3.2 The unit cell and fractional coordinates

For the box we used to construct our ideal crystal, we give the name the unit cell. We still need not consider exactly how the atoms are arranged within that box, but since every box (or equivalently unit cell) is identical, once we have described the contents of the unit cell, we can construct an exact description of the entire crystal. This leads to a very convenient mechanism for describing the coordinates of atoms within the unit cell using crystallographic axes, which are also sometimes called the crystal axes. We label the three sides of the unit cell as a, b & c, and define the angles between the sides as \(\alpha \), \(\beta \), & \(\gamma \), where we define as \(\gamma \) the angle between the a & b sides, \(\beta \) between a & c, and \(\alpha \) between b & c. We then define axes, x, y & z along a, b & c directions, respectively. Note that the angles between these axes are dictated by \(\alpha \), \(\beta \) & \(\gamma \), so for systems other than cubic, tetragonal and orthorhombic, angles between axes may not be 90 degrees. Such axes systems are non-orthogonal (also called oblique) which makes performing three-dimensional algebra more difficult, but will so greatly simplify our descriptions of the crystal that this mechanism is much preferred over orthogonal coordinate systems. Table 3.2 outlines the common conventions used for assigning axes for the seven crystal systems.

Finally, we scale each axis so that one unit of length corresponds to the length of the unit cell in that direction. This means that for most crystal systems the per unit along each axis may differ, so This means that we can label any point inside the unit cell with coordinates (x,y,z) where \(0\leq \rm x<1\), \(0\leq \rm y<1\) and \(0\leq \rm z<1\). Locations on the faces of the unit cell will have \(\rm x=0\) or \(\rm y=0\) or \(\rm z=0\) and the coordinates for the eight corners of the unit cell are (0,0,0), (1,0,0), (0,1,0), (0,1,0), (1,1,0), (1,0,1), (0,1,1) and (1,1,1). The center of the unit cell is (0.5, 0.5, 0.5). This system of coordinates is known as fractional coordinates. We can also consider the location of a point inside the cell with location (x,y,z) as a vector from the origin to the atom position, \(\vec {x_{xyz}}=\ x\hat {a}+\ y\hat {b}+z\hat {c}\) where \(\hat {a}\), \(\hat {b}\) and \(\hat {c}\) are unit vectors (i.e. with length in the same units, for example, Å) in the appropriate direction.

Note that since the contents of every unit cell is the same, adding or subtracting any multiple of 1 to x, y and/or z results in a location that is indistinguishable from the initial point. We can write a rule that point (x,y,z) is equivalent to (x+n_x, y+n_y, z+n_z) where n_x, n_y and n_z can each be any integer to express the superposition of the infinite lattice replicating the contents of the unit cell in all directions.