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\usepackage[OT2,OT1,T1]{fontenc}
\usepackage[authoryear,sectionbib]{natbib}
\usepackage{makeidx}
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\usepackage{xspace}
\DeclareMathAlphabet\mathbfcal{OMS}{cmsy}{b}{n}

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% Definitions added in CE for 3rd edn:
\newcommand{\e}{\mathrm{e}}         % for Euler's number
\newcommand{\dd}{\mathrm{d}}       % for differential operator
\newcommand{\dV}{\dd V}                % for differential volume
\newcommand{\im}{\mathrm{i}}         % for imaginary unit
\DeclareMathOperator{\grad}{grad}   % gradient operator
\DeclareMathOperator{\curl}{curl}      % curl operator
\DeclareMathOperator{\diverg}{div}   % divergence operator
\newcommand{\Laplacian}{\Delta}     % Laplace operator
\newcommand{\muB}{\ensuremath{\mu_\mathrm{B}}\xspace}           % Bohr magneton
\newcommand{\kB}{\ensuremath{k_\mathrm{B}}\xspace}                   % Boltzmann constant
\newcommand{\Cur}{\mathrm{C}}                                                      % abbr. for Curie
\newcommand{\CurT}{\ensuremath{\varTheta_\Cur}\xspace}             % Curie temperature
\newcommand{\Deb}{\mathrm{D}}                                                     % abbr. for Debye
\newcommand{\DebT}{\ensuremath{\varTheta_\Deb}\xspace}            % Debye temperature
\newcommand{\DebF}{\ensuremath{\omega_\Deb}\xspace}               % Debye frequency
\newcommand{\Einst}{\mathrm{E}}                                                     % abbr. for Einstein
\newcommand{\Fermi}{\mathrm{F}}                                                    % abbr. for Fermi
\newcommand{\EF}{\ensuremath{E_\Fermi}\xspace}                            % Fermi energy
\newcommand{\gap}{\mathrm{g}}                                                       % abbr. for gap
\newcommand{\Eg}{\ensuremath{E_\gap}\xspace}                               % gap energy
\newcommand{\con}{\mathrm{c}}                                                        % abbr. for conduction (band)
\newcommand{\val}{\mathrm{v}}                                                         % abbr. for valence (band)
\newcommand{\crit}{\mathrm{c}}                                                         % abbr. for critical
\newcommand{\Tc}{\ensuremath{T_\crit}\xspace}                                % critical Temperature
\newcommand{\vecr}{\ensuremath{\mathbf{r}}\xspace}                       % vector r
\newcommand{\vecR}{\ensuremath{\mathbf{R}}\xspace}                     % vector R
\newcommand{\veck}{\ensuremath{\mathbf{k}}\xspace}                      % vector k
\newcommand{\el}{\mathrm{e}}                                                           % abbr. for 'electron'
\newcommand{\mel}{\ensuremath{m_\el}\xspace}                               % electron mass
\newcommand{\chim}{\ensuremath{\chi_\mathrm{m}}\xspace}            % magnetic susceptibilty
\newcommand{\BW}{\ensuremath{\mathbf{B}_\mathrm{W}}\xspace}   % Weiss field
% End definitions added CE for 3rd edn

\begin{document}
%\addtolength{\voffset}{-2.0cm}
%\tableofcontents
%================================================================== 

\section{Crystal Structures}

 \begin{equation}
      \vecR=m\mathbf{a_{1}}+n\mathbf{a_{2}},
  \end{equation}
  \begin{equation}
    \vecR=m\mathbf{a_{1}}+n\mathbf{a_{2}}+o\mathbf{a_{3}}.
  \end{equation}
\begin{equation}\label{equ:bragg1}
n\lambda = 2d \sin \theta.
\end{equation}
\begin{equation}
 \mathbfcal{E}(\vecr,t) = \mathbfcal{E}_0 \:\e^{\im\veck \cdot \vecr-\im\omega t}.
\end{equation}
\begin{equation}
  I(\vecr) = \left|\mathbfcal{E}_0 \:\e^{\im\veck  \cdot \vecr-\im\omega t}\right|^2
                        = \left|\mathbfcal{E}_0\right|^2.
\end{equation}
\begin{equation}
 \mathbfcal{E}(\vecr,t) = \mathbfcal{E}_0\:\e^{\im\veck \cdot (\vecr- \vecR)-\im\omega t}.
\end{equation}
\begin{equation}\label{equ:amplitudeattheposition}
 \mathbfcal{E}(\vecr,t) \propto  \e^{\im\veck \cdot (\vecr-
\vecR)} \: \e^{-\im\omega t}.
\end{equation}
\begin{equation}
 \mathbfcal{E}(\vecR',t) \propto \mathbfcal{E}(\vecr,t) \rho(\vecr) \:\e^{\im \veck' \cdot (\mathbf{R'} - \vecr)}.
\end{equation}
\begin{equation}
\mathbfcal{E}(\vecR',t) \propto  \e^{\im\veck \cdot
(\vecr- \vecR)} \rho(\vecr) \: \e^{\im \veck' \cdot
(\mathbf{R'}-\vecr)} \: \e^{-\im\omega t} =  \e^{\im(\veck' \cdot
\mathbf{R'}-\veck  \cdot \vecR)} \rho(\vecr)\: \e^{\im
(\veck-\veck') \cdot \vecr} \: \e^{-\im\omega t}.
\end{equation}
\begin{equation}
 \mathbfcal{E}(\vecR',t) \propto \e^{-\im\omega t} \int_V \rho(\vecr) \: \e^{\im (\veck-\veck') \cdot \vecr}\:\dV.
\end{equation}
\begin{equation}\label{equ:diffractedintensity}
I(\mathbf{K}) \propto \left| \e^{-\im\omega t} \int_V \rho(\vecr)
\:\e^{\im (\veck-\veck') \cdot \vecr} \:\dV \right|^{2} =
\left| \int_V \rho(\vecr) \: \e^{-\im \mathbf{K} \cdot \vecr}
\:\dV \right|^{2},
\end{equation}
  \begin{equation}
    \vecR=m\mathbf{a_{1}}+n\mathbf{a_{2}}+o\mathbf{a_{3}},
  \end{equation}
\begin{equation}\label{equ:laue1}
\vecR  \cdot \mathbf{G}=2\uppi\, l,
\end{equation}
  \begin{equation}\label{equ:definepropreclatt}
\e^{\mathbf{\im G} \cdot \vecR}=1.
  \end{equation}
  \begin{equation}\label{equ:definitionofthereciprocallattice}
    \mathbf{G}=m'\,\mathbf{b_{1}}+n'\,\mathbf{b_{2}}+o'\,\mathbf{b_{3}},
  \end{equation}
  \begin{equation}\label{equ:explicitreclatvec}
     \mathbf{b}_{1}=2 \uppi \frac{\mathbf{a}_{2} \times  \mathbf{a}_{3}}{\mathbf{a}_{1}\cdot (\mathbf{a}_{2} \times  \mathbf{a}_{3})},  \quad
      \mathbf{b}_{2}=2 \uppi \frac{\mathbf{a}_{3} \times  \mathbf{a}_{1}}{\mathbf{a}_{1}\cdot (\mathbf{a}_{2} \times  \mathbf{a}_{3})},  \quad
      \mathbf{b}_{3}=2 \uppi \frac{\mathbf{a}_{1} \times  \mathbf{a}_{2}}{\mathbf{a}_{1}\cdot (\mathbf{a}_{2} \times  \mathbf{a}_{3})}.
   \end{equation}
  \begin{equation}\label{equ:perp}
     \mathbf{a}_{i} \cdot  \mathbf{b}_{j}=2 \uppi\, \updelta_{ij},
   \end{equation}
\begin{equation}
  \e^{\im\mathbf{G} \cdot \vecr} =
  \e^{\im\mathbf{G} \cdot \vecr} \:\e^{\im\mathbf{G} \cdot \vecR} =
  \e^{\im\mathbf{G} \cdot (\vecr + \vecR)}.
\end{equation}
\begin{equation}\label{equ:fu1}
\rho(x)=C+ \sum_{n=1}^{\infty} \Big\{ C_n \cos(x \, 2\uppi\, n /a)+S_n
\sin(x\, 2\uppi\, n /a)    \Big\}
\end{equation}
\begin{equation}\label{equ:fu1a}
\rho(x)=\sum_{n=-\infty}^{\infty} \rho_n\:\e^{\im x n \,2\uppi /a}.
\end{equation}
\begin{equation}\label{equ:coesym}
 \rho_{-n}^*=\rho_n,
\end{equation}
\begin{equation}\label{equ:reclattice1d}
g=n\frac{2\uppi}{a},
\end{equation}
\begin{equation}\label{equ:latticeperiodicchargedensity}
\rho(\vecr)=\sum_{\mathbf{G}}^{}\rho_{\mathbf{G}}\:\e^{\im\mathbf{G}
\cdot\vecr},
\end{equation}
\begin{equation}\label{equ:intensityfordiffractionfromperiodic}
I(\mathbf{K}) \propto  \left| \sum_{\mathbf{G}}   \rho_{\mathbf{G}}
\int_V  \e^{\im (\mathbf{G}-\mathbf{K}) \cdot \vecr} \:\dV \right|^2.
\end{equation}
\begin{equation}\label{equ:laue3}
\mathbf{K}=\veck'-\veck=\mathbf{G},
\end{equation}
\begin{equation}\label{equ:whatinunitcell1}
I(\mathbf{G}) \propto \left| \int_V \rho(\vecr) \:\e^{-\im \mathbf{G} \cdot \vecr} \:\dV \right|^2.
\end{equation}
\begin{equation}\label{equ:whatinunitcell2}
 I(\mathbf{G}) \propto \left|\sum_{\vecR} \int_{V_{\text{cell}}} \rho(\vecr+\vecR) \:\e^{-\im \mathbf{G} \cdot (\vecr+\vecR)} \:\dV \right|^2 =
 \left|N \int_{V_{\text{cell}}} \rho(\vecr) \:\e^{-\im \mathbf{G} \cdot \vecr} \:\dV \right|^2,
\end{equation}
\begin{equation}
  \rho(\vecr) = \sum_i \rho_i(\vecr-\vecr_i),
\end{equation}
\begin{equation}
 \int_{V_\text{cell}} \rho(\vecr)\:\e^{-\im\, \mathbf{G} \cdot \vecr} \:\dV = \sum_i \e^{-\im\, \mathbf{G} \cdot \vecr_i}  \int_{V_{\text{atom}}} \rho_{i}(\vecr^{\prime})\:\e^{-\im \mathbf{G} \cdot \vecr^{\prime}} \:\dV^{\prime},
\end{equation}
\begin{equation}\label{equ:bragglaue}
{{k}^{\prime}}_{\perp}-{k}_{\perp}=2 {k}_{\perp} = 2 \frac{2
\uppi}{\lambda} \sin \varTheta = G_{\perp},
\end{equation}

%================================================================== 

\section{Bonding in Solids}
\begin{equation}\label{equ:21}
 \phi(r)=\frac{A}{r^{n}}-\frac{B}{r^{m}},
\end{equation}
\begin{equation}\label{equ:energygainonemadelung}
E_{\mathrm{Na}}=-1.748\frac{e^{2}}{4\uppi\,\varepsilon_{0}a}=-\alpha\frac{e^{2}}{4\uppi\,\varepsilon_{0}a}.
\end{equation}
\begin{align}
H & =   -\frac{\hbar^2 \nabla^2_1}{2\mel} -\frac{\hbar^2
\nabla^2_2}{2\mel}
 +\frac{e^{2}}{4\uppi\,\varepsilon_{0}} \left\{ \frac{1}{R}
 + \frac{1}{|\vecr_1-\vecr_2|} \right.  \nonumber \\
 &  \left. - \frac{1}{|\vecr_1-\vecR_\text{A}|}-  \frac{1}{|\vecr_2-\vecR_\mathrm{B}|} -  \frac{1}{|\vecr_2-\vecR_\text{A}|}-  \frac{1}{|\vecr_1-\vecR_\mathrm{B}|}\right\}, \label{hyd1}
\end{align}
\begin{equation}\label{hyd_ion}
    H' =  -\frac{\hbar^2 \nabla^2}{2\mel}
     +\frac{e^{2}}{4\uppi\,\varepsilon_{0}} \left\{ \frac{1}{R}-  \frac{1}{|\vecr-\vecR_\text{A}|}-  \frac{1}{|\vecr-\vecR_\mathrm{B}|}\right\}.
\end{equation}
\begin{equation}
    H' \varPsi (\vecr) = H' \big( c_1 \phi_\text{A}(\vecr) + c_2 \phi_\mathrm{B}(\vecr) \big) = E' \big( c_1 \phi_\text{A}(\vecr) + c_2 \phi_\mathrm{B}(\vecr) \big),
\end{equation}
\begin{equation}\label{equ:algeb1}
\begin{aligned}
    c_1 H'_\text{AA} + c_2 H'_\text{AB} &= c_1 E' + c_2 S E'   \\
    c_1 H'_\text{BA} + c_2 H'_\text{BB} &= c_2 E' + c_1 S E',
\end{aligned}
\end{equation}
\begin{equation}\label{equ:algeb2}
\begin{aligned}
    c_1 (H'_\text{AA} - E') + c_2 (H'_\text{AB} - S E') &= 0   \\
    c_1 (H'_\text{AB} - S E') + c_2 (H'_\text{AA} - E') &= 0.
\end{aligned}
\end{equation}
\begin{equation}\label{equ:bondantiH2ion}
    E'_{\pm}=\frac{H'_\text{AA} \pm H'_\text{AB}}{1 \pm S}.
\end{equation}
\begin{align}
\varPsi_{\uparrow \downarrow}(\vecr_1,\vecr_2) &\propto \phi_\text{A}(\vecr_1)\phi_\mathrm{B}(\vecr_2) + \phi_\text{A}(\vecr_2)\phi_\mathrm{B}(\vecr_1), \label{equ:sym}\\
\varPsi_{\uparrow \uparrow}(\vecr_1,\vecr_2) &\propto \phi_\text{A}(\vecr_1)\phi_\mathrm{B}(\vecr_2) - \phi_\text{A}(\vecr_2)\phi_\mathrm{B}(\vecr_1). \label{equ:asym}
\end{align}
\begin{equation}\label{equ:approxcalculationenergylevels}
E=\frac{\int \varPsi^*(\vecr_1,\vecr_2) H  \varPsi(\vecr_1,\vecr_2) \:\dd\vecr_1 \:\dd\vecr_2}{\int \varPsi^*(\vecr_1,\vecr_2)  \varPsi(\vecr_1,\vecr_2) \:\dd\vecr_1 \:\dd\vecr_2}.
\end{equation}
 \begin{align}
E _{\mathrm{singlet}} &= 2 E_0+\Delta E_{\uparrow \downarrow} \label{equ:energysym}, \\
E _{\mathrm{triplet}}  &= 2 E_0+\Delta E_{\uparrow \uparrow}\label{equ:energyasym}.
\end{align}
\begin{equation}\label{equ:heitlerlondonresult}
E=2E_0+C \pm X,
\end{equation}
%================================================================== 
\section{Mechanical Properties}
\begin{equation}\label{equ:definitionofyoungsmodulus}
 Y=\frac{\sigma}{\varepsilon}=\frac{F}{A}\frac{l}{\Delta{l}}.
\end{equation}
\begin{equation}
 \sigma=Y \varepsilon.
\end{equation}
\begin{equation}\label{equ:youngspring}
F= \frac{Y A}{l} \Delta l,
\end{equation}
\begin{equation}\label{equ:modrid}
 G=\frac{\tau}{\alpha}.
\end{equation}
\begin{equation}\label{equ:bulkmodulus}
K=-p \frac{V}{\Delta V},
\end{equation}
\begin{equation}\label{equ:definepoisson}
 \frac{\Delta{l_{2}}}{l_{2}}=\frac{\Delta{l_{3}}}{l_{3}}=-\nu\frac{\Delta{l_{1}}}{l_{1}}=-\nu \varepsilon.
\end{equation}
\begin{equation}
(l_1+\Delta l_1)(l_2+\Delta l_2)(l_3+\Delta l_3).
\end{equation}
\begin{equation}\label{equ:smallvolumechange1}
 l_1 l_2 l_3 + \Delta l_1 l_2 l_3+l_1 \Delta l_2 l_3 + l_1 l_2 \Delta l_3.
\end{equation}
\begin{align}
\Delta l_1 l_2 l_3+l_1 \Delta l_2 l_3 + l_1 l_2 \Delta l_3 &= \Delta l_1 l_2 l_3 + l_1 \bigg(-\nu \frac{\Delta l_1}{l_1} l_2\bigg) l_3+ l_1 l_2 \bigg(-\nu \frac{\Delta l_1}{l_1} l_3\bigg)\nonumber\\
& =(1-2\nu) \Delta l_1 l_2 l_3.\label{equ:smallvolumechange2}
\end{align}
\begin{equation}\label{equ:youngridid}
 G=\frac{Y}{2(1+\nu)}.
 \end{equation}
\begin{equation}\label{equ:taylorpot}
 \phi(x) = \phi(a)+ \frac{\phi'(a)}{1!} (x-a) + \frac{\phi''(a)}{2!}
(x-a)^2 + \frac{\phi'''}{3!}  (x-a)^3 + \dots
\end{equation}
\begin{equation}
 \alpha=\tan^{-1}\left(\frac{x}{a}\right)\approx\frac{x}{a}.
\end{equation}
\begin{equation}\label{equ:yieldstr1}
 \tau=G\alpha\approx\frac{Gx}{a},
\end{equation}
\begin{equation}\label{equ:yieldperiod}
 \tau=C\sin\left(\frac{2\uppi\, x}{b}\right),
\end{equation}
\begin{equation}
C\frac{2\uppi\,x}{b}=  \frac{Gx}{a}
\end{equation}
\begin{equation}\label{equ:stressest}
 C=\tau_\text{y}= \frac{Gb}{2\uppi\, a}.
\end{equation}

%================================================================== 
\section{Thermal Properties of the Lattice}

