1
Review of Crystallography
1 Crystal Basics
2 Symmetry
3 Crystal Structure Analysis
4 Crystal Chemistry
5 Some Important Crystal Structures
1 Crystal Basics
1.Crystal
2.Fundamental Characteristics of Crystals
Why Solids?
μ ALL Compounds are solids under suitable
conditions of temperature and pressure. Many exist
only as solids.
μ atoms in ~fixed position
“simple”case ? crystalline solid ? Crystal Structure
Why study crystal structures?
μ description of solid
μ comparison with other similar materials ?
classification
μ correlation with physical properties
Early Ideas
Crystals are solid ? but solids are not
necessarily crystalline
Crystals have symmetry (Kepler, 1611) and long
range order
Spheres and small shapes can be packed to
produce regular shapes (Hooke; Hauy,1812)
?
Definition ? Crystal
Crystals ? A homogenous solid formed by a
repeating, three-dimensional pattern of
atoms, ions, or molecules and having fixed
distances between constituent parts.
Crystallinity
A crystal may be defined as a collection of atoms
arranged in a pattern that is periodic in 3D.
Crystals are necessarily solids, but not all solids are
crystalline (amorphous solids lack long range periodic
order).
In a perfect single crystal, all atoms in the crystal are
related either through translational symmetry or point
symmetry.
Polycrystalline materials are made up of a great
number of tiny (μm to nm) single crystals
Crystalline solids can be divided into two categories
extended and molecular
2
Single Crystal and
Polycrystalline Materials
§ Single crystal:atoms are in a repeating or periodic
array over the entire extent of the material
§ Polycrystalline material:comprised of many small
crystals or grains. The grains have different
crystallographic orientation.There exist atomic
mismatch within the regions where grains meet.
These regions are called grain boundaries.
Single Crystals
repeated arrangement of atoms extends
throughout the specimen
all unit cells have the same orientation
exist in nature
can also be grown (eg. Si)
without external constraints, will have flat,
regular faces
Beautiful Crystals Polycrystalline Materials
Crystals of different
sizes
orientations
shapes
Grain Boundaries
?mismatch between two
neighboring crystals
Polycrystalline Materials
Most crystalline materials are composed of many
small crystals called grains
Crystallographic directions of adjacent grains are
usually random
There is usually atomic mismatch where two
grains meet ?? this is called a grain boundary
Most powdered materials have many randomly
oriented grains
Basic Characteristic of Crystals
Homogeneity ?? Under macroscopic
observation,the physics effect and chemical
composition of a crystal are the same.
Anisotropy ?? Physical properties of a
crystal differ according to the direction of
measurement.
3
Anisotropy
Different directions in a crystal have
different packing. For instance, atoms
along the edge of FCC unit cell are
more separated than along the face
diagonal. This causes anisotropy in the
properties of crystals, for instance, the
deformation depends on the direction
in which a stress is applied.
In some polycrystalline materials, grain
orientations are random, so bulk material
properties are isotropic.
Some polycrystalline materials have grains
with preferred orientations (texture), so
properties are dominated by those relevant to
the texture orientation and the material
exhibits anisotropic properties.
Law of Constancy of Interfacial Angle
The interfacial angles are constant for all
crystals if a given mineral with identical
composition at the same temperature.
Since all crystals of the same substance will
have the same spacing between lattice points
(they have the same crystal structure), the
angles between corresponding faces of the same
mineral will be the same.
The symmetry of the lattice will determine the
angular relationships between crystal faces.
Crystal Shape
The external shape of a crystal is referred to as its
Habit
Not all crystals have well defined external faces
Typically see faces on crystals grown from
solution
Natural faces always have low indices (orientation
can be described by Miller indices that are small
integers)
The faces that you see are the lowest energy faces
Surface energy is minimized during growth
This is a term that refers to the form that a crystal
takes as it grows.
