Prof. Dr. Robert Hofstadter > Research Profile
by Luisa Bonolis
Robert Hofstadter
Nobel Prize in Physics 1961 together with Rudolf Mössbauer
"for his pioneering studies of electron scattering in atomic nuclei and for his thereby achieved discoveries concerning the structure of the nucleons".
Between 1909 and 1911, Ernest Rutherford and his collaborators Hans Geiger and Ernst Marsden in Manchester bombarded very thin metal foils with alpha particles in order to measure the pattern of scattered particles, which was expected to provide information about the distribution of charge within the atom. According to J. J. Thomson's model of the atom, in which thousands of tiny, negatively charged corpuscles were swarming inside a cloud of massless positive charge, many of the alpha particles would go through the target virtually unaffected or be deflected by, at most, a few degrees by encounters with the negatively charged electrons. However, the Manchester team discovered that a surprisingly large number of alpha particles bombarding a gold foil were scattered through wide angles, much larger than 90 degrees. Rutherford, realized that a “smeared out” charge would not be able to produce such results, and that they could only be explained by Coulomb interaction with a heavy, very small and concentrated positively charged object, which he called the nucleus, a term already used by Michael Faraday in 1844 to refer to the “central point of the atom.”
Since then, one of the most fundamental problems in physics has been to investigate how the nucleus itself is constituted. Some 40 years later at Stanford, Robert Hofstadter performed Rutherford-like experiments by scattering high-energy electrons from thin targets and measuring the distribution of the number of these electrons as a function of angle. He showed that the nuclei have an internal structure, extending over a small but measurable distance, and also provided the first evidence of the finite size of the proton and neutron, opening the way for future investigations on the substructure of nucleons. As a direct outgrowth of Hofstadter's pioneering work, that was rewarded with the 1961 Nobel Prize in Physics, later experiments carried out by his former collaborators Jerome Friedman, Henry Kendall and Richard Taylor proved of essential importance for the development of the quark model in particle physics.
Learning How to Build Crystal Detectors
Robert Hofstadter was born in Manhattan on February 5, 1915. He majored in physics and mathematics at the City College of New York and received a B.S. degree, magna cum laude in 1935. A Coffin Fellowship from the General Electric Company enabled him to attend Princeton University, where he received his Ph.D. in 1938. After receiving his doctorate, he continued doing research at Princeton on the photoconductivity in crystals, a work that aroused his interest in the use of crystals as electron detectors and in the general problem of scintillation counters. In 1939 Hofstadter received the Harrison Fellowship from the University of Pennsylvania, where he participated in the construction of a large Van de Graaff machine, an experience which stimulated his interest in nuclear physics. The advent of World War II interrupted his career in pure research, and he went on to industrial scientific work arising from military needs. He acquired in particular good knowledge of solid-state physics at the General Electric Laboratory and between 1942 and 1945 he continued to be involved in war research.
Hofstadter returned to academic life in 1946, becoming assistant professor of physics at Princeton. His research there centred on the development of counters made from very well-formed crystals of sodium iodide doped with thallium. When the crystal is struck by an energetic atomic particle or by a photon, it emits a burst of light whose intensity is proportional to the energy of the particle or photon. In 1949, Hofstadter showed how thallium-activated sodium iodide crystals could be used as an efficient detector for gamma rays and in 1950, with Jack McIntyre, he developed the technique of single-crystal gamma-spectrometry with NaI(Tl) crystals, a new powerful detection and spectroscopy technique. Until 1949 the Geiger counter had been the most commonly used detector for gamma rays. Within the space of a couple of years it was replaced by the sodium iodide detectors, and counting efficiencies were increased by a factor of 1000. All this became the basis of the scintillation spectrometer, one of the basic measuring tools for nuclear radiation studies, starting the mass production of gamma-spectra. Since then this material has been widely used as a gamma-ray spectrometer in all branches of nuclear and high-energy physics and astrophysics, as well as in medicine, biology, chemistry, geology and other fields. Hofstadter's long-standing interest in the development of large crystal detectors for the measurement of high-energy gamma rays became instrumental for the study of elementary particles and in his later activity within space-based gamma-ray astronomy, in particular when he became principal investigator in a NASA project for the development of the Energetic Gamma-Ray Experiment Telescope (EGRET) a gamma-ray telescope launched in 1991, which provided an unprecedented view of the gamma-ray sky and data on incredibly energetic gamma-ray bursts.