\begin{equation}\label{equ:oneharm}
 M\frac{\dd^{2}x}{\dd t^{2}}=-\gamma {x},
\end{equation}
\begin{equation}\label{equ:fundamentalvibrequency}
 \omega=\sqrt{\frac{\gamma}{M}}.
\end{equation}
\begin{equation}
 E=\frac{1}{2}Mv^{2}+\frac{1}{2}\gamma{x^{2}}.
\end{equation}
\begin{equation}
 \frac{1}{2} \gamma x_{\mathrm{max}}^2=\kB T,
\end{equation}
\begin{equation}\label{equ:xmaxestimate}
  x_{\text{max}}=\left(\frac{2\kB T}{\gamma}\right)^{1/2}.
\end{equation}
\begin{equation}
M\frac{\dd^2u_n}{\dd t^2}=-\gamma (u_n-u_{n-1})+\gamma(u_{n+1}-u_n),
\end{equation}
\begin{equation}\label{equ:oneatommotion}
M\frac{\dd^2u_n}{\dd t^2}=-\gamma [2u_n-u_{n-1}-u_{n+1}].
\end{equation}
\begin{equation}\label{equ:movementoneatom}
  u_n(t) = u\:\e^{\im(kan-\omega t)},
\end{equation}
\begin{align}
 -M\omega^2 \e^{\im(kan-\omega t)} &=-\gamma \left[2-\e^{-\im ka}-\e^{\im ka}\right]\e^{\im(kan-\omega t)} \nonumber \\
 & =-2 \gamma (1-\cos ka) \e^{\im(kan-\omega t)},
\end{align}
\begin{equation}\label{equ:dispchain}
\omega(k)=\sqrt{\frac{2\gamma (1-\cos ka)}{M}}=2\sqrt{\frac{\gamma}{M}}\left|\sin\frac{k a}{2}\right|.
\end{equation}
\begin{equation}\label{equ:sound}
 \omega(k)=\sqrt{\frac{\gamma}{M}} a k=\nu k,
\end{equation}
 \begin{equation}\label{equ:movementoneatomphase}
  u_{n+1}(t) = u\:\e^{\im(k a (n+1)-\omega t)} = \e^{\im k a}u_n(t).
 \end{equation}
\begin{equation}
\begin{split}
 M_1\frac{\dd^2u_n}{\dd t^2}=-\gamma [2u_n-v_{n-1}-v_{n}], \\
 M_2\frac{\dd^2v_n}{\dd t^2}=-\gamma [2v_n-u_{n}-u_{n+1}],
\end{split}
\end{equation}
\begin{equation}\label{equ:motionfortwoatoms}
\begin{split}
 u_n(t)=u\:\e^{\im (k b n-\omega t)},\\
 v_n(t)=v\:\e^{\im(k b n-\omega t)}.
\end{split}
\end{equation}
\begin{equation}\label{equ:motionfortwoatoms2}
\begin{split}
 -\omega^2 M_1 u = \gamma v (1+\e^{-\im kb})-2\gamma u, \\
 -\omega^2 M_2 v = \gamma u (\e^{\im kb}+1)-2\gamma v.
\end{split}
\end{equation}
\begin{equation}\label{equ:quasidynmatrix}
\left | \begin{array}{cc}
2\gamma-\omega^2 M_1 & - \gamma (\e^{-\im kb}+1)\\
- \gamma (1+\e^{\im kb}) & 2\gamma -\omega^2 M_2\\
 \end{array}
\right | = 0.
\end{equation}
\begin{equation}\label{equ:dispersiononedchaintwoatoms}
\omega^2=\gamma \left( \frac{1}{M_1}+\frac{1}{M_2} \right) \pm \gamma \left[\left( \frac{1}{M_1}+\frac{1}{M_2} \right)^2-\frac{4}{M_1 M_2} \sin^2 \frac{kb}{2} \right]^{1/2},
\end{equation}
\begin{equation}\label{equ:bornkarman1d}
 u_{N+n}(t) = u_{n}(t).
\end{equation}
\begin{equation}
\e^{\im kan}=\e^{\im ka(N+n)},
\end{equation}
\begin{equation}
\e^{\im kNa}=1,
\end{equation}
\begin{equation}\label{equ:konedim}
k=\frac{2\uppi}{aN}m,
\end{equation}
\begin{equation}
 E_{l} = \left(l+\frac{1}{2}\right)\hbar\omega
\end{equation}
\begin{equation}\label{equ:phononenergies}
 E_{l}(k) = \left(l+\frac{1}{2}\right)\hbar\omega(k).
\end{equation}
\begin{equation}\label{equ:threedwavevec}
 \veck = (k_x,k_y,k_z)
                  = \frac{2\uppi}{aN}(n_x,n_y,n_z)
                  = \left(\frac{n_x 2 \uppi}{L},\frac{n_y 2 \uppi}{L}\frac{n_z 2 \uppi}{L}\right),
\end{equation}
\begin{equation}
 \sigma=\frac{F}{a^2}.
\end{equation}
\begin{equation}
 F=\gamma{\Delta a}
\end{equation}
\begin{equation}
 \sigma=\frac{\gamma \Delta a}{a^2}.
\end{equation}
\begin{equation}\label{equ:youngspringlattice}
 Y = \frac{\sigma}{\varepsilon}
    =\frac{\gamma \Delta a}{a^2}\frac{a}{\Delta a}
    =\frac{\gamma}{a}.
\end{equation}
\begin{equation}
 E_n=\left(n+\frac{1}{2}\right)\hbar\omega_\Einst.
\end{equation}
\begin{equation}
\langle E\rangle = 3 N_\mathrm{\!A} \left(\langle{n}\rangle+\frac{1}{2}\right)\hbar\omega_\Einst.
\end{equation}
\begin{equation}\label{equ:boseeinstein}
\langle n\rangle=\frac{1}{\e^{\hbar\omega_\Einst/\kB T}-1}.
\end{equation}
\begin{equation}\label{equ:meaneeinstein}
\langle{E}\rangle=3 N_\mathrm{\!A} \left(\frac{1}{\e^{\hbar\omega_\Einst/\kB T}-1}+\frac{1}{2}\right)\hbar\omega_\Einst.
\end{equation}
\begin{equation}
 C = \frac{\partial \langle{E}\rangle}{\partial T} =3R\left(\frac{\hbar\omega_\Einst}{\kB T}\right)^{2}\frac{\e^{\hbar\omega_\Einst/\kB T}}{(\e^{\hbar\omega_\Einst/\kB T}-1)^{2}}. \end{equation}
\begin{equation}
p_1 \propto {\e^{-\hbar\omega_\Einst/\kB T}},
\end{equation}
\begin{equation}
\langle{E}\rangle= \frac{\hbar\omega}{\e^{\hbar\omega/\kB T}-1}
\end{equation}
\begin{equation}
\langle{E}\rangle=3 \int_{0}^{\DebF} \frac{\hbar\omega}{\e^{\hbar\omega/\kB T}-1}\:\dd\omega,
\end{equation}
\begin{equation}\label{equ:debyebase}
 \langle{E}\rangle=3 \int_{0}^{\DebF} \frac{g(\omega) \hbar\omega}{\e^{\hbar\omega/\kB T}-1}\:\dd\omega
\end{equation}
\begin{equation}
N=  \frac{4}{3} \uppi\, n^3,
\end{equation}
\begin{equation}\label{equ:phononnumbstates}
N = \frac{4}{3} \uppi \left(\frac{Lk}{2 \uppi} \right)^3.
\end{equation}
\begin{equation}
N(\omega)=\frac{4}{3} \uppi \left(\frac{L \omega}{2 \uppi\, \nu} \right)^3=\frac{V}{6 \uppi^2\, \nu^3} \omega^3,
\end{equation}
\begin{equation}\label{equ:phonondensityofstates}
g(\omega) = \frac{\dd N}{\dd\omega} = \frac{\omega^{2}V}{2\uppi^{2}\,\nu^{3}}.
\end{equation}
\begin{equation}\label{equ:limitdebye}
3N=3\int_{0}^{\DebF}g(\omega)\:\dd\omega.
\end{equation}
\begin{equation}\label{equ:calcomegadebye}
\DebF^3=6 \uppi^2 \frac{N}{V} \nu^3.
\end{equation}
\begin{equation}\label{equ:debyebase3}
\langle{E}\rangle = 3 \int_{0}^{\DebF}\frac{\omega^{2}V}{2\uppi^{2}\,\nu^{3}} \frac{ \hbar\omega}{\e^{\hbar\omega/\kB T}-1}\:\dd\omega
      =\frac{3 V \hbar}{2\uppi^2\,\nu^3}  \int_0^{\DebF}\frac{ \omega^3}{\e^{\hbar\omega/\kB T}-1}\:\dd\omega,
\end{equation}
\begin{equation}\label{equ:debyeenergy}
\langle{E}\rangle=\frac{3 V \kB ^{4} T^4}{2\uppi^{2}\,\nu^{3}\hbar^3}  \int_{0}^{x_\Deb }\frac{ x^{3}}{\e^{x}-1}\:\dd x = 9 N \kB  T  \left(\frac{T}{\DebT} \right)^3 \int_{0}^{x_\Deb }\frac{ x^{3}}{\e^{x}-1}\:\dd x.
\end{equation}
\begin{equation}\label{equ:lowtemperatureresultdebyemodel}
 C= \frac{12 \uppi^4}{5}N \kB   \left(\frac{T}{\DebT} \right)^3.
\end{equation}
\begin{equation}
\kappa=\kappa_\mathrm{p}+\kappa_\mathrm{e}.
\end{equation}
\begin{equation}
 \kappa=\frac{1}{A}\frac{\partial Q}{\partial t} \frac{\Delta x}{ \Delta T}.
\end{equation}
\begin{equation}\label{equ:thermalcondphon}
 \kappa_\mathrm{p}=\frac{1}{3} c \lambda_\mathrm{p}v_\mathrm{p},
  \end{equation}
\begin{equation}\label{equ:thermalexpansioncoef}
\frac{\Delta l}{l}=\alpha \Delta{T},
\end{equation}
\begin{equation}
 G = U + PV - TS.
\end{equation}
\begin{equation}
 T_\mathrm{m} = \frac{(0.05a)^2 \gamma}{2\kB }
                       = \frac{(0.05a)^2 \omega^2 M}{2\kB }.
 \end{equation}
\begin{equation}\label{equ:lindemann}
 T_\mathrm{m}=\frac{(0.05a)^2 \DebT^2  \kB M}{2 \hbar^2}.
 \end{equation}
%================================================================== 
\section{Electronic Properties of Metals: Classical Approach}
\begin{equation}\label{equ:meankindrude}
 \frac{1}{2}\mel v_\mathrm{t}^{2} = \frac{3}{2}\kB T.
\end{equation}
\begin{equation}\label{equ:eldrift}
 \mel\frac{\dd\mathbf{v}}{\dd t}= -e{\mathbfcal{E}}
\end{equation}
\begin{equation}
 \mathbf{v}(t)= \frac{-e{\mathbfcal{E}}{t}}{\mel},
\end{equation}
\begin{equation}\label{equ:drift}
 \overline{\mathbf{v}}=\frac{-e{\mathbfcal{E}}\tau}{\mel}.
\end{equation}
\begin{equation}
 n|\overline{\mathbf{v}}| A,
\end{equation}
\begin{equation}
 -en |\overline{\mathbf{v}}| A.
\end{equation}
\begin{equation}\label{equ:defcurrent}
 \mathbf{j}=-en \overline{\mathbf{v}},
\end{equation}
\begin{equation}\label{equ:ohm}
 \mathbf{j} = \frac{n e^{2} \tau}{\mel}\mathbfcal{E}
                 = \sigma\mathbfcal{E}
                 = \frac{\mathbfcal{E}}{\rho},
\end{equation}
\begin{equation}\label{equ:elconduct}
 \sigma=\frac{n e^{2} \tau}{\mel},
\end{equation}
\begin{equation}
 \rho=\frac{\mel}{n e^{2} \tau}.
\end{equation}
\begin{equation}\label{equ:mob1}
 \mu = \frac{e\tau}{\mel},
\end{equation}
\begin{equation}\label{equ:mob2}
 \sigma=n\mu{e}, \qquad \rho=\frac{1}{n\mu{e}}.
\end{equation}
\begin{equation}\label{equ:defhall}
 \mathcal{E}_\mathrm{H} = R_\mathrm{H} j_x B_z,
\end{equation}
\begin{equation}
|-e\mathcal{E}_\mathrm{H}| = |-eB_z v_x|.
\end{equation}
\begin{equation}
R_\mathrm{H} = \frac{\mathcal{E}_\mathrm{H}}{j_x B_z}
                      = \frac{\mathcal{E}_\mathrm{H}}{-en v_x B_z}
                      = \frac{v_x B_z}{-en v_x B_z}
                      = \frac{-1}{ne}.
\end{equation}
\begin{equation}
 R_\mathrm{H} = \frac{1}{pe},
\end{equation}
\begin{equation}\label{equ:wave1}
 \mathbfcal{E}(z,t) = \mathbfcal{E}_0 \:\e^{\im(kz-\omega t)},
\end{equation}
\begin{equation}\label{equ:wave1a}
 k = \frac{2 \uppi\, N}{\lambda_0},
\end{equation}
\begin{equation}\label{equ:definitioncomplexindexofrefraction}
 N = n + \im \kappa
\end{equation}
\begin{equation}\label{equ:max}
N = \sqrt{\varepsilon}
    = \sqrt{\varepsilon_\mathrm{r} + \im \varepsilon_\mathrm{i}}
\end{equation}
\begin{equation}\label{equ:wave2}
 \mathbfcal{E}(z,t) = \mathbfcal{E}_0 \:\e^{\im((2\uppi\, N / \lambda_0)z-\omega t)}
                             = \mathbfcal{E}_0 \:\e^{\im((\omega \sqrt{\varepsilon}/c)z-\omega t)}.
\end{equation}
\begin{equation}\label{equ:motionplasmon}
 \mel\frac{\dd^2x(t)}{\dd t^2}=-e \mathcal{E}_0 \:\e^{-\im\omega t}.
\end{equation}
\begin{equation}\label{equ:solutionequationplasmon}
 x(t) = A \:\e^{-\im\omega{t}},
\end{equation}
\begin{equation}
 A = \frac{e \mathcal{E}_0}{\mel \omega^2}.
\end{equation}
\begin{equation}
 P(t) = -nex(t)
       = -neA \e^{-\im\omega{t}}
       = -\frac{ne^2\mathcal{E}_0\:\e^{-\im\omega{t}}}{\mel \omega^2}. \label{equ:pol}
\end{equation}
\begin{equation}
 D = \varepsilon \varepsilon_0 \mathcal{E}
    = \varepsilon_0 \mathcal{E} +P,
\end{equation}
\begin{equation}
 \varepsilon = 1+\frac{P(t)}{\varepsilon_0 \mathcal{E}_0 \:\e^{-\im\omega{t}}}.
\end{equation}
\begin{equation}\label{equ:dieldrude}
 \varepsilon = 1-\frac{ne^2}{\varepsilon_0 \mel \omega^2}
                   = 1-\frac{\omega_\mathrm{p}^2}{\omega^2},
\end{equation}
\begin{equation}
\omega_\mathrm{p}^2=\frac{n e^2}{\mel \varepsilon_0}.
\end{equation}
\begin{equation}\label{equ:wiedemanngeneral}
 \frac{\kappa}{\sigma}=LT,
\end{equation}
\begin{equation}\label{equ:wiedemannclassic}
 \frac{\kappa}{\sigma} = \frac{3}{2}\frac{\kB ^2}{e^2}T = LT,
\end{equation}

