Crystalline Structures and Material Properties

A comprehensive set of flashcards summarizing key vocabulary and definitions related to crystalline structures and material properties.

Crystalline

A solid material whose atoms, ions, or molecules are arranged in a highly ordered, repeating pattern in three dimensions.

Unit Cell

The smallest repeating structural unit of a crystal lattice that shows the full symmetry of the crystal structure.

Atomic Packing Factor (APF)

The fraction of volume in a crystal structure that is occupied by constituent particles (atoms). APF = (Volume of atoms in a unit cell) / (Total volume of the unit cell).

Coordination Number

The number of nearest neighbor atoms surrounding a central atom in a crystal structure.

BCC (Body-Centered Cubic)

A crystal structure where atoms are located at each of the eight corners and a single atom at the center of the cube.

FCC (Face-Centered Cubic)

A crystal structure where atoms are located at each of the eight corners and the centers of all six faces of the cube.

Polycrystalline

A material composed of many small crystalline regions (grains) with different orientations, separated by grain boundaries.

Vacancy

A crystalline defect where an atom is missing from a lattice site that would normally be occupied.

Interstitial

A crystalline defect where an atom occupies a site between the normal lattice positions.

Substitutional Impurity

An impurity atom that replaces a host atom in its lattice site.

Hume-Rothery rules

A set of guidelines for the solid solubility of one element in another, based on factors like atomic size, crystal structure, electronegativity, and valence.

Vacancy Diffusion

The mechanism of atomic migration where an atom moves into an adjacent vacant lattice site.

Interstitial Diffusion

The mechanism of atomic migration where an atom moves from one interstitial site to a neighboring one.

Edge Dislocation

A linear crystalline defect where an extra half-plane of atoms is inserted into the crystal structure.

Screw Dislocation

A linear crystalline defect where the atomic planes are connected in a helical ramp.

Grain Boundary

The interface between two grains (crystallites) in a polycrystalline material.

Phase Boundary

The boundary between two different phases within a material.

Twin Boundary

A planar defect where the lattice on one side of the boundary is a mirror reflection of the lattice on the other side.

Interdiffusion

Diffusion of different atomic species in opposite directions, typically in substitutional solid solutions.

Self-Diffusion

The migration of atoms in a pure material.

Stress (σ)

The internal force per unit area acting on a material.

Strain (ε)

The measure of the deformation of a material.

Elastic Deformation

A non-permanent deformation where the material returns to its original shape upon removal of the load.

Elastic Modulus (E)

A measure of a material's stiffness, defined as the slope of the stress-strain curve in the elastic region.

Poisson’s Ratio (ν)

The negative ratio of lateral strain to axial strain in elastic deformation.

Atomic Packing Factor (APF) Definition

The fraction of volume in a unit cell occupied by atoms.

APF Equation

APF = (Volume of atoms in unit cell) / (Total unit cell volume).

FCC APF

The atomic packing factor for Face-Centered Cubic is 0.74.

BCC APF

The atomic packing factor for Body-Centered Cubic is 0.68.

Atoms per Unit Cell (BCC)

There are 2 atoms per unit cell in a Body-Centered Cubic structure.

Coordination Number (BCC)

The coordination number for Body-Centered Cubic is 8.

Atoms per Unit Cell (FCC)

There are 4 atoms per unit cell in a Face-Centered Cubic structure.

Coordination Number (FCC)

The coordination number for Face-Centered Cubic is 12.

Miller Indices Purpose

A notation system to identify specific planes and directions within a crystal lattice.

6-STEP PROCESS for Planes (hkl)

Identify intercepts with x, y, z axes, take reciprocals, clear fractions, and write the results in parentheses (hkl). Negative numbers are written with a bar over them.

6-STEP PROCESS for Directions [uvw]

Determine coordinates of two points, subtract coordinates, clear fractions, and write in square brackets [uvw]. Negative numbers are written with a bar over them.

Theoretical Density (ρ) Equation

ρ = (n * A) / (V꜀ * Nₐ).

Density Variables

n = number of atoms per unit cell, A = atomic weight (g/mol), V꜀ = unit cell volume (cm³), Nₐ = Avogadro's number.

Number of Vacancies (Nv) Definition

The equilibrium number of vacancies in a solid.

Number of Vacancies Equation

Nv = N * exp(-Qv / (k * T)).

Number of Lattice Sites (N) Definition

The total number of atomic sites in a crystal, including occupied sites and vacancies.

Diffusion Flux (J) Definition

The mass (or number of atoms) moving through a unit area per unit time.

Flux Units

Units for flux can be atoms/(m²·s) or kg/(m²·s).

Fick’s First Law Definition

The flux of atoms is proportional to the concentration gradient.

Fick’s First Law Equation

J = -D * (dC/dx).

Fick’s Second Law Definition

Describes how the concentration of a diffusing species changes with time at a given point.

Fick’s Second Law Equation (1D)

∂C/∂t = D * (∂²C/∂x²).

Diffusion Coefficient (D) Definition

A measure of the rate of atomic diffusion.

Diffusion Coefficient Equation (Arrhenius)

D = D₀ * exp(-Qd / (R * T)).

Mechanical Properties Stress (σ) Definition

Force applied per unit area.

Mechanical Properties Stress Equation

σ = F / A₀.

Stress Variables

F = Applied Force (N), A₀ = Original Cross-Sectional Area (m²).

Stress Units

Units include Pascals (Pa), Megapascals (MPa), and Gigapascals (GPa).

Strain (ε) Definition

The measure of deformation representing the displacement between particles in a material.

Strain (ε) Equation

ε = (lᵢ - l₀) / l₀.

Strain Variables

lᵢ = instantaneous length, l₀ = original length.

Strain Units

Dimensionless (often expressed as m/m or in/in).

Hooke’s Law Definition

In the elastic region, stress is proportional to strain.

Hooke’s Law Equation

σ = E * ε.

Poisson’s Ratio (ν) Definition

The ratio of transverse contraction strain to axial extension strain in the elastic region.

Poisson’s Ratio (ν) Equation

ν = - (εₓ / ε₂) = - (εy / εz).

Poisson’s Ratio Units

Dimensionless; typical values range from 0.2 to 0.5 for most engineering materials.