The document discusses different types of crystalline solids and their structures. It defines key terms like unit cell, lattice, space lattice, and basis. It describes one, two and three dimensional lattices and the different types of unit cells in two and three dimensions. It also discusses crystal structures, Miller indices for planes and directions, Bravais lattices, and provides examples of rock salt and zinc blende structures.
3. 3
SOLIDS
can be divided into two catagories.
Crystalline
Amorphous
Crystalline has long range order
Amorphous materials have short range order
Effect of Crystallinity on Physical properties - ex. Polyethylene
4. 4
Crystal
Type
Particles Interparticle
Forces
Physical Behaviour Examples
Atomic
Molecular
Metallic
Ionic
Network
Atoms
Molecules
Atoms
Positive and
negative
ions
Atoms
Dispersion
Dispersion
Dipole-dipole
H-bonds
Metallic bond
Ion-ion
attraction
Covalent
• Soft
• Very low mp
• Poor thermal and electrical
conductors
Fairly soft
Low to moderate mp
Poor thermal and electrical
conductors
Soft to hard
Low to very high mp
Mellable and ductile
Excellent thermal and
electrical conductors
Hard and brittle
High mp
Good thermal and electrical
conductors in molten condition
• Very hard
• Very high mp
• Poor thermal and electrical
conductors
Group 8A
Ne to Rn
O2, P4, H2O,
Sucrose
Na, Cu, Fe
NaCl, CaF2,
MgO
SiO2(Quartz)
C (Diamond)
TYPES OF CRYSTALLINE SOLIDS
7. 7
CRYSTAL STRUCTURE
Crystal structure is the periodic arrangement of atoms in the
crystal. Association of each lattice point with a group of
atoms(Basis or Motif).
Lattice: Infinite array of points in space, in which each point has
identical surroundings to all others.
Space Lattice Arrangements of atoms
= Lattice of points onto which the atoms are hung.
Elemental solids (Argon): Basis = single atom.
Polyatomic Elements: Basis = two or four atoms.
Complex organic compounds: Basis = thousands of atoms.
+
Space Lattice + Basis = Crystal Structure
=
• • •
• • •
• • •
14. The “unit cell” is the basic repeating unit of the
arrangement of atoms, ions or molecules in a
crystalline solid.
The “lattice” refers to the 3-D array of particles in a
crystalline solid. One type of atom occupies a
“lattice point” in the array.
19. 19
COUNTING ATOMS IN THE THREE DIMENTIONAL
UNIT CELL
Vertex(corner) 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
Atoms in different positions in a cell are shared by
differing numbers of unit cells
21. Contributions of Atoms to Cubic Unit Cells
Position of Atoms
in Unit Cell
Contribution to
Unit Cell
Unit-Cell Type
Center 1 bcc
Face 1/2 Fcc
Corner 1/8
fcc, bcc, simple
cubic
How many total atoms are found in a simple cubic
unit cell? Face centered cube? Body centered cube?
22. 22
Number of Atoms per unit cell in cubic
crystal system(z)
1.For SCC( Simple Cubic Crystal Lattice)
Vertex(corner) atom shared by 8 cells 1/8 atom per cell
Z = 1/8 × 8 = 1
2. For FCC (Face Centred Cubic lattice)
Vertex(corner) atom shared by 8 cells 1/8 atom per cell
Face atom shared by 2 cells 1/2 atom per cell
Z = 1/8 × 8 + 1/2 × 6 = 1+3 = 4
23. Unit Cells
A body-centered
cubic (bcc) unit cell
has atoms at the 8
corners of a cube and
at the center of the
cell
A simple cubic unit
cell has atoms only at
the 8 corners of a
cube.
24. Number Atoms in a Unit Cell
In the simple cubic cell
there are only the 8
atoms at the corners.
1/8 x 8 = 1 atom in cell
In bcc, 8 atoms at the
corners and 1 in center.
1/8 x 8 + 1 x 1 = 2 atoms
in the cell
25. 25
3.For BCC(Body Centred Cubic lattice)
Vertex(corner) atom shared by 8 cells 1/8 atom per cell
And Body unique to 1 cell 1 atom per cell
Z = 1/8 × 8 + 1 × 1 = 1+1 = 2
26. 26
Density of crystal matter (D)
Density = Mass of the Unit cell
………………………………
Volume of the Unit cell
Mass of unit cell = Z × mass of each atom
= Z × M/N0
Volume of unit cell = a3 = where a is edge length is in pm
1pm = 10-12 m.
