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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 Enthalpy: $dH=TdS+Vdp$
94 \frac{\partial}{\partial S}
95 \left(\left.\frac{\partial H}{\partial p}\right|_S\right)_p=
96 \frac{\partial}{\partial p}
97 \left(\left.\frac{\partial H}{\partial S}\right|_p\right)_S
99 \left.\frac{\partial V}{\partial S}\right|_p=
100 \left.\frac{\partial T}{\partial p}\right|_S
102 Helmholtz free energy: $dF=-pdV-SdT$
104 \frac{\partial}{\partial V}
105 \left(\left.\frac{\partial F}{\partial T}\right|_V\right)_T=
106 \frac{\partial}{\partial T}
107 \left(\left.\frac{\partial F}{\partial V}\right|_T\right)_V
109 \left.-\frac{\partial S}{\partial V}\right|_T=
110 \left.-\frac{\partial p}{\partial T}\right|_V
112 \item For a thermodynamic potential $\Phi(X,Y)$ the following identity
113 expressing the permutability of derivatives holds:
115 \frac{\partial^2 \Phi}{\partial X \partial Y} =
116 \frac{\partial^2 \Phi}{\partial Y \partial X}
118 Derive the Maxwell relations by taking the mixed derivatives of the
119 potentials in (b) with respect to the variables they depend on.
120 Exchange the sequence of derivation and use the identities gained in (b).
123 \section{Thermal expansion of solids}
125 It is well known that solids change their length $L$ and volume $V$ respectively
126 if there is a change in temperature $T$ or in pressure $p$ of the system.
127 The following exercise shows that
128 thermal expansion cannot be described by rigorously harmonic crystals.
131 \item The coefficient of thermal expansion of a solid is given by
132 $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$.
133 Show that the coefficient of thermal expansion of the volume
134 $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
135 equals $3\alpha_L$ for isotropic materials.
136 \item Find an expression for the pressure as a function of the free energy
138 Rewrite this equation to express the pressure entirely in terms of
139 the internal energy $E$.
140 Evaluate the pressure by using the harmonic form of the internal energy.
142 Step 2 introduced an integral over the temperature $T'$.
143 Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
144 Use integration by parts with respect to $x$.
145 \item The normal mode frequencies of a rigorously harmonic crystal
146 are unaffected by a change in volume.
147 What does this imply for the pressure
148 (Which variables does the pressure depend on)?
149 Draw conclusions for the coefficient of thermal expansion.
150 \item Find an expression for $C_p-C_V$ in terms of temperature $T$,
151 volume $V$, the coefficient of thermal expansion $\alpha_V$ and
152 the inverse bulk modulus (isothermal compressibility)
153 $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
154 $C_p=\left.\frac{\partial E}{\partial T}\right|_p$ is the heat capacity
155 for constant pressure and
156 $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity