\section{Introduction}
Study of exact solutions of Einstein's Field Equations is an
important part of the theory of General Relativity. This importance
is not only from more formal mathematical aspects associated with
the theory (e.g.  the classification of space-times) but  also from
the  growing importance  of the  application of general relativity
to astrophysical phenomena.  For example, exact solutions may offer
physical insights that numerical solutions cannot.  Present time
trend of analyzing different aspects of black hole (BH) solutions
did lead us to grow our interests in cleaner $(2+1)$ dimensional
gravity. Discovery of BTZ BH~\cite{Bandos1} ignited the light first.
Through this $(2+1)$ dimensional model if we need to explore the
foundations of classical and quantum gravity we would not find any
Newtonian limit and no propagating degrees of freedom will arise. In
literature, very easy-to-find works in this aspect comprise the
study of quasi normal modes of charged dilaton BHs in $(2+1)$
dimensional solutions in low energy string theory with asymptotic
anti de-Sitter space times~\cite{Fernanado1}. Hawking radiation from
covariant anomalies in $(2+1)$ dimensional BHs~\cite{Nam1} is
another beautiful example. Lastly, we must also name the study of
branes with naked singularities analogous to linear or planar
defects in crystals and showing that zero branes in AdS space times
are ``negative mass BHs!"~\cite{Zanelli1}. Taking charged gravastars
as an alternative to charged BHs in $(2+1)$ AdS space times is
already investigated~\cite{Rahman1}. Extensions of BTZ BH solutions
with charge are also available in the literature. These are obtained
by employing nonlinear Born Infield electrodynamics to eliminate the
inner singularity~\cite{Mazharimousavi1}. The non-static charged BTZ
like BHs in $(N+1)$ dimensions have also  been studied \cite{Ghosh1}
which in its static limit, for $N=2$, reduces to $(2+1)$ BTZ BH
solutions.

Study of interior solutions in $(2+1)$ dimension~\cite{Rahman2}
shows that even the noncommutative-geometry-inspired BTZ BH is not
free from any singularity. Study of interior solutions are farely
found in literature. For example, solutions of C. Wolf~\cite{wolf}
and S. Yazadjiev~\cite{yaz}, solutions in te framework of
Brans-Dicke theory of gravity by S.M.Kozyrev \cite{kozyrev} and  new
class of solutions corresponding to BTZ exterior spacetime by Sharma
{\it et. al.}~\cite{Sharma}, which is regular at the centre and it
satisfies all the physical requirements except at the boundary where
the authors propose a thin ring of matter content with negative
energy density so as to prevent collapsing. The discontinuity of the
affine connections at the boundary surface provide the above matter
confined to the ring. Such a stress-energy tensor is not ruled out
from the consideration of Casimir effect for massless fields.

The purpose of the present work is to find exact interior solutions for
perfect fluid model both with and without cosmological constant, $\Lambda$.
The motivation for doing so is provided by the
fact that the assumption of equation of state (EoS), $p=m\rho$, which
seems to be very reasonable for describing the matter distribution
in the study of relativistic   objects like stars~\cite{ayan,cruz},
wormholes~\cite{fr1,kim} and gravastars~\cite{Rahman1,fr3}.

The structure of our work is as follows: In sec~(\ref{Ein_Field}),
we derive  required Einstein equations. Sec.~ \ref{Interior})
constitutes of different interior solutions for various cases of EoS.
Lastly, in sec.~(\ref{conclusion}) a brief conclusion is provided.

\section{Einstein field equations in  $(2 + 1)$ dimension}\label{Ein_Field}
 We  take the static metric to describe the interior region  of a $(2+1)$
dimensional space time as
 \begin{equation}\label{metric}
 ds^{2} = -e^{2\nu(r)}dt^{2}+ e^{2\mu(r)}dr^{2} + r^{2} d\theta^{2},
 \end{equation}
where $\nu(r)$ and $\mu(r)$ are the two unknown metric functions. We
take the perfect fluid   form of the energy momentum tensor
\begin{equation}\label{stress}
T_{ij}=  diag(-\rho,p,p),
\end{equation}
where $\rho$ is energy density and  $p$ is pressure.
Einstein's field equations with a
cosmological constant, $\Lambda$, for the space-time metric
(Eq.(\ref{metric})) together with the energy momentum tensor given in
Eq.(\ref{stress}) may be written as
\begin{eqnarray}\label{field_eq_1}
 2\pi\rho +\Lambda&=& \mu^{\prime}e^{-2\mu(r)}/r,\\
 2\pi p -\Lambda&=& \nu^{\prime}e^{-2\mu(r)}/r,\\
 2\pi p -\Lambda&=& e^{-2\mu}\left({\nu^{\prime\prime}}+{\nu^{\prime 2}}-\nu^{\prime}\mu^{\prime}\right).
 \end{eqnarray}
Here superscript `$\prime$' denotes the derivative with respect to
$r$. Assuming $G=c=1$, the generalized Tolman-Oppenheimer-Volkov
(TOV) equation may be written as
\begin{equation}\label{continuity}
\left(\rho+p\right)\nu^{\prime }+{p^{\prime }}   =0,
 \end{equation}
which represents conservation equations in $(2+1)$ dimensions.

We take the EoS of the form
\begin{equation}\label{EoS_1}
p  =m \rho,
 \end{equation}
where $m$ is EoS parameter.

\section{Interior solutions }\label{Interior}
We first obtain interior solutions without any cosmological
constant, thereby taking $\Lambda$=0.  Latter on, we generalize our
study to non-zero value of $\Lambda$. We choose various cases of EoS
parameter for both the choices of $\Lambda$.

\subsection{With no cosmological constant ($\Lambda=0$)}
\subsubsection{$~0~< m~< ~1~$}
For $\Lambda= 0$, the field equations (3)-(6) become
 \begin{eqnarray}\label{8}
  2 \pi \rho&=&\mu'e^{-2\mu}/r,\\
  2 \pi m\rho&=&\nu'e^{-2\mu}/r,\\
  2 \pi m\rho&=& e^{-2\mu}(\nu'^{2}+\nu''-\mu'\nu').
 \end{eqnarray}
The TOV equation (11)  takes the form,
\begin{equation}\label{EoS2}
(\rho+m \rho)\nu'+m \rho'=0,
\end{equation}
Equation (\ref{EoS2}) yields
\begin{equation}\label{12}
\rho^{m}e^{(1+m)\nu}=C.
\end{equation}
Solving equations (8) and (9),  we get
\begin{equation}\label{13}
 \nu=m\mu +A~~.
 \end{equation}
 Equating Eq.(9) with Eq.(10),  we get
\begin{equation}\label{14}
e^{\mu}=e^{\nu}\nu'/r.
\end{equation}
Now, solving equations (\ref{13}) and (\ref{14}),  we obtain
\begin{eqnarray}
\nu&=&
\frac{A}{1-m}-\frac{m}{1-m}ln\left\{\frac{1-m}{m}\left(B-\frac{r^{2}}{2}
\right)  \right\},\\
\mu&=&
\frac{1}{1-m}\left[A-ln\left\{\frac{1-m}{m}\left(B-\frac{r^{2}}{2}
\right)  \right\}  \right],\\
\rho&=& \frac{1}{2 \pi m} e^{-\frac{2A}{1-m}}\left[\frac{1-m}{m}\left(
B-\frac{r^{2}}{2} \right)\right]^{\frac{1+m}{1-m}}.
\end{eqnarray}
Here $C$, $A$ and $B$ are integration constants.

