Formulas and Math (Mat Sci 8)

Atomic Structure and Crystallinity

Atomic Packing Factor (APF):

  • Formula: APF = (Volume of atoms in a unit cell) / (Volume of the unit cell)

  • FCC APF: 0.74

  • BCC APF: 0.68

  • SC APF: 0.52

Density (ρ\rhoρ) for Crystals:

  • Formula: ρ=n⋅AVc⋅NA\rho = \frac{n \cdot A}{V_c \cdot N_A}ρ=Vc​⋅NA​n⋅A​

    • nnn: Number of atoms per unit cell

    • AAA: Atomic mass

    • VcV_cVc​: Volume of the unit cell

    • NAN_ANA​: Avogadro's number (6.022×10236.022 \times 10^{23}6.022×1023)

Miller Indices:

  • Formula: (h,k,l)=reciprocal of intercepts of crystal plane with axes(h, k, l) = \text{reciprocal of intercepts of crystal plane with axes}(h,k,l)=reciprocal of intercepts of crystal plane with axes

Grain, Weight, and Atom Properties

Grain Size Number (G):

  • Formula: G=−log⁡2(N)+1G = -\log_2(N) + 1G=−log2​(N)+1

    • NNN: Number of grains per unit area

Number of Atoms (nnn):

  • Formula: n=MassAtomic mass×NAn = \frac{\text{Mass}}{\text{Atomic mass}} \times N_An=Atomic massMass​×NA​

Atomic Radius:

  • BCC: R=34aR = \frac{\sqrt{3}}{4} aR=43​​a

  • FCC: R=24aR = \frac{\sqrt{2}}{4} aR=42​​a

Phase Rules and Composition

Gibbs Phase Rule:

  • Formula: P+F=C+2P + F = C + 2P+F=C+2

    • PPP: Phases

    • FFF: Degrees of freedom

    • CCC: Components

Lever Rule (Phase Fraction):

  • Formula:

    • fL=CS−C0CS−CLf_L = \frac{C_S - C_0}{C_S - C_L}fL​=CS​−CL​CS​−C0​​

    • fS=C0−CLCS−CLf_S = \frac{C_0 - C_L}{C_S - C_L}fS​=CS​−CL​C0​−CL​​

Stress, Strength, and Strain

Stress (σ\sigmaσ):

  • Formula: σ=FA\sigma = \frac{F}{A}σ=AF​

Strain (ϵ\epsilonϵ):

  • Formula: ϵ=ΔLL0\epsilon = \frac{\Delta L}{L_0}ϵ=L0​ΔL​

Young's Modulus (EEE):

  • Formula: E=σϵE = \frac{\sigma}{\epsilon}E=ϵσ​

Poisson’s Ratio (ν\nuν):

  • Formula: ν=−ϵlateralϵlongitudinal\nu = -\frac{\epsilon_{\text{lateral}}}{\epsilon_{\text{longitudinal}}}ν=−ϵlongitudinal​ϵlateral​​

Failure

Fracture Toughness (KcK_cKc​):

  • Formula: Kc=YσπaK_c = Y \sigma \sqrt{\pi a}Kc​=Yσπa​

    • YYY: Geometry factor

    • σ\sigmaσ: Applied stress

    • aaa: Crack length

Safety Factor:

  • Formula: Factor of Safety=Yield StressWorking Stress\text{Factor of Safety} = \frac{\text{Yield Stress}}{\text{Working Stress}}Factor of Safety=Working StressYield Stress​

Thermal Properties

Thermal Expansion:

  • Formula: ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta TΔL=αL0​ΔT

    • α\alphaα: Coefficient of thermal expansion

Thermal Conductivity (kkk):

  • Formula: Q=−kdTdxQ = -k \frac{dT}{dx}Q=−kdxdT​

Electric and Dielectric Properties

Resistivity (ρ\rhoρ):

  • Formula: ρ=RAL\rho = R \frac{A}{L}ρ=RLA​

Capacitance (CCC):

  • Formula: C=ϵAdC = \epsilon \frac{A}{d}C=ϵdA​

Dielectric Constant (κ\kappaκ):

  • Formula: κ=ϵϵ0\kappa = \frac{\epsilon}{\epsilon_0}κ=ϵ0​ϵ​

Ceramic Properties

Porosity (PPP):

  • Formula: P=1−ρmeasuredρtheoreticalP = 1 - \frac{\rho_{\text{measured}}}{\rho_{\text{theoretical}}}P=1−ρtheoretical​ρmeasured​​

Flexural Strength (σf\sigma_fσf​):

  • Formula: σf=3FL2bd2\sigma_f = \frac{3F L}{2bd^2}σf​=2bd23FL​

Heat Capacity (CpC_pCp​):

  • Formula: Q=mCpΔTQ = m C_p \Delta TQ=mCp​ΔT