Page updated: November 1, 2020
Author: Collin Roesler
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# Measurement of Absorption

### From Theory to Reality

Consider a scenario where the goal is to measure the absorption spectrum of a thin layer of material (Figure 1A). The incident radiant power is given by ${\Phi }_{o}$, in the form of a collimated beam. The radiant power transmitted through the layer,${\Phi }_{t}$, is detected. If ${\Phi }_{t}={\Phi }_{o}$, there is no loss of radiant power and therefore no attenuation. If however the medium absorbs some quantity of radiant power, ${\Phi }_{a}$, then ${\Phi }_{t}<{\Phi }_{o}$, and ${\Phi }_{o}={\Phi }_{t}+{\Phi }_{a}$ (Figure 1B). In the case of material that both absorbs and scatters (Figure 1C), the scattered radiant power is given by ${\Phi }_{b}$, and ${\Phi }_{o}={\Phi }_{t}+{\Phi }_{a}+{\Phi }_{b}$. Figure 1: Diagrammatic representation of theoretical attenuation by a thin layers of non-attenuating (panel A), absorbing (panel B), and absorbing and scattering (panel C) material. The thickness of the layer is given by $\Delta x$.

To quantify the absorbed radiant power only, it is necessary to measure both the transmitted and scattered radiant power. This is a requirement for an absorption meter. Consider ﬁrst a nonscattering material. The measured dimensionless transmittance, $T$, is the fraction of incident power transmitted through the layer:

 $T=\frac{{\Phi }_{t}}{{\Phi }_{o}}\phantom{\rule{0.3em}{0ex}}.$

The absorptance, $A$, is the fraction of incident radiant power that is absorbed ($1-T$):

 $A=\frac{{\Phi }_{a}}{{\Phi }_{o}}=\frac{{\Phi }_{o}-{\Phi }_{t}}{{\Phi }_{o}}\phantom{\rule{0.3em}{0ex}}.$

The absorption coeﬃcient $a$ (with units of ${m}^{-1}$) is the absorptance per unit distance

 $a=\frac{A}{\Delta x}$

which, for an inﬁnitesimally thin layer can be expressed as:

 $a=\frac{\frac{\Delta \Phi }{\Phi }}{\Delta x}=\frac{\Delta \Phi }{\Phi \Delta x},.$

Rearranging this expression and taking the limit as $\Delta x\to 0$ yields:

 $a\Delta x=\underset{\Delta x\to 0}{lim}\left(\frac{\Delta \Phi }{\Phi }\right)$

Assuming that the absorption coeﬃcient is constant over the layer of thickness $x$ and integrating gives

$\begin{array}{llll}\hfill {\int }_{0}^{x}a\phantom{\rule{0.3em}{0ex}}dx=& -{\int }_{{\Phi }_{o}}^{{\Phi }_{t}}\frac{d\Phi }{\Phi }\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ax{|}_{0}^{x}=& -ln\Phi {|}_{{\Phi }_{o}}^{{\Phi }_{t}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ax=& -\left[ln\left({\Phi }_{t}\right)-ln\left({\Phi }_{o}\right)\right]=-ln\left(\frac{{\Phi }_{t}}{{\Phi }_{o}}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill a=& -\frac{1}{x}ln\left(\frac{{\Phi }_{t}}{{\Phi }_{o}}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

This equation provides a guide toward designing instruments to accurately measure absorption. The Level 2 pages beginning at Benchtop Spectrometry of Solutions give the speciﬁcs on techniques to measure absorption by dissolved and particulate constituents in seawater.