Prismatic
Pyramidal
Tabular
Rhombohedra
Dodecahedral
Acicular
Bladed
Crystal Habits
Crystal Habits
4
Law of Symmetry
Law of Symmetry: Only 1,2,3,4,6-fold rotation
axis can exist in crystal.
Why snowflakes have 6 corners, never 5 or 7?
By considering the packing of polygons in 2
dimensions, demonstrate why pentagons and
heptagons shouldn ’t occur.
Empty space
not allowed
Quasicrystal: AlFeCu
Allowed
rotation axis:
1, 2, 3, 4, 6
NOT 5, > 6
Quasicrystal Structures (First in 1984)
R0.09Mg0.34Zn0.57
Dodecahedral morphology
R0.1Mg0.4Cd0.5
Rhombic triacontahedral
morphology
ED:
5 fold axis
Face-centred icosahedral R-Mg-Zn Primitive icosahedral R-Mg-Cd
Non?periodic long?range ordered structures
Rotational symmetry of diffraction patterns (e.g.
5?fold, 10?fold) impossible for periodic crystals
Quasicrystalline Materials
Quasi-unit cells
Amorphous Solids
Ideal solid crystals exhibits structural long range
order (LRO)
Real crystals contain imperfections, i.e., defects
and impurities, which spoil the LRO
Amorphous solids lack any LRO [though may
exhibit short range order (SRO)]
Crystal Glass(amorphous) Gas
5
Quartz Crystal and Quartz Glass
Quartz GlassQuartz Crystal
Glass
§ Transparent, amorphous solid
–Composition almost all silicon dioxide (SiO2 –
Quartz sand)
§Ordinary glass à75% SiO2
§Pyrex glass àSiO2 with B2O3
§Lead glass àSiO2 + PbO, and K2O
§Green glass (cheap bottles) àFeO + SiO2
§Blue glass àCobalt oxide + SiO2
§Violet glass àManganese + SiO2
§Yellow glass àUranium oxide + SiO2
§Red glass àGold and copper + SiO 2
Liquid Crystal
Liquid crystals are a phase
of matter whose order is
intermediate between that
of a liquid and that of a
crystal. The molecules are
typically rod-shaped
organic moieties about 25
Angstroms in length and
their ordering is a function
of temperature.
From Crystal to Liquid Crystal
to Liquid
(a)crystal， (b)、 (c) anisotropic liquids， (d) isotropic
liquid
This is the structure change process of some molecules
with long chains when increasing temperatures
Nematic
Entropy driven formation of liquid crystals of rod?like colloids
Isotropic
Crystal Smectic
= Direction of increasing density
Principles of Liquid Crystal Displays
No voltage voltage
6
Smaller, lighter, with no radiation problems. Found
on portables and notebooks, and starting to appear
more and more on desktops.
Less tiring than c.r.t. (Cathode?ray tube) displays,
and reduce eye?strain, due to reflected nature of light
rather than emitted. Use of super?twisted crystals
have improved the viewing angle, and response rates
are improving all the time (necessary for tracking
cursor accurately).
Liquid Crystal Displays 2 Symmetry
Symmetry: Point Symmetry
Space Symmetry
32 Point Groups of Crystals
Unit Cell, 7 Crystal Systems, Lattice Planes,
Miller indices
Lattices and 14 Bravias Types of Lattices
230 Space Groups
Crystal Symmetry
§ Mathematics of Symmetry
§ Crystal’s Symmetry
§ Physical Properties caused by Symmetry
Symmetry in Nature, Art and Math
Symmetry is one idea by which man through the
ages has tried to comprehend and create order,
beauty and perfection. Hermann Weyl
Eiffel tower in Paris, France is a
wonderful example of symmetry
Macroscopic Symmetry Elements
(Point Symmetry Elements)
Point symmetry elements operate to change the
orientation of structural motifs
A point symmetry operation does not alter at
least one point that it operates on
Symmetry Elements and Symmetry Operations:
1. Mirror Planes ——Reflection or Mirror
2. Center of Symmetry ——Inverse
3. Rotation Axis ——Rotate
4. Rotoinversion Axis ——Rotate and inverse
Mirror Plane Symmetry
Mirror plane symmetry arises
when one half of an object is
the mirror image of the other
half
?Can be folded in half
?Seen externally with animals
σ
σ
7
This molecule has two mirror planes:
One is horizontal, in the plane of the
paper and bisects the H-C-H bonds
Other is vertical, perpendicular to
the plane of the paper and bisects the
Cl-C-Cl bonds
A crystal has reflectional symmetry if an
imaginary plane can divide the crystal into halves,
each of which is the mirror image of the other.
