Development of atomic theory

The concept of the atom that Western scientists accepted in broad outline from the 1600s until about 1900 originated with Greek philosophers in the 5th century BC. Their speculation about a hard, indivisible fundamental particle of nature was replaced slowly by a scientific theory supported by experiment and mathematical deduction. It was 2,000 years before modern physicists realized that the atom is indeed divisible and that it is not hard, solid, or immutable.

The atomic philosophy of the early Greeks

Leucippus of Miletus (5th century BC) is thought to have originated the atomic philosophy. His famous disciple, Democritus of Abdera, named the building blocks of matter atomos, meaning literally “indivisible,” about 430 BC. Democritus believed that atoms were uniform, solid, hard, incompressible, and indestructible and that they moved in infinite numbers through empty space until stopped. Differences in atomic shape and size determined the various properties of matter. In Democritus’s philosophy, atoms existed not only for matter but also for such qualities as perception and the human soul. For example, sourness was caused by needle-shaped atoms, while the colour white was composed of smooth-surfaced atoms. The atoms of the soul were considered to be particularly fine. Democritus developed his atomic philosophy as a middle ground between two opposing Greek theories about reality and the illusion of change. He argued that matter was subdivided into indivisible and immutable particles that created the appearance of change when they joined and separated from others.

The philosopher Epicurus of Samos (341–270 BC) used Democritus’s ideas to try to quiet the fears of superstitious Greeks. According to Epicurus’s materialistic philosophy, the entire universe was composed exclusively of atoms and void, and so even the gods were subject to natural laws.

Most of what is known about the atomic philosophy of the early Greeks comes from Aristotle’s attacks on it and from a long poem, De rerum natura (“On the Nature of Things”), which the Latin poet and philosopher Titus Lucretius Carus (c. 95–55 BC) wrote to popularize its ideas. The Greek atomic theory is significant historically and philosophically, but it has no scientific value. It was not based on observations of nature, measurements, tests, or experiments. Instead, the Greeks used mathematics and reason almost exclusively when they wrote about physics. Like the later theologians of the Middle Ages, they wanted an all-encompassing theory to explain the universe, not merely a detailed experimental view of a tiny portion of it. Science constituted only one aspect of their broad philosophical system. Thus, Plato and Aristotle attacked Democritus’s atomic theory on philosophical grounds rather than on scientific ones. Plato valued abstract ideas more than the physical world and rejected the notion that attributes such as goodness and beauty were “mechanical manifestations of material atoms.” Where Democritus believed that matter could not move through space without a vacuum and that light was the rapid movement of particles through a void, Aristotle rejected the existence of vacuums because he could not conceive of bodies falling equally fast through a void. Aristotle’s conception prevailed in medieval Christian Europe; its science was based on revelation and reason, and the Roman Catholic theologians rejected Democritus as materialistic and atheistic.

The emergence of experimental science

De rerum natura, which was rediscovered in the 15th century, helped fuel a 17th-century debate between orthodox Aristotelian views and the new experimental science. The poem was printed in 1649 and popularized by Pierre Gassendi, a French priest who tried to separate Epicurus’s atomism from its materialistic background by arguing that God created atoms.

Soon after the Italian scientist Galileo Galilei expressed his belief that vacuums can exist (1638), scientists began studying the properties of air and partial vacuums to test the relative merits of Aristotelian orthodoxy and the atomic theory. The experimental evidence about air was only gradually separated from this philosophical controversy.

The Anglo-Irish chemist Robert Boyle began his systematic study of air in 1658 after he learned that Otto von Guericke, a German physicist and engineer, had invented an improved air pump four years earlier. In 1662 Boyle published the first physical law expressed in the form of an equation that describes the functional dependence of two variable quantities. This formulation became known as Boyle’s law. From the beginning, Boyle wanted to analyze the elasticity of air quantitatively, not just qualitatively, and to separate the particular experimental problem about air’s “spring” from the surrounding philosophical issues. Pouring mercury into the open end of a closed J-shaped tube, Boyle forced the air in the short side of the tube to contract under the pressure of the mercury on top. By doubling the height of the mercury column, he roughly doubled the pressure and halved the volume of air. By tripling the pressure, he cut the volume of air to a third, and so on.

This behaviour can be formulated mathematically in the relation PV = PV′, where P and V are the pressure and volume under one set of conditions and P′ and V′ represent them under different conditions. Boyle’s law says that pressure and volume are inversely related for a given quantity of gas. Although it is only approximately true for real gases, Boyle’s law is an extremely useful idealization that played an important role in the development of atomic theory.

Soon after his air-pressure experiments, Boyle wrote that all matter is composed of solid particles arranged into molecules to give material its different properties. He explained that all things are

made of one Catholick Matter common to them all, and…differ but in the shape, size, motion or rest, and texture of the small parts they consist of.

In France Boyle’s law is called Mariotte’s law after the physicist Edme Mariotte, who discovered the empirical relationship independently in 1676. Mariotte realized that the law holds true only under constant temperatures; otherwise, the volume of gas expands when heated or contracts when cooled.



By the end of the 18th century, chemists were just beginning to learn how chemicals combine. In 1794 Joseph-Louis Proust of France published his law of definite proportions (also known as Proust’s law). He stated that the components of chemical compounds always combine in the same proportions by weight. For example, Proust found that no matter where he got his samples of the compound copper carbonate, they were composed by weight of five parts copper, four parts oxygen, and one part carbon.

The beginnings of modern atomic theory
Experimental foundation of atomic chemistry

The English chemist and physicist John Dalton extended Proust’s work and converted the atomic philosophy of the Greeks into a scientific theory between 1803 and 1808. His book A New System of Chemical Philosophy (Part I, 1808; Part II, 1810) was the first application of atomic theory to chemistry. It provided a physical picture of how elements combine to form compounds and a phenomenological reason for believing that atoms exist. His work, together with that of Joseph-Louis Gay-Lussac of France and Amedeo Avogadro of Italy, provided the experimental foundation of atomic chemistry.

On the basis of the law of definite proportions, Dalton deduced the law of multiple proportions, which stated that when two elements form more than one compound by combining in more than one proportion by weight, the weight of one element in one of the compounds is in simple, integer ratios to its weights in the other compounds. For example, Dalton knew that oxygen and carbon can combine to form two different compounds and that carbon dioxide (CO2) contains twice as much oxygen by weight as carbon monoxide (CO). In this case the ratio of oxygen in one compound to the amount of oxygen in the other is the simple integer ratio 2:1. Although Dalton called his theory “modern” to differentiate it from Democritus’s philosophy, he retained the Greek term atom to honour the ancients.

Dalton had begun his atomic studies by wondering why the different gases in the atmosphere do not separate, with the heaviest on the bottom and the lightest on the top. He decided that atoms are not infinite in variety as had been supposed and that they are limited to one of a kind for each element. Proposing that all the atoms of a given element have the same fixed mass, he concluded that elements react in definite proportions to form compounds because their constituent atoms react in definite proportion to produce compounds. He then tried to figure out the masses for well-known compounds. To do so, Dalton made a faulty but understandable assumption that the simplest hypothesis about atomic combinations was true. He maintained that the molecules of an element would always be single atoms. Thus, if two elements form only one compound, he believed that one atom of one element combined with one atom of another element. For example, describing the formation of water, he said that one atom of hydrogen and one of oxygen would combine to form HO instead of H2O. Dalton’s mistaken belief that atoms join together by attractive forces was accepted and formed the basis of most of 19th-century chemistry. As long as scientists worked with masses as ratios, a consistent chemistry could be developed because they did not need to know whether the atoms were separate or joined together as molecules.

