transition metal ions. Utilizing a theoretical approach developed by Hans Bethe (1929), they were able to account for hitherto unexplained differences between the susceptibilities of the free ions and the susceptibilities of the corresponding ions in crystalline environments.

The year 1932 also marked the publication by Oxford University Press of Van Vleck’s classic monograph, The Theory of Electric and Magnetic Susceptibilities, which had a major impact on the investigation and understanding of the properties of magnetic materials. At the time of his death, Van Vleck left an incomplete set of notes for a second edition of his book which were given to Chun Lin by his widow Abigail Van Vleck. Although the notes are largely fragmentary, there is an entirely new chapter on the local field that would have replaced the chapter contrasting the susceptibilities in the old and new quantum mechanics. A PDF of the new chapter is available on the University of Wisconsin Physics Department web page,
http://www.physics.wisc.edu/vanvleck/.

Nicolaas Bloembergen, who for many years was Van Vleck’s colleague at Harvard, spoke about Van Vleck’s role in the development of the theory of magnetic resonance line widths in solids when there are significant exchange interactions between the resonating spins. Van Vleck showed that the presence of the exchange interactions led to a decrease in the dipolar line width relative to the value it would have in the absence of such interactions – a phenomenon known as “exchange narrowing”. This result was obtained by an analysis of the second and fourth moments of the line shape function. Van Vleck found that the exchange interaction did not contribute to the second moment, only to the fourth. Bloembergen also spoke of the important role that Van Vleck played in his own career at Harvard and his interactions with Dutch physicists such as C. J. Gorter and H. A. Kramers. John Van Vleck was awarded the Lorentz Medal by the Royal Netherlands Academy of Arts and Sciences in 1974.

The title of Charles Slichter’s lecture was “Remembering Van: Three Madison Families and other Tales.” In it, he spoke about the influential roles played at the University of Wisconsin by his grandfather, Charles S. Slichter, and the fathers of John Bardeen and John Van Vleck. The tale begins in 1903 when Charles Van Hise, a distinguished geologist, was named President of the University of Wisconsin. In 1904, Van Hise recruited Charles Bardeen, John’s father, to found a medical school at the University. In 1906, Van Hise appointed Slichter to head the mathematics department. Slichter’s first action as department head was to recruit Edward B. Van Vleck to bring strength in pure mathematics. John Bardeen and John Van Vleck did their undergraduate work at Wisconsin, finishing in 1920 and 1926, respectively. After Bardeen finished his Master’s in engineering, Van Vleck provided guidance and help, recommending him to Trinity College, Cambridge University for a fellowship and later to Harvard University for appointment as a Junior Fellow. Charles Slichter, who did his undergraduate work at Harvard, had Van Vleck as an advisor. Van Vleck recommended he remain at the university for his Ph. D. and later suggested that he do his doctoral research on magnetic resonance with Edward Purcell.

Horst Meyer spoke about Van Vleck’s work on the magnetic susceptibility of the clathrate compounds and how it was stimulated by the measurements carried out at the Clarendon Laboratories, Oxford University by A. H. Cooke, W. P. Wolf and himself. By trapping paramagnetic molecules such as O₂ in clathrate cages, they were able to extend measurements of the susceptibility temperatures below 1 K. The data from these measurements were the first test of earlier theoretical predictions of Van Vleck on the low temperature behavior of the susceptibilities of free molecules. Not surprisingly, it was found that there was a discrepancy between experiment and theory. This discrepancy stimulated collaboration between Van Vleck and Mary O’Brien from Oxford who was on sabbatical at Harvard at that time. O’Brien and Van Vleck showed that the discrepancy was due to the free molecule approximation and disappeared when a ‘hindering potential’ was taken into effect. Later measurements by Meyer and colleagues involving electron spin resonance and infrared absorption confirmed the magnitude of the hindering potential inferred by Van Vleck and O’Brien in O2 and NO. In later years, Van Vleck became interested in the properties of rare-earth iron garnets and worked with Meyer on the interpretation of his calorimetric and NMR experiments.

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