From 2f9bb6d4e443e47ae15727d9150c184b092b179d Mon Sep 17 00:00:00 2001 From: vladomiro Date: Tue, 7 Nov 2023 14:26:05 -0600 Subject: [PATCH] pres 7 modificados: .gitignore nuevos archivos: Mossbauer.jpg modificados: notas5.pdf modificados: notas5.tex modificados: pres7.pdf modificados: pres7.tex --- .gitignore | 0 .ç | Bin Lunch_Nuclear_220923.jpg | Bin Mossbauer.jpg | Bin 0 -> 72245 bytes README.md | 0 area.png | Bin axarinp.py | 0 axarinp.py~ | 0 azarin.jpg | Bin azarin.py | 0 beta_parabola.png | Bin beta_parabola2.png | Bin bethe.png | Bin binding.png | Bin bohm_aharanov.jpg | Bin bubble.jpg | Bin cascada.py | 0 cascada.py~ | 0 catodicos.png | Bin chamber.jpg | Bin cherenkov.jpg | Bin ciclotron.png | Bin compton.png | Bin cuad_nuclei.jpg | Bin deca2.py | 0 deca2.py~ | 0 decaa.py | 0 decaa.py~ | 0 decay.jpg | Bin decay2.jpg | Bin dispersion.eps | 0 elip.py | 0 estabilidad.png | Bin excitados.jpg | Bin fase.jpg | Bin fermi.jpg | Bin fotoelectrico.jpg | Bin frenamiento.png | Bin 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zUjMMVjnEF8)xAGwds|=^V|!bn19m_c?1g=>9}d7l=!QdZ7>+;>9ED?W98ReF zHEHd8HK=}N>7)j2BbNF!xcUFj?$_Xd$x}`PlZqv3F^rJ{xp z$1I)I@SP1!EE?S#_JcmvWJ}d3bKg=zqn4bd5{=|Kk^hld5V*s^M5=S zraS|Uo=Zi~uNL$^KG3!YogtRLeTGmbHH8drhMA5#}Im~j>pY~UEe ixK5+HdEa1kT>b%v?P&s+YM2Ng3NSe{3MC~)PeuylTw{j- diff --git a/notas5.tex b/notas5.tex old mode 100644 new mode 100755 index 2c8ba87..fb64175 --- a/notas5.tex +++ b/notas5.tex @@ -205,25 +205,25 @@ De hacer un análisis parecido al del decaimiento alfa, y si despreciamos al neu Podemos hacer el análisis de energía tomando en cuenta las tres partículas del lado derecho \begin{align*} - M_Pc^2 &= T_H + M_Hc^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_Pc^2 - M_Hc^2 - m_ec^2 - m_{\bar{\nu}_e}c^2 + 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*} Analizando este término se pueden ver las distintas opciones. Para que la emisión de un electrón suceda debe cumplirse que \begin{align*} - (M(A,Z)-M(A,Z+1)-m_{\nu_e})c^2 &\geq 0 \\ - \approx (M(A,Z)-M(A,Z+1))c^2 &\geq 0. + (M_P-M_H-m_{\nu_e})c^2 &\geq 0 \\ + \approx (M_P-M_H)c^2 &\geq 0. \end{align*} -Con la masa la del átomo completo. Dado que el núcleo hijo es muy pesado su energía cinética será muy reducida, también se puede omitir, quedando que toda la energía liberada corresponde a las energías cinéticas del electrón y el neutrino, $E\approx T_e+T_{\nu}$. +Con la masas $M_P$ y $M_H$ las de los átomos completos, incluyendo electrones. Dado que el núcleo hijo es muy pesado su energía cinética será muy reducida, también se puede omitir, quedando que toda la energía liberada corresponde a las energías cinéticas del electrón y el neutrino, $E\approx T_e+T_{\nu}$. La energía de desintegración para la emisión del positrón \begin{align*} - E &= (M_P - M_H - m_e - m_{\nu})c^2 \\ - E &= (M(A,Z) - M(A,Z-1) - 2m_e -m_{\nu_e})c^2 \\ - &\approx (M(A,Z) - M(A,Z-1) - 2m_e)c^2 + 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*} Con las mismas condiciones del caso pasado. Para captura electrónica @@ -234,13 +234,13 @@ Con las mismas condiciones del caso pasado. Para captura electrónica &\approx (M(A,Z) - M(A,Z-1))c^2 \end{align*} -En todos estos casos se desprecian las energías de ligadura (en $eV$) de los electrones en el átomo. El espectro de energías nos puede dar una idea de la masa de los neutrinos, pero por su pequeña masa no es una tarea fácil. +En todos estos casos se desprecian las energías de ligadura (en $eV$) de los electrones en el átomo. El espectro de energías nos puede dar una idea de la masa de los neutrinos, pero por su valor tan pequeño no es una tarea fácil. Si regresamos a la consideración del pozo de potencial, sabemos que hay una barrera centrífuga que de igual manera evita que ciertas configuraciones puedan salir del pozo. Si el momento angular orbital de la partícula que sale es $\ell=1$ será más complicado que escape. En cambio si $\ell=0$ será más sencillo. Entonces se dividen los decaimientos: \begin{itemize} -\item $L=0$, decaimiento beta permitido -\item $L>0$, decaimientos beta prohibidos ($L=1$ primero prohibido, $L=2$ segundo prohibio, etc.) +\item $L=0$, decaimiento $\beta$ permitido +\item $L>0$, decaimientos $\beta$ prohibidos ($L=1$ primero prohibido, $L=2$ segundo prohibio, etc.) \end{itemize} En todo esto hemos considerado la interacción del núcleo con el campo