Chapter 3 Lattice, Basis, Unit Cells, and Crystal Structures

Packing Factor Equation

((# of atoms or cells)(Volume of each atom))/Volume of unit cell

Density Equation

((# of cells or atoms)(atomic mass))/((Volume of unit cell)(Avogadro constant))

Planes of a form or planes of a family

Crystallographic planes that all have the same characteristics, although their orientations are different. Denoted by { } braces.

Diffraction

The constructive interference, or reinforcement, of a beam of x-rays or electrons interacting with a material. The diffracted beam provides useful information concerning the structure of the material.

A crystal structure is characterized by the

lattice parameters of the unit cell

Interstitial sites, or holes between the normal atoms in a crystal structure, can be filled by______________.

other atoms and ions

The effect of crystallographic anisotropy may be masked in a___________. Why?

polycrystalline material, because of the random orientation of grains.

Atoms or ions may be arranged in solid materials with either a ______ or ____ order.

short-range or long-range

Amorphous materials, such as silicate glasses, metallic glasses, amorphous silicon, and many polymers, have only a

short-range order

If mechanical, magnetic, optical, and dielectric properties are the same in all directions what type of crystal is it?

the crystal is isotropic

unit cell

the smallest subdivision of the crystal structure that still describes the lattice

Long-range order (LRO)

A regular repetitive arrangement of atoms in a solid which extends over a very large distance.

Transmission electron microscopy (TEM)

A technique for imaging and analysis of microstructures using a high energy electron beam.

Rapid solidification

A technique used to cool metals and alloys very quickly.

When do amorphous materials form?

Amorphous materials form whenever the kinetics of a process involved in the fabrication of a material do not allow the atoms or ions to assume the equilibrium positions.

Metallic glass

Amorphous metals or alloys obtained using rapid solidification.

Grain

A small crystal in a polycrystalline material.

Miller-Bravais indices

A special shorthand notation to describe the crystallographic planes in hexagonal close-packed unit cells.

Unit cell

A subdivision of the lattice that still retains the overall characteristics of the entire lattice.

X-ray diffraction (XRD)

A technique for analysis of crystalline materials using a beam of x-rays.

Cubic site

An interstitial position that has a coordination number of eight. An atom or ion in the cubic site has eight nearest neighbor atoms or ions.

Tetrahedral site

An interstitial position that has a coordination number of four. An atom or ion in the tetrahedral site has four nearest neighbor atoms or ions.

Octahedral site

An interstitial position that has a coordination number of six. An atom or ion in the octahedral site has six nearest neighbor atoms or ions.

What plays an important role in strengthening materials, influencing the physical properties of materials, and controlling the processing of materials?

Atoms or ions located in interstitial sites

Polymorphism

Compounds exhibiting more than one type of crystal structure.

Directions of a form or directions of a family

Crystallographic directions that all have the same characteristics. Denoted by < > brackets.

Interstitial sites

Locations between the "normal" atoms or ions in a crystal into which another—usually different—atom or ion is placed. Typically, the size of this interstitial location is smaller than the atom or ion that is to be introduced.

Density

Mass per unit volume of a material, usually in units of g / cm^3.

Crystalline materials

Materials comprising one or many small crystals or grains.

Amorphous materials

Materials, including glasses, that have no long-range order or crystal structure.

What properties may differ when measured along different directions or planes within a crystal that is anisotropic?

Mechanical, magnetic, optical, and dielectric properties

Lattice points

Points that make up the lattice. The surroundings of each lattice point are identical.

Bravais lattice grouped in crystal systems

There are three Bravais lattices with a cubic symmetry. One distinguishes the simple/primitive cubic (sc), the body centered cubic (bcc) and the face centered cubic (fcc)lattice. There are two tetragonal Bravais lattices with a=b≠ca=b≠c and α=β=γ=90∘α=β=γ=90∘. One is primitive and the other body centered. There are four orthorhombic Bravais lattices with a≠b≠ca≠b≠c and α=β=γ=90∘α=β=γ=90∘: Primitive, body centered, face centered and base centered. When two sides are of equal length with an enclosed angle of 120∘120∘ the crystal has a hexagonal structure and thus a 6-fold rotary axis. As in the orthohombric structure, all edges are of unequal length. However, one of the three angles is ≠90∘≠90∘. The trigonal (or rhombohedral) lattice has three edges of equal length and three equal angles (≠90∘≠90∘). In the triclinic lattice, all edges and angles are unequal.

What can be used for imaging of microstructural features in materials at smaller length scales?

Transmission electron microscopy

Crystal structure

The arrangement of the atoms in a material into a regular repeatable lattice. A crystal structure is fully described by a lattice and a basis.

Crystallography

The formal study of the arrangements of atoms in solids.

The long-range periodicity in crystalline materials is described by the__________.

crystal structure

Miller indices

A shorthand notation to describe certain crystallographic directions and planes in a material. A negative number is represented by a bar over the number.

Packing factor

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

Lattice

A collection of points that divide space into smaller equally sized segments.

