Crystal structures Unit-I Hari Prasad Assistant Professor MVJCE-Bangalore Hari Prasad

Learning objectives After the chapter is completed, you will be able to answer: Difference between crystalline and noncrystalline structures Different crystal systems and crystal structures Atomic packing factors of different cubic crystal systems Difference between unit cell and primitive cell Difference between single crystals and poly crystals Hari Prasad

What is space lattice? Space lattice is the distribution of points in 3D in such a way that every point has identical surroundings, i.e., it is an infinite array of points in three dimensions in which every point has surroundings identical to every other point in the array. Hari Prasad

Common materials: with various ‘viewpoints’ Graphite Glass: amorphous Ceramics Crystal Metals Polymers

Common materials: examples Metals and alloys  Cu, Ni, Fe, NiAl (intermetallic compound), Brass (Cu-Zn alloys) Ceramics (usually oxides, nitrides, carbides)  Alumina (Al2O3), Zirconia (Zr2O3) Polymers (thermoplasts, thermosets) (Elastomers) Polythene, Polyvinyl chloride, Polypropylene Based on Electrical Conduction Conductors  Cu, Al, NiAl Semiconductors  Ge, Si, GaAs Insulators  Alumina, Polythene* Based on Ductility Ductile  Metals, Alloys Brittle  Ceramics, Inorganic Glasses, Ge, Si * some special polymers could be conducting

MATERIALS SCIENCE & ENGINEERING The broad scientific and technological segments of Materials Science are shown in the diagram below. To gain a comprehensive understanding of materials science, all these aspects have to be studied. MATERIALS SCIENCE & ENGINEERING Science of Metallurgy PHYSICAL MECHANICAL ELECTRO- CHEMICAL TECHNOLOGICAL Extractive Casting Metal Forming Welding Powder Metallurgy Machining Structure Physical Properties Deformation Behaviour Thermodynamics Chemistry Corrosion

Crystal = Lattice + Motif Definition 1 Crystal = Lattice + Motif Motif or Basis: typically an atom or a group of atoms associated with each lattice point Lattice  the underlying periodicity of the crystal Basis  Entity associated with each lattice points Lattice  how to repeat Motif  what to repeat Lattice Crystal Translationally periodic arrangement of points Translationally periodic arrangement of motifs

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An array of points such that every point has identical surroundings Space Lattice A lattice is also called a Space Lattice An array of points such that every point has identical surroundings In Euclidean space  infinite array We can have 1D, 2D or 3D arrays (lattices) or Translationally periodic arrangement of points in space is called a lattice

Unit cell: A unit cell is the sub-division of the space lattice that still retains the overall characteristics of the space lattice. Primitive cell: the smallest possible unit cell of a lattice, having lattice points at each of its eight vertices only. A primitive cell is a minimum volume cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell. Hari Prasad

Materials and Packing Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 (Quartz) Si Oxygen Non-crystalline materials... noncrystalline SiO2 (Glass) • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline Hari Prasad

Crystal Systems Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems 14 crystal lattices a, b, and c are the lattice constants Hari Prasad

A single crystal can have many unit cells. The Unite Cell is the smallest group of atom showing the characteristic lattice structure of a particular metal. It is the building block of a single crystal. A single crystal can have many unit cells. Hari Prasad

Crystal systems Cubic Three equal axes, mutually perpendicular a=b=c ===90˚ Tetragonal Three perpendicular axes, only two equal a=b≠c ===90˚ Hexagonal Three equal coplanar axes at 120˚ and a fourth unequal axis perpendicular to their plane a=b≠c == 90˚ =120˚ Rhombohedral Three equal axes, not at right angles a=b=c ==≠90˚ Orthorhombic Three unequal axes, all perpendicular a≠b≠c ===90˚ Monoclinic Three unequal axes, one of which is perpendicular to the other two a≠b≠c ==90˚≠  Triclinic Three unequal axes, no two of which are perpendicular a≠b≠c ≠ ≠≠90˚ Hari Prasad

