History of atomic theory

Atomic theory is the scientific theory that matter is composed of particles called atoms. The definition of the word "atom" has changed over the years in response to scientific discoveries. Initially, it referred to a hypothetical concept of there being some fundamental particle of matter, too small to be seen by the naked eye, that could not be divided. Then the definition was refined to being the basic particles of the chemical elements, when chemists observed that elements seemed to combine with each other in ratios of small whole numbers. Then physicists discovered that these particles had an internal structure of their own and therefore perhaps did not deserve to be called "atoms", but renaming atoms would have been impractical by that point.

Atomic theory is one of the most important scientific developments in history, crucial to all the physical sciences. At the start of The Feynman Lectures on Physics, physicist and Nobel laureate Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept.

Philosophical atomism
The basic idea that matter is made up of tiny indivisible particles is an old idea that appeared in many ancient cultures. The word atom is derived from the ancient Greek word atomos, which means "uncuttable". This ancient idea was based in philosophical reasoning rather than scientific reasoning. Modern atomic theory is not based on these old concepts. In the early 19th century, the scientist John Dalton noticed that chemical substances seemed to combine with each other by discrete and consistent units of weight, and he decided to use the word atom to refer to these units.

Groundwork
Near the end of the 18th century, a number of important developments in chemistry emerged without referring to the notion of an atomic theory. The first was Antoine Lavoisier redefining an element as a substance which scientists could not decompose into simpler substances by experimentation. This brought an end to the ancient idea of the elements of matter being fire, earth, air, and water, which had no experimental support. Lavoisier showed that water can be decomposed into hydrogen and oxygen, which in turn he could not decompose into anything simpler, thereby proving these are elements. Lavoisier also defined the law of conservation of mass, which states that in a chemical reaction, matter does not appear nor disappear into thin air; the total mass remains the same even if the substances involved were transformed. Finally, there was the law of definite proportions, established by the French chemist Joseph Proust in 1797, which states that if a compound is broken down into its constituent chemical elements, then the masses of those constituents will always have the same proportions by weight, regardless of the quantity or source of the original compound. This definition distinguished compounds from mixtures.

Dalton's law of multiple proportions
John Dalton studied data gathered by himself and by other scientists. He noticed a pattern that later came to be known as the law of multiple proportions: in compounds which contain two particular elements, the amount of Element A per measure of Element B will differ across these compounds by ratios of small whole numbers. This suggested that each element combines with other elements in multiples of a basic quantity.

In 1804, Dalton explained his atomic theory to his friend and fellow chemist Thomas Thomson, who published an explanation of Dalton's theory in his book A System of Chemistry in 1807. According to Thomson, Dalton's idea first occurred to him when experimenting with "olefiant gas" (ethylene) and "carburetted hydrogen gas" (methane). Dalton found that "carburetted hydrogen gas" contains twice as much hydrogen per measure of carbon as "olefiant gas", and concluded that a molecule of "olefiant gas" is one carbon atom and one hydrogen atom, and a molecule of "carburetted hydrogen gas" is one carbon atom and two hydrogen atoms. In reality, an ethylene molecule has two carbon atoms and four hydrogen atoms (C2H4), and a methane molecule has one carbon atom and four hydrogen atoms (CH4). In this particular case, Dalton was mistaken about the formulas of these compounds, and it wasn't his only mistake. But in other cases, he got their formulas right. The following examples come from Dalton's own books A New System of Chemical Philosophy (in two volumes, 1808 and 1817):

Example 1 — tin oxides: Dalton identified two types of tin oxide. One is a grey powder that Dalton referred to as "the protoxide of tin", which is 88.1% tin and 11.9% oxygen. The other is a white powder which Dalton referred to as "the deutoxide of tin", which is 78.7% tin and 21.3% oxygen. Adjusting these figures, in the grey powder there is about 13.5 g of oxygen for every 100 g of tin, and in the white powder there is about 27 g of oxygen for every 100 g of tin. 13.5 and 27 form a ratio of 1:2. These compounds are known today as tin(II) oxide (SnO) and tin(IV) oxide (SnO2). In Dalton's terminology, a "protoxide" is a molecule containing a single oxygen atom, and a "deutoxide" molecule has two. The modern equivalents of his terms would be monoxide and dioxide, but these are not used for tin oxides as they are actually crystals; they do not exist in molecular form.

