Pyotr Kapitsa

How to use this course

Pyotr Kapitsa (often spelled Kapitza in English) was an experimental physicist with an engineer’s instincts. He solved problems by building machines nobody else thought could be built: generators that produced the strongest magnetic fields of their day, a helium liquefier without liquid hydrogen, a small turbine that changed how oxygen is made. With one of those machines he found that liquid helium below 2.17 K flows without measurable viscosity, and he named the effect superfluidity. He was also one of the very few Soviet scientists who argued with Stalin in writing and survived.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow his work in order: magnets (1924–1930), cryogenics (1934), superfluidity (1937–1941), oxygen engineering (1939–1946), and the dynamical problems he studied in exile at his dacha (1946–1955).

Timeline

Kronstadt and Petrograd, 1894–1921

Pyotr Leonidovich Kapitsa was born on 9 July 1894 (27 June in the Julian calendar then used in Russia) in Kronstadt, the naval fortress on an island in the Gulf of Finland. His father, Leonid Petrovich Kapitsa, was a military engineer who built the fortifications; his mother, Olga Ieronimovna, from the Stebnitsky family, worked on children’s literature and folklore. As a boy he liked clocks, tools and taking machines apart.

He entered the Petrograd Polytechnical Institute and joined the electromechanical section where Abram Ioffe was building the school that would produce much of Soviet physics. During the First World War he interrupted his studies to serve as an ambulance driver on the Polish front. He completed his studies in 1918 and stayed on to work with Ioffe. With Nikolay Semenov, later a Nobel laureate in chemistry, he proposed a method for measuring the magnetic moment of an atom by deflecting a beam in an inhomogeneous field, the idea that Otto Stern and Walther Gerlach carried out in 1922.

The civil war years were catastrophic for him personally. In 1920 his young wife, their two small children and his father died in the epidemics that swept famine-stricken Petrograd. Ioffe, worried for him, included him in a 1921 scientific mission to Western Europe to buy equipment and restore contacts. Kapitsa went to England and asked Ernest Rutherford to take him into the Cavendish Laboratory.

Rutherford’s Cambridge, 1921–1930

Rutherford at first said the laboratory was full. In the version Kapitsa told, he asked what accuracy Rutherford aimed for in his experiments; on hearing a few per cent, he pointed out that one more researcher among some thirty would not be noticed within that error. Rutherford laughed and took him on. Kapitsa started in the summer of 1921 with a study of the energy loss of alpha particles at the end of their range, and took his Cambridge PhD in 1923. He was a Clerk Maxwell Student from 1923 to 1926.

He quickly showed a talent the Cavendish lacked: designing large, bold apparatus. In 1923 he placed a Wilson cloud chamber in a strong magnetic field and photographed the curved tracks of alpha particles. To go further he needed far stronger fields, and he invented a way to get them for a hundredth of a second (Lesson 04). From 1924 to 1932 he was Assistant Director of Magnetic Research at the Cavendish. His young engineering assistant John Cockcroft, later a Nobel laureate himself, worked on the magnet.

In 1922 he founded the Kapitza Club, an informal evening seminar where young Cambridge physicists presented new ideas, often their own. Paul Dirac became a close friend. Kapitsa gave Rutherford a nickname, the Crocodile, which stuck. In 1927 he married Anna Alekseevna Krylova, daughter of the applied mathematician and shipbuilder Aleksey Krylov; their sons Sergey (born 1928) and Andrey (born 1931) became a physicist and a geographer. In 1929 he was elected a Fellow of the Royal Society, a rare honour for a foreigner, and in the same year a corresponding member of the USSR Academy of Sciences.

The Mond Laboratory and the trap, 1930–1935

In 1930 the Royal Society appointed Kapitsa its Messel Research Professor and used funds from the bequest of the chemist Ludwig Mond to build him a laboratory for strong fields and low temperatures in the Cavendish courtyard. The Royal Society Mond Laboratory opened on 3 February 1933. A crocodile carved by Eric Gill on the outside wall was a joke on Rutherford’s nickname. There, in 1934, Kapitsa built a helium liquefier that needed no liquid hydrogen (Lesson 05), which made liquid helium a routine laboratory material.

Throughout these years Kapitsa had visited the Soviet Union regularly and returned without difficulty. In September 1934, at the end of a summer visit, he was told that he could not leave. Soviet leaders wanted his talent at home. Rutherford and others protested, and Kapitsa refused for months to work. The deadlock ended in a compromise: the Soviet government founded an Institute for Physical Problems in Moscow for him to direct, and bought the equipment of the Mond Laboratory from Cambridge, which Rutherford agreed to sell. Two of his Cambridge technicians came to Moscow to help install it. Anna Kapitsa and the boys later joined him in Moscow.

