\begin{spacing}{0}
\chapter{Interaction of an Intense Proton Beam with Hg Jet in a Magnetic Field}
\label{chap:exp_result_beam}
\end{spacing}
\
\par

In this chapter, the jet's interacting characteristics in magnetic fields are investigated.
The disruption of the jet interacting with various beam intensities and beam energy is
observed and the magnetic suppression to it is discussed. The captured images show the mechanism
of the beam-jet interaction, and the qualitative consistency with the distribution of calculated
energy deposition is discussed. The energy deposition induced by the proton beam generates filaments
on the Hg jet surface due to thermal stress.
\ \\
\ \\

\section{High Energy Proton Beam Structure}

\subsection{Proton synchrotron machine}

Neutrino factories requires a large number of muons, which are obtained from the decay of pions. Efficient production of pions can be achieved by colliding an intense proton beam with a high-Z target.
An important consideration is the problem of removing the power deposited by the proton beam without
interfering with the process of extracting the end-product, which is the muon beam.
Experiments on the interaction of a 14~GeV and 24~GeV proton beam with
pulse structures of 4 to 16~bunches per pulse and the spot sizes in the order of
2 to 10~mm$^{2}$ RMS up to 30 tera-protons(Tp) per pulse in magnetic fields up to 15~T have been
carried out at CERN. Figure~\ref{fig:psmachine} (a) shows the infrastructures for experiment
at CERN. All equipments for experiment are installed at tunnel TT2/TT2A and these are
controlled remotely from control room. The proton beam is delivered from proton synchrotron
ring and the beam setup is schematically shown in Fig.~\ref{fig:psmachine} (b).
The PS machine is set up in harmonic 16 bunches and the extracted protons fill the machine in
bunch pairs. A bunch in harmonic 8 mode is consisted of a bunch pair.
Therefore, a bunch period in harmonic 8 mode is two times of a bunch period in harmonic 16 mode.
Each bunch can fill protons up to 2 ~ 2.5 $\times$ 10$^{12}$. Therefore, the maximum beam
intensity can be achieved up to ~ 32 $\times$ 10$^{12}$ protons.
Figure~\ref{fig:tt2layout} shows the layout of tunnel at CERN, where equipments for experiment are
installed. Electronic equipments for optical diagnostics, hydraulic power unit, and cryogenic
system are positioned at tunnel TT2. Hg loop system, solenoid magnet, and beam diagnostic system
are positioned at tunnel TT2A. The fibers for optical diagnostics of Hg target in solenoid magnet
and cables for controlling the Hg loop system and solenoid magnet are connected between TT2 and
TT2A passing through an artificially drilled hole.

\subsection{Proton beam pulse length}

In order to produce the design number of 10$^{21}$ muons/year in muon storage ring,
4~MW of proton beam power is desired. For our experiment, the CERN PS ran typically in a harmonic
16 mode. Hence, it is possible to fill with 2 $\times$ 10$^{12}$ protons/bunch and therefore up to
32 $\times$ 10$^{12}$ protons/spill. One beam pulse consists of several beam bunches.
The bunch lengths for harmonic 16 mode are 50~ns and 30~ns at full width at half maximum (FMWH) respectively. The bunch lengths for harmonic 8 mode are 70~ns and 40~ns at full width at half maximum (FMWH) respectively.
The bunch-to-bunch differences for harmonic 16 mode and harmonic 8 mode are multiples of
131~ns and 262~ns respectively.
The proton beam pulse structure of harmonic 8 and harmonic 16 in 14~GeV, 6~Tp is shown in
Fig.~\ref{fig:beamstructure}.
The spot size of beam at the experiment is in the order of 2 to 10 mm$^{2}$ RMS.
This allows to place up to 32 $\times$ 10$^{12}$ protons on the Hg target, generating a peak
energy deposition of $\sim$ 150~J/g.

\subsection{Proton beam envelope by optics and camera screen}

The proton beam with 14~GeV and 24~GeV beam energy is employed in the experiment.
As the number of protons in a beam pulse increases, it is reported \cite{Efthymiopoulos2008}
that the beam spot size increases. The beam spot size is calculated by CERN using the measured beam
emittance, dispersion, and the momentum spread of the beam particle. The emittance is measured by measuring the beam profile in a position of known beam parameters based on optics.
The geometrical emittance $\varepsilon$ is defined by
\begin{equation}
 \label{e:eq51a}
\varepsilon = \frac{(2 \sigma)^{2}}{\beta}  .
\end{equation}

The normalized emittance $\varepsilon^{*}$ is defined by

\begin{equation}
 \label{e:eq51b}
\varepsilon^{*} = (\beta \gamma) \varepsilon   \ \  , \ \  \beta \gamma = \frac{P}{M} \ ,
\end{equation}

where $\beta$ and $\gamma$ are the relativistic functions at the measurement point, and $P$ and $M$ are
momentum and mass of a proton respectively.
Considering dispersion and emittance of beam, the spot size is estimated as follow:

\begin{equation}
 \label{e:eq51c}
\sigma = \frac{1}{2} \sqrt{\varepsilon \cdot \beta + (D_{p} \frac{\delta p}{p})^{2}} \ ,
\end{equation}

where $D_{p}$ is the dispersion function at the measurement point and $\frac{\delta p}{p}$ is the momentum
spread of the particle beam. The measured $\frac{\delta p}{p}$ for 14~GeV beam is 1.66 (2~$\sigma$, 0.1~$\%$)
and 1.1 for 24~GeV beam \cite{Efthymiopoulos2008}.
The normalized emittance is directly measured as a function of beam intensity from accelerator machine
as follow:

\begin{equation}
 \label{e:eq51d}
\varepsilon^{*}_{2 \sigma} = f ( width_{4 \sigma}, \ \ \frac{\delta p}{p}_{2 \sigma} )
= \frac{(\frac{width_{4 \sigma}}{2})^{2} - ( |D_{p}| \frac{\delta p}{p}_{2 \sigma} )^{2}}{\beta} .
\end{equation}

The beta function and dispersion function from optics are calculated using parameters of quadrupole
strengths and locations, which are the values at the position of Hg target \cite{Efthymiopoulos2008}.
The estimated 1 $\sigma$ beam spot at Hg target position is given in Table~\ref{tab:beam_size}.
Figure~\ref{fig:beamsize_ilias} shows the estimated 1~$\sigma$ beam spot size at the center of target
based on optics \cite{Efthymiopoulos2008}.
Figure~\ref{fig:beamsize_goran} shows the measured 1~$\sigma$ beam spot size at the phosphor camera screen monitor
installed $\sim$ 4.2~m away from the center of magnet before entering the magnet \cite{Skoro}. It is
also reported \cite{Skoro} that the beam spot size increases as the number of protons increase.
Due to the saturation of image, the measured beam spot size is shown as $\sim$ 2 times larger than the estimated
beam spot size from optics.
Figure~\ref{fig:beamsize_goran_timehistory} (c) shows the beam sizes distribution measured by phosphor
screen monitor as a function of recorded time interval between each beam shot, where the histogram for events of beam size in horizontal plane is shown in Fig.~\ref{fig:beamsize_goran_timehistory} (a) and the histogram for
events of beam size in vertical plane is shown in Fig.~\ref{fig:beamsize_goran_timehistory} (b).
This plots show that the possible residual saturation of image by phosphor screen monitor is not related
with recorded time interval between each beam shot. The distribution of beam spot size is uniform regardless of
the possible residual saturation by the screen monitor.

\section{MARS Simulation for Energy Deposition to Mercury Jet by Proton Beam}

\subsection{Physics model}

MARS is a Monte Carlo code for inclusive and exclusive simulation of 3D hadronic and electromagnetic
cascades, muon and heavy ion transport in accelerator, detector, and shielding components in the energy
range from a fraction of an eV up to 100~TeV \cite{Mokhov2000}. In MARS code, hadron production, neutrino
interactions, electromagnetic interactions of heavy particles, and electromagnetic showers are considered.
For hadron production, information on the nuclides generated in nuclear collisions is scored, or reported
in the results of the simulation, which covers a hadron kinetic energy range up to 100~TeV.
For neutrino interactions, the model permits the selection of the energy and angle of each particle
($ \nu, e, \mu $ )emanating from a simulated interaction. These particles, and the showers initiated by
them, are then further processed in the code. Four types of neutrino interactions are distinguished
($ \nu_{\mu}, \bar{\nu_{\mu}} , \nu_{e}, \bar{\nu_{e}} $) and the model identifies all possible types
of neutrino interactions with nuclei. The corresponding formulas for these processes as well as results
of Monte Carlo simulations are considered. For electromagnetic interactions of heavy particles,
electromagnetic interactions of muons and charged hadrons in arbitrary materials are simulated.
Radiative processes and atomic excitation and ionization with energy transfer are considered.
The electromagnetic showers are based on the physics of electromagnetic interactions and it gives
electron and photon interactions in composite solid, liquid, and gaseous materials \cite{Mokhov2000}.

\subsection{Mercury jet modeling in MARS code}

Using MARS code, calculation of energy deposition is performed at Fermi National Accelerator
Laboratory \cite{Striganov20092}. For the modeling of jet in MARS, the experimentally measured
Hg jet size and trajectory in magnetic field with assumption of sectionally elliptic jet shape and
circular jet shape with equivalently reduced mass density to the initial flow rate from nozzle are employed.
The proton beam is passing through the center of magnetic axis. For simplicity, the z coordinate of
modeling in MARS defines as 0 at the center of magnetic axis along the direction of magnetic field.
Accordingly, the x coordinate of modeling in MARS defines as the vertical direction perpendicular
to the direction of magnetic field.
The measurement of vertical distance between magnetic axis and
the center of jet is given in Table~\ref{tab:jetsize_mars}, where the measured jet size
as well as approximated mass density for case simulation of assumed circular jet shape are also given.
The vertical distance in cm in MARS code between center of jet and magnetic axis is employed as follow:

\begin{equation}
 \label{e:eq55b}
x_{vert} = -1.4522 - 3.65 \times 10^{-2} \times zz - 3.1672 \times 10^{-4} \times zz^{2} +
5.4206 \times 10^{-9} \times zz^{4} ,
\end{equation}

where $zz = z - 46$ in cm.
Figure~\ref{fig:modeling_mars} (a) and (b) show the sectional view of elliptic and circular jet, and
Fig.~\ref{fig:modeling_mars} (c) shows the sectional side view of jet interacting with proton beam in
a magnetic field, where the magnetic field direction and strength are indicated as arrows.
Number of meshes and sizes are given in Table~\ref{tab:jetsize_mars}.
Using MARS code, calculation of energy deposition in $GeV/g/proton$ with various magnetic field strengths
and beam intensities is performed at Fermi National Accelerator Laboratory \cite{Striganov20091}.
Note that $1~eV = 1.6022 \time 10^{-19}~J$ is used for conversion of energy unit between $eV$ and $J$.

\subsection{Energy deposition to mercury jet}

\subsubsection{\textit{effect of magnetic field on energy deposition}}

Figure~\ref{fig:energydep_magnetic} shows the averaged energy deposition in $J/g$ along the jet axis
for the case of 2~Tp according to the variation of magnetic field. Calculated energy deposition
of each meshed volume is averaged along the jet axis with respect to azimuthal angle and radial
distance using Eqn.~(\ref{e:eq58}):

\begin{equation}
 \label{e:eq58}
E_{p, \ average} (z) = \frac{\sum_{r} \sum^{2 \pi}_{\theta = 0} \rho \ V_{\theta}^{r} (r, \theta, z) E^{r}_{p,  \theta} (r, \theta, z)}{\sum_{r} \sum^{2 \pi}_{\theta = 0} \rho \ V_{\theta}^{r} (r, \theta, z)} \ ,
\end{equation}

where $V^{r}_{\theta}$ and $E^{r}_{p, \theta}$ represent the volume of each mesh along azimuthal angle
at each radial distance and its energy deposition respectively.
As the magnetic field increases, the distribution of energy deposition over the jet increases dependent of jet size.
This indicates interaction of charged particles with magnetic field, so that more atomic excitation and
ionization with energy transfer occurs in higher magnetic field. Also, the electromagnetic shower
produced by a particle that interacts via the electromagnetic force gives electron and photon interactions
in Hg. From the equation of particle motion and Lorentz force in Eqn.~(\ref{e:eq59}), the momentum
of charged particle has an influence of the intensity of magnetic field followed by Maxwell's equations:

\begin{equation}
 \label{e:eq59}
\frac{d \textbf{p}}{d t} = \textit{e} [ \textbf{E} + \textbf{v} \times \textbf{B} ] ,
\end{equation}

where \textit{e} is the charge on the particle and \textbf{v} is the particle velocity.
Figure~\ref{fig:energydep_angle_24gev} (a) and Figure~\ref{fig:energydep_angle_14gev} (a) show the azimuthal averaged energy deposition over the orientation in sectional jet area along the jet axis for the case of
2~Tp with 24~GeV and 14~GeV in 10~T.
Calculated energy deposition of each meshed volume is averaged along the jet axis and azimuthal angle
with respect to radial distance using Eqn.~(\ref{e:eq510}):

\begin{equation}
 \label{e:eq510}
E_{p, \ average} (\theta, z) = \frac{\sum_{r} \rho \ V_{\theta}^{r} (r, \theta, z) E^{r}_{p, \theta} (r, \theta, z)}{\sum_{r} \rho \ V_{\theta}^{r} (r, \theta, z)} \ .
\end{equation}

Larger distribution of energy deposition occurs at bottom ($\sim$ 270 $^{\circ}$)of jet where
the beam enters. Gradually the larger distribution moves to the top ($\sim$ 90 $^{\circ}$) of jet where the beam leaves. Figure~\ref{fig:energydep_angle_24gev} (b) and Figure~\ref{fig:energydep_angle_14gev} (b) show the
azimuthal averaged energy deposition according to the variation of magnetic field along the orientation in
sectional jet area for the case of 2~Tp with 24~GeV and 14~GeV in 10~T.
Calculated energy deposition of each meshed volume is averaged along the azimuthal angle with respect
to jet axis and radial distance using Eqn.~(\ref{e:eq511}):

\begin{equation}
 \label{e:eq511}
E_{p, \ average} (\theta) = \frac{\sum_{z} \sum_{r} \rho \ V_{z}^{r} (r, \theta, z) E^{r}_{p, z} (r, \theta, z)}{\sum_{z} \sum_{r} \rho \ V_{z}^{r} (r, \theta, z)}  \ ,
\end{equation}

where $V_{z}^{r}$ and $E^{r}_{p, z}$ represent the volume of each mesh along axial jet axis at
each radial distance and its energy deposition respectively. Larger axially averaged energy
deposition is at bottom ($\sim$ 270 $^{\circ}$) of jet and the distribution of energy
deposition increases as the magnetic field increases. The geometrical distribution of energy
deposition depending on the applied magnetic field does not changes but keeps uniform profile
of distribution, which indicates that the profile of distribution is most likely dependent on jet shape.

\subsubsection{\textit{effect of proton beam spot size on energy deposition}}

Figure~\ref{fig:energydep_intensity} shows the averaged energy deposition per proton along the jet axis
using Eqn.~(\ref{e:eq58}) according to the variation of incident number of protons in 5~T.
As shown in Fig.~\ref{fig:beamsize_ilias}, the beam spot size increases as the incident number of protons
increase, where beam spot area is defined as $\sigma_{x} \sigma_{y}$.
We speculate on peak energy deposition and total energy deposition in Hg jet.
Peak energy deposition is the largest energy found in entire meshes on the jet model and total energy deposition
is the summed energy in entire meshes on the jet model.
Figure~\ref{fig:energydep_beamarea} shows the variation of peak energy deposition per proton and total
energy deposition per proton to Hg jet with respect
to the incident number of protons and proton beam spot size from Fig.~\ref{fig:beamsize_ilias} at both 14~GeV and
24~GeV beam energy in magnetic fields. The total energy deposition amounts to $\sim$ 6
$\sim $ 8~$\%$ of the incident beam energy and the total energy deposition is slightly decreasing depending on the variation of beam spot size. However, the total energy deposition increases as the magnetic field
increases. As discussed in Fig.~\ref{fig:energydep_magnetic}, it again indicates interaction of
charged particles with magnetic field, so that more atomic excitation and ionization with energy
transfer occurs in higher magnetic field. However, the peak energy deposition is independent of magnetic
field strength for given protons but influenced by beam parameters.
The lines in Fig.~\ref{fig:energydep_beamarea} represent the global fit of calculated peak energy deposition per
proton and calculated total energy deposition per proton using Eqn.~(\ref{e:eq512}) and Eqn.~(\ref{e:eq513})
respectively, shown as

\begin{equation}
 \label{e:eq512}
E_{p} = (a_{1} + b_{1} N_{p}^{b_{2}}) E_{b}^{c_{1}} \ , \ \  E_{p} = (a_{1} + b_{1} A_{b}^{b_{2}}) E_{b}^{c_{1}}
\end{equation}

and

\begin{equation}
 \label{e:eq513}
E_{p} = a_{1} (B - b_{1})^{b_{2}} E_{b}^{c_{1}} + N_{p} b_{3} E_{b}^{c_{2}} \ , \ \  E_{p} = a_{1} (B - b_{1})^{b_{2}} E_{b}^{c_{1}} + A_{b} b_{3} E_{b}^{c_{2}} \ ,
\end{equation}

where $N_{p}$, $A_{b}$, $B$, $E_{p}$, and $E_{b}$ denote number of protons, beam spot area, magnetic field,
energy deposition, and beam energy respectively.
Note that the parameterized values of coefficients and errors of the fit functions are given
in Table~\ref{tab:fitdata_ch5}.
The peak energy deposition as a function of incident number of protons decreases with square rooted power of
number of protons, and it increases with $\sim$ 1.5 power of beam energy between
14~GeV and 24~GeV. The ratio of beam energy between 14~GeV and 24~GeV is $\sim$ 1.7. The total energy
deposition decreases slightly linearly with number of protons and increases with 0.06 power of magnetic
field strength. Thus, the total energy deposition as a function of incident number of protons
has an increase with $\sim $ 1.4 power of beam energy
as an offset between 14~GeV and 24~GeV, and $\sim$ 0.9 power of beam energy as an slope in fit function
, which indicates possibly that the absolute ratio of power $\sim$ 1.5 due to the beam energy difference
is separated into two coefficient terms ratio of $c_{1}$ to $c_{2}$ in fit function.

Based on the result in Fig.~\ref{fig:energydep_beamarea}, the incident number of protons are multiplied to the
peak energy deposition per proton, which yields results in Fig.~\ref{fig:energydep_nproton}
on a logarithmic scale.
The peak energy deposition with respect to the number of protons increases parabolically due to
the increase of parabolically approximated beam cross sectional area, which directly influences
to the peak energy deposition to Hg jet. Also, the higher magnetic field again results in larger total
energy deposition to Hg jet.

The solid line in Fig.~\ref{fig:energydep_nproton} (a) and (b) represent the fit of calculated peak energy deposition using Eqn.~(\ref{e:eq514}) and Eqn.~(\ref{e:eq515}) respectively, shown as

\begin{equation}
 \label{e:eq514}
E_{p} = a_{1} N_{p}^{b_{1}} E_{b}^{c_{1}}
\end{equation}

and

\begin{equation}
 \label{e:eq515}
E_{p} = a_{1} N_{p}^{b_{1} + b_{2} B^{b_{3}}} E_{b}^{c_{1}} .
\end{equation}

The fit result from Eqn.~(\ref{e:eq514}) shows that the peak energy increases with $\sim$ 0.8 power of beam 
intensity on a linear scale, which can be a linear relation on a logarithmic scale with $\sim$ 0.8 between
beam intensity and peak energy deposition, and $\sim$ 1.6 between beam energy and number of protons. The fit result from Eqn.~(\ref{e:eq515}) shows that the total energy deposition increases with $\sim$ 0.9 power of beam 
intensity, but it slightly increases with $\sim $ 0.4 power of magnetic field. On a logarithmic scale, 
total energy deposition increases linearly with $\sim$ 1.4 times of beam energy.
This study is useful since it allows one to extrapolate the trend for estimation of profile of energy
deposition, so that one can approximate the profile of energy deposition over all of the region of
Hg jet based on the characteristic relations in energy deposition to magnetic field, beam intensity,
and Hg jet shape.

\section{Observation of Proton Beam Interaction and Hg Jet Response by Energy Deposition}

\subsection{Hg jet pressurization by energy deposition of proton beam}

The energy deposition $E_{p}$ due to ionization losses of the
protons is $\sim$ 33~J/g and additional ionization due to secondary particles
from interactions of the protons in the target raises this to a peak of $\sim$
100~J/g at 10~cm into the target \cite{McDonald20005}. The energy
deposition, $E_{p}$, leads to peak pressure $P_{p}$ that can be estimated as follow:

\begin{equation}
 \label{e:eq55}
P_{p} \approx K \alpha_{v} \Delta T  = \frac{\alpha_{v} K E_{p}}{c_{p}} \ ,
\end{equation}

where $\alpha_{v}$ is the thermal volumetric expansion coefficient, which corresponds to
3 times of thermal linear expansion coefficient, $K$ is
the bulk modulus, $E_{p}$ is the energy deposition, and $c_{p}$ is the specific heat capacity.
For Hg, $\alpha_{v}$ = 180 $\times$ 10$^{-6}$/K , \ \
$K$ = 25~GPa, \ \ $c_{p}$ = 138~J/(K \ kg).  A peak value of
$E_{p}$=100~J/g corresponds to a peak stress of $\sim$ 3000~MPa.
The Hg target is disrupted by the proton beam, leading to a breakup
into droplets. The strain energy is built up in the jet due to compression \cite{Sievers}.

\subsection{Observation of proton beam interaction and jet breakup}

Figure~\ref{fig:jet_response} is the photographs of the typical Hg jet interacting mechanism with
a 16~Tp, 14~GeV proton beam at 5~T captured at Viewport 3 at a 500~$\mu$s frame rate,
which shows clearly how the Hg jet is
responding from the sudden energy deposition by the proton beam. The beam hits the Hg jet at the
bottom surface, passing through the center of jet at Viewport 2, leaving the Hg jet
on the top surface. The captured photos show the response of the Hg jet at upstream,
midstream, and downstream with the interaction of the proton beam.
There are filaments on the top surface of jet at downstream, where the
beam is leaving, and on the bottom surface of the jet at upstream, where the proton beam is entering.
The jet break up voids at midstream, where the beam is passing through and possibly caused
by the cavitations from energy deposition.

\subsubsection{\textit{energy deposition calculation with low intensity of proton beam and its observation}}

Figure~\ref{fig:energydep_3tp} (a) shows the distribution of energy deposition by 24~GeV, 3~Tp intensity
of proton beam in 5~T. Calculated energy deposition of each meshed volume is averaged along jet axis
and vertical radius of jet with respect to azimuthal angle using Eqn.~(\ref{e:eq56}):

\begin{equation}
 \label{e:eq56}
E_{p, \ average} (r, z) = \frac{\sum^{2 \pi}_{\theta = 0} \rho \ V_{\theta}^{r} (r, \theta, z) E^{r}_{p, \theta} (r, \theta, z)}{\sum^{2 \pi}_{\theta = 0} \rho \ V_{\theta}^{r} (r, \theta, z)}  \ .
\end{equation}

The energy density distribution is plotted according to the radial position of Hg jet from jet center.
Therefore, the peak of energy deposition density exists respectively depending on the radial position.
It shows that the maximum energy deposition density is obtained at the bottom surface of
jet at $\sim$ 13~cm from the center of magnet, where Viewport 1 is actually positioned, and the peak
energy density moves to the center of Hg jet followed by the larger energy density located at the top
surface of the Hg jet.
The peak energy deposition density is moving corresponding to the beam interacting trajectory in Hg jet.
The most dense energy deposition is distributed at the center of Hg jet between upstream and midstream,
where Hg jet breaks up. The collected photos in Fig.~\ref{fig:energydep_3tp} (b) clearly supports
these simulation results, where the frame rate is 2~ms and measured disruption length at Viewport 3
is 11~cm.