%================================================================== 
\section{Electronic Properties of Solids: Quantum Mechanical Approach}
\begin{equation}\label{equ:schrsolid}
 -\frac{\hbar^2 \nabla^2}{2\mel } \psi(\vecr)+ U(\vecr) \psi(\vecr)
      = E \psi(\vecr).
\end{equation}
\begin{equation}\label{equ:ulattper}
 U(\vecr) = U(\vecr+\vecR),
\end{equation}
\begin{equation}
 \psi(\vecr) = \psi(x,y,z) = \psi(x+L,y,z) = \psi(x,y+L,z) = \psi(x,y,z+L).
\end{equation}
\begin{equation}\label{equ:singlefree}
     \psi(\vecr) = \frac{1}{\sqrt{V}} \:\e^{\im \veck \cdot \vecr}.
\end{equation}
\begin{equation}\label{equ:kallowedborn}
 \mathbf k =(k_x, k_y, k_z) = \left(\frac{n_x 2 \uppi}{L},\frac{n_y 2 \uppi}{L},\frac{n_z 2 \uppi}{L}\right),
\end{equation}
\begin{equation}\label{equ:freeelenergies}
E(\veck)=\frac{\hbar^{2}k^{2}}{2\mel }=\frac{\hbar^{2}}{2\mel }(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}).
\end{equation}
\begin{equation}
\frac{\hbar^{2}}{2\mel }\left(\frac{2 \uppi}{L}\right)^{2}
\end{equation}
\begin{equation}\label{equ:calcnmaxinfreeelectronmodel}
  \frac{N}{2} = \frac{4}{3} \uppi \left(\frac{Lk}{2 \uppi} \right)^3,
  \end{equation}
\begin{equation}\label{equ:calcn_k_F_freeelectronmodel}
k_\Fermi = \left( \frac{3 \uppi^2 N}{L^3} \right)^{1/3}=\left( 3 \uppi^2 \,n \right)^{1/3}.
\end{equation}
\begin{equation}\label{equ:newc}
  \EF= \frac{\hbar^{2}k_\Fermi^2}{2\mel }=  \frac{\hbar^{2}}{2\mel } \left( \frac{3 \uppi^2 N}{L^3} \right)^{2/3} = \frac{\hbar^{2}}{2\mel }\left( 3 \uppi^2 n\right)^{2/3}.
 \end{equation}
\begin{equation}\label{equ:basisfordoscalc2}
 N(E) = \frac{V}{3 \uppi^2}\left(\frac{2\mel }{\hbar^{2}}\right)^{3/2} E^{3/2}.
\end{equation}
\begin{equation}\label{equ:freeeldos}
 g(E) = \frac{\dd N}{\dd E}
        = \frac{V}{2 \uppi^2}\left(\frac{2\mel }{\hbar^{2}}\right)^{3/2} E^{1/2}.
\end{equation}
\begin{equation}
 f(E,T) = \frac{1}{\e^{(E-\mu)/{\kB T}}+1},
\end{equation}
\begin{equation}
 N = \int_0^{\infty} g(E) f(E,T) \:\dd E.
\end{equation}
\begin{equation}
 \langle{E}\rangle = \frac{3}{2}{\kB  T}\,g(\EF) \kB  T
\end{equation}
 \begin{equation}
 C = \frac{\partial \langle{E}\rangle}{\partial T}
    = 3 \kB ^2 T \,g(\EF).
\end{equation}
\begin{equation}\label{equ:heatcapacityoffreeelectrons}
 C = \frac{\uppi^2}{3}\kB ^2 T\, g(\EF)
\end{equation}
\begin{equation}\label{equ:wiedemannfranzquantum}
 \frac{\kappa}{\sigma} = \frac{\uppi^2}{3}\frac{\kB ^2}{e^2}T = LT.
\end{equation}
\begin{equation}\label{equ:screen1}
 \phi_0(r) = \frac{1}{4 \uppi\,  \varepsilon_0}\frac{q}{r},
\end{equation}
\begin{equation}\label{equ:tfcd}
 \rho(r)=-\frac{e^2}{V} g(\EF) \phi(r).
\end{equation}
\begin{equation}\label{equ:thomasfermiscreening1}
 \nabla^2 \phi(\vecr) = \frac{\partial^2 \phi(r)}{\partial r^2} +
                                             \frac{2}{r}\frac{\partial  \phi(r)}{\partial r}
                                         = \frac{e^2}{V \varepsilon_0} g(\EF) \phi(r).
\end{equation}
\begin{equation}\label{equ:thomasfermiscreening}
 \phi(r) = c \frac{1}{r}\:\e^{-r/r_{\mathrm{TF}}},
\end{equation}
\begin{equation}\label{equ:thomasfermiscreeninglength}
 r_\mathrm{TF} = \sqrt{\frac{V \varepsilon_0}{e^2 g(\EF)}}.
\end{equation}
\begin{equation}\label{equ:thomasfermiscreening3}
 \phi(r) = \frac{1}{4 \uppi\,  \varepsilon_0} \frac{q}{r} \:\e^{-r/r_\mathrm{TF}},
\end{equation}
\begin{equation}\label{equ:bloch}
 \psi_{\veck}(\vecr) = \e^{\im\veck \cdot \vecr}u_{\veck}(\vecr),
\end{equation}
\begin{equation}
 u_{\veck}(\vecr) = u_{\veck}(\vecr+\vecR),
\end{equation}
\begin{equation}\label{equ:bloch2}
 \psi_{\veck}(\vecr + \vecR) = \e^{\im\veck \cdot \vecR} \psi_{\veck}(\vecr).
\end{equation}
\begin{equation}\label{equ:planewaveexp}
 \psi(\vecr) = \sum_{\veck} c_{\veck}\:\e^{\im\veck \cdot \vecr}.
\end{equation}
\begin{equation}\label{equ:potentialexpan}
  U(\vecr) = \sum_{\mathbf{G}} U_\mathbf{G}\:\e^{\im\mathbf{G} \cdot \vecr}.
\end{equation}
\begin{equation}
  U_{-\mathbf{G}} = U_{\mathbf{G}}^*.
\end{equation}
\begin{equation}
  -\frac{\hbar^2 \nabla^2}{2\mel } \psi(\vecr)
  =\sum_\veck \frac{\hbar^2 k^2}{2\mel } c_\veck\:\e^{\im\veck \cdot \vecr}.
\end{equation}
\begin{align}
  U(\vecr) \psi(\vecr) &= \left(\sum_\mathbf{G} U_\mathbf{G}\:\e^{\im\mathbf{G} \cdot \vecr} \right)\left(\sum_\veck c_\veck\:\e^{\im\veck \cdot \vecr} \right) \nonumber\\
   &= \sum_\veck \sum_\mathbf{G} U_\mathbf{G} c_\veck\:\e^{\im(\mathbf{G}+\veck) \cdot \vecr} \nonumber\\
  &= \sum_\mathbf{k'} \sum_\mathbf{G} U_\mathbf{G} c_{\mathbf{k'}-\mathbf{G}} \:\e^{\im\mathbf{k'} \cdot \vecr}.
\end{align}
\begin{equation}\label{equ:schrfouriercomplete}
  \sum_\veck \e^{\im\veck \cdot \vecr} \left\{\left( \frac{\hbar^2 k^2}{2\mel }  -E  \right) c_\veck +\sum_\mathbf{G} U_\mathbf{G} c_{\veck-\mathbf{G}}\right\} = 0.
  \end{equation}
\begin{equation}\label{equ:schrfourier}
  \left( \frac{\hbar^2 k^2}{2\mel }  - E  \right) c_\veck + \sum_\mathbf{G} U_\mathbf{G} c_{\veck-\mathbf{G}}= 0.
 \end{equation}
\begin{equation}\label{equ:blochfinal}
  \psi_\veck(\vecr) = \sum_\mathbf{G} c_{\veck-\mathbf{G}}\:\e^{\im(\veck-\mathbf{G}) \cdot \vecr}.
  \end{equation}
\begin{equation}
  \psi_\veck(\vecr) = \e^{\im\veck \vecr} \left(\sum_\mathbf{G} c_{\veck-\mathbf{G}}\:\e^{-\im\mathbf{G} \cdot \vecr} \right).
  \end{equation}
\begin{equation}
 \psi_{\veck+\mathbf{G}'}(\vecr) = \sum_\mathbf{G} c_{\veck-\mathbf{G}+\mathbf{G}'}\:\e^{\im(\veck-\mathbf{G}+\mathbf{G}') \cdot\vecr} = \sum_\mathbf{G''} c_{\veck-\mathbf{G''}}\:\e^{\im(\veck-\mathbf{G''}) \cdot\vecr}
\end{equation}
\begin{equation}\label{equ:degenwave}
 \psi_{\veck+\mathbf{G}'} = \psi_\veck(\vecr),
\end{equation}
\begin{equation}\label{equ:degenener}
 E(\veck+\mathbf{G}') = E(\veck).
\end{equation}
\begin{equation}\label{equ:schrfourier1D_1}
  \left( \frac{\hbar^2 k^2}{2\mel }  -E  \right) c_{{k}} + \sum_{{g}} U_{{g}} c_{{k}-{g}}= 0.
 \end{equation}
\begin{equation}\label{equ:fourierpotential1D}
 U(x)=\sum_g U_g \,\e^{\im gx}.
\end{equation}
\begin{equation}\label{equ:schrfourier1D_2}
  \left( \frac{\hbar^2 (k-g_1)^2}{2\mel } -E  \right) c_{k-g_1} + \sum_g U_g c_{k-g-g_1}= 0.
 \end{equation}
 \begin{align}
  \left( \frac{\hbar^2 k^2}{2\mel }  -E  \right) c_{{k}} + U c_{{k}+{g_1}} + U c_{{k}-{g_1}} &= 0,
 \nonumber \\
 \left( \frac{\hbar^2 (k-g_1)^2}{2\mel }  -E  \right) c_{{k}-g_1} + U c_{{k}} + U c_{{k}-{g_2}} &= 0.
 \label{equ:BZboundnearlyfree}
\end{align}
\begin{equation}\label{equ:blochfinal1D}
  \psi_k(x) = \sum_g c_{k-g} \: \e^{\im(k-g) x} \approx c_k \:\e^{\im k x} +  c_{k-g_1}\:\e^{\im(k-g_1) x}.
\end{equation}
\begin{equation}\label{equ:BZboundnearlyfree2}
\begin{split}
          \left( \frac{\hbar^2 k^2}{2\mel }  -E  \right) c_k  + U c_{k-g_1} &= 0, \\
  U c_k + \left( \frac{\hbar^2 (k-g_1)^2}{2\mel }  -E  \right) c_{k-g_1}  &= 0,
\end{split}
\end{equation}
\begin{equation}
  E_0 = \frac{\hbar^{2}}{2\mel } \left( \pm \frac{\uppi}{a}\right)^2.
\end{equation}
 \begin{equation}
\left| \begin{array}{cc}
E_0 - E & U\\
U & E_0-E
\end{array}\right|=0.
\end{equation}
 \begin{align}
 \psi(+) &\propto \e^{\im(\uppi/a)x}+\e^{-\im(\uppi/a)x} = 2 \cos\left(\frac{\uppi}{a}x\right),\\
 \psi(-)  &\propto \e^{\im(\uppi/a)x}-\e^{-\im(\uppi/a)x} = 2\im\sin\left(\frac{\uppi}{a}x\right).
\end{align}
\begin{equation}
 -\im \hbar \:\nabla\psi_{\veck}(\vecr) =
 \hbar\veck\,\psi_{\veck}(\vecr) - \e^{\im\veck \cdot
\vecr} \:\im\: \hbar \:\nabla u_{\veck}(\vecr).
\end{equation}
\begin{equation}\label{equ:groupvelo}
 v_\mathrm{g}=\frac{\dd \omega(k)}{\dd k} = \frac{1}{\hbar}\frac{\dd E(k)}{\dd k}.
\end{equation}
 \begin{equation}
 H_\mathrm{at}  = -\frac{\hbar^2 \nabla^2}{2\mel } + V_\mathrm{at} (\vecr),
\end{equation}
\begin{align}
 H_\mathrm{sol}  &=
     -\frac{\hbar^2 \nabla^2}{2\mel } + \sum_\vecR V_\mathrm{at}(\vecr-\vecR) \nonumber \\
     &= -\frac{\hbar^2 \nabla^2}{2\mel } +  V_\mathrm{at}(\vecr) + \sum_{\vecR \ne 0}
        V_\mathrm{at}(\vecr-\vecR).\label{equ_tb1}
\end{align}
\begin{equation}\label{equ_tb6}
  H_\mathrm{sol}  = -\frac{\hbar^2 \nabla^2}{2\mel } + V_\mathrm{at}(\vecr) + v(\vecr)
                            = H_\mathrm{at}+ v(\vecr),
\end{equation}
\begin{equation}
 v(\vecr) = \sum_{\vecR \ne 0} V_\mathrm{at}(\vecr-\vecR).
\end{equation}
\begin{equation}\label{equ:tbwithonlyoneorbital}
 \int  \phi_n^\ast(\vecr) H_\mathrm{sol} \phi_n(\vecr) \:\dd\vecr
       = E_n + \int  \phi_n^\ast(\vecr) v(\vecr) \phi_n(\vecr) \:\dd\vecr
       = E_n -\beta,
\end{equation}
 \begin{equation}\label{equ_tb2}
 \psi_\veck(\vecr) = \frac{1}{\sqrt{N}} \sum_\vecR c_{\veck,\vecR} \phi_n(\vecr-\vecR).
\end{equation}
\begin{equation}
\psi_\veck(\vecr) = \frac{1}{\sqrt{N}}\sum_{\vecR} \:\e^{\im\veck \cdot \vecR} \phi_n(\vecr-\vecR),
 \label{equ_tb10}
\end{equation}
\begin{align}
 \psi_\veck(\vecr+\vecR')
   &= \frac{1}{\sqrt{N}}\sum_{\vecR} \:\e^{\im\veck \cdot \vecR} \phi_n(\vecr-\vecR+\vecR') \nonumber \\
   &= \frac{1}{\sqrt{N}} \:\e^{\im\veck\vecR'} \sum_{\vecR} \:\e^{\im\veck \cdot (\vecR-\vecR')} \phi_n\big(\vecr-(\vecR-\vecR')\big)\nonumber \\
   &= \frac{1}{\sqrt{N}} \:\e^{\im\veck\vecR'} \sum_{\vecR''} \:\e^{\im\veck \cdot \vecR''} \phi_n(\vecr-\vecR'')
      =  \e^{\im\veck \cdot \vecR'} \psi_{\veck}(\vecr),
\end{align}
\begin{align}
 E(\veck) &=\int \psi^\ast_\veck(\vecr) H_\mathrm{sol} \psi_\veck(\vecr)\:\dd\vecr \nonumber\\
                      &=\frac{1}{{N}}  \sum_\vecR \sum_\mathbf{R'} \e^{\im\veck \cdot (\vecR-\vecR')} \int  \phi^\ast_n(\vecr-\mathbf{R'}) H_\mathrm{sol} \phi_n(\vecr-\vecR) \:\dd\vecr,
 \label{equ_tb4}
\end{align}
\begin{equation}\label{equ_tb4a}
 E(\veck) = \sum_\vecR \e^{\im\veck \cdot \vecR} \int  \phi^\ast_n(\vecr) H_\mathrm{sol} \phi_{n}(\vecr-\vecR) \:\dd\vecr.
\end{equation}
\begin{equation}\label{equ_tb4b}
 E(\veck) = E_n -\beta + \sum_{\vecR \ne 0} \e^{\im\veck \cdot \vecR} \int  \phi^\ast_n(\vecr) H_\mathrm{sol} \phi_n(\vecr-\vecR) \:\dd\vecr.
\end{equation}
\begin{multline}\label{equ_tb4c}
 \int  \phi^\ast_n(\vecr) H_\mathrm{sol} \phi_n(\vecr-\vecR) \:\dd\vecr
 = E_n \int \phi^\ast_n(\vecr) \phi_n(\vecr-\vecR) \:\dd\vecr \\
        + \int \phi^\ast_n(\vecr) v(\vecr) \phi_n(\vecr-\vecR) \:\dd\vecr.
\end{multline}
\begin{equation}\label{equ:gammadef}
 \gamma(\vecR) =- \int \phi^\ast_n(\vecr) v(\vecr) \phi_n(\vecr-\vecR) \:\dd\vecr,
\end{equation}
 \begin{equation}
 E(\veck)= E_n - \beta - \sum_{\vecR \ne 0} \gamma(\vecR) \;\e^{\im\veck \cdot \vecR}.
\label{equ:tbbands}
\end{equation}
\begin{equation}\label{equ_tb12}
E_\mathrm{s}(k) = E_\mathrm{s} - \beta_\mathrm{s} - \gamma_\mathrm{s} (\e^{\im ka} + \e^{-\im ka})
                         = E_\mathrm{s} - \beta_\mathrm{s} - 2  \gamma_\mathrm{s}   \cos ka,
\end{equation}
\begin{equation}\label{equ:motionelectrons1}
\dd E = -e \mathcal{E} v_\mathrm{g}\:\dd t .
\end{equation}
\begin{equation}\label{equ:motionelectrons3}
 \frac{\dd E}{\dd t} = \frac{\dd E}{\dd k} \frac{\dd k}{\dd t},
\end{equation}
\begin{equation}\label{equ:motionelectrons}
 \hbar \frac{\dd k}{\dd t} = -e\mathcal{E}.
\end{equation}
\begin{equation}
a = \frac{\dd v_\mathrm{g}}{\dd t}
   = \frac{1}{\hbar}\frac{\dd}{\dd t}\frac{\dd E(k)}{\dd k}
   =\frac{1}{\hbar}\frac{\dd^2E(k)}{\dd k^2}\frac{\dd k}{\dd t}.
\end{equation}
\begin{equation}
a=-\frac{1}{\hbar^2}\frac{\dd^2E(k)}{\dd k^2}e \mathcal{E}.
\label{equ:semiclass}
\end{equation}
\begin{equation}
 m^\ast = \hbar^2\left(\frac{\dd^2E(k)}{\dd k^2}\right)^{-1}.
\label{equ:effmass}
\end{equation}

%================================================================== 
\section{Semiconductors}
\begin{equation}\label{equ:eldens}
 n = \frac{1}{V} \int_{E_\con}^{\infty} g_\con(E) f(E,T)\:\dd E,
\end{equation}
\begin{equation}\label{equ:hodens}
  p = \frac{1}{V}\int_{-\infty}^{E_\val} g_\val(E)[1-f(E,T)]\:\dd E,
\end{equation}
\begin{equation}
 E = \Eg  + \frac{\hbar^{2}k^{2}}{2\mel^\ast}.
\end{equation}
\begin{equation}
 g_\con(E) = \frac{V}{2\uppi^{2}}\left(\frac{2\mel^\ast}{\hbar^{2}}\right)^{3/2} (E-\Eg )^{1/2}.
\end{equation}
\begin{equation}
 E=-\frac{\hbar^{2}k^{2}}{2m_\mathrm{h}^\ast}
\end{equation}
\begin{equation}
 g_\val(E) = \frac{V}{2\uppi^{2}}\left(\frac{2m_\mathrm{h}^\ast}{\hbar^{2}}\right)^{3/2}(-E)^{1/2}.
\end{equation}
\begin{equation}\label{equ:fermi1}
 f(E,T) = \frac{1}{\e^{(E-\mu)/\kB T}+1} \approx \e^{-(E-\mu)/\kB T}.
\end{equation}
\begin{equation}\label{equ:fermi2}
 1 - f(E,T) = 1-\frac{1}{\e^{(E-\mu)/\kB T}+1} \approx \e^{(E-\mu)/\kB T}
\end{equation}
\begin{align}
 n &= \frac{1}{V} \int_{\Eg }^{\infty} \frac{V}{2\uppi^2} \left(\frac{2\mel^\ast}
         {\hbar^2}\right)^{3/2} (E-\Eg )^{1/2} \: \e^{-(E-\mu)/\kB T}\:\dd E \nonumber \\
    & = \frac{(2\mel^\ast)^{3/2}}{2 \uppi^2\hbar^3}  \e^{\mu/\kB T}
           \int_{\Eg }^{\infty} (E-\Eg )^{1/2}  \:\e^{-E/\kB T} \:\dd E.
\end{align}
\begin{equation}
 n = \frac{(2\mel^\ast)^{3/2}}{2 \uppi^2\hbar^{3}}  (\kB T)^{3/2} \: \e^{-(\Eg -\mu)/\kB T} \int_0^\infty X_\gap^{1/2} \:\e^{-X_\gap}\:\dd X_\gap.
\end{equation}
\begin{equation}\label{equ:finaleldens}
 n = \frac{1}{\sqrt{2}}\left(\frac{ \mel^\ast \kB T}{\uppi\, \hbar^{2}}\right)^{3/2}\:\e^{-(\Eg -\mu)/\kB T}
    =N_\mathrm{eff}^C \:\e^{-(\Eg -\mu)/\kB T},
\end{equation}
\begin{equation}\label{equ:finalhodens}
 p = \frac{1}{\sqrt{2}}\left(\frac{m_\mathrm{h}^\ast \kB T}{\uppi\, \hbar^2} \right)^{3/2}
       \:\e^{-\mu/\kB T}
    = N_\mathrm{eff}^V \:\e^{-\mu/{\kB T}}.
\end{equation}
\begin{equation}\label{equ:massaction}
 np = 4\left(\frac{\kB T}{2\uppi\,\hbar^{2}}\right)^{3}(\mel^\ast m_\mathrm{h}^\ast)^{3/2}\:\e^{-\Eg /\kB T},
\end{equation}
\begin{equation}
 n_\mathrm{i} = p_\mathrm{i}
                      = \sqrt{np}
                      = 2\left(\frac{\kB T}{2\uppi\,\hbar^2}\right)^{3/2}(\mel^\ast m_\mathrm{h}^\ast)^{3/4}\:\e^{-\Eg /2 \kB T}.
\end{equation}
\begin{equation}\label{equ:chempotpos}
 \mu = \frac{\Eg }{2}+\frac{3}{4}\kB T \ln \left(\frac{m_\mathrm{h}^\ast}{\mel^\ast}\right).
\end{equation}
\begin{equation}
 \omega_c = \frac{B e}{\mel^\ast}.
\end{equation}
\begin{equation}\label{equ:ionizationimpurity}
E_n=-\frac{\mel e^{4}}{8\varepsilon_{0}^{2}h^{2}}\frac{1}{n^2}.
\end{equation}
\begin{equation}
 a_0 = \frac{4 \uppi\, \varepsilon_0 \hbar^2}{\mel e^2}.
\end{equation}
\begin{equation}\label{equ:semiconductorconduction}
 \sigma=e(n\mu_\mathrm{e}+p\mu_\mathrm{h}),
\end{equation}
\begin{equation}\label{equ:poissonpnjunction}
 \frac{\dd^2 U}{\dd x^2} = -\frac{\rho}{\varepsilon\varepsilon_0},
\end{equation}
\begin{equation}\label{equ:potentialdifferencepnjunctionbc}
 \left. \frac{\dd U}{\dd x}\right|_{x=-d_\mathrm{p},d_\mathrm{n}} = 0.
\end{equation}
\begin{equation}\label{equ:potentialdifferencepnjunction}
\Delta U = \frac{e}{2\varepsilon\varepsilon_0} \left(N_\mathrm{d}d_\mathrm{n}^{2} + N_\mathrm{a}d_\mathrm{p}^2\right).
\end{equation}
\begin{equation}
 d_\mathrm{p}=\left(\frac{\Delta U{2}\varepsilon\varepsilon_{0}}{eN_\mathrm{a}}\frac{N_\mathrm{d}}{N_\mathrm{a}+N_\mathrm{d}}\right)^{1/2}, \qquad d_\mathrm{n}=\left(\frac{\Delta U{2}\varepsilon\varepsilon_{0}}{eN_\mathrm{d}}\frac{N_\mathrm{a}}{N_\mathrm{a}+N_\mathrm{d}}\right)^{1/2}.
\end{equation}
\begin{align}
 n_\mathrm{p} &= N_{\mathrm{eff}}^\con \:\e^{(\mu-\Eg  - e\Delta U) / {\kB T}}, &
 p_\mathrm{p} &= N_{\mathrm{eff}}^\val \:\e^{(e\Delta U-\mu)/{\kB T}},
                                        \label{equ:carrierdensityonthepside}\\
n_\mathrm{n} &= N_{\mathrm{eff}}^\con \:\e^{(\mu-\Eg )/{\kB T}}, &
p_\mathrm{n} &= N_{\mathrm{eff}}^\val \:\e^{-\mu/{\kB T}}.
\end{align}
\begin{equation}
  |I_{\mathrm{diffusion}}| = |I_{\mathrm{drift}}| = |I_0|
                                      = C\:\e^{(\mu-\Eg -e\Delta U)/{\kB T}},
\end{equation}
 \begin{equation}
 |I_{\mathrm{diffusion}}|=C\:\e^{[(\mu+eV)-\Eg -e\Delta U]/{\kB T}}.
\end{equation}
\begin{equation}
  I = I_{\mathrm{diffusion}}-I_{\mathrm{drift}}
     =I_0 \left(\e^{eV/{\kB T}}-1\right).
\end{equation}
\begin{equation}
  |I_{\mathrm{diffusion}}| = C\:\e^{[(\mu-eV)-\Eg -e\Delta U]/{\kB T}},
\end{equation}
\begin{equation}
  I = I_{\mathrm{diffusion}}-I_{\mathrm{drift}} = I_0\left(\e^{-eV/{\kB T}}-1\right).
\end{equation}

