Discrete Rotational Symmetry

The operators of discrete rotational groups, best described in terms of both proper and improper symmetry axes, have the special property that they leave one point in space unmoved hence the term point group. Proper rotations, like translation, do not affect the internal symmetry of an asymmetric motif on which they operate and are referred to as operators of the first kind. The three-dimensional operators of improper rotation are often subdivided into inversion axes, mirror planes and centres of symmetry. These operators of the second kind have the distinctive property of inverting the handedness of an asymmetric unit. This means that the equivalent units of the resulting composite object, called left and right, cannot be brought into coincidence by symmetry operations of the first kind. This inherent handedness is called chirality. 

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Discrete Rotational Symmetry This is a subset of continuous rotations and reflections in three-dimensional space. Since rotation has no translational components their symmetry groups are known as point groups. Point groups are used to specify the symmetry of isolated objects such as molecules. 

The three ordered stales of the Potts model correspond to a preferential occupation of one of the three sublattices a,b,c into which the triangular lattice is split in the (-/3x-v/3)R30° structure. In the order parameter plane (0x.0r), the minima of F occur at positions (1, 0)MS, (—1/2, i/3/2)yWs, (—1/2, -yf3/2)Ms, where Ms is the absolute value of the order parameter, i.e. they are rotated by an angle of 120° with respect to each other. The phase transition of the three-state Potts model hence can be interpreted as spontaneous breaking of the (discrete) Zj symmetry. While Landau s theory implies [fig. 13 and eqs. (20), (21)] that this transition must be of first order due to the third-order invariant present in eq. (34), it actually is of second order in d = 2 dimensions (Baxter, 1982, 1973) in agreement with experimental observations on monolayer ( /3x /3)R30o structures (Dash, 1978 Bretz, 1977). The reasons why Landau s theory fails in predicting the order of the transition and the critical behavior that results in this case will be discussed in the next section. 

This drift velocity field is shown in Fig. 9.2 corresponding to r = 0. The constant ip is taken as (/> == —1.8 in accordance with the chosen parameters for the Oregonator model (9.1). The field has rotational symmetry, but the drift angle 7 monotonously increases with the distance z from the detector point. Hence, there is a discrete set of sites along any radial direction, where the drift direction is orthogonal to the radial one i.e. 

It is the mentioned symmetry properties additional to the discrete translational symmetry that lead to a classification of the various possible point lattices by five Bravais lattices. Like the translations, these symmetry operations transform the lattice into itself They are rigid transforms, that is, the spacings between lattice points and the angles between lattice vectors are preserved. On the one hand, there are rotational axes normal to the lattice plane, whereby a twofold rotational axis is equivalent to inversion symmetry with respect to the lattice point through which the axis runs. On the other hand, there are the mirror lines (or reflection lines), which he within the lattice plane (for the three-dimensionaUy extended surface these hnes define mirror planes vertical to the surface). Both the rotational and mirror symmetry elements are point symmetry elements, as by their operation at... 

Here, as in the discussion of the Smoluchowski effect, we have made expHdt reference to the discrete, periodic stracture of the lattice. The symmetry ofa periodic lattice is substantially lower than the continuous translational and rotational symmetry of the uniform jeUium model. Only translations involving a Bravais lattice vector and a discrete number of point symmetry operations remain and the constant potential has to be replaced by a crystal potential, which reflects the lowered symmetry. This has profound consequences for the electronic structure. 

A quasicrystal (QC) is a type of solid that is well-ordered but not periodic. QCs are often associated with classically forbidden rotational symmetries, although strictly speaking, this is not necessary [1]. The discovery of QCs led to a refinement of the definition of a crystal as any soHd with an essentially discrete diffraction pattern. [2]. This transferred the definition of the concept from real to reciprocal space and in doing so broadened the scope of the term to encompass both periodic and quasiperiodic materials. Thus a QC is a nonperiodic crystalline material. 

As a consequence of the rotation symmetry around the 2-axis, the T matrix is invariant with respect to discrete rotations of angles Ofc = 2irk/N, k = 1,2,..., N - 1. A necessary and sufHcient condition for the equation... 

There is another symmetry element or operation needed for discrete molecules, the improper rotation, S . A tetrahedron (the structure of... 

Figure 5.8 Energy eigenvalues (dots) of the discretized and complex rotated one-particle Dirac Hamiltonian with s-symmetry. Bound states that are well represented in the box and on the grid are lying on the x-axis. The low-energy pseudo-continuum states are rotated with an angle of approximately 20 (where 0 is the complex rotation angle) down from the real axis, as indicated with the straight solid line.

The hands of analogue and digital clocks rotate in two fundamentally different ways, characterized by continuous and discrete symmetry groups respectively. The continuous rotation during a complete cycle of 27t is isomorphic with translations on the real line and has an infinite number of equivalent positions. The moving pointer touches all of these points during... 

Altmann considered two types of operations that belong to the Schrodinger subgroup the Euclidean and the discrete symmetry operations. Euclidean operations are those that change the laboratory axes, leaving the Hamiltonian operator invariant. They are translations and rotations of the whole molecule, in free space, in which the x,y,z molecular axes are kept constant. A discrete symmetry operation is a change of the molecular axes in such away as to induce permutations of the coordinates of identical particles [10]. 

In the case of symmetry operations the product is the successive performance of these operations. If a group has a finite number of elements it is said to be a discrete group. The point groups, which arise firom the rotations and reflections of bodies such as isolated molecules are discrete groups. 

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