Solid State Physics (1) Phys3710 Crystal Structure 3 Lecture 3 Department of Physics Dr Mazen Alshaaer Second semester 2013/2014 Ref.: Prof. Charles W. Myles, Department of Physics, Texas Tech University

Wigner-Seitz Method to Construct a Primitive Cell A simple, geometric method to construct a Primitive Cell is called the Wigner-Seitz Method. The procedure is: Choose a starting lattice point. Draw lines to connect that point to its nearest neighbors. At the mid-point & normal to these lines, draw new lines. The volume enclosed is called a Wigner-Seitz cell. Illustration for the 2 dimensional parallelogram lattice. 2

3 Dimensional Wigner-Seitz Cells Face Centered Cubic Wigner-Seitz Cell Body Centered Cubic Wigner-Seitz Cell 3

Basic definitions – Lattice sites Define basic terms and give examples of each: Points (atomic positions) Vectors (defines a particular direction - plane normal) Miller Indices (defines a particular plane) relation to diffraction 3-index for cubic and 4-index notation for HCP

Lattice Sites in a Cubic Unit Cell 1) To define a point within a unit cell…. The standard notation is shown in the figure. It is understood that all distances are in units of the cubic lattice constant a, which is the length of a cube edge for the material of interest. Figure 35-18. Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

2) Directions in a Crystal: Standard Notation See Figure Choose an origin, O. This choice is arbitrary, because every lattice point has identical symmetry. Then, consider the lattice vector joining O to any point in space, say point T in the figure. As we’ve seen, this vector can be written T = n1a1 + n2a2 + n3a3 [111] direction In order to distinguish a Lattice Direction from a Lattice Point, (n1n2n3), the 3 integers are enclosed in square brackets [ ...] instead of parentheses (...), which are reserved to indicate a Lattice Point. In direction [n1n2n3], n1n2n3 are the smallest integers possible for the relative ratios. Figure 35-18. Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

Directions in a Crystal Procedure: Any line (or vector direction) is specified by 2 points. The first point is, typically, at the origin (000). Determine length of vector projection in each of 3 axes in units (or fractions) of a, b, and c. X (a), Y(b), Z(c) 1 1 0 Multiply or divide by a common factor to reduce the lengths to the smallest integer values, u v w. Enclose in square brackets: [u v w]: [110] direction. a b c DIRECTIONS will help define PLANES (Miller Indices or plane normal). 5. Designate negative numbers by a bar Pronounced “bar 1”, “bar 1”, “zero” direction. 6. “Family” of [110] directions is designated as <110>.

Examples X = 1 , Y = ½ , Z = 0 X = ½ , Y = ½ , Z = 1 [1 ½ 0] [2 1 0] 210 X = 1 , Y = ½ , Z = 0 [1 ½ 0] [2 1 0] Figure 35-18. Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still. X = ½ , Y = ½ , Z = 1 [½ ½ 1] [1 1 2]

Negative Directions R = n1a1 + n2a2 + n3a3 When we write the direction [n1n2n3] depending on the origin, negative directions are written as R = n1a1 + n2a2 + n3a3 To specify the direction, the smallest possible integers must be used. (origin) O - Y direction X direction - X direction Z direction - Z direction Y direction 9 9

Examples of Crystal Directions X = 1 , Y = 0 , Z = 0 [1 0 0] X = -1 , Y = -1 , Z = 0 [110] 10 10

Examples A vector can be moved to the origin. X =-1 , Y = 1 , Z = -1/6 [-1 1 -1/6] [6 6 1] 11

These are called lattice planes. Crystal Planes Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are called lattice planes. In the figure, the density of lattice points on each plane of a set is the same & all lattice points are contained on each set of planes. b a b a The set of planes for a 2D lattice. 12

Miller Indices Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice & are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. To find the Miller indices of a plane, take the following steps: Determine the intercepts of the plane along each of the three crystallographic directions. Take the reciprocals of the intercepts. If fractions result, multiply each by the denominator of the smallest fraction. 13

Example 1 Axis 1 ∞ 1/1 1/ ∞ Miller İndices (100) X Y Z (1,0,0) Axis X Y Z Intercept points 1 ∞ Reciprocals 1/1 1/ ∞ Smallest Ratio Miller İndices (100) 14

Example 2 Axis 1 ∞ 1/1 1/ 1 1/ ∞ Miller İndices (110) X Y Z (1,0,0) (0,1,0) Axis X Y Z Intercept points 1 ∞ Reciprocals 1/1 1/ 1 1/ ∞ Smallest Ratio Miller İndices (110) 15

Example 3 Axis 1 1/1 1/ 1 Miller İndices (111) X Y Z Intercept points (1,0,0) (0,1,0) (0,0,1) Axis X Y Z Intercept points 1 Reciprocals 1/1 1/ 1 Smallest Ratio Miller İndices (111) 16