Mirror Plane Symmetry Symmetry Operation
Reflection
qflips all points in the
asymmetric unit over a
line, which is called the
mirror and thereby
changes the handedness of
any figures in the
asymmetric unit. The
points along the mirror
line are all invariant
points under a reflection.
Rotational Symmetry
Rotated about a point
Allows chirality
In crystals limited to
1,2,3,4, and 6
rotations
Symmetry Operation
Rotation
turns all the points in the
asymmetric unit around one
point, the center of rotation.
A rotation does not change
the handedness of figures in
the plane. The center of
rotation is the only invariant
point.
Symmetry Axis of Rotation
We say a crystal has a symmetry
axis of rotation when we can turn it
by some degree about a point and
the pattern looks exactly the same.
Think of the center of a pizza. If it
is made so that all the pieces are
the same size and have the same
ingredients in the same places,
then the pizza could be turned and
you couldn't tell the difference.
This means the pizza has
rotational symmetry. The pizza
below has rotational symmetry of
60 degrees.
Rotational Symmetry
coincidence upon rotation
about the axis of 360°/n
? n-fold axis of rotational
symmetry
graphite
O
H
Symbol for a symmetry
element for which the
operation is a rotation of 360°/n
C2 = 180°, C3=120°, C4 = 90°,
C5 = 72°, C6 = 60°, etc.
8
Can rotate by 120° about the C-Cl
bond and the molecule looks identical
? the H atoms are indistinguishable.
This is called a rotation axis
? in particular, a three fold
rotation axis, as rotate by 120° (=
360°/3) to reach an identical
configuration
Rotation Axis (Cn)
In general:
n-fold rotation axis = rotation by (360°/n)
“present if you can draw a straight line from any
point, through the center, to an equal distance the
other side, and arrive at an identical point”.
Center of
symmetry at S
No center of
symmetry
( x,y,z)
( - x, - y, - z)
Center of Symmetry
(Inversion symmetry)
i
Symmetry Operation
Inversion
every point on one
side of a center of
symmetry has a
similar point at an
equal distance on
the opposite side of
the center of
symmetry.
Rotoinversion Axis ?? Symmetry
Axis of Rotary Inversion
Rotoinversion Axis (Sn or ) : n-fold rotation
combined with an inversion.
n
i1 = m2 = 3 = 3?fold rotation + inversion
4 6
=3?fold rotation
with perpendicular
mirror plane
Macroscopic Symmetry Elements:
Point Groups
Electrical resistance
Thermal expansion
Magnetic susceptibility
Elastic constants
Macroscopically measured
properties
? Macroscopic symmetry
?X
Translation symmetry
Combination of mirror, center of symmetry,
rotational symmetry, center of inversion ? point
groups
9
Point Groups
Point groups have symmetry about a single point at
the center of mass of the system.
Symmetry elements are geometric entities about
which a symmetry operation can be performed. In a
point group, all symmetry elements must pass
through the center of mass (the point). A symmetry
operation is the action that produces an object
identical to the initial object.
Group theory is a very powerful mathematical tool
that allows us to rationalize and simplify many
problems in chemistry. A group consists of a set of
symmetry elements (and associated symmetry
operations) that completely describe the symmetry
of an object.
