The application of a set of rules leads to the assignment of the Miller Indicesa set of numbers which quantify the intercepts and thus may be used to uniquely identify the plane or surface. ... So the surface/plane illustrated is the plane of the cubic crystal.Miller indices, group of three numbers that indicates the orientation of a plane or set of parallel planes of atoms in a crystal. If each atom in the crystal is represented by a point and these points are connected by lines, the resulting lattice may be divided into a number of identical blocks, or unit cells; the intersecting edges of one of the unit cells defines a set of crystallographic axes, and the Miller indices are determined by the intersection of the plane with these axes. The reciprocals of these intercepts are computed, and fractions are cleared to give the three Miller indices lengths equal to the edges of a unit cell has Miller indices of (111). This scheme, devised by British mineralogist and crystallographer William Hallowes Miller, in 1839, has the advantage of eliminating all fractions from the notation for a plane. In the hexagonal system, which has four crystallographic axes, a similar scheme of four Bravais-Miller indices is used.Phase, in thermodynamics, chemically and physically uniform or homogeneous quantity of matter that can be separated mechanically from a nonhomogeneous mixture and that may consist of a single substance or of a mixture of substances. The three fundamental phases of matter are solid, liquid, and gas (vapour), but others are considered to exist, including crystalline, colloid, glassy, amorphous, and plasma phases. When a phase in one form is altered to another form, a phase change is said to have occurred.• A vector r passing from the origin to a lattice point can be written as: r = r1 a + r2 b + r3 c where, a, b, c → basic vectors and Miller Indices for Planes: Procedure 1. Identify the plane intercepts on the x, y and z-axes. 2. Specify intercepts in fractional coordinates. 3. Take the reciprocals of the fractional intercepts. In Materials Science it is important to have a notation system for atomic planes since these planes influence • Optical properties • Reactivity • Surface tension • Dislocations
We've now seen how crystallographic axes can be defined for the various crystal systems. Two important points to remember are that
The lengths of the crystallographic axes are controlled by the dimensions of the unit cell upon which the crystal is based.
The angles between the crystallographic axes are controlled by the shape of the unit cell.
We also noted last time that the relative lengths of the crystallographic axes control the angular relationships between crystal faces. This is true because crystal faces can only develop along lattice points. The relative lengths of the crystallographic axes are called axial ratios, our first topic of discussion.
Axial ratios are defined as the relative lengths of the crystallographic axes. They are normally taken as relative to the length of the b crystallographic axis. Thus, an axial ratio is defined as follows:
Axial Ratio = a/b : b/b : c/b
where a is the actual length of the a crystallographic axis, b, is the actual length of the b crystallographic axis, and c is the actual length of the c crystallographic axis.
For Triclinic, Monoclinic, and Orthorhombic crystals, where the lengths of the three axes are different, this reduces toModern crystallographers can use x-rays to determine the size of the unit cell, and thus can determine the absolute value of the crystallographic axes. For example, the mineral quartz is hexagonal, with the following unit cell dimensions as determined by x-ray crystallography:
a1 = a2 = a3 = 4.913Å
c = 5.405Å
where Å stands for Angstroms = 10-10 meter.
Thus the axial ratio for quartz is
1 : 1 : 1 : 5.405/4.913
1: 1 : 1 : 1.1001
which simply says that the c axis is 1.1001 times longer than the a axes.
For orthorhombic sulfur the unit cell dimensions as measured by x-rays are:
a = 10.47Å
b = 12.87Å
c = 24.39Å
Thus, the axial ratio for orthorhombic sulfur is:
10.47/12.87 : 12.87/12.87 : 24.39/12.87
0.813 : 1 : 1.903
Because crystal faces develop along lattice points, the angular relationship between faces must depend on the relative lengths of the axes. Long before x-rays were invented and absolute unit cell dimensions could be obtained, crystallographers were able to determine the axial ratios of minerals by determining the angles between crystal faces. So, for example, in 1896 the axial ratios of n name, or index faces of crystals and define directions within crystals.Crystal faces can be defined by their intercepts on the crystallographic axes. For non-hexagonal crystals, there are three cases