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-\frac{a_1}{c^2}A \\ - &+ \frac{a_2}{c^2} + \frac{a_3}{c^@} \frac{Z^2}{A^{\frac{1}{3}}} + \frac{a_4}{c^2}\frac{(N-Z)^2}{A} \pm \frac{a_5}{c^2}A^{-\frac{3}{4}} + &+ \frac{a_2}{c^2} + \frac{a_3}{c^2} \frac{Z^2}{A^{\frac{1}{3}}} + \frac{a_4}{c^2}\frac{(N-Z)^2}{A} \pm \frac{a_5}{c^2}A^{-\frac{3}{4}} \end{align*} \end{frame} diff --git a/pres_nuc1.pdf b/pres_nuc1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..42b943afe9e5bc8543dc3ad5b65e06f782693ef0 GIT binary patch literal 887649 zcmdRVbzIbI^XPYXS-PZ~r9-;gr8@+qMY5Q}Pg`h>U!p+vQO979|+T%G-0T!~dwh=W6dTpfLhL!Ugh z)Hh}pr|f%aITaWj|K{MSvV~x_oH^(A;wQ=+?Q~2c%KYQ=%OmzzWcAC41lm#Q);EEOI?x*?sRRX%vw~_#?x@DkNXrJM3PA`l@AwKx0}R!nOk*zYAJ(P?49*zu4W$)(#f)N$AQo8aR2!+Jucz~<*V;h%c$qxjFFBRZik&BbVFrxBxIS* zd6f6y-AX)4@XAaa3^Hq#ztn&#GN_)YY)YyK(A%QE;2~R=M&K@}pJxA2uOl)uJ7-Yq zJ!^;aAOlhUo}kH!@al0%BB68LIs$0pAI@YMzRhfM6)T}Bc#?6bUKDn*h!=xAe7{O{gM>R`T z+5R-K!b5Osb5m7NIHD=?>rraL`=J4wElz6=M2)1CnvKBx!F>Jm>JPq;P26;yGxsRA 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+\setbeamertemplate{footline}[frame number] + +\newcommand{\backupbegin}{ + \newcounter{finalframe} + \setcounter{finalframe}{\value{framenumber}} +} + +\newcommand{\backupend}{ + \setcounter{framenumber}{\value{finalframe}} +} + +\author{Física Nuclear y subnuclear } +\title{Física Nuclear: Radiación} +%\setbeamercovered{transparent} +%\setbeamertemplate{navigation symbols}{} +%\logo{} +%\institute{} +%\date{} +%\subject{} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +\begin{frame}{Fallos del modelo de capas} + \begin{itemize} + \item Momentos cuadrupolares mucho mayores que los predichos por el modelo + \item Deformando se pueden obtener tales momento cuadrupolares + \item Modos colectivos de excitación: oscilaciones + \item Modelo nuclear unificado + \end{itemize} +\end{frame} + +\begin{frame}{Momento cuadrupolar} + \begin{equation*} + \mathbb{Q} = Z\int d^3 r(3z^2-r^2)\rho(r) + \end{equation*} + Si es un elipsoide uniformemente cargado con $Ze$ + \begin{equation*} + \mathbb{Q} = \frac{2}{5}Z(b^2-a^2),\ b\parallel z + \end{equation*} + Con: + \begin{align*} + \overline{R} =& (1/2)(a+b)\\ + \Delta R =& b-a\\ + \delta =& \overline{R}/\Delta R\text{ tenemos }\\ + \mathbb{Q} =& \frac{4}{5}ZR^2\delta + \end{align*} +\end{frame} + +\begin{frame}{Momentos cuadrupolares en el experimento} + \begin{multicols}{2} + \begin{align*} + \mathbb{Q}_{red} =& \frac{\mathbb{Q}}{ZR^2}\\ + \mathbb{Q}_{red} =& \frac{4}{5}\delta + \end{align*} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.7\linewidth]{cuad_nuclei.jpg} + %\caption{Momentos cuadrupolares reducidos como función del número de nucleones impar. Las flechas muestran los lugares de capa cerrada. Imagen tomada y adaptada del libro de Henley con fines educativos.} + \label{fig:shell} + \end{center} + \end{figure} + \end{multicols} +\end{frame} + +\begin{frame}{Espectro rotacional} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.5\linewidth]{rot_spectrum.jpg} + \caption{Espectro rotacionel del núcleo deformado ${}^{170}Hf$, con valores de energía rotacionales obtenidos experimentalmente y teóricamente. Imagen tomada de \cite{Henley} con fines educativos.