Heike Kamerlingh Onnes

How to use this course

Heike Kamerlingh Onnes was an experimentalist and an organizer. He did not invent a single equation that carries his name in daily use, apart from the expansion in which he wrote the equation of state of a real gas. What he built was a method: a large laboratory, run like a factory, with trained glassblowers and instrument makers, that could produce cold on demand. With it he liquefied helium, the last of the permanent gases, and opened the temperature range below 5 K, where quantum mechanics shows itself on the scale of a laboratory bench.

Part II follows the logic of his programme. Lessons 04 and 05 cover the theory of real gases that told him how cold helium would have to be. Lesson 06 explains the cooling method. Lesson 07 is the liquefaction itself. Lessons 08 and 09 cover superconductivity as he found it, and Lesson 10 the superfluid transition that happened in his apparatus without being recognized. Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions.

Timeline

Groningen and Heidelberg, 1853–1873

Heike Kamerlingh Onnes was born in Groningen, in the north of the Netherlands, on 21 September 1853. His father Harm owned a brickworks near the town; his mother, Anna Gerdina Coers, came from Arnhem. The family was comfortable and artistic. His younger brother Menso became a painter, and Menso’s son Harm followed him. A sister, Jenny, married the painter Floris Verster. Another brother, Onno, would later help the physicist find the raw material for his helium.

Heike attended the Hoogere Burgerschool, the modern secondary school created in the 1860s, which taught science and languages but not the classics. University entrance required Latin and Greek, so he was tutored privately. He entered the University of Groningen in 1870 and took the candidaats degree, roughly a bachelor’s, the next year. As a student he won a gold medal in a competition set by the science faculty at Utrecht (1871) and a silver medal in a similar competition at Groningen (1872).

Heidelberg

From October 1871 to April 1873 he studied at Heidelberg. The chemist Robert Bunsen and the physicist Gustav Kirchhoff had made the university the centre of spectroscopy. Kirchhoff’s style left its mark: a physics in which careful measurement and exact theory are two halves of the same work.

Delft and the rotating Earth, 1873–1882

Back in Groningen he passed the doctoraal examination in 1878. That year he became assistant to Johannes Bosscha, director of the Polytechnic School at Delft, and in 1881 and 1882 he lectured in Bosscha’s place.

His doctorate came in 1879 under Rudolf Adriaan Mees, with the thesis Nieuwe bewijzen voor de aswenteling der aarde (New proofs of the rotation of the Earth). The subject was the Foucault pendulum, whose plane of swing turns relative to the ground at a rate \(\Omega\sin\varphi\) at latitude \(\varphi\). The thesis analysed the pendulum of Léon Foucault in theory and in experiment, with attention to the small disturbing effects that spoil real measurements. It is a mathematical-physics thesis, and it shows an early taste for chasing systematic errors.

Meeting van der Waals

In Amsterdam, Johannes Diderik van der Waals had published his thesis on the continuity of the gaseous and liquid states in 1873, and in 1880 he formulated the law of corresponding states. Kamerlingh Onnes met him and became a lifelong follower. In 1881 he published a paper on the general theory of liquids, which argued for corresponding states from the mechanical similarity of molecular motions. His low-temperature programme was built to test van der Waals’s theory over the widest possible range of temperature (Lesson 04).

Building a laboratory, 1882–1906

In 1882, at 29, Kamerlingh Onnes became professor of experimental physics and meteorology at Leiden, succeeding P. L. Rijke. His inaugural lecture on the role of quantitative research in physics is remembered for its motto, Door meten tot weten: through measurement to knowledge. A year later he was elected to the Royal Academy of Sciences in Amsterdam. In 1887 he married Betsy Bijleveld, who ran a hospitable household and appears in laboratory lore urging her husband to eat during long experiments. Their only son, Albert, became a senior civil servant in The Hague.

The cascade

His first major goal was a reliable supply of liquid oxygen and air. He adopted the cascade method used by Raoul Pictet: each liquefied gas, boiled under reduced pressure, cools the next gas below its critical temperature. The Leiden cascade ran methyl chloride, ethylene and oxygen, and from 1893 it produced liquid oxygen in quantity. The emphasis was on industrial reliability. The laboratory had pumps, compressors and gas holders, and it kept them running for hours.

Then came a setback. Work with large volumes of compressed hydrogen alarmed the town. The laboratory stood near the site of the 1807 gunpowder-ship explosion that had devastated the centre of Leiden, and the authorities stopped the dangerous experiments. After long negotiations, work resumed on 4 June 1898.

A school of craftsmen

Kamerlingh Onnes understood that cryogenics depended on glass, solder and leak-tight joints. In 1901 he founded a school for instrument makers and glassblowers attached to the laboratory, and he relied on outstanding technicians: Gerrit Flim, who became technical manager of the cryogenic plant, and the master glassblower Oskar Kesselring. The laboratory’s results appeared in its own series, the Communications from the Physical Laboratory of the University of Leiden, which became required reading for anyone working at low temperatures.

Meanwhile James Dewar in London liquefied hydrogen in 1898. Leiden’s hydrogen liquefier, slower to arrive but larger, ran in 1906 and produced about four litres an hour. With liquid hydrogen at 20 K available in bulk, helium became the next target.

Corresponding states

For the general reader. Different gases look very different: carbon dioxide liquefies under pressure at room temperature, while helium resists liquefaction to within a few degrees of absolute zero. Van der Waals discovered that, once temperature, pressure and volume are measured in units set by each gas’s critical point, all gases follow nearly the same law. Kamerlingh Onnes used this “law of corresponding states” as a map: it told him in advance how cold, and how large, his machines had to be.

The van der Waals equation

For one mole of gas with molar volume \(V\),

where \(b\) accounts for the finite size of molecules and \(a\) for their mutual attraction. Below a critical temperature the isotherms have a maximum and a minimum, and the fluid separates into liquid and vapour. The critical point is the inflection point where the extremum and minimum merge: \(\partial P/\partial V=\partial^2P/\partial V^2=0\).

Derivation. Write \(P=RT/(V-b)-a/V^2\). The two conditions are \(RT/(V-b)^2=2a/V^3\) and \(RT/(V-b)^3=3a/V^4\). Dividing gives \(V-b=2V/3\), so \(V_c=3b\); then \(RT_c=2a(2b)^2/(27b^3)=8a/27b\), and substituting gives \(P_c\). Replacing \(P,V,T\) by \(pP_c,vV_c,tT_c\) and dividing through by \(P_cV_c\) yields the reduced form.

The modern proof is a scaling argument, close in spirit to Kamerlingh Onnes’s 1881 mechanical similarity. In the classical configuration integral \(\int e^{-\sum u(r_{ij})/kT}d^{3N}r\), substitute \(r=\sigma x\). The result depends on the gas only through \(kT/\varepsilon\) and \(V/N\sigma^3\), so pressure in units of \(\varepsilon/\sigma^3\) is a universal function of these two numbers. The critical point is then at universal values of all three, and reduced variables collapse every such fluid onto one surface. Light gases deviate: for helium and hydrogen, quantum effects (the de Broglie wavelength compared with \(\sigma\)) break the classical scaling, a correction Kamerlingh Onnes saw in his own isotherms.

The virial expansion

For the general reader. No simple formula describes a real gas exactly. In 1901 Kamerlingh Onnes proposed writing the deviation from the ideal gas law as a series in the density. The coefficients, called virial coefficients, can be measured, and statistical mechanics later showed that each one encodes interactions among a definite number of molecules: pairs, triples, and so on.

The expansion

In the Leiden form, the compressibility factor of one mole is expanded as

Kamerlingh Onnes fitted his isotherms with empirical temperature dependences for \(B, C, \dots\) The approach suited an experimental laboratory: it made no theoretical commitment, it reproduced measurements over a wide range, and it isolated the leading correction to the ideal gas, which is exactly what a gas thermometer needs.

Derivation sketch. Write \(e^{-u_{ij}/kT}=1+f_{ij}\), with the Mayer function \(f\) that vanishes at large separation. The configuration integral becomes \(Q=\int\prod_{i<j}(1+f_{ij})\,d^{3N}r\approx V^N\big(1+\tfrac{N^2}{2V}\int f(r)\,d^3r\big)\) to first order in density. Then \(P=kT\,\partial\ln Q/\partial V\) gives \(PV/NkT=1-\tfrac{N}{2V}\int f\,d^3r+\cdots\). Converting to molar volume and writing \(d^3r=4\pi r^2dr\) gives the formula. The third coefficient \(C\) involves triangles of \(f\)-bonds among three molecules, which is the cluster expansion of Joseph Mayer (1937).

Boyle temperature

At low temperature attraction dominates and \(B<0\); at high temperature repulsion dominates and \(B>0\). The Boyle temperature \(T_B\) is where \(B(T_B)=0\): there the gas follows Boyle’s law to second order in density.

