# Opacity (optics)

**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). An opaque substance transmits very little light, and therefore reflects, scatters, or absorbs most 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. In general, a material tends to emit different colors in the same proportions as it absorbs it.

## Definition

The opacity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_\nu**
gives the rate of absorption (or extinction), which is the fraction of the intensity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): I_\nu**
, of the radiation that is absorbed or scattered per unit distance along a ray of propagation:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\partial I_\nu\over\partial x}=-I_\nu\kappa_\nu**.

For a given medium it has a numerical value that may range between 0 and infinity.
It is also called the absorption coefficient (see also extinction coefficient).
In general **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_\nu**
depends on the frequency **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \nu**
of the radiation, as well as the density, temperature, and composition of the medium.
The mean free path is the distance a photon travels in the medium before absorption or scattering is defined as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/(\kappa_\nu \rho)**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \rho**
is the density of the material. The notation **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_\lambda**
is the opacity described as a function of wavelength **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda**
.
While many materials are very opaque (steel in visible light having near-infinite opacity), and others very transparent (air in visible light having near-zero opacity), so that opacity often seems to be a boolean property, many others (such as water) have intermediate opacity.

In astronomy and planetary imaging fields, tau, the optical depth, defines the opacity: zero indicates transparent and higher numbers indicate more and more opaque in an inverse exponential fashion, for example a tau of 1 indicates 36 percent of the light passes (e^{-1} = 0.36), and a Tau of 5 indicates less than 1 percent passes (e ^{-5} = 0.0067).^{[1]}

In astrophysics and plasma physics "opacity", or absorption coefficient, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_\nu**
is defined so that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_\nu \rho I_\nu d\nu d\Omega**
gives the corresponding energy absorbed per unit volume per unit time from a beam of given intensity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): I_\nu**
in a medium of density **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \rho**
(thus **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\rm cm}^2 {\rm g}^{-1}**
).
The optical depth **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tau_\nu**
along the propagation direction is then **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): d \tau_\nu = \kappa_\nu \rho ds**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): ds**
is the distance along this direction. 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 **Rosseland opacity**, on the other hand, uses a temperature derivative of Planck distribution (normalized) as the weighting function, and averages **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_\nu^{-1}**
,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_{\rm es} = 0.40 {\rm cm}^2 {\rm g}^{-1}**
and for nonrelativistic thermal bremsstrahlung, or free-free transitions, it is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \kappa_{\rm ff}(\rho, T) = 0.64 \times 10^{23} (\rho[ {\rm g}~ {\rm cm}^{-3}])(T[{\rm K}])^{-7/2} {\rm cm}^2 {\rm g}^{-1}**
.^{[2]}
The Rosseland mean absorption coefficient including both scattering and absorption (also called the extinction coefficient) is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \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}**.^{[3]}

## Applications

In astrophysics, the variations in opacity within a star are important to the understanding of radiation transfer in stellar atmospheres and the spectra we observe.

In several types of chemical analysis, the concentration of a sample in a transparent medium (typically air or water) is determined via measuring its opacity or absorbance. In spectrophotometry the device identifies the sample's constituent substances from their absorbances.

Opacity is also used as a measurement of particulate emissions.

## Extinction coefficient

The **extinction coefficient** for a particular substance is a measure of how well it scatters and absorbs electromagnetic radiation (EM waves). If the EM wave can pass through very easily, the material has a low extinction coefficient. Conversely, if the radiation hardly penetrates the material, but rather quickly becomes "extinct" within it, the extinction coefficient is high.

A material can behave differently for different wavelengths of electromagnetic radiation. Glass is transparent to visible light, but many types of glass are opaque to ultra-violet wavelengths. In general, the extinction coefficient for any material is a function of the incident wavelength. The extinction coefficient is used widely in ultraviolet-visible spectroscopy.

## Physical definitions

The parameter used to describe the interaction of electromagnetic radiation with matter is the complex index of refraction, *ñ*, which is a combination of a real part and an imaginary part:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tilde{n}=n-ik.**

Here, *n* is also called the *index of refraction*, which sometimes leads to confusion. *k* is the *extinction coefficient*, which represents the damping of an EM wave inside the material. Both depend on the wavelength.

An EM wave travels in the material with velocity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): v**
and angular frequency **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \omega**
. The time-varying electric field of the wave is described by

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \mathbf{E}(z,t) = \mathbf{E}_0 e^{i\omega(t - \frac{z}{v})},**

where only the real part of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \mathbf E**
has physical significance. For simplicity, the radiation is assumed to be a plane wave, and its direction of propagation is denoted **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): z**
.

The index of refraction is *defined* to be the ratio of the speed of light in a vacuum to the speed of the EM wave in the medium:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tilde{n} = \frac{c}{v}.**

Substituting for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tilde{n}**
in the expression above gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{v} = \frac{n}{c} - i\frac{k}{c}.**

Substituting this in the expression for the EM wave's electric field gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \mathbf E(z, t) = \mathbf E_0 e^{i\omega(t - z(\frac{n}{c}))} e^{-(\frac{k \omega}{c})z}.**

This expression describes a propagating electromagnetic wave with an exponentially damped amplitude due to the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): k**
term. This term causes the EM wave to "die out" as it travels further into the material. The intensity of the wave, which corresponds to the energy it carries with it, is simply the square of the magnitude of the wave's electric field. The intensity of the wave is therefore

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): I(z) = I_0 e^{-\frac{2\omega k}{c}z}.**

A law called the Beer-Lambert law states that in any medium that is absorbing light, the decrease in intensity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): I**
per unit length **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): z**
is proportional to the instantaneous value of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): I**
. In mathematical form this is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{dI\left( z \right)}{dz}={-\alpha I\left(z\right)},**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \alpha**
is the absorption coefficient of the material for that wavelength of EM radiation. This equation has the solution

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {I\left(z \right)}={I_0 e^{-\alpha z}}**,

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): I_0**
is the intensity of the electromagnetic radiation at the surface of the absorbing medium. Comparing the two expressions for intensity obtained above gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \alpha = \frac{2\omega k}{c}.**

Since **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): c**
here denotes the speed of the EM wave in vacuum,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): c= \frac{\omega}{2\pi}\lambda**.

Substituting this in the expression above and rearranging shows that the extinction coefficient and the absorption coefficient are related by

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): k={\frac{\lambda}{4\pi}}\alpha**,

where λ is the vacuum wavelength (not the wavelength of the EM wave in the material).

## See also

## References

- ↑ Are The Mars Rovers Doomed? 'A tau of five means that less than one percent of direct sunlight is reaching the Mars surface'
- ↑ Stuart L. Shapiro and Saul A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars" 1983, ISBN 0-471-87317-9.
- ↑ George B. Rybicki and Alan P. Lightman, "Radiative Processes in Astrophysics" 1979 ISBN 0-471-04815-1.

ca:Opacitat de:Opazität eo:Opakeco it:Diafanità nl:Opaciteit no:Opasitet sk:Opacita