The Reciprocal Lattice
Outline
• Definitions and Examples
• First Brillouin Zone
• Lattice Planes and Miller Indices
Questions:
What is “Reciprocal Lattice”?
What’s the relationship between reciprocal lattice
and Bravais lattice?
Definition of Reciprocal lattice
Consider a set of points 𝑹 constituting a Bravais lattice and a
plane wave 𝒆𝒙𝒑(𝒊𝒌 ⋅ 𝒓).
The set of all wave vectors 𝒌 that yield plane waves with the
periodicity of a given Bravais lattice is known as its reciprocal
lattice:
𝒆𝒊𝑲⋅(𝒓+𝑹) = 𝒆𝒊𝑲⋅𝒓
for any 𝒓, and for all 𝑹 in the Bravais lattice
𝒆𝒊𝒌⋅𝑹 = 𝟏
mapping
Direct Lattice
Reciprocal Lattice
How can we get Reciprocal Lattice basis vector from Bravais Lattice
basis vector?
Let 𝒂𝟏 , 𝒂𝟐 , 𝒂𝟑 be a set of primitive vectors for the direct lattice.
Then the reciprocal lattice can be generated by the three primitive
vectors
𝒂𝟐 × 𝒂𝟑
𝒃𝟏 = 𝟐𝝅
𝒂𝟏 ⋅ (𝒂𝟐 × 𝒂𝟑 )
𝒃𝟐 = 𝟐𝝅
𝒂𝟑 × 𝒂𝟏
𝒂𝟏 ⋅ (𝒂𝟐 × 𝒂𝟑 )
𝒃𝟑 = 𝟐𝝅
𝒂𝟏 × 𝒂𝟐
𝒂𝟏 ⋅ (𝒂𝟐 × 𝒂𝟑 )
Note that:
𝒃𝒊 ⋅ 𝒂𝒋 = 𝟐𝝅𝜹𝒊𝒋
Kronecker delta symbol
𝜹𝒊𝒋 =
𝟎
𝟏
(𝒊 ≠ 𝒋)
(𝒊 = 𝒋)
For any vector 𝒌,
𝑲 = 𝒌𝟏 𝒃𝟏 + 𝒌𝟐 𝒃𝟐 + 𝐤 𝟑 𝐛𝟑
𝑹 = 𝒏𝟏 𝒂𝟏 + 𝒏𝟐 𝒂𝟐 + 𝒏𝟑 𝒂𝟑
𝒏𝒊 ∈ 𝒁
It follows from the relationship between basis vector 𝒂 and 𝒃 that
𝒌 ⋅ 𝑹 = 𝟐𝝅 𝒌𝟏 𝒏𝟏 + 𝒌𝟐 𝒏𝟐 + 𝒌𝟑 𝒏𝟑
= 𝟐𝝅 𝒕𝒊𝒎𝒆𝒔 𝒂𝒏 𝒊𝒏𝒕𝒆𝒈𝒆𝒓 𝒇𝒐𝒓 𝒂𝒏𝒚 𝒏𝒊
This requires 𝒌𝟏 𝒌𝟐 𝒌𝟑 to be integers.
K reciprocal lattice
𝒌𝟏 𝒌𝟐 𝒌𝟑 are integers
Thus the reciprocal lattice is a Bravais lattice and 𝒃𝒊 can be taken as
primitive vectors.
What is the reciprocal lattice of the reciprocal lattice?
Bravais Lattice
(r space)
Reciprocal Lattice
(k space)
Proof: Read P87 in textbook A&M.
Examples:
Cubic Bravais lattice with cubic primitive cell of side 𝒂
𝒂𝟏 = 𝒂𝒙,
𝒃𝟏 =
𝒂𝟐 = 𝒂𝒚,
𝟐𝝅
𝒙,
𝒂
𝒃𝟐 =
𝟐𝝅
𝒚,
𝒂
𝒂𝟑 = 𝒂𝒛
𝒃𝟑 =
𝟐𝝅
𝒛
𝒂
FCC (conventional unit cell of side a)
𝒂𝟏 =
𝒃𝟏 =
𝒂
(𝒚
𝟐
𝟒𝝅 𝟏
(𝐲
𝒂 𝟐
+ 𝒛),
𝒂𝟐 =
+ 𝐳 − 𝐱), 𝒃𝟐 =
𝒂
(𝒛 +
𝟐
𝟒𝝅 𝟏
(𝒛
𝒂 𝟐
𝒙),
𝒂𝟑 =
𝒂
(𝒙
𝟐
+ 𝒙 − 𝒚), 𝒃𝟑 =
+ 𝒚)
𝟒𝝅 𝟏
(𝒙
𝒂 𝟐
+ 𝒚 − 𝒛)
Volume of the reciprocal lattice primitive cell
Direct lattice 𝑽
Reciprocal lattice
𝟐𝝅 𝟑
𝑽
The obvious primitive cell 𝒓 = 𝒙𝟏 𝒂𝟏 + 𝒙𝟐 𝒂𝟐 + 𝒙𝟑 𝒂𝟑 , 𝒙𝒊 ∈ [𝟎. 𝟏)
The parallelipiped spanned by 𝒂𝟏 , 𝒂𝟐 , 𝒂𝟑 .
Outline
• Definitions and Examples
• First Brillouin Zone
• Lattice Planes and Miller Indices
Questions:
What is “First Brillouin Zone”?
The Wigner-Seitz primitive cell of the reciprocal lattice
is known as the 1st Brillouin zone. ( in k-space)
Example: The first Brillouin zone of 2D square lattice
Outline
• Definitions and Examples
• First Brillouin Zone
• Lattice Planes and Miller Indices
Questions:
What is “Lattice Planes”?
Which “language” shall we use when we talk about
lattice plane?
Lattice Planes
Vectors in Reciprocal lattice
Planes of points in Direct lattice
How to classify all possible families of lattice planes?
Read P90 in textbook A&M.
We use Miller Indices to describe Lattice Planes.
(010)
(110)
(111)
?
Summary
• Definitions and Examples
– What is “Reciprocal Lattice”?
– What’s the relationship between RL and BL?
• First Brillouin Zone
– What is “FBZ”?
• Lattice Planes and Miller Indices
– What is “lattice planes”?
– Which “language” shall we use to describe lattice planes?