The crystal structures of metallic solids are usually composed of three types of unit cells that are formed due to the way metal atoms are packed in a crystal.
Consider the atoms as hard spheres touching each other to form a closely packed single layer (see diagram above). To maintain the close packed arrangement, a second layer (orange) is placed above the bottom layer (blue) with each second-layer atom sitting in the notches of the bottom layer. There are two ways to position atoms for the third layer (yellow), starting with notch A or notch B. If the first atom in the third layer starts at notch A, all atoms of the third layer are directly above atoms in the first layer, with the layers having an XYXYXY… pattern, giving the crystal (e.g. magnesium) a structure called hexagonal close-packed (hcp). However, if the first atom in the third layer starts at notch B, all atoms of the third layer are staggered relative to atoms in the first layer, resulting in the layers having an XYZXYZXYZ… pattern. A crystal with such an arrangement of layers (e.g. copper) has a structure called cubic close-packed (ccp).
Another common crystal structure of metals, shown in the diagram below, is the body centred cubic (bcc). The bottom layer (blue) is less closely packed than hcp and ccp, with each atom touching only four neighbouring atoms in the same layer instead of six. The second layer of atoms (orange) sits in the notches of the bottom layer and the third layer fits into the notches of the second layer, giving an XYXYXY… pattern. Iron and potassium have bcc structures.
Having seen how metal atoms are packed in a crystal, let’s find the respective unit cells. A crystal with a ccp structure has a face-centred cubic unit cell (see below diagram) where a = b = c and α = β = γ = 90°. There is one lattice point at each corner of the cube and one lattice point in the middle of each surface plane. Each lattice point meets the criteria of having the same environment by having six and eight neighbouring atoms at distances a and (√2/2)a respectively (it is easier to visualise the neighbouring atoms by replicating the unit cell around itself).
A crystal with a bcc structure has a unit cell of the same name (see below diagram) where a = b = c and α = β = γ = 90°. There is one lattice point at each corner of the cube and one lattice point in the middle of the cube. Each lattice point meets the criteria of having the same environment by having six and eight neighbouring atoms at distances a and (√3/2)a respectively (using Pythagoras’ theorem).
To map out the unit cell of a hcp structure, we let a pair of atoms be a lattice point (denoted by a yellow oval in the 3rd diagram below). The atom pair or basis consists of an atom from the bottom layer (blue) and one from the second layer (orange). Connecting the centres of four of the ovals (numbered 1 to 4), we get a rhombus with two opposite angles of 120°; and linking two adjacent rhombi, we obtain the unit cell called the hexagonal unit cell. The reason it is called a hexagonal unit cell is the combination of three such unit cells form a hexagon.
Finally, the different forms of packing and hence, unit cells, are a result of the different electron densities of atoms and molecules and their interactions to produce energetically stable arrangements in the solid state.
Question
Calculate the packing fraction of Cu (i.e. the fraction of volume occupied by Cu spheres).
Answer
Cu has a fcc unit cell and the volume of an fcc unit cell is a3. A fcc unit cell contains 4 Cu spheres since one-eighth of a sphere resides in each corner of the unit cell and half of a sphere remains on each surface of the unit cell. The length of a diagonal across a surface of the unit cell is determined using Pythagoras’ theorem as a√2 and is equivalent to exactly four radii of a Cu sphere. Hence, the radius of a Cu sphere is and the volume of 4 Cu spheres is
. Therefore, the packing fraction of Cu is
, which computes to 0.740.
