\magnification=\magstep1

$$
{\sigma} \quad = \quad {\rm reaction\ cross\ section\ } ({\rm cm}^2)
$$


$$
{\rm number\ density\ of\ protons\ } \quad = \quad { {\rho} \over {m_p} } 
$$

$$
\eqalign{
L \quad &= \quad { {1} \over ({\rm number\ density}) \thinspace ({\rm cross\ section}) } \cr
  &\phantom{=} \cr
        &= \quad \quad { {m_p} \over { \rho \sigma } } \cr
}
$$

$$
\eqalign{
  \sigma(\nu,{\rm p^+}) \quad &\sim \quad 10^{-50} \thinspace {\rm m}^2 \cr
  \sigma(\gamma,{\rm p^+}) \quad &\sim \quad 10^{-29} \thinspace {\rm m}^2 \cr
}
$$


$$
^{37}{\rm Cl} \thinspace + \thinspace \nu \quad \rightarrow \quad
        ^{37}{\rm Ar} \thinspace + \thinspace e^-
$$ 


$$
^{71}{\rm Ga} \thinspace + \thinspace \nu \quad \rightarrow \quad
        ^{71}{\rm Ge} \thinspace + \thinspace e^-
$$ 

$$
\eqalign{
  \nu_e \thinspace + \thinspace n \quad &= \quad
         p^{+} \thinspace + \thinspace e^{-} \thinspace + \thinspace \gamma \cr
  \nu_{\mu} \thinspace + \thinspace n \quad &= \quad
         p^{+} \thinspace + \thinspace {\mu}^{-} \thinspace + \thinspace \gamma \cr
}
$$


$$
{\rm p^{+}} \thinspace + \thinspace {\rm e^{-}} \quad \rightarrow \quad
     {\rm n} \thinspace + \thinspace {\rm \nu_e} 
$$


$$
\eqalign{
  \sigma(\nu_e,{\rm e^-} \rightarrow \nu_e,{\rm e^-}) \quad &= \quad 
      9.33 \times 10^{-48} \thinspace {\rm m}^2 \thinspace [E / {\rm 10\  MeV}] \cr
  \sigma(\bar{\nu}_e,{\rm e^-} \rightarrow \bar{\nu}_e,{\rm e^-}) \quad &= \quad
      3.88 \times 10^{-48} \thinspace {\rm m}^2 \thinspace [E / {\rm 10\  MeV}] \cr
  \sigma(\bar{\nu}_e,{\rm p^+} \rightarrow {\rm e^+},{\rm n}) \quad &= \quad
      9.77  \times 10^{-46} \thinspace {\rm m}^2 \thinspace {[E / {\rm 10\  MeV}]}^2 \cr
}
$$


$$
{\rm GPE} \quad = \quad - \thinspace {{3}\over{5}} \thinspace {{G M^2}\over{R}} 
$$




\bye
