The Path to the Quantum Atom John Heilbron Describes the Route That Led Niels Bohr to Quantize Electron Orbits a Century Ago

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Niels Bohr and his wife Margrethe around 1930. The path to the quantum atom John Heilbron describes the route that led Niels Bohr to quantize electron orbits a century ago.

n the autumn of 1911, the Danish from the Royal Danish Academy of Sciences so advanced that no one in Denmark could physicist Niels Bohr set sail for a post- in 1908, at the age of 23, for a theoretical and evaluate it fully. doctoral year in England inflamed with experimental study of water jets published Bohr went to the University of Cam- I“all my stupid wild courage”, as he expressed by the Royal Society of London. His doctoral bridge, UK, to work with Joseph John (J. J.) his state of mind in a letter to his fiancée, thesis on the electron theory of metals was Thomson, famous as the discoverer of the Margrethe Nørlund1. Bohr would need that electron and recipient of the Nobel Prize in courage on his route to his revolutionary THE QUANTUM ATOM Physics for 1906. For Bohr, Thomson was “a quantum atom of 1913. A Nature special issue genius who showed the way to everyone”. But Bohr had reason to think himself designed nature.com/bohr00 Thomson was too full of his own ideas to lis- for great things. He had won a gold medal ten to those of a foreigner whose English

NUCLEAR MODEL In February 1912, Bohr went to the Victoria University of Manchester, UK, to arrange to work on radioactivity in Ernest Ruther- ford’s laboratory. He looked forward to it in his understated way: “My courage is ablaze, so wildly, so wildly”1. Rutherford satisfied his expectations: “a really first-rate man and

extremely capable, in many ways more able BOHR ARCHIVE, COPENHAGEN NIELS than Thomson, even though perhaps he is not as gifted”1. Rutherford certainly surpassed Thomson as a research director. When Bohr arrived, several men in the laboratory were work- ing on implications of the nuclear model of the atom that Rutherford had introduced in 1911. To explain the unexpected reflection of α-particles from thin metal foils, detected by his research students, Rutherford had found it necessary to collect all the positive charge in Thomson’s spheres into a tiny kernel at the atom’s centre. Soon Bohr joined in via his natural route: criticism. In calculating the transfer of energy from an α-particle to atomic electrons, Rutherford’s theorist Charles Galton Darwin had not taken into account the reso- nance that occurs when the time of passage of the particle past the atom coincides with the natural frequency at which the perturbed electrons oscillate. In improving the calculations, Bohr discovered that some modes of oscillation of a ring of electrons in the plane of their orbit grow until they tear the atom apart. This mechanical instability could not be mended by deploying accepted physical concepts. Bohr’s thesis work had familiarized him with more general examples of failure in theo- ries of heat radiation and magnetism that allowed electrons all the freedom that statis- tical mechanics granted them. To his unique way of thinking, the nuclear model appealed to Bohr precisely because it expressed this failure so conspicuously. Niels Bohr (left) with Albert Einstein in 1925. The model had further advantages. It made a clear distinction between radioactive he had to struggle to understand. “They charged. In this picture, Thomson elucidated and chemical phenomena, which in Bohr’s say he would walk away from the king,” Niels the periodic properties of the elements, the view derived from the nucleus and the elec- wrote to his brother Harald, “which means formation of simple molecules, radioactivity, tronic structure, respectively. This inference more in England than in Denmark”1. the scattering of X-rays and β-particles, and was not as evident then as it is now. Even Even if Thomson had been interested, he the ratio between the weight of an atom and Rutherford had not yet grasped the distinc- would have had trouble perceiving that his the number of its electrons. tion, and assigned the origin of β- and γ-rays postdoc was a mature mathematical physi- Bohr spent much of his time at Cam- to extra-nuclear electrons. cist. Furthermore, Bohr’s speciality was criti- bridge attending talks and reading widely. Most importantly, the nuclear model, cism. In his thesis work, he had discovered He extolled Thomson’s lectures and found combined with Rutherford’s conception errors in Thomson’s papers, which he tried much to admire in the treatise Aether and of the α-particle as a bare nucleus, almost to bring to the professor’s attention. That was Matter (1900), in which Joseph Larmor, the thrust the concept of atomic number on not the right gambit. occupant of the chair of mathematics once physicists. They knew that the α-particle Thomson was preoccupied with devel- held by Isaac Newton, developed a world was a helium atom minus two electrons; its oping consequences of the model atom he system based on electrons conceived as nucleus must therefore have a charge of two, had proposed in 1903. Later inappropriately permanent twists in the ether. “When I read implying that hydrogen’s has a charge of one, and derisively nicknamed a ‘plum pudding’, something that is so good and grand as that,” lithium’s a charge of three, and so on. it consisted of concentric rings of electrons Bohr wrote to Margrethe1, “then I feel such His confidence replenished by Ruther- rotating through a resistanceless spheri- courage and desire to try whether I too could ford’s interest, Bohr drew up a memoran- cal space that acted as if it were positively accomplish a tiny bit.” dum in June or July 1912 to show how Max

