Rotational symmetry INDEX

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

Low Miller index surfaces of metallic single crystals are the most commonly used substrates in LEED investigations. The reasons for their widespread use are that they have the lowest surface free energy and therefore are the most stable, have the highest rotational symmetry and are the most densely packed. Also, in the case of transition metals and semiconductors they are chemically less reactive than the higher Miller index crystal faces. 

The metal substrates used in the LEED experiments have either face centered cubic (fee), body centered cubic (bcc) or hexagonal closed packed (hep) crystal structures. For the cubic metals the (111), (100) and (110) planes are the low Miller index surfaces and they have threefold, fourfold and twofold rotational symmetry, respectively. 

Identity element, 387-388 Identity operation, 54, 395 Improper axis of symmetry, 53 Improper rotation, 396 Index of refraction, 132 INDO method, 71, 75-76 and ESR coupling constants, 380 and force constants, 245 and ionization potentials, 318 and NMR coupling constants, 360 Induced dipole moment, 187 Inertial defect, 224-225 Inertia tensor, 201... 

The rotational symmetry breaking cannot be detected in such a way there are no orientational Goldstone modes. One could look at Raman spectra or neutron diffraction experiments that are sensitive to the molecular orientations. The order parameter field for an orientation order in molecular systems can be chosen to be a three-component field of the cosine distribution of the mutual orientations of molecular axes. This index reveals the continuous, low-temperature transition. 

In Tables 6-8 we have listed the submonolayer and monolayer stractures of metals on the principal low index surfaces of metal substrates from 2- to 4-fold rotational symmetry. This compilation comprises heteroepitaxial systems only since the structure of homoepitaxial systerrrs is in most cases trivial. For umeconstructed surfaces the bulk stacking is pseudomorphically continued and for reconstructed ones the reconstruction is lifted below the adlayer and at the same time taken on by the adsorbate layer. Only few homoepitaxial cases are worth mentioning since their reconstructions can metastably be lifted, as seen for Au/Au(110)-(lx2) [97Giin], or a reconstruction can be induced at a lower tenqreratrrre by homoepitaxial adsorption, as seen for Pt/Pt( 111) [93Bot]. 

Second, we consider the optical anisotropy of the blue phases. Generally speaking, the refractive indices of a crystal form an ellipsoid, as discussed in Chapter 2. Now the blue phases have cubic symmetries. On a macroscopic scale, the refractive index ellipsoid must have the same cubic symmetries. Cubic symmetries contain four-fold rotational symmetry around three orthogonal axes. Therefore the refractive index ellipsoid must be a sphere, that is, the refractive index in any direction is the same at macroscopic scale. Due to this optical isotropy, when a blue phase sample is sandwiched between two crossed polarizers, the transmittance is zero. This is the dark state of the blue phase display based on field induced birefringence. 

For imiaxial crystals, two of the n ( i =ri2) are equal and the index ellipsoid has rotational symmetry around the principal axis, called the optical axis of the uniaxial crystal, which we choose to be the z-axis of our laboratory coordinate system (Fig. 5.107a). 

Equations (7.53) simplify if the system has point-group or translational symmetry. Then local basis functions of equivalent cells are related to those of a smaller number of generating cells by phase factors and elementary rotations. Considering only translational symmetry, it is convenient to index cells in a reference translational cell by indices /x, etc. and to index translated equivalent cells by the corresponding indices jl, etc., such that the displacement of cell if, relative to cell t,... 

By convention, surface superlattices are given names like 31/2 x 1 R 30° this is Wood s 20 notation of 1964 [8], [9]. This notation describes the symmetry of the superlattice, but does not identify its origin—that is, the exact point where the superlattice is anchored (even though this is a very desirable datum for aficionados of chemisorption). The 3m x 1 R 30° superlattice means that (1) the fls and b axes of the superlattice crystal lie in the plane of the surface of index hkl (whose axes we call abM and bbki), (2) the first superlattice axis as is 31/2 times longer than the first surface axis fl(3) the second superlattice axis bs is equal in length to the second surface axis b/ h (4) these two superlattice axes fls and bs are then rotated by 30° clockwise. An alternative notation uses... 

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