Miller Indices in Solid State Physics

Dr. Umayal Ramanathan College for Women, Karaikudi. Accredited with B+ Grade by NAAC Affiliated to Alagappa University Class: III B.Sc., Physics 7BPHE2C ELECTIVE COURSE II (C) SOLID STATE PHYSICS Dr. M.Meenakshi Associate Professor, Department of Physics & EC URCW Department of Physics, URCW

UNIT -1 CRYSTAL STRUCTURE Miller Indices Important Features of Miller Indices Department of Physics, URCW

Miller Indices Definition: 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 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. E.g. in 1D there will be 1 index, 2D there will be two indices and in 3D there will be 3 indices, etc. Department of Physics, URCW

Representation (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 <hkl> represents a family of directions (hkl) represents a plane {hkl} represents a family of planes Department of Physics, URCW

Procedure for finding Miler Indices The intercepts of the plane along the axes X, Y and Z in terms of the Lattice Constant a, b, c are determined. The reciprocals of these numbers are found. The least common denominator (LCD) is determined and each number is multiplied by this LCD. The result is written in the form (hkl) and is called the Miller Indices of the plane. Department of Physics, URCW

Miller Indices for Directions 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 (r1 r2 r3 ) miller indices Fractions in (r1 r2 r3 ) are eliminated by multiplying all components by their common denominator. [e.g. (1, , ) will be expressed as (432)] Department of Physics, URCW

Miller Indices for Directions - Example This direction is [42] MillerIndiceswithmagnitude MillerIndices 2[21] Miller indices [53] [21] Department of Physics, URCW

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. Department of Physics, URCW

Miller Indices for Planes: Illustration Consider the plane in pink, which is one of an infinite number of parallel plane each a consistent distance ( a ) away from the origin (purple planes) The plane intersects the x-axis at point a. It runs parallel along y and z axes. Thus, this plane can be designated as (1, , ) The yellow plane can be designated as ( ,1, ) and the green plane can be written as ( , ,1) Department of Physics, URCW

Contd Miller Indices are the reciprocals of the parameters of each crystal face. Thus: Pink Face = (1/1, 1/ , 1/ ) = (100) Green Face = (1/ , 1/ , 1/1) = (001) Yellow Face = (1/ , 1/1, 1/ ) = (010) Department of Physics, URCW

Department of Physics, URCW

Department of Physics, URCW

Procedure for drawing the given plane having Miller indices (hkl) 1. A unit cell is drawn with the given lattice parameters. After taking any convenient point as the origin O, the OX, OY and OZ crystallographic axes are to be marked. If lattice parameters are not given, then unit cubic cell of arbitrary lattice constant will be taken. The reciprocals of the Miller indices 1/h, 1/k, 1/l are to be taken. These values provide the intercepts of the given plane on OX, OY and OZ axes, respectively. The intercepts are marked in the unit cell and the plane is drawn. If the intercept is on any axis, the plane drawn will be parallel to that axis. 2. 3. 4. 5. Department of Physics, URCW

Department of Physics, URCW

Department of Physics, URCW

Important Features of Miller Indices 1. A plane which is parallel to any one of the co-ordinate axes has an intercept of infinity ( ) and therefore, the Miller index for that axis is zero. 2. All equally spaced parallel planes with a particular orientation have same index number (h k I). 3. Miller indices do not only define particular plane but a set of parallel planes. 4. It is the ratio of indices which is only of importance. The planes (211) and (422) are the same. 5. A plane passing through the origin is defined in terms of a parallel plane having nonzero intercepts. Department of Physics, URCW

Cont 6. All the parallel equidistant planes have the same Miller indices. Thus the Miller indices define a set of parallel planes. 7. A plane parallel to one of the coordinate axes has an intercept of infinity. 8. If the Miller indices of two planes have the same ratio (i.e., 844 and 422 or 211), then the planes are parallel to each other. 9. If (h k I) are the Miller indices of a plane, then the plane cuts the axes into a/h, b/k and c/l equal segments respectively. 10. When the integers used in the Miller indices contain more than one digit, the indices must be separated by commas for clarity, e.g., (3, 11, 12). Department of Physics, URCW

