# Brillouin, Léon Nicolas

# BRILLOUIN, LéON NICOLAS

(*b*. Sèvres, France, 7 August 1889; *d*. New York City. 4 October 1969)

*quantum theory, solid-state physics, electronics, information theory*.

Brillouin was the son of Charlotte Mascart and Marcel Brillouin. His father held the chair of general physics and mathematics at the Collége de France; his maternal grandfather. Éleuthère Mascart, had held the chair of experimental physics in the same institution; and his great-grandfather, Charles Briot, had been a professor of mechanics at the Sorbonne. Nevertheless, the physics taught at the lycée failed to interest him until, following his father’s suggestion, he read some of Pascal’s letters on atmospheric pressure.

From 1908 to 1912. Brillouin joined the elite of French students at the École Normale Supérieure. where he attended the best physics and mathematics lectures available in France at that time. The most influential of his teachers were Paul Langevin. Henri Poincaré, and Jean Perrin. Unlike most of their French colleagues, they were exploring the newest and still controversial fields of physics. such as quantum theory and relativity, and maintained strong intellectual ties with the leading foreign physicists.

Brillouin’s early research was inspired by some of Einstein’s successes in molecular and quantum physics. In 1911, under Perrin’s supervision, he extracted a new value of Avogadro’s number from his measurements of the blue light of the sky. He did not publish his results (they depended too much on the state of the weather), but in his subsequent work he returned to the basic process involved, the scattering of light in a thermally fluctuating medium.

In 1912 Brillouin began an academic year in Munich under Arnold Sommerfeld, to observe the spectacular progress made there in X-ray studies and, above all, to learn more about the methods of theoretical physics. One of his first publications (1914) resulted from a fruitful application of a mathematical tool taught by Sommerfeld, the method of steepest descents, to the propagation of light in dispersive media. This study generalized Rayleigh’s results on group and phase velocities, and explained such subtle physical effects as the fast and weak undulations preceding the’ front’ of a light signal (which travels at group velocity).

Back in Paris, Brillouin started his dissertation work on the quantum theory of solids but was soon halted by World War I. Like Maurice de Broglie, he joined the military research on radio transmission in the laboratory of General Ferrié. His main contributions, a new type of amplifier (the’ resistance amplifier’) and a remote-control system for ships and planes, won him the Legion of Honor. This work was not a mere interlude; it determined his lifelong interest in radio engineering problems.

In the two years following the war, Brillouin completed his dissertation work, mainly built on the idea that Einstein’s theory of the quantum solid (or the improved versions by Peter Debye and by Max Born and Theodor von Kármán) should be to real solids what the theory of the ideal gas is to real gases. As the counte)t of gas pressure, he defined a pressure (in tensor form) of the solid vibration (also in analogy with the pressure of electromagnetic waves). The main interest of this work was a derivation of several important characteristics of the perfect solid independent of the quantum hypothesis (which restricts the energy of a monochromatic component of the vibrations to an integral multiple of its frequency times *h*). For instance, the form F(v/T) of the thermal average of the energy per vibration of frequency *v* at temperature *T* results from a purely classical theorem on abiabatic transformations by Ludwig Boltzmann and Paul Ehrenfest. The quantum hypothesis, Brillouin emphasized, remained compatible with many classical arguments; it just completed them.

In harmony with this judgment, the main new effect that Brillouin predicted, the’ Brillouin doublet,’ was essentially classical; the (requeues of the light scattered in a given direction by a vibrating transparent medium is shifted negatively or positively, according to the modulation by the reflecting elastic plane wave. Although the experimental confirmation of this effect had to wait until 1932, it is now commonly observed in laser experiments for both supersonic and thermal vibrations of the medium. In the first case the absolute value of the frequency shift equals the supersonic frequency: in the second case it is close to the Debye cutoff frequency.

In 1922 Brillouin published *La théorie des quanta et l’atome de Bohr*, the first textbook covering all quantum topics to be written by a French scholar. It shows some of his strongest qualities: an exceptional clarity of exposition, an analysis in depth of the physical foundation of reasoning, and a thorough consideration of the mathematical methods involved. The study of Einstein’s solid had inspired Brillouin to consolidate the foundation of statistical mechanics: the calculation of the propagation and diffraction of light had caused him to develop the method of steepest descent: and the calculation of the pressure of solid vibrations brought him to extend the applications of tensor algebra to physics. Langevin recognized his student’s accomplishments, and from 1923 to 1928 kept him as assistant director of his laboratory at the College de Franee.

In 1925 Brillouin was the only French theoretician to react competently to Werner Heisenberg’s new matrix mechanics. In two papers published in 1926, he contributed to the exploration of the mathematical content of Heisenberg’s theory. As an expert in tensor algebra, he was not intimidated by matrices. In his book on quantum theory, he had speculated, as had Born, on a new discontinuous mechanics of quantum jumps. Nevertheless, as soon as he knew of Erwin Schrödinger’s theory and its formal equivalence to matrix mechanics, he favored the wave point of view, which he could easily connect to his earlier studies on wave propagation.

Brillouin’s first contribution in this field was important. Through a new method of semiclassical approximation, he discovered the relation between Schrodinger’s mechanics and the quantum theory of Niels Bohr and Sommerfeld. Presumably inspired by de Broglie’s early analogies between mechanics and optics, he found this approximation as the quantum mechanical counte)t of the approximation
of geometrical optics (in respect to the more exact wave theory). In this procedure, stationary solutions of the Schrödinger equation are sought in the form e^{iS/h} (as in the eikonal approximation of optics). In the first approximation (h small), S must be a solution of the Hamilton-Jacobi equation of classical mechanics, and the Bohr-Sommerfeld conditions (*S* = 2*πnh* on a closed trajectory) must be satisfied for the ψ function to be defined and singlevalued in all space. Subsequent corrections are proportional to successive powers of h. They intermix the various Bohr trajectories. thereby reintroducing the complex interplay of quantum states found in matrix mechanics, This method, published by Brillouin in July 1926—anticipated by Harold Jeffreys in 1923 in a purely mathematical context, reinvented by Gregor Wentzel in September 1926. and perfected by Hendrik Kramers in November 1926—is now called the (J) BWK method and is widely used in many quantum mechanical problems.

In spite of a sympathy for de Broglies pilot waves. Brillouin did not dwell much on the interpretation of quantum mechanics. Instead, he labored to apply it to concrete situations, especially to the many-particle statistical problems with which he was most familiar. For instance, in 1927 he derived (as Wolfgang Pauli did independently) the quantum mechanical expression for paramagnetic susceptibility. In this expression appear the’ Brillouin functions,’ generalizing the function earlier introduced by Langevin for the same problem.

In statistical matters the most novel result brought by quantum theory was the new method of treating identical particles, the Bose-Einstein and the Fermi-Dirac statistics. In 1930 Brillouin published *Les statistiques quantiques et leurs applications* soon translated into several languages. He first demystified the new statistics, arguing that, from the elementary point of view of the distribution of identical particles over quantum cells, they were no less natural than Boltzmann’s statistics. Then he fitted them into a common formal framework, starting from the expression 1-pb for the probability of a particle to go into a cell already occupied p = +1 times; b = -1 gives the Bose statistics, and b = + 1 (never more than one particle in one cell) gives the Fermi-Dirac statistics.

More profoundly innovative were applications to the electron theory of metals. A brief survey of the work of other pioneers in this field will help in the understanding of Brillouin’s contribution. In 1927. through a clever application of the Fermi-Dirac distribution, Sommerfeld managed to remove the main paradoxes in the old model of a free electron gas (for instance, the fact that only a small proportion of the’ free’ electrons contribute to the thermal and electrical conductivity). But his expressions for the thermal and electric conductivity still contained the mean free path, which required further quantum mechanical calculations of the electron-lattice interaction. In 1928, as a first step toward these calculations, Felix Bloch proved a very important theorem, according to which any Schrödinger eigenfunction for an electron in a periodic potential has the form *e*^{ik-r}u_{k}(r), where u_{k} has the periodicity of the potential, and he described the first band of the energy spectrum in the strong-binding approximation (wherein the interaction with the lattice is regarded as a small perturbation to atomic bound states). In the same year Yakov Frenkel and Hans Bethe discovered the energy gaps of total reflection of electron waves by a periodic potential (in analogy with the diffraction of X rays by crystals). In 1929 Rudolf Peierls analyzed the properties of the energy bands in the approximation of weakly bound electrons, and M. J. O. Strunt discussed the exact solution of the one-dimensional problem (through the Mathicu-Hill equation). In early 1930 Philip M. Morse explained the relation between the gaps of total reflection and the band spectrum.