Probing the Nucleus
In 1950, Hofstadter left Princeton to become an associate professor of physics at Stanford University, where a high energy linear accelerator which could accelerate electrons to energies of 100 to 500 million electron volts was soon to be made available. Since 1947 a linear accelerator called Mark I had been built at Stanford. It was a tube about one meter long and 3 1/2 inches in diameter. Its successor, the Mark II, completed in 1949, was 4.2 meters long. Upon Hofstadter's arrival Mark III was being built. It started out as a 9.1-meter machine, which by 1953 had grown to 64 meters, accelerating electrons to an energy of 400 million electron volts.
When leaving Princeton Hofstadter had only a vague idea of what research he might do in Stanford. Along the way, he visited Eugene Feenberg in St. Louis and on that occasion he discussed with him the work he was planning to do using NaI(Tl) crystals for the detection of high-energy electrons and gamma rays, and for studies of electromagnetic showers. Feenberg, well known for his contributions to quantum mechanics and nuclear physics, is said to have remarked: “Why not do electron diffraction (on nuclei) like the earlier work on atoms?” Feenberg's suggestion about doing scattering experiments immediately aroused Hofstadter's interest, and by the time he arrived at Stanford he had decided to build the equipment that he would need to study the structure of the nucleus using high-energy electrons. However, he realized that in order to detect the individual particles within short bursts of electrons arriving at the same time, he needed a tool to separate the elastically scattered particles from those that were scattered inelastically, and to measure their angle of scattering. He continued to improve his scintillation counters while also developing new detectors for neutrons and X rays, and within a couple of years of his arrival at Stanford he had designed and developed a big magnet for a large magnetic spectrometer with which he embarked on a systematic program to study elastic and inelastic scattering of high-energy electrons by atomic nuclei.
Traditionally, physicists have used two methods to investigate the structure of atoms and nuclei: the scattering and absorption of bombarding particles from these systems and the emission of radiation or material particles from such systems. Most of our knowledge about the outer electronic structure, and thus about the energy levels of the atom, has come from an analysis of the spectral lines emitted by electrons which can be easily excited to higher levels in ordinary flames, discharge tubes and in sparks. On the other hand, knowledge of the structure and behaviour of atomic nuclei came from studies of radioactive and nuclear reactions. The study of radioactive decay is invaluable and has many practical applications as well, but it is limited because radioactive decay is essentially immutable and gives us no control over our key parameters. Nuclear reactions, on the contrary, allow for great variation. In essence almost any type of event we can imagine will actually occur in nuclear reactions: elastic and inelastic scattering, transfer, fusion, fission, etc. For any given pair of interacting objects and energy of interaction, it is only a question of which of these processes predominates.
Physicists have known about the scattering technique for nearly half a century, ever since the alpha particle experiments of Ernest Rutherford and his collaborators in Manchester.
In 1909, when Rutherford suggested to his undergraduate assistant Ernest Marsden, who was working under Hans Geiger, to study the alpha-particle scattering at large angles, he did not see it as very likely, considering the then current views on atomic structure. According to Thomson's “raisin pudding” model of the atom, a positive charge was evenly distributed throughout a sphere, with sprinkled negatively charged electrons in it, so that all of the alpha particles should have passed straight through the target or been found within a small fraction of a degree from the beam. However, in the case of gold, they were astonished to find that alpha particles occasionally scattered at angles greater than 90 degrees (around 1 in 8000). Rutherford stated that the observation of such huge deflections was the most incredible event that ever happened to him in his life: “It was almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you.” Based on the evidence of such a small percentage of large-angle scattered particles, Rutherford deduced that all the positive charge in an atom is concentrated into a small positively charged region, the nucleus, which strongly repelled the incoming positively charged alpha particles. The small size of the nucleus actually explained the small number of alpha particles that were repelled in this way. He derived a scattering formula modelling the interaction between alpha particles from nuclei (considering both to be pointlike charges) from the classical Coulomb force (i.e. electrostatic force between charged particles) and treated it as an orbit. In April 1911 Rutherford theorised that “the atom consists of a central charge supposed concentrated at a point, and that the large passage through the large single deflexions of the alpha and beta particles are mainly due to their passage through the strong central field.”
