Proceedings of the 13th Annual Meeting of Particle Accelerator Society of Japan
August 8-10, 2016, Chiba, Japan

PASJ2016 TUP017

INTER-DIGITAL H-MODE DRIFT-TUBE LINAC DESIGN WITH
ALTERNATIVE PHASE FOCUSING FOR MUON LINAC
M. Otani, T. Mibe, M. Yoshida, KEK, Oho, Tsukuba, 305-0801, JapanK. Hasegawa, Y. Kondo, JAEA, Tokai, Naka, Ibaraki, 319-1195, Japan
N. Hayashizaki, Tokyo Institute of Technology, Tokyo, 152-8550, JapanY. Iwashita, Kyoto University, Kyoto, 611-0011, JapanY. Iwata, NIRS, Chiba, 263-8555, Japan R. Kitamura, University of Tokyo, Hongo, 113-8654, Japan
N. Saito, J-PARC Center, Tokai, Naka, Ibaraki, 319-1195, Japan
Abstract
We have developed an inter-digital H-mode (IH) drifttube linac (DTL) design with an alternative phase focusing (APF) scheme for a muon linac, in order to measure
the anomalous magnetic moment and electric dipole moment (EDM) of muons at the Japan Proton Accelerator Research Complex (J-PARC). The IH-DTL accelerates muons
from β = v/c = 0.08 to 0.28 at an operational frequencyof 324 MHz. The synchronized phase array is optimized
with analytical calculation of the beam dynamics to get
smaller emittance growth during acceleration. Then an IHDTL cavity is modeled in the 3D EM simulation. Finallythe beam dynamics in the cavity are evaluated numerically.
The output beam emittances are calculated as 0.315π and
0.195π mm mrad in the horizontal and vertical directions,
respectively, which satisfies the experimental requirement.
Additional emittance growth due to an error field caused
by fabrication error is evaluated to be less than 10% witheight slug tuners. The design and results are described inthis paper.

INTRODUCTION

region (less than 0.28), in order to avoid muon decay loss
due to the muon finite lifetime of 2.2 µs. Moreover, theconstruction cost is desirable to be reduced. Based on these
considerations, an IH-DTL incorporating the APF method is
employed for the acceleration stage from β = 0.08 to 0.28.
This paper focuses on the IH-DTL [6] and details ofother structures and preparation status can be found in elsewhere [7–9]. In following sections, the APF IH-DTL design
and results are described. After that, an error field study ispresented.

APF IH-DTL DESIGNIn the APF scheme, the gap-to-gap synchronous phases
are varied in order to achieve longitudinal and transverse
focusing. However, the synchronous phase array determines
each cell length and changes the average on-axis field E0 ateach gap, which in turn affects the particle motion. Hence,the phase array optimization is strongly correlated to thebeam dynamics and the cavity design. In order to solve
this convoluted problem, the procedure is divided into thefollowing three steps:A) Synchronous phase array optimization

The low emittance muon beam has been discussed in several scientific fields [1–3]. One of those is the quest for
hunting beyond the Standard Model (SM) of elementary
particle physics. In the muon anomalous magnetic moment(g − 2) µ , there is about three standard deviation between
the SM prediction and the measured value with a precisionof 0.54 ppm [4]. This discrepancy is thought to be due tointeractions or particles that are unknown to the SM; therefore, further investigations are desired. The low emittancemuon beam will provide more precise measurement since
the dominant systematic uncertainties in the previous experiment resulted from the muon beam dynamics in the muon
storage ring.
We are developing a muon linac for the (g − 2) µ experiment [5] at Japan Proton Accelerator Research Complex(J-PARC) to realize the low emittance muon beam. Figure 1
shows the muon linac configuration. In order to satisfy theexperimental requirement of an extremely small transverse
divergence angle of 10−5 , the muon should be accelerated toa momentum of 300 MeV/c (212 MeV), without substantial
emittance growth. In addition, the effective accelerating
gradient should be relatively high, especially in the low βB) Cavity optimization
C) Particle tracking
In first step, the particle dynamics are calculated analytically using certain approximations and for a particularsynchronous phase array. These calculations are performed
using “LINACSapf” [10], with some modifications for the
dynamics calculations and the synchronized phase array definition to accommodate the π–mode acceleration, whereas
2π–mode acceleration is assumed in the original code. In“LINACSapf”, the beam dynamics is calculated by so called
drift-kick-drift method [11]. One notable difference to softwares for an ion-linac design such as PARMILA [12] isthe implementation of the transverse focus and de-focusing
due to the radial displacement and the electro-static effect;
because velocity evolution in muon acceleration is larger,
those effect has about 30% of RF de-focusing only. E0 isdetermined from the Kilpatrick criterion [13, 14]. The Kilpatrick limit Ekilpat at 324 MHz is 17.8 MV/m and a valueof approximately 1.8 × Ekilpat is employed for the maximum
surface field, based on the empirical value applying to RFQs.

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Proceedings of the 13th Annual Meeting of Particle Accelerator Society of Japan
August 8-10, 2016, Chiba, Japan

PASJ2016 TUP017

ultra-cold muon
generation

conventional surface
muon beammuon linac

W = 4 MeV
0.34 MeV
0.08

W = 5.6 keV
β = 0.01

4.5 MeV
0.27

45 MeV
0.7

212 MeV
0.95

Injection to
storage ringW = 30 meV
Thermal muonium
generation

Laser
dissociation

Electro-static

Low β: IH

RFQ

Middle β: DAWHigh β: Disk loaded

Figure 1: Configuration of low-emittance muon beam.
Assuming that the peak-to-average ratio is three, E0 is set to
10 MV/m. The maximum surface field will be confirmed
following discussion of the cavity design.
In the code, the dimensions of the drift tubes and gaps arecalculated with assigned E0 , the particle velocity of the central orbit, and assigned synchronous phase array φ n . Thenthe dynamics of all the particles are calculated. The initial
settings of the synchronized phase array are assigned basedon those for the HIMAC accelerator [15]. All the φ n are
optimized using the minimization function f , which is defined based on the emittance growth (∆ε), the energy spreadof the output beam (∆E), and the loss particle efficiency(ε loss ), such that f = ∆ε + ∆E + ε loss .
Table 1 shows details of the optimized parameters. Gap
numbers 1–2, 6–9, 15 and 16 have negative synchronous
phases, during which time the beam is longitudinally focused.
However, gap numbers 3–5 and 10–14 have positive, during
which time the beam is transversely focused. Because theelectrostatic focusing is stronger in the lower-beta part, thefirst collection of positive phase groups has a smaller numberof gaps. The output energy is 4.5 MeV with total length of
1.3 m.
Table 1: Cell Parameters for Optimized Phase Array
cell
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
exit

W [MeV]
0.34
0.43
0.57
0.74
0.92
1.14
1.38
1.63
1.86
2.16
2.49
2.82
3.10
3.50
3.95
4.30
4.50

β
0.08
0.09
0.10
0.12
0.13
0.15
0.16
0.17
0.19
0.20
0.21
0.23
0.24
0.25
0.27
0.28

φ
[degrees]
-35.9
-14.9
12.9
32.9
15.4
-13.8
-31.4
-44.3
-18.8
12.5
27.6
47.6
23.2
10.8
-34.6
-15.6

cell length
[mm]
29.5
46.0
54.9
60.3
54.4
56.0
66.4
74.1
97.2
108
106
116
94.2
108
91.5
142

total
[mm]
29.5
75.4
130
191
245
301
367
442
539
646
753
868
963
1070
1160
1300

As a second step, the IH cavity is modeled in a threedimensional EM simulation. Because the IH cavity is notaxially symmetric, a three-dimensional model is necessary
in order to evaluate the electro-magnetic field. In addition,

the electro-magnetic field and the resonant frequency depend on the entire structure of the IH cavity, and the detailof the overall structure (including the ridges, etc.) should
be incorporated in the calculation model. Therefore, the entire IH cavity is modeled using the CST Micro Wave (MW)Studio [16] three-dimensional field solver, in order to calculate the electro-magnetic field. Figure 2 shows the threedimensional model of the IH cavity in CST MW Studio.

Ridge

Drift
tube

Stem

Ridge
tuner

Figure 2: Three-dimensional model of IH cavity in CST
MW Studio calculation.

