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Suggested reading material

CHEM 410/713/7130-01 - Prof. Holger Kleinke

Chemistry of Inorganic Solid State Materials

- A. R. West, Solid State Chemistry and its Applications, John Wiley.

- R. Hoffmann, Solids and Surfaces, VCH.

- J. K. Burdett, Chemical Bonding in Solids, Oxford University Press.

- C. Kittel, Introduction to Solid State Physics, John Wiley.

- M. O'Keeffe, B. G. Hyde, Crystal structures: Patterns and Symmetry.

gg g

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- H. F. Franzen, Physical Chemistry of Solids.

- plus review articles, TBA

Course web page: https://uwaterloo.ca/learn

Periodic Table

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From molecules to the SOLID: is the solid state special?

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

3

) ( g ) y

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 ...

Crystalline Solids and CrystallographyMechanical Properties

Electrical Properties

Metals/Alloys, e.g. Titanium for aircraft Cement/Concrete Ca3SiO5

'Ceramics', e.g. clays, BN, SiCLubricants, e.g. Graphite, MoS2

Abrasives, e.g. Diamond, Corundum

Magnetic Properties

Optical Properties

Metallic Conductors, e.g. Cu, Ag ... Semiconductors, e.g. Si, GaAs Superconductors, e.g. Nb3Sn, YBa2Cu3O7

Electrolytes, e.g. LiI in pacemakers Piezoelectrics, e.g. -Quartz in watches

CrO2, Fe3O4 for recording technology

Pigments, e.g. TiO2 in paints

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Catalysts

g g 2 pPhosphors, e.g. Eu3+ in Y2O3 is red on TV Lasers, e.g. Cr3+ in Al2O3 is ruby

Zeolite ZSM-5 (an aluminosilicate) - Petroleum refining - methanol to octane

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Solid State Chemistry - Permutations

ExampleSuperconducting Tc (K)

Elements - A 80-106 (known) Nb (9)

Binary - AB (80 x 79)/2 = 3160 (known) Nb3Ge (23)Mg3B2 (39)

Ternary - ABC (80 x 79 x 78)/6 (5% known) La2CuO4 (40)

Quaternary - ABCD 1.5 M (surface scratched) YBa2Cu3O7 (93)

ABCDE Gazillions (trace) HgBa2Ca2Cu3O8 (134)

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DON’T mix up atoms (the motif) with

CRYSTAL STRUCTURE

UNIT CELL

The periodic arrangement of atoms in the crystal.

The smallest component of the crystal, which when replicated in (one, two or) three dimensions reproduces the whole crystal.

The solid state: some basic definitions

DON T mix up atoms (the motif) withlattice points

Lattice points are infinitesimal pointsin space

Atoms are physical objects

Lattice Points do not necessarily lie

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Lattice Points do not necessarily lie at the centre of atoms

Primitive (P) unit cells contain only one lattice pointNon-primitive (NP) contains more than one lattice point

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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

(parallelepipedon) with:

• 3 sides - a, b, c

• 3 angles - , ,

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Atomic position: (x, y, z)(fractional coordinates)


Atoms in different positions in a unit cell are shared by differing

numbers of unit cells

Counting Atoms in 2D unit cellsCounting Atoms in 2D unit cells

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

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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

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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

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http://www.mcescher.com/Gallery/gallery-symmetry.htm

2D-Lattices, e.g., graphene

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3D-Lattices, e.g., graphite

Clinographicview

Planview

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Note staggering of consecutive layers(contrast with Boron Nitride)

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

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P = PrimitiveI = Body-centredF = Face-centeredC = Side-centered…plus 7 crystal classes 14 Bravais Lattices

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Bravais Lattice EquivalenciesThere are fourteen Bravais lattices. It easy to see that a cubic system cannot be side centered (because it would no longer be cubic). But how about these possibilities....?

Show that an I-centered monoclinicb

ac

lattice should be transformed into a C-centered one.

c

13

Show that an F-centered tetragonallattice must be transformed into a smaller I-centered.

How about monoclinic B - or tetragonal C?

Close Packing of Spheres

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 displayse.g. have you noticed how oranges are most effectively packed in displays at your local shop?

