X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F2_04s.tex;h=b03977cab07536a0db1e5f909f0e5ae4e24e524f;hp=287c275bde4c50d7960e253074f52cd985143810;hb=fec7e52be07515b5dac392facfa0b2022d002c21;hpb=9c6ed4d9ce5cdc917ceab10ae57a50ba2891f9fd diff --git a/solid_state_physics/tutorial/2_04s.tex b/solid_state_physics/tutorial/2_04s.tex index 287c275..b03977c 100644 --- a/solid_state_physics/tutorial/2_04s.tex +++ b/solid_state_physics/tutorial/2_04s.tex @@ -89,6 +89,16 @@ \left.\frac{\partial G}{\partial T}\right|_p=-S \] \item Maxwell relations:\\ + Internal energy: $dE=TdS-pdV$ + \[ + \frac{\partial}{\partial S} + \left(\left.\frac{\partial E}{\partial V}\right|_S\right)_V= + \frac{\partial}{\partial V} + \left(\left.\frac{\partial E}{\partial S}\right|_V\right)_S + \Rightarrow + \left.-\frac{\partial p}{\partial S}\right|_V= + \left.\frac{\partial T}{\partial V}\right|_S + \] Enthalpy: $dH=TdS+Vdp$ \[ \frac{\partial}{\partial S} @@ -109,52 +119,167 @@ \left.-\frac{\partial S}{\partial V}\right|_T= \left.-\frac{\partial p}{\partial T}\right|_V \] - \item For a thermodynamic potential $\Phi(X,Y)$ the following identity - expressing the permutability of derivatives holds: + Gibbs free energy: $dG=Vdp-SdT$ \[ - \frac{\partial^2 \Phi}{\partial X \partial Y} = - \frac{\partial^2 \Phi}{\partial Y \partial X} + \frac{\partial}{\partial p} + \left(\left.\frac{\partial G}{\partial T}\right|_p\right)_T= + \frac{\partial}{\partial T} + \left(\left.\frac{\partial G}{\partial p}\right|_T\right)_p + \Rightarrow + \left.-\frac{\partial S}{\partial p}\right|_T= + \left.\frac{\partial V}{\partial T}\right|_p \] - Derive the Maxwell relations by taking the mixed derivatives of the - potentials in (b) with respect to the variables they depend on. - Exchange the sequence of derivation and use the identities gained in (b). \end{enumerate} \section{Thermal expansion of solids} -It is well known that solids change their length $L$ and volume $V$ respectively -if there is a change in temperature $T$ or in pressure $p$ of the system. -The following exercise shows that -thermal expansion cannot be described by rigorously harmonic crystals. - \begin{enumerate} - \item The coefficient of thermal expansion of a solid is given by - $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$. - Show that the coefficient of thermal expansion of the volume - $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$ - equals $3\alpha_L$ for isotropic materials. - \item 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 $T$, - volume $V$, the coefficient of thermal expansion $\alpha_V$ and - the inverse bulk modulus (isothermal compressibility) - $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\ - $C_p=\left.\frac{\partial E}{\partial T}\right|_p$ is the heat capacity - for constant pressure and - $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity - for constant volume. + \item Coefficients of thermal expansion:\\ + Consider a cube with side lengthes $L_1,L_2,L_3$. + Isotropic material: $\frac{1}{L_1}\frac{\partial L_1}{\partial T}= + \frac{1}{L_2}\frac{\partial L_2}{\partial T}= + \frac{1}{L_3}\frac{\partial L_3}{\partial T}= + \alpha_L$. + \begin{eqnarray} + \alpha_V&=&\frac{1}{V}\frac{\partial V}{\partial T}= + \frac{1}{L_1L_2L_3}\frac{\partial}{\partial T}(L_1L_2L_3)= + \frac{1}{L_1L_2L_3}\left(L_2L_3\frac{\partial L_1}{\partial T}+ + L_1L_3\frac{\partial L_2}{\partial T}+ + L_1L_2\frac{\partial L_3}{\partial T}\right) + \nonumber\\ + &=&\frac{1}{L_1}\frac{\partial L_1}{\partial T}+ + \frac{1}{L_2}\frac{\partial L_2}{\partial T}+ + \frac{1}{L_3}\frac{\partial L_3}{\partial T}=3\alpha_L\nonumber + \end{eqnarray} + \item \[ + dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial F}{\partial V}\right|T + \] + \[ + \left.