To optimize the IH cavity, the drift tubes and the acceleration gaps are first arranged according to the previously
determined optimized parameters shown in Table 1. Because
the drift tube and gap length depend on both the synchronous
phase and beta, the inductance and capacitance along thecavity are not constant, which distorts the acceleration field.
In order to achieve flat electric field on the beam axis Ez ,the following dimensions are adjusted: the cavity radius, theridge tuner length, the cavity taper, the stem radius, and thetube radius.
Note that adjustment of the cavity radius is used to tune theresonant frequency, and the radius tuning does not strongly
influence the Ez flatness. The flatness is first optimized
through tuning of the other parameters; the resonant frequency is then adjusted by changing the cavity radius.
The Ez flatness is primarily tuned using the ridge tuner
and the cavity taper. The black and blue lines in Fig. 3 show
the longitudinal electric field along the beam axis before andafter these optimizations. Before the optimizations, the fieldin the downstream region is lower than that in the upstream
region, because the acceleration gaps in the downstream
region are smaller than those upstream, and the equivalentcapacitance gradient results in a field gradient. The tilting
field is prescribed by adjusting the ridge tuner length andthe cavity taper.

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Proceedings of the 13th Annual Meeting of Particle Accelerator Society of Japan
August 8-10, 2016, Chiba, Japan

PASJ2016 TUP017

Fine tuning of Ez in each gap is achieved by changing thestem radius and the drift tube radius, as shown by the red line
in Fig. 3. By first adjusting the radius and then varying theequivalent inductive load at each gap, the field differences
between the gaps can be corrected. Because the stem radius
and the drift tube radius are correlated with the feasibilityof the manufacturing process, the range of adjustment isfeasible in terms of manufacturing.
Before optimization
After RT&CT optimization

After Rstem and RDT optimization

250 mm and 650–1070 mm, where the transverse focusing is
implemented. During these periods, the horizontal velocity
is decreased.
Figure 4 (bottom) shows the normalized velocity in they-direction along the beam axis. The vertical trajectory isdominated by the finite value of the vertical electric field. Inorder to reduce the additional growth, some conventional
solutions, such as the use of drift tube bulges [18], have been
considered. However, this approach only reduces the verticalfield of the IH structure with few percents because the cell
length is relatively large (because of the application of the IH
structure to yield a higher β region). As a result, no sufficient
suppression of the extra growth is achieved using these theadditional structures. Because the extra growth is acceptably
small and the output beam satisfies the requirement of the
J-PARC (g − 2) µ experiment , no additional structures are
implemented in this design, so as to avoid additional power
loss at these structures.
60

Z [mm]

0.002

Figure 3: Effects of ridge tuner (RT), cavity taper (CT),stem radius (Rstem ), and drift tube radius (RDT ) adjustment.
Horizontal axis is the distance from the cavity start on the
beam axis, and vertical axis is the Ez normalized at thepeak value. Black: longitudinal electric field along z before
optimizations, blue: field after optimizations with RT andCT, red: field after optimizations with Rstem and RDT .

βx

φ [degree]
00-0.001-60 60-0.002
0.002

βy

0.001

φ [degree]
00-0.001-0.002

0

700

1400

-60

The variation in the electric field in the gaps after the
optimization is approximately 10%, excluding the first andlast cells. The resonant frequency is tuned to a slightly
lower value than the 324-MHz operation frequency, in orderto leave room for the tuner knobs with the inductive tuner
installed on the cavity side wall. The quality factor (Q0 ) is
calculated to be 1.07×104 . The effective shunt impedance iscalculated to be 92 MΩ/m, and the operation power power
is required to be 250 kW. The effective shunt impedanceis competitive to those of other IH structures given our IH
application to relatively higher velocity region.
The maximum surface field is evaluated to be 34 MV/m
at the outer surface of the most downstream drift tube, corresponding to 1.9 times the Kilpatrick limit. This value isreasonable based on the experiences in RFQs.
In third step, the beam particle trajectory is simulatedusing the General Particle Tracer (GPT) [17]. In GPT, thedynamics is calculated with an embedded fifth order RungeKutta driver with the sufficiently small step size. The electricand magnetic fields calculated using CST MW Studio are
implemented in the code and the particle dynamics are calculated numerically. The number of simulated particles is
105 that corresponds to designed muon beam intensity perbunch.
Figure 4 (top) shows the normalized velocity in the xdirection along the beam axis (z) with overwriting of thesynchronous phase (φ). As shown in the red hatched box
in Fig. 4, the synchronous phases are positive for z = 130–0.001

Z [mm]

