Structures of Simple Inorganic Solids

Dr S.J. Heyes

First of Four Lectures in the 1st Year Inorganic Chemistry Course

Michaelmas Term 1999

If you have any comments please contact stephen.heyes@chem.ox.ac.uk 

Lecture 1. 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. Representations of structures

• Perspective (Clinographic)
• Projection (Plan) diagrams

4. Close-Packing of spheres

• hexagonal close packing (hcp)
• cubic close packing (ccp)

5. Structures of metallic elements

6. Interstitial sites in close-packed arrangements

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Aims of this Lecture

After studying this lecture you should be able to:-

1. Define the Terms

• 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 ccp, hcp 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

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Why Study Solids?

1. ALL Compounds are Solids under suitable conditions of temperature and pressure. Many exist only as solids.

2. Solids are of immense Technological Importance

Appearance

• Precious and Semi-precious Gemstones of many varieties

  Mechanical Properties

• Metals/Alloys, e.g. Titanium for aircraft
• Cement/Concrete Ca₃SiO₅
• 'Ceramics', e.g. clays, BN, SiC
• Lubricants, e.g. Graphite, MoS₂
• Abrasives, e.g. Diamond, Quartz (SiO₂), Corundum

  Electrical Properties

• Metallic Conductors, e.g. Cu, Ag...
• Semiconductors, e.g. Si, GaAs
• Superconductors, e.g. Nb₃Sn, YBa₂Cu₃O₇
• Electrolytes, e.g. LiI in pacemaker batteries
• Piezoelectrics, e.g. a Quartz (SiO₂)in watches

  Magnetic Properties

• e.g. CrO₂, Fe₃O₄ for recording technology

  Optical Properties

• Pigments, e.g. TiO₂ in paints
• Phosphors, e.g. Eu³⁺ in Y₂O₃ is red on TV
• Lasers, e.g. Cr³⁺ in Al₂O₃ is ruby
• Frequency-doubling of light, e.g. LiNbO₃

  Catalysts

• Zeolite ZSM-5 (an aluminosilicate)

