by Manjit Kumar
In a clear sign of the high regard in which he held Pauli, Sommerfeld asked him to help write a major article on relativity for the Encyklopädie der Mathematischen Wissenschaffen. Sommerfeld had accepted the task of editing the fifth volume of the Encyklopädie that dealt with physics. After Einstein declined, Sommerfeld decided to write on relativity himself but found he had little time to do so. He needed help and turned to Pauli. When Sommerfeld saw the first draft, ‘it proved to be so masterly that I renounced all collaboration’.13 It was not only a brilliant exposition of the special and general theories of relativity, but an unrivalled review of the existing literature. It remained for decades the definitive work in the field and drew Einstein’s wholehearted praise. The article appeared in 1921, two months after Pauli received his doctorate.
As a student, Pauli preferred to spend his evenings enjoying the Munich nightlife in some café or other, returning to his lodgings to work through much of the night. He rarely attended lectures the following morning, turning up only around noon. But he attended enough to be drawn to the mysteries of quantum physics by Sommerfeld. ‘I was not spared the shock which every physicist accustomed to the classical way of thinking experienced when he came to know Bohr’s basic postulate of quantum theory for the first time’, Pauli said more than 30 years later.14 But he quickly got over it as he set about tackling his doctoral thesis.
Sommerfeld had set Pauli the task of applying the quantum rules of Bohr and his own modifications to the ionised hydrogen molecule, in which one of the two hydrogen atoms that make up the molecule has had its electron ripped off. As expected, Pauli produced a theoretically impeccable analysis. The only problem was that his results did not agree with the experimental data. Used to one success after another, Pauli was despondent at this lack of agreement between theory and experiment. However, his thesis was regarded as the first strong evidence that the outer limits of the Bohr-Sommerfeld quantum atom had been reached. The ad hoc way in which quantum physics was bolted onto classical physics had always been unsatisfactory, and now Pauli had shown that the Bohr-Sommerfeld model could not even deal with the ionised hydrogen molecule, let alone more complex atoms. In October 1921, armed with his doctorate, Pauli left Munich for Göttingen to take up the post of assistant to the professor of theoretical physics.
Max Born, 38, a key figure in the future development of quantum physics, had arrived in the small university town from Frankfurt just six months before Pauli. Growing up in Breslau, capital of the then Prussian province of Silesia, it was mathematics, not physics, that attracted Born. His father, like Pauli’s, was a highly cultured medical man and academic. A professor of embryology, Gustav Born advised his son not to specialise too early once he enrolled at Breslau University. Dutifully, Born settled on astronomy and mathematics only after having attended courses in physics, chemistry, zoology, philosophy and logic. His studies, including time at the universities of Heidelberg and Zurich, ended in 1906 with a doctorate in mathematics from Göttingen.
Immediately afterwards he began a year of compulsory military service that was cut short because of asthma. After spending six months in Cambridge as an advanced student, where he attended the lectures of J.J. Thomson, Born returned to Breslau to begin experimental work. But quickly discovering that he possessed neither the patience nor the skills required to be even a competent experimenter, Born turned to theoretical physics. By 1912 he had done enough to become a privatdozent in the world-renowned mathematics department at Göttingen, where they believed that ‘physics is much too hard for physicists’.15
Born’s success in tackling a string of problems by harnessing the power of mathematical techniques unknown to most physicists led in 1914 to an extraordinary professorship in Berlin. Just before war broke out, another newcomer arrived at the epicentre of German science: Einstein. Before long the two men, who shared a passion for music, became firm friends. When war came, Born was called up for military service. After a spell as a radio operator with the air force, he spent the rest of the war conducting artillery research for the army. Fortunately stationed near Berlin, Born was able to attend seminars at the university, meetings of the German Physical Society, and musical evenings at Einstein’s home.
After the war, in the spring of 1919, Max von Laue, an ordinary professor at Frankfurt, suggested to Born that they swap posts. Laue had won the 1914 Nobel Prize for the theory behind the diffraction of X-rays by crystals, and wanted to work with Planck, his former supervisor and a scientist he idolised. Born, encouraged by Einstein to ‘definitely accept’, agreed, as the exchange meant promotion to a full professorship and independence.16 Less than two years later, he moved to Göttingen to head the university’s institute of theoretical physics. It consisted of one small room, one assistant, and a part-time secretary, but Born was determined to build on these humble beginnings an institute to rival Sommerfeld’s in Munich. High on his list of priorities was getting Wolfgang Pauli, whom he described as ‘the greatest talent in the physics area that has emerged in the last years’.17 Born had already tried once before and failed, as Pauli opted to stay in Munich to finish his doctorate. This time he got his man.
‘W. Pauli is now my assistant; he is amazingly intelligent, and very able’, Born wrote to Einstein.18 Soon he discovered that the hired help had his own way of doing things. Pauli might have been brilliant, but he put in long hours of hard thinking as he continued his practice of working into the middle of the night and sleeping late. Whenever Born was unable to give his eleven o’clock lecture, the only way he could ensure Pauli would be there to teach in his place was by sending the maid to wake him up at 10.30 am.
It was clear from the beginning that Pauli was an ‘assistant’ in name only. Born admitted later that he learnt more from Pauli, despite his bohemian ways and poor time-keeping, than he was able to teach the ‘infant prodigy’. He was sad to see him go when in April 1922 Pauli left to become an assistant at Hamburg University. Swapping the quiet life of the small university town that he could hardly bear for the bustling nightlife of the big city was not the only reason he left so quickly. Pauli trusted his sense of physical intuition in pursuit of a logically flawless argument when tackling any physics problem. Born, however, turned much more readily to mathematics and allowed it to lead his search for a solution.
Two months later, in June 1922, Pauli was back in Göttingen to hear Bohr’s celebrated lecture series and met the great Dane for the first time. Impressed, Bohr asked Pauli if he would come to Copenhagen for a year as his assistant to help edit work in progress for publication in German. Pauli was taken aback by the offer. ‘I answered with that certainty of which only a young man is capable: “I hardly think that the scientific demands which you will make on me will cause me any difficulty, but the learning of a foreign tongue like Danish far exceeds my abilities.” I went to Copenhagen in the fall of 1922, where both my contentions were shown to be wrong.’19 It was also, he recognised later, the beginning of ‘a new phase’ in his life.20
Aside from helping Bohr, Pauli made a serious effort in Copenhagen to explain the ‘anomalous’ Zeeman effect – a feature of atomic spectra that the Bohr-Sommerfeld model could not explain. If atoms were exposed to a strong magnetic field, then the resulting atomic spectra contained lines that were split. It was quickly shown by Lorentz that classical physics predicted a splitting of a line into a doublet or a triplet: a phenomenon known as the ‘normal’ Zeeman effect which Bohr’s atom could not accommodate.21 Fortunately, Sommerfeld came to the rescue with two new quantum numbers and the modified quantum atom resolved the problem. It involved a series of new rules governing electrons jumping from one orbit (or energy level) to another based on three ‘quantum numbers’, n, k, and m, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing. But the celebrations were short-lived when it was discovered that the splitting of the red alpha line in the spectrum of hydrogen was smaller than expected. The situation grew worse with the
confirmation that some spectral lines actually split up into a quartet or more instead of just two or three lines.
Although called the ‘anomalous’ Zeeman effect because the extra lines could not be explained using either existing quantum physics or classical theory, it was in fact far more common than the ‘normal’ effect. For Pauli it signalled nothing less than the ‘deep seated failure of the theoretical principles known till now’.22 Having set himself the task of rectifying this miserable state of affairs, Pauli could not come up with an explanation. ‘Up till now I have thoroughly gone wrong’, he wrote to Sommerfeld in June 1923.23 Consumed by the problem, Pauli later admitted that he was in complete despair for some time.
One day another physicist from the institute met him while strolling around the streets of Copenhagen. ‘You look very unhappy’, said his colleague. Pauli turned on him: ‘How can one look happy when he is thinking about the anomalous Zeeman effect?’24 The use of ad hoc rules to describe the complex structure of atomic spectra was just too much for Pauli. He wanted a deeper, more fundamental explanation of the phenomena. Part of the problem, he believed, was the guesswork involved in Bohr’s theory of the periodic table. Did it really describe the correct arrangement of electrons inside atoms?
By 1922 the electrons in the Bohr-Sommerfeld model were believed to move in three-dimensional ‘shells’. These were not physical shells, but energy levels within atoms around which electrons seemed to cluster. A vital clue in helping Bohr construct this new electron shell model was the stability of the so-called noble gases: helium, neon, argon, krypton, xenon and radon.25 With atomic numbers of 2, 10, 18, 36, 54 and 86, the relatively high energies required to ionise any noble gas atom – to rip away an electron and turn it into a positive ion – together with their reluctance to chemically bond with other atoms to form compounds, suggested that the electron configurations in these atoms were extremely stable and consisted of ‘closed shells’.
The chemical properties of the noble gases were in stark contrast to the elements that preceded them in the periodic table – hydrogen and the halogens: fluorine, chlorine, bromine, iodine, and astatine. With atomic numbers 1, 9, 17, 35, 53 and 85, all of these elements easily formed compounds. Unlike the chemically inert noble gases, hydrogen and the halogens united with other atoms because in the process they picked up another electron and thereby filled the single vacancy in the outermost electron shell. By doing so, the resulting negative ion had a completely full or ‘closed’ set of electron shells and acquired the highly stable electronic configuration of a noble gas atom. Mirroring the halogens, the alkalis group – lithium, sodium, potassium, rubidium, caesium and francium – were quick to lose an electron as they formed compounds and became positive ions with the electron distribution of a noble gas.
The chemical properties of these three groups of elements formed part of the evidence that led Bohr to propose that the atom of each element in a row of the periodic table is built up from the previous element by the addition of another electron to the outer electron shell. Each row would end with a noble gas in which the outer shell was full. Since only electrons outside the closed shells, called valence electrons, took part in chemical reactions, atoms with the same number of valence electrons shared similar chemical properties and occupied the same column in the periodic table. The halogens all have seven electrons in the outermost shell, requiring just one more electron to close it and acquire an electron configuration of a noble gas. The alkalis, on the other hand, all have one valence electron.
It was these ideas that Pauli heard Bohr outline during the Göttingen lectures in June 1922. Sommerfeld had greeted the shell model as ‘the greatest advance in atomic structure since 1913’.26 If he could mathematically reconstruct the numbers 2, 8, 18…of the elements in the rows of the periodic table, then it would be, Sommerfeld told Bohr, ‘the fulfilment of the boldest hopes of physics’.27 In truth, there was no hard mathematical reasoning to back up the new electron shell model. Even Rutherford told Bohr that he was struggling ‘to form an idea of how you arrive at your conclusions’.28 Nevertheless, Bohr’s ideas had to be taken seriously, especially after the announcement in his Nobel lecture in December 1922 that the unknown element with atomic number 72, later called hafnium, did not belong to the ‘rare earth’ group of elements was later confirmed to be correct. However, there was no organising principle or criteria behind Bohr’s shell model. It was an ingenious improvisation based on an array of chemical and physical data that could in large part explain the chemical properties of the various groupings of elements in the periodic table. Its crowning glory was hafnium.
As he continued to fret over the anomalous Zeeman effect and the shortcomings of the electron shell model, Pauli’s time in Copenhagen came to an end. In September 1923 he returned to Hamburg, where the following year he was promoted from assistant to privatdozent. But with Copenhagen a short train journey and a ferry across the Baltic Sea, Pauli was still a regular visitor to the institute. He concluded that Bohr’s model could work only if there was a restriction on how many electrons could occupy any given shell. Otherwise, in contradiction of the results of atomic spectra, there seemed nothing to prevent all the electrons in any atom from occupying the same stationary state, the same energy level. At the end of 1924 Pauli discovered the fundamental organising rule, the ‘exclusion principle’, that provided the theoretical justification that had been missing in Bohr’s empirically devised electron shell atomic model.