During the period 1911-1913 in a table-top apparatus, Geiger and Marsden continued to bombard the metal foils with high energy alpha particles and observed the number of scattered alpha particles as a function of angle. The large angle scattering corresponded to the closer approach to the nucleus. This meant that the size of the nucleus could be worked out by finding the maximum angle for which the inverse square scattering formula worked, and finding how close to the centre of the nucleus such an alpha came. Rutherford estimated the radius of the aluminium nucleus to be less than about 10^-13 cm, that is 10^5 times smaller than an atomic radius. Rutherford's alpha scattering experiments, which disproved Thomson's pudding model of the atom, were the first experiments in which individual particles were systematically scattered and detected. This is now the standard operating procedure of particle physics. The nuclear model of the atom was born and very soon adopted by Niels Bohr on a quantum theoretical basis, launching a new generation of studies of the atom.
Knowledge of the structure of the nucleus came almost entirely from subsequent scattering experiments, since the nucleus is a small, very tightly bound structure which cannot be easily excited, like electrons in the outer shells of the atom. However, using alpha particles to obtain a detailed picture of the structure of the nucleus itself, that is of the distribution of the electric charge and magnetic moments, is not feasible, because of the large Coulomb repulsion suffered by an alpha particle as it approaches the nucleus. In addition, only a small range of alpha particle energies is available from natural sources. With the development of the first particle accelerators, high-speed protons became available. The scattering of these particles from light nuclei revealed many features of nuclear forces and nuclear structure. In the meantime, with the discovery of the neutron, a new era had begun in nuclear theory. In the hands of physicists like Fermi the neutron, too, became a powerful analytical tool. The principal information concerning geometric details of nuclear structure was derived from experiments on comparative energy releases in mirror nuclei, on fast neutron capture (and total) cross sections, on binding energies, and, in the case of the heaviest elements, on the energies and half-lives of alpha activities. All approaches led to the same general range of values of the nuclear radii for a uniformly charged sphere, which was taken universally as the appropriate model of the nucleus. The results can be summarised in a well-known formula for the radius of a uniform sphere R=r0A^1/3 m, where A is the mass number of the nucleus and r0 is a constant having the value of about 1 fermi, that is 10^-15 m, the natural unit on the nuclear scale. This model gives a uniform mass density to all nuclei and a variable charge density, proportional to the nuclear charge and inversely varying with the mass number. Later, the value of r0 resulted no longer in a constant, as in these old models, but in a variable, still of the order of 1 fermi.
But why not use electrons for probing nuclei? It may appear that the electron is the natural particle since it is negatively charged and therefore is not electrically repelled by nuclei. This would be the case if the electron were a classical particle. The de Broglie wavelength of a particle lambda=h/mv (being m the relativistic mass of the particle) varies inversely as the product of its mass and speed. Because of the small mass of the electron, its de Broglie wavelength at ordinary speeds is much larger than the diameter of a nucleus, and in a sense it does not “see” the fine details of the nucleus any more than we can see the fine details of a molecule with radio waves, whose wavelength is larger than 1 millimetre.
Electrons could thus be used as nuclear probes if their speed was increased so that their de Broglie wavelengths would be reduced below the diameters of nuclei, that is, a few ten-trillionths of a centimetre. This means they must have energies in the range 200-500 MeV. After higher energy electrons became available from accelerators, interest in their use as probes of the nucleus increased. The advantages of electron scattering are due to the fact that the electron is a pointlike particle, and probes nuclei through the well-understood electromagnetic interaction. Since the electron-scattering process of a nucleus is well described by quantum electrodynamics, extraction of internal structure information from experimental data is straightforward and precise. In addition, the electromagnetic interaction is weak enough that the electron probes the whole volume of a target nucleus without serious distortion. Early electron-scattering experiments were carried out at the University of Illinois in 1951 at an incident electron energy of about 15.7 MeV, however, few details of nuclear shape or size could be discerned because the energy of the electrons was relatively low and the corresponding de Broglie wavelength of the electrons was larger than a typical size of the nucleus.
This was the point from which Hofstadter studies began in 1953, when he embarked in a wide program to study elastic and inelastic scattering of high-energy electrons by nuclei using the Mark III linear electron accelerator in the High Energy Physics Laboratory (HEPL) at Stanford, which was built in stages and eventually reached the maximum energy of its design value of 1 GeV. Rutherford and Chadwick had discovered the constituents of the nucleus by knocking protons and neutrons out of the nuclei. The experiments planned by Hofstadter, were conceptually different: the idea was to shine a beam of electrons into a nucleus, in order to reveal the contents by the
scattering of the beam.
