diff --git a/src/10_xray_solutions.md b/src/10_xray_solutions.md index b4f3649747a48216f2561c1f46b4a50aebdeccf8..8d9d9b92d0e4e9cfe5dd9e07802603f0a04d6cc3 100644 --- a/src/10_xray_solutions.md +++ b/src/10_xray_solutions.md @@ -142,19 +142,32 @@ plt.show() ## Exercise 4: Structure factors 1. -$$ -S(\mathbf{G}) = \sum_j f_j e^{i \mathbf{G} \cdot \mathbf{r_j}} = f(1 + e^{i \pi (h+k+l)}) -$$ +$S(\mathbf{G}) = \sum_j f_j e^{i \mathbf{G} \cdot \mathbf{r_j}} = f(1 + e^{i \pi (h+k+l)})$ + 2. + Solving for $h$, $k$, and $l$ results in -$$ +$ S(\mathbf{G}) = \begin{cases} 2f, \: \text{if $h+k+l$ is even}\\ 0, \: \text{if $h+k+l$ is odd}. \end{cases} -$$ -Thus if $h+k+l$ is odd, diffraction peaks disappear. +$ + +Thus if $h+k+l$ is odd, diffraction peaks dissapear + +3. + +Let $f_1 \neq f_2$, then + +$ +S(\mathbf{G}) = \begin{cases} +f_1 + f_2, \text{if $h+k+l$ is even}\\ +f_1 - f_2, \text{if $h+k+l$ is odd} +\end{cases} +$ + 4. Due to bcc systematic absences, the peaks from lowest to largest angle are: