Structure of Solids
Objectives
By the end of this section you should be able to:
•
Compare bcc,
fcc
and
hcp
crystal structures
•
Identify structure in common ionic and non

closed packed materials
•
Construct a reciprocal lattice
•
Interpret points in reciprocal space
•
Compare reciprocal space to a Fourier transform
Groups: Fill in this Table
for Cubic Structures
SC
BCC
FCC
Volume of conventional cell
a
3
a
3
a
3
# of atoms
per cubic cell
1
2
4
Volume, primitive
cell
a
3
½ a
3
¼ a
3
#
of nearest neighbors
6
8
12
Nearest

neighbor
distance
a
½ a
3
a/
2
# of second neighbors
12
6
6
Second neighbor distance
a
2
a
a
Coordination
# = 8
Adapted from Fig. 3.2,
Callister
6e.
(Courtesy P.M. Anderson)
• Close packed directions are cube diagonals.

Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
BODY CENTERED CUBIC STRUCTURE (BCC)
How would we calculate
the atomic packing factor?
a
R
• APF for a body

centered cubic structure =
p
3/8 = 0.68
Unit cell c
ontains:
1 + 8 x 1/8
=
2 atoms/unit cell
Adapted from
Fig. 3.2,
Callister 6e.
ATOMIC PACKING FACTOR: BCC
Better
packing than SC
•
In the body

centred
cubic (bcc) structure 68% of the total
volume is occupied
.
•
Next

nearest neighbors relatively close by
–
make structure
stable in some instances. Examples: Alkali metals, Ba, V,
Nb
,
Ta, W, Mo, Cr, Fe
•
Is this cube a primitive lattice?
•
No. The
bcc structure
is a
Bravais
lattice but the edges of the
cube are not the
primitive
lattice vectors.
Not smallest Vol.
• Coordination # = 12
Adapted from Fig. 3.1(a),
Callister 6e.
(Courtesy P.M. Anderson)
• Close packed directions are face diagonals.

Note: All atoms are identical; the face

centered atoms are shaded
differently only for ease of viewing.
FACE CENTERED CUBIC STRUCTURE (FCC)
Unit cell c
ontains:
6 x 1/2 + 8 x 1/8
=
4
atoms/unit cell
a
• APF for a body

centered cubic structure =
p
/(3
2) = 0.74
(best possible packing of identical spheres)
Adapted from
Fig. 3.1(a),
Callister 6e.
ATOMIC PACKING FACTOR: FCC
Simple Crystal FCC
In the face

centred
cubic (
fcc
) structure 74% of the total
volume is occupied (slightly better than bcc with 68%)
Another view
Close packed crystals
A plane
B plane
C plane
A plane
…ABCABCABC… packing
[Face Centered Cubic (FCC)]
…ABABAB… packing
[Hexagonal Close Packing (HCP)]
Close

packed structures:
fcc
and
hcp
hcp
ABABAB...
fcc
ABCABCABC...
In groups, build these two
differing crystal structures.
• ABAB... Stacking Sequence
APF
=
0.74 (same as
fcc
)
What is the packing direction?
• 3D Projection
• 2D Projection
A sites
B
sites
A sites
HEXAGONAL CLOSE

PACKED STRUCTURE
(HCP)
For
ideal
packing, c/a
ratio of 1.633
However
, in most
metals, ratio
deviates from this value
Lattice Planes and Miller Indices
Hexagonal structure:
a

b
plane (2D hexagon) can be
defined by 3 vectors in plane (
hkl
)
3D structure can be defined by 4
miller indices (
h k l m
)
Third miller index not
independent:
h + k =

l
H
ave more on HCP planes in the Additional
M
aterials tab of website
e
h
k
l
m
The Crystal Lattice
–
3D
There are 7
(instead of 5)
possibilities
to define
basis vectors
Seven Lattice Systems and Fourteen Bravais Lattices
•
In ionic materials, different
considerations can be important
(electrostatics, different size of ions)
•
F
igure shows the crystal structure of
Cs
+
Cl

. The
lattice constant is 4.12
Å and all the bonds shown
have the same length. The grey atoms are Cs and
the green ones are Cl.
•
Group: Define
crystal structure
: meaning what are
the
primitive
Bravais
lattice
and the associated
basis
for this crystal (
including the locations
of
these atoms in terms of lattice parameter
a
)?
•
What is the angle between the chemical bonds?
Ionic materials
(Transferred Electron)
Cesium Chloride Structure Cs
+
Cl

•
S
imple
cubic
lattice
with
a
basis
consisting
of
a
cesium
ion
at
the
origin
0
and
a
chlorine
ion
at
the
cube
center
•
CsBr
and
CsI
crystallize
in
this
structure
.
The
lattice
constants
are
in
the
order
of
4
angstroms
.
)
(
2
/
z
y
x
a
NaCl
(Salt) Structure
•
In
NaCl
the small Na are in
interstitial positions
between the
Cl
ions
•
Group: Define the crystal
structure
•
This one is harder.
Cs
+
Cl

