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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 \begin{itemize}
154 \item Total energy contribution of a particular normal mode:
155 $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$
156 with $n_{{\bf k}s}=0,1,2,\ldots$
157 \item A state of the crystal is specified by the excitation numbers
158 of the 3N normal modes.
159 \item The total energy is the sum of the energies of the individual
161 $E=\sum_{{\bf k}s}(n_{{\bf k}s}+
162 \frac{1}{2})\hbar\omega_s({\bf k})$
166 w&=&-\frac{1}{V}\frac{\partial}{\partial \beta} ln\left(
167 \prod_{{\bf k}s}(\exp(-\beta\hbar\omega_s({\bf k})/2)+
168 \exp(-3\beta\hbar\omega_s({\bf k})/2)+
169 \exp(-5\beta\hbar\omega_s({\bf k})/2)+
172 &=&-\frac{1}{V}\frac{\partial}{\partial \beta} ln \prod_{{\bf k}s}
173 \frac{\exp(-\beta\hbar\omega_s({\bf k})/2)}
174 {1-\exp(-\beta\hbar\omega_s({\bf k}))}\nonumber
177 \item Evaluate the expression of the energy density.
179 The energy levels of a harmonic crystal of N ions
180 can be regarded as 3N independent oscillators,
181 whose frequencies are those of the 3N classical normal modes.
182 The contribution to the total energy of a particular normal mode
183 with angular frequency $\omega_s({\bf k})$
184 ($s$: branch, ${\bf k}$: wave vector) is given by
185 $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$ with the
186 excitation number $n_{{\bf k}s}$ being restricted to integers greater
188 The total energy is given by the sum over the energies of the individual
190 Use the totals formula of the geometric series to expcitly calculate
191 the sum of the exponential functions.
192 \item Separate the above result into a term vanishing as $T$ goes to zero and
193 a second term giving the energy of the zero-point vibrations of the
195 \item Write down an expression for the specific heat.
196 Consider a large crystal and thus replace the sum over the discrete
197 wave vectors with an integral.
198 \item Debye replaced all branches of the vibrational spectrum with three
199 branches, each of them obeying the dispersion relation
201 Additionally the integral is cut-off at a radius $k_{\text{D}}$
202 to have a total amount of N allowed wave vectors.
203 Determine $k_{\text{D}}$.
204 Evaluate the simplified integral and introduce the
205 Debye frequency $\omega_{\text{D}}=k_{\text{D}}c$
206 and the Debye temperature $\Theta_{\text{D}}$ which is given by
207 $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
208 Write down the resulting expression for the specific heat.