Example 2 — iron oxides: Dalton identified two oxides of iron. There is one type of iron oxide that is a black powder which Dalton referred to as "the protoxide of iron", which is 78.1% iron and 21.9% oxygen. The other iron oxide is a red powder, which Dalton referred to as "the intermediate or red oxide of iron" which is 70.4% iron and 29.6% oxygen. Adjusting these figures, in the black powder there is about 28 g of oxygen for every 100 g of iron, and in the red powder there is about 42 g of oxygen for every 100 g of iron. 28 and 42 form a ratio of 2:3. These compounds are iron(II) oxide and iron(III) oxide and their formulas are Fe2O2 and Fe2O3 respectively (iron(II) oxide's formula is normally written as FeO, but here it is written as Fe2O2 to contrast it with the other oxide). Dalton described the "intermediate oxide" as being "2 atoms protoxide and 1 of oxygen", which adds up to two atoms of iron and three of oxygen. That averages to one and a half atoms of oxygen for every iron atom, putting it midway between a "protoxide" and a "deutoxide".

Example 3 — nitrogen oxides: Dalton was aware of three oxides of nitrogen: "nitrous oxide", "nitrous gas", and "nitric acid". These compounds are known today as nitrous oxide, nitric oxide, and nitrogen dioxide respectively. "Nitrous oxide" is 63.3% nitrogen and 36.7% oxygen, which means it has 80 g of oxygen for every 140 g of nitrogen. "Nitrous gas" is 44.05% nitrogen and 55.95% oxygen, which means there is 160 g of oxygen for every 140 g of nitrogen. "Nitric acid" is 29.5% nitrogen and 70.5% oxygen, which means it has 320 g of oxygen for every 140 g of nitrogen. 80 g, 160 g, and 320 g form a ratio of 1:2:4. The formulas for these compounds are N2O, NO, and NO2.

Dalton defined an atom as being the "ultimate particle" of a chemical substance, and he used the term "compound atom" to refer to "ultimate particles" which contain two or more elements. This is inconsistent with the modern definition, wherein an atom is the basic particle of a chemical element and a molecule is an agglomeration of atoms. The term "compound atom" was confusing to some of Dalton's contemporaries as the word "atom" implies indivisibility, but he responded that if a carbon dioxide "atom" is divided, it ceases to be carbon dioxide. The carbon dioxide "atom" is indivisible in the sense that it cannot be divided into smaller carbon dioxide particles.

Dalton made the following assumptions on how "elementary atoms" combined to form "compound atoms" (what we today refer to as molecules). When two elements can only form one compound, he assumed it was one atom of each, which he called a "binary compound". If two elements can form two compounds, the first compound is a binary compound and the second is a "ternary compound" consisting of one atom of the first element and two of the second. If two elements can form three compounds between them, then the third compound is a "quaternary" compound containing one atom of the first element and three of the second. Dalton thought that water was a "binary compound", i.e. one hydrogen atom and one oxygen atom. Dalton did not know that in their natural gaseous state, the ultimate particles of oxygen, nitrogen, and hydrogen exist in pairs (O2, N2, and H2). Nor was he aware of valencies. These properties of atoms were discovered later in the 19th century.

Because atoms were too small to be directly weighed using the methods of the 19th century, Dalton instead expressed the weights of the myriad atoms as multiples of the hydrogen atom's weight, which Dalton knew was the lightest element. By his measurements, 7 grams of oxygen will combine with 1 gram of hydrogen to make 8 grams of water with nothing left over, and assuming a water molecule to be one oxygen atom and one hydrogen atom, he concluded that oxygen's atomic weight is 7. In reality it is 16. Aside from the crudity of early 19th century measurement tools, the main reason for this error was that Dalton didn't know that the water molecule in fact has two hydrogen atoms, not one. Had he known, he would have doubled his estimate to a more accurate 14. This error was corrected in 1811 by Amedeo Avogadro. Avogadro proposed that equal volumes of any two gases, at equal temperature and pressure, contain equal numbers of molecules (in other words, the mass of a gas's particles does not affect the volume that it occupies). Avogadro's hypothesis, now usually called Avogadro's law, provided a method for deducing the relative weights of the molecules of gaseous elements, for if the hypothesis is correct relative gas densities directly indicate the relative weights of the particles that compose the gases. This way of thinking led directly to a second hypothesis: the particles of certain elemental gases were pairs of atoms, and when reacting chemically these molecules often split in two. For instance, the fact that two liters of hydrogen will react with just one liter of oxygen to produce two liters of water vapor (at constant pressure and temperature) suggested that a single oxygen molecule splits in two in order to form two molecules of water. The formula of water is H2O, not HO. Avogadro measured oxygen's atomic weight to be 15.074.

Opposition to atomic theory
Dalton's atomic theory attracted widespread interest but not everyone accepted it at first. The law of multiple proportions was shown not to be a universal law when it came to organic substances, whose molecules can be quite large. For instance, in oleic acid there is 34 g of hydrogen for every 216 g of carbon, and in methane there is 72 g of hydrogen for every 216 g of carbon. 34 and 72 form a ratio of 17:36, which is not a ratio of small whole numbers. We know now that carbon-based substances can have very large molecules, larger than any the other elements can form. Oleic acid's formula is C18H34O2 and methane's is CH4. The law of multiple proportions by itself was not complete proof, and atomic theory was not universally accepted until the end of the 19th century.

One problem was the lack of uniform nomenclature. The word "atom" implied indivisibility, but Dalton defined an atom as being the ultimate particle of any chemical substance, not just the elements or even matter per se. This meant that "compound atoms" such as carbon dioxide could be divided, as opposed to "elementary atoms". Dalton disliked the word "molecule", regarding it as "diminutive". Amadeo Avogadro did the opposite: he exclusively used the word "molecule" in his writings, eschewing the word "atom", instead using the term "elementary molecule". Jöns Jacob Berzelius used the term "organic atoms" to refer to particles containing three or more elements, because he thought this only existed in organic compounds. Jean-Baptiste Dumas used the terms "physical atoms" and "chemical atoms"; a "physical atom" was a particle that cannot be divided by physical means such as temperature and pressure, and a "chemical atom" was a particle that could not be divided by chemical reactions.

The modern definitions of atom and molecule—an atom being the basic particle of an element, and a molecule being an agglomeration of atoms—were established in the late half of the 19th century. A key event was the Karlsruhe Congress in Germany in 1860. As the first international congress of chemists, its goal was to establish some standards in the community. A major proponent of the modern distinction between atoms and molecules was Stanislao Cannizzaro.

"The various quantities of a particular element involved in the constitution of different molecules are integral multiples of a fundamental quantity that always manifests itself as an indivisible entity and which must properly be named atom."

Cannizzaro criticized past chemists such as Berzelius for not accepting that the particles of certain gaseous elements are actually pairs of atoms, which led to mistakes in their formulation of certain compounds. Berzelius believed that hydrogen gas and chlorine gas particles are solitary atoms. But he observed that when one liter of hydrogen reacts with one liter of chlorine, they form two liters of hydrogen chloride instead of one. Berzelius decided that Avogadro's law does not apply to compounds. Cannizzaro preached that if scientists just accepted the existence of single-element molecules, such discrepancies in their findings would be easily resolved. But Berzelius did not even have a word for that. Berzelius used the term "elementary atom" for a gas particle which contained just one element and "compound atom" for particles which contained two or more elements, but there was nothing to distinguish H2 from H since Berzelius did not believe in H2. So Cannizzaro called for a redefinition so that scientists could understand that a hydrogen molecule can split into two atoms in the course of a chemical reaction.

A second objection to atomic theory was philosophical. Scientists in the 19th century had no way of directly observing atoms. They inferred the existence of atoms through indirect observations, such as Dalton's law of multiple proportions. Some scientists, notably those who ascribed to the school of positivism, argued that scientists should not attempt to deduce the deeper reality of the universe, but only systemize what patterns they could directly observe. The anti-atomists argued that while atoms might be a useful abstraction for predicting how elements react, they do not reflect concrete reality.

Such scientists were sometimes known as "equivalentists", because they preferred the theory of equivalent weights, which is a generalization of Proust's law of definite proportions. For example, 1 gram of hydrogen will combine with 8 grams of oxygen to form 9 grams of water, therefore the "equivalent weight" of oxygen is 8 grams. This position was eventually quashed by two important advancements that happened later in the 19th century: the development of the periodic table and the discovery that molecules have an internal architecture that determines their properties.

