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34 {\LARGE {\bf Materials Physics II}\\}
39 {\Large\bf Tutorial 4 - proposed solutions}
44 \section{Legendre transformation and Maxwell relations}
47 \item Legendre transformation:
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
54 \Rightarrow g=g(x_1,\ldots,x_r,u_{r+1},\ldots,u_n)
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 $E=E(S,V)$:
60 \Rightarrow dE=\frac{\partial E}{\partial S}dS +
61 \frac{\partial E}{\partial V}dV =
66 \Rightarrow dH=dE+Vdp+pdV=TdS-pdV+Vdp+pdV=TdS+Vdp
70 \left.\frac{\partial H}{\partial S}\right|_p=T \textrm{ and }
71 \left.\frac{\partial H}{\partial p}\right|_S=V
73 Helmholtz free energy $F=E-TS$:
75 \Rightarrow dF=dE-SdT-TdS=TdS-pdV-SdT-TdS=-pdV-SdT
79 \left.\frac{\partial F}{\partial V}\right|_T=-p \textrm{ and }
80 \left.\frac{\partial F}{\partial T}\right|_V=-S
82 Gibbs free energy $G=H-TS=E+pV-TS$:
84 \Rightarrow dG=dH-SdT-TdS=TdS+Vdp-SdT-TdS=Vdp-SdT
88 \left.\frac{\partial G}{\partial p}\right|_T=V \textrm{ and }
89 \left.\frac{\partial G}{\partial T}\right|_p=-S
91 \item Maxwell relations:\\
92 Internal energy: $dE=TdS-pdV$
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
99 \left.-\frac{\partial p}{\partial S}\right|_V=
100 \left.\frac{\partial T}{\partial V}\right|_S
102 Enthalpy: $dH=TdS+Vdp$
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
109 \left.\frac{\partial V}{\partial S}\right|_p=
110 \left.\frac{\partial T}{\partial p}\right|_S
112 Helmholtz free energy: $dF=-pdV-SdT$
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
119 \left.-\frac{\partial S}{\partial V}\right|_T=
120 \left.-\frac{\partial p}{\partial T}\right|_V
122 Gibbs free energy: $dG=Vdp-SdT$
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
129 \left.-\frac{\partial S}{\partial p}\right|_T=
130 \left.\frac{\partial V}{\partial T}\right|_p
134 \section{Thermal expansion of solids}
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}=
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)
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
155 dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial F}{\partial V}\right|T
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
163 \left.\frac{\partial S}{\partial T}\right|_V=
164 \frac{1}{T}\left.\frac{\partial E}{\partial T}\right|_V
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 $T=0$)}
173 p=-\frac{\partial}{\partial V}\left(
174 E-T\int_0^T\frac{dT'}{T'}\frac{\partial E}{\partial T'}
177 Harmonic approximation of the internal energy:
179 E=E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})+
181 \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
187 x=\hbar\omega_s({\bf k})/T'
194 p=-\frac{\partial}{\partial V}\left(
195 E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
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}
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.\\
205 The pressure solely depends on the volume.\\
207 The pressure required to maintain a given volume
208 does not vary with temperature.
210 \left.\frac{\partial p}{\partial T}\right|_V=0
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
217 \item $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$
218 and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$\\
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
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
233 T\left.\frac{\partial S}{\partial T}\right|_p-
234 T\left.\frac{\partial S}{\partial T}\right|_V=
236 \left.\frac{\partial S}{\partial T}\right|_p-
237 \left.\frac{\partial S}{\partial T}\right|_V
242 dS=\left.\frac{\partial S}{\partial T}\right|_p dT
243 +\left.\frac{\partial S}{\partial p}\right|_T dp
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
250 and the Maxwell relation
252 \left.\frac{\partial S}{\partial p}\right|_T=
253 -\left.\frac{\partial V}{\partial T}\right|_p
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
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}
271 -\left.\frac{\partial S}{\partial p}\right|_T
272 \left.\frac{\partial p}{\partial T}\right|_V
274 \left.\frac{\partial V}{\partial T}\right|_p
275 \left.\frac{\partial p}{\partial T}\right|_V
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)=
282 For a rigorously harmonic potential $C_p=C_V$.