#This file was created by <fpedrera> Sun Feb 11 22:59:24 2001
#LyX 0.12 (C) 1995-1998 Matthias Ettrich and the LyX Team
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\language spanish
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\paperorientation portrait
\leftmargin 15pt
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\rightmargin 10pt
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\secnumdepth 3
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\quotes_language english
\quotes_times 2
\papercolumns 2
\papersides 1
\paperpagestyle default

\layout Section*

tema 1.
 Estructura Cristalina.
\layout Standard
\align left 
Posicion de un punto de la red: 
\begin_inset Formula \( \vec{r}=n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Volumen de la celda unidad: 
\begin_inset Formula \( V=\left| \vec{a}\cdot \vec{b}\times \vec{c}\right|  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Distancia entre planos (SC, BCC, FCC): 
\begin_inset Formula \( d_{hkl}=\frac{a}{\sqrt{h^{2}+k^{2}+l^{2}}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Red Ortorrómbica: 
\begin_inset Formula \( d_{hkl}=\frac{1}{\sqrt{\frac{h^{2}}{a^{2}}+\frac{k^{2}}{b^{2}}+\frac{l^{2}}{c^{2}}}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Estructuras cristalinas simples.
 
\series default 
i) NaCl (FCC).
 Cl: 
\begin_inset Formula \( 000,\frac{1}{2}\frac{1}{2}0,\frac{1}{2}0\frac{1}{2},0\frac{1}{2}\frac{1}{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Na: 
\begin_inset Formula \( \frac{1}{2}\frac{1}{2}\frac{1}{2},00\frac{1}{2},0\frac{1}{2}0,\frac{1}{2}00 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 ii) CsCl (SC) Cs: 
\begin_inset Formula \( 000 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Cl: 
\begin_inset Formula \( \frac{1}{2}\frac{1}{2}\frac{1}{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 iii) hcp: Factor de empaquetamiento: 
\begin_inset Formula \( c/a=(8/3)^{1/2} \)
\end_inset 


\begin_inset Formula \( \bullet  \)
\end_inset 

 iv) Estructura diamante (FCC).
 
\begin_inset Formula \( 000,\frac{1}{2}\frac{1}{2}\frac{1}{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 v) ZnS (2FCC).
 Zn: 
\begin_inset Formula \( 000,0\frac{1}{2}\frac{1}{2},\frac{1}{2}0\frac{1}{2},\frac{1}{2}\frac{1}{2}0 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 S: 
\begin_inset Formula \( \frac{1}{4}\frac{1}{4}\frac{1}{4},\frac{1}{4}\frac{3}{4}\frac{3}{4},\frac{3}{4}\frac{1}{4}\frac{3}{4},\frac{3}{4}\frac{3}{4}\frac{1}{4} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Volúmenes celdas primitivas: i) SC: 
\begin_inset Formula \( V=a^{3} \)
\end_inset 

 ii) BCC: 
\begin_inset Formula \( V=a^{3}/2 \)
\end_inset 

 iii) FCC: 
\begin_inset Formula \( V=a^{3}/4 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Coefiiente de dilatación lineal: 
\begin_inset Formula \( L'=L_{0}(1+\alpha \Delta T) \)
\end_inset 


\layout Section*

tema 2.
 Difracción.
\layout Standard
\align left 
Ley de Bragg: 
\begin_inset Formula \( 2d\sin \theta =n\lambda  \)
\end_inset 

, intensidad máxima 
\begin_inset Formula \( n=1 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Red Real 
\begin_inset Formula \( \rightarrow  \)
\end_inset 

 
\begin_inset Formula \( a \)
\end_inset 

, Red Recíproca 
\begin_inset Formula \( \rightarrow  \)
\end_inset 

 
\begin_inset Formula \( 2\pi /a \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Número de electrones en la red 
\layout Standard
\align left 
periódica: 
\begin_inset Formula \( n(\vec{r})=\sum _{G}n_{G}\exp (-i\vec{G}\cdot \vec{r}) \)
\end_inset 

 con 
\begin_inset Formula \( n_{G}=\frac{1}{V_{c}}\int _{c}dVn(\vec{r})\exp (-i\vec{G}\cdot \vec{r}) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Vector de la red recíproca: 
\begin_inset Formula \( G=h\vec{b}_{1}+k\vec{b}_{2}+l\vec{b}_{3} \)
\end_inset 

 con 
\begin_inset Formula \( \vec{b}_{1}=2\pi \frac{\vec{a}_{2}\times \vec{a}_{3}}{\vec{a}_{1}\cdot \vec{a}_{2}\times \vec{a}_{3}} \)
\end_inset 

