From d269907fc4b926ccfc21abca7db0c7c2dd3b02a3 Mon Sep 17 00:00:00 2001 From: Vladimir Date: Tue, 31 Oct 2023 15:20:44 -0600 Subject: [PATCH] notas 5 --- cuad_nuclei.jpg | Bin 0 -> 86535 bytes elip.py | 23 +++++ notas4.pdf | Bin 470236 -> 667213 bytes notas4.tex | 88 +++++++++++++++- notas5.pdf | Bin 550233 -> 562880 bytes pres7.pdf | Bin 0 -> 397731 bytes pres7.tex | 256 +++++++++++++++++++++++++++++++++++++++++++++++ pres7.tex~ | 46 +++++++++ rot_spectrum.jpg | Bin 0 -> 84786 bytes roti.jpg | Bin 0 -> 16016 bytes 10 files changed, 409 insertions(+), 4 deletions(-) create mode 100644 cuad_nuclei.jpg create mode 100644 elip.py create mode 100644 pres7.pdf create mode 100644 pres7.tex create mode 100644 pres7.tex~ create mode 100644 rot_spectrum.jpg create mode 100644 roti.jpg diff --git a/cuad_nuclei.jpg b/cuad_nuclei.jpg new file mode 100644 index 0000000000000000000000000000000000000000..1a6769631722049596cb22c21c154a4c8383a9be GIT binary patch literal 86535 zcmdS9by(Ej(=hy5mZiI!r5ou6mu{q_QDEsVB}838C8WDkN?{jXyG=L&LHw-7W0!H}RDg$@i31-KKy<-$S(v3M@P>Cm3ep#QLX zfQtr$4g!1u%Xc}!_prDJ4#xv1L>Fvnt_j9Q2XLgvzi^j-;b_lbKOiFmWO!Zu{DFNu zn-km>i$`H`Ki>>2-@gy} zAc!ynf_|p{{{6k|_wS$O5CmU=pdSDK=m(TSkisXxPyG*%w*-PHuR~D#z<+oyIS_O| z7J`@-1D%7NaqYlCiRbPPK?l_kL~aW~v{Mj7Vt+DkpblFPRm?+>C73ItClHjE4?+AM zz_!ExGHxQE@Xx&cUuFKx-|uBe6N2O6VgCT&gFgfj0)dZ@AR#0qAR;3nBO@gtB_*ez zp(3ZCrXVGyVxXd?rK6{(C!=I!VxVK9p`)k68iB!q4nBeyfgq+MCncx*e+|F;AZjA~ zIyfyHMh)Rn!{F4g-vbakaD31;Y#{vw7!d&>=p`O8sHXdO#fiZ0C5Q|Tl&IiTz=_+C zf7Sl~_=yh@gE73yQaMasUqEiFJi8ZEocBUaUnQENcSUJfYmX*`opwbndGsKZqV^4Y zMfv4I-QK>7CcA{hJ8ZG8M@o&<-xam2y)9OEBvlA19G)HvO|D&*vAzE^^jI-DZBEf-cfzvo zX`#m*^`6iay*Ihc7}kls64zq&MYT!p)@-uRl!9N?D77-`o%y@vN9TFJscI%3P;hrK znC>W0X4r7Rf>6*OAk(_5-OLNA3&74Ay1?);B(nh_s}MvaEaZQt6m-YfJD`skpN2?8 z$oY(AItK;1O| zp;`|FAC~ocfPwJH8T?w7K=d%pxZQxL^pN##g}m@{StFfmCuZxwV!%2>&{ttD@39eET5UGIG{WgIz)?q15rclCyv<+y z6@{484ak>*7@R&(>7DS9sVI;yku)$1@_Or#ORO?;RWv~ywJs(jpf8|?