Cont 11. The crystal directions of a family are not necessarily parallel to one another. Similarly, not all members of a family of planes are parallel to one another. 12. By changing the signs of all the indices of a crystal direction, we obtain the antiparallel or opposite direction. By changing the signs of all the indices of a plane, we obtain a plane located at the same distance on the other side of the origin. 13. The normal to the plane with indices (hkl) is the direction [hkl]. Department of Physics, URCW

Cont 14.The distance d between adjacent planes of a set of parallel planes of the indices (h k I) is given by- Where a is the edge of the cube. Normally the planes with low index numbers have wide interplanar spacing compared with those having high index numbers. Moreover, low index planes have a higher density of atoms per unit area than the high index plane. In fact, it is the low index planes which play an important role in determining the physical and chemical properties of solids. Department of Physics, URCW

Cont 15.The angle between the normal to the two planes (h1 k1 l1) and (h2 k2 l2) is- 16. A negative Miller index shows that the plane (hkl) cuts the axis on the negative side of the origin. 17. Miller indices are proportional to the direction cosines of the normal to all corresponding plane. Department of Physics, URCW

Some Important Planes of a Cube Department of Physics, URCW

Calculate the Miller Indices Intercepts Fraction Reciprocal Multiply by LCD hkl 2a b c 2/1 1/1 1/1 1/1 1/1 2/2 2/1 2/1 (122) 2a b 2/1 1/1 /1 1/1 1/ 2/2 2/1 2/ (120) a -b 2c 1/1 -1/1 2/1 1/1 -1/1 1/2 2/1 -2/1 2/2 (2-21) -a 2b c -1/1 2/1 -1/1 -1/1 -1/1 -2/1 2/2 -2/1 (-21-2) Department of Physics, URCW

Problems 1. For the intercepts x, y, and, z with values of 3,1, and 2 respectively, find the Miller indices. Compute the Miller Indices for a plane intersecting at x= , y=1, and x=1/2 Calculate the miller indices for the plane with intercepts 2a, - 3b and 4c the along the crystallographic axes. 2. 3. Intercept Fraction Reciprocal Multiply by LCD hkl 3,1,2 3/1,1/1,2/1 1/3,1/1,1/2 6/3,6/1,6/2 (263) , 1, 1/2 4/1,1/1, 2/1 , 1/ 1, 1/2 4/4,4/1, 4/2 (142) 2,-3,4 2/1, -3/1, 4/1 , -1/3/ 1/4 12/2,-12/3, 12/4 (6-43) Department of Physics, URCW

Determine the Miller indices (hkl) of the shaded planes below. Show your work on each step to determine the plane Intercept Fraction Reciprocal Multiply by LCD hkl ,1, /1,1/1, /1 1/ ,1/1,1/ (010) 1,2,3 1/1,2/1, 3/1 1/1,1/2, 1/3 6/1,6/2,6/3 (632) 2, ,2 2/1, /1,2/1 ,1/ , 1/2 2/2,2/ ,2/2 (101) Department of Physics, URCW

Which one is showing the plane (221) Department of Physics, URCW

Graph the plane and determine the axis intersects of a surface with the Miller Index (013). Intersects at y = 1, z = 1/3, plane does not intersect the x-axis Intercept Fraction Reciprocal Multiply by LCD hkl , 1, 1/3 /1, 1/1, 1/3 1/ , 1/1. 3/1 -------- (013) Department of Physics, URCW

Problem A certain crystal has lattice parameters of 4.24, 10 and 3.66 on X, Y, Z axes respectively. Determine the Miller indices of a plane having intercepts of 2.12, 10 and 1.83 on the X, Y and Z axes. Department of Physics, URCW

Solution: Lattice parameters are = 4.24, 10 and 3.66 The intercepts of the given plane = 2.12, 10 and 1.83 i.e. The intercepts are, 0.5, 1 and 0.5. Step 1: The Intercepts are 1/2, 1 and 1/2. Step 2: The reciprocals are 2, 1 and 2. Step 3:Therefore the Miller indices of the given plane is (2 1 2). Department of Physics, URCW

The lattice constant for a unit cell of aluminum is 4.031 Calculate the interplanar space of (2 1 1) plane. ? ? ??= ( 2+ ?2+ ?2 Answer: d = 1.6456 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 . Ans: d = d1- d2 = 0.0749 a Department of Physics, URCW

Department of Physics, URCW