Max von Laue

How to use this course

Max von Laue is remembered for one experiment, and he did not perform it himself. In 1912 he proposed that a crystal should act as a three-dimensional diffraction grating for X-rays. Two experimenters in Munich tried it, and within weeks the result settled two questions at once: X-rays are waves, and crystals are lattices of atoms. Crystallography, solid-state physics and structural biology all start there.

Laue was also one of the first physicists to take up Einstein’s relativity. He wrote its first textbook, gave the first relativistic treatment of extended bodies, and later contributed to the theory of superconductivity. Under the Nazi regime he was among the very few German physicists who opposed the attacks on Einstein openly. This course follows both the science and the life.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 and 05 cover relativity (1907–1911), Lessons 06 to 09 the diffraction of X-rays (1912–1949), and Lesson 10 superconductivity (1932–1947). Formulas use SI units unless stated otherwise.

Timeline

A wandering childhood, 1879–1903

Max Laue was born on 9 October 1879 in Pfaffendorf, a village on the Rhine opposite Koblenz. His father, Julius Laue, was a civil official in the military administration, and his postings moved the family across the German Empire: Brandenburg, Altona, Posen, Berlin and finally Strassburg. Max went to school in Posen, Berlin and Strassburg. At the Protestant gymnasium in Strassburg a teacher named Goering introduced him to the exact sciences, and he never looked back.

He took the Abitur in 1898 and began his studies of mathematics, physics and chemistry at the University of Strassburg during his year of military service. In 1899 he moved to Göttingen. There Woldemar Voigt, the theorist of crystal physics, drew him to theoretical physics, and Max Abraham, the electron theorist, made a lasting impression. After a semester in Munich he went to Berlin in 1902 to study with Max Planck. He also attended Otto Lummer’s lectures on interference spectroscopy and heat radiation, which shaped his first research topic.

His doctoral thesis, Über die Interferenzerscheinungen an planparallelen Platten, dealt with the interference fringes produced by plane-parallel glass plates, the basis of the Lummer–Gehrcke interferometer. Planck accepted it in July 1903. Laue later called Berlin his true intellectual home, and Planck remained the central figure of his scientific life.

Planck’s assistant, 1903–1909

After the doctorate Laue spent two years in Göttingen and passed the state examination to teach in secondary schools. In 1905 Planck called him back to Berlin as his assistant at the Institute for Theoretical Physics. His habilitation of 1906 addressed a question at the edge of Planck’s radiation theory: what is the entropy of light beams that can interfere? Laue showed that the entropies of two coherent beams cannot simply be added, because the beams are not statistically independent. Thermodynamics, he argued, must take coherence into account. The problem kept him close to optics, which would matter in 1912.

The year 1905 also brought Einstein’s paper on the electrodynamics of moving bodies. Planck took it up at once, and Laue, after initial hesitation, became one of its first convinced supporters. In 1907 he published a short paper showing that Einstein’s rule for adding velocities explains Fizeau’s 1851 measurement of the speed of light in flowing water (Lesson 04). This was one of the first experimental arguments for the new theory, and it was achieved without any ether.

Between 1907 and 1911 he wrote eight papers on relativity. They established him as one of the few physicists who could use the theory as a working tool.

Munich and the idea of 1912

In 1909 Laue moved to Munich as Privatdozent at Arnold Sommerfeld’s Institute for Theoretical Physics, where he taught optics, thermodynamics and relativity. In 1910 he married Magdalene Degen. In 1911 he published Das Relativitätsprinzip, the first textbook of special relativity. It used Minkowski’s four-dimensional formalism throughout and went through many editions; a volume on general relativity followed in 1921.

Munich in 1912 held three ingredients in one building complex. Wilhelm Röntgen, discoverer of X-rays, directed the experimental institute. Sommerfeld had worked on the theory of X-ray emission, and his group believed X-rays were very short electromagnetic pulses or waves, with wavelengths estimated at roughly \(10^{-9}\) to \(10^{-8}\) cm. Crystallographers in Munich, following Paul Groth, took for granted that crystals were regular arrays of atoms, with spacings of about \(10^{-8}\) cm, though nobody had seen the lattice.

Early in 1912, Sommerfeld’s doctoral student Paul Peter Ewald consulted Laue about his thesis on the optics of a lattice of dipoles. While discussing it, Laue asked what would happen if the waves were much shorter than the lattice spacing. The thought followed: if X-rays have wavelengths comparable to atomic spacings, a crystal should diffract them as a grating diffracts light. Sommerfeld was sceptical, partly because thermal motion of the atoms was expected to wash out the pattern. Walter Friedrich, Sommerfeld’s assistant, and Paul Knipping, a young physicist from Röntgen’s institute, agreed to try.

The first exposure, through a crystal of copper sulphate, showed spots around the direct beam. A thin plate of zinc blende cut perpendicular to a cube axis then gave a sharp, fourfold-symmetric pattern. In June 1912 the results went to the Bavarian Academy of Sciences as Interferenz-Erscheinungen bei Röntgenstrahlen, by Friedrich, Knipping and Laue, with Laue’s theory alongside. Laue was by then leaving for an extraordinary professorship in Zurich.

Fizeau’s experiment and velocity addition

For the general reader. In 1851 Hippolyte Fizeau sent light through moving water and found that the water drags the light along, but only partly. For fifty years the explanation required an ether that was partly carried by matter. Laue showed in 1907 that Einstein’s rule for adding velocities gives the measured result directly, with no ether at all.

The experiment

Fizeau split a beam, sent the two halves through parallel tubes of water flowing in opposite directions, and recombined them. The fringe shift measured the difference in light speed with and against the flow. Augustin Fresnel had predicted in 1818 that a medium of refractive index \(n\) moving at speed \(v\) changes the light speed by \(f v\), with the drag coefficient \(f=1-1/n^2\). Fizeau confirmed this, and Albert Michelson and Edward Morley repeated it with greater precision in 1886.

Derivation

Expand the denominator: \(\big(1+\tfrac{v}{nc}\big)^{-1}=1-\tfrac{v}{nc}+O(v^2/c^2)\). Then

The expansion parameter \(v/c\) is about \(10^{-8}\) for Fizeau’s apparatus, so the second-order terms are far below any measurement. The ether theories needed a physical hypothesis about how matter drags the ether. Relativity needs only the transformation law of velocities, and it treats source, medium and observer symmetrically. Lorentz had already derived the drag from his electron theory, and later added a correction for dispersion: since the light frequency in the moving water is Doppler-shifted, \(n\) should be evaluated at the frequency seen by the water, giving \(f=1-\frac1{n^2}-\frac{\lambda}{n}\frac{dn}{d\lambda}\). Zeeman’s measurements of 1914–15 confirmed this term.

Relativistic continua and Laue’s theorem

For the general reader. Relativity says energy has inertia. But a body is made of parts under stress: a gas pushing on its container, a charge held together against its own repulsion. Laue showed in 1911 that stresses carry momentum when the body moves, and that the familiar rule \(E=mc^2\) holds for a body as a whole only when its internal stresses balance out. This solved a puzzle that had dogged the theory of the electron.

The energy-momentum tensor

Minkowski had combined the electromagnetic energy density, momentum density and Maxwell stresses into one four-dimensional object. Laue’s 1911 paper Zur Dynamik der Relativitätstheorie extended this to any continuous medium. Using units with \(c=1\), the symmetric tensor \(T^{\mu\nu}\) has \(T^{00}\) the energy density, \(T^{0i}\) the momentum density (equal to the energy flux), and \(T^{ij}\) the stress, the flux of \(i\)-momentum across a surface normal to \(j\). For a complete system with no external forces,

The four integrals \(P^\nu=\int T^{0\nu}\,d^3x\) are the total energy and momentum. They are conserved, but whether they form a four-vector is a separate question.

Proof

For static fields the conservation law reduces to \(\partial_jT^{jk}=0\). Then

so \(\int T^{ik}\,d^3x\) is the integral of a divergence. By Gauss’s theorem it equals a surface integral that vanishes for a bounded system. For the second part, boost the body to velocity \(v\) along \(x\). The tensor transforms as \(T'^{00}=\gamma^2(T^{00}+\beta^2T^{11})\) and \(T'^{01}=\gamma^2\beta(T^{00}+T^{11})\) (with \(T^{01}=0\) in the rest frame), while the volume element at fixed laboratory time contracts by \(1/\gamma\). Writing \(S=\int T^{11}\,d^3x\),

These are the components of a four-vector of mass \(M\) exactly when \(S=0\), which the first part guarantees.

The 4/3 problem

Before relativity, Max Abraham and Hendrik Lorentz had computed the momentum of the electric field of a moving charged sphere and found an electromagnetic mass \(\tfrac43\,E_{\text{em}}/c^2\) rather than \(E_{\text{em}}/c^2\). The electromagnetic field alone is not a closed system: its stresses push the charge outward, and the field tensor is traceless, \(-T^{00}+\sum_iT^{ii}=0\). For a spherically symmetric field each diagonal stress then integrates to \(S=E_{\text{em}}/3\), and the boost formula gives \(P'=\gamma\beta\cdot\tfrac43E_{\text{em}}\). Laue showed that once the cohesive stresses that hold the electron together, proposed by Henri Poincaré in 1905–06, are included, the total system satisfies his theorem and the factor disappears. The same argument handles a gas in a box, black-body radiation in a cavity and stressed solids. It was a general demonstration that the inertia of every form of energy obeys \(E=Mc^2\) for a closed system.

A note on the twin paradox

Laue also wrote on the clock problem. In 1913, in the second edition of his book, he explained the asymmetric ageing of the travelling twin by the fact that the traveller occupies two different inertial frames, outward and back, while the stay-at-home twin occupies one. The point is that proper time \(\int\sqrt{1-v^2/c^2}\,dt\) is a property of the world-line, and the inertial world-line between two events is the longest.

The Laue equations and the reciprocal lattice

For the general reader. A ruled grating spreads light into a few sharp directions because the waves from each groove add up in step only in those directions. A crystal is a grating in three dimensions. Laue wrote down the three conditions that the scattered X-rays must satisfy to add up in step, one for each lattice direction. All of crystallography rests on these conditions.

Scattering from a lattice

Let a plane wave \(e^{i\mathbf k_0\cdot\mathbf r}\) fall on point scatterers at the lattice sites \(\mathbf R=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3\). Far away in the direction of wavevector \(\mathbf k\), with \(|\mathbf k|=|\mathbf k_0|=2\pi/\lambda\) (elastic scattering), the amplitude is proportional to the lattice sum

For a crystal of \(N_1\times N_2\times N_3\) cells the sum factorizes into three geometric series:

Each factor has sharp maxima of height \(N_j^2\) where \(\mathbf q\cdot\mathbf a_j\) is a multiple of \(2\pi\), and is small elsewhere.

Laue himself wrote the conditions with unit vectors \(\mathbf s_0,\mathbf s\) as \(\mathbf a_1\cdot(\mathbf s-\mathbf s_0)=h\lambda\), and so on. Each condition alone fixes a cone of directions around \(\mathbf a_j\); a diffracted beam appears only where three cones meet in a common line, which for a fixed wavelength is exceptional. This is why a single-wavelength beam on a fixed crystal usually gives no spots, and why the white (continuous) spectrum of an X-ray tube was essential in 1912. The reciprocal basis is \(\mathbf b_1=2\pi\,\mathbf a_2\times\mathbf a_3/V_c\) and cyclically, with \(V_c=\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3)\).

Equivalence with Bragg’s law

The vector \(\mathbf G_{hkl}\) is perpendicular to the lattice planes with Miller indices \((hkl)\), and if \(h,k,l\) have no common factor, \(|\mathbf G_{hkl}|=2\pi/d_{hkl}\), with \(d_{hkl}\) the spacing of these planes. For elastic scattering through angle \(2\theta\), \(|\mathbf k-\mathbf k_0|=2k\sin\theta=(4\pi/\lambda)\sin\theta\). Setting this equal to \(|\mathbf G_{nh,nk,nl}|=2\pi n/d_{hkl}\) gives

with \(\mathbf k-\mathbf k_0\) normal to the planes, so the beam is mirror-reflected by them. This is the form that Lawrence Bragg found in the autumn of 1912. The two descriptions are the same mathematics; Bragg’s picture made the geometry intuitive and opened the way to structure determination.

Reading a Laue photograph

For the general reader. The 1912 photographs show a central black spot surrounded by smaller spots arranged with the symmetry of the crystal. Each spot is a beam reflected from one family of atomic planes, picking out from the white X-ray beam just the wavelength those planes can reflect. Reading the pattern means working back from spot positions to plane orientations.

The Ewald construction

Draw \(\mathbf k_0\) ending at the origin of reciprocal space. The condition \(\mathbf k=\mathbf k_0+\mathbf G\) with \(|\mathbf k|=|\mathbf k_0|\) says that \(\mathbf G\) lies on a sphere of radius \(k\) centred at the tail of \(\mathbf k_0\). Ewald introduced this construction in 1913. A monochromatic beam gives a single sphere, which rarely passes through reciprocal lattice points. A white beam with \(\lambda_{\min}\le\lambda\le\lambda_{\max}\) gives a family of spheres filling a crescent-shaped region, and every lattice point inside the region produces a spot.

All harmonics \((nh,nk,nl)\) fall on the same spot, since the plate sees only the direction. Each spot is thus a superposition of orders, which made intensities hard to interpret.

Zinc blende and Laue’s analysis

The zinc blende photograph, with the beam along a cube axis, showed fourfold symmetry. Laue indexed the spots on a simple cubic lattice with one ZnS unit per cube, whose edge he computed from the density. He found he could account for the observed spots only by supposing that the beam contained a few discrete wavelengths, as if the crystal emitted characteristic radiation. In November 1912 Lawrence Bragg showed that a continuous spectrum and a face-centred cubic lattice explain the pattern: the missing spots are exactly those forbidden by the fcc lattice (Exercise 6.1). Laue’s geometric insight was right; his model of the crystal and of the beam needed correction.

Structure factors and Friedel’s law

For the general reader. The lattice decides where the spots are. The arrangement of atoms inside each cell decides how bright they are. Laue’s theory of 1912 already contained this split, and it is the basis of every crystal structure determination since, from salt to proteins.

The kinematic theory

Suppose each wave is scattered once and the incident beam is not depleted. If the cell contains atoms at positions \(\mathbf r_j\) with scattering factors \(f_j(\mathbf q)\), the amplitude factorizes as \(A(\mathbf q)=F(\mathbf q)\sum_{\mathbf R}e^{-i\mathbf q\cdot\mathbf R}\), with

The scattering factor \(f_j\) is the Fourier transform of the atom’s electron density: it equals the number of electrons at \(\mathbf q=0\) and decreases with angle. Thermal vibration further reduces intensities by the Debye–Waller factor \(e^{-2W}\), where \(2W\) is proportional to the mean-square displacement times \(q^2\). This is why Sommerfeld’s worry about thermal motion was misplaced: vibrations weaken the spots, especially at large angles, but do not blur them away.

Friedel’s law

The proof is one line: complex conjugation reverses the sign of every exponent. As a consequence, X-ray diffraction alone, in this approximation, cannot distinguish a crystal from its inverted image, and a Laue photograph shows one of the 11 centrosymmetric Laue classes rather than one of the 32 point groups. Near an absorption edge, \(f_j\) acquires an imaginary part and Friedel’s law fails; this “anomalous scattering” is now used to determine absolute configurations and to solve protein phases.

The phase problem

Since the electron density is \(\rho(\mathbf r)=\frac1{V_c}\sum_{hkl}F_{hkl}\,e^{i\mathbf G\cdot\mathbf r}\), knowing every complex \(F_{hkl}\) would give the structure at once. Experiments measure only \(|F_{hkl}|^2\). Recovering the lost phases is the central problem of crystallography, solved by trial structures in the Braggs’ era, then by Patterson maps, direct methods and heavy-atom substitution.

Dynamical theory and anomalous transmission

For the general reader. In a large perfect crystal a reflected X-ray beam is strong enough to be reflected again, back into the forward direction. The two beams trade energy back and forth as they travel. Laue reformulated this “dynamical” theory in 1931, and after the war he explained a striking effect: under exact reflection conditions, a thick crystal can become much more transparent to X-rays.

From Darwin and Ewald to Laue

Charles Galton Darwin gave the first dynamical theory in 1914, treating the crystal as a stack of reflecting planes. Ewald, in his work of 1916–17, solved the problem with a lattice of dipoles. In 1931 Laue recast the theory with a continuous, periodic dielectric susceptibility, \(\chi(\mathbf r)=\sum_{\mathbf G}\chi_{\mathbf G}e^{i\mathbf G\cdot\mathbf r}\), which is the formulation used today. For X-rays \(\chi_{\mathbf G}=-\dfrac{r_e\lambda^2}{\pi V_c}F_{\mathbf G}\), of order \(10^{-5}\) to \(10^{-6}\), where \(r_e\) is the classical electron radius.

Two-beam theory

Inside the crystal, write the field as a Bloch wave \(\mathbf D=\sum_{\mathbf G}\mathbf D_{\mathbf G}e^{i(\mathbf K_0+\mathbf G)\cdot\mathbf r}\). When only \(\mathbf G=0\) and one reflection \(\mathbf G=\mathbf h\) are near the Ewald sphere, Maxwell’s equations reduce, for one polarization, to

with \(\mathbf K_h=\mathbf K_0+\mathbf h\). A solution exists only if the determinant vanishes, which defines the dispersion surface, a hyperboloid with two branches separated near the Bragg condition. Each incident wave excites one point on each branch.

The name, “pendulum solution”, is Ewald’s, by analogy with two coupled pendulums exchanging energy.

The Borrmann effect

In 1941 Gerhard Borrmann observed that a thick, nearly perfect crystal, set at a Bragg angle in transmission, lets through far more X-rays than its absorption coefficient allows. Laue gave the explanation in a paper he wrote while interned at Farm Hall and published in Acta Crystallographica. At the Bragg condition the two wavefields are standing waves across the reflecting planes. In one, the nodes sit on the atomic planes, where absorbing electrons are concentrated, so it is barely absorbed. In the other, the antinodes sit on the planes, and it is absorbed strongly. Through a thick crystal only the first survives. At the Fritz Haber Institute in the 1950s Laue continued this work with Borrmann. Anomalous transmission is now a standard tool in X-ray topography and standing-wave methods.

Superconductors in a magnetic field

For the general reader. A superconductor carries current with no resistance, but a strong enough magnetic field destroys the effect. In the early 1930s experimenters found that the destroying field seemed to depend on the shape of the sample, which made no sense for a property of the material. Laue explained it: the field at the surface of a sample is not the applied field, and it depends on the shape. He spent much of the next fifteen years on the electrodynamics of superconductors.

Shape and the critical field

In 1932 Laue, in contact with Walther Meissner at the Reichsanstalt, interpreted a puzzling measurement by Wander de Haas. He pointed out that what matters is the field at the surface of the sample, which differs from the applied field by an amount set by the shape. The argument helped prepare the discovery by Meissner and Robert Ochsenfeld in 1933 that a superconductor expels magnetic flux: inside, \(\mathbf B=0\). A sample in an applied field \(\mathbf H_0\) therefore distorts the field around it, concentrating it at some points of the surface. Superconductivity first fails where the local surface field reaches the critical value \(H_c\).

Derivation. Outside, \(\mathbf H=-\nabla\Phi\) with \(\Phi=-H_0\big(r+\frac{R^3}{2r^2}\big)\cos\vartheta\). This satisfies Laplace’s equation and gives \(H_r=-\partial_r\Phi=H_0\big(1-\frac{R^3}{r^3}\big)\cos\vartheta\), which vanishes at \(r=R\), as required by \(B_r\) continuous and zero inside. The tangential component at \(r=R\) is \(H_\vartheta=-\frac1R\partial_\vartheta\Phi=-\frac32H_0\sin\vartheta\). Between \(\tfrac23H_c\) and \(H_c\) the sphere is in an “intermediate state” of normal and superconducting domains.

The London equations

In 1935 Fritz London and Heinz London proposed that the supercurrent density \(\mathbf J_s\) obeys

Combined with Ampère’s law \(\nabla\times\mathbf B=\mu_0\mathbf J_s\), the second equation gives \(\nabla^2\mathbf B=\mathbf B/\lambda_L^2\), with

so fields decay exponentially within a penetration depth of the surface. The same year Laue published a joint paper with the London brothers in the Zeitschrift für Physik. From 1937 to 1947 he wrote twelve papers and the book Theorie der Supraleitung (1947), treating superconductors within Maxwell’s phenomenological electrodynamics and thermodynamics. The microscopic explanation came with the theory of John Bardeen, Leon Cooper and Robert Schrieffer in 1957.

Berlin under two regimes, 1919–1945

Laue’s Zurich post lasted two years. In 1914 he took the chair of theoretical physics at the newly founded University of Frankfurt, and he received the Nobel Prize in Physics for 1914, for the discovery of the diffraction of X-rays by crystals. In 1913 his father had been ennobled, and from then on he was Max von Laue. During the war he worked at Würzburg on vacuum tubes for military telephony and radio. In 1917 he joined the new Kaiser Wilhelm Institute for Physics in Berlin, where he took over most of the administration from its director, Albert Einstein, a close friend. In 1919 he and Max Born exchanged chairs, so that Laue could return to Berlin and to Planck. He held the Berlin professorship until 1943, and in 1925 he became theoretical adviser to the Physikalisch-Technische Reichsanstalt. Leo Szilard took his doctorate under him in 1922.

When the Nazis came to power in 1933, Laue resisted openly and often alone. When Einstein resigned from the Prussian Academy, Laue protested in the Academy and found almost no support. In September 1933, as chairman of the German Physical Society, he opened its Würzburg meeting by comparing the persecution of Galileo with the campaign against relativity. He blocked the election of Johannes Stark, champion of “German physics”, to the Prussian Academy. Stark, now president of the Reichsanstalt, removed him as adviser in December 1933.

In 1934 Laue published an obituary of the chemist Fritz Haber, who had died in exile; it was read as a provocation. On 29 January 1935 he, Planck and Otto Hahn held a memorial for Haber in Berlin-Dahlem although the ministry had forbidden university staff to attend. With Hahn he quietly helped persecuted colleagues to emigrate. In 1937 his son Theodor left for the United States. When Germany occupied Denmark in 1940, George de Hevesy dissolved the Nobel medals of Laue and James Franck in aqua regia at Niels Bohr’s institute to keep them from the Nazis; the Nobel Foundation later recast them from the recovered gold.

Laue remained deputy director of the Kaiser Wilhelm Institute for Physics, where Peter Debye was director from 1935 to 1939. Laue did not take part in the uranium project. In 1943 he retired early from his chair and moved with the institute to Hechingen in Württemberg, where he wrote a short history of physics.

Farm Hall, rebuilding, legacy

French troops reached Hechingen on 23 April 1945. The Allied Alsos mission took Laue into custody, and he was interned with nine other German scientists, among them Werner Heisenberg and Otto Hahn, at Farm Hall, near Huntingdon in England. Their conversations were secretly recorded, including their reactions to the news of Hiroshima. Laue, who had no part in nuclear research, used the time to write his paper on the absorption of X-rays under diffraction conditions (Lesson 09).

He returned to Germany early in 1946 and settled in Göttingen, where he helped re-establish the Kaiser Wilhelm Institute for Physics as a Max Planck Institute. In July 1946 he was the only German invited to an international crystallography conference in England. He helped found the German Physical Society in the British zone, helped reunite the Reichsanstalt as the Physikalisch-Technische Bundesanstalt in Braunschweig, and in 1950 helped re-establish the federation of German physical societies. In 1948 the new International Union of Crystallography made him its honorary president. In 1951, aged 71, he became director of the Max Planck Institute for Physical Chemistry and Electrochemistry in Berlin-Dahlem; at his request it was renamed the Fritz Haber Institute in 1953. He retired in the late 1950s.

Laue loved fast driving. On 8 April 1960, on his way to the institute, his car was struck by a motorcyclist. The motorcyclist was killed and the car overturned. Laue died of his injuries on 24 April 1960, aged 80, and was buried in Göttingen.

