Absorption length

(Redirected from Absorption lengths)

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.