Niels Bohr

How to use this course

Niels Bohr made three kinds of contribution. In 1913 he gave the first quantum theory of an atom and explained the hydrogen spectrum. Through the 1920s he guided the creation of quantum mechanics from his institute in Copenhagen, more as a questioner and interpreter than as a calculator. From 1936 he applied a simple picture of the nucleus to nuclear reactions and, in 1939, to fission. He then spent his last two decades arguing for openness between nations about atomic weapons. This course covers the physics in detail and the life alongside it.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Formulas are in SI units unless marked as Gaussian. Bohr’s own work used Gaussian units; for those lessons both forms are given where they differ.

Timeline

A Copenhagen childhood, 1885–1903

Niels Henrik David Bohr was born on 7 October 1885 in Copenhagen. His father, Christian Bohr, was professor of physiology at the University of Copenhagen and is remembered for the Bohr effect, the way carbon dioxide lowers the oxygen affinity of haemoglobin. His mother, Ellen Adler, came from a wealthy Jewish family of bankers and politicians. Niels had an older sister, Jenny, and a younger brother, Harald, born in 1887.

The two brothers were inseparable, and Niels called Harald his best friend for life. The home was a meeting place for the Copenhagen intelligentsia: the philosopher Harald Høffding, and the physicist Christian Christiansen came regularly, and the boys listened to their conversations. Niels attended Gammelholm Latin School from 1891, where he was a good but not outstanding pupil, slow to write and fond of argument. He passed the school-leaving examination in 1903.

Student, prize-winner, doctor, 1903–1911

Bohr entered the University of Copenhagen in 1903 to study physics under Christian Christiansen, the only professor of physics in Denmark. He also attended Høffding’s philosophy lectures. Both brothers played football for Akademisk Boldklub; in 1905 Niels was in goal. Harald was the stronger player and won a silver medal with the Danish team at the 1908 Olympic Games.

In 1905 the Royal Danish Academy of Sciences set a prize problem: to measure the surface tension of liquids using the oscillations of a jet, following a method of Lord Rayleigh. Bohr built the apparatus in his father’s physiology laboratory, made glass tubes himself, and extended Rayleigh’s theory to finite amplitudes and to the effect of viscosity. He won the gold medal, announced in 1907, and published the work in the Philosophical Transactions of the Royal Society in 1909. It was his only experimental research.

He took his master’s degree in 1909 and turned to the electron theory of metals for his doctorate. Christian Bohr died suddenly in February 1911. On 13 May 1911 Niels defended his thesis, Studier over metallernes elektrontheori. It showed, among other things, that classical physics cannot explain magnetism in metals (Lesson 04). The conclusion pointed to the limits of classical mechanics, and Bohr took it with him to England.

Cambridge and Manchester, 1911–1913

With a grant from the Carlsberg Foundation, Bohr went to Cambridge in September 1911 to work with J. J. Thomson, discoverer of the electron. The visit disappointed him. His English was weak, Thomson did not read his thesis, and Bohr’s habit of pointing out errors in Thomson’s work did not help. In late 1911 he met Ernest Rutherford, who had just discovered the atomic nucleus. In March 1912 Bohr moved to Manchester.

Rutherford’s atom had a problem. A classical electron circling a nucleus radiates and spirals inward in a tiny fraction of a second. In Manchester Bohr worked on the absorption of alpha particles in matter (Lesson 06) and began to think that Planck’s constant must fix the size of atoms. He returned to Copenhagen in July 1912 and married Margrethe Nørlund on 1 August; she would be his closest collaborator in writing for the rest of his life. They had six sons.

In early 1913 the spectroscopist Hans Marius Hansen asked Bohr how his ideas explained the Balmer formula for hydrogen. Bohr later said that once he saw the formula, everything became clear to him. Within weeks he had the theory. Rutherford communicated the first paper to the Philosophical Magazine, where it appeared in July 1913, followed by parts II and III in September and November.

Electrons in metals and the absence of classical magnetism

For the general reader. Many materials are magnetic, and by 1910 physicists assumed this came from moving electrons. Bohr’s thesis proved that this cannot work in classical physics. If electrons obey Newton’s laws and the rules of thermal equilibrium, the magnetic effects of their motion cancel exactly. Magnetism is a quantum phenomenon.

