# Orbit Error Correction on the High Energy Beam Transport Line at the KHIMA
Accelerator System

Chawon Park<EMAIL_ADDRESS>Heejoong Yim  Garam Hahn  Dong Hyun An
<EMAIL_ADDRESS>Division of Heavy Ion Accelerator, Korea Institute of
Radiological & Medical Sciences, Seoul 139-706

For the purpose of treatment of various cancer and medical research, the
synchrotron based medical machine under the Korea Heavy Ion Medical
Accelerator (KHIMA) project have been conducted and is going to treat the
patient at the beginning of 2018. The KHIMA synchrotron is designed to
accelerate and extract the carbon ion (proton) beam with various energy range,
110 up to 430 MeV/u (60 up to 230 MeV). A lattice design and beam optics
studies for the High Energy Beam Transport (HEBT) line at the KHIMA
accelerator system have been carried out with WinAgile and the MAD-X codes.
Because the magnetic field errors and the mis-alignments introduce to the
deviations from the design parameters, these error sources should be treated
explicitly and the sensitivity of the machine’s lattice to different
individual error sources is considered. Various types of errors which are
static and dynamic one have been taken into account and have been
consequentially corrected with a dedicated correction algorithm by using the
MAD-X program. As a result, the tolerances for the diverse error contributions
have been specified for the dedicated lattice components in the whole HEBT
lines.

KIRAMS, KHIMA, HEBT, orbit, error, correction

###### pacs:

29.27.Bd, 29.27.Fh

##  I INTRODUCTION

The Korea Heavy Ion Medical Accelerator (KHIMA) project at the Korean
Institute of Radiological And Medical Sciences (KIRAMS) has carried out the
development of an accelerator based on synchrotron with multi-ion sources for
various cancer treatement. The designed synchrotron accelerates the proton
beam (the carbon ion, ${}^{12}C^{6+}$, beam) from 60 MeV (110 MeV/$u$) to 230
MeV (430 MeV/$u$). Those energy ranges correspond to the penetration depth of
3.0 cm to 31.0 cm in water. A schematic layout of the accelerator center is
shown in Fig. 1.

Figure 1: (Color online) Schematic layout of KHIMA synchrotron system
including each treatment and research rooms.

At the Electron Cyclotron Resounce Ion Source (ECRIS), ions with a charge to
mass ratio q/m = 1/3, either $H^{+}_{3}$ or ${}^{12}C^{4+}$, are generated up
to 8.0 keV/$u$. These ions are accelerated up to 7 MeV/$u$ through Radio
Frequency Quadrupole (RFQ) linac and Interdigital H-mode Drift Tube Linac (IH-
DTL). At the beginning of Medium Energy Beam Transport (MEBT) two
corresponding ions are stripped and fully ionized to either proton or
${}^{12}C^{6+}$, then are transported to the synchrotron being accelerated up
to designed energies. Each ion is injected into the synchrotron through a
multi-turn injection mechanism, accelerated by switching the RF system and
then extracted into High Energy Beam Transport (HEBT) line by slow resonance
extraction scheme Extract ; HJYim .

##  II Beam characteristic on the HEBT

In each HEBT line, the beam is defined by the medical specifications and thus
there is a definite range of vertical Twiss functions,
$2.0<\beta_{y}<27~{}\rm{m},~{}\alpha_{y}=0$. For adjusting the beam size in
the vertical plane, the straightforward method is to be used. The beam size is
determined by the traditional geometrical emittance of the beam originated
from synchrotron and the beta function at the iso-center. The vertical beam
size is calculated as $y=\sqrt{\epsilon_{y}\times\beta_{y}}$ PIMMS . On the
other hand the peculiar distribution in the horizontal plane is appeared so
called ’bar of charge’ in phase space, because the extracted beam is the
segment of the extraction separatrix that could be represented as the diameter
of an unfilled ellipse. Thus by varying the phase advance ($\Delta\mu_{x}$),
namely by changing the orientation of the charge bar and consequently its
projection to the x-axis determines the horizontal beam size PIMMS ; MBenedikt
; SRossi .

##  III Beam optics simulation

The design concept of KHIMA extraction, HEBT, lines is based on the Proton-Ion
Medical Machine Study (PIMMS) PIMMS . Three medical treatment rooms and one
research oriented irradiation room are prepared for the center. The HEBT lines
compose the 6 different transport branches with 4 horizontal- and 2 vertical-
lines as shown in Fig 1. A slowly extracting beam through electrostatic septum
(ES) in the synchrotron ring is to be selectively transported into each
treatment room. An integrated system is designed with telescope modules with
integer $\pi$ phase advances ($\Delta\mu$). The HEBT lines are also based on a
modular design taking into account the strong asymmetry between two transverse
beams. The trapezoidal distribution of the horizontal beam is considered. For
horizontal beam in phase space, the bar of charge is applied to create an
independent control of the horizontal beam size by rotating the bar in an
unfilled ellipse MBenedikt01 . On the other hand, the Gaussian shaped beam
distribution is taken into account the vertical beam.

