Extinction coefficient

Not to be confused with Molar extinction coefficient.

The extinction coefficient for a particular substance is a measure of how well it 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:

${\tilde {n}}=n-ik.$

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 $v$ and angular frequency $\omega$. The time-varying electric field of the wave is described by

$\mathbf {E} (z,t)=\mathbf {E} _{0}e^{i\omega (t-{\frac {z}{v}})},$

where only the real part of $\mathbf {E}$ has physical significance. For simplicity, the radiation is assumed to be a plane wave, and its direction of propagation is denoted $\mathbf {z}$.

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:

${\tilde {n}}={\frac {c}{v}}.$

Substituting for ${\tilde {n}}$ in the expression above gives

${\frac {1}{v}}={\frac {n}{c}}-i{\frac {k}{c}}.$

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

$\mathbf {E} (z,t)=\mathbf {E} _{0}e^{i\omega (t-z({\frac {n}{c}}))}e^{-({\frac {k\omega }{c}})z}.$

This expression describes a propagating electromagnetic wave with an exponentially damped amplitude due to the $k$ 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

$I(z)=I_{0}e^{-{\frac {2\omega k}{c}}z}.$

A law called the Beer-Lambert law states that in any medium that is absorbing light, the decrease in intensity $I$ per unit length $z$ is proportional to the instantaneous value of $I$. In mathematical form this is

${\frac {dI\left(z\right)}{dz}}={-\alpha I\left(z\right)},$

where $\alpha$ is the absorption coefficient of the material for that wavelength of EM radiation. This equation has the solution

${I\left(z\right)}={I_{0}e^{-\alpha z}}$ ,

where $I_{0}$ is the intensity of the electromagnetic radiation at the surface of the absorbing medium. Comparing the two expressions for intensity obtained above gives

$\alpha ={\frac {2\omega k}{c}}.$

Since $c$ here denotes the speed of the EM wave in vacuum,

$c={\frac {\omega }{2\pi }}\lambda$.

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

$k={\frac {\lambda }{4\pi }}\alpha$ ,

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