Kapitsa never accepted that he had been treated rightly. He did not leave the Soviet Union again until 1965, when he was allowed to travel to Denmark to receive the Niels Bohr International Gold Medal.

Pulsed magnetic fields

For the general reader. An electromagnet strong enough to bend fast particles or reveal new properties of metals would melt or tear itself apart if run continuously. Kapitsa’s idea was to run it for only a hundredth of a second. In that time the coil does not have time to overheat, and with a powerful generator briefly short-circuited through it he reached fields above 30 tesla, several times beyond any steady magnet of the day.

Pressure and energy

A field \(B\) stores energy density \(u=B^2/(2\mu_0)\), and exerts an equal outward magnetic pressure on the windings of a solenoid. In SI units:

At 32 T this is about \(4\times10^8\) Pa, roughly 4000 atmospheres. The coil must be built like a pressure vessel, and Kapitsa reinforced his with a heavy steel jacket.

Heating and the pulse

A coil of resistance \(R\) carrying current \(I\) for time \(\tau\) dissipates \(I^2R\tau\). If the pulse is short compared with the time for heat to leave the conductor, the heating is adiabatic: per unit volume of copper with resistivity \(\rho_e\) and current density \(j\), the temperature rise is

where \(c_V\) is the heat capacity per unit volume. Holding \(\Delta T\) fixed, the allowed \(j\), and hence the field, grows like \(\tau^{-1/2}\). Kapitsa’s first pulses in 1924 came from discharging a lead-acid battery. He then had a large alternating-current generator built with a heavy rotor that stored kinetic energy. At the right instant the generator was short-circuited through the coil for about a hundredth of a second, delivering power of the order of megawatts. Fields of about 32 T (320 kilogauss) were reached in a volume of about 2 cm³.

The linear law

The simplest free-electron model predicts no magnetoresistance at all, and a two-carrier model predicts a quadratic rise that saturates. Kapitsa’s data, taken mostly on polycrystalline samples, did neither. A satisfactory explanation had to wait for the theory of electron orbits on real Fermi surfaces developed in the 1950s by Ilya Lifshitz and his school, in which open orbits and averaging over crystal grains play the key role.

Liquefying helium with an engine

For the general reader. Helium boils at 4.2 K. Heike Kamerlingh Onnes first liquefied it in 1908 by precooling the gas with liquid hydrogen and then letting it expand through a valve. Kapitsa replaced most of that chain with a small engine in which the cold gas pushes a piston and does work. A gas that does work cools far more than one that merely leaks through a valve. His 1934 machine made liquid helium cheaper, safer and much more widely available.

Two ways to expand a gas

In a throttling (Joule–Thomson) expansion through a valve, enthalpy is conserved, and the temperature change per unit pressure drop is the Joule–Thomson coefficient

with \(\alpha\) the thermal expansion coefficient. For an ideal gas \(T\alpha=1\) and \(\mu_{\mathrm{JT}}=0\). Cooling occurs only below an inversion temperature, which for helium is about 40 K. That is why Kamerlingh Onnes had to precool with liquid hydrogen. In an isentropic expansion against a piston, entropy is conserved and

always. For an ideal gas this gives \(T\propto P^{(\gamma-1)/\gamma}\), a large drop at any temperature.

Subtract the two formulas above. The second follows from \(dS=(C_P/T)dT-V\alpha\,dP\) and the Maxwell relation \((\partial S/\partial P)_T=-(\partial V/\partial T)_P\).

Kapitsa’s machine (1934)

Expansion engines had been proposed before, but at 10 K any lubricating oil freezes. Kapitsa let the piston run in its cylinder with a very small clearance and no lubricant at all; the helium leaking through the narrow gap acted as the lubricant and the leak was small because the stroke was fast. Helium compressed to about 30 atmospheres was precooled by liquid nitrogen, cooled further by expansion in the engine to below 10 K, and then liquefied in a final Joule–Thomson stage, which works well that far below the inversion temperature. The design idea, a reciprocating or turbine expander, is used in large helium liquefiers today.

The discovery of superfluidity

For the general reader. Below 2.17 K liquid helium changes into a strange second liquid, helium II. It was known to conduct heat astonishingly well. In 1937 Kapitsa forced it through a gap half a micrometre wide between two polished glass disks and found that it flowed through almost as if it had no viscosity at all. He called the new state superfluid, by analogy with superconductors, which carry current without resistance.

