The pesky anomalous Zeeman effect
IN THE AUTUMN of 1923 Pauli left Copenhagen to go back to Hamburg. He had still made no progress with the anomalous Zeeman effect. The problem, to recap, was to find the correct equation to describe the spectral lines of an atom placed in a weak magnetic field. He worried at it like a dog with a bone. But no matter how hard he tried, he just couldn’t crack it.
Pauli was growing more and more despondent. He had become friendly with Bohr’s assistant, Hendrik Kramers, who promised to visit him in Hamburg to cheer him up. Then Bohr decided that Kramers must go to England with him. Pauli wrote telling Bohr how deeply offended he was by this decision and how much he had looked forward to seeing his friend, whose presence “would mean a great deal for me psychically.” “I feel myself so unwell,” he added. He had just returned and already he was writing to Bohr about how unhappy he was. His letter was in effect a cry for help.
Not long after he had settled back in Hamburg he gave his inaugural lecture. His subject was the periodic table of chemical elements, but his heart was not in it. He was all too aware that the most basic problem in understanding it had yet to be resolved: what was the reason that the shells of electrons in each atom filled in the way they did? He had a hunch that it was related in some way to the multiplets in the anomalous Zeeman effect. Surely it was all tied together. After all, the way in which the shells filled up with electrons determined the numbers of spectral lines.
In struggling to find a mathematical description for the effect, he began by reworking equations from the normal Zeeman effect—where Bohr’s theory produced equations that agreed reasonably closely with the spectral lines that had been observed. Pauli’s goal was to apply the new equations to the anomalous Zeeman effect. Sommerfeld had already made some progress along these lines.
To study the anomalous Zeeman effect, physicists focused on alkali atoms—primarily sodium, potassium, and cesium—which displayed behavior similar to the hydrogen atom for which Bohr’s theory seemed to work. Like the hydrogen atom, alkali atoms have only one electron in their outer shell and this is the only electron that can bond with other chemical elements. The other electrons are in the inner shells, which have already been filled and thus cannot react.
Sommerfeld set Heisenberg, who was just nineteen, to work on the problem. Since the Bohr theory now dealt with an electron in an atom moving on a three-dimensional shell, in addition to a principal quantum number each electron had two more associated with it. Locating an object in a room requires three numbers, two to give its location from the walls and the third its height from the floor. An electron can be located in an atom in a similar way, with three numbers identifying its position within the Bohr atom relative to the nucleus. These are taken to be whole numbers and are called quantum numbers.
Sommerfeld supplied Heisenberg with the newest data as well as his own unpublished research, including speculation on new ways to combine the three quantum numbers at the very basis of Bohr’s theory of the atom. “All right, you have an interest in mathematics; it may be that you know something; it may be that you know nothing…. We will see what you can do,” he said. Heisenberg quickly came up with his own ideas on how to tackle the anomalous Zeeman effect. He rewrote one of Sommerfeld’s equations using half figures—1/2, 3/2, and so on—and discovered he could produce an equation that described most of the observed multiplets.
Then he turned to Bohr’s model of the atom—with the nucleus as a rigid core surrounded by filled shells of electrons, the whole thing spinning like a ball. Heisenberg made the audacious assumption that the core and the surrounding electron shared a half unit of angular momentum by means of an interaction that he left unspecified. (An object moving in a line has linear momentum [mass times velocity]. Similarly an object spinning like a top has angular momentum [which is related to mass times angular velocity].) The mysterious interaction between the core and the lone electron could be the explanation for the anomalous Zeeman effect. Sommerfeld was stunned, as was Pauli. Surely this would result in an atom emitting a half quanta of energy. But that had to be wrong because quanta were assumed to be indivisible. This was a basic postulate of the quantum theory. All the same, Heisenberg’s equation produced multiplets for the alkalis which precisely duplicated data from experiments. “Success sanctifies the means,” Heisenberg wrote to Pauli.
Sommerfeld was astonished that this novice dared take such a dramatically different approach to problems with which experienced scientists had struggled. Instead of getting tied up in endless complicated calculations, Heisenberg came up with instant solutions. Eventually Sommerfeld had to give in and accepted that there had to be half quantum numbers. After all, he reasoned, classical physics was frequently proved wrong. Why not atomic physics, too?
Bohr, however, insisted that while breakdowns in classical physics were fine, it was not acceptable when it came to his own theory of the atom. At the Bohr-Festspiele in Göttingen, he had discussed Heisenberg’s new approach and referred to it as “very interesting,” by which he meant that it was almost certainly wrong. Although it happened to fit existing data, Bohr argued, it was not an end in itself. Bohr was more interested in unraveling a problem than in instant solutions.
Bohr now suggested that there might be a force that linked the core and the lone outer electron in an alkali atom and that this force might distort the core in two different ways, giving rise to a “double-valuedness,” which he, too, was willing to include as a half quantum number. Thus Bohr was able to reproduce the required multiplets, while avoiding the other half quantum numbers that were essential to Heisenberg’s model. But what was this strange force? Pauli couldn’t accept it and argued tooth and nail with Bohr. He continued to torture himself over the problem of the anomalous Zeeman effect but could make no sense of it. Bohr insisted that Pauli publish his own contribution to these mathematical models and he did so “with a tear in my eye,” as he wrote to Sommerfeld. As for Heisenberg’s theory of the anomalous Zeeman effect, Pauli found it “unsightly” and “monstrous.” “I am deeply insulted by it,” he wrote to Bohr.
Ten days later Pauli wrote to Bohr again, offering his own deeply critical assessment of the situation: “The atomic physicists in Germany can now be divided into two classes. Some work out a given problem first with half quantum numbers, and if it doesn’t agree with experiment, they do it again with integral ones. The others calculate first with integral values, and if it doesn’t work, do it again with halves.” In other words, they had all been reduced to desperate measures.
As far as he was concerned the problem of the anomalous Zeeman effect was far from solved. He was becoming convinced that “there is no [satisfactory] model for the anomalous Zeeman effect and that we have to create something fundamentally new.”
But he had no idea what this might be. The whole farrago was getting him down. “I myself have no taste at all for this sort of theoretical physics,” he wrote to Bohr, and wanted to withdraw from it. Atomic physics had all become “too difficult.”
Dr. Jekyll and Mr. Hyde
Physics was Pauli’s heart and soul. His physics research gave definition to his life and his fruitless attempts to solve the anomalous Zeeman effect, on top of what he regarded as his lack of success with the hydrogen-molecule ion and helium atom, began to take a heavy toll on his already fragile psyche. His early successes—his maiden papers on relativity theory—suddenly seemed in the distant past. He began drinking more and more heavily. “I have noticed that wine agrees very well with me,” he wrote to a friend. “After the second bottle of wine or champagne I usually adopt the manners of a good companion (which I never have in the sober state) and then may under these circumstances enormously impress the surroundings, particularly if they are women.”
By day he behaved like a staid Germanic professor. By night he roamed the Sankt Pauli, Hamburg’s notorious red-light district full of risqué cabarets and bars catering largely to a rough clientele. He described his life to a friend: “During the day, calming work, in the night, sexual excitement in the underworld—without feeling, without love, indeed without humanity.” Much later, writing to Jung, he recalled “the complete split between my day life and my night life in my relations with women.” He seemed to have split into Dr. Jekyll and Mr. Hyde. Robert Louis Stevenson’s celebrated novel had been published some forty years earlier. No doubt Pauli had read the story of the scientist who is taken over and destroyed by his darker impulses. Perhaps he saw a parallel between Dr. Jekyll and his own increasingly erratic behavior.
Hamburg was a vibrant city that welcomed all comers and in which one could savor the steamy side of postwar Germany. Munich banned the American cabaret performer Josephine Baker, who was famous for her nude dancing; Hamburg welcomed her with open arms.
The real action was on the side streets off the main Sankt Pauli avenue, particularly on a street called Grosse Freiheit. Even during the day it was difficult to see inside the bars there. The odor of spilled beer and the sticky unmopped floors made the interiors stifling. When Pauli walked in in his fine suit and went to the bar, no doubt in the early days at least conversation would grind to a halt and everyone would stare until he had finished his drink and left. But he soon became a regular. To make things worse, the more he drank, the more obnoxious he became.
Often he ended up getting beaten in a brawl. Once he was eating in one of his favorite restaurants in the area. A row broke out and Pauli found himself right in the middle of it. He only pulled himself together when someone threatened to throw him out of a second-floor window. Afterward, he said, he could not understand how he had gotten into such a situation.
He began to feel as if he were losing control. He was frightened of the person he was becoming. “[I] tended toward being a criminal, a thug (which could have degenerated into my becoming a murderer),” he later recalled. By day, immersed in his research, he felt “detached from the world—a totally unintellectual hermit with outbursts of ecstasy and visions.” His two parallel worlds were in danger of colliding with potentially fatal effect.
