Hans Bethe

How to use this course

Hans Bethe worked as a theoretical physicist for more than seventy-five years. His first paper appeared in 1928 and his last research papers in the 2000s. In between he contributed to solid-state physics, atomic physics, nuclear physics, quantum electrodynamics, astrophysics and neutrino physics. His style was distinctive. He picked a concrete problem, chose the simplest model that captured it, and pushed the calculation through to a number that could be compared with experiment. Colleagues called the method “Bethe’s way”, and several of his papers became the standard reference for decades.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow the order of his work: crystals and stopping power (1928–1932), exactly solvable chains (1931), the nucleus (1934–1937), stellar energy (1938–1939), quantum electrodynamics (1947), bound states and nuclear matter (1951–1970), and supernovae and neutrinos (1979–2000).

Timeline

Strasbourg, Kiel, Frankfurt, 1906–1926

Hans Albrecht Bethe was born on 2 July 1906 in Strasbourg, then the capital of the German imperial territory of Alsace-Lorraine. His father, Albrecht Bethe, was a physiologist at the university. His mother, Anna Kuhn, was the daughter of a Strasbourg professor of medicine. Her family was Jewish; Hans was raised as a Protestant, like his father. He was an only child.

The boy loved numbers. Family stories describe him computing with fractions and powers at an age when other children were learning to read, and filling notebooks with tables. In 1912 the family moved to Kiel, where his father directed the Institute of Physiology, and in 1915 to Frankfurt, where Albrecht Bethe took a chair at the new university. Hans entered the Goethe-Gymnasium, but tuberculosis in 1916 interrupted his schooling. He spent periods at a school in Bad Kreuznach and at the progressive Odenwaldschule before returning to the Goethe-Gymnasium, from which he graduated in 1924. By his own account he taught himself calculus as a teenager.

At the University of Frankfurt he began in chemistry and mathematics. He found the experimental work clumsy and the physics teaching weak, apart from the lectures of Walter Gerlach, who left in 1925. A lecturer, Karl Meissner, advised him that the place to learn theoretical physics was Munich. In April 1926 he moved there.

Sommerfeld’s Munich and the wander years, 1926–1933

Arnold Sommerfeld’s institute in Munich had already trained Werner Heisenberg, Wolfgang Pauli and Peter Debye. Bethe arrived in the year Erwin Schrödinger published wave mechanics, and he learned the new quantum mechanics as it was being made. Sommerfeld set him a problem prompted by the experiments of Clinton Davisson and Lester Germer on electron diffraction. Bethe’s thesis, Theorie der Beugung von Elektronen an Kristallen, earned a summa cum laude in July 1928. It introduced the dynamical theory of electron diffraction and showed that the mean electrostatic potential inside a crystal shifts the Bragg peaks.

He then held one-semester posts at Frankfurt and at the Technische Hochschule Stuttgart, where he worked with Paul Ewald, the X-ray crystallographer whose daughter Rose he would later marry. Back in Munich he wrote two papers that would each found a subject: crystal field theory (1929) and the theory of stopping power (1930). He became a Privatdozent at Munich in 1930.

A Rockefeller Foundation fellowship took him to Cambridge in the autumn of 1930, where he worked in the circle of Ralph Fowler, and to Rome in the spring terms of 1931 and 1932. In Rome he learned from Enrico Fermi a lighter, more intuitive style, one that looked for the essential physics before the mathematics. He later said that Sommerfeld taught him rigour and Fermi taught him simplicity. In Rome he also solved the one-dimensional Heisenberg chain exactly (Lesson 05). While in Cambridge he wrote, with Guido Beck and Wolfgang Riezler, a mock paper deriving absolute zero from the fine-structure constant, a parody of Arthur Eddington’s numerology. Die Naturwissenschaften printed it, and the authors later had to publish a retraction.

With Sommerfeld he wrote the long Handbuch der Physik article on the electron theory of metals (1933), and alone the article on one- and two-electron atoms, which remained a standard reference for decades. In the winter of 1932–33 he became acting assistant professor at Tübingen, under Hans Geiger.

