# Bravais lattice - Why is every reflex allowed in Bravais lattice?

I cannot wrap my head around why every reflex in a Bravais lattice (3-Dimensional) is allowed, according to my book, it's because there is only one term in the expression for the structure factor, but I still don't understand.

I must've missed something important involving Bravais lattice but I cannot wrap my head around what that would be.

## 1 Answer

I think maybe you have misunderstood what every reflex is allowed means. In every 3-Dimensional Lattice reflexes are possible if the von Laue condition

$$\vec{k'} = \vec{k} + \vec{G}$$

is satisfied. Here $\vec{G}$ denotes a vector in the reciprocal lattice and $\vec{k}$ and $\vec{k'}$ are the incoming and reflected wave vectors.

Now on to the structure factor. The scatered wave is given by $$\psi(t) = e^{-i\omega t}\int\rho(r)e^{i (k-k')}d^3r$$

where $\rho$ denotes the scattering density. We can write this scattering density as a convolution of the Lattice with the scattering density inside a primitive cell. The convolution of these two is then basically a repetition of the unit cell scattering density in the same manner as the lattice points.

We obtain

$$\psi \propto N \int\rho_b(r)e^{-ikG}d^3r = N * S_G$$

where $\rho_b$ is the scattering density inside a unit cell. If we calculate S for a primitive cubic lattice with two different Basis Atoms for example CsCl with a Caesium at $(000)$ and a Cl at ($\frac{1}{2}\frac{1}{2}\frac{1}{2})$ we get

$$S_{hkl} = f_1 + f_2 e^{-i\pi(h+k+l)} = \left\{\begin{array}{lr} f_1 + f_2, & \text{for } h+k+l \ even\\ f_1 - f_2, & \text{for } h+k+l \ odd\\ \end{array}\right\}$$

So we see that the reflexes corresponding to some reciprocal lattice vectors are suppressed if $f_1$ and $f_2$ are close to each other.

This happens because whe have a multi atom base. In the simple bravais lattice this cannot happen however and thats what the author of your text had in mind.

• Thank you for this answer, this is probably a dumb question, but how do you know that a simple bravais lattice only contains one "kind" of atoms, couldnt it be repeating itself with, as you mention, Cs and Cl to give an example? Feb 24, 2018 at 11:36
• i was just talking about scattering on the bravais lattice points, excluding the base. Maybe it was a little unclear what i meant . Feb 24, 2018 at 12:10
• I realize when I speak to you that I've obviously missed a lot of theory, just one more question, if you say that you were simply talking about scattering on the bravais lattice points, excluding the base, I mean, I thought you had to have atoms or a basis if you so will to even have an atomic formfactor, am I right or wrong? Feb 24, 2018 at 12:17
• you are right. I usually tend to picture the bravais lattice as a collection of dirac deltas at the different lattice points. So when no base is specified this is what i have in my head and what i picture light scattering off of. But maybe this is just an inaccuracy i tend to make. I should have said that i am talking about a Bravais lattice with a $\delta(x-0)$ base. Feb 24, 2018 at 12:25