Wilhelm Wien

How to use this course

Wilhelm Wien received the Nobel Prize in Physics for 1911 for his discoveries on the laws of heat radiation. Two of them frame the work of Max Planck: the displacement law of 1893, which fixed how the black-body spectrum depends on temperature, and the distribution law of 1896, which was right at short wavelengths and wrong at long ones. Planck’s formula of 1900 was built as a correction to Wien’s, and Albert Einstein found the light quantum by reading the entropy hidden in it. Wien was also an experimentalist of the first rank: his work on canal rays identified the positive particle with the mass of hydrogen and led to mass spectrometry. This course covers both sides, alongside the life of a conservative nationalist who ran the leading German physics journal for over twenty years.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 to 07 form a sequence on radiation; read them alongside Lessons 05 to 07 of the Planck course. Lessons 08 to 10 treat particle beams and atomic radiation. Formulas use modern notation; where Wien’s differs, the text says so.

Timeline

East Prussia, 1864–1882

Wilhelm Carl Werner Otto Fritz Franz Wien was born on 13 January 1864 at Gaffken, an estate near Fischhausen on the Samland peninsula west of Königsberg. His father, Carl Wien, was a landowner. In 1866 the family moved to Drachenstein, near Rastenburg in the lake district of East Prussia, where his father had bought a larger estate. Wien grew up in the country, expecting to take over the farm.

In 1879 he entered the gymnasium at Rastenburg. His results there were poor and he left for private tutoring. He completed his schooling between 1880 and 1882; the Nobel Foundation’s biography places him at a city school in Heidelberg, while German sources name the Altstädtisches Gymnasium in Königsberg.

Helmholtz and the farm, 1882–1890

In 1882 Wien began to study mathematics and natural science at Göttingen, and moved to Berlin the same year. From 1883 to 1885 he worked in the physics laboratory of Hermann von Helmholtz, then the most powerful figure in German science. In 1886 he received his doctorate with a thesis on the diffraction of light at the edges of metal screens and on how the material of the screen affects the colour of the diffracted light. Helmholtz was his advisor.

His father then fell ill, and Wien returned to Drachenstein to help run the estate. He spent the next four years as a farmer, keeping up physics on the side. In 1887 he made experiments on the permeability of thin metal films to light and heat rays. In 1890 the estate was sold, and Wien was free to return to science.

The Reichsanstalt, 1890–1900

Around 1890 Wien became an assistant to Helmholtz at the Physikalisch-Technische Reichsanstalt in Charlottenburg, the imperial institute for physics and technology founded in 1887 with Helmholtz as its first president. The PTR set standards for German industry, and one of its tasks was to measure high temperatures and the light output of lamps. Wien worked with Ludwig Holborn on the measurement of high temperatures. The practical question of how a glowing body radiates led him to the theoretical question of what law governs radiation in equilibrium.

He habilitated at the University of Berlin in 1892 and lectured there as a Privatdozent. Between 1893 and 1896 he published the papers that made his name: the displacement law (1893), the paper on the temperature and entropy of radiation (1894), the proposal with Otto Lummer to realize a black body as a heated cavity (1895), and the distribution law (1896). The PTR group of Lummer, Ernst Pringsheim and later Ferdinand Kurlbaum and Heinrich Rubens tested his formulas over the next four years with ever greater precision.

In 1896 Wien moved to the Technische Hochschule in Aachen, succeeding Philipp Lenard, and worked in Adolf Wüllner’s institute on discharge tubes. There he married Luise Mehler of Aachen in 1898; they had four children. In 1899 he was called to a chair at Giessen, and almost at once to Würzburg, where he took up the chair of physics on 1 April 1900 as successor to Wilhelm Conrad Röntgen, who had discovered X-rays there in 1895.

Temperature and entropy of radiation

For the general reader. A beam of light can carry heat, and so it should be possible to speak of its temperature and its entropy, just as for a gas. In 1894 Wien worked out what this means. He defined the ideal black body precisely, and with Otto Lummer he showed how to build one: a closed oven with a small hole. The light coming out of the hole is the standard against which every hot surface is measured.

