Bravais cell

An appreciation of the crystal field effect on the vibrations of the Bravais cell which is repeated to build the crystal is extremely important when interpreting the vibrational spectra of many substances, since in the presence of a crystal field influence the number of observed bands in the spectrum cannot be directly determined from the formula unit which goes to make up the unit cell. In other words, there is almost always a larger number of bands to account for when investigating solid state samples. The solid state effects often cause degenerate bands to split in the same degree as symmetric and antisymmetric stretching modes split. 

One of the most important points, adequately emphasized by Fateley, McDevitt and Bently u) in their excellent paper, is that the spectroscopic unit cell must be chosen, and not the crystallographic unit cell. The spectroscopic unit cell is identical with the Bravais cell the Bravais cell is identical with the crystallographic unit cell in the P (primitive) cells, but the other structures, designated by B, C, I, F.in the crystallographic tables 14) contain 2, 2,3,4.Bravais unit cells. Fateley, McDevitt and Bentley 10 de-... 

The actual infinite lattices are obtained by parallel translations of the Bravais lattices as unit cells. Some Bravais cells are also primitive cells, others are not. For example, the body-centered cube is a unit cell but not a primitive cell. The primitive cell in this case is an oblique parallelepiped constructed by using as edges the three directed... 

When a crystalline solid is considered, the N atoms present in the smallest (primitive) Bravais cell must be taken into consideration to count the fundamental vibrational modes. They give rise to 3N total degrees of freedom, three of which... 

If the solid is molecular, the molecules (considered to be formed by M atoms, where M = N/r and r is the number of molecules in the smallest Bravais cell) can be treated as for the gas phase, so giving rise to 3M- 6 (or 3M- 5 if linear) vibrations for each molecule. The degrees of freedom associated with the external modes of every molecular unit (6r for non-linear molecules and 5r for linear molecules) give rise to lattice vibrations ( frustrated translations and rotations ) and to three acoustic modes. On the other hand, the internal vibrations of each molecules should in principle give rise to r-fold splitting, owing to the coupling of the vibrations within its primitive unit cell as a whole. 

Besides, several lattice energy minimisations, using the primitive unit cell (Bravais cell, 156 atoms) instead of the cristallographic unit cell (cubic unit cell, 622 atoms) are in preparation. The Bravais cell with a smaller number of atoms is the relevant system for phonon frequency calculations. 

At room temperature Mg2[FeHg] crystallizes in the cubic space group FmSmsC (number 225) with four molecules in the unit cell. There is one formula unit in the Bravais cell and the ions all lie on special sites the Mg " ions on tetrahedral sites and the [FeHe] ions on octahedral Ob sites. Using the correlation method [77] we can classify the vibrations as in Table 6.11. In Fig. 6.24 we show the infi-ared, Raman and INS spectra of Mg2[FeHe]. 

A more complete analysis including lattice modes can be made by the method of factor group analysis developed by Bhagavantam and Venkatarayudu [140]. In this method, we consider all the normal vibrations for an entire Bravais cell. Figure 1.42 illustrates the Bravais cell of calcite, which consists of the following symmetry elements ... 

V, total number of monoatomic molecules (ions) in the Bravais cell. 

Table 1.24 shows the correlation diagram for lattice vibrations of the ion. The variables V and / denote the degrees of translational freedom of the CO3 ion for each ion and for the Bravais cell, respectively. The same result is obtained for the rotatory lattice vibrations. Table 1.25 shows the correlation diagram for translatory... 

Table 1.26 summarizes the results obtained in Tables 1.23-1.25. The total number of vibrations including the acoustical modes should be 30 since the Bravais cell contains 10 atoms. These results are in complete agreement with that obtained by factor group analysis (Table 1.21). 

In the preceding section, the 30 normal vibrations of the Bravais cell of calcite crystal have been classified into symmetry species of the factor group The results (Table 1.26) show that three intramolecular (A2m + 2 J, three translatory lattice (A2u + 2 ) and two rotatory lattice (A2 + vibrations are IR-active, whereas three intramolecular (Aig + 2Eg), one translatory lattice (Eg), and one rotatory lattice (Eg) vibrations are Raman-active. In order to classify the observed bands into these symmetry species, it is desirable to measure infrared dichroism and polarized Raman spectra using single crystals of calcite. 

Since the Bravais cell contains lO atoms, it has 3x10 — 3 = 27 normal vibrations, excluding three translational motions of the cell as a whole.l These 27 vibrations can be classified into various symmetry species of the factor group Djd, using a procedure similar to that described in Sec. 1-7 for internal vibrations. First, we calculate the characters of representations corresponding to the entire freedom possessed by the Bravais primitive cell [a r( )] translational motions of the whole cell [ (T)], translatory lattice modes [ f(T )],... 

The correlation method developed by Fateley et al. is simpler than facioi group analysis and gives the same results. A complete normal coordinate analysis on the whole Bravais cell of calcite-type crystals was carried out by Nakagawa and Walter. Similar treatments have been extended to nitro and aquo complexes. For a review of lattice vibrations of inorganic and coordination compounds, see Ref. lOl. 

To characterize arrangement of the adsorbate head groups on the substrate surface the lUPAC nomenclature for two dimensional Bravais cells is used [A. M. Bradshaw and N. V. Richardson, Pure Appl. Chem. 68, 457 (1996)]. 

An ideal surface is a surface of a half-crystal in which the atoms are held in their original positions. The structure of an ideal surface is identical to that of a parallel crystallographic plane in the bulk. For a 2-D lattice, the elementary Bravais cell can have only one of the five structures shown in Fig. 5.2-1. 

Coordinates are referred either to crystal axes (Oxyz) or to a system O A7Z, in which A7 are axes in the surface plane X is parallel to a side of the 2-D Bravais cell and Z is perpendicular to the surface. 

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