Group theory crystal symmetry

Polarized Raman measurements and special crystal orientations have been used by Pofahl et al. (1986) to analyze the mode symmetries as shown in fig. 15. It is possible to determine by group theory the symmetry of the different CEF transitions ... 

This is an immediate consequence of the lowering of the symmetry as, even in the regular octahedral geometry, group theory tells us that the highest dimension of the irreducible representation is three. This is the basis of Crystal Field Theory, whose deeply symmetry-based formalism was developed by Bethe in 1929 [16]. 

The previous example has shown how group theory can be used in a symmetry reduction problem. This symmetry reduction also occurs when an ion is incorporated in a crystal. We will now treat how to predict the number of energy levels of the ion in the crystal (the active center) and how to properly label these levels by irreducible representations. 

Thus, the expected energy-level scheme for the Eu + ions in the crystal O symmetry) is that displayed in Figure 7.7. However, it should be recalled here that by group theory we cannot know neither the energy location of each level nor the energy order of these levels. 

Yttrium aluminum borate, YAlj (603)4 (abbreviated to YAB), is a nonlinear crystal that is very attractive for laser applications when doped with rare earth ions (Jaque et al, 2003). Figure 7.9 shows the low-temperature emission spectrum of Sm + ions in this crystal. The use of the Dieke diagram (see Figure 6.1) allows to assign this spectrum to the " Gs/2 Hg/2 transitions. The polarization character of these emission bands, which can be clearly appreciated in Figure 7.9, is related to the D3 local symmetry of the Y + lattice ions, in which the Sm + ions are incorporated. The purpose of this example is to use group theory in order to determine the Stark energy-level structure responsible for this spectrum. 

Symmetry considerations derived from group theory predict three main absorption-bands for Cr + in an octahedral environment and a number of low-intensity quartet-doublet-transitions in addition. The energies of the corresponding levels are calculated by means of crystal-field theory to be those of table 2 for the special choices AjB = 20 and 30 respectively ). 

Symmetry also plays an important part in the determination of the structure of molecules. Here, a great deal of the evidence comes from the measurement of crystal structures, infra-red spectra, ultra-violet spectra, dipole moments, and optical activities. All of these are properties which depend on molecular symmetry. In connection with the spectroscopic evidence, it is interesting to note that in the preface to his famous book on group theory, Wigner writes ... 

A group is a collection of elements that are interrelated according to certain rules. We need not specify what the elements are or attribute any physical significance to them in order to discuss the group which they constitute. In this book, of course, we shall be concerned with the groups formed by the sets of symmetry operations that may be carried out on molecules or crystals, but the basic definitions and theorems of group theory are far more general. 

Crystal symmetries that entail centering translations and/or those symmetry operations that have translational components (screw rotations and glides) cause certain sets of X-ray reflections to be absent from the diffraction pattern. Such absences are called systematic absences. A general explanation of why this happens would take more space and require use of more diffraction theory than is possible here. Thus, after giving only one heuristic demonstration of how a systematic absence can arise, we shall go directly to a discussion of how such absences enable us to take a giant step toward specifying the space group. 

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... 

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

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