For the consistency of solutions, the constants should follow the
constraint equation,
\begin{equation}
A=m \ln(2 \pi m)+\ln C~~.
\end{equation}
These solutions  are regular at the center. The central density
is given by
\begin{equation}
\rho_c= 1/(2 \pi m) e^{-\frac{2A}{1-m}}\left[B(1-m)/m
\right]^{\frac{1+m}{1-m}}.
\end{equation}
The interior solution is valid up to the radius $r<\sqrt{2B}$.
For a physically meaningful solution the radial and tangential
pressure should be decreasing function of r.   From  equation (17),
we find
  \begin{equation}
\frac{d\rho}{dr}=\frac{1}{m} \frac{dp}{dr}
 < 0  ,
\end{equation}
which gives density and pressure as decreasing functions of r.
 At $r=0$, one can get
 \begin{equation}
\frac{dp}{dr}=0,~~ \frac{d\rho}{dr}=0 ~~{\rm and}~~
\frac{d^2\rho}{dr^2}=\left[\frac{(1-m)B}{m}\right]
^{\frac{2m}{1-m}} <0
\end{equation}
which support maximality of central density and radial central
pressure. Here, density and pressure decrease radially outward as
shown in FIG.~1.\\
\\

\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{rho1.eps}
       \caption{  Variation of the energy-density($\rho$)    in the
  interior region. Description of curves are as follows:
The red, brown and black colors represent $m$=1/3, $m$=1/2 and
$m$=2/3, respectively.  For all these, solid, dashed and chain lines
represent $C=0.25$, 0.5 and 0.75, respectively. Thin, thick and
thickest lines correspond to B=5, 10, and 20, respectively. }
    \label{fig:1}
\end{figure}
The above TOV (Eq.~11) may be re-written as
\begin{eqnarray}
 \frac{M_G\left(\rho+p\right)}{r}e^{\frac{\mu-\nu}{2}}+\frac{dp}{dr}
=0,
\end{eqnarray}
where $M_G=M_G(r)$ is the gravitational mass inside a sphere of
radius $r$ and is given by Tolman-Whittaker formula, which may be derived
from  field equations,
\begin{equation}
M_G(r)=re^{\frac{\nu-\mu}{2}}\nu^{\prime}\label{eq26}.
\end{equation}

This modified form of  TOV equation indicates  the equilibrium
condition for the fluid sphere  subject to the gravitational and
hydrostatic  forces,
\begin{equation}
 F_g+ F_h =0,
\end{equation}
where
\begin{small}
\begin{eqnarray}
F_g &=&  \nu^{\prime}\left(\rho+p\right)
=r\frac{1+m}{2\pi m}e^{-\frac{2A}{1-m}}\left[\frac{1-m}{m}\left(B-\frac{r^{2}}{2}\right)\right]^{\frac{2m}{1-m}}, \\
F_h &=&  \frac{dp}{dr}=-F_g.
\end{eqnarray}
\end{small}
  The profiles of $F_g$ and  $F_h$   for the specific values of the parameters
are shown in FIG.~2 which provides the
information  about the static equilibrium   due to the combined
effect of   gravitational and hydrostatic forces.

Mass, $M(r)$, within a radius $r$, is calculated as
\begin{small}
\begin{eqnarray}
M(r) &=& \int_0^r 2\pi \rho \tilde{r}d\tilde{r} =
\frac{1}{2}e^{-\frac{2A}{1-m}}\left[B\left(\frac{1-m}{m}\right)\right]^{\frac{2}{1-m}} \nonumber \\
&-&\frac{1}{2}e^{-\frac{2A}{1-m}}\left[\left(B-\frac{r^{2}}{2}\right)\left(\frac{1-m}{m}\right)\right]^{\frac{2}{1-m}}.
\end{eqnarray}
\end{small}
\\
\\
\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{u1.eps}
       \caption{    Variation of compactness( $u=\frac{M(r)}{r}$  )  in the
  interior region. Description of curves is the same as in FIG.~1. }
    \label{fig:2}
\end{figure}

\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{f1.eps}
       \caption{  Variation of the forces    in the
  interior region. Description of curves is the same as in FIG.~\ref{fig:1}.
}
    \label{fig:2}
\end{figure}

The compactness of the fluid sphere, $u(r)$, is thus defined as be
found as
\begin{equation}
\label{eq30} u(r)=  M(r)/r.
\end{equation}
This is an increasing function of the radial parameter ( see figure
3). Correspondingly,
 the surface redshift ($Z_s$)  is given by
\begin{equation}
\label{eq34} Z_s(r)= ( 1-2 u(r))^{-\frac{1}{2}} - 1.
\end{equation}

FIG.~4 provides variation of $Z_s$ against r for different values of
the parameters.

In (2+1) dimensional spacetime, the vacuum solution does not exist
without cosmological constant. Thus it is not possible
to match our interior solution with the BTZ black hole as it is the
vacuum solution with non zero $\Lambda$. However, if one takes B as
large as possible, then the  solution is valid for the infinite large
fluid sphere. This means that  we don't have the vacuum region left.
\\
\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{z1.eps}
       \caption{  Variation of redshift in the
  interior region.  Description of curves is same as in FIG.~\ref{fig:1}.}
    \label{fig:5}
\end{figure}

\subsubsection{$~m=1~$}
For stiff  fluid model, $p=\rho$, and with $\Lambda= 0$, the field
equations (3)-(6) yield  following solutions
 \begin{eqnarray}
\nu &=& e^{-D}(r^{2}/2)+E,\\
\mu &=& -D + e^{-D}(r^{2}/2) + E, \\
\rho&=& F e^{-2E-r^{2}e^{-D}}.
\end{eqnarray}
Here $D$, $E$ and $F$ are integration constants.

For the consistency of the solutions, the constants should follow
the following constraint equation.
\begin{equation}\label{r40} D= \ln(2 \pi F)~~. \end{equation}
This ensures that $F>0$. \\The solutions are regular at the center
and are valid for infinite large sphere. The central density is
$\rho_c= F e^{-2E}$.
\\
\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{rho2.eps}
       \caption{  Variation of the energy-density($\rho$)    in the
  interior region for m=1. Description of curves are as follows:
   $ F=1,2,4,6$ and $8$  correspond to black, brown, blue, orange and red colors, respectively.
$E=0, -1, -2$ and $-3$  correspond to solid, dotted, dashed and
dot-dashed lines, respectively. For $ E\geq0$ values are too small.
}
    \label{fig:5}
\end{figure}
\\
\\
\\

From  Eq.~(32), we find at $r=0$,
\begin{equation}
\frac{dp}{dr}=0,~~ \frac{d\rho}{dr} = 0~~{\rm and}~~
\frac{d^{2}\rho}{dr^{2}}=\frac{1}{\pi}\left[-e^{-2E}\right] <0
\end{equation}
Thus central density is maximum.
\\

The mass, $M(r)$,  within a  radial distance $r$ is given by

\begin{equation}
M(r)=e^{D}\left[e^{D-2E}-e^{D-2E-r^{2}e^{-D}}\right]/2~~.
\end{equation}
The compactness of the fluid sphere  is thus, $u(r)= M(r)/r$. Having
$u$, the $Z_s$ is determined using Eq.~(\ref{eq34}). The important
physical characteristics such as density, compactness  and redshift
are shown in FIGs. 5-7.\\
\\
\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{u2.eps}
       \caption{    Variation of compactness( $u=\frac{M(r)}{r}$  )  in the
  interior region. Description of curves is the same as in FIG.~5. }
    \label{fig:2}
\end{figure}
\\

 The TOV equation  yields
\begin{equation}
 F_g+ F_h =0,
\end{equation}
where
 \begin{equation}
F_g=-F_h=(r/\pi)e^{-2E-r^{2}e^{-D}}.\end{equation}
\\
\\
\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{z2.eps}
       \caption{  Variation of redshift in the
  interior region.  Description of curves is the same as in FIG.~4. }
    \label{fig:9}
\end{figure}

 The profiles of
$F_g$ and  $F_h$  for the specific values of parameters are shown in
FIG.~8, which provides information about the static equilibrium
due to gravitational and hydrostatic forces combined. As before, we
can not match our interior solution with BTZ exterior vacuum
solution.
\\
\\
\begin{figure}[htbp]
    \centering
        \includegraphics[scale=.24]{f2.eps}
       \caption{   Variation of the forces    in the
  interior region for $m$=1. Description of curves is the same as in
  FIG.~4.
}
    \label{fig:10}
\end{figure}

 \subsubsection{$~m=-1~$}
The equation of state of the kind, $p =-\rho$   is related to the
$\Lambda-dark \:\: energy$, an agent responsible for the second
phase of the inflation of Hot Big Bang theory. Using  the equation
of state of the kind, $p =-\rho$, and with $\Lambda= 0$, the field
equations (3)-(6)  yield following solutions

\begin{eqnarray}
\rho&=& -p = J,\\
\nu &=& \left[H+ \ln (r^{2}+K)\right]/2,\\
\mu &=& \left[H- \ln (r^{2}+K)\right]/2.
\end{eqnarray}

Here $J$, $K$  and $H$ are integration constants.
Solutions hold good for the following constraint equation
\begin{equation}2 \pi J + e^{-H}=0~~.
\end{equation}
These  are regular at the center if $K$ is positive and the solution
is valid for the infinite large fluid sphere. However, for $K<0$,
solution is valid for $r>\sqrt{-K}$ up to infinite large radius.

\subsubsection {$~m=0~$}
For the dust case i.e. when $p=0$ and $\rho \neq 0$, the field
equations (3)-(6) reduce to
\begin{equation}\nu = \nu_0
\end{equation}
and
\begin{equation}e^{-2\mu} = \mu_0 - \int 4 \pi r \rho dr.
\end{equation}
Here, $\nu_0 ~and~ \mu_0$ are integration constants.

Unless   specifying the energy density, one can not get
exact analytical solution of the field equations. Thus dust model in
(2+1) dimensional space time is   possible for known energy density.
\subsection{With cosmological constant $(\Lambda\neq 0)$}
 \subsubsection{$~m=-1~$}
As before for  the equation of state of the kind $p =-\rho$ with non
zero $\Lambda$, the field equations (3)-(6) yield $\rho = c_4$. The
metric coefficients  may be obtained as
  \begin{equation}\nu = \ln(r^{2}+B_5)/2+D_5 \end{equation}
and
 \begin{equation}\mu = -\ln (r^{2}+B_5)/2 -D_5 +A_4. \end{equation}
 Here, $c_4$, $A_4$, $B_5$ and $D_5$ are integration constants.

These solutions are consistent if
 \begin{equation}2 \pi c_4 +\Lambda + e^{2D_5-2A_4}=0~~.\end{equation}
These solutions  are regular at the center if $B_5$ is positive and
the solution is valid for the infinite large fluid sphere. However,
for $ B_5 <0$, the   solution is valid for $r>\sqrt{-B_5}$ up to
infinite large radius. The nature of the solutions of the metric
potentials is independent of the sign of  $\Lambda$. However, sign
of $\Lambda$
plays a crucial role to get positive energy density.
For positivity of energy density, one should take negative $\Lambda$.

Note that without any loss of generality, we can take $D_5 =0$ as it
can be absorbed by re-scaling the coordinates.
We match the interior solution to the exterior BTZ black hole metric
\[ds^{2}=-(-M_0-\Lambda r^{2})dt^{2}+(-M_0-\Lambda r^{2})^{-1}dr^{2}+r^{2}d\theta^{2},\]
at the boundary $r=R$, which yield
  \begin{equation}\left(-M_0-\Lambda R^{2}\right)=R^{2}+B_5 ,\end{equation}
 \begin{equation}\left(-M_0-\Lambda
R^{2}\right)^{-1}=e^{2A_4}(R^{2}+B_5)^{-1}.\end{equation} Solving
these two  equations, we get
 \begin{equation}B_5=-M_0-(\Lambda +1 )R^{2}~~and ~~A_4=0~~.\end{equation}
The consistency relation assumes the form
 \begin{equation}2 \pi c_4 +\Lambda + 1=0~~.\end{equation}

\subsubsection{ $0<m\leq 1$}
From the field equations (3)-(6) after some manipulation, we arrive
at
\begin{equation}\label{51}2 \pi (1+m)A_1
e^{\frac{m-1}{m}\nu}\nu'^{3}=2r\nu'^{2}+r\nu''-\nu'~~.\end{equation}

One can observe   that $\nu'=0$ will be a particular solution of
this equation. This yields $\nu=constant$. Equation (4) implies $p =
\Lambda/2 \pi$. Finally, we  get the following solution for  $\mu$
as
\begin{equation}
\mu=-\ln\left[A_5-Nr^2\right]/2~~.
\end{equation}
Here, $N=\Lambda (1+m)/m$ and  $A_5$ is integration constant.

For positivity of energy density, one should take positive
$\Lambda$. The solutions  are regular at the center if $A_5>0$ and
valid up to $r< \sqrt{A_5/N}$.
In this case, we can not match the interior solution to the exterior
BTZ black hole metric which is vacuum solution with negative
$\Lambda$.

\subsubsection{$~m=0~$}
For the dust case i.e. when $p=0$ and $\rho \neq 0$,  one can not
obtain the exact analytical solution of the field equations. Thus
dust model in $(2+1)$ dimensional spacetime with non zero $\Lambda$
is not possible.

 \section{ Conclusion }\label{conclusion}

In this paper we have obtained a new class of exact  interior
solution of Einstein field equation in $(2+1)$ dimensional space
time   assuming the equation of state  $p=m \rho $ ( where $ m $ is
the equation of state parameter).  The interior solutions obtained without
cosmological, $\Lambda$,  are physically acceptable for the
following reasons:\\
  (i) the solutions are  regular at the origin,\\
  (ii)  both the
pressure (p) and energy density ($\rho$) are  positive definite at
the origin,\\
 (iii)  the pressure reduce to zero at some finite
boundary radius $r_b > 0$,\\
 (iv) both the pressure and energy density
are monotonically decreasing to the boundary,\\
 (v) the subluminal
sound speed ($v^2_s =\frac{ dp }{d \rho} = m \leq 1$)\\
(vi) and Ricci scalar is non zero i.e. spacetime is non flat.

 It is to be noted that at very high densities the adiabatic sound
speed may not equal the actual propagation speed of the signal. By
studying TOV equation, we have shown that  equilibrium stage of the
interior region without  $\Lambda$ can be achieved due to the
combined effect of gravitational and hydrostatic forces. We know BTZ
exterior vacuum solution in (2+1) dimension is valid only for non
zero $\Lambda$. Therefore, it is not possible to match our interior
solution ( without $\Lambda$ ) with BTZ spacetime at some boundary.
We emphasis the following fact that any interior solution in four
dimensional space  made with a perfect fluid must be glued with an
exterior vacuum solution only at a regular surface p = 0 ( this is
consequence of the well-known Israel matching conditions for the
related problem ).  For barotropic equation of state the
configurations present p = 0 surfaces at the same location where
$\rho = 0$. For the solutions (15)-(17) with $0 < m < 1$  this
occurs at $r = \sqrt{2B}$. However, the metric coefficients are
singular at the same locus, in fact, there is a curvature
singularity at $r = \sqrt{2B}$.  Hence, this is not a regular region
where spacetime can be continuously glued with other spacetime.
Hence,  one should take B as large as possible so that  the
solution is valid for the infinite large fluid sphere and  we don't
have the vacuum region left. For m=1 case, there does not exist any
radius for which p or $\rho=0$, hence, Israel matching condition
does not occur.

While finding  interior solution with non zero $\Lambda$, we note that
density and pressure remain constant. Interestingly, we observe that
it is not possible to get dust model in $(2+1)$ dimensional
spacetime with non zero $\Lambda$. Investigation on full collapsing
model of a (2 + 1) dimensional configuration will be a future
project.

\section{Acknowledgements}
FR , AAU and RB would like to thank the Inter-University Centre for
Astronomy and Astrophysics (IUCAA), Pune, India, for research
facility.  FR is also grateful to UGC, Govt. of  India, for
financial support under its Research Award Scheme. PB is  thankful
to CSIR, Govt. of  India for providing JRF. RB thanks CSIR for
awarding Research Associate fellowship. We are thankful to Dr. R.
Sharma for valuable discussion.


\section{Conclusion}
\label{Sec:Conclusion}

\par  The removal of radon-born lead\ contamination from organic liquid scintillator was studied in preparation
of a low background phase of the KamLAND experiment. Reduction factors were measured for various purification
methods: water extraction with purified water and ion-enhanced water, isotope
exchange, filtration, adsorption, and distillation.

