Thomas G. Mayerhofer

The Bouguer-Beer-Lambert law is frequently applied in spectroscopy and spectrophotometry. Often it is assumed that it accurately describes the observed changes induced by light-matter interactions and properly reflects the physical phenomena at play. For most cases, however, this is not true, and issues can arise when the Bouguer-Beer-Lambert law is applied uncritically. In this short article, we will comment on its fundamental limits and their physical background.

The Bouguer-Beer-Lambert (BBL) law has one big flaw, and that is its name. A more appropriate denomination, in our opinion, would be the Ideal Absorption law. In analogy to the ideal gas law, everybody would be well aware of the fact that it is just a (sometimes very good) approximation. Consequently, we would be much more cautious when applying it. Also, authors of basic textbooks and university teachers would spend more effort to explain its limits. Since completely renaming a well-known relation is usually not a good idea, we will call it the “BBL Approximation” in this article, where we will make you familiar with its limitations and pitfalls.

For truly understanding the BBL approximation, it is helpful to have some knowledge about its origin. When Bouguer and Lambert found the expression (1,2):

they were studying the absorbing properties of the atmosphere. 

 is the intensity of the light before and 

 the intensity after the traverse through the atmosphere, 

, a constant. As a side note, in the original expression the natural logarithm came up, but since it was much easier at that time to compute decadic logarithms, and since both differ just by a constant, the decadic logarithm was used later on.

Now, what are the limitations of equation 1? In general, it is always a good start to recall the particular use case—the experimental conditions (see Figure 1). If we acquire a spectrum of the atmosphere, the light source and the detector would actually be within the medium that is investigated. Consequently, we are not dealing with transmission of light through a sample, but with light that travels within a medium. Accordingly, in contrast to transmission, there is no reflection, since there is no interface between the incidence medium and the sample, as the sample is in fact the incidence medium. In addition, there is no interface between the sample and an exit medium, because the sample is also the exit medium in this special case. So, again, reflection at this second interface also does not occur. If a sample is put in the light path that has well-defined interfaces perpendicular to this path, then the light transmitted into the sample is not only reflected at the second interface and traveling back, but a part is then reflected at the first interface. This part is then traveling along the original light path until it hits the second interface where again a part is reflected and traveling back (3). These multiple reflections are not considered in equation 1. If the thickness of the sample is the same over the whole area where the light beam hits it, then something happens which is not at all taken into account in equation 1. Light behaves as a wave and the forward and backward traveling waves interfere with each other. Depending on the thickness of the sample, the wavelength of the light and the refractive index of the sample at that wavelength, the light intensity fluctuates (4). If the interference is destructive, the intensity will be lower than expected and higher for constructive interference. Accordingly, for such samples one would not obtain a straight line, if the intensity was plotted over the thickness, but it would in fact fluctuate. This is of great importance for interpreting infrared (IR) absorbance spectra, since the described effects will affect band intensities in corresponding spectra. In addition, band shapes can change and shifts can occur. Typical samples, where this can happen, are films on IR transparent substrates like CaF

, ZnSe, or Si (see Figure 2). The latter two often lead to distinct interference fringes which boldly remind you that equation 1 is just a first approximation (5). For CaF

, visible fringes are absent, but interference effects are still at play, although more subtly (6). Thin films on metals are another case where the wave nature of light leads to pronounced effects: one measures the reflectance, but often obtains spectra very similar to transmission experiments. Constructive and destructive interference is in particular strong for such samples, and bands can even occur in spectral regions where there is no absorption of the material the film is made of (7,8). Many methods have been devised to deal with interference effects. Unfortunately, most of them just aim at removing the fringes visually without considering their physical origin. Consequently, other interference-based changes remain in the spectra, which makes their correct interpretation difficult. To achieve more than cosmetics, a wave optics-based approach is required (5,6,8).

FIGURE 1: Schematic display of the different conditions of the experiments as performed (a) by Bouguer and Lambert and (b) the way spectroscopy and spectrophotometry are usually performed today.

FIGURE 2: Comparison of the influence of sample geometry, measurement technique and substrate on absorbance spectra. Bottom shows light interactions with different substrates. Adapted with permission from reference (9).

