Absorption length

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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 = 2 r_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{alignat}{2}
\mu & = \frac{\rho N_a}{m_a} \sigma \\
    & = \frac{\rho N_a}{m_a} 2 r_e \lambda f_2
\end{alignat}

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.

Elemental dependence

Elements-abs.pngElements-mu.png

Energy dependence

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

Examples

silicon

Silicon-AttLen.pngSilicon-mu.png

gold

Gold-AttLen.pngGold-mu.png

Elemental/Energy dependence

Elements2D-abs.pngElements2D-mu.png

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 2 r_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} 2 r_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} 2 r_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 2 r_e \lambda  \epsilon} f_2 = \frac{M_a }{\rho N_a 2 r_e \lambda} \mu f_2 = \frac{M_a }{ N_a 2 r_e \lambda} \frac{\mu}{\rho} f_2 f_2 = \frac{\sigma}{2 r_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 = 2 r_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.

See Also