\section{Calculations of Pion Yield and Radiation Dose Using MARS}
%Nikolai Mokhov
Detailed MARS14(2000)~\cite{muc0175,mars14} simulations have been performed for the optimized
Study-II target-capture system configuration. 
A 24-GeV kinetic energy proton beam ($\sigma_x = 
\sigma_y = 1.5$~mm, $\sigma_z = 3$~ns, 67~mrad) interacts with a 5~mm radius mercury 
jet tilted by 100~mrad, which is ejected from the nozzle at $z=-60$ cm,
crosses the $z$-axis at $z= 0$~cm, and hits a mercury pool at $z=220$ cm, 
$x=-25$ cm. 

Results~\cite{muc0194} are based on two runs of 400,000 protons on target each,
including energy deposition in the mercury jet, the yield of captured pions,
fluxes of charged and neutral particles and the consequent radiation dose
in the materials of the target system. 
For example, the total power dissipation in the jet 
at $-60 < z < 0$ cm is 100 kW for 6 bunches at 2.5 Hz and $1.7 \times 10^{13}$
protons per bunch.  Preliminary results were given in
Refs.~\cite{feas00,muc0169}.

As noted earlier, to be conservative, we estimate radiation effects based on $2\times 10^7$~s per 
operating year, though we estimate physics production based on a standard 
$1\times 10^7$~s year.

\subsection[Captured Pion/Muon Beam \textit{vs.} Target and Beam Parameters]{Captured $\pi/\mu$ Beam $\textit{vs.}$ Target and Beam Parameters}

Realistic 3-D geometry based on Fig.~\ref{tgtc},
together with material and magnetic field distributions
based on the solenoid magnet design optimization, have been implemented into
\textsc{mars}.  The level of detail in the model is illustrated by 
Fig.~\ref{Tgt:geo3}, which shows a transverse section at $z = 5.2$~m that
includes the mercury pool that serves as the proton beam absorber. 
\begin{figure}[!bht] %nm3
\begin{center}
\includegraphics*[width=3in]{../template/report/ps-and-eps/geo3.eps}
\caption[Transverse section of the target system at $ z = 5.2$ m ]
{Transverse section of the target system at $z = 5.2$ m, showing the mercury 
pool that serves as the proton beam absorber.} 
\label{Tgt:geo3}
\end{center}
\end{figure}
  
The use of a 3-D magnetic field map
results in the reduction of the $\pi /\mu$-yield in the decay
channel by about 7\% for C and by 10-14\% for Hg targets, compared with the
assumption that $B_z(r,z)$ obeys Eq.~(\ref{TgtP.1}).

Both graphite (C) and mercury (Hg) tilted targets were studied. 
A two-interaction-length target (80~cm for C of radius $R_T=7.5$~mm, 
and 30~cm for Hg of $R_T=5$~mm) is found to be optimal in most cases, and we keep $R_T \geq 2.5~\sigma_{x,y}$, where $\sigma_{x,y}$ are the beam rms
spot sizes.


Results of a detailed optimization of the particle yield $Y$ are presented 
below, in most cases for a sum of the numbers of $\pi$ 
and $\mu$ of a given sign and energy interval at a fixed distance $z = 9$~m
from the target. 
For proton energies $E_p$ from a few GeV to about 30~GeV,
the shape of the low-momentum spectrum of such a sum is energy-independent
and peaks around 250 MeV/$c$ momentum (145 MeV kinetic energy), as
illustrated in Fig.~\ref{Tgt:e910data}.  Moreover, the sum is practically independent of $z$
at $z \geq 9$~m--confirming a good matching and capturing--with a growing
number of muons and proportionately decreasing number of pions along the
decay channel.
For the given parameters, the $\pi /\mu$ kinetic energy interval of 
$30~\textrm{MeV}\leq E\leq 230~\textrm{MeV},$ around
the spectrum maximum is considered as the one to be captured 
by the downstream phase rotation system. 


