George Gamow

How to use this course

George Gamow had a gift for seeing the simple physical picture behind a hard problem, and for being early. He was the first to apply quantum mechanics to the nucleus, among the first to treat the early universe as a nuclear furnace, and the first physicist to frame the genetic code as a problem in information. He was often wrong in the details and usually right in the essentials. He left the heavy calculations to others, and several of his ideas were completed by collaborators: Edward Teller, Ralph Alpher, Robert Herman and, at a distance, Hans Bethe.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order follows his career: the nucleus (1928–1936), the stars (1938–1944), the early universe (1946–1953) and molecular biology (1953–1955). Readers who have taken the Bethe course on this site will meet the Gamow peak again in Lesson 05, seen from the other side.

Timeline

Odessa and Leningrad, 1904–1928

Georgiy Antonovich Gamov was born in Odessa on 4 March 1904 (20 February in the Julian calendar then used in Russia). Both parents were teachers: his father, Anton, taught Russian language and literature at a secondary school, and his mother, Aleksandra, taught geography and history at a school for girls. She died in 1913, when he was nine. His schooling was disrupted by the First World War, the revolutions of 1917 and the civil war, during which Odessa changed hands several times.

He began studying mathematics and physics at Novorossiya University in Odessa and in the early 1920s moved to Petrograd, renamed Leningrad in 1924. There he heard Alexander Friedmann, who had just found the expanding-universe solutions of Albert Einstein’s equations. Gamow later said he had hoped to work on relativistic cosmology with Friedmann, but Friedmann died in 1925 and Gamow turned to the new quantum theory.

With Lev Landau and Dmitri Ivanenko he formed a trio of young theorists who read every new paper from Göttingen and Copenhagen as it arrived. They called themselves the Three Musketeers; with Matvei Bronstein and others they were also known as the “Jazz Band”. They wrote joke papers and argued about quantum mechanics late into the night. In 1926 he became a graduate student at Leningrad University, and in 1928 he was sent abroad to spend the summer at Göttingen.

Göttingen, Copenhagen, Cambridge, 1928–1931

In Göttingen, the centre of Max Born’s quantum mechanics, most theorists were working on atoms and molecules. Gamow chose the nucleus. Within weeks he had explained alpha decay as quantum tunnelling through the Coulomb barrier (Lesson 04). His paper appeared in the Zeitschrift für Physik that year. Ronald Gurney and Edward Condon at Princeton found the same explanation independently.

On his way home he stopped in Copenhagen and asked Niels Bohr for a place. Bohr arranged a fellowship on the spot, and Gamow spent the following year at Bohr’s institute. From 1929 he spent a year at the Cavendish Laboratory in Cambridge as a Rockefeller fellow, with Rutherford. His calculations showed that protons of modest energy could penetrate light nuclei, which encouraged John Cockcroft and Ernest Walton to build the accelerator with which they split lithium in 1932. In 1929 he also proposed treating the nucleus as a drop of nuclear fluid held together by surface tension, the first version of the liquid-drop model.

Gamow was a cheerful, irreverent presence at these institutes, known for practical jokes and cartoons. He drew cartoons and illustrated his own writing, a habit he kept all his life. His Constitution of Atomic Nuclei and Radioactivity (Oxford, 1931) was the first textbook on theoretical nuclear physics.

Return and escape, 1931–1934

In 1931 Gamow returned to Leningrad. He held posts at the Radium Institute, the university and the Academy’s Physico-Mathematical Institute, and helped plan an early cyclotron. In 1932, aged 28, he was elected a corresponding member of the USSR Academy of Sciences. He married the physicist Lyubov Vokhmintseva, whom he called Rho, in November 1931.

The political climate was closing in. Theories that seemed to conflict with dialectical materialism were attacked, and in 1931 he was refused permission to attend a conference in Rome. In 1932 he and Rho made two attempts to escape by kayak, once across the Black Sea toward Turkey and once from the Murmansk area toward Norway. Bad weather defeated both.

