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34 {\LARGE {\bf Materials Physics II}\\}
39 {\Large\bf Tutorial 3}
44 The specific heat (capacity) is the measure of the energy
45 required to increase the temperature of a unit quantity of a substance
46 by a certain temperature interval.
47 Thus, the specific heat at constant volume $V$ is given by
49 c_V = \frac{\partial w}{\partial T}
51 in which $w$ is the energy density of the system.
53 \section{Specific heat in the classical theory of the harmonic crystal -\\
54 The law of Dulong and Petit}
56 In the classical theory of the harmonic crystal equilibrium properties
57 can no longer be evaluated by simply assuming that each ion sits quitly at
58 its Bravais lattice site {\bf R}.
59 From now on expectation values have to be claculated by
60 integrating over all possible ionic configurations weighted by
61 $\exp(-E/k_{\text{B}}T)$, where $E$ is the energy of the configuration.
62 Thus, the energy density $w$ is given by
64 w=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)},
65 \qquad \beta=\frac{1}{k_{\text{B}}T}
67 in which $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$
68 is the volume elemnt in crystal phase space.
69 ${\bf u}({\bf R})$ and ${\bf P}({\bf R})$ are the 3N canonical coordinates
70 (here: deviations from equlibrium sites)
71 and 3N canonical momenta
72 of the ions whose equlibrium sites are ${\bf R}$.
74 \item Show that the energy density can be rewritten to read:
76 u=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
78 \item Show that the potential contribution to the energy
79 in the harmonic approximation is given by
81 U&=&U_{\text{eq}}+U_{\text{harm}} \nonumber \\
82 U_{\text{eq}}&=&\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'})
84 U_{\text{harm}}&=&\frac{1}{4}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
85 [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'})
86 [u_v({\bf R})-u_v({\bf R'})] \nonumber
89 $\Phi_{\mu v}({\bf r})=
90 \frac{\partial^2 \Phi({\bf r})}{\partial r_{\mu}\partial r_v}$
91 and $\Phi({\bf r})$ is the potential contribution of two atoms
92 separated by ${\bf r}$.
94 Write down the potential energy for the instantaneous positions
95 ${\bf r}({\bf R})$, with ${\bf u}({\bf R})={\bf r}({\bf R})-{\bf R}$.
96 Apply Taylor approximation to $\Phi({\bf r}+{\bf a})$ with
97 ${\bf r}={\bf R}-{\bf R'}$ and
98 ${\bf a}={\bf u}({\bf R})-{\bf u}({\bf R'})$
99 and only retain terms quadratic in $u$.
100 \item Use the evaluated potential to calculate the energy density
101 (do not forget the kinetic contribution to energy) and
102 the specific heat $c_{\text{V}}$.
104 Use the following change of variables
106 {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad
107 {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R})
109 to extract the temperature dependence of the integral.
110 Does this also work for anharmonic terms?
111 Which parts of the integral do not contribute to $w$ and why?
115 \section{Specific heat in the quantum theory of the harmonic crystal -\\
116 Models of Debye and Einstein}