Showing reciprocol lattice is perpenducular to a plane

[TFM]

Homework Statement

If $$a_1 a_2 a_3$$ are the unit vectors of a real space lattice, then the so-called “reciprocal lattice” is defined by the unit vectors $$b_1 b_2 b_3$$ where:

$$b_1 = \frac{2\pi a_2 \times a_3}{a_1 \cdot a_2 \times a_3}$$

$$b_2 = \frac{2\pi a_3 \times a_1}{a_1 \cdot a_2 \times a_3}$$

$$b_3 = \frac{2\pi a_1 \times a_2}{a_1 \cdot a_2 \times a_3}$$


Consider a plane hkl in a crystal lattice.

(a) Prove that the reciprocal lattice vector $$G = hb_1 + kb_2 + lb_3$$ is perpendicular to this plane.

(b) Prove that the distance between two adjacent parallel planes of the lattice is

$$d(hkl) = \frac{2\pi}{|G|}$$

Homework Equations

$$a \cdot b = |A||B| cos \theta$$
$$a \times b = |A||B| sin \theta$$

The Attempt at a Solution

So far I have subsituted in the values of a1/2/3 into the given equation to give:

$$G = h\frac{2\pi a_2 \times a_3}{a_1 \cdot a_2 \times a_3} + k\frac{2\pi a_3 \times a_1}{a_1 \cdot a_2 \times a_3} + l\frac{2\pi a_1 \times a_2}{a_1 \cdot a_2 \times a_3}$$

but I am not quite sure how to show that this is perpendicular.

I have added the dot and cross product as I feel these may be useful?

 

[physicsworks]

Think of reciprocal lattice as a set of wave vectors $$\mathbf{G}$$ such that the plane wave $$e^{i \mathbf{k}\mathbf{r}}$$ with $$\mathbf{k = G}$$ has a translation symmetry of Bravais lattice.
So, for any vector $$\mathbf{R}$$ of Bravais lattice ($$\mathbf{R}= n_1 \mathbf{a_1} + n_2 \mathbf{a_2} + n_3 \mathbf{a_3}$$) you should have:
$$e^{i \mathbf{G} \mathbf{r}} = e^{i \mathbf{G (r + R)}}$$
and
$$e^{i \mathbf {GR}}=1$$
This is far more physical definition of the reciprocal lattice (RL). From this, you can easily show that three vectors $$\math{b_1}, \math{b_2}, \math{b_3}$$ as you defined them form a reciprocal lattice and every vector of RL has a form $$G=k_1 \mathbf{b_1} + k_2 \mathbf{b_2} + k_3 \mathbf{b_3}$$ where $$k_1, k_2, k_3 \in Z$$.
Now you can proof that:
1) for any set of lattice planes separated by the distance d, there are reciprocal vectors perpendicular to them and the shortest vector has a length $$2 \pi/d$$.
2) for any reciprocal lattice vector $$\mathbf{G}$$ there is a set of planes normal to $$\mathbf{G}$$ separated by the distance d and the the shortest reciprocal vector parallel to $$\mathbf{G}$$ has a length $$2 \pi/d$$.
To do this just use the fact that $$e^{i\mathbf{G r}}$$ is constant value in lattice planes perpendicular to $$\mathbf{G}$$ and separated by the distance $$\lambda = 2 \pi/K$$, so given
$$\mathbf{K}=\frac{2 \pi \mathbf{s}}{d}$$
where $$\mathbf{s}$$ is a unit vector perpendicular to the planes
you have $$\lambda = d$$.