The Structure of Crystalline
Solids
Atomic Arrangement
Crystal structures have short-range order and
long-range order (on a lattice - see CDROM)
unit cell = smallest grouping of atoms that show
the geometry of the structure and be repeated to
form the structure
Seven unit cell geometries possible
Atomic Arrangement
When a=b=c and a=b=g=90 the crystal
system is cubic
lattice points = atom positions within a unit
cell (e.g., at the corners, center of the cube,
center of the faces, etc.)
14 types of unit cells (Bravis lattices) in 7
crystal systems are possible
Atomic Arrangement
lattice parameter (for cubic crystals) = a0
measured at room T and given in
nm = 1x10-9 m
you should remember
or
these unit conversions
Å = 1x10-10 m
Why do we care?
To study properties of crystalline materials
(such as strength or electrical conductivity)
we work with unit cells.
Therefore we need to know how many
atoms/unit cell there are for a particular
material
Atoms/unit cell
Depending on where the
atoms are located within
the unit cell they belong to
it completely or are shared
with one or more other
cells.
Remember! These cells
are replicated in 3D to
generate the macroscopic
material
Atom in center of cube is not shared with any
other unit cell = 1 atom/unit cell
Atoms/unit cell
Atom in the face of the cube Atom in the corner of the
is shared with 1 neighboring cube is shared among 8
unit cell = 1/2 atom/unit cell neighboring unit cells = 1/8
atom/unit cell
This is called a Simple
Cubic (SC) crystal structure
a0
1 atom/unit cell
Volume = V = a03
Coordination number = 6
Coordination number = number of nn of each atom in a
perfect crystal
This is called a Body
Centered Cubic (BCC or
bcc) crystal structure
a0
2 atom/unit cell
V = a03
Coordination number = 8
examples: Fe, W
This is called a Face
Centered Cubic (FCC or
fcc) crystal structure
a0
4 atom/unit cell
V = a03
Coordination number = 12
examples: Ag, Pt, Au
Close-Packed Directions
In SC:
Directions in which atoms
are in continuous contact
with one another
Front of
the cube
a0 = 2r
lattice constant = 2 x atomic radius
In BCC:
3 a0 = 4r
a0
a0 =
4r
3
In FCC:
2 a0 = 4r
a0
a0 =
4r
2
= 2 2 r
Derivation given in example
problem 3.1 in the book, p. 36
This is called a Hexagonal
Close Packed (HCP or hcp)
crystal structure
c0
2 atom/unit cell (3 unit cells
shown in figure)
V=a02 c0 cos 30 per unit cell
a0
a0
Coordination number = 12
examples: Cd, Zn
angles in hexagon = 120
Ideally c0/a0 = 1.633
Atomic Packing Factor
If we assume each atom is a hard sphere the atomic
packing factor (APF) is the fraction of the unit cell
occupied by atoms
APF = Vatoms/Vcell = (#atoms)(Vatom)/(Vcell)
Vatom = 4pr3
where r = radius of atom
3
Vcell given for each type of unit cell; r and a are related
to each other by close-packed directions (examples)
Density
Mass/Volume
Theoretical density of a crystalline material assumes
the unit cell propagate in 3D space perfectly
r = (mass of atoms in unit cell)/(Vcell)
= (atoms/cell)(atomic mass)
(Vcell)(Avagodros number)
= (1 cell)(atoms/cell)(g/mol)
= g/cm3
(cm3)(atoms/mol)
(examples)
Polymorphic Structures
Same material, more than one crystal structure
possible.
When an elemental material is polymorphic,
this is called allotropy.
Crystalline and Noncrystalline
Materials
Single crystal materials =
crystalline solid where the
periodic arrangement of the
crystalline unit cell extends in
all directions without
interruption
Polycrystalline materials =
crystalline solids made up of
many single crystals
separated by grain boundaries
Crystalline and Noncrystalline
Materials
Amorphous materials = non-crystalline materials
that have short-range order (to satisfy local
bonding requirements) but no long-range order
Crystalline and Noncrystalline
Materials
Isotropic materials => measured properties are
independent of direction of measurement (e.g.,
cubic single crystals, polycrystalline materials)
Anisotropic materials => measured properties
depend on direction of measurement (e.g., HCP
single crystals)
Points, Directions, and Planes
We have already discussed close-packed directions in
some unit cells. Materials cleave, deform, etc.
preferentially along certain planes or in certain
directions.
Therefore, we need clear, unambiguous language to
discuss directions and planes in crystalline unit cells.
z
Points - Cubic Unit Cells
0, 0, 1
1, 0, 1
Distance is
measured in
terms of the
number of
lattice
parameters to
0, 0, 0
move in x, y,
1/3, 0, 0
and z to get to 1, 0, 0
a point from
the origin
x
1, 1, 1
0, 1, 0
y
Miller Indices
Short-hand notation to unambiguously
describe directions and planes in crystalline
materials
There is a specific procedure for finding miller
indices for Directions
Directions - Cubic Unit Cells




Determine coordinates of 2 points that form
a line in the direction of interest
Subtract the coordinates of the tail point
from the coordinates of the head point = #
lattice parameters traveled along each axis
Reduce to smallest integers
Put final numbers in [uvw] with any
negative numbers shown as “bar” numbers
(e.g., -2 is written as 2)
(examples)
Notes


Because directions are vectors, a direction
and its negative are not identical. Therefore,
[100] is not the same direction as [100].
A direction and its multiple are identical.
[100] is the same direction as [200]; it has
just not been reduced to the smallest
integers.
(examples)
More Notes

Certain groups of directions are equivalent;
they are only different because of the way
we constructed the coordinates. In cubic
lattices, [100] is equivalent to [010] if we
redefine the coordinate system (or rotate the
unit cell). Equivalent groups of equivalent
directions are written in <uvw>
(examples)
Planes - Cubic Unit Cells

Find the x, y, z intercepts of the plane within the
unit cell in terms of the number of lattice
parameters. Note: if the plane passes through the
point you have designated as the origin, the origin
must be moved.
 Take the reciprocals of the intercepts
 Clear all fractions but do not reduce to lowest
integers
 Write inside (uvw); negative numbers written as
“bar” numbers
(examples)
c
Points - Hexagonal Unit Cells a3
a1
a3 is redundant:
a1+a2 = -a3
a2
0, 0, 1
0, 1, 1
can use (u’v’w’) = (a1, a2, c)
or
0, 0, 0
can use (uvtw) = (a1,a2,a3,c)
1, 0, 0
0, 1, 0
1, 1, 0
Directions - Hexagonal Unit
Cells
Determine
the number of lattice parameters to
move in each direction to get from tail to head and
keep u + v = -t
Can be expressed in terms of [u’v’w’] or [uvtw]
To convert between 3-coordinate and 4-coordinate
system, follow these rules:
u = 1/3 (2u’-v’)
Then clear fractions and reduce
v = 1/3 (2v’-u’)
to the smallest integers
t = -1/3(u’+v’) = -(u+v)
(examples)
w = w’
Planes - Hexagonal Unit Cells

Express in 3-coordinate system or 4-coordinate
system
 Use u + v = -t to convert back and forth between
the two
 Otherwise, no different from planes in cubic
crystals
(examples)
Linear Density
Number of atoms along a particular directions.
1D version of the packing factor
Fraction of the line length intersected by
atoms
(examples)
Planar Density
Number of atoms per unit area in the plane.
2D version of the packing factor
Fraction of area of the plane covered by atoms
(examples)