4.a A tale from the atomic nucleus, ca. 1930s–1950s

Historical work on early nuclear models has drawn attention to the unexpected role played by geochemistry, earth sciences, and cosmochemistry (see Johnson 2004; Kragh 2000). I draw on this work, retrace some of these surprising connections, and look into new ones too. These multi-disciplinary contributions should not be consigned to the infancy of nuclear physics. I see them as pivotal to its successful historical development in the 1930s–1950s. They furnish a perfect example of a plurality of scientific perspectives in dialogue, and of perspectival modelling in action.

What chemical elements compose the earth’s crust and atmosphere? Why are some but not others abundant? How did chemical elements form in the universe? Which new chemical elements could be found in nature or created in a lab? The story of the atomic nucleus begins with these questions. And answers to them around the 1900s–1930s did not necessarily come from the burgeoning quantum theory of Max Planck and Bohr’s model of the atom, but instead from the daily work of petrologists, mineralogists, meteorologists, and geochemists.

The Italian region from Naples to north of Rome (near Lake Bolsena), with its volcanic origins and active Phlegraean Fields, has always been of great interest for volcanologists, geologists, mineralogists, and petrologists. In 1894, a young American petrologist, Henry Stephens Washington¹—just graduated from the University of Leipzig—embarked on a journey in the area to study the mineral composition of volcanic rocks. Samples were carefully labelled according to their geographical origin and classified by chemical composition. Igneous rocks were divided into class, order, range, and subrange, following what became known as the ‘Cross, Iddings, Pirsson and Washington (CIPW) norm classification’ (see Cross et al. 1902) marking the beginning of normative mineralogy. The idea behind the CIPW norm was to offer chemical analysis of the elements and their estimated percentages in rocks that had formed via complex geochemical processes.

The outcome of long years of petrological fieldwork in Central Italy was a volume that Washington published in 1906, The Roman Comagnatic Region. The work established a common origin for the volcanoes of the area and defined the region as ‘potassic’ owing to the high percentage of potash. Percentages for silica, aluminia, lime, and others were carefully calculated. Via petrological studies of rocks, the relative abundances of chemical elements in the earth’s crust entered the public domain. But it was not just the study of the earth’s crust and volcanism that provided data about elements and their abundances.

Data about the percentages of chemical elements in the atmosphere increased at the start of the twentieth century. In 1895, William Ramsay had discovered helium in a radioactive mineral called cleveite (Ramsay 1895). Samples of cleveite placed in an exhausted glass flask with boiling acid yielded a gas whose spectrum had a bright yellow line. The gas was eventually named ‘helium’ after Lockyer’s first identification of it in the spectrum of the sun’s (‘helios’ in Greek) chromosphere a few years earlier. This marked the beginning of a long series of experiments extracting gases from radioactive substances.

Rutherford, Barnes, and Soddy ran experiments to extract inert gases like helium and argon from radium (see Ramsay 1904) and thorium. By 1908, Ramsay had published estimates of percentages of krypton and xenon in the atmosphere (Ramsay 1908). Over the following two decades, percentages of chemical elements present in the earth’s atmosphere became of increasing interest for meteorologists all over the world to better understand the troposphere and the stratosphere. Data about percentages of oxygen, hydrogen, helium, neon, krypton, and argon came in from Bavaria, Paris, and Moscow using balloon flights able to go as high as 8–9 km in the atmosphere. Refinements of these percentages continued into the late 1930s to eliminate errors due to contamination from the balloon gas. By 1937, the Austrian-British chemist Friedrich Paneth announced that the percentage composition of the air was independent of height throughout the whole troposphere and in the first kilometres of the stratosphere (Paneth 1937).

These data about percentages of chemical elements in the earth’s crust and atmosphere raised new questions. Was any such pattern specific to the terrestrial distribution? Or was it identifiable also in outer space and in meteorites? Between 1915 and 1921, the American chemist William Draper Harkins concluded that elements with even atomic number Z were predominant. He concluded that 89% of atoms on the surface of the earth and 98% in meteorites had an even atomic number (Harkins 1921), with even-Z atoms having on average twice as many isotopic varieties as odd-Z atoms.

But these were figures from massive aggregates of atoms and molecules. What about a more precise estimate of the percentages of individual atomic species with their isotopic varieties? How many isotopes are there for each chemical element? Answering this question requires going beyond geochemistry and studying the atomic mass of individual atoms. A new instrument was necessary: the mass spectrograph.

Working on the same principles as cathode rays (ionization at low pressure in a strong field), the mass spectrograph earned Francis Aston the Nobel Prize for chemistry in 1922 (Aston 1922/1966). By ionizing a sample of a chemical element and using a strong magnetic field to deflect the ions, Aston was able to measure individual isotopic varieties in samples of non-radioactive elements. This reopened the search for new patterns.

In a 1924 paper, Francis Aston referred to the work of Henry S. Washington on composition of igneous rocks and Ramsay on the atmosphere, and plotted the relative abundances of atomic species for the first 39 elements of the periodic table (Aston 1924). The table showed on the x-axis the so-called mass number (or atomic weight) of each atom (which Aston identified with protons since the neutron had not yet been discovered) and on the y-axis the logarithm to base 10 of the total number of gram-atoms on earth.