, 
\begin_inset Formula \( \vec{b}_{2}=2\pi \frac{\vec{a}_{3}\times \vec{a}_{1}}{\vec{a}_{1}\cdot \vec{a}_{2}\times \vec{a}_{3}} \)
\end_inset 

, 
\begin_inset Formula \( \vec{b}_{3}=2\pi \frac{\vec{a}_{1}\times \vec{a}_{2}}{\vec{a}_{1}\cdot \vec{a}_{2}\times \vec{a}_{3}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \vec{b}_{i}\cdot \vec{a}_{j}=2\pi \delta _{ij} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Amplitud de 
\emph on 
Scattering
\emph toggle 
: 
\begin_inset Formula \( F=\sum _{G}\int dVn_{g}e^{i(\vec{G}-\Delta \vec{k})\vec{r}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Condición de difracción: 
\begin_inset Formula \( \Delta \vec{k}=\vec{G} \)
\end_inset 

, si elástica 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( 2\vec{k}\vec{G}=G^{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( d_{hkl}=2\pi /\left| \vec{G}\right|  \)
\end_inset 

.
 Si 
\begin_inset Formula \( \Delta \vec{k}=\vec{G} \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( F_{G}=N\int _{c}dVn(\vec{r})e^{-i\vec{G}\cdot \vec{r}}=NS_{G} \)
\end_inset 

, con 
\begin_inset Formula \( S_{G}\equiv  \)
\end_inset 

 Factor de Esctructura 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( n(\vec{r})=\sum ^{s}_{j=1}n_{j}(\vec{r}-\vec{r}_{j})=\sum ^{s}_{j=1}n_{j}(\vec{\rho }) \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( S_{G}=\sum _{j}\int dVn_{j}(\vec{r}-\vec{r}_{j})\exp (-i\vec{G}\cdot \vec{r}) \)
\end_inset 


\begin_inset Formula \( =\sum _{j}\exp (-i\vec{G}\cdot \vec{r}_{j})\int dvn_{j}(\vec{\rho })\exp (-i\vec{G}\cdot \vec{\rho }) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Factor atómico de forma: 
\begin_inset Formula \( f_{j}=\int dVn_{j}(\vec{\rho })\exp (-i\vec{G}\cdot \vec{\rho }) \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( S_{G}=\sum _{j}f_{j}\exp (-i\vec{G}\cdot \vec{r}) \)
\end_inset 

 y también 
\begin_inset Formula \( S_{G}=\sum _{j}f_{j}\exp \{-i(hn_{1}+kn_{2}+ln_{3}) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Anillos de difracción: 
\begin_inset Formula \( d_{hkl}=\lambda /2 \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( \sin \theta _{max}=\pi /2 \)
\end_inset 

, aparecen anillos para 
\begin_inset Formula \( d_{hkl}\geq \lambda /2 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Factor atómico de forma, distribución esférica de carga: 
\begin_inset Formula \( f_{j}=4\pi \int _{0}^{\infty }drr^{2}n_{j}(r)\frac{\sin Gr}{Gr} \)
\end_inset 

 
\begin_inset Formula \( \bullet \int _{0}^{\infty }dxx^{n}e^{-ax}=\frac{\Gamma (n+1)}{a^{n+1}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \Gamma (n+1)=n\Gamma (n)=n! \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 BCC: 
\begin_inset Formula \( S_{G,BCC}=f_{BCC}\left( 1+(-1)^{h+k+l}\right)  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 FCC: 
\begin_inset Formula \( S_{G,FCC}=f_{FCC}\left( 1+(-1)^{h+k}+(-1)^{k+l}+(-1)^{h+l}\right)  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Intensidad de difracción: 
\begin_inset Formula \( I=S_{G}\cdot S_{G}^{*} \)
\end_inset 


\layout Section*

tema 3.
 Enlace Cristalino.
\layout Standard
\align left 

\series bold 

\begin_inset Formula \( U_{T}=U_{atrac}+U_{repuls} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Potenciales repulsivos:
\series default 
 i) Potencial de Lenard-Jones: 
\begin_inset Formula \( U_{R}=B/R^{12} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 ii) Potencial de Born-Mayer: 
\begin_inset Formula \( U^{'}_{R}=\lambda \exp (-R/\rho ) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Cristales inertes (
\emph on 
Van der Waals
\emph toggle 
):
\series default 
 Potencial de interacción de una pareja de átomos: 
\begin_inset Formula \( U_{ij}=B/R_{ij}^{12}-A/R_{ij}^{6}=4\epsilon \left[ \left( \sigma /R_{ij}\right) ^{12}-\left( \sigma /R_{ij}\right) ^{6}\right]  \)
\end_inset 