Cooling by expansion

For the general reader. When a compressed real gas leaks through a valve into a region of lower pressure, it usually cools, because its molecules do work against their mutual attraction as they spread out. James Prescott Joule and William Thomson (Kelvin) measured this in 1852. Recycling the cooled gas to precool the incoming stream makes the effect cumulative. This is the principle behind every liquefier Kamerlingh Onnes used for hydrogen and helium.

Throttling

Push one mole steadily through a porous plug from pressure \(P_1\) to \(P_2\), with no heat exchange. The work done on the gas is \(P_1V_1-P_2V_2\), and it equals the change in internal energy, so \(U_1+P_1V_1=U_2+P_2V_2\). The enthalpy \(H=U+PV\) is conserved. (The word enthalpy itself is generally credited to Kamerlingh Onnes.)

Derivation. From \(dH=T\,dS+V\,dP\) and the Maxwell relation \((\partial S/\partial P)_T=-(\partial V/\partial T)_P\), we get \((\partial H/\partial P)_T=V-T(\partial V/\partial T)_P\). The cyclic relation gives \(\mu_{JT}=-(\partial H/\partial P)_T/C_P\). For \(V=RT/P+B(T)\), \(T(\partial V/\partial T)_P-V=TB'-B\).

For the van der Waals gas, \(B=b-a/RT\) gives \(TB'-B=2a/RT-b\). Cooling occurs below the inversion temperature

Measured low-pressure inversion temperatures are roughly 600 K for nitrogen, about 200 K for hydrogen and about 40 K for helium. Nitrogen and air therefore cool when throttled from room temperature, which is how Carl von Linde and William Hampson built their air liquefiers in 1895. Hydrogen warms unless precooled with liquid air, and helium warms unless precooled with liquid hydrogen. Kamerlingh Onnes’s helium liquefier was a Linde–Hampson circuit inside baths of liquid air and liquid hydrogen.

Liquid helium, 1908

For the general reader. Helium was the last gas no one had liquefied. Dewar in London and Karol Olszewski in Kraków had tried and failed. On 10 July 1908, after a day that began before dawn, Kamerlingh Onnes saw a liquid surface in his apparatus: about 60 millilitres, enough for a small teacup, at roughly 4 degrees above absolute zero.

The helium

William Ramsay had found terrestrial helium in 1895. Kamerlingh Onnes needed hundreds of litres of gas. Through his brother Onno he obtained monazite sand from North Carolina; heating it released helium, and purifying the gas took chemists months. About 200 litres of gas were used in the 1908 run, with 160 litres in reserve.

The day

Liquid air (75 litres) had been prepared the day before. Liquid hydrogen was ready by 1:30 pm, and helium cooling began at 2:30. The circulation started at 4:20, and by evening the gas was colder than liquid hydrogen. Around 7:30 the thermometer stopped falling at about 4 K. A colleague watching the thermometer suggested the reason was a liquid that could not be seen, because its refractive index is so close to that of its vapour. Lighting it from below revealed the meniscus. His first estimate of the boiling point was about 4.5 K; the modern value is 4.22 K at one atmosphere, and the critical point is at 5.2 K. He then pumped on the liquid, trying to freeze it. It would not solidify.

Pumping on the bath lowers the pressure above it. Liquid evaporates to restore coexistence, each mole carrying away \(L\), and the bath cools until \(P(T)\) matches what the pump can hold. Because the vapour pressure falls exponentially, each further step down in temperature costs an exponentially larger pumping speed. With large pumps Kamerlingh Onnes reached about 1.1 K in 1910.

The same curve served as a thermometer. Once \(P(T)\) of helium was calibrated against a gas thermometer (itself corrected with the virial coefficients of Lesson 05), reading the pressure above the bath gave its temperature.

The vanishing resistance, 1911

For the general reader. What happens to the electrical resistance of a metal as it approaches absolute zero? Some physicists expected it to rise, as the electrons froze onto their atoms. Others expected it to fall smoothly to zero. In 1911 Kamerlingh Onnes’s team found something nobody predicted: at 4.2 K the resistance of mercury dropped abruptly to an unmeasurably small value. They had found superconductivity.

Expectations

In the electron theory of metals of Paul Drude (1900), resistance comes from collisions of conduction electrons. Kelvin proposed in 1902 that at very low temperatures electrons would condense onto atoms, so resistance would pass through a minimum and rise to infinity. Early Leiden data on gold and platinum, taken with Jacob Clay, seemed to hint at a minimum, and Kamerlingh Onnes at first leaned toward Kelvin.

On 2 December 1910 a platinum wire in liquid helium settled at a constant resistance below 4.25 K. It neither rose nor vanished. Kamerlingh Onnes read the plateau as the effect of impurities, and concluded that a perfectly pure metal should reach zero resistance at helium temperatures.