The task was nothing less than trying to identify ‘the relative stability of nuclei during the evolution of the atoms’, assuming ‘a lithosphere of mass 5.98 × 10²⁷ gm having the average composition of the igneous rocks, a hydrosphere of mass 1.45 × 10²⁴ gm of water and an atmosphere of mass 5.29 × 10²¹ gm of ordinary air’ (Aston 1924, p. 394). The graph did not reveal any regular pattern but a stark abundance of elements of even-atomic number Z, and peaks around oxygen (O) with Z = 8, silicon (Si) with Z = 14, calcium (Ca) with Z = 20, and iron (Fe) with Z = 26.

That tables like Aston’s could provide evidence for the stability of the inner nucleus—even in the absence of an explanation—became key to an entire programme of cosmochemistry that flourished between 1926 and 1937 thanks to the work of Victor Moritz Goldschmidt, among others.² Like Washington before him, Goldschmidt was not a nuclear physicist. He was the Director of the Mineralogical Institute of the University of Oslo and Head of the Mineralogical Institute in Göttingen from 1929 before returning to Norway and eventually having to flee to Sweden during World War II. Mineralogy was a thriving field in Norway. As Chairman of the Norway Government Commission for Raw Materials (Rosbaud 1961), Goldschmidt had an important task, for establishing the relative abundances of elements such as nickel (Z = 28) in rocks had far-reaching economical-industrial consequences for the production of alloys and for minting coins.

But there was a more far-reaching interest as well. What chemical elements are most concentrated in the silicate crust of the earth? And what can the relative abundances of the same elements inside meteorites tell us about the origins of chemical elements in the primordial universe? To answer these questions, X-ray crystallography offered a powerful new instrument, for it allowed the gathering of data about the crystallization of molten rocks and silicate melts and which atoms and ions might have escaped the process. Goldschmidt saw the basic problem of geochemistry as that of determining ‘the quantitative chemical composition of the earth and to find the laws which underlie the frequency and distribution of the various elements in nature’ (Rosbaud 1961, p. 361). When Goldschmidt plotted the data about relative abundances, now against neutron number N rather than the proton number Z, a regular pattern began to emerge around N = 2, 8, 20, 50, 82, 126 (see Figure 4.a.1).

‘Much of what we know today about the origin of the elements has been derived from chemical analysis of meteorites performed by Goldschmidt and his students’ (as summarized in Figure 4.a.1), writes physical chemist Hans E. Suess (1988, p. 385). While the considerations behind Goldschmidt’s analysis—like Washington’s and Ramsay’s before him and Suess’s after him—were mostly geochemical and cosmochemical, the numbers so identified had already attracted attention elsewhere, among different epistemic communities.

From the atomic theory point of view, analogies between the structure of nucleons and that of electrons had been explored since the discovery of the Pauli exclusion principle in 1924. In 1930–1932, James Bartlett (1932) had suggested that nucleons might be arranged a bit like electrons in s, p, and d shells (or orbitals) with closed shells of 2, 8, and 18 nucleons. Some evidence for these speculations came from light nuclei with mass number up to 36, but the data were lacking for heavier nuclei and the nature of the nuclear force binding protons and neutrons remained to be explored.

In 1933–1934, Walter Elsasser published two papers (Elsasser 1933, 1934; for a discussion, see Mladjenovic 1998, pp. 287–305). Using Aston’s data for light nuclei and invoking Pauli’s principle, Elsasser assumed that protons and neutrons moved independently as particles in a field. Nucleons would occupy independent closed shells, whose energy levels were defined by a set of quantum numbers including azimuthal quantum number l and spin quantum number s. Pauli’s principle dictated that there could not be more than 2(2l + 1) nucleons for each shell and hence 2, 6, 10 . . . nucleons; but Bartlett’s data for light nuclei suggested a higher number of nucleons (2, 8, 18 . . . ) per closed shell.³ Moreover, the nature of the potential in which the allegedly independent nucleons were moving proved a stumbling block because it was not a Coulomb potential (see Johnson 1992, pp. 166–167).

A separate line of inquiry—this time from physical chemistry—offered some pointers for shell closure in heavier nuclei. Plotting the number of isotopes (elements with the same atomic number Z) over the number of isotones (elements with the same neutron number N), in 1934 Karl Guggenheimer found evidence of discontinuity in binding energy around N = 50 and N = 82. A cluster of 6 isotones and 11 isotopes were identified around number 50 and 7 isotones and 9 isotopes around number 82: this was the first evidence of abundances in heavier nuclei suggesting a possible shell closure around numbers 50 and 82 (Guggenheimer 1934; Mladjenovic 1998, p. 289).

But it took 13 years for the nuclear physics community to realize that there was an important phenomenon in those plots and numbers: namely that abundances of heavier chemical elements were clustered around particular neutron numbers N and proton numbers Z such as 20, 28, 50, 82, and that those numbers were special (ausgezeichneten Zahlen, as Hans Suess called them) or ‘magic numbers’, as they became known. It was Hans Suess, with the help of Otto Haxel, who in 1947 realized the importance of these special numbers as key to the atomic structure (Suess 1947a, 1947b).