Gay-Lussac soon took the relationship between chemical masses implied by Dalton’s atomic theory and expanded it to volumetric relationships of gases. In 1809 he published two observations about gases that have come to be known as Gay-Lussac’s law of combining gases. The first part of the law says that when gases combine chemically, they do so in numerically simple volume ratios. Gay-Lussac illustrated this part of his law with three oxides of nitrogen. The compound NO has equal parts of nitrogen and oxygen by volume. Similarly, in the compound N2O the two parts by volume of nitrogen combine with one part of oxygen. He found corresponding volumes of nitrogen and oxygen in NO2. Thus, Gay-Lussac’s law relates volumes of the chemical constituents within a compound, unlike Dalton’s law of multiple proportions, which relates only one constituent of a compound with the same constituent in other compounds.

The second part of Gay-Lussac’s law states that if gases combine to form gases, the volumes of the products are also in simple numerical ratios to the volume of the original gases. This part of the law was illustrated by the combination of carbon monoxide and oxygen to form carbon dioxide. Gay-Lussac noted that the volume of the carbon dioxide is equal to the volume of carbon monoxide and is twice the volume of oxygen. He did not realize, however, that the reason that only half as much oxygen is needed is because the oxygen molecule splits in two to give a single atom to each molecule of carbon monoxide. In his Mémoire sur la combinaison des substances gazeuses, les unes avec les autres (1809; “Memoir on the Combination of Gaseous Substances with Each Other”), Gay-Lussac wrote:

Thus it appears evident to me that gases always combine in the simplest proportions when they act on one another; and we have seen in reality in all the preceding examples that the ratio of combination is 1 to 1, 1 to 2 or 1 to 3.…Gases…in whatever proportions they may combine, always give rise to compounds whose elements by volume are multiples of each other.…Not only, however, do gases combine in very simple proportions, as we have just seen, but the apparent contraction of volume which they experience on combination has also a simple relation to the volume of the gases, or at least to one of them.

Gay-Lussac’s work raised the question of whether atoms differ from molecules and, if so, how many atoms and molecules are in a volume of gas. Amedeo Avogadro, building on Dalton’s efforts, solved the puzzle, but his work was ignored for 50 years. In 1811 Avogadro proposed two hypotheses: (1) The atoms of elemental gases may be joined together in molecules rather than existing as separate atoms, as Dalton believed. (2) Equal volumes of gases contain equal numbers of molecules. These hypotheses explained why only half a volume of oxygen is necessary to combine with a volume of carbon monoxide to form carbon dioxide. Each oxygen molecule has two atoms, and each atom of oxygen joins one molecule of carbon monoxide.

Until the early 1860s, however, the allegiance of chemists to another concept espoused by the eminent Swedish chemist Jöns Jacob Berzelius blocked acceptance of Avogadro’s ideas. (Berzelius was influential among chemists because he had determined the atomic weights of many elements extremely accurately.) Berzelius contended incorrectly that all atoms of a similar element repel each other because they have the same electric charge. He thought that only atoms with opposite charges could combine to form molecules.

Because early chemists did not know how many atoms were in a molecule, their chemical notation systems were in a state of chaos by the mid-19th century. Berzelius and his followers, for example, used the general formula MO for the chief metallic oxides, while others assigned the formula used today, M2O. A single formula stood for different substances, depending on the chemist: H2O2 was water or hydrogen peroxide; C2H4 was methane or ethylene. Proponents of the system used today based their chemical notation on an empirical law formulated in 1819 by the French scientists Pierre-Louis Dulong and Alexis-Thérèse Petit concerning the specific heat of elements. According to the Dulong-Petit law, the specific heat of all elements is the same on a per atom basis. This law, however, was found to have many exceptions and was not fully understood until the development of quantum theory in the 20th century.

To resolve such problems of chemical notation, the Sicilian chemist Stanislao Cannizzaro revived Avogadro’s ideas in 1858 and expounded them at the First International Chemical Congress, which met in Karlsruhe, Germany, in 1860. Lothar Meyer, a noted German chemistry professor, wrote later that when he heard Avogadro’s theory at the congress, “It was as though scales fell from my eyes, doubt vanished, and was replaced by a feeling of peaceful certainty.” Within a few years, Avogadro’s hypotheses were widely accepted in the world of chemistry.

Atomic weights and the periodic table

As more and more elements were discovered during the 19th century, scientists began to wonder how the physical properties of the elements were related to their atomic weights. During the 1860s several schemes were suggested. The Russian chemist Dmitry Ivanovich Mendeleyev based his system (see photograph) on the atomic weights of the elements as determined by Avogadro’s theory of diatomic molecules. In his paper of 1869 introducing the periodic law, he credited Cannizzaro for using “unshakeable and indubitable” methods to determine atomic weights.

The elements, if arranged according to their atomic weights, show a distinct periodicity of their properties.…Elements exhibiting similarities in their chemical behavior have atomic weights which are approximately equal (as in the case of Pt, Ir, Os) or they possess atomic weights which increase in a uniform manner (as in the case of K, Rb, Cs).

Skipping hydrogen because it is anomalous, Mendeleyev arranged the 63 elements known to exist at the time into six groups according to valence (see figure). Valence, which is the combining power of an element, determines the proportions of the elements in a compound. For example, H2O combines oxygen with a valence of 2 and hydrogen with a valence of 1. Recognizing that chemical qualities change gradually as atomic weight increases, Mendeleyev predicted that a new element must exist wherever there was a gap in atomic weights between adjacent elements. His system was thus a research tool and not merely a system of classification. Mendeleyev’s periodic table raised an important question, however, for future atomic theory to answer: Where does the pattern of atomic weights come from?

Kinetic theory of gases

Whereas Avogadro’s theory of diatomic molecules was ignored for 50 years, the kinetic theory of gases was rejected for more than a century. The kinetic theory relates the independent motion of molecules to the mechanical and thermal properties of gases—namely, their pressure, volume, temperature, viscosity, and heat conductivity. Three men—Daniel Bernoulli in 1738, John Herapath in 1820, and John James Waterston in 1845—independently developed the theory. The kinetic theory of gases, like the theory of diatomic molecules, was a simple physical idea that chemists ignored in favour of an elaborate explanation of the properties of gases.

Bernoulli, a Swiss mathematician and scientist, worked out the first quantitative mathematical treatment of the kinetic theory in 1738 by picturing gases as consisting of an enormous number of particles in very fast, chaotic motion. He derived Boyle’s law by assuming that gas pressure is caused by the direct impact of particles on the walls of their container. He understood the difference between heat and temperature, realizing that heat makes gas particles move faster and that temperature merely measures the propensity of heat to flow from one body to another. In spite of its accuracy, Bernoulli’s theory remained virtually unknown during the 18th century and early 19th century for several reasons. First, chemistry was more popular than physics among scientists of the day, and Bernoulli’s theory involved mathematics. Second, Newton’s reputation ensured the success of his more-comprehensible theory that gas atoms repel one another. Finally, Joseph Black, another noted British scientist, developed the caloric theory of heat, which proposed that heat was an invisible substance permeating matter. At the time, the fact that heat could be transmitted by light seemed a persuasive argument that heat and motion had nothing to do with each other.

