Solid State Chemistry Pdf 283h1d

Introduction

Introduction to Solid State Chemistry Anorganische Chemie III (Festkörperchemie) International Graduate Studies Chemistry of Materials

SS 2002

(Wednesday 08-09 and Thursday 08-10 Room SR 01.132) Tremel: Solid State Chemistry

Introduction Textbooks: Introduction to Solid State Chemistry (English) 1. Doughlas, McDaniel, Alexander, Concepts and Models of Inorganic Chemistry, Wiley, 1992 2. Shriver, Atkins, Inorganic Chemistry (3rd ed, 1999) W.H. Freeman and Company (Chs. 3, 18 ...) 3. L. Smart, E. Moore, Solid State Chemistry, 2nd Ed. Chapman & Hall, London, 1995 4. P.A. Cox, The Electronic Structure and Chemistry of Solids, Oxford University Press, 1987 5. U. Müller, Inorganic Structural Chemistry Wiley, Chichester, 1993 6. A.R. West, Solid State Chemistry and its Applications Wiley, New York, 1984 Tremel: Solid State Chemistry

Introduction Textbooks II: Solid State Chemistry (German) 1. E. Riedel: Moderne Anorganische Chemie, de Gruyter 1999, S. 329 – 523 („Festkörperchemie“) 2. L. Smart, E. Moore: Einführung in die Festkörperchemie, Vieweg 1997 3. A.R. West: Grundlagen der Festkörperchemie, VCH 1992 4. U. Müller: Anorganische Strukturchemie Teubner Studienbücher 1991 5. E. Riedel: Moderne Anorganische Chemie de Gruyter 1999, S. 329 – 523 („Festkörperchemie“)

Tremel: Solid State Chemistry

Contents Fundamental Aspects of Solids & Sphere Packing 1. Why Study Solids? 2. Some crystallographic ideas •lattice (lattice types) •motif (basis) •crystal structure •unit cell (counting atoms in unit cells) •fractional coordinates •coordination number 3. Representation of structures •Perspective (Clinographic) •Projection (Plan) diagrams 4. Close packing of spheres •hexagonal close packing (h) •cubic close packing (c) 5. Structures of metallic elements 6. Interstitial sites in close-packed arrangements Tremel: Solid State Chemistry

Aims of the lecture After studying this lecture you should be able to: 1. Define the • Lattice, motif, crystal structure, unit cell, coordination number 2. Interpret a Plan or Perspective structure diagram of a Unit Cell & determine • Lattice type, number of atoms in unit cell, coordination number/geometry of atoms 3. For c, h and bcc structures • Describe the Sphere Packing • Identify symmetry of unit cell • Draw unit cell in plan or perspective • State Coordination Number/Geometry of a sphere & the degree of space-filling • Give some examples of metallic elements which adopt each structure • Draw on a plan or perspective view of the unit cell the positions of any octahedral &/or tetrahedral interstitial holes


What is special about the solid compared to molecules? Discrete molecules

Covalent solids

H H

C

H

conducting

H

aromatic

aliphatic

insulating, thermally conducting hard C60 aromatic Tremel: Solid State Chemistry

What is special about the solid compared to molecules? Molecules such as HCl, C6H6 or C60 are identical and stochiometric Solid state compounds may be nonstochiometric Example: „FeO“ is really Fe1-xO ! Fe1-xO ≠ FeO1+x Other examples: NaxWO3, WO3-x


What is special about the solid compared to molecules? Solid solutions: composition determines the properties Examples:

„blue NaCl“

Doped semiconductors n-Si/p-Si LASERs Al2O3(Cr3+) Dielectrics Ba1-xCaxTiO3 Pigments (TiO2-xFx, CdS/CdSe) Steel (Fe/C) „mailbox yellow“


What is special about the solid compared to molecules? Periodicity leads to Order-disorder effects (e.g. in intermetallic compounds)

FeAl or CuZn

Localization-delocalization of charge carriers (metallic conductivity, polarons ...) Collective properties such as magnetism (e.g. Fe, Co, NiO, CoFe2O4)


What is special about the solid compared to molecules? Extended structures ⇒ no localized chemical reactivity ⇒ low mobility of constituents large activation energies for reactions ⇒ high reaction temperatures/thermodynamic control ⇒ phase diagrams important for chemical reactions ⇒ experiments under extreme physical conditions: very high pressures and temperatures ⇒ Description in of structure types Packing/coordination polyhedra vs. molecular units


What is special about the solid compared to molecules? Frequently used classifications for solids a) elements: new modifications (e.g. fullerenes), structure comparisons, growth of high purity crystals ... b) crystalline materials: synthesis and crystal growth, defects, structure – chemical bonding – physical properties ... c) noncrystalline/amorphous materials: special physical and chemical properties ... d) (quasi)molecular solids (e.g. phosphides): structural relations ... e) covalent solids (e.g. diamond, boron nitride): extreme hardness, very high melting points ... f) ionic solids (e.g. NaCl): structural chemistry, ionic conductivity ... g) metals, semiconductors, isolators: close packings of spheres, alloys, mechanical and electrical properties, chemical bonding ...


Some Basic Definitions LATTICE An infinite array of points in space, in which each point has identical surroundings to all others.

CRYSTAL STRUCTURE The periodic arrangement of atoms in the crystal. It can be described by associating with each lattice point a group of atoms called the MOTIF (BASIS)

UNIT CELL The smallest component of the crystal, which when stacked together with pure translational repetition reproduces the whole crystal, Primitive (P)unit cells contain only a single lattice point Don't mix up atoms with lattice points Lattice points are infinitesimal points in space Atoms are physical objects Lattice Points do not necessarily lie at the centre of atoms Tremel: Solid State Chemistry

Some Basic 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 - α, β, γ


Some Basic Definitions


Some Basic Definitions Counting Lattice Points/Atoms in 2D Lattices Unit cell is Primitive (1 lattice point) but contains TWO atoms in the motif Atoms at the corner of the 2D unit cell contribute only 1/4 to unit cell count Atoms at the edge of the 2D unit cell contribute only 1/2 to unit cell count Atoms within the 2D unit cell contribute 1 (i.e. uniquely) to that unit cell

Counting Atoms in 3D Cells Atoms in different positions in a cell are shared by differing numbers of unit cells Vertex atom shared by 8 cells ⇒ 1/8 atom per cell Edge atom shared by 4 cells ⇒ 1/4 atom per cell Face atom shared by 2 cells ⇒ 1/2 atom per cell Body unique to 1 cell ⇒ 1 atom per cell




Some Basic Definitions 1-dimensional lattice symmetries are found in many ornaments 2-dimensional lattice symmetries were famously exploited by the artist M.C. Escher in many patterns






Close packing 1926 Goldschmidt proposed atoms could be considered as packing in solids as hard spheres This reduces the problem of examining the packing of like atoms to that of examining the most efficient packing of any spherical object - e.g. have you noticed how oranges are most effectively packed in displays at your local shop?


Principles of close packings of spheres A single layer of spheres is closest-packed with a HEXAGONAL coordination of each sphere

A second layer of spheres is placed in the indentations left by the first layer •space is trapped between the layers that is not filled by the spheres •TWO different types of HOLES (INTERSTITIAL sites) are left •OCTAHEDRAL (O) holes with 6 nearest sphere neighbours •TETRAHEDRAL (T±) holes with 4 nearest sphere neighbours Tremel: Solid State Chemistry

Principles of close packings of spheres When a third layer of spheres is placed in the indentations of the second layer there are TWO choices 1. The third layer lies in indentations directly in line (eclipsed) with the 1st layer ⇒ Layer ordering may be described as ABA 2. The third layer lies in the alternative indentations leaving it staggered with respect to both previous layers ⇒ Layer ordering may be described as ABC


Principles of close packings of spheres

A second layer of spheres is placed in the indentations left by the first layer

(polytypism !!)

h e xagonalcl ose pack ing (h )and cubic cl ose pack ing(c) Tremel: Solid State Chemistry

Principles of close packings of spheres Common unit cells for h and c

hexagonal close packing (h) Be, Mg

cubic close packing (c or fcc) Ca, Sr, Cu, Ag, Au


Principles of close packings of spheres Holes in sphere packings

Calculation of the size of holes Tremel: Solid State Chemistry

Principles of close packings of spheres Close-Packed Structures The most efficient way to fill space with spheres Is there another way of packing spheres that is more space-efficient? In 1611 Kepler asserted that there was no way of packing equivalent spheres at a greater density than that of a face-centred cubic arrangement. This is now known as the Kepler Conjecture. This assertion has long remained without rigorous proof. In 1998 Hales announced a computer-based solution. This proof is contained in over 250 manuscript pages and relies on over 3 gigabytes of computer files and so it will be some time before it has been checked rigorously by the scientific community to ensure that the Kepler Conjecture is indeed proven! (Simon Singh, Daily Telegraph, Aug. 13th 1998)


Principles of close packings of spheres Features of Close-Packing 1.

