Crystal Structures of Solids. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Transcription

1 Crystal Structures of Solids 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

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3 Table of metals, metalloids, and nonmetals 3

4 Solids, Liquids and Gases 4 1A 8A H 2A 3A 4A 5A 6A 7A He Li Be B C N O F Ne Na Mg 3B 4B 5B 6B 7B 8B 1B 2B Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Rf Db Sg Bh Hs Mt Ds S L G Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

5 The Solid State 1. Classification of Solid Structures a. Crystalline Solids = regular arrangement of components in 3 dimensions b. Amorphous Solids = disordered arrangement of components 2. Crystal Structure Basics a. Crystal = a piece of a crystalline solid b. Lattice = 3-dimensional system of designating where components are c. Unit cell = smallest repeating unit of the lattice d. Examples: simple cubic, body-centered cubic, face-centered cubic 5

6 Early Crystallography 6 René-Just Haüy (1781): cleavage of calcite Common shape to all shards: rhombohedral How to model this mathematically? What is the maximum number of distinguishable shapes that will fill three space? Mathematically proved that there are only 7 distinct space-filling volume elements

7 Christian Huygens Studying calcite crystals made drawings of atomic packing and bulk shape.

8 Beryl Be 3 Al 2 (SiO 3 ) 6 8

9 Anisotropy 9 The physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are taken. For example, the elastic modulus, electrical conductivity, and the index of refraction may have different values in the [100] and [111] directions. The directionality of the properties is termed anisotropy and is associated with the atomic spacing.

10 Isotropic 10 If measured properties are independent of the direction of measurement then they are isotropic. For many polycrystalline materials, the crystallographic orientations of the individual grains are totally random. So, though, a specific grain may be anisotropic, when the specimen is composed of many grains, the aggregate behavior may be isotropic.

11 Single Crystals 11 For a crystalline solid, when the periodic and repeated arrangement of atoms extends throughout without interruption, the result is a single crystal. The crystal lattice of the entire sample is continuous and unbroken with no grain boundaries. For a variety of reasons, including the distorting effects of impurities, crystallographic defects and dislocations, single crystals of meaningful size are exceedingly rare in nature, and difficult to produce in the laboratory under controlled conditions. Huge KDP (monopotassium phosphate) crystal grown from a seed crystal in a supersaturated aqueous solution at LLNL. Below, silicon boule.

12 Single Crystal A garnet single crystal found in Tongbei, Fujian Province, China. If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume its geometric shape, with flat surfaces as shown in the figure. 12 The periodic and repeated arrangements of atoms is perfect or extends throughout the entirety of the specimen without interruption. All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but they may also produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled. Single crystals are needed for modern technologies today. Electronic micro-chips uses single crystals of silicon and other semiconductors.

13 Polycrystalline Materials 13 Composed of a collection of many small crystals or grains. Stages in the solidification of a polycrystalline material: a. Crystallite Nuclei b. Growth of the Crystallites c. Formation of grains d. Microscopic view

14 Anisotropy 14 Physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are made. This directionality of properties is termed anisotropy, and it is associated with the variance of atomic or ionic spacing with crystallographic direction. Substances in which measured properties are independent of the direction of measurement are isotropic.

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16 16 Structures The properties of some materials are directly related to their crystal structures. Significant property differences exist between crystalline and noncrystalline materials having the same composition.

17 Crystals as Building Blocks Some engineering applications require single crystals: -- diamond -- turbine blades Single crystals for abrasives Properties of crystalline materials often related to crystal structure. -- Ex: Quartz fractures more easily along some crystal planes than others. 17

18 Atomic Arrangement Minerals must have a highly ordered atomic arrangement The crystal structure of quartz is an example 18

19 olivine epidote Examples of silicate minerals Mineral pictures from: mindat.org 19 augite hornblende beryl muscovite quartz

20 Quartz Varieties 20 Pink (Rose) : due to traces of iron, manganese or titanium. Amethyst : May be manganese but some believe it could be organic, iron or even aluminum. Citrine : iron Aventurine : inclusion of green mica (fushite) Tiger's eye : inclusion of fiber of silicified crocidolite (variety of asbestos) Prasiolite : Iron or copper Milk quartz : gas and liquid inclusions Smoky : Radioactivity on quartz containing aluminium Blue : pressure. Chalcedony is a variety of quartz with micro-crystals. Agate is a multicolor variety of chalcedony and onyx is a variety of agate with parallel strips of various nuances of black.

21 Quartz Crystals The external appearance of the crystal may reflect its internal symmetry 21

22 Quartz Blob Or the external appearance may show little or nothing of the internal structure 22

23 Building Blocks A cube may be used to build a number of forms 23

24 Fluorite Fluorite may appear as octahedron (upper photo) Fluorite may appear as a cube (lower photo), in this case modified by dodecahedral crystal faces 24

25 Crystal Growth 25 Ways in which a crystal can grow: Dehydration of a solution Growth from the molten state (magma or lava) Direct growth from the vapor state

26 Types of Solids 26 Single crsytal, polycrystalline, and amorphous, are the three general types of solids. Each type is characterized by the size of ordered region within the material. An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.