, con 
\begin_inset Formula \( \sigma =(B/A)^{1/6}\equiv  \)
\end_inset 

 parámetro de distancia; 
\begin_inset Formula \( \epsilon =A^{2}/4B\equiv  \)
\end_inset 

 parámetro de energía 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía total: 
\begin_inset Formula \( U_{T}=2N\epsilon \left[ \sum _{j\neq i}\left( \sigma /R_{ij}\right) ^{12}-\sum _{j\neq i}\left( \sigma /R_{ij}\right) ^{6}\right]  \)
\end_inset 

 
\begin_inset Formula \( \rightarrow  \)
\end_inset 

 
\begin_inset Formula \( R_{ij}=\rho _{ij}R \)
\end_inset 

.
 
\begin_inset Formula \( R\equiv  \)
\end_inset 

 distancia entre vecinos más próximos 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( U_{T}(R)=2N\epsilon \left[ A_{12}(\sigma /R)^{12}-A_{6}(\sigma /R)^{6}\right] \equiv  \)
\end_inset 

 Energía total de interacción o Energía de Cohesión.
 
\begin_inset Formula \( \bullet  \)
\end_inset 

 FCC: 
\begin_inset Formula \( A_{12}=12.13188 \)
\end_inset 

, 
\begin_inset Formula \( A_{6}=14.45392 \)
\end_inset 

.
 HCP: 
\begin_inset Formula \( A_{12}=12.13229 \)
\end_inset 

, 
\begin_inset Formula \( A_{6}=14.4589 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Parámetro de red: 
\begin_inset Formula \( \left. \frac{dU_{T}(R)}{dR}\right\rfloor _{R=R_{0}}=0\Rightarrow R_{0}=2\left( \frac{A_{12}}{A_{6}}\right) ^{1/6}\sigma  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Cristales iónicos: 
\series default 
Interacción electrostática: 
\begin_inset Formula \( U_{elec}=\sum _{j\neq i}\frac{\pm Q^{2}}{4\pi \epsilon _{0}R_{ij}} \)
\end_inset 

 (SI), 
\begin_inset Formula \( U_{elec}=\sum _{j\neq i}\frac{\pm Q^{2}}{R_{ij}} \)
\end_inset 

 (CGS) 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía total: 
\begin_inset Formula \( U_{Total}(R)=N\left[ -\frac{Q^{2}}{4\pi \epsilon _{0}R}\sum _{j\neq i}\frac{\pm 1}{P_{ij}}+z\lambda e^{-R/\rho }\right]  \)
\end_inset 

 con 
\begin_inset Formula \( \alpha =\sum _{i\neq j}\frac{\pm 1}{P_{ij}}\equiv  \)
\end_inset 

 Constante de Madelung, tb 
\begin_inset Formula \( \frac{\alpha }{R}=\sum _{j}\frac{1}{r_{ij}} \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( U_{TOT}(R)=-\frac{NQ^{2}\alpha }{4\pi \epsilon _{0}R}+zN\lambda e^{-R/\rho } \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía de Cohesión: 
\begin_inset Formula \( U(R_{0})=-\frac{NQ^{2}\alpha }{4\pi \epsilon _{0}R_{0}}\left[ 1-\rho /R_{0}\right]  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( Na+PI\rightarrow Na^{+}+e^{-} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( e^{-}+Cl\rightarrow Cl^{-}+AE \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( Na+Cl\rightarrow NaCl+E_{cohesi\acute{o}n} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Cristal iónico lineal: 
\begin_inset Formula \( U_{Tot}=N\left[ \frac{A}{R^{N}}-\frac{\alpha q^{2}}{R}\right]  \)
\end_inset 


\series bold 
 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Cristales covalentes:
\series default 
 Singlete-Enlazante.
 Triplete-Antienlazante.
 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Cristales Metálicos.
\layout Section*

Tema 4.
 Fonones.
 Vibraciones de la red.
 