Background

In 1932 Willem Keesom and his collaborators at Leiden had found a sharp peak in the heat capacity of liquid helium at 2.17 K, the lambda point, so called because the curve looks like the Greek letter. Below it, helium II stopped boiling visibly and conducted heat millions of times better than above. A Toronto measurement by rotating cylinders gave a viscosity that dropped at the lambda point but did not vanish.

Flow through a slit

For steady laminar flow of a viscous fluid between parallel plates a distance \(h\) apart, driven by a pressure difference \(\Delta P\) over a length \(L\), the Navier–Stokes equations give a parabolic profile and a volume flow per unit width

The flow rate is inversely proportional to the viscosity \(\eta\). Kapitsa measured the rate at which helium drained through the gap under a small hydrostatic head and found a flow so fast that, if the formula held, \(\eta\) would have to be below about \(10^{-9}\) poise, tens of thousands of times smaller than for helium I. He concluded that the liquid was in a special state and proposed the word superfluid. His letter appeared in Nature on 8 January 1938. In the same issue, on the facing page, John F. Allen and Don Misener at Cambridge reported equivalent results for flow through fine capillaries. The two groups had worked independently.

The two contradictory viscosity measurements were the clue that led László Tisza and Lev Landau to the two-fluid picture of Lesson 07.

Two fluids and Landau’s criterion

For the general reader. Kapitsa’s experiment needed a theory, and his institute supplied it. Lev Landau, whom Kapitsa had just rescued from prison, explained in 1941 that helium II behaves as two interpenetrating fluids: a superfluid that carries no entropy and flows without friction, and a normal fluid made of thermal excitations. Friction is impossible at low speed because the liquid cannot create an excitation cheaply enough. The theory predicted a second kind of sound, a wave of temperature, which was found at Kapitsa’s institute in 1944.

Landau’s criterion

Let a liquid at zero temperature flow with velocity \(\mathbf v\) past a wall. In the liquid’s rest frame, an excitation of momentum \(\mathbf p\) and energy \(\varepsilon(p)\) can be created only if energy and momentum allow it. Transforming to the wall frame, the energy change is \(\varepsilon(p)+\mathbf p\cdot\mathbf v\), which must be negative for the process to lower the energy.

For helium-4 the spectrum (hero figure) has a phonon branch \(\varepsilon\approx cp\) with \(c\approx238\) m/s, and a roton minimum \(\varepsilon\approx\Delta+(p-p_0)^2/(2\mu)\) with \(\Delta/k_B\approx8.6\) K and \(p_0/\hbar\approx1.9\ \text{Å}^{-1}\). The minimum of \(\varepsilon/p\) is near the roton minimum, giving \(v_c\approx\Delta/p_0\approx60\) m/s. Observed critical velocities in flow are usually much lower because quantized vortices form first, a mechanism found later by Lars Onsager and Richard Feynman.

Two fluids

At \(T>0\) the gas of excitations moves with its own velocity \(\mathbf v_n\) and carries all the entropy. Total density and mass current split as

At low temperature the phonons dominate and give \(\rho_n=\dfrac{2\pi^2k_B^4T^4}{45\,\hbar^3c^5}\). The superfluid component, having no entropy, can pass through a superleak while the normal component is held back. Equilibrium of chemical potential across a superleak gives the fountain pressure

with \(s\) the entropy per unit mass (Heinz London, 1939). Linearized two-fluid hydrodynamics has two sound modes. Ordinary sound is a pressure wave; second sound is a wave in which \(\mathbf v_s\) and \(\mathbf v_n\) oscillate in opposition at nearly constant density, so temperature and entropy oscillate. Its speed is

with \(c\) the specific heat per unit mass. Vasily Peshkov measured second sound at the Institute for Physical Problems in 1944, at about 20 m/s near 1.5 K, confirming Landau’s theory. Landau received the 1962 Nobel Prize in Physics for this work.

Kapitza resistance

For the general reader. Helium II conducts heat so well that one expects no temperature difference anywhere inside it. In 1941 Kapitsa found that there is a sudden temperature jump right at the boundary between the helium and a solid. The heat carriers, sound waves of very different speeds in the two materials, mostly bounce back at the interface. This boundary resistance, now called Kapitza resistance, limits cooling in every dilution refrigerator and matters in nanoscale electronics.