Exile and Cornell, 1933–1938

In April 1933 the Nazi Law for the Restoration of the Professional Civil Service removed from state employment anyone with a Jewish grandparent. Bethe’s mother came from a Jewish family, and he lost his post at Tübingen. Sommerfeld found him a fellowship in Munich for the summer. William Lawrence Bragg offered a temporary lectureship at Manchester, and in October 1933 Bethe left Germany.

In Manchester he shared lodgings with Rudolf Peierls, another refugee theorist. When James Chadwick and Maurice Goldhaber disintegrated the deuteron with gamma rays in 1934, Bethe and Peierls worked out the theory within weeks (Lesson 06). In the same year Bethe and Walter Heitler published the quantum theory of bremsstrahlung and pair production, which became the basis of the theory of cosmic-ray showers. After a semester at Bristol with Nevill Mott, he accepted an offer from Cornell University and arrived in Ithaca in February 1935. He remained at Cornell for the rest of his life.

Cornell had little theoretical physics. Bethe built a programme by teaching, and by writing. With Robert Bacher and Milton Stanley Livingston he produced three long articles in Reviews of Modern Physics (1936–37) that surveyed all of nuclear physics, much of it rederived by Bethe himself. A generation of physicists learned the subject from “Bethe’s Bible”. In 1938 George Gamow and Edward Teller persuaded a reluctant Bethe to attend a small Washington conference on the energy of stars. Within months he had answered the question (Lesson 07).

Crystals and fast particles

For the general reader. Put an atom inside a crystal and its energy levels split, because the surrounding ions break the symmetry of empty space. Shoot a fast charged particle into a block of matter and it slows down by kicking electrons out of their atoms. Bethe gave the theory of both effects between 1929 and 1932. The first underlies the colours of gemstones and the magnetism of transition-metal compounds. The second is still used every day in radiation therapy, detector design and nuclear physics.

Crystal field theory (1929)

An ion in a crystal feels an electrostatic potential with the point symmetry \(G\) of its site, a finite subgroup of \(\mathrm{O}(3)\). A free-ion level of angular momentum \(l\) carries the \((2l+1)\)-dimensional representation \(D^{(l)}\) of \(\mathrm{SO}(3)\). Restricted to \(G\), it is in general reducible.

For a \(d\) level (\(l=2\)) in octahedral symmetry the five orbitals split into a triplet \(t_{2g}\) and a doublet \(e_g\). This is the starting point of ligand-field theory in chemistry. Bethe’s paper also treated half-integer \(j\) through double groups.

Stopping power (1930–1932)

A heavy particle of charge \(ze\) and speed \(v\) crosses a material with \(n\) electrons per unit volume. Classically, Niels Bohr had computed the momentum given to an electron at impact parameter \(b\): \(\Delta p = 2ze^2/(bv)\) in Gaussian units. The energy lost per unit path is then

The logarithm depends on cutoffs. Bethe replaced the classical collision by quantum transitions in the first Born approximation, summed over all final states of the atom using a sum rule, and found that the cutoffs combine into a single property of the material.

The formula depends on the projectile only through \(z\) and \(v\), and on the material only through \(n\) and \(I\). Empirically \(I\approx 10\,\text{eV}\times Z\) for heavier elements. Later refinements add shell corrections at low speed and the density-effect correction at very high energy. Because \(-dE/dx\propto1/v^2\), the loss rises sharply as the particle slows, producing the Bragg peak near the end of the track. Proton therapy uses that peak to deposit dose in a tumour while sparing tissue beyond it.

The Bethe ansatz

For the general reader. A chain of quantum spins, each coupled to its neighbours, is one of the simplest models of magnetism. In 1931 Bethe guessed the exact form of every energy eigenstate: a superposition of plane waves, with the waves scattering off each other only in pairs. The guess, now called the Bethe ansatz, was the first exact solution of an interacting quantum many-body system. Its descendants run through statistical mechanics, cold-atom physics and string theory.

The model

Take \(N\) spin-\(\tfrac12\) sites on a ring and the Heisenberg Hamiltonian

where \(P_{n,n+1}\) exchanges the two spins. The state with all spins up has energy 0. Since \(H\) commutes with \(S^z_{\text{tot}}\), we work in sectors with \(M\) flipped spins at positions \(n_1<\dots<n_M\).

