- The basic equations • Interpretation of the Faraday A-, B- and C-terms. - 8.3 A vibronic development of the vibrational Raman. - One could say that optical activity provides a peephole into the fabric of the universe!. - As well as using the results of the. - η ellipticity of the polarization ellipse θ azimuth of the polarization ellipse S 0 , S 1 , S 2 , S 3 Stokes parameters. - D ( j) irreducible representation of the proper rotation group R 3 + T q k irreducible spherical tensor operator. - g i g-value of the ith particle spin. - e i electric charge of the i th particle. - A α,βγ real part of the electric dipole–electric quadrupole optical activity tensor. - A α,βγ imaginary part of the electric dipole–electric quadrupole optical activity tensor. - A α,βγ real part of the electric quadrupole–electric dipole optical activity tensor. - A α,βγ imaginary part of the electric quadrupole–electric dipole optical activity tensor. - The propagation direction is out of the plane of the paper. - the circular dichroism of the medium.. - 1.4 The ellipticity and anomalous optical rotatory dispersion in the region of the electronic absorption wavelength λ j . - which is the ratio of the circular dichroism to the conventional absorption. - For light passing through the medium in the direction of the magnetic field (north pole to south pole) most diamagnetic materials rotate the plane of polarization. - 1.6 The normal Zeeman effect (a) for light emitted perpendicular to the magnetic field and (b) for light emitted in the direction of the magnetic field.. - The sign shown here obtains when the magnetic field is in the direction of propagation of the light beam.. - it provides a measure of the anisotropy in the molecular optical activity using an isotropic sample such as a liquid or solution.. - to the direction of propagation of the beam.. - The effects originate mainly in a partial orientation of the molecules in the medium. - glass plate in the presence of a magnetic field perpendicular to the direction of propagation of the light (Williams, 1968). - Under T , the direction of propagation of the light beam is reversed, but its sense of optical rotation is preserved. - Since T reverses the direction of the magnetic field but does not affect the medium (if nonmagnetic in the absence of the applied magnetic field) (Fig. - The tip of the electric field. - Reversing the handedness of the ellipticity reverses the sense of optical rotation. - 1 2 corresponding to the two opposite projections of the spin angular momentum onto a space-fixed axis. - Of the experiments they suggested, that performed by Wu et al. - The conceptual basis of the theory rests on two pillars. - Instead, an inversion of just one of the two axes is required (Halperin, March-Russel and Wilczek, 1989). - is the refractive index of the medium.. - (2.2.8) which is conveniently written as the real part of the complex expression. - (2.2.9) where ω , the angular frequency of the wave, is related to the wavelength by ω = 2 πv/ λ . - These two equations are uncoupled by exploiting the arbitrariness in the definition of the potentials.. - where Λ is an arbitrary function of the coordinates and time. - The prop- agation direction z is out of the plane of the paper. - Measured intensities are time averages of real quadratic functions of the fields.. - A convenient representation of the polarization tensor is ρ αβ = 1 2 [i α i β + j α j β + (i α i β − j α j β ) P cos 2 η cos 2 θ. - (2.3.20) where i α and j α are the α components of the unit vectors i and j.. - where r i is the position vector of the ith charge. - where m i and p i are the mass and linear momentum of the ith charge. - The vector product r × p is the orbital angular momentum l of the particle. - (2.4.15) where g i is the g-value of the i th particle spin (g = 2 . - which change during the motion of the charges. - The meaning of the terms μ (0) α − n α n β. - v is the generalized momentum p of the particle. - From (2.5.6) the potential energy of the ith charge at r i in a static electric field characterized by a scalar potential is e i φ (r i. - 1) n T 12 , where n is the order of the tensor.. - The antisymmetric part of the second term of (2.5.19) already has the. - the second of which makes use of the Maxwell equation. - and the A 2 term has no components at the frequency of the incident light wave.. - We require approximate solutions of the time-independent Schr¨odinger equation. - In the absence of V , the general solution of (2.6.17) is the stationary state. - (2.6.28b) Similarly for the other molecular property tensors involving products of