###  III.1 Electrostatic Septum to Matching Section

On the base of the PIMMS report, the dispersion vector at the ES is determined
by appearance of the charge bar for different momenta of the particle beam. In
Table 1, the data for lowest extraction energy are summarized and the
dispersion ($Dx$) and its derivative (${Dx}^{\prime}$) values are listed. At
higher energies, the corresponding values are not so different and not
significantly gotten out.

Furthermore, the $Dx$ and ${Dx}^{\prime}$ from the synchrotron are not
representative ones as for the betatron amplitude function. The $Dx$ and
${Dx}^{\prime}$ were calculated as listed in Table 1, thus it could be
utilized universally up to matching section. In principle any values can be
selected as an initial values at the ES, but the useful one should be selected
depending on the dedicated machine. Because the dispersion function represents
the displacement of the off-momentum particles, it is natural to be selected
as an initial value corresponding to the distance between the center of
gravity of the on-momentum particles and the center of gravity of the off-
momentum ones divided by the relative momentum difference as described and
calculated in Table 1.

Table 1: Dispersion of the extracted beam at the entry to the ES. | On resonance particles |  | Off resonace particles |
---|---|---|---|---
| $\Delta p/p=0$ |  | $\Delta p/p=0.0010$ |
| Position(m) | Angle(rad) | Position(m) | Angle(rad)
Inner edge of segment | 0.050000 | 0.001952 | 0.050000 | 0.002048
Outer edge of segment | 0.057023 | 0.002164 | 0.054715 | 0.002188
Average radial position/angle | 0.053115 | 0.002058 | 0.052358 | 0.002118
Radial shift(m) | -0.000757
Angluar shift(rad) | 0.00013
Dispersion, $Dx$(m) | 0.7407
Derivative of dispersion, ${Dx}^{\prime}$(m) | 0.1272
Figure 2: (Color online) Betatron amplitude and dispersion function at the
matching section for the horizontal and vertical plane, where half height
rectangular box corresponds to the quadrupole magnet having their polarity
while the full height one for the dipole magnet.

According to the determined values in Table 1, the boundary conditions at the
ES are selected. Fig. 2 shows the distribution of Twiss parameters at the
matching module.

###  III.2 Horizontal Beam Lines

Figure 3: (Color online) Betatron amplitude and the dispersion functions for
the 3rd Horizontal (H3) line, where $\Delta\mu_{x}=~{}6.0~{}\pi$ and
$\beta_{y}=27~{}m$ at the iso-center are set.

The common module downstream matching section in the horizontal common line
(HC) is interleaved for controlling beam size at the iso-center. This module
has an important advantage for commissioning and operation. As described in
previous section, the vertical beam size at the iso-center is controlled by
the vertical $\beta$ function. In order to fulfil the requirements, the module
can be tuned to provide the values among $2<\beta_{y}<27$ m with
$\alpha_{y}=~{}0$. A minimum of five components are needed to control the five
Twiss variables, i.e., the horizontal, vertical $\beta$’s and the $\alpha$’s
at the exit and the horizontal phase advance. As the variable component, each
quadrupole magnet in the common line is used. At least six quadrupoles are
needed to facilitate the matching and to have some redundancy. This
composition of consecutive six quadrupole magnets, which functions as the
phase shifter on the horizontal plane and also as the stepper on the vertical
plane, is named simply as the stepper.

As one of the horizontal lines, Fig. 3 shows the Twiss function distributions
up to 3rd treatment room through the horizontal H3 line.

##  IV Error analysis

Because various distortion making elements in the transport line are exist,
the closed orbit control is a basic constituent for the efficient performance
in the beam line. A large closed orbit distortion minimizes the available
aperture and also affect the dynamics of the beam via non-linear elements.
Various error sources contribute to the closed orbit distortion. Most of
errors are random in nature, while others are systematic and some may be time
or field dependent Badano .

The results have been calculated by using the accelerator codes, which are the
WinAGile Agile  or the MAD-X Schmidt ; Herr . Both codes base the correction
process on a least square method. A series of test runs showed that the
results obtained from two program codes are consistent within their
statistical uncertainty. Thus it is determined to use the MAD-X code
representatively in this analysis. One of the simulations is performed with a
statistics for 1000 randomly generated machines in the vertical transport line
(V2).

###  IV.1 Requirements for Closed Orbit Correction

The basic guidelines for the closed orbit correction for the HEBT in the KHIMA
accelerator are that the global closed orbit correction must be within 7.5 mm
for both horizontal and vertical planes. For error analysis in the study, only
the static errors are described, while the dynamic errors originated from the
stability of magnets are small enough to be ignored compared to static ones.

###  IV.2 Alignment and Field Tolerances

The impact of each error source, so called sensitivity study, was evaluated
using the MAD-X simulation code. The accepted tolerances for various errors
resulting in an orbit distortion are listed in Table 2. The dipole field error
is originated from the packing factor tolerance and the its length error. The
correctors and position monitors are considered to be affected by several
errors. The position monitors also include both alignment and reading errors.
Alignment- and field-errors are applied to all the elements according to
Gaussian distribution by cutting 3 standard deviations, while the monitor
reading errors are given with a uniform distribution as described in Table 2.

