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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}\bar{{\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\\
175 &=&-\frac{1}{V}\frac{\partial}{\partial \beta} \sum_{{\bf k}s} ln
176 \frac{\exp(-\beta\hbar\omega_s({\bf k})/2)}
177 {1-\exp(-\beta\hbar\omega_s({\bf k}))}\nonumber\\
178 &=&-\frac{1}{V}\sum_{{\bf k}s}
179 \frac{1-\exp(-\beta\hbar\omega_s({\bf k}))}
180 {\exp(-\beta\hbar\omega_s({\bf k})/2)}\nonumber\\
182 \frac{(1-e^{-\beta\hbar\omega_s({\bf k})})
183 e^{-\beta\hbar\omega_s({\bf k})/2}(-\hbar\omega_s({\bf k})/2)+
184 e^{-\beta\hbar\omega_s({\bf k})/2}
185 e^{-\beta\hbar\omega_s({\bf k})}(-\hbar\omega_s({\bf k}))}
186 {(1-e^{-\beta\hbar\omega_s({\bf k})})^2}\nonumber\\
187 &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
188 \frac{{\color{red}-}e^{-\beta\hbar\omega_s({\bf k})}-
189 \frac{1}{2}(1-e^{-\beta\hbar\omega_s({\bf k})})}
190 {1-e^{-\beta\hbar\omega_s({\bf k})}}\nonumber\\
191 &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
192 \frac{{\color{red}-}\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}}
193 {1-e^{-\beta\hbar\omega_s({\bf k})}}
194 =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
195 \frac{e^{-\beta\hbar\omega_s({\bf k})}+1}
196 {1-e^{-\beta\hbar\omega_s({\bf k})}}\cdot
197 \frac{e^{\beta\hbar\omega_s({\bf k})}}{e^{\beta\hbar\omega_s({\bf k})}}
199 &=&\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
200 \frac{1+e^{\beta\hbar\omega_s({\bf k})}}
201 {e^{\beta\hbar\omega_s({\bf k})}-1}
202 =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
203 \frac{2+e^{\beta\hbar\omega_s({\bf k})}-1}
204 {e^{\beta\hbar\omega_s({\bf k})}-1}\nonumber\\
205 &=&\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
206 (\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}
207 +\frac{e^{\beta\hbar\omega_s({\bf k})}-1}
208 {2(e^{\beta\hbar\omega_s({\bf k})}-1)})
209 =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
210 (\underbrace{\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}}_{n_s({\bf k})}
211 +\frac{1}{2})\nonumber
213 $n_s({\bf k})$: Mean excitation number of the normal mode ${\bf k}s$ at
218 \frac{1}{V}\sum_{{\bf k}s}\frac{1}{2}\hbar\omega_s({\bf k})+
219 \frac{1}{V}\sum_{{\bf k}s}
220 \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
223 c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}\frac{\partial}{\partial T}
224 \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
228 \lim_{v\rightarrow\infty}c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}
229 \frac{\partial}{\partial T}
230 \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
231 =\frac{\partial}{\partial T}
232 \sum_s\int\frac{d{\bf k}}{(2\pi)^3}
233 \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
235 \item \begin{itemize}
236 \item {\color{red}3} branches with Debye dispersion relation
237 $w={\color{green}ck}$
238 \item Volume of $k$-space per wave vector: $(2\pi)^3/V$\\
239 $\Rightarrow (2\pi)^3N/V=4\pi k_{\text{D}}^3/3
240 \Rightarrow n=\frac{N}{V}=\frac{k_{\text{D}}^3}{6\pi^2}$,
241 $k_{\text{D}}^3=6\pi^2 n$
242 \item $dn={\color{blue}\frac{k^2}{2\pi^2}dk}$
243 \item Debye frequency: $\omega_{\text{D}}=k_{\text{D}}c$
244 \item Debye temperature:
245 $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$,
246 $\Theta_{\text{D}}=\hbar ck_{\text{D}}/k_{\text{B}}$,
247 $\Theta_{\text{D}}^3=\frac{\hbar^3c^3k_{\text{D}}^3}
249 \frac{\hbar^3c^3}{k_{\text{B}}^3}6\pi^2n$
253 c_{\text{V}}=\frac{\partial}{\partial T}\, {\color{red}3}\int_0^{k_D}
254 {\color{blue}\frac{k^2}{2\pi^2}dk} \frac{\hbar {\color{green}ck}}
255 {e^{\beta\hbar {\color{green}ck}}-1}=
256 \frac{\partial}{\partial T}\frac{3\hbar c}{2\pi^2}\int_0^{k_D}
257 \frac{k^3}{e^{\beta\hbar ck}-1}dk=
258 \frac{3\hbar c}{2\pi^2}\int_0^{k_D}
259 \frac{k^3e^{\beta\hbar ck}\beta\hbar ck\frac{1}{T}}
260 {(e^{\beta\hbar ck}-1)^2}dk
262 Change of variables: $\beta\hbar ck=x$
265 k=\frac{x}{\beta\hbar c} \quad \textrm{, } \quad
266 dk=\frac{1}{\beta\hbar c} dx
269 c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T}
270 \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}=
271 \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T}
272 \frac{x^4e^x}{(e^x-1)^2}dx
275 \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}=
276 \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}=
277 \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3}
281 c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}}
282 \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx