3.9 Centering and the 14 Bravais lattices

I previously introduced the seven crystal systems and one might think that this is sufficient to describe the symmetry of a lattice, but in fact there is an additional type of symmetry that must be taken into account, which is called lattice centering. With the inclusion of the four types of centering and the seven unit cell types we generate the fourteen lattice symmetry classes, which are known as the Bravais lattice types. To illustrate the concept of centering, let us consider the symmetry of a brick wall, where every other row of the wall has the bricks shifted by a half unit. As seen in Fig. 3.1, as a guide for a potential lattice, I have placed a dot at the center of each brick. If I use those points, I have the possibility to create many different unit cells, all with caveats; two are shown with lines in Fig. 3.2.

The problem with the cell to the left (in red) is that it is oblique – the angles between sides are not right angles – and thus we have lost some essential symmetry information about the rectangular bricks. The problem with the cell at the right (in green) is that while it preserves the rectangular symmetry of the bricks, it is twice as large as a single brick and thus is less than optimal in providing the simplest possible description of the lattice. Note that we cannot use a single brick as a unit cell, because that would violate the principle that a unit cell translation of applied to a point in any direction provides an identical point. Translation vertically by one unit would take a point from the middle of one unit cell to the edge of another.

Lattice centering provides a secondary rule for lattice generation which places lattice points not only at the corners of unit cells but also at locations within cells. If we label axes above so that we are looking at the x-y plane and call the corners of the green box as (0,0,0), (1,0,0), (0,1,0) and (1,1,0) then we can also label the central point between them as (½, ½, 0). We already have said that adding any number of unit cell translations brings us to a point that is equivalent to our original point, which we can write algebraically as saying that point (x,y,z) is equivalent to (x+n_x,y+n_y,z+n_z), where n_x, n_y, n_z and n_c can each be any integer. To express the idea that the environment of (½, ½, 0) is the same as (0, 0, 0) we can write an additional equivalence, that point (x,y,z) is equivalent to (x+n_x+½ , y+n_y+½, z+n_z), where again n_x, n_y, n_z and n_c can each be any integer.

This brings us to the concept of a symmetry operation, which will be needed when we discuss space groups. We use these only to describe the symmetry within the unit cell, so we do not need to consider the translations above, where we added integers to x, y or z as that relates the contents of one unit cell to a neighbor. However, one requirement of mathematical groups is that they must contain an identity operator, so even if there is pattern of repetition within the cell, the identity symmetry operation x,y,z must be present. However, in the case of the centering operation with our bricks, we can write a symmetry operator ½+x,½+y,z meaning that if we place an atom at coordinates (x,y,z), we automatically generate another a half unit cell away in both x and y. Note that if x¿1/2 then x+½ will be in the next unit cell, but we can add or subtract any integer from x to translate back into the current unit cell, so ½+x+n_x,½+y,z is implied by ½+x,½+y,z and thus is equivalent to x-½,½+y,z.

Lattice centering is most useful for crystal classes where the centered lattice demonstrates some symmetry that the primitive lattice (as we call a lattice without centering) does not. If we place a lattice center into the middle of a triclinic cell, with coordinates (½, ½, ½) or equivalently consider this as the symmetry operation ½+x, ½+y, ½+z, it will have a doubled volume in comparison to any of the smallest possible primitive cells, but provides no extra symmetry information. This does not mean that it is illegal to use a centered description with a triclinic unit cell, in fact it may be useful for understanding what is changing when a material transforms from a structure with a centered cell to one that is triclinic, but the standard descriptions will not include any centered triclinic cells. In contrast, if we take an orthorhombic cell with a body center, we can construct several primitive cells, but none will have the three perpendicular axes found in the centered cell. Thus, centering allows us to demonstrate the full symmetry of that orthorhombic lattice.

There are two types of centers needed to construct the possible symmetry types for all three-dimensional lattices, known as the Bravais lattices. One center type is placed in the side of a unit cell and called a face center. A face center with coordinates (½, ½, 0) is in the face perpendicular to c and is called a C-center. Likewise, centers at (½, 0, ½) and (0, ½, ½) are called B-center and A-centers, respectively. As we saw before, a center at the middle of the unit cell is called a body center or is abbreviated as an I-center (short for the German word Innenzentriert.). Combinations of centers are possible, but the only combination that cannot be simplified without loss of the underlying symmetry is when A-, B- and C-centers are all combined. This combination is referred to as face centering or an F-center. Note that a A-, B-, C- or I-center will double the number of symmetry operations while face centering leads to four symmetry operations x,y,z; ½+x,½+y,z; ½+x,y,½+z; and x,½+y,½+z. Sometimes the expanded volume of a centered lattice is referenced by comparison to the low-symmetry primitive unit cell. Thus, a A-, B-, C- and I-centered lattice is doubly-primitive, meaning the centered cell has double the volume of the primitive cell. An F-centered lattice is quadruply-primitive. It is also possible to consider a lattice center on an edge of a unit cell, such as (½,0,0). Edge centers are used for description of unit cells for magnetic scattering but are not needed for the 14 Bravais lattices.

There is one special case of centering specific to related to rhombohedric unit cells. It is possible to transform any lattice constructed from rhombohedric unit cells (a=b=c, \(\alpha \)=\(\beta \)=\(\gamma \)) into one that has a=b, \(\gamma \)=120° and \(\alpha \)=\(\beta \)=90° unit cells,e.g. the same cell as a hexagonal lattice but this cell has three times the volume of the primitive rhombohedric unit cell and with lattice centers at (2⁄3, 1⁄3, 1⁄3) and (1⁄3, 2⁄3, 2⁄3). We use the symbol R is used for this type of centering. While both types of unit cells are used for this system because the same lattice can be generated either from a primitive rhombohedron or a triply-primitive centered cell in a hexagonal lattice, the hexagonal description tends to be used more commonly because it is usually somewhat more computationally simple.