Update 10_xray.md - polish

docs/10_xray.md

@@ -54,14 +54,12 @@ _based on chapters 13.1-13.2 & 14.1-14.2 of the book_
 In the last lecture, we introduced crystallographic terminology in order to be able to discuss and analyze crystal structures.
 In this lecture, we will 1) study how real-space lattices give rise to lattices in reciprocal space (with the goal of understanding dispersion relations) and 2) consider how to probe crystal structures using X-ray diffraction experiments.
 
-## Reciprocal lattice motivation 1D case
-In [lecture 7](7_tight_binding.md) we discussed the reciprocal space of a simple 1D lattice. 
-To obtain the dispersion relation we considered waves of the form
-
+## Recap: the reciprocal lattice in one dimension
+In [lecture 7](7_tight_binding.md) we discussed the reciprocal space of a simple 1D lattice with lattice points $x_n = na$, where $n$ is an integer and $a$ is the spacing between the lattice points. To obtain the dispersion relation, we considered waves of the form
 $$
 e^{ikx_n} = e^{ikna}, \quad n \in \mathbb{Z},
 $$
-where $x_n = na$ are the lattice points. We then observed that waves with wave vectors $k$ and $k+G$, where $G=2\pi m/a$ with integer $m$, are exactly the same: 
+We then observed that waves with wavevectors $k$ and $k+G$, where $G=2\pi m/a$ with integer $m$, are exactly the same: 
 $$
 e^{i(k+G)na} = e^{ikna+im2\pi n} =  e^{ikna},
 $$
@@ -71,21 +69,20 @@ e^{iGx_n} = e^{i2\pi mn} =  1.
 $$
 The set of points $G=2\pi m/a$ forms the **reciprocal lattice**.
 
-Let us now generalize this idea to describe reciprocal lattices in 3D.
+Let us now generalize this idea to describe reciprocal lattices in three dimensions.
 
-## Extending to higher dimensions
+## Extending to three dimensions
 We start from a lattice in real space:
 $$
-\mathbf{R}=n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_1, \quad \{n_1, n_2, n_3\} \in \mathbb{Z},
+\mathbf{R}=n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3, \quad \{n_1, n_2, n_3\} \in \mathbb{Z},
 $$
 where $\mathbf{a}_1$, $\mathbf{a}_2$ and $\mathbf{a}_3$ are the primitive lattice vectors. The reciprocal lattice is also a lattice, but in $k$-space:
-
 $$
 \mathbf{G}=m_1\mathbf{b}_1+m_2\mathbf{b}_2+m_3\mathbf{b_3}, \quad \{m_1, m_2, m_3\} \in \mathbb{Z}.
 $$
+The vectors $\mathbf{b}_1$, $\mathbf{b}_2$ and $\mathbf{b}_2$ are the primitive lattice vectors of the **reciprocal lattice**.
 
-The vectors $\mathbf{b}_1$, $\mathbf{b}_2$ and $\mathbf{b}_2$ are the primitive vectors of the **reciprocal lattice**.
-Let us now determine the ${\mathbf{b}_i}$ by requiring that waves that differ by a reciprocal lattice vector are indistinguishable:
+We determine the ${\mathbf{b}_i}$ by requiring that waves with wavevectors that differ by a reciprocal lattice vector $\mathbf{G}$ are indistinguishable:
 $$
 e^{i\mathbf{k}\cdot\mathbf{R}} = e^{i(\mathbf{k} + \mathbf{G})\cdot\mathbf{R}},
 $$