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29
30 \begin{document}
31
32 % header
33 \begin{center}
34  {\LARGE {\bf Materials Physics II}\\}
35  \vspace{8pt}
36  Prof. B. Stritzker\\
37  SS 2008\\
38  \vspace{8pt}
39  {\Large\bf Tutorial 3 - proposed solutions}
40 \end{center}
41
42 \vspace{8pt}
43
44 \section{Specific heat in the classical theory of the harmonic crystal -\\
45          The law of Dulong and Petit}
46
47 \begin{enumerate}
48  \item Energy:
49        \begin{eqnarray}
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
58        \end{eqnarray}
59  \item Potential energy:
60        \[
61        U=\frac{1}{2}\sum_{{\bf RR'}}\Phi({\bf r}({\bf R})-{\bf r}({\bf R'}))
62         =\frac{1}{2}\sum_{{\bf RR'}}
63          \Phi({\bf R}-{\bf R'}+{\bf u}({\bf R})-{\bf u}({\bf R'}))
64        \]
65        Using Taylor and
66        $U_{\text{eq}}=\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'})$:
67        \[
68        U=U_{\text{eq}}+
69          \frac{1}{2}\sum_{{\bf RR'}}({\bf u}({\bf R})-{\bf u}({\bf R'}))
70          \nabla\Phi({\bf R}-{\bf R'})+
71          \frac{1}{4}\sum_{{\bf RR'}}
72          [({\bf u}({\bf R})-{\bf u}({\bf R'})) \nabla]^2
73          \Phi({\bf R}-{\bf R'}) + \mathcal{O}(u^3)
74        \]
75        Linear term:\\
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.\\\\
82        Harmonic term:\\
83        $(a\nabla)^2 \Phi=
84         a\nabla a\nabla \Phi=
85         a\nabla \sum_u a_u \frac{\partial\Phi}{\partial r_u}=
86         \sum_v \frac{\partial \sum_u a_u
87         \frac{\partial\Phi}{\partial r_u}}{\partial r_v} a_v=
88         \sum_{uv}\frac{\partial}{\partial r_v} a_u
89         \frac{\partial \Phi}{\partial r_u} a_v=
90         \sum_{uv}a_u \frac{\partial^2\Phi}{\partial r_u \partial r_v} a_v$\\
91        \[\Rightarrow
92        U_{\text{harm}}=\frac{1}{4}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
93        [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'})
94        [u_v({\bf R})-u_v({\bf R'})],
95        \quad \Phi_{\mu v}({\bf r})=
96         \frac{\partial^2 \Phi({\bf r})}{\partial r_{\mu}\partial r_v}.
97        \]
98  \item Change of variables:
99        \[
100        {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad
101        {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R})
102        \]
103        \[
104        \Rightarrow
105        d{\bf u}({\bf R})=\beta^{-3/2}d\bar{{\bf u}}({\bf R}), \qquad
106        d{\bf P}({\bf R})=\beta^{-3/2}d\bar{{\bf P}}({\bf R}), \qquad
107        \]
108        Kinetic energy contribution:
109        \[
110        H_{\text{kin}}=\frac{{\bf P}({\bf R})^2}{2M}
111        \]
112        Integral (using change of variables):
113        \begin{eqnarray}
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\\
117        &=&
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})
120        \nonumber\\
121        &&\times \exp\left[
122        -\sum\frac{1}{2M}\bar{{\bf P}}({\bf R})^2
123        -\frac{1}{4}\sum
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
128        \end{eqnarray}
129        \[
130        \Rightarrow w=-\frac{1}{V}\frac{\partial}{\partial \beta}
131        ln\left((\exp(-\beta U_{\text{eq}})\beta^{-3N} \times \text{const}
132        \right)
133        =\frac{U_{\text{eq}}}{V}+3\frac{N}{V}k_{\text{B}}T
134        =u_{\text{eq}}+3nk_{\text{B}}T
135        \]
136        \[
137        \Rightarrow
138        c_{\text{V}}=\frac{\partial w}{\partial T}=3nk_{\text{B}}
139        \]
140 \end{enumerate}
141
142 \section{Specific heat in the quantum theory of the harmonic crystal -\\
143          The Debye model}
144
145 \[
146 w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
147 \]
148 \begin{enumerate}
149  \item Energy: $\rightarrow$ 1(a)
150        \[
151    w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i).
152        \]
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
160               normal modes:\\
161               $E=\sum_{{\bf k}s}(n_{{\bf k}s}+
162                \frac{1}{2})\hbar\omega_s({\bf k})$
163        \end{itemize}
164        \begin{eqnarray}
165        \Rightarrow
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)+
170                         \ldots)
171        \right)\nonumber\\
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\\
181        &&\times
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})}}
198        \nonumber\\
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
212        \end{eqnarray}
213        $n_s({\bf k})$: Mean excitation number of the normal mode ${\bf k}s$ at
214                        temperature $T$.
215
216  \item \[
217        w=w_{\text{eq}}+
218          \frac{1}{V}\sum_{{\bf k}s}\frac{1}{2}\hbar\omega_s({\bf k})+
219          \frac{1}{V}\sum_{{\bf k}s}
220          \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
221        \]
222  \item \[
223        c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}\frac{\partial}{\partial T}
224        \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
225        \]
226        Large crystal:
227        \[
228        \lim_{v\rightarrow\infty}c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}
229        \frac{\partial}{\partial T}
230        \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
231        =\frac{\partial}{\partial T}
232        \sum_s\int\frac{d{\bf k}}{(2\pi)^3}
233        \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
234        \]
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               $\Theta_{\text{D}}^3=\frac{\hbar^3c^3k_{\text{D}}^3}
248               {k_{\text{B}}^3}=
249               \frac{\hbar^3c^3}{k_{\text{B}}^3}6\pi^2n$
250        \end{itemize}
251        Integral:
252        \[
253        c_{\text{V}}=\frac{\partial}{\partial T}\, {\color{red}3}\int_0^{k_D}
254        {\color{blue}\frac{k^2}{2\pi^2}dk} \frac{\hbar {\color{green}ck}}
255        {e^{\beta\hbar {\color{green}ck}}-1}=
256        \frac{\partial}{\partial T}\frac{3\hbar c}{2\pi^2}\int_0^{k_D}
257        \frac{k^3}{e^{\beta\hbar ck}-1}dk=
258        \frac{3\hbar c}{2\pi^2}\int_0^{k_D}
259        \frac{k^3e^{\beta\hbar ck}\beta\hbar ck\frac{1}{T}}
260        {(e^{\beta\hbar ck}-1)^2}dk
261        \]
262        Change of variables: $\beta\hbar ck=x$
263        \[
264        \Rightarrow
265        k=\frac{x}{\beta\hbar c} \quad \textrm{, } \quad
266        dk=\frac{1}{\beta\hbar c} dx
267        \]
268        \[
269        c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T}
270        \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}=
271        \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T}
272        \frac{x^4e^x}{(e^x-1)^2}dx
273        \]
274        \[
275        \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}=
276        \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}=
277        \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3}
278        \]
279        \[
280        \Rightarrow
281        c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}}
282        \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx
283        \]
284 \end{enumerate}
285
286 \end{document}