Deviations from Beer's law on the microscale – nonadditivity of absorption cross sections

Mayerhöfer, Thomas G.; Höfer, Sonja; Popp, Jürgen

doi:10.1039/c9cp01987a

Cited by 29 publications

(24 citation statements)

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“…Absorbance is additive on the molar level and linearly depending on the concentration because when the absorption cross sections of one mole of molecules are added then the molar absorption coefficient results. This reasoning cannot directly be checked for a single molecule, but what we can do is e. g. to check the additivity of the absorption cross sections of small spherical particles by numerically solving Maxwell's equations with methods like the Finite Difference Time Domain (FDTD) method, [58] as the theoretical results agree very well with experiments. [59]…”

Section: Deviations Between Beer's Law and Dispersion Theory: Non-linmentioning

confidence: 99%

“…E.g., for two spheres of amorphous SiO 2 with a radius of 125 nm, the scattering cross section is about 1 % of the absorption cross section in the range around the SiÀ O stretching vibrations (7.7-10 μm wavelength). [58] In the extreme case, the minimal distance between two spheres is zero, so that they touch each other. In this case, three different principal arrangements have to be considered, cf.…”

Section: Non-additivity Of the Absorption Cross-sections Of Small Sphmentioning

confidence: 99%

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The Bouguer‐Beer‐Lambert Law: Shining Light on the Obscure

2020

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The Beer‐Lambert law is unquestionably the most important law in optical spectroscopy and indispensable for the qualitative and quantitative interpretation of spectroscopic data. As such, every spectroscopist should know its limits and potential pitfalls, arising from its application, by heart. It is the goal of this work to review these limits and pitfalls, as well as to provide solutions and explanations to guide the reader. This guidance will allow a deeper understanding of spectral features, which cannot be explained by the Beer‐Lambert law, because they arise from electromagnetic effects/the wave nature of light. Those features include band shifts and intensity changes based exclusively upon optical conditions, i. e. the method chosen to record the spectra, the substrate and the form of the sample. As such, the review will be an essential tool towards a full understanding of optical spectra and their quantitative interpretation based not only on oscillator positions, but also on their strengths and damping constants.

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Section: Deviations Between Beer's Law and Dispersion Theory: Non-linmentioning

confidence: 99%

Section: Non-additivity Of the Absorption Cross-sections Of Small Sphmentioning

confidence: 99%

The Bouguer‐Beer‐Lambert Law: Shining Light on the Obscure

2020

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“…In addition to chemical interactions and local field effects, nearfield interactions and electromagnetic coupling, which were recently shown to influence the complex index/indices of refraction and cause nonadditivity of the absorption cross sections, [27] are explicitly excluded. Thus, the medium, i. e., the sample, is assumed to be isotropic (scalar dielectric function), not only isotropic in relation to the wavelength but completely homogenous.…”

Section: Beer's Law-why Integrated Absorbance Depends Linearly On Conmentioning

confidence: 99%

“…In particular, any alteration of the electric field intensity inside the medium must be due to absorption. If the electric field intensity changes locally, e. g., by interference effects ("electric field standing wave effect"), [38][39][40] scattering, plasmonic enhancement or electromagnetic coupling [27,41,42] the simple connection between the dielectric function and absorption index is invalidated, and the linear relationship between concentration and (integral) absorbance is therefore revoked. The details of the derivation of the sum rules are discussed in the supporting information and ref.…”

Section: Beer's Law-why Integrated Absorbance Depends Linearly On Conmentioning

confidence: 99%

Beer's Law‐Why Integrated Absorbance Depends Linearly on Concentration

2019

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As derived by Max Planck in 1903 from dispersion theory, Beer's law has a fundamental limitation. The concentration dependence of absorbance can deviate from linearity, even in the absence of any interactions or instrumental nonlinearities. Integrated absorbance, not peak absorbance, depends linearly on concentration. The numerical integration of the absorbance leads to maximum deviations from linearity of less than 0.1 %. This deviation is a consequence of a sum rule that was derived from the Kramers‐Kronig relations at a time when the fundamental limitation of Beer's law was no longer mentioned in the literature. This sum rule also links concentration to (classical) oscillator strengths and thereby enables the use of dispersion analysis to determine the concentration directly from transmittance and reflectance measurements. Thus, concentration analysis of complex samples, such as layered and/or anisotropic materials, in which Beer's law cannot be applied, can be achieved using dispersion analysis.