D = Z × M where M –molecular weight
……….
No × a3 No- Avogadro's Number(6.023×10-23
27. 27
Numerical problem:
1.The length of side of unit cell of a cubic crystal is 4×10-3 m.
the density of crystal matter is 1.2× 10-3 kg m-3.if molar mass
is 2.4×10-2 mole-1 then find out i) type of lattice ii) number of
atoms in each unit cell
Solution :
28. 28
Numerical problem:
1. The corner of a face centred cubic Crystal has atoms
of elements X while at centre of face has of element
Y. find out the formula of crystal compound.
29. The Chemistry of Solids
Miller Indices (l,m,n) are a
way of denoting planes in
crystal lattices.
30. Types of planes of cubic Crystal or indexing the planes
(110) planes (130) planes
a
b
(-210) planes
37. 37
Law of crystallography :
1. Law of constancy of interfacial angles:
2. Law of symmetry :
3. Law of rational indices
38. Face intercepts I
Crystal faces are defined by indicating their intercepts on the crystallographic axes. The
units along the axes is determined by the periodicity along theses axes:
- c
- b
- a
Intercepts: 5a : 3b : 2c = 5 : 3 : 2
+ c
+ b
+ a
2c
3b
5a
39. Face intercepts II
Faces parallel to an axis have an intercept with that axis at infinity
+ c
- c
+ b- b
+ a
- a
a
Intercepts: 3a : b : c = 3 : :
c
b
40. Face intercepts III
Intercepts are always given as relative values, e.g. they are divided until they have no common fac
Parallel faces in the same quadrant have, therefore, the same indices
+ c
+ b- b
+ a
- a
a
4c
2b1b
2a
2c
Intercepts: 4a : 2b : 4c = 4 : 2 : 4
div. by 2
2 : 1 : 2
- c
Intercepts: 2a : 1b : 2c = 2 : 1: 2
Intercept ratios are called Weiss indices
42. Miller indices I
The Miller indices of a face are derived from the Weiss indices by inverting the latter
and, if necessary, eliminating the fractions.
Reason for using Miller indices:- avoiding the index
- simplifies crystallographic calculations
- simplifies the interpretation of x-ray diffraction
Example:
Weiss indices Miller indices
1 1 1
1 1 1 0 0
1 1 1
1 2 3 1 2 3 1 0.5 0.333 x 6 6 3 2
Miller indices are placed in round brackets, e.g. (1 0 0). Commas are only used, if two
digit indices appear, e.g. (1,14,2)
Negative intercepts are indicated by a bar above the number, e.g. (1 0 0).
Indices, which are not precisely known, are replaced by the letters h, k, l. This system
is also used to indcate indices of faces with common orientation properties e.g.
(0, k, 0) all faces parallel to the a- and c-axis
(0, k, l) all faces parallel to the a- axis
43. Find intercepts along axes → 2 3 1
Take reciprocal → 1/2 1/3 1
Convert to smallest integers in the same ratio → 3 2 6
Enclose in parenthesis → (326)
(2,0,0)
(0,3,0)
(0,0,1)
Miller Indices for planes
45. Interplaner Distances for Cubic Crystal or Spacing of planes
222
lkh
a
dhkl
Here a = length of any side of a cube in Angstrom (A0 )
h – intercept made by plane on x- axis
k- – intercept made by plane on y- axis
l - – intercept made by plane on z- axis
46. Index
Number of
members in a
cubic lattice
dhkl
(100) 6
(110) 12 The (110) plane bisects the
face diagonal
(111) 8 The (111) plane trisects the
body diagonal
(210) 24
(211) 24
(221) 24
(310) 24
(311) 24
(320) 24
(321) 48
100d a
110 / 2 2 / 2d a a
111 / 3 3 /3d a a
49. very small wavelengths of
radiation.
Why are electrons go for studying matter?
Why are electrons not ideal?
What could be used instead?
Xrays made by slamming electrons into metals.
50. Bragg’s Law
nλ = 2 d sin θ
Constructive interference only occurs for certain θ’s
correlating to a (hkl) plane, specifically when the path
difference is equal to n wavelengths.
54. path difference=nλ give constructive
interference.
for different angles path difference varies. Some angles give constructive some
destructive.
