tarting of exc 2

[lectures/latex.git] / solid_state_physics / tutorial / 2_03s.tex

diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex
index a8eab9d..fb0e83b 100644 (file)
--- a/solid_state_physics/tutorial/2_03s.tex
+++ b/solid_state_physics/tutorial/2_03s.tex
@@ -109,33 +109,71 @@
        \[
        H_{\text{kin}}=\frac{{\bf P}({\bf R})^2}{2M}
        \]
-       Integral:
-       \[
-       \int d\Gamma \exp(-\beta H)=
+       Integral (using change of variables):
+       \begin{eqnarray}
+       \int d\Gamma \exp(-\beta H)&=&
        \int d\Gamma \exp\left[-\beta\left(\sum \frac{{\bf P}({\bf R})^2}{2M}+
-       U_{\text{eq}} + U_{\text{harm}}\right)\right]
+       U_{\text{eq}} + U_{\text{harm}}\right)\right]\nonumber\\
+       &=&
+       \exp(-\beta U_{\text{eq}})\beta^{-3N}
+       \LARGE(\int\prod_{{\bf R}}d\bar{{\bf u}}({\bf R})d\bar{{\bf P}}({\bf R})
+       \nonumber\\
+       &&\times \exp\left[
+       -\sum\frac{1}{2M}{\bf P}({\bf R})^2
+       -\frac{1}{4}\sum
+       [\bar{u}_{\mu}({\bf R})-\bar{u}_{\mu}({\bf R'})]
+       \Phi_{\mu v}({\bf R}-{\bf R'})
+       [\bar{u}_v({\bf R})-\bar{u}_v({\bf R'})]
+       \right]\LARGE)\nonumber
+       \end{eqnarray}
+       \[
+       \Rightarrow w=-\frac{1}{V}\frac{\partial}{\partial \beta}
+       ln\left((\exp(-\beta U_{\text{eq}})\beta^{-3N} \times \text{const}
+       \right)
+       =\frac{U_{\text{eq}}}{V}+3\frac{N}{V}k_{\text{B}}T
+       =u_{\text{eq}}+3nk_{\text{B}}T
+       \]
+       \[
+       \Rightarrow
+       c_{\text{V}}=\frac{\partial w}{\partial T}=3nk_{\text{B}}
        \]
-
 \end{enumerate}
 
 \section{Specific heat in the quantum theory of the harmonic crystal -\\
          The Debye model}
 
-As found in exercise 1, the specific heat of a classical harmonic crystal
-is not depending on temeprature.
-However, as temperature drops below room temperature
-the specific heat of all solids is decreasing as $T^3$ in insulators
-and $AT+BT^3$ in metals.
-This can be explained in a quantum theory of the specific heat of
-a harmonic crystal, in which the energy density $w$ is given by
 \[
 w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
 \]
 \begin{enumerate}
- \item Show that the energy density can be rewritten to read:
+ \item Energy: $\rightarrow$ 1(a)
        \[
    w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i).
        \]
+ \item \begin{itemize}
+        \item Total energy contribution of a particular normal mode:
+              $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$
+             with $n_{{\bf k}s}=0,1,2,\ldots$
+        \item A state of the crystal is specified by the excitation numbers
+              of the 3N normal modes.
+        \item The total energy is the sum of the energies of the individual
+             normal modes:\\
+             $E=\sum_{{\bf k}s}(n_{{\bf k}s}+
+              \frac{1}{2})\hbar\omega_s({\bf k})$
+       \end{itemize}
+       \begin{eqnarray}
+       \Rightarrow
+       w&=&-\frac{1}{V}\frac{\partial}{\partial \beta} ln\left(
+       \prod_{{\bf k}s}(\exp(-\beta\hbar\omega_s({\bf k})/2)+
+                        \exp(-3\beta\hbar\omega_s({\bf k})/2)+
+                        \exp(-5\beta\hbar\omega_s({\bf k})/2)+
+                       \ldots)
+       \right)\nonumber\\
+       &=&-\frac{1}{V}\frac{\partial}{\partial \beta} ln \prod_{{\bf k}s}
+       \frac{\exp(-\beta\hbar\omega_s({\bf k})/2)}
+            {1-\exp(-\beta\hbar\omega_s({\bf k}))}\nonumber
+       \end{eqnarray}
+
  \item Evaluate the expression of the energy density.
        {\bf Hint:}
        The energy levels of a harmonic crystal of N ions