CHAPTER 4 Crystal Structure Fundamental Crystallography Crystal Structure

CHAPTER 4 Crystal Structure： Fundamental Crystallography, Crystal Structure of Metals and X-Ray Diffraction 1

structure 4. 1 Introduction ◎ In this chapter we expand our view of materials to incorporate larger numbers of atoms ◎Crystallinity: Crystalline solids and Amorphous solids ．Amorphous solids：show SRO (short range order )in three dimensions , but no LRO (long range order) ．Crystalline solids：exhibit both SRO and LRO in three dimensions F 3. 22 2

4. 2 Crystal System to name the crystalline structure of a material, to identify the crystalline structure of a material to view the crystal to be composrd of lattice point (instead of atoms) (If you look at atoms making up a material to identify or to distinguish it from others, you will say every material has its own crystalline structure. But this is not what crystallography tells us. ) ◎ Crystal structure. Lattice = + basis ．Lattice : arrangement of lattice points Basis ．Lattice point≡ basis 3

The Basis(or basic group or lattice point) A crystal contains a structural unit, called the basis (or motif), that is repeated in three dimensions to generate the crystal structure. The environment of each repeated unit is the same throughout the crystal (neglecting surface effects). The basis may be a single atom or molecule, or it may be a small group of atoms, molecules, or ions. Each repeated basis group has the same structure and the same spatial orientation as every other one in the crystal. The basis must have the same stoichiometric composition as the crystal. f 24. 3 (All crystalline solids have ordered structures They all have repeated units (structure units). ) 4

For Na. Cl, the basis consists of one Na+ ion and one Cl- ion. For Cu, the basis is a single Cu atom. For Zn, the basis consists of two Zn atoms. For diamond the basis is two C atoms; the two atoms of the basis are each surrounded tetrahedrally by four carbons, but the four bonds at one basis atom differ in orientation from those at the other atom. For CO 2, the basis is four CO 2 molecules. For benzene, the basis is four C 6 H 6 molecule. f 13. 2 f 24. 15 f 3. 7 -2 f 3. 7 -3 f 3. 7 -4 The Space Lattice. If we place a single point at the same location in each repeated basis group, the set of points obtained forms the (space ) lattice of the crystal. f 24. 3 f 302 5

The Unit Cell. The space lattice of a crystal can be divided into identical parallelepipeds by joining the lattice points with straight lines. (A parallelepiped Is a six-sided geometrical solid whose faces are all parallelograms. ) each such parallelepiped is called a unit cell. In crystallography, one chooses the unit cell so that it has the maximum symmetry and has the smallest volume consistent with the maximum symmetry; the maximum-symmetry requirement implies the maximum number of perpendicular unit-cell edges. f 24. 4 In two dimensions, a unit cell is a parallelogram with sides of length a and b and angle between these sides. In three dimensions, a unit cell is a parallelepiped with edges of length a、b、c and angles α 、β 、 γ where α is the angle between edges b and c , etc. F 3. 4 6

Crystal System In 1848 Bravais showed that there are 14 different kinds of lattices in three dimensions. The 14 Bravais lattices are grouped into seven crystal systems, based on unit-cell symmetry. f 24. 5 Unit cells that have lattice points only at their corners are called primitive (or simple) unit cells. Seven of the Bravais lattices have f 24. 5 T 3. 2 primitive (p or S) unit cells. A body-centered lattice (denoted by the letter I or B, from the German Innenzen-trierte) has a lattice point within the unit cell as well as at each corner of the unit cell. A face-centered (F) lattice has a lattice point on each of the six unit-cell faces as well as at the corners. 7

The Letter c denotes an end-centered lattice with a lattice point on each of the two faces bounded by edges of lengths a and b. Number of lattice point per unit cell. Each point at a unit-cell corner is shared among eight adjacent unit cells in the lattice: four at the same level and four immediately above or below. Therefore a primitive unit cell has 8/8 =1 lattice point and 1 basis group per unit cell. Each point on a unit-cell face is shared between two unit cells, so an F unit cell has 8/8+6/2 =4 lattice points and 4 basis groups per unit cell. F 302 F 301 8

4. 3 CRYSTALS WITH ONE ATOM PER LATTICE SITE AND HEXAGONAL CRYSTALS (usually metals) Lattice ◎ basis Arrangement of Atoms Number of atoms per basis (lattice point) ◎ Number of atoms per lattice site 4. 3. 1 Body-centered Cubic Crystal F 302 f 3. 3 -1 f 3. 3 -2 BCC ) structure : e. g. , tungsten ◎ body-centered cubic ( BCC chromium , iron , molybdenum , and vanadium. ◎ a 0 (a 0 /2=2 r , or a 0 (BCC)=4 r/ a, r (3. 3) R) ◎ The total number of atoms per unit cell is two CN {[8× (1/8)]+(1× 1)} ; CN(BCC)=8 9

4. 3. 2 Face-centered Cubic Crystals F 301 f 3. 3 -3 ◎ Face-centered Cubic ( FCC ) structure : e. g. , aluminum , calcium , cooper , gold , lead , platinum , and silver. ( cubic close – packed structure ) ◎ (FCC)=4 r/ (3. 1) ◎ There are four atoms per FCC cell {[8× (1/8)]+[(6× 1/2)]} , CN(FCC)=12 F 303 f 3. 3 -4 4. 3. 3 Hexagonal Close-packed structure Hexa ◎ Hexagonal close-packed ( HCP ) structure : e. g. , cadmium , cobalt , magnesium , titanium , yttrium , and zinc. 10

◎ The total number of atoms in the large HCP cell is six {[12× (1/6)]+[2× (1/2)]+(3× 1)} , CN(HCP)=12. ◎ =2 r ◎ c=(4/ ◎ ) =1. 633 =3. 266 r (large HCP)=( ) (3. 3 -3) (3. 3 -4) 4. 4 MILLER INDICES 4. 4. 1 Coordinates of Points F 305 ◎ Align the three coordinate axes with the edges of the unit cell , with the origin at a corner of the cell. E 3. 4 E 3. 5 11

4. 4. 2 Indices of Directions F 306 ◎ Miller indices for directions are obtained using the following procedure: ※ Method one (1) Determine the coordinates of two points that lie in the direction of interest：h 1, k 1, l 1, and h 2, k 2, l 2. (2) h = h 2－ h 1; k = k 2 － k 1 ; l = l 2 － l 1. (3) Clear fractions from the differences - h, k , and l , to give indices in lowest integer values nh , nk , nl. • h：k：l determines the direction • h：k：l = nh：nk：nl 12

(4) [nh’ nk’ nl’] (5) For negative integers , as an example , if h<0 , we write [h k l]. E 3. 4 -2 ※ Method two (1) A vector of convenient length is positioned such that it passes through the origin of the coordinate system. (2) The length of the vector projection are measured in terms of the unit cell dimensions a , b , c. (3) These numbers are reduced to the smallest integer values. E 3. 6 E 3. 7 13

(4) The three indices , are enclosed in square brackets , thus : [uvw]. ◎ Families of directions : a cubic crystal as an example all the edges are equivalent, all the face diagonals are equivalent , all the body diagonals are equivalent , thus families of direction : <100> , <111>. <100>: [100], [ī 00], [010], [0 ī 0], [001] and [00 ī] <110>: [110], [101], [011], [ ī 10], [ī 01], [0ī 1], [1ī 0], [10ī], [01ī], [īī 0], [ī 0ī] and [0īī] <111>: A 14

• Directions in cubic crystals having the same indices without regard to order or sign (e. g. , [123], [213], and [123]), are equivalent. • Only arrangement of atoms (or ions) are important for directions. • Directions (with different indices) having similar atomic arrangement are equivalent. E 3. 4 -4 ◎ The angle between direction. If A= ui + vj + wk and B= i+ j+ k , then A·B= A B cos = {(u +v +w (3. 4 -1) )/[( + + ]} (3. 4 -2) 15

Z (h 2, k 2, l 2) a b Y (h 1, k 1, l 1) X b=y 2 -y 1 h=h 2 -h 1 a=x 2 -x 1 k=k 2 -k 1 l=l 2 -l 1 What is projection? What is the projection of OC on X axis? On Y axis? What is the projection of AB on X axis? On Y axis? 16

