The reciprocal lattice

Learn the reciprocal secrets of crystals¹.

Introduction

Armed with our newly minted crystallographic terminology and descriptions of crystal structure, we are going to explore the other side: reciprocal space. We have explored the ideas of reciprocal space throughout this course, but here we are going to put on the spelunking gear and get to the bottom of things.

Expected competencies

It is assumed that you have familiarity with the following concepts/techniques:

• Mathematics: vectors, the Fourier transform

Text reference

The material covered here is discussed in section(s) $\S 13.1$ of The Oxford Solid State Basics

Computational content

The Jupyter notebook associated with this section can be accessed by clicking the icon below:

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Reciprocal lattice motivation 1D case

Previously, we discussed the reciprocal space of a simple 1D lattice. To obtain the dispersion relation we considered waves of the form

$$e^{ikx_n} = e^{ikna}, \quad n \in \mathbb{Z},$$

where $x_n = n \times a$ is the 1D lattice point. We then observed that the waves with wave vectors $k$ and $k+G$ where $G=2\pi m/a$ with integer $m$, are exactly the same:

$$e^{i(k+G)na} = e^{ikna+im2\pi n} = e^{ikna}$$

where we have used

$$e^{iGx_n} = e^{i2\pi mn} = 1.$$

The set of points $G=2\pi m/a$ forms the reciprocal lattice. Let us now generalize the same idea to describe the reciprocal lattice in higher dimensions.

Extending to higher dimensions

We start from a lattice in real space:

$$\mathbf{R}=n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_1, \quad \{n_1, n_2, n_3\} \in \mathbb{Z},$$

where $\mathbf{a}_1$, $\mathbf{a}_2$ and $\mathbf{a}_3$ are the lattice vectors. The reciprocal lattice is also a lattice, but in the $k$-space:

$$\mathbf{G}=m_1\mathbf{b}_1+m_2\mathbf{b}_2+m_3\mathbf{b_3}, \quad \{m_1, m_2, m_3\} \in \mathbb{Z}$$

where the vectors $\mathbf{b}_1$, $\mathbf{b}_2$ and $\mathbf{b}_2$ are the reciprocal lattice vectors. Let us now determine the reciprocal lattice vectors by requiring that waves that differ by a reciprocal lattice vector are indistinguishable. That is, we require that

$$e^{i\mathbf{k}\mathbf{R}} = e^{i(\mathbf{k} + \mathbf{G})\mathbf{R}},$$

for any $\mathbf{R}$ in the lattice. Substituting the definitions of $\mathbf{R}$ and $\mathbf{G}$ we get

$$\mathrm{e}^{i\mathbf{G}\cdot\mathbf{R}} = \mathrm{e}^{i\sum_{\{i,j\}=1}^{3} n_i m_j \mathbf{a}_i \cdot \mathbf{b}_j}=1.$$

This relation only holds if

$$\mathbf{a_i}\cdot\mathbf{b_j}=2\pi\delta_{ij}.$$

Indeed, after expanding the dot products in the exponent, we get

$$\mathrm{e}^{i\mathbf{G}\cdot\mathbf{R}} = \mathrm{e}^{2\pi i(n_1 m_1 + n_2 m_2 + n_3 m_3)}.$$

Because $n_i$ and $m_j$ are both integers, the exponent necessarily evaluates to 1.

The relation $\mathbf{a_i}\cdot\mathbf{b_j}=2\pi\delta_{ij}$ means that if we write the lattice vectors as rows of a matrix, the reciprocal lattice vectors are $2\pi$ times the columns of the inverse of that matrix.

2D example: triangular lattice

In order to gain extra intuition of the properties of the reciprocal lattice, let us study a specific example.

Consider a triangular lattice, which is shown in the figure below. The left panel show the real-space lattice with lattice vectors $\mathbf{a}_1 = a \mathbf{\hat{x}}$ and $\mathbf{a}_2 = a/2\mathbf{\hat{x}} + \sqrt{3}a/2 \mathbf{\hat{y}}$. While the right panel shows the corresponding reciprocal lattice and its reciprocal lattice vectors $\mathbf{b}_1$ and $\mathbf{b}_2$.

