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30 \begin{document}
31
32 % header
33 \begin{center}
34  {\LARGE {\bf Materials Physics II}\\}
35  \vspace{8pt}
36  Prof. B. Stritzker\\
37  SS 2008\\
38  \vspace{8pt}
39  {\Large\bf Tutorial 4 - proposed solutions}
40 \end{center}
41
42 \vspace{4pt}
43
44 \section{Legendre transformation and Maxwell relations}
45
46 \begin{enumerate}
47  \item Legendre transformation:
48        \begin{eqnarray}
49        dg &=& df - \sum_{i=r+1}^{n} d(u_ix_i)\nonumber\\
50           &=& df - \sum_{i=r+1}^{n} (x_idu_i + u_idx_i)\nonumber\\
51           &=& \sum_{i=1}^r u_idx_i - \sum_{i=r+1}^n x_idu_i\nonumber
52        \end{eqnarray}
53        \[
54        \Rightarrow g=g(x_1,\ldots,x_r,u_{r+1},\ldots,u_n)
55        \]
56  \item Use $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
57        $-p=\left.\frac{\partial E}{\partial V}\right|_S$.\\
58        Start with internal energy $E=E(S,V)$:
59        \[
60        \Rightarrow dE=\frac{\partial E}{\partial S}dS +
61                       \frac{\partial E}{\partial V}dV =
62                       TdS - pdV
63        \]
64        Enthalpy $H=E+pV$:
65        \[
66        \Rightarrow dH=dE+Vdp+pdV=TdS-pdV+Vdp+pdV=TdS+Vdp
67        \]
68        \[
69        \Rightarrow
70        \left.\frac{\partial H}{\partial S}\right|_p=T \textrm{ and }
71        \left.\frac{\partial H}{\partial p}\right|_S=V
72        \]
73        Helmholtz free energy $F=E-TS$:
74        \[
75        \Rightarrow dF=dE-SdT-TdS=TdS-pdV-SdT-TdS=-pdV-SdT
76        \]
77        \[
78        \Rightarrow
79        \left.\frac{\partial F}{\partial V}\right|_T=-p \textrm{ and }
80        \left.\frac{\partial F}{\partial T}\right|_V=-S
81        \]
82        Gibbs free energy $G=H-TS=E+pV-TS$:
83        \[
84        \Rightarrow dG=dH-SdT-TdS=TdS+Vdp-SdT-TdS=Vdp-SdT
85        \]
86        \[
87        \Rightarrow
88        \left.\frac{\partial G}{\partial p}\right|_T=V \textrm{ and }
89        \left.\frac{\partial G}{\partial T}\right|_p=-S
90        \]
91  \item Maxwell relations:\\
92        Internal energy: $dE=TdS-pdV$
93        \[
94        \frac{\partial}{\partial S}
95        \left(\left.\frac{\partial E}{\partial V}\right|_S\right)_V=
96        \frac{\partial}{\partial V}
97        \left(\left.\frac{\partial E}{\partial S}\right|_V\right)_S
98        \Rightarrow
99        \left.-\frac{\partial p}{\partial S}\right|_V=
100        \left.\frac{\partial T}{\partial V}\right|_S
101        \]
102        Enthalpy: $dH=TdS+Vdp$
103        \[
104        \frac{\partial}{\partial S}
105        \left(\left.\frac{\partial H}{\partial p}\right|_S\right)_p=
106        \frac{\partial}{\partial p}
107        \left(\left.\frac{\partial H}{\partial S}\right|_p\right)_S
108        \Rightarrow
109        \left.\frac{\partial V}{\partial S}\right|_p=
110        \left.\frac{\partial T}{\partial p}\right|_S
111        \]
112        Helmholtz free energy: $dF=-pdV-SdT$
113        \[
114        \frac{\partial}{\partial V}
115        \left(\left.\frac{\partial F}{\partial T}\right|_V\right)_T=
116        \frac{\partial}{\partial T}
117        \left(\left.\frac{\partial F}{\partial V}\right|_T\right)_V
118        \Rightarrow
119        \left.-\frac{\partial S}{\partial V}\right|_T=
120        \left.-\frac{\partial p}{\partial T}\right|_V
121        \]
122        Gibbs free energy: $dG=Vdp-SdT$
123        \[
124        \frac{\partial}{\partial p}
125        \left(\left.\frac{\partial G}{\partial T}\right|_p\right)_T=
126        \frac{\partial}{\partial T}
127        \left(\left.\frac{\partial G}{\partial p}\right|_T\right)_p
128        \Rightarrow
129        \left.-\frac{\partial S}{\partial p}\right|_T=
130        \left.\frac{\partial V}{\partial T}\right|_p
131        \]
132 \end{enumerate}
133
134 \section{Thermal expansion of solids}
135
136 \begin{enumerate}
137  \item Coefficients of thermal expansion:\\
138        Consider a cube with side lengthes $L_1,L_2,L_3$.
139        Isotropic material: $\frac{1}{L_1}\frac{\partial L_1}{\partial T}=
140                             \frac{1}{L_2}\frac{\partial L_2}{\partial T}=
141                             \frac{1}{L_3}\frac{\partial L_3}{\partial T}=
142                             \alpha_L$.
143        \begin{eqnarray}
144        \alpha_V&=&\frac{1}{V}\frac{\partial V}{\partial T}=
145        \frac{1}{L_1L_2L_3}\frac{\partial}{\partial T}(L_1L_2L_3)=
146        \frac{1}{L_1L_2L_3}\left(L_2L_3\frac{\partial L_1}{\partial T}+
147                                 L_1L_3\frac{\partial L_2}{\partial T}+
148                                 L_1L_2\frac{\partial L_3}{\partial T}\right)
149                                 \nonumber\\
150        &=&\frac{1}{L_1}\frac{\partial L_1}{\partial T}+
151           \frac{1}{L_2}\frac{\partial L_2}{\partial T}+
152           \frac{1}{L_3}\frac{\partial L_3}{\partial T}=3\alpha_L\nonumber
153        \end{eqnarray}
154  \item \[
155        dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial F}{\partial V}\right|T
156        \]
157        \[
158        \left.\frac{\partial E}{\partial T}\right|_V=
159        \left.\frac{\partial E}{\partial S}\right|_V
160        \left.\frac{\partial S}{\partial T}\right|_V=
161        T\left.\frac{\partial S}{\partial T}\right|_V
162        \Rightarrow
163        \left.\frac{\partial S}{\partial T}\right|_V=
164        \frac{1}{T}\left.\frac{\partial E}{\partial T}\right|_V
165        \]
166        \[
167        \textrm{Using } F=E-TS \textrm{ and }
168        TS=T\int_0^T\frac{\partial S}{\partial T'}dT'
169        \textrm{ (Entropy density vanishes at $T=0$)}
170        \]
171        \[
172        \Rightarrow
173        p=-\frac{\partial}{\partial V}\left(
174        E-T\int_0^T\frac{dT'}{T'}\frac{\partial E}{\partial T'}
175        \right)
176        \]
177        Harmonic approximation of the internal energy:
178        \[
179        E=E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})+
180        \sum_{{\bf k}s}
181        \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
182        \]
183        \[
184        \ldots
185        \]
186        \[
187        x=\hbar\omega_s({\bf k})/T'
188        \]
189        \[
190        \ldots
191        \]
192        \[
193        \Rightarrow
194        p=-\frac{\partial}{\partial V}\left(
195        E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
196        \right)+
197        \sum_{{\bf k}s}\left(-\frac{\partial}{\partial V}\hbar\omega_s({\bf k})
198        \right)\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}
199        \]
200  \item The pressure depends on temperature
201        only if the normal mode frequencies depend on the volume.
