Quantum

by Manjit Kumar

  Chapter 4

  THE QUANTUM ATOM

  Slagelse, Denmark, Thursday, 1 August 1912. The cobbled streets of the small, picturesque town some 50 miles south-west of Copenhagen were decked out in flags. Yet it was not in the beautiful medieval church, but in the civic hall that Niels Bohr and Margrethe Nørland were married in a two-minute ceremony conducted by the chief of police. The mayor was away on holiday, Harald was best man, and only close family were present. Like his parents before him, Bohr did not want a religious ceremony. He had stopped believing in God as a teenager, when he had confessed to his father: ‘I cannot understand how I could be so taken in by all this; it means nothing whatsoever to me.’1 Had he lived, Christian Bohr would have approved when, a few months before the wedding, his son formally resigned from the Lutheran Church.

  Originally intending to spend their honeymoon in Norway, the couple were forced to change their plans as Bohr failed to finish a paper on alpha particles in time. Instead the newlyweds travelled to Cambridge for a two-week stay during their month-long honeymoon.2 In between visits to old friends and showing Margrethe around Cambridge, Bohr completed his paper. It was a joint effort. Niels dictated, always struggling for the right word to make his meaning clear, while Margrethe corrected and improved his English. They worked so well together that for the next few years Margrethe effectively became his secretary.

  Bohr disliked writing and avoided doing so whenever he could. He was able to complete his doctoral thesis only by dictating it to his mother. ‘You mustn’t help Niels so much, you must let him learn to write himself’, his father had urged, to no avail.3 When he did put pen to paper, Bohr wrote slowly and in an almost indecipherable scrawl. ‘First and foremost,’ recalled a colleague, ‘he found it difficult to think and write at the same time.’4 He needed to talk, to think aloud as he developed his ideas. He thought best while on the move, usually circling a table. Later, an assistant, or anyone he could find for the task, would sit with pen poised as he paced about dictating in one language or other. Rarely satisfied with the composition of a paper or lecture, Bohr would ‘rewrite’ it up to a dozen times. The end result of this excessive search for precision and clarity was often to lead the reader into a forest where it was difficult to see the wood for the trees.

  With the manuscript finally completed and safely packed away, Niels and Margrethe boarded the train to Manchester. On meeting his bride, Ernest and Mary Rutherford knew that the young Dane had been lucky enough to find the right woman. The marriage indeed proved to be a long and happy one that was strong enough to endure the death of two of their six sons. Rutherford was so taken with Margrethe that for once there was little talk of physics. But he made time to read Bohr’s paper and promised to send it to the Philosophical Magazine with his endorsement.5 Relieved and happy, a few days later the Bohrs travelled to Scotland to enjoy the remainder of their honeymoon.

  Returning to Copenhagen at the beginning of September, they moved into a small house in the prosperous coastal suburb of Hellerup. In a country with only one university, physics posts rarely became vacant.6 Just before his wedding day, Bohr had accepted a job as a teaching assistant at the Lœreanstalt, the Technical College. Each morning, Bohr would cycle to his new office. ‘He would come into the yard, pushing his bicycle, faster than anybody else’, recalled a colleague later.7 ‘He was an incessant worker and seemed always to be in a hurry.’ The relaxed, pipe-smoking elder statesman of physics lay in the future.

  Bohr also began teaching thermodynamics as a privatdozent at the university. Like Einstein, he found preparing a lecture course arduous. Nevertheless, at least one student appreciated the effort and thanked Bohr for ‘the clarity and conciseness’ with which he had ‘arranged the difficult material’ and ‘the good style’ with which it had been delivered.8 But teaching combined with his duties as an assistant left him precious little time to tackle the problems besetting Rutherford’s atom. Progress was painfully slow for a young man in a hurry. He had hoped that a report written for Rutherford while still in Manchester on his nascent ideas about atomic structure, later dubbed the ‘Rutherford Memorandum’, would serve as the basis of a paper ready for publication soon after his honeymoon.9 It was not to be.

  ‘You see,’ Bohr said 50 years later in one of the last interviews he gave, ‘I’m sorry because most of that was wrong.’10 However, he had identified the key problem: the instability of Rutherford’s atom. Maxwell’s theory of electromagnetism predicted that an electron circling the nucleus should continuously emit radiation. This incessant leaking of energy sends the electron spiralling into the nucleus as its orbit rapidly decays. Radiative instability was such a well known failing that Bohr did not even mention it in his Memorandum. What really concerned him was the mechanical instability that plagued Rutherford’s atom.

  Beyond assuming that electrons revolved around the nucleus in the manner of planets around the sun, Rutherford had said nothing about their possible arrangement. A ring of negatively-charged electrons circling the nucleus was known to be unstable due to the repulsive forces the electrons exert on each other because they have the same charge. Nor could the electrons be stationary; since opposite charges attract, the electrons would be dragged towards the positively-charged core. It was a fact that Bohr recognised in the opening sentence of his memo: ‘In such an atom there can be no equilibrium [con]figuration without the motion of electrons.’11 The problems that the young Dane had to overcome were mounting up. The electrons could not form a ring, they could not be stationary, and they could not orbit the nucleus. Lastly, with a tiny, point-like nucleus at its heart, there was no way in Rutherford’s model to fix the radius of an atom.

  Whereas others had interpreted these problems of instability as damning evidence against Rutherford’s nuclear atom, for Bohr they signalled the limitations of the underlying physics that predicted its demise. His identification of radioactivity as a ‘nuclear’ and not an ‘atomic’ phenomenon, his pioneering work on radioelements, what Soddy later called isotopes, and on nuclear charge convinced Bohr that Rutherford’s atom was indeed stable. Although it could not bear the weight of established physics, it did not suffer the predicted collapse. The question that Bohr had to answer was: why not?

  Since the physics of Newton and Maxwell had been impeccably applied and forecast electrons crashing into the nucleus, Bohr accepted that the ‘question of stability must therefore be treated from a different point of view’.12 He understood that to save Rutherford’s atom would require a ‘radical change’, and he turned to the quantum discovered by a reluctant Planck and championed by Einstein.13 The fact that in the interaction between radiation and matter, energy was absorbed and emitted in packets of varying size rather than continuously, was something beyond the realm of time-honoured ‘classical’ physics. Even though like almost everyone else he did not believe in Einstein’s light-quanta, it was clear to Bohr that the atom ‘was in some way regulated by the quantum’.14 But in September 1912 he had no idea how.

  All his life, Bohr loved to read detective stories. Like any good private eye, he looked for clues at the crime scene. The first were the predictions of instability. Certain that Rutherford’s atom was stable, Bohr came up with an idea that proved crucial to his ongoing investigation: the concept of stationary states. Planck had constructed his blackbody formula to explain the available experimental data. Only then did he attempt to derive his equation and in the process stumbled across the quantum. Bohr adopted a similar strategy. He would begin by rebuilding Rutherford’s atomic model so that electrons did not radiate energy as they orbited the nucleus. Only later would he try to justify what he had done.

  Classical physics placed no restrictions on an electron’s orbit inside an atom. But Bohr did. Like an architect designing a building to the strict requirements of a client, he restricted electrons to certain ‘special’ orbits in which they could not continuously emit radiation and spiral into the nucleus. It was a stroke of genius. Bohr believed that certain
laws of physics were not valid in the atomic world and so he ‘quantised’ electron orbits. Just as Planck had quantised the absorption and emission of energy by his imaginary oscillators so as to derive his blackbody equation, Bohr abandoned the accepted notion that an electron could orbit an atomic nucleus at any given distance. An electron, he argued, could occupy only a few select orbits, the ‘stationary states’, out of all the possible orbits allowed by classical physics.

  It was a condition that Bohr was perfectly entitled to impose as a theorist trying to piece together a viable working atomic model. It was a radical proposal, and for the moment all he had was an unconvincing circular argument that contradicted established physics – electrons occupied special orbits in which they did not radiate energy; electrons did not radiate energy because they occupied special orbits. Unless he could offer a real physical explanation for his stationary states, the permissible electron orbits, they would be dismissed as nothing more than theoretical scaffolding erected to hold up a discredited atomic structure.

