more proofreading

src/10_xray.md

@@ -270,14 +270,13 @@ fig.show()
 
 To find the reciprocal lattice vectors, we use the relation
 $$
-\mathbf{a_i}\cdot\mathbf{b_j}=2\pi\delta_{ij}.
-$$ 
-The relation leads to several simple conclusions.
-One such conclusion is the orthogonality between several real-space and reciprocal lattice vectors: 
+\mathbf{a_i}\cdot\mathbf{b_j}=2\pi\delta_{ij},
 $$
-\mathbf{a}_1\cdot\mathbf{b}_2=\mathbf{a}_2\cdot\mathbf{b}_1=0.
+which gives us the following equations:
 $$
-We also find
+\mathbf{a}_1\cdot\mathbf{b}_2=\mathbf{a}_2\cdot\mathbf{b}_1=0,
+$$
+and
 $$
 \mathbf{a}_1\cdot\mathbf{b}_1=\mathbf{a}_2\cdot\mathbf{b}_2=2\pi.
 $$
@@ -287,12 +286,12 @@ $$
 $$
 where $\theta_i$ is the angle between the vectors $\mathbf{a}_i$ and $\mathbf{b}_i$.
 To find the angles $\theta_1$ and $\theta_2$, we use the orthogonality relations above and the fact that the angle between $\mathbf{a}_1$ and $\mathbf{a}_2$ is $60^\circ$.
-From this we conclude that $\theta_1 = \theta_2 = 30^\circ$.```
+From this we conclude that $\theta_1 = \theta_2 = 30^\circ$.
 Because $\lvert \mathbf{a}_1 \rvert = \lvert \mathbf{a}_2 \rvert = a$, we find
 $$
 \lvert \mathbf{b}_1 \rvert = \lvert \mathbf{b}_2 \rvert = \frac{4\pi}{a\sqrt{3}}. 
 $$
-Unsurprisingly, we find that the lengths of the reciprocal lattice vectors are equal and have inverse dependence on the lattice constant $a$.
+Unsurprisingly, we find that the lengths of the reciprocal lattice vectors are equal and inversely proportional to the lattice constant $a$.
 With $\lvert \mathbf{b}_2 \rvert$ and $\mathbf{a}_1 \perp \mathbf{b}_2$, we easily find 
 $$
 \mathbf{b}_2 = \frac{4\pi}{a\sqrt{3}} \mathbf{\hat{y}}.
@@ -305,17 +304,16 @@ $$
 $$
 
 ??? Question "Is the choice of a set of reciprocal lattice vectors unique? If not, which other ones are possible?"
-    No. As is the case for the real-space lattice vectors, the choice of a set of reciprocal lattice vectors is not unique.
-    There are multiple sets of reciprocal lattice vectors that fulfill all criteria.
-    The reciprocal lattice vectors that also fullfil the criteria are
+    There are many equivalent ways to choose lattice vectors of the reciprocal lattice. In the example above we could as well use
     $$
     \mathbf{b}_1 = \frac{4\pi}{a\sqrt{3}} \left(-\frac{\sqrt{3}}{2} \mathbf{\hat{x}} + \frac{1}{2}\mathbf{\hat{y}} \right) \quad \text{and} \quad \mathbf{b}_2 = -\frac{4\pi}{a\sqrt{3}} \mathbf{\hat{y}}.
     $$
+    There is however only one choice that satisfies the relations $\mathbf{a_i}\cdot\mathbf{b_j}=2\pi\delta_{ij}$.
 
+### 3D lattice example
 
-### Generalization to a 3D lattice
-We generalize the procedure above to a 3D lattice.
-The reciprocal lattice vectors can be composed directly from their real-space counterparts:
+Let us now consider a more involved example of the 3D lattice.
+The explicit expression for the reciprocal lattice vectors in terms of their real space counterparts is:
 
 $$
 \mathbf{b}_1=\frac{2\pi(\mathbf{a}_2\times\mathbf{a}_3)}{ \mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})}
@@ -329,10 +327,10 @@ $$
 \mathbf{b_3}=\frac{2\pi(\mathbf{a}_1\times\mathbf{a}_2)}{ \mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})}
 $$
 
-Note that the denominator $V = \mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})$ is the volume of the real-space unit cell spanned by the lattice vectors $\mathbf{a}_1$, $\mathbf{a}_2$ and $\mathbf{a}_3$. 
-The definitions of the reciprocal lattice vectros are cyclic: $\mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})=\mathbf{a}_2\cdot(\mathbf{a_3}\times\mathbf{a}_1)=\mathbf{a_3}\cdot(\mathbf{a}_1\times\mathbf{a}_2)$.
+Note that the denominator $\mathbf{a}_1\cdot(\mathbf{a}_2\times\mathbf{a_3})$ is the volume $V$ of the real-space unit cell spanned by the lattice vectors $\mathbf{a}_1$, $\mathbf{a}_2$ and $\mathbf{a}_3$. 
 
 ### The reciprocal lattice as a Fourier transform
+
 One can also think of the reciprocal lattice as a Fourier transform of the real-space lattice. 
 For simplicity, we illustrate this for a 1D lattice (the same principles apply to a 3D lattice).
 We model the real-space lattice as a density function consisting of delta peaks: