Tetragonal lattices

The radius of the 24-coordinate metal site in MBs is too large (215-225 pm) to be comfortably occupied by the later (smaller) lanthanide elements Ho, Er, Tm and Lu, and these form MB4 instead, where the metal site has a radius of 185-200 pm. The structure of MB4 (also formed by Ca, Y, Mo and W) consists of a tetragonal lattice formed by chains of Bs octahedra linked along the c-axis and joined laterally by pairs of B2 atoms in the xy plane so as to form a 3D skeleton with tunnels along the c-axis that are filled by metal atoms (Fig. 6.11). The pairs of boron atoms are thus surrounded by trigonal prisms of... 

The above technique has the practical inconvenience of requiring as many different sets of Tchebyschev coefficients as the unit cell non equivalent sublattices. Furthermore, for non cubic systems, these coefficients depend on the lattice distortion ratios. Namely, for tetragonal lattices different sets of coefficients are required for each value of c/a. This situation has made difficult the implementation of KKR and KKR-CPA calculations for complex lattice structures as, for example, curates. 

Tetragonal crystal system, 8 114t Tetragonal lattice structure, of silicon, 22 482... 

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]. 

For a number of compounds AsMeFs besides cubic also tetragonal lattice constants were reported, belonging to low temperature modifications which are stable at ordinary temperatures in many cases. 

The similarity of the tetragonal lattice constants a in these and iso-structural compounds follows from the similar dimensions of the octahedral axes F—Me—F (about 4 A), two of which are connected lineary and infinitely with others in the same plane. 

A tetragonal lattice is the combination of two perpendicular one-dimensional lattices with the same lattice constant and having a reflection symmetry, as shown in Fig. 6.5. Therefore, we start with a one-dimensional case. The tunneling conductance from the nth atom is g(x — na, z). The total tunneling conductance from all the atoms is... 

This treatment can be extended immediately to the case of surfaces with tetragonal symmetry. The Fourier coefficients for the tunneling conductance of a tetragonal lattice is... 

F — ferromagnetic A — antiferromagnetic P — paramagnetic imp — impurity phase SR — spin reorientation Tq — Curie temperature Tn — Ndel temperature a and c — tetragonal lattice parameters, z — coordinate of B with c as its unit. 

Fig. 128. To a primitive tetragonal lattice ABCDEFGH add extra lattice points at the face-centres. The new lattice is equivalent to the body-centred lattice BJCIFLGK.

If in addition to orthogonality of the translation vectors we also require two vectors to be of equal length, say a = b, we have a tetragonal lattice. This now has the same mirror planes and twofold axes as an orthorhombic lattice but has fourfold axes parallel to the c direction. In this case there is only one form of centering possible, namely, / centering. 

Worked Example Why is a C-centered tetragonal lattice not possible Because the C faces are square, C centering would be equivalent to redefining the a and b vectors (as a and b ) to produce a primitive tetragonal lattice with a unit cell of one half of the volume, as shown below. 

A C-centered tetragonal lattice would be nothing more than a primitive tetragonal lattice with two incorrectly chosen vectors. 

Worked Example Why is an A-centered tetragonal lattice not possible Because centering only on the A faces (or only on the B faces) would destroy the fourfold symmetry and hence the lattice would not be tetragonal. The question of why centering on both the A and B faces is also disallowed is left as an exercise for the reader. 

For all the tetragonal lattices one of the translation vectors is a C4 axis and the other two are C2 axes. These elements plus the center of inversion give us the point group D4/i. 

Figure 7.9 Energy band structure of YNi2B2C along three different directions of the first Brillouin zone (FBZ) for a body-centered tetragonal lattice.15 T (0,0,0) X (1/2,0,0) M (1/2,1/2,0) and (0,0,1/2). Thus, the line TM is along [110] and Mr is along [001] in the FBZ. (Reprinted wifh permission from the American Chemical Society.)...

The solid solution (Ca, xSrx)NiN with CaNiN type tetragonal lattice was obtained in a limited compositional range up to x = 0.50.14 The interlayer distance can easily expand but the a-axis, corresponding to twice the Ni-N distance in a chain, expands very slightly with the substitution of Ca2+ with Sr2+. CaNiN, SrNiN and their solid solutions (Ca, JCSrx)NiN showed metallic behavior down to 10 K. 

There are two kinds of nickel sites in Ni2N as shown in Figure 9.12. One is in the corner of the tetragonal lattice and is planarly coordinated with four nitrogen atoms. The other is at the body center and is linearly... 

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