# Reciprocal crystal lattice

1. Sep 4, 2006

### big man

I know this might be a really stupid question, but to convert a crystal lattice 2D representation to a 2D reciprocal lattice do you justdo you just invert the scaling. I know this is a pretty poor explanation so I will try and illustrate what I mean.

Let's say that you have a reciprocal lattice like the one below:

. 020 . 120 . 220

. 010 . 110 . 210

. 000 . 100 . 200

|----|
. 25 nm^-1

Is the crystal lattice just a similar drawing with the spacing inverted, that is, 4 nm??? By the way the above diagram is meant to have the vertical and horizontal spacings equal so they are both 0.25 nm^-1.

I'm sorry but I just haven't really found anything on this at all and I'm preparing for my test this week. The only example question I could find was converting from reciprocal lattice to crystal lattice, but I imagine the process will be the same if you were trying to convert from crystal lattice to a reciprocal lattice.

Thanks for any assistance.

2. Sep 4, 2006

### Kurdt

Staff Emeritus
To find the reciprocal lattice vectors you use the following equation.

$$\mathbf{b_1} = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3} {\mathbf {a}_1 \cdot \mathbf{a}_2 \times \mathbf{a}_3}$$

For a 2D case the vector $$\mathbf{a}_3$$ becomes just the z unit vector.

To get the vector $$\mathbf{b}_2$$ you just cyclically permute the numerator.

You are of course correct that the unit of the reciprocal lattice is length-1 but its a little more complicated than simply inverting the lengths.