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Crystal Structure: Materials and Lattice Points

This chapter explores the crystal structure of materials, including crystalline and amorphous solids. It covers topics such as lattice points, basis groups, space lattice, unit cell, crystal systems, and crystals with one atom per lattice site.

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Crystal Structure: Materials and Lattice Points

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

  2. 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 F3.22

  3. 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 ．Lattice point≡ basis Basis

  4. 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. (All crystalline solids have ordered structures They all have repeated units (structure units). ) f24.3

  5. For NaCl, 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 CO2, the basis is four CO2 molecules. For benzene, the basis is four C6H6 molecule. f13.2 f24.15 f3.7-2 f3.7-3 f3.7-4 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. f24.3 f302

  6. 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. 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. f24.4 F3.4

  7. 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. f24.5 Unit cells that have lattice points only at their corners are called primitive (or simple) unit cells. Seven of the Bravais lattices have 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. f24.5 T3.2

  8. 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. F301 F302

  9. 4.3 CRYSTALS WITH ONE ATOM PER LATTICE SITE AND HEXAGONAL CRYSTALS (usually metals) Lattice Arrangement of Atoms ◎ Number of atoms per basis (lattice point) ◎ Number of atoms per lattice site 4.3.1 Body-centered Cubic Crystal ◎ ( BCC ) structure :e.g., tungsten chromium , iron , molybdenum , and vanadium. ◎ a0 /2=2r , or a0 (BCC)=4r/ (3.3) (a0 a , r R) ◎ The total number of atoms per unit cell is two {[8× (1/8)]+(1× 1)} ; CN(BCC)=8 basis F302 f3.3-1 f3.3-2 body-centered cubic BCC CN

  10. 4.3.2 Face-centered Cubic Crystals ◎ Face-centered Cubic ( FCC ) structure : e.g. , aluminum , calcium , cooper , gold , lead , platinum , and silver. ( cubic close – packed structure ) ◎ (FCC)=4r/ (3.1) ◎ There are four atoms per FCC cell {[8× (1/8)]+[(6× 1/2)]} , CN(FCC)=12 4.3.3 Hexagonal Close-packed structure ◎ Hexagonal close-packed ( HCP ) structure : e.g. , cadmium , cobalt , magnesium , titanium , yttrium , and zinc. F301 f3.3-3 Face-centered Cubic FCC cubic close – packed F303 f3.3-4 Hexa Hexagonal close-packed HCP

  11. ◎ The total number of atoms in the large HCP cell is six {[12× (1/6)]+[2× (1/2)]+(3× 1)} , CN(HCP)=12. ◎ =2r ◎ c=(4/ ) =1.633 =3.266r (3.3-3) ◎ (large HCP)=( ) (3.3-4) 4.4 MILLER INDICES 4.4.1 Coordinates of Points ◎ Align the three coordinate axes with the edges of the unit cell , with the origin at a corner of the cell. E3.4E3.5 F305

  12. 4.4.2 Indices of Directions ◎ 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：h1, k1, l1, and h2, k2, l2. (2) h = h2－ h1; k = k2 － k1 ; l = l2－ l1. (3) Clear fractions from the differences - h, k , and l , to give indices in lowest integer valuesnh , nk , nl. F306 • h：k：l determines the direction • h：k：l = nh：nk：nl

  13. (4) [nh’ nk’ nl’] (5) For negative integers , as an example , if h<0 , we write [h k l]. ※ 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. E3.4-2 E3.6 E3.7

  14. (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> , <110> , <111>. <100>: [100], [ī00], [010], [0 ī 0], [001] and [00 ī] <110>: [110], [101], [011], [ ī10], [ī01], [0ī1],[1ī0], [10ī], [01ī], [īī0], [ī0ī] and [0īī] <111>: A

  15. ◎ The angle between direction. If A= ui + vj + wk and B= i+ j+ k , then A·B= A B cos (3.4-1) = {(u +v +w )/[( + + ( + + ]} (3.4-2) • 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. E3.4-4

  16. Z (h2,k2, l2) b Y (h1,k1, l1) a X b=y2-y1 a=x2-x1 h=h2-h1 k=k2-k1 l=l2-l1 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?

  17. For a cubic crystal <100>: [100], [ī00], [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], [ī00], [010], [0 ī 0], [001] and [00 ī]

  18. 4.4.3 Indices of Planes ◎ Miller indices for planes : 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 ) F3.9 E3.9 E3.10 E3.4-6 E3.4-7

  19. ◎ 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 ī1), (ī 1 ī), (1 ī 1), and (111), ◎Only arrangement of atoms are important for planes.

