Structure & Properties - Engineering Materials - Mechanical Engineering

Introduction

Solids in nature occur principally in two structural forms: crystalline and non-crystalline (amorphous). These two classes differ fundamentally in the degree of atomic order and, consequently, in many physical and mechanical properties. Crystalline solids exhibit a periodic, long-range arrangement of atoms that repeats throughout the material, whereas amorphous solids lack such long-range periodicity.

Crystals

Crystalline solids possess long-range order: the arrangement of atoms at one location in the crystal is, apart from defects, identical to that at any other equivalent location. The science that describes the geometric arrangement of atoms, the symmetry and the periodicity of crystals is called crystallography. Many physical and chemical properties - for example, elastic constants, electrical conductivity, optical behaviour and cleavage - depend on crystal structure; understanding a material's crystal structure is therefore essential for predicting and exploiting its properties.

Space Lattice and Basis

Space lattice (or lattice) is an infinite array of points in three dimensions in which every point has an environment identical to that of every other point. It provides the geometric scaffold with reference to which a crystal structure is described.

Basis is the group of one or more atoms associated with each lattice point. A crystal structure is obtained by placing the basis at every lattice point in the space lattice. In symbolic form:

• Space lattice + basis → crystal structure

Unit Cell

The unit cell is the smallest repeating unit of the lattice that, by translation, reconstructs the entire lattice. Its geometry and the positions of atoms inside it define the crystal structure.

The unit cell is described by three edge lengths a, b, c (the lattice constants) measured along mutually defined axes and by the three interfacial angles α (between b and c), β (between a and c) and γ (between a and b). Knowing these parameters gives the shape and size of the unit cell.

Primitive Cell

A primitive cell is a smallest-volume cell that, when translated through all lattice vectors, fills space without overlap or gaps. Every primitive cell contains exactly one lattice point. A primitive cell need not be the conventional unit cell used for convenience.

Types of Crystals (Microstructure)

Most engineering materials are not single crystals but aggregates of many small crystals or grains; this aggregate is called the polycrystalline microstructure. The orientation of neighbouring grains is generally different, and the small regions where orientations change are grain boundaries.

• Single crystals: A specimen that is a single grain throughout. Single crystals can be grown artificially (for example, from the melt or vapour) and show anisotropic properties - properties vary with direction. Examples: natural diamond, single crystals of NaCl, large single crystals grown for electronics or optics.
• Polycrystalline materials: Aggregates of many crystals separated by grain boundaries. Most engineering metals and ceramics are polycrystalline.
• Isotropy and anisotropy: A polycrystalline material with randomly oriented grains typically exhibits nearly isotropic macroscopic properties, whereas a single crystal exhibits anisotropic properties.
• Effect of grain size: Grain boundaries impede dislocation motion and thereby strengthen materials (Hall-Petch effect). Finer grains generally increase yield strength but may reduce ductility.

Crystal Families and Crystal Systems

Crystals are classified by the geometry of their unit cell into crystal systems. The seven crystal systems (often encountered in materials science) are:

• Triclinic (anorthic)
• Monoclinic
• Orthorhombic
• Tetragonal
• Trigonal (rhombohedral)
• Hexagonal
• Cubic (isometric)

Within these systems there are 14 distinct Bravais lattices that represent all possible lattice translations in three dimensions. Crystallographers use a conventional notation to indicate centring of lattices:

• P → primitive
• B or A → base-centred
• C → base-centred (alternate notation)
• I → body-centred
• F → face-centred

Atomic Radius

Atomic radius in a crystal is commonly taken as one half of the distance between nearest neighbour atoms. It is denoted by r. The relation between the atomic radius and the lattice constant a depends on the lattice geometry (simple cubic, BCC, FCC, etc.).

Atomic Packing Factor (APF)

Atomic Packing Factor (APF), also called relative density of packing, is the fraction of unit cell volume that is occupied by constituent atoms (treated as hard spheres). It is defined as:

• APF = (total volume of atoms in one unit cell) / (volume of the unit cell)
• If n is the effective number of atoms per cell, r the atomic radius and Vcell the unit cell volume, APF = n × (4/3)πr3 / Vcell.

Density of a Crystal

The mass density of a crystalline solid can be calculated from the unit cell:

• The mass of atoms in one unit cell is n × (atomic mass) / N_A, where n is the number of atoms per cell and N_A is Avogadro's number.
• The cell volume is Vcell (for cubic cells Vcell = a3).
• Therefore the density ρ is given by:

In symbolic form for a cubic cell:

• ρ = nM / (a3 N_A), where M is the atomic mass and a is the cube edge.

Linear and Planar Density

• Linear density: Number of atoms per unit length along a specified crystallographic direction (important for slip and deformation along that direction).
• Planar density: Number of atoms per unit area on a specified crystallographic plane (planes with low planar density often play roles in fracture and corrosion).

Directions, Lattice Planes and Miller Indices

Properties such as elastic response, plastic slip and diffusion can be strongly anisotropic in crystals and are described using crystallographic directions and planes.

• Crystal directions: A direction is specified by a vector connecting lattice points; crystallographic directions are denoted by square brackets [u v w].
• Crystallographic planes: A set of parallel lattice planes is specified by Miller indices (h k l) - integers that are inversely proportional to the intercepts of the plane with the crystallographic axes.
• Miller indices procedure: Determine intercepts of the plane with the unit cell axes in terms of a, b, c; take reciprocals of these intercepts; clear fractions to obtain smallest integer set (h k l); enclose in parentheses for a plane and in square brackets for a direction.
• Interplanar spacing (cubic): The distance d between adjacent (h k l) planes in a cubic crystal with lattice constant a is given by:

Coordination Number

Coordination number is the number of nearest neighbours surrounding a given atom in the crystal. Typical values for common cubic structures are:

• Simple cubic (SC): coordination number = 6
• Body-centred cubic (BCC): coordination number = 8
• Face-centred cubic (FCC): coordination number = 12
• Close-packed structures (FCC and hexagonal close packed) have coordination number = 12

For example, a carbon atom in diamond has coordination number 4 (tetrahedral bonding); this lower coordination relative to close-packed metals results in a less efficient packing of atoms.