Some engineering applications require single crystals: --diamond single crystals for abrasives --turbine blades Hari Prasad

What is coordination number? The coordination number of a central atom in a crystal is the number of its nearest neighbours. What is lattice parameter? The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c. Hari Prasad

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) Hari Prasad

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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 Hari Prasad

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Atomic Packing Factor: BCC • APF for a body-centered cubic structure = 0.68 a R a 3 a a 2 length = 4R = Close-packed directions: 3 a APF = 4 3 p ( a/4 ) 2 atoms unit cell atom volume a Hari Prasad

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 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 Hari Prasad

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Atomic Packing Factor: FCC • APF for a face-centered cubic structure = 0.74 a 2 a maximum achievable APF Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell APF = 4 3 p ( 2 a/4 ) atoms unit cell atom volume a Hari Prasad

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

Putting atoms in the B position in the II layer and in C positions in the III layer we get a stacking sequence  ABC ABC ABC….  The CCP (FCC) crystal + + = C A B FCC A A B B C C

Hexagonal Close-Packed Structure (HCP) • ABAB... Stacking Sequence • 3D Projection • 2D Projection c a A sites B sites Bottom layer Middle layer Top layer • Coordination # = 12 6 atoms/unit cell • APF = 0.74 ex: Cd, Mg, Ti, Zn • c/a = 1.633 Hari Prasad

APF for HCP c a A sites B sites C=1.633a Number of atoms in HCP unit cell= (12*1/6)+(2*1/2)+3=6atoms Vol.of HCP unit cell= area of the hexagonal face X height of the hexagonal Area of the hexagonal face=area of each triangle X6 a=2r Area of triangle = 𝒃𝒉 𝟐 = 𝒂𝒉 𝟐 = 𝟏 𝟐 𝒂. 𝒂 𝟑 𝟐 Area of hexagon = 𝟔. 𝒂𝟐 𝟑 𝟒 Volume of HCP= 𝟔. 𝒂𝟐 𝟑 𝟒 .𝐂=𝟔. 𝒂𝟐 𝟑 𝟒 .𝟏.𝟔𝟑𝟑𝐚 APF= 6∗ 𝟒𝝅𝒓𝟑 𝟑 /( 𝟑 𝟒 ∗𝟔∗𝟏.𝟔𝟑𝟑∗𝐚𝟑) a h a APF =0.74 Hari Prasad

SC-coordination number 6 Hari Prasad

• Coordination # = 6 (# nearest neighbors) Hari Prasad

BCC-coordination number 8 Hari Prasad

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FCC-coordination number 4+4+4=12 Hari Prasad

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HCP-coordination number 3+6+3=12 Hari Prasad

Theoretical Density, r Cell Unit of Volume Total in Atoms Mass VC NA n A  = where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.023 x 1023 atoms/mol Hari Prasad

Theoretical Density, r  = Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm a = 4R/ 3 = 0.2887 nm  = a 3 52.00 2 atoms unit cell mol g volume 6.023 x 1023 theoretical = 7.18 g/cm3 ractual = 7.19 g/cm3 Hari Prasad

Polymorphism Two or more distinct crystal structures for the same material (allotropy/polymorphism)     titanium   , -Ti carbon diamond, graphite BCC FCC 1538ºC 1394ºC 912ºC -Fe -Fe -Fe liquid iron system Hari Prasad

Miller indices Miller indices: defined as the reciprocals of the intercepts made by the plane on the three axes. Hari Prasad

Procedure for finding Miller indices Determine the intercepts of the plane along the axes X,Y and Z in terms of the lattice constants a, b and c. Step 1 Hari Prasad

Determine the reciprocals of these numbers. Step 2 Hari Prasad

Find the least common denominator (lcd) and multiply each by this lcd Step 3 Find the least common denominator (lcd) and multiply each by this lcd Hari Prasad

