Difference between revisions of "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 ''λ'' is defined as the distance into a material where the beam flux has dropped to 1/''e'' of its incident flux.
+
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==
 
==Absorption==
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\frac{I(x)}{I_0} = e^{ -  x / \epsilon }
 
\frac{I(x)}{I_0} = e^{ -  x / \epsilon }
 
</math>
 
</math>
The '''attenuation coefficient''' (or '''absorption coefficient''') is simply the inverse of the absorption length; <math>\mu = 1/
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The '''attenuation coefficient''' (or '''absorption coefficient''') is simply the inverse of the absorption length; <math>\mu = 1/\epsilon</math>
 +
:<math>
 +
\frac{I(x)}{I_0} = e^{ -  \mu x }
 +
</math>
  
 
==Calculating==
 
==Calculating==
 
The absorption length arises from the imaginary part of the [[atomic scattering factor]], ''f''<sub>2</sub>. It is closely related to the absorption cross-section, and the mass absorption coefficient. Specifically, the '''atomic photoabsorption cross-section''' can be computed via:
 
The absorption length arises from the imaginary part of the [[atomic scattering factor]], ''f''<sub>2</sub>. It is closely related to the absorption cross-section, and the mass absorption coefficient. Specifically, the '''atomic photoabsorption cross-section''' can be computed via:
 
:<math>
 
:<math>
\sigma = 2 r_e \lambda
+
\sigma = 2 r_e \lambda f_2
 
</math>
 
</math>
 +
Where ''λ'' is the x-ray wavelength, and ''r<sub>e</sub>'' is the classical electron radius. The '''attenuation coefficient''' is given by:
 +
:<math>
 +
\begin{alignat}{2}
 +
\mu & = \frac{\rho N_a}{m_a} \sigma \\
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    & = \frac{\rho N_a}{m_a} 2 r_e \lambda f_2
 +
\end{alignat}
 +
</math>
 +
where ''ρ'' is density, ''N<sub>a</sub>'' is the Avogadro constant, and ''m<sub>a</sub>'' is the atomic molar mass. Note that the '''mass attenuation coefficient''' is simply <math>\mu/\rho</math>.
 +
 +
==Elemental dependence==
 +
 +
[[Image:Elements-abs.png|400px]][[Image:Elements-mu.png|400px]]
 +
 +
==Energy dependence==
 +
Notice that the absorption undergoes sharp increases when passing through an absorption edge.
 +
 +
===Examples===
 +
 +
====[[Material:Silicon|silicon]]====
 +
 +
[[Image:Silicon-AttLen.png|400px]][[Image:Silicon-mu.png|400px]]
 +
 +
====[[Material:Gold|gold]]====
 +
 +
[[Image:Gold-AttLen.png|400px]][[Image:Gold-mu.png|400px]]
 +
 +
==Elemental/Energy dependence==
 +
[[Image:Elements2D-abs.png|400px]][[Image:Elements2D-mu.png|400px]]
 +
 +
==Related forms==
 +
As can be seen, there are many related quantities which express the material's absorption:
 +
* '''Absorption length''' <math>\epsilon</math>, the distance over which the intensity falls to 1/''e''.
 +
* '''Attenuation coefficient''' <math>\mu</math>, the characteristic inverse-distance for attenuation.
 +
* '''Mass attenuation coefficient''' <math>\mu/\rho</math>, the density-scaled attenuation.
 +
* '''Absorptive [[atomic scattering factor]]''' <math>f_2</math>, the intrinsic dissipative interaction of the material.
 +
* '''Atomic photoabsorption cross-section''' <math>\sigma</math>, the cross-section ('effective size') of the atom's x-ray absorption (capture) efficiency.
 +
* '''Imaginary [[refractive index]]''' <math>\beta</math>, the resonant component of the refractive index.
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* '''Imaginary [[Scattering Length Density]]''' <math>\mathrm{Im}(\mathrm{SLD})</math>, the absorptive component of the scattering contrast.
 +
 +
{| class="wikitable"
 +
 +
|-
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| <math>\epsilon</math>
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| <math>\epsilon = \frac{1}{\mu}</math>
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| <math>\epsilon = \frac{\rho}{\mu/\rho}</math>
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| <math>\epsilon = \frac{M_a}{\rho N_a 2 r_e \lambda f_2 }</math>
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| <math>\epsilon = \frac{M_a}{\rho N_a \sigma}</math>
 +
| <math>\epsilon = \frac{ \lambda }{4 \pi \beta}</math>
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| <math>\epsilon = \frac{1}{2 \lambda \mathrm{Im}(\mathrm{SLD})} </math>
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|-
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| <math>\mu = \frac{1}{\epsilon}</math>
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| <math>\mu</math>
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| <math>\mu = \frac{\mu/\rho}{\rho}</math>
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| <math>\mu = \frac{\rho N_a}{M_a} 2 r_e \lambda f_2</math>
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| <math>\mu = \frac{\rho N_a}{M_a} \sigma</math>
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| <math>\mu = \frac{4 \pi }{ \lambda } \beta</math>
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| <math>\mu = 2 \lambda\mathrm{Im}(\mathrm{SLD})</math>
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|-
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| <math>\frac{\mu}{\rho} = \frac{1}{\rho\epsilon}</math>
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| <math>\frac{\mu}{\rho} = \mu/\rho</math>
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| <math>\frac{\mu}{\rho}</math>
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| <math>\frac{\mu}{\rho} = \frac{N_a}{M_a} 2 r_e \lambda f_2</math>
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| <math>\frac{\mu}{\rho} = \frac{N_a}{M_a} \sigma</math>
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| <math>\frac{\mu}{\rho} = \frac{4 \pi}{ \rho \lambda  } \beta</math>
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| <math>\frac{\mu}{\rho} = \frac{2 \lambda}{\rho  } \mathrm{Im}(\mathrm{SLD})</math>
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|-
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| <math>f_2 = \frac{M_a }{\rho N_a 2 r_e \lambda  \epsilon} </math>
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| <math>f_2 = \frac{M_a }{\rho N_a 2 r_e \lambda} \mu </math>
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| <math>f_2 = \frac{M_a }{ N_a 2 r_e \lambda} \frac{\mu}{\rho} </math>
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| <math>f_2</math>
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| <math>f_2 = \frac{\sigma}{2 r_e \lambda}</math>
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| <math>f_2 = \frac{2 \pi M_a}{ \rho N_a r_e \lambda^2 } \beta</math>
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| <math>f_2 = \frac{M_a}{\rho N_a r_e } \mathrm{Im}(\mathrm{SLD})</math>
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|-
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| <math>\sigma = \frac{M_a}{\rho N_a \epsilon} </math>
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| <math>\sigma = \frac{M_a}{\rho N_a} \mu</math>
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| <math>\sigma = \frac{M_a}{N_a} \frac{\mu}{\rho}</math>
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| <math>\sigma = 2 r_e \lambda f_2</math>
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| <math>\sigma</math>
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| <math>\sigma = \frac{4 \pi M_a}{ \rho N_a \lambda } \beta</math>
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| <math>\sigma = \frac{2 \lambda M_a}{\rho N_a}\mathrm{Im}(\mathrm{SLD})</math>
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|-
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| <math>\beta = \frac{ \lambda }{4 \pi  \epsilon}</math>
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| <math>\beta = \frac{ \lambda  }{4 \pi } \mu</math>
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| <math>\beta = \frac{ \rho \lambda  }{4 \pi } \frac{\mu}{\rho}</math>
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| <math>\beta = \frac{ \rho N_a r_e \lambda^2 }{2 \pi M_a} f_2</math>
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| <math>\beta = \frac{ \rho N_a \lambda }{4 \pi M_a} \sigma</math>
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| <math>\beta</math>
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| <math>\beta = \frac{\lambda^2}{2 \pi} \mathrm{Im}(\mathrm{SLD})</math>
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|-
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| <math>\mathrm{Im}(\mathrm{SLD}) = \frac{1 }{2 \lambda \epsilon} </math>
 +
| <math>\mathrm{Im}(\mathrm{SLD}) = \frac{\mu}{2 \lambda} </math>
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| <math>\mathrm{Im}(\mathrm{SLD}) = \frac{\rho }{2 \lambda} \frac{\mu}{\rho}</math>
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| <math>\mathrm{Im}(\mathrm{SLD}) = \frac{\rho N_a r_e }{M_a} f_2</math>
 +
| <math>\mathrm{Im}(\mathrm{SLD}) = \frac{\rho N_a}{2 \lambda M_a}\sigma</math>
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| <math>\mathrm{Im}(\mathrm{SLD}) = \frac{2 \pi }{\lambda^2} \beta</math>
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| <math>\mathrm{Im}(\mathrm{SLD})</math>
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 +
|}
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See also '''[[Atomic_scattering_factors#Related_forms|scattering factors]]''' for a comparison of the quantities related to ''f''<sub>1</sub>.
  
 
==See Also==
 
==See Also==
 +
* [[Resonant scattering]]
 +
** [[RSoXS]]
 +
** [[Resonant reflectivity]]
 
* [http://henke.lbl.gov/optical_constants/atten2.html LBL X-Ray Attenuation Length calculator]
 
* [http://henke.lbl.gov/optical_constants/atten2.html LBL X-Ray Attenuation Length calculator]
* [http://11bm.xray.aps.anl.gov/absorb/absorb.php APS absorption calculator
+
* [http://11bm.xray.aps.anl.gov/absorb/absorb.php APS absorption calculator]
 +
* [http://henke.lbl.gov/optical_constants/filter2.html CXRO transmission calculator]
 
* [http://en.wikipedia.org/wiki/Mass_attenuation_coefficient Wikipedia: Mass attenuation coefficient]
 
* [http://en.wikipedia.org/wiki/Mass_attenuation_coefficient Wikipedia: Mass attenuation coefficient]
 +
* [http://en.wikipedia.org/wiki/Absorption_cross_section Wikipedia: Absorption cross sectio]
 
* [http://www.nist.gov/pml/data/xraycoef/ NIST tables of x-ray mass attenuation coefficient]
 
* [http://www.nist.gov/pml/data/xraycoef/ NIST tables of x-ray mass attenuation coefficient]

Latest revision as of 15:43, 29 July 2015

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:

The attenuation coefficient (or absorption coefficient) is simply the inverse of the absorption length;

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:

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

where ρ is density, Na is the Avogadro constant, and ma is the atomic molar mass. Note that the mass attenuation coefficient is simply .

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 , the distance over which the intensity falls to 1/e.
  • Attenuation coefficient , the characteristic inverse-distance for attenuation.
  • Mass attenuation coefficient , the density-scaled attenuation.
  • Absorptive atomic scattering factor , the intrinsic dissipative interaction of the material.
  • Atomic photoabsorption cross-section , the cross-section ('effective size') of the atom's x-ray absorption (capture) efficiency.
  • Imaginary refractive index , the resonant component of the refractive index.
  • Imaginary Scattering Length Density , the absorptive component of the scattering contrast.


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

See Also