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\author{pod}
\title{Física Nuclear i de partícules}
\date{Primavera, 2003}

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\renewcommand{\MM}{\mathcal{M}}

\begin{document}
\maketitle

Alcance $ R \le \frac{c \hbar}{M_x c^2} \ .$ \\
C-paridad $ \hat C \Ket{f \bar f, JLS} = (-1)^{s+\ell} \Ket{f \bar f, JLS} \ .$

Sec. eficaz $ W = J \, N \sigma \ .$ \\
Atenuación $ \mu = n_t \sigma$, $\tilde\mu = \mu / \rho$ , $t = \rho x$: $$\deriv[I]{x} = - \mu I \ , \quad I(x) = I_0 \expon{-\mu x} \ .$$
$$ \lambda = \frac1\mu \ , \quad P(x) \dx = \left| \frac{\dd I}{I_0} \right| = \mu\expon{-\mu x}\dx \ . $$


\sep
\begin{align*}
x'     & = \gamma(x - v t)            \ , & ct' & = \gamma(ct-vx/c)      \ , \\
u'_x   & = \frac{u_x-v}{1-u_x v/c^2}  \ , & \vu & = \frac{\vp c^2}{E}    \ , \\
p'_x   & = \gamma ( p_x - v E / c^2 ) \ , & E'  & = \gamma ( E - v p_x ) \ , \\
E      & = m \gamma(u) c^2            \ , & \vp & = m \gamma(u) \vu      \ , \\
m^2c^4 & = E^2 - p^2c^2               \ , & s   & = E_\text{cm}^2 = E^2 - \vP_\text{cm}^2 \ , \\
\end{align*}


\sep
Rutherford 
$ \deriv[\sigma]{\Omega} = \left( \frac{z_1 Z_2\, e^2}{2 m v_0^2} \right)^{\!\!2} \frac1{\sin^4\frac\theta2}$, $T = \frac{z_1 Z_2\,e^2}{d_\text{min}}$ 

Lippman-Schwinger $\hat T = \hat V + \hat V \hat G_0 \hat T \approx \hat V$,
$$ \deriv[\sigma]{\Omega} = (2\pi\hbar)^4 \left(\frac{p_i}{v_i\hbar}\right)^2 \left| T_{E_i} (\vp_f, \vp_i) \right|^2 \ ,$$
%
$$ T_{E_i} (p_f, p_i) = \Braket{p_f | \hat T | p_i} = \Braket{ \hat V  } = - \frac{Z e^2}{2\pi q^2} F(\vq) \, , $$
%
$$F(\vq) = \int\!\!\dtr' \, \rho(\vr') \expon{i\vq\cdot\vr} = \frac{4\pi}{q} \int_0^\infty\!\!\! \dr\, r \rho(r) \sin qr \ , $$
%
$$ F(\vq) \mathop{\sim}_{\rho(\vr) = \rho(-\vr)} 1 - \frac16 q^2 \Braket{ r^2 } \ , $$
%
$$ \deriv[\sigma]{\Omega} = (2\pi)^4\left(\frac{p_i}{v_i}\right)^2 \left|\frac{Ze^2}{2\pi^2 q^2} F(\vq) \right|^2 \ . \quad q = 2p\sin\frac\theta2 \ .$$
%
$$\text{$e^-$--N:}\quad  \deriv[\sigma]{\Omega} = \left(\frac{Ze^2}{2E_i}\right) \frac{F^2(\vq)}{\sin^4 \frac{\theta}{2}}  \ , $$
%
$$\text{$e^-$--N (spin):}\quad  \deriv[\sigma]{\Omega} = \left(\frac{Ze^2}{2E_i}\right) \frac{\cos^2\frac\theta2}{\sin^4 \frac{\theta}{2}} F^2(\vq) \ , $$
%
$$\text{no rel.:}\quad  \deriv[\sigma]{\Omega} = \left(\frac{Ze^2}{4T}\right) \frac{F^2(\vq)}{\sin^4 \frac{\theta}{2}}  \ , $$

$$ \rho_F(r) = \frac{\rho_0}{1+\expon{(r-R)/a}} \ , \ a = 0.54\femto\metre \ , $$ 
%
$$  t_{0.9\rho_0 \to 0.1\rho_0} = a \ln81 = 2.4\femto\metre \ . $$
%
$$ \int\!\!\dtr \, \rho(r) = 1 \ , \quad \rho(r) = \left( 1 + w \frac{r^2}{R^2} \right) \rho_F \ . $$


