The positions of the atoms in a unit cell describe completely the crystal structure. It is useful to learn now certain rules and notations to describe geometry in and around a unit cell, such as describing lattice positions, lattice directions and planes, because certain planes and directions in metallic crystal structures play very important role in understanding the plastic deformation, the hardening reaction, and other aspects of the behaviour of metals.
These notations act as a vocabulary that allows us to communicate efficiently about crystalline structure. Miller developed a concise and convenient quantitative description given by sets of numbers that identify given planes of atoms, and directions. These numbers are called Miller indices, which are universally used.
Lattice Points:
The coordinates of a lattice point, or of an atom centre within a unit cell are specified in terms of fractions (or multiples) of the lattice parameters a, b and c illustrated in Fig. 1.45 (a). For example, the body-centred position in the unit cell projects midway along each of the three unit cell edges and is designated by the ½, ½ , ½ position. A BCC unit cell has two lattice points, one at ½, ½ , ½ and the other at 0, 0, 0. It is unnecessary to specify 1, 0, 0; 1, 1,0; etc. as the lattice points, since these are equivalent to 0 0, 0.
One important characteristic of the crystalline structure is that a given lattice position in a given unit cell is structurally equivalent to the same position in any other unit cell of the same structure, as illustrated in Fig. 1.45 (b).
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If any lattice point can be connected to another lattice point by a linear, integral combination of vectors defined by the unit cell edges, then the two lattice points are equivalent. Table 1.8 lists unique lattice points in 14 Bravais lattices. Lattice points need not coincide with all atom centres.
Miller Indices of a Lattice Plane:
This shorthand means of denoting various atomic planes in a crystal is based on the equation for plane in three dimensions:
where, A, B and C are numerical intercepts of this plane along three axes x, y, and z respectively, when expressed in terms of the lattice parameter units. These intercepts, when converted to a set of small integers, h = m/A, k = m/B, and l = m/c are called the Miller indices of the plane, and the plane is denoted as (h k I) enclosed in parentheses as a standard notation indicating the plane.
Thus, Miller indices are based on the intercepts of a plane with the three crystal axes, each intercept with an axis being measured in terms of unit cell dimension (a, b, or c) along that axis, and not in terms of centimeters or angstrom units. The following four step procedure is used to determine the indices of a given plane. Let us consider the plane in Fig. 1.46 (any one of the figs)
Step I:
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Find the intercepts of the plane on the three crystal axes in terms of axial lengths from the origin (the intercepts are measured as fraction, or multiples of the fundamental vectors-unit distances). The above plane cuts the positive side of x-axis at 2 units (of a), 3 units (of b) on positive y-axis, and 1 unit (of c) on the positive z-axis.
Thus, we write as:
(If the plane passes through the origin, the origin of the coordinate system must be changed).
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Step II:
Take the reciprocals of these intercepts, such as-
(The reciprocal is taken as 0, for the intercept of ∝ for a plane parallel to an axis).
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Step III:
Change the reciprocals into smallest integers having the same ratio.
(The factor that results in changing the reciprocals to integers may be indicated outside the brackets, but is usually omitted).
Step IV:
Enclose in parentheses as-
(326)
Thus, the planes in Fig. 1.46 are specified by the Miller indices (326). These indices give only geometrical information about the crystal planes, and describe nothing about the distribution, or kinds of atoms in the plane.
The Miller indices (326) describe not merely a single crystal plane of Fig. 1.46, but entire array of planes parallel to the plane of Fig. 1.46.
A parallel plane PQR in Fig. 1.46 (b) having intercepts on X, Y and Z axis as 4, 6, and 2 respectively can be taken as an example:
Thus, all parallel and equally spaced planes have the same indices. If the set of Miller indices is the smallest possible then the planes it represents have identical atomic packing.
Problem 1:
How many planes (326) are there between origin and the first plane with intercepts 2, 3 and 1 as drawn in Fig. 1.46 (b)?
Solution:
Five.
The first plane is through the origin in the parallel set of (326) planes, where 3, 2 and 6 represent three integers of Miller indices. The first plane from the origin has the (fractional) intercepts 1/3, 1/2 and 1/6 on x, y and z axes, measured in units of the lattice constants.
The other four planes have intercepts such as:
Negative intercepts of axes result in Miller indices with a bar above the particular index. For example (1̅00) indicates that the intercept along x-axis is negative. This set of indices (1̅00), is read ‘minus one, zero, zero’, Fig. 1.47 illustrates some common planes in cubic crystals with Miller indices.
The space lattice of atoms is considered to be infinite in all three directions, and since the origin can be made to coincide with any one of the atoms, there is no physical difference between plane such as (100) and the (1̅00) plane the latter happens to be on the other side of the arbitrary origin as illustrated in Fig. 1.47 (b). Suppose we had taken origin at P (instead of at 0), the Miller indices of the two planes (1̅00) would have become (100).
Analysis of planes passing through the origin is fruitless as it has indeterminate indices. However, its Miller indices can be obtained either by shifting the origin to an equivalent lattice point, or by determining the Miller indices of a parallel equivalent plane.
For example, the plane p in Fig. 1.48 passes through the origin. Thus, to obtain Miller indices of this plane, a parallel plane is drawn that makes the smallest integral intercepts on the coordinate axes. The intercepts by this new plane on axes are 3, 2 and ∞.
Thus, the Miller indices are:
On the same basis, (010) plane, Fig. 1.47 (c), is equivalent to (100) plane because of choosing the origin of the coordinate in a different way. Thus, crystallographically equivalent, non-parallel planes are termed planes of a form, or simply a family of planes, and are denoted by the curly brackets, { }. Thus, {100} stands for the family of planes (100), (1̅00), (010), (001), (001̅), (01̅0). Only in cubic system, all planes having same set of Miller indices have same type of atomic packing, and belong to same family.
Interplanar Spacing:
Interplanar spacing between neighbouring planes of Miller indices (hkl), dhkl is defined as the spacing between the first such plane and a parallel plane passing through the origin. It is clear that d100 [the spacing of (100)- type planes] is simply, a, the lattice constant. Using Pythagoras’ theorem, d110 = a/√2 and d111 = a√3. In cubic crystals, the general rule to get interplanar spacing is-
The interplanar spacing is the same for all members of a family of planes (because the indices are squared and then summed in the denominator).
For tetragonal system,
Miller Indices of a Direction:
Directions in a crystal are specified in a shorthand vector notation. Let a vector r represents a direction in a crystal. The Miller indices are simply the vector components of the direction resolved along each of the coordinate axes and reduced to smallest integers, i.e., the components of the vector along the three axes are determined as multiples of the unit vector corresponding to each direction.
For example, the vector r passing through the origin, 0, to a lattice point, can be given in terms of the fundamental translation vectors X, Y and Z which make the crystal axes:
where, r1, r2 and r3 are integers. Fig. 1.49 (a) illustrates that vector, r has a component of 1 along x-axis, 1 along y-axis and 1 along z-axis. It is clear that absolute magnitudes of the components along X, Y and Z are not the same, but are unity when expressed as multiples of the corresponding unit vectors. The Miller indices of this direction is [111], where square bracket [ ] stands for a direction. A negative component is denoted as a bar on the top of the appropriate number.
Fig. 1.49 (b) shows diagonal q and it has direction [111] (one unit along x-axis, one unit along y-axis, and one unit along z-axis). If here the components along the axes had been 2 each, then the Miller indices is 2 [111] as the Miller indices are usually specified as the smallest possible integers.
The magnitude of the vector r gives the magnitude of that crystal direction. For example, the crystal directions [230], [460] and [1 (1 1/2) 0], all have the same direction, but different magnitudes. Thus, the difference in magnitude for same three directions are given as-
If a direction is not through the origin, then either shift the origin so that the direction vector passes through it (as the choice of the origin is arbitrary), or draw a vector parallel to given direction vector passing through the origin.
Then get its components along the three axes to put in a square bracket to obtain the Miller indices of the direction. For example, for getting the direction of the body diagonal, p of Fig. 1.49 (b), draw a vector t starting from origin [Fig. 1.49 (c)] parallel to vector p. The components along three axes are 1, -1, 1, and thus, the direction is given by [11̅1], or start from e in Fig. 1.49 (b) to move along three axes to reach point in.
Thus, move -1 to origin along y-axis, 1 along x-axis and 1 along z-axis to reach point, m. The direction is [11̅1]). The four body diagonals of the cube have indices [111], [1̅11], [11̅1], [11̅1].
These directions are equivalent but have different indices because of the way we put the origin, i.e., these directions are same if we shift the origin, or redefine the coordinate system. These groups of equivalent directions are called directions of a form or a family of directions.
A family of directions is obtained by all possible combinations of the indices, both positive and negative. The family of such equivalent directions is represented by < 111 >, where the pointed brackets of the type < > denote the entire family.
The body diagonals in the cubic system are:
In the same way, the family of cube edges is represented by <100> which thus includes [100], [1̅00], [01̅0], [001], [001̅], [010].
Miller-Bravais Indices:
For hexagonal crystal, two equivalent systems may be used:
1. Miller Indices:
Consider a hexagonal (parallelopiped) unit cell as illustrated in Fig. 1.51 (darker lines). Three axes have been shown as used for Miller indices so far. Three equivalent atomic planes have Miller indices (100), (010), (11̅0) (derived) based on normal method, i.e., Miller indices consists of three integers.
It is desirable and convenient for crystallo- graphically equivalent planes in the hexagonal system to have the same set of indices (this is not true in Miller indices as seen above), and thus, the four indices system to denote a plane (hkil) is used and is called Miller-Bravais indices.
2. Miller Bravais Indices:
Although three noncoplanar vectors are sufficient to describe a plane, or a direction in a crystal, the four indices notation, hkil, called Miller-Bravais indices in used for hexagonal crystals specially.
Fig. 1.52 illustrates a hexagonal prism showing that the three axes a1,a2, and a3 corresponding to three close packed directions, are coplanar, lie in the basal plane of the crystal, making 120° angles with each other. The fourth axis is perpendicular to the basal plane, and is called ‘c’-axis. Here a3 axis is redundant.
Indices of a Plane:
The four digits are enclosed in parentheses (hkil). These indices are the reciprocals of the intercepts on the a1, a2, a3 and c axes respectively and after these reciprocals have been divided by the largest common factor. As only three non-coplanar axes are necessary to specify a plane in space, the fourth indices cannot be independent, i.e., additional condition to be satisfied is-
h + k = – i
Let us determine the Miller-Bravais indices of some of the close- packed planes in the hexagonal crystal lattice. The top surface of the unit cell corresponds to the basal plane of the crystal, because it is parallel to it. The top surface is parallel to the axes, a1, a2 and a3 and thus, it intercepts them at infinity. Its intercept on c-axis is 1. Thus, (0001) is the Miller-Bravais indices of the top surface as well as the basal plane.
The six vertical surfaces of the unit cells are called prismatic planes. The front face makes an intercept of 1 at a1 axis, ∞ at a2 (as it is parallel to it), -1 at a3 and ∞ at c-axis (being parallel). Thus, the indices are (101̅0). Fig. 1.53 (c) illustrates a plane which intercepts with a1 at ∞ a3 at -1 and c at 1/2 . Thus, the Miller-Bravais indices are (101̅2). Plane P in Fig. 1.53 (b) has intercepts, a1 = 1, a2 = 1, a3 = -1/2, c = 1/2. Thus indices are (112̅1).
As by Miller Bravais system, the equivalent planes (100), (010) and (11̅0) are transformed to (101̅0), (011̅0) and (11̅00), and have same set of indices, thus, are members of the {1100} family of planes.
Miller Bravais Indices of a Direction:
The indices of the directions are also designated by four integers but enclosed in square brackets as a convention. The vector components of the direction is resolved along each of the four axes and reduced to smallest integers with the restriction that the third digit must always equal the negative sum of the first two digits, i.e., h + k = – i.
Let us find indices of axis a1, Fig. 1.53(d). It has no component along the c-axis, thus, the fourth digit of the Miller Bravais indices becomes zero. As the direction is parallel to a1-axis, and when resolved along a2 and a3-axis, the components are -1 and -1 on them.
As the value of the first digit corresponding to at axis can be obtained by the restriction rule:
h + k = -1
or h – 1 = -(-1)
or h = 2
Thus, the Miller indices of a1-axis direction is [21̅ 1̅0]. The corresponding indices of a2 and a3 are [1̅21̅0] and [1̅ 1̅20]. The direction of the diagonal axes is illustrated in Fig. 1.53 (d) by vector r. It is equal to vector sum of a unit vector lying on a1, and -1 on a3. Thus, the indices of this diagonal axis are denoted by [101̅0].
Planar Density:
Crystals by nature are anisotropic, i.e., properties are different in various directions. This is because of the different densities of atoms in various planes as well as in directions. For calculating the planar density of atoms, the area occupied by the atoms on a plane is calculated. But as a rule, the plane (or a line in case of linear density) must pass through the centre of an atom, otherwise, the atom shall not be counted. Thus, planar density is the number of atoms per unit area whose centres lie on the plane.