Braggs Law Path difference =2dsinθ=nλ
55. The Braggs realised that by sending the X rays in at different angles they should get periods of constructive
and destructive interference.
X ray tube target
Turntable
Collimating
Slits
The Bragg Spectrometer
This would help them
determine the lattice
structure including d.
56. In two dimensions this type of
pattern is produced.
typical lattice diffraction pattern for iron
57. 57
a) ROCK SALT STRUCTURE (NaCl)
• CCP Cl- with Na+ in all Octahedral
holes
• Lattice: FCC
• Motif: Cl at (0,0,0); Na at (1/2,0,0)
• 4 NaCl in one unit cell
• Coordination: 6:6 (octahedral)
• Cation and anion sites are topologically
identical
STRUCTURE TYPE - AX NaCl
CLOSE PACKED STRUCTURES
58. 58
• CCP S2- with Zn2+ in half Tetrahedral holes ( T+ {or T-}
filled)
• Lattice: FCC
• 4 ZnS in one unit cell
• Motif: S at (0,0,0); Zn at (1/4,1/4,1/4)
• Coordination: 4:4 (tetrahedral)
• Cation and anion sites are topologically identical
b) SPHALERITE OR ZINC BLEND (ZnS) STRUCTURE
59. 59
• HCP with Ni in all Octahedral holes
• Lattice: Hexagonal - P
• Motif: 2Ni at (0,0,0) & (0,0,1/2) 2As at (2/3,1/3,1/4)
& (1/3,2/3,3/4)
• 2 NiAs in unit cell
• Coordination: Ni 6 (octahedral) : As 6 (trigonal
prismatic)
c) NICKELARSENIDE (NiAs)
60. 60
• HCP S2- with Zn2+ in half Tetrahedral holes ( T+ {or T-}
filled )
• Lattice: Hexagonal - P
• Motif: 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
• Coordination: 4:4 (tetrahedral)
d) WURTZITE ( ZnS )
62. 62
STRUCTURE TYPE - AX
NON – CLOSE PACKED STRUCTURES
CUBIC-P (PRIMITIVE) ( eg. Cesium Chloride ( CsCl ) )
• Motif: Cl at (0,0,0); Cs at (1/2,1/2,1/2)
• 1 CsCl in one unit cell
• Coordination: 8:8 (cubic)
• Adoption by chlorides, bromides and iodides of larger cations,
• e.g. Cs+, Tl+, NH4
+
63. 63
• CCP Ca2+ with F- in all Tetrahedral holes
• Lattice: fcc
• Motif: Ca2+ at (0,0,0); 2F- at (1/4,1/4,1/4) & (3/4,3/4,3/4)
• 4 CaF2 in one unit cell
• Coordination: Ca2+ 8 (cubic) : F- 4 (tetrahedral)
• In the related Anti-Fluorite structure Cation and
Anion positions are reversed
STRUCTURE TYPE - AX2
CLOSE PACKED STRUCTURE eg. FLUORITE (CaF2)
64. 64
• CCP Ca2+ with F- in all Tetrahedral holes
• Lattice: fcc
• Motif: Ca2+ at (0,0,0); 2F- at (1/4,1/4,1/4) & (3/4,3/4,3/4)
• 4 CaF2 in one unit cell
• Coordination: Ca2+ 8 (cubic) : F- 4 (tetrahedral)
• In the related Anti-Fluorite structure Cation and
Anion positions are reversed
STRUCTURE TYPE - AX2
CLOSE PACKED STRUCTURE eg. FLUORITE (CaF2)
65. 65
ALTERNATE REPRESENTATION OF FLUORITE
STRUCTURE
Anti–Flourite structure (or Na2O structure) – positions of
cations and anions are reversed related to Fluorite structure
67. 67
• HCP of Iodide with Cd in Octahedral holes of alternate layers
• CCP analogue of CdI2 is CdCl2
STRUCTURE TYPE - AX2
NON-CLOSE PACKED STRUCTURE
LAYER STRUCTURE ( eg. Cadmium iodide ( CdI2 ))
69. 69
HCPANALOGUE OF FLOURITE (CaF2) ?
• No structures of HCP are known with all Tetrahedral sites (T+
and T-) filled. (i.e. there is no HCP analogue of the Fluorite/Anti-
Fluorite Structure).
• The T+ and T- interstitial sites above and below a layer of close-
packed spheres in HCP are too close to each other to tolerate the
coulombic repulsion generated by filling with like-charged species.