The black body and its realization

Gustav Kirchhoff proved in 1859–60 that for any body in equilibrium with radiation, the ratio of emissive power to absorptivity at each frequency is a universal function of frequency and temperature. A body that absorbs all incident radiation, a black body, emits exactly this function. Wien insisted in 1894 that the theory of radiation should start from the black body as a precise concept. In 1895 he and Lummer pointed out that no real surface is perfectly black, but the radiation inside a closed cavity at uniform temperature is black whatever the walls, and a small opening lets a sample of it out without disturbing the equilibrium. Ray-traced through the opening, the inside looks black at room temperature because light entering the hole is almost never reflected back out.

Pressure, energy, entropy

For isotropic radiation the pressure is \(p=u/3\), from James Clerk Maxwell’s stress tensor or by counting momentum transfer. With \(U=uV\) and \(u=u(T)\), the thermodynamic identity \(\big(\partial U/\partial V\big)_T=T\big(\partial p/\partial T\big)_V-p\) gives \(u=aT^4\), the Stefan–Boltzmann law (see Planck, Lesson 05). Wien’s 1894 paper added the entropy.

Derivation. \(dS=\frac{dU+p\,dV}{T}=\frac{1}{T}\big(4aT^3V\,dT+aT^4dV\big)+\frac{aT^3}{3}dV=4aT^2V\,dT+\tfrac43aT^3dV=d\big(\tfrac43aT^3V\big)\).

Wien went further and assigned a temperature and entropy to each spectral component separately. This step, natural for him, prepared the way for Planck’s and Einstein’s later use of the entropy of monochromatic radiation.

The displacement law

For the general reader. As iron is heated it glows dull red, then orange, then yellow-white. In 1893 Wien proved that the whole spectrum of a black body at one temperature determines it at every other temperature by a simple stretching. The colour of peak brightness moves to shorter wavelength in exact inverse proportion to the temperature. This is how astronomers read the temperatures of stars.

The moving mirror

Enclose black-body radiation in a cylinder with perfectly reflecting walls and a reflecting piston. A perfect reflector neither absorbs nor emits, so the only way the spectrum can change is through the Doppler shift on reflection from the moving piston. Wien showed two things. First, a reversible adiabatic compression carries black radiation into black radiation; otherwise one could build a perpetual motion machine of the second kind by letting radiation of different spectra exchange heat through a filter. Second, the Doppler shifts move each frequency in a definite way that he could compute.

Sketch. A mirror moving at speed \(v\ll c\) shifts the frequency of a wave incident at angle \(\theta\) by \(\delta\nu=2\nu(v/c)\cos\theta\). Averaging over the isotropic radiation and over the number of reflections per unit time in a sphere of radius \(R\) gives \(d\nu/\nu=-dR/R\). For each normal mode of a cavity the same result follows from the fact that a slowly changing oscillator conserves \(E/\nu\) (an adiabatic invariant, the same \(E/\nu\) that became Paul Ehrenfest’s key to quantization).

Proof. Follow the radiation in the band \([\nu,\nu+d\nu]\) along an adiabat. By the lemma, \(\nu/T\) is fixed and the band energy \(u\,d\nu\,V\) scales as \(\nu\). Since \(V\propto\nu^{-3}\) and \(d\nu\propto\nu\), \(u\propto\nu^3\). So \(u/\nu^3\) is constant on the curves \(\nu/T=\) const, which means it is a function of \(\nu/T\) alone. Every temperature is reached along some adiabat, so the form holds everywhere.

Wien himself stated the law in wavelength form. The name “displacement law” (Verschiebungsgesetz) refers to the shift of the spectrum with temperature. The value of \(b\) was measured by Lummer and Pringsheim in 1899; it was later expressed through Planck’s constant as \(b=hc/(4.965\,k)\).

The distribution law of 1896

For the general reader. The displacement law left one unknown function. In 1896 Wien guessed it, using the idea that the molecules emitting the light move with the random speeds of a hot gas. His formula matched every measurement then available. Four years later, new measurements far into the infrared showed it falling short, and Planck repaired it.

Wien’s argument

Wien supposed that each emitting molecule radiates at a wavelength determined by its speed, \(\lambda=\lambda(v)\), and that the intensity at that wavelength is proportional to the number of molecules with that speed. By Maxwell’s distribution the number is proportional to \(e^{-mv^2/2kT}\). Taking \(v^2\) inversely proportional to \(\lambda\), the intensity contains a factor \(e^{-c_2/\lambda T}\). The displacement law then fixes the rest: \(u_\lambda=\lambda^{-5}\psi(\lambda T)\) requires the prefactor \(\lambda^{-5}\). The molecular picture is untenable, and Planck later gave a different derivation through the entropy of resonators, but the formula stood.

