Introduction to vibrational spectroscopies

Abstract

Vibrational spectroscopies play, still today, a crucial role in the non-destructive characterization of material having the most varied origins (e.g., environmental, geological, polymeric, artistic, etc.), and, together with UV/VIS spectroscopy, they represent university students' first approach to spectroscopy. Vibrational spectroscopy may be defined as a non-destructive identification tool that measures the vibrational energy in a compound. Each chemical bond corresponds to a specific vibrational energy that can be considered as a distinctive fingerprint, useful to determine compound structures, by comparing it with the fingerprints of known compounds. Several techniques are included under the name of vibrational spectroscopy, but the most important are spectroscopies in the middle and near infrared and Raman. Both mid-infrared and Raman spectroscopies are related to the fundamental vibrations of the molecules, which are then used to determine their structures. Since the vibrational energy levels are unique to each molecule, the infrared and Raman spectra provide a fingerprint of a particular compound. The frequencies of these molecular vibrations depend on the masses of the atoms, their geometric arrangement, and the strength of their chemical bonds. Spectral interpretation thus provides information on the molecular structure, dynamics, and surrounding environment. The aim of this tutorial text is to give a general view of the two main vibrational spectroscopy techniques, namely mid-infrared and Raman spectroscopies. An insight into surface-enhanced Raman spectroscopy, a fundamental technique that aims to overcome the limitations of Raman, is also given. The three techniques are discussed separately, with a brief introductory explanation of the theory behind them, and giving useful practical information about instrumentation, sample preparation, and spectral interpretation. The text can be considered the basis for two or three lectures (4–6 h) in a university course of analytical chemistry/applied spectroscopy.

Graphical abstract

Appendices

Appendix A

A glossary of common concepts related to optical spectroscopy is listed below.

Absorbance

The absorbance is the quantity of light that is absorbed by a body. In the spectroscopic field it is defined as the opposite of the logarithm of the transmittance:

$$A = - \ln T = \ln I_{0} - \ln I_{1}$$

(19)

where I₁ is the intensity of transmitted light and I₀ the intensity of the incident light at a given wavelength. Absorbance is linearly related to the concentration of a sample—for sufficiently low concentrations—according to the Lambert-Beer law.

Attenuation, absorption, and scattering coefficients

During its propagation in a medium other than vacuum, an electromagnetic wave can be attenuated. The intensity I of a wave at a distance x from the source of intensity I₀ is attenuated exponentially according to the following Lambert-Beer law:

$$I = I_{0} {\text{e}}^{ - \alpha x}$$

(20)

where α is called the effective attenuation coefficient of the wave in the medium. The forward propagation of the wave in the medium depends on both absorption and scattering. The absorption coefficient k_a and the scattering coefficient k_s of a radiation at a given wavelength in a medium are defined as the fraction of the power that is respectively absorbed and scattered per unit length of the medium, and they are related to the effective attenuation coefficient according to the following relationship:

$$\alpha = \sqrt {\frac{{k_{a} }}{D}}$$

(21)

where D is the constant of diffusion of the wave in the medium, inversely proportional to both k_a and k_s. The total attenuation coefficient k is defined as the sum of k_a and k_s.

Fourier transform

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes a function into its constituent frequencies. The FT of a signal ƒ is defined by the integral:

$$f\left( \omega \right) = \int_{ - \infty }^{ + \infty } {f\left( t \right)e^{ - i\omega t} {\text{d}}t} = \int_{ - \infty }^{ + \infty } {f\left( t \right)[\cos \left( {\omega t} \right) - i\sin \left( {\omega t} \right)]{\text{d}}t}$$

(22)

If ƒ(ω) is the superposition of ƒ(t) with a wave of frequency ω and we consider only the real part:

$$f\left( \omega \right) = \sum\nolimits_{ - \infty }^{ + \infty } {f\left( t \right)\cos \left( {\omega t} \right)}$$

(23)

Only the frequencies that contribute to the signal give a finite (integral) summation while the remaining ones return a zero (integral) summation. The FT of a signal in the time domain is therefore the decomposition of the signal into its frequency components and their corresponding amplitudes (spectrum). FT is an invertible operation: it is possible to exactly reconstruct the original signal starting from its frequency spectrum (inverse Fourier transform).

