Reciprocal length or inverse length is a quantity or

measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...

used in several branches of

science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

and

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, defined as the reciprocal of

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

. Common units used for this measurement include the reciprocal metre or inverse metre (symbol: m⁻¹), the reciprocal centimetre or inverse centimetre (symbol: cm⁻¹). In

optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

, the dioptre is a unit equivalent to reciprocal metre.

List of quantities

Quantities Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...

measured in reciprocal length include: * absorption coefficient or attenuation coefficient, in

materials science Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials sci ...

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

of a line, in mathematics * gain, in laser physics * magnitude of vectors in

reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray diffraction, X-ray and Electron diffraction, electron diffraction as well as the Electronic band structure, e ...

, in

crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

* more generally any

spatial frequency In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier tra ...

e.g. in cycles per unit length *

optical power In optics, optical power (also referred to as dioptric power, refractive power, focal power, focusing power, or convergence power) is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the ...

of a

lens A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements'') ...

, in optics *

rotational constant In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special r ...

of a rigid rotor, in

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

wavenumber In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of ...

, or magnitude of a wavevector, in

spectroscopy Spectroscopy is the field of study that measures and interprets electromagnetic spectra. In narrower contexts, spectroscopy is the precise study of color as generalized from visible light to all bands of the electromagnetic spectrum. Spectro ...

* density of a linear feature in

hydrology Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...

and other fields; see kilometre per square kilometre * surface area to volume ratio

Measure of energy

In some branches of physics, a set of natural units is adopted, such that the universal constants ''c'', the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

, and ''ħ'', the

reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

, are treated as being unity (i.e. that ''c'' = ''ħ'' = 1), which leads to mass, energy, momentum, frequency and reciprocal length all having the same unit. As a result, reciprocal length is used as a measure of

energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

. The

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

of a

photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...

yields a certain photon energy, according to the

Planck–Einstein relation The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck–Einstein relation, Planck equation, and Plan ...

, and the frequency of a photon is related to its

via the speed of light. Spatial frequency is a reciprocal length, which can thus be used as a measure of energy, usually of a particle. For example, the reciprocal centimetre, , is an energy unit equal to the energy of a photon with a

wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...

of 1 cm. That energy amounts to approximately or . The energy is inversely proportional to the size of the unit of which the reciprocal is used, and is proportional to the number of reciprocal length units. For example, in terms of energy, one reciprocal metre equals (one hundredth) as much as a reciprocal centimetre. Five reciprocal metres are five times as much energy as one reciprocal metre.

List of quantities

Measure of energy

See also

Further reading