The Klein-Nishina Formula

In Chapter 15 of the 4^th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I present the Klein-Nishina formula (Eq. 15.17).

The inclusion of dynamics, which allows us to determine the relative number of photons scattered at each angle, is fairly complicated. The quantum-mechanical result is known as the Klein-Nishina formula.

At first glance, Eq. 15.17 doesn’t look quantum-mechanical, because it does not appear to contain Planck’s constant, h. However, closer inspection reveals that the variable x in the equation, defined on the previous page (Eq. 15.15), does indeed contain h. Russ and I don’t derive the Klein-Nishina formula, nor do we give much background about it. Yet, this equation played an important role in the development of quantum mechanics, and specifically of quantum electrodynamics.

In the book Nishina Memorial Lectures: Creators of Modern Physics, the Nobel Prize winning physicist Chen Ning Yang wrote a chapter about “The Klein-Nishina Formula and Quantum Electrodynamics.”

One of the greatest scientific revolutions in the history of mankind was the development of Quantum Mechanics. Its birth was a very difficult process, extending from Planck’s paper of 1900 to the papers of Einstein, Bohr, Heisenberg, Schrodinger, Dirac, and many others. After 1925–1927, a successful theory was in place, explaining many complicated phenomena in atomic spectra. Then attention moved to higher energy phenomena. It was in this period, 1928–1932, full of great ideas and equally great confusions, that the Klein-Nishina formula played a crucial role. The formula was published in 1929, in the journals Nature and Z. Physik. It dealt with the famous classical problem of the scattering of light rays by a charged particle…

Oskar Klein and Yoshio Nishina derived their formula starting from the Dirac equation, which is a relativistic version of Schrodinger’s equation for an electron, including the effect of spin. During the summer of 1928, Klein and Nishina performed the lengthy calculations necessary to derive their formula. They would work independently during the day, and then compare results each evening (as Russ and I say, the calculation is “fairly complicated”). The final result was published in the German journal Zeit. f. Phys. (“Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac,” Volume 52, Pages 853–868, 1929). I don’t read German, so I can’t enjoy the original.

Later, the theory of Quantum Electrodynamics (QED) was developed to more completely describe the quantum mechanical interactions of electrons and photons. For an elementary introduction to this subject, see Richard Feynmann’s book QED. (Although I took several semesters of quantum mechanics in graduate school, I never really mastered quantum electrodynamics.) When the problem of the scattering of light by electrons was reexamined using QED, the result was identical to the Klein-Nishina formula derived earlier. To learn more about how these results were obtained, see “The Road to Stueckelberg's Covariant Perturbation Theory as Illustrated by Successive Treatments of Compton Scattering,” by J. Lacki, H. Ruegg, and V. Telegdi (http://arxiv.org/abs/physics/9903023). But beware, the paper is quite mathematical and not for the faint of heart.

Who were the two men who derived this formula? Oskar Klein (1894–1977) was a Swedish theoretical physicist. He is known for the Kaluza-Klein theory, the Klein-Gordon equation, and the Klein paradox. Yoshio Nishina (1890–1951) was a Japanese physicist. He was a friend of Niels Bohr, and a close associate of Albert Einstein. The crater Nishina on the Moon is named in his honor. During World War II he was the head of the Japanese atomic program.

Let me share one last anecdote about Klein, Nishina, and Paul Dirac that I find amusing. Gosta Ekspong tells the story in his chapter “The Klein-Nishina Formula,” in the book The Oskar Klein Memorial Lectures.

When Dirac paid a short visit to Copenhagen in 1928, he met Klein and Nishina. The three of them were once conferring in the library of the Bohr Institute. Dirac was a man of few words, so when the remark came from Nishina that he had found an error of sign in the new Dirac paper on the electron, Dirac drily answered: “But the result is correct.” Nishina, in an attempt to be helpful, said: “There must be two mistakes,” only to get Dirac’s reply that “there must be an even number of mistakes.”