Pauli was inspired by the work of a Cambridge postgraduate student. Edmund Stoner, 35, was still working on his doctorate under Rutherford when in October 1924 his paper ‘The Distribution of Electrons Among Atomic Levels’ was published in the Philosophical Magazine. Stoner argued that the outermost or valence electron of an alkali atom has as many energy states to choose from as there are electrons in the last closed shell of the first inert noble gas that follows it in the periodic table. For example, lithium’s valence electron could occupy any one of eight possible energy states, exactly the number of electrons in the corresponding closed shell of the gas neon. Stoner’s idea implied that a given principal quantum number n corresponds to a Bohr electron shell which would be completely full or ‘closed’ when the number of electrons it contains reaches twice its number of possible energy states.
If each electron in an atom is assigned the quantum numbers n, k, m, and each unique set of numbers labels a distinct electron orbit or energy level, then according to Stoner, the number of possible energy states for, say, n=1, 2 and 3 would be 2, 8 and 18. For the first shell n=1, k=1 and m=0. These are the only possible values the three quantum numbers can have and they label the energy state (1,1,0). But according to Stoner, the first shell is closed when it contains 2 electrons, double the number of available energy states. For n=2, either k=1 and m=0 or k=2 and m=–1,0,1. Thus in this second shell there are four possible sets of quantum numbers that can be assigned to the valence electron and the energy states it can occupy: (2,1,0), (2,2,–1), (2,2,0), (2,2,1). Therefore, the shell n=2 can accommodate 8 electrons when it is full. The third shell, n=3, has 9 possible electron energy states: (3,1,0), (3,2,–1), (3,2,0), (3,2,1), (3,3,–2), (3,3,–1), (3,3,0), (3,3,1), (3,3,2).29 Using Stoner’s rule, the n=3 shell can contain a maximum of 18 electrons.
Pauli had seen the October issue of the Philosophical Magazine, but ignored Stoner’s paper. Not known for his athleticism, Pauli ran to the library to read it after Sommerfeld mentioned Stoner’s work in the preface to the fourth edition of his textbook Atomic Structure and Spectral Lines.30 Pauli realised that for a given value of n, the number of available energy states, N, in an atom that an electron could occupy was equivalent to all the possible values that the quantum numbers k and m could take, and was equal to 2n2. Stoner’s rule yielded the correct series of numbers 2, 8, 18, 32…for the elements in the rows of the periodic table. But why was the number of electrons in a closed shell twice the value of N or n2? Pauli came up with the answer – a fourth quantum number h
ad to be assigned to electrons in atoms.
Unlike the other numbers n, k, and m, Pauli’s new number could have only two values, so he called it Zweideutigkeit. It was this ‘two-valuedness’ that doubled the number of electron states. Where there had previously been a single energy state with a unique set of three quantum numbers n, k, and m, there were now two energy states: n, k, m, A and n, k, m, B. These extra states explained the enigmatic splitting of spectral lines of the anomalous Zeeman effect. Then the ‘two-valued’ fourth quantum number led Pauli to the exclusion principle, one of the great commandments of nature: no two electrons in an atom can have the same set of four quantum numbers.
The chemical properties of an element are not determined by the total number of electrons in its atom but only by the distribution of its valence electrons. If all the electrons in an atom occupied the lowest energy level, then all the elements would have the same chemistry.
It was Pauli’s exclusion principle that managed the occupancy of the electron shells in Bohr’s new atomic model and prevented all of them from gathering in the lowest energy level. The exclusion principle provided the underlying explanation for the arrangement of the elements in the periodic table and the closing of shells with chemically inert rare gases. Yet despite these successes, Pauli admitted in his paper, ‘On the Connection between the Closing of Electron Groups in Atoms and the Complex Structure of Spectra’, published on 21 March 1925 in Zeitschrift für Physik: ‘We cannot give a more precise reason for this rule.’31
Why four quantum numbers, and not three, were needed to specify the position of electrons in an atom was a mystery. It had been accepted since the seminal work of Bohr and Sommerfeld that an atomic electron in orbital motion around a nucleus moves in three dimensions and therefore requires three quantum numbers for its description. What was the physical basis of Pauli’s fourth quantum number?