Herapath, an English amateur physicist ignored by his contemporaries, published his version of the kinetic theory in 1821. He also derived an empirical relation akin to Boyle’s law but did not understand correctly the role of heat and temperature in determining the pressure of a gas.

Waterston’s efforts met with a similar fate. Waterston was a Scottish civil engineer and amateur physicist who could not even get his work published by the scientific community, which had become increasingly professional throughout the 19th century. Nevertheless, Waterston made the first statement of the law of equipartition of energy, according to which all kinds of particles have equal amounts of thermal energy. He derived practically all the consequences of the fact that pressure exerted by a gas is related to the number of molecules per cubic centimetre, their mass, and their mean squared velocity. He derived the basic equation of kinetic theory, which reads P = NMV2. Here P is the pressure of a volume of gas, N is the number of molecules per unit volume, M is the mass of the molecule, and V2 is the average velocity squared of the molecules. Recognizing that the kinetic energy of a molecule is proportional to MV2 and that the heat energy of a gas is proportional to the temperature, Waterston expressed the law as PV/T = a constant.

During the late 1850s, a decade after Waterston had formulated his law, the scientific community was finally ready to accept a kinetic theory of gases. The studies of heat undertaken by the English physicist James Prescott Joule during the 1840s had shown that heat is a form of energy. This work, together with the law of the conservation of energy that he helped to establish, had persuaded scientists to discard the caloric theory by the mid-1850s. The caloric theory had required that a substance contain a definite amount of caloric (i.e., a hypothetical weightless fluid) to be turned into heat; however, experiments showed that any amount of heat can be generated in a substance by putting enough energy into it. Thus, there was no point to hypothesizing such a special fluid as caloric.

At first, after the collapse of the caloric theory, physicists had nothing with which to replace it. Joule, however, discovered Herapath’s kinetic theory and used it in 1851 to calculate the velocity of hydrogen molecules. Then the German physicist Rudolf Clausius developed the kinetic theory mathematically in 1857, and the scientific world took note. Clausius and two other physicists, the Scot James Clerk Maxwell and the Austrian Ludwig Eduard Boltzmann (who developed the kinetic theory of gases in the 1860s), introduced sophisticated mathematics into physics for the first time since Newton. In his 1860 paper Illustrations of the Dynamical Theory of Gases, Maxwell used probability theory to produce his famous distribution function for the velocities of gas molecules. Employing Newtonian laws of mechanics, he also provided a mathematical basis for Avogadro’s theory. Maxwell, Clausius, and Boltzmann assumed that gas particles were in constant motion, that they were tiny compared with their space, and that their interactions were very brief. They then related the motion of the particles to pressure, volume, and temperature. Interestingly, none of the three committed himself on the nature of the particles.

Studies of the properties of atoms
Size of atoms

The first modern estimates of the size of atoms and the numbers of atoms in a given volume were made by the German chemist Joseph Loschmidt in 1865. Loschmidt used the results of kinetic theory and some rough estimates to do his calculation. The size of the atoms and the distance between them in the gaseous state are related both to the contraction of gas upon liquefaction and to the mean free path traveled by molecules in a gas. The mean free path, in turn, can be found from the thermal conductivity and diffusion rates in the gas. Loschmidt calculated the size of the atom and the spacing between atoms by finding a solution common to these relationships. His result for Avogadro’s number is remarkably close to the present accepted value of about 6.022 × 1023. The precise definition of Avogadro’s number is the number of atoms in 12 grams of the carbon isotope C-12. Loschmidt’s result for the diameter of an atom was approximately 10−8 cm.

Much later, in 1908, the French physicist Jean Perrin used Brownian motion to determine Avogadro’s number. Brownian motion, first observed in 1827 by the Scottish botanist Robert Brown, is the continuous movement of tiny particles suspended in water. Their movement is caused by the thermal motion of water molecules bumping into the particles. Perrin’s argument for determining Avogadro’s number makes an analogy between particles in the liquid and molecules in the atmosphere. The thinning of air at high altitudes depends on the balance between the gravitational force pulling the molecules down and their thermal motion forcing them up. The relationship between the weight of the particles and the height of the atmosphere would be the same for Brownian particles suspended in water. Perrin counted particles of gum mastic at different heights in his water sample and inferred the mass of atoms from the rate of decrease. He then divided the result into the molar weight of atoms to determine Avogadro’s number. After Perrin, few scientists could disbelieve the existence of atoms.

Electric properties of atoms

While atomic theory was set back by the failure of scientists to accept simple physical ideas like the diatomic molecule and the kinetic theory of gases, it was also delayed by the preoccupation of physicists with mechanics for almost 200 years, from Newton to the 20th century. Nevertheless, several 19th-century investigators, working in the relatively ignored fields of electricity, magnetism, and optics, provided important clues about the interior of the atom. The studies in electrodynamics made by the English physicist Michael Faraday and those of Maxwell indicated for the first time that something existed apart from palpable matter, and data obtained by Gustav Robert Kirchhoff of Germany about elemental spectral lines raised questions that would be answered only in the 20th century by quantum mechanics.

Until Faraday’s electrolysis experiments, scientists had no conception of the nature of the forces binding atoms together in a molecule. Faraday concluded that electrical forces existed inside the molecule after he had produced an electric current and a chemical reaction in a solution with the electrodes of a voltaic cell. No matter what solution or electrode material he used, a fixed quantity of current sent through an electrolyte always caused a specific amount of material to form on an electrode of the electrolytic cell. Faraday concluded that each ion of a given chemical compound has exactly the same charge. Later he discovered that the ionic charges are integral multiples of a single unit of charge, never fractions.

On the practical level, Faraday did for charge what Dalton had done for the chemical combination of atomic masses. That is to say, Faraday demonstrated that it takes a definite amount of charge to convert an ion of an element into an atom of the element and that the amount of charge depends on the element used. The unit of charge that releases one gram-equivalent weight of a simple ion is called the faraday in his honour. For example, one faraday of charge passing through water releases one gram of hydrogen and eight grams of oxygen. In this manner, Faraday gave scientists a rather precise value for the ratios of the masses of atoms to the electric charges of ions. The ratio of the mass of the hydrogen atom to the charge of the electron was found to be 1.035 × 10−8 kilogram per coulomb. Faraday did not know the size of his electrolytic unit of charge in units such as coulombs any more than Dalton knew the magnitude of his unit of atomic weight in grams. Nevertheless, scientists could determine the ratio of these units easily.

More significantly, Faraday’s work was the first to imply the electrical nature of matter and the existence of subatomic particles and a fundamental unit of charge. Faraday wrote:

The atoms of matter are in some way endowed or associated with electrical powers, to which they owe their most striking qualities, and amongst them their mutual chemical affinity.

Faraday did not, however, conclude that atoms cause electricity.

Light and spectral lines

In 1865 Maxwell unified the laws of electricity and magnetism in his publication A Dynamical Theory of the Electromagnetic Field. In this paper he concluded that light is an electromagnetic wave. His theory was confirmed by the German physicist Heinrich Hertz, who produced radio waves with sparks in 1887. With light understood as an electromagnetic wave, Maxwell’s theory could be applied to the emission of light from atoms. The theory failed, however, to describe spectral lines and the fact that atoms do not lose all their energy when they radiate light. The problem was not with Maxwell’s theory of light itself but rather with its description of the oscillating electron currents generating light. Only quantum mechanics could explain this behaviour (see below The laws of quantum mechanics).