Acoustic mismatch model

Isaak Khalatnikov, a student of Landau, gave the theory in 1952. Phonons in the helium (sound speed \(c_{\mathrm{He}}\approx238\) m/s) strike the solid (sound speed \(c_s\) of several km/s). By Snell’s law \(\sin\theta_t/\sin\theta_i=c_s/c_{\mathrm{He}}\), so only phonons within a narrow critical cone \(\sin\theta_c=c_{\mathrm{He}}/c_s\) can enter at all, and of those a fraction of order \(4Z_{\mathrm{He}}/Z_s\) is transmitted, with acoustic impedances \(Z=\rho c\). The phonon energy flux incident on a wall from one side is \(\frac{\pi^2k_B^4T^4}{120\hbar^3c_{\mathrm{He}}^2}\) for one longitudinal branch. Multiplying by the cone fraction \(\sin^2\theta_c\) and the transmission, and linearizing \(T^4-(T-\Delta T)^4\approx4T^3\Delta T\), gives the leading estimate

Khalatnikov’s full result has the same form with a numerical factor of order one that accounts for transverse waves in the solid. The \(T^3\) law is confirmed below about 0.1 K. At higher temperatures measured conductances are much larger than predicted, sometimes by orders of magnitude. The excess comes from phonon scattering at surface defects, the dense adsorbed helium layer, and other channels the simple model leaves out.

The turboexpander and cheap oxygen

For the general reader. Steel making, medicine and munitions all need large amounts of oxygen, which is separated from liquefied air. Before Kapitsa, air was compressed to around 200 atmospheres and cooled by throttling, which wastes most of the work. In 1939 he built a small, very efficient turbine in which cold air at only a few atmospheres spins a rotor and cools by doing work. The low-pressure cycle with a turboexpander became the basis of the modern industrial air separation plant.

Minimum work

This is \(T\) times the entropy of mixing. For air (\(x_{\mathrm{O_2}}\approx0.21\), \(x_{\mathrm{N_2}}\approx0.79\)) at 300 K it is about 1.3 kJ per mole of air. Real plants of the 1930s used tens of times more energy than this, mostly lost in compressing air to high pressure and throttling it.

Kapitsa’s turbine (1939)

Engineers had tried turbines for air liquefaction, but their efficiencies were poor. Kapitsa argued that at low temperature air is dense, closer to a liquid than a gas in its flow behaviour, and so the turbine should be designed on the principles of a water turbine. His radial-inflow reaction turbine, with a rotor only a few centimetres across spinning at tens of thousands of revolutions per minute, reached adiabatic efficiencies around 80 per cent. Since the expander, not the throttle, does the cooling, the cycle could run at about 6 atmospheres. During the war his institute built oxygen plants based on this design, and in 1943 he was put in charge of the new Main Administration for the Oxygen Industry. The design was opposed by established engineers, and his opponents later used it against him (Lesson 11). Low-pressure turboexpander cycles are now standard worldwide.

The Kapitza pendulum and the Kapitza–Dirac effect

For the general reader. Two of Kapitsa’s most-cited ideas are small and elegant. In 1951, working in his home laboratory during his years of disgrace, he showed how a pendulum can be made to stand upside down if its pivot is shaken up and down fast enough, and gave a simple method to compute such effects. In 1933, with Paul Dirac, he predicted that electrons should be diffracted by a standing wave of light, as by a crystal. The prediction was finally verified in 2001.

Fast oscillations

Derivation. Write \(x=X+\xi\), with \(\xi\) small and fast. To leading order \(m\ddot\xi=f(X)\cos\omega t\), so \(\xi=-\frac{f}{m\omega^2}\cos\omega t\). Expanding the forces to first order in \(\xi\) and averaging over a period, \(m\ddot X=-U'(X)+\overline{\xi\,f'(X)\cos\omega t}=-U'-\frac{ff'}{2m\omega^2}\), which is \(-\frac{d}{dX}\big(U+\frac{f^2}{4m\omega^2}\big)\). ∎ The effective potential is the kinetic energy of the fast jiggle, which the particle tries to minimise.

The inverted pendulum

A pendulum of length \(\ell\) has its pivot driven vertically as \(a\cos\omega t\). In the pivot’s frame gravity is replaced by \(g-a\omega^2\cos\omega t\), so the equation of motion is \(m\ell^2\ddot\theta=-m\ell(g-a\omega^2\cos\omega t)\sin\theta\). The fast generalized force is \(f(\theta)\cos\omega t\) with \(f=m\ell a\omega^2\sin\theta\). Applying the theorem with angle as coordinate and inertia \(m\ell^2\) in place of \(m\) gives

The upright position \(\theta=\pi\) is a minimum, so the inverted pendulum is stable, when \(a^2\omega^2>2g\ell\). The same averaging principle underlies the Paul trap for ions and the ponderomotive force in plasmas and lasers.