Planck’s idea that energy came in packets, or quanta, could extend the purview of the THE BALMER FORMULA nuclear model to the problems that Thomson had considered, and to fix the size of atoms. Bohr’s key to the microworld Although most of the memorandum was qualitative, in one essential point Bohr The Balmer formula expresses the nth state with the second term, –Rh/n2. The could be exact where Thomson could only frequencies of some lines in the spectrum first term would then be the negative of estimate. Rutherford’s scattering theory and of hydrogen in simple algebra: the energy for the second state (n = 2), and experiments required that, for helium, the the formula could be read to mean that a 2 2 atomic weight (4) was twice the number νn = R(1/2 – 1/n ) Balmer line originates in a jump of of electrons (2). Thomson could only say, a hydrogen electron from its nth to its after extensive theoretical and experimen- where νn is the nth Balmer line and second state. tal work on the scattering of X-rays and R is the universal Rydberg constant for To calculate R, Bohr equated the energy β-particles, that the number of electrons frequency, named in honour of the Swedish of the nth state, –Rh/n2, with the expression in an element was roughly three times its spectroscopist Johannes Rydberg, who he already had for the kinetic energy Tn of atomic weight. generalized Balmer’s formula to apply to an orbiting electron in his quantized model After these first easy gains, Niels wrote elements beyond hydrogen. of the nuclear atom: 2 4 2 2 to Harald, “Perhaps I have found out a lit- Following Max Planck’s radiation theory, Tn = 2π me /h n tle about the structure of atoms. If I should Niels Bohr converted the equation into where e and m are the charge and mass be right, it wouldn’t be a suggestion of the units of energy, by multiplying both sides of the electron. Equating the two energies, nature of a possibility (i.e., impossibility as by Planck’s constant, h. This allowed him Bohr had R in terms of the fundamental J. J. Thomson’s theory) but perhaps a little to identify the energy of the electron in its constants. bit of reality”1. Nonetheless, Bohr followed Thomson’s Lyman series lead in the other subjects he discussed with HYDROGEN SPECTRUM Rutherford: the periodic properties of the Electrons jumping between energy levels in a Balmer series hydrogen atom give o light at certain frequencies, Paschen series elements, determined by stability require- corresponding to the energy di erence (wavelengths ments imposed on their ring structures, and given in nanometres, nm). Bohr focused on the Balmer series of spectral lines, between the second 94 nm the binding of atoms into simple molecules, and higher energy levels. secured by exchanges of electrons. 95 nm To proceed with his calculations, Bohr laid down the ad-hoc postulate, conceived in 97 nm analogy to Planck’s radiation theory, that if the kinetic energy of each electron is propor- 103 nm tional to the frequency of its orbit, it would 486 nm neither radiate nor succumb to unstable 122 nm 656 nm 434 nm oscillations, and he guessed that the constant 410 nm of proportionality was a fraction of Planck’s h. = 1 n 1,875 nm