For comparison
Simple Crystal Structures
NaCl
•
NaCl
: interpenetrating
fcc
structures
–
One atom at (0,0,0)
–
Second atom displaced by
(1/2,0,0)
•
Majority of ionic crystals
prefer
NaCl
structure despite
lower coordination (fewer NN)
–
Radius of
cations
much smaller
than anions typically
–
For very small
cations
, anions
can not get too close in the
other typical structure (
CsCl
)
–
This favors
NaCl
structure where
anion contact does not limit
structure as much
NaCl
Other non
close

packed structures
•
These are covalent
materials (bond direction
is
more
important than packing)
diamond (only 34 % packing)
graphite
Group Exercise
•
How many
atoms
are in the primitive
unit cell of graphite? Identify a unit cell.
Simple Crystal Structures
Diamond
•
Crystal class
T
d
(tetrahedral)

Each atom has 4 nearest

neighbors (
nn
).
•
Can be interpreted as two
inter

penetrating
fcc
structures
–
One atom at origin
–
Other atom displaced along
diagonal (¼, ¼, ¼)
•
Includes C, Si,
Ge
,
a

Sn
How would we calculate the atomic packing factor?
For ABCABC… stacking it is called zinc
blende
Diamond &
Zincblende
crystals
•
Basis set
:
2 atoms.
Lattice
face centered cubic (
fcc
).
•
The
fcc
primitive lattice is generated by
r = n
1
a
1
+n
2
a
2
+n
3
a
3
w
ith lattice vectors:
a
1
= (½)a(0,1,0), a
2
= (½)a(1,0,1),
a
3
= (½)a(1,1,0)
NOTE:
The
a
i
’s
are
NOT
mutually orthogonal!
Diamond:
2 identical atoms in basis (e.g. 2 C)
fcc
lattice
Zincblende
:
2 different atoms in basis and
fcc
lattice
Many
semiconductors have the
Wurtzite
Structure
Tetrahedral coordination
:
Each atom has 4 nearest

neighbors (
nn
).
Basis set
:
2 atoms.
Lattice
hexagonal close packed (
hcp
).
A Unit Cell looks like
For ABAB… stacking it is called
wurzite
structure (
fcc
zincblende
was ABCABC…)
Some compounds can have either structure (i.e.,
GaN
,
SiC
)
hcp
primitive lattice vectors :
a
1
= c(0,0,1)
a
2
= (½)a[(1,0,0) + (3)
½
(0,1,0)]
a
3
= (½)a[(

1,0,0)
+ (3)
½
(0,1,0)]
Perovskites
•
Superconductors
•
Ferroelectrics
(
BaTiO
3
)
•
Colossal Magnetoresistance
(LaSrMnO
3
)
•
Multiferroics
(BiFeO
3
)
•
High
ε
r
Insulators (SrTiO
3
)
•
Low
ε
r
Insulators (LaAlO
3
)
•
Conductors (Sr
2
RuO
4
)
•
Thermoelectrics
(doped SrTiO
3
)
•
Ferromagnets
(SrRuO
3
)
A

site
(
Ca
)
Oxygen
B

site (Ti)
C
aTiO
3
e
g
t
2g
Formula unit
–
ABO
3
A
atoms
at
the corners
B atoms (smaller) at the body

center
O atoms at the face centers
In groups, define the crystal structure.
•
Lattice
:
Simple
Cubic (
idealized
structure)
•
1 CaTiO
3
per unit cell
•
Cell
Motif
: Ti at (0, 0, 0); Ca at (
1
/
2
,
1
/
2
,
1
/
2
);
3 O
at
(
1
/
2
, 0, 0), (0,
1
/
2
, 0), (0, 0,
1
/
2
)
could label differently
•
Ca 12

coordinate by
O, Ti
6

coordinate by
O, O
distorted octahedral
PEROVSKITES
A

site
(
Ca
)
Oxygen
B

site (Ti)
C
aTiO
3
Is this cube a primitive lattice?
Spinel AB
2
O
4
Structure (~CCP)
Alternate layers parallel (111)
Octahedral &
Octahedral

Tetrahedral
Oct
Oct
Oct & Tet
Perpendicular to (111
)
CCP with
1/8 Tetrahedral = A
¼ Octahedral = B
Normal Spinel: B all Oct
Inverse Spinel: B ½ Tet
8
12
Coordination
number
6
Primitive cubic
Body centered cubic
Face centered cubic
27
Close

packed structures:
fcc
and
hcp
hcp
ABABAB...
fcc
ABCABCABC...
28
Close

packed structures:
fcc
and
hcp
hcp
ABABAB...
fcc
ABCABCABC...
29
Close

packed structures:
fcc
and
hcp
hcp
ABABAB...
fcc
ABCABCABC...
30
Close

packed structures:
fcc
and
hcp
hcp
ABABAB...
fcc
ABCABCABC...
Close

packed structures:
fcc
and
hcp
hcp
ABABAB...
fcc
ABCABCABC...
•
The face

centred
cubic (
fcc
) and hexagonal close

packed (
hcp
) structure have the same packing
fraction
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