Update 10_xray.md - polish

docs/10_xray.md

@@ -352,7 +352,7 @@ A general convention in reciprocal space is to use the Wigner-Seitz cell, which
 Because the Wigner-Seitz cell is primitive, the 1st Brillouin zone (1BZ) contains a set of unique $\mathbf{k}$-vectors.
 This means that any wavevector $\mathbf{k'}$ outside the 1st Brillouin zone is related to a wavevector $\mathbf{k}$ inside the first Brillouin Zone by shifting it by a reciprocal lattice vector: $\mathbf{k'} = \mathbf{k}+\mathbf{G}$. 
 
-In the previous lecture we already discussed how to construct Wigner-Seitz cells. However, here is a reminder of what such a cell can look like:
+In the previous lecture we already discussed how to construct Wigner-Seitz cells. However, here is a reminder of such a cell:
 
 ![](figures/brillouin_mod.svg)
 
@@ -360,26 +360,20 @@ In the previous lecture we already discussed how to construct Wigner-Seitz cells
 
 ### Reciprocal lattice: Laue conditions
 
-The reciprocal lattice manifests directly in the diffraction experiments.
-A diffraction experiment uses the crystal as a target and scatters high energy particles (X-rays, neutrons, or electrons) off of it.
-As a result of interference between mutiple waves, the scattered radiation reveals the reciprocal lattice of the crystal.
-In order to find the relationship between the incoming wave and the scattered one, let us consider a lattice of atoms separated by a lattice vector $\mathbf{R}$.
-An incoming wave with wave vector $\mathbf{k}$ is incident upon the lattice.
-After scattering, the outgoing wave's wave vector is $\mathbf{k'}$.
-We assume that the atomic scattering is elastic (does not cause an energy loss), such that $|\mathbf{k'}|=|\mathbf{k}|$.
-Below we present a simple sketch of two different atoms scattering an incoming wave.
+Another reason for why it is important to understand the reciprocal lattice is that it manifests directly in diffraction experiments. Such experiments are some of our most powerful tools for determining the crystal structure of materials.
+
+In a diffraction experiment, a beam of high-energy waves or particles (e.g. X-rays, neutrons, or electrons) is directed at a material of interest . As a result of interference, the scattered radiation pattern reveals the reciprocal lattice of the crystal. To understand the relationship between the incoming wave with wavevector $\mathbf{k}$ and scattered wave with wavevector $\mathbf{k'}$, let us consider the scattering caused by two atoms in a lattice separated by a lattice vector $\mathbf{R}$ (see figure). We assume that the scattering is elastic (does not cause an energy loss), such that $|\mathbf{k'}|=|\mathbf{k}|$.
 
 ![](figures/scattering.svg)
 
-Observe that the bottom ray travels a larger distance compared to the upper ray. 
-The difference in distance results in a relative phase shift between the rays $\Delta \phi$. 
+Observe that the bottom ray travels a larger distance than the upper ray. This results in a phase shift $\Delta \phi$ between these rays. 
 With a bit of geometry, we find that the extra distance traveled by the lower ray relative to the upper one is 
 
 $$
 x_{\mathrm{extra}} = \Delta x_1+\Delta x_2 = \cos(\theta) \lvert R \rvert + \cos(\theta') \lvert R \rvert.
 $$
 
-As a result of the travel distance, the phase difference is: 
+The corresponding phase difference is: 
 
 \begin{align}
 \Delta \phi &= \lvert\mathbf{k} \rvert(\Delta x_1+\Delta x_2)\\
@@ -388,22 +382,20 @@ As a result of the travel distance, the phase difference is:
 \end{align}
 
 However, that is only a phase difference between waves scattered off of two atoms.
-To find the outgoing wave's amplitude, we must sum over scattered waves from each and every atom in the lattice:
+To find the outgoing wave's amplitude, we must sum over the scattered waves from each and every atom in the lattice:
 
 $$
 A\propto\sum_\mathbf{R}\mathrm{e}^{i\left(\Delta \phi-\omega t\right)} = \sum_\mathbf{R}\mathrm{e}^{i\left((\mathbf{k'}-\mathbf{k})\cdot\mathbf{R}-\omega t\right)}.
 $$
 
-The above sum is non-zero if and only if the scattered waves interfere constructively i.e. the phase difference equals $2\pi n$, where $n$ is an integer.
-Furthermore, we know that real and reciprocal lattice vectors are related by $\mathbf{G} \cdot \mathbf{R} = 2 \pi n$.
-Therefore, we conclude that the difference between incoming and outgoing waves must be:
+This sum is non-zero if and only if the scattered waves interfere constructively, i.e., the phase difference equals $2\pi n$, where $n$ is an integer. Furthermore, we know that real and reciprocal lattice vectors are related by $\mathbf{G} \cdot \mathbf{R} = 2 \pi n$.
+Therefore, we conclude that the difference between the incoming and outgoing waves must be:
 
 $$
 \mathbf{k'}-\mathbf{k}=\mathbf{G}.
 $$
 
-In other words, if the difference of the wavevector between the incoming and outgoing wave vectors coïncides with a reciprocal lattice point, we expect constructive interference. 
-This requirement is known as the _Laue condition_.
+to get constructive interference. In other words, we can only get constructive interference at very specific angles, as determined by the structure of the reciprocal lattice. This requirement is known as the _Laue condition_.
 As a result, the interference pattern produced in diffraction experiments is a direct measurement of the reciprocal lattice!
 
 ### Structure factor