  ‘I hope to be able to finish the paper in a few weeks,’ Bohr wrote to Rutherford at the beginning of November.15 Reading the letter and sensing Bohr’s mounting anxiety, Rutherford replied that there was no reason ‘to feel pressed to publish in a hurry’ since it was unlikely anyone else was working along the same lines.16 Bohr was unconvinced as the weeks passed without success. If others were not already actively engaged in trying to solve the mystery of the atom, then it was only a matter of time. Struggling to make headway, in December he asked for and was granted a few months’ sabbatical by Knudsen. Together with Margrethe, Bohr found a secluded cottage in the countryside as he set about searching for more atomic clues. Just before Christmas he found one in the work of John Nicholson. At first he feared the worst, but he soon realised that the Englishman was not the competitor he dreaded.

  Bohr had met Nicholson during his abortive stay in Cambridge, and had not been overly impressed. Only a few years older at 31, Nicholson had since been appointed professor of mathematics at King’s College, University of London. He had also been busy building an atomic model of his own. He believed that the different elements were actually made up of various combinations of four ‘primary atoms’. Each of these ‘primary atoms’ consisted of a nucleus surrounded by a different number of electrons that formed a rotating ring. Though, as Rutherford said, Nicholson had made an ‘awful hash’ of the atom, Bohr had found his second clue. It was the physical explanation of the stationary states, the reason why electrons could occupy only certain orbits around the nucleus.

  An object moving in a straight line has momentum. It is nothing more than the object’s mass times its velocity. An object moving in a circle possesses a property called ‘angular momentum’. An electron moving in a circular orbit has an angular momentum, labelled L, that is just the mass of the electron multiplied by its velocity multiplied by the radius of its orbit, or simply L=mvr. There were no limits in classical physics on the angular momentum of an electron or any other object moving in a circle.

  When Bohr read Nicholson’s paper, he found his former Cambridge colleague arguing that the angular momentum of a ring of electrons could change only by multiples of h/2, where h is Planck’s constant and (pi) is the well-known numerical constant from mathematics, 3.14….17 Nicholson showed that the angular momentum of a rotating electron ring could only be h/2 or 2(h/2) or 3(h/2) or 4(h/2)…all the way to n(h/2) where n is an integer, a whole number. For Bohr it was the missing clue that underpinned his stationary states. Only those orbits were permitted in which the angular momentum of the electron was an integer n multiplied by h and then divided by 2. Letting n=1, 2, 3 and so on generated the stationary states of the atom in which an electron did not emit radiation and could therefore orbit the nucleus indefinitely. All other orbits, the non-stationary states, were forbidden. Inside an atom, angular momentum was quantised. It could only have the values L=nh/2 and no others.

  Just as a person on a ladder can stand only on its steps and nowhere in between, because electron orbits are quantised, so are the energies that an electron can possess inside an atom. For hydrogen, Bohr was able to use classical physics to calculate its single electron’s energy in each orbit. The set of allowed orbits and their associated electron energies are the quantum states of the atom, its energy levels En. The bottom rung of this atomic energy ladder is n=1, when the electron is in the first orbit, the lowest-energy quantum state. Bohr’s model predicted that the lowest energy level, E1, called the ‘ground state’, for the hydrogen atom would be –13.6eV, where an electron volt (eV) is the unit of measurement adopted for energy on the atomic scale and the minus sign indicates that the electron is bound to the nucleus.18 If the electron occupies any other orbit but n=1, then the atom is said to be in an ‘excited state’. Later called the principal quantum number, n is always an integer, a whole number, which designates the series of stationary states that an electron can occupy and the corresponding set of energy levels, En, of the atom.

  Bohr calculated the values of the energy levels of the hydrogen atom and found that the energy of each level was equal to the energy of the ground state divided by n2, (E1/n2)eV. Thus, the energy value for n=2, the first excited state, is –13.6/4 = –3.40eV. The radius of the first electron orbit, n=1, determines the size of the hydrogen atom in the ground state. From his model, Bohr calculated it as 5.3 nanometres (nm), where a nanometre is a billionth of a metre – in close agreement with the best experimental estimates of the day. He discovered that the radius of the other allowed orbits increased by a factor of n2: when n=1, the radius is r; when n=2, then the radius is 4r; when n=3, the radius is 9r and so on.