\sep
\begin{align*} 
B(A,z) & = \Bigl[ Z M_p + N M_n - M(A,Z) \Bigr] c^2 \\
       & = \Bigl[Z M_{^1H} + N M_n - M(A,z) \Bigr] c^2 \,
\end{align*}
%
$$ M(A,Z) =  (A+\Delta)\! \cdot\! (1\atomicmass) \, , \, \frac{B}{A} \approx 8.5\mega\electronvolt \, (A \geq 12) \, . $$
%
\begin{align*} 
B(A,z) = & a_V A - a_s A^{2/3} - a_c \frac{Z^2}{A^{1/3}} \\
         & - a_A \frac{(A-2Z)^2}{A} + a_P \frac{\delta}{A^{1/2}} \ ,
\end{align*}
%
\begin{align*} 
a_V & = 15.56\mega\electronvolt \ , & a_S & = 17.27\mega\electronvolt  \ ,\\
a_C & = 0.697\mega\electronvolt \ , & a_A & = 23.285\mega\electronvolt \ , \\
a_P & = 12   \mega\electronvolt \ , &
%
\delta & = 
 \begin{cases}
  +1 & \text{si N,Z parells} \\
  -1 & \text{si N,Z senars}  \\
   0 & \text{si A senar}
 \end{cases} \ .
\end{align*}
\\
%
$$ Z_\text{más estable} = \frac A 2 \frac 1 {1+ \frac{a_C}{4a_A} A^{2/3}} = \frac{A/2}{1+0.00748 A^{2/3}} \ . $$


\sep
$$ \phi(\vr) = \int\!\!\dtr' \, \frac{\rho(\vr')}{|\vr - \vr'|} = \sum_{l=0}^\infty \sum_{m=-\ell}^\ell \frac{4\pi q_{\ell m}}{2\ell+1}  \frac{Y_{\ell m}(\theta,\phi)}{r^{\ell+1}} $$
%
$$ q_{lm} = \int\!\! \dtr' \, {r'}^\ell \rho(\vr') Y^*_{\ell m}(\Omega') \ . $$
%
$$ \text{m. eléctric:}\ Q_{\ell m} = e \, \Big< Q_{JM} \Big| \overbrace{\sum_{i=1}^Z r_i^\ell Y^*_{\ell m}(\hat r'_i)}^{\hat Q_{\ell m}} \Big| Q_{JM} \Big> . $$
%
$$ \text{m.quad.el:}\,Q = \sqrt{\frac{16\pi}{5e^2}} Q_{20} \, , \ Y_{20} = \sqrt{ \frac{5}{16\pi} } \frac{3z^2-r^2}{r^2} $$
%
\begin{align*}
\text{m.dip.magn.:} & \vmu = \frac12 \sum_i q_i \vx_i \times \vv_i = \sum_i \frac{q_i}{2m_i} \vl_i \ , \\
& \mu = \Braket{ J M=J | \hat \mu_z | J M=J } \ ,
\end{align*}
%
$$ \mu_\ell = g_\ell \mu_N \vl \ , \quad \mu_s = g_s \mu_N \vS \ , \ \mu = \mu_\ell + \mu_s \ ,$$
%
$$ g_\ell = \begin{cases} 0 & \text{prot} \\ 1 & \text{neut} \end{cases} \ , \
   g_s    = \begin{cases} 5.5856912(22) & \text{prot} \\ -3.8260837(18) & \text{neut} \end{cases} \ , $$