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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

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•OCTAHEDRAL (O) holes with 6 nearest sphere neighbours

•TETRAHEDRAL (T±) holes with 4 nearest sphere neighbours


T

O

P

P

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

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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

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ccp/fcchcp

Close Packing of Spheres Mg, Be, Sc, Ti…

Hexagonal Close Packing

hcp

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Cubic Close PackingCu, Ca, Sr, Ag, Au, …

ccp/fcc

http://kleinke.uwaterloo.ca/simplestruc.html

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Hexagonal Close Packing a = b, c = 1.63a, = = 90°, = 120°

A

B

A

B

A

C

Cubic Close Packing

Coordination number CN. 12Anti-Cuboctahedral

2 atoms per unit cell(1/3, 2/3, 1/4) (2/3, 1/3, 3/4)

a = b = c, = = = 90°

B

C

A

Note the shift in origin! ABABAB (HCP)

B

A

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Coordination number CN. 12Cuboctahedral

4 atoms per unit cell(0, 0, 0) (0, ½, ½)(½, 0, ½) (½, ½, 0)

A

B

C

A

ABCABC (CCP/FCC)

Close Packing of SpheresThe 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 spheresat a greater density than that of a face-centred cubic arrangement. This is nowknown as the Kepler Conjecture.

This assertion has long remained without rigorous proof In 1998 HalesThis assertion has long remained without rigorous proof. In 1998 Halesannounced a computer-based solution. This proof is contained in over 250manuscript pages and relies on over 3 gigabytes of computer files and so it willbe some time before it has been checked rigorously by the scientificcommunity to ensure that the Kepler Conjecture is indeed proven! (SimonSingh, Daily Telegraph, Aug. 13th 1998)

G l S filli R V /V ith

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General Space filling Rfill = Vatoms/Vcell, with

Vatoms = Sum of atom volumes per cell, thus

Vatoms = Z 4/3 r3 with Z = number of atoms per cell;

Vcell = volume of the unit cell. CCP: Vcell = a3, Z = 4

(4r)2 = 2a2

Rfill = 0.74

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Alternatives

a = b = c, = = = 90°

2 atoms in the unit cell:

(0 0 0) (1/2 1/2 1/2)(0, 0, 0) (1/2, 1/2, 1/2)

W, Li, V, Cr…

Packing efficiency Rfill = Vatoms/Vcell

Vatoms = Z 4/3 r3 = 8/3 r3; Vcell = a3

d = 4r

21Hence: not that close-packing!

d2 = a2 + a2 + a2 = 3 a2 => d = 3 a = 4r Pythagorean theorem

<=> a = 4/3 r = 2.31 r

=> Vcell = (4/3 r)3 = (4/3)3 r3

Rfill = Vatoms/Vcell = 8/3 / (4/3)3 = 3 /8

Rfill = 0.68

Primitive Cubic [-Po]Coordination Number 652% Packing Efficiency

Vatoms/Vcell = 0.52, since


Comparison of Packing Efficiencies

atoms cell

Vcell = a3 and Vatoms = Z 4/3 r3

with a = 2r and Z = 1

Body Centered Cubic [W]

Close-Packed (ccp or hcp)Coordination Number 1274% Packing Efficiency

ccp

22(increased pressure favors higher packing efficiency)

Body-Centered Cubic [W]Coordination Number 868% Packing Efficiency

bcchcp

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Periodic Table of Metal Structures


Classification seems unpredictable, but patterns emerge from an understanding of band structures (stability of a particular structure as a function of electron count).

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Other examples


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

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Trend: BCC clearly adopted for low number of valence electrons

Best explanations are based on Band Theory of Metals (later)

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Polymorphism:

• Some metals exist in different structure types at ambient temperature & pressure

pressure > 92 kbarpressure

T < 13.2ºC

92 kbar

Gray tin- semiconductor- diamond structure - CN 4

Å

White tin- metal- tetragonal structure- CN 6

d(S S ) 3 02 3 18 Å

High pressure tin I- metal- tetragonal structure

CN 8

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- d(Sn-Sn): 2.81 Å- density: 5.7 gcm-3