The story goes that Otto Haxel had discussed the matter with nuclear theorist Hans Jensen in Hannover and

Both Haxel and Suess tried to convince Jensen that their ‘special numbers’ were some sort of key to nuclear structure, but he seemed reluctant to pursue the issue. He saw absolutely no theoretical way to account for the regularities. Then, in August 1948, Maria Goeppert Mayer’s paper appeared in The Physical Review, setting out extensive evidence for the same numbers that Haxel and Suess were so excited about, and referring to those numbers as ‘magic numbers’. (Johnson 2004, pp. 303–304)

Hans Jensen (1965) recalls how he came across the work of Maria Goeppert Mayer while visiting Bohr in Copenhagen and how it was Bohr’s interest that encouraged him to pursue the matter further. But what was so special about those ‘magic numbers’ that Maria Goeppert Mayer saw in 1948? The answer is revealing both about the way scientific perspectives intersect, and about the importance of a plurality of perspectival models.

To recap the early part of the story, data from meteorites, ores, silicate melts, and so forth, provided evidence for the relative abundances of some chemical elements. Data plots like Bartlett’s, Aston’s, Goldschmidt’s, or Suess’s gradually revealed the abundance of some nuclides in the earth’s crust and in the universe.

Such plots provided evidence for the phenomenon of ‘nuclear stability’. The inference from data about cosmic abundances to the phenomenon of nuclear stability was perspectival in distinctive ways. It required a number of experimental, theoretical, and technological resources spread out across a number of scientific communities. For example, while Aston’s mass spectrograph was central to the work of atomic theorists like Elsasser, X-ray crystallography was pivotal to Goldschmidt’s mineralogical research building on Washington’s earlier petrological work.

This plurality of scientific perspectives allowed reliable scientific knowledge claims about the percentages of particular elements in rocks, alloys in meteorites, and gases in the atmosphere, among others. The methodological-epistemic principles that justified the reliability of the claims were themselves perspectival, ranging from principles in normative petrology to those of crystal structure and nuclear physics.

This plurality of ‘intersecting scientific perspectives’ made it possible to establish that there was a modally robust phenomenon about the stability of some nuclides with special or magic numbers. This phenomenon could be teased out from Goldschmidt’s data about cosmic abundances as well as from Suess’s later data. Chemical considerations about the periodicity of chemical elements (like inert gases) and analogy with Pauli’s principle for electron shells were important too. And slow neutron-capture data, as we shall see later, played a key role in the rest of this story.

The fact that it took a decade to realize that these data were evidence for nuclear stability is revealing of the role and importance of perspectival modelling. Data D by themselves are not enough to draw conclusions about any specific phenomenon P. Being able to reliably infer that P from D required a plurality of scientific models. What was needed in particular was the shell model of the nucleus introduced independently by Jensen and Goeppert Mayer in 1948. They succeeded where Elsasser’s and Guggenheimer’s shell models had failed in explaining the interactions among nucleons inside the atomic nucleus and shedding light on the ‘magic numbers’ associated with the abundant nuclides.

The main stumbling block for the emergence of the shell model was the popularity of another model of the nucleus: the liquid drop model. Bohr’s ‘papal blessing’ was required for Hans Jensen to take up the modelling challenge and look more closely into Suess and Haxel’s data. In Maria Goeppert Mayer’s case, her experimental training allowed her to see something in the shell model that others had not.

But another (now forgotten) model, the so-called odd-particle model, explored by Theodore Schmidt and Katherine Way in the late 1930s, was also important in bridging the gap between the fashionable liquid drop model and the shell model. In the next section, I take a look at these three models for the nucleus and the reasons why they are good examples of what I call perspectival models.

In the early 1910s, the alpha-particle experiments led by Ernest Rutherford at the University of Manchester provided evidence that almost the entire mass of the atom was compressed within a tiny core, the nucleus, contradicting J.J. Thomson’s earlier ‘plum pudding model’. Yet these experiments did not give any conclusive answer about the nature of the nucleus, its constituents, or the force binding them. Indeed, in the late 1920s, Erwin Schrödinger was still cautioning that: ‘Just because you see alpha particles coming out of the nucleus, you should not necessarily conclude that they exist inside it in the same form!’ (quoted from Jensen 1965, p. 1420).

Roger Stuewer (1994) has reconstructed the development of the liquid drop model, starting with George Gamow’s model first presented at the Royal Society in 1929. Well before the emergence of Planck’s quantum hypothesis, a liquid drop model was originally applied to the study of electrically charged liquid droplets (Rayleigh 1879). A half-century later, Gamow proposed it could also explain the nuclear binding energy.

Taking a cue from Rutherford’s alpha-particle scattering experiments and speculations that those particles must have been inside the nucleus with protons and electrons, Gamow came to conceive of the nucleus as a ‘small drop of water in which the particles are held together by surface tension’ (see Rutherford et al. 1929, p. 386). The main evidence for this model was once again found in the data from Aston’s mass spectrograph.

As Aston himself explained at the same 1929 Royal Society meeting, by measuring accurately the atomic mass number for various elements and plotting against it the ‘percentage excess or defect from a whole number on the oxygen scale’—known as the ‘packing fraction’ (Aston in Rutherford et al. 1929, p. 383)—a curve was found (known as the mass defect curve for it highlighted the discrepancy between the total mass of an atom and the sum of the masses of its alleged constituents). Gamow was able to derive a formula for the total energy of nuclei based on number of alpha particles they presumably contained. But his efforts came to a halt because of poor agreement with Aston’s mass defect curve.

The 1932 discovery of the neutron by Chadwick as detailed in Heisenberg’s (1934) paper for the Solvay Congress laid the foundations for a model of the nucleus as consisting of neutrons and protons. One of the main theoretical problems was to understand the nature of the nuclear binding force as a function of number of nucleons, or atomic mass A. In 1935, Heisenberg’s student Carl Friedrich von Weizsäcker gave a semi-empirical formula for the nuclear binding energy of the liquid drop model, which included a series of terms (volume or total number of particles, surface energy proportional to the surface area of the nucleus, Coulomb repulsive energy acting among protons, among others) showing that for nuclei with Z less than or equal to 20, the greatest stability for any given atomic mass occurred when number of neutrons N equals number of protons Z (see Portides 2011 for a philosophical discussion). For each Z, there was a range of N for which the nucleus was stable. Nuclei outside that range were subject to beta-decay. Understanding nuclear stability versus beta-decay required making a series of assumptions about the shell structure of the nucleus. But after the setback experienced by Elsasser’s and Guggenheimer’s earlier shell models, that took more than a decade.

While Heisenberg and von Weizsäcker were laying foundations for the liquid drop model, Enrico Fermi and colleagues in Rome were irradiating rhodium with a neutron source (Amaldi et al. 1935). They expected that neutrons would undergo scattering (with associated nucleus excitation). But the experiments revealed instead a new phenomenon: neutron capture. The neutrons would attach to the nucleus rather than being scattered.

There was more. If the apparatus was immersed in water, neutrons would interact with the hydrogen of water and be slowed down, and slow neutrons were more easily captured by nuclei. Fermi examined the absorption of slow neutrons in different elements and identified ‘anomalously large absorption coefficients for the slow neutrons’ (Amaldi et al. 1935, p. 525). Collision cross-sections of slow neutrons were much larger than those for fast neutrons and exceeded the expected scattering cross-sections. This raised interesting questions about the nature of the force acting in neutron capture. In 1936, Bohr published a paper in Nature where he referred to Fermi’s results to conclude that

the phenomena of neutron capture thus force us to assume that a collision between a high-speed neutron and a heavy nucleus will in the first place result in the formation of a compound system of remarkable stability. The possible later breaking up of this intermediate system by the ejection of a material particle, or its passing with emission of radiation to a final stable state, must in fact be considered as separate competing processes which have no immediate connection with the first stage of the encounter. (Bohr 1936, p. 344)

This was the beginning of Bohr’s ‘compound nucleus model’—an evolution of Gamow’s 1929 liquid drop model—which treated the incoming neutron hitting the nucleus as if it were absorbed by the nucleus, exciting states of it. As James Rainwater pointed out many decades later in his Nobel Prize speech, Bohr’s compound model

is not necessarily incompatible with a shell model, since the shell model refers mainly to the lowest states of a set of fermions in the nuclear ‘container’. However, when combined with the discouragingly poor fit with experiment of detailed shell model predictions . . . the situation 1948 was one of great discouragement concerning a shell model approach. (Rainwater 1975, p. 262)

Bohr’s attack on the shell model as ‘unsuited to account for the typical properties of nuclei for which . . . energy exchanges between the individual nuclear particles is a decisive factor’ (Bohr 1936, p. 345) played a central role in shifting attention away from the shell model for about a decade. Bohr argued that in the case of the nucleus the ‘procedure of approximation, resting on a combination of one-body problems, . . . loses any validity’ when dealing with ‘essential collective aspects of the interplay between constituent particles’ (p. 345). Bohr referred once more to Aston’s precise measurements of isotopic varieties to conclude that (contrary to the shell model) the excitations of heavy nuclei should be attributed to the ‘quantised collective type of motion of all the nuclear particles’ (p. 346) rather than the excitation of individual nucleons.

In the meantime, Gregory Breit and Eugene Wigner (1936) were also working on slow neutron capture and were able to derive formulas for neutron-capture and neutron-scattering cross-sections, which agreed with Fermi’s results (see Johnson 1992, p. 168). And between 1936 and 1937, Hans Bethe co-authored three substantial review papers that became known as the ‘Bethe Bible’ and contributed to the popularity of the liquid drop model (see Johnson 1992, p. 169). In his first paper co-authored with R. F. Bacher, Bethe pointed out that while neutron and proton shells provided the ‘basis for a prediction of certain periodicity in nuclear structure for which there is considerable experimental evidence’, the ‘assumption can certainly not claim more than moderate success as regards the calculation of nuclear binding energies’ (Bethe and Bacher 1936, p. 171). Referring to the work of Bartlett, Elsasser, and Guggenheimer, they warned against ‘taking the neutron and proton shells too literally . . . with the effect of discrediting the whole concept . . . among physicists’ (p. 176).

In his second paper, Bethe compared the shell model with the liquid drop model. Both offered statistical treatments of the nucleus. The former, Bethe said, started with the assumption of free individual particles and treated the nucleus as if it was a mixture of two Fermi gases of protons and neutrons. The latter did not regard nucleons as individual particles and treated the interaction among nucleons as larger than the individual kinetic energies as if the nucleons behaved like particles in a drop of liquid.⁴ He concluded that the latter model seemed to ‘come nearer the truth’ (Bethe 1937, p. 80) as the estimated nuclear energy levels compared ‘very favorably with the average spacing of neutron levels estimated from experimental data’ from slow neutron experiments (p. 90).