Electrons and a light grating

An electron in an oscillating field \(\mathbf E_0(\mathbf r)\cos\omega t\) feels, by the same averaging, the ponderomotive potential \(U_p=e^2E_0^2/(4m\omega^2)\). In a standing light wave of wavelength \(\lambda\), \(E_0^2\propto\cos^2(2\pi x/\lambda)\), so \(U_p\) is periodic with period \(\lambda/2\). In 1933 Kapitsa and Dirac described this as stimulated Compton scattering and predicted Bragg reflection of electrons from the standing wave. The momentum transferred is a multiple of \(2\hbar k\), with \(k=2\pi/\lambda\). The effect was too weak to see with the light sources of 1933. After lasers made it accessible, Daniel Freimund, Kayvan Aflatooni and Herman Batelaan observed clear diffraction peaks in 2001.

Landau, the war and Beria, 1935–1955

The Institute for Physical Problems, built on Vorobyovy Hills in Moscow, became under Kapitsa one of the best-run laboratories in the world: small, well equipped and free of bureaucracy. In 1937 he invited Lev Landau from Kharkov to head its theory department. On 28 April 1938 Landau was arrested by the NKVD. On the same day Kapitsa wrote to Stalin asking for the case to be handled with care. A year later, in April 1939, he wrote to Vyacheslav Molotov, explaining that he had found phenomena in liquid helium that only Landau could explain. Summoned to the NKVD, he signed a personal guarantee for Landau’s conduct, and Landau was released at the end of April 1939. He also intervened on behalf of the theorist Vladimir Fock and others. Over his life he wrote dozens of letters to Stalin and other leaders, frank in tone, arguing for the independence of science.

War and oxygen

When Germany invaded in 1941 the institute was evacuated to Kazan. Kapitsa’s oxygen turbines went into production, and in 1943 he was made head of the new Main Administration for the Oxygen Industry under the Council of People’s Commissars. He received the Stalin Prize in 1941 and 1943 and the title of Hero of Socialist Labour in 1945.

Disgrace

After Hiroshima, Kapitsa was appointed to the Special Committee on the atomic bomb chaired by Lavrentiy Beria. On 3 October 1945 he wrote to Stalin criticising Beria’s handling of scientists, and on 25 November he asked to be released from the committee, objecting both to Beria and to a plan simply to copy the American bomb. He did not oppose a Soviet bomb as such. A government commission then condemned his oxygen technology, and in August 1946 he was dismissed from his institute and from his oxygen post. Anatoly Aleksandrov replaced him as director. For eight years Kapitsa worked in a laboratory he built in an outbuilding at his dacha at Nikolina Gora outside Moscow. There he studied the dynamics of the driven pendulum, the flow of thin liquid films, and the generation of high-power microwaves.

Return, the Nobel Prize and legacy, 1955–1984

After Stalin’s death and Beria’s arrest in 1953, Kapitsa’s position recovered. In January 1955 he was reinstated as director of the Institute for Physical Problems, which he led until his death. He became editor-in-chief of the Journal of Experimental and Theoretical Physics, a member of the Presidium of the Academy from 1957, and a founder and long-time teacher of the Moscow Institute of Physics and Technology. His research turned to high-power microwave generators and to plasma: he found a stable high-pressure discharge with electron temperatures he estimated at over a million kelvin, which he proposed as a route to controlled fusion.

He used his standing cautiously but persistently. In 1966 he was among the signatories of a letter to the Soviet leadership warning against the rehabilitation of Stalin. He defended Andrei Sakharov and other dissenting scientists in letters to Yuri Andropov, and he took part in the Pugwash movement. In 1965 he was allowed abroad for the first time since 1934, to Copenhagen to receive the Niels Bohr International Gold Medal, and in 1966 he visited Cambridge again.

In 1978 he received half the Nobel Prize in Physics “for his basic inventions and discoveries in the area of low-temperature physics”; the other half went to Arno Penzias and Robert Woodrow Wilson for the cosmic microwave background. Characteristically, his Nobel lecture was not about helium at all but about plasma and controlled thermonuclear fusion. He died in Moscow on 8 April 1984, aged 89, and is buried at Novodevichy Cemetery.

