Materials physics — the microscopic structural and defect-physics foundations of macroscopic materials behaviour. Distinguished from materials science (broader, engineering-adjacent) by its focus on reductive physical mechanism rather…
materials-physics
Defect concentration: c = exp(-E_f/k_B·T) (Arrhenius equilibrium)
The equilibrium vacancy concentration in a crystal at temperature T follows an Arrhenius law c = exp(-E_f/k_B·T), where E_f is the vacancy…
Hall-Petch strengthening: σ_y = σ_0 + k_HP·d^(-1/2)
Hall (1951) and Petch (1953) independently established the inverse-square-root grain-size dependence of yield strength in polycrystalline…
Nabarro-Herring creep: ε̇ ∝ σ·D·Ω/(k_B·T·d²)
Nabarro (1948) and Herring (1950) derived the bulk-diffusion creep law: under applied stress σ at high homologous temperature (T/T_melt ≳…
Defect Arrhenius: c(E_f=0)=1; c(E_f=k_B·T)=exp(-1)
Sympy-exact witness of the Arrhenius defect-concentration limiting cases. Setup: c = exp(-E_f/(T·k_B)). Identity 1 (zero formation…
Hall-Petch limit: σ_y(d=1)=σ_0+k_HP; σ_y(d→∞)=σ_0
Sympy-exact witness of the Hall-Petch limiting behaviour. Setup: σ_y = σ_0 + k_HP/√d. Identity 1 (d=1 reference-grain-size anchor):…
Nabarro-Herring grain ratio: ε̇(2d)/ε̇(d) = 1/4
Sympy-exact witness of the grain-size-quadratic scaling of Nabarro-Herring creep. Setup: ε̇_NH ∝ σ·D·Ω/(T·k_B·d²) — inverse-square in…
Hooke linear elasticity sigma_ij = C_ijkl eps_kl; tensor decomposition into hydrostatic + deviatoric
Hooke's law / linear-elasticity tensor framework (Hooke 1676 De Potentia Restitutiva; Cauchy 1822 Bull Soc Philomath; Lame 1852 Lecons sur…
Born-von Karman 1D chain dispersion omega^2 = (4 K/m) sin^2(ka/2); linear-recurrence
Born-von Karman lattice dynamics framework (Born-von Karman 1912 Phys Z 13, 297). Setup: 1D chain of N identical atoms (mass m, equilibrium…
Sommerfeld electronic heat capacity gamma = (pi^2/3) D(eps_F) k_B^2; symmetric-function expansion
Sommerfeld electronic-heat-capacity framework (Sommerfeld 1928 Z Phys 47, 1). Setup: at T << T_F (Fermi temperature), the Fermi-Dirac…
Theorem: K(nu = 0) - E/3 = 0 (Hooke isotropic bulk-modulus identity at zero Poisson ratio)
Theorem (Hooke-K-zero-Poisson canonical): for isotropic linear elasticity K = E/(3(1 - 2 nu)). At nu = 0 (incompressible-like limit,…
Theorem: omega(k = pi/a)^2 - 4K/m = 0 (Born-von Karman Brillouin-zone-edge max frequency)
Theorem (Born-vK-zone-edge canonical): for the 1D monatomic-chain dispersion omega^2 = (4K/m) sin^2(ka/2), the Brillouin-zone-edge…
Theorem: gamma * 3/(pi^2 D k_B^2) - 1 = 0 (Sommerfeld coefficient identity)
Theorem (Sommerfeld-coefficient canonical): gamma = (pi^2/3) D(eps_F) k_B^2 rearranges to 3 gamma/(pi^2 D k_B^2) = 1, residual 3…
DFT for materials
Kohn-Sham 1965 DFT; modern VASP + Quantum ESPRESSO + ABINIT codes; basis of materials-genome-initiative + ML-DFT hybrids.
Cahn-Hilliard (1958)
J W Cahn-J E Hilliard 1958 phase-field free-energy; modern microstructure simulation + grain-boundary mobility predictions.
Ferroelectric perovskites (Rabe 2007)
K Rabe 2007 'Physics of Ferroelectrics' first-principles approach; modern morphotropic-phase-boundary PMN-PT giant-piezoelectric;…
Topological materials (Fu-Kane 2007)
L Fu-C L Kane 2007 Z2 topological-insulator prediction; Bi2Se3 + Bi2Te3 ARPES validation 2009; modern topological-catalogue…
2D materials boom
Graphene 2004 + transition-metal-dichalcogenides 2010s + boron-nitride; modern moire-superlattice + magic-angle 2018 Tarnopolsky.
Materials genome (2011)
US-MGI 2011 high-throughput-screening; AFLOW + OQMD + Materials Project Jain-Persson 2013; modern ML-accelerated discovery 2020+.
Hall-Petch (1951-1953)
E Hall 1951 + N Petch 1953 σ_y = σ_0 + k/√d; modern modern foundational text + nano-grain breakdown + 2024 ML-prediction.
DFT (Hohenberg-Kohn 1964)
P Hohenberg-W Kohn 1964 + Kohn-Sham 1965 (Nobel 1998) DFT; modern modern foundational text + Materials Project + 2024 ML-functional…
CALPHAD (Kaufman 1970)
L Kaufman 1970 + Saunders-Miodownik 1998 CALPHAD; modern modern foundational text + Thermo-Calc + 2024 ML-CALPHAD high-entropy.
Griffith (1921)
A Griffith 1921 K_c = √(2γE) crack-fracture; modern modern foundational text + Linear-Elastic-Fracture-Mechanics + 2024…
Composite (Voigt 1889 / Reuss 1929)
W Voigt 1889 + A Reuss 1929 + Hashin-Shtrikman 1963 composite-bounds; modern modern foundational text + 2024 ML-property-prediction…
Topological (Haldane 2017)
F D M Haldane 2016 (Nobel 2016) topological-phases; modern modern foundational + Bi2Se3 + 2024 topological-superconductor-cmd Majorana…