Structure of Solids Objectives

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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