The Symmetry

As mentioned above, the symmetry of the ferroelectric smectic C phase corresponds to the polar symmetry group C2, Fig. 7.1, so that when going along the -coordinate parallel to a helix axis and perpendicular to the smectic layers the director L and the polarization vector P, directed along the C2 axis, rotate such as 

In the absence of external fields the equilibrium helix pitch Rq corresponds to the minimum of the FLC free energy, i.e.. 

The necessary conditions for its existence (P 0) are a finite tilt angle 6 0, chirality of molecules, resulting in the hindered rotation of molecules 

FIGURE 7.1. (a) Macroscopic structure of the ferroelectric phase and (b) its microscopic interpretation [23]. 

A classical example of an FLC molecule is p-decyloxybenzylidene-p -amino-2-methyl-butyl cinnamate or DOBAMBC (l.ix), first discovered by Meyer et al. [1]. The chiral fragment of the molecule 

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The measurements were made along the cracks with an average step size of 3 mm. The predictions were calculated from a position -15 mm to + 15 ram for set 1, from -40 mm to + 40 mm for set 2 and from -25 mm to + 25 mm for set 3. The impedance change has been calculated at 1mm intervals in the range. Taking into account the symmetry of the configuration, only half of the predictions need to be calculated. 

For precise 3D-FEM simulations, a huge number of nodes is required (>30,000), which results in calculation times of several hours (sun spare 20) for one model. In order to decrease the number of nodes, we took advantage of the symmetry of the coils and calculated only a quarter or half of the test object. The modelled crack has a lenght of 15 mm, a height of 3 mm and is in a depth of 5 mm. The excitation frequency was 200 Hz. 

Equation XVI-21 provides for the general case of a molecule having n independent ways of rotation and a moment of inertia 7 that, for an asymmetric molecule, is the (geometric) mean of the principal moments. The quantity a is the symmetry number, or the number of indistinguishable positions into which the molecule can be turned by rotations. The rotational energy and entropy are [66,67]... 

The individual values of the exponents are detennined by the symmetry of the Hamiltonian and the dimensionality of the system. 

It is easy to derive the ooexistenoe ourve. Beeause of the symmetry, the double tangent is horizontal and the ooexistent... 

For T shaped curves, reminiscent of the p, isothemis that the van der Waals equation yields at temperatures below the critical (figure A2.5.6). As in the van der Waals case, the dashed and dotted portions represent metastable and unstable regions. For zero external field, there are two solutions, corresponding to two spontaneous magnetizations. In effect, these represent two phases and the horizontal line is a tie-line . Note, however, that unlike the fluid case, even as shown in q., form (figure A2.5.8). the symmetry causes all the tie-lines to lie on top of one another at 6 = 0 B = 0). 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

Often it is possible to resolve vibrational structure of electronic transitions. In this section we will briefly review the symmetry selection rules and other factors controlling the intensity of individual vibronic bands. 

Because of the generality of the symmetry principle that underlies the nonlinear optical spectroscopy of surfaces and interfaces, the approach has found application to a remarkably wide range of material systems. These include not only the conventional case of solid surfaces in ultrahigh vacuum, but also gas/solid, liquid/solid, gas/liquid and liquid/liquid interfaces. The infonnation attainable from the measurements ranges from adsorbate coverage and orientation to interface vibrational and electronic spectroscopy to surface dynamics on the femtosecond time scale. 

All of the symmetry classes compatible with the long-range periodic arrangement of atoms comprising crystalline surfaces and interfaces have been enumerated in table Bl.5,1. For each of these syimnetries, we indicate the corresponding fonn of the surface nonlinear susceptibility With the exception of surfaces... 

For femiions (especially) and bosons diere are additional problems. Let /Jbe one of the pemuitations of particle labels. Then the femiion density matrix has the symmetry... 

Another interesting physical feature relates to the cliromophoric character of fullerenes. Based on the symmetry prohibitions, solutions of [60]fullerene absorb predominantly in the UV region, with distinct maxima at 220, 260 and 330 nm. In contrast to extinction coefficients on the order of 10 cm at these wavelengths, the visible region shows only relatively weak transitions (X at 536 nm s =710 cm ) [142]. 

NMR is not the best method to identify thennotropic phases, because the spectmm is not directly related to the symmetry of the mesophase, and transitions between different smectic phases or between a smectic phase and the nematic phase do not usually lead to significant changes in the NMR spectmm [ ]. However, the nematic-isotropic transition is usually obvious from the discontinuous decrease in orientational order. NMR can, however,... 

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

In these equations, J and M are quantum numbers associated with the angular momentum operators and J, respectively. The number II = 0, 1 is a parity quantum number that specifies the symmetry or antisymmetry of the column vector with respect to the inversion of the nuclei through G. Note that the same parity quantum number II appears for and Also, the... 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electi onic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164-168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

A final point to be made concerns the symmetry of the molecular system. The branching space vectors in Eqs. (75) and (76) can be obtained by evaluating the derivatives of matrix elements in the adiabatic basis... 

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