All Key Formulas for PC-11: Solid State

Unit Cell Atoms per Cell

Edge Length–Radius Relations

$r_{SC} = \frac{a}{2} \qquad r_{BCC} = \frac{a\sqrt{3}}{4} \qquad r_{FCC} = \frac{a\sqrt{2}}{4}$

Packing Efficiency

$PE_{SC} = \frac{\pi}{6} \approx 52.4\% \qquad PE_{BCC} = \frac{\pi\sqrt{3}}{8} \approx 68\% \qquad PE_{FCC} = \frac{\pi\sqrt{2}}{6} \approx 74\%$

Density Formula and Rearrangements

$\rho = \frac{ZM}{a^3 N_A} \qquad a = \left(\frac{ZM}{\rho N_A}\right)^{1/3} \qquad M = \frac{\rho a^3 N_A}{Z}$

Void Counts (for n close-packed atoms)

$N_{oct} = n \qquad N_{tet} = 2n \qquad N_{total} = 3n$

Radius Ratio Limits (Geometry-Derived)

$\left(\frac{r^+}{r^-}\right)_{tet} = \sqrt{\frac{3}{2}} - 1 \approx 0.225$

$\left(\frac{r^+}{r^-}\right)_{oct} = \sqrt{2} - 1 \approx 0.414$

Unit Conversions

$1 \text{ Å} = 10^{-8} \text{ cm} = 10^{-10} \text{ m} = 100 \text{ pm}$ $1 \text{ pm} = 10^{-10} \text{ cm} = 10^{-12} \text{ m}$

Semiconductor Doping

$\text{Group 15 (P, As, Sb)} \rightarrow n\text{-type} \quad (e^- \text{ majority})$ $\text{Group 13 (B, Ga, In)} \rightarrow p\text{-type} \quad (h^+ \text{ majority})$