In the first edition of his book on quantum statistics, Brillouin corrected several mistakes in calculation and in physical interpretation made by his predecessors. Then, in the course of the preparation of the German edition (published in 1931), he greatly advanced the discussion of the band structure. Since, according to Morse, the bands are complementary to the gaps of total reflection, the energy of one-electron states as a function of Bloch’s **k** will experience discontinuities whenever **k** satisfies the Bragg condition for total reflection, **k** = ǀ**k** -gǀ where g belongs to the reciprocal lattice. Whereas his predecessors had been satisfied with one-dimensional simplifications. Brillouin took pleasure in describing the set of mutually intersecting planes in **k** -space defined by this condition (see Figure 1). These planes (the bisecting planes of the vectors of the reciprocal lattice) delimit a sequence of interlocked zones, there being an energy discontinuity at the border between two consecutive zones. Originally Brillouin imagined that these nicely shaped zones would play a role in the calculation of conductivities, for Bragg’s reflections had to interfere with the free propagation of electrons (the correspondence between vanishing conductivity and complete band filling was not known before A. H. Wilson’s work of 1931). This intuition was confirmed when the zones were used to determine the position

and shape of the Fermi surface (delimiting the occupied states in **k** -space). on which most relevant electronic properties of a crystal depend. The’ Brillouin zones’ now play a central part in any account of the electronic properties of solids.

In 1932 the Collège de France certified Brillouin’s excellence by giving him the chair of theoretical physics. During the four preceding years he had taught in the same field at the Institut Henri Poincarè (as a Sorbonne professor). With the benefit of this experience, his lectures at the Collège de France focused on ongoing developments with a great mathematical clarity. His main originality was perhaps an insistence on mathematical methods common to quantum and classical physics. As he had proved in his own research, the transposition of mathematical methods from one subject to the other could help organize and develop both fields. For example, in his hands matrices were helpful in the theory of radiation pressure, in quantum mechanics, and in studies of wave propagation and electric filters (around 1936).

Meanwhile, Brillouin continued research on his favorite topics—conduction theory and wave propagation—with a growing interest in technical applications, for instance, to the telephone. In pure theoretical research some of his best (but little-known) work consisted in comparative studies of the various approximations used in the quantum theory of metals. In 1933 Brillouin introduced a new perturbation method well suited to the calculation of the corrections to the Hartree-Fock approximation fin the latter approximation the electron-electron interaction is replaced with the interaction of an electron with an average “self-consistent” held created by the other electrons). Theresulting’ BrillouinWigner’ formula (which is obtained by replacing the unperturbed energy in the energy denominators of ordinary perturbative expansions with the exact energy) later proved useful in other many-particle problems, mainly in quantum chemistry and nuclear theory. Applying this method to his problem of electrons in a lattice, Brillouin could demonstrate the superiority of the Dirac-Fock approximation, a generalization of the Hartree-Fock approximation giving the best possible antisymmetrized product of individual wave functions for the global wave function. He also proved the’ Brillouin theorem’: the energy variation in the Dirac-Fock approximation when increasing the electron number from *n* to *n* + 1 is given as the (*n* + 1)th energy level of the one-electron problem in the self-consistent field. This theorem is important. for it explains the success of the most widely used approximation, that of weakly bound electrons.

In July 1939 Brillouin accepted the directorship of the French national radio broadcasting system. In the spring of 1940. Germany invaded northern France. In absence of specific military directions, Brillouin ordered the destruction of the broadcasting stations that would fall into enemy hands. He also had to organize the move of his employees to the unoccupied zone. Under the Vichy government he could not immediately resign from his position, and he had the unpleasant experience of being present at anti-Semitic and collaborationist propaganda events, incidents that he could not control. From August 1940 on, he was trying to find a position at an American university and to obtain a border pass; he did not leave France until March 1941, after his life had been threatened by the accusation in French newspapers that he had destroyed public property without orders.

Brillouin had lectured in several American universities in the 1920’s and 1930’s, In the academic year 1941–1942, he taughl at the Universty of Wisconsin, and the next academic year at Brown, In 1943 he joined the military radar project at Columbia University. His research there, with the exploitation of the’ Brillouin flow’ of electrons in a magnetron (that is, a global rotation, as if it were a fluid mass, of the space charge in the magnetic field) won him a prize of the National Electronic Conference (1957) and later proved to be relevant to plasma theory. He also found time to promote Franco-American exchanges, as vice president of the American Institute of France, and professor and vice president of the newly created École Libre des Hautes Eacutetudes de New York.

At the end of the war, uninformed about Brillouin’s patriotic and scientific activities, the French Commission d’ Épuration published a decree forbidding him access to any profession related to radio broadcasting. He and his friends protested vigorously and successfully: not only had he not collaborated in any political actions of the Vichy government, but he had clandestinely—and competently—sabotaged German radio-jamming devices directed against London.

In 1945 and 1946 Brillouin resumed his teaching at the Collège de France, but in 1947 he decided to emigrate to the United States. There he had a brilliant career, first as a Harvard professor (1947–1949), then as a research director at IBM (1949–1954, partly to organize a new teaching program in electronics). and as a professor at Columbia University from 1954 until his death. He was elected to the National Academy of Sciences in 1953. He had become a United States citizen in 1949.

