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**THE STRUCTURE OF CRYSTALLINE SOLIDS**

CHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS

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**Atoms in an amorphous solid are arranged randomly- No Order**

3.2 FUNDAMENTAL CONCEPTS SOLIDS AMORPHOUS CRYSTALLINE Atoms in a crystalline solid are arranged in a repetitive three dimensional pattern Long Range Order Atoms in an amorphous solid are arranged randomly- No Order All metals are crystalline solids Many ceramics are crystalline solids Some polymers are crystalline solids

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**LATTICE Lattice -- points arranged in a pattern that**

LATTICE Lattice -- points arranged in a pattern that repeats itself in three dimensions The points in a crystal lattice coincides with atom centers

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3-D view of a lattice

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3.3 UNIT CELL Unit cell -- smallest grouping which can be arranged in three dimensions to create the lattice. Thus the Unit Cell is basic structural unit or building block of the crystal structure

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Unit cell & Lattice

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Unit Cell Lattice

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**3.4 METALLIC CRYSTALS • Tend to be densely packed.**

• Have several reasons for dense packing: -Typically, only one element is present, so all atomic radii are the same. -Metallic bonding is not directional. -Nearest neighbor distances tend to be small in order to have lower bonding energy. • Have the simplest crystal structures. Let us look at three such structures... 4

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**FACE CENTERED CUBIC STRUCTURE (FCC)**

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**FACE CENTERED CUBIC STRUCTURE (FCC)**

Al, Cu, Ni, Ag, Au, Pb, Pt

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**BODY CENTERED CUBIC STRUCTURE (BCC)**

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**BODY CENTERED CUBIC STRUCTURE (BCC)**

Cr, Fe, W, Nb, Ba, V

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**HEXAGONAL CLOSE-PACKED STRUCTURE HCP**

Mg, Zn, Cd, Zr, Ti, Be

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**SIMPLE CUBIC STRUCTURE (SC)**

Only Pd has SC structure

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**Number of atoms per unit cell**

BCC 1/8 corner atom x 8 corners + 1 body center atom =2 atoms/uc FCC 1/8 corner atom x 8 corners + ½ face atom x 6 faces =4 atoms/uc HCP 3 inside atoms + ½ basal atoms x 2 bases + 1/ corner atoms x 12 corners =6 atoms/uc

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**Relationship between atomic radius and edge lengths**

For FCC: a = 2R√2 For BCC: a = 4R /√3 For HCP a = 2R c/a = (for ideal case) Note: c/a ratio could be less or more than the ideal value of 1.633

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**Face Centered Cubic (FCC)**

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**Body Centered Cubic (BCC)**

a0

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**Coordination Number The number of touching or nearest neighbor atoms**

SC is 6 BCC is 8 FCC is 12 HCP is 12

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ATOMIC PACKING FACTOR • APF for a simple cubic structure = 0.52 6

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**ATOMIC PACKING FACTOR: BCC**

• APF for a body-centered cubic structure = 0.68 a = 4R /√3 8

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**FACE CENTERED CUBIC STRUCTURE (FCC)**

• Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. • Coordination # = 12

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**ATOMIC PACKING FACTOR: FCC**

• APF for a face-centered cubic structure = 0.74 a = 2R√2

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3.5 Density Computations Density of a material can be determined theoretically from the knowledge of its crystal structure (from its Unit cell information) Density= mass/Volume Mass is the mass of the unit cell and volume is the unit cell volume. mass = ( number of atoms/unit cell) “n” x mass/atom mass/atom = atomic weight “A”/Avogadro’s Number “NA” Volume = Volume of the unit cell “Vc”

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THEORETICAL DENSITY

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**Example problem on Density Computation**

Problem: Compute the density of Copper Given: Atomic radius of Cu = nm (1.28 x 10-8 cm) Atomic Weight of Cu = 63.5 g/mol Crystal structure of Cu is FCC Solution: = n A / Vc NA n = 4 Vc= a3 = (2R√2)3 = 16 R3 √2 NA = x 1023 atoms/mol = 4 x 63.5 g/mol / 16 √2(1.28 x 10-8 cm)3 x x 1023 atoms/mol Ans = 8.98 g/cm3 Experimentally determined value of density of Cu = 8.94 g/cm3

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**3.6 Polymorphism and Allotropy**

Polymorphism The phenomenon in some metals, as well as nonmetals, having more than one crystal structures. When found in elemental solids, the condition is often called allotropy. Examples: Graphite is the stable polymorph at ambient conditions, whereas diamond is formed at extremely high pressures. Pure iron is BCC crystal structure at room temperature, which changes to FCC iron at 912oC.