(DW{0uJiC6 z2M`b^04I_0@=D3TW_29kgINV*JQlv@LQdp|o|NOdO{TZN@9O0;UVIPi8o-D`oNRo> zc-&tN$oXcb=>9(x5<6lDnNEyMZ_B%u!4$EbQjF3P9kOpp()3)dPd@ zuFxuS=_^n8}9Pm!x8Aj{1i@K?q^TKm4O7OPv1jeQ3{QcbwpOFeKbQ zm)U9bPt`vkRA#}w1-XAnSUtJ7EpA@14EQ;giKewbRewH^sp`CRxyy*?zicP%P;nHF zfI%5sF?+{qiO6R+@^~Q#AJPLJ4^hF8xN3aJoF}LNxBgfUA;7l|-oW5$Hbo*{h)UKx zvGKJa7FCbSA800={tTQRh6K(Cq3|&f4%D^DAK2_;QLtS2a6FOGAb%Od>c;PRG2y!3Gfa!X9SFkcVa5Mhkm&2InaxgE7Kks3Ci`SKo@(DZ1}8@qI<^w| zsy*z9yK8m24K0=Iu1g%~=6?Zv>&H)dX3xLtdrWXo#u?iOtS2i~OCIQ{C3h~%a;+6? z{iF6bU=&g&V=iZ3$MJ#d6Fd~z{i_LhGSl<&p8x2msUN8$u@n)|YPpo1h{XZGFj-q2 z7#s=;f_w$zxxWN>^6gIugyyEJ_Q4k=BL~NUqM{l3887H4| zaX%M=0X38o)x$LoCnBIZmLHAVBPSFnWzDu&@10N}^Tep<#TT%M_D(i=C0L#3$z4}_ z+KhtmC}nC17lY#eD8cRX@jH6{5D;=kfa}2t1+pEH%yBx$#{p-7RDX?}oY2Fu2d-6{ zQl${Y0kcVw=p4X60Dzwmw8_ANQCI>Ct32m(mW1Dv={S0WRm;(pz+23 zE+gDBU*&J-@nA?WVB1#Ai84MO3Y#v{v3?|bErw;{0R~`LAb0X~Iq@BA@IaoLW_dxY zq7f@6wb(LdP}uo3!^Arb2u}kxf)epmEEjkgJC7De-Zc)hz#c3EyS$JLEVTf2_|)?| z!2bZ;Ab)?Kl*6MG(9?v?UK(2%fB^{1EgSKXrVum5QDE#4vSDaKP6C%1aXbjNMkn~m zaPJ}tG)E9G1=nHxr&9J>HL2@~6ez8EPo>RJQLhg92kNBRguSe4^I#r}FvE zZ84iaI0UnQlI)8c4R&(CDFly1&DHH8w(s=5FDkvo&5r(v5){a;d-eR)BU$l#CJkW| zICj|hsl*Vcz^myN(%{QjMlAzXo%Wa_t`-zCJN&k&zXX*PZAkw0AyS7h%ztKRJ;tJ3V1Xkmc1 zWwW-9IP5h*u>h+v*0C?ep`gHPVKuGuNOV9e1L9BP@FzlLE8-E@QIbU4l~-qXe?kEq z3(tm1|7jQx#^)v18_yx+j6(iLms7ZxZxCxJx`5%RX8Vrn-vyBAjsBB%K(543wc-iM zW$p_|MPX<71m@jSRQj-Pjz{9_AXUP!a0GB1Oc8eYOMvVML*kK%&Q%?&1h*C3o+B8Q z3o<8_@;c;_+y5g7K3Ht_pImV;1#tm8u)uqjMb`Ez3b3(`NJJ2$FpdYj6yQ-=Mx#bY ziv+n6(&Dm<0VG%a;q0mz7ye@hR$CiiqQ@Iun8S7lks<|ZF~mqb6rs__Yn5k= zUShD~Foc)#aE)*J?Si?UWKKn>uc5m&_&?TyoMk%|?7?hcvy z{WIlwCE2&{?o7)8uL0`~Lt-b1qU^ZY77d9-WMP1tY|U;LiFmb;Cj$ZO^g;9sljP-q zI01|1=lS@H1W$42Auu5I7>>o~syuV}m{1Z|UuiwgQe9vh*6_sN=HOvod>hchO=Gk% zQ{>H;f2}~mc`iw6uKepK3_^$7e6_OFI}GmAF7pJ;&%Exjx|^_f7czc~uGf5HfDp5KJR&OmLkQ_#TJzP(g-4%2)1Ryr{+ z+^U$m>xn|B@di+aVbf8J7Ds0X3Tq0x7BIwVG5y;{14|f)7?(D~b@JsU?-;c%T*r_H z`e=;Om3Ddr9vJ-c<1s9dSb-Er39oCg_#c&8oOIj>SR`98EtJ{&+`_ek(kDQ#+Q1}>FDP`yi z3W~}{&Bxk&BBgN!`r!C>X57`fUrnwQM}e(O=c$wnuqfEHV`QGbQ5>g`Fkw4`Le}wr zmZl|?r3TRot|eeV5IN*Jv7-Lib5cukF@)h-x?P!