%================================================================== 
\section{Magnetism}
\begin{equation}
 \oint \mathbf{B} \:\dd\mathbf{a} = 0, \qquad  \diverg \mathbf{B} = 0
\end{equation}
\begin{equation}
 \mathbf{B} = \mu_0 \mathbf{H},
\end{equation}
\begin{equation}\label{equ:fund1}
 \mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M}) = \mathbf{B}_0+\mu_0\mathbf{M},
\end{equation}
\begin{equation}\label{equ:macroscopicmagnetization}
 \mathbf{M} = \boldsymbol{\upmu}\frac{N}{V}.
\end{equation}
\begin{equation}\label{equ:linearsus}
 \mu_0\mathbf{M} = \chim \mathbf{B}_0,
\end{equation}
\begin{equation}
 U = -V \int_0^{B_0} {M}\:\dd B'_0
    = -V \int_0^{B_0} \frac{\chim}{\mu_0} B'_0 \:\dd B'_0
    = -V \frac{ \chim}{2 \mu_0} B_0^2,
\label{equ:dipolefield}
\end{equation}
\begin{equation}
 \mathbf{B} = \curl\mathbf{A}.
\end{equation}
\begin{equation}\label{equ:vetorpotentialmomentum}
\mathbf{p} \rightarrow \mathbf{p}-q\mathbf{A}.
\end{equation}
\begin{equation}
 \mathbf{A}= - \frac{1}{2}\vecr \times \mathbf{B}_0,
\end{equation}
\begin{align}
H_{\mathrm{kin}} &\rightarrow H'_{\mathrm{kin}}, \nonumber \\
\frac{\mathbf{p}^2}{2\mel} & \rightarrow  \frac{1}{2\mel}\left( \mathbf{p} +e\mathbf{A} \right) ^2=\frac{1}{2\mel}\left( \mathbf{p} - e \frac{\vecr \times \mathbf{B}_0}{2} \right) ^2.
\end{align}
\begin{equation}
 H'_{\mathrm{kin}} = \frac{1}{2\mel}\left( \mathbf{p}^2 +e \mathbf{B}_0 \cdot (\vecr \times \mathbf{p})+\frac{e^2}{4} (\vecr \times \mathbf{B}_0)^2 \right),
\end{equation}
\begin{equation}
 H'_\mathrm{kin} =H_\mathrm{kin} +H'
                           = \frac{\mathbf{p}^2}{2\mel} +
                              \frac{e}{2\mel} B_0  (\vecr \times \mathbf{p})_z +
                              \frac{e^2}{8\mel} B_0^2(x^2+y^2).
\end{equation}
\begin{equation}\label{equ:suscepenergycorr}
 E'=\frac{e}{2\mel} B_0  \langle \psi|(\vecr \times \mathbf{p})_z|\psi \rangle +
      \frac{e^2}{8\mel} B_0^2 \langle \psi |(x^2+y^2)|\psi \rangle.
\end{equation}
\begin{equation}\label{equ:spincorrection}
 g_\mathrm{e} m_\mathrm{s} \frac{e \hbar}{2\mel} {B}_0 =
 g_\mathrm{e} m_\mathrm{s} \muB  B_0,
\end{equation}
\begin{equation}
 \boldsymbol{\upmu}= -\frac{e \hbar}{2\mel} \mathbf{L} =-\muB  \mathbf{L}.
\end{equation}
\begin{equation}
 \mu_l = -\frac{em_{l}\hbar}{2\mel}=-m_{l}\muB .
\end{equation}
\begin{equation}\label{equ:spinmoment}
\boldsymbol{\upmu} =-g_\mathrm{e} \muB  \mathbf{S},
\end{equation}
\begin{equation}
\mu_\mathrm{s}=- g_\mathrm{e} m_\mathrm{s}\muB ,
\end{equation}
\begin{equation}\label{equ:jmoment}
\mu_J=- g m_{J}\muB ,
\end{equation}
\begin{equation}
 L=\sum{m_{l}} \quad \text{and} \quad S=\sum{m_\mathrm{s}},
\end{equation}
\begin{equation}
 g_J = \frac{3J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.
\end{equation}
\begin{equation}\label{equ:magmomex}
 \mu = -\frac{\partial E'}{\partial B_0}
        = -\frac{e^2}{4\mel} B_0 \langle  \psi |(x^2+y^2)|\psi \rangle.
\end{equation}
\begin{equation}\label{equ:diamag}
 \mu = -\frac{Ze^2}{6\mel} r_\mathrm{a}^2 {B_0}.
\end{equation}
\begin{equation}
 \chim = \mu_0 \frac{M}{B_0}
                            =-\frac{\mu_0  ZNe^2}{6V\mel} r_\mathrm{a}^2.
\end{equation}
\begin{equation}\label{equ:diamagelectron}
\chim = -\frac{1}{3V}\muB ^{2}\,\mu_0\,g(\EF ) \left(\frac{\mel}{m^{*}}\right)^{2}.
\end{equation}% units checked for this one: no units
\begin{equation}\label{equ:meanmoment}
 \bar{\mu} = \frac{1}{Z}\sum_{m_{\!J}=-J}^J g_J \muB  m_{\!J}
                    \:\e^{g_{\!J} \muB  m_{\!J} B_0/\kB T},
\end{equation}
\begin{equation}
 Z = \sum_{m_{\!J}=-J}^J \e^{-g_J \muB  m_{\!J} B_0/\kB T}.
\end{equation}
\begin{equation}\label{equ:curie}
 \chim=\frac{C}{T},
\end{equation}
\begin{equation}\label{equ:curieconstant}
 C =\frac{\mu_0 Ng_J^2 \muB ^2 J(J+1)}{3V\kB }.
\end{equation}
\begin{equation}
 N_{\downarrow\downarrow B_0}-N_{\downarrow\uparrow B_0} = g(\EF ) \muB  B_0,
\end{equation}
\begin{equation}
 M = \frac{1}{V}(N_{\downarrow\downarrow B_0}-N_{\downarrow\uparrow B_0})\muB
     = \frac{1}{V}g(\EF )\muB ^2 B_0,
\end{equation}
\begin{equation}\label{equ:suspauli}
 \chim = \frac{1}{V}\mu_{0}\,\muB ^2\,g(\EF ),
\end{equation}
\begin{equation}
E_{\uparrow \uparrow}-E_{\uparrow \downarrow}=-2X.
\label{equ:singlettripletsplitting}
\end{equation}
\begin{equation}
 H = -2X  \mathbf{S}_1 \cdot  \mathbf{S}_{2}.
\end{equation}
\begin{equation}\label{equ:heisenbergmagnet1}
 H = -\sum_i \sum_{j\neq i}  X_{ij} \mathbf{S}_i \cdot  \mathbf{S}_{j} + g_\mathrm{e} \muB  \mathbf{B}_0  \cdot \sum_i  \mathbf{S}_i,
\end{equation}
\begin{equation}\label{equ:heisenbergmagnet2}
 H = -X \sum_i \sum_\mathrm{n.n.}  \mathbf{S}_i \cdot  \mathbf{S}_\mathrm{n.n.} + g_\mathrm{e} \muB  \mathbf{B}_0  \cdot \sum_i  \mathbf{S}_i,
\end{equation}
\begin{equation}
 H = \sum_i \mathbf{S}_i   \cdot \left(- \sum_\mathrm{n.n.} X \langle \mathbf{S} \rangle + g_\mathrm{e} \muB  \mathbf{B}_0 \right)
    =  \sum_i \mathbf{S}_i   \cdot \left( -n_\mathrm{n.n.} X \langle \mathbf{S} \rangle + g_\mathrm{e} \muB  \mathbf{B}_0 \right),
\label{equ:hheisenbergmeanfield}
\end{equation}
\begin{equation}
\mathbf{M} = - g_\mathrm{e} \muB  \langle \mathbf{S} \rangle \frac{N}{V},
\end{equation}
\begin{equation}\label{equ:heisenberg3}
 H = \sum_i \mathbf{S}_i   \cdot \left( \frac{ n_\mathrm{n.n.} XV}{g_\mathrm{e} \muB  N} \mathbf{M}  + g_\mathrm{e} \muB  \mathbf{B}_0 \right)
    = g_\mathrm{e} \muB  \sum_i \mathbf{S}_i   \cdot \left( \BW  + \mathbf{B}_0 \right)
\end{equation}
\begin{equation}\label{equ:weissfield}
 \BW = \mathbf{M} \frac{n_\mathrm{n.n.} X V }{g_\mathrm{e}^2 \muB ^2 N}.
 \end{equation}
\begin{equation}
 M(T) = \frac{\muB  N }{V} \frac{\e^x- \e^{-x} }{\e^{-x} + \e^x}= M(0) \tanh (x),
\label{equ:heisenbergtempdep}
\end{equation}
\begin{equation}
\frac{M(T)}{ M(0)} = \tanh \left( \frac{M(T)}{ M(0)} \frac{\CurT}{T} \right),
\label{equ:heisenbergtempdep2}
\end{equation}
\begin{equation}
\CurT=\frac{n_\mathrm{n.n.} X }{g_\mathrm{e}^2 \kB }
\label{equ:curietemperature}
\end{equation}
\begin{equation}\label{equ:curieweisslaw}
\chim=\frac{C}{T-\CurT}.
\end{equation}