\par Among these methods, distillation was identified as the
most efficient method. It provides the reduction factors
of 10$^3$ -- 10$^4$ required for KamLAND's solar neutrino detection phase.
We obtained reduction factors for all components of the
KamLAND LS: $n$-dodecane, pseudocumene and PPO.
The studies performed on a scaled-up system provide
the basic data for the design and operation of a purification
plant for continuous circulation and purification of 1000~tons of KamLAND LS.

\par The second most effective procedure was silica gel
extraction, which provided highly reproducible results.
However, a steady decline observed in lead reduction
factors after successive extractions cannot be explained
by simple plate theory. One possible explanation
is that a portion of the dissolved lead exists
in non-polar organometallic compounds, which can be
degraded by heat and/or chemically targeted on the lead-carbon\
bond. Our studies on silica gel extraction with preheating
and processing with FeCl$_3$ showed improved reduction
factors consistent with the existence of such non-polar
lead compounds in LS.

\section{Introduction}
\label{Sec:Intro}

\par Sensitive detection of trace radioactivity is important for
low energy solar neutrino, dark matter, and neutrinoless double beta decay experiments.
Due to demanding constraints on detector backgrounds it has become essential to better
understand the mechanisms that can be used to remove trace-level radioactivity from detector
components, specifically from the active volume.   This study was motivated by the goal of
expanding the physics capability of the KamLAND experiment to include detection of low energy
solar neutrinos and other events from rare physics phenomena.

\par The KamLAND detector is situated in the Kamioka mine, Gifu prefecture, Japan and
was commissioned in 2002 to test the large mixing angle (LMA) solution of the solar
neutrino problem~\cite{bib:BahcallSolarTheory,bib:SolarNeutrinoProblem}.  KamLAND
demonstrated the validity of the LMA solution by observing the disappearance of
anti-neutrino s produced in nuclear reactors in
Japan~\cite{bib:KamLAND_PRL_1,bib:KamLAND_PRL_2,bib:KamLAND_PRL_3,bib:KamLAND_PRD_4}.
KamLAND's active target is 1 kton of organic liquid scintillator (LS), composed of
80.2\% $n$-dodecane, 19.8\% 1,2,4-trimethylbenzene (PC) and~$1.36 \pm 0.03$ g/L of
2,5-diphenyloxazole (PPO).

\par  The statistical analysis of KamLAND's low energy event distribution is performed
by fitting summed beta spectra of different radionuclides to KamLAND's singles
data (events not requiring delayed coincidence) and
allowing their normalizations to float freely.
The result of this procedure is
depicted in Fig.~\ref{Fig:SolarRegionWithBackgrounds} together with the data.
The mass concentrations of the contaminants appearing in
Fig.~\ref{Fig:SolarRegionWithBackgrounds} are listed along with several others
in Table~\ref{tab:KamLANDConc}~\cite{bib:KeeferThesis}.
Analysis of the singles data yields an integral event rate
of~$3.7 \times 10^{7}$~kton$^{-1}$~day$^{-1}$ in the energy window of interest ($\left[0.25,0.8\right]$ MeV).
This rate is to be compared to the expected $^{7}$Be\ solar neutrino elastic
scattering signal of
291~events~kton$^{-1}$~day$^{-1}$.
Spectral analysis allows the identification of the principal contributors to KamLAND's
background to be $^{85}$Kr, $^{210}$Bi and $^{210}$Po, the latter two supported
by the decay of long-lived $^{210}$Pb~\cite{bib:KeeferThesis}.
$^{85}$Kr\ is a fission product, found in the atmosphere mainly due to nuclear fuel
re-processing~\cite{bib:TraceNobleGases}, while $^{210}$Bi and $^{210}$Po are $^{222}$Rn\ decay daughters.
As both $^{85}$Kr\ and $^{222}$Rn\ are airborne, it was assumed that these contaminants were introduced
into the KamLAND LS\ during filling from exposure to air.
The initial growth of $^{210}$Po, observed in early KamLAND data,
supports this hypothesis~\cite{bib:DjurcicThesis}. After the initial growth the $^{210}$Po
activity leveled off, indicating that it was supported
by its long-lived parent $^{210}$Pb. The analysis of delayed $^{214}$Bi-$^{214}$Po
beta-alpha coincidences, which occur above $^{210}$Pb\ in the U-decay chain,
provided an effective $^{238}$U
decay rate of $3.7\pm 0.5$ kton$^{-1}$ day$^{-1}$. This excludes the possibility
that $^{210}$Po, observed with an event rate of $4.6\times 10^6$ kton$^{-1}$ day$^{-1}$,
was supported by either $^{238}$U or $^{226}$Ra.
Although there was no direct spectroscopic evidence for the presence of
\argon activity (an atmospheric spallation product), its existence was
suspected because of the air exposure hypothesis, and a limit on its
concentration is
included for completeness in Table~\ref{tab:KamLANDConc}.

\begin{figure}[t!]
\centering
\includegraphics[keepaspectratio,width=0.45\textwidth,trim= 15mm 5mm 25mm 0mm]{Figure01.pdf}
\caption[]{KamLAND low energy singles spectrum. The data is displayed together
with a background model. The major background components are indicated along with
their integral rates inside the energy interval~$\left[0.25,0.8\right]$ MeV (hatched
region).  The expected $^{7}$Be\ solar neutrino recoil event spectrum (calculated using BS05 solar
rates~\cite{bib:BS05OP})  is overlaid to provide the appropriate scale. All spectra
are converted to visible energy with KamLAND's energy response function and
folded with a $\sigma = 7.8 \%/ \sqrt{ E~[{\rm MeV}]}$
Gaussian energy resolution~\cite{bib:KeeferThesis}.}
\label{Fig:SolarRegionWithBackgrounds}
\end{figure}

\input{Table01}

\par The studies presented in this paper were undertaken to understand
if and how the internal background rate,
five orders of magnitude larger than the solar neutrino signal,
can be sufficiently reduced by means of chemical purification.
At the fundamental level, removal of impurities
can be achieved by differences in chemical potentials between different media and phases.
The diffusion of impurities from one medium to another is required to achieve
equilibrium~\cite{bib:Crank}.
Consequently we studied the distribution of impurities identified to be responsible
for KamLAND's low energy background with regards to their behavior in
different media and phases.  These included:
\begin{itemize}
\item vapor-liquid phases in distillation
\item solid-liquid phases in adsorption, as well as filtering and isotope
exchange
\item liquid-liquid phases in water extraction.
\end{itemize}
In addition we studied the impact of heating on the lead\ removal efficiency to target possible
organometallic lead\ compounds formed after alpha decay and known to be prone to heat-induced break up. This
paper reports measured reduction factors for these different purification methods.
Studies done on the removal of radioactive gases ($^{85}$Kr\ and $^{222}$Rn) are described in ~\cite{bib:TakeuchiMastersThesis}
and are not covered in this work.
Final reduction factors achieved from full-scale scintillator purification in the KamLAND
detector are not within the scope of this paper.

\par The experimental procedures used throughout this study are outlined in Sec.~\ref{Sec:ExpMethods}.
In Sec.~\ref{Sec:RnDaughterRemoval} we discuss the analytical techniques and methods
used to quantify and remove radon\ decay products from~LS\ and its
constituents  PC, $n$-dodecane, and PPO. Finally, in
Sec.~\ref{Sec:Conclusion}, we discuss the implications and significant findings of this work.

\section{Experimental Procedures and Detection Methods}
\label{Sec:ExpMethods}

\par Measuring the large reductions in $^{210}$Pb\ needed for KamLAND's
low background phase required the quantitative determination of lead\ concentrations
at a level of $10^{-24}$ to $10^{-20}$~g/g. The starting value
corresponds to approximately $4.6\times 10^{4}$ $^{210}$Pb\ atoms per liter of LS,
while the smaller concentration represents the goal for
adequately purified LS.  Clearly this is a challenging analytical
problem.

\par The measurement of lead\ concentrations at such extremely low levels
is best achieved using radioactive tracers. The isotope of concern,
$^{210}$Pb, is a low Q-value beta emitter. Its daughters, $^{210}$Bi and
$^{210}$Po, are pure beta and alpha emitters, respectively. None of these
decays offer a convenient experimental signature suited to measure small
concentrations in lab scale experiments. Furthermore, $^{210}$Pb\ has
a half life of 22.3~y resulting in a relatively
small specific activity of $4.5\times 10^{-5}$~Bq/L at the starting
concentration of $10^{-20}$~g/g.
To overcome this experimental challenge, we
assume that all isotopes of lead\ born via alpha decay
have similar chemical characteristics in organic media,
and thus, similar responses to purification techniques.
Isotope-specific studies showed a factor of 3 higher $^{212}$Pb\ reduction than observed for $^{214}$Pb
~\cite{bib:KeeferThesis}. This might be attributed to differences
in nuclear recoil energies and lead\ residence times in the LS.  Given the interest
in large reduction factors, and not analytic precision, the possible impact of these effects are neglected in this work.

\par The short lived isotopes of $^{212}$Pb ($T_{1/2}=10.64$~h),
born via $^{216}$Po ($^{220}$Rn daughter), and $^{214}$Pb ($T_{1/2}=26.8$~min),
born via $^{218}$Po ($^{222}$Rn daughter),
can be used to study lead\ removal within our target concentration range of
$10^{-20}$ to $10^{-24}$~g/g.
At a concentration of $10^{-20}$~g/g the specific
$^{212}$Pb activity is 0.83~Bq/L, a factor $1.8\times 10^4$
larger than that of $^{210}$Pb\ at equal concentration.
Characteristic gamma radiation, of energies 238~keV and 351~keV, are emitted
after the beta decays
of $^{212}$Pb and $^{214}$Pb, respectively. This allows for convenient detection with a
low-background germanium (Ge) detector. The subsequent fast Bi-Po beta-alpha decay sequences
following these lead\ decays allow the further utilization of delayed
coincidence counting with LS\ for high-sensitivity measurements.

\subsection{Determination of the Radio-nuclide Reduction Factors}
\label{RnSoures}

\par The effectiveness of various lead\ removal techniques was tested
by loading LS\ with trace amounts of radioactive $^{212}$Pb or $^{214}$Pb.
This was achieved by first dissolving $^{220}$Rn or $^{222}$Rn,
exploiting the excellent solubility of radon\ in organic solvents,
and then allowing it to decay into the meta stable lead\ isotopes.
The dissolved lead\ activity, $\it{A_{i}} $, was measured after loading.
The remaining activity, $\it{A_{f}}$, was measured after the loaded solution had been
subjected to a specific purification procedure. Both activities were
corrected to correspond to a common reference time.
The reduction factor $R$ for a given species is defined
as:
\begin{equation}
\it{ R \; =\; \frac{A_i}{A_f}}
\label{eq:eff}
\end{equation}

\par Two detector types were used to determine the lead\ activities.
For samples with specific activities of more than 1~Bq/L,
Ge detectors were utilized to measure
the gamma radiation emitted in $^{214}$Pb, $^{214}$Bi, $^{212}$Pb,
$^{212}$Bi and $^{208}$Tl beta decays.
For low activity samples, liquid scintillation counting was employed, utilizing
the delayed beta-alpha coincidences from the
$^{212}$Bi-$^{212}$Po ($^{220}$Rn decay chain) and $^{214}$Bi-$^{214}$Po
($^{222}$Rn decay chain) decay sequences.  The latter technique allowed the
nearly background-free measurement of specific activities down to 10~mBq/L.
Together, these detection techniques provided a means to perform studies of
purification reduction factors spanning five orders of magnitude
and reaching $^{212}$Pb concentrations as low as $10^{-22}$~g/g. Further sensitivity to
the target concentration of $10^{-24}$~g/g can only be achieved using the KamLAND
detector itself.

\subsection{Scintillator Loading and Purification Procedures for Radon Daughters}
\label{Sec:PurProcedure}

\par Gas flow-through sources from Pylon Electronics Inc. (models TH-1025 and RN-1025)
were used to load LS\ with $^{220}$Rn or $^{222}$Rn, respectively. A second
$^{220}$Rn source was
made using a ThO$_2$ powder.  In all instances N$_{2}$ was utilized as
the carrier gas. To remove dust particles from the gas stream, the N$_2$ was
passed through 100~$\mu$m and
0.8~$\mu$m filters prior to a flow-through source.
The radon-loaded N$_{2}$ gas was then passed through 500~cm$^{3}$ of LS\ via a
glass bubbler, fitted with a sparger to break up the gas for increased radon\ absorption.
Flow meters were attached to the input and output gas stream to monitor flow rates.

\par Our studies showed that the use of a filtered carrier gas was important for achieving
reproducible results. Particulates carried by the gas stream readily collect charged radon\
decay products and are efficiently removed by filtering, thus ``faking'' a
removal mechanism not present in KamLAND scintillator.

\par Lead removal experiments were performed with LS,  $n$-dodecane, PC, and PPO.
All samples were handled using the following standardized procedure (unless otherwise
noted):

\begin{enumerate}
\item  Sparge liquid with radon-loaded N$_2$\ for a maximum of 24 hours.\footnote{
Care must be taken when loading LS\ or pure PC.  PC has a high
vapor pressure causing it to evaporate during bubbling, resulting in a
lowering of the PC concentration as well as corrosion of filters and membranes.}
\label{En:Bubbling}

\item  Transfer the $^{212,214}$Pb-loaded liquid into a counting container
to determine the initial activity $\it{A_{i}}$.
For counting with Ge detectors, two 125~cm$^{3}$
Nalgene$^{\textrm{\textregistered}}$ bottles were used.
If liquid scintillation counting was to be used, half of the sample was
transferred to a 10.80~cm diameter, 4.44~cm long acrylic
cell, fitted with a 7.62~cm diameter photo-multiplier tube (PMT).
The counted LS\ volume was typically 200~cm$^{3}$.

\item  When using the Ge detectors, the counted liquid was transferred into the
purification system. When using liquid scintillation counting,
sufficient liquid was prepared to
allow parallel counting and purification.  This method assumes
a homogeneous distribution of lead\ in the LS.  \label{En:BottleCount}

\item  After purification, the liquid was counted using the appropriate
detector to determine the final activity $A{_{f}}$.  \label{En:PurCount}

\end{enumerate}
\vskip 0.1in

\par  The ``before'' and ``after'' counting was always performed using the same detector
to allow for cancellation of systematic errors.  In some instances it was necessary to use
only a single component of the LS\ during the purification process. If an LS\ component was analyzed
by liquid scintillation counting a scintillating cocktail had to be
made prior to loading the counting cell. This dilution was taken into account
in the reported reduction factor.  Furthermore, measurements were carried out to cross
calibrate the two different detector types.

\par Detailed studies of systematic errors were performed on every aspect of the
purification processes.  Thorough cleaning of all containers used in the loading and
counting was imperative when dealing with extremely low levels
of radioactivity. Cross contamination had to be avoided in order to achieve
reproducible results.
When chemically compatible, all components were cleaned with high purity solvents (analytic grade or better).
Each part was cleaned with acetone, then with 1 molar HNO$_3$, followed by a rinsing with de-ionized (DI) water and ethanol.

\par In the case of Ge detector counting, it was found that a substantial fraction
(up to 50\%) of the lead\ activity contained in the LS\ could adhere to the
walls of the Nalgene bottles used for the counting. The bottles were
thus counted again after the lead\ loaded scintillator had been transferred
into the purification setup to account for this loss mechanism.

\section{Removal of Radon-Born Lead from Liquid Scintillator}
\label{Sec:RnDaughterRemoval}

\par In any purification process, the difference in concentrations and solubilities
between phases drives the separation of impurities between them.  In this work
these phases were realized in the form of water-oil
and silica-oil systems or gas-liquid phases.  Processes which rely primarily on diffusion of the
impurities between phases require initial concentrations in the transient phase to be less than those
listed in Table~\ref{tab:KamLANDConc}.
The mass transfer involved in such diffusive processes can be modeled assuming thermodynamic
equilibrium. The work presented here does not report the reduction in terms of a Gibbs potential,
as done in previous publications~\cite{bib:Borexino-NIM-SilicaGel}. Instead,
we report the results in the form of a reduction factor as defined in Eq.~\ref{eq:eff}.
We found this to be sufficient for the purpose of identifying the most effective method of purification.