%\begin{figure}[hbt!]
%\begin{minipage}[t!]{.48\linewidth} % fig 
%\includegraphics[width=\textwidth]{mok-targ1-fg3a.eps}
%\end{minipage}\hfill
%\begin{minipage}[t!]{.48\linewidth} % fig 
%\includegraphics[width=\textwidth]{mok-targ1-fg3b.eps}
%\end{minipage}
%\vspace{-2mm}
%\caption[Pion yield in Hg and C targets \textit{vs.}\ $E_p$ and $\theta_p$]
%{Pion yield from Hg and C targets \emph{vs} $E_p$ (left) and
%yield from a Hg target at $E_p$=16~GeV \emph{vs} tilt angle (right).}
%\label{fg:hg-c}
%\end{figure}

The yield $Y$ grows with the proton energy $E_p$, is almost material-independent
at low energies and grows with target $A$ at high energies, being almost a factor
of two higher for Hg than for C at $E_p$=16-30~GeV (Fig.~\ref{fg:mok4b}).
To avoid absorption of spiraling pions by target material, the target and beam 
are tilted
by an angle $\alpha$ with respect to the solenoid axis. The yield is higher by 
10-30\% 
for the tilted target. For a short Hg target, $\alpha$=150~mrad seems to be the 
optimum, as shown in Fig.~\ref{fg:tilt_r} (left).
%Fig.~\ref{fg:radius} shows the dependence of the yield on Hg and C target radii
%under the baseline $R_{T}=2.5\sigma_{x,y}$ condition. 
The maximum yield occurs at target radius $R_{T} = 5$~mm for 
Hg with $R_{T} = 2.5 \sigma_{x,y}$ 
 ($R_{T}$=7.5~mm and $R_{T}=3.5\sigma_{x,y}$ for C),
as shown in Figs.~\ref{fg:tilt_r} (right) and \ref{fg:sigma} (left).
The yield with mercury could be further increased by increasing the target
radius to gain secondary pion production, but the target heating would also
be increased significantly, as shown in Fig.~\ref{fg:sigma} (right). 

%\begin{figure}[hbt!]
%\begin{minipage}[t!]{.48\linewidth} % fig 
%\includegraphics[width=\textwidth]{mok-targ1-fg4a.eps}
%\end{minipage}\hfill
%\begin{minipage}[t!]{.48\linewidth} % fig 
%\includegraphics[width=\textwidth]{mok-targ1-fg4b.eps}
%\end{minipage}
%\vspace{-2mm}
%\caption[Pion yield \textit{vs.}\ target radius]
%{Pion yield as a function of a target radius, Hg (left) and C (right),
%for a 16-GeV proton beam and several tilt angles.}
%\label{fg:radius}
%\end{figure}

\begin{figure}[hbt!]
\begin{minipage}[t!]{.48\linewidth} % fig 
\includegraphics[width=\textwidth]{../template/report/ps-and-eps/mok-targ1-fg5a.eps}
\end{minipage}\hfill
\begin{minipage}[t!]{.48\linewidth} % fig 
\includegraphics[width=\textwidth]{../template/report/ps-and-eps/mok-targ1-fg5b.eps}
\end{minipage}
\vspace{-2mm}
\caption[Pion yield and $\Delta T$ \textit{vs.}\ $r_{\rm target}/r_{\rm beam}$]
{Pion yield (left) and maximum instantaneous temperature rise (right)
as a function of the ratio of target radius to RMS beam spot size.
}
\label{fg:sigma}
\end{figure}

%The ratio of Hg to C yields varies with the beam energy, as well as
%with other beam/target parameters. At 16~GeV it is in the range of 1.5-1.7 for positives 
%and 1.7-2.2 for negatives. Optimizing beam/target parameters, it is found that 
%the best results for the particle yield in the decay channel at 16~GeV with the given cut are:
%$Y_{\pi^+ + \mu^+}$ = 0.182 and $Y_{\pi^- + \mu^-}$ = 0.153 for the 80-cm C 
%target and 
%$Y_{\pi^+ + \mu^+}$ = 0.309 and $Y_{\pi^- + \mu^-}$ = 0.315 for the 30-cm Hg
%target, i.e., at 16 GeV (best Hg)/(best C) = 1.7 (+) and 2.06 (-).