Rutherford's alpha particles had not been able to approach the nucleus closely enough to measure its size. Hofstadter's electrons were more than fifty times as energetic as the alpha particles. In 1953, the commissioning of the first half of the new Mark III linac provided an external electron beam of unprecedented intensity at energies up to 225 MeV. Complementing this advance in accelerator technology, Hofstadter and his collaborators constructed a quasi-permanent scattering facility based on a 180° magnetic spectrometer (radius of bending = 18 inches). The spectrometer could be rotated about the target to measure different scattering angles. The excitation of the magnet could be varied to change the energy of the electrons detected.
The first experiments were made with 116 MeV electrons. This was soon increased to 180 MeV and 550 MeV electrons were available by 1956. They had enough energy to penetrate the nucleus. When an electron penetrates a nucleus, electric and magnetic forces deflect it from its original path, a deflection that depends on the initial speed and on the structure of the nucleus. Elastic scattering, as in a billiard ball collision, is a process in which no energy is lost (to other processes) and the electron's kinetic energy is shared between itself and the target nucleus after the collision. Momentum is of course always conserved, but their direction of propagation is modified by interaction with the nucleus. These investigations are of an experimental and theoretical nature. If scattered electrons are separated into groups according to velocity, the quantum-mechanical laws of scattering can be used to analyse the structure of the nucleus. The angular distributions of high-energy electrons scattered elastically from the atomic nuclei was obtained by employing the narrow momentum selection permitted by the use of the magnetic spectrometer, which is an instrument that sorts out electrons according to their energy and their angle of deflection from their original path.
As the electron is a fundamental particle (so it can be considered as a point without any structure), the effects of any size are not there to complicate results and only the target is observed. If a nucleus were just a point charge, calculations based on Dirac theory and quantum electrodynamics lead to a fairly simple expression for the scattering cross section. This is the target area around the nucleus that the electron must hit to be deflected through a certain angle from its original direction of motion. But the nucleus is not a point charge and its finite size must be taken into account in analysing the electron scattering. In the case of elastic scattering from a fixed charge distribution, this is done simply by multiplying the scattering cross section for a point charge by the so called form factor, a structure factor which depends on the size of the nucleus and the charge distribution inside it. It takes account of the interference between scattered wavelets arising from different parts of the same, finite, nucleus, and therefore is responsible for diffraction effects observed in the angular distribution. Since the form factor depends only on the amount of momentum transferred by the scattered electron to the nucleus, which is always smaller than one, the effect of the finite size of the nucleus is to reduce the scattering cross section below its value for a point charge. All of our ignorance is subsumed in the form factor, which contains an electric charge density function, depending on the distance from the centre of the nucleus. By using a wide range of electron energies, a comparison can be made between the observed scattering cross section and the theoretical cross section for a point charge. The difference gives the form factor for some particular value of the momentum transfer. A mathematical inversion of the form-factor equation thus allows one to deduce the form of the charge density function, once the form factor is known over a large enough range of values of the momentum transfers. A simple mathematical device can be used to calculate the charge distribution inside the nucleus.
The first experiments performed by the research team led by Hofstadter measured the angular distribution of 116-MeV electrons undergoing elastic scattering on beryllium tantalum and lead. The data did not reveal a precise picture of the charge distribution in the nuclei, but suggested a smooth surface, the charge density gradually falling from a constant density inside to a vanishing density outside. A more precise picture of the charge density was obtained in 1954, when the electron beam in Stanford reached an energy of 186 MeV.
Moreover, if atomic nuclei were infinitely small, Hofstadter should have seen the same angular distribution of scattered particles as Rutherford. When he compared his measurements to the theoretical prediction, he found that fewer electrons were scattered at large angles. Some of the electrons were going right through the atomic nuclei. That meant the nucleus was not an infinitely small point, but rather a “fuzzy ball.”