Figure 4: Beam velocity distribution as a function of distancealong the IH structure (black) and synchronous phase in eachposition (red). β x (top) and β y (bottom) along beam axis.
Finally, the output beam is evaluated using realistic input beam distributions. The input beam was obtained fromsimulations of the surface muon beamline, the ultra slow
muon system, and the RFQ [19]. From the results, the normalized root mean square (rms) emittances of the input
beam were evaluated as 0.297π mm mrad in the x-direction,
0.168π mm mrad in the y-direction, and 0.0181π MeV deg
in the z-direction. Because the structure is not periodic
due to the APF method and rapidly changing velocity profile, the twiss parameters (α and β) at the IH entrance were
scanned in order to obtain a matching condition to the IH,- 860 -

Proceedings of the 13th Annual Meeting of Particle Accelerator Society of Japan
August 8-10, 2016, Chiba, Japan

PASJ2016 TUP017

(B)

(C)

(D)

Δ

0.1

εx
εy

0.08
0.06
0.04
0.02
0

2

4

6

Random range[%]

Figure 6: Additional emittance growth as a function of the
field scale factor.
One possible cause of an error field is a fabrication error.
Because our operational frequency is slightly higher thanexisting IH-DTL, the fabrication error may be significant impact. In order to estimate the fabrication error effect, somedimensions in the IH model in CST MW Studio are artificially changed and the axial field variations are investigated.
Figure 7 shows the field variation when one of the drift tube
outer radius is changed. The change is 2% in maximum withgeneral fabrication error of 100 um. Other dimensions suchas the drift tube inner radius, stem radius, and incline of
the tube and stem are also investigated. It reveals that thefield error is about 2% when the fabrication error is less than
100 um.
∆E(A)error is required to be less than 2% to suppress additional
emittance growth with less than 10%.
∆ε [π mm mrad]instead of the conventional method solving a periodic solution of the transfer matrix. Based on the scan results, thetransport line from the RFQ to the IH was designed using
TRACE3D [20], and the beam distributions at the RFQ exit
were then transported using PARMILA [12]. Figure 5 shows
the calculated phase-space distributions of the output beam.
The emittance growth was calculated to be 0.018π (6.1%)
and 0.027π mm mrad (16%) in the x- and y-directions, respectively. This is consistent with the evaluations using the
wb distribution to within a few percent, and the small discrepancy is due to the difference in the distribution shapein the z-direction. The transmission efficiency without any
selections in output beam was calculated to be 99.9%. The
beam transit time t tran. was 25 ns and the muon survival rateis calculated to be exp(t tran. /τµ γ) = 98.9%, where the average Lorentz factor during acceleration is labeled γ. The totaltransmission is expected to be 98.7%, which is sufficient for
the J-PARC (g − 2) µ experiment.

out

R DT#6+40um

0.04

out

R DT#6+80um[degree]out

R DT#6+100um

0.02

Figure 5: Calculated phase space distributions at IH exit. (A)the horizontal divergence angle x 0 vs x, (B) the vertical divergence angle y 0 vs y, (C) y vs x, and (D) ∆W (W −4.5 MeV)vs ∆φ.

out

R DT#6+200um

0
−0.02
−0.04

In conclusion, the beam emittance will meet the requirement for the J-PARC g − 2/EDM experiment.

0

5

10

15

gap#

ERROR FIELD STUDY

Figure 7: Variation of the axial field peak in each gap when
one of the drift tube outer radius is changed.

Since beam stability relies on the RF field in the APF
method, the beam dynamics may be strongly affected byan error field. In order to estimate the error field effect tothe emittance growth, the RF field in each gap is scaled bya range of few percents independently and then the beam
dynamics in the scaled fields are calculated numerically.
The emittances of the output beam are calculated with 50
sets of the pseudo RF field and sum of the average shift
and rms is evaluated as the additional emittance growth ∆ε.
Figure 6 shows ∆ε as a function of the scaled factor. Because
fluctuation of the emittance is larger in the y-direction due tothe vertical field, ∆ε y is slightly larger than ∆ε x . The field

Since the error field due to the general fabrication errorof 100 um is expected to be comparable to the requirements
for 10% additional emittance growth, a slug tuner is studied
as the field tuner. Figure 10 shows the field variation when
the slug tuner is inserted. The error field distribution isconsistent with the mode mixing of higher mode of TE111
whose resonant frequency is higher with about 6 MHz thanoperational mode of TE110. Because there is a clause atz = 800 mm in the TE111 resonant field, it is important to
set more tuners around that position and mix higher modeof TE112, in order to get higher tuning capability. The- 861 -

Proceedings of the 13th Annual Meeting of Particle Accelerator Society of Japan
August 8-10, 2016, Chiba, Japan

PASJ2016 TUP017

∆E

0.15

# of points

calculation with several setting of the slug tuner shows good
linearity of the field variation to the tuner length.