Point Group
Point group (point symmetry) All molecules
characterized by 32 different combinations of
symmetry elements
There are two naming systems commonly used
in describing symmetry elements
1. The Schoenflies notation used extensively by
spectroscopists
2. The Hermann-Mauguinor international
notation preferred by crystallographers
All Combinations of Point
Symmetry Elements Are Not Possible
The allowed combinations of point symmetry
elements are called point groups
Point Symmetry Elements
Compatible With 3D Translations
Deduction of 32 Point Groups
Rotation: 1?, 2?, 3?, 4?, 6?fold? 5 point group
among 32
Rotation?inversion:
? 5 point group among 32
Combinations of rotation: 222, 223, 224, 226, 23,
432 ? 6 point group among 32
Combinations of rotation and an inversion or a
mirror:15 point group among 32
mirror: 1 point group among 32
Deduction of 32 Point Groups
3L44L36L23L24L3L66L2 L22L2
1、 2、 3、 4、 6 fold rotation axis exist in crystal, which
are marked as C1、 C2、 C3、 C4 and C6.
The combination of rotation axis can deduce： 222、 223、
224、 226、 23、 432， which are marked as D2、 D3、 D4、
D6、 T and O.
The above 11 point groups consists of only rotation axis
and do not have inversion axis.
Schoenflies Symbols
vCn: cyclic, the point group which only one
rotation axis,n is the order of the rotation axis.
vDn: dihedral, the group point which generated
from the combination of 2-fold axis, n is the
order of the main rotation axis).
vT: tetrahedral
vO: octahedral
The combination of rotation axis
with higher order
10
The 32 Point Groups
C1 C2 C3 C4 C6 D2 D3 D4 D6 T O 11
+Ph Cs C2h C3h C4h C6h D2h D3h D4h D6h Th Oh 22
+Pv -- C2v C3v C4v C6v -- -- -- -- -- -- 26
+Pd -- -- -- -- -- D2d D3d -- -- Td -- 29
+C Ci -- C3i -- -- -- -- -- -- -- -- 31
n S4 32
Add mirror plane to the above 11 basic point groups, the
adding mirror plane intersect at one point with other
symmetry elements, and in addition, no new symmetry
types are formed, thus there are three ways:
1)Mirror plane is horizontal with the main rotation axis, Ph
2)Mirror plane is vertical to the main rotation axis, Pv
3)Mirror plane is vertical to the main rotation axis,and is
diagonal to the neighboring 2-fold axis), Pd
vAttention: adding Pv to Dn、 T and O is equal to Dnh、 Th
and Oh respectively, Od is equal to Oh.
32 Crystallographic Point Groups
Crystal System Number of
Point Groups
Herman-Mauguin
Point Group
Schoenflies
Point Group
Triclinic 2 1, ‘1 C1, Ci
Monoclinic 3 2, m, 2/m C2, C s, C2h
Orthorhombic 3 222, mm2, mmm D2, C2v, D2h
Trigonal 5 3,‘3, 32,
3m, ‘3m
C 3, S6, D3,
C 3v, D3d
Hexagonal 7 6,‘6, 6/m, 622,
6mm, ‘62m, 6mm
C 6, C3h, C4h, D6,
C6v, D3h , D6h
Tetragonal 7 4,‘4, 4/m, 422,
4mm,‘42m, 4/mmm
C 4, S4, C4h, D4,
C 4v , D2d, D4h
Cubic 5 23, m3, 432,
‘432, m‘3m
T, Th, O,
Td, Oh
We can use a flow chart such
as this one to determine the
point group of any object. The
steps in this process are:
1. Determine the symmetry is
special.
2. Determine if there is a
principal rotation axis.
3. Determine if there are
rotation axes perpendicular to
the principal axis.
4. Determine if there are
mirror planes.
5. Assign point group.
Identifying Point Groups Identifying Point Groups (1)
Identifying Point Groups (2) Identifying Point Groups (3)
11
Microscopic Symmetry Elements
(Space Symmetry Elements)
1. Lattice —— its corresponding operation is
translational symmetry.
2. screw axes —— combination of a rotation and
a translational symmetry.
