Update 11_nearly_free_electron_model.md - typo

src/11_nearly_free_electron_model.md

@@ -35,21 +35,21 @@ Let's summarize what we learned about electrons so far:
 
 
 
-In this lecture, we will analyze how electrons behave in solids using the *nearly-free electron model*. This model considers electrons as plane waves (as in the free electron model) that are only weakly perturbed by the periodic potential associated with the atoms in a solid. This approach is opposite to the tight-binding model, where our starting point was that the electrons are strongly bound to the individual atoms and we considered the hopping to other atoms to be a small effect. Perhaps surprisingly, we will find that the nearly-free electron model gives very similar results to the tight binding model: the nearly-free electron model also leads to the formation of energy bands, and these bands are separated by *band gaps* - regions in the band structure where there are no allowed energy states.  
+In this lecture, we will analyze how electrons behave in solids using the *nearly-free electron model*. This model considers electrons as plane waves (as in the free electron model) that are weakly perturbed by the periodic potential associated with the atoms in a solid. This approach is opposite to that of the tight-binding model, where our starting point was that the electrons are strongly bound to the individual atoms and we included hopping to other atoms as a small effect. Perhaps surprisingly, we will find that the nearly-free electron model gives very similar results to the tight binding model: it also leads to the formation of energy bands, and these bands are separated by *band gaps* - regions in the band structure where there are no allowed energy states.  
 
 The different models can be organized as a function of the strength of the lattice potential $V(x)$:
 
 ![](figures/models.svg)
 
-Why would it be possible to describe an electron that can scatter off all the atoms in a solid by something that even remotely looks like a plane wave? The answer lies in that the periodic potential created by the atoms can only scatter an electron between momentum states $|\mathbf{k}$ and $|\mathbf{k'}$ if these momenta differ by a reciprocal lattice vector. This condition is very similar to the Laue condition of X-ray scattering. In this lecture we will explicitly analyze it in the context of the nearly-free electron model. The condition is known as the *conservation of crystal momentum* and is central to Bloch's theorem, which provides a general framework for computing band structures in crystals.
+Why would it be possible to describe an electron that can scatter off all the atoms in a solid by something that even remotely looks like a plane wave? The answer lies in that the periodic potential created by the atoms can only scatter an electron between momentum states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$ *if these momenta differ by a reciprocal lattice vector*. This condition is very similar to the Laue condition of X-ray scattering. In this lecture we will explicitly analyze it in the context of the nearly-free electron model. The condition is known as the **conservation of crystal momentum** and is central to Bloch's theorem, which provides a general framework for computing band structures in crystals.
 
 ## Bloch theorem
 
 > All Hamiltonian eigenstates in a crystal have the form
-> $$ \psi_n(\mathbf{r}) = u_n(\mathbf{r})e^{i\mathbf{kr}} $$
+> $$ \psi_n(\mathbf{r}) = u_n(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{r}} $$
 > with $u_n(\mathbf{r})$ having the same periodicity as the lattice potential $V(\mathbf{r})$, and index $n$ labeling electron bands with energies $E_n(\mathbf{k})$.
 
-In other words: any electron wave function in a crystal is a product of a periodic part that describes electron motion within a unit cell and a plane wave. We will see that both the wavefunctions in the tight-binding and the nearly-free electron models are consistent Bloch's theorem.
+In other words: any electron wave function in a crystal is a product of a periodic part that describes electron motion within a unit cell and a plane wave. We will see that both the wavefunctions in the tight-binding and the nearly-free electron models are consistent with Bloch's theorem.
 
 ### Extra remarks
 
@@ -71,13 +71,13 @@ Within the **nearly free electron model** we start from the dispersion relation
 
 ![](figures/nearly_free_electron_bands.svg)
 
-In this figure, the red curves represent the nearly-free electron dispersion, which differs from the free-electron dispersion (black curves) because of the interaction with the lattice. We see that **band gaps** (regions in the band structure where there are no allowed energy states) open where two copies of the free electron dispersion cross. A key goal of this lecture is to understand how the weak interaction with the lattice leads to this modified band structure.
+In this figure, the red curves represent the nearly-free electron dispersion, which differs from the free-electron dispersion (black curves) because of the interaction with the lattice. We see that **band gaps** open where two copies of the free-electron dispersion cross. A key goal of this lecture is to understand how the weak interaction with the lattice leads to this modified band structure.
 
 ### Analyzing the avoided crossings
 
 *Remark: An avoided crossing is an important concept in quantum mechanics that can be analyzed using **perturbation theory**. You will only learn this theory later in QMIII, so we will need to postulate some important facts here.*
 
-To analyze what happens near the crossings, we first neglect the lattice potential and consider the free-electron energy dispersion near the crossing at $k=\pi/a$ in 1D. Near this crossing, we see that two copies of the free-electron dispersion come together (one copy centered at $k=0$, the other at $k=2\pi/a$). We call the corresponding plane-wave eigenfunctions $|k\rangle$ and $|k'\rangle =|k-2\pi/a\rangle$. We now express the wavefunction near this crossing as a linear superposition $|\psi\rangle = \alpha |k\rangle + \beta |k'\rangle$. Note that this wave function is very similar to that used in the LCAO model, except there we used linear combinations of the orbitals $|1\rangle$ and $|2\rangle$ instead of the plane waves $|k\rangle$ and $|k'\rangle$.
+To analyze what happens near the crossings, we first neglect the lattice potential and consider the free-electron dispersion near the crossing at $k=\pi/a$ in 1D. Near this crossing, we see that two copies of the dispersion come together (one copy centered at $k=0$, the other at $k=2\pi/a$). We call the corresponding plane-wave eigenfunctions $|k\rangle$ and $|k'\rangle =|k-2\pi/a\rangle$. We now express the wavefunction near this crossing as a linear superposition $|\psi\rangle = \alpha |k\rangle + \beta |k'\rangle$. Note that this wave function is very similar to that used in the LCAO model, except there we used linear combinations of the orbitals $|1\rangle$ and $|2\rangle$ instead of the plane waves $|k\rangle$ and $|k'\rangle$.
 
 We express the Hamiltonian near the crossing as a matrix, using $| k \rangle$ and $| k' \rangle$ as the basis states. The matrix elements are given by $\langle k |H|k\rangle = E_0 + v \hbar \delta k$ and $\langle k' |H|k'\rangle = E_0 - v \hbar \delta k$, where $\delta k = k-\pi/a$ is the distance from the center of the crossing and we approximated the dispersion near the crossing by a linear term. In matrix form, this yields