27 Crystalline Solid 27 Single crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry Single Pyrite Crystal Single Crystal Amorphous Solid

28 Ionic Crystals 28 Examples include sodium chloride, cupric sulfate.

29 Some Factors Affecting Crystalline Structure 29 Size of atoms or ions involved Stoichiometry of salt Materials involved Some substances do not form crystalline solids

30 Crystals 30 The periodic array of atoms, ions, or molecules that form the solids is called Crystal Structure Crystal Structure = Space (Crystal) Lattice + Basis Space (Crystal) Lattice is a regular periodic arrangement of points in space, and is purely mathematical abstraction Crystal Structure is formed by putting the identical atoms (group of atoms) in the points of the space lattice This group of atoms is the Basis

31 31 The definition of crystals are based on symmetry and not on the geometry of the unit cell Our choice of unit cell cannot alter the crystal system a crystal belongs to Crystals based on a particular lattice can have symmetry equal to or lower than that of the lattice When all symmetry (including translation) is lost the construct is called amorphous

32 Ideal versus Real crystals 32 Ideal crystals may have perfect positional and orientational order with respect to geometrical entities and physical properties In (defining) real crystals some of these strict requirements may be relaxed: the order considered may be only with respect to the geometrical entity the positional order may be in the average sense the orientational order may be in the average sense In addition real crystals: are finite may contain other defects.

33 Why study crystal structures? 33 When we look around us many of the organic materials are non-crystalline But, many of the common inorganic materials are usually* crystalline: Metals: Cu, Zn, Fe, Cu-Zn alloys Semiconductors: Si, Ge, GaAs Ceramics: Alumina (Al 2 O 3 ), Zirconia (Zr O 2 ), SiC, SrTiO 3 Also, the usual form of crystalline materials (say a Cu wire or a piece of alumina) is polycrystalline and special care has to be taken to produce single crystals Polymeric materials are usually not fully crystalline The crystal structure directly influences the properties of the material (as we have seen in the Introduction chapter many additional factors come in) Why study crystallography? Gives a terse (concise) representation of a large assemblage of species Gives the first view towards understanding of the properties of the crystal * Many of the materials which are usually crystalline can also be obtained in an amorphous form

34 What is crystallography? Crystallography 34 The branch of science that deals with the geometric description of crystals and their internal arrangement. Crystallography is essential for solid state physics Symmetry of a crystal can have a profound influence on its properties. Any crystal structure should be specified completely, concisely and unambiguously. Structures should be classified into different types according to the symmetries they possess.

35 35 Elementary Crystallography A basic knowledge of crystallography is essential for solid state physicists; to specify any crystal structure and to classify the solids into different types according to the symmetries they possess. Symmetry of a crystal can have a profound influence on its properties. We will concern in this course with solids with simple structures.

36 Crystal Lattice 36 What is crystal (space) lattice? In crystallography, only the geometrical properties of the crystal are of interest, therefore one replaces each atom by a geometrical point located at the equilibrium position of that atom. Platinum Platinum surface Crystal lattice and (scanning tunneling microscope) structure of Platinum

37 Crystal Lattice 37 An infinite array of points in space, y B b C α D E Each point has identical surroundings to all others. O a A x Arrays are arranged exactly in a periodic manner.

38 38 The unit cell is the basic repeating unit of the arrangement of atoms, ions or molecules in a crystalline solid. The lattice refers to the 3-D array of particles in a crystalline solid. One type of atom occupies a lattice point in the array.

39 Examples of Unit Cells 39

40 Crystal Systems 40 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

41 Unit Cell Concept 41 The unit cell is the smallest structural unit or building block that uniquely can describe the crystal structure. Repetition of the unit cell generates the entire crystal. By simple translation, it defines a lattice. Lattice: The periodic arrangement of atoms in a crystal. a Lattice Parameter : Repeat distance in the unit cell, one for in each dimension b

42 The Unit Cell Concept 42 The simplest repeating unit in a crystal is called a unit cell. Opposite faces of a unit cell are parallel. The edge of the unit cell connects equivalent points. Not unique. There can be several unit cells of a crystal. The smallest possible unit cell is called primitive unit cell of a particular crystal structure. A primitive unit cell whose symmetry matches the lattice symmetry is called Wigner-Seitz cell.

43 43 Each unit cell is defined in terms of lattice points. Lattice point not necessarily at an atomic site. For each crystal structure, a conventional unit cell, is chosen to make the lattice as symmetric as possible. However, the conventional unit cell is not always the primitive unit cell. A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage (tendency to split along certain planes with smooth surfaces), electronic band structure and optical properties.

