The Structure of Crystalline Solids

Vocabulary flashcards reviewing core definitions, crystal structures, unit cell formulas, crystallographic coordinates, indices, and characterization techniques from Chapter 3 of Materials Science & Engineering.

Crystalline Materials

Materials in which atoms are arranged in periodic, three-dimensional arrays possessing long-range atomic order, typical of metals, many ceramics, and some polymers.

Non-crystalline Materials

Materials in which atoms have no periodic, long-range arrangement, also referred to as amorphous materials; typically formed in complex structures or upon rapid cooling.

Coordination Number

The number of nearest-neighbor, or touching, atoms surrounding a specific central atom in a crystal lattice structure.

Atomic Packing Factor (APF)

The fraction of a unit cell's volume occupied by hard-sphere atoms, calculated as $\text{APF} = \frac{\text{volume of atoms in unit cell}}{\text{volume of unit cell}}$.

Hard-Sphere Assumption

The structural modeling assumption that atoms are represented as touching, incompressible spheres of fixed radius $R$.

Simple Cubic (SC) Crystal Structure

A rare metallic crystal structure with atom centers located at the 8 corners of a cube, featuring a coordination number of 6, 1 atom per unit cell, an APF of 0.52, and close-packed directions along cube edges (exemplified by polonium, Po).

Body-Centered Cubic (BCC) Crystal Structure

A metallic crystal structure with atoms located at eight cube corners and one atom at the cube center, featuring 2 atoms per unit cell, a coordination number of 8, an APF of 0.68, and close-packed directions along body diagonals.

Face-Centered Cubic (FCC) Crystal Structure

A metallic crystal structure with atoms at eight cube corners and one atom at the center of each of the six faces, featuring 4 atoms per unit cell, a coordination number of 12, an APF of 0.74, and close-packed directions along face diagonals.

FCC Plane Stacking Sequence

The $ABCABC...$ repeating layer sequence of close-packed atomic planes, where the third layer sits in a set of hollows distinct from both the first ($A$) and second ($B$) layers.

Hexagonal Close-Packed (HCP) Crystal Structure

A metallic crystal structure featuring an $ABAB...$ plane stacking sequence, 6 atoms per unit cell, a coordination number of 12, an APF of 0.74, and an ideal $c/a$ ratio of 1.633.

Theoretical Density (\rho)

The mass per unit volume of a crystal calculated via $\rho = \frac{n A}{V_c N_A}$, where $n$ is atoms per unit cell, $A$ is atomic weight, $V_c$ is unit cell volume, and $N_A$ is Avogadro's number ($6.022 \times 10^{23}\,\text{atoms/mol}$).

Material Class Density Spectrum

The general density ordering among material classes, given by $\rho(\text{metals}) > \rho(\text{ceramics}) > \rho(\text{polymers})$, driven by metallic packing efficiency, atomic weights, and bonding types.

Single Crystal

A crystalline material in which the periodic arrangement of unit cells extends without interruption across the entire specimen.

Polycrystalline Material

A crystalline material composed of many small individual single crystals called grains, with grain sizes typically ranging from $1\,\text{nm}$ to $2\,\text{cm}$.

Grain Boundary

The boundary region separating adjacent single-crystal grains of differing crystallographic orientations in a polycrystalline material.

Anisotropy

The condition in which a material's measured property value depends on the crystallographic direction along which the measurement is made (e.g., single-crystal BCC iron elastic modulus $E_{[100]} = 125\,\text{GPa}$ vs. $E_{[111]} = 273\,\text{GPa}$).

Isotropy

The condition in which material properties are identical in all measurement directions, typically observed in polycrystals with randomly oriented grains.

Texture

A preferential crystallographic orientation of grains within a polycrystalline material, leading to anisotropic macroscopic properties.

Polymorphism (Allotropy)

The phenomenon where a solid material can exist in two or more distinct crystal structures depending on temperature and pressure (e.g., iron shifting between BCC \alpha-Fe, FCC \gamma-Fe, and BCC \delta-Fe).

Point Coordinates

A lattice position specified as fractional distances along the coordinate axes in terms of unit cell edge lengths $a$, $b$, and $c$, written without commas.

Family of Directions (\langle uvw \rangle)

A set of crystallographically equivalent directions that share identical atomic spacing and linear packing density.

Common Crystallographic Directions

The primary low-index direction vectors in a cubic lattice, corresponding to the cube edge [100], face diagonal [110], and body diagonal [111].

Linear Density (LD)

The number of atoms centered per unit length along a specific crystallographic direction vector, defined as $\text{LD} = \frac{\text{number of atoms centered on direction vector}}{\text{length of direction vector}}$.

Miller Indices ((hkl))

A set of three integers enclosed in parentheses designating the orientation of a crystallographic plane, determined by taking the reciprocals of the plane's fractional intercepts with the unit cell axes.

Family of Planes ({hkl})

A collection of crystallographically equivalent planes that share identical atomic arrangements and planar density.

Miller-Bravais Indices ((hkil))

A four-index coordinate system used to specify crystallographic planes in hexagonal lattice systems, where the index $i$ is determined by $i = -(h + k)$.

Planar Density (PD)

The number of atoms centered per unit area on a specific crystallographic plane, calculated as $\text{PD} = \frac{\text{number of atoms centered on plane}}{\text{area of plane}}$.

Bragg's Law

The fundamental relationship governing constructive X-ray diffraction from crystal planes, expressed as $n\lambda = 2d\sin(\theta)$.

Interplanar Spacing (d)

The perpendicular distance between adjacent parallel crystallographic planes of atoms in a crystal lattice.

X-Ray Diffraction Pattern for Polycrystalline \alpha-Iron

An experimental intensity vs. $2\theta$ plot showing distinct diffraction peaks corresponding to reflection planes such as (110), (200), and (211) satisfying Bragg's law.