Example 4 Axis 1/2 1 ∞ 1/(½) 1/ 1 1/ ∞ 2 Miller İndices (210) X Y Z (1/2, 0, 0) (0,1,0) Axis X Y Z Intercept points 1/2 1 ∞ Reciprocals 1/(½) 1/ 1 1/ ∞ Smallest Ratio 2 Miller İndices (210) 17

Example 5 Axis 1 ∞ ½ 2 Miller İndices (102) a b c 1/1 1/ ∞ 1/(½) Intercept points 1 ∞ ½ Reciprocals 1/1 1/ ∞ 1/(½) Smallest Ratio 2 Miller İndices (102) 18

Example 6 Axis -1 ∞ ½ 2 Miller İndices (102) a b c 1/-1 1/ ∞ 1/(½) Intercept points -1 ∞ ½ Reciprocals 1/-1 1/ ∞ 1/(½) Smallest Ratio 2 Miller İndices (102) 19

Examples of Miller Indices Plane intercepts axes at 3 2 [2,3,3] Reciprocal numbers are: Miller Indices of the plane: (2,3,3) Indices of the direction: [2,3,3] (200) (111) (100) (110) (100) 20

Crystal Structure 21

Example 7 22

Indices of a Family of Planes Given any plane in a lattice, there is 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. The Miller indices (hkl) usually refer to the plane that is nearest to the origin without passing through it. You must always shift the origin or move the plane parallel, otherwise a Miller index integer is 1/0, i.e.,∞! Sometimes (hkl) will be used to refer to any other plane in the family, or to the family taken together. Importantly, the Miller indices (hkl) is the same vector as the plane normal!

Indices of a Family of Planes Sometimes. when the unit cell has rotational symmetry, several nonparallel planes may be equivalent by virtue of this symmetry, in which case it is convenient to lump all these planes in the same Miller Indices, but with curly brackets. Thus indices {h,k,l} represent all the planes equivalent to the plane (hkl) through rotational symmetry. 24

Classification of Crystal Structures Crystallographers showed a long time ago that, in 3 Dimensions, there are 14 BRAVAIS LATTICES + 7 CRYSTAL SYSTEMS This results in the fact that, in 3 dimensions, there are only 7 different shapes of unit cell which can be stacked together to completely fill all space without overlapping. This gives the 7 crystal systems, in which all crystal structures can be classified. These are: The Cubic Crystal System (SC, BCC, FCC) The Hexagonal Crystal System (S) The Triclinic Crystal System (S) The Monoclinic Crystal System (S, Base-C) The Orthorhombic Crystal System (S, Base-C, BC, FC) The Tetragonal Crystal System (S, BC) The Trigonal (or Rhombohedral) Crystal System (S) 25 25

26 26

Simple Cubic (SC) coordination number = 6 For a Bravais Lattice, The Coordinatıon Number  The number of lattice points closest to a given point (the number of nearest-neighbors of each point). Because of lattice periodicity, all points have the same number of nearest neighbors or coordination number. (That is, the coordination number is intrinsic to the lattice.) Examples Simple Cubic (SC) coordination number = 6 Body-Centered Cubic coordination number = 8 Face-Centered Cubic coordination number = 12 Crystal Structure 27

Atomic Packing Factor (or Atomic Packing Fraction) For a Bravais Lattice, The Atomic Packing Factor (APF)  the volume of the atoms within the unit cell divided by the volume of the unit cell. When calculating the APF, the volume of the atoms in the unit cell is calculated AS IF each atom was a hard sphere, centered on the lattice point & large enough to just touch the nearest-neighbor sphere. Of course, from Quantum Mechanics, we know that this is very unrealistic for any atom!!

1- CUBIC CRYSTAL SYSTEMS 3 Common Unit Cells with Cubic Symmetry Simple Cubic Body Centered Cubic Face Centered Cubic (SC) (BCC) (FCC)

3 Common Unit Cells with Cubic Symmetry

a- The Simple Cubic (SC) Lattice The SC Lattice has one lattice point in its unit cell, so it’s unit cell is a primitive cell. In the unit cell shown on the left, the atoms at the corners are cut because only a portion (in this case 1/8) “belongs” to that cell. The rest of the atom “belongs” to neighboring cells. The Coordinatination Number of the SC Lattice = 6. a b c 31

Simple Cubic (SC) Lattice Atomic Packing Factor 32

b- The Body Centered Cubic (BCC) Lattice The BCC Lattice has two lattice points per unit cell so the BCC unit cell is a non-primitive cell. Every BCC lattice point has 8 nearest- neighbors. So (in the hard sphere model) each atom is in contact with its neighbors only along the body-diagonal directions. Many metals (Fe,Li,Na..etc), including the alkalis and several transition elements have the BCC structure. b c a 33