Tests

Friedrich Paschen in Hanover, and Lummer and Pringsheim at the PTR, measured cavity spectra between 1896 and 1899. Paschen found good agreement in the near infrared. Lummer and Pringsheim reported in 1899 and early 1900 systematic excesses at longer wavelengths and higher temperatures, where \(\lambda T\) is large. In 1900 Rubens and Kurlbaum measured at wavelengths up to about 50 μm with residual-ray techniques and found the intensity growing linearly with \(T\), while Wien’s law predicts saturation. Planck’s law of October 1900 fitted both regimes.

Wien’s law and the light quantum

For the general reader. Wien’s formula was superseded within four years, but it contained a secret. In 1905 Einstein computed how the entropy of radiation obeying Wien’s law changes when its volume changes, and found exactly the formula for a gas of independent particles. He concluded that light, in this regime, behaves as if it consisted of energy packets \(h\nu\). The road to the photon runs through Wien’s law, not Planck’s.

Entropy of Wien radiation

Consider radiation in a volume \(V\) with energy \(E\) in a narrow band \([\nu,\nu+d\nu]\), so \(u=E/(V\,d\nu)\). The entropy density per unit frequency \(s(u)\) satisfies \(ds/du=1/T\). Wien’s law, inverted, gives

in modern constants (Einstein wrote \(R\beta/N\) for \(h/k\)). Integrating in \(u\):

Derivation. \(s(u)=-\frac{ku}{h\nu}\Big(\ln\frac{c^3u}{8\pi h\nu^3}-1\Big)\). The total entropy is \(S=V\,d\nu\,s(E/(V\,d\nu))\). Differentiate in \(V\) at fixed \(E\): \(\partial S/\partial V=d\nu\,(s-us'(u))\). Here \(s-us'=\frac{ku}{h\nu}\), so \(\partial S/\partial V=\frac{kE}{h\nu V}\), which integrates to the result. For a gas, \(S(V)-S(V_0)=Nk\ln(V/V_0)\), because the probability that all \(N\) independent particles are found in \(V\) out of \(V_0\) is \((V/V_0)^N\).

With Planck’s law the same calculation gives an energy fluctuation with two terms, one from particles and one from waves (Einstein, 1909). Wien’s law is the pure-particle limit. Einstein used the result to explain the photoelectric effect, and his Nobel Prize of 1921 was for that law.

Cathode rays, canal rays and the Wien filter

For the general reader. A discharge tube produces two kinds of rays. Cathode rays stream from the negative electrode; canal rays pass backward through holes drilled in it. In 1897–98 Wien measured how much each kind is bent by electric and magnetic fields. Cathode rays turned out to be very light negative particles, as J. J. Thomson found the same year. Canal rays turned out to be positive and thousands of times heavier, the lightest of them with the mass of a hydrogen atom. Wien’s device of crossed fields is still used to select particles by speed.

Deflection by fields

A particle of charge \(q\), mass \(m\) and speed \(v\) entering a region of length \(\ell\) with a transverse field leaves with a small deflection angle

Measuring both gives \(v=\frac{E\,\theta_B}{B\,\theta_E}\) and \(\frac qm=\frac{v\,\theta_B}{B\ell}\). In 1897 Wien applied this to cathode rays, confirming Jean Perrin’s demonstration that they carry negative charge and finding \(e/m\) about two thousand times that of the hydrogen ion in electrolysis.

Proof. The Lorentz force is \(q(\mathbf E+\mathbf v\times\mathbf B)\). For \(\mathbf v\perp\mathbf B\) and the fields oriented so that \(\mathbf v\times\mathbf B\) is antiparallel to \(\mathbf E\), the transverse force vanishes exactly when \(E=vB\).