Legacy

• Low-temperature physics: superfluidity, the expansion-engine liquefier and the Kapitza resistance are foundations of cryogenics and quantum fluids. The institute he founded now bears his name, the P. L. Kapitza Institute for Physical Problems.
• High magnetic fields: pulsed-field methods descended from his are used at high-field laboratories worldwide.
• Engineering: low-pressure turboexpander cycles for air separation.
• Physics of oscillations: the Kapitza pendulum and the effective-potential method; the Kapitza–Dirac effect in matter-wave optics.
• Honours: Fellow of the Royal Society (1929); full member of the USSR Academy of Sciences (1939); Faraday Medal (1942), Franklin Medal (1944), Lomonosov Gold Medal (1959), Rutherford Medal (1966); twice Hero of Socialist Labour (1945, 1974); Nobel Prize (1978). Minor planet 3437 Kapitsa is named after him.

Connections

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

Teachers and advisors

• Ernest Rutherford 1921-1934
• Abram Ioffe 1914-1921

Students

• David Shoenberg

Influenced by

• Heike Kamerlingh Onnes

Colleagues and collaborators

• Andrei Sakharov
• Isaak Khalatnikov
• John Cockcroft 1924-1928
• Lev Landau 1937-1962
• Nikolay Semenov 1918-1921
• Vladimir Fock

Rivals

• John F. Allen 1938

Friends

• James Chadwick
• Paul Dirac

Family

• Aleksey Krylov
• Sergey Kapitsa

Also mentioned in this life

• Niels Bohr
• Henry Cavendish
• Arthur Compton
• Louis de Broglie
• Peter Debye
• Michael Faraday
• Enrico Fermi
• Benjamin Franklin
• James Prescott Joule
• James Clerk Maxwell
• Claude-Louis Navier
• George Gabriel Stokes
• Johannes Diderik van der Waals

Fields

• magnetism 5 lives

On CoursesHub

• Plasma Physics

Sources and further reading

Biography

1. Pyotr Kapitsa: Biographical, NobelPrize.org.
2. The Nobel Prize in Physics 1978, NobelPrize.org.
3. Pyotr Kapitsa, Wikipedia (with extensive references).
4. D. Holloway, The Scientist and the Tyrant, New York Review of Books, 1 March 1990.
5. P. L. Kapitza, Kapitza Institute for Physical Problems, Russian Academy of Sciences.
6. D. Shoenberg, Piotr Leonidovich Kapitza, Biographical Memoirs of Fellows of the Royal Society 31 (1985) 325–374.
7. J. W. Boag, P. E. Rubinin and D. Shoenberg (eds.), Kapitza in Cambridge and Moscow: Life and Letters of a Russian Physicist, North-Holland, 1990.

Physics

1. P. Kapitza, A method of producing strong magnetic fields, Proc. Roy. Soc. A 105 (1924) 691–710.
2. P. Kapitza, The change of electrical conductivity in strong magnetic fields, Proc. Roy. Soc. A 123 (1929) 292–372.
3. P. L. Kapitza and P. A. M. Dirac, The reflection of electrons from standing light waves, Proc. Camb. Phil. Soc. 29 (1933) 297–300.
4. P. Kapitza, The liquefaction of helium by an adiabatic method without pre-cooling with liquid hydrogen, Proc. Roy. Soc. A 147 (1934) 189–211.
5. P. Kapitza, Viscosity of liquid helium below the λ-point, Nature 141 (1938) 74.
6. J. F. Allen and A. D. Misener, Flow of liquid helium II, Nature 141 (1938) 75.
7. L. Landau, Theory of the superfluidity of helium II, Physical Review 60 (1941) 356–358.
8. P. L. Kapitza, Dynamic stability of a pendulum when its point of suspension vibrates, Zh. Eksp. Teor. Fiz. 21 (1951) 588; see also L. D. Landau and E. M. Lifshitz, Mechanics, §30.
9. D. L. Freimund, K. Aflatooni and H. Batelaan, Observation of the Kapitza–Dirac effect, Nature 413 (2001) 142–143.
10. G. L. Pollack, Kapitza resistance, Reviews of Modern Physics 41 (1969) 48–81.
11. P. L. Kapitza, Plasma and the controlled thermonuclear reaction, Nobel Lecture, 8 December 1978.
12. A. Griffin, New light on the intriguing history of superfluidity in liquid ⁴He, Journal of Physics: Condensed Matter 21 (2009) 164220.