Isomerism
Scientists discovered some substances have the exact same chemical content but different properties. For instance, in 1827, Friedrich Wöhler discovered that silver fulminate and silver cyanate are both 107 parts silver, 12 parts carbon, 14 parts nitrogen, and 16 parts oxygen (we now know their formulas as both AgCNO). In 1830 Jöns Jacob Berzelius introduced the term isomerism to describe the phenomenon. In 1860, Louis Pasteur hypothesized that the molecules of isomers might have the same set of atoms but in different arrangements.

In 1874, Jacobus Henricus van 't Hoff proposed that the carbon atom bonds to other atoms in a tetrahedral arrangement. Working from this, he explained the structures of organic molecules in such a way that he could predict how many isomers a compound could have. Consider, for example, pentane (C5H12). In van 't Hoff's way of modelling molecules, there are three possible configurations for pentane, and scientists did go on to discover three and only three isomers of pentane.

Isomerism was not something that could be fully explained by alternative theories to atomic theory, such as radical theory and the theory of types.

Mendeleev's periodic table
Dmitrii Mendeleev noticed that when he arranged the elements in a row according to their atomic weights, there was a certain periodicity to them. For instance, the second element, lithium, had similar properties to the ninth element, sodium, and the sixteenth element, potassium — a period of seven. Likewise, beryllium, magnesium, and calcium were similar and all were seven places apart from each other on Mendeleev's table. Using these patterns, Mendeleev predicted the existence and properties of new elements, which were later discovered in nature: scandium, gallium, and germanium. Moreover, the periodic table could predict how many atoms of other elements that an atom could bond with — e.g., germanium and carbon are in the same group on the table and their atoms both combine with two oxygen atoms each (GeO2 and CO2). Mendeleev found these patterns validated atomic theory because it showed that the elements could be categorized by their atomic weight. Inserting a new element into the middle of a period would break the parallel between that period and the next, and would also violate Dalton's law of multiple proportions.

In the modern periodic table, the periodicity of the elements mentioned above is eight rather than seven because the noble gases were not known back when Mendeleev devised his table. The rows also now have different lengths (2, 8, 18, and 32) which fits with quantum theory.

The elements on the periodic table were generally arranged in order of increasing atomic weight. However, in a number of places chemists chose to swap the positions of certain adjacent elements so that they appeared in a group with other elements with similar properties. For instance, tellurium is placed before iodine even though tellurium is heavier (127.6 vs 126.9) so that iodine can be in the same column as the other halogens. In 1913, Henry Moseley discovered that atoms of each element, when excited, emit X-rays at a frequency proportional to the element's position on the adjusted periodic table, which validated these adjustments.

Statistical mechanics
In order to introduce the ideal gas law and statistical forms of physics, it was necessary to postulate the existence of atoms. In 1738, Swiss physicist and mathematician Daniel Bernoulli postulated that the pressure of gases and heat were both caused by the underlying motion of molecules.

In 1860, James Clerk Maxwell, who was a vocal proponent of atomism, was the first to use statistical mechanics in physics. Ludwig Boltzmann and Rudolf Clausius expanded his work on gases and the laws of thermodynamics especially the second law relating to entropy. In the 1870s, Josiah Willard Gibbs extended the laws of entropy and thermodynamics and coined the term "statistical mechanics."

At the beginning of the 20th century, Albert Einstein independently reinvented Gibbs' laws, because they had only been printed in an obscure American journal. Einstein later commented that had he known of Gibbs' work, he would "not have published those papers at all, but confined myself to the treatment of some few points [that were distinct]." All of statistical mechanics and the laws of heat, gas, and entropy took the existence of atoms as a necessary postulate.

Brownian motion
In 1827, the British botanist Robert Brown observed that dust particles inside pollen grains floating in water constantly jiggled about for no apparent reason. In 1905, Einstein theorized that this Brownian motion was caused by the water molecules continuously knocking the grains about, and developed a mathematical model to describe it. This model was validated experimentally in 1908 by French physicist Jean Perrin, who used Einstein's equations to measure the size of atoms.

Discovery of the electron
Atoms were thought to be the smallest possible division of matter until 1897 when J. J. Thomson discovered the electron through his work on cathode rays.