A common misbelief is that interpreting equation 1 in the following, different way, can solve the issues mentioned above in general:

 is the transmittance of the sample, and 

 is the transmittance of the substrate. Where does equation 2 come from? It actually reflects the way spectrophotometry is being performed. Accordingly, 

 is the transmittance of the absorbing solute in a non-absorbing solvent and 

 is the transmittance of the neat non-absorbing solvent. However, when certain conditions are met, equation 2 will actually work. As long as the cuvette is thick and has thickness inhomogeneities, these will cause the interference effects to average out (10). Furthermore, as long as the refractive index of the solution is not very different from that of the neat solvent, the way light is multiply reflected is approximately the same, and corresponding effects cancel out as well. Finally, the refractive index of the solvent should be as small as possible, but values up to around 1.5 are all right, if equation 2 is used, in contrast to equation 1, where the refractive indices are assumed to stay close to 1 as those of gas mixtures do (11). So far, we did not discuss the BBL approximation in the form which is commonly used in textbooks (12,13).

 is the molar absorption coefficient and 

 is the molar concentration. Before we discuss the limits of the right side of equation 3, a word of caution is necessary: It seems that using the molar concentration is somewhat arbitrary, and one can use mass or weight equally well. This is a fallacy caused by the assumption that this right part is empiric. Actually, it is not, as recently has been shown (14). It must be a concentration based on the number of molecules in a certain volume. This means that amount or volume fractions for close to ideal mixtures are equally acceptable, but not weight or mass fractions. Overall, you should keep in mind that the concentrations must be very small. Not only because of changing chemical interactions that can lead to changing material properties—this is nowadays a very common misconception. In fact, if you put a dye in different solvents with which it does not chemically interact, it nevertheless changes its color. The reason is that light interacts with matter and polarizes it, which can lead to color changes. The degree of polarization depends on the environment of a molecule (15) and, thereby, the color changes (16). As a consequence, 

 is a constant only for neat substances. As long as you keep the concentration small, a dye molecule will only be influenced by the solvent, but once the concentration becomes too large, the dye molecule is also influenced by others of its kind. If you are interested in larger concentrations or even in liquid or solid mixtures (in gases there is no problem at lower pressures due to the larger intermolecular distances), and you have made sure that chemical interactions do not occur or did not change, we recommend focusing on comparably weak bands (17), because the lower the transition moment, the smaller the polarizability. Focusing on a single band caused by a weak transition can be much more accurate than chemometric methods using the full spectrum (18).

By the way, sometimes you read that shadowing of one molecule by another is a problem at higher concentrations. In our opinion, this is nothing but a hoax, since molecules are so small compared to the wavelength that light acts as a wave and not as a ray.

Coming back to the experiments conducted by Bouguer, Lambert, and Beer, there is one more aspect, we have not discussed yet. Gases as well as diluted solutions are microhomogeneous. This means that if you use a microscope operating at the same frequencies, wavelengths, or wavenumbers and inspect your sample, it will look exactly the same everywhere. If this is not the case, you will be in trouble, if you want to apply the BBL approximation (19,20). As an example, consider a polymer like polytetrafluoroethylene which has pores with diameters in the micrometer range. This might be all right in the terahertz range, because, due to the long wavelengths, it feels an effective refractive index. If the wavelengths are smaller, than light that travels within a pore is not absorbed at all. On the other hand, while entering and leaving the pore, it is reflected, leading to scattering. Depending on the microstructure and the wavelength, one or the other effect will dominate. But, since the microstructure depends, for example, on the thickness of the polymer, simply determining the absorbance and dividing it by the thickness will not lead to the same result as equation 1 would imply.

Certainly, this short article could only touch superficially on this topic. A more detailed discussion of these and other limits can be found in a recent review by us (9). For solutions of some of the related problems, we suggest to consult books about wave optics and dispersion theory (4,21,22).

(2) Lambert, J. H.; Anding, E. Lambert’s photometrie: (photometria sive de mensura et gradibus luminis, colorum et umbrae). (1760). Heft 3, Tl. 6 und 7 -

(3) Airy, G. B. On the Phenomena of Newton’s Rings token formed between two transparent Substances of different refractive Powers. 

The London and Edinburgh Philosophical Magazine and Journal of Science

rinciples of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light; 

(5) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. Removing Interference-Based Effects from Infrared Spectra – Interference Fringes Re-Revisited. 

(6) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. CaF2: An Ideal Substrate Material for Infrared Spectroscopy? 

