Tetragonal and hexagonal crystal systems

The quadratic form of Eq. 5.2 in the hexagonal crystal system is found in Eq. 5.4 and its analogue in the tetragonal crystal system is 

Following the approach illustrated using Eqs. 5.11 to 5.13 and maintaining similar notations, Eqs. 5.4 and 5.18 can be written as 

The solution of Eq. 5.20 could be found after calculating all possible differences between the observed pairs of Qhki- This leads to the following series of equations  

As follows from Eq. 5.21, when two Bragg peaks have the same value of /, e.g. hikfs and hjkfi, or A,A,1 and hjkjX, or A,A,-2 and hjkjl and so on, the resulting difference is only a function oiAhk and a  

Solving the indexing problem becomes a matter of identifying the differences that result in whole numbers when divided by a common divisor and c, respectively). The expected whole numbers are shown in Table 5.14 through Table 5.16 for several small h, k and /. It only makes sense to consider these small values because successful indexing is critically dependent on the availability of low Bragg angle peaks, which usually have small values of indices. 

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Trigonal, tetragonal, and hexagonal crystal systems have three, four and sixfold axes of symmetry, respectively, while the cubic crystal contains four threefold axes along with diagonals of the cube as well as two-fold axes passing through the faces (see Fig. 15-14). 

The next possibility is 0.0524, which occurs three times for the differences between pairs of first seven observed Bragg peaks. Not only this value is itself more frequently occurring than any other smaller quantity found in the table, but when multiplied by two it yields 0.1047, which occurs in the array twice - these nearly identical numbers are shown in italic. Testing 0.0524 multiplied by three (0.1571) has three additional occurrences (all are shown underlined). There is also one occurrence of 4x0.0524 = 0.2095, both are double underlined. Hence, this value seems to be an excellent candidate for one of the reciprocal lattice parameters. After consulting Table 5.14 through Table 5.16 it is clear that 2xc is not expected to be seen but 2xa should be observed quite frequently in both the tetragonal and hexagonal crystal systems. 

Fortunately, but also evidently, many simplifications arise when considering the uniaxial indicatrix of crystals belonging to the trigonal, tetragonal, and hexagonal crystal systems. First, there is only one optic axis which, by convention, always lies along the z crystallographic axis hence, X, Y, and Z disappear, as z suffices to define this direction. The refractive index associated with this direction is called or n. The plane perpendicular to the optic axis is necessarily a circular section whose diameters all have the same refractive index denoted or n. Thus, and Hy disappear. Optically positive means... 

At this point (or after the whole pattern has been indexed) the analysis of the observed values of A enables one to establish whether we deal with the tetragonal or hexagonal crystal systems. As seen in Table 5.19, the whole multipliers of a are 1 and 3, and 3 is only possible in the hexagonal crystal system for A =, k= 1. After a simple calculation using the average values of a and c listed in Table 5.19 we find approximate values of a and c as 5.046 and 4.015 A, respectively. A least squares refinement of the lattice parameters using the entire array of indexed Bragg peaks obviously yields the same lattice parameters as were established before (see Table 5.6). 

The correct unit cell may be identified by associating indices to n inter-planar distances, where n depends on the lattice symmetry. In accordance with Table 7.3, the minimum values of n are n=l for the cubic system, n = 2 for tetragonal and hexagonal crystals, = 3, 4, 6 for orthorhombic, monoclinic and triclinic systems, respectively. 

The list of alternate symbols arises because at an earlier stage in the history of X-ray crystallography, it was the accepted convention to call the unique axis b. The reader can easily see that if this is done, the alternate symbols become correct. Since in the other crystal systems with a unique direction (i.e., tetragonal and hexagonal) the unique direction is called c, the formally correct practice is now to do the same for the monoclinic system. However, the literature is still replete with the old choice of axes and it is necessary to be cognizant of both systems and of their relationship. 

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. 

In addition to different types of crystal system there are also different types of lattice within those crystal systems, which correspond to specific arrangements of the atoms/ions within them. As discussed earlier, the two-dimensional system with the simplest sort of lattice which contains only one lattice point, is termed primitive. Similarly, for each three-dimensional crystal system there is always a primitive unit cell which consists of atoms located at the comers of the particular parallelepiped (i.e. a solid figure with faces which are parallelograms). For example. Figure 1.5 shows primitive lattices (symbol P) for both tetragonal and hexagonal systems. 

Table 2.1 shows the crystal structure data of the phases existing in the Mg-H system. Pnre Mg has a hexagonal crystal structure and its hydride has a tetragonal lattice nnit cell (rutile type). The low-pressure MgH is commonly designated as P-MgH in order to differentiate it from its high-pressure polymorph, which will be discussed later. Figure 2.2 shows the crystal structure of p-MgH where the positions of Mg and H atoms are clearly discerned. Precise measurements of the lattice parameters of p-MgH by synchrotron X-ray diffraction yielded a = 0.45180(6) mn and c = 0.30211(4) nm [2]. The powder diffraction file JCPDS 12-0697 lists a = 0.4517 nm and c = 0.30205 nm. The density of MgH is 1.45 g/cm [3]. 

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