Miller Indices Prof Ajit Surve Introduction Miller indices

Miller Indices Prof. Ajit Surve

Introduction • Miller indices are used to specify directions and planes. • These directions and planes could be in lattices or in crystals. • The number of indices will match with the dimension of the lattice or the crystal. • The crystal lattice may be regarded as made up of an infinite set of parallel equidistant planes passing through the lattice points which are known as lattice planes. • In simple terms, the planes passing through lattice points are called ‘lattice planes’. • For a given lattice, the lattice planes can be chosen in a different number of ways.

Introduction • The orientation of planes or faces in a crystal can be described in terms of their intercepts on the three axes. • Miller introduced a system to designate a plane in a crystal. • He introduced a set of three numbers to specify a plane in a crystal. • This set of three numbers is known as ‘Miller indices’ of the concerned plane. Miller indices is defined as the reciprocals of the intercepts made by the plane on the three axes.

Notations • (h, k, l) represents a point – note the exclusive use of commas. • Negative numbers/directions are denoted with a bar on top of the number. • [hkl] represents a direction. Rectangular Bracket • <hkl> represents a family of directions. Angle bracket. • (hkl) represents a plane. Circular Bracket. • {hkl} represents a family of planes. Curly Bracket.

Procedure for finding Miller Indices for planes 1. Determine the intercepts of the plane along the axes X, Y and Z in terms of the lattice constants a, b and c. 2. Determine the reciprocals of these intercepts. 3. Find the least common denominator (lcd) and multiply each by this lcd. 4. The result is written in paranthesis. This is called the `Miller Indices’ of the plane in the form (h k l).

ILLUSTRATION 1 Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis. X-axis Y-axis Z-axis Intercept 2 3 2 Reciprocal 1/2 1/3 1/2 LCM (6) 3 2 3 Miller Indices 3 2 3 Plane Indices (323)

Plane parallel to axis/axes • A plane which is parallel to any one of the co-ordinate axes has an intercept of infinity (∞). Therefore the Miller index for that axis is zero; i. e. for an intercept at infinity, the corresponding index is zero. • Illustration 2 X-axis Y-axis Z-axis Intercept 1 ∞ ∞ Reciprocal 1/1=1 1/∞=0 LCM Miller Indices Plane 1 0 0 (100)

A plane passing through the origin • All equally spaced parallel planes have same ‘Miller indices’ i. e. The Miller indices do not only define a particular plane but also a set of parallel planes. Thus the planes whose intercepts are 1, 1, 1; 2, 2, 2; -3, -3 etc. , are all represented by the same set of Miller indices. • A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts. • It is only the ratio of the indices which is important in this notation. The (8 4 4) planes are the same as (2 1 1) planes.

A plane passing through the origin • If a plane cuts an axis on the negative side of the origin, corresponding index is negative. It is represented by a bar, like (1 0 0). i. e. Miller indices (1 0 0) indicates that the plane has an intercept in the – ve X –axis. • Illustration 3 Plane ABCD CBEO Origin O Intercept 1 ∞ Reciprocal 1 Miller indices 1 Plane O* 1 -1 ∞ 0 0 1 -1 0 0 0 1 1 0 (100) ∞ (110)

Families of Lattice Planes • Given any plane in a lattice, there is {100}cubic = (100), (010), (001) a infinite set of parallel lattice planes (or family of planes) that are equally spaced from each other. • One of the planes in any family always passes through the origin.

Distance between (110) planes CUBIC CRYSTALS Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell Angle between two directions [h 1 k 1 l 1] and [h 2 k 2 l 2]:

Miller indices for directions • To find the Miller indices of a direction, Choose a perpendicular plane to that direction. • Find the Miller indices of that perpendicular plane. • The perpendicular plane and the direction have the same Miller indices value. • Therefore, the Miller indices of the perpendicular plane is written within a square bracket to represent the Miller indices of the direction like [ ]. • Or the coordinates of the point through which line of direction passes (assuming one end is origin).

Miller indices for directions

Miller indices for directions • Direction A X-axis Y-axis Z-axis Projection 0 a -b -c Projection 0 -1 -1 LCM Not required Miller 0 1 Direction [011] Direction B X-axis Y-axis Z-axis Projection -a b/2 0 c Projection -1 ½ 0 LCM (2) -2 1 0 Miller 2 1 0 Direction 1 [210]

Miller indices for directions Direction C X-axis Y-axis Z-axis Projection a/2 b/2 C Projection ½ ½ 1 LCM(2) 1 1 2 Miller 1 1 2 Direction [112] Direction D X-axis Y-axis Z-axis Projection a/2 b/2 -c Projection ½ ½ -1 LCM (2) 1 1 -2 Miller 1 1 2 Direction [112]

Miller indices for hexagonal system • hexagonal systems, the Miller-Bravais indices are often used. • These have the form (hkil), where h, k, and i are the reciprocals of the plane intercepts for the three coplanar vectors indicated below and ‘i’ is the reciprocal for the intercept in the c direction. • h, k and i are not linearly independent so the rule h+k+i = 0 must always be obeyed. • i. e. i=-(h+k)

Miller indices for hexagonal system -planes x 1 x 2 x 3 C or z Intercept ∞ ∞ ∞ 1 Reciprocal 1/∞=0 1 LCM 0 0 0 1 Miller 0 0 0 1 Plane (0001) Fig. adapted from BASIC CRYSTALLOGRAPHY Rajesh Prasad Department of Applied Mechanics Indian Institute of Technology New Delhi 110016 rajesh@am. iitd. ac. in

Miller indices for hexagonal system -planes Plane A C A B Fig. adapted from BASIC CRYSTALLOGRAPHY Rajesh Prasad Department of Applied Mechanics Indian Institute of Technology New Delhi 110016 rajesh@am. iitd. ac. in Plane B Plane C Intercept 1 ∞ -1 ∞ ∞ 1 -1 ∞ -1 1 ∞ ∞ Reciprocal 1 0 -1 0 LCM Not required Miller Plane 0 1 -1 0 -1 1 0 0 1 0 -1 0 0 1 -1 0 -1 1 0 0 (1010) (0110) (1100)

Directions in HCP Crystals • To emphasize that they are equal, a and b is changed to a 1 and a 2. • 2. The unit cell is outlined in blue. • 3. A fourth axis is introduced (a 3) to show symmetry. Symmetry about c axis makes a 3 equivalent to a 1 and a 2. Vector addition gives a 3 = –( a 1 + a 2). • 4. This 4 -coordinate system is used: [a 1 , a 2, –( a 1 + a 2) c]

The 3 rd index is redundant and is included to bring out the equality between equivalent directions (like in the case of planes). i=-(h+k) In the drawing of the directions we use an additional guide hexagon 3 times the unit basis vectors (ai). Adapted from PPT of Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016

* Trace a path along the basis vectors as required by the direction. In the current example move 1 unit along a 1, 1 unit along a 2 and 2 units along a 3. * Directions are projected onto the basis vectors to determine the components and hence the Miller-Bravais indices can be determined as in the table. a 1 a 2 a 3 Projections a/2 −a Normalized wrt LP 1/2 − 1 Factorization 1 1 − 2 Indices [1 1 2 0]

• Directions in HCP Crystals: 4 -index notation • What is 4 -index notation for vector B? Projecting the vector onto the basal plane, it lies between a 1 and a 2 (vector B is projection). Vector B = (a 1 + a 2), so the direction is [110] in coordinates of [a 1 a 2 c], where c-intercept is 0. Fig. Adapted from MSE 280: Introduction to Engineering Materials ©D. D. Johnson 2004, 2006 -08 In 4 -index notation, because a 3 = –( a 1 + a 2), the vector B is since it is 3 x farther out.

Miller indices for hexagonal system -Directions • [uvw]=>[UVTW] Vectorially a 1+ a 2+ a 3=0 Require that: U +V +T = 0

• What is 4 -index notation for vector D? In 4 -index notation c = [0001], which must be added to get D (reduced to integers) D = [1123] Note: Find the coordinate axes that straddle the vector of interest, and follow along those axes (but divide the a 1, a 2, a 3 part of vector by 3 because you are now three times farther out!) Projection of the vector D in units of [a 1 a 2 c] gives u’=1, v’=1, and w’=1. Already reduced integers. In 4 -index notation: Reduce to smallest integers:

Weiss zone law • Condition for a direction [uvw] to be parallel to a plane or lie in the plane (hkl):

Zone Axis • The direction common to a set of planes is called the zone axis of those planes. • If (h 1 k 1 l 1) & (h 2 k 2 l 2) are two planes having a common direction [uvw] then according to Weiss zone law: u. h 1 + v. k 1 + w. l 1 = 0 & u. h 2 + v. k 2 + w. l 2 =0 • Useful in Selected Area Diffraction Patterns (SADP) in a TEM.

