Menu Outline Exercise Questions Appendices Lectures Student Projects Books Sections Introduction Atoms Molecules Crystal Structure Crystal Diffraction Crystal Binding Photons Phonons Electrons Band Model Crystal Physics Semiconductors

PHY.K02UF Molecular and Solid State Physics

## Reciprocal lattices

Every Bravais lattice has a reciprocal lattice. The Bravias lattice can be specified by giving three primitive lattice vectors $\vec{a}_1$, $\vec{a}_2$, and $\vec{a}_3$. A translation vector is a vector that points from one Bravais lattice point to some other Bravais lattice point. The translation vectors are,

$$\vec{T}_{hkl} = h\vec{a}_1 + k\vec{a}_2 + l\vec{a}_3,$$

where $h$, $k$, and $l$ are integers. The reciprocal lattice is a set of points connected by three primitive reciprocal lattice vectors $\vec{b}_1$, $\vec{b}_2$, and $\vec{b}_3$. The reciprocal lattice vectors point from one reciprocal lattice point to some other reciprocal lattice point,

$$\vec{G}=\nu_1\vec{b}_1+\nu_2\vec{b}_2+\nu_3\vec{b}_3\hspace{1.5 cm}\nu_1,\nu_2,\nu_3 = \cdots ,-2,-1,0,1,2,\cdots.$$

These reciprocal lattice vectors are needed to construct a Fourier series. The primitive reciprocal lattice vectors can be determined from the real space primitive lattice vectors with the formula,

$$\vec{a}_i\cdot \vec{b}_j=2\pi\delta_{ij},$$

where $\delta_{ij}$ is the Kronecker delta,

$$\delta_{ij}= \begin{cases} 1 & \mbox{for } i=j \\ 0, & \mbox{for } i\ne j \end{cases}.$$

Another way to determine the reciprocal primitive lattice vectors from the real space primitive lattice vectors (or vice versa) is,

\begin{eqnarray} \vec{b}_1=2\pi \frac{\vec{a}_2\times\vec{a}_3}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}&\hspace{2cm}&\vec{a}_1=2\pi \frac{\vec{b}_2\times\vec{b}_3}{\vec{b}_1\cdot\left(\vec{b}_2\times\vec{b}_3\right)} \\ \vec{b}_2=2\pi \frac{\vec{a}_3\times\vec{a}_1}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}&\hspace{2cm}&\vec{a}_1=2\pi \frac{\vec{b}_3\times\vec{b}_1}{\vec{b}_1\cdot\left(\vec{b}_2\times\vec{b}_3\right)} \\ \vec{b}_3=2\pi \frac{\vec{a}_1\times\vec{a}_2}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}&\hspace{2cm}&\vec{a}_3=2\pi \frac{\vec{b}_1\times\vec{b}_2}{\vec{b}_1\cdot\left(\vec{b}_2\times\vec{b}_3\right)} \end{eqnarray}

Notice that because of the cross product, $\vec{b}_1$ is perpendicular to $\vec{a}_2$ and $\vec{a}_3$ so $\vec{a}_2\cdot\vec{b}_1 = \vec{a}_3\cdot\vec{b}_1=0$ and $\vec{a}_1\cdot\vec{b}_1= 2\pi \vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)/\left( \vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)\right) = 2\pi$ so that the expressions with the cross products are the same as $a_ib_j=2\pi\delta_{ij}$.

By applying these rules one can show that a simple cubic lattice has a simple cubic reciprocal lattice.

$$\text{sc:}\qquad\vec{a}_1=a\hat{x},\quad \vec{a}_2=a\hat{y},\quad\vec{a}_3=a\hat{z},\\ \vec{b}_1=\frac{2\pi}{a}\hat{k}_x,\quad \vec{b}_2=\frac{2\pi}{a}\hat{k}_y,\quad\vec{b}_3=\frac{2\pi}{a}\hat{k}_z.$$

An fcc lattice has a bcc reciprocal lattice.

$$\text{fcc:}\qquad\vec{a}_1=\frac{a}{2}(\hat{x}+\hat{z}),\quad \vec{a}_2=\frac{a}{2}(\hat{x}+\hat{y}),\quad\vec{a}_3=\frac{a}{2}(\hat{y}+\hat{z}),\\ \vec{b}_1=\frac{2\pi}{a}(\hat{k}_x-\hat{k}_y+\hat{k}_z),\quad \vec{b}_2=\frac{2\pi}{a}(\hat{k}_x+\hat{k}_y-\hat{k}_z),\quad\vec{b}_3=\frac{2\pi}{a}(-\hat{k}_x+\hat{k}_y+\hat{k}_z).$$

A bcc lattice has an fcc reciprocal lattice.

$$\text{bcc:}\qquad\vec{a}_1=\frac{a}{2}(\hat{x}+\hat{y}-\hat{z}),\quad \vec{a}_2=\frac{a}{2}(-\hat{x}+\hat{y}+\hat{z}),\quad\vec{a}_3=\frac{a}{2}(\hat{x}-\hat{y}+\hat{z}),\\ \vec{b}_1=\frac{2\pi}{a}(\hat{k}_x+\hat{k}_y),\quad \vec{b}_2=\frac{2\pi}{a}(\hat{k}_y+\hat{k}_z),\quad\vec{b}_3=\frac{2\pi}{a}(\hat{k}_x+\hat{k}_z).$$

A hexagonal lattice has a hexagonal reciprocal lattice.

$$\text{hex:}\qquad\vec{a}_1=a\hat{x},\qquad\vec{a}_2=\frac{a}{2}\hat{x}+\frac{\sqrt{3}a}{2}\hat{y},\qquad\vec{a}_3=c\hat{z},\\\vec{b}_1=\frac{2\pi}{\sqrt{3}a}\left(\sqrt{3}\hat{k}_x-\hat{k}_y\right),\qquad\vec{b}_2=\frac{4\pi}{\sqrt{3}a}\hat{k}_y,\qquad\vec{b}_3=\frac{2\pi}{c}\hat{k}_z.$$

The form below will calculate the primitive reciprocal lattice vectors given the real space primitive lattice vectors.

Primitive lattice vectors:

 $\vec{a}_1=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m] $\vec{a}_2=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m] $\vec{a}_3=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m]

Primitive reciprocal lattice vectors

 $\vec{b}_1=2\pi\frac{\vec{a}_2\times\vec{a}_3}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1] $\vec{b}_2=2\pi\frac{\vec{a}_3\times\vec{a}_1}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1] $\vec{b}_3=2\pi\frac{\vec{a}_1\times\vec{a}_2}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1]