By far the richest clues about the structure of the atom came from spectral line series. Mounting a particularly fine prism on a telescope, the German physicist and optician Joseph von Fraunhofer had discovered between 1814 and 1824 hundreds of dark lines in the spectrum of the Sun. He labeled the most prominent of these lines with the letters A through G. Together they are now called Fraunhofer lines (see figure). A generation later Kirchhoff heated different elements to incandescence in order to study the different coloured vapours emitted. Observing the vapours through a spectroscope, he discovered that each element has a unique and characteristic pattern of spectral lines. Each element produces the same set of identifying lines, even when it is combined chemically with other elements. In 1859 Kirchhoff and the German chemist Robert Wilhelm Bunsen discovered two new elements—cesium and rubidium—by first observing their spectral lines.

Johann Jakob Balmer, a Swiss secondary-school teacher with a penchant for numerology, studied hydrogen’s spectral lines (see photograph) and found a constant relationship between the wavelengths of the element’s four visible lines. In 1885 he published a generalized mathematical formula for all the lines of hydrogen. The Swedish physicist Johannes Rydberg extended Balmer’s work in 1890 and found a general rule applicable to many elements. Soon more series were discovered elsewhere in the spectrum of hydrogen and in the spectra of other elements as well. Stated in terms of the frequency of the light rather than its wavelength, the formula may be expressed:

Here ν is the frequency of the light, n and m are integers, and R is the Rydberg constant. In the Balmer lines m is equal to 2 and n takes on the values 3, 4, 5, and 6.

Discovery of electrons

During the 1880s and ’90s scientists searched cathode rays for the carrier of the electrical properties in matter. Their work culminated in the discovery by English physicist J.J. Thomson of the electron in 1897. The existence of the electron showed that the 2,000-year-old conception of the atom as a homogeneous particle was wrong and that in fact the atom has a complex structure.

Cathode-ray studies began in 1854 when Heinrich Geissler, a glassblower and technical assistant to the German physicist Julius Plücker, improved the vacuum tube. Plücker discovered cathode rays in 1858 by sealing two electrodes inside the tube, evacuating the air, and forcing electric current between the electrodes. He found a green glow on the wall of his glass tube and attributed it to rays emanating from the cathode. In 1869, with better vacuums, Plücker’s pupil Johann W. Hittorf saw a shadow cast by an object placed in front of the cathode. The shadow proved that the cathode rays originated from the cathode. The English physicist and chemist William Crookes investigated cathode rays in 1879 and found that they were bent by a magnetic field; the direction of deflection suggested that they were negatively charged particles. As the luminescence did not depend on what gas had been in the vacuum or what metal the electrodes were made of, he surmised that the rays were a property of the electric current itself. As a result of Crookes’s work, cathode rays were widely studied, and the tubes came to be called Crookes tubes.

Although Crookes believed that the particles were electrified charged particles, his work did not settle the issue of whether cathode rays were particles or radiation similar to light. By the late 1880s the controversy over the nature of cathode rays had divided the physics community into two camps. Most French and British physicists, influenced by Crookes, thought that cathode rays were electrically charged particles because they were affected by magnets. Most German physicists, on the other hand, believed that the rays were waves because they traveled in straight lines and were unaffected by gravity. A crucial test of the nature of the cathode rays was how they would be affected by electric fields. Heinrich Hertz, the aforementioned German physicist, reported that the cathode rays were not deflected when they passed between two oppositely charged plates in an 1892 experiment. In England J.J. Thomson thought Hertz’s vacuum might have been faulty and that residual gas might have reduced the effect of the electric field on the cathode rays.

Thomson repeated Hertz’s experiment with a better vacuum in 1897. He directed the cathode rays between two parallel aluminum plates to the end of a tube where they were observed as luminescence on the glass. When the top aluminum plate was negative, the rays moved down; when the upper plate was positive, the rays moved up. The deflection was proportional to the difference in potential between the plates. With both magnetic and electric deflections observed, it was clear that cathode rays were negatively charged particles. Thomson’s discovery established the particulate nature of electricity. Accordingly, he called his particles electrons.

From the magnitude of the electrical and magnetic deflections, Thomson could calculate the ratio of mass to charge for the electrons. This ratio was known for atoms from electrochemical studies. Measuring and comparing it with the number for an atom, he discovered that the mass of the electron was very small, merely 1/1,836 that of a hydrogen ion. When scientists realized that an electron was virtually 1,000 times lighter than the smallest atom, they understood how cathode rays could penetrate metal sheets and how electric current could flow through copper wires. In deriving the mass-to-charge ratio, Thomson had calculated the electron’s velocity. It was 110 the speed of light, thus amounting to roughly 30,000 km (18,000 miles) per second. Thomson emphasized that

we have in the cathode rays matter in a new state, a state in which the subdivision of matter is carried very much further than in the ordinary gaseous state; a state in which all matter, that is, matter derived from different sources such as hydrogen, oxygen, etc., is of one and the same kind; this matter being the substance from which all the chemical elements are built up.

Thus, the electron was the first subatomic particle identified, the smallest and the fastest bit of matter known at the time.

In 1909 the American physicist Robert Andrews Millikan greatly improved a method employed by Thomson for measuring the electron charge directly. In Millikan’s oil-drop experiment, he produced microscopic oil droplets and observed them falling in the space between two electrically charged plates. Some of the droplets became charged and could be suspended by a delicate adjustment of the electric field. Millikan knew the weight of the droplets from their rate of fall when the electric field was turned off. From the balance of the gravitational and electrical forces, he could determine the charge on the droplets. All the measured charges were integral multiples of a quantity that in contemporary units is 1.602 × 10−19 coulomb. Millikan’s electron-charge experiment was the first to detect and measure the effect of an individual subatomic particle. Besides confirming the particulate nature of electricity, his experiment also supported previous determinations of Avogadro’s number. Avogadro’s number times the unit of charge gives Faraday’s constant, the amount of charge required to electrolyze one mole of a chemical ion.

Identification of positive ions

In addition to electrons, positively charged particles also emanate from the anode in an energized Crookes tube. The German physicist Wilhelm Wien analyzed these positive rays in 1898 and found that the particles have a mass-to-charge ratio more than 1,000 times larger than that of the electron. Because the ratio of the particles is also comparable to the mass-to-charge ratio of the residual atoms in the discharge tubes, scientists suspected that the rays were actually ions from the gases in the tube.

In 1913 Thomson refined Wien’s apparatus to separate different ions and measure their mass-to-charge ratio on photographic plates. He sorted out the many ions in various charge states produced in a discharge tube. When he conducted his atomic mass experiments with neon gas, he found that a beam of neon atoms subjected to electric and magnetic forces split into two parabolas instead of one on a photographic plate. Chemists had assumed the atomic weight of neon was 20.2, but the traces on Thomson’s photographic plate suggested atomic weights of 20.0 and 22.0, with the former parabola much stronger than the latter. He concluded that neon consisted of two stable isotopes: primarily neon-20, with a small percentage of neon-22. Eventually a third isotope, neon-21, was discovered in very small quantities. It is now known that 1,000 neon atoms will contain an average of 909 atoms of neon-20, 88 of neon-22, and 3 of neon-21. Dalton’s assumptions that all atoms of an element have an identical mass and that the atomic weight of an element is its mass were thus disproved. Today the atomic weight of an element is recognized as the weighted average of the masses of its isotopes.

