Symmetry rotation axes

Quasicrystals or quasiperiodic crystals are metallic alloys which yield sharp diffraction patterns that display 5-, 8-, 10- or 12-fold symmetry rotational axes - forbidden by the rules of classical crystallography. The first quasicrystals discovered, and most of those that have been investigated, have icosahedral symmetry. Two main models of quasicrystals have been suggested. In the first, a quasicrystal can be regarded as made up of icosahedral clusters of metal atoms, all oriented in the same way, and separated by variable amounts of disordered material. Alternatively, quasicrystals can be considered to be three-dimensional analogues of Penrose tilings. In either case, the material does not possess a crystallographic unit cell in the conventional sense. 

Quasi Crystals. Until 1984. one of the most firmly established fundaments of crystallography was the postulate that crystals have three-dimensional translation symmetry, and that only symmetry rotation axes with multiplicities 1,2. 3,4, and 6 can occur because they are the only ones that are consistent with homogeneous space filling. However electron microscopy and electron diffraction have shown that 5-, 8-, 10-, and 12-fold symmetry axes occur in the structure of certain, usually rap-... 

In the crystal structures, neighboring doublehelices have the same rotational orientation and the same translation of half a fiber repeat as in the PARA 1 model. Only the Ax vector is slightly larger in the calculated interaction (1.077 nm) than in the observed ones 1.062 nm and 1.068 nm in the A type and B type, respectively. This may be due to the fact that in the crystal structures the helices depart slightly from perfect 6-fold symmetry. Also, no interpenetation of the van der Waals surfaces is allowed in the calculations, whereas some of them may occur in the cristallographic structure. It is quite interesting to note that the network of inter double-helices hydrogen bonds found in the calculated PARA 1 model reproduces those found in the crystalline structures. 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

For an Abelian group, each element is in a class by itself, since X 1AX = AX IX = A. Since rotations about the same axis commute with each other, the group e is Abelian and has n classes, each class consisting of one symmetry operation. 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

Figure 3.7. Left the symmetry operations of the two-dimensional square lattice the thick straight lines indicate the reflections (labeled ax,ay,ai,as) and the curves with arrows the rotations (labeled C4, C2, C4). Right the Irreducible Brillouin Zone for the two-dimensional square lattice with full symmetry the special points are labeled F, S, A, M, Z, X.

The second term in Eqs. (9.75) and (9.76), die rotational atomic polarizability tensor reflects the contribution of molecular translation and rigid-body rotation to ax- The inclusion of the six external molecular coordinates in those equations - the diree translations Xy and X2, and the three rotations p, Py and P2, completes die set of molecular coordinates up to 3N. In diis vray polarizability dmivatives are transformed into quantities corresponding to a space-fixed Cartesian coordinate system. As already pointed out in section 4.1, the great advantage of such a step is that the imensity parameters defined in terms of a space-fixed coordinate system are independent on isotopic substitutions provided the symmetry of the molecule is preserved. This will be illustrated with an example in the succeeding section. By analogy with Eq. (9.77), die rotational polarizability tensor can be represented as... 

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