24/ Rayleigh and Raman scattering. Stokes and anti-stokes lines. Rotational Raman spectroscopy. Polarizability ellipsoid. Selection rules.
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1 Subject Chemistry Paper No and Title Module No and Title Module Tag 8/ Physical Spectroscopy 24/ Rayleigh and Raman scattering. Stokes and anti-stokes lines. Rotational Raman spectroscopy. Polarizability ellipsoid. Selection rules. CHE_P8_M24
2 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. The Classical Theory of the Raman Effect (Molecular Polarizability) 4. The Quantum Theory of the Raman Effect 5. Pure Rotational Raman Spectra 6. Summary
3 1. Learning Outcomes After going through this module, you should be able to: (a) Explain the classical theory of the Raman effect and polarizability (b) Explain the quantum theory of the Raman effect (c) Understand the rotational Raman spectra 2. Introduction Raman spectroscopy is another form of spectroscopy, which is used for structure determination. Unlike the other techniques, it is based on the scattering of light. When a beam of monochromatic radiation passes through a liquid or gas, it might be transmitted, absorbed or scattered. In the case of light scattering, nearly all of the scattered light is observed at the same frequency as the incident light (a phenomenon known as elastic or Rayleigh scattering). However, the great Indian physicist Sir C.V. Raman observed in 1928 that some of the scattered light also has discrete frequencies above and below the incident frequency. This phenomenon is known as inelastic or Raman Scattering. The Raman Effect comprises a very small fraction, about 1 in 10 7, of the incident photons. A powerful laser source in the visible or UV region is used as the light source. Figure 1 shows Rayleigh and Raman scattering. The lines at lower frequency than the incident radiation are known as Stokes lines, and those at higher frequency are called anti-stokes lines.
4 Figure 1: Rayleigh and Raman scattering Unlike the other techniques, the molecular property that must change for a rotation or vibration to be Raman active is the molecular polarizability, which is a measure of the ease with which the electron cloud in a molecule can be distorted. This rule should be contrasted with that for IR and microwave activity, which states that the molecular motion must produce a change in the electric dipole of the molecule. Thus, Raman spectroscopy can be used on species with no dipole moment, e.g. H 2, Cl 2, etc., thus complementing microwave spectroscopy. 3. The Classical Theory of the Raman Effect (Molecular Polarizability) When a static field is applied on a molecule, a polarization of charge takes place, the positively charged nuclei getting attracted to the negative pole of the field, and the electronic cloud to the positive pole. A dipole is induced in the molecule, even if it has no dipole moment initially, and the molecule is said to be polarized. The induced electric dipole moment, µ ind, depends on both
5 the magnitude of the applied field, E, and the ease with which the electron cloud of the molecule can be distorted. Thus, µ ind = αe (1) where α is the polarizability of the molecule. The polarizability is anisotropic, which means that the induced dipole moment is not the same in all directions. The polarizability is greater along the bond axis than across it since it is difficult to polarize in a direction perpendicular to the internuclear axis. The polarizability in various directions is conveniently represented by drawing a polarizability ellipsoid. The ellipsoid is a three dimensional surface whose distance from the electrical centre of the molecule is proportional to 1/ α i, where α i is the polarizability along the line joining point i on the ellipsoid with the electrical centre. Thus, where the polarizability is greatest, the axis of the ellipsoid is least (it is the minor axis of the ellipsoid). Since the polarizability of a diatomic molecule is the same in all directions at right angles to the bond axis, the ellipsoid has a circular cross section in these directions (Fig. 2). (a) (b) (c) (d) Figure 2: Polarizability ellipsoid of hydrogen molecule
6 When a sample of diatomic molecules is subjected to a beam of radiation of frequency v, the electric field experienced by each molecule varies as follows: E = E 0 sin2πνt, (2) A time-dependent dipole moment is, therefore, induced in the molecule: µ ind = αe = αe 0 sin2πνt (3) where E 0 is the strength of the applied field. According to electromagnetic theory, an oscillating dipole emits radiation whose frequency is the same as its oscillation frequency. Equation (3) is, therefore, the classical explanation of Rayleigh scattering. If, in addition, the molecule undergoes some internal motion, such as a vibration or a rotation, which changes the polarizability periodically, then the oscillating dipole will have superimposed upon the vibrational or rotational oscillation. The oscillating dipole has frequency ν ± ν vib as well as the exciting frequency v. It should be noted that if the vibration or rotation does not change the polarizability of the molecule, the dipole oscillates only at the frequency of the incident radiation. Thus, we have the selection rule: In order to be Raman active, a molecular rotation or vibration must cause some change in a component of the molecular polarizability. A change in polarizability is reflected by a change in either the magnitude or direction of the polarizability ellipsoid. 