In 1933 he was invited to the Solvay Conference in Brussels and insisted that Rho accompany him. With intervention from senior Soviet figures, permission came. At Brussels Marie Curie helped them to extend their stay abroad. After visits to Paris, Cambridge and Copenhagen, and lectures at the summer school of the University of Michigan, Gamow became a professor at George Washington University in Washington, D.C., in 1934. He never returned to the Soviet Union. In 1938 the Academy expelled him; the expulsion was reversed in 1990.

Alpha decay and tunnelling

For the general reader. An alpha particle trapped in a uranium nucleus does not have enough energy to climb over the electric barrier around it. Classically it should stay forever. Quantum mechanically its wave leaks through the barrier, and each time the particle strikes the wall there is a tiny chance it escapes. Gamow showed that this tiny chance depends exponentially on energy, which explains why alpha-emitters with similar energies have half-lives ranging from microseconds to billions of years.

The model

Model the daughter nucleus (charge \(Z_d\)) and the alpha particle (\(z=2\)) by a potential that is attractive inside a radius \(R\) and Coulombic outside:

The alpha particle has kinetic energy \(Q\) far away. The classical turning point is \(b=\dfrac{zZ_de^2}{4\pi\varepsilon_0Q}\), well outside \(R\) (about 60 fm against 9 fm for uranium).

Derivation

Substitute \(r=b\cos^2\theta\). Then \(V-Q=Q\big(\frac br-1\big)=Q\tan^2\theta\) and \(dr=-2b\cos\theta\sin\theta\,d\theta\), so

This gives the bracket in the formula. Expanding for small \(x\), \(\arccos\sqrt x\approx\frac\pi2-\sqrt x\) and \(\sqrt{x(1-x)}\approx\sqrt x\), which gives the two-term approximation.

The liquid drop and thermonuclear reactions

For the general reader. Tunnelling runs in both directions. If an alpha particle can leak out of a nucleus, a proton can leak in. In 1929 Robert Atkinson and Fritz Houtermans used Gamow’s formula to argue that at the temperatures inside stars, light nuclei could fuse often enough to power the Sun. Gamow himself took up the idea in the 1930s. In the same period he gave the first picture of the nucleus as a drop of liquid, a model that later explained fission.

The drop of nuclear fluid

Nuclear radii grow as \(R\approx r_0A^{1/3}\), so nuclear density is constant: nuclei are incompressible. Binding energy per nucleon is also nearly constant. Gamow concluded in 1929 that a nucleus resembles a drop of liquid, held together by short-range forces that act only between neighbours, with a binding deficit at the surface. His version treated the nucleus as a collection of alpha particles; after the neutron was discovered, Carl Friedrich von Weizsäcker (1935) and Bethe turned the picture into the semi-empirical mass formula, and Bohr and John Wheeler (1939) used it to explain fission.

The Gamow window

In a star of temperature \(T\), two nuclei of charges \(Z_1,Z_2\) and reduced mass \(\mu\) fuse at the rate per pair

where \(E_G=2\mu c^2(\pi\alpha Z_1Z_2)^2\) is the Gamow energy and \(S(E)\) is a slowly varying nuclear factor. The Boltzmann factor falls with energy and the tunnelling factor rises. Their product is a narrow peak.

Gamow and Teller (1938) used this formula to compute stellar reaction rates, just before Bethe found the specific reactions that power the stars.

Gamow–Teller transitions

For the general reader. In beta decay a neutron in the nucleus turns into a proton, emitting an electron and an antineutrino. Enrico Fermi’s 1934 theory let the two light particles carry away no spin, which meant the nucleus could not change its spin. Experiments showed that it often did. In 1936 Gamow and Edward Teller added a second kind of interaction in which the leptons carry off one unit of spin. Both kinds occur in nature, and their relative strength is one of the basic numbers of the weak interaction.

Allowed transitions

When the leptons are emitted with zero orbital angular momentum (allowed transitions), the nuclear part of the operator is either the identity in spin space or the spin operator. In modern notation the transition operators are \(\sum_k\tau_k^{\pm}\) (Fermi) and \(\sum_k\boldsymbol\sigma_k\tau_k^{\pm}\) (Gamow–Teller), summed over nucleons.