A turning point in the history of the liquid drop model came in December 1938, when Otto Hahn and Fritz Strassman found that slow neutrons interacting with uranium led to barium. To explain this, Lise Meitner and her nephew Otto Frisch resorted to the liquid drop model. They reasoned that the nucleus might have become deformed after absorbing a neutron, with the surface area increasing and the surface tension opposing this deformation and trying to keep the nucleus spherical. However, Frisch and Meitner realized that under the repulsive force among protons the deformation would eventually split the nucleus in two.

Roger Stuewer (1994) argues that the joint work of Frisch and Meitner represented the coming together of two different strands in the history of the liquid drop model. Meitner, familiar with Heisenberg and von Weizsäcker’s work, approached the problem of the nuclear mass defect. Frisch, familiar with Bohr’s work, focused instead on the dynamic features of the model and how it fared vis-à-vis nuclear excitations. Bohr heard of the Frisch–Meitner interpretation of the phenomenon while visiting Princeton. He began to work there with John Wheeler to develop the liquid drop model into a full-blown theory of nuclear fission—the resulting Bohr–Wheeler paper (1939) laid the foundations for nuclear fission with uranium-235 and plutonium-239.

But, surprisingly, the same neutron-capture phenomenon that had been an incentive for the liquid drop model proved also a key factor in the revival of the shell model. In 1948, Gamow sent a letter to Physical Review jointly written with R.A. Alpher and H. Bethe proposing that all chemical elements could have formed via neutron capture from an overheated primordial ‘neutral nuclear fluid’, which eventually produced protons and electrons via beta decay in an expanding early universe (Alpher et al. 1948). The relative abundances of individual atoms were ascribed to neutron-capture cross-sections rather than mass defect, and once again Victor Goldschmidt’s geochemical data about abundances offered a benchmark.

Nine years later, the same geochemical data—improved by more recent measurements by Suess and Urey—were key for the interpretation of how chemical elements might have formed inside stars in the seminal paper by E.M. Burbidge, G.R. Burbidge, W. A. Fowler, and F. Hoyle (1957). The phenomena of neutron capture which had been the original trigger for Bohr’s ‘compound model’ also prompted a revival of interest in the shell model around 1939–1949 in the continuing attempt to understand nuclear stability and isotopic abundances.

In 1963, Maria Goeppert Mayer shared half of the Nobel Prize in Physics with Hans Jensen (the remaining half went to Eugene Wigner). The prize was given for her discovery concerning the nuclear shell structure. A year later, she wrote a review article in Science where she described two approaches to nuclear physics:

There are essentially two ways in which physicists at present seek to obtain a consistent picture of the atomic nucleus. The first, the basic approach, is to study the elementary particles, their properties and mutual interaction. Thus one hopes to obtain a knowledge of the nuclear forces. If the forces are known, one should in principle be able to calculate deductively the properties of individual complex nuclei. Only after this has been accomplished can one say that one completely understands nuclear structures. . . . But our knowledge of the nuclear forces is still far from complete.

The other approach is that of the experimentalist and consists in obtaining by direct experimentation as many data as possible for individual nuclei. One hopes in this way to find regularities and correlations which give a clue to the structure of the nucleus. . . . The shell model, although proposed by theoreticians, really corresponds to the experimentalist’s approach. It was born from a thorough study of the experimental data, plotting them in different ways and looking for interconnections. This was done on both sides of the Atlantic Ocean and on both sides one found that the data show a remarkable pattern. (Goeppert Mayer 1964, p. 999)

The breakthrough came in 1947 when Goeppert Mayer found an explanation for the surprising stability of certain heavier nuclei. Only a few possible combinations of neutrons and protons exist in nature as stable nuclei that do not decay by beta decay. The most stable nuclei tend to have an even number of protons and neutrons, as Suess had already noted. ‘Eighty-two and fifty are “magic” numbers. That nuclei of this type are unusually abundant indicates that the excess stability must have played a part in the process of the creation of the element’, declared Goeppert Mayer (1964, p. 999). Magic numbers were found elsewhere: 2, 8, 20, 28, 50, 82, and 126 were all magic numbers.

Goeppert Mayer published two articles in Physical Review. Her first, on 1 August 1948, established the stability of nuclei with 20, 50, 82, and 126 neutrons or protons (Goeppert Mayer 1948). Referring to the earlier work of Elsasser, the paper covered experimental evidence for the existence of nuclear stability, including familiar items such as:

A footnote indicated that: ‘The author is indebted to Dr Katherine Way, who pointed out the connection of the closed shells with neutron absorption cross sections’ (Goeppert Mayer 1948, p. 238). This connection was surprising and a welcome addition to the experimental data. More importantly, this was the kind of evidence needed to convince physicists that the shell models had some legs, despite Bohr’s influential opposition and Bethe’s verdict in favour of the liquid drop model.