The ansatz

For \(M=1\), \(H|n\rangle=\frac J2\big(|n+1\rangle+|n-1\rangle-2|n\rangle\big)\), so the plane wave \(\sum_n e^{ikn}|n\rangle\) has energy \(J(\cos k-1)\), with \(e^{ikN}=1\). Bethe proposed that for general \(M\) the amplitudes are

with amplitudes for permutations differing by a transposition related by a two-body phase. Parametrize momenta by rapidities, \(e^{ik}=\dfrac{\lambda+i/2}{\lambda-i/2}\).

Why it works. When two flipped spins are not adjacent, \(H\) acts on each separately and any superposition of plane waves solves the equation. Adjacency imposes one extra condition, which fixes the ratio \(A_{\dots k_l k_j\dots}/A_{\dots k_jk_l\dots}\) to the two-body S-matrix \(S(\lambda_j-\lambda_l)=\frac{\lambda_j-\lambda_l-i}{\lambda_j-\lambda_l+i}\) (up to convention). With three or more flipped spins, consistency requires the S-matrices to compose in either order. This is the Yang–Baxter equation, made explicit by Chen Ning Yang and Rodney Baxter decades later. Periodicity then gives the Bethe equations.

For the antiferromagnet (\(J>0\)) the ground state has \(M=N/2\) real rapidities. In 1938 Lamek Hulthén solved the equations in the limit \(N\to\infty\) and found the ground-state energy per site \(E_0/N=-J\ln2\), equivalently \(J(\tfrac14-\ln2)\) for \(J\sum\mathbf S_n\cdot\mathbf S_{n+1}\).

The nucleus: deuteron, mass formula, the Bible

For the general reader. After the neutron was discovered in 1932, physicists knew what nuclei were made of but not what held them together. Bethe’s approach was to extract as much as possible from the simplest nucleus, the deuteron, and from smooth trends across all nuclei. His three review articles of 1936–37 organized the whole field so thoroughly that it was taught from them for a decade.

The deuteron (Bethe and Peierls, 1935)

The nuclear force has a range of about \(1\) fm, while the deuteron binding energy \(B=2.2246\) MeV is small. Outside the force range the radial wavefunction obeys the free equation with reduced mass \(\mu\approx M/2\):

Bethe and Peierls realized that most low-energy properties depend only on \(\kappa\), not on the shape of the potential. They computed the photodisintegration cross-section and neutron–proton scattering in this “zero-range” approximation.

Bethe’s 1949 paper gave a short derivation from the radial equation and became the standard treatment of low-energy two-nucleon scattering. The same expansion now organizes cold-atom physics near Feshbach resonances.

The semi-empirical mass formula

Carl Friedrich von Weizsäcker proposed in 1935 that nuclear binding energies follow from a liquid-drop picture plus quantum corrections. Bethe and Bacher simplified and refitted the formula in the 1936 Bible, and it is often called the Bethe–Weizsäcker formula:

Typical modern fits give \(a_V\approx15.8\), \(a_S\approx18.3\), \(a_C\approx0.714\), \(a_A\approx23.2\) MeV, and a pairing term \(\delta\approx+12/\sqrt A\) MeV for even–even nuclei, \(-12/\sqrt A\) for odd–odd, and 0 for odd \(A\).

Energy production in stars

For the general reader. Arthur Eddington argued in 1920 that the Sun shines by turning hydrogen into helium, but no one knew how. In 1938 Bethe found the two answers: a chain of reactions starting with two protons fusing, and a cycle in which carbon acts as a catalyst. The Nobel Prize of 1967 was given for this work and for his contributions to the theory of nuclear reactions.

The two routes

Both routes convert four protons into one helium nucleus, releasing \(26.7\) MeV, part of it carried away by neutrinos.

Bethe reached the cycle by elimination. He checked systematically which light nuclei could react with protons fast enough at stellar temperatures without being destroyed, and found that only a closed loop through carbon and nitrogen survived. He also argued, correctly, that no stable nucleus of mass 5 or 8 exists, so helium could not be built up further in main-sequence stars by adding protons or alpha particles.