202        However, the normal mode frequencies of a rigorously harmonic crystal
203        are unaffected by a change in volume.\\
204        $\Rightarrow$
205        The pressure solely depends on the volume.\\
206        $\Rightarrow$
207        The pressure required to maintain a given volume
208        does not vary with temperature.
209        \[
210        \left.\frac{\partial p}{\partial T}\right|_V=0
211        \]
212        \[
213        \left.\frac{\partial V}{\partial T}\right|_p=
214        -\frac{\left.\frac{\partial p}{\partial T}\right|_V}
215              {\left.\frac{\partial p}{\partial V}\right|_T}=0
216        \]
217  \item $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$
218        and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$\\
219        \[
220        C_p=\left.\frac{\partial H}{\partial T}\right|_p=
221            \left.\frac{\partial H}{\partial S}\right|_p
222            \left.\frac{\partial S}{\partial T}\right|_p=
223            T\left.\frac{\partial S}{\partial T}\right|_p
224        \]
225        \[
226        C_V=\left.\frac{\partial E}{\partial T}\right|_V=
227            \left.\frac{\partial E}{\partial S}\right|_V
228            \left.\frac{\partial S}{\partial T}\right|_V=
229            T\left.\frac{\partial S}{\partial T}\right|_V
230        \]
231        \[
232        \Rightarrow C_p-C_V=
233        T\left.\frac{\partial S}{\partial T}\right|_p-
234        T\left.\frac{\partial S}{\partial T}\right|_V=
235        T\left(
236        \left.\frac{\partial S}{\partial T}\right|_p-
237        \left.\frac{\partial S}{\partial T}\right|_V
238        \right)
239        \]
240        Using the equality
241        \[
242        dS=\left.\frac{\partial S}{\partial T}\right|_p dT
243        +\left.\frac{\partial S}{\partial p}\right|_T dp
244        \Rightarrow
245        \left.\frac{\partial S}{\partial T}\right|_V=
246        \left.\frac{\partial S}{\partial T}\right|_p+
247        \left.\frac{\partial S}{\partial p}\right|_T
248        \left.\frac{\partial p}{\partial T}\right|_V
249        \]
250        and the Maxwell relation
251        \[
252        \left.\frac{\partial S}{\partial p}\right|_T=
253        -\left.\frac{\partial V}{\partial T}\right|_p
254        \]
255        and the equality
256        \[
257        dV=\left.\frac{\partial V}{\partial T}\right|_p dT+
258        \left.\frac{\partial V}{\partial p}\right|_T dp
259        \stackrel{\left.\frac{\partial}{\partial T}\right|_V}{\Rightarrow}
260        0=\left.\frac{\partial V}{\partial T}\right|_p+
261          \left.\frac{\partial V}{\partial p}\right|_T
262          \left.\frac{\partial p}{\partial T}\right|_V
263        \Rightarrow
264        \left.\frac{\partial p}{\partial T}\right|_V=
265        -\frac{\left.\frac{\partial V}{\partial T}\right|_p}
266        {\left.\frac{\partial V}{\partial p}\right|_T}
267        \]
268        we obtain:
269        \[
270        C_p-C_V=T\left(
271        -\left.\frac{\partial S}{\partial p}\right|_T
272        \left.\frac{\partial p}{\partial T}\right|_V
273        \right)=T\left(
274        \left.\frac{\partial V}{\partial T}\right|_p
275        \left.\frac{\partial p}{\partial T}\right|_V
276        \right)=T\left(
277        \frac{\left.\left.\frac{\partial V}{\partial T}\right|_p\right.^2}
278        {-\left.\frac{\partial V}{\partial p}\right|_T}
279        \right)=T\left(\frac{V^2\alpha_V^2}{V\frac{1}{B}}\right)=
280        TVB\alpha_V^2
281        \]
282        For a rigorously harmonic potential $C_p=C_V$.
283 \end{enumerate}
284
285 \end{document}