Paul Langevin

How to use this course

Paul Langevin is a name most physicists meet in an equation before they meet the man. The Langevin equation is the starting point of stochastic physics. The Langevin function describes paramagnets and polar liquids. The Langevin rate governs reactions between ions and molecules in interstellar clouds. The twin paradox was his. So was the first practical use of quartz to send ultrasound under water, the ancestor of sonar and medical echography.

He was also one of the most public French scientists of the twentieth century: a teacher of generations at the Collège de France and the École de physique et chimie, a defender of human rights, an antifascist arrested by the Gestapo, and the architect of a post-war plan for French education. This course presents the physics in seven lessons and the life around them.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Formulas are in SI units unless a lesson says otherwise; Lesson 04 uses Gaussian units for the capture rate because that is how the result is quoted in the literature.

Timeline

Montmartre to the École normale, 1872–1897

Paul Langevin was born on 23 January 1872 in Montmartre, then a working-class district of Paris, a few months after the suppression of the Commune. His father, Victor-Charles Langevin, was a métreur-vérificateur, a surveyor of building works. His mother, Marie-Adèle Pinel, had been a schoolteacher; she was a great-niece of the alienist Philippe Pinel. The family was modest and republican, and the boy’s schooling followed the route open to such families: a higher primary school, then the “special” baccalauréat in 1888, without Latin and without a lycée.

That route led him to the new École municipale de physique et chimie industrielles of the City of Paris, founded in 1882 to train engineers for industry. Its laboratories were run by working scientists, and one of them was Pierre Curie, then the school’s chef de travaux. Langevin completed the course in 1891, and Curie remained his mentor.

Without the classical training of the lycées, Langevin still sat the entrance examination of the École normale supérieure. He was admitted first in 1893 and studied there from 1894 to 1897, taking licences in physical and mathematical sciences at the Sorbonne and the agrégation in physical sciences in 1897. The combination of an engineering school and the ENS gave him an unusual profile: a theorist with a laboratory instinct, comfortable with apparatus and with mathematics.

Cambridge and ionized gases, 1897–1902

With a scholarship from the City of Paris, Langevin spent the academic year 1897–98 at the Cavendish Laboratory in Cambridge, working near J. J. Thomson in the months when Thomson identified the electron. The laboratory was full of research students working on the conduction of electricity through gases made conducting by X-rays and by the new radioactivity; Ernest Rutherford was among them. Langevin took up the same subject: how ions are produced in a gas, how fast they drift in an electric field, and how they disappear by recombining.

On his return in 1898 he joined the physics teaching laboratory of the Paris faculty under Edmond Bouty. The same year he married Jeanne Desfosses; they had four children, Jean, André, Madeleine and Hélène. In 1902 he defended his doctoral thesis, Recherches sur les gaz ionisés, prepared under Pierre Curie. It contained a method for measuring ion mobilities and the first sound theory of recombination, which Lesson 04 develops. The thesis already showed his style: a clear physical picture, reduced to a formula, then tested against measurement.

Collège de France and the year of scandal, 1902–1911

In 1902 Langevin began to substitute for Éleuthère Mascart in the chair of physics at the Collège de France, and he succeeded to the chair in 1909. In 1905, after Pierre Curie moved to the Sorbonne, he succeeded him as professor at the EPCI. These were his most productive years. In 1904 he reported on the physics of electrons at the international congress in Saint Louis, where Henri Poincaré also spoke. In 1905 he published his theory of magnetism; in 1908 the Langevin equation; in 1910–11 he taught special relativity at the Collège de France, one of the first courses on the subject in France.

Pierre Curie died in a street accident in April 1906. Langevin, already estranged from his wife, grew close to Curie’s widow, Marie Curie. In mid-1910 he rented an apartment near her laboratory. In 1911 private letters between them reached the nationalist press while both were at the first Solvay Conference in Brussels. From 4 November 1911 newspapers ran the story, with attacks on Marie Curie as a foreigner. Langevin challenged one of the editors to a duel; it took place, and neither man fired. Weeks later Marie Curie received her second Nobel Prize, in chemistry. Jeanne Langevin had begun separation proceedings that year. The episode is documented here because it shaped both careers and the public image of French science in those years.

At the Brussels conference itself Langevin was a full participant. With Maurice de Broglie he served as a scientific secretary, and he joined the discussions on radiation and quanta with Max Planck, Einstein, Hendrik Lorentz and Jean Perrin.

Ions in gases: mobility, recombination, capture

For the general reader. When X-rays pass through air, they knock electrons out of molecules and leave pairs of positive and negative ions. Left alone, the ions find each other and recombine. Langevin worked out how fast this happens from a simple picture: each ion drifts toward its partner in the partner’s electric field, slowed by collisions. Two years later he showed how the attraction between an ion and a neutral molecule controls how ions move, an idea that now governs much of the chemistry of interstellar space.

Mobility

In a gas at ordinary pressure an ion in a field \(\mathbf E\) quickly reaches a drift velocity proportional to the field, \(\mathbf v=k\mathbf E\). The constant \(k\) is the mobility. For small ions in air near atmospheric pressure, \(k\) is of order \(10^{-4}\ \mathrm{m^2\,V^{-1}\,s^{-1}}\), with negative ions somewhat more mobile than positive ones. If ions of both signs are produced at a rate \(q\) per unit volume and recombine, their densities obey

Derivation. Sit on a positive ion. At distance \(r\) its field is \(E=e/(4\pi\varepsilon_0 r^2)\). A negative ion there drifts inward with speed \(k_-E\), and the positive ion drifts toward it with \(k_+E\), so the relative inward speed is \((k_++k_-)E\). The inward flux of negative ions through a sphere of radius \(r\) is

independent of \(r\), as Gauss’s law guarantees. Every ion that crosses the sphere is captured, so the rate per positive ion is \(\alpha n_-\) with \(\alpha\) as stated. At low pressure, where ions travel freely over the capture distance, J. J. Thomson’s three-body theory applies instead, and Langevin’s value is an upper bound. The same formula reappeared a century later as “Langevin recombination” of electrons and holes in disordered semiconductors.

Capture by polarization

In 1905 Langevin gave a kinetic theory of ion mobility in which the ion attracts the neutral molecule by polarizing it. In Gaussian units, a molecule of polarizability \(\alpha_p\) (a volume) at distance \(r\) from a charge \(e\) has interaction energy \(V(r)=-\alpha_p e^2/(2r^4)\). The capture rate that follows from this potential is now called the Langevin rate.

Derivation. With collision energy \(\mathcal E=\tfrac12\mu v^2\) and angular momentum \(\mu vb\), the effective radial potential is \(V_{\rm eff}(r)=\mathcal E b^2/r^2-\alpha_p e^2/(2r^4)\). Setting \(dV_{\rm eff}/dr=0\) gives \(r_*^2=\alpha_p e^2/(\mathcal E b^2)\) and a barrier height \(V_{\max}=\mathcal E^2b^4/(2\alpha_p e^2)\). The particles pass the barrier when \(\mathcal E\ge V_{\max}\), that is when \(b^4\le2\alpha_p e^2/\mathcal E=4\alpha_p e^2/(\mu v^2)\). Then \(\sigma=\pi b_c^2\).

Paramagnetism and diamagnetism

For the general reader. Some substances are weakly drawn into a magnet, others weakly pushed out. In 1895 Pierre Curie found that the first effect weakens as temperature rises and the second does not. Ten years later his former student explained both. Paramagnetic molecules carry tiny permanent magnets which the field tries to align and heat tries to scramble. Diamagnetism comes from the field changing the orbital motion of electrons, an effect that heat cannot touch.

Paramagnetism

Let each molecule carry a permanent moment of magnitude \(\mu\), free to point in any direction. In a field \(\mathbf B\) its energy is \(-\mu B\cos\theta\). With the Boltzmann weight and the solid-angle measure \(\sin\theta\,d\theta\), the partition function per molecule is

The result turned Curie’s empirical law into a measurement of the molecular moment: the Curie constant gives \(\mu\). Pierre Weiss used Langevin’s values to propose a natural unit of magnetic moment, the “Weiss magneton”, before the Bohr magneton replaced it.

Diamagnetism

For diamagnetism Langevin used the electron theory of Lorentz. When a field is switched on, the induced electric field changes the angular velocity of each orbiting electron by the Larmor frequency \(\omega_L=eB/(2m)\). The extra circulation produces a moment opposed to \(\mathbf B\):

where \(\rho\) is the distance from the axis through the nucleus along \(\mathbf B\) and \(\langle\rho^2\rangle=\tfrac23\langle r^2\rangle\) for a spherically symmetric atom.

A puzzle remained. In 1911 Niels Bohr, and in 1919 Hendrika van Leeuwen, proved that in purely classical statistical mechanics the total magnetization in thermal equilibrium is exactly zero. Langevin’s results survive because he assumed rigid moments and fixed orbits, which classical physics cannot supply. His theory was quantum mechanics in disguise, and the quantum treatment of the 1920s reproduces both formulas with \(\mu\) and \(\langle r^2\rangle\) given by atomic structure.

From Langevin to Weiss and Debye

For the general reader. Langevin’s function was quickly put to work elsewhere. Pierre Weiss added an internal field that makes neighbouring magnets help each other, and obtained the first theory of ferromagnets such as iron. Peter Debye replaced magnetic moments by electric ones and explained why water and other polar liquids respond so strongly to electric fields, and why that response falls with temperature.

The molecular field (Weiss, 1907)

Weiss assumed that each moment feels, besides the applied field, an internal field proportional to the magnetization, \(B_{\rm eff}=B+wM\). Langevin’s result then becomes a self-consistency condition:

Derivation. For small argument, \(M=n\mu\cdot\mu(B+wM)/(3k_BT)=C(B+wM)/T\), so \(M(T-wC)=CB\). At \(B=0\), plot \(y=L(x)\) against the line \(y=(k_BT/n\mu^2w)\,x\) obtained from \(x=\mu wM/k_BT\). Since \(L\) is concave for \(x>0\) with slope \(1/3\) at the origin, a second intersection exists exactly when the line’s slope is below \(1/3\), that is when \(T<n\mu^2w/(3k_B)=\theta\).

Polar molecules (Debye, 1912)

A molecule with permanent electric dipole \(p\) in a field \(E\) behaves like Langevin’s magnetic moment. Its mean dipole along the field is \(pL(pE/k_BT)\approx p^2E/(3k_BT)\). Adding the induced polarizability \(\alpha\) and using the Lorentz local field gives the Debye equation for a gas or dilute solution of polar molecules:

A plot of the left side against \(1/T\) is a straight line whose slope gives \(p\). Dipole moments measured this way became a standard tool of structural chemistry; the unit of dipole moment, the debye, records the debt.

The quantum version

In quantum mechanics a moment with angular momentum quantum number \(J\) takes only \(2J+1\) orientations. The average is then the Brillouin function \(B_J(x)\), introduced by Léon Brillouin, Langevin’s student, in 1927. As \(J\to\infty\) with \(\mu\) fixed, \(B_J\to L\); for \(J=\tfrac12\), \(B_{1/2}(x)=\tanh x\).

The Langevin equation

For the general reader. A pollen-sized grain in water is battered by molecules from every side. Einstein in 1905 and Marian Smoluchowski in 1906 predicted how far such a grain wanders. In 1908 Langevin derived the same result in a few lines by writing Newton’s law for the grain with two forces: a steady drag and a rapidly fluctuating push. That equation, the first stochastic differential equation in physics, is now used for everything from molecular motors to stock prices.

