Multiple stem spoke cavity design to minimize transverse field asymmetry

shootceaselessUrban and Civil

Nov 16, 2013 (3 years and 8 months ago)

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TD
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Multiple stem spoke cavity design to minimize transverse field
asymmetry

Paolo Berrutti, Timergali Khabiboulline, Leonardo Ristori, Vyacheslav Yakovlev

Introduction

SSR2 is a spoke resonator under development at Fermilab, it is part of Project X and it is meant
to accelerate particle in the low
β

range. SSR2 has been initially designed as a single spoke
resonator, which means that the cavity geometry has an inner elec
trode shaped as straight spoke.
That is the simplest design for a cavity derived from a coaxial line approximately half wave
length long. The
electric field lines of the accelerating mode are directed from the electrode to
the cavity walls, the magnetic li
nes are spinning around the central post
.

In

figure

1
electric field
lines are represented in green (left) and magnetic ones in orange (right).



Figure 1: on the left the electric field lines are sketched in green, the magnetic field lines are
shown on
the right in orange color.

Let’s assume the particle
s

are traveling
along z axis, the spoke is lying on y axis

and it is
perpendicular to x. T
he symmetry on the x
-
y plane has been broken by the
electrode;

the
electromagnetic field

will reflect this asymmet
ry and the transverse

kick will show a quadrupole
component
.
The electric field asymmetry can be mitigated using a d
onut

shape for the central
part
of the spoke electrode,
since
the electric field lines are directed from the inner conductor to
the outer wa
lls. The magnetic lines spin around the cen
tral electrode, it implies that the magnetic
field will have asymmetric transverse component
s
; consequently the magnetic kick will have
different X and Y components.
If the magnetic field contribution to the trans
verse momentum
gain is not
negligible,

the only
possible way to get

symmetr
ic transverse kick

is
by

modif
ication
of

the single stem electrode into
an

X
-
shaped or a Y
-
shaped spoke.

These modifications will
provide
different

cavity symmetry, reducing the qu
a
drupole effect on the transverse plane.


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SSR2 Cavity



SSR2 will be the third family of spoke resonators used in Project X front
-
end: this
accelerator section will accelerate the bunches from the end of RFQ section (2.1 MeV) to
the beginning of the five
-
cell

cavity part (160 MeV)



SSR2
will

be inserted after HWR
(
β
=0.11 design by Argonne) and SSR1
(spoke

resonator
β

optimal=0.21 by Fermilab), the particle velocity range covered by SSR2
cavity goes from 40 to 160 MeV.



The single spoke resonator having
β

optimal 0.47 is the best choice for Fermilab Project
X front
-
end because it minimizes the number of cavities needed.



Double and triple spoke have too narrow transit time factor vs
β
, they would require a
higher number of cavities for Project X front end.



All coaxial resonators (
qu
a
rter wave, half wave and spoke
resonators)

show transverse
field asymmetry in the region where the beam travels.

Field asymmetry



The nature of the fundamental TEM mode implies that: a quadrupole component is
always present in the
se cavities because the inner post breaks the azimuthal symmetry of
the structure.



T
he electric field can be symmetrized easily modifying either the shape of the central
electrode or the beam tubes, for example HWR resonator by Argonne has a ring shaped
e
lectrode to compensate the electric field asymmetry.



SSR2 single spoke cavity has a high quadrupole component due to the
magnetic field

which spins around the spoke electrode and interacts with particles traveling through the
structure.



SSR2 single spoke s
hows a considerable transverse field asymmetry, due to the magnetic
field
.



This field asymmetry leads to an asymmetry in the transverse kick

(quadrupole)
, which
increases the beam envelope along the accelerator length

in SSR2 section
.



The only solution to
symmetrize the transverse magnetic field is to modify the spoke
post, giving symmetry to the cavity: instead of a straight spoke one can use X or Y
shaped inner conductors.


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Single spoke cavity:









Figure 2: SSR2 single spoke cavity (
left
) and
transverse electric and magnetic field on a
10 mm offset

(right)
.

SSR2 single spoke transverse fields were evaluated with a 10 mm offset
, in both X and Y
directions,

with respect to the center of the beam pipe (right side of the figure above). The
electric

field components (first column in the plot) look quite similar one to the other:
pattern
s

and
amplitudes |E
x
| and |E
y
| are

very close
. The transverse magnetic fields H
x

and
H
y

differ quite significantly by pattern and amplitude: the
ir pattern
s

along Z axi
s are
substantially different,
and H
y

component is three times smaller than

H
x
.




