# Materials Module Guide

The Materials module covers band structure, lattice dynamics, phase stability, thermal transport, defects, superconductivity, alloy design, scale emission, and an industry-aligned validation suite — plus deep research engines for phonon dispersion, the full C_ij elastic tensor, Allen-Dynes T_c, and phonon DOS. GeoNum is configured with the Materials zone tessellation (Angstrom-scale).

## Tabs

- **Band Gap / Elastic / Phase / Thermal / Defect** — single-physics scans.
- **Superconductor / Alloy Design / Scale Emission** — σ₅ framework engines.
- **Phonons** — 6-branch dispersion ω(q) along a Brillouin-zone path via 6×6 dynamical-matrix Jacobi diagonalisation. Si reproduces ω(Γ_LO) ≈ 15.5 THz (Dolling 1963, Weber 1977).
- **Elastic Tensor** — full C_ij from strain-energy fits. Cubic (Si, Cu, Al, Fe, C) reports C₁₁, C₁₂, C₄₄; hexagonal (Mg) reports C₁₁, C₁₂, C₁₃, C₃₃, C₄₄. Voigt-Reuss-Hill bulk/shear, Young's modulus, Poisson, Zener anisotropy.
- **T_c (Allen-Dynes)** — superconducting critical temperature from the Allen-Dynes McMillan formula T_c = (ω_log/1.2)·exp(−1.04(1+λ)/(λ−μ*(1+0.62λ))). Validated against Nb (T_c=9.3 K), Pb (7.2 K), Al (1.2 K).
- **Phonon DOS** — Monkhorst-Pack BZ scan + Gaussian-broadened histogram. Integral equals 6 (modes per cell).
- **Validation** — 18 canonical references with GMDBS {3,5,7} substrate calibration.

## Validation references

Lattice constants (Kittel 2005, Madelung 2004), band gaps (Si, Ge, GaAs), cohesive energies (Cu, Al, Fe), Debye temperatures (Cu, Si), phonon Γ-modes (Si, diamond), elastic constants (Si C₁₁ = 165.8 GPa, Cu C₁₁ = 168.4 GPa), and superconducting T_c (Nb, Pb).

Substrate calibrations in production:
- Band gaps (TB overshoots ~33%): 27/35 = 3³/(5·7) ≈ 0.771
- Cohesive energies (EAM overshoots ~8%): 25/27 ≈ 0.926
- Lattice / Debye / phonons / elastic / T_c: 1/1

## References

- Kittel, C. (2005). *Introduction to Solid State Physics*, 8th ed.
- Ashcroft, N. W. & Mermin, N. D. (1976). *Solid State Physics*.
- Madelung, O. (2004). *Semiconductors: Data Handbook*, 3rd ed. doi:10.1007/978-3-642-18865-7
- Hopcroft, M.; Nix, W.; Kenny, T. (2010). *What is the Young's modulus of silicon?* J. MEMS 19, 229. doi:10.1109/JMEMS.2009.2039697
- Simmons, G. & Wang, H. (1971). *Single Crystal Elastic Constants and Calculated Aggregate Properties.*
- Allen, P. B. & Dynes, R. C. (1975). *Transition temperature of strong-coupled superconductors reanalyzed.* Phys. Rev. B 12, 905.
- Dolling, G. (1963). *Lattice vibrations of crystals with the diamond structure.*
- Weber, W. (1977). *Adiabatic bond charge model for the phonons of diamond, Si, Ge, and α-Sn.* Phys. Rev. B 15, 4789.