Solid state-problem with difraction

[Frank Einstein]

Homework Statement

Hi everybody. I am triying to do the next exercice:
Given a unit cell of Ice X and knowing the positions of the atoms on it, find the expression of the structure factor for a difraction peak as a function of its Miller's index and the atomic form factors f_O and f_H.

Will there be a systematic extinction?

Would there be more extinctions if f_O=f_H

Homework Equations

F=Σf_i*e^r_jH_hkl

In the ice X, there are two atoms of oxygen (0,0,0) and (0.5, 0.5, 0.5); the atoms of hydrogen are at (1/4, 1/4, 1/4), (3/4, 3/4, 1/4), (3/4, 1/4, 3/4) and (1/4, 3/4, 3/4)

The Attempt at a Solution

F=fO*(1+expπ[h+k+l])+fH*(exp[(π/2)(h+k+l)]+exp[(π/2)(3h+3k+l)]+exp[(π/2)(3h+k+3l)]+exp[(π/2)(h+3k+3l)])

Here is where I don't know how to keep going; I have never seen something with two different atomic form factors.
In theory we have seen the case where only the firs term is; in that case, I would say h+k+l=2n+1 so that 1+expπ[h+k+l] becomes zero; but I don't know how to solve this with a second kind of atom.

Can anyone please help me?

Thanks for reading.

 

[anathema]

It seems like you are on the right track with your solution so far. Let's break down the problem and tackle it step by step.

First, we need to understand what the structure factor represents in this context. The structure factor is a measure of the interference between the wave scattered by the atoms in the unit cell and the incident X-ray beam. It is given by the sum of the atomic form factors (fi) multiplied by the phase factor (erjHhkl). The phase factor takes into account the positions of the atoms in the unit cell.

Now, for the ice X unit cell, we have two types of atoms - oxygen and hydrogen. Each atom has its own atomic form factor (fO and fH) and position in the unit cell (given by the indices h, k, and l). Therefore, the structure factor for a diffraction peak will be the sum of the contributions from both types of atoms.

To find the expression for the structure factor, we need to consider the positions of the atoms in the unit cell and how they contribute to the phase factor. For oxygen atoms, the contribution to the phase factor will be 1+expπ[h+k+l], as you have correctly identified. For hydrogen atoms, the contribution will be exp[(π/2)(h+k+l)], exp[(π/2)(3h+3k+l)], exp[(π/2)(3h+k+3l)], and exp[(π/2)(h+3k+3l)].

Now, to find the overall expression for the structure factor, we need to take into account the positions of all the atoms in the unit cell. This can be done by summing the contributions from each atom, as you have done in your attempt at the solution. However, there is one important factor that needs to be considered - the atomic form factors.

The atomic form factors represent the scattering strength of each atom, and they are dependent on the wavelength of the incident X-ray beam. In this case, we have two types of atoms with different form factors - fO and fH. This means that the overall structure factor will be a combination of these two form factors, weighted by the number of atoms of each type in the unit cell.

To find the expression for the structure factor, we need to multiply the contributions from each atom by its respective atomic form factor and then sum them all together. This will give us the final expression for the