For a cubic crystal <100>: [100], [ī 00], [010], [0 ī 0], [001] and [00 ī] How many edges does a cubic unit cell have? The same: Each set of the four parallel edges are the same and can only be counted as one member of the family. But its opposite direction is counted as another family member. Equivalent: [100], [ī 00], [010], [0 ī 0], [001] and [00 ī] 17

4. 4. 3 Indices of Planes ◎ Miller indices for planes : F 3. 9 1. Identify the coordinates at which the plane intersects the x , y , and z axes 2. Take the reciprocal of the intercepts 3. Clear fractions 4. Cite planes in parentheses : (h k l ) E 3. 9 E 3. 10 E 3. 4 -6 E 3. 4 -7 18

◎ Families of planes , {h k l} : all planes in a family are equivalent in that they contain exactly the same arrangement of atoms. In cube systems, for example , the members of {1 0 0} are (1 0 0) , (0 1 0) , (0 0 1) , and their negatives (ī 0 0) , (0 ī 0) , and (0 0 ī); and {111}: (ī ī ī), (ī 11), (1 ī ī), (11 ī), (ī ī 1), (ī 1 ī), (1 ī 1), and (111), ◎ Only arrangement of atoms are important for planes. 19

※ Several important features and relationships : 1. Planes and their negatives are equivalent. The negatives of directions are not equivalent but rather point in opposite directions. 2. Planes are not necessarily equivalent to their multiples. Directions are invariant to a multiplier. 3. In cubic crystals , a plane and a direction with the same indices are orthogonal. F 19. 9 20

4. 4. 4 Indices in the Hexagonal System Miller-Bravais coordinate system : F 3. 3 F 3. 7 f 3. 16 Utilizing a four-axis, or Miller-Bravais coordinate system The three a 1, a 2, and a 3 axes are within a single plane (called the basal plane). The z axis is perpendicular to this basal plane. Unit Cell ◎ Direction Four indices, as [uvtw]; the first three indices pertain to projections along the respective a 1, a 2, and a 3 axes in the basal plane. 21

Example a 1 : projections: 1, -½, 0 [2ī ī 0] 22

Conversion from the three-index system to the four-index system, (3. 6 a) (3. 6 b) (3. 6 c) (3. 6 d) E 3. 8 n is a factor that may be required to reduce u, v, t, and w to the smallest integers. 23

◎ Planes four-index (hkil) scheme, Example Basal planes: (0001) intercepts: a 1, ∞； a 2, ∞； a 3, ∞；C, 1. h = 0, k = 0, i = 0, l = 1 24

Prism plane(ABCD): (10ī 0) intercepts: a 1, +1; a 2, ; a 3, -1; C, . h = 1, k = 0, i = -1, l = 0 prism plane (DCGH): (01ī 0) family of prism planes: {10ī 0} (3. 7) E 3. 11 The three h, k, and l indices are identical for both indexing systems. 25

4. 5 DENSITIES AND PACKING FACTORS OF CRYSTALLINE STRUCTURES 4. 5. 1 Linear Density ◎ linear density ( or LD) is the number of equivalent lattice points per unit length along a direction. = ( LD ) 26

As an example , consider the [1 1 0] direction in an FCC crystal , the number of atoms is 2 [i. e. , 2×(1/2)+(1× 1)] , the length of the line is 4 r. for [1 1 0] in FCC is 1/(2 r). F 312 ◎ The <110> family of directions has special significance in the FCC structure , since these are the directions in which atoms are in direct contact. As such , <110> directions have the highest of any directions in the FCC system. In any crystal system the directions with the highest are termed the close-packed directions. 27

4. 5. 2 Planar Density ◎ Planar density ( or PD) is the number of atoms per unit area on a plane of interest. = ◎ Consider the planar density of the (1 1 1) plane in a FCC crystal. The length of the side of each triangle is 4 r. The area is 4 (i. e. , ; the three atoms at the corners , 1/6 ) the three atoms along the edges , fraction of ½. The total number of atoms on the plane is two. Planar density is 1/(2 ( PD ) ). f 3. 5 -2 28

◎ This value of represents the highest possible planar density for spherical atoms , any plane in any crystal system that has a value of =1/(2 ) will be referred to as a close-packed plane. ◎ For any specific crystal system , a family of planes with a maximum value are referred to as highest -density planes. 29

4. 5. 3 Volumetric Density ◎ Volumetric density ( or VD ) is the number of atoms per unit volume. ◎ for FCC is 4/(16 )=1/(4 ). This is the highest volumetric density possible for spherical atoms. This type of structure will be referred to as close-packed structures. HCP crystals also have = 1/(4 ) f 3. 3 -3 30

4. 5. 4 Atomic Packing Factors and Coordination Numbers ◎ The ratio of the volume occupied by the atoms to the total available volume is defined to be the atomic packing factor (APF) (3. 2) APF = ◎ APF(SC)=0. 52 , APF(BCC)=0. 68 , and APF(FCC)= APF(HCP)=0. 74. F 3. 2 ◎ CN=12 for APF=0. 74. 31

4. 5. 5 Theoretical Density , A knowledge of the crystal structure of a metallic solid permits computation of its theoretical density through the relationship (3. 5) where n = number of atoms associated with each unit cell A = atomic weight F 4. 12 Vc = volume of the unit cell NA = Avogadro’s number (6. 023 × 1023 atoms/mol) 4. 5. 6. % Theoretical density ( = / 32 100% )

4. 5. 6 Close-Packed Structures ◎ Both FCC and HCP structures are characterized by an APF 74% , a CN of 12 , the close-packed planes , the close-packed directions and the difference is the arrangement of their close-packed planes. ◎ FCC：cubic close-packed structure ◎ HCP：hexagonal close-packed structure f 3. 3 -4 117 f 3. 5 -3 112 f 3. 5 -4 67 122 33

4. 8 CRYSTALLINE AND NONCRYSTALLINE MATERIALS Solids: F 3. 22 166 crystalline : with long rang order (of course , also with short rang order) semicrystalline : with some long range order amorphous : with short range order only 34

Crystalline solid: (1)sharp melting point f 4 -2. 2 f 4 -2. 3 f 3 -18 FACET (2)well-developed faces and a characteristic shape (3)Characteristic X-ray diffraction pattern (peaks) shows a regular , ordered structure composed of identical repeating units having the same orientation throughout the crystal (with short and long-range order) motion 35

Amorphous solid: f 6 -2. 1 (1)When heated , it softens and melts over a wide temperature range (glass-transition temperature) (2)does not have a characteristic crystal shape (3)No characteristic X-ray diffraction peaks. Semicrystalline solids: (1)crystalline polymers (2)Liquid crystal : fluids that show some degree of long-range order. f 24. 8 F 15 -13 motion 36

4. 9 SINGLE CRYSTALS AND POLYCRYSTALLINE MATERIALS: Microstructure ◎Crystal: Microstructure • Single crystal The whole piece (or body) of material has the same crystal structure and orientation. • Polycrystalline made up of a number of crystals (small single)with identical structures but different orientations. ※ grains : small (single)crystals 0. 5 – 50 m. F 3. 17 f 3. 9 -1 ※Grain boundaries : internal surfaces of finite thickness where crystals of different orientations meet. f 4. 12 37

◎Many ceramic materials are in the form of polycrystalline solids. ◎Noncrystalline structure (amorphous) can be formed by cooling a material sufficiently quickly (quenching) that crystal formation does not occur. ◎Polymers are usually either semicrystalline or amorphous. ◎Few materials are used in single-crystal form, single-crystal materials have no grain boundaries, so they offer unique mechanical, optical, and electrical properties. ◎Amorphous materials do not have grain boundaries either. ＊Grain boundaries can scatter photons (light), phonons (heat) and electrons, rendering the materials to be less transparent, less thermally and electrically conductive. 38

Examples : 1. Single-crystal quartz (Si. O ) and perovskites are used as transducers, such as : telephone receivers, and phonograph cartridges. 2. Single-crystal germanium and silicon : microelectronics industry. 3. Single-crystal nickel alloys : turbine blades. 4. Sapphire (A 1 O ) and diamond (C) single crystals : precious stones. 39

4. 10 POLYMORPHISM ◎ Materials whose crystal structures change from one unit cell to another at specific temperatures are termed polymorphic. Iron, for example, Fe (BCC, room temp) 1394 ℃ 912 ℃ Fe ( FCC at 912 ℃) (BCC , 1394 ℃) Other examples : Silica carbon, (Si. O 2), alumina (A 12 O 3), and titania (Ti. O 2) ◎ Property changes accompany structural changes : e. g. , volume, density, may increase or decrease. 40

◎ Many brittle materials cannot withstand the internal forces that develop as a result of these volume changes. These materials fail at the transformation temperature. An example is zirconia, Zr. O 2 : cooling, ~ 1000 ℃ p. 48 Tetragonal monoclinic crack !! f 3. 10 -1 ◎ Another polymorphic materials is carbon. 1. The diamond structure has a 3 -D tetrahedral network of covalent bonds with CN = 4, the highest melting temperature of any of the elements. 2. Graphite has a hexagonal two-dimensional layered structure. The bonds within each layer are covalent bonds, but the interlayer are weak secondary bonds which are easily broken giving graphite its excellent lubricating properties. 41

4. 11 ANISOTROPY Isotropic : properties of a material are independent of direction : Anisotropic : properties of a material depend on direction : ◎ properties, such as modulus of elasticity and coefficient of thermal expansion can be estimated from the bond-energy curve. But the curve is derived in the close-packed directions. The separation distance between atoms in any direction other than the close packed directions will be greater than . ◎ Single crystals exhibit some degree of anisotropy. (because the atomic packing depends on direction. ) F 2. 8 f 0103 42

◎ An example of a highly anisotrophic crystalline material is graphite. Its coefficient of thermal expansion along the c axis is more than 25 times greater than that in any direction parallel to the basal planes. f 3. 10 -1 ◎ While single crystals are anisotrophic, most polycrystals are nearly isotropic on the macroscopic scale. F 3. 17 F 4. 12 ◎ Anisotropic can also occur on a larger structural scale examples : wood, steel-reinforced concrete, carbon-fiberreinforced epoxy, oriented polymers. 43

4. 12 DETERMINATION OF CRYSTAL STRUCTURES X-Ray Diffraction f 3 -17 f 24. 17 f 3. 15 F 3. 18 F 3. 19 X-Ray Diffraction is generated by interaction between X-ray photons and electrons in the material. Figure 24. 17 b shows the x-ray beam incident at angle to one of the (210) planes. Most of the x-ray photons will pass through this plane with no change in direction, but a small fraction will collide with electrons in the atoms of this plane and will be scattered, that is, will undergo a change in direction. The x-ray photons are scattered in all directions. The zero-path-difference condition gives 0 = pq (cos - cos ), so =. Thus, waves scattered from a plane of lattice points at an angle equal to the angle of incidence are in phase with one another. Waves scattered at other angles will generally be out phase with one another and will give destructive interference. diffraction f 3. 18 44

The single plane of lattice points acts as a “mirror” and “reflects” a small fraction of the incident x-rays. The x-ray beam will penetrate the crystal to a depth of millions of layers of planes. For constructive interference between x-rays scattered by the entire set of planes, the waves constructively scattered (“reflected”) by two adjacent planes must have a path difference of an integral number of x-ray wavelengths. 2 d sin = n n = 1, 2, 3, ……. f 3. 12 -2 The Bragg equation is the fundamental equation of x-ray crystallography. n=1 1 st order diffraction n=2 2 nd order diffraction 45

F 19. 6 F 21 -3 F 19. 9 XRD-1 46 Only planes with low indices generate XRD peaks

47

For the special case that the unit cell axes are mutually perpendicular (i. e. , for orthorhombic, tetragonal, and cubic unit cells), i. e. , α=β=γ=90° = f 19. 9 2 d sin = n 48

For cubic unit cells: a = b = c (for n = 1) Interplanar Spacing (dh, k, l) of Cubic Lattice (1) Simple cubic For primitive cubic crystals dhkl may have the following values: (100) (111) (200) (120) (112) θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 (220) 49