Brillouin s postwar activities extended from consulting in radio engineering to theoretical criticism, for instance, regarding the structure of classical dynamical systems. His most important work dealt with information theory and its application to various subjects, including thermodynamics.

This last preoccupation originated in Brillouin’s broader interest in such philosophical questions as the epistemological status of physical theories and the relation between physics and life. A joint meeting of physicists and philosophers that he had organized in 1938 at the Collège de France gave a first stimulus. In 1946, at Harvard, he had the opportunity for two other excursions to the frontiers of physics; he marveled at the new mathematical machines built there, and he participated in discussions on thermodynamics and life. Percy W. Bridgman impressed him with the operationalist argument forbidding the use of the concept of entropy in analyzing living organisms. On the one hand, Brillouin later commented, the usual definition of entropy through reversible transformations could not be applied in this case. On the other hand, the energy transformations produced by living organisms did not violate the second principle of thermodynamics (once it was properly applied to the organism plus its surroundings); they just suggested an intimate relation between life and the ability to exploit the energy from metastable reservoirs in complex catalytic processes, Like Schrödinger in *What Is Life*? (1945), Brillouin believed that a proper understanding of life would require a new principle, different from the known principles of physics but still a physical principle. Unable to say much positive about the nature of this principle, he contented himself with the hasty elimination of a few possibilities: It could not be a molecular principle, access to molecular constitution being complementary lin Bohr’s sense) lo life, nor could it be an algorithmic principle, mathematical machines (computers) being irremediably un-intelligent.

Brillouin was better when he did not venture so far from mathematical physics, for instance, in his widely acclaimed *Science and Information Theory* (1956). His deepest insights concerned the relation between entropy and information. In two remarkable works of that time, Norbert Wiener’s *Cybernetics* (1948) and Claude K. Shannon’s *Mathematical Communication Theory* (1949). this relation had already appeared in contexts of particular interest to Brillouin: control mechanisms in the behavior of men and machines (Wiener), and electronic transmission of information (Shannon). Brillouin made the relation quantitative and general by the following argument.

Consider a physical system in a given macrostate corresponding to *P* equiprobable miciostates (complexions). Entropy is defined as S = klnP, while the information that would be needed to completely specify the microstate is, according to Shannon’s general definitions, I = ClnP (the choice of the constant *C* being conventional). Therefore, entropy is identical to the lack of information about the physical system under consideration, if only information is measured in proper units (C = k). In other words, negative entropy is identical to -l, the information’ bound’ to a physical system: this is Brillouin’s “negentropy principle of information.”

As a corollary, a simple way to diminish the entropy of a system is to increase information about it through more detailed measurement by its configuration. This procedure cannot be used to build cyclic machines violating the second principle of thermodynamics (as anticipated by Leo Szilard in 1929). Brillouin gave many instructive illustrations of this impossibility, the most famous of which concerned Maxwell’s demon. By hypothesis, the demon is able to get information on the speeds of gas molecules, and to use this information to operate a trapdoor to separate fast molecules from slow ones. In this process the total entropy of the gas is indeed decreased. But the necessary information—here is the essential point—cannot be obtained by the demon without perturbing the gas is a way that increases the total entropy (of the system gas + demon + observation device) more than the information about the state of the gas. In general, Brillouin proved, the minimal amount of entropy increase required by an observation is kln2, and the entropy increase during any observation is always higher than the amount of information obtained (expressed in entropy units). From this fundamental law Brillouin also derived new limits for measurements of lengths and times beyond Heisenberg’s uncertainty relations.

Altogether Brillouin published some two hundred papers and two dozen books of high quality. Except for his tendency to work alone, his way of doing physics was strikingly modern. More than most of his French colleagues, he was open to new ideas and theories, and did not hesitate to cross disciplinary borders, addressing mathematical, engineering, biophysical, even philosophical problems. He constantly fought national isolationism. Against some deleterious trends of French theoretical physics before World War II, he stood out as an inspiring example.

## BIBLIOGRAPHY

I, Original Works. An extensive bibiliography is in the Brillouin archive and in the biography by Hilleth Thomas (see below). See also Brillouin’s *Titres et travaux scientifiques de Léon Brillouin* (Niort, 1931). Important books are *La théorie des quanta et l’ atome de Bohr* (Paris, 1922); *Les statistiques quantiques et leurs applications aux électrons* (Paris, 1930); *Les tenseurs en mécanique et en élasticité* (Paris, 1937; 2nd ed., 1949). trans, by R. Brennan as *Tensors in Mechanics and Elasticity* (New York, 1963); *Wave Propagation in Periodic Structures* (New York, 1946: 2nd ed 1953); *Science and Information Theory* (New York. 1956; 2nd ed., 1961); *Vie, matiére et observation* (Paris. 1959).