\layout Standard
\align left 
Ecuación de movimiento: 
\begin_inset Formula \( m\frac{d^{2}u_{n}}{dt^{2}}=\sum _{p}C_{p}(u_{n+p}-u_{n}) \)
\end_inset 

, soluciones: Modos normales:
\series bold 
 
\series default 

\begin_inset Formula \( u_{n}(x,t)=u_{0}e^{i(kx-\omega t)} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Relación de dispersión: 
\begin_inset Formula \( \omega ^{2}=\sum _{p}2C_{p}\frac{1}{m}\left[ 1-\cos kpa\right]  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Interacción a primeros vecinos, 
\begin_inset Formula \( p=1 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( x=sa \)
\end_inset 

 con 
\begin_inset Formula \( s=0,1,... \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( u_{n+1}=u_{n}(x+a) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( m\frac{d^{2}u_{s}}{dt^{2}}=C(u_{s+1}+u_{s-1}-2u_{s}) \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( \omega =\sqrt{\frac{4C_{1}}{m}}\left| \sin \frac{ka}{2}\right|  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \omega _{max}=\omega (k=\pm \pi /a)=\sqrt{4C_{1}/m} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 1ª Zona de Brillouin: 
\begin_inset Formula \( -\pi \leq ka\leq \pi  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Condición de Bragg: 
\begin_inset Formula \( k_{max}=\pm k/a \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Velocidad de grupo: 
\begin_inset Formula \( v_{g}=\frac{d\omega }{dk} \)
\end_inset 

 (1D), 
\begin_inset Formula \( v_{g}=\nabla _{\vec{k}}\omega (\vec{k}) \)
\end_inset 

 (3D).
 
\begin_inset Formula \( v_{g}=\sqrt{\frac{C_{1}a^{2}}{m}}\left| \cos \frac{ka}{2}\right|  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Aproximación de onda larga: 
\begin_inset Formula \( ka\ll 1\Rightarrow \sin \frac{ka}{2}\simeq \frac{ka}{2} \)
\end_inset 

.
 Así pues, 
\begin_inset Formula \( \omega ^{2}=\frac{C_{1}}{m}k^{2}a^{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Dos átomos por celda primitiva: 
\series default 
Ecuaciones de movimiento: 
\begin_inset Formula \( m_{1}\frac{d^{2}u_{s}}{dt^{2}}=C(v_{s}+v_{s-1}-2u_{s}) \)
\end_inset 

, 
\begin_inset Formula \( m_{2}\frac{d^{2}v_{s}}{dt^{2}}=C(u_{s+1}+u_{s}-2v_{s}) \)
\end_inset 

.
 Soluciones (modos normales): 
\begin_inset Formula \( u_{s}=u_{0}e^{iska}e^{-i\omega t} \)
\end_inset 

, 
\begin_inset Formula \( v_{s}=v_{0}e^{iska}e^{-i\omega t} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Relación de dispersión: 
\begin_inset Formula \( \omega _{\pm }^{2}=C\left( \frac{m_{1}+m_{2}}{m_{1}m_{2}}\right) \pm C\sqrt{\left( \frac{m_{1}+m_{2}}{m_{1}m_{2}}\right) ^{2}-\frac{4}{m_{1}m_{2}}\sin ^{2}\frac{ka}{2}} \)
\end_inset 

.
 
\begin_inset Formula \( \omega _{+}\rightarrow  \)
\end_inset 

 Rama óptica, 
\begin_inset Formula \( \omega _{-}\rightarrow  \)
\end_inset 

 Rama acústica.
 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Aproximación de onda larga 
\begin_inset Formula \( (ka\ll 1) \)
\end_inset 

: 
\begin_inset Formula \( \omega _{+}^{2}\simeq 2C\frac{m_{1}+m_{2}}{m_{1}m_{2}} \)
\end_inset 

.
 
\begin_inset Formula \( \omega _{-}^{2}\simeq \frac{C}{2(m_{1}+m_{2})}k^{2}a^{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( k_{max}=\pm \frac{\pi }{a} \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( \omega _{+}^{2}=2C/m_{1} \)
\end_inset 

.
 
\begin_inset Formula \( \omega _{-}^{2}=2C/m_{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Fonones: 
\series default 

\begin_inset Formula \( E=(n+1/2)\hbar \omega  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \vec{p}=\hbar \vec{k} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Scattering de un fonón: 
\begin_inset Formula \( \vec{k}'=\vec{k}+\vec{k}_{fon\acute{o}n}+\vec{G} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía total de los fonones: 
\begin_inset Formula \( U=\sum _{k}\sum _{p}\left\langle n_{k,p}\right\rangle \hbar \omega _{k,p} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Número medio de fonones: 
\begin_inset Formula \( \left\langle n\right\rangle =\frac{1}{e^{\hbar \omega /k_{B}T}-1} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( U=\sum _{p}\int d\omega D_{p}(\omega )\frac{\hbar \omega }{e^{\hbar \omega /k_{B}T}-1} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( C_{V}=\frac{\partial U}{\partial T} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Densidad de estados:
\series default 
 (1D) Un estado permitido por cada intervalo 
\begin_inset Formula \( \left( \frac{2\pi }{L}\right)  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( D(\omega )=\frac{L}{\pi }\frac{dk}{d\omega }=\frac{dN}{d\omega } \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 (2D) 1 estado por área 
\begin_inset Formula \( \left( \frac{2\pi }{L_{x}}\right) \left( \frac{2\pi }{Ly}\right)  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( D(\omega )=\frac{L^{2}}{2\pi }k\frac{dk}{d\omega } \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 (3D) 1 estado por volumen 
\begin_inset Formula \( \left( \frac{2\pi }{L_{x}}\right) \left( \frac{2\pi }{L_{y}}\right) \left( \frac{2\pi }{L_{z}}\right)  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( D(\omega )=\frac{V}{2\pi ^{2}}k^{2}\frac{dk}{d\omega } \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Modelo de Debye: 
\series default 
Dispersion lineal 
\begin_inset Formula \( \omega =vk \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( D(\omega )=\frac{V\omega ^{2}}{2\pi ^{2}v^{3}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( N=\int _{0}^{\omega _{D}}D(\omega )d\omega  \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 Freciencia de Debye (frecuencia máx.): 
\begin_inset Formula \( \omega _{D}^{3}=\frac{6\pi ^{2}v^{3}}{V}N \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( k_{D}=\omega _{D}/v \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía térmica (1 polarización): 
\begin_inset Formula \( U_{p}=\int d\omega D(\omega )f(\omega )E_{\omega } \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( U=U_{p}\times 3 \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( U=9Nk_{B}T\left( \frac{T}{\theta _{D}}\right) ^{3}\int _{0}^{x_{D}}dx\frac{x^{3}}{e^{x}-1} \)
\end_inset 

 con 
\begin_inset Formula \( x=\beta \hbar \omega  \)
\end_inset 

, 
\begin_inset Formula \( \theta _{D}=x_{D}T=\frac{\hbar }{k_{B}}v\left( \frac{6\pi ^{2}}{V}\right)  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( C_{V}=9Nk_{B}\left( \frac{T}{\theta _{D}}\right) ^{3}\int _{0}^{x_{D}}dx\frac{x^{4}e^{x}}{(e^{x}-1)^{2}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Ley 
\begin_inset Formula \( T^{3} \)
\end_inset 

 de Debye: 
\series default 
i) Altas temperaturas: 
\begin_inset Formula \( k_{B}T\gg \hbar \omega \Rightarrow x\ll 1 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( U=3Nk_{B}T \)
\end_inset 

, 
\begin_inset Formula \( C_{V}=3Nk_{B} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 ii) Bajas temperaturas: 
\begin_inset Formula \( k_{B}T\ll \hbar \omega \Rightarrow x\gg 1 \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( U=9Nk_{B}T^{4}\frac{\pi ^{4}}{15\theta _{D}^{3}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( C_{V}=\frac{12\pi ^{4}}{5}\frac{Nk_{B}}{\theta _{D}^{3}}T^{3} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Modelo de Einstein: 
\series default 

\begin_inset Formula \( U=3N\frac{\hbar \omega }{e^{\hbar \omega /k_{B}T}-1} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( C_{V}=\frac{3Nk_{B}(\hbar \omega /k_{B}T)^{2}}{(e^{\hbar \omega /k_{B}T}-1)^{2}}e^{\hbar \omega /k_{B}T} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 i) Altas temperaturas: 
\begin_inset Formula \( C_{V}\simeq 3Nk_{B} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 ii) Bajas temp.: 
\begin_inset Formula \( C_{V}\simeq 3Nk_{B}\left( \frac{\hbar \omega }{k_{B}T}\right) ^{3}e^{-\frac{\hbar \omega }{k_{B}T}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \theta _{E}=\frac{\hbar \omega _{E}}{k_{B}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Expansión térmica: 
\series default 