The result is written in parenthesis. Step 4 The result is written in parenthesis. This is called the `Miller Indices’ of the plane in the form (h k l). Hari Prasad

Miller Indices for planes (0,0,1) (0,3,0) (2,0,0) Find intercepts along axes → 2 3 1 Take reciprocal → 1/2 1/3 1 Convert to smallest integers in the same ratio → 3 2 6 Enclose in parenthesis → (326)

Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis. Hari Prasad

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) Hari Prasad

IMPORTANT FEATURES OF MILLER INDICES For the cubic crystal especially, the important features of Miller indices are, 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. 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. Hari Prasad

The intercepts are 2, - 3 and 4 Worked Example: Calculate the miller indices for the plane with intercepts 2a, - 3b and 4c the along the crystallographic axes. The intercepts are 2, - 3 and 4 Step 1: The intercepts are 2, -3 and 4 along the 3 axes Step 2: The reciprocals are Step 3: The least common denominator is 12. Multiplying each reciprocal by lcd, we get 6 -4 and 3 Step 4: Hence the Miller indices for the plane is Hari Prasad

Intercepts → 1   Plane → (100) Family → {100} → 3 Intercepts → 1 1  Plane → (110) Family → {110} → 6 Intercepts → 1 1 1 Plane → (111) Family → {111} → 8 (Octahedral plane)

Miller Indices :   (100) Hari Prasad

Fractional intercepts : 1 , 1 , ∞ Miller Indices : (110) Intercepts :   a , a , ∞ Fractional intercepts :   1 , 1 , ∞ Miller Indices :   (110) Hari Prasad

Fractional intercepts : 1 , 1 , 1 Miller Indices : (111) Intercepts :   a , a , a Fractional intercepts :   1 , 1 , 1 Miller Indices :   (111) Hari Prasad

Fractional intercepts : ½ , 1 , ∞ Miller Indices : (210) Intercepts :   ½ a , a , ∞ Fractional intercepts :   ½ , 1 , ∞ Miller Indices :   (210) Hari Prasad

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Z (101) Y X Hari Prasad

(122) Hari Prasad

(211) Hari Prasad

Crystallographic Directions The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, the crystallographic planes are fictitious planes linking nodes. The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a, b, and c. Hari Prasad

Find the Miller indices of that perpendicular plane. 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 [ ]. Hari Prasad

Summary of notations Symbol Alternate symbols Direction [ ] [uvw] → Particular direction < > <uvw> [[ ]] Family of directions Plane ( ) (hkl) Particular plane { } {hkl} (( )) Family of planes Point . . .xyz. Particular point : : :xyz: Family of point *A family is also referred to as a symmetrical set

The above image shows [100], [110], and [111] directions within a For each of the three axes, there will exist both positive and negative coordinates. Thus negative indices are also possible, which are represented by a bar over the appropriate index. For example, the 1 The above image shows [100], [110], and [111] directions within a unit cell Hari Prasad

The vector, as drawn, passes through the origin of the coordinate system, and therefore no translation is necessary. Projections of this vector onto the x, y, and z axes are, respectively,1/2, b, and 0c, which become 1/2, 1, and 0 in terms of the unit cell parameters (i.e., when the a, b, and c are dropped). Reduction of these numbers to the lowest set of integers is accompanied by multiplication of each by the factor 2.This yields the integers 1, 2, and 0, which are then enclosed in brackets as [120]. Hari Prasad

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The two directions are [2 1 1] and [1 1 2] Worked Example Find the angle between the directions [2 1 1] and [1 1 2] in a cubic crystal. The two directions are [2 1 1] and [1 1 2] We know that the angle between the two directions, Hari Prasad

In this case, u1 = 2, v1 = 1, w1 = 1, Type equation here In this case, u1 = 2, v1 = 1, w1 = 1, Type equation here.u2 = 1, v2 = 1, w2 = 2 (or) cos  = 0.833  = 35° 3530. Hari Prasad

Reference http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swf http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swf http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swf Hari Prasad