Читать книгу Spectroscopy for Materials Characterization - Группа авторов - Страница 13

An introductory experiment that highlights the effect of absorption of light is represented in Figure 1.1.

A parallel beam of lightwave with wavelength λ and intensity I ₀(λ) impinges perpendicularly on a face of a parallelepiped specimen of matter. It is useful to recall that λ and ν are connected by the speed of light c (2.9979 × 10⁸ m⋅s⁻¹ in vacuum): λ = c/ν [1]. Passing through the sample, the light intensity could be reduced, and at the exit, the amount I measured at λ could be diminished to the value I _t(λ) [2–4]. The eventual intensity reduction inside the specimen increases continuously on increasing the size L. If a portion of thickness dx of the sample is considered, it is expected that passing through it, I decreases by a quantity dI (for simplicity, λ will be omitted henceforth). This effect depends on the presence of absorbing centers in the volume considered, on the one side, and on their physical properties, on the other. These features are taken into account by the concentration of centers, N, and by their cross section, σ. Larger is the concentration of centers, larger absorption will take place. On increasing the probability of radiation–matter interaction, σ increases too, as well as the absorption effect. If a homogeneous and isotropic distribution of absorbing centers inside the volume of the sample is considered, it can be assumed that [3]:

(1.1)

where the units are J (cm²⋅s)⁻¹ for I, centers cm⁻³ for N, cm² for σ, and cm for dx. By considering the entire sample, Eq. (1.1) can be integrated to obtain:

(1.2)

and determine the solution

(1.3)

In general, from solution (1.3), it is found that at a position x inside the sample

(1.4)

this is the Lambert–Beer law that expresses the attenuation of light intensity as a function of the thickness of the sample traversed [2, 3]. A typical expected profile of light intensity in traversing a sample is reported in Figure 1.1. It is useful to introduce some quantities commonly associated to the absorption effect. The empirical one is the absorption coefficient defined by the experimental macroscopic measurement of attenuation:

(1.5)

It is easy to show that α = σN, connecting the macroscopic quantities α and N to the microscopic one σ (see Section 1.2 to find the relation to atomic and molecular properties). Then, we report the instrumental quantity, the optical density (OD), also called absorbance (A) [5, 6]:

(1.6)

and the transmittance, T:

(1.7)

Figure 1.1 Schematic representation of a beam of light at wavelength λ passing through a parallelepiped of matter. In the bottom, the qualitative decrease of intensity is reported. I ₀ 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):

(1.8)

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:

(1.18)

and the conversion equation

(1.19)

that enables to correlate a value of energy with the wavelength and vice versa. Another useful quantity in spectroscopy is the wavenumber, . This is defined by (wavelength)⁻¹, 1/λ, and, using Eq. (1.18), it is shown that

Figure 1.2 Bottom: Typical absorption spectrum registered as a function of wavelength. Top: Representative experimental absorption (continuous line) and emission (dashed line) spectra registered as a function of wavelength.

(1.20)

The wavenumber is usually reported in units of cm⁻¹. Combining Eqs. (1.19) and (1.20), it is found that

(1.21)

Concluding this paragraph, it is worth mentioning that the absorption phenomenon is one of the basic processes of the radiation–matter interaction and it is extended in a wide range of energy of the electromagnetic spectrum. The underling physical process is related to the specific atomic or molecular species absorbing the energy from the electromagnetic wave [8, 9]. The frequency range of interest for this chapter includes the visible (Vis) radiation and goes from the near infrared (NIR) to the ultraviolet (UV). In particular, the visible range in vacuum extends in frequency from about 3.8⋅10¹⁴ to 7.5⋅10¹⁴ Hz, in wavelength from 800 to 400 nm, and in energy from 1.6 to 3.1 eV [1].