%================================================================== 
\section{Dielectrics}
\begin{equation}\label{equ:polari1}
 \mathbf{P} =\chi_\mathrm{e} \varepsilon_0 \mathbfcal{E},
\end{equation}
\begin{equation}\label{equ:poldef}
 \mathbf{P} = \frac{N}{V} \mathbf{p} = \frac{N}{V} q \boldsymbol{\updelta}.
\end{equation}
\begin{equation}\label{equ:polatom}
 \mathbf{p}=\alpha \mathbfcal{E},
\end{equation}
\begin{equation}\label{equ:pol_extra}
 \mathbf{P}=(\varepsilon-1) \varepsilon_0 \mathbfcal{E}=\frac{N}{V}
 \mathbf{p}=\frac{N}{V} \alpha \mathbfcal{E},
\end{equation}
\begin{equation}
 \alpha= \frac{(\varepsilon-1) \varepsilon_0 V}{N}.
\end{equation}
\begin{equation}
 \mathbfcal{E}_{\mathrm{loc}}=\frac{1}{3}(\varepsilon+2)\mathbfcal{E}.
\end{equation}
\begin{equation}
\mathbf{P}  = \frac{N}{V}\alpha \mathbfcal{E}_{\mathrm{loc}}
                  = \frac{{N}\alpha}{3 V}(\varepsilon+2)\mathbfcal{E}.
\end{equation}
\begin{equation}
 \alpha=\frac{\varepsilon-1}{\varepsilon+2} \frac{3 \varepsilon_0 V}{N}.
\end{equation}
\begin{equation}\label{equ:dielosci}
 \frac{\dd^2x}{\dd t^2} + \eta\frac{\dd x}{\dd t} + \omega_0^2x = \frac{e\mathcal{E}_0}{M}\:\e^{-\im\omega t}.
\end{equation}
\begin{equation}
x(t) = A\:\e^{-\im\omega{t}},
\end{equation}
\begin{equation}\label{equ:amplitude0}
 A = \frac{e \mathcal{E}_0}{M}  \frac{1}{\omega_0^2-\omega^2-\im\eta \omega}.
\end{equation}
\begin{equation}\label{equ:amplitude}
 A = \frac{e{\mathcal{E}_0}}{M} \left(  \frac{\omega_0^2 -\omega^2}{(\omega_0^2 -\omega^2)^2+\eta^2\omega^2} + \frac{\im\eta \omega}{(\omega_0^2 -\omega^2)^2+\eta^2\omega^2}    \right).
\end{equation}
\begin{equation}
 P(t) = P_\mathrm{i}(t) + P_\mathrm{e}(t)
       = \frac{N}{V} eA \:\e^{-\im\omega{t}}+  \frac{N}{V} \alpha \mathcal{E}_0 \:\e^{-\im\omega{t}}.
 \end{equation}
\begin{equation}
 \varepsilon = \frac{P(t)}{\varepsilon_0 \mathcal{E}_0 \:\e^{-\im\omega{t}}} +1
                   = \frac{N eA} {V \varepsilon_0 \mathcal{E}_0}+\frac{ N \alpha} {V\varepsilon_0} +1.
\end{equation}
\begin{equation}
  \varepsilon_\mathrm{opt} = \frac{ N \alpha} {V \varepsilon_0} +1,
\end{equation}
\begin{equation}\label{equ:splitoscopt}
 \varepsilon(\omega)= \frac{N eA} {V\varepsilon_0 \mathcal{E}_0}+\varepsilon_{\mathrm{opt}}.
\end{equation}
\begin{equation}\label{equ:epsilonrealharm}
 \varepsilon_\mathrm{r}(\omega) = \frac{Ne^{2}}{V \varepsilon_0M} \frac{\omega_0^2-\omega^2}{(\omega_0^2-\omega^2)^2+\eta^2\omega^2} +\varepsilon_{\mathrm{opt}}
\end{equation}
\begin{equation}\label{equ:epsilonimagharm}
 \varepsilon_\mathrm{i}(\omega) = \frac{Ne^{2}}{V \varepsilon_0M}\frac{\eta\omega}{(\omega_0^2-\omega^2)^2+\eta^2\omega^2}.
\end{equation}
\begin{equation}
  p(t) = j(t) \mathcal{E}(t),
\end{equation}
\begin{equation}\label{equ:dissipatecurrent}
 j(t) = -\frac{\partial D}{\partial t}
      = -\frac{\partial }{\partial t} \varepsilon \varepsilon_0 \mathcal{E}(t)
      = \varepsilon_0 \mathcal{E}(t) \left(\im \omega \varepsilon_\mathrm{r} - \omega \varepsilon_\mathrm{i} \right).
\end{equation}
\begin{equation}
 \overline{p}=\frac{1}{T} \int_0^T \mathcal{E}(t) j(t) \:\dd t,
\end{equation}
\begin{equation}
 \frac{1}{2} \varepsilon_0 \varepsilon_\mathrm{i} \omega\mathcal{E}_0^2.
\end{equation}
\begin{equation}\label{equ:epsilonimicr}
\varepsilon_\mathrm{i} (h \nu) \propto \sum_\veck M^2 \updelta\big[E_\con(\veck) - E_\val(\veck)-h \nu\big],
\end{equation}