Francis William Aston, an English physicist, improved Thomson’s technique when he developed the mass spectrograph in 1919. This device spread out the beam of positive ions into a “mass spectrum” of lines similar to the way light is separated into a spectrum. Aston analyzed about 50 elements over the next six years and discovered that most have isotopes.

Discovery of radioactivity

Like Thomson’s discovery of the electron, the discovery of radioactivity in uranium by the French physicist Henri Becquerel in 1896 forced scientists to radically change their ideas about atomic structure. Radioactivity demonstrated that the atom was neither indivisible nor immutable. Instead of serving merely as an inert matrix for electrons, the atom could change form and emit an enormous amount of energy. Furthermore, radioactivity itself became an important tool for revealing the interior of the atom.

The German physicist Wilhelm Conrad Röntgen had discovered X-rays in 1895, and Becquerel thought they might be related to fluorescence and phosphorescence, processes in which substances absorb and emit energy as light. In the course of his investigations, Becquerel stored some photographic plates and uranium salts in a desk drawer. Expecting to find the plates only lightly fogged, he developed them and was surprised to find sharp images of the salts. He then began experiments that showed that uranium salts emit a penetrating radiation independent of external influences. Becquerel also demonstrated that the radiation could discharge electrified bodies. In this case discharge means the removal of electric charge, and it is now understood that the radiation, by ionizing molecules of air, allows the air to conduct an electric current. Early studies of radioactivity relied on measuring ionization power (see figure) or on observing the effects of radiation on photographic plates.

In 1898 the French physicists Pierre and Marie Curie discovered the strongly radioactive elements polonium and radium, which occur naturally in uranium minerals. Marie coined the term radioactivity for the spontaneous emission of ionizing, penetrating rays by certain atoms.

Experiments conducted by the British physicist Ernest Rutherford in 1899 showed that radioactive substances emit more than one kind of radiation. It was determined that part of the radiation is 100 times more penetrating than the rest and can pass through aluminum foil one-fiftieth of a millimetre thick. Rutherford named the less-penetrating emanations alpha rays and the more-powerful ones beta rays, after the first two letters of the Greek alphabet. Investigators who in 1899 found that beta rays were deflected by a magnetic field concluded that they are negatively charged particles similar to cathode rays. In 1903 Rutherford found that alpha rays were deflected slightly in the opposite direction, showing that they are massive, positively charged particles. Much later Rutherford proved that alpha rays are nuclei of helium atoms by collecting the rays in an evacuated tube and detecting the buildup of helium gas over several days.

A third kind of radiation was identified by the French chemist Paul Villard in 1900. Designated as the gamma ray, it is not deflected by magnets and is much more penetrating than alpha particles. Gamma rays were later shown to be a form of electromagnetic radiation, like light or X-rays, but with much shorter wavelengths. Because of these shorter wavelengths, gamma rays have higher frequencies and are even more penetrating than X-rays.

In 1902, while studying the radioactivity of thorium, Rutherford and the English chemist Frederick Soddy discovered that radioactivity was associated with changes inside the atom that transformed thorium into a different element. They found that thorium continually generates a chemically different substance that is intensely radioactive. The radioactivity eventually makes the new element disappear. Watching the process, Rutherford and Soddy formulated the exponential decay law (see decay constant), which states that a fixed fraction of the element will decay in each unit of time. For example, half of the thorium product decays in four days, half the remaining sample in the next four days, and so on.

Until the 20th century, physicists had studied subjects, such as mechanics, heat, and electromagnetism, that they could understand by applying common sense or by extrapolating from everyday experiences. The discoveries of the electron and radioactivity, however, showed that classical Newtonian mechanics could not explain phenomena at atomic and subatomic levels. As the primacy of classical mechanics crumbled during the early 20th century, quantum mechanics was developed to replace it. Since then experiments and theories have led physicists into a world that is often extremely abstract and seemingly contradictory.

Models of atomic structure

J.J. Thomson’s discovery of the negatively charged electron had raised theoretical problems for physicists as early as 1897, because atoms as a whole are electrically neutral. Where was the neutralizing positive charge and what held it in place? Between 1903 and 1907 Thomson tried to solve the mystery by adapting an atomic model that had been first proposed by the Scottish scientist William Thomson (Lord Kelvin) in 1902. According to the Thomson atomic model, often referred to as the “plum-pudding” model, the atom is a sphere of uniformly distributed positive charge about one angstrom in diameter (see figure). Electrons are embedded in a regular pattern, like raisins in a plum pudding, to neutralize the positive charge. The advantage of the Thomson atom was that it was inherently stable: if the electrons were displaced, they would attempt to return to their original positions. In another contemporary model, the atom resembled the solar system or the planet Saturn, with rings of electrons surrounding a concentrated positive charge. The Japanese physicist Nagaoka Hantaro in particular developed the “Saturnian” system in 1904. The atom, as postulated in this model, was inherently unstable because, by radiating continuously, the electron would gradually lose energy and spiral into the nucleus. No electron could thus remain in any particular orbit indefinitely.

Rutherford’s nuclear model

Rutherford overturned Thomson’s model in 1911 with his famous gold-foil experiment, in which he demonstrated that the atom has a tiny, massive nucleus (see figure). Five years earlier Rutherford had noticed that alpha particles beamed through a hole onto a photographic plate would make a sharp-edged picture, while alpha particles beamed through a sheet of mica only 20 micrometres (or about 0.002 cm) thick would make an impression with blurry edges. For some particles the blurring corresponded to a two-degree deflection. Remembering those results, Rutherford had his postdoctoral fellow, Hans Geiger, and an undergraduate student, Ernest Marsden, refine the experiment. The young physicists beamed alpha particles through gold foil and detected them as flashes of light or scintillations on a screen. The gold foil was only 0.00004 cm thick. Most of the alpha particles went straight through the foil, but some were deflected by the foil and hit a spot on a screen placed off to one side. Geiger and Marsden found that about one in 20,000 alpha particles had been deflected 45° or more. Rutherford asked why so many alpha particles passed through the gold foil while a few were deflected so greatly. “It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper, and it came back to hit you,” Rutherford said later.

On consideration, I realized that this scattering backwards must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then that I had the idea of an atom with a minute massive centre carrying a charge.

Many physicists distrusted the Rutherford atomic model because it was difficult to reconcile with the chemical behaviour of atoms (see figure). The model suggested that the charge on the nucleus was the most important characteristic of the atom, determining its structure. On the other hand, Mendeleyev’s periodic table of the elements had been organized according to the atomic masses of the elements, implying that the mass was responsible for the structure and chemical behaviour of atoms.

Moseley’s X-ray studies

Henry Gwyn Jeffreys Moseley, a young English physicist killed in World War I, confirmed that the positive charge on the nucleus revealed more about the fundamental structure of the atom than Mendeleyev’s atomic mass. Moseley studied the spectral lines emitted by heavy elements in the X-ray region of the electromagnetic spectrum. He built on the work done by several other British physicists—Charles Glover Barkla, who had studied X-rays produced by the impact of electrons on metal plates, and William Bragg and his son Lawrence, who had developed a precise method of using crystals to reflect X-rays and measure their wavelength by diffraction. Moseley applied their method systematically to measure the spectra of X-rays produced by many elements.