Papers of special importance are’ Über die Fortpflanzungdes Liehtes in dispergierenden Medien,’ in *Annalen der Physik* 4th ser., **44** (1914), 203–240;’ La théorie ties solides et les quanta,’ in *Annates scientifiques*, **37** (1920). 357–459, his doctoral dissertation;’ Thcrmodynamique et probabilife: Revision des hypothéses fondamentales,” in *Journal de physique et le Radium*, 6th ser., **2** (1921), 65–84;’ Les amplilkateurs à résistance,’ in *L’onde électrique*, **1** (1922), 7–17, 101–123:’ Diffusion de la lumiére et des rayons X par un corps transparent homogène. Influence de ragitation thermique/* in Annates de physique, 17 (1922), 88–122;’ Les lois de Lelasticite sous forme tensorielle valable pour des coordonnées quelconques,” *ibid*., **3** (1925), 251–298;’ Sur les tensions de radiation,’ **4** (1925), 528–586: La nouvelle mécanique atomique,’ in *Journal de physique et le* Radium, 6th ser., **7** (1926), 135–160:’ La mécanique ondulatoire de Schrödinger: Une méthode générale de résolution par approximations successive,’ in *Compies rendus de l’A cadémie des sciences*, **183** (1926), 24–26;’ Les moments de rotation et le magnétisme dans la mécanique ondulatoire,’ *in Journal de physique et le Radium*, 6th ser., 8 (1927), 74–84:’ Comparaison des différentes statistiques quantique appliquées au problemes des quanta.’’ in *Annales de physique*, **7** (1927), 315–331.

“Les électrons dans les métaux et le role des conditions de réflexion selective de Bragg,’ in *Comptes rendus de I’Académie des sciences*. **191** (1930), 198–200:”Les électrons libres dans les métaux et le rô;le des ré;flexions de Bragg,” in *Journal de physique et le Radium*, 7th ser., **1** (1930), 377–400;’ Champs self-consistents et électrons métalliques III,” *ibid*., 7th ser., **4** (1933 J. 1–9;’ Les bases de la théorie électronique des métaux et la méthode du champ self-consistent,’ in *Helvetica physica acta*. **7** supp. 2 (1934), 33–46:’ La théorie des matrices et la propagation des ondes,’ in *Journal de physique et le Radium*, 7th ser, **7** (1936), 401–410:’ La théorie du magnétron, *ibid*., 8th ser,. **I** (1940), 233–241:’ Inlluenee of Space Charge on the Bunching of Electron Beams,” in *Physical Review*, **70** (1946), 187–196:’ Lesgrandes machines mathémaliques amcriéainces,’ in *Annales de télécommunication’s*, **2** (1947), 329: “Life, Thermodynamics and Cybernetics,” in *American Scientist*, **37** (1949), 554–568:’ Thermodynamics and Information Theory.’ *ibid*., **38** (1950). 594: “Maxwell”s Demon Cannot Operate: Information and Entropy, I,’ in *Journal of Applied Physics*. **22** (1951), 334–337; and’ Poincare and the Shortcomings of the Hamilton-Jacobi Method for Classical and Quantized Mechanics,’ In *Archive fof Rational Mechanics and Analysis*, **5** (1960), 76–94.

For the American Institute of Physics, Brillouin wrote extensive’ Notes sur une carriére scientifique,’ included in the Brillouin papers deposited at the Center for History of Physics in New York City. Most of Brillouin’s prewar papers have disappeared, except for the documents preserved at the College de Prance.

II. Secondary Literature. The most detailed published biography is L. Hilleth Thomas, “Leéon Nicolas Brillouin,” in *Biographical Memoirs. National Academy of Sciences*, **55** (1985), 69–92. See also the introduction by A. Georges in Brillouin’s *Vie*, matiére et observation: D. W, Harding,’ Names in Physics: Brillouin,’ in *Physics Education*, **4** (1969). 46–48; and A. Kastler.’ La vie et l’oeuvre de Léon Brillouin,’ in *L’ onde éectrique*, **50** (1970) 269–280. Information im Brillouin’s contributions to solid-state physics are in L. Hoddeson, G. Baym, and M. Eckert,’ The Development of the (Quantum Mechanical Electron Theory of Metals, 1928–1933,’ in *Review of Modern Physics*, **59** (1987), 287–3:7.

Olivier Darrigol