(7) Mayerhöfer, T. G.; Popp, J. The Electric Field Standing Wave Effect in Infrared Transflection Spectroscopy. 

(8) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. Removing Interference-Based Effects from the Infrared Transflectance Spectra of Thin Films on Metallic Substrates: A Fast and Wave Optics Conform Solution. 

(9) Mayerhöfer, T. G.; Pahlow, S.; Popp, J. The Bouguer-Beer-Lambert Law: Shining Light on the Obscure. 

(10) Hawranek, J. P.; Neelakantan, P.; Young, R. P.; Jones, R. N. The Control of Errors in I.R. Spectrophotometry—III. Transmission Measurements Using Thin Cells. 

(11) Mayerhöfer, T. G.; Mutschke, H.; Popp, J. Employing Theories Far beyond Their Limits—The Case of the (Bouguer-) Beer–Lambert Law. 

(14) Mayerhöfer, T. G.; Popp, J. Beer’s Law Derived from Electromagnetic Theory. 

(15) Lorentz, H. A. Ueber die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte. 

(16) Spange, S.; Mayerhöfer, T. G. The Negative Solvatochromism of Reichardt‘s Dye B30 – A Complementary Study. 

(17) Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Popp, J. Beyond Beer’s Law: Quasi-Ideal Binary Liquid Mixtures. 

(18) Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Popp, J. Infrared spectroscopy of quasi-ideal binary liquid mixtures: The challenges of conventional chemometric regression. 

(19) Jones, R. N. The Absorption of Radiation by Inhomogeneously Dispersed Systems. 

(20) Mayerhöfer, T. G.; Popp, J. Beyond Beer’s Law: Spectral Mixing Rules. 

Wave Optics in Infrared Spectroscopy - Theory, Simulation and Modelling;

 are with the Leibniz Institute of Photonic Technology (IPHT), in Jena, Germany, and the Institute of Physical Chemistry and Abbe Center of Photonics at Friedrich Schiller University, in Jena, Germany. Direct correspondence to: 

Articles

The Bouguer-Beer-Lambert law has its limitations and it doesn't always properly reflect the physical phenomena at play. This article examines the law's limitations. 

Understanding the Limits of the Bouguer-Beer-Lambert Law

Spectroscopic literature often conveys the impression that peak shifts are generally related to structural changes on the molecular or unit cell level. Here, we discuss several situations in which peaks shift for reasons unrelated to chemical structural changes. To avoid misinterpretations, a practitioner should be aware of these reasons.

When you are a student and learn about spectroscopic theory, you are told that every peak shift in an infrared (IR) or ultraviolet-visible (UV-vis) spectrum is related to changes in chemical structure or chemical composition. However, this important lesson sometimes does not hold up in the first few months that you start working on your bachelors, masters, or PhD thesis. The latter happened to one of us when working on two isostructural compounds for a thesis. In one of these isostructural compounds, a heavy element was replaced by a much lighter one. Based on the accepted theory, the expectation was that only some peaks should be blueshifted, because they are related to vibrations of the heavier and lighter element. However, when the spectra were recorded, it was a shock to see that nearly all peaks were blueshifted, and X-ray diffraction (XRD) confirmed that the two samples where chemically isostructural. This raises an important question: What was the reason behind this phenomenon?

Peak Shifts Are Consequences of Increased Oscillator Strength and Molar Concentration

Spectroscopic theory, as we know it today, is a simplified version of a merger of wave optics and dispersion theory, where the wave properties of light were removed to make it easily digestible (1). When Hendrik Antoon Lorentz and Max Planck worked to understand dispersion based on the damped harmonic oscillator model, it was clear to them, for example, that peaks shift with oscillator strength (2,3). The damped harmonic oscillator model, often called the 

, generates a peak in the absorption index spectrum that blue-shifts with increasing oscillator strength and molar concentration, whereas the oscillator position (the band position) remains fixed (4). The Lorentz oscillator model is not common knowledge because Lorentz also derived the Lorentz profile, the peak of which does not shift because of some approximations (2,5). In academic curriculum, only the Lorentz profile is consistently taught. The approximations are usually valid for weak oscillators, but easily noticeable, for example, for the C=O vibration, where it is of the order of 5 cm

. This might seem hardly worth mentioning, but the oscillators in inorganic materials are much stronger and as a result, so are the corresponding shifts.