A Crookes tube is a sealed glass container in which two electrodes are separated by a vacuum. When a voltage is applied across the electrodes, cathode rays are generated, creating a glowing patch where they strike the glass at the opposite end of the tube. Through experimentation, Thomson discovered that the rays could be deflected by electric fields and magnetic fields, which meant that these rays were not a form of light but were composed of very light charged particles, and their charge was negative. Thomson called these particles "corpuscles". He measured their mass-to-charge ratio to be several orders of magnitude smaller than that of the hydrogen atom, the smallest atom. This ratio was the same regardless of what the electrodes were made of and what the trace gas in the tube was.

In contrast to those corpuscles, positive ions created by electrolysis or X-ray radiation had mass-to-charge ratios that varied depending on the material of the electrodes and the type of gas in the reaction chamber, indicating they were different kinds of particles.

In 1898, Thomson measured the charge on ions to be roughly 6 × 10-10 electrostatic units (2 × 10-19 Coulombs). In 1899, he showed that negative electricity created by ultraviolet light landing on a metal (known now as the photoelectric effect) has the same mass-to-charge ratio as cathode rays; then he applied his previous method for determining the charge on ions to the negative electric particles created by ultraviolet light. By this combination he showed that electron's mass was 0.0014 times that of hydrogen ions. These "corpuscles" were so light yet carried so much charge that Thomson concluded they must be the basic particles of electricity, and for that reason other scientists decided that these "corpuscles" should instead be called electrons following an 1894 suggestion by George Johnstone Stoney for naming the basic unit of electrical charge.

In 1904, Thomson published a paper describing a new model of the atom. Electrons reside within atoms, and they transplant themselves from one atom to the next in a chain in the action of an electrical current. When electrons do not flow, their negative charge logically must be balanced out by some source of positive charge within the atom so as to render the atom electrically neutral. Having no clue as to the source of this positive charge, Thomson tentatively proposed that the positive charge is everywhere in the atom, the atom being shaped like a sphere. The balance of electrostatic forces would distribute the electrons throughout this sphere in a more or less even manner. Thomson further explained that ions are atoms that have a surplus or shortage of electrons.

Thomson's model is popularly known as the plum pudding model, based on the idea that the electrons are distributed throughout the sphere of positive charge with the same density as raisins in a plum pudding. Neither Thomson nor his colleagues ever used this analogy. It seems to have been a conceit of popular science writers. The analogy suggests that the positive sphere is like a solid, but Thomson likened it to a liquid, as he proposed that the electrons moved around in it in patterns governed by the electrostatic forces. Thomson's model was incomplete, it could not predict any of the known properties of the atom such as emission spectra or valencies.

In 1906, Robert A. Millikan and Harvey Fletcher performed the oil drop experiment in which they measured the charge of an electron to be about -1.6 × 10-19, a value now defined as -1 e. Since the hydrogen ion and the electron were known to be indivisible and a hydrogen atom is neutral in charge, it followed that the positive charge in hydrogen was equal to this value, i.e. 1 e.

Discovery of the nucleus
Thomson's plum pudding model was supplanted in 1911 by one of his former students, Ernest Rutherford, who discovered that the positive charge and most of the mass of an atom is concentrated in a very small fraction of its volume, which he assumed to be at the very center.

Ernest Rutherford and his colleagues Hans Geiger and Ernest Marsden came to have doubts about the Thomson model after they encountered difficulties when they tried to build an instrument to measure the charge-to-mass ratio of alpha particles (these are positively-charged particles emitted by certain radioactive substances such as radium). The alpha particles were being scattered by the air in the detection chamber, which made the measurements unreliable. Thomson had encountered a similar problem in his work on cathode rays, which he solved by creating a near-perfect vacuum in his instruments. Rutherford didn't think he'd run into this same problem because alpha particles are much heavier than electrons. According to Thomson's model of the atom, the positive charge in the atom is not concentrated enough to produce an electric field strong enough to deflect an alpha particle, and the electrons are so lightweight they should be pushed aside effortlessly by the much heavier alpha particles. Yet there was scattering, so Rutherford and his colleagues decided to investigate this scattering carefully.