The argument is that independent scattering rates add, \(1/\tau=1/\tau_{\text{imp}}+1/\tau_{\text{ph}}\), and Drude’s formula \(\rho=m/(ne^2\tau)\) is linear in \(1/\tau\). Mercury was the natural test: it is liquid at room temperature and can be purified by repeated distillation, which Leiden had long practised.

8 April 1911

A cryostat able to hold liquid helium around a sample was ready in early April. Gilles Holst ran the Wheatstone bridge and galvanometer; Cornelis Dorsman controlled and measured temperatures; Flim and Kesselring kept the plant and glassware working. On 8 April the notebook records, in Dutch, that the resistance of mercury was practically zero. On 23 May the team found the resistance at 3 K below \(10^{-7}\) of its value at 0 °C, and saw it return near 4.2 K on warming. The famous plot of 26 October 1911 shows the resistance jumping to about 0.1 Ω within 0.01 K at 4.20 K. Kamerlingh Onnes wrote that mercury had passed into a new state, which he called the supraconductive state. He reported it at the first Solvay Conference in Brussels that autumn.

Critical fields and persistent currents

For the general reader. Kamerlingh Onnes immediately imagined superconducting magnets that would need no power. He soon found the catch: a modest magnetic field, or a large enough current, destroys superconductivity. He also showed that a current started in a closed superconducting ring keeps flowing with no battery at all.

New superconductors and the critical field

In 1913 the Leiden group found that tin and lead also superconduct (modern transition temperatures 3.7 K and 7.2 K). Lead, easy to work into coils, made magnet experiments possible. On 17 January 1914 they showed that a magnetic field destroys the state; for lead at 4.25 K the critical field was about 600 gauss.

Rule (ii) follows from Ampère’s law \(\oint H\cdot dl=I\) around the surface. Francis Silsbee, of the US National Bureau of Standards, proposed it from the Leiden data. It explained why Kamerlingh Onnes’s hopes for high-field magnets failed with lead and tin: the currents needed for strong fields drove the coils normal.

Persistent currents

In April 1914 Kamerlingh Onnes reported that a current of about 0.6 A, induced in a closed lead coil, showed no noticeable decrease over an hour. Paul Ehrenfest watched and wrote excitedly to Lorentz. Later Leiden rings carried persistent currents of hundreds of amperes; in 1932 Flim carried one, still circulating in its cryostat, from Leiden to London.

What came later

Zero resistance alone would trap whatever field was present when the metal cooled. In 1933 Walther Meissner and Robert Ochsenfeld found that a superconductor actually expels a weak field. Fritz and Heinz London (1935) described this with the equation \(\nabla^2\mathbf B=\mathbf B/\lambda_L^2\), \(\lambda_L=\sqrt{m/\mu_0ne^2}\), so the field and the supercurrent live in a surface layer of thickness \(\lambda_L\). Kamerlingh Onnes had computed current densities assuming the current filled the wire, and so greatly underestimated them. The microscopic theory came with John Bardeen, Leon Cooper and Robert Schrieffer in 1957.

The transition nobody saw

For the general reader. On the day mercury lost its resistance, the liquid helium around it went through a second, stranger transformation. Pumped below about 2.2 K, it stopped bubbling. Leiden noticed the change and did not pursue it. Two decades later it was recognized as the onset of superfluidity, a quantum state of a whole liquid.

From Leiden to Moscow and Cambridge

In April 1911 the team saw an abrupt change in the boiling of the helium bath as they pumped it down. Later work at Leiden showed that the density of liquid helium has a maximum near 2.2 K. Willem Keesom and Mieczysław Wolfke named the two phases He I and He II in 1927, and Keesom and Klaus Clusius found a sharp peak in the specific heat at 2.19 K in 1932. The curve’s resemblance to the Greek letter λ gave the “lambda point” its name; the modern value at the vapour pressure is 2.17 K. The bath stops boiling below it because He II conducts heat so well that bubbles cannot form in its bulk. In 1938 Pyotr Kapitsa in Moscow, and John Allen and Don Misener in Cambridge, found that He II flows through fine channels with no measurable viscosity.

Derivation sketch. The number of particles in excited states is at most \(V\int\frac{d^3p}{(2\pi\hbar)^3}\frac{1}{e^{p^2/2mkT}-1}=V\,\zeta(3/2)/\lambda_T^3\), with thermal wavelength \(\lambda_T=\sqrt{2\pi\hbar^2/mkT}\). When \(n\lambda_T^3\) exceeds \(\zeta(3/2)\), the excess must occupy the ground state. Setting \(n\lambda_T^3=\zeta(3/2)\) gives \(T_{\text{BE}}\). In 1938 Fritz London proposed that the lambda transition is this condensation, strongly modified by interactions.

