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The dilution factors $df$ are needed to correct the raw asymmetries for the contributions of  the unpolarized nucleons in the target. %get a handle on the target polarization.   It is the kinematics dependent   ratio of cross-sections from the protons to cross-sections from all the materials in the $NH_3$   target sample. A dilution factor is necessary for each and every target load used during running of SANE.   In order to get dilution factors one must first get packing fractions. A packing fraction is essentially  the amount of target material in the target cup. Similar to dilution factors, a packing fraction is  needed for each target load used during the experiment. However a packing fraction is a percentage and  independent of kinematics. The packing fractions are obtained from HMS (High Momentum Spectrometer) data   where a carbon target was used. The carbon target is used because its cross-sections are well known and   it is very similar to nitrogen, which is a significant part of the $NH_3$ target sample. A detailed description   of how the packing fractions were obtained is given in the section on packing fractions.  The dilution factors originate from measuring the counts $N^{+,-}$ for each helicity from all materials in the target  region, seen by the detectors, forming a raw asymmetry  \begin{equation}\label{asymraw}  A_{raw} =   \frac{N^{+} - N^{-}}{N^{+} + N^{-}} = \frac{N^{+} - N^{-}}{sum_{all}} \nonumber  \end{equation}  Unpolarized counts cancel in the numerator, but not in the denominator, so the numerator is actually   $N^{+}_{proton} - N^{-}_{proton}$. We want the asymmetry for the proton only   \begin{equation}\label{prasymraw}  A_{proton} =   \frac{N^{+}_{proton} - N^{-}_{proton}}{sum_{proton}} \nonumber  \end{equation}  This can then be written as   \begin{equation}\label{prasymrawdf}  A_{raw} =   \frac{N^{+}_{proton} - N^{-}_{proton}}{sum_{proton}} * \frac{sum_{proton}}{sum_{all}} = A_{proton}*df \nonumber  \end{equation}  The role the dilution factors play is that they are the ratio of rates of free polarizable nucleons (proton)  to all nucleons composing the total target (helium, nitrogen, ...). This is kinematics dependant. For each target load,   used during the running of the experiment, a dilution factor is needed. The amount of time needed to obtain a  statistical error $\Delta A$ has a time dependence $t \propto \frac{1}{df}$.   The dilution factor, dependent on the 4-momentum transfer $Q^{2}$ and invariant mass of final states $W$, is defined  \begin{equation}\label{dilfac}  df(Q^2,W) =   %\frac{C_{1}\,\sigma_{1}(Q^2,W)}{C_{1}\,\sigma_{1}(Q^2,W) + C_{4}\,\sigma_{4}(Q^2,W) + C_{14}\,\sigma_{14}(Q^2,W)   %\nonumber  %+ \sum_i{N_{A}\,\sigma_{A}}(Q^2,W)}\nonumber  \frac{C_{1}\,\sigma_{1}(Q^2,W)}{\sum_{A}{C_{A}\,\sigma_{A}}(Q^2,W)} \nonumber  \end{equation}  Here, $A$ is the atomic number for all the nucleons making up the target sample and $C_{A}$ a constant that accounts for   the packing-fraction dependent density, effective length, and $A$. This calculation was previously used for the RSS   analysis ~\cite{rondon_df}.  The dilution factors were obtained in several steps. MC events are generated, on the order of $10^{6}$,   using the cross-section model F1F209 ~\cite{pbec_F1F21,pbec_F1F22}. The MC is GEANT based ~\cite{gwarren_mc}, applying both external   and internal radiative corrections ~\cite{motsai_rc}. The kinematic quantities are calculated using energy and position   reconstructed via artificial neural network. The standard set of cuts are applied to the MC, which are also used for   data. The $df(Q^{2},W),df(Q^{2},x)$ ratios are then calculated using the packing fractions that were obtained with the   HMS information. These ratios are binned similar to the data, based on the reconstructed energy resolution ~\cite{jrm_bin}.   The obtained $df$ ratios have an average value of 0.18, which is close to the expected value (Fig.~\ref{fig:dfWQsq23}). As   a check, some $df$ were calculated with the F1F209. There is very good agreement between these. The radiative corrections   don't seem to be significant in the kinematic regions of interest. This was checked by comparing the radiative corrected   cross-sections with the uncorrected Born cross-sections. The expected contribution to systematics is approximately 5\%.   \begin{figure}[htb*]  %\centering  %\hskip-5mm  \includegraphics[width=\columnwidth,clip]{../measured_a/dil_f/dfF1F209W_para_Qsq3.00_BornRC_limstats_021012_enhanced.eps}  \caption{\label{fig:dfWQsq23}  Dilution factor for one of the $NH_3$ target loads as a function of $W$(GeV). This is for a parallel target magnet field   configuration, for both energies. The apparent structures for $W < 2$ represent some of the resonances for the proton.}  \end{figure}