\begin_inset Formula \( \left\langle x\right\rangle =\frac{3g}{4c^{2}}k_{B}T \)
\end_inset 

, con 
\begin_inset Formula \( U(x)=cx^{2}-gx^{3}-fx^{4} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Conductividad térmica: 
\series default 

\begin_inset Formula \( K=\frac{1}{3}Cvl \)
\end_inset 

 con 
\begin_inset Formula \( C=C_{V}/V \)
\end_inset 

, 
\begin_inset Formula \( v= \)
\end_inset 

 velocidad media, y 
\begin_inset Formula \( l= \)
\end_inset 

 recorrido libre medio.
\layout Section*

Tema 5.
 Gas de electrones libres de Fermi.
\layout Standard
\align left 

\series bold 
Niveles de energía (1D): 
\series default 

\begin_inset Formula \( \psi _{n}=A\sin \frac{2\pi n}{L} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \epsilon _{n}=\frac{\hbar ^{2}}{2m}\left( \frac{n\pi }{L}\right) ^{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía de Fermi: 
\begin_inset Formula \( \epsilon _{F}=\frac{\hbar ^{2}}{2m}\left( \frac{N\pi }{2L}\right) ^{2} \)
\end_inset 

, con 
\begin_inset Formula \( 2n_{F}=N \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Distribución de Fermi-Dirac: 
\begin_inset Formula \( f(E)=\frac{1}{e^{(\epsilon -\mu )/k_{B}T}-1} \)
\end_inset 

.
 
\series default 
A
\series bold 
 
\begin_inset Formula \( T=0K\Rightarrow \mu =\epsilon _{F} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Gas de Electrones libres (3D): 
\series default 

\begin_inset Formula \( \psi _{k}(\vec{r})=e^{i\vec{k}\vec{r}} \)
\end_inset 

, con 
\begin_inset Formula \( k_{x}=0,\pm 2\pi /L,\pm 4\pi /L... \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \epsilon _{k}=\frac{\hbar ^{2}k^{2}}{2m} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \vec{p}=\hbar \vec{k} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( k_{F}=\left( \frac{3\pi ^{2}N}{V}\right) ^{1/3} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía de Fermi: 
\begin_inset Formula \( \epsilon _{F}=\frac{\hbar ^{2}}{2m}k_{F} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

Velocidad de Fermi: 
\begin_inset Formula \( v_{F}=\frac{\hbar }{m}\left( \frac{3\pi ^{2}N}{V}\right) ^{1/3} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Densidad de estados: 
\begin_inset Formula \( N(\epsilon )=\frac{V}{3\pi 2}\left( \frac{2m\epsilon }{\hbar ^{2}}\right) ^{3/2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( D(\epsilon )=\frac{dN}{d\epsilon }=\frac{V}{2\pi ^{2}}\left( \frac{2m}{\hbar ^{2}}\right) ^{3/2}\epsilon ^{1/2}=\frac{3N}{2\epsilon } \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Capacidad calorífica de un gas de electrones: 
\begin_inset Formula \( C_{e^{-}}=\frac{d}{dT}\left\langle U\right\rangle =\frac{d}{dT}\int _{0}^{\infty }d\epsilon \epsilon D(\epsilon )f(\epsilon ) \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( C_{e^{-}}\simeq \frac{1}{3}\pi ^{2}\frac{3N}{2\epsilon _{F}}k_{B}^{2}T \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 Capacidad calorífica experimental: 
\begin_inset Formula \( C=\gamma T \)
\end_inset 

(
\begin_inset Formula \( e^{-} \)
\end_inset 

 libres)
\begin_inset Formula \( +\alpha T^{3} \)
\end_inset 

 (Debye) 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Ley de Ohm: 
\series default 

\begin_inset Formula \( \vec{F}=m\frac{d\vec{v}}{dt}=\hbar \frac{dk}{dt}=-e(\vec{E}+\vec{v}\wedge \vec{B}) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Desplazamiento de la esfera de Fermi: 
\begin_inset Formula \( \delta \vec{k}=\frac{-e\vec{E}}{\hbar }t \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Densidad de corriente: 
\begin_inset Formula \( \vec{j}=nq\vec{v}=\sigma \vec{E} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Conductividad eléctrica: 
\begin_inset Formula \( \sigma =\frac{ne^{2}\tau }{m} \)
\end_inset 