Katharine Way was a former PhD student of John Wheeler and a later member of the Manhattan Project, who had herself worked on the liquid drop model. In a 1939 paper (Way 1939), she pointed out that the liquid drop model was in ‘very poor’ agreement with experimental evidence about nuclear magnetic moments found by spectroscopist Theodore Schmidt in 1937 (Schmidt 1937; see also Schüler and Schmidt 1935).⁵ Moreover, Way pointed out, using data for heavy nuclei from Wheeler and Teller, that even the largest nuclear spin I would be too small to justify the identification with a uniformly charged spinning drop. A better agreement with the measured data, Way concluded, was given by the so-called odd-particle model (known also as single-particle model; see Figure 4.a.2)

In this alternative model, the magnetic moments for nuclei with odd-Z and odd-N were non-zero and were attributed entirely to the single extra odd nucleon ‘moving outside a central momentless core’ (Way 1939, p. 964). Now long forgotten, the odd-particle model was not a full-blown model as such and it cannot be regarded as a shell model either,⁶ but it provided nonetheless an important bridge between the early shell models pre-1947 and the fashionable liquid drop model of the late 1930s. It showed the epistemic limit of the liquid drop model. And it offered reasons as to why a quasi-shell model could provide better agreement with experimental data on nuclear magnetic moments and neutron absorption cross-sections. The model allowed scientists to make relevant and appropriate inferences about features (e.g. the nuclear magnetic moments and neutron absorption cross-sections) of the phenomena under study (e.g. stable nuclides). Such inferences were in turn pivotal to establish clear shifts in the stability line (and hence binding energy) of nuclei, as Maria Goeppert-Mayer concluded in her 1948 paper:

Between Z = 50 and N = 82, however, the experimental values of Z seem to be below the theoretical curve. The disagreement can be explained by a definite shift of the stability line at 82 neutrons. This shift of the stability line can be explained by a change in binding energy of about 2 MeV. . . . Whereas these calculations are undoubtedly very uncertain, they may serve as an estimate of the order of magnitude of the discontinuities in the binding energies. Since the average neutron binding energy in this region of the periodic table is about 6 MeV, the discontinuities represent only a variation of the order of 30 percent. This situation is very different from that encountered at the closed shells of electrons in atoms where the ionization energy varies by several hundred percent. Nevertheless, the effect of closed shells in the nuclei seems very pronounced. (Goeppert-Mayer 1948, p. 239)

Despite the limits of the analogy with electronic shells, clear indications of shifts in the stability of nuclei had been found by 1948. That 50, 82, and 126 were special or ‘magic’ numbers for nuclear stability was now established. What was still missing was an explanation for what made those numbers ‘magic’.

In her second Physical Review article, Goeppert Mayer stated that the magic numbers occur ‘at the place of the spin-orbit splitting of levels of higher angular momentum’ (Goeppert Mayer 1949, p. 1969). The idea that spin-orbit coupling might explain the stability of heavy nuclei was suggested to her by Fermi. But what was needed was a model that could explain the magic numbers along the lines of how the spin-orbit coupling had been helpful to explain the closure of electronic groups. This was the shell model for which Goeppert Mayer and Jensen shared half of the Nobel Prize for Physics in 1963.

The model treated each proton or neutron as a fairly independent particle occupying orbitals, rather like those of electrons in atoms, whose orbital angular momentum (indicated by the quantum number l) is quantized so that for each l, there was a discrete number of states of different orientation in space given by the magnetic quantum number m_l. The only problem was that following the atomic prescription for the magic numbers led to particularly stable nuclei for heavier nuclei that had the wrong numbers of protons and neutrons, as Elsasser and Guggenheimer had already found. Goeppert Mayer and Jensen saw that, assuming an additional degree of freedom with the spin (quantum number m_s which can be m_s = ½ for spin up and m_s = –½ for spin down), Pauli’s principle could be applied to the structure of the nucleus and dictate the maximum number of nucleons that could be sitting in any shell. Goeppert Mayer postulated a particularly strong spin-orbit interaction that led to a reordering of the energies of the proton and neutron orbitals.

As the protons or neutrons increased to fill orbitals to capacity, energy gaps appeared (see Figure 4.a.3). This led to stability so that orbitals below the energy gaps were full while orbitals above the energy gaps were empty. The numbers of protons or neutrons are the magic numbers. Nuclei with magic numbers of both protons and neutron, like ²⁰⁸Pb, are said to be doubly magic. Such nuclei are not only markedly more stable than those with more or fewer nucleons, but they are also always spherical, meaning that they never show evidence of rotational properties.⁷

As we now know, not every conceivable combination of protons and neutrons can exist in nature (e.g. ⁴⁰C or ¹⁰⁰H). The limits to the number of protons or neutrons for any given mass number A correspond to the driplines in the Segrè chart that maps atomic nuclei on the basis of their proton number Z and their neutron number N. Iron, for example (with Z = 26 and an average atomic mass A of 55.8 due to the various isotopes ⁵⁴Fe, ⁵⁶Fe, ⁵⁷Fe, ⁵⁸Fe, among others), is one of the most stable elements in nature—and one of the most abundant elements in the earth’s core and stars.

The heavier and larger the nuclei, the more sensitive they become to the electrostatic repulsions among protons. The radioactive element barium Ba (Z = 56) and its isotope ¹³⁷Ba mark (with A = 137) the bottom of what physicists call the energy valley. Any element to the left or the right of this valley is subject to beta decay. The farther out one moves along the walls of the valley, the more unstable the nuclei become, and if one tries to add more protons or neutrons to create new combinations, the driplines mark the boundaries beyond which any further addition of nucleons would be impossible. But they also mark the space within which new nuclei can be discovered in nature or created in a lab.

