@@ -268,8 +268,8 @@ where the plus sign corresponds to the optical branch and the minus sign to the
4. Show that the group velocity at $k=0$ for the _optical_ branch is zero.
5. Derive an expression of the density of states $g(\omega)$ for the _acoustic_ branch and small $k$. Make use of your expression of the group velocity.
Compare this expression with the Debye density of states derived in [exercise 1](2_debye_model/#Exercise 1*: Deriving the density of states for the linear dispersion relation of the Debye model) of the Debye lecture.
6. Show that the size of the band gap is $\frac{\kappa}{m}\Delta\omega \approx \sqrt{1-\tfrac{4\mu^2}{m_1m_2}}$ when $m_1\approx m_2$
Compare this expression with the 1D Debye density of states derived in exercise 1 of the Debye lecture.
6. Show that the size of the band gap is $\Delta\omega \approx \sqrt{\frac{\kappa}{\mu}}\frac{|m_1-m_2|}{m_1+m_2}$ when $m_1\approx m_2$.
### Exercise 2: Analyzing the LCAO chain with alternating hoppings
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@@ -339,6 +339,6 @@ Suppose we have a vibrating 1D atomic chain with 3 different spring constants al
6. What will happen to the periodicity of the band structure if $\kappa_1 = \kappa_2 = \kappa_3$? Sketch the band structure for this case. Now suppose $\kappa_3$ becomes slightly different from $\kappa_1$ and $\kappa_2$. Sketch the band structure for this case in the same figure. Is your sketch consistent with your answers to the previous two subquestions?
7. Suppose we set $\kappa_3 = 0$. What set of simpler systems do we obtain? (it always helps to make a drawing!). What are the eigenfrequencies and their degeneracies of this set of simpler systems when $\kappa_1=\kappa_2$? (you can refer to the first exercise of the lecture on Bonds and Spectra)
7. Suppose we set $\kappa_3 = 0$. What set of simpler systems do we obtain? (it always helps to make a drawing!). What are the eigenfrequencies and their degeneracies of this set of simpler systems when $\kappa_1=\kappa_2$? (you can refer to the first exercise of the Bonds and Spectra lecture)