} + \label{fig:rot} + \end{center} + \end{figure} + \begin{equation*} + \Delta\phi \delta L_{\phi} \geq \hbar + \end{equation*} +\end{frame} + +\begin{frame}{Rotaciones} + \begin{multicols}{2} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.7\linewidth]{roti.jpg} + %\caption{Espectro rotacionel del núcleo deformado ${}^{170}Hf$, con valores de energía rotacionales obtenidos experimentalmente y teóricamente. Imagen tomada de \cite{Henley} con fines educativos.} + \label{fig:roti} + \end{center} + \end{figure} + Rotación alrededor del eje 1 + \begin{equation*} + H_{rot}= \frac{R^2}{2\mathbb{I}} + \end{equation*} + Traduciendo a mecánica cuántica: + \begin{align*} + \hat{H}_{rot}\psi=& \frac{\hat{R}^2}{2\mathbb{I}}\psi = E\psi\\ + \hat{R}^2Y_J^M =& J(J+1)\hbar^2Y_J^M, \\ J= 0,1,2,... + \end{align*} + Con la paridad dada por $(-1)^J$, sólo se aceptan valore par de $J$ + \begin{equation*} + E_J= \frac{\hbar^2}{2\mathbb{I}}J(J+1),\ J=0,1,2,... + \end{equation*} + \end{multicols} +\end{frame} + +\section*{Radiación nuclear} +\begin{frame}{Lo que sabemos hasta ahora} + \begin{itemize} + \item Los núcleos están compuestos porpprotones y neutrones + \item Protones y neutrones sienten las fuerzas: electromagnética, nuclear fuerte y nuclear débil + \item Núcleos de helio, electrones y fotones los hemos tratado, pero no hemos hablado más de ellos como radiación + \end{itemize} +\end{frame} + +\begin{frame}{Decaimiento alfa} + \begin{equation*} + {}^AX^Z \rightarrow {}^{A-4}Y^{Z-2} + {}^4He^2 + \end{equation*} + + \begin{equation*} + M_P c^2 = M_Hc^2 + T_H + M_{\alpha}c^2 + T_{\alpha}, + \end{equation*} +\end{frame} + +\begin{frame}{Análisis de energía} + \begin{equation*} + T_H + T_{\alpha} = (M(A,Z) - M(A-4,Z-2) - M(4,2))c^2 + \end{equation*} + + \begin{align*} + T_H =& \frac{1}{2} M_H v_H^2, \notag \\ + T_{\alpha} =& \frac{1}{2} M_{\alpha} v_{\alpha}^2, + \end{align*} + +\end{frame} + +\begin{frame}{Conservaciones} + \begin{align*} + M_H v_H =& M_{\alpha} v_{\alpha}, \notag \\ + \text{despejando, } v_H =& \frac{M_{\alpha}}{M_H} v_{\alpha} + \label{ec:vel} + \end{align*} + Por lo regular $M_H \gg M_{\alpha}$, entonces $v_H\ll v_{\alpha}$. + + \begin{align*} + T_H + T_{\alpha} =& \frac{1}{2}M_H \left( \frac{M_{\alpha}}{M_H} v_{\alpha} \right)^2 + \frac{1}{2} M_{\alpha} v_{\alpha}^2 \\ + =& \frac{1}{2} M_{\alpha} v_{\alpha}^2 \left( \frac{M_{\alpha}}{M_H} +1 \right) \\ + =& T_{\alpha}\frac{M_{\alpha} + M_H}{M_H} + \end{align*} +\end{frame} + +\begin{frame}{Liberación de energía} + \begin{align*} + T_H &= T_{\alpha}\left( \frac{M_{\alpha} + M_H}{M_H}\right) - T_{\alpha} \\ + &= T_{\alpha}\left( \frac{M_{\alpha} + M_H}{M_H} - 1\right) \\ + &= T_{\alpha} \frac{M_{\alpha} + M_H - M_H}{M_H} = \frac{M_{\alpha}}{M_H}T_{\alpha}\ll T_{\alpha} + \end{align*} +\end{frame} + +\begin{frame}{Diversas energías} + \begin{equation*} + {}^AX^Z \rightarrow {}^{A-4}{Y^{*}}^{Z-2} + {}^4He^2 + \end{equation*} + + \begin{equation*} + {}^{A-4}{Y^{*}}^{Z-2} \rightarrow {}^{A-4}Y^{Z-2} + \gamma + \end{equation*} + + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.6\linewidth]{excitados.jpg} + \caption{Decaimineto por emisión $\alpha$ del ${}^{228}Th^{90}$ al ${}^{224}Ra^{88}$. Imagen tomada de Das y Ferbel.