3. glide planes —— combination of a refection
and a translational symmetry.
All these action are space symmetry. Every point
in the space is changed, but the space do not
change after the action. So, their symmetry is
called space groups.
Symmetry Elements:
Translation
moves all the points in
the asymmetric unit the
same distance in the
same direction. This has
no effect on the
handedness of figures in
the plane. There are no
invariant points under a
translation.
A C
ar
ar
b
r
Glide Reflection reflects the asymmetric
unit across a mirror line
and then translates
parallel to the mirror. A
glide reflection changes the
handedness of figures in
the asymmetric unit. There
are no invariant points
under a glide reflection.
Glide Reflection: A glide reflection combines a
reflection with a translation along the direction of
the mirror line.
A glide plane is a combination of a reflection and
a translation.
The orientation of the plane and its symbol
determine what sort of translation is involved.
b glide parallel to (001)
Glide Planes
Diagonal Glide
the diagonal glide (n?glide) have a displacement
vector of ?(a+b).
Diamond Glides
the diamond glide (d?glide) have a displacement
vector of ?(a+b).
12
A screw axis with symbol nm is a combination of
an n?fold rotation followed by a translation of
m/n of the unit cell repeat parallel to the axis.
e.g. a 41 axis parallel to z axis involves rotation of
90° followed by translation of 1/4 c.
Screw Axes
the operation of 31 axes
the operation of 61 axes
Symmetry Elements:
Screw axes rotation about the axis of symmetry by
360°/n, followed by a
translation parallel to
the axis by b/n of the
unit cell length in
that direction.
A Floor?Tiling Problem
Seven Types of Symmetry
Points symmetry
mirror
inversion
rotation
rotoinversion
translation
screw axes
glide planes
Space symmetry
Can not restore
the left-handed
and the right-
handed, only
return the
equivalent figures
Can restore the
left-handed and
right-handed
13
Space Group
Space group ( point & translational symmetry)
There are 230 possible arrangements of
symmetry elements in the solid state. Any
crystal must belong to one (and only one) space
group.
Definition of Crystal Structures
Crystal Structure: The spatial order of the
atoms is called the crystal structure, or 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).
Structure=Lattice+Motif
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)
Lattice: Periodic arrangement of points in space.
Must be one of the 14 Bravais lattices.
Motif: Collection of atoms to be placed
equivalently about each lattice point. Consists of
atomic identities and fractional coordinates.
Structure=Lattice+Motif
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 The unit cell is the basic building block of the
crystal
The unit cell can contain multiple copies of the
same molecule whose positions are governed by
symmetry rules
Unit Cell
14
2D example ? Rocksalt
(sodium chloride, NaCl)
We define lattice points: these are points
with identical environments
Choice of origin is arbitrary: lattice points need
not be atoms, but unit cell size should always be
the same.
This is also a unit cell ?
it doesn't matter if you start from Na or Cl ? or if you don’t start from an atom
This is NOT a unit cell even though
they are all the same ? empty space is
not allowed!
In 2D, this is a unit cell
In 3D, it is NOT
15
Unit Cell ? The smallest volume of a crystal which
can be used to generate the entire crystal by
repetition, through translation only, in three
dimensions.
NaCl has a cubic
unit cell which, if
repeated indefinitely, can
reproduce an entire salt crystal.
Unit cells do not have to be cubic (that is, they do not have
to have the same length of edge in all three directions).
In fact, the unit cells of most real molecules are not cubic,
but belong to less symmetric lattice groups.
However, since cubic cells are relatively easy to visualize,
we will use them for our examples.
The Face?centered Cubic Unit Cell
From Unit Cell To Lattice
If we take the square unit cell and stack it, we produce this
square lattice. Notice that once we begin stacking the unit
cells, we never change the orientation of any subsequent
unit cells as they stack. In other words, once the
orientation of a unit cell is determined, all unit cells within
that lattice have the same orientation.
Unit Cell Symmetries ?