\par The following purification techniques were studied: water extraction,
ion enhanced water extraction, isotope exchange, filtering, adsorption and distillation.  When
applicable, experiments were performed in series with one another to establish if the methods
scale as the product of their individual reduction factors. For each of these techniques, the reduction factor,
time of operation, solute volume, and initial activity were studied for correlations.

\par Among the $^{222}$Rn\ daughters $^{210}$Pb\ is of particular interest because of
its long half life.
Following the assumption discussed in Section~\ref{Sec:ExpMethods} that different
alpha-born isotopes of lead in organic media have similar characteristics,
the purification techniques described below were evaluated using $^{212}$Pb\ and $^{214}$Pb\ as tracers.
However, one cannot assume that all  metallic radon\ daughters are removed with
equal efficiency as they have different chemical characteristics.
The radionuclides $^{208}$Tl, $^{212}$Bi, $^{214}$Bi and $^{218}$Po were analyzed
separately to better understand the variation in the reduction factors across
chemical species. For example, experimental information obtained
on $^{218}$Po was used to gain some insight into how much $^{210}$Po ($T_{1/2}=138\; \rm d$)
would remain in the LS\ after purification.  The results of these non-lead\ purification studies are not reported here.
More details on the purification research performed with KamLAND LS\
can be found in~\cite{bib:KeeferThesis,bib:KishimotoAIPProc,bib:TakeuchiMastersThesis,bib:KosekiMastersThesis,bib:EbiharaMastersThesis,bib:KibeMastersThesis,bib:MomoyoMastersThesis,bib:KeeferNuclPhysBProc,bib:KeeferAIPProc}.

\subsection{Water Extraction}
\label{WaterExt}

\par A water purification system already existed in KamLAND; it had
been used during the initial filling~\cite{bib:OgawaThesis}. We thus first
tried to understand whether this system could be used, perhaps with
modification, to remove dissolved lead\ from KamLAND LS.
A miniature water extraction system was built to circulate LS\ through a glass
purification column filled with DI water.  The LS, which is less
dense than water, was removed from the top of the water column and re-circulated
back to the bottom of the column using a peristaltic pump.  To increase the surface
to volume ratio, the LS\ was broken into small bubbles with a custom-made glass
frit affixed to the bottom of the column.

\par  Water extraction experiments showed weak lead\ reduction factors between
1.02 and 1.10. Using about 500 cm$^3$ of water, 250 cm$^3$ of LS\ and a
flow rate of 200 $\rm cm^3/min$, the observed reduction factors saturated for circulation times longer than 5 hours~\cite{bib:KeeferThesis}.
A second set of independent experiments was performed using 50 cm$^3$ of lead-loaded LS\ with 60 cm$^3$ of
distilled water mixed with 0.01 to 6 mol/L of HNO$_3$, HCl, NaOH~\cite{bib:KosekiMastersThesis}.
Mixing was achieved by vigorously shaking for 5 min.
In these experiments we observed saturation levels consistent with the more controlled mixing noted above.
In all experiments the lead\ reduction factor showed a clear decrease with growing pH.
However, the magnitude of this change over the pH range never exceeded 30\%.
Therefore, water extraction was found to be not well suited for achieving the
very large lead\ reduction factors needed in KamLAND.

\subsection{Isotope Exchange, Filtration, and HTiO Adsorption}
\label{Sec:IsoFiltering}

\par LS\ loaded with radioactive lead\ was circulated through a bed of
granular lead, or lead\ shot. Assuming that the dissolved lead\ forms
an equilibrium with the essentially infinite number of stable lead\
atoms, we tested the possibility of isotope exchange. These experiments
showed no quantifiable lead\ reduction effect beyond the typical loss
to the surfaces of the pumping system.

\par Early studies showed that filtering loaded LS\ removed the
dissolved lead\ in a variable manner. This fluctuation in lead removal efficiency was traced to
the presence of dust particles in the carrier gas stream to which the charged radon decay products readily attach~\cite{bib:Migdal}.
Carefully filtering the gas stream avoided this interference and enhanced the reproducibility
of the experiments.

\par HTiO has successfully been used in the SNO experiment to remove actinides
from light and heavy water~\cite{bib:SNO_HTIO}.
We investigated this method for its suitability in removing lead\ from organic
LS. HTiO pre-loaded HEPA PTFE filters (surface density 0.64 g/cm$^{2}$),
supplied by the Oxford University SNO group, were utilized for these tests.
lead-loaded LS\ was placed into a 60 ml syringe attached to the filter and the
liquid was pushed slowly through the filter. The filter was replaced after every 125 ml
of lead-loaded LS. For flow rates of 3.3 and 16.5~cm$^3$/min we observed
lead\ reduction factors of $1.06 \pm 0.01$ and $1.07 \pm 0.01$, respectively.
No further consideration was given to this method.

\begin{figure}[tbp]
\begin{center}
  \subfigure[]{\includegraphics[keepaspectratio,height=0.65\columnwidth]{Figure02.jpg}\label{Fig:HeatingApp}}
  \subfigure[]{\includegraphics[keepaspectratio,height=0.65\columnwidth]{Figure03.jpg}\label{Fig:AdsorbApp}}
\caption{The stainless steel vessel used to heat the LS\ prior to adsorption is depicted in Fig.~\ref{Fig:HeatingApp}. Fig.~\ref{Fig:AdsorbApp} is the adsorption column in which all silica gel experiments were performed.  At the top one can see the heating vessel and thermocouple.  Furthermore, a vacuum line was used to regulate the flow rate through the silica column.}
\end{center}
\end{figure}

\subsection{Chromatography with Silica Gels (Adsorption)}
\label{Sec:AdsorbentPur}

\par Previous studies identified silica gel extraction as an
efficient means of removing inorganic impurities from LS ~\cite{bib:Borexino-NIM-SilicaGel}.
This method was studied in depth to evaluate its suitability for removing lead\
using different adsorption materials packed in chromatographic columns.
Plate theory~\cite{bib:UngerKK} is commonly used to describe liquid-solid
chromatography, which requires mass and thermodynamic equilibrium between the
mobile (LS) and stationary (silica gel) phases.  One can
divide the column into many distinct sections, referred to as plates.
The concentration of an impurity in successive plates can be derived from mass balance and
equilibrium conditions:
\begin{equation}
\rm X_{M(p)} \; =\; \frac{X_{0}\nu^{p}}{p!} e^{-\nu},  \;\;\;\;\;  p=1,2,3,...
\label{eq:ElutionCurve}
\end{equation}
where $\rm X_{0}$ is the initial concentration of the impurity, $\rm X_{M(p)}$ is the concentration
in the p$\rm ^{th}$ plate, and $\nu$ is the plate volume.  Eq.~\ref{eq:ElutionCurve} is referred
to as the elution curve and
indicates that given enough adsorbent, there is no theoretical limit to the achievable reduction factor.

\par There are two possible methods for purifying LS\ with this technique.  One can assume that the silica gel
is infinitely adsorptive and run the liquid through a column in a loop mode, or
make a single pass through the column in a batch mode.
Experiments showed batch purification through a long column to be the method of
choice, as multiple passes through the same column lead to lower reduction
factors.  Batch mode makes efficient use of the concentration gradient of the
solute developing along the column.
Our observation that batch mode provides higher purification factors
than loop mode agrees well with previous studies~\cite{bib:Borexino-NIM-SilicaGel}.
To illustrate this point, Table~\ref{Ta:BatchLoop} lists the lead\ reduction
factors obtained for three different types of silica gel used in both batch
and loop mode.