A certain crystal has lattice parameters of 4. 24, 10 and 3. 66 Å on X, Y, Z axes respectively. Determine the Miller indices of a plane having intercepts of 2. 12, 10 and 1. 83 Å on the X, Y and Z axes. Lattice parameters are = 4. 24, 10 and 3. 66 Å The intercepts of the given plane = 2. 12, 10 and 1. 83 Å i. e. The intercepts are, 0. 5, 1 and 0. 5. X-axis Y-axis Z-axis Intercept ½ 1 ½ Reciprocal 2 1 2 LCM Miller Indices Plane 2 1 (212) Or (424) 2

Calculate the miller indices for the plane with intercepts 2 a, - 3 b and 4 c the along the crystallographic axes. The intercepts are 2, - 3 and 4 X-axis Y-axis Z-axis Intercept 2 -3 4 Reciprocal 1/2 -1/3 1/4 LCM Miller Indices Plane 6 6 -4 -4 3 3 (643)

Find the angle between the directions [2 1 1] and [1 1 2] in a cubic crystal. In this case, u 1 = 2, v 1 = 1, w 1 = 1, u 2 = 1, v 2 = 1, w 2 = 2 (or) cos = 0. 833 = 35°

The lattice constant for a unit cell of aluminum is 4. 031Å Calculate the interplanar space of (2 1 1) plane. a = 4. 031 Å (h k l) = (2 1 1) Interplanar spacing = 1. 6456 Å

Find the perpendicular distance between the two planes indicated by the Miller indices (1 2 1) and (2 1 2) in a unit cell of a cubic lattice with a lattice constant parameter ‘a’. The perpendicular distance between the origin and the plane is (1 2 1) and the perpendicular distance between the origin and the plane (2 1 2), The perpendicular distance between the planes (1 2 1) and (2 1 2) are, d = d 1 – d 2 = d = 0. 0749 a.

Exercises- Find Miller indices

Find Miller indices for the planes given below.

Find Miller indices

Find Miller indices for directions A, B, C, D, E, F.

1. Which one is showing the plane (221)?

2. On a simple cubic lattice of spacing = 1, draw the [100], [010], [110], and [111] directions. ANS:

3. Draw the (100) and (110) planes of a body centered cubic (bcc) lattice to THE CORRECT scale (Give dimensions). You can assume that the length of the cell is 1. Repeat part (b) for a face centered cubic (fcc) crystal lattice for the (100), (110), and (111) planes. FCC crystals have atoms at each corner and atoms at the center of each face.

4. Determine the Miller indices (hkl) of the shaded planes below. Show your work on each step to determine the plane (IE. Intercept, reciprocals, reduction, etc …) a) (010), b) (632), (c) (101) 5. For the intercepts x, y, and, z with values of 3, 1, and 2 respectively, find the Miller indices. Answer: (263) 6. Compute the Miller Indices for a plane intersecting at x= ¼ , y=1, and x=1/2, Answer: (4, 1, 2) 7. Graph the plane and determine the axis intersects of a surface with the Miller Index (013).

8. Explain the meaning of {100} and it’s importance. Answer: {100} is a short way of referring to 6 different planes. These indices all refer to the same lattice as viewed from different points of reference defined by the axis. 1. (100) 2. (010) 3. (001) 4. (-100) 5. (0 -10) 6. (00 -1) 9. Consider the two following planes that cut the crystallographic axes as indicated. Plane 1 cuts the axes at a / 3, b / 2 and c /4 and plane 2 cuts the axes at a / 2, b /3 and. What are the Miller indices of these planes ? ( b) Draw the planes for which the Miller indices are (112), (200), (120) and (221)

Sketch within a cubic unit cell the following planes:

Determine the Miller indices for the planes shown in the following unit cell: ANS