In 1956 Hofstadter summarised the results of his measurements. Although he found that the average density of all nuclei is approximately the same as expected; the radius of several nuclei had been measured as well as the “nuclear surface thickness”, which according to Hofstadter is the distance in which the charge density drops from 90% to 10% of its central value. He discovered that in all nuclei it is about 1 fermi, which is 10^-15 m. This showed that electrons see the nucleus as an object with a diffuse surface about 1 fm thick. These precision measurements also confirmed that the volume of the nucleus is proportional to the total number of nucleons, which means that these particles are not packed any more tightly in the large and heavy nuclei than they are in the small and light ones. The relatively uniform nuclear density was found to be 150 million kilograms per cubic meter, while water, for example, has a density of 1000 kg per cubic meter.
Probing the Nucleons
The following step was a jump into a brand new realm. The method of elastic electron scattering at high energies used to discover the finite size and internal structure of nuclei could also be used to determine the structure of nucleons. The proton, deuteron and alpha particle were most interesting to study, because they are among the simplest nuclear structures. Furthermore, nuclei are made of protons and neutrons, so it was fascinating to think of what the proton itself is built.
At the root of all elastic scattering processes involving charged particles lies Rutherford's famous formula expressing the differential cross section for scattering incident charged point particles of kinetic energy E against immovable charged point centres, conceived as intense sources of an electric field.
While the Rutherford scattering law applies to alpha particles and protons of medium energies, it is not a relativistic formula and takes no account of the possible spins of the scattering partners. In a well known paper published in 1929, Nevill Mott considered the relativistic scattering of Dirac particles, such as electrons, against point nuclei. In this case the incident particle, the electron, is assumed to have a spin (and a Dirac magnetic moment), although the scattering centre (the nucleus) is assumed to have neither a spin nor a magnetic moment. Considerations of the effects of finite nuclear size on electron scattering were made later by several authors. In particular, the elastic scattering of high-energy electrons by the magnetic moment of the proton had been predicted by Rosenbluth in 1950.
Until Hofstadter's research, there was no evidence that the proton or the neutron had a structure or a finite size, although it was already known that the proton did not obey the Dirac equation because its magnetic moment had been found to be larger than that predicted by the Dirac theory. Now if the proton was neither a point charge nor a point magnetic moment, this would require the existence of two different form factors, taking into account both the electrostatic and magnetic interaction between the proton and the electron. In fact, Rosenbluth had carried out such a calculation, basing on relativistic quantum mechanics, and including such form factors.
In 1955 the linear accelerator achieved reliable operation at 600 MeV. At that time a new spectrometer facility and a liquid hydrogen target were constructed and installed. A large effort was focused on scattering from hydrogen, whose nucleus is a single proton. When Hofstadter and Robert McAllister shot their electrons into hydrogen, their results revealed that the charge distribution of the proton has a finite size, with a radius of the order of 0.7 fermi, about 0.7 ten-trillionths of a centimetre.
In applying the same electron-scattering techniques to study the structure of the neutron Hofstadter demonstrated that, like the proton, it is not a point but is distributed over a finite volume. Since the neutron has no charge distribution, its structure must be expressed in terms of its magnetic moment, which is distributed over a finite radius, as the magnetic moment of the proton.
Hofstadter obtained the most relevant results in the years from 1954 to 1957. These results, achieved with a precision that had scarcely been attained before in high-energy physics, showed persuasively that the proton and the neutron are extended structures. After that time he was concerned with improving the accuracy and obtaining more precise form factors, the mathematical quantities that give the distributions of charge in the nucleons. This fundamental discovery was rapidly accepted by the physics community and in 1959 Hofstadter was named California scientist of the Year.
Two years later, in 1961, Robert Hofstadter was awarded the Nobel Prize in Physics, which he received for his ground-breaking discovery that the proton is in fact not a pointlike particle, but has a substructure and is hence an extended object. He shared the Prize with Rudolf Mössbauer, who received it for detailed investigations into the structure of matter performed through the emission and absorption of gamma radiation by the atomic nuclei. The unexpected effect he had discovered during this research, since then bears the name of Mössbauer effect.
Towards a Sub-Structure of Nucleons: Deep-Inelastic Scattering
Hofstadter's pioneering experiments marked the beginning of the search for sub-structure in the proton. Experiments made possible by increasing electron energy and intensity, along with increasingly sophisticated detectors continued to shed light on the structure of nuclei and nucleons over the years. Additional information also came from experiments using secondary beams of muons and neutrinos from proton accelerators.
Since 1954 Hofstadter had suggested to build a GeV linear accelerator at Stanford. A study group was formed, discussions began and a two-mile accelerator - powered by one thousand klystrons - was planned. Under the direction of Wolfgang Panofsky and R.B. Neal, what came to be called the Stanford Liner Accelerator Center (SLAC) was built. It became one of the great engineering achievements of the early 1960s and one of the most expensive projects ever in high energy physics. The 20 GeV energy of the accelerator presented an unprecedented opportunity to pursue the studies of nucleon structure. In 1966, the Stanford Linear Accelerator was ready to deliver beams to experiments. Among the physicists using the accelerator were Henry Kendall, Jerome Friedman, and Richard Taylor, who had worked closely with Hofstadter. With nineteen other physicists they formed the SLAC-MIT collaboration.
Hofstadter had measured elastic scattering. When two particles scatter elastically, their momentum changes, but their nature does not. An electron remains an electron and a proton remains a proton throughout the interaction. The electron may emerge in a different direction, but it loses little energy. In inelastic scattering, on the other hand, the electron may lose a great deal of energy. They looked for the electron undergoing a large deflection, and where the proton, rather than keeping its identity, seized a lot of the collision energy and broke up into a shower of new particles. Sometimes the energy transformed the proton into a resonance, an extremely short lived particle, with a lifetime on the order of 10^-23 seconds. Sometimes the energy created additional particles, such as pions. Experiments showed that deep inelastic scattering was far more frequent than expected, displaying a totally new facet of proton behaviour. When the SLAC-MIT physicists analysed their data, they noticed a number of electrons were actually being scattered at large angles. It was the same phenomenon Rutherford had observed inside the atom. It suggested that, like the atom, the proton was made of particles much smaller than itself. At that time, physicists at the Italian laboratory of Frascati, working at the electron-positron collider ADONE, had already found that more pions than they had expected were produced in annihilations. The interpretation was given primarily by the theorists James Bjorken, and by Richard Feynman who had already received the 1965 Nobel Prize jointly with Sin-Itiro Tomonaga and Julian Schwinger, “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.” The above mentioned results showed that the electrons ricocheted off hard pointlike objects inside the proton: hadrons were made of pointlike constituents, that Feynman had named
partons
.
Up to that time, at the deepest level of the microscopic world were the electron, the proton and the neutron, particles which for years had been considered to be the fundamental building blocks of matter. An additional, ultimate level, consisting of only a few building blocks, which he called quarks, had also already been hypothesised by George Zweig and Murray Gell-Mann, the future 1969 Nobel Prize laureate in Physics “for his contributions and discoveries concerning the classification of elementary particles and their interactions.” Hardly anyone at the time believed that quarks were real physical entities, but now the Stanford electron scattering data were combined with subsequent neutrino data from CERN and it began to look as if the partons and quarks were one and the same thing, thus simplifying the physicist's picture of the world. A new theory called quantum chromodynamics, or QCD, added neutral particles called gluons to the hadrons constituents. According to QCD, gluons are the particles that bind quarks together.
The team headed by Kendall, Friedman, and Taylor, continuing in Hofstadter's line of investigation, had opened the way to the confirmation of the quark structure of the proton. In 1990, only a couple of weeks before Hofstadter died, the Nobel Prize in Physics was awarded jointly to Friedman, Kendall and Taylor “for their pioneering investigations concerning deep inelastic scattering of electrons on protons and bound neutrons, which have been of essential importance for the development of the quark model in particle physics.”
The Standard Model of elementary particles is now based on three generations of quarks (that make up protons and neutrons), leptons, gauge bosons, and on the Higgs boson, whose existence has been confirmed by experiments at the CERN Large Hadron Collider (LHC). All of them are currently treated as pointlike, but this might not always be true...
Bibliography
Boorse, H. A. et al. (1989) Nuclei and Nucleons: Robert Hofstadter. In: The Atomic Scientists. A Biographical History, Wiley, John & Sons, pp. 442-447
Friedman, I. J. and Little, W. L. (2001) Robert Hofstadter (1915-1990). National Academy of Sciences. Biographical Memoirs 79: 3-25
Herman, R. (1995). Robert Hofstadter (5 February 1915-17 November 1990). Proceedings of the American Philosophical Society 139(3): 277-285
Hofstadter, R. (1956) Electron Scattering and Nuclear Structure. Reviews of Modern Physics 28: 214-254
Wasson, T. (ed.) (1987) Hofstadter, Robert. In Nobel Prize Winners, H. W. Wilson Company, New York, pp. 474-475