# Absorption length

(Redirected from Absorption)

The absorption length or attenuation length in x-ray scattering is the distance over which the x-ray beam is absorbed. By convention, the absorption length ϵ is defined as the distance into a material where the beam flux has dropped to 1/e of its incident flux.

## Absorption

The absorption follows a simple Beer-Lambert law:

${\displaystyle {\frac {I(x)}{I_{0}}}=e^{-x/\epsilon }}$

The attenuation coefficient (or absorption coefficient) is simply the inverse of the absorption length; ${\displaystyle \mu =1/\epsilon }$

${\displaystyle {\frac {I(x)}{I_{0}}}=e^{-\mu x}}$

## Calculating

The absorption length arises from the imaginary part of the atomic scattering factor, f2. It is closely related to the absorption cross-section, and the mass absorption coefficient. Specifically, the atomic photoabsorption cross-section can be computed via:

${\displaystyle \sigma =2r_{e}\lambda f_{2}}$

Where λ is the x-ray wavelength, and re is the classical electron radius. The attenuation coefficient is given by:

{\displaystyle {\begin{alignedat}{2}\mu &={\frac {\rho N_{a}}{m_{a}}}\sigma \\&={\frac {\rho N_{a}}{m_{a}}}2r_{e}\lambda f_{2}\end{alignedat}}}

where ρ is density, Na is the Avogadro constant, and ma is the atomic molar mass. Note that the mass attenuation coefficient is simply ${\displaystyle \mu /\rho }$.

## Energy dependence

Notice that the absorption undergoes sharp increases when passing through an absorption edge.

## Related forms

As can be seen, there are many related quantities which express the material's absorption:

• Absorption length ${\displaystyle \epsilon }$, the distance over which the intensity falls to 1/e.
• Attenuation coefficient ${\displaystyle \mu }$, the characteristic inverse-distance for attenuation.
• Mass attenuation coefficient ${\displaystyle \mu /\rho }$, the density-scaled attenuation.
• Absorptive atomic scattering factor ${\displaystyle f_{2}}$, the intrinsic dissipative interaction of the material.
• Atomic photoabsorption cross-section ${\displaystyle \sigma }$, the cross-section ('effective size') of the atom's x-ray absorption (capture) efficiency.
• Imaginary refractive index ${\displaystyle \beta }$, the resonant component of the refractive index.
• Imaginary Scattering Length Density ${\displaystyle \mathrm {Im} (\mathrm {SLD} )}$, the absorptive component of the scattering contrast.
 ${\displaystyle \epsilon }$ ${\displaystyle \epsilon ={\frac {1}{\mu }}}$ ${\displaystyle \epsilon ={\frac {\rho }{\mu /\rho }}}$ ${\displaystyle \epsilon ={\frac {M_{a}}{\rho N_{a}2r_{e}\lambda f_{2}}}}$ ${\displaystyle \epsilon ={\frac {M_{a}}{\rho N_{a}\sigma }}}$ ${\displaystyle \epsilon ={\frac {\lambda }{4\pi \beta }}}$ ${\displaystyle \epsilon ={\frac {1}{2\lambda \mathrm {Im} (\mathrm {SLD} )}}}$ ${\displaystyle \mu ={\frac {1}{\epsilon }}}$ ${\displaystyle \mu }$ ${\displaystyle \mu ={\frac {\mu /\rho }{\rho }}}$ ${\displaystyle \mu ={\frac {\rho N_{a}}{M_{a}}}2r_{e}\lambda f_{2}}$ ${\displaystyle \mu ={\frac {\rho N_{a}}{M_{a}}}\sigma }$ ${\displaystyle \mu ={\frac {4\pi }{\lambda }}\beta }$ ${\displaystyle \mu =2\lambda \mathrm {Im} (\mathrm {SLD} )}$ ${\displaystyle {\frac {\mu }{\rho }}={\frac {1}{\rho \epsilon }}}$ ${\displaystyle {\frac {\mu }{\rho }}=\mu /\rho }$ ${\displaystyle {\frac {\mu }{\rho }}}$ ${\displaystyle {\frac {\mu }{\rho }}={\frac {N_{a}}{M_{a}}}2r_{e}\lambda f_{2}}$ ${\displaystyle {\frac {\mu }{\rho }}={\frac {N_{a}}{M_{a}}}\sigma }$ ${\displaystyle {\frac {\mu }{\rho }}={\frac {4\pi }{\rho \lambda }}\beta }$ ${\displaystyle {\frac {\mu }{\rho }}={\frac {2\lambda }{\rho }}\mathrm {Im} (\mathrm {SLD} )}$ ${\displaystyle f_{2}={\frac {M_{a}}{\rho N_{a}2r_{e}\lambda \epsilon }}}$ ${\displaystyle f_{2}={\frac {M_{a}}{\rho N_{a}2r_{e}\lambda }}\mu }$ ${\displaystyle f_{2}={\frac {M_{a}}{N_{a}2r_{e}\lambda }}{\frac {\mu }{\rho }}}$ ${\displaystyle f_{2}}$ ${\displaystyle f_{2}={\frac {\sigma }{2r_{e}\lambda }}}$ ${\displaystyle f_{2}={\frac {2\pi M_{a}}{\rho N_{a}r_{e}\lambda ^{2}}}\beta }$ ${\displaystyle f_{2}={\frac {M_{a}}{\rho N_{a}r_{e}}}\mathrm {Im} (\mathrm {SLD} )}$ ${\displaystyle \sigma ={\frac {M_{a}}{\rho N_{a}\epsilon }}}$ ${\displaystyle \sigma ={\frac {M_{a}}{\rho N_{a}}}\mu }$ ${\displaystyle \sigma ={\frac {M_{a}}{N_{a}}}{\frac {\mu }{\rho }}}$ ${\displaystyle \sigma =2r_{e}\lambda f_{2}}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma ={\frac {4\pi M_{a}}{\rho N_{a}\lambda }}\beta }$ ${\displaystyle \sigma ={\frac {2\lambda M_{a}}{\rho N_{a}}}\mathrm {Im} (\mathrm {SLD} )}$ ${\displaystyle \beta ={\frac {\lambda }{4\pi \epsilon }}}$ ${\displaystyle \beta ={\frac {\lambda }{4\pi }}\mu }$ ${\displaystyle \beta ={\frac {\rho \lambda }{4\pi }}{\frac {\mu }{\rho }}}$ ${\displaystyle \beta ={\frac {\rho N_{a}r_{e}\lambda ^{2}}{2\pi M_{a}}}f_{2}}$ ${\displaystyle \beta ={\frac {\rho N_{a}\lambda }{4\pi M_{a}}}\sigma }$ ${\displaystyle \beta }$ ${\displaystyle \beta ={\frac {\lambda ^{2}}{2\pi }}\mathrm {Im} (\mathrm {SLD} )}$ ${\displaystyle \mathrm {Im} (\mathrm {SLD} )={\frac {1}{2\lambda \epsilon }}}$ ${\displaystyle \mathrm {Im} (\mathrm {SLD} )={\frac {\mu }{2\lambda }}}$ ${\displaystyle \mathrm {Im} (\mathrm {SLD} )={\frac {\rho }{2\lambda }}{\frac {\mu }{\rho }}}$ ${\displaystyle \mathrm {Im} (\mathrm {SLD} )={\frac {\rho N_{a}r_{e}}{M_{a}}}f_{2}}$ ${\displaystyle \mathrm {Im} (\mathrm {SLD} )={\frac {\rho N_{a}}{2\lambda M_{a}}}\sigma }$ ${\displaystyle \mathrm {Im} (\mathrm {SLD} )={\frac {2\pi }{\lambda ^{2}}}\beta }$ ${\displaystyle \mathrm {Im} (\mathrm {SLD} )}$

See also scattering factors for a comparison of the quantities related to f1.