44 Crystal Systems 44 Units cells and lattices in 3-D: When translated in each lattice parameter direction, MUST fill 3-D space such that no gaps, empty spaces left. a b c Lattice Parameter : Repeat distance in the unit cell, one for in each dimension

45 The Importance of the Unit Cell 45 One can analyze the crystal as a whole by investigating a representative volume. Ex: from unit cell we can Find the distances between nearest atoms for calculations of the forces holding the lattice together Look at the fraction of the unit cell volume filled by atoms and relate the density of solid to the atomic arrangement The properties of the periodic Xtal lattice determine the allowed energies of electrons that participate in the conduction process.

46 Unit cell 46

47 Unit Cell 47 The unit cell and, consequently, the entire lattice, is uniquely determined by the six lattice constants: a, b, c, α, β and γ. Only 1/8 of each lattice point in a unit cell can actually be assigned to that cell. Each unit cell in the figure can be associated with 8 x 1/8 = 1 lattice point.

48 48 b a Unit cell: Simplest portion of the structure which is repeated and shows its full symmetry. Basis vectors a and b defines relationship between a unit cell and (Bravais) lattice points of a crystal. Equivalent points of the lattice is defined by translation vector. r = ha + kb where h and k are integers. This constructs the entire lattice.

49 49 By repeated duplication, a unit cell should reproduce the whole crystal. A Bravais lattice (unit cells) - a set of points constructed by translating a single point in discrete steps by a set of basis vectors. In 3-D, there are 14 unique Bravais lattices. All crystalline materials fit in one of these arrangements. In 3-D, the translation vector is r = ha + kb + lc

50 Unit Cell in 2D 50 The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal. S S S S S b S S S S S a S S S S S

51 Unit Cell in 2D 51 The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal. The choice of unit cell is not unique. S S b S S a

52 2D Unit Cell example -(NaCl) We define lattice points ; these are points with identical environments 52

53 Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same. 53

54 This is also a unit cell - it doesn t matter if you start from Na or Cl 54

55 - or if you don t start from an atom 55

56 This is NOT a unit cell even though they are all the same - empty space is not allowed! 56

57 In 2D, this IS a unit cell In 3D, it is NOT 57

58 Unit Cell in 3D 58

59 Unit Cell in 3D 59

60 Crystal Structure Crystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif) to the lattice sides of the lattice point. Crystal Structure = Crystal Lattice + Basis 60

61 Lattice Sites in Cubic Unit Cell 61

62 A two-dimensional Bravais lattice with different choices for the basis 62

63 Crystal structure 63 Don't mix up atoms with lattice points Lattice points are infinitesimal points in space Lattice points do not necessarily lie at the centre of atoms Crystal Structure = Crystal Lattice + Basis

64 64 Crystal Lattice Bravais Lattice (BL) Non-Bravais Lattice (non- BL) All atoms are of the same kind All lattice points are equivalent Atoms can be of different kind Some lattice points are not equivalent A combination of two or more BL

65 Types Of Crystal Lattices 65 1) Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appears exactly the same, from whichever of the points the array is viewed. Lattice is invariant under a translation. Nb film

66 Types Of Crystal Lattices 66 2) Non-Bravais Lattice Not only the arrangement but also the orientation must appear exactly the same from every point in a bravais lattice. The red side has a neighbour to its immediate left, the blue one instead has a neighbour to its right. Red (and blue) sides are equivalent and have the same appearance Red and blue sides are not equivalent. Same appearance can be obtained rotating blue side 180º. Honeycomb

67 Translational Lattice Vectors 2D 67 A space lattice is a set of points such that a translation from any point in the lattice by a vector; P R n = n 1 a + n 2 b Point D(n1, n2) = (0,2) Point F (n1, n2) = (0,-1) locates an exactly equivalent point, i.e. a point with the same environment as P. This is translational symmetry. The vectors a, b are known as lattice vectors and (n 1, n 2 ) is a pair of integers whose values depend on the lattice point.

68 Lattice Vectors 2D 68 The two vectors a and b form a set of lattice vectors for the lattice. The choice of lattice vectors is not unique. Thus one could equally well take the vectors a and b as a lattice vectors.

69 Five Bravais Lattices in 2D 69

70 70 Lattice Vectors 3D An ideal three dimensional crystal is described by 3 fundamental translation vectors a, b and c. If there is a lattice point represented by the position vector r, there is then also a lattice point represented by the position vector where u, v and w are arbitrary integers. r = r + u a + v b + w c (1)

71 71 Gold (Why one naturally get Gold in pure form?