, con 
\begin_inset Formula \( \tau \equiv  \)
\end_inset 

 tiempo medio entre choques.
 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Resistividad eléctrica: 
\begin_inset Formula \( \rho =1/\sigma  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Recorrido libre medio: 
\begin_inset Formula \( l=v_{F}\tau  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Resisitividad experimental: 
\begin_inset Formula \( \rho =\rho _{fon\acute{o}n}+\rho _{defectos} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Movimiento en un CEM: 
\series default 
Ecuación de movimiento: 
\begin_inset Formula \( \hbar \left( \frac{d}{dt}+\frac{1}{\tau }\right) \delta \vec{k}=\vec{F} \)
\end_inset 

 tb: 
\begin_inset Formula \( m\left( \frac{d}{dt}+\frac{1}{\tau }\right) \vec{v}=-e(\vec{E}+\vec{v}\wedge \vec{B}) \)
\end_inset 

.
 Soluciones: 
\begin_inset Formula \( v_{x}=-\frac{e\tau }{m}E_{x}-\omega _{c}\tau v_{y} \)
\end_inset 

, 
\begin_inset Formula \( v_{y}=-\frac{e\tau }{m}E_{y}-\omega _{c}\tau v_{x} \)
\end_inset 

, 
\begin_inset Formula \( v_{z}=\frac{-e\tau }{m}E_{z} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Frecuencia de ciclotrón: 
\begin_inset Formula \( \omega _{c}=\frac{eB}{mc} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Efecto Hall: 
\series default 
Campo Hall: 
\begin_inset Formula \( E_{y}=-\omega _{c}\tau E_{x}=-\frac{eB\tau }{m}E_{x} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Coeficiente Hall: 
\begin_inset Formula \( R_{H}=\frac{E_{y}}{j_{x}B}=-\frac{1}{ne} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Resistencia Hall: 
\begin_inset Formula \( \rho _{H}=-\frac{B}{ne}=\frac{E_{y}}{j_{x}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Conductividad térmica en metales: 
\begin_inset Formula \( K=\frac{\pi ^{2}nk_{B}T\tau }{3m} \)
\end_inset 


\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series default 
Número de Lorenz: 
\begin_inset Formula \( \frac{k}{\sigma }=\frac{\pi ^{2}}{3}\left( \frac{k_{B}}{e}\right) ^{2}T \)
\end_inset 

, 
\begin_inset Formula \( L=\frac{k}{\sigma T}=2.44\times 10^{-8}\frac{J^{2}}{K^{2}\cdot C^{2}} \)
\end_inset 


\layout Section*

Tema 6.
 Bandas de energía.
 
\layout Standard
\align left 

\series bold 
Potencial periódico.
 Modelo de 
\begin_inset Formula \( e^{-} \)
\end_inset 

 cuasilibres.

\series default 
 Gap en 
\begin_inset Formula \( k=\pm \pi /a=\pm \frac{1G}{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Ondas en 1ª zona de Brillouin: 
\begin_inset Formula \( \psi _{+}=2\cos \frac{\pi x}{a} \)
\end_inset 

, 
\begin_inset Formula \( \psi _{-}=2i\sin \frac{\pi x}{a} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Magnitud del gap de Energía: 
\begin_inset Formula \( U(x)=U\cos \frac{2\pi x}{a} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( E_{g}=\int _{0}^{1}dxU(x)\left\{ \left| \psi _{+}\right| ^{2}-\left| \psi _{-}\right| ^{2}\right\} =U \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Funciones de Bloch.
 
\series default 

\begin_inset Formula \( \psi _{\vec{k}}(\vec{r})=u_{\vec{k}}(\vec{r})e^{i\vec{k}\vec{r}} \)
\end_inset 

 con 
\begin_inset Formula \( u_{\vec{k}}(\vec{r})=u_{\vec{k}}(\vec{r}+\vec{T}) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Modelo de Kronig-Penney.
 
\series default 
Potencial periódico (PP): función escalón.
 Condiciones de contorno periódicas.
 Aprox: funciones 
\begin_inset Formula \( \delta  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Ecuación de ondas del 
\begin_inset Formula \( e^{-} \)
\end_inset 

 en un PP.
 
