is the impinging intensity and I t the transmitted one.
It is worth noting that when OD ≪ 1, 1 − T = 1 − 10−OD ≈ 1 − (1 − OD) = OD = A, so the absorbance and the optical density can be derived directly from the transmittance [6]. Finally, it is also useful to introduce a quite diffuse alternative to Eq. (1.6):
where ε is the molar extinction coefficient, or molar absorption coefficient, having units liter/(mole⋅cm) [M−1⋅cm−1], and C is the concentration of absorbing centers, in mole/liter [M]. By equating (1.6) and (1.8), it is shown that
(1.9)
having used centers cm−3 for N and cm2 for σ. Then, considering the Avogadro’s number, N A = 6.022 1023 centers mole−1, we obtain the conversion formula
(1.10)
these quantities are related to the electronic states of absorbing centers, as will be shown later.
Concluding, the Lambert–Beer law states that the optical density is proportional to the concentration of absorbing centers and to their electronic properties. All of the above considerations can be extended to any λ and the study of absorption as a function of the wavelength impinging on the sample gives origin to the absorption spectrum.
It is worth noting that some physical phenomena can influence the experimental evaluation of the optical density. The light scattering (both elastic process, Rayleigh scattering, and anelastic process, Raman scattering [7, 8]) can deviate the beam and avoid its exit in the detection direction. This effect could give origin to an inexact estimate of OD and can be evidenced by a λ −4 background dependence of absorbance [1, 7]. In particular, it could be erroneously concluded that photons have been absorbed whereas only their path has been deviated by the matter without any energy transfer from the electromagnetic field to the atoms. A second physical effect is the emission of light from the sample caused by the return of the electron to its thermal equilibrium state after the absorption phenomenon, promoting it to an excited state (see further). The emission is usually at a wavelength different to the impinging one, but if the light exiting from the sample is not recorded identifying the λ, as usually done in a single monochromator spectrometer, a wrong estimate of the optical density can be done. The latter effect could be relevant if absorption is large and photons of impinging light are highly reduced in number through the sample and the exiting counted photons mainly coincide with those emitted. The latter effect can be instrumentally avoided by using a double monochromator setup. Neglecting instrumental effects like stray light, that is parasitic light arriving at the detector not passing through the sample, and signal‐to‐noise limits [2, 3], another physical effect to be taken into account is reflection [1, 4]. When the parallel beam reported in Figure 1.1 impinges perpendicularly on the sample surface, the mismatch of refractive index between the medium (n 1) and the sample (n 2) induces a transmitted and a reflected beam [1, 4]. Introducing the reflectivity r for normal incidence of light:
(1.11)
the light entering the sample has the intensity
(1.12)
This light is attenuated, according to the Lambert–Beer law (1.4). Furthermore, before exiting the sample, the light is reflected again on the exit surface. It is found that the transmitted light for single reflection path is given by
(1.13)
and the absorbance estimation is
(1.14)
This result shows that, due to the reflection effect, an absorption different from zero is experimentally observed even in the absence of absorbing centers, that is when N = 0. Taking the more accurate multiple reflections effect between the two surfaces with refractive index mismatch between the sample and the medium, it is found that [4]:
(1.15)
where p is the reflection factor. When this factor, or the refractive index dependence on the wavelength, is not known, the “parasitic” effect of reflection cannot be estimated. A technical solution is to take the measurement of the same material using two different thicknesses, if possible. In fact, considering two samples of thickness L 1 and L 2, respectively, we obtain:
(1.16)
(1.17)
This way, it is shown that the two measurements enable to find the experimentally relevant features related to the absorbing centers: the cross section and the concentration, or the absorption coefficient.
To conclude this paragraph, in Figure 1.2, a typical absorption spectrum is reported with the absorbance in the vertical axis and the wavelength (in nanometer) in the horizontal axis.
It can be observed that the amount of absorbance (or optical density) changes by changing the wavelength, with a profile depending on the specific features of the investigated material. To carry out a meaningful interpretation of the spectrum, taking in due account the spectral profile and the electronic state distribution, the wavelength axis has to be changed into an axis of energy, E (usually reported in electronvolt, eV; 1 eV = 1.602 ⋅ 10−19 J). To achieve this aim, it is useful to refer to the Planck–Einstein relation [9]: E = hν, where h is the Planck’s constant (6.626 ⋅ 10−34 J⋅s = 4.136 ⋅ 10−15 eV⋅s). To convert the axis from wavelength to energy, one can use the formula:
and the conversion equation
that enables to correlate a value of energy with the wavelength and vice versa.