%================================================================== 
\section{Superconductivity}
\begin{equation}\label{equ:criticalfieldmag}
 B_\crit(T) = B_\crit(0)\left[1-\left(\frac{T}{\Tc}\right)^{2}\right].
\end{equation}
\begin{equation}
 \oint \mathbfcal{E} \:\dd\mathbf{l} = -\frac{\dd\varPhi_B}{\dd t},
\end{equation}
\begin{equation}
 \frac{ \partial \mathbf{j}}{\partial t}=\frac{n_\mathrm{s} q^2}{m_\mathrm{s}}\mathbfcal{E},
\end{equation}
\begin{equation}\label{equ:lon21}
\frac{\partial}{\partial t}\left(\frac{m_\mathrm{s}}{n_\mathrm{s} q^2}\curl  \mathbf{j}+\mathbf{B}\right)=0.
 \end{equation}
 \begin{equation}\label{equ:lon22}
 \frac{\partial}{\partial t} \left(\int \frac{m_\mathrm{s}}{n_\mathrm{s} q^2}\curl  \mathbf{j} \:\dd\mathbf{A} +  \int \mathbf{B} \:\dd\mathbf{A} \right)
 = \frac{\partial}{\partial t}\left(\oint \frac{m_\mathrm{s}}{n_\mathrm{s} q^2} \mathbf{j} \:\dd l+ \int \mathbf{B} \:\dd\mathbf{A} \right) = 0.
 \end{equation}
\begin{equation}\label{equ:london2}
 \frac{m_\mathrm{s}}{n_\mathrm{s} q^2}\curl  \mathbf{j}+\mathbf{B}=0.
\end{equation}
\begin{equation}
 \curl  \mathbf{B}=\mu_0\, \mathbf{j}.
\end{equation}
\begin{equation}
\curl \curl\mathbf{B} = \mu_0 \curl  \mathbf{j}
                                 =-\frac{\mu_0 n_\mathrm{s} q^2}{m_\mathrm{s}}\mathbf{B}.
\end{equation}
\begin{equation}\label{equ:bfielddecay}
 \Laplacian\mathbf{B}=\frac{\mu_0 n_\mathrm{s} q^2}{m_\mathrm{s}}\mathbf{B}.
\end{equation}
\begin{equation}
 \Laplacian\mathbf{j}=\frac{\mu_0 n_\mathrm{s} q^2}{m_\mathrm{s}}\mathbf{j}.
\end{equation}
\begin{equation}\label{equ:londonpenetration}
\lambda_\mathrm{L}=\sqrt{m_\mathrm{s} / \mu_0 n_\mathrm{s} q^2}.
\end{equation}
\begin{equation}\label{equ:ginzlan}
 \varPsi(\vecr)= \varPsi_0(\vecr) \:\e^{\im\phi (\vecr)},
\end{equation}
 \begin{equation}\label{equ:bcstc}
 \Tc=1.13 \DebT\exp \frac{-1}{g(\EF)V},
 \end{equation}
\begin{equation}
 \oint \frac{\mathbf{p}}{h} \:\dd\vecr=n,
\end{equation}
\begin{equation}
 \oint \mathbf{p}-q\mathbf{A} \:\dd\vecr=nh.
\end{equation}
\begin{equation}\label{equ:coherence1}
 \frac{m_\mathrm{s}}{n_\mathrm{s} q} \oint \mathbf{j}\:\dd\vecr - q  \oint \mathbf{A} \:\dd\vecr=nh.
\end{equation}
\begin{equation}
 \oint\mathbf{A} \:\dd\vecr = \int\curl\mathbf{A}\:\dd\mathbf{a}
                                                  = \int\mathbf{B}\:\dd\mathbf{a}
                                                  = \varPhi_B,
 \end{equation}
 \begin{equation}
 \frac{m_\mathrm{s}}{n_\mathrm{s} q^2} \oint \mathbf{j}\:\dd\vecr - \varPhi_B=n\frac{h}{q}.
\end{equation}
\begin{equation}
 \varPhi_B = n\frac{h}{q}.
\end{equation}

%================================================================== 
\section{Finite Solids and Nanostructures}
\begin{equation}
\varPsi(\vecr)=\varPsi(z)\,\varPsi(x,y).
\end{equation}
\begin{equation}\label{eq:exy}
 E_{xy} = \frac{\hbar^2 \,k_{xy}^2}{2\mel},
\end{equation}
\begin{equation}
 \varPsi(z) = A \:\e^{\im k_z z}+B\:\e^{-\im k_z z},
\end{equation}
\begin{equation}\label{eq:kquant}
 k_z d = n\uppi, \qquad n=1,2,3,\dots
\end{equation}
\begin{equation}\label{eq:ez}
 E_z = \frac{\hbar^2 k_z^2}{2\mel},
\end{equation}
\begin{equation}\label{eq:kquant2}
 2 k_z d + \varPhi_\mathrm{i} +\varPhi_\mathrm{v} =2 \uppi\, n, \qquad n=1,2,3,\dots
\end{equation}
\begin{equation}\label{equ:energyquantumdot}
 E_{\mathrm{min}}=\Eg + \frac{\hbar^2 \uppi^2}{2 \mu r^2}-\frac{1.8 e^2}{4 \uppi\, \varepsilon_0 \varepsilon r}
\end{equation}
\begin{equation}
 \psi_{\veck}(\vecr)=\e^{\im\veck\vecr}u_{\veck}(\vecr).
\end{equation}
\begin{equation}\label{equ:sstate}
 \psi_{\veck}(\vecr) = \e^{\Im (k_z) z} \:\e^{\im\mathbf{k'} \cdot \vecr}u_{\veck}(\vecr),
\end{equation}
\begin{equation}\label{eqn:nanodotmagnetis}
 \Delta E \approx V \mu_0 M^2.
\end{equation}




%================================================================== 


 \end{document}