Bravais tetragonal

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The only possible cells in two dimensions are oblique (p only), rectangular (p and c) and hexagonal (p). For each of the seven three-dimensional crystal systems primitive and centred cells can be chosen, but centring is not advantageous in all cases. In the case of triclinic cells no centred cell can have higher symmetry than the primitive and is therefore avoided. In all there are 14 different lattice types, known as the Bravais lattices Triclinic (P), Monoclinic (P,C), Orthorhombic (P,C,I,F), Trigonal (R), Tetragonal (P,I), and Cubic (P,I,F). 

Both are body-centered Bravais lattices and for both the site symmetry of the origin is identical with the short space group symbol. The body-center position is of the lowest multiplicity (two-fold) and highest symmetry, and thus is considered as the origin in the lA/mmm space group. However, in the tetragonal lattice, a = b c. Hence, the body center position is not an inversion center. It possesses four-fold rotational symmetry (the axis is parallel to c) with a perpendicular mirror plane and two additional perpendicular mirror planes that contain the rotation axis. 

The Bravais lattice of BaTiOs at room temperature is primitive tetragonal but because the distortion from cubic is small, the structure can be considered to be primitive cubic with a(pc) = a = 6 c 0.4 nm. 

At first glance, the list of Bravais lattices in Table 2-1 appears incomplete. Why not, for example, a base-centered tetragonal lattice The full lines in Fig. 2-4 delineate such a cell, centered on the C face, but we see that the same array of lattice points can be referred to the simple tetragonal cell shown by dashed lines, so that the base-centered arrangement of points is not a new lattice. However, the base-centered cell is a perfectly good unit cell and, if we wish, we may choose to use it rather than the simple cell. Choice of one or the other has certain consequences, which are described later (Problem 4-3). 

The reader may have noticed in the previous examples that some of the information given was not used in the calculations. In (a), for example, the cell was said to contain only one atom, but the shape of the cell was not specified in (b) and (c), the cells were described as orthorhombic and in (d) as cubic, but this information did not enter into the structure-factor calculations. This illustrates the important point that the structure factor is independent of the shape and size of the unit cell. For example, any body-centered cell will have missing reflections for those planes which have ft + k + 1) equal to an odd number, whether the cell is cubic, tetragonal, or orthorhombic. The rules we have derived in the above examples are therefore of wider applicability than would at first appear and demonstrate the close connection between the Bravais lattice of a substance and its diffraction pattern. They are summarized in Table 4-1. These rules are subject to... 

Not all types of lattice are allowable within each crystal system, because the symmetrical relationships between cell parameters mean a smaller cell could be drawn in another crystal system. For example a C-centred cubic unit cell can be redrawn as a body-centred tetragonal cell. The fourteen allowable combinations for the lattices are given in Table 1.4. These lattices are called the Bravais lattices. 

Fig. 10.25. Change in Bravais lattice and well structure associated with the cubic-tetragonal transformation (adapted from Bhattacharya (1991)). Uj, U2 and U3 are the matrices that map the original cubic Bravais lattice into the three tetragonal variants.

Conventionally, the superconductors that we treat are labelled with four digits in a set that correspond, in order, to the stoichiometric coefficients of Tl, Ba, Cu and Ca. Figure 17-5 shows the known structures for m=. In these structures the Bravais lattice is primitive P, and the number of sheets that occur in one unit cell corresponds to the sum of four stoichiometric coefficients 4, 6, 8, 10 and 12 for n=l, 2, 3, 5 and 5, in order. The structures are tetragonal P4/mmm and the value of cell parameter a, 3.85A, corresponds to twice the length of the Cu—O bond. From member n to n+l, cell parameter c increases by about... 

Fig. 1.52. The Bravais unit cells of the orthorhombic and tetragonal forms of YBa2Cu30y- [ 149].

Figure 1.52 shows the Bravais unit cells of the orthorhombic (Pmmm = D ) and tetragonal (P4lmmm = forms. The former is a distorted, oxygen-deficient form of perovskite. The orthorhombic unit cell contains 13 atoms, and their possible site symmetries can be found from the tables of site symmetries (Appendix X) as follows ... 

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. 

Standard ASTM E157-82a has the Bravais lattices designations as following C - primitive cubic B - body-centered cubic F - face-centered cubic T - primitive tetragonal U - body-centered tetragonal R - rhombohedral H - hexagonal O - primitive orthorhombic P - body-centered orthorhombic Q - base-centered orthorhombic S - face-centered orthorhombic M - primitive monoclinic N - centered monoclinic A - triclinic. 

Notes. The d planes may exist in orthorhombic F, tetragonal I, cubic I and cubic F Bravais lattices (Section 2.6.1). In the tetragonal, trigonal, hexagonal and cubic systems (Sections 2.5.8 and 2.5.9) we find mirror planes which are not parallel to the (100), (010) or (001) planes. The glides for the n and d planes corresponding to these orientations are oblique with respect to the a, b, c axes. More detailed information may be found in the International Tables for Crystallography. 

For example, the crystal class for I4c2 is 4m2, an alternative symbol for 42m. The crystal system is tetragonal and the Bravais lattice is tetragonal I. 

Standard ASTM E157-82ahasthe Bravais lattices designations as following C — primitive cubic B — body-centered cubic F — face-centered cubic T — primitive tetragonal U—body-centered tetragonal R—rhombohedral H — hexagonal O—primitive orthorhombic P — body-centered orthorhombic Q — base-centered orthorhombic S — face-centered orthorhombic M — primitive monoclinic N — centered monoclinic A — triclinic. 

Why is there no Bravais lattice called orthorhombic A, monoclinic B, or tetragonal C ... 

Primitive three-dimensional lattices have been classified into seven crystalline systems triclinic, monoclinic, orthorombic, tetragonal, cubic, trigonal, and hexagonal. They are different in the relative lengths of the basis vectors as well as in the angles they form. An additional seven nonprimitive lattices, belonging to the same crystalline systems, are added to the seven primitive lattices, which thus completes the set of all conceivable lattices in ordinary space. These 14 different types of lattices are known as Bravais lattices (Figure 3). 

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