  ‘I hope very soon to be able to send you my paper on the atoms,’ Bohr wrote to Rutherford on 31 January 1913, ‘it has taken far more time than I had thought; I think, however, that I have made some progress in it in the latest time.’19 He had stabilised the nuclear atom by quantising the angular momentum of the orbiting electrons, and thereby explained why they could occupy only a certain number, the stationary states, of all possible orbits. Within days of writing to Rutherford, Bohr came across the third and final clue that allowed him to complete the construction of his quantum atomic model.

  Figure 6: Some of the stationary states and the corresponding energy levels of the hydrogen atom (not drawn to scale)

  Hans Hansen, a year younger and a friend of Bohr’s from their student days in Copenhagen, had just returned to the Danish capital after completing his studies in Göttingen. When they met, Bohr told him about his latest ideas on atomic structure. Having conducted research in Germany in spectroscopy, the study of the absorption and emission of radiation by atoms and molecules, Hansen asked Bohr if his work shed any light on the production of spectral lines. It had long been known that the appearance of a naked flame changed colour depending upon which metal was being vaporised: bright yellow with sodium, deep red with lithium, and violet with potassium. In the nineteenth century it had been discovered that each element produced a unique set of spectral lines, spikes in the spectrum of light. The number, spacing and wavelengths of the spectral lines produced by the atoms of any given element are unique, a fingerprint of light that can be used to identify it.

  Spectra appeared far too complicated, given the enormous variety of patterns displayed by the spectral lines of the different elements, for anyone to seriously believe that they could be the key to unlocking the inner workings of the atom. The beautiful array of colours on a butterfly’s wing were all very interesting, Bohr said later, ‘but nobody thought that one could get the basis of biology from the colouring of the wing of a butterfly’.20 There was obviously a link between an atom and its spectral lines, but at the beginning of February 1913 Bohr had no inkling what it could be. Hansen suggested that he take a look at Balmer’s formula for the spectral lines of hydrogen. As far as Bohr could remember, he had never heard of it. More likely he had simply forgotten it. Hanson outlined the formula and pointed out that no one kne
w why it worked.

  Johann Balmer was a Swiss mathematics teacher at a girls’ school in Basel and a part-time lecturer at the local university. Knowing that he was interested in numerology, a colleague told Balmer about the four spectral lines of hydrogen after he had complained about having nothing interesting to do. Intrigued, he set out to find a mathematical relationship between the lines where none appeared to exist. The Swedish physicist, Anders Ångström, had in the 1850s measured the wavelengths of the four lines in the red, green, blue and violet regions of the visible spectrum of hydrogen with remarkable accuracy. Labelling them alpha, beta, gamma and delta respectively, he found their wavelengths to be: 656.210, 486.074, 434.01 and 410.12nm.21 In June 1884, as he approached 60, Balmer found a formula that reproduced the wavelengths () of the four spectral lines: = b[m2/(m2–n2)] in which m and n are integers and b is a constant, a number determined by experiment as 364.56nm.

  Balmer discovered that if n was fixed as 2 but m set equal to 3, 4, 5 or 6, then his formula gave an almost exact match for each of the four wavelengths in turn. For example, when n=2 and m=3 is plugged into the formula, it gives the wavelength of the red alpha line. However, Balmer did more than just generate the four known spectral lines of hydrogen, later named the Balmer series in his honour. He predicted the existence of a fifth line when n=2 but m=7, unaware that Ångström, whose work was published in Swedish, had already discovered and measured its wavelength. The two values, experimental and theoretical, were in near-perfect agreement.

  Had Ångström lived (he died in 1874 aged 59), he would have been astounded by Balmer’s use of his formula to predict the existence of other series of spectral lines for the hydrogen atom in the infrared and ultraviolet regions by simply setting n to 1, 3, 4 and 5 while letting m cycle through different numbers, as he had done with n set at 2 to generate the four original lines. For example, with n=3 and m=4 or 5 or 6 or 7…, Balmer predicted the series of lines in the infrared that were discovered in 1908 by Friedrich Paschen. Each of the series forecast by Balmer was later discovered, but no one had been able to explain what lay behind the success of his formula. What physical mechanism did the formula, arrived at through a process of trial and error, symbolise?