Moseley found that each element radiates X-rays of a different and characteristic wavelength. The wavelength and frequency vary in a regular pattern according to the charge on the nucleus. He called this charge the atomic number. In his first experiments, conducted in 1913, Moseley used what was called the K series of X-rays to study the elements up to zinc. The following year he extended this work using another series of X-rays, the L series. Moseley was conducting his research at the same time that the Danish theoretical physicist Niels Bohr was developing his quantum shell model of the atom. The two conferred and shared data as their work progressed, and Moseley framed his equation in terms of Bohr’s theory by identifying the K series of X-rays with the most-bound shell in Bohr’s theory, the N = 1 shell, and identifying the L series of X-rays with the next shell, N = 2.

Moseley presented formulas for the X-ray frequencies that were closely related to Bohr’s formulas for the spectral lines in a hydrogen atom. Moseley showed that the frequency of a line in the X-ray spectrum is proportional to the square of the charge on the nucleus. The constant of proportionality depends on whether the X-ray is in the K or L series. This is the same relationship that Bohr used in his formula applied to the Lyman and Balmer series of spectral lines. The regularity of the differences in X-ray frequencies allowed Moseley to order the elements by atomic number from aluminum to gold. He observed that, in some cases, the order by atomic weights was incorrect. For example, cobalt has a larger atomic mass than nickel, but Moseley found that it has atomic number 27 while nickel has 28. When Mendeleyev constructed the periodic table, he based his system on the atomic masses of the elements and had to put cobalt and nickel out of order to make the chemical properties fit better. In a few places where Moseley found more than one integer between elements, he predicted correctly that a new element would be discovered. Because there is just one element for each atomic number, scientists could be confident for the first time of the completeness of the periodic table; no unexpected new elements would be discovered.

Bohr’s shell model

In 1913 Bohr proposed his quantized shell model of the atom (see Bohr atomic model) to explain how electrons can have stable orbits around the nucleus. The motion of the electrons in the Rutherford model was unstable because, according to classical mechanics and electromagnetic theory, any charged particle moving on a curved path emits electromagnetic radiation; thus, the electrons would lose energy and spiral into the nucleus. To remedy the stability problem, Bohr modified the Rutherford model by requiring that the electrons move in orbits of fixed size and energy (see figure). The energy of an electron depends on the size of the orbit and is lower for smaller orbits. Radiation can occur only when the electron jumps from one orbit to another. The atom will be completely stable in the state with the smallest orbit, since there is no orbit of lower energy into which the electron can jump.

Bohr’s starting point was to realize that classical mechanics by itself could never explain the atom’s stability. A stable atom has a certain size so that any equation describing it must contain some fundamental constant or combination of constants with a dimension of length. The classical fundamental constants—namely, the charges and the masses of the electron and the nucleus—cannot be combined to make a length. Bohr noticed, however, that the quantum constant formulated by the German physicist Max Planck has dimensions which, when combined with the mass and charge of the electron, produce a measure of length. Numerically, the measure is close to the known size of atoms. This encouraged Bohr to use Planck’s constant in searching for a theory of the atom.

Planck had introduced his constant in 1900 in a formula explaining the light radiation emitted from heated bodies. According to classical theory, comparable amounts of light energy should be produced at all frequencies. This is not only contrary to observation but also implies the absurd result that the total energy radiated by a heated body should be infinite. Planck postulated that energy can only be emitted or absorbed in discrete amounts, which he called quanta (Latin for “how much”). The energy quantum is related to the frequency of the light by a new fundamental constant, h. When a body is heated, its radiant energy in a particular frequency range is, according to classical theory, proportional to the temperature of the body. With Planck’s hypothesis, however, the radiation can be emitted only in quantum amounts of energy. If the radiant energy is less than the quantum of energy, the amount of light in that frequency range will be reduced. Planck’s formula correctly describes radiation from heated bodies. Planck’s constant has the dimensions of action, which may be expressed as units of energy multiplied by time, units of momentum multiplied by length, or units of angular momentum. For example, Planck’s constant can be written as h = 6.6 × 10−34 joule∙seconds.

In 1905 Einstein extended Planck’s hypothesis by proposing that the radiation itself can carry energy only in quanta. According to Einstein, the energy (E) of the quantum is related to the frequency (ν) of the light by Planck’s constant in the formula E = hν. Using Planck’s constant, Bohr obtained an accurate formula for the energy levels of the hydrogen atom. He postulated that the angular momentum of the electron is quantized—i.e., it can have only discrete values. He assumed that otherwise electrons obey the laws of classical mechanics by traveling around the nucleus in circular orbits. Because of the quantization, the electron orbits have fixed sizes and energies. The orbits are labeled by an integer, the quantum number n. In Bohr’s model, radius an of the orbit n is given by the formula an = h2n2ε0/π2, where ε0 is the electric constant. As Bohr had noticed, the radius of the n = 1 orbit is approximately the same size as an atom.

With his model, Bohr explained how electrons could jump from one orbit to another only by emitting or absorbing energy in fixed quanta (see figure). For example, if an electron jumps one orbit closer to the nucleus, it must emit energy equal to the difference of the energies of the two orbits. Conversely, when the electron jumps to a larger orbit, it must absorb a quantum of light equal in energy to the difference in orbits.

Bohr’s model accounts for the stability of atoms because the electron cannot lose more energy than it has in the smallest orbit, the one with n = 1. The model also explains the Balmer formula for the spectral lines of hydrogen. The light energy is the difference in energies between the two orbits in the Bohr formula. Using Einstein’s formula to deduce the frequency of the light, Bohr not only explained the form of the Balmer formula but also explained accurately the value of the constant of proportionality R.

The usefulness of Bohr’s theory extends beyond the hydrogen atom. Bohr himself noted that the formula also applies to the singly ionized helium atom, which, like hydrogen, has a single electron. The nucleus of the helium atom has twice the charge of the hydrogen nucleus, however. In Bohr’s formula the charge of the electron is raised to the fourth power. Two of those powers stem from the charge on the nucleus; the other two come from the charge on the electron itself. Bohr modified his formula for the hydrogen atom to fit the helium atom by doubling the charge on the nucleus. Moseley applied Bohr’s formula with an arbitrary atomic charge Z to explain the K- and L-series X-ray spectra of heavier atoms. The German physicists James Franck and Gustav Hertz confirmed the existence of quantum states in atoms in experiments reported in 1914. They made atoms absorb energy by bombarding them with electrons. The atoms would only absorb discrete amounts of energy from the electron beam. When the energy of an electron was below the threshold for producing an excited state, the atom would not absorb any energy.

Bohr’s theory had major drawbacks, however. Except for the spectra of X-rays in the K and L series, it could not explain properties of atoms having more than one electron. The binding energy of the helium atom, which has two electrons, was not understood until the development of quantum mechanics. Several features of the spectrum were inexplicable even in the hydrogen atom (see figure). High-resolution spectroscopy shows that the individual spectral lines of hydrogen are divided into several closely spaced fine lines. In a magnetic field the lines split even farther apart. The German physicist Arnold Sommerfeld modified Bohr’s theory by quantizing the shapes and orientations of orbits to introduce additional energy levels corresponding to the fine spectral lines.