The Gamow peak

Nuclei in a star at temperature \(T\) have Maxwell–Boltzmann energies, and must tunnel through their Coulomb barrier with probability \(e^{-\sqrt{E_G/E}}\), the Gamow factor, where \(E_G=2\mu c^2(\pi\alpha Z_1Z_2)^2\). Writing the cross-section as \(\sigma(E)=\frac{S(E)}{E}e^{-\sqrt{E_G/E}}\), with \(S(E)\) slowly varying, the rate per pair is

Because \(E_G\) grows as \(Z_1^2Z_2^2\), reactions with nitrogen (\(Z=7\)) are far more temperature-sensitive than proton–proton fusion. At low temperatures the pp chain wins; above about 17 million kelvin the CN cycle does. Bethe concluded that the Sun runs mainly on the pp chain and more massive stars on the CN cycle, which is the modern picture.

The Lamb shift and renormalization

For the general reader. Paul Dirac’s theory predicted that two excited levels of hydrogen have exactly the same energy. In 1947 Willis Lamb and Robert Retherford measured a small difference. Days later Bethe explained it, with a calculation he did largely on a train. The interaction of the electron with the quantum vacuum gives infinite answers, but the difference between a bound electron and a free one is finite. This idea, mass renormalization, made quantum electrodynamics a working theory.

The calculation

At the Shelter Island conference (2–4 June 1947), Lamb reported that the \(2s_{1/2}\) level of hydrogen lies about 1000 MHz above \(2p_{1/2}\). Hendrik Kramers had suggested that the electromagnetic mass of the electron should be absorbed into the observed mass. Bethe took the idea literally. In nonrelativistic second-order perturbation theory, the emission and reabsorption of a transverse photon shifts level \(n\) by

with photon energy \(k\) and a cutoff \(K\). This diverges linearly. A free electron has a similar term, which only renormalizes its mass. Subtracting it, Bethe obtained an expression that diverges only logarithmically:

He argued that relativity would cut off the integral at \(K\approx mc^2\). For the sum he used a closure relation, \(\sum_m|\langle m|\mathbf p|n\rangle|^2(E_m-E_n)=\frac{\hbar^2}{2}\langle n|\nabla^2V|n\rangle=2\pi\hbar^2\alpha\hbar c\,Z|\psi_n(0)|^2\), which vanishes unless the state has \(l=0\), and he replaced the energy in the logarithm by a numerically computed average.

Bethe’s three-page paper, The electromagnetic shift of energy levels, appeared in Physical Review in August 1947. It used no relativistic field theory at all. Within two years Julian Schwinger, Richard Feynman, Sin-Itiro Tomonaga and Freeman Dyson, who had come to Cornell as Bethe’s student, gave the fully relativistic, covariant renormalization of QED. The modern value, including vacuum polarization and higher orders, is 1057.8 MHz.

Bound states and nuclear matter

For the general reader. After the Lamb shift, Bethe turned to the hardest problems of bound systems: two relativistic particles held together by exchanging quanta, and the dense sea of nucleons inside a heavy nucleus. In both cases he and his collaborators found equations that sum infinite classes of interactions, and that are still the starting points of their fields.

The Bethe–Salpeter equation (1951)

Edwin Salpeter and Bethe wrote a covariant equation for the two-body bound state in field theory. Let \(G\) be the full two-particle Green’s function and \(K\) the two-particle-irreducible interaction kernel. Then \(G=G_0+G_0KG\), where \(G_0\) is the product of the two dressed one-particle propagators.

The equation includes relative time (or energy) as a variable, which makes it hard to solve and to interpret. It became the basis for the precise theory of positronium and for relativistic models of mesons. Murray Gell-Mann and Francis Low derived the same equation independently.

Nuclear matter and the Bethe–Goldstone equation

Nuclear forces have a strongly repulsive core at short distance, so ordinary perturbation theory fails. Keith Brueckner proposed in the mid-1950s to replace the bare potential \(V\) by an effective interaction summing repeated two-body scatterings inside the medium. Bethe and his student Jeffrey Goldstone (1957) solved the resulting two-body problem.

The healing property explains why the independent-particle shell model works despite strong forces. Over the following fifteen years, with Baird Brandow, Albert Petschek and others, Bethe pushed this theory to a quantitative account of nuclear saturation, summarized in his 1971 review Theory of nuclear matter. The three-body extension led to the Bethe–Faddeev equations. The same machinery later fed his work on the equation of state of neutron stars.