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**POLYMORPHISM AND ALLOTROPY**

BCC (From room temperature to 912 oC) Fe FCC (at Temperature above 912 oC) 912 oC Fe (BCC) Fe (FCC)

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3.7 Crystal Systems Since there are many different possible crystal structures, it is sometimes convenient to divide them into groups according to unit cell configurations and/or atomic arrangements. One such scheme is based on the unit cell geometry, i.e. the shape of the appropriate unit cell parallelepiped without regard to the atomic positions in the cell. Within this framework, an x, y, and z coordinate system is established with its origin at one of the unit cell corners; each x, y, and z-axes coincides with one of the three parallelepiped edges that extend from this corner, as illustrated in Figure.

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**The Lattice Parameters**

a, b, c, , , are called the lattice Parameters.

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**Seven different possible combinations of edge lengths and angles give seven crystal systems.**

Shown in Table 3.2 Cubic system has the greatest degree of symmetry. Triclinic system has the least symmetry.

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**The Lattice Parameters**

Lattice parameters are a, b, c, , , are called the lattice Parameters.

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3.7 CRYSTAL SYSTEMS

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**3.8 Point Coordinates in an Orthogonal Coordinate System Simple Cubic**

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**3.9 Crystallographic Directions in Cubic System**

Determination of the directional indices in cubic system: Four Step Procedure (Text Book Method) Draw a vector representing the direction within the unit cell such that it passes through the origin of the xyz coordinate axes. Determine the projections of the vector on xyz axes. Multiply or divide by common factor to obtain the three smallest integer values. Enclose the three integers in square brackets [ ]. e.g. [uvw] u, v, and w are the integers

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**Crystallographic Directions**

z Algorithm 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvw] y x ex: 1, 0, ½ Lecture 2 ended here => 2, 0, 1 => [ 201 ] -1, 1, 1 where overbar represents a negative index [ 111 ] => families of directions <uvw>

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**Crystallographic Directions in Cubic System**

[111] [120] [110]

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**Crystallographic Directions in Cubic System**

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**Head and Tail Procedure for determining Miller Indices for Crystallographic Directions**

Find the coordinate points of head and tail points. Subtract the coordinate points of the tail from the coordinate points of the head. Remove fractions. Enclose in [ ]

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**Indecies of Crystallographic Directions in Cubic System**

Direction A Head point – tail point (1, 1, 1/3) – (0,0,2/3) 1, 1, -1/3 Multiply by 3 to get smallest integers 3, 3, -1 A = [33Ī] Direction B Head point – tail point (0, 1, 1/2) – (2/3,1,1) -2/3, 0, -1/2 Multiply by 6 to get smallest integers _ _ B = [403] C = [???] D = [???]

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**Indices of Crystallographic Directions in Cubic System**

Direction C Head Point – Tail Point (1, 0, 0) – (1, ½, 1) 0, -1/2, -1 Multiply by 2 to get the smallest integers C = [0I2] Direction D Head Point – Tail Point (1, 0, 1/2) – (1/2, 1, 0) 1/2, -1, 1/2 Multiply by 2 to get the smallest integers D = [I2I] B= [???] A = [???]

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**Crystallographic Directions in Cubic System**

[210]

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**Crystallographic Directions in Cubic System**

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CRYSTALLOGRAPHIC DIRECTIONS IN HEXAGONAL UNIT CELLS Miller-Bravais indices -- same as Miller indices for cubic crystals except that there are 3 basal plane axes and 1 vertical axis. Basal plane -- close packed plane similar to the (1 1 1) FCC plane. contains 3 axes 120o apart.

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Direction Indices in HCP Unit Cells – [uvtw] where t=-(u+v) Conversion from 3-index system to 4-index system:

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**HCP Crystallographic Directions**

Hexagonal Crystals 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows. Fig. 3.8(a), Callister 7e. - a3 a1 a2 z = ' w t v u ) ( + - 2 3 1 ] uvtw [

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**Crystallographic Directions in Cubic System**

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**Indices of a Family or Form **

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**3.10 MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES **

Miller Indices for crystallographic planes are the reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with the crystallographic x,y,z axes of the three nonparallel edges of the cubic unit cell. 4-Step Procedure: Find the intercepts that the plane makes with the three axes x,y,z. If the plane passes through origin change the origin or draw a parallel plane elsewhere (e.g. in adjacent unit cell) Take the reciprocal of the intercepts Remove fractions Enclose in ( )

