An isolated atom can be described in terms of certain discrete states called quantum states. Each quantum state has a definite energy associated with it, but several quantum states can have the same energy. These quantum states and their energy levels are calculated from the basic principles of quantum mechanics. For the simplest atom, hydrogen, which consists of a single proton and a single electron, the energy levels have been calculated and tested to an uncertainty of better than one part in 10¹¹, but for atoms with many electrons, the accuracy of the calculations may not be much better than a few percent of the energy of the levels.

Atomic energy levels are typically measured by observing transitions between two levels. For example, an atom in its lowest possible energy state (called the ground state) can be excited to a higher state only if energy is added by an amount that is equal to the difference between the two levels. Thus, by measuring the energy of the radiation that has been absorbed by the atom, the difference in its energy levels can be determined. The energy levels are identical for atoms of the same type; allowed energies of a particular atom of silver are equal to those for any other atom of the same isotope of silver.

Other isolated systems, including molecules, ions (charged atoms or molecules), and atomic nuclei, have discrete allowed energies. The analysis of these simple systems is carried out with techniques that are analogous to those that were first applied to simple atomic spectra. More complex structures, such as clusters of atoms, and bulk condensed matter, such as solids and liquids, also have energy levels describable by quantum mechanics. The energy levels in these complex systems, however, are so closely spaced that they smear into a continuous band of energies. Transitions between these bands allow researchers to discern many important properties of a given material. The location and properties of the energy states are often referred to as the electronic structure of the material. By comparing spectroscopic measurements to quantum mechanical calculations based on an assumed model of the material, one can use knowledge of a material’s electronic structure to determine its physical structure.

If an atom in its ground state is given some amount of energy so that it is promoted to an excited state, the atom will release that extra energy spontaneously as it moves back into lower states, eventually returning to the ground state. For an isolated atom, the energy is emitted as electromagnetic radiation. The emitted energy E equals the upper-state energy minus the lower-state energy; this energy is usually carried by a single quantum of light (a photon) having a frequency ν in which photon energy (E) is equal to a constant times the frequency, E = hν, where h, Planck’s constant, equals 6.626 × 10⁻³⁴ joule second. This relationship determines the frequencies (and wavelengths, because λ = c/ν) of light emitted by atoms if the energies of the states are known. Conversely, the relationship allows the energy states of an atom to be determined from measurements of its frequency or wavelength spectrum. The analysis of the discrete wavelengths emitted or absorbed by an atom or molecule was historically carried out using prism or grating spectrometers; because of the appearance of the separated light in these instruments, these discrete wavelengths are sometimes called spectral lines.

Historical survey

The basis for analytical spectroscopy is the discovery, made in 1859 by the German physicist Gustav R. Kirchhoff, that each pure substance has its own characteristic spectrum. Another German physicist, Joseph von Fraunhofer, repeating more carefully an earlier experiment by a British scientist, William Wollaston, had shown in 1814 that the spectrum of the Sun’s electromagnetic radiation does not grade smoothly from one colour to the next but has many dark lines, indicating that light is missing at certain wavelengths because of absorption. These dark lines, sometimes called Fraunhofer lines, are also collectively referred to as an absorption spectrum. The spectra of materials that were heated in flames or placed in electric-gas discharges were studied by many scientists during the 18th and 19th centuries. These spectra were composed of numerous bright discrete lines, indicating that only certain wavelengths were present in the emitted light. They are called brightline, or emission, spectra.

Although the possibility that each chemical element has a unique characteristic spectrum had been considered by numerous investigators, the early studies were hampered by the difficulty of obtaining relatively pure substances. Any sample could contain impurities that would result in the simultaneous production of many spectra. By using carefully purified substances, Kirchhoff demonstrated characteristic spectra and initiated the technique of spectroscopic analysis of the chemical composition of matter. The technique was applied by Kirchhoff and his colleague the German chemist Robert Bunsen in 1861 to the analysis of the Sun’s electromagnetic spectrum and the identification of the chemical elements in the Sun.

Before the 20th century, there was no theory that could satisfactorily explain the origin of the spectra of the elements or the reason why different elements have different spectra. The quantitative understanding of the elemental spectra needed the development of a fundamentally new physical theory, and the spectra of the simplest atoms played the key role in the development of this theory. Many of the major developments in 20th-century physics were motivated by an ever-increasing accuracy in the measurement of the spectra of the hydrogen atom; highlights include the discovery in 1885 by the Swiss scientist Johann J. Balmer that the frequency spectrum of hydrogen followed a simple numerical pattern, later revised by the Swedish physicist Johannes R. Rydberg and given in modern notation as 1/λ = R_H (1/2² − 1/n²), where R_H is the so-called Rydberg constant for hydrogen. In 1913 the Danish physicist Niels Bohr presented the first theoretical model that could give quantized energy levels that were in quantitative agreement with measurements of the hydrogen spectrum.

Despite the success of the Bohr theory in describing the hydrogen spectrum, the theory failed badly when applied to the next simplest atom, helium, which contains two electrons. It was also incapable of predicting the likelihood of transitions between energy levels. In 1925–26 a new theory that could explain the discrete, quantum nature of the spectra was developed by the German physicists Werner Heisenberg and Erwin Schrödinger. This theory, known as quantum mechanics, was extended by the Austrian-born Swiss physicist Wolfgang Pauli, the German physicist Max Born, and others. It has been remarkably successful in describing the spectra of complex atoms, ions, simple molecules, and solids.

As the spectral lines of the hydrogen atom were measured with increased accuracy, greater demands were placed on the theoretical understanding of atomic spectra. The British physicist Paul A.M. Dirac combined quantum mechanics with the special theory of relativity in 1928 to describe particles moving close to the speed of light. His formulation of relativistic quantum mechanics provided an explanation for the so-called fine structure of the hydrogen spectrum (see below Foundations of atomic spectra: Hydrogen atom states: Fine and hyperfine structure of spectra). At still higher resolution, two energy levels of the hydrogen atom in the first excited state were predicted by Dirac’s theory to be exactly the same. In 1947, the American physicists Willis Lamb and Robert Retherford discovered that the levels actually differ by roughly 10⁹ hertz (see below X-ray and radio-frequency spectroscopy: Radio-frequency spectroscopy: Methods). In contrast, the transition frequency between the ground state and the first excited states was calculated as approximately 2.5 × 10¹⁵ hertz. Two American physicists, Richard Feynman and Julian Schwinger, and a Japanese physicist, Shinichirō Tomonaga, developed yet another refinement to quantum mechanics to explain this measurement. The theory, known as quantum electrodynamics (QED), had its foundations in the discoveries of Dirac, Heisenberg, and Pauli. It is a complete description of the interaction of radiation with matter and has been used to calculate the energy levels of the hydrogen atom to an accuracy of better than 1 part in 10¹¹. No other physical theory has the ability to predict a measurable quantity with such precision, and, as a result of the successes of quantum electrodynamics, the theory has become the paradigm of physical theories at the microscopic level.