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\sect{Heat capacity of solids}

Consider a piece of solid whose low laying 
excitations are bosonic modes that have 
spectral density $g(\omega)=C\omega^{\alpha-1}$
up to a cutoff frequency ${\omega_c}$, 
as in the well-known Debye model (items 1-5). 
Similar description applies for magnetic materials (item 6).
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In items 7-8 assume that the solid is a "glass", 
whose low laying excitations are like two level entities 
that have a spectral density $g(\omega)$. 


\begin{itemize}

\item[(1)] 
Write a general expression for the energy $E(T)$ of the system.
This expression may involve a numerical prefactor that is defined 
by an $\alpha$ dependent definite integral.   

\item[(2)] 
Write a general expression for the heat capacity $C(T)$.

\item[(3)] 
Write a general expression for the variance $\text{Var}(r)$ 
of an atom that reside inside the solid.

\item[(4)]
Determine what are $\alpha$ and $C$ and $\omega_c$ 
for a piece of solid that consists of $N$ atoms that occupy a 
volume $L^d$ in $d=1,2,3$ dimensions, 
assuming a dispersion relation $\omega=c|k|$, as for "phonons".  

\item[(5)]
Write explicitly what are $C(T)$ and $\text{Var}(r)$ 
for ${d=1,2,3}$. Be careful with the evaluation of $\text{Var}(r)$.  
In all cases consider both low temperatures ($T\ll \omega_c$), 
and high temperatures ($T\gg \omega_c$). 

\item[(6)]
Point out what would be $\alpha$ if the low laying excitations 
had a dispersion relation $\omega=a|k|^2$ as for "magnons".   

\item[(7)]
What is the heat capacity of a "glass" whose two level 
entities have excitation energies $\omega = \Delta$,
where $\Delta$ has a uniform distribution with density $C$.    
 
\item[(8)]
What is the heat capacity of a "glass" whose two level 
entities have excitation energies $\omega = \omega_c \exp(-\Delta)$, 
where that barrier $\Delta>0$ has a uniform distribution with density $D$.   

\end{itemize}