Reciprocal Lattice and Brillouin Zone Construction

From Real-Space Periodicity to Momentum-Space Discreteness

The defining feature of a crystal is that the physical environment is identical at points separated by a lattice vector $\vec{R}$. This means any physical observable, such as the electron charge density $\rho (\vec{r})$ or the crystal potential $V(\vec{r})$, must share this periodicity:

As you know from your mathematics courses, any periodic function can be represented as a Fourier series—a sum of plane waves whose wavevectors are constrained by the periodicity. For a one-dimensional lattice with lattice constant $a$, a general (non-periodic) function can be expressed as a Fourier integral over all possible wavevectors $k$. However, imposing the constraint $f(x)=f(x+a)$ drastically changes the situation. The continuous integral collapses into a discrete sum over a specific set of wavevectors.

The following video provides an excellent introduction to this idea, starting with a simple 1D case and building up the general concept.

The crucial insight is that the set of allowed wavevectors, $\{G\}$, forms a lattice in its own right—the reciprocal lattice. Any function with the periodicity of the real-space lattice can be written as a Fourier series of plane waves, $\exp (i\vec{G}\cdot \vec{r})$, where the $\vec{G}$ are vectors of this reciprocal lattice.

For this expansion to be consistent with the lattice periodicity, each plane wave component must itself be periodic. This requires that for any real-space lattice vector $\vec{R}$,

which simplifies to the fundamental condition:

This equation serves as the defining property of the reciprocal lattice: it is the set of all vectors $\vec{G}$ that satisfy this condition for every vector $\vec{R}$ in the real-space (or direct) lattice.

Formal Construction of the Reciprocal Lattice

With the defining condition established, we can now derive a systematic way to construct the reciprocal lattice for any given direct lattice. Let the direct lattice be generated by primitive vectors $\{{\vec{a}}_1,{\vec{a}}_2,{\vec{a}}_3\}$ and the reciprocal lattice by primitive vectors $\{{\vec{b}}_1,{\vec{b}}_2,{\vec{b}}_3\}$.

Any direct lattice vector is $\vec{R}=n_1{\vec{a}}_1+n_2{\vec{a}}_2+n_3{\vec{a}}_3$ and any reciprocal lattice vector is $\vec{G}=m_1{\vec{b}}_1+m_2{\vec{b}}_2+m_3{\vec{b}}_3$, where $n_i,m_j$ are integers. The condition $e^{i\vec{G}\cdot \vec{R}}=1$ must hold for all choices of integers. This implies that $\vec{G}\cdot \vec{R}$ must be an integer multiple of $2\pi$. This is satisfied if the primitive vectors themselves fulfill the relation:

where ${\delta }_{ij}$ is the Kronecker delta. This orthogonality relation is the cornerstone of the relationship between the direct and reciprocal lattices. From this, we can derive an explicit formula for constructing the reciprocal vectors ${\vec{b}}_j$ from the direct vectors ${\vec{a}}_i$.

The construction shown in the text, for example for ${\vec{b}}_1$, is:

Notice the denominator is the volume $V$ of the real-space primitive unit cell. This makes it clear that the reciprocal lattice vectors have dimensions of inverse length, which is why reciprocal space is also known as momentum space or k-space (since momentum $\vec{p}=\hslash \vec{k}$).

The reciprocal lattice vector $\vec{G}$ has a beautiful geometric interpretation. The direction of $\vec{G}$ is perpendicular to a set of parallel planes in the direct lattice, and its magnitude is $|\vec{G}|=2\pi /d$, where $d$ is the spacing between these planes.

The Brillouin Zone

Just as the direct lattice is tiled by primitive unit cells, the reciprocal lattice can be tiled by its own primitive cells. While any choice of primitive cell would tile the space, there is one particular choice that is exceptionally important due to its symmetry: the Wigner-Seitz cell. The Wigner-Seitz cell of the reciprocal lattice is known as the first Brillouin zone.