Free electrons in a metal

Paul Drude and Hendrik Lorentz had modelled a metal as a gas of free electrons. The model explained Ohm’s law and the Wiedemann–Franz ratio of thermal to electrical conductivity, but not specific heats or magnetic properties. Bohr examined these theories critically and in great generality, allowing arbitrary forces between electrons and metal atoms.

Proof. The classical partition function is \(Z=\int\prod_i d^3p_i\,d^3r_i\;e^{-\beta H}\). For fixed positions, substitute \(\mathbf p_i'=\mathbf p_i-q_i\mathbf A(\mathbf r_i)\). This is a translation in momentum space, with unit Jacobian and infinite range of integration, so \(Z\) equals the partition function with \(\mathbf A=0\). Hence \(F=-kT\ln Z\) is independent of the field \(\mathbf B\), and \(\mathbf M=-\partial F/\partial\mathbf B=0\).

The physical picture: electrons in the interior circle in one sense, but those hitting the boundary skip around it in the opposite sense, and the two currents cancel exactly. Hendrika Johanna van Leeuwen found the theorem independently in her 1919 Leiden thesis. Real diamagnetism, paramagnetism and ferromagnetism need quantization: discrete Landau levels, spin and the exchange interaction.

The 1913 atom

For the general reader. Bohr kept Rutherford’s picture of electrons circling a nucleus, but added two rules that broke with classical physics. Only certain orbits are allowed, and in them the electron does not radiate. Light is emitted only when the electron jumps from one allowed orbit to another, with a frequency set by the energy difference. These rules reproduced the colours of hydrogen’s light to high precision.

Bohr’s first paper derived the levels from a different condition, matching the energy to \(\tfrac12nh\) times the orbital frequency; the angular momentum form appears in the same paper as an equivalent statement.

Derivation

An electron of mass \(m\) circles a nucleus of charge \(Ze\) at radius \(r\). Newton’s law and the quantum condition give

Solving,

The agreement of \(R_\infty\), computed from \(m\), \(e\) and \(h\), with the measured Rydberg constant was the decisive result. Bohr also assigned the Pickering lines, seen in stars and in Alfred Fowler’s laboratory, to ionized helium (\(Z=2\)) rather than hydrogen. Fowler objected that the wavelengths did not quite match. Bohr replied that the nucleus is not infinitely heavy, so \(m\) must be replaced by the reduced mass \(\mu=mM/(m+M)\). With this correction the ratio of the helium and hydrogen constants came out at 4.0016, as measured. Einstein, told of the result, reportedly called it one of the greatest discoveries.

Stopping of charged particles

For the general reader. When an alpha particle passes through matter, it slows down by knocking electrons. How far it travels depends on how strongly the electrons are bound. Bohr worked this out in Manchester in 1912, and the work led him to think about the frequencies of electrons in atoms just before he built his model. His formula is still the starting point for the physics of radiation damage and particle detectors.

Impulse approximation

A heavy particle of charge \(Z_1e\) and speed \(v\) passes an electron at impact parameter \(b\). In Gaussian units the transverse field at the electron integrates to a momentum transfer

With \(n\) electrons per unit volume, the energy lost per unit path is \(\int\Delta E\;2\pi nb\,db\), which diverges logarithmically at both ends. At small \(b\) the impulse approximation fails when \(\Delta E\) approaches the maximum possible transfer. At large \(b\), Bohr saw, an electron bound with frequency \(\omega\) responds adiabatically when the collision time \(b/v\) exceeds \(1/\omega\), and absorbs almost no energy.

The argument of the logarithm is the ratio \(b_{\max}/b_{\min}\), with \(b_{\max}\sim v/\omega\) and \(b_{\min}\sim Z_1e^2/(mv^2)\). Hans Bethe’s quantum version of 1930 replaces \(\hbar\omega\) by a mean excitation energy and \(b_{\min}\) by a quantum length. Bohr returned to the problem repeatedly, writing a long review in 1948.

The correspondence principle

For the general reader. Bohr’s rules contradicted classical physics, yet classical physics works perfectly for large objects. Bohr insisted that the quantum theory must agree with classical theory in the limit of large orbits, and he turned this requirement into a tool for guessing the new laws. Heisenberg built matrix mechanics in 1925 by following this guide.

