Update exercises and add warmup exercises to 'many atoms' lecture

src/8_many_atoms.md

@@ -89,8 +89,6 @@ Looking at the eigenvectors we see that for every $k$ there are now two values o
 
 <!--- 
 Should we mention that we choose omega > 0?
-
-Also I dont know how to fix the "Text(0, -0.55, '1st Brillouin zone')" which appears in a box in the text
 --->
 
 ```python
@@ -118,7 +116,7 @@ pyplot.ylim(bottom=0)
 draw_classic_axes(ax)
 ax.annotate(s='', xy=(-pi, -.3), xytext=(pi, -.3),
             arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
-ax.text(0, -.55, '1st Brillouin zone', ha='center')
+ax.text(0, -.55, '1st Brillouin zone', ha='center');
 #draw_classic_axes(ax, xlabeloffset=.2)
 ```
 The figure above shows a plot of the eigenfrequencies as a function of $ka$.
@@ -227,8 +225,18 @@ And although the band structures are different, working out the density of state
 * Systems with different band structures can have the same density of states.
 
 ## Exercises
-#### Exercise 1: analyzing the diatomic vibrating chain
-Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes with 2 different masses (can be found below [here](#more-degrees-of-freedom-per-unit-cell)).
+
+### Warm-up questions
+1. Verify that the expression for $\omega^2$ is always positive. Why is this important?
+2. Work out the expression for $\omega^2$ in the case $m_1 = m_2$. Compare this to the solution for the monatomic chain.
+3. When calculating the DOS, we only look at the first Brillouin zone. Why?
+
+### Exercise 1: analyzing the diatomic vibrating chain
+As we have derived, the eigenfreqencies of a diatomic vibrating chain with 2 different masses are:
+
+$$\omega^2=\frac{\kappa(m_1+m_2)}{m_1m_2}\pm \kappa\left\{\left(\frac{m_1+m_2}{m_1m_2}\right)^2-\frac{4}{m_1m_2}{\rm sin}^2\left(\frac{1}{2}ka\right)\right\}^{\frac{1}{2}}$$
+
+where the plus sign corresponds to the optical branch and the minus sign to the acoustic branch.
 
 1. Find the magnitude of the group velocity near $k=0$ for the _acoustic_ branch.
 
@@ -236,9 +244,10 @@ Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes w
         Make use of a Taylor expansion.
 
 2. Show that the group velocity at $k=0$ for the _optical_ branch is zero.
-3. Derive an expression for the density of states $g(\omega)$ for the _acoustic_ branch and small $ka$. Make use of your expression for the group velocity in 1.
+3. Derive an expression for the density of states $g(\omega)$ for the _acoustic_ branch and small $k$. Make use of your expression for the group velocity in 1.
+Compare this expression with that of the derived density of states from [exercise 1](2_debye_model/#exercise-1-debye-model-concepts) of the Debye lecture.
 
-#### Exercise 2: the Peierls transition
+### Exercise 2: the Peierls transition
 In the previous lecture, we have derived the electronic band structure of an 1D, equally spaced atomic chain. Such chains, however, are in fact not stable and the equal spacing will be distorted. This is also known as the [Peierls transition](https://en.wikipedia.org/wiki/Peierls_transition).
 
 The spacing of the distorted chain alternates between two different distances and this also causes the hopping energy to alternate between $t_1$ and $t_2$. We further set the onsite energies of the atoms to $\epsilon$. The situation is depicted in the figure below.
@@ -247,7 +256,7 @@ The spacing of the distorted chain alternates between two different distances an
 
 Due to the alternating hopping energies, we must treat two consecutive atoms as two different orbitals ($\left| n,1 \right>$ and $\left| n,2 \right>$ in the figure) from the same unit cell. The corresponding LCAO of this chain is given by $$\left|\Psi \right> = \sum_n \left( \phi_n \left| n,1 \right> + \psi_n \left| n,2 \right>\right)$$ As usual, we assume that all these atomic orbitals are orthogonal to each other.
 
-1. Indicate the length of the unit cell $a$ in the figure.
+1. Indicate the the unit cell $a$ in the figure.
 2. Using the Schrödinger equation, write the equations of motion of the electrons.
 
     ??? hint
@@ -258,12 +267,12 @@ Due to the alternating hopping energies, we must treat two consecutive atoms as
 5. Find an expression of the group velocity $v(k)$ and effective mass $m^*(k)$ of both bands.
 6. Derive an expression for the density of states $g(E)$ of the entire band structure and make a plot of it. Does your result makes sense when considering the band structure?
 
-#### Exercise 3: atomic chain with 3 different spring constants
+### Exercise 3: atomic chain with 3 different spring constants
 Suppose we have a vibrating 1D atomic chain with 3 different spring constants alternating like $\kappa_ 1$, $\kappa_2$, $\kappa_3$, $\kappa_1$, etc. All the the atoms in the chain have an equal mass $m$.
 
 1. Make a sketch of this chain and indicate the length of the unit cell $a$ in this sketch.
 2. Derive the equations of motion for this chain.
-3. By filling in the trial solutions for the equations of motion (which should be similar to the ones in eqs. (10.3) and (10.4) of the book), show that the eigenvalue problem is given by $$ \omega^2 \mathbf{x} = \frac{1}{m} \begin{pmatrix} \kappa_1 + \kappa_ 3 &  -\kappa_ 1 & -\kappa_ 3 e^{i k a} \\ -\kappa_ 1 & \kappa_1+\kappa_2 & -\kappa_ 2 \\  -\kappa_ 3 e^{-i k a} & -\kappa_2 & \kappa_2 + \kappa_ 3 \end{pmatrix} \mathbf{x}$$
+3. By filling in the trial solutions for the equations of motion (which should be similar to Ansazt used in the lecture), show that the eigenvalue problem is given by $$ \omega^2 \mathbf{x} = \frac{1}{m} \begin{pmatrix} \kappa_1 + \kappa_ 3 &  -\kappa_ 1 & -\kappa_ 3 e^{i k a} \\ -\kappa_ 1 & \kappa_1+\kappa_2 & -\kappa_ 2 \\  -\kappa_ 3 e^{-i k a} & -\kappa_2 & \kappa_2 + \kappa_ 3 \end{pmatrix} \mathbf{x}$$
 4. In general, the eigenvalue problem above cannot be solved analytically, and can only be solved in specific cases. Find the eigenvalues $\omega^2$ when $k a = \pi$ and $\kappa_ 1 = \kappa_ 2 = q$.
 
     ??? hint