Figure \ref{Tgt:edjet} shows longitudinal profiles of the energy density 
deposited in the mercury jet target in three radial regions.  The center of
the proton beam enters the jet at $z = - 45$~cm, and the energy deposition
peaks about 12~cm downstream of this point, at $z = - 33$~cm.


\begin{figure}[!bht] %nm4
\begin{center}
\includegraphics*[width=3in]{../template/report/ps-and-eps/edjet.eps}
\caption[Energy deposited in a mercury target \textit{vs.}\ $z$ ]
{Longitudinal profiles of the energy density deposited in the mercury jet
target in three radial regions.} 
\label{Tgt:edjet}
\end{center}
\end{figure}

\subsection{Particle Fluxes, Power Density and Radiation Dose}
Figure~\ref{caprad} shows the radiation per $2 \times 10^{7}$ s 
in the vicinity of the target. 
%Figure \ref{radvsr} shows the power density and radiation dose per 
%$2 \times 10^{7}$ s at a
%transverse section at $z= 20$ cm).
Table~\ref{capturerad} gives the maximum doses per year and expected lifetime 
%at the worst location 
for various components (Note that for assessing radiation effects we take a larger operating year to be conservative). 
%Note that Figs.~\ref{caprad} and \ref{radvsr}
%assume a year of $2 \times 10^7$ s, but Table~\ref{capturerad} gives the doses 
%per year of $1 \times 10^7$ s,
%consistent with the convention of Study-II.

\begin{figure}[htb]
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/fs2-ts-dose.eps}
\caption[Radiation dose in the target system for $-2 < z < 6$ m ]
{Absorbed radiation dose per year of $2 \times 10^7$ s and a 1 MW proton beam
in the target system for $-2 < z < 6$ m and $r < 1.4$ m.} 
\label{caprad}
\end{center}
\end{figure}

\begin{table}[htb]
\begin{center}
\caption[Radiation doses and lifetimes of some components ]
{Radiation doses and lifetimes of some components of the target system.}
%\rr ***needs updating}
\label{capturerad}
%\vspace{2.5mm}
\begin{tabular}{|p{3.2cm}ccccc|}
\hline
Component & Radius & Dose/yr & Max allowed Dose & 1 MW Life & 4 MW life \\
        & (cm) & (Grays/$2 \times 10^7$ s) & (Grays) & (years) & (years) \\
\hline
Inner shielding      & 7.5 & $2 \times 10^{11}$ & $10^{12}$ &   5 & 1.25 \\
Hg containment       & 18  & $2 \times 10^{9}$  & $10^{11}$ &  50 & 12 \\
Hollow conductor     & 18  & $1 \times 10^9$    & $10^{11}$ & 100 & 25 \\
Superconducting coil & 65  & $6 \times 10^6$    & $10^8$    &  16 &  4 \\
\hline
\et
%\vskip.2in
\end{center}
%* Assuming that stainless steel can withstand this dose so long as it is not 
%stressed. Cu is claimed to take it, but is not compatible with Hg. 
%If SS will not, then we should try and find something that would. 
%It is assumed that this shielding would be cooled by mercury.
\end{table}

Figures \ref{Tgt:flux-slice} and \ref{Tgt:pd-slice} illustrate charged and
neutral particle fluxes, and the resulting power deposition and radiation dose, as a function of radius at the downstream end of the target.
Figure~\ref{Tgt:be-window-dose} shows the power density and radiation dose in
the beryllium window at $z = 6.1$~m.

\begin{figure}[!bht] %nm11
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/sc12-flux-slice.eps}
\caption[Particle fluxes in the magnet system \textit{vs.}\ $r$ ]
{Flux of neutral (top) and charged (bottom) particles as a function of radius 
at the downstream end of the target.} 
\label{Tgt:flux-slice}
\end{center}
\end{figure}

\begin{figure}[!bht] %nm12
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/sc12-pd-slice.eps}
\caption[Power density and radiation dose in the target system \textit{vs.}\ $r$ ]
{Power density (top) and total radiation dose (bottom) due to secondary
particles as a function of radius at the downstream end of the target.} 
\label{Tgt:pd-slice}
\end{center}
\end{figure}


\begin{figure}[!bht] %nm5
\begin{center}
\includegraphics*[width=3in]{../template/report/ps-and-eps/be-window-pd.eps}
\includegraphics*[width=3in]{../template/report/ps-and-eps/be-window-dose.eps}
\caption[Power and radiation dose in the Be window at $z = 6.1$ m ]
{Power density (left) and absorbed radiation dose (right) in the beryllium 
window at $z = 6.1$ m.} 
\label{Tgt:be-window-dose}
\end{center}
\end{figure}