Why should we care about shifts that we might not even be able to see in spectra? Because nowadays, techniques are in use like 2D-correlation spectroscopy (2D-COS), principal component analysis (PCA), or multiple curve resolution (MCR) (6,7). Even if you do not see shifts in conventional spectra, these techniques are so sensitive that they will show the resulting nonlinearities (8). Accordingly, in experimental spectra, because you will always get an asynchronous spectrum in 2D-COS, you will always detect more principal components by PCA than you have chemical components, even in ideal systems. Meanwhile, MCR will help you to find components that are often nothing but a mere linear combination of the spectra of the real components, which indicates nothing but nonlinearities caused by peak shifts and deviations from Beer’s law.

Peak Shifts as a Consequence of the Polarization of Matter by Light 

Sometimes these shifts are cancelled by an effect known by Lorentz and Planck (2,9). When light hits matter, the latter becomes polarized. As a consequence, the local electric field is not the one externally applied (it is the same effect you know from capacitors). For example, these local fields lead to increasing redshifts if you put your solute in different solvents with increasing refractive index or dispersion strength (10,11). Figure 1 displays a comparison between the Lorentz profile, a Lorentz oscillator, and this effect for various concentrations. The same is observed when changing from potassium bromide (KBr) to cesium iodide (CsI) and finally to silver bromide (AgBr) as a matrix material for the pellet technique (12). This transformation occurs because of an interaction that is induced by light and vanishes if switched off, which is a macroscopic manifestation of the uncertainty principle: By observation, you change what you want to see.

FIGURE 1: Peak shifts modelled for different concentrations with the constant c·d = 5·10

 cm·mol/L. Shown here is (a) Lorentz profile, (b) Lorentz oscillator, (c) uncoupled oscillator assuming that light induces polarization, and (d) coupled oscillator assuming that induces polarization. The oscillator has an oscillator strength typical of a C=O vibration (10). Note that the black and red trace overlap and all curves in case of (a). Note concentrations shown (in mol/L) are: (black) 0.05, (red) 0.5, (green) 5, and (blue) 50.

Peak Shifts as a Consequence of the Influence of the Refractive Index on Spectra

Comparably strong redshifts that are not related to matrix effects, but are also because of the influence of the refractive index, this time not of the embedding, but of the incidence medium and the sample, occur for attenuated total reflection (ATR) (4,13). In the case of ATR, the absorbance is modulated by the refractive index dispersion of the sample. Because the refractive index has a maximum redshifted relative to that of the maximum of the absorption index, the absorbance maximum is also redshifted, easily by 10 cm

 and more, even for organic substances if you use the very common combination of a zinc selenide (ZnSe) ATR crystal with an angle of incidence of 45° (4). Note that the shift scales again with oscillator strength, which means that for mixtures the shift increases with concentration because it is often combined with the one resulting from local field effects. To avoid corresponding problems, you can either use a high index incidence medium like germanium at a high angle of incidence, which only gives problems with inorganic materials. As long as the critical angle of the sample is always lower than the incidence angle of your accessory, you can also correct the problem with an advanced ATR correction procedure, or, with an even more advanced procedure, which lets you determine the optical constants of your sample, but this requires use of a polarizer (4). Figure 2 compares the influence of the sample geometry, technique, and substrate on absorbance spectra.

FIGURE 2: Comparison of the influence of sample geometry, technique, and substrate on absorbance spectra of (a) poly(methyl methacrylate) (PMMA), and (b) spinel (inorganic material). Film thicknesses for layers on calcium fluoride (CaF

) and silicon substrates are 1 μm and 500 nm for gold (Au); in these cases, wave interference is at play. For attenuated total reflection (ATR) the film thickness is much larger than the effective thickness to reflect only effects introduced by the ATR technique (4). Key to spectra are (black) calcium fluoride, (red) silicon, (green) gold, (blue). (a): ATR ZnSe 45°-s; (b) ATR germanium 60°-s, (orange) true absorbance.

Peak Shifts as a Consequence of the Wave Nature of Light and Interference Effects

Quite often, samples consist of layers on substrates. It seems to be clever to use refractive-index matched substrates to avoid interference effects within the layers (which change the internal fields from the ones applied), but because absorption is accompanied by refractive index changes, this approach can easily fail, particularly for thinner layers (4 μm or less in case of calcium fluoride) (14). If you use a highly reflecting substrate, like a low-e glass or a metal-coated substrate, the interference effects can be so strong that not only the peaks can shift by 25 cm

 or more, but new peaks not directly related to vibrations may emerge (15,16). New peaks emerge because the product of electric field intensity and absorptivity because of a neighboring peak is particularly high. Such interference effects are not restricted to layers—for example, they also exist in spheres (17). Correspondingly, Mie scattering-related effects in heterogeneous samples do not only generate disturbing baselines, but also cause interference-related shifts in spectra. Keep in mind that just getting rid of ripples or fringes in the baseline does not correct the shifts in your spectra!

Peak Shifts as a Consequence of Anisotropy and the Continuity Relations of the Electric Field at Interfaces

A last effect that we will touch upon that causes shifts in spectra is anisotropy. This may be surprising, but the signal in your spectrum does not only depend of the orientation of the transition moment relative to the polarization direction, as you usually hear. The orientation of the transition moment relative to the interface is also important because this orientation can cause shifts (1,18,19). Why? Because if the transition moment is perpendicular to the surface, then the transmittance or reflectance does not depend on the dielectric function, but on its inverse (this is a consequence of the continuity relations of the electric field between two different media). The maximum in the spectrum is, therefore, at the minimum in the dielectric function, which is always blueshifted. For organic materials this shift is usually 5 cm

 or smaller, but for inorganic materials a shift of 50 cm

 is nothing out of the ordinary (1). Actually, this was also the case for the example we mentioned in the beginning—have a look at Figure 3. Funnily enough, these two samples were randomly oriented polycrystalline samples, but it was nevertheless an effect related to the anisotropy of the individual crystallites and how orientation averaging takes place. The latter is different if the crystallites are small or large compared to the resolution limit, which causes the blueshifts. The simple (but not always cheap) solution is to use the corresponding single crystals instead.

FIGURE 3: Reflectance spectra of randomly oriented, polycrystalline fresnoite (Ba

) with crystallites small and large compared to the resolution limit of IR light. Key to spectra are (black) small polycrystalline fresnoite, (red) large polycrystalline strontium fresnoite, (green) large polycrystalline fresnoite.

From the potpourri of examples that we presented, you get a good impression of the many different effects that can be at work, sometimes in combination, and that you may have to correct before you can interpret remaining peak shifts in the classical way. In principle, such a correction is adequate with the determination of the optical constants of a sample, which can require significant effort at times. Before we close, a word of caution: At the moment, many exciting innovations are becoming available, such as new modalities with which you can record spectra via surprising ways like photothermal IR upon quantum cascade laser irradiation, but we are still investigating the resulting consequences of the new measurement schemes or radiation sources. However, we can say that all effects described in this paper will stay, but additional ones will be introduced.

. DOI: 10.13140/ RG.2.2.14293.55520: 2021.

Sitzungsberichte der K ̈oniglich Preussischen Akademie der Wissenschaften

(4) T.G. Mayerhöfer, S. Pahlow, and J. Popp, 

Two-Dimensional Correlation Spectroscopy: Applications in Vibrational and Optical Spectroscopy 

Quantitative Infrared Spectroscopy for Understanding of a Condensed Matter

(8) T.G. Mayerhöfer, O. Ilchenko, A. Kutsyk, and J. Popp, 

Beyond Beer’s law: Ideal binary liquid mixtures

Sitzungsberichte der K ̈oniglich Preussischen Akademie der Wissenschaften zu Berlin

(13) L.A. Averett, P.R. Griffiths, and K. Nishikida, 

(14) T.G. Mayerhöfer, S. Pahlow, U. Hübner, and J. Popp, 

(16) T.G. Mayerhöfer, S. Pahlow, U. Hübner, and J. Popp, 

(17) T.G. Mayerhöfer, S. Höfer, and J. Popp, 

Peak shifts in infrared spectra may occur for many reasons other than structural changes on the molecular or unit cell level. Here, we discuss several examples.

Thomas G. Mayerhofer

Articles

Understanding the Limits of the Bouguer-Beer-Lambert Law

Five Reasons Why Not Every Peak Shift in Infrared (IR) Spectra Indicates a Chemical Structure Change