Continuous Translational Symmetry

Consider a particle located at position xq on the real line and the translation operator T that causes displacement of the particle by a distance x. To avoid confusion the position x0 may be defined as the particle state xo), using Dirac notation, although the particle may be classical. The action of T(x) on xQ is written 

It is easy to see that T x) must have the following properties  

These properties are just those required for T(x) in the interval —oo x +oo to constitute a group. Moreover, since the translation is a continuous operation the homorphism 

The localized states x) and the translational states p) are related by an expression such as 

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It is well known since the pioneering work of Israelachvilli and Tabor [1], that the structure of an ordinary liquid near a solid wall is quite different from that in the bulk. The bi eaking of continuous translational symmetry of free space by a solid wall imposes positional ordering of molecules near that wall. As a result, a sequence of several molecular layers is usually formed, that extends from the surface and vanishes in the isotropic bulk. 

After the isotropic to nematic transition, the next step towards more ordered mesophas-es is the condensation of SmA order when the continuous translational symmetry is broken along the director. The theoretical description of the N-SmA transition begins with the identification of an order parameter. Following de Gennes and McMillan [1, 16], we notice that the layered structures of a SmA phase is characterized by a periodic modulation of all the microscopic properties along the direction z perpendicular to the layers. The electron density for instance, commonly detected by X-ray scattering can be expanded in Fourier series ... 

Symmetry Continued) translational. 74 Symmetry allowed transitions,... 

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... 

In the framework of irreversible thermodynamics (compare, for example, [31, 32]) the macroscopic variables of a system can be divided into those due to conservation laws (here mass density p, momentum density g = pv with the velocity field v and energy density e) and those reflecting a spontaneously broken continuous symmetry (here the layer displacement u characterizes the broken translational symmetry parallel to the layer normal). For a smectic A liquid crystal the director h of the underlying nematic order is assumed to be parallel to the layer normal p. So far, only in the vicinity of a nematic-smectic A phase transition has a finite angle between h and p been shown to be of physical interest [33],... 

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. 

It is reasonable therefore to consider that fused silica resembles liquid water. Just as liquid water retains from the parent shucture (ice) the three-dimensional network but not the long-range periodicity of the network, one would expect that liquid silica also retains the continuity of the tetrahedra, i.e., the space network, but loses much of the periodicity and long-range order that are the essence of the crystalline state. This model of fused silica, based on keeping the extension of the network but losing the translational symmetry of crystalline silica, implies a low concentration of charge... 

It may seem that an extended impurity problem could require an impossibly large cluster size. This is not true, however, basically because of the physical implications of translational symmetry away from the impurity region. In fact, the asymptotic limits of the continued fraction parameters and the cut in the real energy axis are determined by the perfect crystal only this allows a guideline for appropriate extrapolation of the recursion codffidents. 

Whether quasicrystalline structures are limited to alloys remains an open question. It is possible that their occurrence is much more widespread than had been previously thought. Indeed there is evidence for quasicrystallinity in both thermotropic and lyotropic liquid crystals. Diffraction patterns of decagonal symmetry have been recorded in lyotropic liquid crystals [K. Fontell, private communication], (Fig. 2.19), and there is theoretical evidence for the existence of a quasicrystalline structure within the blue phase of cholesterol (Chapters 4, 5). (The decagonal structure has quasisymmetry perpendicular to the tenfold axes, and translation symmetry along them.) Viruses crystallise in icosahedral clusters and the list continues to grow. In addition to five-fold symmetry, it has been shown that eight and ten- fold quasisymmetry is possible. ... 

So far, our discussion of symmetry of the lattice was limited to lattice points and symmetry of the unit cell. The next step is to think about symmetry of the lattice including the contents of the unit cell. This immediately brings translational symmetry into consideration to reflect the periodic nature of crystal lattices, which are continuous or infinite object. As... 

The wave equation is solved subject to the boundary condition that the tangential (x and y) components of E and H are continuous and that the normal z) components of eE and jjiH 2xt continuous at the boundaries between the different regions. In each region, labeled by the subscript j = l,m,2, the solutions that are applicable to systems with translational symmetry in the x and y directions, have the form fj z)tx [i qxX qyy] for each component of the fields the modes are thus labeled by the wavevector q = (qx, qy). The wave equation dictates that the function f z) has the form... 

We see from Eqs. (9.67) and (9.74) that thanks to the translational symmetry, we ma treat each k separately, infinity continues to make us a little nervous. In the expression for F, we... 

It should be noted that cholesteric liquid crystals (chiral nematics) having point group symmetry Dqo are also periodic with flie pitch considerably exceeding a molecular size. The preferable direction of the local molecular orientatiOTi, i.e. the director oriented along the Coo axis, rotates additionally through subsequent infinitesimal angles in the direction perpendicular to that axis. Hence a helical structure forms with a screw axis and continuous translation group. 

The isotropic phase formed by achiral molecules has continuous point group symmetry Kh (spherical). According to the group representations [5], upon cooling, the symmetry Kh lowers, at first, retaining its overall translation symmetry T(3) but reduces the orientational symmetry down to either conical or cylindrical. The cone has a polar symmetry Coov and the cylinder has a quadrupolar one Dooh- The absence of polarity of the nematic phase has been established experimentally. At least, polar nematic phases have not been found yet. In other words, there is a head-to-tail symmetry taken into account by introduction of the director n(r), a unit axial vector coinciding with the preferred direction of molecular axes dependent on coordinate (r is radius-vector). 

It is the less steep front which is of interest here because of its sensitivity to convection velocity v. Therefore we have used the moving wave coordinate transformation and studied the dependence of the wave velocity u on v by means of the continuation techniques. The result is shown in Fig. 2 for several values of the Lewis number which is a measure of the heat capacity of the bed. The dynamics shown in Fig. 1 correspond to Le=l. Because of translational symmetry, in this case the dependence is simply a straight line with slope equal to one and u can be negative or positive, depending on v as observed in Fig. 2. In fact, there is also a reflection symmetry, and since the line does not pass through origin, there is a pair of less steep waves for each v. For Le < (the lower limit for he as defined here is the porosity of the bed e 0.4) the dependence is highly nonlinear with a... 

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