# 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:

${\frac {I(x)}{I_{0}}}=e^{-x/\epsilon }$ The attenuation coefficient (or absorption coefficient) is simply the inverse of the absorption length; $\mu =1/\epsilon$ ${\frac {I(x)}{I_{0}}}=e^{-\mu x}$ ## 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:

$\sigma =2r_{e}\lambda f_{2}$ Where λ is the x-ray wavelength, and re is the classical electron radius. The attenuation coefficient is given by:

{\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 $\mu /\rho$ .

## 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 $\epsilon$ , the distance over which the intensity falls to 1/e.
• Attenuation coefficient $\mu$ , the characteristic inverse-distance for attenuation.
• Mass attenuation coefficient $\mu /\rho$ , the density-scaled attenuation.
• Absorptive atomic scattering factor $f_{2}$ , the intrinsic dissipative interaction of the material.
• Atomic photoabsorption cross-section $\sigma$ , the cross-section ('effective size') of the atom's x-ray absorption (capture) efficiency.
• Imaginary refractive index $\beta$ , the resonant component of the refractive index.
• Imaginary Scattering Length Density $\mathrm {Im} (\mathrm {SLD} )$ , the absorptive component of the scattering contrast.
 $\epsilon$ $\epsilon ={\frac {1}{\mu }}$ $\epsilon ={\frac {\rho }{\mu /\rho }}$ $\epsilon ={\frac {M_{a}}{\rho N_{a}2r_{e}\lambda f_{2}}}$ $\epsilon ={\frac {M_{a}}{\rho N_{a}\sigma }}$ $\epsilon ={\frac {\lambda }{4\pi \beta }}$ $\epsilon ={\frac {1}{2\lambda \mathrm {Im} (\mathrm {SLD} )}}$ $\mu ={\frac {1}{\epsilon }}$ $\mu$ $\mu ={\frac {\mu /\rho }{\rho }}$ $\mu ={\frac {\rho N_{a}}{M_{a}}}2r_{e}\lambda f_{2}$ $\mu ={\frac {\rho N_{a}}{M_{a}}}\sigma$ $\mu ={\frac {4\pi }{\lambda }}\beta$ $\mu =2\lambda \mathrm {Im} (\mathrm {SLD} )$ ${\frac {\mu }{\rho }}={\frac {1}{\rho \epsilon }}$ ${\frac {\mu }{\rho }}=\mu /\rho$ ${\frac {\mu }{\rho }}$ ${\frac {\mu }{\rho }}={\frac {N_{a}}{M_{a}}}2r_{e}\lambda f_{2}$ ${\frac {\mu }{\rho }}={\frac {N_{a}}{M_{a}}}\sigma$ ${\frac {\mu }{\rho }}={\frac {4\pi }{\rho \lambda }}\beta$ ${\frac {\mu }{\rho }}={\frac {2\lambda }{\rho }}\mathrm {Im} (\mathrm {SLD} )$ $f_{2}={\frac {M_{a}}{\rho N_{a}2r_{e}\lambda \epsilon }}$ $f_{2}={\frac {M_{a}}{\rho N_{a}2r_{e}\lambda }}\mu$ $f_{2}={\frac {M_{a}}{N_{a}2r_{e}\lambda }}{\frac {\mu }{\rho }}$ $f_{2}$ $f_{2}={\frac {\sigma }{2r_{e}\lambda }}$ $f_{2}={\frac {2\pi M_{a}}{\rho N_{a}r_{e}\lambda ^{2}}}\beta$ $f_{2}={\frac {M_{a}}{\rho N_{a}r_{e}}}\mathrm {Im} (\mathrm {SLD} )$ $\sigma ={\frac {M_{a}}{\rho N_{a}\epsilon }}$ $\sigma ={\frac {M_{a}}{\rho N_{a}}}\mu$ $\sigma ={\frac {M_{a}}{N_{a}}}{\frac {\mu }{\rho }}$ $\sigma =2r_{e}\lambda f_{2}$ $\sigma$ $\sigma ={\frac {4\pi M_{a}}{\rho N_{a}\lambda }}\beta$ $\sigma ={\frac {2\lambda M_{a}}{\rho N_{a}}}\mathrm {Im} (\mathrm {SLD} )$ $\beta ={\frac {\lambda }{4\pi \epsilon }}$ $\beta ={\frac {\lambda }{4\pi }}\mu$ $\beta ={\frac {\rho \lambda }{4\pi }}{\frac {\mu }{\rho }}$ $\beta ={\frac {\rho N_{a}r_{e}\lambda ^{2}}{2\pi M_{a}}}f_{2}$ $\beta ={\frac {\rho N_{a}\lambda }{4\pi M_{a}}}\sigma$ $\beta$ $\beta ={\frac {\lambda ^{2}}{2\pi }}\mathrm {Im} (\mathrm {SLD} )$ $\mathrm {Im} (\mathrm {SLD} )={\frac {1}{2\lambda \epsilon }}$ $\mathrm {Im} (\mathrm {SLD} )={\frac {\mu }{2\lambda }}$ $\mathrm {Im} (\mathrm {SLD} )={\frac {\rho }{2\lambda }}{\frac {\mu }{\rho }}$ $\mathrm {Im} (\mathrm {SLD} )={\frac {\rho N_{a}r_{e}}{M_{a}}}f_{2}$ $\mathrm {Im} (\mathrm {SLD} )={\frac {\rho N_{a}}{2\lambda M_{a}}}\sigma$ $\mathrm {Im} (\mathrm {SLD} )={\frac {2\pi }{\lambda ^{2}}}\beta$ $\mathrm {Im} (\mathrm {SLD} )$ See also scattering factors for a comparison of the quantities related to f1.