Between 1908 and 1913, Rutherford and his colleagues performed a series of experiments in which they bombarded thin foils of metal with a beam of alpha particles. They spotted alpha particles being deflected by angles greater than 90°. According to Thomson's model, all of the alpha particles should have passed through with negligible deflection. Rutherford deduced that the positive charge of the atom is not distributed throughout the atom's volume as Thomson believed, but is concentrated in a tiny nucleus at the center, and that the nucleus also has most of the atom's mass. Only such an intense concentration of charge, anchored by its high mass, could produce an electric field strong enough to deflect the alpha particles as observed. Rutherford's model is sometimes called the "planetary model".

Bohr model
The planetary model of the atom had two significant shortcomings. The first is that, unlike planets orbiting a sun, electrons are charged particles. An accelerating electric charge is known to emit electromagnetic waves according to the Larmor formula in classical electromagnetism. An orbiting charge should steadily lose energy and spiral toward the nucleus, colliding with it in a small fraction of a second. The second problem was that the planetary model could not explain the highly peaked emission and absorption spectra of atoms that were observed.

Quantum theory revolutionized physics at the beginning of the 20th century, when Max Planck and Albert Einstein postulated that light energy is emitted or absorbed in discrete amounts known as quanta (singular, quantum). This led to a series of quantum atomic models such as the quantum model of Arthur Erich Haas in 1910 and the 1912 John William Nicholson quantum atomic model that quantized angular momentum as h/2$\pi$. In 1913, Niels Bohr incorporated this idea into his Bohr model of the atom, in which an electron could only orbit the nucleus in particular circular orbits with fixed angular momentum and energy, its distance from the nucleus (i.e., their radii) being proportional to its energy. Under this model an electron could not spiral into the nucleus because it could not lose energy in a continuous manner; instead, it could only make instantaneous "quantum leaps" between the fixed energy levels. When this occurred, light was emitted or absorbed at a frequency proportional to the change in energy (hence the absorption and emission of light in discrete spectra).

Bohr's model was not perfect. It could only predict the spectral lines of hydrogen, not those of multielectron atoms. Worse still, it could not even account for all features of the hydrogen spectrum: as spectrographic technology improved, it was discovered that applying a magnetic field caused spectral lines to multiply in a way that Bohr's model couldn't explain. In 1916, Arnold Sommerfeld added elliptical orbits to the Bohr model to explain the extra emission lines, but this made the model very difficult to use, and it still couldn't explain more complex atoms.

Discovery of isotopes
While experimenting with the products of radioactive decay, in 1913 radiochemist Frederick Soddy discovered that there appeared to be more than one variety of some elements. The term isotope was coined by Margaret Todd as a suitable name for these varieties.

That same year, J. J. Thomson conducted an experiment in which he channeled a stream of neon ions through magnetic and electric fields, striking a photographic plate at the other end. He observed two glowing patches on the plate, which suggested two different deflection trajectories. Thomson concluded this was because some of the neon ions had a different mass. The nature of this differing mass would later be explained by the discovery of neutrons in 1932: all atoms of the same element contain the same number of protons, while different isotopes have different numbers of neutrons.

Discovery of the proton
Back in 1815, William Prout observed that the atomic weights of the known elements were multiples of hydrogen's atomic weight, so he hypothesized that all atoms are agglomerations of hydrogen, a particle which he dubbed "the protyle". Prout's hypothesis was put into doubt when some elements were found to deviate from this pattern—e.g. chlorine atoms on average weigh 35.45 daltons—but when isotopes were discovered in 1913, Prout's observation gained renewed attention.

In 1898, J. J. Thomson found that the positive charge of a hydrogen ion was equal to the negative charge of a single electron.

In an April 1911 paper concerning his studies on alpha particle scattering, Ernest Rutherford estimated that the charge of an atomic nucleus, expressed as a multiplier of hydrogen's nuclear charge (qe), is roughly half the atom's atomic weight.

In June 1911, the Dutch physicist Antonius van den Broek noted that on the periodic table, each successive chemical element increased in atomic weight on average by 2, which in turn suggested that each successive element's nuclear charge increased by 1 qe.

In 1913, Henry Moseley measured the X-ray emissions of all the elements on the periodic table and found that the frequency of the X-ray emissions was a mathematical function of the element's atomic number and the charge of a hydrogen nucleus (see Moseley's law).