Coordination Number = 12

2.

74% of space is occupied Space filling Rfill = 4/3 π(ΣiZiri3)/Vcell Zi = No. atoms/unit cell, Vcell = unit cell volume example: 4r = v2 aVcell=a3=(4/v2 r)3=16 v2 r3 a = 4/ v2 r

Vatom = Zi 4/3 pr3

RE=Vatom/Vcell=(16 pr3)/ (3·16 v2 r3)= p/3 v2 = 0.74 3.

octahedral (O) ( r = 0.414) ~ 1 per sphere tetrahedral (T±) (r = 0.225) ~ 2 per sphere


Principles of close packings of spheres Density Calculations In the fcc cell the atoms touch along the face diagonals, but not along the cell edge: length face diagonal = a(2)1/2 = 4r . Al has a c arrangement of atoms. The radius of Al = 1.423Å. Calculate the lattice parameter of the unit cell and the density of solid Al. Solution: fcc lattice ⇒ 4 atoms/cell [8 at corners (each 1/8), 6 in faces (each 1/2)] Lattice parameter: atoms in along face diagonal, therefore 4rAl = a(2)1/2 ⇒ a = 4(1.432Å)/(2)1/2 = 4.050Å. ρAl = Mcell/Vcell Mcell/ 4 x mAl = (26.98)(g/mol)(1mol/6.022x1023atoms)(4 atoms/unit cell) = 1.792 x 10-22 g/unit cell Vcell = a3 = (4.05x10-8cm)3 = 66.43x10-24 cm3/unit cell ρAl = {1.792x10-22g/unit cell}/{66.43x10-24 cm3/unit cell} = 2.698 g/cm3 Tremel: Solid State Chemistry




ABABAB.... repeat gives Hexagonal Close-Packing (H) Unit cell showing the full symmetry of the arrangement is Hexagonal 2 atoms in the unit cell: (0, 0, 0) (2/3, 1/3, 1/2) Tremel: Solid State Chemistry


ABCABC.... repeat gives Cubic Close-Packing (C) Unit cell showing the full symmetry of the arrangement is Face-Centred Cubic 4 atoms in the unit cell: (0, 0, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0) Tremel: Solid State Chemistry




68% of space is occupied Coordination Number ? 8 Nearest Neighbours at 1 2 3 a =0.87a 6 Next-Nearest Neighbours at 1a



1

2

3


Principles of close packings of spheres Polymorphism: •Some metals exist in different structure types at ambient temperature & pressure •Many metals adopt different structures at different temperature/pressure •Not all metals are close-packed •Why different structures? residual effects from some directional effects of atomic orbitals •Difficult to predict structures BCC clearly adopted for low number of valence electrons Best explanations are based on Band Theory of Metals (later) In cases of polymorphism BCC is the structure adopted at higher temperatures


Principles of close packings of spheres More Complex close-packing sequences than simple H & C are possible •H & C are merely the simplest close-packed stacking sequences, others are possible! All spheres in an H or C structure have identical environments •Repeats of the form ABCB.... are the next simplest •There are two types of sphere environment •surrounding layers are both of the same type (i.e. anti-cuboctahedral coordination) like H, so labelled h •surrounding layers are different (i.e. cuboctahedral coordination) like C, so labelled c •Layer environment repeat is thus hchc...., so labelled hc •Unit cell is alternatively labelled 4 H •has 4 layers in the c-direction •hexagonal •The hc (4 H) structure is adopted by early lanthanides •Samarium (Sm) has a 9-layer chh repeat sequence Tremel: Solid State Chemistry

Principles of close packings of spheres •Non-Ideality of Structures •Cobalt metal that has been cooled from T > 500°C has a close-packed structure with a random stacking sequence •"Normal" H cobalt is actually 90% AB... & 10% ABC... - i.e. non-ideal H •Many metals deviate from perfect H by „axial compression" •e.g. for Beryllium (Be) c/a = 1.57 (c.f. ideal c/a = 1.63) •Coordination is now [6 + 6] with slightly shorter distances to neighbours in adjacent layers


Principles of close packings of spheres Other systems may be classified as having similar structures


Principles of close packings of spheres Location of interstitial holes in close-packed structures The HOLES in close-packed arrangements may be filled with atoms of a different sort. It is therefore important to know: •How holes are displaced in space relative to the positions of the spheres •How holes are displaced relative to each other








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