Decays such as \(^6\)He \(\to\,^6\)Li, a \(0^+\to1^+\) transition, are forbidden in Fermi’s original theory but proceed rapidly. The later V−A theory (1957–58) fixed the relativistic form of both terms.

Stars: the Urca process and red giants

For the general reader. Once nuclear reactions were known to power stars, Gamow asked what happens when the fuel runs out. With the Brazilian physicist Mário Schenberg he found a way for a dense stellar core to lose energy invisibly, by emitting neutrinos that escape at once. They named it after a Rio de Janeiro casino where, Gamow joked, money disappeared just as fast. He also argued that red giants are stars burning hydrogen in a shell around an exhausted core, a view that became the basis of stellar evolution theory.

The Urca process (Gamow and Schenberg, 1940–41)

In matter dense enough that electrons are degenerate with high Fermi energy, a nucleus \((A,Z)\) can capture an electron, and the daughter can then beta-decay back:

Gamow and Schenberg argued that Urca losses could remove pressure support so fast that a stellar core would collapse in a fraction of a second, a possible trigger for supernovae. The precise mechanism they proposed was not the one now accepted, but neutrino cooling of dense stars, through Urca and related processes, is central to the theory of stellar collapse and of cooling neutron stars, and to Bethe’s later supernova work.

Red giants

In the early 1940s Gamow argued that when hydrogen is exhausted in a star’s centre, burning continues in a shell around the helium core, and the envelope expands. In 1944 he connected Walter Baade’s two stellar populations with this evolutionary picture. Detailed computations by Martin Schwarzschild, Fred Hoyle and others in the 1950s confirmed that red giants are evolved stars with inert cores and burning shells.

The hot Big Bang and the αβγ paper

For the general reader. Friedmann and Georges Lemaître had shown that Einstein’s equations allow an expanding universe, and Edwin Hubble had observed the expansion. Gamow, Friedmann’s former student, asked what the universe was like when it was very young and very dense. He concluded that it must have been hot, a furnace of neutrons, protons and radiation, and that the chemical elements might have been cooked in its first minutes. The idea was only partly right: the early universe makes hydrogen, helium and a little lithium, while heavier elements come from stars. The helium prediction is one of the great successes of cosmology.

Radiation domination

For a flat universe with energy density \(\rho c^2\), the Friedmann equation is \(H^2=\big(\dot a/a\big)^2=\frac{8\pi G}{3}\rho\). Radiation density scales as \(a^{-4}\).

Proof. Write \(\rho=Ca^{-4}\). Then \(\dot a/a=\sqrt{8\pi GC/3}\,a^{-2}\), so \(a\,\dot a=\) const and \(a^2\propto t\). Thus \(H=1/(2t)\) and \(\rho=\frac{3H^2}{8\pi G}=\frac{3}{32\pi Gt^2}\). The temperature formula follows by equating the two expressions for \(\rho\). ∎

The αβγ paper

In 1946 Gamow proposed in Physical Review that the elements were made by rapid neutron capture in the early, expanding universe. His graduate student Ralph Alpher worked out the details for his doctoral thesis. The resulting letter, The Origin of Chemical Elements, appeared in Physical Review on 1 April 1948 under the names Alpher, Bethe and Gamow. Bethe had not taken part; Gamow added his name for the pun on the first three Greek letters. Alpher later felt that the famous names obscured his own role.

The model assumed a hot gas of neutrons (Gamow’s “ylem”) that decay and capture to build heavier nuclei step by step. Its fatal problem, pointed out by Enrico Fermi and Anthony Turkevich among others, is that no stable nuclei exist with mass 5 or 8. Neutron capture cannot bridge these gaps quickly. In the modern theory, nearly all neutrons end up in \(^4\)He, and heavier elements are made in stars.