72 Hydrated and Unhydrated Crystals Hydrated Crystals - Water molecules become chemically bonded to ions in the crystal. Anhydrous Crystals Crystal without water CuSO 4 5 H 2 O a hydrated crystal 72

73 Hydrated Crystals Ex: CuSO 4 5H 2 O + heat CuSO H 2 O (Blue) (White) Hydrate Anhydrous Hydrate

74 Ionic solids Group 1A (alkali metals) contains lithium (Li), sodium (Na), potassium (K),..and these combine easily with group 7A (halogens) of fluorine (F), chlorine (Cl), bromine (Br),.. and produce ionic solids of NaCl, KCl, KBr, etc. Rare (noble) gases Group 8A elements of noble gases of helium(he), neon (Ne), argon (Ar), have a full complement of valence electrons and so do not combine easily with other elements. Elemental semiconductors Silicon(Si) and germanium (Ge) belong to group 4A. Compound semiconductors The periodic table 1) III-V compound s/c s; GaP, InAs, AlGaAs (group 3A-5A) 2) II-VI compound s/c s; ZnS, CdS, etc. (group 2B-6A) 74

75 Molecular Solids 75 H 2 O, S 8, P 4 Molecules occupy positions in crystal lattice Melting points increase with size and polarity

76 Classification of solids SOLID MATERIALS CRYSTALLINE POLYCRYSTALLINE AMORPHOUS (Non-crystalline) Single Crystal 76

77 Crystalline Solid 77 Crystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimension.

78 Polycrystalline Solids 78 Atomic order present in sections (grains) of the solid. Different order of arrangement from grain to grain. Grain sizes = hundreds of m. An aggregate of a large number of small crystals or grains in which the structure is regular, but the crystals or grains are arranged in a random fashion.

79 Polycrystalline Solid Polycrystal is a material made up of an aggregate of many small single crystals (also called crystallites or grains). The grains are usually 100 nm microns in diameter. Polycrystals with grains that are <10 nm in diameter are called nanocrystalline Polycrystalline Pyrite form (Grain) Polycrystal 79

80 Amorphous Solids 80 No regular long range order of arrangement in the atoms. Eg. Polymers, cotton candy, common window glass, ceramic. Can be prepared by rapidly cooling molten material. Rapid minimizes time for atoms to pack into a more thermodynamically favorable crystalline state. Two sub-states of amorphous solids: Rubbery and Glassy states. Glass transition temperature Tg = temperature above which the solid transforms from glassy to rubbery state, becoming more viscous.

81 Amorphous Solid 81 Amorphous (non-crystalline) Solid is composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures.

82 Single- Vs Poly- Crystal 82 Properties of single crystalline materials vary with direction, ie anisotropic. Properties of polycrystalline materials may or may not vary with direction. If the polycrystal grains are randomly oriented, properties will not vary with direction i.e isotropic. If the polycrystal grains are textured, properties will vary with direction i.e anisotropic

83 Single- Vs Poly- Crystal 83

84 Single- Vs Poly- Crystal m -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.

85 Physical Properties Related to Solid Structure 85 Density Luster Hardness Electrical Properties Melting Point Magnetic Properties

86 Crystal Shape Examples of few crystal shapes (cubic crystal system) 86 Cube Tetrakaidecahedron (Truncated Octahedron) Octahedron Tetrahedron

87 87 The shape of the crystal (Eumorphicwell formed) will reflect the point group symmetry of the crystal

88 Early ideas 88 Crystals are solid - but solids are not necessarily crystalline Crystals have symmetry (Kepler) and long range order Spheres and small shapes can be packed to produce regular shapes (Hooke, Hauy)?

89 89 Kepler wondered why snowflakes have 6 corners, never 5 or 7. By considering the packing of polygons in 2 dimensions, it can be shown why pentagons and heptagons shouldn t occur. Empty space not allowed

90 Crystal Types 90 Three types of solids, classified according to atomic arrangement: (a) crystalline and (b) amorphous materials are illustrated by microscopic views of the atoms, whereas (c) polycrystalline structure is illustrated by a more macroscopic view of adjacent single-crystalline regions, such as (a).

91 91 Crystal structure Amorphous structure quartz

92 Crystals 92 The periodic array of atoms, ions, or molecules that form the solids is called Crystal. Crystal Structure = Space (Crystal) Lattice + Basis Space (Crystal) Lattice is a regular periodic arrangement of points in space, and is purely mathematical abstraction Crystal Structure is formed by putting the identical atoms (group of atoms) in the points of the space lattice This group of atoms is the Basis

93 Crystal System 93 The crystal system: Set of rotation and reflection symmetries which leave a lattice point fixed. There are seven unique crystal systems: the cubic (isometric), hexagonal, tetragonal, rhombohedral (trigonal), orthorhombic, monoclinic and triclinic.

94 Bravais Lattice and Crystal System 94 Crystal structure: contains atoms at every lattice point. The symmetry of the crystal can be more complicated than the symmetry of the lattice. Bravais lattice points do not necessarily correspond to real atomic sites in a crystal. A Bravais lattice point may be used to represent a group of many atoms of a real crystal. This means more ways of arranging atoms in a crystal lattice.