- d(Sn-Sn): 3.02 - 3.18 Å- density 7.2 gcm-3

- CN 8d(Sn-Sn): 3.11 - 3.70 Ådensity 8.4 gcm-3

High pressure tin II- Metal- bcc- CN (8+6)

p > 500 kbar - d(Sn-Sn) 2.85 - 3.29 Å- density 10.9 gcm-3

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 ABCBABCBABCB.... are the next simplest

• There are two types of sphere environment

• surrounding layers are both of the same type (i.e. anti-cuboctahedralcoordination) like HCP, so labelled h

• surrounding layers are different (i.e. cuboctahedral coordination) like CCP, so labelled c

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• Layer environment repeat is thus hchc...., so labeled hc

• Alternative: unit cell is alternatively labeled 4 H,

because it has 4 layers in the c-direction and is hexagonal

• The hc (4 H) structure is adopted by early lanthanides

• Samarium (Sm) has a 9-layer chh repeat sequence

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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)

C di ti i [6 + 6]

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• Coordination is now [6 + 6]

with slightly shorter distances to neighbors in adjacent layers

Layers and Interstitial Cavities in Close packed Structures

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There are one Oh and two Td holes per site in CCP - hence K3C60.

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Location of interstitial holes in close-packed structures

The HOLES in close-packed arrangements may be filled with atoms of a

different kind.

It is therefore important to know:

• where holes are placed in space relative to the positions of the spheres• where holes are placed in space relative to the positions of the spheres,

• where holes are placed relative to each other.

fcc packing

fcc packing with Oh holes (+)

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Different cavity sizes: the octahedral hole

r: radius of packing atom;rh: radius of hole

ccp/fcc

M: metal atom (hole);X: anion (packing atom)

a = rX + 2 rM + rX

½ a = rM + rX (I)

d2 = a2 + a2 => d = 2 a

d = 4 rX = 2 a

½ a = 2/2 rX = rX 2 (II)

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(I) = (II):

rM + rX = rX 2

rM = (2 – 1) rX

rM/rX = 2 – 1 = 0.414

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Different cavity sizes: the tetrahedral hole

d2 = a2 + a2 => d = 2 a

d = 4 rX = 2 a <=> ¼ a = ½ rX 2

=> rX = ¼ a 2

g = rX + 2 rM + rX

g/2 = rM + rX

(g/2)2 = (a/4)2 + (b/2)2

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(g/2)2 = (a/4)2 + (b/2)2

(rM + rX)2 = (½ rX 2)2 + rX2

(rM + rX)2 = ½ rX2 + rX

2 = 3/2 rX2

rM + rX = (3/2) rX = ½ 6 rX

rM = (½ 6 – 1) rX

rM/rX = ½ 6 – 1 = 0.225

Ionic (and other) structures may be derived from the occupation ofinterstitial sites in close packed arrangements.


34http://kleinke.uwaterloo.ca/simplestruc.html

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ccp Li3Bi typefilling of all Oh and Td voids


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K C3 60C60

Close Packing of Spheres: descriptions based on linking polyhedra; e.g. triangle, tetrahedron,

square pyramid, trigonal bipyramid, octahedron, trigonal prism, pentagonal bipyramid, square antiprism, cube, … and capped variants thereof.

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Overview: coordination numbers and polyhedra

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cf. VSEPR rules (CHEM212): http://www.shef.ac.uk/chemistry/vsepr/

Examples? Point groups? Missing polyhedra?

Perfect packing occurs when anions touch and cations fit perfectly in the pocket. Then, anions touch anions and cations to ch anionstouch anions.

If cations are too large (larger rM/rX), anions cannot touch (not ideal).

If anions are too large (smaller rM/rX), cations “ ttl ” i k t ( )

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“rattle” in pocket (worse).

Conclusion: The listed rM/rX ratios are the minima, not maxima.

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Limiting Radius Ratios

What's the Numerical Value of a Specific Ionic Radius?•Ionic radii in most scales do not generally meet at experimental electron density minima, because of polarization of the anion by the cation

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•The various scales are designed to be selfconsistent in reproducing ro = rM + rX [ or: ro = r+ + r– ]

{Use the same scale for cation and anion!} •Ionic radii change with coordination number r8 > r6 > r4

{Use the appropriate one!}

What's the Numerical Value of a Specific Ionic Radius?