Cubic
4 fold rotation axes
passing through pairs of
opposite face centers,
parallel to cell axes)
TOTAL = 3
Unit Cell Symmetries ?
Cubic
4 fold rotation axes
TOTAL = 3
3-fold rotation axes
(passing through cube
body diagonals)
TOTAL = 4
16
Unit Cell Symmetries ?
Cubic
4-fold rotation axes
TOTAL = 3
3-fold rotation axes
TOTAL = 4
2-fold rotation axes
(passing through diagonal
edge centers)
TOTAL = 6
Mirror Planes ?
Cubic
3 equivalent planes
in a cube
6 equivalent planes
in a cube
Cubic Unit Cell
a=b=c, α=β=γ=90°
Many examples of cubic unit cells:
e.g. NaCl, CsCl, ZnS, CaF2, BaTiO3
All have different arrangements of atoms within
the cell. So to describe a crystal structure we need
to know:
nthe unit cell shape and dimensions
nthe atomic coordinates inside the cell
γβ
α
a
c
b
Tetragonal Unit Cell
a = b ≠ c α = β = γ = 90°
c < a, b c > a, b
elongated / squashed cube
One 4-axis
No 3-axes
Two 2-axes
Five mirrors
Example
CaC2 ? has a rocksalt-like structure but
with non-spherical carbides
2 -C
C
Carbide ions are
aligned parallel to c
∴ c > a,b ?
tetragonal
symmetry
Reduction in Symmetry
Cubic Tetragonal
Three 4-axes One 4-axis
Four 3-axes No 3-axes
Six 2-axes Two 2-axes
Nine mirrors Five mirrors
17
γ≠90°a≠bOblique
γ=120°a=bHexagonal
γ=90°a≠bCentered Rectangular
γ=90°a≠bRectangular
γ=90°a=bSquare
5 Bravais Lattice in 2D
P P NP
Orthorhombic: P, I, F, C
Symmetry in Crystals
C F
Primitive
(P)
Body-centered
(I)
Side-centered
(C)
Face-centered
(F)
Hexagonal
Monoclinic Triclinic
Unit Cells of the Fourteen Bravais Lattices
a
b
c
αβ
γ
1. Primitive
Triclinic
a
b
c
αβ
γ
2. Primitive
Monoclinic
a
b
c
αβ
γ
3. Side(or C) - centered
Monoclinic
a
b
c
4. Primitive
orthorhombic 5. C - centered orthorhombic
a
b
c
a
b
c
6. Body - centered orthorhombic
a
b
c
7. Face - centered
orthorhombic
a
c
a
8. Primitive
Tetragonal
a
c
a
9. Body - centered
Tetragonal
a
c
a
120 °
10. Primitive
hexagonal
11. Primitive
rhombohedral
( trigonal )
a a
a
α
12. Primitive cubic
a
a
a
13. Body - centered
cubic
a
a
a
a
a
a
14. Face - centered
cubic
Definition:
Bravais Lattice: an infinite array of discrete points with
an arrangement and orientation that appears exactly the
same from whichever of the points the array is viewed.
3D: 14 Bravais Lattice, 7 Crystal System
Name Number of Bravais
lattices
Conditions
Triclinic 1 (P) a? b? c
a ?b ?g
Monoclinic 2 (P, C) a? b?c
a = b = 90 ? g
Orthorhombic 4 (P, F, I, A) a ?b ?c
a = b = g = 90°
Tetragonal 2 (P, I) a = b? c
a = b = g = 90°
Cubic 3 (P, F, I) a = b = c
a = b = g = 90°
Trigonal 1 (P) a = b = c
a = b = g < 120° ? 90°
Hexagonal 1 (P) a = b ?c
a = b = 90° g = 120°
18
The 14 possible
Bravais Lattices
14 Bravais
Lattices
connect the
macroscopic
morphology
of the
crystals and
their inner
periodic
structure
Seven Unit Cell Shapes
Seven Crystal Systems
Cubic a=b=c α=β=γ=90°
Tetragonal a=b≠c α=β=γ=90°
Orthorhombic a≠b≠c α=β=γ=90°
Monoclinic a≠b≠c α=γ=90°, β≠90°
Triclinic a≠b≠c α≠β≠γ≠90°
Hexagonal a=b≠c α=β=90°, γ=120°
Trigonal
(Rhombohedral) a=b=c α=β=γ≠90°
Trigonal P : 3-fold rotation
Trigonal P
a=b=c α=β=γ≠90°
Crystal Systems
There are seven crystal systems which can be defined either on
the basis of symmetry, or, upon the basic building block of the
crystal. The seven main symmetry groups into which all
crystals, whether natural or artificial, can be classified. All
crystals grow in one of following seven shapes on the
microscopic level.