~P^z`xL@+4Z|b>s;Khyh1|YIJ;UFp!GIog6y4sWyHsb#s!5a~2n=^bsQBX`jO@ZLvF| z%S|Jn!+UN7avn$s67ec#wSg$EU1y0Ke3FUXmjU4AWq)m`nwAR(cUoT`_?6-{z?9{B zTR;Vo1=b__5SH?zd2ocF**A?ajGBScIe6-uEV7Os2xei(;j~kraxlb77L9`Fzm6ug zLxkDk{4PNpLa7jpfX?gi631>?(vJ1vDYw!Ep2zS&#sgdY7_U|2zS)S7Xb{(4B$pTsQqPI#=5bPXf_zNWDbVFTC zahFN|AqI-Rn>YR0U#hxbVa}b#zAu+SqE=^Kj2H=B7*mMkeNt(+b0GFJSR}{hKQoZojBf zF*^*-4on9}+qH@+J2LziRk5d0l%>Gmyfc(lV_v615^x7L_X;ss69#&lo0pw6wIvz| zo-&HL-qS=7Hlv5zFB-Vmx)iZ(?<*#$S@&NpNMuSo9iq4a7|W~G0-T4_mFw983XMWd%aH$HOYIAH2?{% z2PAp=m_Ok}kcpMq-fAZI1k==Xa~kKe5W-eozbJp8R%cc$2<(l+2712z&y{xMyv?Ur zRyuhxDBinOFQsT-149i2i_&Y5!q^IWxaN}14`mG2I>|YQ;QLIA{zg!hN1li-X%U-PNp}*-z#7HSdW)VPo%Sb;$AA~d0T40Pw>2B**k$+js-Vt0F~O4lHBz5~J>8AQ)k>MM%8^dYKp8 zn?K8Q)!o`M;|XM`=^jXj5Ps4zeh^ywtav!s=|$E%&tNxQddrstt{}#&Xt(Zp2xW1i zu;#&5cKAdXc7Y_)R;q&mHQwt;X*6H?01T%ti6K(wsXUS}^=$bg2IMely_JPFgUUTu z3?!`YlsE-nE3aHtUc)0s_{ksdV!%|_WkVp_JmZgy*p5);xx#!68I8j3q>2|PI*o3% zT1t&%{dnQtgK(0Qygi-(u2@y(A+Xb$XDsm^aj4glQuxKpsmx^N6oU5$kdt(@z!_A! 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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: extra modelos nucleares} +%\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:rot} + \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}{Contenido} +% \tableofcontents +%\end{frame} + +\end{document} diff --git a/pres7.tex~ b/pres7.tex~ new file mode 100644 index 0000000..3017174 --- /dev/null +++ b/pres7.tex~ @@ -0,0 +1,46 @@ +\documentclass[12pt]{beamer} +\usetheme{Berlin} +\usepackage[utf8]{inputenc} +\usepackage[spanish]{babel} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{braket} + +\usepackage{appendixnumberbeamer} + +%\setbeamerfont{page number in head}{size=\large} +%\setbeamertemplate{footline}{Diapositiva} +\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: extra modelos nucleares} +%\setbeamercovered{transparent} +%\setbeamertemplate{navigation symbols}{} +%\logo{} +%\institute{} 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