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Limiting Radius Ratios: do they work?

•For Li+ and Na+ salts, ratios calculated from both r6 and r4 are indicated •Radius ratios suggest adoption of C Cl t t th iCsCl structure more than is observed in reality •NaCl structure is observed morethan is predicted •Radius ratios are only correct ca. 50% of the time, not very good for a family of archetypal ionic solids -random choice might be just as

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random choice might be just as successful as radius ratio rules and saying that all adopt the NaCl structure more so! •Is the Goldschmidt cation-anion Contact Criterion all there is to it? No!

too simplifying! Lattice energy must be considered!

Limiting Radius Ratios: why don’t they work better?

U A/r

•Energy rises as r+/r- falls until the structure can no longer support cation-anion contact.•At the geometric limiting radius ratio the -UL A/r0•At the geometric limiting radius ratio, the energy remains constant as r+/r- falls, because ro = r+ + r- but instead is limited by "anion-anion contact".•NaCl-CsCl transition at r+/r- = 0.71•ZnS-NaCl transition at r+/r- = 0.32•Taking r8 > r6 into account indicates that the CsCl structure is never favorable

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(dotted line)•CsCl structure is only adopted where 8:8 coordination maximizes Dispersion Forces•NaCl structure is highly favored by covalency (best utilization of Cl- p3

orbitals)

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When the radius ratio rules work, do they do so for the right reasons?• Radius ratios might appear to work well for predicting structures of oxides, e.g. TiO2 (consider r(Ti4+)/r(O2-) )• 75 pm / 126 pm (Shannon & Prewitt) = 0.6061 pm / 140 pm (Pauling) = 0.44

Limiting Radius Ratios

• 6:3 Coordination, as is observed in practice for TiO2

• But...• Ti4+ is very polarizing - covalency in bonding?!• r(O2-) was determined in systems like MgO, CaO etc...• Perhaps we really should use covalent radii?• rcov(O)/rcov(Ti) = 73 pm / 95 pm = 0.77• 8:4 Coordination, i.e. fluorite structure - NOT what's observed!

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,Alternative reasoning?Ti in TiO2 is 6-coordinate because this maximizes covalent bonding, not

because the radius ratio rules are necessarily correct! Beware! Use radius ratios as a simple predictional tool, not as a means of rationalizing structures.

Ternaries

Binaries AB

Structure Maps Is there any general influence of ionic radii on structure?

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Separation of structure types is achieved on structure diagrams,... but the boundaries are complex

Conclusion: Size matters, but not necessarily in any simple way!

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Mooser-Pearson Plots

Combine bond directionality,size and electronegativity

x-axis is , (bond ionicity) y axis is the average PQNy-axis is the average PQN

larger orbitals (larger PQN)

lead to less directionality

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-Sharp borderlines, a critical ionicity which determines structure

-Boundaries are curved - not so helpful!

- Indicates the increased ionicity of Wurtzite over Zinc Blende

(higher Madelung Constant of Wurtzite)

Unit cell

Sodium ChlorideCl with Na in all Oh holesLattice: fccMotif: Cl at (0,0,0);

Na at (1/2,0,0) 4 NaCl in unit cell (Z = 4)Coordination: 6:6 (octahedral)

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[NaCl6] and [ClNa6] octahedra

Linking of [NaCl6] octahedra

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Structure variants of NaCl

NaSbS2: Na+ and Sb3+ randomly on the Na+ sites; S2- on Cl-.

Long term annealing: Na and Sb orderLong term annealing: Na and Sb order. d(Na-S): 2.85, 2.88, 2x 2.92, 2.97, 3.00 Å;d(Sb-S): 2.44, 2.73, 2.74, 2x 2.92, 3.38 Å.

=> Monoclinic distortion.

FeS2: Fe2+ on Na+;

a

b

c

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CaC2: Ca2+ on Na+ ; C2

2- on Cl-.

S22- on Cl-.