\frac{\partial E}{\partial T}\right|_V= + \left.\frac{\partial E}{\partial S}\right|_V + \left.\frac{\partial S}{\partial T}\right|_V= + T\left.\frac{\partial S}{\partial T}\right|_V + \Rightarrow + \left.\frac{\partial S}{\partial T}\right|_V= + \frac{1}{T}\left.\frac{\partial E}{\partial T}\right|_V + \] + \[ + \textrm{Using } F=E-TS \textrm{ and } + TS=T\int_0^T\frac{\partial S}{\partial T'}dT' + \textrm{ (Entropy density vanishes at $T=0$)} + \] + \[ + \Rightarrow + p=-\frac{\partial}{\partial V}\left( + E-T\int_0^T\frac{dT'}{T'}\frac{\partial E}{\partial T'} + \right) + \] + 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 $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$\\ + \[ + C_p=\left.\frac{\partial H}{\partial T}\right|_p= + \left.\frac{\partial H}{\partial S}\right|_p + \left.\frac{\partial S}{\partial T}\right|_p= + T\left.\frac{\partial S}{\partial T}\right|_p + \] + \[ + C_V=\left.\frac{\partial E}{\partial T}\right|_V= + \left.\frac{\partial E}{\partial S}\right|_V + \left.\frac{\partial S}{\partial T}\right|_V= + T\left.\frac{\partial S}{\partial T}\right|_V + \] + \[ + \Rightarrow C_p-C_V= + T\left.\frac{\partial S}{\partial T}\right|_p- + T\left.\frac{\partial S}{\partial T}\right|_V= + T\left( + \left.\frac{\partial S}{\partial T}\right|_p- + \left.\frac{\partial S}{\partial T}\right|_V + \right) + \] + Using the equality + \[ + dS=\left.\frac{\partial S}{\partial T}\right|_p dT + +\left.\frac{\partial S}{\partial p}\right|_T dp + \Rightarrow + \left.\frac{\partial S}{\partial T}\right|_V= + \left.\frac{\partial S}{\partial T}\right|_p+ + \left.\frac{\partial S}{\partial p}\right|_T + \left.\frac{\partial p}{\partial T}\right|_V + \] + and the Maxwell relation + \[ + \left.\frac{\partial S}{\partial p}\right|_T= + -\left.\frac{\partial V}{\partial T}\right|_p + \] + and the equality + \[ + dV=\left.\frac{\partial V}{\partial T}\right|_p dT+ + \left.\frac{\partial V}{\partial p}\right|_T dp + \stackrel{\left.\frac{\partial}{\partial T}\right|_V}{\Rightarrow} + 0=\left.\frac{\partial V}{\partial T}\right|_p+ + \left.\frac{\partial V}{\partial p}\right|_T + \left.\frac{\partial p}{\partial T}\right|_V + \Rightarrow + \left.\frac{\partial p}{\partial T}\right|_V= + -\frac{\left.\frac{\partial V}{\partial T}\right|_p} + {\left.\frac{\partial V}{\partial p}\right|_T} + \] + we obtain: + \[ + C_p-C_V=T\left( + -\left.\frac{\partial S}{\partial p}\right|_T + \left.\frac{\partial p}{\partial T}\right|_V + \right)=T\left( + \left.\frac{\partial V}{\partial T}\right|_p + \left.\frac{\partial p}{\partial T}\right|_V + \right)=T\left( + \frac{\left.\left.\frac{\partial V}{\partial T}\right|_p\right.^2} + {-\left.\frac{\partial V}{\partial p}\right|_T} + \right)=T\left(\frac{V^2\alpha_V^2}{V\frac{1}{B}}\right)= + TVB\alpha_V^2 + \] + For a rigorously harmonic potential $C_p=C_V$. \end{enumerate} \end{document}