To find the reciprocal lattice vectors, we use the relation

$$\mathbf{a_i}\cdot\mathbf{b_j}=2\pi\delta_{ij},$$

which gives us the following equations:

$$\mathbf{a}_1\cdot\mathbf{b}_2=\mathbf{a}_2\cdot\mathbf{b}_1=0,$$

and

$$\mathbf{a}_1\cdot\mathbf{b}_1=\mathbf{a}_2\cdot\mathbf{b}_2=2\pi.$$

We substitute $\mathbf{a_i}\cdot\mathbf{b_i} = \lvert \mathbf{a_i} \rvert \lvert \mathbf{b_i} \rvert \cos(\theta_i)$:

$$\lvert \mathbf{a}_1 \rvert \lvert \mathbf{b}_1 \rvert =\frac{2\pi}{\cos(\theta_1)} \:\: \text{and} \:\: \lvert \mathbf{a}_2 \rvert \lvert \mathbf{b}_2 \rvert =\frac{2\pi}{\cos(\theta_2)},$$

where $\theta_i$ is the angle between the vectors $\mathbf{a}_i$ and $\mathbf{b}_i$.

To find the angles $\theta_1$ and $\theta_2$, we use the orthogonality relations above and the fact that the angle between $\mathbf{a}_1$ and $\mathbf{a}_2$ is $60^\circ$. From this we conclude that $\theta_1 = \theta_2 = 30^\circ$. Because $\lvert \mathbf{a}_1 \rvert = \lvert \mathbf{a}_2 \rvert = a$, we find $$\lvert \mathbf{b}_1 \rvert = \lvert \mathbf{b}_2 \rvert = \frac{4\pi}{a\sqrt{3}}.$$ Unsurprisingly, we find that the lengths of the reciprocal lattice vectors are equal and inversely proportional to the lattice constant $a$. With $\lvert \mathbf{b}_2 \rvert$ and $\mathbf{a}_1 \perp \mathbf{b}_2$, we easily find $$\mathbf{b}_2 = \frac{4\pi}{a\sqrt{3}} \mathbf{\hat{y}}.$$

We follow the same procedure to find $\mathbf{b}_1$:

$$\mathbf{b}_1 = \frac{4\pi}{a\sqrt{3}} \left(\frac{\sqrt{3}}{2} \mathbf{\hat{x}} - \frac{1}{2}\mathbf{\hat{y}} \right).$$

4.2.1 Is the choice of a set of reciprocal lattice vectors unique? If not, which other ones are possible?

3D lattice example

Let us now consider the more involved example of a 3D lattice. The explicit expression for the reciprocal lattice vectors in terms of their real space counterparts is:

$$\mathbf{b}_1=\frac{2\pi(\mathbf{a}_2\times\mathbf{a}_3)}{ \mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})}$$

$$\mathbf{b}_2=\frac{2\pi(\mathbf{a_3}\times\mathbf{a}_1)}{ \mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})}$$

$$\mathbf{b_3}=\frac{2\pi(\mathbf{a}_1\times\mathbf{a}_2)}{ \mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})}$$

Note that the denominator $\mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})$ is the volume $V$ of the real-space unit cell spanned by the lattice vectors $\mathbf{a}_1$, $\mathbf{a}_2$ and $\mathbf{a}_3$.

