# Absorption length

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 $\mu/\rho$.

## 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.
 $\epsilon$ $\epsilon = \frac{1}{\mu}$ $\epsilon = \frac{\rho}{\mu/\rho}$ $\epsilon = \frac{M_a}{\rho N_a 2 r_e \lambda f_2 }$ $\epsilon = \frac{M_a}{\rho N_a \sigma}$ $\epsilon = \frac{ \lambda }{4 \pi \beta}$ $\epsilon = \frac{1}{2 \lambda \mathrm{Im}(\mathrm{SLD})}$ $\mu = \frac{1}{\epsilon}$ $\mu$ $\mu = \frac{\mu/\rho}{\rho}$ $\mu = \frac{\rho N_a}{M_a} 2 r_e \lambda f_2$ $\mu = \frac{\rho N_a}{M_a} \sigma$ $\mu = \frac{4 \pi }{ \lambda } \beta$ $\mu = 2 \lambda\mathrm{Im}(\mathrm{SLD})$ $\frac{\mu}{\rho} = \frac{1}{\rho\epsilon}$ $\frac{\mu}{\rho} = \mu/\rho$ $\frac{\mu}{\rho}$ $\frac{\mu}{\rho} = \frac{N_a}{M_a} 2 r_e \lambda f_2$ $\frac{\mu}{\rho} = \frac{N_a}{M_a} \sigma$ $\frac{\mu}{\rho} = \frac{4 \pi}{ \rho \lambda } \beta$ $\frac{\mu}{\rho} = \frac{2 \lambda}{\rho } \mathrm{Im}(\mathrm{SLD})$ $f_2 = \frac{M_a }{\rho N_a 2 r_e \lambda \epsilon}$ $f_2 = \frac{M_a }{\rho N_a 2 r_e \lambda} \mu$ $f_2 = \frac{M_a }{ N_a 2 r_e \lambda} \frac{\mu}{\rho}$ $f_2$ $f_2 = \frac{\sigma}{2 r_e \lambda}$ $f_2 = \frac{2 \pi M_a}{ \rho N_a r_e \lambda^2 } \beta$ $f_2 = \frac{M_a}{\rho N_a r_e } \mathrm{Im}(\mathrm{SLD})$ $\sigma = \frac{M_a}{\rho N_a \epsilon}$ $\sigma = \frac{M_a}{\rho N_a} \mu$ $\sigma = \frac{M_a}{N_a} \frac{\mu}{\rho}$ $\sigma = 2 r_e \lambda f_2$ $\sigma$ $\sigma = \frac{4 \pi M_a}{ \rho N_a \lambda } \beta$ $\sigma = \frac{2 \lambda M_a}{\rho N_a}\mathrm{Im}(\mathrm{SLD})$ $\beta = \frac{ \lambda }{4 \pi \epsilon}$ $\beta = \frac{ \lambda }{4 \pi } \mu$ $\beta = \frac{ \rho \lambda }{4 \pi } \frac{\mu}{\rho}$ $\beta = \frac{ \rho N_a r_e \lambda^2 }{2 \pi M_a} f_2$ $\beta = \frac{ \rho N_a \lambda }{4 \pi M_a} \sigma$ $\beta$ $\beta = \frac{\lambda^2}{2 \pi} \mathrm{Im}(\mathrm{SLD})$ $\mathrm{Im}(\mathrm{SLD}) = \frac{1 }{2 \lambda \epsilon}$ $\mathrm{Im}(\mathrm{SLD}) = \frac{\mu}{2 \lambda}$ $\mathrm{Im}(\mathrm{SLD}) = \frac{\rho }{2 \lambda} \frac{\mu}{\rho}$ $\mathrm{Im}(\mathrm{SLD}) = \frac{\rho N_a r_e }{M_a} f_2$ $\mathrm{Im}(\mathrm{SLD}) = \frac{\rho N_a}{2 \lambda M_a}\sigma$ $\mathrm{Im}(\mathrm{SLD}) = \frac{2 \pi }{\lambda^2} \beta$ $\mathrm{Im}(\mathrm{SLD})$

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