BCC Structure

Body Centered Cubic (BCC) Lattice Atomic Packing Factor 2 (0,433a) Crystal Structure 35

Elements with the BCC Structure Note: This was the end of lecture 1

c- The Face Centered Cubic (FCC) Lattice In the FCC Lattice there are atoms at the corners of the unit cell and at the center of each face. The FCC unit cell has 4 atoms so it is a non-primitive cell. Every FCC Lattice point has 12 nearest-neighbors. Many common metals (Cu,Ni,Pb..etc) crystallize in the FCC structure. Crystal Structure 37

FCC Structure

Face Centered Cubic (FCC) Lattice Atomic Packing Factor 0,74 4 (0,353a) Crystal Structure 39

Elements That Have the FCC Structure

FCC & BCC: Conventional Cells With a Basis Alternatively, the FCC lattice can be viewed in terms of a conventional unit cell with a 4-point basis. Similarly, the BCC lattice can be viewed in terms of a conventional unit cell with a 2- point basis.

Comparison of the 3 Cubic Lattice Systems Unit Cell Contents Counting the number of atoms within the unit cell Atom Position Shared Between: Each atom counts: corner 8 cells 1/8 face center 2 cells 1/2 body center 1 cell 1 edge center 2 cells 1/2 Lattice Type Atoms per Cell P (Primitive) 1 [= 8  1/8] I (Body Centered) 2 [= (8  1/8) + (1  1)] F (Face Centered) 4 [= (8  1/8) + (6  1/2)] C (Side Centered) 2 [= (8  1/8) + (2  1/2)] Crystal Structure 42

2- HEXAGONAL CRYSTAL SYSTEMS In a Hexagonal Crystal System, three equal coplanar axes intersect at an angle of 60°, and another axis is perpendicular to the others and of a different length. The atoms are all the same. Crystal Structure

Simple Hexagonal Bravais Lattice

Hexagonal Close Packed (HCP) Lattice

Hexagonal Close Packed (HCP) Lattice This is another structure that is common, particularly in metals. In addition to the two layers of atoms which form the base and the upper face of the hexagon, there is also an intervening layer of atoms arranged such that each of these atoms rest over a depression between three atoms in the base. Crystal Structure 46

Hexagonal Close Packed (HCP) Lattice The HCP lattice is not a Bravais lattice, because orientation of the environment of a point varies from layer to layer along the c-axis.

Hexagonal Close Packed (HCP) Lattice Bravais Lattice : Hexagonal Lattice He, Be, Mg, Hf, Re (Group II elements) ABABAB Type of Stacking  a = b Angle between a & b = 120° c = 1.633a,  basis: (0,0,0) (2/3a ,1/3a,1/2c) Crystal Structure 48

Comments on Close Packing B A B C Sequence AAAA… - simple cubic Sequence ABABAB.. hexagonal close pack Sequence ABAB… - body centered cubic Sequence ABCABCAB.. -face centered cubic close pack Crystal Structure 50

Hexagonal Close Packing

HCP Lattice  Hexagonal Bravais Lattice with a 2 Atom Basis

Comments on Close Packing

Close-Packed Structures ABCABC… → fcc ABABAB… → hcp

Crystal Structure 57

3 - TRICLINIC & 4 - MONOCLINIC CRYSTAL SYSTEMS Triclinic crystals have the least symmetry of any crystal systems. The three axes are each different lengths & none are perpendicular to each other. These materials are the most difficult to recognize. Triclinic (Simple) a ¹ ß ¹ g ¹ 90 oa ¹ b ¹ c Monoclinic (Base Centered) a = g = 90o, ß ¹ 90o a ¹ b ¹ c, Monoclinic (Simple) a = g = 90o, ß ¹ 90o a ¹ b ¹c 58

5 - ORTHORHOMBIC CRYSTAL SYSTEM Orthorhombic (BC) a = ß = g = 90o a ¹ b ¹ c Orthorhombic (Simple) a = ß = g = 90o a ¹ b ¹ c Orthorhombic (Base-centred) a = ß = g = 90o a ¹ b ¹ c Orthorhombic (FC) a = ß = g = 90o a ¹ b ¹ c Crystal Structure 59

6 – TETRAGONAL CRYSTAL SYSTEM Tetragonal (P) a = ß = g = 90o a = b ¹ c Tetragonal (BC) a = ß = g = 90o a = b ¹ c Crystal Structure 60

7 - RHOMBOHEDRAL (R) OR TRIGONAL CRYSTAL SYSTEM Rhombohedral (R) or Trigonal (S) a = b = c, a = ß = g ¹ 90o Crystal Structure 61