\section{Neutrinos from reactors}
\label{sec:Reactor}
%\REDBLA{Version 0 by JEC 2/3/06}
%\REDBLA{updated by A. Bueno 23/3/06}
%\REDBLA{updated by JEC 16/10/06: this is a section now}
%\REDBLA{updated by JEC 20/10/06}
%\REDBLA{updated by JEC 24/10/06}
%
 %T. Marrodan Undagoitia  10/12/06 START Small corrections
%\REDBLA{
The KamLAND 1~kT liquid scintillator detector located at Kamioka in
Japan had measured the flux of 53 power reactors corresponding to
701~Joule/cm${}^{2}$ \cite{Araki:2004mb}. An event rate of
$365.2\pm23.7$ above 2.6~MeV for an exposure of 766~ton.y from this
nuclear power reactors was expected. The observed rate was 258 events
with a total of background of $17.8\pm7.3$. The clear deficit
interpreted in terms of neutrino oscillation enables a measurement
of $\theta_{12}$, the neutrino 1-2 family mixing angle
($\sin^2\theta_{12} = 0.31^{+0.02}_{-0.03}$) as well as the mass
squared difference $\Delta m^2_{12} = 7.9\pm0.3~10^{-5}$eV${}^2$
(error quoted at $1~\sigma$).

Future precision measurements are currently been
investigated. Running KamLAND
for 2-3 more years would gain 30\% (4\%) reduction in the spread of
$\Delta m^2_{12}$ ($\theta_{12}$).
%JEC 20/10/06 START include text from Schwetz & Petcov {Petcov:2006gy}
 Although it has been shown in sections \ref{sec:SN} and \ref{sec:Geo}
that $\bar{\nu}_e$ originated from nuclear reactors can be a serious
background for diffuse supernova neutrino and geoneutrino detections,
the Fréjus site can take benefit of the nuclear reactors located in
the Rh\^one valley to measure $\Delta m_{21}^2$ and $\sin^2\theta_{12}$.
%}
 %T. Marrodan Undagoitia  10/12/06 END Small corrections



In fact approximately 67\% of the total reactor
$\bar{\nu}_e$ flux at Fréjus originates from four nuclear power plants
in the Rhone valley, located at distances between 115~km and 160~km. 
The indicated baselines are particularly suitable for
the study of the $\bar{\nu}_e$ oscillations driven by $\Delta m_{21}^2$---they
are similar to those exploited in the KamLAND experiment.
%
The authors of \cite{Petcov:2006gy} have investigated the possibility to use
one module of MEMPHYS (147~kt fiducial mass) doped with Gadolinium (MEMPHYS-Gd) or the LENA detector, updating the previous work of \cite{Choubey:2004bf}. 
%T. Marrodan Undagoitia  10/12/06 START Small corrections
%\REDBLA{ 
 Above 3~MeV (2.6~MeV) the event rate is 59,980 (16,670) events/yr for
MEMPHYS-Gd (LENA), which is 2 orders of magnitude larger than the
KamLAND event rate.  
%}
%T. Marrodan Undagoitia  10/12/06 END Small corrections

  
\begin{figure}
\includegraphics[width=\columnwidth]{./figures/LENAMEMPHYS-reac-histogram.eps}
%
  \caption{The ratio of the event spectra in positron energy 
  in the case of oscillations with $\Delta m_{21}^2 = 7.9\times 10^{-5}$~eV$^2$ and
  $\sin^2\theta_{12} = 0.30$ and in the absence of oscillations, 
  determined using one year data of MEMPHYS-Gd and LENA located at Frejus. 
  The error bars correspond to $1\sigma$ statistical error.}
%  
\label{fig:LENAMEMPHYS-reac-histo}
\end{figure}

To test the sensitivity of the experiments the prompt energy 
spectrum is divided into 20 bins between 3~MeV
and 12~MeV for MEMPHYS-Gd and SK-Gd, and into 25 bins between 2.6~MeV and
10~MeV for LENA (\refFig{fig:LENAMEMPHYS-reac-histo}). 

A $\chi^2$ analysis taking into account the statistical and systematical errors shows that each of the two
detectors, MEMPHYS-Gd and LENA, if placed at Fréjus, would allow a
very precise determination of the solar neutrino oscillation
parameters $\Delta m_{21}^2$ and $\sin^2\theta_{12}$: with one year, the
3$\sigma$ uncertainties on $\Delta m_{21}^2$ and $\sin^2\theta_{12}$ can be
reduced respectively to less than 3\% and to approximately 20\% (see also \refFig{fig:reactor-sensitivities}). In comparison, the
Gadolinium doped Super-Kamiokande detector (SK-Gd) that might be envisaged in a near future can reach
a similar precision if the SK/MEMPHYS fiducial mass ratio of 1 to 7 is
compensated by a longer SK-Gd data taking time.
%
\begin{figure}
\includegraphics[width=\columnwidth]{./figures/memphys-sk-sol-lena.eps}
%
  \caption{The accuracy of the determination of $\Delta m_{21}^2$ and
  $\sin^2\theta_{12}$, which can be obtained using one year of data
  from MEMPHYS-Gd and LENA at Frejus, and from SK-Gd at Kamioka,
  compared to the current precision from solar neutrino and KamLAND
  data. We show the allowed regions at $3\sigma$ (2 d.o.f.) in the
  $\Delta m_{21}^2-\sin^2\theta_{12}$ plane, as well as the projections of the
  $\chi^2$ for each parameter.}
%  
\label{fig:reactor-sensitivities}
\end{figure}
%
Several years of reactor $\bar{\nu}_e$ data collected by 
MEMPHYS-Gd or LENA would allow a determination 
of $\Delta m_{21}^2$ and $\sin^2\theta_{12}$ with
uncertainties of approximately 1\% and 10\% at 3$\sigma$,
respectively. 

However, some caveat are worth to be mentioned. The prompt energy trigger of 3~MeV requires a very low PMT dark current rate in case of MEMPHYS detector. If the energy threshold is higher then the parameter precision decreases as can be seen on \refFig{fig:reactor-MEMPHYS-threshold} \cite{Schwetz:2006private}. The systematic uncertainties are also an important factor in the experiments under consideration, especially the determination of the
mixing angle (eg. the energy scale and the overall normalization).
%
\begin{figure}
\includegraphics[width=\columnwidth]{./figures/MEMPHYSGdreactorthreshold.eps}
%
  \caption{The accuracy of the determination of $\Delta m_{21}^2$ and
  $\sin^2\theta_{12}$, which can be obtained using one year of data
  from MEMPHYS-Gd as a function of the prompt energy threshold.}
%  
\label{fig:reactor-MEMPHYS-threshold}
\end{figure}
%
Anyhow the accuracies on the solar oscillation parameters, which can be
obtained in the high statistics experiments considered here are
comparable to those that can be reached for the atmospheric neutrino
oscillation parameters $\Delta m_{31}^2$ and $\sin^2\theta_{23}$ in future
long-baseline superbeam experiments like T2HK in Japan or SPL from
CERN to MEMPHYS. Hence, such reactor measurements would complete the
program of the high precision determination of the leading neutrino
oscillation parameters. 
%JEC 20/10/06 AND

%It has been shown that using \WC\ loaded with Gadolinium to increase by a factor 10 the neutron capture  one can expect 80\% (34\%) reduction of the spread of  $\Delta m^2_{12}$ ($\theta_{12}$) in 110~kT.y exposure at Kamioka using SuperKamiokande. 
%