Unknown HCP
analogue of FluoriteFluorite
71. 71
Formula Type and fraction of sites
occupied
CCP HCP
AX All octahedral
Half tetrahedral (T+ or T-)
Rock salt (NaCl)
Zinc Blend (ZnS)
Nickel Arsenide (NiAs)
Wurtzite (ZnS)
AX2 All Tetrahedral
Half octahedral (ordered
framework)
Half octahedral (Alternate
layers full/ empty)
Fluorite (CaF2),
Anti-Fluorite (Na2O)
Anatase (TiO2)
Cadmium Chloride
(CdCl2)
Not known
Rutile (TiO2)
Cadmium iodide (CdI2)
A3X All octahedral & All
Tetrahedral
Li3Bi Not known
AX3 One third octahedral YCl3 BiI3
SUMMARY OF IONIC CRYSTAL STRUCTURE TYPES
72. 72
Rock salt(NaCl) – occupation of all octahedral holes
• Very common (in ionics, covalents & intermetallics )
• Most alkali halides (CsCl, CsBr, CsI excepted)
• Most oxides / chalcogenides of alkaline earths
• Many nitrides, carbides, hydrides (e.g. ZrN, TiC, NaH)
Fluorite (CaF2) – occupation of all tetrahedral holes
• Fluorides of large divalent cations, chlorides of Sr, Ba
• Oxides of large quadrivalent cations (Zr, Hf, Ce, Th, U)
Anti-Fluorite (Na2O) – occupation of all tetrahedral holes
• Oxides /chalcogenides of alkali metals
Zinc Blende/Sphalerite ( ZnS ) – occupation of half tetrahedral holes
• Formed from Polarizing Cations (Cu+, Ag+, Cd2+, Ga3+...) and
Polarizable Anions (I-, S2-, P3-, ...)
e.g. Cu(F,Cl,Br,I), AgI, Zn(S,Se,Te), Ga(P,As), Hg(S,Se,Te)
Examples of CCP Structure Adoption
73. 73
Examples of HCP Structure Adoption
Nickel Arsenide ( NiAs ) – occupation of all octahedral holes
• Transition metals with chalcogens, As, Sb, Bi e.g. Ti(S,Se,Te);
Cr(S,Se,Te,Sb); Ni(S,Se,Te,As,Sb,Sn)
Cadmium Iodide ( CdI2 ) – occupation half octahedral (alternate) holes
• Iodides of moderately polarising cations; bromides and chlorides of
strongly polarising cations. e.g. PbI2, FeBr2, VCl2
• Hydroxides of many divalent cations. e.g. (Mg,Ni)(OH)2
• Di-chalcogenides of many quadrivalent cations . e.g. TiS2, ZrSe2, CoTe2
Cadmium Chloride CdCl2 (CCP equivalent of CdI2) – half octahedral holes
• Chlorides of moderately polarising cations e.g. MgCl2, MnCl2
• Di-sulfides of quadrivalent cations e.g. TaS2, NbS2 (CdI2 form as well)
• Cs2O has the anti-cadmium chloride structure
74. 74
PEROVSKITE STRUCTURE
Formula unit – ABO3
CCP of A atoms(bigger) at the corners
O atoms at the face centers
B atoms(smaller) at the body-center
75. 75
• Lattice: Primitive Cubic (idealised structure)
• 1 CaTiO3 per unit cell
• A-Cell Motif: Ti at (0, 0, 0); Ca at (1/2, 1/2, 1/2);
3O at (1/2, 0, 0), (0, 1/2, 0), (0, 0, 1/2)
• Ca 12-coordinate by O (cuboctahedral)
• Ti 6-coordinate by O (octahedral)
• O distorted octahedral (4xCa + 2xTi)
PEROVSKITE
• Examples: NaNbO3 , BaTiO3 ,
CaZrO3 , YAlO3 , KMgF3
• Many undergo small distortions:
e.g. BaTiO3 is ferroelectric
76. 76
SPINEL STRUCTURE
Formula unit AB2O4 (combination of Rock Salt
and Zinc Blend Structure)
Oxygen atoms form FCC
A2+ occupy tetrahedral holes
B3+ occupy octahedral holes
INVERSE SPINEL
A2+ ions and half of B3+ ions
occupy octahedral holes
Other half of B3+ ions occupy
tetrahedral holes
Formula unit is B(AB)O4