Choice of unit cell

The concept of the unit cell has been important throughout the present chapter. The unit cell represents an object that, when repeated by translations, gives an infinite crystal. In this simple definition almost every word can be a trap. 

Is it feasible Is the choice unique If not, then what are the differences among them How is the motif connected to the unit cell choice Is the motif unique Which motifs may we think about  

As we have already noted, the choice of unit cell as well as of motif is not unique. This is easy to see. Indeed Fig. 9.21 shows that the unit cell and the motif can be chosen in many different and equivalent ways. 

Moreover, there is no chance of telling, in a responsible way, which of the choices are reasonable and which are not. And it happens that in this particular case we really have a plethora of choices. Putting no limits to our fantasy, we may choose a unit cell in a particularly capricious way. Figs. 9.21.b and 9.22. 

We may say there may be many legal choices of motif, but this is without any theoretical meaning, because all the choices lead to the same infinite i stem. Well, this is true with respect to theory, but in practical applications the choice of motif may be of prime importance. We can see this from Table 9.2, which corresponds to Fig. 9.22. 

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Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

There are several ways to draw unit cells that will repeat to generate the entire lattice. The choice of unit cell is determined by conventions that are beyond the scope of this text (the smallest unit cell that indicates all of the symmetry present in the lattice is typically the one of choice). 

FIGURE 1.16 A one-dimensional lattice (a,b) and the choice of unit cells (c). 

FIGURE 1.17 Choice of unit cell in a square two-dimensional lattice. 

Figure 11.3. The five distinct plane (2D) lattices (a) oblique, (b) primitive rectangular, (c) square, (d) and (e) are both centered rectangular but show alternative choices of unit cell, (/) hexagonal.

Consider two different choices of unit cells a F-centered unit cell with axes (ax, b, ci) and a primitive one with axes (fl2, b2, C2), as shown in Fig. 9.2.4. We can write... 

The monoclinic crystals now are listed with the b axis as the unique axis, but prior to 1940, another popular "setting" used c as the unique axis. Of the 230 space groups, 7 have two choices of unit cell, a primitive rhombohedral one (R) and, for convenience, a nonprimitive hexagonal one (H), with three times the volume of the rhombohedral cell. The 3x3 transformation matrices from rhombohedral (obverse, or positive, or direct) cipbj, Cr to hexagonal axes aH, bur Ch and vice versa are shown in the caption to Fig. 7.17. 

As we stressed earlier, relative phases are a function of the choice of unit cell origin. A Bragg reflection does not have an absolute phase angle, but only one relative to the chosen origin. If another origin were chosen, the intensity of the Bragg reflection would remain the same, but the relative phase angle would be different. 

We must imagine that the pattern extends indefinitely (to the end of the wall). In each pattern two of the many possible choices of unit cells are outlined. Once we identify a unit cell and its contents, repetition by translating this unit generates the entire pattern. In (a) the unit cell contains only one cat. In (b) each cell contains two cats related to one another by a 180° rotation. Any crystal is an analogous pattern in which the contents of the three-dimensional unit cell consist of atoms, molecules, or ions. The pattern extends in three dimensions to the boundaries of the crystal, usually including many thousands of unit cells. 

The dashed lines outline an alterative choice of the unit cell. The entire pattern is generated by repeating either unit cell (and its contents) in all three directions. Several such choices of unit cells are usually possible, (d) A stereoview of the sodium chloride strucmre, extending over several unit cells. 

The point H = 0 represents Schwarz s primitive minimal surface, the only case for which the two choices of unit cell are congruent by virtue of the fact that this minimal surface contains straight lines. 

It follows from these conventions governing the choice of unit cell that the most convenient cell is not necessarily the smallest possible cell. Cubic structures, for example, are always described in terms of a unit cell which is a cube, even when, as is sometimes the case, a rhombohedral cell of smaller volume could be selected. 

The choice of unit cell shape and volume is arbitrary but there are preferred conventions. A unit cell containing one motif and its associated lattice is called primitive. Sometimes it is convenient, in order to realise orthogonal basis vectors, to choose a unit cell containing more than one motif, which is then the non-primitive or centred case. In both cases the motif itself can be built up of several identical component parts, known as asymmetric units, related by crystallographic symmetry internal to the unit cell. The asymmetric unit therefore represents the smallest volume in a crystal upon which the crystal s symmetry elements operate to generate the crystal. 

By analogy with what we did in two dimensions, we ean see that another choice of unit cell would include one-eighth of each of eight spheres at the comers of a cube. This is actually the preferred unit eell and is referred to as a simple cubic unit cell. 

The results, without taking into account the long-range interactions, depend very strongly on the choice of unit cell motif. 

Use of the multipole expansion greatly improves the results and, lo very good accuracy, makes them independent of the choice of unit cell motif as it should be. 

Table 9.3. Total energy per unit cell Ej- in the pofymer LiH as a function of unit cell definition (Fig. 9.23). For each choice of unit cells, this energy is computed in four ways (1) without long-range forces (long range = 0) i.e.. unit cell 0 interacts with N = 6 unit cells on its right-hand side and N unit cells on its left-hand-side (2), (3), (4) with the long range computed with multipole interactions up to the and a terms, respective. The...

There are an infinite number of ways to reconstruct the same system from parts. These ways are not equivalent in practical calculations if, for any reason, we are unable to compute all the interactions in the system. However, if we have a theory (in our case the multipole method) that is able to compute the interactions, including the long-range forces, then it turns out the final result is virtually independent of t he choice of unit cell motif. This arbitrariness of choice of subsystem looks analogous to the arbitrariness of the choice of coordinate system. The final results do not depend on the coordinate system used, but still the numerical results (as well as the effort to get the solution) do. 

The choice of unit cell motif is irrelevant from the theoretical point of view, but leads to different numerical results when the long-range interactions are omitted. When taking into account the long-range interactions, the theory becomes independent of the division of the whole system into arbitrary motifs. 

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