$ B(np)  = 2.22463(4)\mega\electronvolt$ , $ R = \sqrt{ \Braket{r^2} } = 2.1\femto\meter$, $V_0 = -35\mega\electronvolt$, $\pi(np) = +1$, $L=0$ ($96\%$) i $L=2$ ($4\%$), $Q = 0.00288(2)\barn$
%
C. spin $ 3 \frac{1}{r^2} (\vsigma_1 \vr) (\vsigma_2 \vr) - \vsigma_1 \vsigma_2 $, $\Delta L = \Delta S = 2 $.
%
\begin{multline*} 
V_{NN} (\vr) = V(r) + V_s(\vr)\vsigma_1 \vsigma_2 \\ + V_{LS}(r)\vL\vS + V_T S_{12}(\hat r)\ .
\end{multline*}

$\Ket{p} = \Ket{t_z = 1/2} \ , \ \Ket{n} = \Ket{t_z = 1/2} \ , q_i = \frac12(1+t_{z_i})$
%
$ \frac12 |N-Z| \leq T \leq \frac12 A $

\sep
$\dd N = - N w \dt$ , $w = \sum w_i$, $\tau = 1/w$, $T_{1/2} = \tau \, \ln 2$, $\Gamma = \hbar / \tau$, $B_i = \Gamma_i / \Gamma$
%
$$ N(t) = N_0 \, \expon{-w t} = N_0 \ , \expon{-t/\tau} \ , $$

\sep
$$ h\nu = \Delta E \frac{1+\frac{\Delta E}{2M_f c^2}}{1+\frac{\Delta E}{M_f c^2}} \approx \Delta E \left( 1 - \frac{\Delta E}{2 M_f c^2} \right) $$
%
$ \Delta S \uparrow \, \Rightarrow \tau \uparrow $ , $ J\ne0$, $\pi_{\gamma,\text{e}} = (-1)^J$,  $\pi_{\gamma,\text{m}} = (-1)^{m+1}$.


\sep
\begin{align*}
Q_{\beta^-} & = \bigl( M(A,Z) - M(A, Z+1) - M_e  \bigr) c^2     \\
            & = \bigl(\MM(A,Z)-\MM(A,Z+1)        \bigr) c^2 \ , \\
Q_{\beta^+} & = \bigl( M(A,Z) - M(A, Z-1) - M_e  \bigr) c^2     \\
            & = \bigl(\MM(A,Z)-\MM(A,Z-1) - 2 M_e\bigr) c^2 \ , \\
Q_{EC}      & = \bigl( M(A,Z) - M(A, Z-1) + M_e  \bigr) c^2     \\
            & = \bigl(\MM(A,Z)-\MM(A,Z-1)        \bigr) c^2 \ .
\end{align*}


\sep
\begin{align*}
Q_{\alpha} & = \left( M(A,Z)-M(A-4,Z-2)-M(4,2)   \right) c^2     \\
            & = \left( B(A-4,Z-2)+B(4,2) - B(A,Z) \right) c^2 \ ,
\end{align*}
%
\begin{equation*}
\frac B A < \frac{B(4,2)}{4} - \parcial[(B/A)]{A}  A = 7.075+7.6\E{-3} A \ .
\end{equation*}

\sep
Fusión $$V_\text{barrera} = Z_1 Z_2 e^2 / 2R \ , $$ 
%
$$ \frac 3 2 k T = V_\text{barrera}/ 2 \ . $$


\sep
\begin{align*}
 S_n & = \left( M(A-1, Z) + M_n - M(A,Z) \right) c^2  \ , \\
 S_p & = \left( M(A, Z-1) + M_p - M(A,Z) \right) c^2  \ ,
\end{align*}
%
Campo central $\psi_{n\ell m} (\vr) = R_{n\ell}(r) Y_{\ell m}(\hat r)$:
%
\begin{multline*}
 \Big\{ -\frac{\hbar^2}{2\mu} \left( \deriv[^2]{r^2} + \frac2r \deriv{r} \right) + \frac{\hbar \ell ( \ell+1 )}{2 \mu \, r^2} \\+ U(r) \Big\} R_{n\ell}(r) = E R_{n\ell}(r) \ , 
\end{multline*}
%
Oscil. $U(r) = \frac12 \mu\omega^2 r^2 - V_0$ , \\ 
$\ E_{n\ell m} = \hbar\omega \left( 2(n-1)+\ell \frac32 \right) - V_0$,  $deg = 2\ell+1$ \\
%
Wood-Saxon $$ U(r) = \frac{-U_0}{1+\expon{(r-R)/a}} \ , $$ $U_0 = 50\mega\electronvolt$,
$a = 0.65\femto\meter$, $R = A^{1/3}1.25\femto\meter$. \\
%
s-o: $$V_{L-S} = - V_1(r) \vl \vs = -\frac12 V_1(r) \left( \vj^2 - \vl^2 - \vs^2 \right) \ . $$