Helium will not freeze

Kamerlingh Onnes pumped helium to the lowest temperatures he could reach and never saw a solid. The reason is quantum zero-point motion. Helium atoms are light and attract each other only weakly, so the kinetic energy of localizing an atom in a lattice exceeds the binding it would gain. Keesom solidified helium in 1926, months after Kamerlingh Onnes’s death, by applying about 25 atmospheres of pressure.

The Nobel Prize and the Leiden school, 1913–1924

The Nobel Prize in Physics for 1913 went to Kamerlingh Onnes for his investigations on the properties of matter at low temperatures, which had led, among other things, to the production of liquid helium. He had already received the Matteucci Medal (1910) and the Rumford Medal of the Royal Society (1912). The Franklin Medal followed in 1915, Foreign Membership of the Royal Society in 1916, and membership of the US National Academy of Sciences in 1920.

For about fifteen years Leiden was the only place in the world with liquid helium. John McLennan in Toronto produced it in 1923, using Leiden’s designs. The monopoly made the laboratory an international centre. Scientists came to Leiden to measure magnetism, specific heats and optical spectra at helium temperatures, and Kamerlingh Onnes was generous with time on his plant. Helium itself remained scarce: from 1911 the Welsbach company supplied gas in exchange for Leiden’s monazite residues, and later the US Bureau of Mines sent helium extracted from natural gas.

Doctoral students who passed through the laboratory included Pieter Zeeman (1893), Johannes Kuenen, Wander de Haas, Jacob Clay and Frans Penning. Willem Keesom, who had studied with van der Waals, became his chief collaborator on equations of state and his successor. Leiden became a physics capital with Lorentz, and after 1912 Ehrenfest, on the theoretical side.

Kamerlingh Onnes retired in 1924. De Haas succeeded him as professor, and the laboratory was run jointly by de Haas and Keesom.

Death and legacy

Kamerlingh Onnes died in Leiden on 21 February 1926, after a short illness, aged 72. Later that year Keesom solidified helium. In 1932 the laboratory was renamed the Kamerlingh Onnes Laboratorium.

Legacy

• Method: the industrial cryogenic laboratory, with its dedicated engineers, glassblowers and an instrument-makers’ school that still exists in Leiden.
• Equations of state: the virial expansion is the standard way to report gas data and the starting point for the cluster expansions of statistical mechanics.
• Superconductivity: from mercury to the high-temperature cuprates of Georg Bednorz and Alex Müller (1986). Superconducting magnets, made from type II alloys that tolerate high fields, now run MRI scanners and particle accelerators, fulfilling the hope Kamerlingh Onnes had in 1913.
• Quantum liquids: superfluid helium became the laboratory for macroscopic quantum phenomena, recognized by the Nobel Prizes of Lev Landau (1962) and Kapitsa (1978).
• Recognition: the IEEE dedicated a Milestone to the discovery of superconductivity in 2011, its centenary.

Sources and further reading

Biography

1. Heike Kamerlingh Onnes: biographical, NobelPrize.org.
2. The Nobel Prize in Physics 1913, NobelPrize.org.
3. Heike Kamerlingh Onnes, Wikipedia.
4. D. van Delft, Freezing Physics: Heike Kamerlingh Onnes and the Quest for Cold, Edita KNAW, 2007.
5. D. van Delft, Little cup of helium, big science, Physics Today 61 (3), 2008.
6. W. J. de Haas, KNAW biographical sketch (succession at Leiden).

Physics

1. D. van Delft and P. Kes, The discovery of superconductivity, Physics Today 63 (9), 2010.
2. H. Kamerlingh Onnes, Expression of the equation of state of gases and liquids by means of series, Proc. KNAW 4 (1901–02).
3. H. Kamerlingh Onnes, The liquefaction of helium, Proc. KNAW 11 (1908).
4. H. Kamerlingh Onnes, Further experiments with liquid helium, Proc. KNAW 13–14 (1911).
5. H. Kamerlingh Onnes, Investigations into the properties of substances at low temperatures, Nobel lecture, 1913.
6. Remarks on the first hundred years of superconductivity, arXiv:1111.5318.
7. J. Rowlinson, Cohesion: A Scientific History of Intermolecular Forces, Cambridge University Press, 2002.
8. M. Tinkham, Introduction to Superconductivity, 2nd ed., McGraw-Hill, 1996.
9. D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, 3rd ed., IOP, 1990.