Methodology & Equations
A transparent record of formulas, approximations, and selection rules used across all tools.
1.Diffraction & Bragg's Law
Diffraction peaks are computed using Bragg's law:
Where λ is the X-ray wavelength (Cu Kα = 1.5406 Å), d is the interplanar spacing, and θ is the Bragg angle.
For cubic crystal systems, the interplanar spacing d is calculated as:
Selection rules govern which planes diffract constructively based on the unit cell Bravais lattice:
- Simple Cubic (SC): All planes allowed.
- Body-Centered Cubic (BCC): Allowed if h + k + l is even.
- Face-Centered Cubic (FCC): Allowed if h, k, l are all odd or all even.
2.Crystallite size (Scherrer Equation)
Crystallite size is estimated using peak broadening:
Where K is the shape factor (standard default = 0.9), β is the FWHM in radians, and θ is the Bragg angle. To account for instrumental broadening, FWHM is corrected as:
3.Thermally Activated Diffusion
Diffusion coefficients are thermally activated and behave according to the Arrhenius relation:
Concentration depth profiles are modeled using the one-dimensional semi-infinite boundary solution:
Where erf is the error function, solved locally in TypeScript using the Abramowitz and Stegun numerical approximation.
4.Point Defects (Vacancy Concentration)
The equilibrium concentration of vacancies in a lattice changes with temperature according to:
Where Qv is the vacancy activation energy, T is absolute temperature, and N is total atomic site density per unit volume.
5.Mechanical Stress-Strain properties
Young's Modulus is computed via linear regression fit of the initial linear portion of the stress-strain curve. Yield strength is estimated using the 0.2% offset method: a line is drawn parallel to the elastic slope starting at 0.002 strain. The intersection with the experimental curve is identified as the offset yield stress. Toughness is estimated by integrating stress-strain data up to the final fracture point via the trapezoidal rule: