Based on Beer's law, it is assumed that the absorbance of a mixture is that of the neat materials weighted by their relative amounts (linear mixing rule). In this contribution, we show that this is an assumption that holds only under various approximations for which no change of the chemical interactions is just one among several. To understand these approximations, which lead incrementally to different well known mixing rules, we finally derive the linear mixing rule from the Lorentz–Lorenz relation, with the first approximation that the local electric field is correctly described in this relation. Further levels of approximation are that the local field equals the applied field (Newton–Laplace mixing rule) and that the change of the index of refraction and, equivalently, absorption is weak (Gladstone–Dale/Arago–Biot mixing rule). Even then the linear mixing rule is only strictly valid if the indices of refraction in the transparency region at higher frequency than the absorption have the same value and the mixing is homogeneous relative to the resolving power of the light (“micro-homogeneous”). Under these preconditions, linear mixing of the individual absorbances is established. We illustrate the spectral differences between the different mixing rules, all of which are based on volume and not on mass fractions, with examples. For micro-heterogeneous samples, a different linear mixing rule governs the optical properties, which refers to the experimental quantities, reflectance, and transmittance. As a result, for such samples, mixtures of already comparably high content give only weak signals due to band flattening, which are hard to distinguish from baseline effects.

中文翻译：

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EXPRESS：超越啤酒定律：光谱混合规则

根据比尔定律，假设混合物的吸光度是纯材料的吸光度，按其相对量加权（线性混合规则）。在这个贡献中，我们表明这是一个假设，仅在各种近似值下才成立，其中化学相互作用的变化只是几个近似值之一。为了理解这些逐渐导致不同的众所周知的混合规则的近似值，我们最终从洛伦兹-洛伦兹关系中推导出线性混合规则，第一个近似是在这种关系中正确描述了局部电场。进一步的近似水平是局部场等于外加场（牛顿-拉普拉斯混合规则）并且折射率的变化和等效的吸收很弱（Gladstone-Dale/Arago-Biot 混合规则）。即使这样，线性混合规则也只有在高于吸收频率的透明区域中的折射率具有相同值并且混合相对于光的分辨能力是均匀的（“微均匀”）时才严格有效。在这些前提条件下，建立了各个吸光度的线性混合。我们举例说明了不同混合规则之间的光谱差异，所有这些都基于体积而不是质量分数。对于微异质样品，不同的线性混合规则控制着光学特性，即实验量、反射率和透射率。因此，对于此类样品，由于频带平坦化，已经相当高含量的混合物仅给出微弱信号，这很难与基线效应区分开来。