The 1908 argument

For a sphere of radius \(a\) and mass \(m\) in a fluid of viscosity \(\eta\), Langevin wrote

with Stokes drag and a “complementary force” \(X\) that is as often positive as negative and independent of the grain’s position. Multiply by \(x\) and use \(x\ddot x=\tfrac12\frac{d^2}{dt^2}x^2-\dot x^2\):

Average over many grains. Equipartition gives \(m\langle\dot x^2\rangle=k_BT\), and \(\langle Xx\rangle=0\). With \(z=\frac{d}{dt}\langle x^2\rangle\),

The exponential dies out in a time of order \(m/(6\pi\eta a)\), about \(10^{-7}\) s for a micrometre grain. Afterwards

Langevin remarked that his derivation was “infinitely more simple” than Einstein’s. It also exposes the physics: diffusion is the long-time consequence of inertia, friction and thermal kicks. Jean Perrin’s measurements of \(\langle x^2\rangle\) in the same year used this formula to obtain Avogadro’s number.

The modern form

Writing \(\gamma=6\pi\eta a/m\) and treating the force as white noise, the velocity obeys

The link between the friction \(\gamma\) and the strength of the noise is the first fluctuation–dissipation theorem. Leonard Ornstein and George Uhlenbeck worked out the full process in 1930; Kubo’s linear response theory of the 1950s generalized the relation. The equation, with a potential force added, is also the basis of Langevin dynamics in molecular simulation and of stochastic thermodynamics.

The traveller of 1911

For the general reader. Einstein had shown in 1905 that a moving clock runs slow. In April 1911, at a philosophy congress in Bologna, Langevin made the consequence vivid. A traveller leaves Earth in a projectile at nearly the speed of light, turns round and comes home. Two years have passed for the traveller. On Earth, two centuries have gone by. The story, retold with twins, became the most famous thought experiment of relativity.

Proper time

Langevin’s traveller moves at a speed differing from \(c\) by one part in 20 000. The Lorentz factor is then \(\gamma=1/\sqrt{1-\beta^2}\approx1/\sqrt{2\times5\times10^{-5}}=100\). One year out and one year back of traveller’s time correspond to 200 years on Earth.

Why it is not symmetric

Each twin sees the other’s clock run slow while both move inertially. The two worldlines are nevertheless different: Earth’s stays in one inertial frame, while the traveller’s changes frame at the turn-around. Langevin stressed that the traveller alone undergoes acceleration and so can tell, by experiments inside the projectile, that the motion is not uniform. The asymmetry also shows up in an exchange of signals: each sends a light flash once a year of their own time, and each counts what arrives. The relativistic Doppler factor for recession at speed \(\beta c\) is

and \(1/f\) for approach. The traveller switches from the low rate to the high rate at the turn-around; Earth receives the high rate only near the end, when light from the turn-around finally arrives.

The inertia of energy

For the general reader. Before relativity, Langevin and others tried to explain the electron’s mass as the inertia of its own electric field. Einstein’s 1905 relation \(E=mc^2\) turned the idea round: any energy has inertia. Langevin took this seriously for atoms. If an atom is built from lighter parts that release energy when they bind, the atom must weigh less than its parts. He suggested in 1913 that this explains why atomic weights are not exact multiples of hydrogen’s.

Electromagnetic mass

Around 1904 Langevin, like Lorentz, Max Abraham and Alfred Bucherer, built models of the electron as a small charged body whose mass is the inertia of its field. In Langevin’s and Bucherer’s model the electron contracts in the direction of motion while keeping its volume. Experiments by Bucherer and others favoured Lorentz’s model, which agrees with relativity. The common thread was that the field energy \(U\) of a charge contributes a mass of order \(U/c^2\).

Mass of a bound system

Langevin’s 1913 paper on the inertia of energy argued that deviations of atomic masses from whole-number multiples of hydrogen measure the binding energy of atoms, and that these energies must be enormous compared with chemical ones. Precise mass spectrometry by Francis Aston after 1919 made the argument quantitative. In 1919–1920 Jean Perrin and Arthur Eddington each used the mass defect of helium to propose that stars shine by converting hydrogen into helium.

Quartz, ultrasound and sonar

For the general reader. In 1915 German submarines were sinking ships faster than they could be built. A Russian engineer in Paris, Constantin Chilowsky, proposed to find them with echoes of sound. Langevin made it work. He used a property of quartz that Pierre and Jacques Curie had discovered: it produces a voltage when squeezed, and changes shape when a voltage is applied. Driven at high frequency, a quartz sandwich sends a narrow beam of ultrasound through water and hears the echo. Modern sonar and medical ultrasound scanners descend from this device.