The reciprocal lattice as a Fourier transform

One can also think of the reciprocal lattice as a Fourier transform of the real-space lattice. For simplicity, we illustrate this for a 1D lattice (the same principles apply to a 3D lattice). We model the real-space lattice as a density function consisting of delta peaks:

$$\rho(x)=\sum_{n} \delta(x-na)$$

We take the Fourier transform of this function to find:

$${\mathcal F}\left\{\rho(x); x \rightarrow k \right\}=\int_{-\infty}^\infty \mathrm{d}x\ \mathrm{e}^{ikx} \rho(x)=\sum_{n} \int_{-\infty}^\infty \mathrm{d}x\ \mathrm{e}^{ikx} \delta(x-na)=\sum_{n} \mathrm{e}^{ikna}$$

This sum is non-zero only if $k=2\pi m/a$. If we recall the beginning of the lecture, then these points correspond to reciprocal lattice points $G$. Therefore, we rewrite this into the form

$${\mathcal F}\left\{\rho(x); x \rightarrow k \right\}=\frac{2\pi}{|a|}\sum_{m} \delta\left(k-G\right).$$ Therefore, we see that the Fourier transform is non-zero only at reciprocal lattice points. In other words, Fourier transforming a real-space lattice yields a reciprocal lattice! The above result generalizes directly to three dimensions:

$${\mathcal F}\left\{\rho(x); x \rightarrow k \right\}=\int \mathrm{d}\mathbf{r}\ \mathrm{e}^{i\mathbf{k}\cdot\mathbf{r}} \rho(\mathbf{r}) = \sum_\mathbf{G}\delta(\mathbf{k}-\mathbf{G}).$$

Periodicity of the reciprocal lattice

In order to describe a reciprocal lattice, we need to define a primitive unit cell in reciprocal space. Previously, we learned that the choice of a primitive unit cell is not unique. However, a general convention in reciprocal space is to use the Wigner-Seitz cell which is called the 1st Brillouin zone. Because the Wigner-Seitz cell is primitive, the 1st Brillouin zone contains a set of unique $\mathbf{k}$ vectors. This means that all $\mathbf{k}$ vectors outside the 1st Brillouin zone are a copy of those inside the 1st Brillouin zone. For example, any $\mathbf{k'}$ outside the 1st Brillouin zone is related to a wave vector inside 1st Brillouin zone $\mathbf{k}$ by shifting it by reciprocal lattice vectors: $\mathbf{k'} = \mathbf{k}+\mathbf{G}$

We have already learned how to construct Wigner-Seitz cells, however here is a reminder of how a Brillouin zone looks like:

Miller planes

When fabricating crystals it is important to know both the orientation and the surface of the crystal. Different cuts of a crystal lead to different surfaces. In the chemical industry, this is especially significant because different surfaces lead to different chemical properties and thus is one of the foundations of research in catalysts. Therefore, we seek a way to describe different planes of a crystal within our developed framework. This leads us to a very important concept - Miller planes. To explain Miller planes, let's start off with a simple cubic lattice:

Where $|{\bf a}_1|=|{\bf a}_2|=|{\bf a}_3|\equiv a$.

We can cut multiple planes through the cubic lattice. Miller planes describe such planes with a set of indices. The plane designated by Miller indices $(u,v,w)$ intersects lattice vector ${\bf a}_1$ at $\frac{|{\bf a}_1|}{u}$, ${\bf a}_2$ at $\frac{|{\bf a}_2|}{v}$ and ${\bf a}_3$ at $\frac{|{\bf a}_3|}{w}$.

A Miller index 0 means that the plane is parallel to that axis (or equivalently, the intersection is at "$\frac{|{\bf a}_3|}{0}\rightarrow\infty$"). A bar above a Miller index means intersection at a negative coordinate.

If a crystal is symmetric under $90^\circ$ rotations, then $(100)$, $(010)$ and $(001)$ are physically indistinguishable. Therefore, we use the notation $\{100\}$ to indicate a whole family of these symmetry-related planes. In a cubic crystal, $[100]$ (this is a vector) is perpendicular to $(100)$ $\rightarrow$ proof in problem set.