\subsubsection{\textit{energy deposition calculation with high intensity of proton beam and its observation}}

Figure~\ref{fig:energydep_10tp} (a) shows the distribution of energy deposition by 24~GeV, 10~Tp intensity
proton beam in 5~T. Averaged energy deposition is also calculated using Eqn.~(\ref{e:eq56}).
The distribution profile of energy deposition throughout Hg jet is similar with low intensity of beam.
The collected photos in Fig.~\ref{fig:energydep_10tp} (b) clearly supports
these simulation results again, where the frame rate is 2~ms and measured disruption length at
Viewport 3 is 17~cm. However, the jet breakup voids the midstream where the beam is passing
through, which is different comparing with the observation of the case of low intensity beam.
These voids are not observed at 3~Tp intensity of beam, possibly indicates threshold of the
existence of cavitation or breakup induced by energy deposition.

\subsection{Hg jet disruption and magnetic suppression of the disruption}

The disruption length is determined by counting the number of frames at Viewport 3, where
the complete disruption of the jet is observed. The time delay between Viewport 2 and
Viewport 3 is $\sim$ 10~ms. Thus, the disruption generated at Viewport 2 by the beam could
be observed at Viewport 3 after 10~ms, where the jet is moving with typically a velocity of 15~m/s.
Each image is separated into 10 segments vertically in order to locate the position of disruption.
Thus, the accuracy of the measurement to define the location of starting(ending) disruption
in measurement could be increased.
The disruption length is given by multiplying the frame rate by the counted number of images and
investigated according to the beam energy, beam intensity, and magnetic field. 230 events out of
360 beam shots are evaluated for the disruption length. About 130 events out of 360 beam shots
are evaluated for the detection of particles without Hg jet. Thus, the images for these events are not
collected. Figure~\ref{fig:estimation_error} shows the standard deviation of the evaluated
disruption lengths with respect to the disruption length.
The solid line represents the curve fitted approximation of the reduced data distribution, where
the line asymptote logarithmic. This curve fitted line is used for estimation of the standard deviation
of the measured disruption length for the case that only one shot is available for disruption length
measurement. Correspondingly, the error bar is determined by dividing the the estimated standard deviation
by the root square of the number of samples N for each data point.

\subsubsection{\textit{characteristics of beam structure in disruption length, harmonic 8 and 16}}

The proton beam pulse structure of harmonic 8 and harmonic 16 in 14~GeV,
6~Tp is shown in Fig.~\ref{fig:beamstructure}. A pulse carries same number of protons with doubled
bunch structures. Figure~\ref{fig:harmonics} shows the dependence of the disruption length of the Hg jet
on the proton beam pulse structure with a 14~GeV beam in 5~T.
The solid line in Fig.~\ref{fig:beamstructure} (a) and (b) show the global fit and independent fit of
disruption length with both harmonic 8 and harmonic 16 bunches as a function of total energy deposition
respectively. A liner fit function is used as follow:

\begin{equation}
 \label{e:eq516}
L^{disruption} = a_{1} (E_{p} - b_{1}) ,
\end{equation}

where $E_{p}$ and $L^{disruption}$ denote the total energy deposition and disruption length respectively.
The $\chi^{2}$ probability of global fit in Fig.~\ref{fig:beamstructure} (a) is 0.056.
The $\chi^{2}$ probability of each independent fit in Fig.~\ref{fig:beamstructure} (b)
by using the sum of $\chi^{2}$ and degrees of freedom of each independent fit yields 0.051.
From this point of view, there is no statistical difference between the two ways of fitting, so
that one could conclude that the disruption length does not depend on harmonic number.
The disruption of Hg jet is affected by the number of protons, resulted from energy deposition by
the interacting number of protons. The short time difference in each bunch structure is negligible.
The disruption on the Hg jet surface disappears when the beam intensity is less than $\sim$ 4~Tp according
to Fig.~\ref{fig:disrupt_intensity_14GeV}. The threshold of beam intensity is $\sim$ 4~Tp for 14~GeV in 5~T.

\subsubsection{\textit{disruption length with 14~GeV proton beam}}

Figure~\ref{fig:disrupt_intensity_14GeV} shows the disruption length with beam intensities up to
30~Tp for a 14~GeV beam. The peak and total energy deposition to Hg with 14~GeV beam energy at 30~Tp in
15~T is $\sim$ 52~J/g and 3700~J by approximating it from Fig.~\ref{fig:energydep_nproton}, where the
disruption length corresponds to $\sim$ 23~cm $\pm$ 5~cm for 10~T to $\sim$ 18~cm $\pm$ 5~cm for 15~T respectively.
At high intensities of beam, the disruption length appears to be approaching an asymptotic level.
The magnetic field suppresses weak disruption such as onset of generation of the filaments
on the jet surface. The threshold of the disruption for beam intensity is around 4~Tp in 5~T and
the magnetic field can increase it, though the effect is not clear in Fig.~\ref{fig:disrupt_intensity_14GeV} due to
the difficulty in quantifying and judging to measure the small amount of the disruption length.

\subsubsection{\textit{disruption length with 24~GeV proton beam}}

Figure~\ref{fig:disrupt_24GeV_compare} shows the disruption length with the beam intensities up to
30~Tp for a 24~GeV proton beam. The estimation of disruption length is performed by estimating the extent
of energy in Hg jet along jet axis larger than the energy deposition threshold experimentally determined by
threshold intensity of beam as follow :

\begin{equation}
 \label{e:eq57}
L^{disruption}_{E_{p}} (z) = z_{2} - z_{1},  \ E_{p} (z) \geq E^{peak}_{p, \ threshold} (z) , \ \ \  L^{disruption}_{E^{peak}_{p, \ threshold}} (z) = 0 ,
\end{equation}

where $L^{disruption}$ and $E^{peak}_{p, \ threshold} $ represent the length of disruption and peak energy
of threshold intensity of beam experimentally determined for jet disruption.
For example, Fig.~\ref{fig:energydep_10tp} (a) shows the profile of energy deposition along jet axis.
Therefore, energy in Hg jet is known.
By using Eqn.~(\ref{e:eq514}), one can estimate peak energy deposition at
3.7~Tp , which is the experimentally determined threshold intensity of beam.
Now, in Fig.~\ref{fig:energydep_3tp} (a), find the extent of length along jet axis where the energy in Hg jet
is larger than the peak energy at threshold intensity of beam. The length along jet axis determined here is judged
as disruption length of jet for estimation purpose and it is plotted in Fig.~\ref{fig:disrupt_24GeV_compare}.

According to Fig.~\ref{fig:energydep_nproton}, the peak and total energy
deposition to Hg with 24~GeV beam energy at 30~Tp in 10~T is $\sim$ 125~J/g and 8200~J, where the
disruption length corresponds to $\sim$ 22~cm $\pm$ 5~cm for 10~T to $\sim$ 17~cm $\pm$ 5~cm for
15~T respectively. The results again show that the magnetic field suppresses the disruption length.
The disruption length appears to be approaching an asymptotic level.
If there is no magnetic field, the disruptions are always generated by proton beam regardless of the
beam intensities, though weak disruptions on the Hg jet surface are observed with low beam
intensities. The threshold of the disruption for beam intensity is $\sim$ 1~Tp at 5~T but
the higher magnetic field increases it. The estimation of disruption length in 10~T based on the
calculation of energy deposition using the beam spot size from optics is somewhat qualitatively agreed with
the experimental measurement, but the estimation in 0~T based on the beam spot size from optics
underestimates the experimental results. Possibly, the difference in MARS model may cause the
difference of energy deposition calculation and the beam spot size is more likely to be larger at 0~T.
Therefore, possibly the estimation by energy deposition from larger beam spot size is more likely to
be fit to the experimental measurement for the case of 0~T.
For theses estimations, the independent threshold of beam intensity is chosen individually
from the experimental results depending on the conditions of individual cases for estimation.
Therefore, the energy for threshold is differently used for each case of estimation using the
beam size from optics. For the case of estimation of 0~T, 5~T, and 10~T, 0.8~Tp, 1.5~Tp, and 3.7~Tp of threshold
beam intensities are chosen respectively.