1 Cubic or Isometric (3 axes of equal length intersect at 90°)
2 Tetragonal (2 axes of same length, all at 90°)
3 Orthorhombic (3 axes of different length at 90°)
4 Hexagonal (3 horizontal axis at 60°. vertical axis at 90°)
5 Monoclinic (3 axes of different length, 2 intersect at 90°,
the other is oblique to the others)
6 Triclinic (3 axes of different length are all oblique to one
another)
7. Trigonal(Rhombohedral) (3 axes of equal length)
Simple Cubic Lattice
Caesium Chloride (CsCl) is
primitive cubic
Different atoms at corners
and body center. NOT body
centered, therefore.
Lattice type P
Also CuZn, CsBr, LiAg
BCC Lattice
α-Iron is body-
centered cubic
Identical atoms at
corners and body
center (nothing at
face centers)
Lattice type I
Also Nb, Ta, Ba, Mo...
19
FCC Lattice
Copper metal is face-
centered cubic
Identical atoms at
corners and at face
centers
Lattice type F
also Ag, Au, Al, Ni...
FCC Lattices
Sodium Chloride (NaCl) ?
Na is much smaller than Cs
Face Centered Cubic
Rocksalt structure
Lattice type F
Also NaF, KBr, MgO… .
Density Calculation
ACNV
nA=ρ n: number of atoms/unit cellA: atomic mass
VC: volume of the unit cell
NA: Avogadro’s number
(6.023x1023 atoms/mole)
Calculate the density of copper.
RCu =0.128nm, Crystal structure: FCC, ACu= 63.5 g/mole
n = 4 atoms/cell 333C R216)2R2(aV ===
3
2338 cm/g89.8]10023.6)1028.1(216[
)5.63)(4( =
×××=ρ
8.94 g/cm3 in the literature
Unit Cell Contents
Counting the number of atoms within the unit cell
Many atoms are shared between unit cells
Atoms Shared Between: Each atom counts:
corner 8 cells 1/8
face center 2 cells 1/2
body center 1 cell 1
edge center 4 cells 1/4
lattice type cell contents
P 1 [=8 x 1/8]
I 2 [=(8 x 1/8) + (1 x 1)]
F 4 [=(8 x 1/8) + (6 x 1/2)]
C 2 [=(8 x 1/8) + (2 x 1/2)]
20
Space Groups in 3 Dimension
14 Bravais lattices + 32 point groups 72 space groups
+ screw axes
+ glide planes
230 space groups
Space group symbol Bravais lattice + basis symmetry
Ex) Fm3m? Cubic face-centered lattice + m3m (point group)
F (face-centered)
I (body-centered)
C (side-centered)
P (primitive)
Three Translational vectors
Five rotation axes Four lattice types
32 symmetry point groups
11 basic symmetry elements
Seven crystal systems
14 Bravais lattices
translational
screw axes
glide planes
230 space groups
Symmetry
combinations
Three simple
symmetry elements
230 Space Groups
1-2 : Triclinic, classes 1 and –1
3-15 : Monoclinic, classes 2, m and 2/m
16-24 : Orthorhombic, class 222
25-46 : Orthorhombic, class mm2
47-74 : Orthorhombic, class mmm
75-82 : Tetragonal, classes 4 and -4
83-88 : Tetragonal, class 4/m
89-98 : Tetragonal, class 422
99-110 : Tetragonal, class 4mm
111-122 : Tetragonal, class -42m
123-142 : Tetragonal, class 4/mmm
143-148 : Trigonal, classes 3 and -3
149-155 : Trigonal, class 32
156-161 : Trigonal, class 3m
162-167 : Trigonal, class -3m
168-176 : Hexagonal, classes 6, -6 and 6/m
177-186 : Hexagonal, classes 622 and 6mm
187-194 : Hexagonal, classes -6m2 and 6/mmm
195-206 : Cubic, classes 23 and m-3
206-230 : Cubic, classes 432, -43m and m-3m
2
13
59
68
25
27
36
Lattice Planes and Miller Indices
Imagine representing a crystal structure on a
grid (lattice) which is a 3D array of points
(lattice points).