The relic radiation

For the general reader. If the universe was once a hot furnace, the light from that furnace should still be here, cooled by the expansion to a few degrees above absolute zero. Alpher and Robert Herman made the first such prediction in 1948, about 5 kelvin, and Gamow gave his own estimates in the following years. The prediction was largely forgotten. When Arno Penzias and Robert Wilson found the cosmic microwave background at about 3 K in 1965, the hot Big Bang became the standard model of cosmology.

Proof. The number of photons per unit volume in \([\nu,\nu+d\nu]\) is \(n(\nu)\,d\nu=\frac{8\pi\nu^2}{c^3}\frac{d\nu}{e^{h\nu/kT}-1}\). Under expansion by a factor \(s\), \(\nu\to\nu/s\) and number density \(\to\) density\(/s^3\). Substituting shows the new distribution is the same Planck form with \(T\to T/s\). ∎

Gamow’s estimate

The logic of Gamow, Alpher and Herman was this. For nucleosynthesis to work, a neutron must meet a proton and form deuterium within the time available when \(T\approx10^9\) K, so \(n_b\langle\sigma v\rangle t\approx1\). This fixes the baryon density at that moment. Today’s baryon density is known roughly from astronomy. Since both \(n_b\) and \(T^3\) scale as \(a^{-3}\),

Different choices of the inputs gave Alpher and Herman about 5 K in 1948; Gamow quoted 7 K in 1953 and other values in later writings. The measured temperature is \(2.7255\) K.

The coding problem

For the general reader. When James Watson and Francis Crick published the double helix in 1953, Gamow read the paper and saw a puzzle in information. DNA is a sequence written in four letters; proteins are sequences in twenty letters, the amino acids. How does one translate into the other? Gamow proposed a specific code within months. It was wrong, but it turned the question into a precise combinatorial problem and drew physicists and mathematicians into molecular biology.

Gamow’s diamond code (1954)

In a letter to Nature in February 1954, Gamow suggested that proteins are assembled directly on the double helix. Four bases bounding each diamond-shaped cavity in the groove would select an amino acid. Counting the distinct cavities, allowing for the symmetries of base pairing, he found exactly 20 types, matching the number of standard amino acids. The code was overlapping: neighbouring amino acids would share bases.

The number 20 appears in all three schemes, which seemed too striking to be a coincidence. Nature uses none of them. Sydney Brenner showed in 1957 that every overlapping triplet code is incompatible with the observed sequences of amino-acid neighbours in proteins. From 1961 Marshall Nirenberg, Heinrich Matthaei, Har Gobind Khorana and others determined the actual code experimentally: 64 triplet codons read in a non-overlapping way from a fixed start, with redundancy and three stop codons.

In 1954 Gamow and Watson founded the RNA Tie Club: twenty members, one for each amino acid, each with a tie and tiepin designed by Gamow. Members circulated unpublished notes. Crick’s 1955 note proposing “adaptor” molecules, later identified as transfer RNA, was written for the club.

Washington and Mr Tompkins, 1934–1956

At George Washington University Gamow built a small centre of theoretical physics. He made it a condition of his appointment that the university hire a second theorist, and in 1935 Edward Teller arrived from London. From 1935 Gamow organized, with GWU and the Carnegie Institution, the annual Washington Conferences on Theoretical Physics. These small meetings shaped American physics. The 1938 conference on stellar energy set Bethe on the path to the CN cycle. At the 1939 conference, on 26 January, Bohr and Fermi discussed the newly discovered fission of uranium. His son Igor was born in 1935, and Gamow became an American citizen in 1940.

During the Second World War Gamow was not brought into the Manhattan Project. He taught at GWU and consulted for the U.S. Navy. In the decade after the war he produced his cosmology with Alpher and Herman and his work on the genetic code. He was elected to the U.S. National Academy of Sciences in 1953.