Intensity

Better called radiant flux (or radiant power) is $I = \Phi_{{\text{e}}} = \frac{{\partial Q_{{\text{e}}} }}{\partial t}$, where Q_e is the radiant energy and t time. Its SI unit is watts or J/s.

Reflectance

The reflectance of the surface of a material is the proportion of directly incident light, conventionally expressed as a percentage, reflected at its boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light and is in general a function of the frequency of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.

Scattering

The scattering (or diffusive dispersion) of electromagnetic radiation is defined as a process that does not involve any type of absorption or emission, according to which there is a dispersion of the radiation by macroscopic or microscopic objects. Generally, in physics the term scattering distinguishes a large class of phenomena in which one or more particles are deflected (i.e., they change their trajectory) because of collisions with other particles. At the microscopic level, the scattering process of an electromagnetic wave originates from the interaction between the incident wave and electron cloud of an atom. In particular, the electrons are placed in oscillation in phase with the applied wave; the result is an electronic oscillator capable of emitting radiation in all directions.

Transmittance

Transmittance (T) is the fraction of incident electromagnetic power that is transmitted through a sample, expressing its effectiveness in transmitting radiant energy. It is defined as T = I₁/I₀, where I₁ is the intensity of transmitted light and I₀ the intensity of the incident light at a given wavelength, and it is commonly expressed as %T (i.e., I₁/I₀ × 100).

Appendix B

Drude-Lorentz model

The optical properties of bulk materials are characterized by their relative permittivity, or dielectric function ϵ_r(ω), which is related to the refraction index n(ω) by:

$$n\left( \omega \right) = \sqrt { \in_{r} \left( \omega \right)}$$

(24)

with both n(ω) and ϵ_r(ω) depending on the light frequency ω.⁴ For a small list of transparent objects, such as lenses and prisms, both n(ω) and ϵ_r(ω) are positive real numbers. More often than not, ϵ_r(ω) at a given wavelength is a complex number, and the material is not transparent, such as in the case of metals.

The electronic structure of metals explains their optical properties. In 1900, Paul Drude (1863–1906) proposed a model to describe the transport properties of electrons in materials, especially in metals.

The Drude model is based on the Lorentz model for atomic polarizability, which describes the optical response of an electron bound to an atom with a restoring force with frequency ω₀. In a metal the conduction electrons are free to move; thus, the restoring force due to the attraction of the nuclei can be ignored in a first approximation (ω ₀ ≈ 0) without introducing significant errors. In this model, when an electric field E₀ is applied to the metal, each electron is accelerated along the direction of the field, colliding with the atoms of the metal. The overall effect of these collisions is equivalent to a viscous force that counterbalances the electric force, so that, on average, the speed of the electrons (along the direction of the field) is constant:

$$m\ddot{x} = - m\gamma_{0} \dot{x} - eE_{0} {\text{e}}^{ - i\omega t}$$

(25)

where m is the electron mass, γ₀ is a damping term due to collisions, and i is the imaginary unit. By solving the differential equation that describes the free electrons' motion (hint: look for a solution to Eq. (25) of the form $x = x_{0} {\text{e}}^{ - i\omega t}$; see [24] for an exhaustive explanation), it is possible to obtain the displacement of the electric charge from its equilibrium position under the action of the electric field. The overall effect of the radiation is the system polarization:

$$x = - \frac{e}{m}\frac{1}{{(\omega^{2} - i\gamma_{0} \omega )}} E_{0} {\text{e}}^{ - i\omega t} = x_{0} {\text{e}}^{ - i\omega t}$$

(26)

whereas the polarization P is

$$P = nex_{0} {\text{e}}^{ - i\omega t}$$

(27)

The electric displacement field or electric induction D is defined as:

$$D = \in_{0} E + P = \in \left( \omega \right)E$$

(28)

with ${\epsilon }_{0}$ the dielectric function in the vacuum. Using the previous equations, the dielectric function can be expressed as:

$$\in_{r} \left( \omega \right) = \frac{ \in \left( \omega \right)}{{ \in_{0} }} = 1 - \frac{{ne^{2} }}{{m \in_{0} }}\;\frac{1}{{\omega^{2} + i\gamma_{0} \omega }}$$

(29)

where n is the number of free electrons per unit of volume and e the elementary charge. If we introduce the plasma frequency ω_p, i.e., the oscillation frequency of the charge density of a free electron plasma, given by the expression:

$$\omega_{p} = \sqrt {\frac{{ne^{2} }}{{m \in_{0} \in_{\infty } }}}$$

(30)

Equation 29 can be written as:

$$\in_{r} \left( \omega \right) = 1 - \frac{{\omega_{{\text{p}}}^{2} }}{{\omega^{2} + i\gamma \omega }}$$

(31)

Since the dielectric function is a complex number, its real and imaginary part can be expressed individually:

$$\in^{\prime}\left( \omega \right) = 1 - \frac{{\omega_{{\text{p}}}^{2} }}{{\omega^{2} + \gamma_{0}^{2} }} \in^{\prime\prime}\left( \omega \right) = \frac{{\omega_{{\text{p}}}^{2} \gamma_{0} }}{{\omega \left( {\omega^{2} + \gamma_{0}^{2} } \right)}}$$

(32)

For most metals, at room temperature the damping term γ₀ is negligible compared to ω, so we can reformulate the real and imaginary part of the dielectric function:

$$\in^{\prime}\left( \omega \right)\sim \left( {1 - \frac{{\omega_{{\text{p}}}^{2} }}{{\omega^{2} }}} \right) \in^{\prime\prime}\left( \omega \right)\sim \frac{{\omega_{{\text{p}}}^{2} }}{{\omega^{3} }}$$

(33)

When ω >  > ω_p, then ${\epsilon }_{r}\left(\omega \right)\cong 1$, and the metal is transparent to the radiation. When the frequency ω < ω_p, then ϵ’(ω) < 0, and, if ω is not too small, also ϵ’’(ω) will be small in this region. When these two conditions are satisfied, i.e., ϵ’(ω) < 0 and ϵ’’(ω) are small, particular optical effects arise, such as plasmonic resonances or the high reflectivity of metals. For metals such as gold or silver, the plasma frequency is in the UV range, corresponding to a working frequency, when nanoparticles are involved, lying in the visible or near-IR region (hint: to obtain the relation between the working frequency and the plasma frequency, calculate the frequency of the laser that induces the largest electric field inside a spherical metallic nanoparticle).

In a real situation, the optical phenomena must also be associated with inter-band transitions, which change the dielectric function into:

$$\in_{r} \left( \omega \right) = \in_{{\text{b}}} \left( \omega \right) - \frac{{\omega_{{\text{p}}}^{2} }}{{\omega^{2} + i\gamma_{0} \omega }}$$

(34)

where ϵ_b(ω) is a coefficient associated with the inter-band transitions affecting both real and imaginary parts of ϵ_r. In most cases, inter-band transitions occur for energies in the UV, where ω >  > ω_p, and their contribution to ϵ_r in the visible is constant and real. For noble metals, inter-band transitions are due to transitions between the d-bands and the empty states in the s-p conduction band. In this case, the threshold of the inter-band transitions is located in the visible-ultraviolet region of the spectrum. The resulting increase in absorption is at the origin of the characteristic coloration of noble metals: red for copper, yellow for gold, and metallic for silver, which is the only one to have an absorption peak shifted in the UV area.