Cubic crystal systems

When the crystal system is cubic, the general form of equation 5.2 is simplified to 

Assume that we have a set of experimental data where the observed Bragg angles have been converted into an array of g-values. Then, if the crystal lattice is cubic, the following system of simultaneous equations can be written to associate each Bragg peak with a certain combination of hkl triplets  

As follows from the second form of Eq. 5.15, the observed array of Q-values should have a common divisor, which results in the array of integers or nearly integers, considering the finite accuracy of the measured Bragg angles, This common divisor is nothing else than the inverse of the square of the edge of the cubic unit cell in the direct space since a = 1/a. 

The numbers in parenthesis indieate the simplest sequence of integers, which describes the relationships between the sums of l, k, and P in the body-centered cubic lattice. The presence of forbidden integers (e.g. 7, which is highlighted in bold) enables one to differentiate between the primitive and body-centered cubic lattices during the ab initio indexing. 

Second, the values from this column are normalized by dividing them by the smallest observed Q, i.e. by 0.05740. This yields the column of data marked as 

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In general, CijU is a 9 x 9 tensor with 81 terms, but symmetry reduces this considerably. Thus, for the cubic crystal system, it has only three terms (Cmi, Cm2, and C4444) and for an isotropic material only two terms remain B = bulk modulus and G = shear modulus. A further simplification is that the bulk modulus, B for the cubic system is given by (Cmi + 2Ci2i2)/3, and the two shear moduli are C44 and (Cmi - Ci2i2)/2. 

Crystal Systems. The cubic crystal system is composed of three space lattices, or unit cells, one of which we have already studied simple cubic (SC), body-centered cubic (BCC), anA face-centered cubic (FCC). The conditions for a crystal to be considered part of the cubic system are that the lattice parameters be the same (so there is really only one lattice parameter, a) and that the interaxial angles all be 90°. 

Any planes that have common factors are parallel. For example, a (222) and a (111) plane are parallel, as are (442) and (221) planes. As with cell directions, a minns sign (in this case, indicating a negative intercept) is designated by an overbar. The (221) plane has intercepts at 1/2, —1/2, and 1 along the x, y, and z axes, respectively. Some important planes in the cubic crystal system are shown in Figure 1.25. 

The pattern of observed lines for the two other cubic crystal systems, body-centred and face-centred is rather different from that of the primitive system. The differences arise because the centring leads to destructive interference for some reflections and these extra missing reflections are known as systematic absences. 

What are the spacings of the 100, 110, and 111 planes in a cubic crystal system of unit cell dimension a In what sequence would you expect to find these reflections in a powder diffraction photograph ... 

Nickel crystallizes in a cubic crystal system. The first reflection in the powder pattern of nickel is the 111. What is the Bravais lattice ... 

Figure 1.44. Constraints imposed on the coordinates of atoms located on the diagonal twofold and three-fold rotation axes in cubic crystal system.

We will begin with the cubic crystal system, where the assignment of indices is nearly transparent and then consider the theory behind the ab initio indexing in crystal systems with tetragonal and hexagonal symmetry. Indeed, as with any kind of experimental work, experience is paramount, and we hope that the contents of this section may help the reader to achieve accurate solutions of real life indexing tasks successfully. 

Cobalt oxide, CoO, crystallizes in the cubic crystal system, space group Fm3m, a = 4,26 A. The measured gravimetric density of the oxide is... 

A four-component capsule with a structure that conforms to a tetrahedron has been recently described by Venkataraman66 Specifically, triphenylamine ortho-tricarboxylic acid self-assembled via 12 O H-O hydrogen bonds to form a molecular tetrahedron (Fig. 36). The hydroxyl groups of the polyhedron participated in hydrogen bonds with single ethanol molecules embedded within each triangular face of each tetrahedron. The assembly crystallized in the rare cubic crystal system. The ability of the acid to form a tetrahedron was reminiscent of the ability of triphenyl-methanol to form a tetrahedron, which also forms inclusion compounds with solvent guests that occupy voids between the polyhedra.32... 

The Miller indices (hkl) represent a series of parallel planes in a crystal with spacing of d i-Combining Equations 2.3 and 2.4, we obtain the following relationship between diffraction data and crystal parameters for a cubic crystal system. 

If Miller indices can be assigned to the various reflections in the powder pattern, it becomes possible to determine the cell constants. This assignment is readily achieved for cubic crystal systems with a simple relationship between the diffraction angles and lattice parameters. 

Ceramics in aluminate systems are usually formed from cubic crystal systems and this includes spinel and garnet. Rare earth aluminate garnets include the phase YAG (yttrium aluminium garnet), which is an important laser host when doped with Nd(III) and more recently Yb(III). Associated applications include applications as scintillators and phosphors. 

Bravais lattices - The 14 distinct crystal lattices that can exist in three dimensions. They include three in the cubic crystal system, two in the tetragonal, four in the orthorhombic, two in the monoclinic, and one each in the triclinic, hexagonal, and trigonal systems. 

Compounds belonging to the cubic crystal system have only a single refractive index value, but other systems are anisotropic, so that the crystal is characterized by two or three unique indexes. Hexagonal, rhombohedral, and tetragonal crystals have two unique indexes which are traditionally labeled and for ordinary ray and extraordinary ray . Orthorhombic, monoclinic, and triclinic crystals are characterized by three indexes which are here called n, and n. The table indicates the crystal system for each entry in order to identify the material uniquely. 

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