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34 {\LARGE {\bf Materials Physics II}\\}
39 {\Large\bf Tutorial 4}
44 \section{Legendre transformation and Maxwell relations}
47 \item Consider the total differential
49 df= \sum_{i=1}^{n} u_i dx_i
51 with the state function $f=f(x_1,\ldots,x_n)$ and its partial derivatives
52 $u_i=\frac{\partial f}{\partial x_i}$.
53 Rewrite the total differential of the function $g$ defined as
55 g=f-\sum_{i=r+1}^{n} u_i x_i
57 in such a way that $g$ is immediately identified to be a function of
58 the variables $x_1,\ldots,x_r$ and $u_{r+1},\ldots,u_n$,
59 where $u_i$ is called the conjugate variable of $x_i$.
60 This transformation is called Legendre transformation.
61 \item By taking the derivatives of transformed thermodynamic potentials
62 with respect to the variables they depend on,
63 relations between intensive and extensive variables can be gained.
65 Start with the internal energy $E=E(S,V)$.
66 Write down the total differential using the equalities
67 $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
68 $-p=\left.\frac{\partial E}{\partial V}\right|_S$.
69 Apply Legendre transformation to the following potentials
71 \item $H=E+pV$ (Enthalpy)
72 \item $F=E-TS$ (Helmholtz free energy)
73 \item $G=H-TS=E+pV-TS$ (Gibbs free energy)
75 and find more relations by taking the appropriate derivatives.
76 \item For a thermodynamic potential $\Phi(X,Y)$ the following identity
77 expressing the permutability of derivatives holds:
79 \frac{\partial^2 \Phi}{\partial X \partial Y} =
80 \frac{\partial^2 \Phi}{\partial Y \partial X}
82 Derive the Maxwell relations by taking the mixed derivatives of the
83 potentials in (b) with respect to the variables they depend on.
84 Exchange the sequence of derivation and use the identities gained in (b).
87 \section{Thermal expansion of solids}
89 It is well known that solids change their length $L$ and volume $V$ respectively
90 if there is a change in temperature $T$ or in pressure $p$ of the system.
91 The following exercise shows that
92 thermal expansion cannot be described by rigorously harmonic crystals.
95 \item The coefficient of thermal expansion of a solid is given by
96 $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$.
97 Show that the coefficient of thermal expansion of the volume
98 $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
99 equals $3\alpha_L$ for isotropic materials.
100 \item Find an expression for the pressure as a function of the free energy
102 Rewrite this equation to express the pressure entirely in terms of
103 the internal energy $E$.
104 Evaluate the pressure by using the harmonic form of the internal energy.
106 Step 2 introduced an integral over the temperature $T'$.
107 Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
108 Use integration by parts with respect to $x$.
109 \item The normal mode frequencies of a rigorously harmonic crystal
110 are unaffected by a change in volume.
111 What does this imply for the pressure
112 (Which variables does the pressure depend on)?
113 Draw conclusions for the coefficient of thermal expansion.
114 \item Find an expression for $C_p-C_V$ in terms of temperature $T$,
115 volume $V$, the coefficient of thermal expansion $\alpha_V$ and
116 the inverse bulk modulus (isothermal compressibility)
117 $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
118 $C_p=\left.\frac{\partial H}{\partial T}\right|_p$ is the heat capacity
119 for constant pressure and
120 $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity