Solid State - Formula Sheet

Standard Unit Cell Formulas

• Density of a Unit Cell: \(\rho = \frac{n M}{a^3 N_A}\)
  \(\rho\) = Density, \(n\) = Number of atoms per cell, \(M\) = Molar mass, \(a\) = Edge length, \(N_A\) = Avogadro's constant.
• Packing Efficiency: \(\text{Efficiency} = \frac{\text{Volume occupied by spheres}}{\text{Total volume of unit cell}} \times 100\)
• Radius Ratio: \(\text{Ratio} = \frac{r^+}{r^-}\)
  \(r^+\) = Cation radius, \(r^-\) = Anion radius.

Cubic System Geometry

• Simple Cubic (SC): \(a = 2r\); \(n = 1\)
• Body Centered Cubic (BCC): \(a = \frac{4r}{\sqrt{3}}\); \(n = 2\)
• Face Centered Cubic (FCC): \(a = \sqrt{8}r\); \(n = 4\)

Void Relationships

• Number of Octahedral Voids: \(= N\) (where \(N\) is the number of spheres)
• Number of Tetrahedral Voids: \(= 2N\)