(2) Face-centered cubic (FCC) The diffraction patterns of face-centered crystals show an absence of all reflections for which the indices hkl are not all even or all odd. Hence, only the reflections 111, 200, 311, 222, 400, 331, 420, etc. , are observed. Consequently, the spacing corresponding to the hkl reflections that may appear on a powder diagram of a face-centered cubic crystal are: f 19. 10 50

(3) Body-centered cubic (BCC) An examination of the diffraction data for body ecentered crystals shows that hkl reflections for which the sum h+k+l is odd are not observed. Accordingly, the interplanar spacing found for a body-centered cubic lattice are which are the distances between the (110), (200), (211), and (220) planes. Other XRD Phenomena and Techniques F 3. 20 f 19. 8 f 19. 11 F 3. 6 f 3. 22 f 3. 13 f 3. 12 -3 F 3. 21 51

The locations of the spots in the diffraction pattern depend only on the nature of the spae lattice and the lengths and angles of the unit cell and that the intensities of the spots depend on the kinds of atoms and the locations of atoms (interactomics distances) within the basis of the crystal structure. Analysis of the locations of the diffraction spots gives the space lattice and the lengths and angles of the unit cell. The unit-cell content z is calculated using the know crystal density. Analysis of the electron probability density (x, y, z) as a function of position in the unit cell. From contours of , the locations of the nuclei are obvious, and bond lengths and angles can be found. 52

The Use of x-Ray Diffraction (1) Identify the crystalline structure of a sample with known chemical composition. Arrangement and the spacing of atoms in crystalline materials (2) Identify the compound (chemical composition) and crystalline structure of an unknown sample. x-Ray diffraction also provides compounds, an x-ray diffraction pattern is unique for each crystalline substance. Thus, if an exact match can be found between the pattern of unknown and an standard sample, chemical identity can be assumed. (3) Estimate the relative amounts of different phases Quantitative information concerning a crystalline compound in a mixture, for example, the percentage of graphite in a graphite charcoal mixture. 53

Types of motion and energy (sensitive to T : motion and energy (unit of motion or particles under consideration) gas single atoms (di- or many atomic) molecules Translation (atoms or molecules) ○ Rotation (atoms) T ) liquid solid ○ ○ × × ○ ○ × Vibration (atoms or bondings) × ○ ○ ○ Electronic (electrons) ○ ○ ○ ○ 164

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Structure and properties of Materials raw materials synthesis shape forming Processing techniques and operation conditions sintering fabrication Material products Structure Properties (結構) (性質) Structure of Materials： atomic scale structure (~0. 1 nm) structure types of atoms, types of bondings, atomic packing (coordination number) scale 小 scale 小 Crystalline structure (>~10 nm) 大 back Microscopic (scale) structure (~0. 1µm) ：SEM 大 166

Basis: Lattice(or space lattice) (arrangement of lattice point: )

A. Metallic Bonding p CN is primarily determined by geometric arrangement (consideration) ＊all atoms：spherical , same size. ＊CN： 8 -12. ＊usually described by crystalline structure. F 3. 1 F 3. 2 T 3. 1 169

XRD a particular material different atoms sets of crystalline structure planes (lattice) (a, b, c, (h, k, l) α, β, γ) relative intensity of peaks d-spacing(or interplanar distance) Bragg Eg. λ=constant standard XRD patterns Or data or software 170

atomic scale structure crystalline structure micro-stucture single crystal same different no grains, no grain boundaries polycrystalline same different made up of grains and grain boundaries Microstructure 172

Basis(group) a, b, c, straight lines Unit Cell α, β, γ Lattice point (Space)Lattice Crystal Systems Bravais Lattices (Type of lattice Actual Materials (arrangement of atoms) lattice points complexity difficulty (arrangement of lattice point basis(group) Number of atoms (or ions)in the basis.

atoms lattice Basis(group) Lattice point (Space)Lattice Actual Materials (arrangement of atoms) difficulty Unit Cell Crystal Systems Bravais Lattices (Type of lattice) lattice points complexity straight lines a, b, c, α, β, γ (arrangement of lattice points) basis(group) Number of atoms (or ions)in the basis.