Canal rays

Eugen Goldstein had discovered canal rays in 1886. Wien showed in 1898 that they are deflected by magnetic and electric fields in the sense of positive charges, and measured their \(q/m\). The values varied with the gas in the tube. In further measurements, reported by 1904, the largest \(q/m\) he found was close to that of the hydrogen ion, about \(10^8\ \mathrm{C/kg}\) in SI units. This was the positive particle later named the proton by Ernest Rutherford. Thomson refined the method into the parabola spectrograph (1907–13), and Francis Aston and Arthur Jeffrey Dempster built the first mass spectrographs on these foundations.

Electromagnetic mass

For the general reader. A moving charge drags its electric field along, and the field carries momentum. So a charged body is harder to accelerate than an uncharged one of the same material. In 1900 Wien proposed that all mass might be of this kind, and that mechanics could be derived from electromagnetism instead of the other way round. The programme failed, but it fed the questions that relativity answered.

In Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik (1900), Wien suggested reversing the usual order: instead of explaining the electromagnetic field by mechanical models of the ether, take the field as fundamental and derive inertia from it. This “electromagnetic world view” was developed by Max Abraham, who in 1902–03 computed the velocity dependence of the mass of a rigid spherical electron, and was tested by Walter Kaufmann’s deflection experiments on fast electrons.

Derivation. To first order in \(v/c\) the electric field is the Coulomb field and the magnetic field is \(\mathbf B=\mathbf v\times\mathbf E/c^2\). The momentum density is \(\varepsilon_0\mathbf E\times\mathbf B=\frac{\varepsilon_0}{c^2}\mathbf E\times(\mathbf v\times\mathbf E)=\frac{\varepsilon_0}{c^2}\big(E^2\mathbf v-(\mathbf E\cdot\mathbf v)\mathbf E\big)\). Averaging over directions, \(\langle(\hat{\mathbf r}\cdot\mathbf v)\hat{\mathbf r}\rangle=\mathbf v/3\), so \(\mathbf p=\frac{2\varepsilon_0}{3c^2}\mathbf v\int E^2dV\). With \(\int\frac{\varepsilon_0}{2}E^2dV=U_{\text{es}}\) the result follows.

The factor \(\tfrac43\) instead of 1 showed that the field alone is not a consistent relativistic object: Henri Poincaré’s non-electromagnetic stresses of 1905–06 are needed to hold the charge together, and with them the total inertial mass is \(U/c^2\). Einstein’s \(E=mc^2\) of 1905 made the electromagnetic origin of mass unnecessary. Wien accepted relativity, and he later nominated Hendrik Lorentz and Einstein for the Nobel Prize.

How long an atom shines

For the general reader. When an atom is excited it does not radiate forever. How long does it glow? Wien found an ingenious way to measure it. Atoms in a canal-ray beam move at a known, high speed. If they enter a good vacuum where nothing re-excites them, the glow fades along the beam, and the fading length divided by the speed is the lifetime. He found times of order a hundred-millionth of a second.

The method

In canal-ray experiments published from 1918 onward, Wien sent a canal-ray beam excited in a discharge through a slit into a high-vacuum chamber. With a beam speed \(v\) of order \(10^5\)–\(10^6\) m/s and a lifetime \(\tau\), the intensity falls as \(I(x)=I_0e^{-x/v\tau}\). The speed is known from the Doppler shift of the emitted lines, and the decay length is measured photographically. The method needs no fast electronics; the beam itself converts time into distance.

Derivation. The time-averaged Larmor power for an oscillator \(x=x_0\cos\omega t\) is \(P=\frac{e^2\omega^4x_0^2}{12\pi\varepsilon_0c^3}\), and the energy is \(E=\tfrac12m_e\omega^2x_0^2\). So \(1/\tau=P/E=\frac{e^2\omega^2}{6\pi\varepsilon_0m_ec^3}\). In quantum mechanics the lifetime of an excited state is \(1/A\), with \(A\) the Einstein coefficient; for strong lines it equals the classical value times the oscillator strength. Wien’s measurements thus gave an independent check of the atomic theory of radiation.

Würzburg and Munich, 1900–1928

Wien spent twenty years in Würzburg as professor of physics and director of the institute where Röntgen had found X-rays. He declined calls to Leipzig in 1902, as successor to Ludwig Boltzmann, and to Berlin in 1906, after the death of Paul Drude. That year he became editor of the Annalen der Physik, the most important physics journal in the German language, and he edited it until his death, with Max Planck responsible for theoretical papers. Much of the new quantum physics appeared in its pages.

