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.
z
z=3
1. Determine intercepts (x, y, z)
of the plane with the coordinate
axes
y=2
y
x=1
x
C. Miller Indices,cont’d.
2. Express the intercepts as multiples of the base vectors of the lattice
In this example, let’s assume that the lattice is given by: a = 1iˆ
Then the intercept ratios become:
3. Form reciprocals:
a 1
= =1
x 1
x 1
= =1
a 1
b 1
=
y 2
y 2
= =2
b 1
b = 1 ˆj
c = 3kˆ
z 3
= =1
c 3
c 1
= =1
z 1
4. Multiply through by the factor that allows you to express these indices as
the lowest triplet of integers:
2 × (1 12 1) = (212)
We call this the (212) plane.
Another example
z
Find the Miller indices of
the shaded plane in this
simple cubic lattice:
a
y
a
x
y=a
a = aiˆ
b = aˆj
c = akˆ
a
Intercepts:
x=∞
z=∞
Intercept ratios:
x
=∞
a
y
=1
a
z
=∞
a
Reciprocals:
a
=0
x
a
=1
y
a
=0
z
non-intersecting → intercept at ∞
We call this the (010) plane.
Note: (hkl) = a single plane; {hkl} = a family of symmetry-equivalent planes
Crystal Planes and Directions
z
Crystal directions are
specified [hkl] as the
coordinates of the lattice
point closest to the origin
along the desired direction:
[001]
y
x
Note: [hkl] = a specific direction;
<hkl> = a family of symmetryequivalent directions
[100]
[00 1 ]
Note that for cubic lattices, the direction
[hkl] is perpendicular to the (hkl) plane
[010]
D. The Reciprocal Lattice
Crystal planes (hkl) in the real-space or direct lattice are characterized by the
normal vector n̂hkl and the interplanar spacing d hkl :
z
y
n̂hkl
x
d hkl
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
2πnˆhkl
Ghkl ≡
d hkl
This vector is parallel to the
[hkl] direction but has
magnitude 2π/dhkl, which is a
reciprocal distance
The Reciprocal Lattice, cont’d.
The reciprocal lattice is composed of all points lying at positions Ghkl 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?
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.)
Definition of Reciprocal Lattice Base Vectors
These reciprocal lattice base vectors are defined:
2π b × c
A≡ a ⋅ b ×c
(
)
2π c × a
B≡ a ⋅ b ×c
(
)
2π a × b
C≡ a ⋅ b ×c
(
)
Which have the simple dot products with the direct-space lattice vectors:
A ⋅ a = B ⋅ b = C ⋅ c = 2π
A⋅b = A⋅c = 0 = B ⋅c = B ⋅ a = C ⋅ a = C ⋅b
Ghkl ⋅ a = 2πh
Ghkl ⋅ b = 2πk
Ghkl ⋅ c = 2πl
Reciprocal lattice ↔direct lattice