Solutions lecture 7

src/8_many_atoms_solutions.md

@@ -128,13 +128,25 @@ pyplot.suptitle('Density of states for 2 atom unit cell and 1 atom unit cell');
 
 ### Subquestion 1
 
+The unit cell should contain exactly one spring of $\kappa_1$, $\kappa_2$ and $\kappa_3$ and exactly three atoms.
+
 ### Subquestion 2
 
+Use the same procedure as before to find: 
+
+$$ m \begin{pmatrix} u_{1,n} \\ u_{2,n} \\ u_{3,n} \end{pmatrix} = \begin{pmatrix} -\kappa_1(u_{n,2}-u_{n,1}) -\kappa_3(u_{n-1,3}-u_{n,1}) \\ -\kappa_2(u_{n,3}-u_{n,2}) -\kappa_1(u_{n,3}-u_{n,2}) \\ -\kappa_3(u_{n+1,1}-u_{n,3}) -\kappa_2(u_{n,2}-u_{n,3}) \end{pmatrix} $$
+
 ### Subquestion 3
 
+Follow the same procedure as before but now using Ansatz $$ \begin{pmatrix} u_{1,n} \\ u_{2,n} \\ u_{3,n} \end{pmatrix} = e^{i\omega t - ikx_n} \begin{pmatrix} A_1 \\ A_2 \\ A_3 \end{pmatrix} $$
+
 ### Subquestion 4
 
+
+
 ### Subquestion 5
 
+If $\kappa_1 = \kappa_2 = \kappa_3$ then we have the uniform mono-atomic chain. If the length of the 3 spring constant unit cell is $a$, then the length of the mono-atomic chain is $a/3$.
+