Fig. 1.54 (a) illustrates a face of FCC unit cell. In area a2, counting can be done of atoms- One atom for the centre and one- quarter atom for each corner atom on this face. Thus,
Linear Density:
The line must pass through the centre of the atom for it to be counted. The length of the line occupied by the atom becomes equal to the diameter of the atom. Fig. 1.54 (b) illustrates a face of FCC unit cell. The linear density in the direction of line AB, [110] is to be calculated. The coiner atoms occupy one half diameters each, and centre atom one full diameter. Thus,
In BCC crystal, the body diagonal in the [111] direction has-
Repeat Distance:
It is the distance between lattice points (centres of the atoms) along the direction. For example, [110] direction in FCC unit cell, Fig. 1.54 (a). Starting from 0, 0, 0, the next lattice point is at the centre of a face, or at 1/2, 1/2, 0 site. The distance between the lattice points is therefore one- half of the face diagonal, or ½ √2a. Copper has a = 0.36151 nm, and thus the repeat distance in copper
Linear density is number of repeat distances (no. of lattice points) per unit length along the direction. In copper, face diagonal
Packing Fraction of a Direction:
It is the fraction actually occupied by the atoms. Copper has one atom at each lattice point. Thus, packing fraction is equal to product of the linear density and twice the atomic radius. For [110] direction, as r = √2 a/4 = 0.12781 nm,
Packing fraction = Linear density x 2 r
= 3.91 x 2 x 0.12781 = 1.0
In FCC, [110] is the close packed direction and atoms touch each other.
Problem 2:
Simple cubic polonium has lattice parameter, 0.334 nm. Calculate the planar density and planar packing fraction for (010) and (020) planes.
Solution:
Both planes are shown. On (010), atoms are centred at each corner of the cube face. Total atoms on each face is one (1/4 + 1/4 + 1/4 + 1/4)
As no atoms are centred on (020) planes, the planar density as well as planar packing fraction are both zero. Thus, (010) and (020) are not equivalent planes.
Planar Density on Some Other Planes:
I. BCC Crystal Structure:
(i) On (100) plane:
The plane (100), the face OPQR has 1/4 + 1/4 + 1/4 + 1/4 = 1 atom in an area of a2, where a is the lattice parameter in nm
∴ No. of atoms per (nm)2 = 1/a2
(ii) On (110) Plane:
OPQR is (110) plane.
It has 1/4 + 1/4 + 1/4 + 1/4 + 1 = 2 atoms and has area OPQR = a x √2a where a is lattice parameter in mn.
II FCC Crystal Structure:
(i) On (100) Plane:
OPQR is (100) plane. This plane has ¼ + ¼ + ¼ + ¼ 1 = 2 atoms. Its area is a2 if ‘a’ is the lattice parameter in nm.
(ii) On (110) Plane:
OPQR is (110) plane. This plane has ¼ + ¼ + ¼ + ¼ + ½ + ½ = 2 atoms
Area of this plane = a. a √2
= a2 √2.
where ‘a’ is lattice parameter in nm.
Thus, planar density
Problem 3:
The atomic radius of copper (FCC) is 1.28 A°. Calculate the planar density, i.e., number of copper atoms per square meter on (110) plane.
Solution:
III. HCP Crystal Structure:
(i) On (0001) Plane:
Let it be either basal plane, or top plane of hexagonal prism. This plane has 1/3 x 6 + 1 =3 atoms. Its area is 3 a2 sin 60°.Thus,
Angles between Directions (Cubic Crystals):
If θ is the angle between two directions [h1 k1 l1] and [h2 k2 l2], then
Intersection of Planes:
The line of intersection [hkl] of two planes (h1 k1 l1) and (h2 k2 l2) can be determined as:
Directions within a Plane:
There are unlimited number of directions in a given plane, but the direction [h1 k1 l1] lies in the plane (h2k2l2), if
Problem 4:
What are the angles between following directions of cubic crystal?
(a) Between [001] and [O11]
(b) Between [001] and [111]
(c) Between [011] and [101]
Solution:
Problem 5:
In a cubic lattice of ‘a’ lattice parameter, (111) and (222) planes. What is their distance from a parallel plane through origin?
Solution:
Important point to be noted is that inter planar spacing for (222) planes is only half of that for (111) planes.