# How to use the condition that $\vec a_i \cdot \vec b_j=2\pi\delta_{ij}$ to find reciprocal lattice vectors, $b_j$, for this rectangular lattice?

Consider a rectangular lattice in two dimensions with primitive lattice vectors $$(a,0)$$ and $$(0,2a)$$.

Which of the following are reciprocal lattice vectors for this lattice?

(a) $$\quad\dfrac{\pi}{a}\left(1,\frac12 \right)$$

(b) $$\quad\dfrac{\pi}{a}\left(1,2 \right)$$

(c) $$\quad\dfrac{\pi}{a}\left(2,-1 \right)$$

(d) $$\quad\dfrac{\pi}{a}\left(0,2 \right)$$

(e) $$\quad\dfrac{\pi}{a}\left(\frac12,-2 \right)$$

My attempt is:

Let the reciprocal lattice vectors $$\vec b_1$$ & $$\vec b_2$$ be $$\vec b_1=\begin{bmatrix}x_1\\ y_1\\ \end{bmatrix}\qquad \text{and}\qquad\vec b_2=\begin{bmatrix}x_2\\ y_2\\ \end{bmatrix}$$ respectively.

and denote the primitive lattice vectors $$\vec a_1$$ & $$\vec a_2$$ as $$\vec a_1=\begin{bmatrix}a\\ 0\\ \end{bmatrix}\qquad \text{and}\qquad\vec a_2=\begin{bmatrix}0\\ 2a\\ \end{bmatrix}$$ respectively

Now using the condition, $$\vec a_i \cdot \vec b_j=2\pi\delta_{ij}\tag{1}$$

$$\vec a_1 \cdot \vec b_1=\begin{bmatrix}a\\ 0\\ \end{bmatrix}\cdot\begin{bmatrix}x_1\\ y_1\\ \end{bmatrix}= 2\pi$$

So $$x_1 =\frac{2\pi}{a}$$

$$\vec a_2 \cdot \vec b_1=\begin{bmatrix}0\\ 2a\\ \end{bmatrix}\cdot\begin{bmatrix}\frac{2\pi}{a}\\ y_1\\ \end{bmatrix}= 0\implies$$ $$y_1=0$$ and hence $$\vec b_1 = \begin{bmatrix}\frac{2\pi}{a}\\ 0\\ \end{bmatrix}=\color{red}{\frac{\pi}{a}\left(2,0\right)}$$

Now to find $$\vec b_2$$,

$$\vec a_2 \cdot \vec b_2=\begin{bmatrix}0\\ 2a\\ \end{bmatrix}\cdot\begin{bmatrix}x_2\\ y_2\\ \end{bmatrix}= 2\pi\implies y_2 = \frac{\pi}{a}$$

$$\vec a_1 \cdot \vec b_2=\begin{bmatrix}a\\ 0\\ \end{bmatrix}\cdot\begin{bmatrix}x_2\\ \frac{\pi}{a}\\ \end{bmatrix}= 0\implies x_2 = 0$$ and hence $$\vec b_2= \begin{bmatrix}0\\ \frac{\pi}{a}\\ \end{bmatrix}=\color{red}{\frac{\pi}{a}\left(0,1\right)}$$

The two vectors calculated (in red) do not correspond to any of the answers above, from which I can tell you that 2 of them are correct. I thought that I was applying the condition $$(1)$$ correctly.

Could someone please explain to me what I'm doing wrong?

I have just realised that the vectors calculated $$\vec b_1=\color{red}{\frac{\pi}{a}\left(2,0\right)}$$ & $$\vec b_2=\color{red}{\frac{\pi}{a}\left(0,1\right)}$$ are still correct answers but they are also the primitive reciprocal lattice vectors.
$$2 \vec b_2=\frac{\pi}{a}\left(0,2\right)$$ and $$\vec b_1 - \vec b_2 = \frac{\pi}{a}\left(2,-1\right)$$
So the correct answers are $$(\rm{c})$$ and $$(\rm{d})$$