\subsubsection{\textit{comparison of measurement at Viewport 3 and Viewport 4}}

In order to check measurement of the disruption length at
Viewport 3, measurement of disruption length at Viewport 4 is also performed.
Fig.~\ref{fig:vp3vp4difference} (a) shows the disruption length at Viewport 3 for 23 events with a
harmonic 16 beam structure, 16~Tp, 14~GeV beam energy in 5~T.
Figure~\ref{fig:vp3vp4difference} (b) shows the disruption length at Viewport 4 for the same events.
Figure~\ref{fig:vp3vp4difference} (c) shows the difference of disruption length between Viewport 3 and
Viewport 4 from Fig.~\ref{fig:vp3vp4difference} (a) and (b). The solid line represents the average and
distribution of the disruption length difference based on gaussian distribution approximation.
The difference of measured disruption length at Viewport 3 and Viewport 4 is 1.3 $\pm$ 3.5~cm.
It seems that the reason for the difference of the disruption length measurement between Viewport 3 and Viewport 4
is mainly caused by the fluctuation of the proton beam and the Hg jet in a magnetic field.
Thus, slightly larger measurement of disruption length is observed at Viewport 4.
The disrupted surface shape seems to be constrained in magnetic fields.
Therefore, the same disrupted shape on jet surface at Viewport 3 is observed at Viewport 4
(30~cm apart from Viewport 3) without variation of the disruption length.

\subsubsection{\textit{measurement of disruption length in pump-probe condition as a check}}

Figure~\ref{fig:disrupt_pumpprobe} shows the measured disruption length of multiple events with pump-probe
condition as a check. The conditions of each group in pump-probe events are given in
Table~\ref{tab:measure_pumpprobe}. There are 4 groups at 14~GeV and each group has different number
of bunches and time delay between pump and probe. Figure~\ref{fig:disrupt_pumpprobe} (a) shows the histogram of disruption length and Fig.~\ref{fig:disrupt_pumpprobe} (b) shows statistics summary such as average, minimum,
maximum, and median value. In group 2, qualitatively meaningful distribution of measurements is given,
which is 19.8 $\pm$ 6.1~cm. In sub-category of group 2, various time delay between 6 bunches and 2
bunches does not show significant difference in disruption length. This check is agreed with the result
provided in both Fig.~\ref{fig:disrupt_intensity_14GeV} and Fig.~\ref{fig:disrupt_totalenergy}.

\section{Hg Jet Disruption by Energy Deposition}

Figure~\ref{fig:disrupt_totalenergy} shows the disruption of Hg jet in magnetic fields as a function of total energy deposition and fit of model using Eqn.~(\ref{e:eq512}) up to 25~T.
Figure~\ref{fig:disrupt_fluence} shows the disruption of Hg jet in magnetic fields as a function of fluence and fit of model using Eqn.~(\ref{e:eq512}) up to 25~T, where the fluence is defined as the normalized beam intensity by
beam spot area $Tp/(\sigma_{x} \sigma{y})$.
Figure~\ref{fig:disrupt_peakenergy} shows the disruption of Hg jet in magnetic fields as a function of peak energy
deposition and fit of model using Eqn.~(\ref{e:eq512}) up to 25~T.
$\chi^{2}$ values indicate comparison of goodness of fit for Fig.~\ref{fig:disrupt_totalenergy}, Fig.~\ref{fig:disrupt_fluence}, and Fig.~\ref{fig:disrupt_peakenergy}, where the fit of model as a function of
total energy deposition yields the lowest $\chi^{2}$ value.
The extent of disruption of jet is dominated by the distribution of
energy deposition over the entire jet interacting with proton beam.
Therefore, the total energy deposition is more likely to play a role in determining of the extent
of disruption of Hg jet. The total energy deposition in magnetic fields is investigated.
The total energy deposition depending on interacting number of protons at both 14~GeV and
24~GeV beam energy is calculated from Fig.~\ref{fig:energydep_nproton} (b). Thus, Fig.~\ref{fig:disrupt_intensity_14GeV} and Fig.~\ref{fig:disrupt_24GeV_compare} are combined as a function
of total energy deposition, which shows the results of experiment in disruption length at a glance.
As a final important result for experiment, Fig.~\ref{fig:disrupt_totalenergy} shows the disruption of
Hg jet in magnetic fields as a function of total energy deposition and its extrapolation up to 25~T.
The global fit with multi-variables for disruption length using the measured disruption length is:

\begin{equation}
 \label{e:eq517}
L^{disruption} = a_{1} (E_{p} - (b_{1} + b_{2} B^{b_{3}}))^{\frac{1}{c_{1} + c_{2} B + c_{3} B^{2}}} ,
\end{equation}

where $E_{p}$ and $B$ are energy deposition and magnetic field respectively. The parameterized
values of coefficients and errors of the fit functions are given in Table~\ref{tab:fitdata_ch5}.
The threshold of disruption increases in 0.8 power of magnetic field, and it is 338~J of total energy
deposition with no magnetic field. Also, the threshold of disruption is $\sim$ 10~J of peak energy
deposition with no magnetic field, and it increases in 1.2 power of magnetic field.
The disruption length approximately increases in square root power of total energy deposition with no
magnetic field, but it is suppressed in $\sim  \ 1/(2 + 0.04 B)$ power of total energy deposition
with magnetic field. Power consumption is dominated by the repetition rate. Thus, the capability to
replace the disrupted jet determines the beam power.
The optimal interaction length for the 24~GeV beam energy is in the region of 30~cm which corresponds to approximately 2 interaction length for Hg \cite{Kirk20081}.
In Fig.~\ref{fig:disrupt_totalenergy}, the disruption length in 15~T is less than 20~cm and
the total energy deposition is $\sim$ 8000~J.
According to Fig.~\ref{fig:energydep_nproton} (b), approximately 6 $\sim$ 8~$\%$ of beam energy is
deposited into Hg target. Therefore, 100 $\sim$ 133~kJ of beam energy can be recycled with a 70~Hz
repetition rate for 20~m/s jet. The disruption length at 30 $\times$ 10$^{12}$ protons with 24~GeV beam energy
in a magnetic field of 15~T is less than 20~cm at 24~GeV beam energy in Fig.~\ref{fig:disrupt_24GeV_compare},
thus preserving the 70~Hz beam repetition rate option.
This result validates that a target system capable of supporting proton beams with powers of up to 8~MW,
which is a key result for this experiment \cite{Kirk20081}.