= 2 n BALMER’S NUMEROLOGY 1,282 nm

Bohr’s three-part paper on the constitution = 3 of atoms and molecules was published in n the London-based Philosophical Magazine 1,094 nm = 4 between July and November 1913. The n second and third parts, which consider = 5 the periodic arrangements of the elements n and molecular binding, record Bohr’s debt to Thomson. Alone they would not have = 6 attracted attention or affected a revolution. n What made Bohr’s ‘trilogy’ memorable was its first part2, on the spectrum of hydrogen, a subject he did not confront until in which, by definition, the electrons have plane, perpendicular ones can be stable. By February 1913. radiated away all the energy that nature calculating the frequencies of rotation of his A colleague asked him how he explained allows them to dispose of. It could not electrons from the spectra, he could com- the formula for the frequencies of a series explain why many frequencies are emitted. pute their angular momenta. He found that, of spectral lines emitted by hydrogen, for Bohr could see the bearing of the Balmer very closely, the angular momentum of each which Johann Jakob Balmer had devised a formula so quickly because, around New of his electrons was a small integral multiple simple arithmetical formula in 1885. Bohr Year, he had extended his model in response of h/2π. replied that spectra were too complicated to a remarkable series of papers by John Nicholson’s finding followed the lead of for his model, but had a look anyway. He William Nicholson, a mathematical physicist the Conseil de Physique Solvay of 1911, the saw immediately, so he later said, how he had met in Cambridge. conference at which Planck, Rutherford, to calculate the ratio of kinetic energy to Nicholson had matched the frequencies Albert Einstein, Hendrik Antoon Lorentz orbital frequency for the model he had of many unattributed lines in solar and and other luminaries considered problems presented to Rutherford six months earlier nebular spectra with the oscillations of elec- in the theory of radiation. Discussion cen- (see ‘Bohr’s key to the microworld’). trons in a nuclear atom perpendicular to the tred on Planck’s concept of energy quanta: That early model had only a ground state, plane of their ring. Unlike oscillations in the the simple harmonic oscillators by which he