Definition: The first Brillouin zone (BZ) is the set of points in reciprocal space that are closer to the origin ($\vec{k}=0$) than to any other reciprocal lattice point.

The construction is straightforward:

1. Start at a reciprocal lattice point (the origin).
2. Draw vectors from the origin to all neighboring reciprocal lattice points.
3. Construct the planes that are the perpendicular bisectors of these vectors.
4. The smallest volume enclosed by these planes around the origin is the first Brillouin zone.

The concept of the Brillouin zone is essential because, as we will see, any plane wave $\exp (i\vec{k}\cdot \vec{r})$ in the crystal can be described by a wavevector $\vec{k}$ within the first Brillouin zone, plus a reciprocal lattice vector $\vec{G}$. This means the BZ contains all the unique wavevectors needed to describe physical processes in the crystal.

The Brillouin Zone and the Translation Group

We can now connect these constructions back to the central theme of our course: group theory. In the previous lesson, we identified the translation group of the lattice. Now, we will see that the Brillouin zone is, in fact, the space of irreducible representations (irreps) of this group.

Consider the action of a translation operator $T_{\vec{R}}$ on a wavefunction $\psi (\vec{r})$. Since the translation operators for different $\vec{R}$ all commute, we can find a basis of wavefunctions that are simultaneous eigenfunctions of all of them. For such an eigenfunction:

Here, $C(\vec{R})$ is the eigenvalue, which is a 1-dimensional representation of the translation group. The group multiplication law $T_{{\vec{R}}_1}{T}_{{\vec{R}}_2}=T_{{\vec{R}}_1+{\vec{R}}_2}$ requires that the eigenvalues satisfy $C({\vec{R}}_1)C({\vec{R}}_2)=C({\vec{R}}_1+{\vec{R}}_2)$. For unitary representations (which correspond to physical states), the solution must be of the form:

for some vector $\vec{k}$. This vector $\vec{k}$ labels the irreducible representation of the translation group.

However, this label is not unique. If we choose a different label ${\vec{k}}^{\prime }=\vec{k}+\vec{G}$, where $\vec{G}$ is any reciprocal lattice vector, the eigenvalue becomes:

The representation is identical! This means that all wavevectors $\vec{k}$ that differ by a reciprocal lattice vector $\vec{G}$ are equivalent and label the same irrep. The set of all unique, inequivalent irreducible representations is therefore given by the set of $\vec{k}$ vectors within one primitive cell of the reciprocal lattice—the Brillouin zone.

Conclusion

In this lesson, we have made the pivotal transition from the real-space description of a crystal to its counterpart in momentum space. By investigating the consequences of the lattice's translational symmetry group, we have constructed the reciprocal lattice and its fundamental domain, the Brillouin zone.

Key Takeaways:

• The periodicity of any function in a crystal restricts its Fourier components to a discrete set of wavevectors, which form the reciprocal lattice.
• The reciprocal lattice ${\Lambda }^{\ast }$ is defined as the set of all vectors $\vec{G}$ such that $e^{i\vec{G}\cdot \vec{R}}=1$ for all vectors $\vec{R}$ in the direct lattice $\Lambda$.
• The primitive vectors of the direct and reciprocal lattices are related by ${\vec{a}}_i\cdot {\vec{b}}_j=2\pi {\delta }_{ij}$.
• The Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice. It contains all the unique wavevectors needed to describe waves in the crystal.
• From a group-theoretic standpoint, the Brillouin zone is the space of all unique irreducible representations of the crystal's translation group, with each vector $\vec{k}$ in the BZ labeling a distinct irrep.

We have established that the states in a crystal can be labeled by a wavevector $\vec{k}$ from the Brillouin zone. In our next lesson, we will use this fact to derive Bloch's theorem, a cornerstone of solid-state physics that dictates the general form of quantum mechanical wavefunctions in a periodic potential. This will lead us directly to the concept of electronic band structure, where the energy of an electron, $E(\vec{k})$, is a periodic function across the Brillouin zone.