Table 2: Accepted tolearances for the magnet elements and position monitors. Error type | Tolerance | Distribution
---|---|---
Qaudrupole alignment (x, y, z) | 0.3 mm | Truncated Gaussian
Qaudrupole tilt | 0.3 mrad | Truncated Gaussian
Dipole alignment (x, y, z) | 0.3 mm | Truncated Gaussian
Dipole tilt | 0.3 mrad | Truncated Gaussian
Integrated dipole field error ($\Delta BL/BL$) | 0.001 | Truncated Gaussian
Integrated quadrupole field error | 0.001 | Truncated Gaussian
Monitor reading error |  $\pm 0.5$ mm | Flat

###  IV.3 Monitor and Corrector System

The monitor system is correlated with the dual corrector which is the steering
activated as a pair in the horizontal and vertical plane. The precision and
reliability of the measurement system are extremely important, because they
determine the quality of the corrector together with the precision of the
corrector and their maximum magnetic strength. The layout through V2 line for
optimizing the orbit correction is shown in Fig. 4.

Figure 4: (Color online) Beam line layout through V2 with the beam position
monitor and corrector.

###  IV.4 Closed Orbit Estimation and Correction

The orbits have been calculated with MAD-X code on the HEBT line through V2.
The mechanism of the transformation is provided by the technique of singular
value decomposition (SVD) SVD  of matrices. For testing what various sources
of errors have a sensitivity, the execursed quantity with respect to the
sequential beam line is evaluated before and after correction as shown in Fig.
5. In the beginning, the total number of monitors used is 15, while 9 for
correctors.

Figure 5: (Color online) Excursion distributions through the V2 line before
and after correction.

One thousand of machines with random errors have been analysed before and
after the correction on the vertical line through V2. At iso-center, the beam
position is accurate dramatically after correction as shown in Fig. 6.

Figure 6: (Color online) Excursion distributions of closed orbit at iso-center
before and after correction.

The absolute maximum excursions are quoted, because these values are of more
direct interest for the aperture than the peak-to-peak values.

Figure 7: (Color online) (Top) Maximum absolute excursions of the closed orbit
before and after correction in both plane. (Bottom) Zoomed distributions of
the excursions after correction within the global closed orbit margin, 7.5 mm,
in horizontal and vertical planes respectively.

The requirement of closed orbit correction was set to be within 7.5 mm for
both horizontal and vertical planes. With a standard alignment technique, a
100.0 % of the machines after correction in horizontal plane and a 99.7 % in
the vertical plane could be within the allowed global closed orbit margin, 7.5
mm, as shown in Fig. 7. After correction, thus almost all the machines could
be well within the stricter tolerances for the line. Fig. 8 shows the maximum
absolute kick angle for both horizontal and vertical planes at highest carbon
beam energy ( E = 430 MeV/u). As a result, the maximum required kick angle is
foreseen to have the value less than 3.5 mrad in both plane. An average
steering power for the orbit correction of all static errors can be estimated
as one root mean square (rms) and is less than 0.5 mrad. Accordingly the
specification of correcting magnet is set to 5 mrad including margin as shown
in Table 3.

Figure 8: (Color online) Maximum absolute steering angle for the horizontal
and vertical planes at highest carbon beam energy, E = 430 MeV/u. Figure 9:
(Color online) Ratio of the machines within tolerance with respect to the
number of beam position monitor in the V2 line.

In order to optimize the alignment of corrector and beam position monitor,
various combinations of corrector and monitor are disposed with respect to the
number of monitor as shown in Fig. 9. The optimized number of monitor in the
beam line is determined to use 9 monitors.

##  V Conclusion

The extracted transfer lines are based on the modular design taking into
account the strong asymmetry between two transverse emittances and depend on
the trapezoidal distribution of the beam in horizontal phase space. The KHIMA
HEBT line has been fundamentally designed by using the WinAgile/Mad-X code for
the whole beam lines up to each treatment room. Based on the error analysis
for every single line, the optimized correction setup is settled and the
specifications for the correcting magnets of the HEBT lines are determined. A
specification of corrector must fulfil the requirement for highest carbon beam
energy, 430 MeV/u. Table 3 summarizes the specification of corrector used in
the HEBT beam lines.

Table 3: Specification for the correcting magnet of the HEBT line. Parameters | Values | Unit
---|---|---
Maximum rigidity (B$\rho$) | 6.62 | Tm
Max. deflection angle ($\theta$) | 5 | mrad
Effective length ($L_{eff}$) | 0.3 | m
Max. magnetic field | 0.11 | T
( B = $\theta/L_{eff}$ $\times$ B$\rho$) |  |
$\Delta$BL/BL | $<2\times 10^{-3}$ |
Field stability | 500 | ppm
No. of corrector | 23 | ea.

###### Acknowledgements.

This work was supported by the National Research and Development Program
through the Korea Institute of Radiological and Medical Sciences funded by the
Ministry of Science, ICT & Future Planning (NRF-2015M2C3A1001637).

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