# Reciprocal length

**Reciprocal length** or **inverse length** is a measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the **reciprocal metre** or **inverse metre** (m^{−1}), the **reciprocal centimetre** or **inverse centimetre** (cm^{−1}), and, in optics, the dioptre.

Quantities measured in reciprocal length include:

- absorption coefficient or attenuation coefficient, in materials science
- curvature of a line, in mathematics
- gain, in laser physics
- magnitude of vectors in reciprocal space, in crystallography
- more generally any spatial frequency e.g. in cycles per unit length
- optical power of a lens, in optics
- rotational constant of a rigid rotor, in quantum mechanics
- wavenumber, or magnitude of a wavevector, in spectroscopy
- density of a linear feature in hydrology and other fields; see kilometre per square kilometre

## Measure of energy

Reciprocal length is used as a measure of energy. The frequency of a photon yields a certain photon energy, according to the Planck-Einstein relation. Therefore, as reciprocal length is a measure of frequency, it can also be used as a measure of energy. For example, the reciprocal centimetre, cm^{−1}, is an energy unit equaling the energy of a photon with 1 cm wavelength. That energy amounts to approximately ×10^{−4} eV or 1.24×10^{−23} J. 1.986

The higher the number of inverse length units, the lower the energy. For example, in terms of energy, one reciprocal metre equals ^{−2} (one hundredth) as much as a reciprocal centimetre. Five reciprocal metres are one-fifth as much energy as one reciprocal metre. 10

## Further reading

- 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).