# Encyclopedia of Biophysics

Living Edition
| Editors: Gordon Roberts, Anthony Watts, European Biophysical Societies

# Absorption Spectroscopy, the Beer-Lambert Law, and Transition Polarizations

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-35943-9_782-1

## Definition

Absorption can be pictorially viewed as either the electric field or the magnetic field (or both) of the radiation pushing the molecular electron density from a starting arrangement to a higher-energy final one (Atkins and de Paula 2006; Atkins and Friedman 2005; Hollas 2004). The electric field is far more effective than the magnetic field in achieving the push that gets a photon absorbed. The direction of net linear displacement of charge is known as the polarization of the transition. The polarization and intensity of a transition are summarized by its electric dipole transition moment, which is a vectorial property having a well-defined direction (the transition polarization) within each molecule and a well-defined length (which is proportional to the square root of the absorbance). The transition moment may be regarded as an antenna by which the molecule absorbs light. Each transition thus has its own antenna, and the maximum probability of absorbing light is obtained when the molecular antenna and the electric field of the light are parallel. Conversely, the absorption is zero when the light polarization and antenna are perpendicular to one another.

The Beer-Lambert law (Hollas 2004; Atkins and Friedman 2005; Atkins and de Paula 2006)
$$A=\varepsilon c\mathrm{\ell}$$
(1)
is arguably the single most useful spectroscopy equation. It follows from considering what happens to a beam of photons propagating along the X direction as it passes through a uniform sample of absorbing molecules. If light of intensity I(X) enters a slice of the sample at position X, then the intensity I(X + δX) after traveling an infinitesimal distance δX is
$$\begin{array}{ll}I\left(X+\delta X\right)=& I(X)\\ {}& -\left(\mathrm{number}\ \mathrm{of}\ \mathrm{molecules}\ \mathrm{hit}\ \mathrm{by}\ \mathrm{a}\ \mathrm{photon}\right)\times \Pi \end{array}}$$
(2)
where Π is the probability that a photon that hits a molecule is absorbed. Now
$$\begin{array}{l}\mathrm{Number}\ \mathrm{of}\ \mathrm{molecules}\ \mathrm{hit}\ \mathrm{by}\ \mathrm{a}\ \mathrm{photon}=I(X)\\ {}\frac{N_{\mathrm{slice}}\times {\sigma}_{\mathrm{molecule}}}{\mathrm{volume}\ \mathrm{of}\ \mathrm{slice}}\delta X\, =I(X)c{\sigma}_{\mathrm{molar}}\delta X\end{array}}$$
(3)
where Nslice is the number of molecules in the slice of the sample of width δX, σmolecule is the cross-sectional area of the molecules facing the incident light beam, σmolar is the corresponding molar quantity, and c is the concentration of the analyte. Thus:
$$\frac{I\left(X+\delta X\right)-I(X)}{\delta X}=-I(X)c{\sigma}_{\mathrm{molar}}\delta X$$
(4)
In the limit of small δX, this becomes
$$\frac{dI(X)}{dX}=-I(X)c{\sigma}_{\mathrm{molar}}\Pi =-I(X)\varepsilon c$$
(5)
where ε is the wavelength-dependent extinction coefficient. Upon integrating across a sample of path length , this gives the Beer-Lambert law. The Beer-Lambert law is not valid if too few photons emerge from the sample for the photomultiplier tube to count them accurately. It also breaks down when the sample is inhomogeneous or if there are concentration-dependent intermolecular interactions that affect the spectroscopy.
Usually the absorbance is dominated by the electric dipole transition moment. Within a quantum electrodynamic formalism, the transition rate from ground state to final state, |0〉 → |f〉, follows from the Fermi golden rule (Tokmakoff 2009; Craig and Thirunamachandran 1984):
$$\Gamma = \frac{{N{\mathcal{B}}I\left( \omega \right)}} {{3c}}$$
(6)
where
$$\mathcal{B}=\frac{1}{2{\varepsilon}_0{\mathrm{\hslash}}^2}{\left|\widehat{\boldsymbol{e}}\cdot {\boldsymbol{\mu}}^{f0}\right|}^2$$
(7)
is an analogue of the Einstein B-coefficient which allows for polarized light. Here, N is the number of absorbers, $$\widehat{\boldsymbol{e}}$$ is the unit vector parallel to the electric field of the light, and $${\boldsymbol{\mu}}^{f\mathsf{0}}$$ is the electric dipole transition moment from the ground state to the final state. By considering a sample of small optical density (short path length or low concentration), in a transmission experiment, Craig and Thirunamachandran 1984 equated the total energy absorbed by an isotropic sample at molecular (Eq. 6) and macroscopic (Beer-Lambert Law) levels to say that the area under the decadic extinction coefficient plot is related to a transition density by
$$\int \varepsilon d\omega =\frac{2\pi {N}_A\mathrm{\hslash}{\omega}_{f0}}{3000c\ln 10}\mathbf{\mathcal{B}}$$
(8)

## References

1. Atkins PW, de Paula J (2006) Physical chemistry. Oxford University Press, OxfordGoogle Scholar
2. Atkins PW, Friedman RS (2005) Molecular quantum mechanics. Oxford University Press, OxfordGoogle Scholar
3. Craig DP, Thirunamachandran T (1984) Molecular quantum electrodynamics: an introduction to radiation-molecule interactions. Academic, LondonGoogle Scholar
4. Hollas JM (2004) Modern spectroscopy, 4th edn. Wiley, ChichesterGoogle Scholar
5. Tokmakoff A (2009) Introductory quantum mechanics II. In: Tokmakoff A, L. C. C. B.-N.-S. (ed) Massachusetts Institute of Technology: MIT OpenCourseWare. MIT, Boston. https://ocw.mit.edu