Reciprocal length orinverse length is aquantity ormeasurement used in several branches ofscience andmathematics, defined as thereciprocal oflength.

Common units used for this measurement include thereciprocal metre orinverse metre (symbol:m⁻¹), thereciprocal centimetre orinverse centimetre (symbol:cm⁻¹).Inoptics, thedioptre is a unit equivalent to reciprocal metre.

Quantities measured in reciprocal length include:

• absorption coefficient orattenuation coefficient, inmaterials science
• curvature of aline, in mathematics
• gain, inlaser physics
• magnitude ofvectors inreciprocal space, incrystallography
• more generally anyspatial frequency e.g. in cycles per unit length
• optical power of alens, in optics
• rotational constant of arigid rotor, inquantum mechanics
• wavenumber, or magnitude of awavevector, inspectroscopy
• density of a linear feature inhydrology and other fields; seekilometre per square kilometre
• surface area to volume ratio

In some branches of physics, a set ofnatural units is adopted, such that the universal constantsc, thespeed of light, andħ, thereduced Planck constant, are treated as being unity (i.e. thatc =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 ofenergy. Thefrequency of aphoton yields a certainphoton energy, according to thePlanck–Einstein relation, and the frequency of a photon is related to itsspatial frequency 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, thereciprocal centimetre,cm⁻¹, is an energy unit equal to the energy of a photon with awavelength of 1 cm. That energy amounts to approximately1.24×10⁻⁴ eV or1.986×10⁻²³ J.

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 equals10⁻² (one hundredth) as much as a reciprocal centimetre. Five reciprocal metres are five times as much energy as one reciprocal metre.

• Lineic quantity
• Reciprocal area
• Reciprocal second
• Reciprocal volume

• Barrett, A. J. (11 July 1983). "A two-parameter perturbation series for the reciprocal length of polymer chains and subchains".Journal of Physics A: Mathematical and General.16 (10):2321–2330.Bibcode:1983JPhA...16.2321B.doi:10.1088/0305-4470/16/10/027.