95 What is a lattice? 95 A lattice is a 3-D system of points designating the positions of the components (atoms, ions, or molecules) that make up the substance Unit Cell: The smallest repeating unit of the lattice. Eg: simple cubic body-centered cubic face-centered cubic

96 Lattice Example 96 We can pick out the smallest repeating unit..

97 UNIT CELL 97 We can pick out the smallest repeating unit called Unit Cell...

98 Definitions The unit cell: The smallest repeat unit of a crystal structure, in 3D, which shows the full symmetry of the structure The unit cell is a box with: 3 sides - a, b, c 3 angles -,, 14 possible crystal structures (Bravais lattices)

99 Unit Cell 99 Simplest (smallest) parallel piped outlined by a lattice Lattice: a two or three (space lattice) dimensional array of points Environment about all lattice points must be identical Unit cell must fill all space, with no holes

100 Physical Properties & Structure 100 Hardness and Structure Hardness depends on how easily structural units can be moved relative to one another Molecular solids with weak intermolecular attractions are rather soft compared with ionic compounds, where forces are much stronger Covalent network solids are quite hard because of the rigidity of the covalent network structure Molecular and ionic crystals are generally brittle because they fracture easily along crystal plane Metallic solids, by contrast, are malleable

101 Physical Properties 101 Electrical Conductivity and Structure Molecular and ionic solids are generally considered nonconductors Ionic compounds conduct in their molten state, as ions are then free to move Metals are all considered conductors Of the covalent network solids, only graphite conducts electricity This is due to the delocalization of the resonant p electrons in graphite s sp 2 hybridization

102 102 UNIT CELL Primitive Conventional & Non-primitive Single lattice point per cell Smallest area in 2D, or Smallest volume in 3D More than one lattice point per cell Integral multiples of the volume of primitive cell Simple cubic(sc) Conventional = Primitive cell Body centered cubic(bcc) Conventional Primitive cell

103 The Conventional Unit Cell 103 A unit cell just fills space when translated through a subset of Bravais lattice vectors. The conventional unit cell is chosen to be larger than the primitive cell, but with the full symmetry of the Bravais lattice. The size of the conventional cell is given by the lattice constant a.

104 104 Primitive and conventional cells of FCC

105 105 Primitive and conventional cells of BCC Primitive Translation Vectors: 1 a1 ( x ˆ y ˆ z ˆ) 2 1 a ˆ ˆ 2 ( x y zˆ ) 2 1 a3 ( x ˆ y ˆ z ˆ) 2

106 Primitive and conventional cells 106 Body centered cubic (bcc): conventional cell b c Fractional coordinates of lattice points in conventional cell: a 000,100, 010, 001, 110,101, 011, 111, ½ ½ ½ b c a Simple cubic (sc): primitive cell=conventional cell Fractional coordinates of lattice points: 000, 100, 010, 001, 110,101, 011, 111

107 Primitive and conventional cells 107 c b Face centered cubic (fcc): primitive (rombohedron) cell a Fractional coordinates: 000, 100, 101, 110, 110,101, 011, 211, 200 b c a Face centered cubic (fcc): conventional cell Fractional coordinates: 000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½,½1 ½, 1 ½ ½, ½ ½ 1

108 Primitive and conventional cells-hcp 108 points of primitive cell c a b Hexagonal close packed cell (hcp): conventional =primitive cell Fractional coordinates: 100, 010, 110, 101,011, 111,000, 001

109 Primitive Unit Cell and vectors A primitive unit cell is made of primitive translation vectors a 1,a 2, and a 3 such that there is no cell of smaller volume that can be used as a building block for crystal structures. A primitive unit cell will fill space by repetition of suitable crystal translation vectors. This defined by the parallelpiped a 1, a 2 and a 3. The volume of a primitive unit cell can be found by V = a 1.(a 2 x a 3 ) (vector products) Cubic cell volume = a 3

110 Primitive Unit Cell 110 The primitive unit cell may have only one lattice point. There can be different choices for lattice vectors, but the volumes of these primitive cells are all the same. a 1 P = Primitive Unit Cell NP = Non-Primitive Unit Cell

111 Wigner-Seitz Method 111 A simply way to find the primitive cell which is called Wigner-Seitz cell can be done as follows; 1. Choose a lattice point. 2. Draw lines to connect these lattice point to its neighbours. 3. At the mid-point and normal to these lines draw new lines. The volume enclosed is called as a Wigner-Seitz cell.

