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 1.24×10⁻⁴ eV or 1.986×10⁻²³ J.

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

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).