\input{Table02}

\par All results presented from here on are based on batch mode operation.
A wide variety of silica gels are commercially available. For this study
a $32-63 \; \mu$m (grain size) silica gel from Selecto Scientific was used as the standard.
Our laboratory chromatographic lead\ purification system, shown in Fig.~\ref{Fig:AdsorbApp},
was designed and operated in the following way: a slightly pressurized stainless
steel reservoir, holding approximately 10 L of LS, was coupled
to a 35~cm long stainless steel chromatographic column with 0.97~cm inner diameter.
Valves installed in the LS\
path allowed flow regulation through the column.  The
column was fitted with a $10 ~\mu$m stainless steel mesh filter at the
outlet to sequester the adsorbent.

\input{Table03}

\par  Prior to runs with LS, the silica gel was first conditioned (compacted). This was
done to keep eddies and voids from developing inside the column.
Sufficient compactification was achieved by providing constant N$_2$\
pressure to the top of the column while pulling partial
vacuum at the bottom (without LS).
Our standard purification procedure gave consistent and reproducible lead\ removal
factors, as verified by repeated measurements with the standard gel. The procedure
consisted of the following steps:
\vskip 0.1in
\begin{enumerate}
  \item Clean all parts with acetone, then with 0.1 - 1.0 molar HNO$_{3}$, followed by a rinse with DI water and ethanol.
  \item Add the desired amount of silica gel to the column, close all valves, and pump down
        to 100~hPa. Assist setting of the silica granules by knocking the column during pump-down.
  \item Apply 1000~hPa N$_2$\ pressure to the top of the column and continue to knock the column
        to set the silica in place.
        The pressure at the outlet rises to approximately 300~hPa when
        equilibrium is reached.
  \item Remove the vacuum pump and slowly release the pressure from the inlet side of the system.
  \item Add the lead-loaded LS\ to the reservoir with all valves closed.  Pressurize the reservoir
        with N$_2$\ to 1000~hPa.
  \item Slowly open the valve between the reservoir and the column to wet the gel.
        Controlled release of LS\ prevents cavities from forming at the top of the silica column.
        The flow rate of LS\ through the column was 50 ml/min.
  \item After purification, the LS\ is passed through a 0.2~$\mu$m PTFE HEPA filter to remove
         silica particulates suspended in the LS.
\end{enumerate}

\par Further conditioning tests were performed by heating the silica gel to
100 - 120~$^{\circ}{\rm C}$\ to remove adsorbed water from its
surface.  Water molecules are known to adhere to the adsorption sites and reduce the
effectiveness of the silica gel~\cite{bib:UngerKK}.  However, this conditioning was not
found to provide any benefit in terms of boosting lead\ removal.

\par Using the procedure outlined above, lead\ removal studies were performed with many different
types of adsorbents. The results of these experiments are given in Table~\ref{Ta:AdsorbentEff}.
Our procedure provided reproducible results, and
typical statistical uncertainties on individual runs were of order 0.1\%.
Repeated runs with the $32-63$~$\mu$m silica gel yielded a standard deviation
of $7.2\%$ for the lead\ reduction factor, which we interpret as a measure of the systematic uncertainty.

\par Since, as already mentioned, the elution curve (Eq.~\ref{eq:ElutionCurve}) indicates that
in principle the reduction factor has no limit, making the chromatographic column arbitrarily long should allow
one to achieve any desired reduction factor. This behavior was not observed, as a maximal reduction factor of 29 was
achieved for a single pass purification.  While this factor is substantial, it
falls short of the required value for KamLAND, so
studies were performed to determine whether this limitation could be overcome.

\label{OrganoMettalicComp}

\par We considered three hypotheses to explain the observed limit in the lead\
reduction factor during silica gel extraction:

\begin{Hypothesis}\label{Hyp:CrossContam}
There exists lead\ cross-contamination between runs (insufficient cleaning).
\end{Hypothesis}

\begin{Hypothesis}\label{Hyp:SiContam}
The intrinsic uranium and thorium concentrations of the silica gel, and the resulting out-gassing radon, re-introduces
$^{212}$Pb\ into the LS. The observed limit may be determined by the equilibrium between the removal and addition of Pb.
\end{Hypothesis}

\input{Table04}

\begin{Hypothesis}\label{Hyp:OrganContam}
Some fraction of the $^{212}$Pb\ may exist in the LS\ in a form without affinity towards the
silica gel.
\end{Hypothesis}

\par Blank runs performed with unloaded LS\ following a purification run with lead-loaded LS\
did not show any carry-over activities, thus excluding
Hypothesis~\ref{Hyp:CrossContam} and constraining also
Hypothesis~\ref{Hyp:SiContam}. Furthermore, bottles and
containers were not re-used whenever
possible, and all re-used equipment underwent vigorous cleaning, as described in Sec.~\ref{Sec:PurProcedure}.

\par Hypothesis~\ref{Hyp:SiContam} was further refuted by measuring the
radioactive content of the gel.
The $32-63$~$\mu$m Selecto Scientific silica gel was measured
to have an intrinsic uranium and thorium content of 167 ppb and 152 ppb,
respectively~\cite{bib:EXO_Materials}, resulting in 1.7~mBq of $^{212}$Pb\ in 2.5~g of silica gel.
Typical loading activities for the LS\ were 5~kBq. This silica gel had an average reduction factor of 19,
resulting in 250~Bq of residual $^{212}$Pb\ activity.
The total intrinsic activity of the silica gel is thus much smaller than the
measured activity after silica gel purification, implying that the former
contributes negligibly to the observed limit in the reduction factor.

\par  Hypothesis~\ref{Hyp:OrganContam} was also tested experimentally.  Chromatographic impurity removal
techniques selectively address polar molecules or ionic states~\cite{bib:UngerKK,bib:SurfaceConc-OH}.
It is the excess charge on the molecule, or atom, which draws it to the hydroxyl group (--OH) on
the active adsorption sites of the gel.  The excellent reproducibility of our lead\ removal experiments
suggests that about 3-5\% of the lead\ atoms were present in a non-polar, molecular state,
without affinity to the active hydroxyl groups of silica gel.

\par In order to understand why this species has no affinity to the silica gel,
it is of particular interest to examine a study performed with $n$-dodecane exposed to a $\rm ^{60}Co$
source~\cite{bib:DodecaneRadiation}.  This study showed that the exposure to ionizing radiation produced
fragmented carbon chains. We note that the $^{212}$Pb\ and $^{214}$Pb\ isotopes used in our studies,
as well as the $^{210}$Pb\ present in KamLAND's LS, are born via alpha decay.
A recoiling lead\ nucleus, with typical energies of a few hundred keV, can easily fragment the carbon-carbon
bonds in $n$-dodecane, having typical binding energies of order 10 eV~\cite{bib:CRCChemPhys}.  These fragmented molecular states
could form stable, electrically neutral, non-polar compounds. The most notable of these are the lead-carbon\ bonds which
tend to be covalent, and thus non-polar.  The decay of the parent nuclei into lead\ occurs in a carbon rich environment
and this bonding is energetically favored over many other available chemical
bonds (e.g.~C--C bonds)~\cite{bib:CRCChemPhys}.
The non-polar lead-carbon\ compounds would have no affinity to the polar reaction sites
on silica gel, supporting Hypothesis~\ref{Hyp:OrganContam}.

\par  Table~\ref{Tab:OrganoPbProp} lists the relevant properties of the known
organic compounds containing lead. All exhibit lower boiling points than those of PC and $n$-dodecane.
It is of interest to note $\rm C_{8}H_{20}Pb$ is thermally unstable and will decompose at 200~$^{\circ}{\rm C}$.
This process would leave behind a polar or ionic lead\ species.

\par To obtain supporting evidence for Hypothesis~\ref{Hyp:OrganContam}, we experimented with
chemical procedures targeting the organic lead-carbon\ bonds.
It is known that the reagents: $\rm FeCl{_3}$~\cite{bib:FeCl3}, $\rm SnCl_{4}$~\cite{bib:TinCl4},
$\rm MoS_{2}$~\cite{bib:MOS} and thiol resin~\cite{bib:ThiolResin} target these lead-carbon\ bonds directly.

