chem 1101 introduction to solid state chemistry

8/7/2019 CHEM 1101 Introduction to Solid State Chemistry

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CHEM 1101 Introduction to Solid State Chemistry

Definitions

y A lattice is an infinite array of points where each point has identical surroundings

y A unit cell is a building block, which when repeated in all directions, gives the lattice

y A lattice point specifies the location of a structural motif

y A crystal lattice is the array of lattice points

y A crystal structure is the collection of structural motifs arranged according to thelattice

Unit Cells

y The position of a unit cell is not unique

y 4 types of unit cells:

y Primitive (P) ± 1 lattice point (8 x 1/8)

y Body-centred (I) ± 2 lattice points (1 + 8 x 1/8)

y Face-centred (F) ± 4 lattice points (1/2 x 6 + 1/8 x 8)

y C-centred ± 2 lattice points (1/2 x 2 + 1/8 x 8)

y Any unit cell can be defined by a parallelpiped with 6 parameters a, b, c and E, F, K

y There are only 7 possible unit cell shapes, which when combined with the 4 possiblelattices, give 14 Bravais lattices (CTORHMT)

Miller Indices

y Refer to the reciprocals of intersection distances used to denote planes in crystals

y Has the form (hkl) corresponding to the no of divisions on the a, b and c axesrespectively

y The separation between planes is represented by d hkl

Crystal Structure Determination

y To generate the X-ray radiation, high energy electrons are fired at metal samples likeCu. The electrons excite core electrons, leaving a vacancy. The excited electrondrops back into the vacancy, emitting X -ray radiation. The emitted quantisedradiation is usually filtered to produce monochromatic X -rays.



Bragg¶s Law

For constructive interference, AB+BC must be an integer multiple of P

nP = AB + BC

AB = BC = d sin U

Path difference = 2dsin U = nP

Note: P is the wavelength of the incident monochromatic radiation (usually 1.542 Angstrom)

Pack ing and St ack ing

y For atoms or molecules with isotropic interactions, optimal packing results in minimalvolume and maximal density

y A single layer of spheres is closest-packed with a hexagonal coordination of eachsphere

y A second layer of spheres is placed over the top of µholes¶ of the first layer (generating octahedral holes with 6 nearest sphere neighbours and tetrahedral holes



with 4 nearest sphere neighbours)

y When the third layer is placed, 2 choices:

y The third layer lies eclipsed with the first layer (directly in line) ABABAB (HCP)

y OR It lies in the alternative holes leaving it staggered with respect to both layers ABCABC (CCP)

y Since the difference between the 2 structures arises in the 3rd

coordination shell,HCP vs CCP is usually very finely energetically balanced

y CCP/HCP have 74% packing efficiency compared to BCC of 64% (less favourable)

y CN=12 for close-packed structures (6 in the sa me layer, 3 above and 3 below)

y Note: HCP has 6 atoms per unit cell (3 + ½ x 2 + 12 x 1/6) while CCP has 4 atomsper unit cell (8 x 1/8 + 6 x ½)

y Polymorphism refers to the ability of a mate rial to adopt different packingarrangements (CCP p HCP p BCC)

y CN remains the same, very small energy difference between structures

y Polytypism refers to the variation in stacking sequences within a structure

y E.g. ABABABC ABAB



Simple Ioni c Solids

Compound Description Lattice Coordination

STRUCTURES DERIVED FROM CC P

NaClCCP Cl- with Na+ in

all O holesFCC

6:6 (octahedral;topologically

identical)

CaF2CCP Ca2+ lattice

with F- in all T holes.4 CaF2 per unit cell

FCC

Ca2+ is 8coordinated (cubic)F- is 4 coordinated

(tetrahedral)

ZnS (zinc blende)

CCP S2- lattice withZn2+ in half the T

holes.4 ZnS per unit cell

FCC4:4 (tetrahedral;

topologicallyidentical)

STRUCTURES DERIVED FROM HC P

ZnS (Wurtzite)

HCP array of S2-

with Zn2+ in half theT holes

2 ZnS per unit cell

Hexagonal Primitive 4:4 (tetrahedral)

NiAsHCP As with Ni in all

O holes2 NiAs per unit cell

Hexagonal PrimitiveNi: 6 (octahedral) As: 6 (trigonal

prismatic)

CdI2HCP I with Cd in Oholes of alternate

layersHexagonal Primitive

Cd: 6 (octahedral)I: 3 (base pyramid)

TiO2 (rutile)(compare with CdI2 ±

half of O holes inTiO2 are full but all

holes are full inalternate layers of

CdI2)

Distorted HCP Olattice with Ti in half

the O holes

Ti: 6 (octahedral)

O: 3 (trigonal planar)

NON-CLOSE PACKED STRUCTURE

CsCl

1 CsCl per unit cellCan be regarded as2 interpenetrating

primitive cubic cells

Cubic Primitive8:8 (large cations, somore anions can be

packed around )



R adius R atio

8 coordination Cuboidal CsCl structure

0.732

6-coordination Octahedral NaCl structure

0.414

4 coordination Tetrahedral ZnS structure

0.225

Latti c e Enthalpies

y LE is the energy change when 1 mole of gaseous ions, which are infinitely separated,form a crystal at 0K

y Always exothermic so (H < 0

y Long range attractive forces between unlike charges

y Short range repulsion at close range between electrons of one ion and that of neighbouring ions

Coulomb attraction

y Where (U = change in internal energy (in J)Z A = charge on ion AZB = charge on ion Br = internuclear distance between ions (in m)

I0 = permittivity of a vacuum (8.854 x 10 -12 Fm-1)e = charge on an electron (1.602 x 10 -19 Cm-1)



y Individual interactions will be negative (attractive) between unlike charges andpositive (repulsive) between like charges

y Need to sum up all the pairwise interactions

Madelung constants

y Where NA = Avogadro¶s no and A = Madelung constant (structure specific)

y E.g. for NaCl, sit on central Na+ and consider neighbour distances

6 Cl- (face of cube) at distance r

12 Na+

at distance

8 Cl-(corner of cube) at distance

y Sum up all interactions and it can be shown that the sum converges to a constantvalue

y A = sum of [charge x 1/distance relative to r]

= (6 x 1/1) ± (12 x 1/ ) + (8 x 1/ ) ± (6 x ½) + ...

Repulsive Component

y Where n = Born exponent (obtaine d from experimental compressibility data) andB = repulsive constant

Overall,

Find and rearrange to find B and substitute B into original expression for U

Born Lande equation

Hess¶ Law

y The enthalpy change for a reaction is independent of the pathway it takes providedthe initial and final states of the reaction remain the same



Kapustinskii equation

y Where d = distance between cation and anion (r+ + r -), assuming touching spheres(in Angstrom)

y Compare ionisation energy with lattice enthalpy to predict which compound formed ismore favourable

chem 1101 introduction to solid state chemistry

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