$$ \Braket{ \ell_z } = \Braket{ \frac{\vl \vj}{\vj^2} j_z } \ , \ \Braket{ S_z } = \Braket{ \frac{\vS \vj}{\vj^2} j_z } \ ,$$
%
\begin{align*}
\mu/\mu_N = &  g^{(\ell)} \frac{\frac12 \left[ j(j+1)+\ell(\ell+1)-3/4\right]}{j(j+1)} \\
           & + g^{(s)} \frac{\frac12 \left[ j(j+1)-\ell(\ell+1)+3/4\right]}{j(j+1)} \ , 
\end{align*}
%
\begin{align*}
\text{n}\quad \mu & = \mu_N \begin{cases} -1.913 & j=\ell+1/2 \\ 1.913\frac{j}{j+1} & j = \ell-1/2  \end{cases} \ , \\
\text{p}\quad \mu & = \mu_N \begin{cases} j+2.293 & j=\ell+1/2 \\ \frac{j(j-1.293)}{j+1} & j = \ell - 1/2 \end{cases} \ .
\end{align*}


\sep
$ \Ket{u} = \ket{t_z = 1/2}$, $\Ket{d} = \ket{t_z =-1/2}$,
%
$$ Y = B+S+C+\tilde B + T \ , \ I_3 = Q - Y\big/2 \ , $$
%
$$ \mu_q = 2\frac{q e}{2m_q\hbar} S_{q,z} = q \frac{m-p}{m-q} \mu_N  , $$
%
Gluons $\Ket{r\bar g}$, $\Ket{r\bar b}$, $\Ket{g\bar r}$, $\Ket{g\bar b}$, $\Ket{b\bar r}$, $\Ket{b\bar g}$, $\frac1{\sqrt{2}}\Ket{r\bar r-g\bar g}$, $\frac1{\sqrt2}\Ket{r\bar r - g\bar g}$.
%
$$ V(r) \sim - \frac{\alpha_s(r)}{r} + \lambda r \ , \ \alpha_s(r<0.1\femto\meter) \sim 0.15\pm0.03 \ . $$


\sep
$$ \Ket{K^0_1, \vp=0} = \frac1{\sqrt2} \left( \Ket{\bar K^0, \vp=0} + \Ket{K^0 , \vp=0} \right) \ ,$$
$$ \Ket{K^0_2, \vp=0} = \frac1{\sqrt2} \left( \Ket{\bar K^0, \vp=0} - \Ket{K^0 , \vp=0} \right) \ ,$$
%
$$ K^0_S = \frac1{\sqrt{1+|\varepsilon|^2}} \left( K_1^0 - \varepsilon K_2^0 \right) \ , $$
$$ K^0_L = \frac1{\sqrt{1+|\varepsilon|^2}} \left( K_1^0 + \varepsilon K_2^0 \right) \ , $$
$$ \varepsilon \approx 2.3\E{-3} \ . $$


\hrule\sep
$ \hbar c = 197.327 \mega\electronvolt \femto\meter $, $e^2 = \alpha \, \hbar c $, $\alpha = \frac{e^2}{\hbar c} = \frac1{137} \ .$

$\mu_N = \frac{e\hbar}{2m_p c} \ .$

$ R \sim 1.2\femto\metre \ A^{1/3} $ , $ R \approx (1.12 A^{1/3} - 0.86 A^{-1/3})\femto\metre \ .$

$ 1 \barn = 10^{-28}\squaren\meter = 10^{-24}\centi\squaren\meter \ .$

\end{document}
