#10H.  The Motion of Magnetically
            Trapped Particles -- History

Going in circles

    The fact charged particles circle around field lines was well known to J.J. Thomson when he experimented with electron and ion beams in a magnetic field. In 1930 Ernest Lawrence of Berkeley applied such circular motion to a machine in which he accelerated ions to high energies and which he named "cyclotron." Higher energy particles describe bigger circles, so over the years, as accelerators achieved higher and still higher energies, cyclotrons and their descendants greatly increased in size.

    The particles inside the Tevatron at Fermilab in Illinois need a diameter of about a mile, and a much bigger machine, the SSC or "Superconducting Super-Collider", was started in Texas but left unfinished when its funds were cut off. Giant accelerators of this form, in tunnels deep underground, also exists in the European CERN facility near Geneva, where they stretch across the French-Swiss border.

   The Fermilab magnet is shaped like a huge ring whose cross-section resembles the letter "c". Inside the "c", where the magnetic field is strong, is the pipe in which protons (and antiprotons) are accelerated (the "c" cradles it the way the rim of a bicycle wheel cradles the inner tube). That pipe also forms a ring about a mile across, with a vacuum on the inside. The magnet is really an electromagnet, and as the accelerated protons gain speed and energy, its electric current is gradually increased, strengthening the magnetic field in a way that keeps the orbits of the protons within the pipe.

Early History of Adiabatic Invariance:
        A Detour on the Road to Explaining the Atoms

    The notion of adiabatic invariance is tied with the early years of quantum theory. Light emitted from atoms (or absorbed by them) has a very well defined pattern of colors--only a few well-defined colors are involved. (For more about this, see here.) By about1910, physicists realized that these patterns indicated changes in the laws of physics as one approached atomic dimensions.

   Atoms consisted of negative light electrons and heavier positive nuclei, and their electric attraction fell with distance at the same rate as gravity, suggesting that electrons orbited in ellipses the way planets orbited the Sun. However, an additional effect was predicted: electro-magnetic processes would also make the electrons constantly lose energy by "broadcasting" it into space, like a miniature radio stations. Apparently, certain orbits were immune to such losses, and light was only emitted when an electron jumped from one to the other.

    The simplest pattern of emitted colors was that of hydrogen, for which a remarkably accurate formula actually existed, discovered around 1885 by a Swiss high school teacher named Johannes Balmer. In 1914 the young Danish physicist Niels Bohr (he and his brother Harald, a mathematician, were the stars of Denmark's soccer team) discovered what seemed like an explanation for the formulas. Bohr showed that Balmer's formula was obtained naturally and accurately, if one assumed a new law of nature. By that law, electron orbits were stable if the "action variables" associated with their periodic motion were an integral multiple (i.e. 1,2,3... times) of a new physical constant, one previously known from other "quantum" effects on the atomic scale. Paul Ehrnfest proposed that that rule extended to other atoms, whose multiple electrons behaved like multiple planets.

   At this point Albert Einstein called attention to the pendulum whose string was gradually shortened: its "adiabatic invariant", the product E times T, was almost constant. Could it be, he suggested, that any quantity that was adiabatically conserved in large-scale nature, was exactly conserved on the atomic scale?

   That led to the early quantum theory of Sommerfeld, for hydrogen and hydrogen-like atoms. However, when Max Born tried it on helium (two electrons) his results disagreed with observed colors of helium light. The successful "wave mechanics" theory of Schroedinger, Heisenberg and Born, which in 1925-6 replaced Bohr's naive (and unexplained) principle, used a completely different approach.

Re-emergence of Adiabatic Invariance:
        A Useful Tool to understanding Plasmas

    He apparently did not realize that this was an adiabatic invariant of the sort defined by Ehrenfest: this was pointed out at about the same time by the Russian physicists Lev Landau (Nobel, 1962) and Solomon Lifshitz, as a worked-out example for the student in their textbook on the theory of fields.

    A "second" adiabatic invariant, also important in the theory of radiation trapped in the Earth's field, was derived by Grad, Longmire and Rosenbluth while studying the confinement of laboratory plasma, and a related "third" invariant was introduced shortly afterwards by Northrop and Teller.