CRYSTALLINE STATE

INTRODUCTION Electronic structure Bonding State of aggregation Primary: Ionic Covalent Metallic Van der Waals Gas Liquid Solid Octet stability Secondary: Dipole-dipole London dispersion Hydrogen

STATE OF MATTER GAS LIQUID SOLID The particles are arranged in tight and regular pattern The particles move very little Retains its shape and volume The particles move past one another The particles close together Retains its volume The particles move rapidly Large space between particles

CRYSTAL SYSTEM CLASSIFICATION OF SOLID BY ATOMIC ARRANGEMENT Ordered regular long-range crystalline “crystal” transparent Disordered random* short-range* amorphous “glass” opaque Atomic arrangement Order Name

Early crystallography Robert Hooke (1660) : canon ball Crystal must owe its regular shape to the packing of spherical particles (balls)  packed regularly, we get long- range order. Neils Steensen (1669) : quartz crystal All crystals have the same angles between corresponding faces, regardless of their sizes  he tried to make connection between macroscopic and atomic world. If I have a regular cubic crystal, then if I divide it into smaller and smaller pieces down to an atomic dimension, will I get a cubic repeat unit?

7 crystal systems Renė-Just Haūy (1781): cleavage of calcite Common shape to all shards: rhombohedral Mathematically proved that there are only 7 distinct space-filling volume elements 7 crystal systems

3 AXES 4 AXES yz =  xz =  xy =  yz = 90 xy = yu = ux = 60 CRYSTALLOGRAPHIC AXES

The Seven Crystal Systems

(rombhohedral)

SPACE FILLING TILING

August Bravais (1848): more math How many different ways can we put atoms into these 7 crystal systems and get distinguishable point environment? He mathematically proved that there are 14 distinct ways to arrange points in space 14 Bravais lattices

The Fourteen Bravais Lattices

1 2 3 Face-centered cubic Simple cubic Body-centered cubic

4 5 Simple tetragonal Body-centered tetragonal

6 7 Simple orthorhombic Body-centered orthorhombic

8 9

10 11 12

13 14 Hexagonal

Repeat unit A point lattice

a, b, c z Lattice parameters , ,  b c   O y  a x A unit cell

Crystal structure Bravais lattice Basis (Atomic arrangement in 3 space) Bravais lattice (Point environment) Basis (Atomic grouping at each lattice point)

EXAMPLE: properties of cubic system*) BRAVAIS LATTICE BASIS CRYSTAL STRUCTURE EXAMPLE FCC atom Au, Al, Cu, Pt molecule CH4 ion pair (Na+ and Cl-) Rock salt NaCl Atom pair DC (diamond crystal) Diamond, Si, Ge C 109 *) cubic system is the most simple most of elements in periodic table have cubic crystal structure

CRYSTAL STRUCTURE OF NaCl

CHARACTERISTIC OF CUBIC LATTICES SC BCC FCC Unit cell volume a3 Lattice point per cell 1 2 4 Nearest neighbor distance a a3 / 2 a/2 Number of nearest neighbors (coordination no.) 6 8 12 Second nearest neighbor distance a2 Number of second neighbor a = f(r) 2r 4/3 r 22 r or 4r = a4 a3 Packing density 0.52 0.68 0.74

EXAMPLE: FCC 74% matter (hard sphere model) FCC 26% void

In crystal structure, atom touch in one certain direction and far apart along other direction. There is correlation between atomic contact and bonding. Bonding is related to the whole properties, e.g. mechanical strength, electrical property, and optical property. If I look down on atom direction: high density of atoms  direction of strength; low density/population of atom  direction of weakness. If I want to cleave a crystal, I have to understand crystallography.

CRYSTALLOGRAPHIC NOTATION POSITION: x, y, z, coordinate, separated by commas, no enclosure O: 0,0,0 A: 0,1,1 B: 1,0,½ z Unit cell A B O y a x

DIRECTION: move coordinate axes so that the line passes through origin Define vector from O to point on the line Choose the smallest set of integers No commas, enclose in brackets, clear fractions z OB 1 0 ½ [2 0 1]  Unit cell AO 0 -1 -1  A O B y x

Denote entire family of directions by carats < > e.g. all body diagonals: <1 1 1>

all face diagonals: <0 1 1> all cube edges: <0 0 1>

MILLER INDICES For describing planes. Equation for plane: where a, b, and c are the intercepts of the plane with the x, y, and z axes, respectively. Let: so that No commas, enclose in parenthesis (h k l) denote entirely family of planes by brace, e.g. all faces of unit cell: {0 0 1} etc.

(h k l)  [h k l] c Miller indices: (h k l) (2 1 0) (2 1 0) b a Intercept at  c Miller indices: (h k l) (2 1 0) (2 1 0) b Intercept at b a Parallel to z axes [2 1 0] Intercept at a/2 MILLER INDICES

Miller indices of planes in the cubic system (0 1 0) (0 2 0) Miller indices of planes in the cubic system

CRYSTAL SYMMETRY Many of the geometric shapes that appear in the crystalline state are some degree symmetrical. This fact can be used as a means of crystal classification. The three elements of symmetry: Symmetry about a point (a center of symmetry) Symmetry about a line (an axis of symmetry) Symmetry about a plane (a plane of symmetry)

 Symmetry about a point A crystal possesses a center of symmetry when every point on the surface of the crystal has an identical point on the opposite side of the center, equidistant from it. Example: cube 

Symmetry about a LINE If a crystal is rotated 360 about any given axis, it obviously returns to its original position. If the crystal appears to have reached its original more than once during its complete rotation, the chosen axis is an axis of symmetry.

AXIS OF SYMMETRY DIAD AXIS TRIAD AXIS TETRAD AXIS HEXAD AXIS Rotated 180 Twofold rotation axis DIAD AXIS TRIAD AXIS Rotated 120 Threefold rotation axis AXIS OF SYMMETRY TETRAD AXIS Rotated 90 Fourfold rotation axis Rotated 60 Sixfold rotation axis HEXAD AXIS

The 13 axes of symmetry in a cube

Symmetry about a plane A plane of symmetry bisects a solid object in a such manner that one half becomes the mirror image of the other half in the given plane. A cube has 9 planes of symmetry: The 9 planes of symmetry in a cube

Cube (hexahedron) is a highly symmetrical body as it has 23 elements of symmetry (a center, 9 planes, and 13 axis). Octahedron also has the same 23 elements of symmetry.

ELEMENTS OF SYMMETRY

Combination forms of cube and octahedron

SOLID STATE BONDING IONIC COVALENT SOLID STATE BONDING MOLECULAR Composed of ions Held by electrostatic force Eg.: NaCl IONIC COVALENT Composed of neutral atoms Held by covalent bonding Eg.: diamond SOLID STATE BONDING MOLECULAR Composed of molecules Held by weak attractive force Eg.: organic compounds Comprise ordered arrays of identical cations Held by metallic bond Eg.: Cu, Fe METALLIC

ISOMORPH Two or more substances that crystallize in almost identical forms are said to be isomorphous. Isomorphs are often chemically similar. Example: chrome alum K2SO4.Cr2(SO4)3.24H2O (purple) and potash alum K2SO4.Al2(SO4)3.24H2O (colorless) crystallize from their respective aqueous solutions as regular octahedral. When an aqueous solution containing both salts are crystallized, regular octahedral are again formed, but the color of the crystals (which are now homogeneous solid solutions) can vary from almost colorless to deep purple, depending on the proportions of the two alums in the solution.

CHROME ALUM CRYSTAL

POLYMORPH A substance capable of crystallizing into different, but chemically identical, crystalline forms is said to exhibit polymorphism. Different polymorphs of a given substance are chemically identical but will exhibit different physical properties, such as density, heat capacity, melting point, thermal conductivity, and optical activity. Example:

ARAGONITE

CRISTOBALITE

Polymorphic Forma of Some Common Substances

Polymorphic transition Material that exhibit polymorphism present an interesting problem: It is necessary to control conditions to obtain the desired polymorph. Once the desired polymorph is obtained, it is necessary to prevent the transformation of the material to another polymorph. Poly- morph 1 Polymorph 1 Polymorphic transition

In many cases, a particular polymorph is metastable Transform into more stable state Relatively rapid infinitely slow Carbon at room temperature Diamond (metastable) Graphite (stable)

POLYMORPH MONOTROPIC ENANTIOTROPIC Different polymorphs are stable at different temperature One of the polymorphs is the stable form at all temperature The most stable is the one having lowest solubility

CRYSTAL HABIT Crystal habit refers to external appearance of the crystal. A quantitative description of a crystal means knowing the crystal faces present, their relative areas, the length of the axes in the three directions, the angles between the faces, and the shape factor of the crystal. Shape factors are a convenient mathematical way of describing the geometry of a crystal. If a size of a crystal is defined in terms of a characteriza- tion dimension L, two shape factors can be defined: Volume shape factor : V =  L3 Area shape factor : A =  L3

External shape of hexagonal crystal displaying the same faces ? = Internal structure External habit Tabular Prismatic Acicular External shape of hexagonal crystal displaying the same faces

Crystal habit is controlled by: Internal structure The conditions at which the crystal grows (the rate of growth, the solvent used, the impurities present) Variation of sodium chlorate crystal shape grown: (a) rapidly; (b) slowly

(a) (b) Sodium chloride grown from: (a) pure solution; (b) Solution containing 10% urea