} + \label{fig:excitados} + \end{center} + \end{figure} +\end{frame} + +\begin{frame}{Un ejemplo} + \begin{equation*} + {}^{240}Pu^{94} \rightarrow {}^{236}U^{92} + {}^4He^2 + \end{equation*} +\end{frame} + +\begin{frame}{Ejemplo} + \begin{align*} + E &= (M(240,94) - M(236,92) - M(4,2))c^2 \\ + &= 94m_p + 146m_n +B.E.(240,94) -92m_p-144m_n \\ + &- B.E.(236,92) - 2m_p -2m_n -B.E.(4,2) \\ + &= B.E.(240,94) - B.E.(236,92) - B.E.(4,2) \\ + &= -1813.4501\ MeV + 1790.4103\ MeV + 28.2956 \\ + &\approx 5.2558\ MeV + \end{align*} +\end{frame} + +\begin{frame}{Penetración de barrera} + \begin{itemize} + \item Para $A\approx 200$ barrera coulombiana de $\sim 20-25\ Mev$ + \item La energía cinética del $\alpha$ es $\sim 5\ MeV$ + \item Decaimiento alfa es un fenómeno de tunelaje + \end{itemize} +\end{frame} + +\begin{frame}{Penetración de barrera} + \begin{equation*} + {}^{232}Th^{90} \rightarrow {}^{228}Ra^{88} + {}^4He^2 + \end{equation*} + \begin{itemize} + \item $\tau = 1.39\times 10^{10}\ \text{años}$ + \item $R=r_0(232)^{1/3} fm. \approx 7.37 \times 10^{-15}m.$ + \item + \end{itemize} + +\end{frame} + +\begin{frame}{Coeficiente de transmisión} + \begin{align*} + T =& \frac{\frac{4k_1k}{(k_1+k)^2}}{1+\left[ 1 + \left( \frac{\kappa^2 - k_1k}{\kappa(k_1+k)}\right)^2 \right]} \\ + \text{con } k_1 =& \left[ \frac{2M_{\alpha}}{\hbar^2} (E+U_0) \right]^{\frac{1}{2}} \\ + k =& \left[ \frac{2M_{\alpha}}{\hbar^2} E \right]^{\frac{1}{2}} \\ + \kappa =& \left[ \frac{2M_{\alpha}}{\hbar^2} (V_0 - E) \right]^{\frac{1}{2}} + \end{align*} +\end{frame} + +\begin{frame}{Posibilidad de penetración de la barrera} + De afuera hacia adentro + \begin{equation*} + T\approx 4\times 10^{40} + \end{equation*} + De adentro hacia afuera (constante de decaimiento $\lambda$) + \begin{align*} + P(\text{emisión }\alpha) &\approx \frac{v_{\alpha}}{R}T \approx 6\times 10^{21}\frac{1}{seg} \times 4\times 10^{-40} \\ + &\approx 2.4\times 10^{-18}seg. + \end{align*} + +\end{frame} + +\begin{frame}{Decaimineto Beta} + \begin{itemize} + \item Fuerza nuclear débil + \item Conservaciones de número bariónico y leptónico + \item Características del neutrino + \item Núcleo con exceso de neutrones + \end{itemize} +\end{frame} + +\begin{frame}{Decaimiento Beta menos} + \begin{equation*} + {}^AX^Z \rightarrow {}^AY^{Z+1} + e^- +\bar{\nu_e} + \end{equation*} + + \begin{equation*} + n\rightarrow p + e^- + \bar{\nu_e} + \end{equation*} +\end{frame} + +\begin{frame}{Decaimineto Beta más} + \begin{equation*} + {}^AX^Z \rightarrow {}^AY^{Z-1} + e^+ +\nu_e + \end{equation*} + + \begin{equation*} + p\rightarrow n+e^+ + \nu_e + \end{equation*} +\end{frame} + +\begin{frame}{Captura electrónica} + \begin{equation*} + {}^AX^Z + e^- \rightarrow {}^AY^{Z-1} +\nu_e + \end{equation*} + + \begin{equation*} + p+e^- \rightarrow n + \nu_{e} + \end{equation*} + + La constante en todos: $\Delta A = 0$ y $|\Delta Z| = 1$ +\end{frame} + +\begin{frame}{Conservación de energía} + \begin{align*} + M(A,Z)c^2 &= T_H + M(A,Z-1)c^2 + T_{e^-} + m_ec^2 + T_{\bar{\nu}_e} + m_{\bar{\nu}_e}c^2 \\ + T_H + T_{e^-} + T_{\bar{\nu}_e} =& M(A,Z)c^2 - M(A,Z-1)c^2 - m_ec^2 - m_{\bar{\nu}_e}c^2 + \end{align*} + De esta forma + \begin{align*} + (M_P-M_H-m_{\nu_e})c^2 &\geq 0 \\ + \approx (M_P-M_H)c^2 &\geq 0. + \end{align*} + Decaimineto $\beta^+$ + \begin{align*} + E &= (M(A,Z) - M(A,Z-1) - m_e - m_{\nu})c^2 \\ + E &= (M_P - M_H - 2m_e -m_{\nu_e})c^2 \\ + &\approx (M_P - M_H - 2m_e)c^2 + \end{align*} +\end{frame} + +\begin{frame}{Conservación de energía} + Captura electrónica + \begin{align*} + E &= (M_P + m_e - M_H - m_{\nu})c^2 \\ + E &= (M(A,Z) - M(A,Z-1) -m_{\nu_e})c^2 \\ + &\approx (M(A,Z) - M(A,Z-1))c^2 + \end{align*} + + No se toman en cuenta las energías de ligadura de los electrones en las capas atómicas. + +\end{frame} + +\begin{frame}{Barrera centrífuga de potencial} + \begin{itemize} + \item $L=0$, decaimiento $\beta$ permitido + \item $L>0$, decaimientos $\beta$ prohibidos ($L=1$ primero prohibido, $L=2$ segundo prohibio, etc.) + \end{itemize} + Un ejemplo + \begin{equation*} + {}^3H^1 \rightarrow {}^3He^2 + e^- + \bar{\nu_e},\ \Delta L = 1 + \end{equation*} + +\end{frame} + +\begin{frame}{Reglas de selección} + \begin{itemize} + \item $J_f = J_i + L$, es una transición de Fermi + \item $J_f = J_i + L + 1$, es una transición de Gamow-Teller + \end{itemize} + Ejemplo + \begin{equation*} + {}^{14}O^6 \rightarrow {}^{14}Ni^{*7} + e^- + \bar{\nu_e},\ \Delta I = 0 + \end{equation*} +\end{frame} + +\begin{frame}{Estabilidad} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.8\linewidth]{estabilidad.png} + \caption{Tabla de nucleones. Imagen de Hiroyuki Koura en el dominio público} + \label{fig:excitados} + \end{center} + \end{figure} +\end{frame} + +\begin{frame}{Esquema de decaimientos $\beta$} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.7\linewidth]{beta_parabola2.png} + \caption{Excesos de masa para los isóbaros con $A= 76$ que tienen decaiminetos $\beta$. Imagen adaptada de \cite{Poves} con licencia CC-BY 3.0} + \label{fig:parabola} + \end{center} + \end{figure} +\end{frame} + +\section*{Decaimiento Gama} +\begin{frame}{Decaimineto $\gamma$} + \begin{itemize} + \item Decaimiento a núcleos excitados + \item Regresado a estado base emitiendo $\gamma$ + \item Espacio entre niveles de $\sim 50\ keV$ + \end{itemize} +\end{frame} + +\begin{frame}{Características del decaimiento $\gamma$} + \begin{itemize} + \item El fotón con energía en el orden de $MeV$ + \item Puede llevarse al menos una unidad de $L$ + \item El núcleo pasa de un estado inicial $E_i$ a uno final $E_f$ + \end{itemize} +\end{frame} + +\begin{frame}{Análisis decaimiento $\gamma$} + \begin{equation*} + h\nu = E_i - E_f + \end{equation*} + + La energía del foton $=$ espaciamiento en niveles, pero qué sucede con la conservación de momento + \begin{equation*} + \frac{h\nu}{c} = Mv, + \end{equation*} +\end{frame} + +\begin{frame}{Análisis de energía} + \begin{align*} + E_i-E_f =& h\nu + \frac{1}{2}Mv^2 \\ + =& h\nu +\frac{1}{2M}\left( \frac{h\nu}{c} \right)^2 \\ + \text{reacomodando } h\nu =& \left( E_i - E_f - \frac{h^2 \nu^2}{2Mc^2} \right) = E_i - E_f - \Delta E_R, + \end{align*} +\end{frame} + +\begin{frame}{Niveles de energía} + $\partial E = \Gamma$ + \begin{align*} + \tau \Gamma &\approx \hbar \\ + \text{o diciéndolo de otra forma } \Gamma &\approx \frac{\hbar}{\tau} \approx \text{incertidumbre en }(E_i-E_f) + \end{align*} + + $\Delta E_R \ll \Gamma$ +\end{frame} + +\begin{frame}{Un caso} + \begin{itemize} + \item ${}^{50}Ti^{22}$ + \item $M\approx 46512.11\ MeV/c^2$ + \item $h\nu\gtrsim 100keV = 10^5 eV$ + \end{itemize} +\end{frame} + +\begin{frame}{Un caso} + \begin{equation*} + \Delta E_R = \frac{(h\nu)^2}{2Mc^2} = \frac{(10^5 eV)^2}{2(46.512\times 10^9 eV)} \approx 0.215\ eV + \end{equation*} + Considerando $\tau = 10^{-12}seg$ + \begin{equation*} + \Gamma \approx \frac{\hbar}{\tau} \approx \frac{6.582\times 10^{-22}MeV\cdot seg}{10^{-12}seg} = 6.582 \times 10^{-4} eV + \end{equation*} +\end{frame} + +\begin{frame}{Efecto Mössbauer} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.4\linewidth]{Mossbauer.jpg} + \caption{Rudolf Mössbauer} + \end{center} + \end{figure} +\end{frame} + +\begin{frame}{Niveles de energía y decaimiento $\gamma$} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.6\linewidth]{nivelesse.jpg} + \caption{Niveles de energía para el ${}^{72}Se^{34}$. Tomado de \cite{Krane}} + \label{fig:niveles} + \end{center} + \end{figure} +\end{frame} + +\begin{frame}{Conversión interna} + \begin{itemize} + \item Sale un rayo $\gamma$ del núcelo y excita un electrón del átomo + \item Electrón de alta energía + \item Espectro de energía cuantizado + \end{itemize} +\end{frame} + +\begin{frame}{Leyes de decaimiento} + \begin{itemize} + \item Tres tipos de decaimientos + \item Tiempo tratado estadísticamente + \item Probabilidad constante de decaimiento por segundo $\lambda$ + \end{itemize} +\end{frame} + +\begin{frame}{Ley de decaimiento} + \begin{equation*} + dN = N(t+dt)- N(t) = -N(t)\lambda dt + \end{equation*} + \begin{align*} + \frac{dN}{N} =& -\lambda dt,\\ + \int_{N_0}^N \frac{dN}{N} =& -\lambda \int_0^t dt, \\ + ln\frac{N(t)}{N_0} =& -\lambda t \\ + N(t) =& N_0 e^{-\lambda t} + \end{align*} +\end{frame} + +\begin{frame}{Escala de tiempo} + \begin{itemize} + \item Tiempo de vida media $t_{\frac{1}{2}}$ + \end{itemize} + \begin{align*} + N(t_{\frac{1}{2}}) =& \frac{N_0}{2} = N_0e^{-\lambda t_{\frac{1}{2}}} \\ + \text{de otra forma } \lambda t_{\frac{1}{2}} =& ln2 \\ + \text{entonces } t_{\frac{1}{2}} =& \frac{ln2}{\lambda} + \end{align*} +\end{frame} + +\begin{frame}{Tiempo de vida media y tiempo promedio} + \begin{align*} + \langle t \rangle = \tau =& \frac{\int_0^{\infty} t N(t) dt}{\int_0^{\infty} N(t) dt} \\ + =& \frac{N_0 \int_0^{\infty} t e^{-\lambda t} dt}{N_0\int_0^{\infty} e^{-\lambda t} dt} \\ + =& \frac{\lambda^{-2}}{\lambda^{-1}} = \frac{1}{\lambda} + \end{align*} + De esta forma $t_{\frac{1}{2}} = \tau (ln2)$. +\end{frame} + +\begin{frame}{Actividad} + \begin{equation*} + \mathcal{A} = | \frac{dN}{dt} | = \lambda N(t) = \lambda N_0 e^{-\lambda t} + \end{equation*} + \begin{itemize} + \item $1$ desintegración por segundo $= 1 Bq$ + \item La actividad de ${}^{226}Ra^{88}$, $3.7 \times 10^{10}\ Bq = 1Ci$ + \item Muestras con actividad en los $mCi$ y $\mu Ci$ + \item $1rd=10^6Bq$ + \end{itemize} +\end{frame} + +\begin{frame}{Varios proceso} + \begin{equation*} + \lambda = \lambda_1 + \lambda_2 + \lambda_3 + ... + \end{equation*} + \begin{equation*} + \frac{1}{t_{\frac{1}{2}}} = \frac{1}{(t_{\frac{1}{2}})_1} + \frac{1}{t_{(\frac{1}{2}})_2}+ \frac{1}{(t_{\frac{1}{2}})_3} + ... + \end{equation*} +\end{frame} + +\begin{frame}{Decaimienots en dos pasos} + \begin{align*} + -\frac{dN_1}{dt} &= \lambda_1 N_1 \\ + \frac{dN_2}{dt} &= \lambda_1 N_1 - \lambda_2 N_2 + \end{align*} + + \begin{align*} + N_1 =& N_{10}e^{-\lambda_1 t}\\ + N_2 =& N_{10}\frac{\lambda_1}{\lambda_2 - \lambda_1} (e^{-\lambda_1 t} - e^{-lambda_2 t}) + \end{align*} + $(t_{\frac{1}{2}})_2 \ll (t_{\frac{1}{2}})_1$ +\end{frame} + +\begin{frame}{Ejemplo} + \begin{itemize} + \item ${}^{226}Ra^{88}$ + \item Actividad inicial $3.7 \times 10^{10}\ Bq$ + \item Tiempo de vida media $t_{\frac{1}{2}} = 1600\text{ años} = 5.04576\times 10^{10}seg.$ + \item Actividad tras $500\text{ años} = 1.5768\times 10^{10} seg.$ + \end{itemize} +\end{frame} + +\begin{frame}{Calculo de la actividad} + \begin{equation*} + \mathcal{A}(t=1.5768\times 10^{10}seg.) = \lambda N_0 e^{-\lambda t} + \end{equation*} + \begin{equation*} + \mathcal{A}(t=1.5768\times 10^{10}seg.) = \mathcal{A}_0 e^{-\lambda t} + \end{equation*} + \begin{equation*} + \mathcal{A}(t=1.5768\times 10^{10}seg.) = (3.7\times 10^{10} Bq) e^{-\frac{ln2}{5.04\times 10^{10}seg.} (1.57\times 10^{10} seg.)} + \end{equation*} + $\mathcal{A}(t=1.5768\times 10^{10}seg.) \approx 2.3\times 10^{10}Bq$ +\end{frame} + +\begin{frame}{Radiación natural y artificial} + \begin{itemize} + \item ${}^{238}U$ y ${}^{232}Th$ con vidas medias en el orden de la edad del universo. + \item $4.5\times 10^9$ años y $1.4\times 10^{10}$ años + \item ¿Qué pasaría si tuvieran vidas medias mucho más cortas? + \item 1934 Pierre Joliot e Irene Curie bombardean $\alpha$'s del decaimiento del polonio bombardeando $Al$, producen ${}^{30}P$ + \item ${}^{30}P$ decae por emisión de positrones con $t_{1/2}= 2.5$ minutos. + \end{itemize} +\end{frame} + +\begin{frame}{Envenenamiento por Polonio} + \begin{itemize} + \item Alexander Litvinenko, miembro de la KGB + \item 1998 acusó publicamente a sus superiores por el intento de asesinato a Boris Berezovski + \item Berezovski era doctor en matemáticas aplicadas (1983) + \item Importación de Mercedes, dueño de la cadena ORT + \item Litvinenko noviembre del 2006, ${}^{210}Po$ + \end{itemize} +\end{frame} + +%\begin{frame}{Contenido} +% \tableofcontents +%\end{frame} + +\end{document}

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