Hydrogen

With \(E_n=-hcR/n^2\), the transition frequency \(\nu_{n\to n-1}=cR\big(\frac{1}{(n-1)^2}-\frac{1}{n^2}\big)\approx\frac{2cR}{n^3}\) for large \(n\). The classical orbital frequency in the orbit of energy \(E_n\) is \(\nu_{\text{cl}}=v/(2\pi r)=2cR/n^3\) exactly. The two agree as \(n\to\infty\). In 1913 Bohr used this agreement to fix the factor in his quantum condition.

Action variables

For a system with separable periodic coordinates, the rule generalizes to the quantization of action variables, found by Sommerfeld and William Wilson in 1915–16:

Paul Ehrenfest showed that these are adiabatic invariants: they are unchanged by slow changes of external parameters. This is why the quantum conditions are consistent; a slow change cannot carry an allowed state into a forbidden one. Since \(\nu_{\text{cl}}=\partial H/\partial J\), the classical frequency is the derivative of energy with respect to action, while the quantum frequency is the difference quotient \(\Delta E/h\) with \(\Delta J=h\). Correspondence is the statement that these agree when \(J\gg h\).

Building the periodic table

For the general reader. Why do lithium, sodium and potassium behave alike? Bohr explained the periodic table by imagining building atoms one electron at a time, each electron settling into the lowest available orbit. He used spectra, X-rays and chemistry to decide how the shells fill, and correctly predicted that the unknown element 72 would be like zirconium, not like the rare earths.

Moseley’s law

In 1913 Henry Moseley, in Manchester, measured X-ray lines of the elements and found a simple pattern that Bohr’s model explained. The K\(\alpha\) line comes from a jump \(n=2\to1\) of an electron seeing the nucleus screened by the one remaining K electron.

Moseley’s plot showed that the periodic table is ordered by nuclear charge, not atomic weight, and revealed gaps at \(Z=43,61,72,75\).

The building-up principle (1921–1922)

In lectures in Copenhagen and in a celebrated series at Göttingen in June 1922, Bohr assigned electrons to orbits labelled \(n_k\) (principal and azimuthal quantum numbers, in the notation of the time). His key physical idea was penetration: elongated orbits of low angular momentum dip inside the inner shells and feel more nuclear charge, so they are more tightly bound. This explains why the 4s orbit fills before the 3d in potassium and calcium, and why the transition metals and rare earths form long periods, with inner shells completed after outer ones have started. The modern rule, with the exclusion principle of Wolfgang Pauli (1925), fills subshells in order of increasing \(n+\ell\), and for equal \(n+\ell\), increasing \(n\).

The rare earths end with lutetium at \(Z=71\), Bohr argued, because the \(4f\) shell holds 14 electrons. Element 72 must therefore have four outer electrons, like zirconium. In 1923 Dirk Coster and George de Hevesy, working at Bohr’s institute, found it in zirconium ores by its X-ray spectrum. They named it hafnium, after Hafnia, the Latin name of Copenhagen.

Complementarity and the debate with Einstein

For the general reader. Light acts like a wave in some experiments and like a stream of particles in others. Bohr argued in 1927 that these pictures are complementary: both are needed, but no single experiment can display both at once, because the apparatus that reveals one excludes the other. For eight years Einstein invented thought experiments to escape this limit, and Bohr answered each one.

Uncertainty and complementarity

In 1927 Werner Heisenberg, working at Bohr’s institute, derived his uncertainty relations. Bohr disagreed with Heisenberg’s first explanation, which blamed the disturbance caused by measurement. In his lecture at Como in September 1927 Bohr presented a broader view: the experimental arrangement must be described in classical terms, and arrangements suited to defining position and those suited to defining momentum are mutually exclusive.

Einstein’s challenges

At the Fifth Solvay Conference in October 1927, Einstein proposed measuring the recoil of a screen with slits to learn which slit a particle passed through while still seeing interference. Bohr showed that knowing the screen’s momentum well enough smears its position enough to wash out the fringes (Exercise 9.1). At the Sixth Solvay Conference in 1930, Einstein proposed a box of radiation with a shutter run by a clock. Weigh the box before and after one photon escapes: the mass loss gives the energy exactly, while the clock gives the time exactly. The next morning Bohr answered by using Einstein’s own general relativity: the clock’s rate depends on its height in the gravitational field (Exercise 9.2).

