measurement of the rf field on rfq linac model cavities
TRANSCRIPT
INS-NUMA-30
MEASUREMENT OF THE RF FIELD ON RFQ
LINAC MODEL CAVITIES
T. Nakanishi, N. Ueda, S. Arai, T. Hor+rM. Takanaka,
A. Noda and T. Kratayama
January, 1982
STUDY GROUP OF NUMATRON ANDHIGH-ENERGY HEAVY-ION PHYSICSINSTITUTE FOR NUCLEAR STUDYUNIVERSITY OF TOKYO
Midori-Cho 3-2-1, Tanashi-Shi,Tokyo 188, Japan
INS-NUMA-3 0
January 1982
MEASUREMENT OF THE RF FIELD ON RFQ LINAC MODEL CAVITIES
T. Nakanishi , N. Ueda, S. Ara i , T. Hori , M. Takanaka,
A. Noda and T. Katayama
ABSTRACT
The results of experimental studies on rf characteristics of a four
vane RFQ cavity are described. Two model cavities were fabricated. With
the first model, which has a set of straight vanes, has given a guide
line for the loop coupling and the tuning method to get a uniform field.
The second model, manufactured with closer tolerance, has two kinds
of vanes, straight and modulated. For the straight vanes, the measured
resonant frequency is 296.0 MHz for the TE210 mode and agrees well with
the calculated value 296.5 MHz using of the computer code SUPERFISH.
The measured one is 293.5 MHz for the modulated vanes which have the same
cross section as the straight vane at their quadrupole symmetry plane.
The measured electric field distribution in the acceleration bore agrees
with the calculated one within the measurement error. A sufficient mode
separation bigger than 1 % and uniform field distribution within a few
percent have been obtained with a single loop coupler which matches the
cavity to the feeder line.
- 1 -
TABLE OF CONTENTS
I1. Introduction '
(
2. Four vane cavity
2.1. General concept I
2.2. End tuner /
!
2.3. Vane extender I
3. Cold model I !
3.1. rf measurement
3.2. Relation between the end space of the cavity and the longitudinal '(
field distribution
4. Cold model II
4.1. Structure of the second model cavity
4.2. Field distribution with the straight vanes j
4.3. Field distribution with the modulated vanes I
;4.4. End tuning to obtain the desired field distribution '
I4.5. Field distribution with a displaced vane j
i
4.6. Mode separation
4.7. The Q value and rf coupling |
4.8. Field penetration into a hole and slots on the cavity wall ;
5. Calculation with SUPERFISH I
6. Concluding remarks |i
Appendix A: Equivalent circuit analysis of a four vane cavity jB: The field measurement system of the RFQ cavity '
i
C: An rf coupling loop for a four vane cavity j
D: Formulae of RFQ beam dynamics
- 2 -
1. Introduction
The injector of the NUMATRON is required to accelerate various
heavy ions up to uranium from several keV to 10 MeV per nucleon. Already
has been proposed a complex consisting of Cockcroft-Walton (C-W) injectors,
WiderOe and Alvarez linacs. ' Thereabout it was demonstrated that a
Radio Frequency Quadrupole (RFQ) linac is an effective accelerator in a
3 4)low energy region. ' ' The POP test linac at LASL has accelerated 38 mA
protons from 100 keV to 640 keV through 1.1 m with a transmission higher
than 70 %. In the POP cavity protons are focused and accelerated by the
TE210 like electric field of 425 MHz generated with four modulated vanes
(Fig. 1.1). An RFQ linac is considered to be preferable at the lowest
stage of the NUMATRON injector linacs for the following reasons:
(1) An RFQ linac can accelerate ions with lower velosity owing to its
shorter cell length than that of a conventional drift tube linac.
Therefore it becomes possible to make use of a lower C-W voltage
and/or a lower charge state beam, which has a higher intensity.
(2) An electric focusing force is
independent of the particle ve-
locity and has an advantage to
magnetic one at a lower velocity.
Therefore higher intensity beams
can be acceptable.
(3) The accelerating and focusing
forces have the same dependence
on the charge to mass ratio q/A. Fig. 1.1. Modulated vanes.
V
-y COS Ut
- 3 -
For various ion species it is enough to tune only the rf power level
for an intervane voltage proportional to q/A.
(4) An RFQ linac can capture more than 80 % of the injected dc beam and
accelerate it to the energy required for injection into the following
drift tube linac.
On application of an RFQ linac for heavy ions, however, should be
studied the following subjects on the structure and the rf power feed:
(1) In order to get sufficient focusing force for ions with lower q/A,
an RFQ linac should be operated at an lower frequency, for example
25 MHz for U . The diameter of a four vane cavity will be about
2.5 m for this frequency. . One problem is how to mount the vanes
to the tank with closer tolerances and good electric contact.
