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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 internal energy density of the system.
52 In the following the contribution to the specific heat due to the
53 degrees of freedom of the lattice ions is calculated.
55 \section{Specific heat in the classical theory of the harmonic crystal -\\
56 The law of Dulong and Petit}
58 In the classical theory of the harmonic crystal equilibrium properties
59 can no longer be evaluated by simply assuming that each ion sits quietly at
60 its Bravais lattice site {\bf R}.
61 From now on expectation values have to be claculated by
62 integrating over all possible ionic configurations weighted by
63 $\exp(-E/k_{\text{B}}T)$, where $E$ is the energy of the configuration.
64 Thus, the energy density $w$ is given by
66 w=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)},
67 \qquad \beta=\frac{1}{k_{\text{B}}T}
69 in which $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$
70 is the volume elemnt in crystal phase space.
71 ${\bf u}({\bf R})$ and ${\bf P}({\bf R})$ are the 3N canonical coordinates
72 (here: deviations from equlibrium sites)
73 and 3N canonical momenta
74 of the ions whose equlibrium sites are ${\bf R}$.
76 \item Show that the energy density can be rewritten to read:
78 w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
80 \item Show that the potential contribution to the energy
81 in the harmonic approximation is given by
83 U&=&U_{\text{eq}}+U_{\text{harm}} \nonumber \\
84 U_{\text{eq}}&=&\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'})
86 U_{\text{harm}}&=&\frac{1}{4}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
87 [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'})
88 [u_v({\bf R})-u_v({\bf R'})] \nonumber
91 $\Phi_{\mu v}({\bf r})=
92 \frac{\partial^2 \Phi({\bf r})}{\partial r_{\mu}\partial r_v}$
93 and $\Phi({\bf r})$ is the potential contribution of two atoms
94 separated by ${\bf r}$.
96 Write down the potential energy for the instantaneous positions
97 ${\bf r}({\bf R})$, with ${\bf u}({\bf R})={\bf r}({\bf R})-{\bf R}$.
98 Apply Taylor approximation to $\Phi({\bf r}+{\bf a})$ with
99 ${\bf r}={\bf R}-{\bf R'}$ and
100 ${\bf a}={\bf u}({\bf R})-{\bf u}({\bf R'})$
101 and only retain terms quadratic in $u$.
102 \item Use the evaluated potential to calculate the energy density
103 (do not forget the kinetic energy contribution) and
104 the specific heat $c_{\text{V}}$.
106 Use the following change of variables
108 {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad
109 {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R})
111 to extract the temperature dependence of the integral.
112 Does this also work for anharmonic terms?
113 Which parts of the integral do not contribute to $w$ and why?
116 \section{Specific heat in the quantum theory of the harmonic crystal -\\
119 As found in exercise 1, the specific heat of a classical harmonic crystal
120 is not depending on temeprature.
121 However, as temperature drops below room temperature
122 the specific heat of all solids is decreasing as $T^3$ in insulators
123 and $AT+BT^3$ in metals.
124 This can be explained in a quantum theory of the specific heat of
125 a harmonic crystal, in which the energy density $w$ is given by
127 w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
130 \item Show that the energy density can be rewritten to read:
132 w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i).
134 \item Evaluate the expression of the energy density.
136 The energy levels of a harmonic crystal of N ions
137 can be regarded as 3N independent oscillators,
138 whose frequencies are those of the 3N classical normal modes.
139 The contribution to the total energy of a particular normal mode
140 with angular frequency $\omega_s({\bf k})$
141 ($s$: branch, ${\bf k}$: wave vector) is given by
142 $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$ with the
143 excitation number $n_{{\bf k}s}$ being restricted to integers greater
145 The total energy is given by the sum over the energies of the individual
147 Use the totals formula of the geometric series to expcitly calculate
148 the sum of the exponential functions.
149 \item Separate the above result into a term vanishing as $T$ goes to zero and
150 a second term giving the energy of the zero-point vibrations of the
152 \item Write down an expression for the specific heat.
153 Consider a large crystal and thus replace the sum over the discrete
154 wave vectors with an integral.
155 \item Debye replaced all branches of the vibrational spectrum with three
156 branches, each of them obeying the dispersion relation
158 Additionally the integral is cut-off at a radius $k_{\text{D}}$
159 to have a total amount of N allowed wave vectors.
160 Determine $k_{\text{D}}$.
161 Evaluate the simplified integral and introduce the
162 Debye frequency $\omega_{\text{D}}=k_{\text{D}}c$
163 and the Debye temperature $\Theta_{\text{D}}$ which is given by
164 $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
165 Write down the resulting expression for the specific heat.