Continuous Rotational Symmetry

A system symmetric under rotations around a fixed point in a plane is described by the rotation group 0(2). An arbitrary vector x in the plane transforms under rotation / ( / ) according to 

The length of the vector a 2 remains invariant under rotation and it is easy to show that R((j )RT((f ) = E V / , where RT is the transpose of R and E is the unit matrix. Real matrices that satisfy this condition are known as orthogonal matrices. The condition implies that [detil( / )]2 = 1 or that detfi( / ) = 1. Matrices with determinant equal to —1 correspond to rotations combined with spatial inversion or mirror reflection. For pure rotations detR = 1, for all f . 

Two rotations in succession result in an equivalent single rotation, with the obvious law of composition 

The group elements of 0(2) are all 0 j 27r, the points on the unit circle that defines the topology of the group parameter space. 

An infinitesimal rotation by the angle dcj) for analytic R( j ) is defined, as before by 

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The continuous rotational symmetry shown by the APES is related to the constant of motion of the total angular momentum operator Jz... 

Circularly polarized linear combinations f =(fx if )/-Jl. suggested by the APES continuous rotational symmetry, allow cry to be put in diagonal form. 

Just below the freezing point the solid clusters have continuous rotational symmetry, which breaks at lower temperatures because of the spontaneous alignment of the molecules along preferred orientations, as can be seen in Fig. 6. This is a plot of the mutual orientations of the molecular axes of symmetry in the disordered and ordered solid state. 

The simplest phase, which contains only molecular-orientational ordering, is the nematic. The term "nematic" means thread in Greek. All known nematics have one symmetry axis, called the director, n, and are optically uniaxial with a strong birefringence. The continuous rotational symmetry of the isotropic liquid phase is broken when the molecules choose a particular direction to orient along in the nematic phase. Since the nematics scatter light intensively, the nematic phase appears turbid. 

Fig. 1.2. (a) in a bulk nematic liquid crystal, the director can point in an arbitrary direction in space. This is a signature of a broken continuous rotational symmetry of the isotropic phase. The mode that restores the broken. symmetry is the Goldstone mode. It represents a homogeneous and coherent rotation of aU molecules, (b) the homogeneous surface couples to the Goldstone mode and pins the director in a certain direction in space. 

Smectic-A liquid crystals, besides the orientational order as nematics, possess onedimensional positional order. They have a layered structure. The liquid crystal director n is perpendicular to the smectic layers. The symmetry of smectic-A is Do h if the constituent molecule is achiral or D if the constiment molecule is chiral. It is invariant for any rotation around n. It is also invariant for the two-fold rotation around any axis perpendicular to n. The continuous rotational symmetry is around n and therefore there is no spontaneous polarization in any direction perpendicular to n. Hence it is impossible to have spontaneous polarization in smectic-A, even when the constiment molecule is chiral. 

The first important characteristic is that living systems are liquid crystalline. Liquid crystals are orientationally ordered molecular liquids. When a molecular liquid is in the isotropic liquid state (which models the primordial soup), it has neither translational nor orientational order. At the phase transition to the nematic state, the most basic liquid crystal transition, the system selects a special direction for long-range orientational order. This direction is called the director and denoted by a unit vector, n. By selecting a special direction, the nematic breaks the continuous rotational symmetry of the isotropic liquid. A phase boundary intervenes between the nematic and the isotropic liquid when they coexist. 

This completes the derivation of the point groups that are important in molecular symmetry, with the exception of the two continuous rotation groups and L) x h, which apply to linear molecules. 

It is immediately evident that in this same limit (1) and (2) possess both continuous translational and rotational symmetry. Yet it is a matter of common experimental experience that at sufficiently low temperature (with... 

Character of Classes of Cubic Symmetry in the (2L + 1)-Dimensional Representation DL of the Continuous Rotation Group and Their Resolution into Irreducible Representations of Cubic Symmetry... 

During the past half a century, fundamental scientific discoveries have been aided by the symmetry concept. They have played a role in the continuing quest for establishing the system of fundamental particles [7], It is an area where symmetry breaking has played as important a role as symmetry. The most important biological discovery since Darwin s theory of evolution was the double helical structure of the matter of heredity, DNA, by Francis Crick and James D. Watson (Figure 1-2) [8], In addition to the translational symmetry of helices (see, Chapter 8), the molecular structure of deoxyribonucleic acid as a whole has C2 rotational symmetry in accordance with the complementary nature of its two antiparallel strands [9], The discovery of the double helix was as much a chemical discovery as it was important for biology, and lately, for the biomedical sciences. 

As far as the continuous symmetry breaking is concerned, the Goldstone theorem states [4] that this will generate hydrodynamic modes, that is, gapless excitations. The order parameter is multicomponent (n > 1) a vector i for magnetism breaking the rotational symmetry, a complex variable i j = ip e/e for charge-density waves and for superconductivity which... 

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... 

The points on the dial are said to be related by rotational symmetry, referred to an axis of order n, corresponding to the number of points allowed on the circumference of the circle of rotation. The various dials considered above, have rotational symmetry axes of order 4, 6, 12, 60 and oo. The allowed rotations in each case constitute a group of which only the last may be continuous. 

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. 

Assume, however, that the asymmetric unit is composed of two or more identical copies of a molecule. These may be related by exact rotational symmetry, such as a dyad or triad, that is not coincident with any crystallographic operator and, hence, is not a part of the space group symmetry. The molecules may also be related by some completely general rotation plus translation. The asymmetric unit will give rise, in either case, to a continuous transform that is essentially a superposition of the transforms of the two independent molecules. This is illustrated schematically in Figure 8.9. The two molecular transforms are identical because the two molecular structures are the same, but the two transforms will be rotated relative to... 

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. 

For the most simple case of fast continuous rotation around a well-defined axis, /3 represents the unique angle between the C—bond and the motional symmetry axis. The narrowing factor is (3 cos p - l)/2, because a tensor rather than a vector is being averaged. For example, rotation of a tetrahedral... 

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