The writer

Gamow is best known to the public as a writer. Mr Tompkins in Wonderland (1940) sends a bank clerk into worlds where the speed of light is a few kilometres per hour or Planck’s constant is large, so that relativity and quantum mechanics become visible in daily life. Mr Tompkins Explores the Atom followed in 1944. One Two Three… Infinity (1947) ranged from Georg Cantor’s infinities to the expanding universe, and The Birth and Death of the Sun (1940) and The Creation of the Universe (1952) brought stellar physics and cosmology to general readers. He illustrated many of the books himself. His textbooks and his history Thirty Years That Shook Physics (1966) reached students as well.

Boulder, death and legacy, 1956–1968

In 1956 Gamow’s marriage to Rho ended, and he left GWU for the University of Colorado in Boulder, where he spent the rest of his career. The same year UNESCO awarded him the Kalinga Prize for the popularization of science. He was a founding member of the Physical Science Study Committee, which redesigned high-school physics teaching in the United States. In 1958 he married Barbara Perkins, an editor at one of his publishers.

His health declined in the 1960s. He died in Boulder on 19 August 1968, aged 64, after months of illness affecting his liver and circulation, and was buried in Green Mountain Cemetery. The physics building at Boulder, Gamow Tower, is named after him, and the university holds an annual Gamow Memorial Lecture.

The detection of the cosmic microwave background in 1965 came three years before his death. Gamow, Alpher and Herman were disappointed that their earlier prediction was little cited at first. Recognition grew later: Alpher and Herman received prizes in the 1990s, and the hot Big Bang is now often traced to Gamow’s papers of 1946–48.

Legacy

• Nuclear physics: tunnelling theory of alpha decay, the Gamow factor, the liquid-drop picture and Gamow–Teller transitions.
• Astrophysics: the Gamow peak in every thermonuclear rate, neutrino cooling of stars, and the shell-burning picture of red giants.
• Cosmology: the hot Big Bang, primordial nucleosynthesis and the prediction of relic radiation.
• Biology: the coding problem as a problem of information.
• Public understanding: Mr Tompkins and One Two Three… Infinity introduced generations of readers to modern physics.
• The term: “Gamow states” (decaying resonances with complex energies) recall his 1928 treatment of quasi-stationary states.

Sources and further reading

Biography

1. George Gamow, Wikipedia (with extensive references).
2. George Gamow 1904–1968, Biographical Memoirs of the National Academy of Sciences.
3. The distinguished life and career of George Gamow, University of Colorado Boulder.
4. G. Gamow, My World Line: An Informal Autobiography, Viking, 1970.
5. E. Harper, George Gamow: scientific amateur and polymath, Physics in Perspective 3 (2001).

Physics and biology

1. G. Gamow, Zur Quantentheorie des Atomkernes, Z. Phys. 51 (1928) 204–212.
2. R. W. Gurney and E. U. Condon, Wave mechanics and radioactive disintegration, Nature 122 (1928) 439.
3. R. Atkinson and F. G. Houtermans, Zur Frage der Aufbaumöglichkeit der Elemente in Sternen, Z. Phys. 54 (1929).
4. G. Gamow, Constitution of Atomic Nuclei and Radioactivity, Oxford, 1931.
5. G. Gamow and E. Teller, Selection rules for the β-disintegration, Phys. Rev. 49 (1936) 895.
6. G. Gamow and M. Schoenberg, Neutrino theory of stellar collapse, Phys. Rev. 59 (1941) 539.
7. G. Gamow, Expanding universe and the origin of elements, Phys. Rev. 70 (1946) 572.
8. R. A. Alpher, H. Bethe and G. Gamow, The origin of chemical elements, Phys. Rev. 73 (1948) 803.
9. R. A. Alpher and R. Herman, Evolution of the universe, Nature 162 (1948) 774.
10. G. Gamow, The evolution of the universe, Nature 162 (1948) 680.
11. G. Gamow, Possible relation between deoxyribonucleic acid and protein structures, Nature 173 (1954) 318.
12. F. H. C. Crick, J. S. Griffith and L. E. Orgel, Codes without commas, PNAS 43 (1957) 416.
13. S. Brenner, On the impossibility of all overlapping triplet codes, PNAS 43 (1957) 687.
14. H. Kragh, Cosmology and Controversy, Princeton University Press, 1996.