General Objective. To develop the knowledge of crystal structure and their properties.

Transcription

1 CRYSTAL PHYSICS 1

2 General Objective To develop the knowledge of crystal structure and their properties. 2

3 Specific Objectives 1. Differentiate crystalline and amorphous solids. 2. To explain nine fundamental terms of crystallography. 3. To discuss the fourteen Bravais lattice of seven crystal system. 3

4 Crystal Physics or Crystallography is a branch of physics that deals with the study of all possible types of crystals and the physical properties of crystalline solids by the determination of their actual structure by using X-rays, neutron beams and electron beams. 4

5 Solids can broadly be classified into two types based on the arrangement of units of matter. The units of matter may be atoms, molecules or ions. They are, Crystalline solids Non-crystalline (or) Amorphous solids 5

6 Ex: Metallic and non-metallic NaCl, Ag, Cu, Au Ex: Plastics, Glass and Rubber 6

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8 Grains Grain boundaries 8

9 SPACE LATTICE A lattice is a regular and periodic arrangement of points in three dimension. It is defined as an infinite array of points in three dimension in which every point has surroundings identical to that of every other point in the array. The Space lattice is otherwise called the Crystal lattice

10 BASIS A crystal structure is formed by associating every lattice point with an unit assembly of atoms or molecules identical in composition, arrangement and orientation. This unit assembly is called the `basis. When the basis is repeated with correct periodicity in all directions, it gives the actual crystal structure. The crystal structure is real, while the lattice is imaginary. 10

11 UNIT CELL A unit cell is defined as a fundamental building block of a crystal structure, which can generate the complete crystal by repeating its own dimensions in various directions. Y B C O A X Z 11

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14 Inter axial lengths: OA = a, OB = b and OC = c Inter axial angles: α,β and γ Primitives: The intercepts OA, OB, OC are called Primitives 7 Crystal systems: 1. Cubic 2. Orthorhombic 3. Monoclinic 4. Triclinic 5. Hexagonal 6. Rhombohedral 7. Tetragonal

15 Fourteen Bravais Lattices in Three Dimensions

16 Fourteen Bravais Lattices

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18 MILLER INDICES 18

19 MILLER INDICES d DIFFERENT LATTICE PLANES 19 Chapter 3 -

20 MILLER INDICES 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. 20

21 MILLER INDICES Miller indices is defined as the reciprocals of the intercepts made by the plane on the three axes. 21 Chapter 3 -

22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine the intercepts of the plane along the axes X,Y and Z in terms of the lattice constants a,b and c. Step 2: Determine the reciprocals of these numbers. 22 Chapter 3 -

23 MILLER INDICES Step 3: Find the least common denominator (lcd) and multiply each by this lcd. Step 4:The result is written in paranthesis.this is called the `Miller Indices of the plane in the form (h k l). This is called the `Miller Indices of the plane in the form (h k l). 23 Chapter 3 -

24 ILLUSTRATION PLANES IN A CRYSTAL Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis. 24

25 ILLUSTRATION DETERMINATION OF MILLER INDICES Step 1:The intercepts are 2,3 and 2 on the three axes. Step 2:The reciprocals are 1/2, 1/3 and 1/2. Step 3:The least common denominator is 6. Multiplying each reciprocal by lcd, we get, 3,2 and 3. Step 4:Hence Miller indices for the plane ABC is (3 2 3) 25 Chapter 3 -

26 MILLER INDICES IMPORTANT FEATURES OF MILLER INDICES A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts. 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,-3 etc., are all represented by the same set of Miller indices. 26 Chapter 3 -

27 MILLER INDICES IMPORTANT FEATURES OF MILLER INDICES It is only the ratio of the indices which is important in this notation. The (6 2 2) planes are the same as (3 1 1) planes. 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. 27 Chapter 3 -

28 MILLER INDICES OF SOME IMPORTANT PLANES 28

29 Simple Cubic Structure (SC) 29

30 Simple Cubic Structure (SC) Rare due to low packing density (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) 30

31 Atomic Packing Factor (APF):SC APF = Volume of atoms in unit cell* Volume of unit cell APF for a simple cubic structure = 0.52 a *assume hard spheres close-packed directions contains 8 x 1/8 = 1 atom/unit cell R=0.5a atoms unit cell APF = 1 volume 4 3 p (0.5a) 3 atom a 3 volume unit cell 31

32 Body Centered Cubic Structure (BCC) Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Cr, W, Fe ( ), Tantalum, Molybdenum Coordination # = 8 2 atoms/unit cell: 1 center + 8 corners x 1/8 32

33 Atomic Packing Factor: BCC APF for a body-centered cubic structure = a a 2 a Adapted from Fig. 3.2(a), Callister & Rethwisch 8e. atoms unit cell APF = R 2 a 4 3 p ( 3a/4)3 a 3 Close-packed directions: length = 4R = 3 a volume unit cell volume atom 33

34 Face Centered Cubic Structure (FCC) Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordination # = 12 Click once on image to start animation (Courtesy P.M. Anderson) Adapted from Fig. 3.1, Callister & Rethwisch 8e. 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 34

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36 Atomic Packing Factor: FCC APF for a face-centered cubic structure = 0.74 maximum achievable APF 2 a a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e. atoms unit cell APF = Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell p ( 2a/4)3 a 3 volume atom volume unit cell 36

37 ABCABC... Stacking Sequence 2D Projection A sites B sites C sites FCC Stacking Sequence A B B C B C B B C B B FCC Unit Cell A B C 37

38 Hexagonal Close-Packed Structure (HCP) 38

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42 ABAB... Stacking Sequence 3D Projection A sites 2D Projection Top layer c B sites Middle layer a Coordination # = 12 APF = 0.74 c/a = A sites Adapted from Fig. 3.3(a), Callister & Rethwisch 8e. Bottom layer 6 atoms/unit cell ex: Cd, Mg, Ti, Zn 42

43 Crystal structure coordination # packing factor close packed directions Simple Cubic (SC) cube edges Body Centered Cubic (BCC) body diagonal Face Centered Cubic (FCC) face diagonal Hexagonal Close Pack (HCP) hexagonal side 43

44 Thank You 44