It turns out that the energy valley is marked by grooves or ‘gullies’ that correspond to Goeppert Mayer’s magic numbers. The most stable nuclides tend to line up along grooves in the energy valley that correspond to magic numbers 2, 8, 20, 28, 50, 82, and 126. When the shells/orbitals are complete with those numbers, they do not easily pick up additional neutrons. Nuclei that tend to capture neutrons over time become unstable and subject to beta decay in the so-called slow neutron-capture process (or s-process). In neutron-rich stars (mostly red giant stars), heavier isotopes form all the time via beta decay, descending the energy valley until they reach a groove. This is how elements heavier than iron and up to atomic number 80 are formed.⁸

But the story does not end with the 1949 shell model (also known as the independent-particle model). For spectroscopic evidence about the hyperfine structure of many nuclei revealed that they had quadrupole moments much larger than could be explained by the shell model under the assumption of a single (odd) nucleon orbiting around the atomic core. Thus, the physics community continued to puzzle about nuclear phenomena and the seeming coexistence of two very different models, as Ben Mottelson recalls:

The situation in 1950, when I first came to Copenhagen, was characterized by the inescapable fact that the nucleus sometimes exhibited phenomena characteristic of independent-particle motion, while other phenomena, such as the fission process and the large quadrupole moments, clearly involved a collective behaviour of the whole nucleus. . . . I had given a report on our work and in the discussion Rosenfeld ‘asked how far this model is based in first principles’. N. Bohr ‘answered that it appeared difficult to define what one should understand by first principle in a field of knowledge where our starting point is empirical evidence of different kinds, which is not directly combinable’. (Mottelson 1975, pp. 236–237)

A solution to the problem of large quadrupole moments was glimpsed in 1949 by John Wheeler, who ‘realized that in big nuclei, a single nucleon, constrained by liquid-drop tension, could travel around the rest of the nucleus in a large orbit, deforming the nucleus substantially’ (Thorne 2019, p. 9). Wheeler sent the paper to Bohr, and while waiting for comments, the same idea was discovered independently by James Rainwater at Columbia University. Rainwater understood that the large nuclear quadrupole moments could be explained if the nucleus (and hence nuclear charge) could be deformed and take the shape of a spheroidal liquid drop under the action of the outer nucleons orbiting the atomic core.

Rainwater shared the 1975 Nobel Prize for Physics with Aage Bohr (Niels Bohr’s son) and Ben Mottelson for the ‘discovery of the connection between collective motion and particle motion in atomic nuclei and the development of the theory of the structure of the atomic nucleus based on this connection’.⁹ Rainwater’s contribution consisted in working out the exact physical details of the spheroidal distortion of the atomic nucleus with a prolate potential (Rainwater 1950). Bohr and Mottelson (1953, 1969, 1975), in turn, assumed a non-spherical potential in which particles moved and were able to show how nuclear rotational spectra were the outcome of the coupling between the outer particles’ motion and the motion of the deformed nucleus, offering in this way a ‘unified model’ that combined features of the liquid drop model and key insights of the shell model (for a recent review of these developments, see Caurier et al. 2005; see also Mackintosh 1977).

All of the nuclear models discussed here are examples of perspectival models in being exploratory. They allowed nuclear physicists to gain knowledge of the nuclear structure at a time (in the early 1930s) when neutrons had just been discovered; speculations still abounded that the nucleus might consist of alpha particles; the quantum chromodynamic nature of the strong interaction binding nucleons was still unknown. The exploratory nature of the 1930s–1950s nuclear models is rooted in their historical evolution in response to new data becoming available over time (e.g. from neutron-capture cross-sections to large quadrupole moments) and new phenomena (e.g. nuclear fission, nuclear rotational spectra) being inferred from these data over time.

These nuclear models are, then, not perspectival₁ in representing the nucleus from different points of view. They are not perspectival₁ representations in offering incompatible or inconsistent images of the nucleus with conflicting essential properties ascriptions. They offer instead perspectival₂ representations in opening up a ‘window’ on the reality of nuclear structure despite the partial, limited, and inevitably piecemeal epistemic access to it. They delivered knowledge of what is possible about the nucleus, its internal structure, isotopic stability, nuclear spectra, and the range of possible combinations of protons and neutrons (either to be found in nature or to be created in a lab).

Could alpha-particle natural radioactive chains end with thallium (Z = 81) rather than lead (Z = 82)? No, because lead has proton number Z = 82 (magic number), marking a groove in the energy valley (and making lead a stable and abundant element on earth). Could there be in nature (or be artificially produced) a nucleus like, for example, ¹⁰⁰H? No, because it would fall out of the dripline of the energy valley. Could there be new very short-lived nuclei with a very large neutron excess along the neutron dripline? Yes, there could be such nuclei, and large investments have gone into searching for them.¹⁰

To be a perspectival realist about the atomic nucleus is, then, to engage with an open-ended series of modally robust phenomena (e.g. nuclear stability, neutron capture, nuclear rotational spectra) at the experimental level and with the many exploratory models that over time have allowed physicists to gain knowledge about what is possible concerning each of these phenomena.

Perspectival models of the atomic nucleus, ca. 1930–1950, were therefore exploratory in enabling a variety of epistemic communities to make relevant and appropriate inferences about the nucleus. Elsasser’s early shell model allowed inferences from data for light nuclei via Aston’s mass spectrograph to the possible number of nucleons per shell. Schmidt’s odd-particle model enabled inferences from data about nuclear magnetic moments to the possible zero-moment atomic core, which in turn informed Way’s research about the unusually low neutron absorption cross-sections for nuclei with 50, 82, or 126 neutrons. Goeppert Mayer’s shell model in turn allowed inferences from low neutron absorption cross-sections (traditionally within the remit of the liquid drop model) and Goldschmidt’s plot of isotopic abundances to the possible existence of shifts in the stability lines (or grooves in the energy valley) corresponding to the magic numbers. Perspectival models of the nucleus around the 1930s–1950s allowed exploration of what is possible about nuclear structure by acting as inferential blueprints—a notion I elaborate in Chapter 5.