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“…In contrast to Raman spectroscopy, however, in UV-Vis and IR-spectroscopy we have to invoke Maxwell's wave equations to understand the changes of the intensity due to absorption. Assuming that our medium is not only scalar but perfectly homogenous (in particular nearfield interactions/electromagnetic coupling are excluded [7] ), then the simple and well-known relation e r ¼n 2 follows, [8] whereinn is the complex index of refraction. Accordingly, it follows:…”

Section: Beyond Beer's Law: Why the Index Of Refraction Depends (Almomentioning

confidence: 99%

Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration

et al. 2020

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Beer's empiric law states that absorbance is linearly proportional to the concentration. Based on electromagnetic theory, an approximately linear dependence can only be confirmed for comparably weak oscillators. For stronger oscillators the proportionality constant, the molar attenuation coefficient, is modulated by the inverse index of refraction, which is itself a function of concentration. For comparably weak oscillators, the index of refraction function depends, like absorbance, linearly on concentration. For stronger oscillators, this linearity is lost, except at wavenumbers considerably lower than the oscillator position. In these transparency regions, linearity between the change of the index of refraction and concentration is preserved to a high degree. This can be shown with help of the Kramers-Kronig relations which connect the integrated absorbance to the index of refraction change at lower wavenumbers than the corresponding band. This finding builds the foundation not only for refractive index sensing, but also for new interferometric approaches in IR spectroscopy, which allow measuring the complex index of refraction function.The exact origin of the form of Beer's law, as we employ it nowadays [1,2] AðṽÞ ¼ e * ðṽÞ � c � dwherein e * ðṽÞ is the molar attenuation coefficient, c the concentration and d the sample thickness, remains unclear. It seems that it was not employed in this form before about 1900, as it was not discussed nor provided in this form in Kayser's handbook of spectroscopy, [3] which was the reference work at this time. When Max Planck derived his particular kind of dispersion theory and employed it to compare the results with Beer's law in 1903, [4] he was most probably unaware of Equation (1), otherwise he probably would have derived it from dispersion theory. Recently, this derivation was carried out, first from dispersion theory [5] and, subsequently, from simple electromagnetic theory. [6] One of the main results of the derivation is that the molar attenuation coefficient is inversely proportional to the index of refraction, which is itself a function of concentration. Concerning the nature of this dependence, as was already stated in Ref.[5], a law comparable to Beer's law, but for the index of refraction function, can be derived. In ref. [5] the nature of this dependence was not investigated in detail. As we will show in the following, the use of the index of refraction variation instead of the absorbance to investigate concentration may have several advantages. To derive the concentration dependence of the complex index of refraction, it is possible to start from simple electromagnetic theory. Accordingly, the macroscopic polarizationP is related to the dipole momentp induced in an atom or a molecule by the following relation:Here, N is the number of dipole moments per unit volume (which is equal to the molar concentration multiplied by Avogadro's constant, N = N A ·c). Inherent to Equation (2) is the assumption that there are no interactions between the microscopic dipoles, i....

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Deviations from Beer's law on the microscale – nonadditivity of absorption cross sections

Abstract: Beer's law on the micro- and nanoscale only holds for vanishing nearfield effects and large distances between the absorbing moieties.

Cited by 29 publications

References 19 publications

The Bouguer‐Beer‐Lambert Law: Shining Light on the Obscure

The Bouguer‐Beer‐Lambert Law: Shining Light on the Obscure

Beer's Law‐Why Integrated Absorbance Depends Linearly on Concentration

Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration

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