Formulas and Math (Mat Sci 8)

Atomic Packing Factor (APF)

A measure of the density of atoms in a unit cell, calculated as the volume of atoms in a unit cell divided by the volume of the unit cell. APF is an important property in crystallography and materials science.

What is the APF for FCC structures?

0.74, calculated using the formula APF = (number of atoms in the unit cell × volume of an atom) / volume of the unit cell.

What is the APF for BCC structures?

0.68, similarly calculated as the number of atoms in the unit cell times the volume of a BCC atom divided by the volume of the unit cell.

What is the APF for SC structures?

0.52, determined using the same approach, considering that simple cubic structures contain one atom per unit cell.

Density formula for crystals

ρ = (n ⋅ A) / (V_c ⋅ N_A), where n is the number of atoms per unit cell, A is atomic mass (in grams per mole), Vc is volume of the unit cell (in cm³), and N_A is Avogadro's number, approximately 6.022 x 10²³.

Avogadro's number (NA)

6.022 x 10^23, the number of atoms in one mole of a substance, which is crucial for converting between atomic scale quantities and macroscopic amounts.

Miller Indices

A notation system in crystallography for planes in a crystal, represented as (h,k,l), which are the reciprocals of intercepts of the crystal plane with the axes, useful for describing crystal orientation.

Grain Size Number (G)

G = -log2(N) + 1, where N is the number of grains per unit area measured in a sample, indicating the average size of grains in crystalline materials.

Number of atoms formula

n = (Mass / Atomic mass) × NA, used to determine the number of atoms in a given mass of a substance based on its atomic mass.

Atomic radius of BCC

R = (sqrt(3) / 4) a, where a is the lattice parameter, determining the size of the spheres that are packed in the body-centered cubic (BCC) lattice.

Gibbs Phase Rule

P + F = C + 2, where P is the number of phases in a system, F is the degrees of freedom (variables that can be changed independently), and C is the number of components in the system, guiding the study of phase behavior.

Lever Rule for phase fraction (fL)

fL = (CS - C0) / (CS - CL), used to determine the phase fraction in two-phase systems, where CS is the composition of the phase in equilibrium, C0 is the overall composition, and CL is the composition of the other phase.

Stress formula (σ)

σ = F / A, where σ is the stress applied to a material, F is the applied force, and A is the cross-sectional area over which the force is applied.

Strain formula (ϵ)

ϵ = ΔL / L0, where ϵ is the strain experienced by a material, ΔL is the change in length, and L0 is the original length of the material.

What is Poisson's Ratio (ν)?

ν = - (εlateral / εlongitudinal), the ratio of lateral strain to longitudinal strain, indicating how much a material deforms in directions perpendicular to the applied load.