\setlength{\baselineskip}{12pt}

\newpage
\clearpage

\begin{landscape}
\begin{table}
\centering
\tablecaption{\label{tab:beam_size} Estimated 1 $\sigma$ beam spot size at the target \cite{Efthymiopoulos2008}. The beam spot size is plotted in Fig.~\ref{fig:beamsize_ilias}.}
\begin{supertabular}{|l|l|l|l|l|l|l|l|l|}
\hline
\hline
\textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7}  & \textbf{8} & \textbf{9} \\
(GeV/c) & (Tp) & - & (mm.mrad) & (mm) & (mm) &  (mm.mrad) &  (mm) &  (mm)  \\
\hline
 14    & 1   & 14.925  & 0.0456  & 1.508193 & 1.5951 & 0.1047 & 0.1708 & 0.7178   \\
 14    & 5   & 14.925  & 0.1208  & 1.508193 & 1.7290 & 0.1453 & 0.1708 & 0.8391   \\
 14    & 10  & 14.925  & 0.2149  & 1.508193 & 1.8830 & 0.1961 & 0.1708 & 0.9695   \\
 14    & 15  & 14.925  & 0.3090  & 1.508193 & 2.0253 & 0.2469 & 0.1708 & 1.0844   \\
 14    & 20  & 14.925  & 0.4030  & 1.508193 & 2.1583 & 0.2977 & 0.1708 & 1.1883   \\
 14    & 25  & 14.925  & 0.4971  & 1.508193 & 2.2836 & 0.3485 & 0.1708 & 1.2837   \\
 14    & 30  & 14.925  & 0.5911  & 1.508193 & 2.4023 & 0.3993 & 0.1708 & 1.3726   \\
 24    & 1   & 25.586  & 0.0266  & 0.999405 & 1.0753 & 0.0610 & 0.1132 & 0.5444   \\
 24    & 5   & 25.586  & 0.0705  & 0.999405 & 1.1899 & 0.0848 & 0.1132 & 0.6376   \\
 24    & 10  & 25.586  & 0.1254  & 0.999405 & 1.3192 & 0.1144 & 0.1132 & 0.7377   \\
 24    & 15  & 25.586  & 0.1802  & 0.999405 & 1.4369 & 0.1440 & 0.1132 & 0.8257   \\
 24    & 20  & 25.586  & 0.2351  & 0.999405 & 1.5457 & 0.1737 & 0.1132 & 0.9052   \\
 24    & 25  & 25.586  & 0.2899  & 0.999405 & 1.6474 & 0.2033 & 0.1132 & 0.9783   \\
 24    & 30  & 25.586  & 0.3448  & 0.999405 & 1.7431 & 0.2329 & 0.1132 & 1.0463   \\
%     &    &   &   &  &  &  &  &    \\
\hline
\hline
\end{supertabular} \\
\ \\
\begin{flushleft}
%\parbox[b]{3.5in}{
\shortstack[l]{\vrule width 3.5in height 0.4pt\\
   \ \ \ \ \ $\textbf{1}$ : Beam momentum  \\
   \ \\
   \ \ \ \ \ $\textbf{2}$ : Beam intensity  \\
   \ \\
   \ \ \ \ \ $\textbf{3}$ : $\beta \gamma$  \\
   \ \\
   \ \ \ \ \ $\textbf{4}$ : Emittance, horizontal plane ($1 \sigma$)  \\
   \ \\
   \ \ \ \ \ $\textbf{5}$ : $D_{p} \frac{\delta p}{p} \ (1 \sigma)$, horizontal plane \\
   \ \\
   \ \ \ \ \ $\textbf{6}$ : Beam size, horizontal plane ($1 \sigma$) \\
   \ \\
   \ \ \ \ \ $\textbf{7}$ : Emittance, vertical plane ($1 \sigma$) \\
   \ \\
   \ \ \ \ \ $\textbf{8}$ : $D_{p} \frac{\delta p}{p} \ (1 \sigma)$, vertical plane \\
      \ \\
   \ \ \ \ \ $\textbf{9}$ : Beam size, vertical plane ($1 \sigma$)   }
\end{flushleft}

\end{table}
\end{landscape}

\newpage
\clearpage

\begin{landscape}
\begin{table}
\centering
\tablecaption{\label{tab:jetsize_mars} Measurement of vertical distance of jet centroid from magnetic axis and jet size for modeling in MARS code for the cases of sectionally elliptic and circular jet shape.}
\begin{supertabular}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\hline
\textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7}  & \textbf{8} & \textbf{9} & \textbf{10} & \textbf{11} & \textbf{12}\\
(T) & (mm) & (mm) & (mm) & (mm) & (mm) &  (mm) &  ( $g/cm^{3}$ ) &  (degree) &  - &  (cm) &  - \\
\hline
 \multicolumn{12}{|c|} {Elliptic jet shape}  \\
\hline
0   &  7.11 & 4.46 & 4.28 & 5.01 & 8.65  & 2.9   &  13.55  & 62,80,90,100,118,179,242,260,270,280,298,360 & 12  & 2  & 50  \\
5   &  7.1  & 4.52 & 3.7  & 4.38 & 8.4   & 3.0   &  13.55   & 62,80,90,100,118,179,242,260,270,280,298,360 & 12  & 2  & 50  \\
10  &  6.57 & 4.08 & 3.66 & 3.71 & 7.95  & 3.15  &  13.55  & 62,80,90,100,118,179,242,260,270,280,298,360 & 12  & 2  & 50   \\
15  &  5.45 & 3.6  & 3.24 & 3.11 & 9.05  & 2.76  &  13.55   & 62,80,90,100,118,179,242,260,270,280,298,360 & 12  & 2  & 50   \\
\hline
 \multicolumn{12}{|c|} {Circular jet shape}   \\
\hline
0   &   7.11 & 4.46 & 4.28 & 5.01 & 8.65  & 8.65  & 4.50 & 30,60,90,120,150,180,210,240,270,300,330,360  & 12   &  2  & 50  \\
5   &   7.1  & 4.52 & 3.7  & 4.38 & 8.4   & 8.4   & 4.77  &  30,60,90,120,150,180,210,240,270,300,330,360  & 12   &  2  & 50   \\
10  &   6.57 & 4.08 & 3.66 & 3.71 & 7.95  & 7.95  & 5.32  &   30,60,90,120,150,180,210,240,270,300,330,360  & 12   &  2  & 50  \\
15  &   5.45 & 3.6  & 3.24 & 3.11 & 9.05  & 9.05  & 4.11 &   30,60,90,120,150,180,210,240,270,300,330,360  & 12   &  2  & 50    \\
\hline
\hline
\end{supertabular} \\
\ \\
\begin{flushleft}
%\parbox[b]{3.5in}{
\shortstack[l]{\vrule width 3.5in height 0.4pt\\
   \ \ \ \ \ $\textbf{1}$ : Magnetic field  \\
   \ \\
   \ \ \ \ \ $\textbf{2}$ : Vertical distance at Viewport1  \\
   \ \\
   \ \ \ \ \ $\textbf{3}$ : Vertical distance at Viewport2  \\
   \ \\
   \ \ \ \ \ $\textbf{4}$ : Vertical distance at Viewport3  \\
   \ \\
   \ \ \ \ \ $\textbf{5}$ : Vertical distance at Viewport4  \\
   \ \\
   \ \ \ \ \ $\textbf{6}$ : Vertical radius of jet \\
   \ \\
   \ \ \ \ \ $\textbf{7}$ : Horizontal radius of jet \\
   \ \\
   \ \ \ \ \ $\textbf{8}$ : Hg density  \\
      \ \\
   \ \ \ \ \ $\textbf{9}$ : Azimuthal meshed angle  \\
      \ \\
   \ \ \ \ \ $\textbf{10}$ : Number of azimuthal mesh  \\
      \ \\
   \ \ \ \ \ $\textbf{11}$ : Axial meshed size  \\
      \ \\
   \ \ \ \ \ $\textbf{12}$ : Number of axial mesh  }
\end{flushleft}