The Nobel Prize in Physics for 1911 was awarded to him “for his discoveries regarding the laws governing the radiation of heat”. He gave his Nobel lecture in Stockholm on 11 December 1911. In 1913 he gave the Ernest Kempton Adams Lectures at Columbia University in New York, and in 1913–14 he served as rector of the University of Würzburg. He was chairman of the Society of German Natural Scientists and Physicians in 1910.

In 1920 Wien moved to Munich, again succeeding Röntgen, as professor of experimental physics. He was chairman of the German Physical Society from 1920 to 1922. In Munich he was the experimental counterpart to Arnold Sommerfeld, and the two disagreed often about the value of theory. In July 1923 he examined Sommerfeld’s student Werner Heisenberg at the doctoral oral, asking about the resolving power of optical instruments, which Heisenberg had not studied; Heisenberg nearly failed and passed with a low grade. Wien’s doctoral students included Lars Vegard, Gabriel Holtsmark and Eduard Rüchardt (1914), who followed him to Munich and continued the canal-ray work. His cousin Max Wien, inventor of the Wien bridge, was professor of physics at Jena.

Politics, death and legacy

Wien was a conservative nationalist. In October 1914 he signed the Manifesto of the 93, in which German scholars denied German war crimes in Belgium. In December 1914 he circulated to leading German physicists a proposal that they cite English work no more often than German, stop publishing in English journals and keep German as their language of publication. Many of the addressees declined to support it: Emil Warburg refused to mix politics and science, Friedrich Paschen rejected such rules in principle, and Planck thought it a matter for peacetime. Wien’s politics did not extend to the “Deutsche Physik” of Philipp Lenard and Johannes Stark. He respected Einstein’s work and supported relativity.

Wien died in Munich on 30 August 1928, aged 64, and is buried in the Waldfriedhof there. His autobiographical retrospect, written in 1927, was published in 1930 with letters and other material as Aus dem Leben und Wirken eines Physikers. His son Karl Wien became a noted mountaineer.

Legacy

• Radiation: the displacement law \(\lambda_{\max}T=b\), still the quickest way to read the temperature of a star, a furnace or the cosmic background; Wien’s distribution law as the high-frequency limit of Planck’s; the cavity radiator as a temperature standard.
• Quantum theory: his law was the starting point of Planck’s interpolation and of Einstein’s light-quantum argument.
• Particle beams: the Wien filter, used in ion implanters, electron microscopes and mass spectrometers; the identification of the positive hydrogen ion in canal rays; lifetimes of excited atoms.
• Institutions: two decades as editor of the Annalen der Physik. A Mars crater and the asteroid 48456 Wilhelmwien bear his name.

Sources and further reading

Biography

1. Wilhelm Wien, biographical, Nobel Prize website.
2. Wilhelm Wien, Wikipedia; German Wikipedia.
3. University of Würzburg archive, Wilhelm Wien.
4. Wilhelm Wien, Encyclopaedia Britannica.
5. S. L. Wolff, Physiker im „Krieg der Geister“, Deutsches Museum.
6. W. Wien, Aus dem Leben und Wirken eines Physikers, Barth, Leipzig, 1930.

Physics

1. W. Wien, Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie, Sitzungsber. Preuss. Akad. Wiss. (1893) 55–62.
2. W. Wien, Temperatur und Entropie der Strahlung, Ann. Phys. 52 (1894) 132–165.
3. W. Wien, O. Lummer, Methode zur Prüfung des Strahlungsgesetzes absolut schwarzer Körper, Ann. Phys. 56 (1895) 451–456.
4. W. Wien, Über die Energievertheilung im Emissionsspectrum eines schwarzen Körpers, Ann. Phys. 58 (1896) 662–669.
5. W. Wien, Die elektrostatischen Eigenschaften der Kathodenstrahlen, Ann. Phys. 65 (1898) 440–452.
6. W. Wien, Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik, Ann. Phys. 5 (1901) 501–513.
7. W. Wien, On the laws of thermal radiation, Nobel Lecture, 11 December 1911.
8. A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt, Ann. Phys. 17 (1905) 132–148.
9. T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912, Oxford University Press, 1978.
10. Wien’s displacement law, Wien approximation and Wien filter, Wikipedia.