Second, these models show the collaborative and social nature of scientific knowledge production, the seamless flow through which model-based knowledge claims are historically put forward, modified, corrected, and re-enacted. To what extent was the Nobel Prize-winning 1949 shell model an evolution of (instead of an abrupt shift from) the 1934 shell models? How to classify Schmidt’s odd-particle model in this lineage? (Technically, as Johnson (1992) remarks, it was not a full-blown nuclear model, yet it assumed that the atomic core consisted of shells.) What about the relation between Gamow’s 1929 liquid drop model and Bohr’s 1936 compound model? How to locate the Rainwater–Bohr–Mottelson ‘unified model’—with its combination of liquid drop model and shell model—in this model genealogy?

One thing is clear. Models at play in perspectival pluralism—as in this historical case study—are not the static entities representing-qua-mapping one-on-one relevant aspects of the target system, as a somewhat impoverished picture of them (often found in philosophy of science) has suggested. These models are dynamic evolving tools with a history of their own, which is often intertwined with the history of other scientific models. Having a history means that perspectival models are also often the battleground for scientific rivalries and questions about co-authorship. Why did Bethe celebrate Bohr’s 1936 compound model without giving credit to Gamow’s 1929 liquid drop model? And what about Wheeler, who missed out on the opportunity of sharing the 1975 Nobel Prize due to a delay in the publication of his insight?¹¹ This is without mentioning Lise Meitner, who was not given the Nobel Prize for her crucial work on nuclear fission.

This inevitable aspect of perspectival models’ authorship should not, however, detract from appreciating their by and large social and collaborative function. Models make it possible for teams of scientists to work together over time, make changes to and tweak an original model and eventually deliver on the task of advancing scientific knowledge about the phenomena of interest by making relevant and appropriate inferences. This is evident in the history of nuclear models around the 1930s–1950s. For they offered perspectival representations for a number of phenomena (nuclear stability, nuclear fission, nuclear rotational spectra) not in the sense of ascribing inconsistent essential properties to the same target system (the nucleus as ‘a given’). Instead, they enabled model-based inferences offered by various authors over time.

The 1949 Nobel Prize-winning shell model by Jensen and Goeppert Mayer is the final output of a long tradition of earlier models offered over time by Elsasser, Guggenheimer, Schmidt and Way. The 1975 Nobel Prize-winning ‘unified model’ by Rainwater–Bohr–Mottelson is itself the final product of the long historical intertwining of the Gamow–Bohr research on liquid drop models and the Elsasser–Guggenheimer–Schmidt–Way–Jensen–Goeppert Mayer studies on shell models.

But there is more. As outlined already in Chapter 1, I understand ‘perspectival modelling’ in a broad sense, rather than in an exclusively narrow one confined to the actual models. Perspectival modelling is an integral part of a scientific perspective in being embedded into historically and culturally situated scientific practices. This is evident in the history of nuclear models if one considers the complex historical intertwining of data-to-phenomena inferences behind plots of isotopic abundances—from Washington’s petrological studies to Aston’s mass spectrograph to Goldschmidt’s cosmochemistry—that made it possible in the first instance to identify the phenomenon of nuclear stability that the shell models were designed to explain. It is through this plurality of intersecting scientific perspectives and ever-evolving perspectival₂ representations offered by perspectival models that knowledge claims about which nuclear phenomena might be possible (and which might not) were advanced.

To conclude, perspectival models are exploratory in offering blueprints with instructions that enable various epistemic communities over time to come together and make relevant and appropriate inferences for the phenomena under study. Much as each model bears someone’s name and authorship, and Nobel Prizes are given on such a basis, the role of these models is in fact to facilitate knowledge production over time among very diverse epistemic communities. The nuclear models of the early 1930s were the arrival point of a number of perspectival data-to-phenomena inferences that saw petrologists, volcanologists, mineralogists, spectroscopists, meteorologists, and physical chemists robustly identify the phenomenon of nuclear stability across rocks, atmospheric gases, and meteorites.

The very same models were also the starting point for a number of further inferences from the identified phenomenon of nuclear stability to the underlying nuclear structure that might be responsible for it. The liquid drop model and the shell model provided the inferential blueprints that enabled physical chemists, atomic theorists, spectroscopists, and nuclear physicists to collaborate and make relevant and appropriate inferences from the phenomena of interest (e.g. nuclear fission, nuclear stability) to what nuclear structure might be like. The overall scientific knowledge delivered by them is knowledge ultimately produced by and shared among a great number of epistemic communities that are historically and culturally situated across scientific perspectives (I return to this point in Chapter 11).

The constraints within which this modelling exercise took place included, in this particular example, lawlike dependencies such as Coulomb’s law of electrostatic repulsion at work among the protons; Pauli’s principle, which guided the analogy with the closure of electronic shells; and von Weiszäcker’s semi-empirical formula, among others. And I will return to the role of these laws for perspectival models qua inferential blueprints in more detail in Chapter 5. But next I turn my attention to two more case studies, which probe a little deeper into the collaborative and social nature of the scientific knowledge produced via perspectival modelling, and shed light also on the semantic nature of the associated inferences.