Defects or Imperfections in Crystals

Real crystals contain defects that strongly influence mechanical, electrical and diffusion properties. Defects are classified by their geometry:

• Point defects- lattice errors at or around individual lattice sites:
  • Vacancies - missing atoms from lattice sites.
  • Interstitials - atoms occupying normally unoccupied interstitial positions.
  • Substitutional defects - foreign atoms replace host atoms at lattice sites.
  • Frenkel defect - an atom (usually a small cation) moves from its normal site to an interstitial site leaving behind a vacancy (vacancy-interstitial pair).
  • Schottky defect - paired vacancies of cations and anions maintain overall electrical neutrality (common in ionic crystals).

Key:

• a = vacancy (Schottky defect)
• b = interstitial
• c = vacancy-interstitial pair (Frenkel defect)
• d = split interstitial

• Line defects- one-dimensional defects such as dislocations. Dislocations are the carriers of plastic deformation:
  • Edge dislocation - extra half plane of atoms inserted, characterised by a Burgers vector perpendicular to the dislocation line.
  • Screw dislocation - the lattice planes form a helical ramp around the dislocation line; Burgers vector parallel to the dislocation line.
  • Dislocation climb - motion out of the slip plane assisted by diffusion (a thermally activated process).

• Surface and grain boundary imperfections - two-dimensional defects where atomic arrangement is disrupted, e.g., free surfaces, internal boundaries between grains, twin boundaries.
• Volume imperfections - three-dimensional defects such as voids, inclusions and second-phase particles.

Simple Cubic Cell (SCC)

In the simple cubic lattice there is one atom at each corner of the cube; the atoms at corners are shared among eight adjacent unit cells, so the effective number of atoms per unit cell is 1.

• Number of atoms per unit cell, n = 1.
• Relationship between lattice parameter and atomic radius: a = 2r.
• Atomic Packing Factor: APF = π/6 ≈ 0.524 (≈ 52%).
• Fractional void space = 100% - 52% ≈ 48%.

Body-Centred Cubic (BCC) Structure

The BCC unit cell has atoms at the eight corners and a single atom at the body centre. The body-centred atom is wholly within the cell while each corner atom is shared by eight cells.

• Number of atoms per unit cell, n = 2.
• Relation between lattice constant and atomic radius: along the body diagonal 4r = √3 a, so a = 4r / √3.
• Atomic Packing Factor: APF = √3π / 8 ≈ 0.68 (≈ 68%).
• Fractional void space = 100% - 68% ≈ 32%.

Face-Centred Cubic (FCC) and Close-Packed Structures

In an FCC unit cell, atoms are located at all eight corners and at the centres of the six faces. Each face-centred atom is shared between two adjacent cells, giving an effective count of four atoms per unit cell.

• Number of atoms per unit cell, n = 4.
• Relation between lattice constant and atomic radius: atoms touch along the face diagonal, so 4r = a√2 and hence a = 2√2 r.
• Atomic Packing Factor: APF = π/(3√2) ≈ 0.74048 (≈ 74%).
• Fractional void space = 100% - 74% ≈ 26%.
• Coordination number = 12 (each atom has 12 nearest neighbours).
• FCC and hexagonal close-packed (HCP) are the two close-packed arrangements of equal spheres; both have the same APF = 0.74 but different unit cell geometries.

Metals that commonly adopt the FCC structure include aluminium, copper, silver and gold; many transition metals such as iron may exist in BCC or FCC forms depending on temperature (allotropy).

Hexagonal Close Packed (HCP)

The HCP structure is another close-packed arrangement with coordination number 12 and APF ≈ 0.74. Its conventional unit cell is hexagonal and described by parameters a and c with an ideal c/a ratio ≈ 1.633 for closest packing. Typical HCP metals include magnesium, titanium and zinc.

Practical Consequences and Applications

• Crystal structure affects mechanical behaviour: slip systems (combination of slip plane and slip direction) determine ductility; FCC metals have multiple closely packed slip systems and are generally more ductile than BCC metals at low temperature.
• Defects control strength and diffusion: dislocations enable plasticity; point defects and grain boundaries control diffusion rates and influence high-temperature creep.
• Density and packing factor calculations are used to relate measured lattice parameters (from X-ray diffraction) to atomic weights and to identify phases.
• Understanding crystallography is essential for materials selection and processing (forging, rolling, heat treatment, alloying) as well as for microstructure design in engineering components.

Worked Relationships (useful formulae)

• Density (cubic): ρ = nM / (a3 N_A).
• APF: APF = [n × (4/3)π r3] / Vcell.
• SCC: n = 1, a = 2r, APF ≈ 0.524.
• BCC: n = 2, a = 4r / √3, APF ≈ 0.680.
• FCC: n = 4, a = 2√2 r, APF ≈ 0.740.
• Interplanar spacing (cubic): d(hkl) = a / √(h2 + k2 + l2).

Summary

A clear understanding of crystal structures - lattices, unit cells, coordination, packing factors and defects - is foundational for predicting and controlling material behaviour. The common cubic lattices (SC, BCC and FCC), close-packed structures (FCC and HCP), and the types of defects govern density, mechanical strength, ductility and diffusion. These concepts are applied in phase identification, alloy design, heat-treatment practice and failure analysis in engineering.