Crystal symmetries restrictions

The result is that Factor III of 2.2.6. given above imposes further symmetry restrictions on the 32 point groups and we obtain a total of 231 space groups. We do not intend to delve further into this aspect of lattice contributions to crystal structure of solids, and the factors which cause them to vary in form. It is sufficient to know that they exist. Having covered the essential parts of lattice structure, we will elucidate how one goes about determining the structure for a given solid. 

The effects of crystal tilt on phases is quite different. The phases are practically unaffected for small tilts and thin crystals. However, as long as the product t-siny is small, the phases are unchanged. Both phase relations and phase restrictions are still valid. Thus, it is possible to determine the (projected) crystal symmetry also from an image of a tilted crystal, using the phases. 

Symmetry restrictions exist for tensors describing macroscopic physical properties of all but triclinic crystals, and for tensors describing the local properties of atoms at sites with point-group symmetries higher than I. 

Symmetry restrictions for a number of crystal systems are summarized in Table B.l. The local symmetry restrictions for a site on a symmetry axis are the same as those for the crystal system defined by such an axis, and may thus be higher than those of the site. This is a result of the implicit mmm symmetry of a symmetric second-rank tensor property. For instance, for a site located on a mirror plane, the symmetry restrictions are those of the monoclinic crystal system. 

The vibrations of an individual molecule in the gas phase are subject only to the symmetry restrictions based on its own intrinsic point symmetry, and this chapter has so far been concerned exclusively with symmetry conditions of that kind. When a molecule resides in a crystal it is, in principle, subject only to the symmetry restrictions arising out of its crystalline environment. To be entirely rigorous, the molecule cannot even be treated as a discrete entity instead the entire array of molecules must be analyzed. However, such a completely rigorous approach is essentially impossible for practical reasons and unnecessary for most purposes, and therefore approximations are justifiably made. Two levels of approximation have frequently been used (1) the... 

The reduction of the free-ion parameters has been ascribed to different mechanisms, where in general two types of models can be distinguished. On the one hand, one has the most often used wavefunction renormalisation or covalency models, which consider an expansion of the open-shell orbitals in the crystal (Jprgcnscn and Reisfeld, 1977). This expansion follows either from a covalent admixture with ligand orbitals (symmetry-restricted covalency mechanism) or from a modification of the effective nuclear charge Z, due to the penetration of the ligand electron clouds into the metal ion (central-field covalency mechanism). 

Not all the tensor components are independent. Between Eqs (6.29a) and (6.29b) there are 45 independent tensor components, 21 for the elastic compliance sE, six for the permittivity sx and 18 for the piezoelectric coefficient d. Fortunately crystal symmetry and the choice of reference axes reduces the number even further. Here the discussion is restricted to poled polycrystalline ceramics, which have oo-fold symmetry in a plane normal to the poling direction. The symmetry of a poled ceramic is therefore described as oomm, which is equivalent to 6mm in the hexagonal symmetry system. 

In an actual crystal the atoms are in permanent motion. However, this motion is much more restricted than that in liquids, let alone gases. As the nuclei of the atoms are much smaller and heavier than the electron clouds, their motion can be well described by small vibrations about the equilibrium positions. In our discussion of crystal symmetry, as an approximation, the structures will be regarded as rigid. However, in modem crystal molecular structure determination atomic motion must be considered [19], Both the techniques of structure determination and the interpretation of the results must include the consequences of the motion of atoms in the crystal. 

The crystal axes for a given crystal may be chosen in many different ways however, they are conventionally chosen to yield a coordinate system of the highest possible symmetry. It has been found that crystals can be divided into six possible systems on the basis of the highest possible symmetry that the coordinate system may possess as a result of the symmetry of the crystal. This symmetry is best described in terms of symmetry restrictions governing the values of the axial lengths a, b, and c and the interaxial angles a, (3, and y. 

