Tensor polarisability

A sequence of calculations can be performed with various applied electric fields in which the dipole moment of the molecule is evaluated, as described above. The 3x3 polarisability tensor, can therefore be constructed. 

Eo and E (Afi(i)) are respectively the electric fields generated by the permanent and induced multipoles moments. a(i) represents the polarisability tensor and Afi(i) is the induced dipole at a center i. This computation is performed iteratively, as Epoi generally converges in 5-6 iterations. It is important to note that in order to avoid problems at the short-range, the so-called polarization catastrophe, it is necessary to reduce the polarization energy when two centers are at close contact distance. In SIBFA, the electric fields equations are dressed by a Gaussian function reducing their value to avoid such problems. 

Here, cfy (co) is the linear polarisability tensor at frequency co and describes the linear variation of the induced dipole with the electric field. The resulting oscillation in the polarisation (oscillating dipole) is, in effect, a moving charge, and will therefore emit radiation of the same frequency as the oscillation. This gives rise to linear optical effects such as birefringence and refraction. 

Using the polarisation tensors (B.16, B.17), the free photon propagator (A.9) and the longitudinal and transverse polarisation functions /7 /r( ) the Dyson equation for reads... 

Guha and Chase reported that an electronic Raman spectrum could be observed only in experiments that select a component of the polarisability tensor transforming as Eo or Eg. The electronic matrix elements in these polarisation geometries are. 

The polarisability tensor is given by a second-order perturbation expression known as the Kramers-Heisenberg dispersion formula (4)... 

To assist in the evaluation of the invariants in terms of elements of the cartesian polarisability tensor, it is convenient to define the symmetric and antisymmetric tensors Spfj — 2 (ttpa pa T ( pa op) Now... 

Polymers are not, however, always randomly oriented and the phenomenon of orientation is discussed in the next two chapters, chapters 10 and 11. One of the ways used to obtain information about the degree of orientation of the polymer is to measure either the refractive indices of the polymer for light polarised in different directions or its birefringences, i.e. the differences in refractive index for light polarised in different directions. In chapter 11 the theory of the method is described in terms of the polarisability of structural units of the polymer. This polarisability is a second-rank tensor like the molecular polarisability referred to in the earlier sections of this chapter and, insofar as the assumption of additivity in section 9.2.3 holds, it is in fact the sum of the polarisability tensors of all the bonds in the unit. Since, however, the whole basis of the method is that the structural units are anisotropic, the tensors must be added correctly, taking account of the relative orientations of the bonds, unlike the treatment used to calculate the refractive index of PVC in example 9.1, where scalar bond refractions are used. 

OX 1X2X2 so that corresponding components for all bonds can be added. It is usual to assume that the polarisability tensor for a bond is cylindrically symmetric around the bond axis. This means (see the appendix) that, if the bond axis for a particular bond in the unit is chosen as the 0x3 axis of a set of axes OX1X2X2 the tensor takes the form an = a22 = oii and = a, with all other components zero, where at represents the component transverse to the bond and represents the component parallel to the bond. Values of at and ap are given for various types of bond in table 9.5. 

The off-diagonal components ai2 etc. are not zero, but, if it is assumed that the axes OX1X2X2 are symmetry axes of the unit, these off-diagonal components will sum to zero when the components for all the bonds in the unit are added to obtain the components a, q ]2, etc. of the polarisability tensor for the unit. The assumption that the axes OT1T2X3 are symmetry axes for the unit means that the unit must have at least orthotropic (statistical orthorhombic) symmetry. Units of lower symmetry are not considered here. 

The calculation of the principal refractive indices for non-orthorhombic crystals is a little more complicated because the axes of the indicatrix, or refractive-index ellipsoid (see section 2.8.1), carmot be predicted in advance of the calculation. It is therefore necessary to calculate the values of all six independent components of the polarisability tensor of the crystal with respect to arbitrarily chosen axes and then to find the principal axes of the resulting tensor. 

The refractive index thus depends on the average value of a,-,-, which depends on the anisotropy of the polarisability and the degree of orientation of the sample. It is shown below that, for uniaxial orientation with respect to OZ3 (for which wj = 2 = t) and a cylindrical polarisability tensor with low asymmetry. 

Equation (10.6) is derived as follows. Assume that the principal axes of the polarisability tensor coincide with the axes OX1X2X3 of the stmetural unit and that, with respect to these axes, the polarisability tensor [ y] takes the form... 

Consequently, the Raman scattered light emanating from even a random sample is polarised to a greater or lesser extent. For randomly oriented systems, the polarisation properties are determined by the two tensor invariants of the polarisation tensor, i.e., the trace and the anisotropy. The depolarisation ratio is always less than or equal to 3/4. For a specific scattering geometry, this polarisation is dependent upon the symmetry of the molecular vibration giving rise to the line. 

The magnitude of the spin polarisation tensor, A x> must be related to the rhombic distortion in the molecule, since it disappears in axially symmetrical molecules. The amount of rhombic distortion may be calculated from the cobalt hyperfine and g- tensors. Some of the deduced parameters in solving the McGarvey(2) equations are related to the energy of excited states. In particular, the excited doublet state (in symmetry) is split in rhombic symmetry to B (d 2) (C2v(Z))- The energy above the ground... 

Fig. 10. Plot relating the axial distortion A against the P polarising tensor (-Axx) ...

In these expressions u is a unit vector along the direction of the transition dipole moment (6i /6vibrational mode concerned and is the anistropic part of the transition polarisability tensor o/j (t)(s 5Qfjj (t)/bq (t) q =o) which can be conveniently divided into an isotropic part... 

It may be shown that the Raman measurements are capable of yielding information on both < cos 0 > and < cos d >. The availability of < cos 0 > data can be valuable is distinguishing between the differing types of stress deformation mechanisms that have been proposed. However, an interpretation of the band intensities in terms of and is possible only when the principal components of the derived polarisability tensor are known. This information is often not available and assumptions must then be made these then render the method non-absolute. Examples of this approach will be considered briefly below. 

Robinson, F.N.H. (1968) Relations between the components of the nonlinear polarisability tensor in cubic and hexagonal II-VI compounds. Physics Letters A, 26, 435. 

---