Can imagine dividing the grid into sets of
“planes”in different orientations?
It is possible to describe certain directions and
planes with respect to the crystal lattice using a
set of three integers referred to as Miller Indices.
Lattice Planes Miller Index
x
y
z
O A
B
C
a
bc
x
y
z
a b
c
- y
Miller indices describe
the orientation and
spacing of a family of
planes
21
(100) (111)
(200) (110)
Examples of Miller Indices
Miller indices describe the orientation and
spacing of a family of planes.
?the spacing between adjacent planes in a
family is referred to as a “d-spacing”
Three different
families of planes
D-spacing between
(300) planes is one
third of the (100)
spacing
(100) (200) (300)
Families of Planes
All planes in a set are identical
The planes are “imaginary”
The perpendicular distance between pairs of
adjacent planes is the d-spacing
Find intercepts on
a,b,c: 1/4, 2/3, 1/2
Take reciprocals 4,
3/2, 2
Multiply up to
integers:(8 3 4) [if
necessary]
Exercise
?What is the Miller index of the plane below?
Find intercepts on
a,b,c: 1/2, 1, 1/2
Take reciprocals
2, 1, 2
Multiply up to
integers: (2, 1, 2)
Plane perpendicular to y
cuts at ∞, 1, ∞
→ (0 1 0) plane
General label is (h k l) which intersects at a/h, b/k, c/l
(hkl) is the Miller Index of that plane.
This diagonal cuts at 1, 1, ∞
→ (1 1 0) plane
(0 means that the plane is
parallel to that axis)
Indexing in the Hexagonal System
In hexagonal unit cells
it is common to refer the
orientation of planes and
lines to four coordinate
axes
The fourth axis a3 is
just = -a2-a1 . This
approach reflects the
three fold symmetry
associated with the unit
cell
22
Indices are expressed as (hkil)
h + k = ?i
All cyclic permutations of h, k and i are symmetry
equivalent
So (1010), (1100), (0110) are equivalent
Properties of Hexagonal Indices Crystallographic Directions and Planes
Miller?Bravais indices
[uvtw], (hkil)
i=-(h+l)t=-(u+v)
d-spacing Formula
For orthogonal
crystal systems
(i.e. α=β=γ=90°):
For cubic crystals
(special case of
orthogonal) a=b=c:
e.g. for (100) d = a
(200) d = a/2
(110) d = a/ etc.
2
2
2
2
2
2
2 c
l
b
k
a
h
d
1 ++=
2
222
2 a
lkh
d
1 ++=
2
A tetragonal crystal has a=4.7 ?, c=3.4 ?.
Calculate the separation of the:
(100) 4.7 ?
(001) 3.4 ?
(111) Planes 2.4 ?
A cubic crystal has a=5.2 ? (=0.52nm). Calculate
the d-spacing of the (110) plane
o2
22
222
2
A7.322.5d
2.5
11
a
lkh
d
1
==?
+=++=
]ba[cla khd1 2
2
2
22
2 =+
+=
Indexing of Planes and Directions