Before20

After

Tuner@z=60mm@z=3000.1

10@z=540@z=700

0.05
0
−0.05

0
−8

@z=760@z=840@z=1000@z=1240

−0.1
−0.15
0

5

10

−6

−4

−2

0

2

4

6

8

∆E/E0[%]

Figure 10: Difference to the design value of the electricfield on each gap. Blue: difference without the tuners, red:difference with the tuners.

15

gap#

Figure 8: Field variation with the slug tuner at several positions.

ACKNOWLEDGMENT
Finally we perform a case study assuming the fabrication
errors of 100 um. First the electro-magnetic field is calculated with the IH model in which the drift tube inner andouter radii, stem radius and its inclination are independently
changed within assigned fabrication error. Assuming thelinearity of the length to the field variation, we can write∆Ei =tuners
X
j

∂Ei
Lj
∂L j

(1)

REFERENCES[1] M. Palmer et al., ICFA Beam Dynamics Newsletter,
No.55, 2011; http://icfa-usa.jlab.org/archive/
newsletter/icfa_bd_nl_55.pdf[2] Y. Miyake et al., Hyperfine Interact, 216, p.79, 2013.
[3] M. Bogomilov et al., JINST 7, P05009, 2012.

where Ei is the axial field in i’th gap, L j is the j’th tuner∂E ilength, and ∂Lis the response matrix for the j’th tunerjwhich was estimated by CST MW Studio. The set of L j is
optimized to correct the error field and the electro-magnetic
field is calculated again with the tuners. Figure 9 shows oneof the results. The difference between design and calculated
value is less than 2% with the tuners whereas it is 6% without
tuners. The case studies are performed five times and thedifference to the design value is almost within 2% with thetuners, as shown in Fig. 10.
E/E0This work was supported by JSPS KAKENHI Grant Numbers 15H03666, 16H03987.

[4] G.W. Bennett et al., Phys. Rev. D73, 072003, 2006.
[5] J-PARC E34 conceptual design report, 2011. (unpublished); https://kds.kek.jp/indico/event/8711/
material/2/0.pdf[6] M. Otani et al., Phys. Rev. Accel. Beams, 19, 040101, 2016.
[7] M. Otani et al., Proc. of IPAC2016, TUPMY003.
[8] Y. Kondo et al., PASJ2016 Proc., MOOM04.
[9] R. Kitamura et al., PASJ2016 Proc., MOP050.
[10] R.A. Jameson, arXiv:1404.5176 [physics.acc-ph].
[11] T.P. Wangler, “RF Linear Accelerators”, WILEY-VCH Verlag,
GmbH&Co, 2011.

0.1

0.05

[12] Los Alamos Accelerator Code Group (LAACG), LANL, Los
Alamos; http://www.laacg.lanl.gov

0target for correction∆ E/E0

−0.05

0

0.04

with tuners
5[13] W.D. Kilpatrick, Rev. Sci. Instr. 28, 824, 1957.
10[14] T.J. Boyd Jr., Los Alamos Group AT-1 report AT-1:82-28,
1982.

15

0.02

[15] Y. Iwata et al., Nucl. Instr. Meth. A569, 685, 2006.

0− 0.02[16] CST Studio Suite, Computer Simulation Technology (CST);
https://www.cst.com/products/CSTMWS− 0.04
0

5

10

15

gap#

Figure 9: Example of the case study result. Red cross: target
value to correct the field error due to the fabrication error,
black: actual field with optimized slug tuner length.

[17] GPT, Pulsar Physics; http://www.pulsar.nl/gpt/[18] S.S. Kurennoy et al., Phys. Rev. ST Accel. Beams 15, 090101,
2012.
[19] Y. Kondo et al., Proc. of IPAC2015, 2015, THPF045, 2015.

In conclusion, the tuning capability against nominal fabrication error with the slug tuners is enough.

[20] K.R. Crandall and D.P. Rustoi, Los Alamos Report, No. LAUR-97-886, 1997.

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