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Crystal Structures & Properties

Problems 1001–1010 from Problems & Solutions on Solid State Physics, Relativity and Miscellaneous Topics (Lim, ed.)—sourced from SUNY Buffalo, Wisconsin, Columbia, and Princeton qualifying exams.

Problem 1001SUNY Buffalo

Two-Dimensional Square Lattice: Primitive Cell, Reciprocal Lattice, and Bloch's Theorem

Problem Statement

Figure 1.1 shows a hypothetical two-dimensional crystal consisting of atoms arranged on a square grid.
  1. a. Show an example of a primitive unit cell.
  2. b. Define "the reciprocal lattice" and explain its relation to Bragg reflection.
  3. c. Show the reciprocal lattice and the first Brillouin zone. How is this zone related to Bragg reflection?
  4. d. State and explain the theorem due to Bloch that says an electron moving in the potential of this lattice has traveling-wave functions. What boundary conditions must be used with this theorem?

Step-by-Step Solution

Part (a): Primitive Unit Cell

A primitive unit cell is a unit cell that contains lattice points at corners only. For the square lattice with edge length $a$, we can choose the primitive cell with basis vectors rotated 45° from the square axes:

$$\mathbf{a}_1 = a(\mathbf{i} - \mathbf{j}), \qquad \mathbf{a}_2 = a(\mathbf{i} + \mathbf{j})$$

where $a$ is the edge of the square lattice. This primitive cell is a rhombus tilted at 45° containing exactly one lattice point.

Part (b): Reciprocal Lattice and Bragg Reflection

If $\mathbf{a}_i$ ($i = 1, 2$) are the basis vectors of the direct lattice, the reciprocal lattice vectors $\mathbf{b}_j$ ($j = 1, 2$) satisfy the relation:

$$\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi\delta_{ij} = \begin{cases} 2\pi, & i = j \\ 0, & i \neq j \end{cases}$$

These are the basis vectors of the reciprocal lattice. In the reciprocal space, the condition for Bragg reflection is that the difference between the reflected wave vector $\mathbf{k}$ and the incident wave vector $\mathbf{k}_0$ is an integer multiple $n$ of a reciprocal lattice vector $\mathbf{k}^*$:

$$\mathbf{k} - \mathbf{k}_0 = n\mathbf{k}^*$$

Part (c): Reciprocal Lattice Vectors and First Brillouin Zone

From the direct basis vectors, the reciprocal basis vectors are obtained as:

$$\mathbf{b}_1 = \frac{\pi}{a}(\mathbf{i} - \mathbf{j}), \qquad \mathbf{b}_2 = \frac{\pi}{a}(\mathbf{i} + \mathbf{j})$$

The reciprocal lattice is also a square lattice rotated 45° relative to the direct lattice. The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice -- a square with side $\frac{2\pi}{a}$ centered at the origin. Bragg reflection takes place at the boundaries of the Brillouin zone.

Part (d): Bloch's Theorem

The wave representing an electron moving in the periodic potential field $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$, where $\mathbf{R}$ is a lattice vector, has the form of a Bloch function:

$$\psi_\mathbf{k}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}\,u_\mathbf{k}(\mathbf{r})$$

where the function $u_\mathbf{k}(\mathbf{r})$ has the same translational symmetry as the lattice:

$$u_\mathbf{k}(\mathbf{r}) = u_\mathbf{k}(\mathbf{r} + \mathbf{R})$$

It is a plane wave modulated by the periodic potential field. This is Bloch's theorem. The exponential part of the Bloch wave is a plane wave which describes the global behavior of electrons in a crystal lattice, while the periodic function describes the local motion of those electrons around the nuclei. The Born--von Kármán periodic boundary condition must be employed with Bloch's theorem.

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