The quantization of the orientation of the angular momentum vector was confirmed in an experiment in 1922 by other German physicists, Otto Stern and Walther Gerlach. Their experiment took advantage of the magnetism associated with angular momentum; an atom with angular momentum has a magnetic moment like a compass needle that is aligned along the same axis. The researchers passed a beam of silver atoms through a magnetic field, one that would deflect the atoms to one side or another according to the orientation of their magnetic moments. In their experiment Stern and Gerlach found only two deflections, not the continuous distribution of deflections that would have been seen if the magnetic moment had been oriented in any direction. Thus, it was determined that the magnetic moment and the angular momentum of an atom can have only two orientations. The discrete orientations of the orbits explain some of the magnetic field effects—namely, the so-called normal Zeeman effect, which is the splitting of a spectral line into three separate subsidiary lines. These lines correspond to quantum jumps in which the angular momentum along the magnetic field is increased by one unit, decreased by one unit, or left unchanged.

Spectra in magnetic fields displayed additional splittings that showed that the description of the electrons in atoms was still incomplete. In 1925 Samuel Abraham Goudsmit and George Eugene Uhlenbeck, two graduate students in physics at the University of Leiden in The the Netherlands, added a quantum number to account for the division of some spectral lines into more subsidiary lines than can be explained with the original quantum numbers. Goudsmit and Uhlenbeck postulated that an electron has an internal spinning motion and that the corresponding angular momentum is one-half of the orbital angular momentum quantum. Independently, the Austrian-born physicist Wolfgang Pauli also suggested adding a two-valued quantum number for electrons, but for different reasons. He needed this additional quantum number to formulate his exclusion principle, which serves as the atomic basis of the periodic table and the chemical behaviour of the elements. According to the Pauli exclusion principle, one electron at most can occupy an orbit, taking into account all the quantum numbers. Pauli was led to this principle by the observation that an alkali metal atom in a magnetic field has a number of orbits in the shell equal to the number of electrons that must be added to make the next noble gas. These numbers are twice the number of orbits available if the angular momentum and its orientation are considered alone.

In spite of these modifications, by the early 1920s Bohr’s model seemed to be a dead end. It could not explain the number of fine spectral lines and many of the frequency shifts associated with the Zeeman effect. Most daunting, however, was its inability to explain the rich spectra of multielectron atoms. In fact, efforts to generalize the model to multielectron atoms had proved futile, and physicists despaired of ever understanding them.

The laws of quantum mechanics

Within a few short years scientists developed a consistent theory of the atom that explained its fundamental structure and its interactions. Crucial to the development of the theory was new evidence indicating that light and matter have both wave and particle characteristics at the atomic and subatomic levels. Theoreticians had objected to the fact that Bohr had used an ad hoc hybrid of classical Newtonian dynamics for the orbits and some quantum postulates to arrive at the energy levels of atomic electrons. The new theory ignored the fact that electrons are particles and treated them as waves. By 1926 physicists had developed the laws of quantum mechanics, also called wave mechanics, to explain atomic and subatomic phenomena.

The duality between the wave and particle nature of light was highlighted by the American physicist Arthur Holly Compton in an X-ray scattering experiment conducted in 1922. Compton sent a beam of X-rays through a target material and observed that a small part of the beam was deflected off to the sides at various angles. He found that the scattered X-rays had longer wavelengths than the original beam; the change could be explained only by assuming that the X-rays scattered from the electrons in the target as if the X-rays were particles with discrete amounts of energy and momentum (see figure). When X-rays are scattered, their momentum is partially transferred to the electrons. The recoil electron takes some energy from an X-ray, and as a result the X-ray frequency is shifted. Both the discrete amount of momentum and the frequency shift of the light scattering are completely at variance with classical electromagnetic theory, but they are explained by Einstein’s quantum formula.

Louis-Victor de Broglie, a French physicist, proposed in his 1923 doctoral thesis that all matter and radiations have both particle- and wavelike characteristics. Until the emergence of the quantum theory, physicists had assumed that matter was strictly particulate. In his quantum theory of light, Einstein proposed that radiation has characteristics of both waves and particles. Believing in the symmetry of nature, Broglie postulated that ordinary particles such as electrons may also have wave characteristics. Using the old-fashioned word corpuscles for particles, Broglie wrote,

For both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words, the existence of corpuscles accompanied by waves has to be assumed in all cases.

Broglie’s conception was an inspired one, but at the time it had no empirical or theoretical foundation. The Austrian physicist Erwin Schrödinger had to supply the theory.

Schrödinger’s wave equation

In 1926 the Schrödinger equation, essentially a mathematical wave equation, established quantum mechanics in widely applicable form. In order to understand how a wave equation is used, it is helpful to think of an analogy with the vibrations of a bell, violin string, or drumhead. These vibrations are governed by a wave equation, since the motion can propagate as a wave from one side of the object to the other. Certain vibrations in these objects are simple modes that are easily excited and have definite frequencies. For example, the motion of the lowest vibrational mode in a drumhead is in phase all over the drumhead with a pattern that is uniform around it; the highest amplitude of the vibratory motion occurs in the middle of the drumhead. In more-complicated, higher-frequency modes, the motion on different parts of the vibrating drumhead are out of phase, with inward motion on one part at the same time that there is outward motion on another.

Schrödinger postulated that the electrons in an atom should be treated like the waves on the drumhead. The different energy levels of atoms are identified with the simple vibrational modes of the wave equation. The equation is solved to find these modes, and then the energy of an electron is obtained from the frequency of the mode and from Einstein’s quantum formula, E = hν. Schrödinger’s wave equation gives the same energies as Bohr’s original formula but with a much more-precise description of an electron in an atom (see figure). The lowest energy level of the hydrogen atom, called the ground state, is analogous to the motion in the lowest vibrational mode of the drumhead. In the atom the electron wave is uniform in all directions from the nucleus, is peaked at the centre of the atom, and has the same phase everywhere. Higher energy levels in the atom have waves that are peaked at greater distances from the nucleus. Like the vibrations in the drumhead, the waves have peaks and nodes that may form a complex shape. The different shapes of the wave pattern are related to the quantum numbers of the energy levels, including the quantum numbers for angular momentum and its orientation.

The year before Schrödinger produced his wave theory, the German physicist Werner Heisenberg published a mathematically equivalent system to describe energy levels and their transitions. In Heisenberg’s method, properties of atoms are described by arrays of numbers called matrices, which are combined with special rules of multiplication. Today physicists use both wave functions and matrices, depending on the application. Schrödinger’s picture is more useful for describing continuous electron distributions because the wave function can be more easily visualized. Matrix methods are more useful for numerical analysis calculations with computers and for systems that can be described in terms of a finite number of states, such as the spin states of the electron.

In 1929 the Norwegian physicist Egil Hylleraas applied the Schrödinger equation to the helium atom with its two electrons. He obtained only an approximate solution, but his energy calculation was quite accurate. With Hylleraas’s explanation of the two-electron atom, physicists realized that the Schrödinger equation could be a powerful mathematical tool for describing nature on the atomic level, even if exact solutions could not be obtained.

Antiparticles and the electron’s spin

The English physicist Paul Dirac introduced a new equation for the electron in 1928. Because the Schrödinger equation does not satisfy the principles of relativity, it can be used to describe only those phenomena in which the particles move much more slowly than the velocity of light. In order to satisfy the conditions of relativity, Dirac was forced to postulate that the electron would have a particular form of wave function with four independent components, some of which describe the electron’s spin. Thus, from the very beginning, the Dirac theory incorporated the electron’s spin properties. The remaining components allowed additional states of the electron that had not yet been observed. Dirac interpreted them as antiparticles, with a charge opposite to that of electrons (see animation). The discovery of the positron in 1932 by the American physicist Carl David Anderson proved the existence of antiparticles and was a triumph for Dirac’s theory.

