# Can the lattice of an element with face-centered cubic (FCC) crystal structure be regarded as simple cubic (SC)?

According to Introduction to Solid State Physics by C. Kittel,

An ideal crystal is constructed by the infinite repetition of identical groups of atoms. A group is called the basis. The set of mathematical points to which the basis is attached is called the lattice.

The crystal structure of copper is face-centered cubic (fcc). So we can say the lattice is fcc, and the basis is composed of one single copper atom.

I was wondering if it is wrong to say, instead, that the lattice is simple cubic (sc), and the basis is composed of four copper atoms.

Both give the same arrangement of atoms in space, but in the second there are fewer lattice points.

The fcc structure can be generated from a sc lattice with a four-atom basis.

The sentence above is from lecture notes by Phil Ahrenkiel of South Dakota School of Mines and Technology.

So, this means that if four atoms with positions

$$\mathbf{d}_1=(0,0,0)$$

$$\mathbf{d}_2=(0.5,0.5,0)$$

$$\mathbf{d}_3=(0.5,0,0.5)$$

$$\mathbf{d}_4=(0,0.5,0.5)$$

are added to each point of a simple-cubic lattice, the resulting arrangement of the whole atoms in space forms a face-centered cubic structure.

• Kittel's book admittedly does not state this explicitly, but it can be inferred from noting that the conventional cells shown in Fig. 10 for the cubic space lattices are all contained within a cube with side length $a$, and that the fcc lattice has four lattice points per conventional cell, as stated both in the text and in Table 2. Nov 23, 2023 at 17:06