Piezoelectricity

Jacques and Pierre Curie discovered in 1880 that certain crystals, quartz among them, develop surface charges under mechanical stress. Gabriel Lippmann predicted from thermodynamics in 1881 that the converse must hold, and the Curies confirmed it.

The Langevin sandwich

Chilowsky and Langevin filed patents in 1916 and 1917. Their first emitters were electrostatic; the decisive step was to use quartz, driven at resonance by the new triode valve oscillators. Because quartz strains so little, Langevin placed a mosaic of thin quartz plates between steel plates, forming a composite resonator vibrating in its fundamental thickness mode. If the stack behaves as a uniform bar of length \(L\) and sound speed \(c_s\), free at both ends, its fundamental frequency is

The steel lowers the frequency to tens of kilohertz and couples the vibration efficiently into water. The same “Langevin transducer” design, now with piezoceramics, drives ultrasonic cleaners, welders and sonar today. Sea trials continued through 1917 and 1918, but the war ended before the system was in service. Parallel British work under Robert Boyle led to ASDIC.

Directivity and ranging

A circular piston of diameter \(D\) radiating wavelength \(\lambda\) produces a beam whose first null is at angle \(\theta\) with \(\sin\theta=1.22\,\lambda/D\). Directional beams need \(\lambda\ll D\), hence high frequencies; but absorption in seawater rises steeply with frequency. Ranging uses the round-trip time: \(d=c_wt/2\) with \(c_w\approx1500\) m/s.

A physicist of the Republic, 1919–1939

After the war Langevin became the central figure of French theoretical physics. From 1920 he directed the Journal de physique et le Radium. He became director of the EPCI in 1925 and turned it into a research school as well as an engineering one. Frédéric Joliot, one of its graduates, went to work for Marie Curie on his recommendation in 1925. Léon Brillouin took his doctorate with him in 1920. In 1924 Louis de Broglie’s thesis on matter waves came before a jury that included Langevin; unsure what to make of it, Langevin sent a copy to Einstein, whose favourable reply helped the idea spread. Others shaped by his teaching include Edmond Bauer, Fernand Holweck, Jacques Solomon and Francis Perrin.

He was a tireless intermediary between French and international science. In 1922 he brought Einstein to the Collège de France, against strong nationalist hostility so soon after the war. He sat on the scientific committee of the Solvay Institutes and presided over the Solvay Conferences of 1930, on magnetism, and 1933, on the atomic nucleus. The Royal Society, which had given him the Hughes Medal in 1915, elected him a Foreign Member in 1928. The Académie des sciences elected him only in 1934.

His public life widened. He was active in the Ligue des droits de l’homme from the 1920s, co-founded the Union rationaliste in 1930 and presided over the French group of the New Education movement. After the right-wing riots of 6 February 1934 he co-founded, with the philosopher Alain and the ethnologist Paul Rivet, the Comité de vigilance des intellectuels antifascistes. He supported the Popular Front and the Spanish Republic, and in 1939 co-founded the review La Pensée.

Occupation, liberation, legacy, 1940–1946

Paris fell in June 1940. On 30 October the German authorities arrested Langevin at the EPCI and held him in La Santé prison. The arrest of so prominent a scientist provoked student protests in the Latin Quarter, which fed into the demonstration of 11 November 1940, one of the first open acts of defiance against the occupation. The Vichy government removed him from his posts, and he was placed under house arrest in Troyes. The war reached his family directly. Jacques Solomon, a theoretical physicist and his son-in-law, was arrested for Resistance work in March 1942 and shot as a hostage at Mont-Valérien on 23 May 1942. His daughter Hélène Solomon-Langevin was deported to Auschwitz in 1943 and survived. In May 1944, warned that he was in danger, Langevin crossed into Switzerland with false papers obtained through Frédéric Joliot.

