Transcript Structure of Solids - West Virginia University

Structure of Solids
Objectives
By the end of this section you should be able to:
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Calculate atomic packing factors (HW)
Compare bcc, fcc and hcp crystal structures
Determine/understand coordination numbers
Identify primitive unit cell lattice parameters
Be able to build the Wigner-Seitz cell for a lattice
Solid Models: Close-Packed Spheres
• Many atoms or ions forming solids have spherical
symmetry (e.g. noble gases and simple metals)
• Considering the atoms or ions as solid spheres we
can imagine crystals as closely packed spheres
• How can we pack them?
ATOMIC PACKING FACTOR (APF)
a
R=0.5a
close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
Lattice constant
APF for the simple cubic structure = 0.52
Simple cubic
• A simple cubic structure is not efficient at packing
spheres (atoms occupy only 52% of the total
volume). Marbles will not resemble.
• Only two elements crystallize in the simple cubic
structure (F and O).
Another Reason Simple Cubic
Structure is Rare
Groups: Using the spheres (like atoms) and
magnetic sticks (like bonds between atoms),
create a simple cubic lattice.
How does this
compare to a
triangular pyramid
structure?
Three Cubic Unit Cell Types in 3D
 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
BODY CENTERED CUBIC STRUCTURE (BCC)
What is the close packed direction?
How would we calculate
the atomic packing factor?
--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
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.
FACE CENTERED CUBIC STRUCTURE (FCC)
What is the close packed direction?
APF = 0.74
--Note: All atoms are identical;
the face-centered atoms are
shaded differently only for ease
of viewing.
What are the lattice
directions of the
primitive unit cell?
Simple Crystal FCC
Another view
Homework 4.1
C. simple cubic with
additional points at the
A. simple cubic with
midpoints of lines
additional points in the
joining nearest
horizontal faces
neighbors
What does it mean to ask if it’s
a primitive Bravais lattice?
One lattice point (not atom) per unit cell
Note: green and orange are same atoms
(only different colors for clarity)
Groups: Fill in this Table
for Cubic Structures
SC
BCC
FCC
Volume of conventional cell
a3
a3
a3
# of atoms per cubic cell
1
2
4
Volume, primitive cell
a3
½ a3
¼ a3
# 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
Wigner-Seitz Method for Defining a
Primitive Unit Cell
(points are closest to each other)
Always a
hexagon in
2D, unless the
lattice is
rectangular.
1. Pick a center atom (origin) within the lattice
2. Draw perp. bisector to all neighbors of reciprocal lattice
3. Draw smallest polyhedron enclosed by bisectors
Wigner-Seitz for BCC & FCC
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...
If time allows:
In groups, build these two
differing crystal structures.
HEXAGONAL CLOSE-PACKED STRUCTURE
(HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection
A sites
B sites
A sites
APF = 0.74 (same as fcc)
What is the packing direction?
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)
m
3D structure can be defined by 4
miller indices (h k l m)
e
k
l
h
Third miller index not
independent:
h + k = -l
Have more on HCP planes in the Additional Materials tab of website
Packing isn’t the only consideration when building a lattice.
The Crystal Lattice – 3D
There are 7
(instead of 5)
possibilities
to define
basis vectors
Other non close-packed structures
• In covalently bonded materials, bond direction is
more important than packing
graphite
diamond (only 34 % packing)
Simple Crystal Structures
Diamond
• Crystal class Td (tetrahedral)
- Each atom has 4 nearestneighbors (nn).
• Can be interpreted as two
combined fcc structures
– One atom at origin
– Other atom displaced along
diagonal (¼, ¼, ¼)
• Includes C, Si, Ge, a-Sn
Diamond & Zincblende Crystal
Structure
• Basis set: 2 atoms. Lattice  face centered cubic (fcc).
• The fcc primitive lattice is generated by r = n1a1+n2a2+n3a3
with lattice vectors:
a1 = a(0,1,0)/2, a2 = a(1,0,1)/2, a3 = a(1,1,0)/2
NOTE: The ai’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
For FCC 2 atom ABCABC stacking, it is called zinc blende
Primitive cubic
Coordination
number
6
Body centered cubic Face centered cubic
8
12
Close-packed structures: fcc and hcp
hcp
ABABAB...
24
fcc
ABCABCABC...
Close-packed structures: fcc and hcp
hcp
ABABAB...
25
fcc
ABCABCABC...
Close-packed structures: fcc and hcp
hcp
ABABAB...
26
fcc
ABCABCABC...
Close-packed structures: fcc and hcp
hcp
ABABAB...
27
fcc
ABCABCABC...
Close-packed structures: fcc and hcp
hcp
ABABAB...
fcc
ABCABCABC...
• The face-centred cubic (fcc) and hexagonal closepacked (hcp) structure have the same packing
fraction