  20. ※ 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

  21. 4.4.4 Indices in the Hexagonal System Miller-Bravais coordinate system : F3.3 F3.7 f3.16 Utilizing a four-axis, or Miller-Bravais coordinate system The three a1, a2, and a3 axes are within a single plane (called the basal plane). The z axis is perpendicular to this basal plane. Four indices, as [uvtw]; the first three indices pertain to projections along the respective a1, a2, and a3 axes in the basal plane. Unit Cell ◎ Direction

  22. Example a1 : projections: 1, -½, -½, 0 [2ī ī0]

  23. Conversion from the three-index system to the four-index system, (3.6a) (3.6b) (3.6c) (3.6d) E3.8 n is a factor that may be required to reduce u, v, t, and w to the smallest integers.

  24. ◎ Planes four-index (hkil) scheme, Example Basal planes: (0001) intercepts: a1, ；a2, ；a3, ；C, 1. h = 0, k = 0, i = 0, l = 1 ∞ ∞ ∞

  25. Prism plane(ABCD): (10ī0) intercepts: a1,+1; a2,; a3,-1; C,. h = 1, k = 0, i = -1, l = 0 prism plane (DCGH): (01ī0) family of prism planes: {10ī0} (3.7) E3.11 The three h, k, and l indices are identical for both indexing systems.

  26. 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 )

  27. 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 4r. for [1 1 0] in FCC is 1/(2r). ◎ 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. F312

  28. 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 4r . The area is 4 ; the three atoms at the corners , 1/6 (i . e ., ) the three atoms along the edges , fraction of ½ . The total number of atoms on the plane is two . Planar density is 1/(2 ). f3.5-2 ( PD )

  29. ◎ 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.

  30. 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 ) f3.3-3

  31. 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) APF = ◎ APF(SC)=0.52 , APF(BCC)=0.68 , and APF(FCC)= APF(HCP)=0.74. ◎ CN=12 for APF=0.74. (3.2) F3.2

  32. 4.5.5 Theoretical Density , A knowledge of the crystal structure of a metallic solid permits computation of its theoretical density through the relationship where n = number of atoms associated with each unit cell A = atomic weight Vc = volume of the unit cell NA = Avogadro’s number (6.023 × 1023 atoms/mol) 4.5.6. % Theoretical density (= / 100% ) (3.5) F 4.12

  33. 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 f3.3-4f3.5-3f3.5-4 117 112 67 122

  34. 4.8 CRYSTALLINE AND NONCRYSTALLINE MATERIALS Solids: F3.22 166 crystalline : with long rang order (of course , also with short rang order) semicrystalline : with some long range order amorphous : with short range order only

  35. Crystalline solid:f4-2.2f4-2.3f3-18 (1)sharp melting point (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 units having the same orientation throughout the crystal (with short and long-range order) FACET motion

  36. Amorphous solid:f6-2.1 (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. f24.8 F15-13 motion

  37. 4.9 SINGLE CRYSTALS AND POLYCRYSTALLINE MATERIALS: Microstructure ◎Crystal: • 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. F3.17f3.9-1 ※Grain boundaries : internal surfaces of finite thickness where crystals of different orientations meet. Micro-structure f4.12

  38. ◎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.

  39. Examples : • Single-crystal quartz (SiO ) 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 (A1 O ) and diamond (C) single crystals : precious stones.

  40. 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) Fe (FCC at 912 ℃) (BCC , 1394 ℃) Other examples : Silica carbon, (SiO2), alumina (A12O3), and titania (TiO2) ◎ Property changes accompany structural changes : e.g., volume, density, may increase or decrease. 912 ℃ 1394 ℃

  41. ◎ 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, ZrO2 : Tetragonal monoclinic crack !! ◎ 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. cooling, ~ 1000 ℃ p.48 f3.10-1

  42. 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.) f0103 F2.8

  43. ◎ 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. ◎ While single crystals are anisotrophic, most polycrystals are nearly isotropic on the macroscopic scale. ◎ Anisotropic can also occur on a larger structural scale examples : wood, steel-reinforced concrete, carbon-fiber- reinforced epoxy, oriented polymers. f3.10-1 F3.17 F4.12

  44. 4.12 DETERMINATION OF CRYSTAL STRUCTURES F3.18 F3.19 X-Ray Diffraction f3-17f24.17f3.15 X-Ray Diffraction is generated by interaction between X-ray photons and electrons in the material. Figure 24.17b 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. f3.18 diffraction

  45. 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. f3.12-2 2d sin = n n = 1,2,3,……. The Bragg equation is the fundamental equation of x-ray crystallography. n=1 1st order diffraction n=2 2nd order diffraction

  46. F19.6 F21-3 F 19.9 XRD-1 XRD 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° = 2d sin = n f19.9

  48. For cubic unit cells: a = b = c (for n = 1) (1) Simple cubic For primitive cubic crystals dhkl may have the following values: (100) (110) (111) (200) (120) (112) (220)  θ1 θ2 θ3 θ4 θ5 θ6 θ7 Interplanar Spacing (dh,k,l) of Cubic Lattice

  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: f19.10

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