\par To test the effect of these chemicals on the lead\ reduction factor, we
first passed $^{212}$Pb-loaded LS\
through a silica gel column to remove the ionic lead\ component. In a second step, 150 ml of the
treated LS\ was stirred magnetically, with one of
the reagents listed above, for 45 min. Next, 15 ml of 0.5 M EDTA ($\rm [CH_2N(CH_2CO_2H)_2]_2$)
solution (pH of 8) was added to the mixture and stirred for an additional 15 min.  The mixture was filtered
under partial vacuum and passed over a molecular sieve to remove any water.  The resulting LS\ was
filtered a second time to remove any particulates of the molecular sieve. The liquid was
counted to determine the remaining amount of $^{212}$Pb.
The lead\ reduction factors obtained this way are tabulated in Table~\ref{tab:OrganicData}.

\input{Table05}

\par LS\ treated with $\rm FeCl{_3}$ before silica gel extraction showed a
50-fold enhanced lead\ reduction factor compared to plain chromatography.
We interpret this boost in effectiveness as supporting evidence for Hypothesis~\ref{Hyp:OrganContam}.
However, the $\rm FeCl{_3}$ treatment of the LS\ was found to permanently degrade its optical properties.
The measured light attenuation length at 436 nm was $< 10^{-4}$ m compared to approximately 13 m for
untreated LS.  Optical transparency could not be restored by means of filtration or repeated
treatment with silica gel.

\par The successful experiments utilizing reagents targeting lead-carbon\ bonds
prompted us to consider heating the lead-loaded LS\ to 100-200~$^{\circ}{\rm C}$\
to remove the tetraethyllead which decomposes~\cite{bib:CRCChemPhys}.
As a test, $^{212}$Pb-loaded LS\ was pre-heated in a stainless steel vessel, as shown in Fig.~\ref{Fig:HeatingApp}, before sending it
through the ion-exchange column, depicted in Fig.~\ref{Fig:AdsorbApp}. The results from these heating experiments are provided
in Table~\ref{tab:HeatingTable}.  The first three entries show a substantial
enhancement of the lead\ reduction factor for pre-heated LS\ compared to the
experiments with silica gel extraction only.
The following three entries give a factor 5 to 10 reduction obtained by just heating the LS\ and
not using silica gel.
Lead adherence to the container walls was excluded by experiments performed at room temperature.

\input{Table06}

\par We consider these results to be further evidence for
 Hypothesis~\ref{Hyp:OrganContam}.  However, the data are not specific enough as to allow us to
identify which compounds are present and in what proportion. This is nevertheless an important
finding, indicating that the limitless reduction factors predicted by plate theory,  Eq.~\ref{eq:ElutionCurve},
are not to be expected for alpha decay products in an organic medium.

\begin{figure*}[tbp]
\begin{center}
  \subfigure[]{\includegraphics[height=0.73\columnwidth, width=0.9\columnwidth]{Figure04.jpg}\label{Fig:UADistillation}}
  \subfigure[]{\includegraphics[height=0.73\columnwidth]{Figure05.jpg}\label{Fig:TohDistillation}}
  \subfigure[]{\includegraphics[height=0.73\columnwidth]{Figure06.jpg}\label{Fig:TohLareDistllation}}
\caption{Size evolution of distillation systems utilized during the purification experiments.
Fig.~\ref{Fig:UADistillation} and Fig.~\ref{Fig:TohDistillation} show the
bench-top systems used at the University of Alabama and Tohoku University,
respectively.  Together they provided a complete analysis of systematic studies
on distillation. Fig.~\ref{Fig:TohLareDistllation} is the scaled-up distillation
system at Tohoku University, allowing us to increase the amount of distillate by a factor 20.}
\label{Fig:AllDistillation}
\end{center}
\end{figure*}

\input{Table07}

\subsection{Distillation}
\label{Sec:Distillation}

\par The final method studied for KamLAND's LS\ purification addresses the removal of all lead\ species
in an integrated way: distillation.  During distillation, the distillate is heated to induce a phase
change. Differences in vapor pressure result in fractionation between the constituents.
As a by-product, the heating further breaks organo-metallic bonds and transforms organometallic lead\ compounds,
that may be in solution, into a polar or ionic form. These, in turn, can be effectively removed by distillation.

\par Distillation experiments primarily utilized lead-loaded $n$-dodecane, which makes up 80\% of the
KamLAND LS.  The distillation apparatuses utilized in these studies are
pictured in Fig.~\ref{Fig:AllDistillation}.   The system in
Fig.~\ref{Fig:UADistillation} consisted of a three-neck flask (500 ml), a silvered
fractionating column packed with stainless steel wool, and a condenser.  A thermocouple was used to regulate the distillate
temperature to within a few~$^{\circ}{\rm C}$. A mercury thermometer was placed on top of the fractionation column
to monitor the temperature of the vapor prior to condensation.  The stainless steel wool
increased the effective number of plates.  In essence, it
allows for multiple distillations of the liquid as the vapor rises
through the column and re-condenses on the steel wool.  Typical operating
conditions were 170~hPa, with a $160 \pm 1$~$^{\circ}{\rm C}$\ set-point for
the distillate in the boiling flask.  The flask temperature was maintained by the thermocouple but the temperature measured by the mercury thermometer
 at the top of the fractionating column varied over time from 125-145~$^{\circ}{\rm C}$.  Temperatures exceeding 145~$^{\circ}{\rm C}$\ the top of the column
resulted in a lower reduction factor due to the boil-over of contaminants.  The rate of distillation at these operating conditions was 10 ml/min.
Operating the distillation system at a reduced pressure is important to minimize the amount of
dissolved radioactive gases ($^{222}$Rn, $^{85}$Kr, and \argon) as well as heat-induced changes in the optical properties of the distillate.

\par  Repeated distillation tests with $^{212}$Pb-loaded $n$-dodecane provided a handle on the variance associated with
maintaining consistent operating procedures. To maintain reproducibility of the distillation results, the following procedures were utilized:

\begin{itemize}
  \item[-] All internal surfaces of the distillation system were cleaned with acetone, then
  etched with 0.1 - 1.0 molar HNO$_{3}$, and rinsed with DI water and ethanol.
  This eliminated carry-over between subsequent distillations.
  \item[-] All distillate collected before the top thermometer reached the boiling point of the distillate was discarded.
    This removes all impurities with a low vapor pressure. Some lead\ candidates, addressed in this way, can be found in Table~\ref{Tab:OrganoPbProp}.
  \item[-] Collection of the distillate was stopped after approximately 90\% of the volume was distilled.
    This operation reduces boil-over effects and re-contamination with
    high-boiling-point compounds.
  \item[-] A constant rate of distillation was maintained.  Rapid boiling at increased distillation rates directly
    contributes to the transfer of impurities into the distillate.
\end{itemize}

\par Distillation was performed under different pressure and temperature
conditions for $n$-dodecane, PC, and PPO.  Similar measurements taken with the
different distillation systems depicted in Fig.~\ref{Fig:AllDistillation}
allowed for the investigation of instrumental variations in the procedure that
could impact the reduction factor.
To ensure that the reduction factors observed in the modestly sized table-top experiments
would apply to larger volumes, a scaled-up distillation system, shown in
Fig.~\ref{Fig:TohLareDistllation}, was designed to process 5~L at a time.
Up to seven distillations were repeated in series to assess saturation of the lead\ reduction
factor.  Distillate samples were taken after consecutive
distillations during a single experiment.  The measured $^{212}$Pb\ reduction factors are
summarized in Table~\ref{tab:DistRedFact}. Beyond three successive distillations, no further lead\ reduction could be observed.
We also tested the combined effect of silica gel extraction and 5-times distillation
resulting in a factor 1.1 improvement compared to distillation only, as can be
seen in Table~\ref{tab:DistRedFact}.

\par Distillation proved to be the most effective means of removing radioactive
lead\ from the KamLAND LS\ components. The reduction factors observed during
the study of distillation ranged from a few hundred to several thousand,
depending on the operating conditions.  These results provided the necessary
technical base  required to achieve the high reduction factors needed for the
KamLAND purification campaign.  They further provided essential procedural
development and initial parameters for the full-scale KamLAND distillation system.