Legacy

• Diffraction: the 1912 experiment founded X-ray crystallography and, through it, solid-state physics, mineralogy, metallurgy and structural biology. The Laue method is still used with white synchrotron and neutron beams, notably for fast time-resolved studies.
• Relativity: the first textbook, the derivation of Fresnel drag, and Laue’s theorem on the inertia of stressed systems.
• Dynamical diffraction: his 1931 formulation and his explanation of the Borrmann effect underlie X-ray topography and the study of perfect crystals.
• Honours: the Nobel Prize in Physics for 1914, the Max Planck Medal (1932), foreign membership of the Royal Society, the Pour le Mérite for Sciences and Arts, and many academies. The Institut Laue–Langevin in Grenoble, the European neutron source, bears his name together with that of Paul Langevin.

Connections

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

Teachers and advisors

• Max Planck 1903

Students

• Leo Szilard 1922

Influenced by

• Hermann Minkowski
• Charles Galton Darwin
• Max Abraham
• Woldemar Voigt

Influenced

• Henry Moseley
• Lawrence Bragg

Colleagues and collaborators

• Arnold Sommerfeld 1909-1912
• Max Born
• Peter Debye 1935-1939
• Fritz London 1935
• Gerhard Borrmann 1950s
• Heinz London 1935
• Paul Knipping 1912
• Paul Peter Ewald 1912
• Walter Friedrich 1912
• Walther Meissner
• Wilhelm Röntgen 1909-1912

Rivals

• Johannes Stark

Friends

• Albert Einstein 1907-1955
• Lise Meitner
• Otto Hahn

Also mentioned in this life

• André-Marie Ampère
• Amedeo Avogadro
• Niels Bohr
• Christian Doppler
• Hippolyte Fizeau
• Joseph Fourier
• Augustin-Jean Fresnel
• Galileo Galilei
• Carl Friedrich Gauss
• Oliver Heaviside
• Werner Heisenberg
• Paul Langevin
• Pierre-Simon Laplace
• Hendrik Antoon Lorentz
• James Clerk Maxwell
• Albert A. Michelson
• Henri Poincaré

Fields

• crystallography 3 lives
• special relativity 3 lives
• optics 38 lives

On CoursesHub

• Special Relativity
• Optics & Photonics

Sources and further reading

Biography

1. Max von Laue: Biographical, NobelPrize.org.
2. Max von Laue, Wikipedia (with extensive references).
3. A. Hermann, Laue, Max von, Neue Deutsche Biographie 13 (1982).
4. International Union of Crystallography, Max von Laue, including his own account of 1912.
5. P. P. Ewald, Fifty Years of X-ray Diffraction, IUCr, 1962.
6. P. P. Ewald, Max von Laue 1879–1960, Biographical Memoirs of Fellows of the Royal Society 6 (1960).
7. K. Hentschel (ed.), Physics and National Socialism: An Anthology of Primary Sources, Birkhäuser, 1996.

Physics

1. M. Laue, Die Mitführung des Lichtes durch bewegte Körper nach dem Relativitätsprinzip, Annalen der Physik 23 (1907).
2. M. Laue, Das Relativitätsprinzip, Vieweg, Braunschweig, 1911.
3. M. Laue, Zur Dynamik der Relativitätstheorie, Annalen der Physik 35 (1911).
4. D. Giulini, Laue’s theorem revisited: energy-momentum tensors, symmetries, and the habitat of globally conserved quantities, arXiv:1808.09320.
5. W. Friedrich, P. Knipping, M. Laue, Interferenz-Erscheinungen bei Röntgenstrahlen, Sitzungsberichte der Bayerischen Akademie der Wissenschaften (1912) 303.
6. M. von Laue, Concerning the detection of X-ray interferences, Nobel lecture, 1920.
7. M. von Laue, Die dynamische Theorie der Röntgenstrahlinterferenzen in neuer Form, Ergebnisse der exakten Naturwissenschaften 10 (1931).
8. M. von Laue, Röntgenstrahl-Interferenzen, Akademische Verlagsgesellschaft, Leipzig, 1941.
9. F. London, H. London, The electromagnetic equations of the supraconductor, Proc. R. Soc. A 149 (1935).
10. M. von Laue, Theorie der Supraleitung, Springer, 1947.
11. A. Authier, Dynamical Theory of X-ray Diffraction, Oxford University Press, 2001.