The neutron flux in the target system is shown in Fig.~\ref{Tgt:tardec-nflux},
and the absorber radiation dose is shown in Figs.~\ref{caprad} and
\ref{Tgt:tardec-dose}.  Even at the end of the decay channel, at $z = 36$~m, the
radiation levels remain high.

\begin{figure}[!bht] %nm8
\begin{center}
\includegraphics*[width=3in]{../template/report/ps-and-eps/fs2-ts-nflux.eps}
\includegraphics*[width=3in]{../template/report/ps-and-eps/fs2-tardec-nflux.eps}
\caption[Flux of neutrons with $E > 100$ keV in the target system ]
{Flux of neutrons with $E > 100$ keV in the target system and decay channel for
$-2 < z < 6$~m and $r<1.4$~m (left) and $-2 < z < 36$~m (right) and $r < 0.8$ m.} 
\label{Tgt:tardec-nflux}
\end{center}
\end{figure}


\begin{figure}[!bht] %nm7
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/fs2-tardec-dose.eps}
\caption[Radiation dose in the target system for $-2 < z < 36$ m ]
{Absorbed radiation dose in the target system and decay channel for
$-2 < z < 36$~m and $r < 0.8$ m.} 
\label{Tgt:tardec-dose}
\end{center}
\end{figure}

%\begin{figure}[!bht] %nm9
%\begin{center}
%\includegraphics*[width=4in]{fs2-ts-dose.eps}
%\caption[Radiation dose in the target system for $-2 < z < 6$ m ]
%{Absorbed radiation dose in the target system for
%$-2 < z < 6$ m and $r < 1.4$ m.} 
%\label{Tgt:ts-dose}
%\end{center}
%\end{figure}




%\begin{figure}[!bht] %nm10
%\begin{center}
%\includegraphics*[width=4in]{fs2-ts-nflux.eps}
%\caption[Total flux of neutrons in the target system ]
%{Total flux of neutrons (with $E > 0$) in the target system for
%$-2 < z < 6$ m and $r < 1.4$ m.} 
%\label{Tgt:ts-nflux}
%\end{center}
%\end{figure}





%\subsection{Beam power considerations}

%The yield per beam power is almost independent of $E_p$ for 
%high-$Z$ targets at 6$<E_p<$24~GeV and drops by 30\% at 16~GeV from a 6-GeV 
%peak for graphite (Fig.~\ref{fg:yield} (left)). The higher $E_p$ reduces the 
%number of protons on target.
%To provide $\mathcal{N}_{\mu}$=2$\times$10$^{20}$ muon decays per year in 
%the straight section at 15~Hz,
%one needs to have 6$\times$10$^{12}$ muons per pulse in the decay channel, 
%assuming
%a factor of 3 total loss on the way from the decay channel to the ring. 
%With that,
%needed are 3.30$\times$10$^{13}$ and 3.92$\times$10$^{13}$ protons per pulse at
% 16 GeV
%on the optimal C target for positives and negatives, respectively. 
%This corresponds to 1.27 and 1.51 MW beams. 





%\begin{figure}[!bht]  %nm1
%\begin{center}
%\includegraphics*[width=4in]{geo1.eps}
%\caption[Geometry of the target system for $-2 < z < 36$ m ]
%{Geometry of the target system and decay channel used in the MARS
%calculations for $-2 < z < 36$ m.} 
%\label{Tgt:geo1}
%\end{center}
%\end{figure}

%\begin{figure}[!bht] %nm2
%\begin{center}
%\includegraphics*[width=4in]{geo2.eps}
%\caption[Geometry of the target system for $-2 < z < 6$ m ]
%{Geometry of the target system used in the MARS
%calculations for $-2 < z < 6$ m.} 
%\label{Tgt:geo2}
%\end{center}
%\end{figure}






