X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F2_04s.tex;fp=solid_state_physics%2Ftutorial%2F2_04s.tex;h=7391ad03ebcc7da137b4e0309d53929564a3d67b;hp=11e16f1fd48fe7af94d000aa7d570ded2e8cd23e;hb=75000006ad7eda5e22f1ca8fb47b619eb92c3cac;hpb=d4433169938510c3c6e8898649c0586cb6c48013

diff --git a/solid_state_physics/tutorial/2_04s.tex b/solid_state_physics/tutorial/2_04s.tex
index 11e16f1..7391ad0 100644
--- a/solid_state_physics/tutorial/2_04s.tex
+++ b/solid_state_physics/tutorial/2_04s.tex
@@ -152,26 +152,81 @@
           \frac{1}{L_3}\frac{\partial L_3}{\partial T}=3\alpha_L\nonumber
        \end{eqnarray}
  \item \[
-       dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial}{\partial V}\right|T
-       \]
-       \[
-       dE=TdS-pdV \Rightarrow 
-       \]
-       Find an expression for the pressure as a function of the free energy
-       $F=E-TS$.
-       Rewrite this equation to express the pressure entirely in terms of
-       the internal energy $E$.
-       Evaluate the pressure by using the harmonic form of the internal energy.
-       {\bf Hint:}
-       Step 2 introduced an integral over the temperature $T'$.
-       Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
-       Use integration by parts with respect to $x$.
- \item The normal mode frequencies of a rigorously harmonic crystal
-       are unaffected by a change in volume.
-       What does this imply for the pressure
-       (Which variables does the pressure depend on)?
-       Draw conclusions for the coefficient of thermal expansion.
- \item Find an expression for $C_p-C_V$ in terms of temperature $T$,
+       dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial F}{\partial V}\right|T
+       \]
+       \[
+       \left.\frac{\partial E}{\partial T}\right|_V=
+       \left.\frac{\partial E}{\partial S}\right|_V
+       \left.\frac{\partial S}{\partial T}\right|_V=
+       T\left.\frac{\partial S}{\partial T}\right|_V
+       \Rightarrow
+       \left.\frac{\partial S}{\partial T}\right|_V=
+       \frac{1}{T}\left.\frac{\partial E}{\partial T}\right|_V
+       \]
+       \[
+       \textrm{Using } F=E-TS \textrm{ and }
+       TS=T\int_0^T\frac{\partial S}{\partial T'}dT'
+       \textrm{ (Entropy density vanishes at $T=0$)}
+       \]
+       \[
+       \Rightarrow
+       p=-\frac{\partial}{\partial V}\left(
+       E-T\int_0^T\frac{dT'}{T'}\frac{\partial E}{\partial T'}
+       \right)
+       \]
+       Harmonic approximation of the internal energy:
+       \[
+       E=E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})+
+       \sum_{{\bf k}s}
+       \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
+       \]
+       \[
+       \ldots
+       \]
+       \[
+       x=\hbar\omega_s({\bf k})/T'
+       \]
+       \[
+       \ldots
+       \]
+       \[
+       \Rightarrow
+       p=-\frac{\partial}{\partial V}\left(
+       E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
+       \right)+
+       \sum_{{\bf k}s}\left(-\frac{\partial}{\partial V}\hbar\omega_s({\bf k})
+       \right)\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}
+       \]
+ \item The pressure depends on temperature
+       only if the normal mode frequencies depend on the volume.
+       However, the normal mode frequencies of a rigorously harmonic crystal
+       are unaffected by a change in volume.\\
+       $\Rightarrow$
+       The pressure solely depends on the volume.\\
+       $\Rightarrow$
+       The pressure required to maintain a given volume
+       does not vary with temperature.
+       \[
+       \left.\frac{\partial p}{\partial T}\right|_V=0
+       \]
+       \[
+       \left.\frac{\partial V}{\partial T}\right|_p=
+       -\frac{\left.\frac{\partial p}{\partial T}\right|_V}
+             {\left.\frac{\partial p}{\partial V}\right|_T}=0
+       \]
+ \item $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$
+       and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
+       \[
+       C_p-C_V=\left.\frac{\partial E}{\partial T}\right|_p-
+       \left.\frac{\partial E}{\partial T}\right|_V=
+       \frac{\partial E}{\partial S}
+       \left.\frac{\partial S}{\partial T}\right|_p-
+       \frac{\partial E}{\partial S}
+       \left.\frac{\partial S}{\partial T}\right|_V=
+       T\left.\frac{\partial S}{\partial T}\right|_p-
+       T\left.\frac{\partial S}{\partial T}\right|_V
+       \]
+       Find an expression for $C_p-C_V$ in terms of temperature $T$,
        volume $V$, the coefficient of thermal expansion $\alpha_V$ and
        the inverse bulk modulus (isothermal compressibility)
        $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\