112 Wigner-Seitz Cell - 3D 112

113 Wigner-Seitz primitive unit cell and first Brillouin zone The Wigner Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to any of the other lattice points. 113 The cell may be chosen by first picking a lattice point. Then, lines are drawn to all nearby (closest) lattice points. At the midpoint of each line, another line (or a plane, in 3D) is drawn normal to each of the first set of lines. 1D case 2D case Important 3D case: BCC

114 The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice 114 1D Real space Reciprocal space 2D

115 3D: Recall that the reciprocal lattice of FCC is BCC X =??? 4 /a Why is FCC so important?

116 The Unit Cell Concept summary 116 The simplest repeating unit in a crystal is called a unit cell. Opposite faces of a unit cell are parallel. The edge of the unit cell connects equivalent points. Not unique. There can be several unit cells of a crystal. The smallest possible unit cell is called primitive unit cell of a particular crystal structure. A primitive unit cell whose symmetry matches the lattice symmetry is called Wigner-Seitz cell.

117 117 Each unit cell is defined in terms of lattice points. Lattice point not necessarily at an atomic site. For each crystal structure, a conventional unit cell, is chosen to make the lattice as symmetric as possible. However, the conventional unit cell is not always the primitive unit cell. A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage (tendency to split along certain planes with smooth surfaces), electronic band structure and optical properties.

118 Crystal System 118 The crystal system: Set of rotation and reflection symmetries which leave a lattice point fixed. There are seven unique crystal systems: the cubic (isometric), hexagonal, tetragonal, rhombohedral (trigonal), orthorhombic, monoclinic and triclinic.

119 Bravais Lattice and Crystal System 119 Crystal structure: contains atoms at every lattice point. The symmetry of the crystal can be more complicated than the symmetry of the lattice. Bravais lattice points do not necessarily correspond to real atomic sites in a crystal. A Bravais lattice point may be used to represent a group of many atoms of a real crystal. This means more ways of arranging atoms in a crystal lattice.

120 Geometry Of Crystals Space Lattices Motifs Crystal Systems Advanced Reading Elementary Crystallography M.J. Buerger John Wiley & Sons Inc., New York (1956) The Structure of Materials Samuel M. Allen, Edwin L. Thomas John Wiley & Sons Inc., New York (1999)

121 We shall consider two definitions of a crystal: 1) Crystal = Lattice + Motif 2) Crystal = Space Group + Asymmetric unit The second definition is the more advanced one (the language of crystallographers) and we shall only briefly consider it in this introductory text The second definition becomes important as the classification of crystals (7 crystal systems) is made based on symmetry and the first definition does not bring out this aspect Note: Since we have this precise definition of a crystal, loose definitions should be avoided (Though often we may live with definitions like: a 3D translationally periodic arrangement of atoms in space is called a crystal) Initially we shall start with ideal mathematical crystals and then slowly we shall relax various conditions to get into practical crystals The language of crystallography is one shortness 121

122 Definition 1 Crystal = Lattice + Motif 122 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 Translationally periodic arrangement of points Crystal Translationally periodic arrangement of motifs

123 As mentioned before crystals are understood based on the language of symmetry 123 Symmetry Symmetry is perhaps the most important principle of nature: though often you will have to dig deeper to find this statement The analogous terms to symmetry are: Symmetry Conservation Invariance The kind of symmetry of relevance to crystallography is geometrical symmetry The kind of symmetry we encountered in the definition of a lattice is TRANSLATIONAL SYMMETRY (t)

124 Auguste Bravais Found fourteen unique lattices which satisfy the requirements Published Études Crystallographiques in

125 Isometric (cubic) Lattices P = primitive I = body-centered (I for German innenzentriate) F = face centered a = b = c, α = β = γ = 90

126 Tetragonal Lattices a = b c α = β = γ =

127 Tetragonal Axes The tetragonal unit cell vectors differ from the cubic one by either stretching the vertical axis, so that c > a (upper image) or compressing the vertical axis, so that c < a (lower image) 127

128 Orthorhombic Lattice a b c α = β = γ = 90 C - Centered: additional point in the center of each end of two parallel faces 128

129 Orthorhombic Axes The axes system is orthogonal Common practice is to assign the axes so the the magnitude of the vectors is c > a > b 129

130 Monoclinic Lattice a b c α = γ = 90 (β 90 ) 130

131 Monoclinic Axes The monoclinic axes system is not orthogonal 131

132 Triclinic Lattice a b c α β γ

133 Triclinic Axes None of the axes are at right angles to the others Relationship of angles and axes is as shown 133

134 Hexagonal 134 Some crystallographers call the hexagonal group a single crystal system, with two divisions Rhombohedral division Hexagonal division Others divide it into two systems, but this practice is discouraged

135 Hexagonal Lattice a = b c α = γ = 90 β =

136 Rhombohedral Lattice a = b = c α = β = γ

137 What are the symmetries of the 7 crystal systems? 137 Cubic Hexagonal Tetragonal Trigonal Characteristic symmetry Four 3-fold rotation axes (two will generate the other two) One 6-fold rotation axis (or roto-inversion axis) (Only) One 4-fold rotation axis (or roto-inversion axis) (Only) One 3-fold rotation axis (or roto-inversion axis) Orthorhombic (Only) Three 2-fold rotation axes (or roto-inversion axis) Monoclinic Triclinic (Only) One 2-fold rotation axis (or roto-inversion axis) None We have stated that basis of definition of crystals is symmetry and hence the classification of crystals is also based on symmetry The essence of the required symmetry is listed in the table more symmetries may be part of the point group in an actual crystal Note: translational symmetry is always present in crystals (i.e. even in triclinic crystal)

138 Crystal Structures - Cubic a a a Simple Face-Centered Body-Centered

139 Crystal Structures - Monoclinic c a b Simple End Face-Centered

140 Crystal Structures - Tetragonal c a a Simple Body-Centered

141 Crystal Structures - Orthorhombic c a b Simple End Face-Centered Body Centered Face Centered

142 Emphasis 142 For a well grown crystal (eumorphic crystal) the external shape reflects the point group symmetry of the crystal the confluence of the mathematical concept of point groups and practical crystals occurs here! The unit cell shapes indicated are the conventional/preferred ones and alternate unit cells may be chosen based on need It is to be noted that some crystals can be based on all possible lattices (Orthorhombic crystals can be based on P, I, F, C lattices); while others have a limited set (only P triclinic lattice)

143 Crystal system 143 Lattices can be constructed using translation alone The definition (& classification) of Crystals is based on symmetry and NOT on the geometry of the unit cell (as often one might feel after reading some books!) Crystals based on a particular lattice can have symmetry: equal to that of the lattice or lower than that of the lattice Based on symmetry crystals are classified into seven types/categories/systems known as the SEVEN CRYSTAL SYSTEMS We can put all possible crystals into 7 boxes based on symmetry Alternate view Symmetry operators acting at a point can combine in 32 distinct ways to give the 32 point groups Lattices have 7 distinct point group symmetries which correspond to the SEVEN CRYSTAL SYSTEMS

144 14 Bravais Lattices divided into 7 Crystal Systems 144 A Symmetry based concept Translation based concept Crystal System Shape of Unit Cell Bravais Lattices P I F C 1 Cubic Cube 2 Tetragonal Square Prism (general height) 3 Orthorhombic Rectangular Prism (general height) 4 Hexagonal 120 Rhombic Prism 5 Trigonal Parallopiped (Equilateral, Equiangular) 6 Monoclinic Parallogramic Prism 7 Triclinic Parallopiped (general) Why are some of the entries missing? Why is there no C-centred cubic lattice? Why is the F-centred tetagonal lattice missing?.? P I F C Primitive Body Centred Face Centred A/B/C- Centred

145 Crystal System Bravais Lattices Cubic P I F? 2. Tetragonal P I 3. Orthorhombic P I F C 4. Hexagonal P 5. Trigonal P 6. Monoclinic P C 7. Triclinic P Why so many empty boxes? E.g. Why cubic C P: Simple; I: body-centred; F: Face-centred; C: End-centred is absent?

146 End-centred cubic not in the Bravais list? 146 a 2 a 2 End-centred cubic = Simple Tetragonal

147 14 Bravais lattices divided into seven crystal systems 147 Crystal system Bravais lattices 1. Cubic P I F C 2. Tetragonal P I 3. Orthorhombic P I F C 4. Hexagonal P 5. Trigonal P 6. Monoclinic P C 7. Triclinic P

148 148 THE 7 CRYSTAL SYSTEMS

149 1. Name of crystal system lattice parameters and relationship amongst them (preferred Unit Cell) Possible Bravais lattices Diagram of preferred Unit Cell 149 Point groups belonging to the crystal system

150 1. Cubic Crystals 150 a = b= c = = = 90º Simple Cubic (P) - SC Body Centred Cubic (I) BCC Face Centred Cubic (F) - FCC SC, BCC, FCC are lattices while HCP & DC are crystals! Point groups 23, 43m, m3, 432, Elements with Cubic structure SC: F, O, Po BCC: Cr, Fe, Nb, K, W, V FCC: Al, Ar, Pb, Ni, Pd, Pt, Ge 4 m 3 2 m Note the 3s are in the second position

151 Examples of elements with Cubic Crystal Structure 151 Po Fe Cu n = 1 SC n = 2 BCC n = 4 FCC/CCP n = 8 DC C (diamond)

152 Note that cubic crystals can have the shape of a cube, an octahedron, a truncated octahedron etc. (some of these polyhedra have the same rotational symmetry axes; noting that cube and octahedron are regular solids (Platonic) while truncated octahedron with two kinds of faces is not a regular solid) The external shape is a reflection of the symmetry at the atomic level Point groups have be included for completeness and can be ignored by beginners Cubic crystals can be based on Simple Cubic (SC), Body Centred Cubic (BCC) and Face Centred Cubic Lattices (FCC) by putting motifs on these lattices After the crystal is constructed based on the SC, BCC or FCC lattice, it should have four 3-fold symmetry axes (along the body diagonals) which crystals built out of atomic entities will usually have if the crystal does not have this feature it will not be a cubic crystal (even though it is based on a cubic lattice) 152

