Mathematical definitions Attenuation coefficient The attenuation coefficient of a volume, denoted μ, is defined as where Φe is the radiant flux; z is the path length of the beam. Spectral hemispherical attenuation coefficient The spectral hemispherical attenuation coefficient in frequency and spectral hemispherical attenuation coefficient in wavelength of a volume, denoted μν and μλ respectively, are defined as: where Φe,ν is the spectral radiant flux in frequency; Φe,λ is the spectral radiant flux in wavelength. Directional attenuation coefficient The directional attenuation coefficient of a volume, denoted μΩ, is defined as where Le,Ω is the radiance. Spectral directional attenuation coefficient The spectral directional attenuation coefficient in frequency and spectral directional attenuation coefficient in wavelength of a volume, denoted μΩ,ν and μΩ,λ respectively, are defined as where Le,Ω,ν is the spectral radiance in frequency; Le,Ω,λ is the spectral radiance in wavelength. Absorption and scattering coefficients When a narrow (collimated) beam passes through a volume, the beam will lose intensity due to two processes: absorption and scattering. Absorption indicated energy that is lost from the beam, while scattering indicates light that is redirected in a (random) direction, and hence is no longer in the beam, but still present, resulting in diffuse light. The absorption coefficient of a volume, denoted μa, and the scattering coefficient of a volume, denoted μs, are defined the same way as the attenuation coefficient. The attenuation coefficient of a volume is the sum of absorption coefficient and scattering coefficients: Just looking at the narrow beam itself, the two processes cannot be distinguished. However, if a detector is set up to measure beam leaving in different directions, or conversely using a non-narrow beam, one can measure how much of the lost radiant flux was scattered, and how much was absorbed. In this context, the "absorption coefficient" measures how quickly the beam would lose radiant flux due to the absorption alone, while "attenuation coefficient" measures the total loss of narrow-beam intensity, including scattering as well. "Narrow-beam attenuation coefficient" always unambiguously refers to the latter. The attenuation coefficient is at least as large as the absorption coefficient; they are equal in the idealized case of no scattering. Mass attenuation, absorption, and scattering coefficients The mass attenuation coefficient, mass absorption coefficient, and mass scattering coefficient are defined as where ρm is the mass density. Napierian and decadic attenuation coefficients Decibels Engineering applications often express attenuation in the logarithmic units of decibels, or "dB",