Pyrite: fool’s gold

Nickel ArsenideHCP As atoms with Ni atoms in all octahedral holesa = b = 3.57 Å, c = 5.10 ÅPositions:

2 Ni at (0, 0, 0) & (0, 0, 1/2) 2 As at (1/3, 2/3, 1/4) & (2/3, 1/3, 3/4) 0 ½( , , ) ( , , )2 NiAs in unit cell, thus Z = 2

0,½ 0,½

0, ½0,½ ¼

¾

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Face and edge-sharing [NiAs6] octahedra

[NiAs6] octahedra[AsNi6] trigonal prism

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Cadmium IodideHCP I atoms with Cd atoms in all octahedral holes of every other layer.

0 0

00

¼

¾

0 ½

CdI2

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Edge-sharing [CdI6] octahedra

0,½ 0,½

0, ½0,½

¼

¾ NiAs

ZINC BLENDE or SphaleriteCCP S2- with Zn2+

in half of the Td holesLattice: FCC4 ZnS in unit cell (Z = 4)Motif: S at (0, 0, 0);

½

½

½

½

¼

¼ ¾

¾

WURTZITEHCP S2- with Zn2+

in half of the T holes

Motif: S at (0, 0, 0); Zn at (1/4, 1/4, 1/4)

Coordination: 4:4 (tetrahedral)

⅝⅝

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in half of the Td holesLattice: Hexagonal - P a = b, c = (8/3)aMotif: 2 S at (0,0,0) & (2/3, 1/3, 1/2);

2 Zn at (2/3, 1/3, 1/8) & (0, 0, 5/8) 2 ZnS in unit cell (Z = 2)Coordination: 4:4 (tetrahedral)

⅝

⅝⅝

⅛½

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Polymorphs ofZinc Sulfide

WURTZITEZINC BLENDE

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4 nearest neighbors 12 next-nearest neighbors

Cub-octahedral Anti-Cub-octahedral

plus different next-next-nearest neighbor coordination

ccp Ca2+ with F- in all tetrahedral holesLattice: fccMotif: Ca2+ at (0, 0, 0);

2 F- at (1/4, 1/4, 1/4) & (3/4, 3/4, 3/4)4 CaF2 in unit cell (Z = 4)

Fluorite (CaF2)

¼,¾

¼,¾

¼,¾

¼,¾

½

½ ½

½

2 ( )Coordination: Ca2+ 8 (cubic) : F- 4 (tetrahedral) anti-Fluorite: Na2O

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Linking of [FCa4] tetrahedra K2[PtCl6]

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Rutile (TiO2)“HCP” anions (distorted), Ti atoms in ½ of the Oh sitesUnit Cell: Primitive Tetragonal (a = b = 4.59 Å, c = 2.96 Å)Positions: 2 Ti at (0, 0, 0) & (1/2, 1/2, 1/2)

4 O at (0.3, 0.3, 0), (0.7, 0.7, 0), (0.2, 0.8, 1/2), (0.8, 0.2, 1/2)

Ti: 6 (octahedral coordination); O: 3 (trigonal planar coordination)

½

½

½

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Rhenium TrioxideLattice: Primitive Cubic 1ReO3 per unit cell Motif: Re at (0, 0, 0); 3 O at (1/2, 0, 0), (0, 1/2, 0), (0, 0, 1/2) Re: 6 (octahedral coordination) O: 2 (linear coordination) ( )ReO6 octahedra share only vertices (= corners)

½ ½

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½ ½

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CdI2: HCP anions (I), Cd atoms in all Oh sites of every other layerCdCl2: CCP anions (Cl), Cd atoms in all Oh sites of every other layer

Layered AB2 structure types

CdCl2

CdI2

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NiAs

Transition metals with chalcogen and pnictogen atoms

e.g. Ti(S,Se,Te); Cr(S,Se,Te,Sb); Ni(S,Se,Te,As,Sb,Sn)

Examples of Structure Adoption

g ( ) ( ) ( )

CdI2Iodides of moderately polarizing cations; bromides and chlorides

of strongly polarizing cations; e.g. PbI2, FeBr2, VCl2Hydroxides of many divalent cations; e.g. (Mg,Ni)(OH)2

Di-chalcogenides of many quadrivalent cations; TiS2, ZrSe2, CoTe2

CdCl2 (ccp equivalent of CdI2)

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Chlorides of moderately polarizing cations; e.g. MgCl2, MnCl2Di-sulfides of quadrivalent cations; e.g. TaS2, NbS2 (CdI2 form as well)