He returned after the Liberation and resumed his posts. In 1944 he joined the French Communist Party and became president of the Ligue des droits de l’homme. From November 1944 he chaired the ministerial commission on education reform; its report, completed by the psychologist Henri Wallon after Langevin’s death and known as the Langevin–Wallon plan, proposed a unified, democratic school system and influenced French educational debate for decades. He served on the Paris municipal council and as a French delegate to UNESCO. He died in Paris on 19 December 1946. On 17 November 1948 his ashes were placed in the Panthéon together with those of Jean Perrin.

Legacy

• Statistical physics: the Langevin equation is the template for stochastic dynamics, from the Fokker–Planck equation to Langevin dynamics in simulation and modern stochastic thermodynamics.
• Magnetism and dielectrics: the Langevin function, Curie’s law explained, Langevin diamagnetism, and through Weiss and Debye the mean-field and polar-molecule theories.
• Ions: Langevin recombination and the Langevin capture rate, basic to plasma physics, atmospheric electricity, organic electronics and astrochemistry.
• Relativity: the twin paradox and the early case for the inertia of nuclear binding energy.
• Acoustics: the piezoelectric transducer and active sonar, and through them medical ultrasound.
• Honours: Hughes Medal (1915), Foreign Member of the Royal Society (1928), Copley Medal (1940), Grand Cross of the Legion of Honour (1946). The Institut Laue–Langevin in Grenoble, founded in 1967, bears his name with that of Max von Laue.

Connections

People linked to Paul Langevin across this collection. A name without a link has no page here yet. See Paul Langevin in the network of all 337 lives →

Teachers and advisors

• J. J. Thomson 1897-1898
• Pierre Curie 1902

Students

• Frédéric Joliot-Curie
• Louis de Broglie 1924
• Léon Brillouin 1920

Influenced by

• Hendrik Lorentz
• Ludwig Boltzmann

Influenced

• Frédéric Joliot-Curie
• Irène Joliot-Curie
• Peter Debye
• Pierre Weiss

Colleagues and collaborators

• Albert Einstein
• Henri Poincaré
• Marie Curie 1910-1911
• Constantin Chilowsky 1915-1918
• Henri Wallon 1944-1946

Friends

• Jean Perrin

Family

• Jacques Solomon

Also mentioned in this life

• Amedeo Avogadro
• Niels Bohr
• Robert Boyle
• Charles-Augustin de Coulomb
• Christian Doppler
• Arthur Eddington
• Carl Friedrich Gauss
• James Clerk Maxwell
• Isaac Newton
• Max Planck
• Ernest Rutherford
• George Gabriel Stokes
• Max von Laue

Fields

• statistical physics 3 lives
• magnetism 5 lives
• relativity 3 lives
• acoustics 9 lives

Sources and further reading

Biography

1. Paul Langevin, Wikipedia; and the more detailed French article.
2. Paul Langevin, Reflets de la physique 1 (2006), Société française de physique.
3. Marie Curie: Scandal, American Institute of Physics online exhibit.
4. Paul Langevin, bibliography, Collège de France archives.
5. Jacques Solomon, Wikipedia.
6. B. Bensaude-Vincent, Langevin, 1872–1946 : science et vigilance, Belin, 1987.

Physics

1. P. Langevin, Recombinaison et mobilités des ions dans les gaz, Annales de chimie et de physique 28 (1903).
2. P. Langevin, Magnétisme et théorie des électrons, Annales de chimie et de physique 5 (1905); Sur la théorie du magnétisme, Journal de physique 4 (1905).
3. P. Langevin, Une formule fondamentale de théorie cinétique, Annales de chimie et de physique 5 (1905).
4. P. Langevin, Sur la théorie du mouvement brownien, C. R. Acad. Sci. Paris 146 (1908) 530–533. English translation: D. S. Lemons and A. Gythiel, Am. J. Phys. 65 (1997) 1079.
5. P. Langevin, L’évolution de l’espace et du temps, Scientia 10 (1911).
6. P. Weiss, L’hypothèse du champ moléculaire et la propriété ferromagnétique, Journal de physique 6 (1907).
7. G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Physical Review 36 (1930) 823.
8. Langevin equation and Sonar (history section), Wikipedia.
9. J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford, 1932.