material; Φee is the radiant power emitted by that material; T = Φet/Φei is the transmittance of that material; ATT = Φeatt/Φei is the attenuation of that material; E = Φee/Φei is the emittance of that material, and according to the Beer–Lambert law, so: Attenuation coefficient Optical depth of a material is also related to its attenuation coefficient by:where l is the thickness of that material through which the light travels; α(z) is the attenuation coefficient or Napierian attenuation coefficient of that material at z, and if α(z) is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes: Sometimes the relation is given using the attenuation cross section of the material, that is its attenuation coefficient divided by its number density:where σ is the attenuation cross section of that material; n(z) is the number density of that material at z, and if is uniform along the path, i.e., , the relation becomes: Applications Atomic physics In atomic physics, the spectral optical depth of a cloud of atoms can be calculated from the quantum-mechanical properties of the atoms. It is given bywhere d is the transition dipole moment; n is the number of atoms; ν is the frequency of the beam; c is the speed of light; ħ is Planck's constant; ε0 is the vacuum permittivity; σ the cross section of the beam; γ the natural linewidth of the transition. Atmospheric sciences In atmospheric sciences, one often refers to the optical depth of the atmosphere as corresponding to the vertical path from Earth's surface to outer space; at other times the optical path is from the observer's altitude to outer space. The optical depth for a slant path is , where τ′ refers to a vertical path, m is called the relative airmass, and for a plane-parallel atmosphere it is determined as where θ is the zenith angle corresponding to the given path. Therefore,The optical depth of the atmosphere can be divided into several components, ascribed to Rayleigh scattering, aerosols, and gaseous absorption. The optical depth of the atmosphere can be measured with a sun photometer. The optical depth with respect to the height within the atmosphere is given by and it follows that the total atmospheric optical depth is given by In both equations: ka is the absorption coefficient w1 is the mixing ratio ρ0 is the density of air at sea