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**Crystallographic Planes**

z x y a b c example a b c Intercepts Reciprocals 1/ / / Reduction Miller Indices (110) example a b c z x y a b c Intercepts 1/ Reciprocals 1/½ 1/ 1/ Reduction Miller Indices (100)

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**Crystallographic Planes**

z x y a b c example a b c Intercepts 1/ /4 Reciprocals 1/½ 1/ /¾ /3 Reduction Miller Indices (634) (001) (010), Family of Planes {hkl} (100), (001), Ex: {100} = (100),

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**Miller Indecies of Planes in Crystallogarphic Planes in Cubic System**

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**Drawing Plane of known Miller Indices in a cubic unit cell**

Draw ( ) plane

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**Miller Indecies of Planes in Crystallogarphic Planes in Cubic System**

Origin for A Origin for B Origin for A A = (IĪ0) B = (I22) A = (2IĪ) B = (02Ī)

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CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN HEXAGONAL UNIT CELLS Miller-Bravais indices -- same as Miller indices for cubic crystals except that there are 3 basal plane axes and 1 vertical axis. Basal plane -- close packed plane similar to the (1 1 1) FCC plane. contains 3 axes 120o apart.

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**Crystallographic Planes (HCP)**

In hexagonal unit cells the same idea is used a2 a3 a1 z example a a a c Intercepts -1 1 Reciprocals / -1 1 Reduction -1 1 Miller-Bravais Indices (1011) Adapted from Fig. 3.8(a), Callister 7e.

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**Miller-Bravais Indices for crystallographic planes in HCP**

_ (1211)

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**Miller-Bravais Indices for crystallographic directions and planes in HCP**

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**Atomic Arrangement on (110) plane in FCC**

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**Atomic Arrangement on (110) plane in BCC**

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**Atomic arrangement on [110] direction in FCC**

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**3.11 Linear and Planar Atomic Densities**

Linear Density “LD” is defined as the number of atoms per unit length whose centers lie on the direction vector of a given crystallographic direction.

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**Linear Density LD for [110] in BCC.**

# of atom centered on the direction vector [110] = 1/2 +1/2 = 1 Length of direction vector [110] = 2 a a = 4R/ 3 [110] 2a

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**Linear Density LD of [110] in FCC**

# of atom centered on the direction vector [110] = 2 atoms Length of direction vector [110] = 4R LD = 2 /4R LD = 1/2R Linear density can be defined as reciprocal of the repeat distance ‘r’ LD = 1/r

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**Area of the plane Planar Density Planar Density “PD”**

is defined as the number of atoms per unit area that are centered on a given crystallographic plane. No of atoms centered on the plane PD = ————————————— Area of the plane

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**Planar Density of (110) plane in FCC**

# of atoms centered on the plane (110) = 4(1/4) + 2(1/2) = 2 atoms Area of the plane = (4R)(2R 2) = 8R22 (111) Plane in FCC a = 2R 2 4R

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**Closed Packed Crystal Structures**

FCC and HCP both have: CN = 12 and APF = 0.74 APF= 0.74 is the most efficient packing. Both FCC and HCP have Closed Packed Planes FCC ----(111) plane is the Closed Packed Plane HCP ----(0001) plane is the Closed Packed Plane The atomic staking sequence in the above two structures is different from each other

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**Closed Packed Structures**

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**Closed Packed Plane Stacking in HCP**

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**Hexagonal Close-Packed Structure (HCP)**

• ABAB... Stacking Sequence • 3D Projection • 2D Projection c a A sites B sites Bottom layer Middle layer Top layer Adapted from Fig. 3.3(a), Callister 7e. • Coordination # = 12 6 atoms/unit cell • APF = 0.74 ex: Cd, Mg, Ti, Zn • c/a = 1.633

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**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

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**Closed Packed Plane Stacking in FCC**

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**Crystalline and Noncrystalline Materials 3.13 Single Crystals**

For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal. All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled. Example: Electronic microcircuits, which employ single crystals of silicon and other semiconductors.

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**Polycrystalline Materials**

3.13 Polycrytalline Materials Polycrystalline crystalline solids composed of many small crystals or grains. Various stages in the solidification : Small crystallite nuclei Growth of the crystallites. Obstruction of some grains that are adjacent to one another is also shown. Upon completion of solidification, grains that are adjacent to one another is also shown. Grain structure as it would appear under the microscope.

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3.15 Anisotropy The physical properties of single crystals of some substances depend on the crystallographic direction in which the measurements are taken. For example, modulus of elasticity, electrical conductivity, and the index of refraction may have different values in the [100] and [111] directions. This directionality of properties is termed anisotropy. Substances in which measured properties are independent of the direction of measurement are isotropic.

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