4.2.2 Why are these Miller planes usefull?

Examples of crystal planes

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Conclusions

• The reciprocal lattice is constructed from the reciprocal lattice vectors which are turn constructed from the real-space lattice vectors.
• Points in reciprocal space that differ by a reciprocal lattice vector are equivalent.
• The reciprocal lattice is related to the real-space lattice by the Fourier transform

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Exercises

Preliminary provocations

1. Calculate $\mathbf{a}_1 \cdot \mathbf{b}_1$ and $\mathbf{a}_2 \cdot \mathbf{b}_1$ using the definitions of the reciprocal lattice vectors given in the lecture. Is the result what you expected?

Exercise 1: Directions and Spacings of Miller planes

1. Explain the terms Miller planes and Miller indices.

2. Consider a cubic crystal with one atom in the basis and a set of orthogonal primitive lattice vectors $a\hat{x}$, $a\hat{y}$ and $a\hat{z}$. Show that the direction $[hkl]$ in this crystal is normal to the planes with Miller indices $(hkl)$.

3. Show that this is not true in general. Consider for instance an orthorhombic crystal, for which the primitive lattice vectors are still orthogonal but have different lengths.

4. Any set of Miller indices corresponds to a family of planes separated by a distance $d$. Show that the spacing $d$ of the $(hkl)$ set of planes in a cubic crystal with lattice parameter $a$ is $d = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$.

   Hint

   Recall that a family of lattice planes is an infinite set of equally separated parallel planes which taken all together contain all points of the lattice.

   Try computing the distance between the plane that contains the site $(0,0,0)$ of the conventional unit cell and a plane defined by the $(hkl)$ indices.

Exercise 2: The reciprocal lattice of the BCC and FCC lattices

The content on this page investigates how to construct the reciprocal lattice from a real-space lattice. We will now zoom in the properties of the FCC and BCC lattices, which are two of the most common lattices encountered in crystal structures. We analyse the reciprocal lattices and the shape of the first Brillouin zone. This helps us understanding e.g. the periodicity of 3D band structures and the Fermi surface database shown in the Attic.

We consider a bcc lattice of which the conventional unit cell has a side length $a$ and volume $V=a^3$.

1. Write down the primitive lattice vectors of the BCC lattice. Calculate the volume of the primitive unit cell. Is the volume the expected fraction of $V$?

2. Calculate the reciprocal lattice vectors associated with the primitive lattice vectors you found in the previous subquestion.

3. Sketch the reciprocal lattice. Which type of lattice is it? What is the volume of its conventional unit cell?

4. Describe the shape of the 1st Brillouin zone. How many sides does it have? (Note that the Brillouin zones are sketched in the Fermi surface periodic table in the Attic). Calculate the volume of the 1st Brillouin zone and check if it is the expected fraction of the volume you found in the previous subquestion.

5. Based on the insight gained in this question, argue what lattice is the reciprocal lattice of the FCC lattice.

Exercise 3: Miller planes and reciprocal lattice vectors

Consider a family of Miller planes $(hkl)$ in a crystal.

1. Prove that the reciprocal lattice vector $\mathbf{G} = h \mathbf{b}_1 + k \mathbf{b}_2 + l \mathbf{b}_3$ is perpendicular to the Miller plane $(hkl)$.

   Hint

   Choose two vectors that lie within the Miller plane and are not parallel to each other.

2. Show that the distance between two adjacent Miller planes $(hkl)$ of any lattice is $d = 2\pi/|\mathbf{G}_\textrm{min}|$, where $\mathbf{G}_\textrm{min}$ is the shortest reciprocal lattice vector perpendicular to these Miller planes.

3. In exercise 2, you derived the reciprocal lattice vectors of the BCC lattice from a set of primitive lattice vectors. Use these vectors to find the family of Miller planes that has the highest density of lattice points $\rho$. Use that $\rho = d/V$, where $V$ is the volume of the primitive unit cell and $d$ is the distance between adjacent planes derived in the previous subquestion. Formulate the Miller plane indices with respect to the primitive lattice vectors.

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1. This is only funny if you noticed the tagline of the previous section ↩

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Last update: August 11, 2023

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