Cs2O has the anti-cadmium chloride structure

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For these structure types:• NaCl rock salt• CaF2 fluorite• ZnS zinc blende• NiAs nickel arsenide• ZnS wurtzite

Binary Solids ABx

ZnS wurtzite• Na2O sodium oxide• TiO2 rutile• CdCl2, CdI2, MoS2 (cadmium dichloride, cadmium diiodide,

molybdenum dissulfide)• Li3Bi lithium bismuthide• ReO3 rhenium oxide

C

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Can you:0. Give examples? 1. Describe the structure as filling of interstitial holes in close-packing? 2. Draw the unit cell in a plan (layer) or perspective view?3. Recognize the structure from a plan or perspective view of a unit cell?4. Identify coordination numbers & geometries & point groups of atoms?5. Calculate density, distances, and formula units?

1 CCP = ABCABCABC (coord # = 12)

2. HCP = ABABABABAB (coord # = 12)

One Oh hole per lattice point, and two Td holes (hence K3C60)

3. BCC (coord # = 8 or 8+6) [not close packed]

4. Derivative (ABx) Structures

Close packed spheres - a Summary of Geometrical Features

( x)

AB NaCl - interpenetrating CCP, Oh sites

NiAs - HCP As atoms, with Ni atoms in Oh sites

ZnS (Zinc blende) - interpenetrating CCP, ½ Td sites

ZnS (Wurtzite) - interpenetrating HCP, ½ Td sites

AB2 CaF2 - Fluorite - cations CCP, anions fill all Td sites

Na2O - Antifluorite - anions CCP, cations fill all Td sites

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CdCl2 - CCP anions, cations ½ Oh sites (alternate layers) filled

CdI2 - HCP anions, cations ½ Oh sites (alternate layers) filled

TiO2 - Rutile - "HCP" anions, cations fill ½ Oh sites (ordered frame)

AB3 ReO3 - Cubic, with cations at all corners, anions along all edges

[not close packed]

Li3Bi and K3C60 - CCP anions with all Td and Oh holes filled

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MoS2: S anions NOTclose packed; layer sequence AABB,

Mo in half of theMo in half of the trigonal prismatic sites of every other layer.

Lattice: Hexagonal - P Motif: 2 Mo at (2/3,

1/3,3/4), (

1/3,2/3,

1/4) 4 S at (2/3,

1/3,1/8), (

2/3,1/3,

3/8), (1/3,

2/3,5/8), (

1/3,2/3,

7/8) 2 M S i it ll

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2 MoS2 in unit cell Coordination: Mo 6 (Trigonal Prismatic) :

S 3 (base pyramid)


filled

empty

60

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More on layered AB2 structure types and layer sequencies

A

B

B

c

c

CdI2

empty

B

B

A

a

c

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MoS2

A

A

B

b


analogous to CdI

A

A

B

to CdI2

NbSe2

B

B

A

A

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analogous to MoS2

A

B

B

B

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CdI2layer

63-MoTe2

Distortion leads to Mo-Mo bondformation, driven by the d2

configuration.

Ternary SolidsPerovskites, ABO3

Lattice: Primitive Cubic 1 CaTiO3 per unit cell Motif (A): Ca at (1/2, 1/2, 1/2); Ti at (0, 0, 0); 3 O at (1/2 ,0, 0), (0, 1/2, 0), (0, 0, 1/2)

Ca: 12 (cuboctahedral coordination)Ti: 6 (octahedral coordination) O: 6 (4 Ca + 2 Ti) TiO6 octahedra share only vertices

Remember ReO3?

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Remember ReO3?

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La2CuO4 {K2NiF4 structure}: Doped La2CuO4 {La2-xSrxCuO4} was the first (1986) High-Tc Superconducting Oxide (Tc ~ 40 K),

La2CuO4 - the first High-Tc Superconductor

for which Bednorz & Müller were awarded a Nobel PrizeNobel Prize.

La2CuO4 may be viewed as if constructed from an ABAB... arrangement of Perovskite cells known

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Perovskite cells, known as an AB Perovskite .