the same multipole transition moments. - where the complex fields of the plane wave light beam are E ˜ α = E ˜ (0) α e i( κβ r β −ω t). - and the real electric dipole moment in terms of the complex moment (2.6.43a), μ α = 1 2. - We illustrate these properties initially for the case of the symmetric polarizability α αβ. - complete knowledge of the other part at all frequencies. - crossing relations of the form (2.6.51) obtain for ˜ R αβ ( ω ) (but not for ˜ α αβ ( ω. - This leads to the following relations between the dispersive and absorptive parts of the antisymmetric polarizability:. - From (2.7.3), the frequency separation of the perturbed levels is. - where μ j γ = j |μ γ | j is the electric dipole moment of the molecule in the unper- turbed state ψ j . - This can be generated as the matrix element of the following complex scattering operator (Berestetskii, Lifshitz and Pitaevskii, 1982):. - W ¯ is the average of the energies W n and W m of the initial and final states. - For example, (2.8.14h) follows from a consideration of temporal correlations between the electric and magnetic dipole moment operators in the absence of the light beam (Harris, 1966).. - An approximate solution of the Schr ¨odinger equation for the complete molecule, namely. - The electronic eigenfunction ψ e (r , R) is a solution of the Schr¨odinger equation [T (r. - The nuclear eigenfunction ψ en ( R) is a solution of the Schr ¨odinger equation [T ( R. - The complete specification of the vibrational part is rather. - Thus the vibrational part of the jth state is written. - Using (2.8.33), the first part of the vibrational transition polarizability (2.8.32) becomes. - Except when the frequency of the exciting light is in the infrared region or below, this term can be ignored.. - Q p ) 0 Q p in the expansion of H e (r , Q), the electronic degeneracy will be lifted so that Q 0 is not an equilibrium configuration. - This chapter constitutes the heart of the book. - For example, Tyndall’s early investigations with aerosols (1869) showed that linear polarization was an important feature of light scattered at right angles, and he pointed out that (quoted by Kerker, 1969) ‘The blue colour of the sky, and the polarization of skylight. - where n d is the propagation vector in the direction of the detected wave. - E (0) β , (3.3.2c) where n i is the propagation vector of the incident wave. - The thickness of the lamina is infinitesimal relative to the wavelength of the light. - From (3.3.3) the electric field of the scattered wave at f from a volume element dx dydz at (x , y, 0) in the lamina is. - The propagation vector in the direction of the detected wave may be written. - and ˜ a αβ f , the forward part of the scattering tensor, is given by (3.3.4) with n d = n i . - (3.4.13b) The forward part of the scattering tensor (3.3.4) in the present approximation now simplifies to. - (3.4.14) where n is the unit vector in the propagation direction of the incident light beam.. - A further contribution to the Kerr effect originates in the perturbation of the dynamic molecular polarizability by the electric field, in accordance with (2.7.1):. - is the corresponding potential energy of the molecule in the field. - We therefore require the following components of the scattering tensor:. - dx dy dz where N is the number density of molecules in the absence of the field. - The required electric fields of the scattered waves at f are now. - (3.4.35) By allowing for the perturbation of the dynamic molecular property tensors in the scattered electric fields (3.4.32) by the electric field and field gradient,. - (3.4.51) The macroscopic change in the degree of polarization is obtained from an integral of the form. - In the Faraday effect, parity arguments (Section 1.9.3) require the magnetic field to be applied along the direction of propagation of the light beam. - The Stokes parameters of the scattered electric vector E ˜ d in the x d , y d , z d system are. - (3.5.8) and P, θ and η specify the polarization of the incident beam.. - In the Raman case ρ depends on the effective symmetry of the molecule and the symmetry species of the molecular vibration. - Another quantity of interest is the circularly polarized component of the scattered light. - (4.2.9) Tensor manipulations are simplified considerably by the use of the following notation. - Thus contraction is the tensor equivalent of the scalar product in vector analysis. - (4.2.12) Hence contraction of the second-rank tensor VW (a dyadic product) has given a tensor of rank zero (a scalar).. - Thus, concentrating on each of the axes x , y , z we obtain