After Anderson’s discovery, subatomic particles could no longer be considered immutable. Electrons and positrons can be created out of the vacuum, given a source of energy such as a high-energy X-ray or a collision (see photograph). They also can annihilate each other and disappear into some other form of energy. From this point, much of the history of subatomic physics has been the story of finding new kinds of particles, many of which exist for only fractions of a second after they have been created.

Advances in nuclear and subatomic physics

The 1920s witnessed further advances in nuclear physics with Rutherford’s discovery of induced radioactivity. Bombardment of light nuclei by alpha particles produced new radioactive nuclei. In 1928 the Russian-born American physicist George Gamow explained the lifetimes in alpha radioactivity using the Schrödinger equation. His explanation used a property of quantum mechanics that allows particles to “tunnel” through regions where classical physics would forbid them to be.

Structure of the nucleus

The constitution of the nucleus was poorly understood at the time because the only known particles were the electron and the proton. It had been established that nuclei are typically about twice as heavy as can be accounted for by protons alone. A consistent theory was impossible until the English physicist James Chadwick discovered the neutron in 1932. He found that alpha particles reacted with beryllium nuclei to eject neutral particles with nearly the same mass as protons. Almost all nuclear phenomena can be understood in terms of a nucleus composed of neutrons and protons. Surprisingly, the neutrons and protons in the nucleus move to a large extent in orbitals as though their wave functions were independent of one another. Each neutron or proton orbital is described by a stationary wave pattern with peaks and nodes and angular momentum quantum numbers. The theory of the nucleus based on these orbitals is called the shell nuclear model. It was introduced independently in 1948 by Maria Goeppert Mayer of the United States and Johannes Hans Daniel Jensen of West Germany, and it developed in succeeding decades into a comprehensive theory of the nucleus.

The interactions of neutrons with nuclei had been studied during the mid-1930s by the Italian-born American physicist Enrico Fermi and others. Nuclei readily capture neutrons, which, unlike protons or alpha particles, are not repelled from the nucleus by a positive charge. When a neutron is captured, the new nucleus has one higher unit of atomic mass. If a nearby isotope of that atomic mass is more stable, the new nucleus will be radioactive, convert the neutron to a proton, and assume the more-stable form.

Nuclear fission was discovered by the German chemists Otto Hahn and Fritz Strassmann in 1938 during the course of experiments initiated and explained by Austrian physicist Lise Meitner. In fission a uranium nucleus captures a neutron and gains enough energy to trigger the inherent instability of the nucleus, which splits into two lighter nuclei of roughly equal size. The fission process releases more neutrons, which can be used to produce further fissions. The first nuclear reactor, a device designed to permit controlled fission chain reactions, was constructed at the University of Chicago under Fermi’s direction, and the first self-sustaining chain reaction was achieved in this reactor in 1942. In 1945 American scientists produced the first fission bomb, also called an atomic bomb, which used uncontrolled fission reactions in either uranium or the artificial element plutonium. In 1952 American scientists used a fission explosion to ignite a fusion reaction in which isotopes of hydrogen combined thermally into heavier helium nuclei. This was the first thermonuclear bomb, also called an H-bomb, a weapon that can release hundreds or thousands of times more energy than a fission bomb.

Quantum field theory and the standard model

Dirac not only proposed the relativistic equation for the electron but also initiated the relativistic treatment of interactions between particles known as quantum field theory. The theory allows particles to be created and destroyed and requires only the presence of suitable interactions carrying sufficient energy. Quantum field theory also stipulates that the interactions can extend over a distance only if there is a particle, or field quantum, to carry the force. The electromagnetic force, which can operate over long distances, is carried by the photon, the quantum of light. Because the theory allows particles to interact with their own field quanta, mathematical difficulties arose in applying the theory.

The theoretical impasse was broken as a result of a measurement carried out in 1946 and 1947 by the American physicist Willis Eugene Lamb, Jr. Using microwave techniques developed during World War II, he showed that the hydrogen spectrum is actually about one-tenth of one percent different from Dirac’s theoretical picture. Later the German-born American physicist Polykarp Kusch found a similar anomaly in the size of the magnetic moment of the electron. Lamb’s results were announced at a famous Shelter Island Conference held in the United States in 1947; the German-born American physicist Hans Bethe and others realized that the so-called Lamb shift was probably caused by electrons and field quanta that may be created from the vacuum. The previous mathematical difficulties were overcome by Richard Feynman, Julian Schwinger, and Tomonaga Shin’ichirō, who shared the 1965 Nobel Prize for Physics, and Freeman Dyson, who showed that their various approaches were mathematically identical. The new theory, called quantum electrodynamics, was found to explain all the measurements to very high precision. Apparently, quantum electrodynamics provides a complete theory of how electrons behave under electromagnetism.

Beginning in the 1960s, similarities were found between the weak force and electromagnetism. Sheldon Glashow, Abdus Salam, and Steven Weinberg combined the two forces in the electroweak theory, for which they shared the Nobel Prize for Physics in 1979. In addition to the photon, three field quanta were predicted as additional carriers of the force—the W particle, the Z particle, and the Higgs particle. The discoveries of the W and Z particles in 1983, with correctly predicted masses, established the validity of the electroweak theory. Physicists are still searching for the much heavier Higgs particle, whose exact mass is not specified by the theory.

In all, hundreds of subatomic particles have been discovered since the first unstable particle, the muon, was identified in cosmic rays in the 1930s. By the 1960s patterns emerged in the properties and relationships among subatomic particles that led to the quark theory. Combining the electroweak theory and the quark theory, a theoretical framework called the Standard Model was constructed; it includes all known particles and field quanta. In the Standard Model there are two broad categories of particles, the leptons and the quarks. Leptons include electrons, muons, and neutrinos, and, aside from gravity, they interact only with the electroweak force.

The quarks are subject to the strong force, and they combine in various ways to make bound states. The bound quark states, called hadrons, include the neutron and the proton. Three quarks combine to form a proton, a neutron, or any of the massive hadrons known as baryons. A quark combines with an antiquark to form mesons such as the pion. Quarks have never been observed, and physicists do not expect to find one. The strength of the strong force is so great that quarks cannot be separated from each other outside hadrons. The existence of quarks has been confirmed indirectly in several ways, however. In experiments conducted with high-energy electron accelerators starting in 1967, physicists observed that some of the electrons bombarded onto proton targets were deflected at large angles. As in Rutherford’s gold-foil experiment, the large-angle deflection implies that hadrons have an internal structure containing very small charged objects. The small objects are presumed to be quarks. To accommodate quarks and their peculiar properties, physicists developed a new quantum field theory, known as quantum chromodynamics, during the mid-1970s. This theory explains qualitatively the confinement of quarks to hadrons. Physicists believe that the theory should explain all aspects of hadrons. However, mathematical difficulties in dealing with the strong interactions in quantum chromodynamics are more severe than those of quantum electrodynamics, and rigorous calculations of hadron properties have not been possible. Nevertheless, numerical calculations using the largest computers seem to confirm the validity of the theory.