6 JUNE 2013 | VOL 498 | NATURE | 29 © 2013 Macmillan Publishers Limited. All rights reserved COMMENT represented material particles able to emit OPEN TO AMBIGUITY and absorb radiation possess energies only Bohr’s ability to entertain several conflicting in integral multiples of their frequencies. ideas, and his courage in demanding sacri- An oscillator could emit or absorb radiation fices of physicists like Einstein, Planck and when its natural frequency equalled that of Lorentz, are breathtaking. We know that he the radiation (ν), and then only in energy did not lack confidence. Blazing courage is increments (E = hν), or quanta. one thing, but tolerance of ambiguity is quite Because Nicholson’s matches were another. astonishingly exact, and his model, like Correspondence with his immediate BOHR ARCHIVE, COPENHAGEN NIELS Bohr’s, was both nuclear and quantized, family, especially Margrethe, suggests Bohr had to take it seriously. Still reluctant sources for this tolerance. Well before he to investigate spectra, he compromised, became entangled in the quantum atom, imagining that a captured electron occu- Bohr had developed a doctrine of multi- pied a sequence of excited states, radiating ple partial truths, each of which contained energy from each in Nicholson’s manner as some bit of reality, and all of which together it descended towards the nucleus and the might exhaust it. “There exist so many dif- ground state. ferent truths,” he wrote to Margrethe. “I can Bohr introduced a running integer (n) almost call it my religion, that I think that into his model to handle the ascending everything that is of value is true.”1 scale of electron energies. By making the Bohr’s seminal analysis of the Balmer kinetic energy of the nth orbit proportional spectrum expressed the partial truth of to n times the orbital frequency, Bohr easily To develop his model, Bohr followed an analogy Planck’s radiation theory and the partial obtained Nicholson’s result about angular to the radiation theory of Max Planck (right). truth of classical physics. Bohr may have momentum and the additional informa- owed his notion of partial truth at least in tion that the constant of proportionality was to produce it, contrary to the concepts by part to ideas he found in the writings of his h/2. Thus Bohr had an integer-based series which physicists usually dealt with radiation. professor of philosophy, Harald Høffding, in his mind when he glimpsed the Balmer Bohr’s sense of responsibility directed and in William James’s Pragmatism, pub- formula. him to attempt to anchor the basic postu- lished in 1907, which Bohr may have Using the relation E = hν, Bohr trans- late of his quantum atom — that the ratio known from Høffding. formed arithmetic into physics by multiply- of kinetic energy to orbital frequency in the Family letters hitherto unavailable, which ing Balmer’s formula by Planck’s constant, h. nth state is proportional to nh/2 — in deeper will be published in part next month by Finn Making it into an energy equation allowed foundations. He did not find the job easy. Aaserud and myself in Love, Literature, and Bohr to identify the kinetic energies of the The first instalment of the trilogy contains the Quantum Atom1, open new directions various states with corresponding terms in four distinct, and largely contradictory, in which to explore this connection, about the altered formula. That enabled him to attempts. which historians and philosophers have spec- derive the parameter in the Balmer formula, Two of them develop the analogy to ulated on the basis of the later and slighter evi- known as the Rydberg constant in terms of Planck’s radiation theory that provided the dence of Bohr’s principle of complementarity. Planck’s constant and the charge and mass form of Bohr’s postulate. The third founda- Between periods in which his courage of the electron. tion is altogether different. It requires that blazed and his blood boiled, Bohr was sub- The successful computation of the Rydberg in jumps between very large neighbouring ject to the sorts of self-doubt that ordinary constant demanded serious sacrifices from orbits, where the electron is almost free people have. As their correspondence shows, physicists. It made a Balmer line originate from the nucleus, the radiation frequency Margrethe played an important, perhaps an in a jump of an electron to the second orbit is asymptotically equal to the frequency of essential, part in smoothing out Niels’ mood from a higher one, and put the explanation the orbits, which are asymptotically equal to swings and reassuring him that he was the of such jumps beyond the reach of physics. one another. This anticipates Bohr’s corre- great man his Danish support system took Rutherford spotted this immediately: in order spondence principle, according to which, at him to be. to ‘vibrate’ at the appropriate frequency, an an appropriate limit, calculations of a physi- In many letters he asks her to help him electron would have to know where it would cal quantity must give the same numerical pay his debts, by which he meant the obliga- stop before it leapt. He was unwilling to con- result in ordinary physics and in quantum tions he felt he owed for his great gifts, for the cede foreknowledge to electrons or admit theory. encouragement he had received to develop frequencies without vibrations. By the end of 1913, Bohr had given up the them and, perhaps, for the wider perspec- Bohr replied that physicists must Planck pedigree as “misleading” (the nuclear tives he gained in England. He could dis- “renounce” — a word he came to use fre- atom is not a simple harmonic oscillator) charge these debts only by great deeds. He quently — the possibility of exact descrip- and adopted the correspondence principle made a huge down payment to these imagi- tions of certain processes in the microworld. as his preferred foundation. He also retained nary creditors — including Thomson and Einstein perceived a greater loss. Planck the fourth formulation, the only one now Rutherford — with his revolutionary quan- had equated the frequencies of radiated remembered: the quantization of the angular tum atom of 1913. ■ light and mechanical oscillation. This was momentum (which follows from the basic possible because the frequency of a simple postulate by replacing the ratio of kinetic John L. Heilbron is emeritus professor harmonic oscillator is the same regard- energy to orbital frequency by its mechani- of history at the University of California, less of its energy. The oscillations of the cal equivalent, π times angular momentum). Berkeley, USA. radiator directly excited the ‘ether’, or the As a condition on the orbit, the fourth foun- e-mail: [email protected] radiation field. But Bohr’s jumps involved dation differs conceptually from the other 1. Aaserud, F. & Heilbron, J. L. Love, Literature, and two orbits of different periods. The fre- three, which relate the orbit to the radia- the Quantum Atom: Niels Bohr’s 1913 Trilogy quency of light emitted did not correspond tion emitted by an electron undergoing a Revisited (Oxford University Press, 2013). with the motions of the electron supposed quantum jump. 2. Bohr, N. Philos. Mag. 26, 1–25 (1913).