Rotational symmetry axes

There are excellent monographs on regular figures, of which we single out those by Coxeter and by Laszlo Fejes Toth as especially noteworthy [91], The Platonic solids have very high symmetries and one especially important common characteristic. None of the rotational symmetry axes of the regular polyhedra is unique, but each axis... 

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. 

In real space, by critically examining the electron density map, we can also evaluate quality. The questions we ask should test the physical reality of the map. For example, are rotational symmetry axes free of density Are the boundaries of individual molecules clearly marked by low-density solvent regions Can we follow segments of the polypeptide chain Is the direction of chain evident from visible carbonyl oxygen positions Can we identify characteristic secondary structural features such as alpha helices and beta sheets like those in Figures 10.12 and 10.13 Can individual amino acid side chains like those in Figure 10.14 be recognized ... 

Note also the (100), (110) and (111) planes are illustrated. Planes are important in soUds because, as we will see, they are used to locate atom positions within the lattice structure. The TETRAD, TRIAD, AND DIAD AXE ARE ALSO SHOWN IN Part D. These are rotational symmetry axes. That is, the triad axis must be rotated 3-times in order to bring a given corner back to its original position. 

Another property of each crystal system that distinguishes one system from another is called symmetry. There are four types of symmetry operations reflection, rotation, inversion, and rotation-inversion. If a lattice has one of these types of symmetry, it means that after the required operation, the lattice is superimposed upon itself. This is easy to see in the cubic system. If we define an axis normal to any face of a cube and rotate the cube about that axis, the cube will superimpose upon itself after each 90 of rotation. If we divide the degrees of rotation into 360°, this tells us that a cube has three fourfold rotational symmetry axes (on axes normal to three pairs of parallel faces). Cubes also have threefold rotational symmetry using an axis along each body diagonal (each rotation is... 

Often a symmetry plane is coincident with a rotation symmetry axis. Which two dotted lines that you drew in Figure L3.7 (to represent mirror planes) are also rotation symmetry axes ... 

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... 

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.

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