Supernovae and solar neutrinos

For the general reader. In his seventies Bethe started a third career in astrophysics. With Gerald Brown he worked on why the collapsing core of a massive star bounces and explodes as a supernova. With James Wilson he identified the mechanism still favoured today: neutrinos from the hot new neutron star reheat the stalled shock. He also gave a clear explanation of why detectors saw only a third of the expected solar neutrinos: the neutrinos change type on their way out of the Sun.

Core collapse

The iron core of a massive star, about 1.4 solar masses, collapses when electron captures and photodisintegration remove its pressure support. In 1979 Bethe, Brown, Jim Applegate and James Lattimer showed that the collapsing matter has low entropy, so nuclei survive until nuclear density, and the core bounces stiffly. The bounce launches a shock. Detailed simulations showed that the shock loses energy dissociating iron and stalls a few hundred kilometres out.

The neutron star’s binding energy, about \(3\times10^{53}\) erg, is radiated almost entirely as neutrinos of all flavours over roughly ten seconds. About 1% reaches the ejecta as kinetic energy. The detection of about two dozen neutrinos from SN 1987A confirmed the energy scale and the duration.

The MSW effect (Bethe, 1986)

Lincoln Wolfenstein (1978) showed that electron neutrinos moving through matter acquire an extra potential \(V=\sqrt2G_Fn_e\) from forward scattering on electrons. Stanislav Mikheyev and Alexei Smirnov (1985) found that this can cause resonant conversion. Bethe’s short 1986 paper gave the adiabatic picture in a form that made it widely understood. For two flavours with mixing angle \(\theta\) and mass-squared difference \(\Delta m^2\), the effective mixing angle in matter satisfies

Bethe used the deficit to estimate \(\Delta m^2\) and argued that solar neutrino data required physics beyond the Standard Model. The Sudbury Neutrino Observatory confirmed flavour conversion in 2001–02; the large-mixing-angle solution is the one realized in nature.

Los Alamos, the H-bomb and arms control, 1941–1995

Bethe married Rose Ewald, Paul Ewald’s daughter, on 13 September 1939; they had two children, Henry and Monica. When war came he looked for a way to help his new country. He became an American citizen in 1941 and joined the MIT Radiation Laboratory, where he worked on microwave radar and invented a waveguide directional coupler still called the Bethe-hole coupler. In the summer of 1942 he joined Oppenheimer’s study group in Berkeley on the design of a fission bomb. When Edward Teller raised the possibility that a nuclear explosion could ignite the atmosphere, Bethe’s calculations helped show that it could not.

In 1943 Oppenheimer made Bethe head of the Theoretical Division at Los Alamos. The appointment hurt Teller, who had hoped for the post, and the rift between them never fully healed. Bethe’s division calculated critical masses, the efficiency of the explosion and the hydrodynamics of implosion. With Richard Feynman he derived a formula for the yield of a fission weapon. He witnessed the Trinity test in July 1945.

After the war Bethe helped found the Federation of American Scientists. In 1950 he argued in Scientific American against a crash programme to build a hydrogen bomb. After the Korean War began and Truman ordered the programme, and after Teller and Stanislaw Ulam found a workable design in 1951, he went back to Los Alamos to help, saying later that he had hoped to show it could not be done. In 1954 he testified for Oppenheimer at the security hearing that removed Oppenheimer’s clearance.

From 1957 he served on President Eisenhower’s Science Advisory Committee. He chaired an interagency panel on whether a nuclear test ban could be verified and took part in the 1958 Geneva conference of experts on detection. These efforts fed into the Partial Test Ban Treaty of 1963. In 1968 he and Richard Garwin argued in Scientific American that anti-ballistic-missile defences could be defeated cheaply, a case that helped shape the ABM Treaty of 1972. In the 1980s he opposed the Strategic Defense Initiative. In 1995, on the fiftieth anniversary of Hiroshima, he called on scientists everywhere to stop working on new nuclear weapons. He remained a strong supporter of civil nuclear power.

Cornell, honours and legacy

Bethe was Cornell’s theoretical physics programme for decades. He brought Feynman there after the war, and he attracted students who became leaders in many fields. His doctoral students included Robert Marshak, Francis Low, Jun John Sakurai, Jeffrey Goldstone, David Thouless, Roman Jackiw, Michel Baranger and Peter Carruthers. Freeman Dyson came to work with him in 1947 without ever taking a doctorate. Bethe was famous for his ability to estimate anything quickly and for writing carefully worked lecture notes. He continued to publish into his nineties, much of it with Gerald Brown on supernovae, neutron stars and black-hole binaries.

The Nobel Prize in Physics for 1967 cited his contributions to the theory of nuclear reactions, especially concerning energy production in stars. Other honours included the Henry Draper Medal (1947), the Max Planck Medal (1955), Foreign Membership of the Royal Society (1957), the Eddington Medal (1961), the Enrico Fermi Award (1961), the Rumford Prize (1963), the National Medal of Science (1975), the Lomonosov Gold Medal (1989), the Oersted Medal for teaching (1993) and the Bruce Medal (2001).

Bethe died of congestive heart failure at his home in Ithaca on 6 March 2005, aged 98. Rose Bethe survived him and lived until 2019.

Legacy

• Methods: the Bethe ansatz and its descendants, the Bethe lattice and Bethe–Peierls approximation in statistical mechanics, effective-range theory, and the Bethe–Salpeter equation.
• Applications: the Bethe formula underlies particle detectors, shielding and proton therapy; crystal field theory underlies ligand-field chemistry.
• Astrophysics: the theory of stellar energy and the delayed neutrino mechanism for supernovae.
• Public role: five decades of technical advice on nuclear weapons, test bans and missile defence.

Sources and further reading

Biography

1. Hans Bethe, Wikipedia (with extensive references).
2. Hans Bethe, Biographical, NobelPrize.org.
3. J. J. O’Connor and E. F. Robertson, Hans Albrecht Bethe, MacTutor History of Mathematics.
4. In memoriam: Hans Bethe, 1906–2005, Arms Control Today, April 2005.
5. S. S. Schweber, Nuclear Forces: The Making of the Physicist Hans Bethe, Harvard University Press, 2012.
6. J. Bernstein, Hans Bethe, Prophet of Energy, Basic Books, 1980.
7. G. E. Brown and S. Lee, Hans Albrecht Bethe 1906–2005, Biographical Memoirs of the National Academy of Sciences, 2009.

Physics

1. H. Bethe, Theorie der Beugung von Elektronen an Kristallen, Ann. Phys. 87 (1928).
2. H. Bethe, Termaufspaltung in Kristallen, Ann. Phys. 3 (1929).
3. H. Bethe, Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie, Ann. Phys. 5 (1930).
4. H. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931).
5. H. Bethe and R. Peierls, Quantum theory of the diplon, Proc. Roy. Soc. A 148 (1935).
6. H. Bethe, R. F. Bacher and M. S. Livingston, Nuclear physics, Rev. Mod. Phys. 8 (1936) and 9 (1937).
7. H. Bethe and C. L. Critchfield, The formation of deuterons by proton combination, Phys. Rev. 54 (1938).
8. H. Bethe, Energy production in stars, Phys. Rev. 55 (1939).
9. H. Bethe, The electromagnetic shift of energy levels, Phys. Rev. 72 (1947).
10. H. Bethe, Theory of the effective range in nuclear scattering, Phys. Rev. 76 (1949).
11. E. E. Salpeter and H. A. Bethe, A relativistic equation for bound-state problems, Phys. Rev. 84 (1951).
12. H. A. Bethe and J. Goldstone, Effect of a repulsive core in the theory of complex nuclei, Proc. Roy. Soc. A 238 (1957).
13. H. A. Bethe and J. R. Wilson, Revival of a stalled supernova shock by neutrino heating, Astrophys. J. 295 (1985).
14. H. A. Bethe, Possible explanation of the solar-neutrino puzzle, Phys. Rev. Lett. 56 (1986).
15. H. A. Bethe, Supernova mechanisms, Rev. Mod. Phys. 62 (1990).
16. H. A. Bethe, Nobel Lecture: Energy production in stars, 1967, NobelPrize.org.
17. M. Karbach and G. Müller, Introduction to the Bethe ansatz I, arXiv:cond-mat/9809162.