\end{table}
\end{landscape}


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\begin{landscape}
\begin{table}[!hbt]
\footnotesize
\centering
\tablecaption{\label{tab:fitdata_ch5} Parameterized coefficients, its error, and statistics summary of fit function in figures.}
\begin{supertabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\hline
 $\textbf{Figure}$ & $\textbf{1}$ & $\textbf{2}$ & $\textbf{3}$ & $\textbf{4}$ & $\textbf{5}$ & $\textbf{6}$ & $\textbf{7}$ & $\textbf{8}$ & $\textbf{9}$  & $\textbf{10}$ \\
\hline
~\ref{fig:energydep_beamarea}(a)  & 0.74078 & 0.03855 & -0.06864 & 0.01598 & 0.50641 & 0.05307 & - & - & 1.48078 & 0.0158    \\
~\ref{fig:energydep_beamarea}(b)  &  0.02228 & 8.60E-04 & -1.09835  & 0.36388 & 0.0613 & 0.00759 & -5.49E-04 & 1.62E-04 & 1.36185 & 0.01097  \\
~\ref{fig:energydep_beamarea}(c)  & 0 & 0 & 4.10076 &  0.37658 & -0.56357 & 0.01667 & - & - & 0.821 & 0.03034 \\
~\ref{fig:energydep_beamarea}(d)  & 0.02814  & 0.00155 & -1.11612 & 0.38081 & 0.05682 &  0.00729 & -4.71E-04 & 1.41E-04 & 1.31199 & 0.01724 \\
~\ref{fig:energydep_nproton}(a)  & 0.06023 & 0.0073 & 0.80386 & 0.0105 & - & - & - & - & 1.5568 & 0.04025   \\
~\ref{fig:energydep_nproton}(b)  & 3.52931 & 0.3187 & 0.88872 & 0.01003 & 0.02553 & 0.01138 & 0.3758 & 0.16582 & 1.4208 & 0.02953    \\
~\ref{fig:harmonics}(a) & 1.43E-04 & 1.86E-05  & 647.56071 & 89.38814 & - & - & - & - & - & -   \\
~\ref{fig:harmonics}(b)(H8) & 1.70E-04  & 3.77E-05 & 638.26526 & 126.57444 & - & - & - & - & - & -  \\
~\ref{fig:harmonics}(b)(H16) & 1.39E-04  & 2.18E-05 & 680.28969 & 113.41709 & - & - & - & - & - & -   \\
~\ref{fig:disrupt_totalenergy} & 0.00649 & 0.00348 & 338.24297 & 15.76037 & 115.38009 & 47.56862  & 0.82899 & 0.22938 & 1.92463  & 0.29005  \\
~\ref{fig:disrupt_fluence}  & 0.09242  & 0.01457 & 1.56733 & 0.12275  & 0.66907 & 0.18602 & 0.71351  & 0.14258 & 1.59393 & 0.26343 \\
~\ref{fig:disrupt_peakenergy}  & 0.04119 & 0.01018 & 9.93998  & 0.48595  & 0.98744 & 0.0975 & 1.21081 & 0.07709 & 1.74961 & 0.25844 \\
\hline
\hline
 $\textbf{Figure}$ & $\textbf{11}$ & $\textbf{12}$ & $\textbf{13}$ & $\textbf{14}$ & $\textbf{15}$ & $\textbf{16}$ & $\textbf{17}$ & $\textbf{18}$ & $\textbf{19}$ &  \\
\hline
~\ref{fig:energydep_beamarea}(a)  & - & - & - & - & 32 & 28 & 14.67464 & 0.99691  & 0 &  \\
~\ref{fig:energydep_beamarea}(b)  &  0.91711  & 0.10273 & - & - & 32  & 26 & 256.24604 & 0.99909 &  0 &  \\
~\ref{fig:energydep_beamarea}(c)  & - & - & - & - & 32 & 29 & 29.85134 & 0.99371 & 0  &   \\
~\ref{fig:energydep_beamarea}(d)  & 1.96014  & 0.10329 & - & - & 32 & 26 & 271.9889 & 0.99904 & 0 &   \\
~\ref{fig:energydep_nproton}(a)  & - & - & - & - & 32 &  29 & 95.44974 & 0.99168  & 0 &   \\
~\ref{fig:energydep_nproton}(b)  &  - & - & - & - & 32 & 27 & 3972.28821 & 0.99628  & 0 &   \\
~\ref{fig:harmonics}(a)  &  - & - & - & - & 11 & 9 & 1.84 & 0.85406  &  0.056 &    \\
~\ref{fig:harmonics}(b)(H8)  &  - & - & - & - & 5 & 3 & 1.97369  & 0.82927 & 0.1155 &  \\
~\ref{fig:harmonics}(b)(H16)  &  - & - & - & - & 6 & 4 & 1.77779 & 0.88853  & 0.1301 &  \\
~\ref{fig:disrupt_totalenergy}  & 0.03939 & 0.01079 &  0 & 0 & 36 & 30 & 1.82037  &  0.88724  &  0.0039 &  \\
~\ref{fig:disrupt_fluence} & 0.06785 & 0.03317 & 0 & 0 & 36 & 30 & 2.18746 & 0.86451  & 0.0001 &  \\
~\ref{fig:disrupt_peakenergy} & 0.05655  &  0.02131 &  0 & 0 & 36 & 30 & 2.86591 & 0.82248 & 2.6019e-7 &  \\
\hline
\hline
\end{supertabular} \\
%  &  &  &  &  &  &  &  &  &  & \\
%\ \\
%\normalsize
\begin{flushleft}

\shortstack[l]{\vrule width 8.5in height 0.4pt\\
        \ \ $\textbf{1}$ : a1 value, $\textbf{2}$ : a1 standard deviation, $\textbf{3}$ : b1 value, $\textbf{4}$ : b1 standard deviation, $\textbf{5}$ : b2 value,  \\
       \ \\
        \ \  $\textbf{6}$ : b2 standard deviation, $\textbf{7}$ : b3 value, $\textbf{8}$ : b3 standard deviation , $\textbf{9}$ : c1 value, $\textbf{10}$ : c1 standard deviation, \\
       \ \\
        \ \ $\textbf{11}$ : c2 value, $\textbf{12}$ : c2 standard deviation, $\textbf{13}$ : c3 value, $\textbf{14}$ : c3 standard deviation, $\textbf{15}$ : Number of points, \\
         \ \\
        \ \  $\textbf{16}$ : Degrees of freedom, $\textbf{17}$ : Reduced $\chi^{2}$, $\textbf{18}$ : Adjusted $\textbf{R}^{2}$, $\textbf{19}$ : $\chi^{2}$ probability.  }
       \ \\
\end{flushleft}

\end{table}
\end{landscape}

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\begin{figure}
\centering
\includegraphics[width=8in]{psmachine2}
\caption{Infrastructures for experiment at CERN. a.) CERN accelerator complex and TT2 tunnel for experiment. b.) 16 harmonics of beam extraction in proton synchrotron.} \label{fig:psmachine}
\end{figure}
\end{landscape}
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\begin{figure}
\centering
\includegraphics[width=8in]{tt2layout2}
\caption{Installation of integrated experimental components in tunnel TT2/TT2A for Hg
target experiment. Extracted proton beam comes from left to right in tunnel TT2A.} \label{fig:tt2layout}
\end{figure}
\end{landscape}
\clearpage