SSR2 Y
-
spoke cavity

design
:









Figure 3
:

SSR2 Y
-
spoke cavity (left) and transverse electric and magnetic field on a 10
mm offset (right).

Y
-
spoke cavity design has been made replicating the spoke inner electrode every 120
degrees to get a Y shape (left in figure 3). This
geometry
show
s
negligible transverse
field asymmetry, compared to the single spoke geometry.

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SSR2 X
-
spoke cavity design
:








Figure 4: SSR2 Y
-
spoke cavity (left) and transverse electric and magnetic field on a 10
mm offset (right).

The left picture in figure 4 shows the X
-
spoke cavity design
, which

has been made
replicating t
he inner electrode every 90 degrees to get an

X

shape
. The

transverse field
asymmetry

looks negligible
.

Transverse Kick Calculation



Field quadrupole asymmetry can be evaluated by calculating the kick in transverse
directions, it has been introduced a parameter





(



)





(




)




(



)





(




)





the smaller the Q
parameter is the better beam quality can be achieved with the resonator.



One can look at the plot of Q vs particle
β

to understand the difference among the various
geometries and their capability of providing low quadrupole effect in the full energy
range of usage
.



Single spoke Q vs
β

(
35
-
160MeV)
:

Figure 5 shows the amplitude of the
Q parameter as a function of particle

β, particles in
SSR2 section of Project X will be accelerated from 35 MeV to 160 MeV, so the full β
domain has been covered by the calculation. Q parameter is normalized so it does not
depend on the peak field, gradient or stored energy in the cavity. The

maximum value of
Q for the single spoke design occurs at the very beginning of the β range, and it is pretty
high 0.4.

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Figure 5: SSR2 single spoke cavity Q vs
β

plot for the full particle energy range.





Y
-
spoke

Q vs
β

(35
-
160MeV)
:

The Y
-
spoke design Q vs β plot shows a much lower quadrupole asymmetry. In the
whole β domain the Y
-
spoke design has lower Q
parameter;

the maximum of the Q
parameter is now 3% of the single spoke highest point.


Figure 6: SSR2 Y
-
spoke cavity Q vs
β

plot
for the full SSR2
β

domain.






X
-
spoke

Q vs
β

(35
-
160MeV)
:

Q asymmetry parameter as a function of
β drops even more for the X
-
spoke design. The
maximum of Q for the X
-
spoke design 0.006% of the peak for the single spoke cavity.

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Figure 7: SSR2 X
-
spoke cavity Q vs
β

plot for the full particle energy range.




Field expansion into multipoles confirms the results obtained using th
e Q parameter

simplified approach, the table below report the RF field expansion coefficients for all of
the cavities

calculate at r=10 mm and at particle
β

equal to 0.47
.










S楮i汥l
獰潫s

夠獰潫Y

X
-
獰潫s

1
st

[keV]

5.812e
-
14

0.2501582

2.63E
-
14

2
nd

[keV/mm]

3.391

0.0170884

0.006208

3
rd

[keV/mm
2
]

4.331e
-
16

0.2362148

3.43E
-
14

4
th

[keV/mm
3
]

4.747e
-
4

0.0009250

0.083717

5
th

[keV/mm
4
]

4.957e
-
18

0.0034837

4.91E
-
14

6
th

[keV/mm
5
]

2.338e
-
08

0.0149121

0.00251

7
th

[keV/mm
6
]

1.367e
-
19

0.0041916

2.63E
-
14

8
th

[keV/mm
7
]

2.719e
-
10

0.0033935

0.005494




X and Y spoke geometries have been optimized and their performances look
comparable
to

the ones of the single spoke cavity


Single spoke

Cross spoke

Y spoke

G

[Ohm]

112.98

122.67

119.94

R/Q

[Ohm]

289.94

272.26

269

B/Eacc

[mT/(MV/m)]

6.107

6.13

6.07

E/E
acc

3.45

3.34

3.48

β
opt

0.4714

0.479

0.48






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Conclusions

Despite the more complicated design multi
-
stem spoke resonators allow to overcome the
transverse field asymmetry. Both magnetic and electric fields are
symmetrized

using X or
Y shaped

inner electrode
,

consequently quadrupole and octupole amplitudes drop to
negligible levels. This happens because the cavity geometry recovers azimuthal
symmetry, o
r

transverse plane

symmetry
, which
would be

broken by the sin
gle spoke
inner electrode. Instead of using a quadrupole corrector inside the cryomodule it is
possible to modify the cavity geometry to obtain zero transverse asymmetry
.

A
preliminary RF optimization

showed that the performances achievable with all the
ca
vities are very similar, the choice of a multi
-
stem spoke does not influence the
electromagnetic parameters.