La2CuO4 - the first High-TC Superconductor

Alternative descriptions

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One may view the structure as based on:

1. Sheets of elongated CuO6 octahedra, sharing only vertices;

2. Layered networks of CuO46-, connected only by La3+ ions.

La2CuO4 - the first High-TC Superconductor

Comparison of La2CuO4/Nd2CuO4

C t t l tif f t li k d C OCommon structural motif of vertex-linked CuO4 squares.

This motif occurs in all the high TC copper oxides.

Differences are in the structure of the 'filling' in

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the 'sandwich' of Cu-O layers.

- Intergrowth Structures

YBa2Cu3O7 - the 1:2:3 SuperconductorThe first material to superconduct at above liquid N2 temperature, TC > 77 K. YBa2Cu3O7 can be viewed as an Oxygen-Deficient Perovskite

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YBa2Cu3O7 - the 1:2:3 Superconductor

Two types of Cu sites:

• Layers of CuO5 square pyramids

(elongation => essentially vertex-

linked CuO4 squares again)

• Chains of vertex-linked CuO4

squares

indicated in a

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indicated in a

Polyhedral Representation

Cooper Pairs

In Ogg's theory it was his intentThat the current keep flowing, once sent;So to save himself trouble

The first man to suggest the paring of electrons was R. A. Ogg,an experimental chemist. No-one believed him.

Superconductivity

The Meissner Effect

So to save himself trouble,He put them in double,and instead of stopping, it went.

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The Meissner Effect

Superconductors are diamagnetic to the total exclusion of all magnetic fields. This is a classic hallmark of superconductivity and can be used to

levitate a magnet.

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Spinels - AB2O4 e.g., MgAl2O4

1. Take eight cubes based on CCP of A.

2. Add B4O4 cubes to half of Td holes, i.e., AB4O4

3. Add AO4 cubes to other half of Td holes, i.e.,

AB4O4 + AO4.

4 STOICHIOMETRY of mother cube =

AO

BT

T

TT

CCP O atoms, A atoms in 1/8 of the Td holes, B atoms in 1/2 of the Oh holes.

4. STOICHIOMETRY of mother cube =

AB4O4 + AO4 = A2B4O8 = AB2O4

T

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Eight sub-cells inside fcc array of A atoms

FRONT BACK

Fe3O4 – magnetite – is ferromagnetic

More on Spinels

Oxidation state

Normal spinel

Inverse spinel

II, III MgAl2O4 MgIn2O4

Atomic positions:Mg (0, 0, 0),Al (x, x, x); x = 0.625O (x x x); x = 0 375

II, IIIIV, IIII, IVI, I

2 4

Co3O4

GeNi2O4

ZnK2(CN)4

WNa2O4

2 4

Fe3O4

TiMg2O4

NiLi2F4

Ionic radii: Mg2+ 72pm Fe2+ 78pm Co2+ 75pm

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O (x, x, x); x = 0.375Mg2+ 72pm Al3+ 54pm

Fe2+ 78pmFe3+ 65pm

Co2+ 75pmCo3+ 61pm

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Rule 1: A coordinated polyhedron of anions is formed about each cation, the cation-

anion distance determined by the sum of ionic radii and the coordination number by

the radius ratio.

Rule 2: The Electrostatic Valency Principle

An ionic structure will be stable to the extent that the sum of the strengths of the

Pauling rules for ionic crystals - J. Am. Chem. Soc. 51, 1010 (1929)

An ionic structure will be stable to the extent that the sum of the strengths of the

electrostatic bonds that reach an ion equal the charge on that ion.

When Mm+ is surrounded by n Xx–: ebs = m/n and x = ebs

with ebs = electrostatic bond strength of M–X bond

Rule 3: The presence of shared edges, and particularly shared faces decreases the

stability of a structure. This is particularly true for cations with large valences and

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small coordination number.

Rule 4: In a crystal structure containing several cations, those of high valency and

small coordination number tend not to share polyhedral elements.

Rule 5: The Principle of Parsimony

The number of different kinds of constituents in a crystal tends to be small.

Crystalline Defects

Schottky defects

Entropy and defects in crystal lattices

74Frenkel Defects

How will these defects affect density?

In ionic conductors, e.g. in battery applications (LiCoO2), small ionsmigrate from site to site via eitheror both mechanisms.

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