It can be seen from Eq. 1.7 that for all 4> 180°, the result will be an antisymmetric matrix (also called skew-symmetric matrices), for which = — J (or, in component form, Jij = —Jij for all i and j). If 4>= 180°, the matrix will be symmetric, in which = J. The lattice stmcture of a crystal, however, restricts the possible values for . In a symmetry operation, the lattice is mapped onto itself. Hence, each matrix element -and thus the trace of R (/ n + 22 + 33) - must be an integer. From Eq. 1.9, it is obvious that the trace is an integer equal to +(1 +2cos(f>). Thus, only one-fold (360°), two-fold (180°), three-fold (120°), four-fold (90°), and six-fold (60°) rotational symmetry are allowed. The corresponding axes are termed, respectively, monad, diad, triad, tetrad, and hexad. 

The hrst suffix of each tensor component gives the row and the second the column in which the component appears. The xis term, for example, measures the component of the polarization parallel to X2 (usually the y direction in a Cartesian coordinate system) when a field is applied parallel to X3 (the z direction). The susceptibility tensor must conform to any restrictions imposed by crystal symmetry, see Eqs. 6.5-6.9. 

Sofar we have not said anything about the number n. For a free molecule it may indeed be any integer number, including infinity. If the molecule is part of a crystal, however, the translational symmetry of the crystal imposes restrictions on n. From simple geometrical arguments one sees, that in this case only the values 1, 2, 3, 4 and 6 are allowed for n. 

The expression for Po oM in the case of mechanical excitons has the same form (2.57), but the functions u ),(()) must be replaced by u (0), obtained by neglecting the effects of the long-wavelength field. Since the operator P° is transformed like a polar vector, and the wavefunction To is invariant under all crystal symmetry transformations, the matrix element (2.57) will be nonzero only for those excitonic states whose wavefunctions are transformed like the components of a polar vector. If, for example, the function ToM transforms like the x-component of a polar vector, the vector Po o will be parallel to the x-axis. Thus the symmetry properties of the excitonic wavefunctions determine the polarization of a light wave which can create a given type of exciton. In the above example only a light wave polarized in the x-direction will be absorbed, obviously, if we restrict the consideration to dipole-type absorption. In a similar way, for example, the quadrupole absorption in the excitonic region of the spectrum can be discussed (for details see, for example, 8 in (12)). 

Molecular symmetry and ways of specifying it with mathematical precision are important for several reasons. The most basic reason is that all molecular wave functions—those governing electron distribution as well as those for vibrations, nmr spectra, etc.—must conform, rigorously, to certain requirements based on the symmetry of the equilibrium nuclear framework of the molecule. When the symmetry is high these restrictions can be very severe. Thus, from a knowledge of symmetry alone it is often possible to reach useful qualitative conclusions about molecular electronic structure and to draw inferences from spectra as to molecular structures. The qualitative application of symmetry restrictions is most impressively illustrated by the crystal-field and ligand-field theories of the electronic structures of transition-metal complexes, as described in Chapter 20, and by numerous examples of the use of infrared and Raman spectra to deduce molecular symmetry. Illustrations of the latter occur throughout the book, but particularly with respect to some metal carbonyl compounds in Chapter 22. 

Although the word crystal in its everyday usage is almost synonymous with symmetry, there are severe restrictions on crystal symmetry. While there are no restrictions in principle on the number of symmetry classes of molecules, this is not so for crystals. AH crystals, as regards their form, belong to one or another of only 32 symmetry classes. They are also called the 32 crystal point groups. Figures 9-12 and 9-13 illustrate them by examples of actual minerals and by stereographic projections with symmetry elements, respectively. 

The deformation potential is subject to restrictions required by crystal symmetry (Gray and Gray 1976). One has... 

Certain materials, such as many phthalocyanines [45], can only be deposited during each upstroke (Z-type deposition) and others only during each insertion (X-type). The type of deposition followed by a particular material affects the properties of the resulting LB films due to symmetry restrictions for second-order physical effects such as piezoelectricity or optical second-harmonic generation. The bilayer unit (equivalent to the unit cell in a crystal) is symmetric for Y-type deposition but non centric for Z- and X-type modes, as indicated in Figure 12.12. Unfortunately, several researchers have found that Z- and X-mode multilayers are often temporally unstable, although the zwitterionic dyes of Ashwell prove to be one of the exceptions [46]. 

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