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34 {\LARGE {\bf Materials Physics II}\\}
39 {\Large\bf Tutorial 3 - proposed solutions}
44 \section{Specific heat in the classical theory of the harmonic crystal -\\
45 The law of Dulong and Petit}
50 w&=&-\frac{1}{V}\frac{\partial}{\partial \beta}
51 ln \int d\Gamma \exp(-\beta H)
52 =-\frac{1}{V}\frac{1}{\int d\Gamma \exp(-\beta H)}
53 \frac{\partial}{\partial \beta} \int d\Gamma \exp(-\beta H)\nonumber\\
54 &=&-\frac{1}{V}\frac{1}{\int d\Gamma \exp(-\beta H)}
55 \int d\Gamma \frac{\partial}{\partial \beta} \exp(-\beta H)\nonumber\\
56 &=&-\frac{1}{V}\frac{1}{\int d\Gamma \exp(-\beta H)}
57 \int d\Gamma \exp(-\beta H) (-H) \qquad \textrm{ q.e.d.} \nonumber
59 \item Potential energy:
61 U=\frac{1}{2}\sum_{{\bf RR'}}\Phi({\bf r}({\bf R})-{\bf r}({\bf R'}))
62 =\frac{1}{2}\sum_{{\bf RR'}}
63 \Phi({\bf R}-{\bf R'}+{\bf u}({\bf R})-{\bf u}({\bf R'}))
66 $U_{\text{eq}}=\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'})$:
69 \frac{1}{2}\sum_{{\bf RR'}}({\bf u}({\bf R})-{\bf u}({\bf R'}))
70 \nabla\Phi({\bf R}-{\bf R'})+
71 \frac{1}{4}\sum_{{\bf RR'}}
72 [({\bf u}({\bf R})-{\bf u}({\bf R'})) \nabla]^2
73 \Phi({\bf R}-{\bf R'}) + \mathcal{O}(u^3)
76 The coefficient of ${\bf u}({\bf R})$ is
77 $\sum_{\bf R'}\nabla\Phi({\bf R}-{\bf R'})$
78 which is minus the force excerted on atom ${\bf R}$
79 by all other atoms in equlibrium positions.
80 There is no net force on any atom in equlibrium.
81 The linear term is zero.\\\\
85 a\nabla \sum_u a_u \frac{\partial\Phi}{\partial r_u}=
86 \sum_v \frac{\partial \sum_u a_u
87 \frac{\partial\Phi}{\partial r_u}}{\partial r_v} a_v=
88 \sum_{uv}\frac{\partial}{\partial r_v} a_u
89 \frac{\partial \Phi}{\partial r_u} a_v=
90 \sum_{uv}a_u \frac{\partial^2\Phi}{\partial r_u \partial r_v} a_v$\\
92 U_{\text{harm}}=\frac{1}{4}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
93 [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'})
94 [u_v({\bf R})-u_v({\bf R'})],
95 \quad \Phi_{\mu v}({\bf r})=
96 \frac{\partial^2 \Phi({\bf r})}{\partial r_{\mu}\partial r_v}.
98 \item Change of variables:
100 {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad
101 {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R})
105 d{\bf u}({\bf R})=\beta^{-3/2}d\bar{{\bf u}}({\bf R}), \qquad
106 d{\bf P}({\bf R})=\beta^{-3/2}d\bar{{\bf P}}({\bf R}), \qquad
108 Kinetic energy contribution:
110 H_{\text{kin}}=\frac{{\bf P}({\bf R})^2}{2M}
112 Integral (using change of variables):
114 \int d\Gamma \exp(-\beta H)&=&
115 \int d\Gamma \exp\left[-\beta\left(\sum \frac{{\bf P}({\bf R})^2}{2M}+
116 U_{\text{eq}} + U_{\text{harm}}\right)\right]\nonumber\\
118 \exp(-\beta U_{\text{eq}})\beta^{-3N}
119 \LARGE(\int\prod_{{\bf R}}d\bar{{\bf u}}({\bf R})d\bar{{\bf P}}({\bf R})
122 -\sum\frac{1}{2M}{\bf P}({\bf R})^2
124 [\bar{u}_{\mu}({\bf R})-\bar{u}_{\mu}({\bf R'})]
125 \Phi_{\mu v}({\bf R}-{\bf R'})
126 [\bar{u}_v({\bf R})-\bar{u}_v({\bf R'})]
127 \right]\LARGE)\nonumber
130 \Rightarrow w=-\frac{1}{V}\frac{\partial}{\partial \beta}
131 ln\left((\exp(-\beta U_{\text{eq}})\beta^{-3N} \times \text{const}
133 =\frac{U_{\text{eq}}}{V}+3\frac{N}{V}k_{\text{B}}T
134 =u_{\text{eq}}+3nk_{\text{B}}T
138 c_{\text{V}}=\frac{\partial w}{\partial T}=3nk_{\text{B}}
142 \section{Specific heat in the quantum theory of the harmonic crystal -\\
146 w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
149 \item Energy: $\rightarrow$ 1(a)
151 w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i).
153 \item Evaluate the expression of the energy density.
155 The energy levels of a harmonic crystal of N ions
156 can be regarded as 3N independent oscillators,
157 whose frequencies are those of the 3N classical normal modes.
158 The contribution to the total energy of a particular normal mode
159 with angular frequency $\omega_s({\bf k})$
160 ($s$: branch, ${\bf k}$: wave vector) is given by
161 $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$ with the
162 excitation number $n_{{\bf k}s}$ being restricted to integers greater
164 The total energy is given by the sum over the energies of the individual
166 Use the totals formula of the geometric series to expcitly calculate
167 the sum of the exponential functions.
168 \item Separate the above result into a term vanishing as $T$ goes to zero and
169 a second term giving the energy of the zero-point vibrations of the
171 \item Write down an expression for the specific heat.
172 Consider a large crystal and thus replace the sum over the discrete
173 wave vectors with an integral.
174 \item Debye replaced all branches of the vibrational spectrum with three
175 branches, each of them obeying the dispersion relation
177 Additionally the integral is cut-off at a radius $k_{\text{D}}$
178 to have a total amount of N allowed wave vectors.
179 Determine $k_{\text{D}}$.
180 Evaluate the simplified integral and introduce the
181 Debye frequency $\omega_{\text{D}}=k_{\text{D}}c$
182 and the Debye temperature $\Theta_{\text{D}}$ which is given by
183 $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
184 Write down the resulting expression for the specific heat.