4. The Quantum Theory of the Raman Effect According to the Quantum Theory, radiation of frequency v can be pictured as a stream of particles (called photons) each having energy hv. These photons are imagined to undergo either inelastic or elastic collisions with molecules. Inelastic collisions result in energy exchange between the photon and the molecule, while elastic collisions result in no energy exchange at all. If the molecule gains energy ΔE from the photon, the photon will be scattered with energy (hv - ΔE) and with frequency (v - ΔE/h). If, however, the molecule loses energy ΔE to the incident radiation, the scattered radiation will have a frequency of (v + ΔE/h). Radiation scattered with a frequency lower than that of the incident radiation is known as Stokes radiation, while that of a higher frequency is anti-stokes radiation. Rayleigh scattering is the
7 situation where the scattered radiation has the same frequency as the incident radiation (see Fig. 1). 5. Pure Rotational Raman Spectra We restrict ourselves to linear molecules. From Figure 2, it is clear that the polarizability ellipsoid of a diatomic molecule changes in appearance when the molecule rotates, and for the first time we are able to get rotational constants for homonuclear diatomic molecules like H 2 and N 2. However, symmetrical molecules like CCl 4 and SF 6 have spherical polarizability ellipsoids and they are neither microwave active nor Raman active, since their polarizability ellipsoids neither change size nor shape during rotations. It is also clear from Figure 2 that the polarizability ellipsoid of a linear molecule presents the same appearance twice during a complete rotation. The selection rule for rotational transitions for Raman spectroscopy is: ΔJ = 0, ± 2 (4) The ΔJ = 0 selection rule refers to Rayleigh transitions, ΔJ = 2 to emissions, and we may ignore both of these in case of absorptions (Stokes lines). The rotational energy levels of linear molecules are given by: F ( J ) = BJ ( J + 1) J = 0, 1, 2, (5) Here, we have not considered centrifugal distortion since the accuracy of Raman spectroscopy is not too high to warrant its inclusion. The energy difference between the excited and ground state is given by Δε = F( J + 2) F( J ) = B( J + 2)( J ) BJ ( J + 1) = B(4J + 6) (6)
8 where J is the rotational quantum number of the lower state (= 0, 1, 2, ). The lines resulting from these energy changes are referred to as the S branch line since we have considered only ΔJ = +2. Thus, if the molecule gains rotational energy from the photon during collision, we have a series of S branch lines on the low wavenumber side of the exciting line (Stokes lines), while if the molecule loses energy to the photon, the S branch lines appear on the high wavenumber side (anti-stokes lines). The wave numbers of the corresponding spectral lines are thus given by: ν = ± B(4J + 6) (7) s ν ex where the minus refers to the Stokes lines and the plus to the anti-stokes lines. The allowed transitions and the Raman spectrum arising is shown in Figure 3. The spacing between adjacent lines is 4 B and the first line to each side of the exciting line is at 6 B. Thus, the Raman spectra yield values of B and hence the moments of inertia and bond lengths for small molecules. Raman spectroscopy is supplementary to microwave and infrared methods. It is a method for obtaining molecular parameters of homonuclear diatomic molecules which are IR and microwave inactive due to the absence of permanent dipole moments in the molecules.
9 Figure 3: Pure rotational Raman transitions and spectrum
10 --- Example 1 The wavenumber of the incident radiation is cm -1. What is the wavenumber of the scattered Stokes radiation for the J = 0 2 transition of 14 N 2. The equilibrium bond length is 110 pm. Solution For a homonuclear diatomic molecule, the reduced mass µ = /2/ = kg. The moment of inertia I = µ r = ( ) = kg m 2 The rotational constant B = h/8π Ic = /8/ / / = cm -1 The Stokes lines occur towards the low wavenumber side and the first transition is at 6 B. Therefore, the line will appear at = cm Summary Raman spectroscopy is based on the scattering of light. Most of the scattered radiation has the same frequency as the incident radiation and this is called Rayleigh scattering. A small part has discrete frequencies at the lower (Stokes lines) and higher (anti-stokes) sides. For a species to be Raman active, the vibration or rotation must cause some change in the polarizability of the molecule. A change in polarizability is reflected by a change in either the magnitude or the direction of the polarizability ellipsoid. Raman spectroscopy can give information about species with no dipole moments, hence complementing microwave and IR spectroscopy.
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