Kepler's conjecture

A conjecture made by Johannes Kepler in 1611 that stated that the maximum packing fraction with spheres of uniform size could not exceed pi/sqrt(18). In 1998, Thomas Hales proved this to be true.

Glass-ceramics

A family of materials typically derived from molten inorganic glasses and processed into crystalline materials with very fine grain size and improved mechanical properties.

Motif

A group of atoms affiliated with a lattice point (same as basis).

Basis

A group of atoms associated with a lattice point (same as motif).

Polycrystalline material

A material comprising many grains.

Electron diffraction

A method to determine the level of crystallinity at relatively small length scales. Usually conducted in a transmission electron microscope.

Defect

A microstructural feature representing a disruption in the perfect periodic arrangement of atoms ions in a crystalline material. This term is not used to convey the presence of a flaw in the material.

Blow-stretch forming

A process used to form plastic bottles.

The fourteen Bravais lattices

Cubic - Cubic system shows three types of Bravais lattices - Primitive, base centered and face centered. Tetragonal - Tetragonal system shows two types of Bravais lattices - Primitive, body centered. Orthorhombic - Orthorhombic system shows four types of Bravais lattices - Primitive, body centered, base centered and face centered. Hexagonal - Hexagonal system shows one type of Bravais lattice which is Primitive. Rhombohedral - Rhombohedral system shows one type of Bravais lattice which is Primitive. Monoclinic - Monoclinic system shows two types of Bravais lattices - Primitive, base centered. Triclinic - Triclinic system shows one type of Bravais lattice which is Primitive.

Crystal systems

Cubic, tetragonal, orthorhombic, hexagonal, monoclinic, rhombohedral and triclinic arrangements of points in space that lead to fourteen Bravais lattices and hundreds of crystal structures.

Atomic level defects

Defects such as vacancies, dislocations, etc., occurring over a length scale comparable to a few interatomic distances.

Close-packed directions

Directions in a crystal along which atoms are in contact.

Interplanar spacing

Distance between two adjacent parallel planes with the same Miller indices.

Anisotropic

Having different properties in different directions.

Isotropic

Having the same properties in all directions.

Liquid crystals (LCs)

Polymeric materials that are typically amorphous but can become partially crystalline when an external electric field is applied. The effect of the electric field is reversible. Such materials are used in liquid crystal displays.

Grain boundaries

Regions between grains of a polycrystalline material.

Glasses

Solid, non-crystalline materials (typically derived from the molten state) that have only short-range atomic order.

Close-packed (CP) structure

Structures showing a packing fraction of 0.74 (FCC and HCP).

Which structures achieve the closest packing by different stacking sequences of close-packed planes of atoms?

The FCC and HCP structures

Atomic radius

The apparent radius of an atom, typically calculated from the dimensions of the unit cell, using close-packed directions (depends upon coordination number).

Allotropy

The characteristic of an element being able to exist in more than one crystal structure, depending on temperature and pressure.

Diamond cubic (DC)

The crystal structure of carbon, silicon, and other covalently bonded materials.

Repeat distance

The distance from one lattice point to the adjacent lattice point along a direction.

Bravais lattices

The fourteen possible lattices that can be created in three dimensions using lattice points.

Packing fraction

The fraction of a direction (linear-packing fraction) or a plane (planar packing factor) that is actually covered by atoms or ions. When one atom is located at each lattice point, the linear packing fraction along a direction is the product of the linear density and twice the atomic radius.

Lattice parameters

The lengths of the sides of the unit cell and the angles between those sides. The lattice parameters describe the size and shape of the unit cell.

Planar density

The number of atoms per unit area whose centers lie on the plane.

Linear density

The number of lattice points per unit length along a direction.

Coordination number

The number of nearest neighbors to an atom in its atomic arrangement.

Stress-induced crystallization

The process of forming crystals by the application of an external stress. Typically, a significant fraction of many amorphous plastics can be crystallized in this fashion, making them stronger.

Crystallization

The process responsible for the formation of crystals, typically in an amorphous material.

Short-range order

The regular and predictable arrangement of the atoms over a short distance—usually one or two atom spacings.

Bragg's law

The relationship describing the angle at which a beam of x-rays of a particular wavelength diffracts from crystallographic planes of a given interplanar spacing.

Stacking sequence

The sequence in which close-packed planes are stacked. If the sequence is ABABAB, a hexagonal close-packed unit cell is produced; if the sequence is ABCABCABC, a face centered cubic structure is produced.

Sodium Chloride Structure

The sodium chloride structure, a FCC unit cell with two ions (Na+ and Cl-) per lattice point.

Basal plane

The special name given to the close-packed plane in hexagonal close-packed unit cells.

Tetrahedron

The structure produced when atoms are packed together with a four-fold coordination.

What can be used for the determination of the crystal structure of crystalline materials?

XRD and electron diffraction

The seven crystal systems

are known as cubic, tetragonal, orthorhombic, rhombohedral (also known as trigonal), hexagonal, monoclinic, and triclinic.

Crystalline materials, including metals and many ceramics, have_______________ order.

both long- and short- range

Many amorphous materials can be crystallized in a________ fashion.

controlled