the mpi publication, finally!

[lectures/latex.git] / solid_state_physics / tutorial / 1_01s.tex

diff --git a/solid_state_physics/tutorial/1_01s.tex b/solid_state_physics/tutorial/1_01s.tex
index 57d822d..ef49e3c 100644 (file)
--- a/solid_state_physics/tutorial/1_01s.tex
+++ b/solid_state_physics/tutorial/1_01s.tex
@@ -79,7 +79,7 @@
          \[
          - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} F_x(x) = E_x F_x(x), \quad
          - \frac{\hbar^2}{2m} \frac{d^2}{dy^2} F_y(y) = E_y F_y(y),\quad
-         - \frac{\hbar^2}{2m} \frac{d^2}{dz^2} F_z(z) = E_x F_z(z).
+         - \frac{\hbar^2}{2m} \frac{d^2}{dz^2} F_z(z) = E_z F_z(z).
         \]
         \[
          \Rightarrow \Big[E_x + E_y + E_z\Big] F_x(x) F_y(y) F_z(z) =
@@ -123,9 +123,9 @@
        \]
  \item $n_x,n_y,n_z=1,2,3\ldots$\\
        Allowed $k_{x,y,z}$ values located in positive octant only.
-       \begin{center}
+       \begin{flushleft}
        \includegraphics[width=10cm]{feg_kvals.eps}
-       \end{center}
+       \end{flushleft}
 
 \end{enumerate}
 
@@ -139,21 +139,36 @@ Convention:
 Prove:
 \[
 V_{real}=a_1(a_2 \times a_3)
+\]\[
+b_1=\frac{2\pi(a_2 \times a_3)}{a_1(a_2 \times a_3)}
+\]\[
+b_2=\frac{2\pi(a_3 \times a_1)}{a_1(a_2 \times a_3)}
+\]\[
+b_3=\frac{2\pi(a_1 \times a_2)}{a_1(a_2 \times a_3)}
 \]
 \[
-V_{rec}=b_1 ( b_2 \times b_3)
-       =\frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} (a_2 \times a_3) [
+V_{rec}=b_1 ( b_2 \times b_3)=
+       \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} (a_2 \times a_3) [
        (a_3 \times a_1) \times (a_1 \times a_2) ]
 \]
 \[
-\textrm{hint 1: }
+\textrm{Hint 1: }
 (a_3 \times a_1) \times (a_1 \times a_2) =
-a_1((a_3 \times a_1)a_2) - a_2((a_3 \times a_1)a_1) =
-a_1((a_3 \times a_1)a_2)
+a_1((a_3 \times a_1)a_2) - \underbrace{a_2((a_3 \times a_1)a_1)}_{=0}
 \]
 \[
 \Rightarrow V_{rec}= \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} 
-(a_2 \times a_3) (a_1(a_3 \times a_1) a_2)
+(a_2 \times a_3) (a_1((a_3 \times a_1) a_2))
+\]
+\[
+\textrm{Hint 2: }
+(a_2 \times a_3) (a_1((a_3 \times a_1) a_2)) =
+(a_2 \times a_3) (a_1((a_2 \times a_3) a_1)) =
+(a_1 (a_2 \times a_3))^2
+\]
+\[
+\Rightarrow V_{rec}=\frac{(2\pi)^3}{a_1(a_2 \times a_3)}=
+\frac{(2\pi)^3}{V_{real}}
 \]
 
 \end{document}