In May 1935 Einstein, Boris Podolsky and Nathan Rosen argued that quantum mechanics is incomplete, using a pair of correlated particles. Bohr’s reply, with the same title, appeared in Physical Review in October 1935. He argued that the EPR criterion of reality was ambiguous, because the experimental conditions define what can be predicted. John Bell’s theorem of 1964 and the experiments that followed showed that no local hidden-variable theory can reproduce quantum predictions.

The compound nucleus and fission

For the general reader. In 1936 Bohr pictured the nucleus as a drop of liquid. A neutron that enters it shares its energy among all the particles, and the drop lives a long time before something boils off. In January 1939 news came that uranium nuclei split in two. Within weeks Bohr explained why, and why only the rare isotope uranium-235 splits with slow neutrons. With John Archibald Wheeler he worked out the theory.

The compound nucleus (1936)

Enrico Fermi’s group in Rome had found that slow neutrons are captured with very large cross-sections at sharp resonances. Bohr explained this with a strongly interacting nucleus. An incoming neutron quickly shares its energy with many nucleons, forming a compound state that lives much longer than the transit time. Its decay is independent of how it was formed, so formation and decay factorize. Narrow resonances are long-lived compound states.

The liquid drop and fission

The binding energy of a nucleus with \(A\) nucleons and charge \(Z\) contains a surface term \(E_S=a_SA^{2/3}\) and a Coulomb term \(E_C=a_CZ^2/A^{1/3}\), with \(a_S\approx17.8\) MeV and \(a_C\approx0.71\) MeV. Deform the sphere into a spheroid at constant volume, with small deformation parameter \(\varepsilon\). To second order

For uranium-238, \(Z^2/A\approx35.6\) against a critical value near 50, so \(x\approx0.7\). The nucleus is stable but has a barrier of about 6 MeV. Bohr realized that the barrier explains the isotope effect. Adding a neutron to U-235 makes the even–even nucleus U-236, gaining the pairing energy, so the excitation is about 6.5 MeV, enough to fission. Adding one to U-238 gives U-239 with only about 4.8 MeV, below the barrier; it needs fast neutrons. He published this in February 1939, and the Bohr–Wheeler paper The Mechanism of Nuclear Fission appeared on 1 September 1939. Separated-isotope experiments by Alfred Nier and John Dunning confirmed the U-235 assignment in 1940.

The Copenhagen institute, 1916–1940

After two years as reader in Manchester, Bohr became professor of theoretical physics in Copenhagen in 1916. He proposed an institute; the university agreed in 1918, and with private and Carlsberg funding the Institute for Theoretical Physics opened on 3 March 1921, later expanded with Rockefeller money. It became the meeting place of the new physics. Hendrik Kramers was Bohr’s first long-term assistant. Werner Heisenberg, Wolfgang Pauli, Paul Dirac, Oskar Klein, Lev Landau, George Gamow, Léon Rosenfeld, John Slater and many others passed through. Discussions with Bohr were long, slow and relentless; he thought by talking, and he drafted papers through many versions dictated to Margrethe or to a collaborator.

The 1922 Nobel Prize in Physics recognized his investigation of the structure of atoms and of the radiation they emit. In 1924, with Kramers and Slater, he proposed that energy might be conserved only statistically in individual atomic processes. Experiments by Walther Bothe and Hans Geiger and by Arthur Compton and Alfred Simon refuted the idea within a year, and Bohr accepted the verdict quickly. Matrix mechanics (1925), wave mechanics (1926) and the uncertainty relations (1927) followed, and the interpretation Bohr and Heisenberg gave them became known as the Copenhagen interpretation, though the two never fully agreed.

In 1932 the Bohrs moved into the Carlsberg Honorary Residence, a mansion given by the Carlsberg Foundation to a leading Danish figure in science or culture. In 1934 their eldest son Christian drowned in a sailing accident. After 1933 the institute became a way station for refugee physicists from Germany, and Bohr worked with Danish committees to find them positions abroad.

Occupation, the bomb, the open world, 1940–1962

Germany occupied Denmark in April 1940. Bohr stayed and kept the institute working. In September 1941 Heisenberg, now leading German work on nuclear fission, visited him in Copenhagen. Their conversation went badly, and accounts of what was said differ; drafts of letters Bohr never sent, released by his family in 2002, show that he remembered it with anger.

In late September 1943, warned that the Germans planned to arrest Denmark’s Jews and that he himself was at risk because of his mother’s Jewish ancestry, Bohr escaped with Margrethe by boat to Sweden on 29 September. In Stockholm he pressed the Swedish government to announce that it would receive Danish Jewish refugees; nearly all of Denmark’s Jews reached Sweden. In early October he was flown to Britain, then went on to the United States. He visited Los Alamos under the name Nicholas Baker and contributed to the physics of the bomb, but his main aim became political. He believed that the bomb could not stay secret and that only openness, including early approaches to the Soviet Union, could prevent an arms race. He met Churchill on 16 May 1944 and Roosevelt on 26 August 1944. Churchill rejected his ideas and briefly suspected him.

After the war Bohr returned to Copenhagen. In 1947 he received the Order of the Elephant, Denmark’s highest order; for his coat of arms he chose the yin–yang symbol and the motto contraria sunt complementa, opposites are complementary. In June 1950 he published an open letter to the United Nations calling for an open world. He supported the founding of CERN in 1952, chaired Nordita from 1957 and the Danish Atomic Energy Commission, and received the first Atoms for Peace Award in 1957. He died of heart failure at home on 18 November 1962, aged 77. His son Aage shared the 1975 Nobel Prize in Physics for work on nuclear structure.

Legacy

• Atomic physics: stationary states and quantum jumps \(h\nu=E_n-E_m\) remain the language of spectroscopy, lasers and atomic clocks.
• Quantum theory: the correspondence principle guided the discovery of quantum mechanics; complementarity framed the interpretation debate that continues today.
• Nuclear physics: the compound nucleus and the liquid-drop theory of fission underlie reactor physics.
• Institutions: the Niels Bohr Institute, CERN and Nordita; the model of a small international institute built around discussion.
• Honours: element 107 is named bohrium. The Bohr radius and the Bohr magneton carry his name.

Sources and further reading

Biography

1. J. J. O’Connor, E. F. Robertson, Niels Bohr, MacTutor History of Mathematics.
2. Niels Bohr, Wikipedia.
3. The Nobel Prize in Physics 1922, Nobel Prize website.
4. A. Pais, Niels Bohr’s Times, in Physics, Philosophy, and Polity, Oxford University Press, 1991.
5. R. Aaserud, J. L. Heilbron, Love, Literature, and the Quantum Atom: Niels Bohr’s 1913 Trilogy Revisited, Oxford University Press, 2013.
6. Niels Bohr Archive, Copenhagen: documents on the 1941 meeting with Heisenberg, released 2002.

Physics

1. N. Bohr, On the constitution of atoms and molecules, Phil. Mag. 26 (1913) 1–25, 476–502, 857–875.
2. N. Bohr, On the theory of the decrease of velocity of moving electrified particles on passing through matter, Phil. Mag. 25 (1913) 10–31.
3. N. Bohr, On the quantum theory of line-spectra, Kgl. Danske Vid. Selsk. Skr. (1918–1922).
4. N. Bohr, H. A. Kramers, J. C. Slater, The quantum theory of radiation, Phil. Mag. 47 (1924) 785–802.
5. N. Bohr, The quantum postulate and the recent development of atomic theory, Nature 121 (1928) 580–590.
6. N. Bohr, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 48 (1935) 696–702.
7. N. Bohr, Neutron capture and nuclear constitution, Nature 137 (1936) 344–348.
8. N. Bohr, J. A. Wheeler, The mechanism of nuclear fission, Phys. Rev. 56 (1939) 426–450.
9. N. Bohr, Discussion with Einstein on epistemological problems in atomic physics, in P. A. Schilpp (ed.), Albert Einstein: Philosopher-Scientist, 1949.
10. Bohr–Einstein debates and Hafnium, Wikipedia.