MEC 2309 – Properties of Engineering Materials I: Crystallographic Points, Directions, and Planes

Vocabulary and definition flashcards covering crystallographic points, directions, planes, Miller and Miller-Bravais indices, and material densities based on lecture MEC 2309.

Miller Index Notation

A shorthand notation used to describe certain crystallographic directions and planes in a material.

Lattice Vectors

The vectors denoted as $a$, $b$, and $c$ used to define the position of any point in a unit cell.

Point Coordinates

The distances on the $x$, $y$, and $z$ axes in terms of the lattice vectors used to identify positions in a unit cell.

Indices [12, 13, 12 ]

The coordinates for a point located at $\frac{a}{2}$ along the $x$-axis, $\frac{b}{3}$ along the $y$-axis, and $\frac{c}{2}$ on the $z$-axis.

Crystallographic Direction

Defined as successive motion parallel to each of the three axes necessary to move from the origin to another point.

Direction Notation

Lattice directions in direct space denoted by square brackets, e.g., $[uvw]$.

Family of Directions

Crystallographic directions that all have the same characteristics, although their sense may be different, denoted by caret brackets $<uvw>$.

Negative Index Representation

Represented by a bar over the number, such as $[\bar{1}10]$.

Family of Major Diagonals

Represented by the notation $<111>$ in a cubic system.

Family of Face Diagonals

Represented by the notation $<110>$ in a cubic system.

Direction Determination Rule

To find the indices, find the coordinates of the two ends of the line and subtract them (Head to Tail).

Indices of Planes

A surface defined by the reciprocal of its intercepts on the three crystal axes, reduced to the smallest three integers.

Reciprocal Space

A mathematical convenience where lattice planes are represented by the vector that is normal (perpendicular) to them.

Family of Planes

Lattice planes that all have the same characteristics and are symmetrically equivalent, denoted by curly brackets $\text{\{hkl\}}$.

(100) Plane

A plane that intersects the $X$-axis at $1$ and is parallel to the $Y$ and $Z$ axes.

{110} Family of Planes

In a cubic crystal, this includes six sets of planes: $(110)$, $(101)$, $(011)$, $(\bar{1}10)$, $(10\bar{1})$, and $(01\bar{1})$.

(hkl) and (̄h̄k̄l)

Represents the same plane, but signifies opposite sides of the parallel plane.

Cubic System Perpendicularity Rule

The condition where planes and directions having the same indices are perpendicular to each other, meaning $h=u$, $k=v$, and $l=w$.

Direction Parallel to a Plane

A direction $[uvw]$ that lies in the plane $(hkl)$, satisfying the condition $hu + kv + lw = 0$.

Miller–Bravais Coordinate System

A four-axis system utilized for crystals having hexagonal symmetry.

Basal Plane

A single plane in the hexagonal system containing the three axes $a_1$, $a_2$, and $a_3$ at $120^{\circ}$ angles.

Hexagonal Index Equation

The relationship where the first three indices must satisfy $h + k + i = 0$.

Miller-Bravais Plane Notation

Denoted by four indices as $(hkil)$, where $i = -(h + k)$.

Miller-Bravais Direction Notation

Denoted by four indices as $[uvtw]$, where $t = -(u+v)$.

[hkil]

A convention where the first three indices pertain to projections along the respective $a_1$, $a_2$, and $a_3$ axes.

(0001) Plane

A specific plane in the hexagonal system that intercepts only the $c$ axis.

Linear Density (LD)

The number of atoms per unit length whose centers lie on the direction vector for a specific crystallographic direction.

LD Formula

$$LD = \frac{\text{Number of atoms centered on direction vector}}{\text{Length of Direction Vector}}$$

Linear Density Units

Expressed in reciprocal length, such as $\text{nm}^{-1}$ or $\text{m}^{-1}$.

Planar Density (PD)

The number of atoms per unit area that are centered on a particular crystallographic plane.

PD Formula

$$PD = \frac{\text{Number of atoms centered on a plane}}{\text{Area of Plane}}$$

Planar Density Units

Expressed in reciprocal area, such such as $\text{nm}^{-2}$ or $\text{m}^{-2}$.

Interplanar Spacing

The spacing between planes in a crystal, denoted as $d_{hkl}$.

Cubic System Spacing Equation

$$\frac{1}{d_{hkl}^2} = \frac{h^2 + k^2 + l^2}{a^2}$$

Tetragonal System Spacing Equation

$$\frac{1}{d_{hkl}^2} = \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}$$

Hexagonal System Spacing Equation

$\frac{1}{d_{hkl}^2} = \frac{h^2}{a^2} + \frac{k^2}{a^2} + \frac{l^2}{c^2}$ or alternatively $\frac{1}{d_{hkl}^2} = \frac{h^2 + hk + k^2}{3a^2} + \frac{l^2}{c^2}$

Orthorhombic System Spacing Equation

$\frac{1}{d_{hkl}^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}$.

Anisotropic

Materials whose properties vary with single crystal orientation.

Isotropic

Materials whose properties are non-directional, often found in polycrystals with randomly oriented grains.

Amorphous

Materials such as glass in which atoms do not assemble into crystals.