\series default 
Ec.
 de Schrödinger: 
\begin_inset Formula \( \left[ \frac{p^{2}}{2m}+U(x)\right] \psi (x)=\epsilon \psi (x) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( U(x)=\sum _{G}U_{G}e^{iGx} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \psi (x)=\sum _{k}C(k)e^{ikx} \)
\end_inset 

 con 
\begin_inset Formula \( k=\frac{2\pi n}{L} \)
\end_inset 

, 
\begin_inset Formula \( L \)
\end_inset 

 longitud del cristal.
 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Ecuación de ondas (Fourier) (Ecuación central): 
\begin_inset Formula \( (\lambda _{k}-\epsilon )C(k)+\sum _{G}U_{G}C(k-G)=0 \)
\end_inset 

 con 
\begin_inset Formula \( \lambda _{k}=\hbar ^{2}k^{2}/2m \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \psi _{k}(x)=\sum _{G}C(k-G)e^{i(k-g)x} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( u_{k}(x)\equiv \sum _{G}C(k-G)e^{-iGx} \)
\end_inset 

 Invariante bajo traslaciones de cristal 
\begin_inset Formula \( T \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \vec{k}\equiv  \)
\end_inset 

 Momento cristalino del 
\begin_inset Formula \( e^{-} \)
\end_inset 

: 
\begin_inset Formula \( \vec{k}+\vec{q}=\vec{k}'+\vec{G} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Energía del 
\begin_inset Formula \( e^{-} \)
\end_inset 

 libre: 
\begin_inset Formula \( \epsilon (k_{x},k_{y},k_{z})=(\hbar ^{2}/2m)(\vec{k}+\vec{G})^{2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Solución aprox.
 en la frontera: 
\begin_inset Formula \( k=\pm \frac{1}{2}G \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( \epsilon =\lambda \pm U=\frac{\hbar ^{2}}{2m}(\frac{1}{2}G)^{2}\pm U \)
\end_inset 

 
\begin_inset Formula \( \Rightarrow  \)
\end_inset 

 
\begin_inset Formula \( \epsilon =\frac{1}{2}(\lambda _{k-G}+\lambda _{k})\pm \left[ \frac{1}{4}(\lambda _{k-G}-\lambda _{k})^{2}+U^{2}\right] ^{1/2} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Nº de orbitales en una banda.
 
\begin_inset Formula \( 2N \)
\end_inset 

 
\series default 
con 
\begin_inset Formula \( N= \)
\end_inset 

 nº de celdas primitivas de parámetro 
\begin_inset Formula \( a \)
\end_inset 

.
\layout Section*

Tema 7.
 Cristales semiconductores.
\layout Standard
\align left 

\series bold 
Gap.
 
\series default 
Transiciones directas: 
\begin_inset Formula \( E_{g}=\hbar \omega _{g} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Transiciones indirectas: 
\begin_inset Formula \( E_{g}=\hbar \omega -\hbar \Omega  \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\series bold 
Ecuaciones de movimiento.
 
\series default 
Velodidad de grupo: 
\begin_inset Formula \( v_{g}=\frac{d\omega }{dk}=\frac{1}{\hbar }\frac{d\epsilon }{dk} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \vec{v}=\frac{1}{\hbar }\nabla _{\vec{k}}\epsilon (\vec{k}) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Fuerza externa aplicada sobre el 
\begin_inset Formula \( e^{-} \)
\end_inset 

: 
\begin_inset Formula \( \vec{F}=\hbar \frac{d\vec{k}}{dt} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Campo magnético: 
\begin_inset Formula \( \hbar \frac{d\vec{k}}{dt}=-e\vec{v}\times \vec{B} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 huecos: 
\begin_inset Formula \( \vec{k}_{h}=-\vec{K}_{e} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \epsilon _{h}(\vec{k}_{h})=-\epsilon _{e}(\vec{k}_{e}) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \vec{v}_{h}=\vec{v}_{e} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( m_{h}=-m_{e} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \hbar \frac{d\vec{k}_{h}}{dt}=e(\vec{E}+\vec{v}_{h}\times \vec{B}) \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Masa efectiva: 
\begin_inset Formula \( \frac{1}{m^{*}}=\frac{1}{\hbar ^{2}}\frac{d^{2}\vec{\epsilon }}{dk^{2}} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 
\begin_inset Formula \( \left( \frac{1}{m^{*}}\right) _{\mu \nu }=\frac{1}{\hbar ^{2}}\frac{d^{2}\vec{\epsilon }_{k}}{dk_{\mu }dk_{\nu }} \)
\end_inset 

 
\begin_inset Formula \( \bullet  \)
\end_inset 

 Frecuencia de ciclotrón: 
\begin_inset Formula \( \omega _{c}=\frac{eB}{m^{*}} \)
\end_inset 


\the_end