The women Pauli found in the bars there offered a way to forget his growing frustration and anger. Typical of his Sankt Pauli girlfriends was a beautiful blond woman some two years younger than he. They had a short and passionate affair that Pauli broke off when he discovered she was a morphine addict. Then one day she turned up at his office at the university. Somehow she had found him, despite his secrecy and desperate attempts to keep his night and day lives separate. Pauli was horrified. Poor, sick, and stick thin from her continuing morphine abuse, she stood like a specter, begging him for help. Pauli threw her out, and told her never to come back—and she disappeared back into the Sankt Pauli. He forgot about her, hoping she was gone forever. Little did he guess that in later years she would come back to haunt him.
Pauli always kept his visits to the Sankt Pauli secret, even from his closest friends and colleagues. These included the always upbeat Otto Stern, Emil Artin, Walter Baade, and Gregor Wentzel. Wilhelm Lenz, director of the Institute for Theoretical Physics, joined them from time to time, particularly for departmental lunches, which were always held in top restaurants scouted out by Stern. Like Pauli they were all bachelors.
Lenz was a man of some means who lived in a fashionable area of Hamburg at 18 Armgartstrasse, on a beautiful canal with grassy banks and the city’s largest lake, the Aussenalster, glittering in the distance. When Pauli first arrived Lenz offered him a room in his house. Pauli later moved around the corner to 16 Papenhude, where he had an apartment on the second floor. Miraculously the area escaped damage in World War II and remains today much as it was then. Lenz was noted for his reserve.
Otto Stern, an unusually gifted experimental physicist, was another recent addition to Hamburg. Like Lenz, Stern was a rather wealthy man. But unlike Lenz he was outgoing—a bon vivant who sometimes flew to Vienna just for lunch. Artin was a mathematician who specialized in number theory, Baade an astronomer, and Wentzel a physicist. These last three were Pauli’s exact contemporaries.
Wentzel was Pauli’s closest friend. Not only did their research interests overlap but so did their idea of a good time. Wentzel frequently went to Paris on the slightest pretext. On one occasion he sent Pauli two of his papers to comment on and signed the letter giving his address simply as “Paris.” Pauli swiftly replied, “The question is this, whether indicating Paris at the end of your work suffices at least to justify all this psychologically and whether in the corrections you should not change it more specifically into Paris, Moulin-Rouge, or something analogous.”
Pauli enjoyed visiting Baade and the astronomers at their observatory in Bergedorf. On full-moon nights it was impossible to observe the stars and they would have a party instead. On one occasion Pauli was present at the observatory when it was discovered that a terrible accident had befallen the great refractor telescope. It was almost destroyed. Naturally everyone chalked it up to the Pauli effect.
Cases of the dreaded Pauli effect were beginning to pile up. Physicists at the university became convinced that Pauli’s presence in or even near a laboratory led to severe breakdowns in the equipment. Stern was reduced to desperate measures. He recalled that the only way he could protect his laboratory from the Pauli effect was that Pauli “was not allowed to enter.” The Hamburg scientists were surprisingly superstitious. One brought a flower and gave it to his apparatus every day. Stern kept a hammer lying next to his as a veiled threat to it not to break down. Pauli himself fervently believed in the Pauli effect and began to wonder whether he emanated powers.
Pauli’s exclusion principle: Four quantum numbers instead of three
Pauli had given up trying to solve the anomalous Zeeman effect, but he kept up with the flood of papers that poured out on the subject. Then in autumn 1924 two ideas suddenly came together for him.
The first was this: Suppose relativity had something to contribute on the subject. Pauli looked into it. He found that in the models of the atom proposed by Bohr and Heisenberg, electrons within the core moved at speeds comparable to that of light. They should therefore be expected to display variations in mass consistent with Einstein’s equation E = mc2. These variations should show up in the spacing between the multiplets, but experiments had not revealed any such effect and could mean only one thing: the core in these models had to be inert; it did not interact and so played no role at all. In other words, every model of the atom that featured a core was wrong.
Then he came across a paper by Edmund Stoner, a twenty-five-year-old physicist at Leeds University. Stoner went far beyond the anomalous Zeeman effect, although he himself had not realized the full significance of what he had found. It had to do with the problem constantly on Pauli’s mind: What stopped every electron in an atom from falling into the atom’s lowest energy level—its ground state?
By clever manipulation of the three quantum numbers for an electron in an atom, Stoner had succeeded in calculating the total number of multiplets of an alkali atom undergoing the anomalous Zeeman effect (that is, when it is placed within a weak magnetic field). He did this, as Heisenberg and Bohr had, by imagining the alkali atom to be made up of a closed core—made of shells filled to their maximum with electrons and so inactive chemically—with a single lone electron revolving around it. From this he was able to show that the total number of electrons in each closed shell was related to twice the total amount of angular momentum of the closed core with the lone electron.
What struck Pauli was the appearance of the number two. Bohr had inserted this number into his model of the core simply so that only halves would appear in formulas for the anomalous Zeeman effect. In other words, when the atom was in a magnetic field, the core containing the closed shells full of electrons could be distorted in two ways, which would give one of its quantum numbers a value of plus or minus a half.
But Pauli had established that the core was inert and that only the lone electron played any role in the chemical activity of an alkali atom. So why not transfer the two possible values of the core to this electron? Pauli began to suspect that Stoner’s work contained the seeds of something new and exciting. He decided to see what would happen if he extended Stoner’s method of manipulating quantum numbers to include a fourth quantum number that had the values of plus and minus a half for the lone electron. The result was astounding. He figured out that the total number of electrons in each closed shell was twice the principal quantum number of that shell squared. It was 2n2, the same number that Bohr had proposed with no basis from his theory of the atom. Now there was one.
Pauli went yet further, proposing that the two possible values for the fourth quantum number be assigned to every electron in every atom, regardless of whether the atom was in a magnetic field.
The conclusion had to be that each electron in an atom required four not three quantum numbers, and, to explain the periodic table of chemical elements, that no two electrons in an atom could have the same four quantum numbers. Basically, two electrons with the same quantum numbers cannot occupy the same shell. (This is Pauli’s famous exclusion principle. The name was given it by Paul Dirac, a physicist at Cambridge University.) This was the reason why Bohr’s building-up principle for atoms worked—why there are precisely two electrons in the inner shell, eight in the next, then eighteen, and so on. This was also why every electron in an atom did not fall into its lowest stationary state. They were prohibited from doing so.
To get a grip on this complicated concept, imagine that an atom is an apartment building with many rooms on many different floors and no elevators. In fact, it is an upside-down pyramid with two rooms at the bottom, eight on the second floor, eighteen on the third, and so on. To avoid overcrowding, the local housing authority passes a law that only one electron can occupy a room. A crowd of electrons enters the building and jostles around, trying to occupy as low a floor as possible. No electron wants to be the only one on a floor—the lone electron, as in an alkali atom. Such an electron cannot relax because on its shoulders rests the chemical activity for the entire atom. This is a very simplified description of the way that Pauli’s exclusion principle works.
Pauli’s paper on the exclusion principle contained none of the mathematical fireworks for which he had become famous. Rather, it was the fruit of his patient examination of data. By searching out patterns among numbers, he came up with what scientists call a restrictive or prohibitive principle. Another example is the principle of relativity, which asserts that the laws of physics must be the same in every laboratory, regardless of its motion. There is no reason for this to be so. Yet it must be, to formulate a systematic theory of how objects from the size of basketballs to planets move. It also enables scientists to predict numerous phenomena, such as the bending of starlight by massive objects, a prediction of relativity theory that was later proved to be true in real life. There is no way of deriving the principle of relativity mathematically. It is simply an axiom.
But what about the exclusion principle? Could it be derived? Pauli was not sure, nor was anyone else.
Hard though it was to understand its deeper meaning, scientists quickly realized the exclusion principle’s importance in explaining the periodic table of chemical elements and thus, also, atomic structure. It also helped clarify why metals are hard and what the fate of stars might be. Pauli had made a discovery that would shape the path of physics in the future and change our understanding of the cosmos.
The search for the meaning of the exclusion principle
Pauli immediately notified Bohr and Heisenberg of his discovery and sent them the draft manuscript of his paper. It was, he wrote them firmly, at the very least “not a bigger nonsense” than the schemes other scientists proposed for understanding the structure of the atom. At least Pauli had avoided hypotheses with no basis such as Bohr’s force of unknown origin, which distorted a core in two different ways. He suspected that his exclusion principle could not be derived from Bohr’s theory of the atom. Understanding it lay rather, he suggested, in the as-yet-unknown properties of “motion and force in quantum theory.” Remembering his many attempts to support Bohr’s theory—the hydrogen-molecule ion, the helium atom, and the core models of atomic structure—all of which ended in failure, he wrote that he would prefer to interpret the exclusion principle free of any model of the atom, especially a model containing the concept of electron orbits. He was sure that the key factors in describing the characteristics of an electron had to be its energy and momentum. Those were real because they were measurable; electron orbits and shells were not. In this Pauli was true to his godfather Ernst Mach’s philosophy—to avoid any unmeasurable concepts in a theory of physics, for those were purely metaphysical.
Heisenberg and Bohr were amused by Pauli’s exclusion principle. Here was proof, Heisenberg wrote to Pauli, that Pauli was entering the “land of the formalist philistines,” practicing a style of physics “of which you had insulted me. [In fact, you] had broken all hitherto existing records [in rising] to an unimagined, giddy height (by introducing individual electrons with 4 degrees of freedom).” Everyone knew that electrons had to move in three-dimensional space like everything else in the universe, and therefore three quantum numbers should surely suffice. Pauli had frequently accused Bohr and Heisenberg of coming up with “swindles.” Now Heisenberg accused Pauli of coming up with a swindle of his own; “swindle x swindle does not yield something correct,” he wrote.
A week later, Bohr had clearly thought more seriously about Pauli’s new theory. He wrote to him, “I have the impression that we stand at a decisive turning point, now that the extent of the whole swindle has been so exhaustively characterized.” What struck him about Pauli’s proposal, he said, was its “complete insanity.” Bohr always condemned new proposals with the words “interesting but not crazy enough.” Saying that Pauli’s was completely insane meant he thought it was most probably right.
Pauli had still not solved the anomalous Zeeman effect, but he had accomplished something far more important. He suspected that the full significance of the exclusion principle would not become clear until there was a deeper understanding of quantum theory. “I will wait patiently; and be satisfied if I live to see the solution,” he wrote to Bohr. He hoped his “insane idea” would help toward understanding the structure of atoms made up of many electrons. If it did, “I would be the happiest man on earth.”
The fourth quantum number
The problem people had with understanding the exclusion principle was the lack of a visual model for the fourth quantum number. Pauli was well aware that it was essential for physicists to be able to visualize a theory—which was what made Bohr’s image of an atom as a miniscule solar system so pleasing. Nevertheless, he wrote to Bohr, although this need for visual images was “in part legitimate and healthy, it should never count as an argument for retaining systems of concepts. Once the systems of concepts are settled, then will visualizability be regained.” In effect, Pauli was suggesting strongly to Bohr that he should drop all visual images from his theory because they had proved to be incorrect and misleading. It was only once a new theory of atomic physics had emerged that it would be possible to develop a visual language to describe the atomic world.
Then Ralph Kronig, a twenty-year-old German American, noticed something Pauli had overlooked. The fourth quantum number of an electron has the mathematical properties of angular momentum, the momentum of an object moving in a circle. Every electron has an angular momentum from its orbit around the nucleus of the atom, like the earth revolving around the sun. Perhaps, thought Kronig, each electron also has an angular momentum of its own, like the earth spinning on its axis.
But whereas the earth’s spin is variable, the electron’s always remains the same. Kronig gave electron “spin” a value of a half, using the units of angular momentum. To explain spectral lines physicists had always assumed that the electron acted like a tiny magnet that could align itself in a magnetic field. Pauli’s and Kronig’s discovery had transformed it into a spinning magnet that could align itself along a magnetic field in one of two directions, depending on whether it had a spin of plus or minus a half. These were precisely the two values that Pauli had transferred from the inert closed core to the lone electron in the outermost shell of an alkali atom. Thus spin was recognized as a distinguishing feature of an electron. Every electron has its own spin, just as each of us has a nose, eyes, and lips that distinguish us from one another. Spin is an intrinsic property of an electron, and no matter where the electron is located it has a spin of a half.
Initially Pauli dismissed Kronig’s proposal, saying merely that it was “indeed a witty idea.” For imagining an electron as a spinning top, as Pauli and everyone else did, led to a serious conflict with relativity theory. It meant that a point on the surface of the electron might move with the velocity of light, which according to relativity theory is impossible. When Kronig visited Copenhagen, Bohr dismissed his proposal with the words “very interesting” Kronig dropped the idea.
Then, nine months later, two Dutch physicists, George Uhlenbeck and Samuel Goudsmit, rediscovered spin and staked their claim in print, warning that one should not visualize the electron as a spinning top. Pauli was deeply embarrassed at having discouraged Kronig from publishing his idea and thereafter always spoke highly of him.
Spin was undeniably a property of an electron but it was entirely impossible to visualize it in a way consistent with relativity theory. Scientists had to accept that the fourth quantum number had no accompanying visual image. It was time for atomic physics to move on from trying to visualize everything in images relating to the world in which we live.