del electrón y el neutrino, a partir de sus valores podemos saber cuanto momento angular es emitido. El momento angular que se llevan es el que corresponde al momento angular orbital $L$ de ambas partículas. Considerando el momento angula inicial y final del núcleo tenemos dos posibilidades diff --git a/notas6.md b/notas6.md old mode 100644 new mode 100755 diff --git a/notas6.pdf b/notas6.pdf old mode 100644 new mode 100755 diff --git a/notas6.tex b/notas6.tex old mode 100644 new mode 100755 diff --git a/optica.jpg b/optica.jpg old mode 100644 new mode 100755 diff --git a/pmt_es.png b/pmt_es.png old mode 100644 new mode 100755 diff --git a/potencial_nuclear.jpg b/potencial_nuclear.jpg old mode 100644 new mode 100755 diff --git a/pres0.pdf b/pres0.pdf old mode 100644 new mode 100755 diff --git a/pres0.tex b/pres0.tex old mode 100644 new mode 100755 diff --git a/pres0.tex~ b/pres0.tex~ old mode 100644 new mode 100755 diff --git a/pres1.pdf b/pres1.pdf old mode 100644 new mode 100755 diff --git a/pres1.tex b/pres1.tex old mode 100644 new mode 100755 diff --git a/pres2.pdf b/pres2.pdf old mode 100644 new mode 100755 diff --git a/pres2.tex b/pres2.tex old mode 100644 new mode 100755 diff --git a/pres2.tex~ b/pres2.tex~ old mode 100644 new mode 100755 diff --git a/pres3.pdf b/pres3.pdf old mode 100644 new mode 100755 diff --git a/pres3.tex b/pres3.tex old mode 100644 new mode 100755 diff --git a/pres3.tex~ b/pres3.tex~ old mode 100644 new mode 100755 diff --git a/pres4.pdf b/pres4.pdf old mode 100644 new mode 100755 diff --git a/pres4.tex b/pres4.tex old mode 100644 new mode 100755 diff --git a/pres4.tex~ b/pres4.tex~ old mode 100644 new mode 100755 diff --git a/pres5.pdf b/pres5.pdf old mode 100644 new mode 100755 diff --git a/pres5.tex b/pres5.tex old mode 100644 new mode 100755 diff --git a/pres5.tex~ b/pres5.tex~ old mode 100644 new mode 100755 diff --git a/pres6.pdf b/pres6.pdf old mode 100644 new mode 100755 diff --git a/pres6.tex b/pres6.tex old mode 100644 new mode 100755 diff --git a/pres6.tex~ b/pres6.tex~ old mode 100644 new mode 100755 diff --git a/pres7.pdf b/pres7.pdf old mode 100644 new mode 100755 index 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{}^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 estadśiticamente + \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}{Contenido} % \tableofcontents %\end{frame} diff --git a/pres7.tex~ b/pres7.tex~ old mode 100644 new mode 100755 diff --git a/reactor.jpg b/reactor.jpg old mode 100644 new mode 100755 diff --git a/regiones.jpg b/regiones.jpg old mode 100644 new mode 100755 diff --git a/rot_spectrum.jpg b/rot_spectrum.jpg old mode 100644 new mode 100755 diff --git a/roti.jpg b/roti.jpg old mode 100644 new mode 100755 diff --git a/rutherford.jpg b/rutherford.jpg old mode 100644 new mode 100755 diff --git a/shells.png b/shells.png old mode 100644 new mode 100755 diff --git a/sincrotron.jpg b/sincrotron.jpg old mode 100644 new mode 100755 diff --git a/slac.png b/slac.png old mode 100644 new mode 100755 diff --git a/tarea1.pdf b/tarea1.pdf old mode 100644 new mode 100755 diff --git a/tarea1.tex b/tarea1.tex old mode 100644 new mode 100755 diff --git a/tarea1.tex~ b/tarea1.tex~ old mode 100644 new mode 100755 diff --git a/tarea2.pdf b/tarea2.pdf old mode 100644 new mode 100755 diff --git a/tarea2.tex b/tarea2.tex old mode 100644 new mode 100755 diff --git a/tarea2.tex~ b/tarea2.tex~ old mode 100644 new mode 100755 diff --git a/tarea3.pdf b/tarea3.pdf old mode 100644 new mode 100755 diff --git a/tarea3.tex b/tarea3.tex old mode 100644 new mode 100755 diff --git a/tarea3.tex~ b/tarea3.tex~ old mode 100644 new mode 100755 diff --git a/tarea4.pdf b/tarea4.pdf old mode 100644 new mode 100755 diff --git a/tarea4.tex b/tarea4.tex old mode 100644 new mode 100755 diff --git a/tarea4.tex~ b/tarea4.tex~ old mode 100644 new mode 100755 diff --git a/tarea5.pdf b/tarea5.pdf old mode 100644 new mode 100755 diff --git a/tarea5.tex b/tarea5.tex old mode 100644 new mode 100755 diff --git a/tarea5.tex~ b/tarea5.tex~ old mode 100644 new mode 100755 diff --git a/te-he-fallado-oppi.jpg b/te-he-fallado-oppi.jpg old mode 100644 new mode 100755 diff --git a/tevatron.jpg b/tevatron.jpg old mode 100644 new mode 100755 diff --git a/vandegraff.jpg b/vandegraff.jpg old mode 100644 new mode 100755

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