In 1917 Rutherford bombarded nitrogen gas with alpha particles and observed hydrogen ions being emitted from the gas. Rutherford concluded that the alpha particles struck the nuclei of the nitrogen atoms, causing hydrogen ions to split off.

These observations led Rutherford to conclude that the hydrogen nucleus was a singular particle with a positive charge equal to that of the electron's negative charge. The name "proton" was suggested by Rutherford at an informal meeting of fellow physicists in Cardiff in 1920. All atomic nuclei contain a number of protons equal to the respective element's atomic number. The atomic number had up to that point been defined as an element's position on the periodic table.

Discovery of the neutron
Physicists in the 1920s believed that the atomic nucleus contained protons plus a number of "nuclear electrons" that reduced the overall charge. These "nuclear electrons" were distinct from the electrons that orbited the nucleus. This incorrect hypothesis would have explained why the atomic numbers of the elements were less than their atomic weights, and why radioactive elements emit electrons (beta radiation) in the process of nuclear decay. Rutherford even hypothesized that a proton and an electron could bind tightly together into a "neutral doublet". Rutherford wrote that the existence of such "neutral doublets" moving freely through space would provide a more plausible explanation for how the heavier elements could have formed in the genesis of the Universe, given that it is hard for a lone proton to fuse with a large atomic nucleus because of the repulsive electric field.

In 1928, Walter Bothe observed that beryllium emitted a highly penetrating, electrically neutral radiation when bombarded with alpha particles. It was later discovered that this radiation could knock hydrogen atoms out of paraffin wax. Initially it was thought to be high-energy gamma radiation, since gamma radiation had a similar effect on electrons in metals, but James Chadwick found that the ionization effect was too strong for it to be due to electromagnetic radiation, so long as energy and momentum were conserved in the interaction. In 1932, Chadwick exposed various elements, such as hydrogen and nitrogen, to the mysterious "beryllium radiation", and by measuring the energies of the recoiling charged particles, he deduced that the radiation was actually composed of electrically neutral particles which could not be massless like the gamma ray, but instead were required to have a mass similar to that of a proton. Chadwick called this new particle "the neutron" and believed that it to be a proton and electron fused together because the neutron had about the same mass as a proton and an electron's mass is negligible by comparison. Neutrons are not in fact a fusion of a proton and an electron.

Modern quantum mechanical models
In 1924, Louis de Broglie proposed that all particles—particularly subatomic particles such as electrons—have an associated wave. Erwin Schrödinger, fascinated by this idea, developed an equation that describes an electron as a wave function instead of a point. This approach predicted many of the spectral phenomena that Bohr's model failed to explain, but it was difficult to visualize, and faced opposition. One of its critics, Max Born, proposed instead that Schrödinger's wave function did not describe the physical extent of an electron (like a charge distribution in classical electromagnetism), but rather gave the probability that an electron would, when measured, be found at a particular point. This reconciled the ideas of wave-like and particle-like electrons: the behavior of an electron, or of any other subatomic entity, has both wave-like and particle-like aspects, and whether one aspect or the other is observed depend upon the experiment.

A consequence of describing particles as waveforms rather than points is that it is mathematically impossible to calculate with precision both the position and momentum of a particle at a given point in time. This became known as the uncertainty principle, a concept first introduced by Werner Heisenberg in 1927.

Schrödinger's wave model for hydrogen replaced Bohr's model, with its neat, clearly defined circular orbits. The modern model of the atom describes the positions of electrons in an atom in terms of probabilities. An electron can potentially be found at any distance from the nucleus, but, depending on its energy level and angular momentum, exists more frequently in certain regions around the nucleus than others; this pattern is referred to as its atomic orbital. The orbitals come in a variety of shapes—sphere, dumbbell, torus, etc.—with the nucleus in the middle. The shapes of atomic orbitals are found by solving the Schrödinger equation. Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the hydrogen atom and the hydrogen molecular ion. Beginning with the helium atom—which contains just two electrons—numerical methods are used to solve the Schrödinger equation.

Qualitatively the shape of the atomic orbitals of multi-electron atoms resemble the states of the hydrogen atom. The Pauli principle requires the distribution of these electrons within the atomic orbitals such that no more than two electrons are assigned to any one orbital; this requirement profoundly affects the atomic properties and ultimately the bonding of atoms into molecules.