II.

Crystal Structure
A. Lattice, Basis, and the Unit Cell
B. Common Crystal Structures
C. Miller Indices for Crystal Directions and
Planes
D. The Reciprocal Lattice



A. Lattice, Basis, and Unit Cell
An ideal crystalline solid is an infinite repetition of identical
structural units in space. The repeated unit may be a single atom
or a group of atoms.
An important concept:
crystal structure = lattice + basis
= +
lattice: a periodic array of points in space. The environment
surrounding each lattice point is identical.
basis: the atom or group of atoms attached to each lattice
point in order generate the crystal structure.
The translational symmetry of a lattice is given by the base
vectors or lattice vectors . Usually these vectors
are chosen either:

1. to be the shortest possible vectors, or
2. to correspond to a high symmetry unit cell
c b a

, ,
Example: a 2-D lattice
These two choices of lattice vectors illustrate two types of unit cells:
a

Conventional (crystallographic) unit cell:

larger than primitive cell; chosen to
display high symmetry unit cell
Primitive unit cell: has minimum volume
and contains only one lattice point
A lattice translation vector connects two points in the lattice that have
identical symmetry:
a

b a

+
b a

2 +
c n b n a n r

3 2 1
+ + = integers
3 2 1
e n n n
In our 2-D lattice:
B. Common Crystal Structures
2-D only 5 distinct point lattices that can fill all space
3-D only 14 distinct point lattices (Bravais lattices)
The 14 Bravais lattices can be subdivided into 7 different crystal classes,
based on our choice of conventional unit cells (see text, handout).

Attaching a basis of atoms to each lattice point introduces new types of
symmetry (reflection, rotation, inversion, etc.) based on the arrangement of
the basis atoms. When each of these point groups is combined with the
14 possible Bravais lattices, there are a total of 230 different possible
space groups in 3-D. We will focus on the few that are common for
metals, semiconductors, and simple compounds.
Crystal Structure Diagrams
(a) NaCl
(b) CsCl
(c) fluorite
(d) perovskite
(e) Laves phase
(f) A15
Crystal Structure Diagrams
(continued)
diamond structure
(C, Si, Ge)
hexagonal close
packed (Be, Mg, Zn)
Analysis of Common Crystal Structures
1. NaCl structure (many ionic solids)
2. CsCl structure (some ionic solids and intermetallic alloys)
lattice: face-centered cubic (fcc)
basis: Na at 000, Cl at
lattice: simple cubic (sc)
basis: Cs at 000, Cl at
Common Crystal Structures, contd
3. hexagonal-close-packed (divalent metals)
4. diamond structure (C, Si, Ge)
lattice: hexagonal
basis: 000, 2/3 1/3 1/2
lattice: face-centered-cubic (fcc)
basis: 000,
(see text for an alternate choice of lattice and basis)
5. zincblende structure (ZnS, GaAs, InP, compound semiconds)
lattice: face-centered-cubic (fcc)
basis: Zn at 000, S at
C. Miller Indices for Crystal Directions & Planes
Because crystals are usually anisotropic (their properties differ
along different directions) it is useful to regard a crystalline
solid as a collection of parallel planes of atoms.
Crystallographers and CM physicists use a shorthand notation
(Miller indices) to refer to such planes.
1. Determine intercepts (x, y, z)
of the plane with the coordinate
axes
x
y
z
x = 1
y = 2
z = 3
C. Miller Indices,contd.
2. Express the intercepts as multiples of the base vectors of the lattice
In this example, lets assume that the lattice is given by: k c j b i a

1 = = =

Then the intercept ratios become:

1
3
3
2
1
2
1
1
1
= = = = = =
c
z
b
y
a
x
3. Form reciprocals: 1
1
1
2
1
1
1
1
= = = = =
z
c
y
b
x
a
4. Multiply through by the factor that allows you to express these indices as
the lowest triplet of integers:
) 212 ( ) 1 1 ( 2
2
1
=
We call this the (212) plane.
Another example
Find the Miller indices of
the shaded plane in this
simple cubic lattice:
k a c j a b i a a

= = =

Intercept ratios:
= = =
a
z
a
y
a
x
1
Reciprocals:
0 1 0 = = =
z
a
y
a
x
a
We call this the (010) plane.
a
a
a
Intercepts:
= = = z a y x
x
y
z
non-intersecting intercept at
Note: (hkl) = a single plane; {hkl} = a family of symmetry-equivalent planes
Crystal Planes and Directions
Crystal directions are
specified [hkl] as the
coordinates of the lattice
point closest to the origin
along the desired direction:
Note that for cubic lattices, the direction
[hkl] is perpendicular to the (hkl) plane
x
y
z
] 1 00 [
] 001 [
] 100 [
] 010 [
Note: [hkl] = a specific direction;
<hkl> = a family of symmetry-
equivalent directions
D. The Reciprocal Lattice
Crystal planes (hkl) in the real-space or direct lattice are characterized by the
normal vector and the interplanar spacing :
Long practice has shown CM physicists the usefulness of defining a different
lattice in reciprocal space whose points lie at positions given by the vectors
hkl
n
hkl
d
x
y
z
hkl
d
hkl
n
hkl
hkl
hkl
d
n
G
2t

This vector is parallel to the

[hkl] direction but has
magnitude 2t/d
hkl
, which is a
reciprocal distance

The Reciprocal Lattice, contd.
The reciprocal lattice is composed of all points lying at positions from
the origin, so that there is one point in the reciprocal lattice for each set of
planes (hkl) in the real-space lattice.
This seems like an unnecessary abstraction. What is the payoff for defining such
a reciprocal lattice?
hkl
G

1. The reciprocal lattice simplifies the interpretation of x-ray diffraction from

crystals (coming soon in chapter 3)
2. The reciprocal lattice facilitates the calculation of wave propagation in
crystals (lattice vibrations, electron waves, etc.)
The Reciprocal Lattice: An Analogy
Waves of lattice vibrations or electron waves moving through a crystal with a
periodicity specified by base vectors can likewise be decomposed into a
sum of plane waves:
In the analysis of electrical signals that are periodic in time, we use Fourier
analysis to express such a signal in the frequency domain:
t i
e C t f
e
e
e

= ) (
If f(t) has period T, then the coefficient C
e
is
nonzero only for frequencies given by
T
n t
e
2
=
n = integer
c b a

) (
) , (
t r k i
k
k
e C t r
e

Here, the coefficient C

k
is nonzero only
when the vector k is a reciprocal lattice
translation vector:
C l B k A h G k
hkl

+ + = =
A, B, and C are the base vectors of the
reciprocal lattice (some books use a*, b*, c*)
Definition of Reciprocal Lattice Base Vectors
These reciprocal lattice base vectors are defined:
Which have the simple dot products with the direct-space lattice vectors:
( ) c b a
c b
A

t 2
So compare, for example:
( ) c b a
a c
B

t 2
( ) c b a
b a
C

t 2
t 2 = = = c C b B a A

b C a C a B c B c A b A

= = = = = = 0
n T t e 2 =
l c G
k b G
h a G
hkl
hkl
hkl
t
t
t
2
2
2
=
=
=

frequency time
Reciprocal lattice direct lattice
Remember:
Problems worthy
of attack
Prove their worth
by hitting back

--Piet Hein