153 Tetragonal Crystals a = b c = = = 90º Simple Tetragonal Body Centred Tetragonal -BCT Point groups 4, 4, 4 m, 422, 4mm, 42m, 4 m 2 m 2 m Elements with Tetragonal structure In, Sn Note the 4 in the first place

154 In Example of an element with Body Centred Tetragonal Crystal Structure Indium 154 [100] views BCT All atoms are In coloured for better visibility In Lattice parameter(s) a = 3.25 Å, c = 4.95 Å Space Group I4/mmm (139) Strukturbericht notation Pearson symbol Other examples with this structure A6 ti2 Pa Wyckoff position Site Symmetry [001] view x y z Occupancy In 2a 4/mmm Note: All atoms are identical (coloured differently for easy visualization)

155 3. Orthorhombic Crystals a b c = = = 90º 155 Simple Orthorhombic Body Centred Orthorhombic Face Centred Orthorhombic End Centred Orthorhombic One convention a b c Point groups 222, 2mm, 2 m 2 m 2 m Elements with Orthorhombic structure Br, Cl, Ga, I, S

156 Ga Example of an element with Orthorhombic Crystal Structure 156 [010] view [001] view Ga Lattice parameter(s) a = 2.9 Å, b = 8.13, c = 3.17 Å Space Group Cmcm (63) Strukturbericht notation Pearson symbol oc4 Wyckoff position Site Symmetry x y z Occupancy Ga 4c m2m Note: All atoms are identical (coloured differently for easy visualization)

157 4. Hexagonal Crystals a = b c = = 90º = 120º 157 Simple Hexagonal Point groups 6, 6, 6 m, 622, 6mm, 6m2, 6 m 2 m 2 m Elements with Hexagonal structure Be, Cd, Co, Ti, Zn

158 Mg Example of an element with Hexagonal Crystal Structure 158 Note: All atoms are identical (coloured differently for easy visualization)

159 5. Trigonal/Rhombohedral Crystals a = b = c = = 90º 159 Rhombohedral (simple) Point groups 3, 3, 32, 3m, 3 2 m Note the 3 s are in the first position Elements with Trigonal structure As, B, Bi, Hg, Sb, Sm

160 -Hg Example of an element with Simple Trigonal Crystal Structure 160 [111] view -Hg Lattice parameter(s) a = Å Space Group R-3m (166) Strukturbericht notation A10 Pearson symbol hr1 Other examples with this structure -Po Wyckoff position Site Symmetry x y z Occupancy Hg 1a -3m

161 6. Monoclinic Crystals a b c = = 90º 161 Simple Monoclinic End Centred (base centered) Monoclinic (A/C) Point groups 2, 2, 2 m Elements with Monoclinic structure P, Pu, Po

162 7. Triclinic Crystals a b c 162 Simple Triclinic Point groups 1, 1

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166 Increasing symmetry Ordering the 7 Crystal Systems: Symmetry 166 Progressive lowering of symmetry amongst the 7 crystal systems Hexagonal(24) Trigonal(12) Cubic(48) Tetragonal(16) Orthorhombic(8) Order Cubic Hexagonal Tetragonal Trigonal Orthorhombic Monoclinic(4) Monoclinic Triclinic(2) Triclinic Arrow marks lead from supergroups to subgroups Superscript to the crystal system is the order of the lattice point group

167 Minimum symmetry requirement for the 7 crystal systems 167 Cubic Crystal system Hexagonal Tetragonal Trigonal Characteric symmetry Point groups Comment Four 3-fold rotation axes One 6-fold rotation axis (or roto-inversion axis) (Only) One 4-fold rotation axis (or roto-inversion axis) (Only) One 3-fold rotation axis (or roto-inversion axis) Orthorhombic (Only) Three 2-fold rotation axes (or roto-inversion axis) Monoclinic (Only) One 2-fold rotation axis (or roto-inversion axis) 23, 6, 4, 3, 6, 4, 3, 222, 2, 2, 43m, 6, m 4, m 32, 2mm, 2 m m3, 432, 622, 6mm, 422, 3m, 2 m 3 2 m 4 m 4mm, 2 m 2 m 3 2 m 6m2, 6 m 4 42m, m 2 m 2 m 2 m 2 m 3 or 3 in the second place Two 3-fold axes will generate the other two 3-fold axes 6 in the first place 4 in first place but no 3 in second place 3 or 3 in the first place Triclinic None 1, 1 1 could be present

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