Kramers’ Law

When preparing the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a homework problem about Kramers’ Law. (We spelled it Kramer’s, but his name is Kramers with an s, so we should have written Kramers’.) Kramers’ law is Eq. 16.3a, the photon energy fluence dΨ/d(hν) as a function of frequency ν for bremsstrahlung radiation

where Z is the atomic number, h is Planck’s constant, ν_o is the frequency of a photon having the same energy as the incident electrons, and C is a constant. In his paper “On the Theory of X-ray Absorption and of the Continuous X-ray Spectrum” (Philosophical Magazine, Volume 46, Pages 836-871, 1923), Kramers writes

“The continuous x-ray spectrum has in the course of the last years been investigated by a number of physicists. The problem is here to determine how, for a given tension [voltage] on the tube and a given anticathode material [typically tungsten], the energy in the continuous spectrum is distributed among different frequencies…

The object of the present paper is to show how it is possible to account theoretically for the main features of the phenomena of x-ray absorption and continuous x-ray emission discussed above. The explanation of these phenomena may be traced back to the determination of the radiation processes which may occur when a free electron of given velocity approaches a positive nucleus with given charge.”

Who was Kramers? According to the Dictionary of Scientific Biography, Hendrik Anthony (Hans) Kramers was born in Rotterdam, the Netherlands in 1894. He joined Niels Bohr’s Institute of Theoretical Physics, and in 1934 he moved to Leiden University, where he remained until his death in 1952. He is known for many contributions to physics, including the Kramers-Kronig Relations. The Dictionary of Scientific Biography article concludes

“Kramers’ work, which covers almost the entire field of theoretical physics, is characterized both by outstanding mathematical skill and by careful analysis of physical principles. It also leaves us with the impression that he tackled problems because he found them challenging, not primarily because they afforded chances of easy success. As a consequence his work is somewhat lacking in spectacular results that can be easily explained to a layman; but among fellow theoreticians he was universally recognized as one of the great masters.”

Here is my favorite Kramers story. Jewish physicist Abraham Pais described in his autobiography A Tale of Two Continents how he spent much of World War II in Holland hiding from the Gestapo. Kramers was one of the few people who knew of his hiding place, and would visit him weekly to talk physics. One day when Kramers was there, Gestapo agents knocked at the door and Pais had to hide in a small enclosure behind a panel in the wall. Pais writes

“I kept sitting in the tiny space, practically bent over double, holding onto the panel, when I heard the door to my room, which lay at the other side of my hiding spot, open softly. Someone entered, I did not at first know who. Then that person sat down on a small bench that stood right at the wall behind which I was folded up. The person began to read, not loud but quite softly. It was Kramers. Earlier he had lent me a volume of Bradley’s Lectures on Shakespeare. What this good man was doing now was reading to me from that book, in order to calm my nerves.”