Absorption coefficient

This article describes the attenuation of electromagnetic radiation in matter. For further meanings, see Absorption Coefficient (disambiguation) .

The articles absorption coefficient and absorption coefficient overlap thematically. Help me to better differentiate or merge the articles (→  instructions ) . To do this, take part in the relevant redundancy discussion . Please remove this module only after the redundancy has been completely processed and do not forget to include the relevant entry on the redundancy discussion page{{ Done | 1 = ~~~~}}to mark. Harro 3:58 pm, Sep 17th 2013 (CEST)

The absorption coefficient , also called the attenuation constant or the linear attenuation coefficient , is a measure of the reduction in the intensity of electromagnetic radiation when it passes through a given material. It is used in optics and in relation to x-rays and gamma rays . Its common formula symbol is in optics or , with X-rays and gamma rays . Its dimension is 1 / length, the usual unit 1 / cm. A large absorption coefficient means that the material shields the radiation in question relatively strongly, while a smaller one means that it is more transparent to the radiation. ${\displaystyle \alpha }$ ${\displaystyle \alpha '}$ ${\displaystyle \mu }$

In the term absorption coefficient, the term absorption is not to be understood in the narrower sense of the transfer of radiation energy to the medium. Rather, scattering processes that only deflect the radiation from its direction also contribute to the decrease in intensity ( extinction ) referred to here .

application

According to Lambert-Beer’s law , the intensity decays exponentially after passing through an absorber of the same thickness or at a penetration depth : ${\displaystyle I}$ ${\displaystyle z}$ ${\displaystyle z}$

${\displaystyle {\begin{aligned}I(z)&=I_{0}\cdot e^{-\alpha z}\\&=I_{0}\cdot \exp \left(-2n''\,{\frac {\omega }{c}}\,z\right)\end{aligned}}}$

With

• the irradiated intensity ${\displaystyle I_{0}}$
• the absorption coefficient ${\displaystyle \alpha =2n''\,{\frac {\omega }{c}}}$
  • the extinction coefficient of the material ${\displaystyle n''}$
  • the angular frequency of the radiation used (depends on its energy ) ${\displaystyle \omega }$
  • the speed of light . ${\displaystyle c}$

Derivation

If you replace in

${\displaystyle {\vec {E}}={\vec {E}}_{0}\cdot e^{i\left[{\vec {k}}\,{\vec {r}}-\omega t\right]}={\vec {E}}_{0}\cdot e^{i\left[kz-\omega t\right]}}$

the circular wavenumber from the wave vector as follows ${\displaystyle k}$ ${\displaystyle {\vec {k}}=k\,{\hat {e}}_{z}}$

${\displaystyle k={\frac {\omega }{c}}n={\frac {\omega }{c}}(n'+\mathrm {i} n'')}$,

(therein is the complex index of refraction ) ${\displaystyle n}$

so you get:

${\displaystyle {\begin{aligned}{\vec {E}}&={\vec {E}}_{0}\cdot e^{\mathrm {i} \left[(n'+\mathrm {i} n''){\frac {\omega }{c}}z-\omega t\right]}\\&={\vec {E}}_{0}\cdot e^{-n''{\frac {\omega }{c}}z}\cdot e^{\mathrm {i} \left[n'{\frac {\omega }{c}}z-\omega t\right]}\end{aligned}}}$

It applies . ${\displaystyle I\propto |E|^{2}}$

Extinction coefficient and absorption index

The extinction coefficient and the absorption index can be calculated from the absorption coefficient of a sample : ${\displaystyle n''}$ ${\displaystyle \kappa ={\frac {n''}{n'}}}$

${\displaystyle \alpha \,{\frac {c}{2\omega }}=n''=n'\cdot \kappa }$

optics

The quotient after crossing a layer thickness is called the degree of transmission : ${\displaystyle I(d)/I_{0}}$ ${\displaystyle d}$ ${\displaystyle T}$

${\displaystyle T={\frac {I}{I_{0}}}=e^{-\alpha d}}$

The inverse transmittance is called opacity : ${\displaystyle O}$

${\displaystyle O=T^{-1}={\frac {I_{0}}{I}}=e^{\alpha d}}$

The negative decadic logarithm of the transmittance, i.e. the decadic logarithm of the opacity, is the extinction : ${\displaystyle E}$

${\displaystyle E=-\lg(T)=\lg(O)=\lg \left({\frac {I_{0}}{I}}\right)=\lg(e)\,\alpha \,d\approx 0{,}434\,\alpha \,d}$

X-rays and gamma rays

The rule of thumb for photon energies above 50  keV applies: The higher the energy, the less dense the material and the smaller the atomic number of the material, the lower the linear attenuation coefficient. Even at lower energies, the atomic number Z of the material increases steeply (proportional to the 4th power ). This is why lead, with its high density, is the preferred material for shielding . ${\displaystyle \mu }$

For practical purposes, the mass attenuation coefficient is often preferred. Multiplied by the density of the material, it gives the linear attenuation coefficient .

See also

• Half-value thickness

literature

• Peter H. Hertrich: X-ray technology: basics and applications . Publicis Publishing, 2004, ISBN 978-3-89578-209-1 , pp. 38–44 ( limited preview in Google Book search).
• Rudolf Nicoletti, Michael Oberladstätter, Franz König: Measurement technology and instrumentation in nuclear medicine: an introduction . facultas.wuv Universitäts, 2006, ISBN 978-3-85076-795-8 , p. 38–39 ( limited preview in Google Book search).