Absorption Coefficient

Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc. An opaque object is neither transparent (allowing all light to pass through) nor translucent (allowing some light to pass through). When light strikes an interface between two substances, in general some may be reflected, some absorbed, some scattered, and the rest transmitted (also see refraction). Reflection can be diffuse, for example light reflecting off a white wall, or specular, for example light reflecting off a mirror. An opaque substance transmits no light, and therefore reflects, scatters, or absorbs all of it. Both mirrors and carbon black are opaque. Opacity depends on the frequency of the light being considered. For instance, some kinds of glass, while transparent in the visual range, are largely opaque to ultraviolet light. More extreme frequency-dependence is visible in the absorption lines of cold gases. Opacity can be quantified in many ways; for example, see the article mathematical descriptions of opacity.

For general information on what makes an object or medium opaque, see the articles on absorption, reflection, and scattering. These are the processes that lead to opacity.

Quantitative definition

The words "opacity" and "opaque" are often used as colloquial terms for objects or media with the properties described above. However, there is also a specific, quantitative definition of "opacity", used in astronomy, plasma physics, and other fields, given here.

In this use, "opacity" is another term for the mass attenuation coefficient (or, depending on context, mass absorption coefficient, the difference is described here) \kappa_\nu at a particular frequency \nu of electromagnetic radiation.

More specifically, if a beam of light with frequency \nu travels through a medium with opacity \kappa_\nu and mass density \rho, both constant, then the intensity will be reduced with distance x according to the formula

I(x) = I_0 e^{-\kappa_\nu \rho x}

where

  • x is the distance the light has traveled through the medium
  • I(x) is the intensity of light remaining at distance x
  • I_0 is the initial intensity of light, at x = 0

For a given medium at a given frequency, the opacity has a numerical value that may range between 0 and infinity, with units of length2/mass.

Planck and Rosseland opacity

It is customary to define the average opacity, calculated using a certain weighting scheme. Planck opacity uses normalized Planck black body radiation energy density distribution as the weighting function, and averages \kappa_\nu directly. Rosseland opacity (after Svein Rosseland), on the other hand, uses a temperature derivative of Planck distribution (normalized) as the weighting function, and averages \kappa_\nu^{-1},

\frac{1}{\kappa} = \frac{\int_0^{\infty} \kappa_{\nu}^{-1} u(\nu, T) d\nu }{\int_0^{\infty} u(\nu,T) d\nu}.

The photon mean free path is \lambda_\nu = (\kappa_\nu \rho)^{-1}. The Rosseland opacity is derived in the diffusion approximation to the radiative transport equation. It is valid whenever the radiation field is isotropic over distances comparable to or less than a radiation mean free path, such as in local thermal equilibrium. In practice, the mean opacity for Thomson electron scattering is:

\kappa_{\rm es} = 0.20(1+X) {\rm\, cm}^2{\rm \,g}^{-1}

where X is the hydrogen mass fraction. For nonrelativistic thermal bremsstrahlung, or free-free transitions, it is:

\kappa_{\rm ff}(\rho, T) = 0.64 \times 10^{23} (\rho[ {\rm g}~ {\rm\, cm}^{-3}])(T)^{-7/2} {\rm\, cm}^2 {\rm\, g}^{-1}.[1]

The Rosseland mean absorption coefficient including both scattering and absorption (also called the extinction coefficient) is:

\frac{1}{\kappa} = \frac{\int_0^{\infty} (\kappa_{\nu, {\rm es}} + \kappa_{\nu, {\rm ff}})^{-1} u(\nu, T) d\nu }{\int_0^{\infty} u(\nu,T) d\nu}.[2]

See also

References

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