  • - Petroleum refining

    - methanol octane

  Sensors

• Oxygen sensor, e.g. ZrO₂/CaO solid solution

3. Solids (especially Crystals) have always been fascinating

Some Historical Landmark Events

Date	X-ray Date	Landmark Event
ca. 6000 BC		Egyptians mine Turquoise
Antiquity		Gemstones (especially diamond, sapphire, emerald & ruby) are much prized. Indeed some still attribute magical properties, such as healing to crystals!
ca. 350 BC	ca. 2310 BX	Theophrastus describes regular form of Garnet crystals
ca. 30 BC	1976 BX	Strabo names Quartz krystlloz (crystallum in Latin), hence our 'crystal'
1597	315 BX	The alchemist Libavius recognizes that the geometrical habit of crystals is characteristic of the salts concerned
17th C	ca. 302 BX	Boyle, Leeuwenhoek, Kepler, Hooke... make numerous observations with the newly invented microscope
1611	301 BX	Kepler suggests that the hexagonal symmetry of snowflakes is due to "regular packing of the constituent particles"
1665	247 BX	Hooke suggests that crystals are composed of "spheroids"
1669	243 BX	Steno observes that Quartz crystals, whatever their origin or state, always preserve the same characteristic interfacial angles
1780	132 BX	Carangeot invents the Contact Goniometer - measures interfacial angles leading to a great mass of crystallographic detail
1783	129 BX	Bergman's studies of crystal cleavage suggest to him that crystals consist of packed rhombohedral units
1783	129 BX	de l'Isle formulates the law of "Constancy of Interfacial Angle"
1801	111 BX	Haüy substantiates the law of "Rational Indices" The Fundamental Laws of Crystal Morphology are established
1808	104 BX	Malus observes the polarization of light by certain crystals
1809	103 BX	Wollaston invents the Reflecting Goniometer - this leads to a massive improvement in the accuracy of interfacial angle data
1815	97 BX	Biot discovers laevo- and dextro-rotatory forms of Quartz
1819-22	93 BX	Mitscherlich discovers Isomorphism (crystals of different composition with the same form) Polymorphism (different crystal forms with the same chemical composition) [= Allotropy in elements]
1839	73 BX	Miller uses his Miller Indices to designate crystal faces
1848	64 BX	Pasteur discovers enantiomorphic crystals
1880s-90s	32 BX	Sohncke, Federov, Schönflies & Barlow develop theories of internal symmetry of crystals - but still no experimental evidence to support these theories
1906-19	6 BX	Groth's "Chemische Krystallographie" tabulates morphological, optical and other properties of 7000 crystalline substances {but it contains no information about internal structures - no experimental techniques!)
1907	5 BX	Barlow & Pope propose that ions in crystals are hard spheres touching each other
1912		Friedrich, Knipping & von Laue discover X-ray diffraction
1913	1 AX	W.H. & W.L. Bragg use orientation dependence of X-ray diffraction from a single crystal to solve the structure of NaCl (& subsequently diamond etc...)
1913	1 AX	Ewald introduces the concept of the Reciprocal Lattice
1914	2 AX	Debye's theory of Thermal Motion of atoms in solids (hence Debye-Waller factors in X-ray structures)
1916	4 AX	Debye & Scherrer's experiments on diffraction by powders
1924	12 AX	Bernal et al. - structure of Graphite
1926	14 AX	Frenkel's investigations of Point Defects in structures
1926	14 AX	Goldschmidt's spherical atom formulation of structures
1927	15 AX	Pauling's formulation of Goldschmidt's Ionic Model into Pauling's Rules
1929	17 AX	Rotating Anode X-ray generator - allows increased X-ray intensities for better diffraction patterns
1934	22 AX	Patterson Function for structure solution from X-ray diffraction
1934	22 AX	Ruska takes images using the first (transmission) electron microscope
1936	24 AX	Halaban & Preiswerk - diffraction of neutrons by crystals
1941	29 AX	Hughes uses Least-Squares refinement to obtain best possible structures from a diffraction data set
1944	32 AX	Buerger invents the Precession Camera
1948	36 AX	Harker & Kasper - Direct Methods for structure solution from X-ray diffraction data
1950s	ca. 38 AX	Automatic diffractometers and computers dramatically increase the ease of solving crystal structures
1951	39 AX	Bijvoet uses anomalous scattering to determine chirality (absolute configuration)
mid 1950s	ca. 43 AX	Computers first used for Structure solution from X-ray data
1955	43 AX	Principles of Laves - space-filling in crystal structures
1956	44 AX	Menter produces first lattice image from Transmission Electron Microscopy (TEM)
1957	45 AX	Müller - Field-Ion Microsopy visualizes individual atoms in metals
1970	58 AX	Crewe, Wall and Langmore - Darkfield Scanning Electron Microscopy (the first general method for imaging individual heavy atoms)
1971	59 AX	Formanek et al. - the first detection of an individual atom by High Resolution Electron Microscopy (HREM)
1974	62 AX	Iijima - the first observation of point defects in structures by electron microscopy
1980s	ca. 68 AX	Synchrotron Radiation Available - massively increased intensity of X-rays (Laue X-ray patterns of crystals obtained on ms timescale)
1982	70 AX	Area detectors for obtaining X-ray diffraction patterns (massive decrease in time taken to obtain a diffraction pattern)
1982	70 AX	Binnig & Rohrer - Scanning Tunnelling Microscopy (STM) images even light atoms at surfaces
1984	72 AX	Schechtman et al. discover Quasi-Crystals
1984	74 AX	Binnig et al. - Atomic Force Microscopy (AFM) images at surfaces (even easier to obtain than STM)
1990s	ca. 80 AX	>200,000 Crystal structures (internal atom coordinates) stored in databases

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)

• 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

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

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2D LATTICES

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

2-Dimensional Lattice Symmetries were famously exploited by the artist Escher in many patterns

• A tutorial on the 2D tessellations of Escher
• A really fun way to create your own 2D patterns with different lattice symmetries is the "Escher Web Sketch" Java program of Wes Hardaker and Gervais Chapuis

Analysing a 3D solid

e.g. Graphite = a staggered arrangement of stacked hexagonal layers

Perspective: Clinographic views of solids

Projection onto a Plane: Plan views of solids

GRAPHITE

Unit Cell Dimensions

• a, b and c are the unit cell edge lengths

• a, b and g are the angles (a between b and c, etc....)

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 Þ ¹/₈ atom per cell
• Edge atom shared by 4 cells Þ ¹/₄ atom per cell
• Face atom shared by 2 cells Þ ¹/₂ atom per cell
• Body unique to 1 cell Þ 1 atom per cell

On combining 7 Crystal Classes with 4 possible unit cell types Symmetry indicates that only 14 3-D lattice types occur

If you have the Chemscape Chime Plug-in you can manipulate the 14 Bravais lattices at the University of Texas, Austin

For applications of different geometry lattice theories to simple structures see:-

• Russell Chu's views of solids as interpenetrating ccp and hcp lattices, including stereoview pictures.
• Scott Childs's application of Synergetic Geometry to crystal structure description

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?

CLOSE-PACKING OF SPHERES

• space is trapped between the layers that is not filled by the spheres
• TWO different types of HOLES (so-called INTERSTITIAL sites) are left
  • OCTAHEDRAL (O) holes with 6 nearest sphere neighbours
  • TETRAHEDRAL (T±) holes with 4 nearest sphere neighbours

{P = sphere, O = octahedral hole, T+ / T- = tetrahedral holes)

• The third layer lies in indentations directly in line (eclipsed) with the 1st layer
  • Layer ordering may be described as ABA
• The third layer lies in the alternative indentations leaving it staggered with respect to both previous layers
  • Layer ordering may be described as ABC

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 Johannes 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, but in August 1998 Prof. Thomas Hales of the University of Michigan 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!

• An article by Dr. Simon Singh © Daily Telegraph, 13th August 1998

Features of Close-Packing

• Coordination Number = 12
• 74% of space is occupied
• Largest interstitial sites are:-
  • octahedral (O) ( r = 0.414) ~ 1 per sphere
  • tetrahedral (T±) (r = 0.225) ~ 2 per sphere

Simplest Close-Packing Structures

• ABABAB.... repeat gives Hexagonal Close-Packing (HCP)
  • Unit cell showing the full symmetry of the arrangement is Hexagonal
    • Hexagonal: a = b, c = 1.63a, a = b = 90°, g = 120°
    • 2 atoms in the unit cell: (0, 0, 0) (²/₃, ¹/₃, ¹/₂)
• ABCABC.... repeat gives Cubic Close-Packing (CCP)
  • Unit cell showing the full symmetry of the arrangement is Face-Centred Cubic
    • Cubic: a = b =c, a = b = g = 90°
    • 4 atoms in the unit cell: (0, 0, 0) (0, ¹/₂, ¹/₂) (¹/₂, 0, ¹/₂) (¹/₂, ¹/₂, 0)

2 atoms in the unit cell (0, 0, 0) (²/₃, ¹ /₃, ¹ /₂)

View a Quicktime HCP Movie or Quicktime HCP VR scene

4 atoms in the unit cell (0, 0, 0) (0, ¹ /₂, ¹ /₂) (¹ /₂, 0, ¹ /₂) (¹ /₂, ¹ /₂, 0)

View a Quicktime CCP Movie or Quicktime CCP VR scene

The most common close-packed structures are METALS

A NON-CLOSE-PACKED structure adopted by some metals is:-

View a Quicktime BCC Movie or Quicktime BCC VR scene

68% of space is occupied

Coordination Number ?

8 Nearest Neighbours at 0.87a

6 Next-Nearest Neighbours at 1a

• 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
• Complex to predict structures
  • BCC clearly adopted for low number of valence electrons
  • Best explanations are based on Band Theory of Metals
    • Initially applied by Coulson & Hume-Rothery in Oxford
    • Still actively researched in Oxford by Prof. D.G. Pettifor using Density Functional Theory (DFT). See his book "Bonding & Structure of Molecules & Solids", OUP, 1995
  • In cases of polymorphism BCC is the structure adopted at higher temperatures
• More Complex close-packing sequences than simple HCP & CCP are possible
  • HCP & CCP are merely the simplest close-packed stacking sequences, others are possible!
    • All spheres in an HCP or CCP 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 HCP, so labelled h
      • surrounding layers are different (i.e. cuboctahedral coordination) like CCP, 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
• 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" HCP cobalt is actually 90% AB... & 10% ABC... - i.e. non-ideal HCP
  • Many metals deviate from perfect HCP 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

• Other Systems may be Classified as having Similar Structures

CrystalMaker file for C₆₀

Further information about Fullerenes

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

The hole positions are shown relative to the unit cells below

The structures possible from filling them are considered in Lecture 2

CCP Octahedral holes------------------------------------------------HCP Octahedral holes

CCP Tetrahedral holes------------------------------------------------HCP Tetrahedral holes

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Solids Page Lecture 1 Lecture 2 Lecture 3 Lecture 4 Problems Set Help