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\begin{figure}
\centering
\includegraphics[width=8in]{beamstructure}
\caption{Proton beam pulse structure of harmonic 8 and harmonic 16 in 6~Tp with 14~GeV. Tp=10$^{12}$ protons.} \label{fig:beamstructure}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{beamsize_ilias}
\caption{1 $\sigma$ proton beam size at the center of magnet by optics \cite{Efthymiopoulos2008}. The beam spot size is given in Table~\ref{tab:beam_size}.} \label{fig:beamsize_ilias}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{beamsize_goran}
\caption{1~$\sigma$ proton beam size by camera screen monitor \cite{Skoro}. a.) 14~GeV beam. b.) 24~GeV beam.} \label{fig:beamsize_goran}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=5.5in]{beamsize_goran_timehistory}
\caption{Beam size measured by phosphor screen monitor as a function of recorded time interval between each beam shot. a.) Histogram of beam size in horizontal plane. b.) Histogram of beam size in vertical plane. c.) Beam size distribution. } \label{fig:beamsize_goran_timehistory}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=5.5in]{modeling_mars}
\caption{Modeling in MARS code for energy deposition calculation \cite{Striganov20092}. a.) Sectional view of elliptic jet. b.) Sectional view of circular jet. c.) Side view of Hg jet interacting with proton beam. Arrow indicates the direction and strength of magnetic field.}
\label{fig:modeling_mars}
\ \\
\begin{flushleft}
\shortstack[l]{\vrule width 3.5in height 0.4pt\\
        \ \  \textcolor{red}{\scriptsize\textcolor{red}{\url{http://www.hep.princeton.edu/~mcdonald/mumu/target/Striganov/edep-grav.pdf}}} }
\end{flushleft}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=6.5in]{energydep_magnetic2}
\caption{Influence of magnetic field on the energy deposition distribution to Hg jet. Beam intensity is 2~Tp and energy deposition in $J/g$ is averaged using Eqn.~(\ref{e:eq58}). }
\label{fig:energydep_magnetic}
\ \\
\begin{flushleft}
\shortstack[l]{\vrule width 3.5in height 0.4pt\\
        \ \  \textcolor{red}{\scriptsize\textcolor{red}{\url{http://www-ap.fnal.gov/~strigano/merit/edep_in_target/gravity/24gev-beam-updated/ellipse/new/}}} \\
        \ \
        \textcolor{red}{\scriptsize\textcolor{red}{\url{http://www-ap.fnal.gov/~strigano/merit/edep_in_target/gravity/24gev-beam-updated/ellipse/}}}     }
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\begin{figure}
\centering
\includegraphics[width=8in]{energydep_angle_24gev2}
\caption{Azimuthal energy deposition distribution along jet axis interacting with 24~GeV proton beam.
Beam intensity is 2~Tp and magnetic field is 10~T. a.) Along jet axis. Energy deposition in $J/g$ is averaged using Eqn.~(\ref{e:eq510}). b.) Along azimuthal angle in jet cross section. Energy deposition in $J/g$ is averaged using Eqn.~(\ref{e:eq511}). } \label{fig:energydep_angle_24gev}
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\begin{figure}
\centering
\includegraphics[width=8in]{energydep_angle_14gev2}
\caption{Azimuthal energy deposition distribution along jet axis interacting with 14~GeV proton beam.
Beam intensity is 2~Tp and magnetic field is 10~T. a.) Along jet axis. Energy deposition in $J/g$ is averaged using Eqn.~(\ref{e:eq510}). b.) Along azimuthal angle in jet cross section. Energy deposition in $J/g$ is averaged using Eqn.~(\ref{e:eq511}). } \label{fig:energydep_angle_14gev}
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\begin{figure}
\centering
\includegraphics[width=6.5in]{energydep_intensity2}
\caption{Energy deposition distribution per proton according to the variation of beam spot size along jet axis. Magnetic field is 5~T and energy deposition in $J/g$ is averaged using Eqn.~(\ref{e:eq58}). $\sigma_{x} \sigma_{y}$ is calculated from Fig.~\ref{fig:beamsize_ilias}. } \label{fig:energydep_intensity}
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\begin{figure}
\centering
\includegraphics[width=7in]{energydep_beamarea2}
\caption{Simulation of peak energy deposition per proton and total energy deposition per proton according to the beam spot size and beam intensity. Fits of model fit to Striganov's calculation results.
Eqn.~(\ref{e:eq512}) and Eqn.~(\ref{e:eq513}) are used for fit of model of peak energy deposition and total
energy deposition, respectively. a.) Peak energy in $J/g$ per proton by beam intensity. b.) Total energy in $J/g$ per proton by beam intensity. c.) Peak energy in $J/g$ per proton by beam spot size. d.) Total energy in $J/g$ per proton by beam spot size.} \label{fig:energydep_beamarea}
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\begin{figure}
\centering
\includegraphics[width=8in]{energydep_nproton}
\caption{Simulation of peak energy deposition and total energy deposition in total number of protons. Fits of model fit
to Striganov's calculation results. a.) Peak energy deposition in $J/g$ and fit of model using Eqn.~(\ref{e:eq514}). b.) Total energy deposition in $J$ and fit of model using Eqn.~(\ref{e:eq515}).} \label{fig:energydep_nproton}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{photo_jet_interact11}
\caption{Photographs of the Hg jet interaction with 16~Tp, 14~GeV proton beam in 5~T. Captured at Viewport 3 at 500~$\mu$s frame rate (continued).} \label{fig:jet_response}
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\begin{figure}
\centering
\includegraphics[width=8in]{photo_jet_interact21}
\caption{Photographs of the Hg jet interaction with 16~Tp, 14~GeV proton beam in 5~T. Captured at Viewport 3 at 500~$\mu$s frame rate (continued).} \label{fig:jet_response}
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\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{photo_jet_interact31}
\caption{Photographs of the Hg jet interaction with 16~Tp, 14~GeV proton beam in 5~T. Captured at Viewport 3 at 500~$\mu$s frame rate.} \label{fig:jet_response}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{energydep_3tp2}
\caption{Qualitative comparison of the jet response by interaction of low intensity (3~Tp) of 24~GeV beam in 5~T. a.) Calculated averaged energy deposition using Eqn.~(\ref{e:eq56}) according to the vertical distance from jet center. b.) Observation of jet response by captured image.} \label{fig:energydep_3tp}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{energydep_10tp2}
\caption{Qualitative comparison of the jet response by interaction of high intensity (10~Tp) of 24~GeV beam in 10~T. a.) Calculated averaged energy deposition using Eqn.~(\ref{e:eq56}) according to the vertical distance from jet center. b.) Observation of jet response by captured image.} \label{fig:energydep_10tp}
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\begin{figure}
\centering
\includegraphics[width=8in]{estimation_error5}
\caption{Standard deviation of disruption length as a function of disruption length and fit of model. The fitted curve is $\sigma_{disruption}$ = 1.9352 L$_{disruption}^{0.3} $. } \label{fig:estimation_error}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{harmonics4}
\caption{Disruption length of Hg jet depending on the beam pulse structure as a function of 14~GeV beam intensity in 5~T. a) Global fit of harmonic 8 and 16 using Eqn.~(\ref{e:eq516}). b.) Independent fit of harmonic 8 and 16 using Eqn.~(\ref{e:eq516}).} \label{fig:harmonics}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{disrupt_intensity_14GeV3}
\caption{Disruption length of Hg jet as a function of 14~GeV beam intensity and magnetic field. Harmonic 16 with 16 bunches is used.} \label{fig:disrupt_intensity_14GeV}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{disrupt_24GeV_compare3}
\caption{Disruption length of Hg jet and its estimation as a function of 24~GeV beam intensity and magnetic field. The estimation of disruption length by energy deposition calculation is compared by using disruption model of Eqn.~(\ref{e:eq57}).} \label{fig:disrupt_24GeV_compare}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=7in]{vp3vp4difference5}
\caption{Comparison of disruption length measurement at Viewport 3 and Viewport 4. a) Disruption length at Viewport 3. b.) Disruption length at Viewport 4. c.) Difference of the disruption length at Viewport 3 and Viewport 4. } \label{fig:vp3vp4difference}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=8in]{disrupt_pumpprobe}
\caption{Multiple disruption length measurements in same condition. Pump-probe conditions with harmonic 8 and 16 bunches are used. The conditions of each group in pump-probe events are given in Table~\ref{tab:measure_pumpprobe}. a.) Histogram of disruption length in each group. b.) Disruption length of each group.} \label{fig:disrupt_pumpprobe}
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\begin{figure}
\centering
\includegraphics[width=7in]{disrupt_totalenergy}
\caption{Disruption of Hg jet in various magnetic fields as a function of total energy deposition and fit of model using Eqn.~(\ref{e:eq517}). } \label{fig:disrupt_totalenergy}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=7in]{disrupt_fluence}
\caption{Disruption of Hg jet in various magnetic fields as a function of fluence and fit of model using Eqn.~(\ref{e:eq517}). } \label{fig:disrupt_fluence}
\end{figure}
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\begin{figure}
\centering
\includegraphics[width=7in]{disrupt_peakenergy}
\caption{Disruption of Hg jet in various magnetic fields as a function of peak energy deposition and fit of model using Eqn.~(\ref{e:eq517}). } \label{fig:disrupt_peakenergy}
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