Another problem is how to tune the cavity to get the required
frequency and field distribution.
(2) The required vane voltage is proportional to A/q, and the rf power
2should be supplied in a wide range proportionally to (A/q) . The
beam intensity limit differs much with ion species, and the ratio
of the beam loading to the wall loss is varied. Single loop coup-
ling is preferable to the slot coupling used on the POP, because
the loop coupling need no outer chamber au? is flexible for changing
the input impedance of the cavity. It is another problem how to
make a uniform field and suppress parasitic excitations with a single
loop coupler.
Two model cavities were manufactured to study these subjects. On
the first cavity with straight vanes the basic characteristics of four
- 4 -
vane cavity have been obtained. The second cavity was made on the basis
of the work on the first cavity. It was made with closer tolerances
in order to reduce the effect of the mechanical errors on the field
distribution. A set of straight vanes was attached to the tank, then
it was replaced by a set of modulated vanes. They were attached in the
ways of setting and electric contacting which are applicable for an
actual acceleration cavity, though the dimension of the cavity is much
smaller than an actual one. For the both, set of vanes the measured
resonant frequencies have been compared with the calculated one by
SUPERFISH. A uniform field has been obtained with a single coupling
loop which matches the cavity to the feeder line. The measured electric
field strength in the acceleration bore has been compared with the cal-
culated one given by the Kapchinskij-Teplyakov potential and with the
magnetic field in the four chambers.
On the basis of the model study and beam dynamic study worked in
parallel, an RFQ Test Linac has been designed and is incourse of con-
struction.1' ' The machine is designed to accelerate heavy ions with
q/A of 1^1/7 from 5 keV/u to the final energy of 138 keV/u and named
'LITL' (Lithium Ion Test Linac).
In this report the results of the model study on the cavities are
given.
- 5 -
2. Four-Vane Cavity
2.1. General concept
An RFQ linac is designed to get a constant pole-tip potential along
the beam axis to obtain a constant focusing force. Several types of
resonant cavities have been proposed; a double H, a four-vane, a clover-
leaf types etc. Owing to easiness of fabrication and close mechanical
tolerances, we have constructed cold model cavities of the four-vane
type. The vanes are attached inside of a cylinder with 90° intervals
each other and with gaps to end walls (Fig. 2.1). The magnetic fluxes
run the length of the cavity inside four chambers partitioned by the
vanes and are in the opposite direction in adjacent chamber, namely,
TE210 mode. The fluxes are connected at the end spaces. When the cavity
radius is much larger than the acceleration bore, the electric field
mainly concentrates arround the bore.
The resonant frequency of the cavity for the TE210 mode is precisely
predicted by a computer code SUPERFISH. It is also roughly calculated
9)with a following equation.
-lnfe = Y + —2-j , (2.1)2 (kRT
where k = 2TT/A,
a; bore radius,
R; cavity radius,
Y; Euler's constant, 0.5772-
- 6 -
It is derived as follows. We assume the electric and magnetic fields
in the chamber as follows: there are field compornents H and E, only
and no variations in the z and ij) directions. The wave impedance in the
chamber is then expressed as Z = E /H , and the boundary and resonant
conditions are given by;
Z = 0 at r = R,
Z = °° at r = a.
Then the following equation is derived.
J (ka) J,(kR)
N (ka) = N. (kR)o 1
For small ka and kR the above equation is approximated by eq.(2.1).
Resonant frequency vs. cavity dimension is shown in Fig. 2.2. The
capacitance per one vane gap is obtained from the resonant frequency.
Co = ^ — (pF/m), (2.2)w u So
where u ; angular frequency 2irf (rad-MHz),
u ; permeability for vacuum, 4ir><10 (H/m),
S ; cross section of one chamber (m ) .
- 7 -
The TElln modes (dipole) exist near the TE21n. The dispersion
curves for the TE21n and TElln families approach each other as the ratio,
R/a, is increased. An overlapping with the resonance curve of the TE110
mode distorts the desired field distribution of the TE210. The mode
separation is increased by adding interstitial vanes on the zero potential
planes of the TE210 mode in both end spaces of the cavity.
2.2. End tuner
Foui end tuners are mounted to each of the end walls, and change
the capacitance between the vanes and the end wall. By regulation with
them, it is possible to produce the uniformity of the longitudinal field
distribution and the symmetry of the azimuthal field distribution.
We consider the distributed constants in the ideal cavity with
vanes which have a quadrupole symmetric and constant geometry along the
axis (Fig. 2.3). The loss of the electric power on the surface is assumed
zero. All the constants are expressed in terms of unit length. The
propagation constants y and y in the end space and vane gap are res-
pectively given by;
Y = j W L Ce e e
= /(L./L Xl-A C ) .
- 8 -
The regulation with the end tuners is equivalent to changing C • When
resonance occures at f = 1/2TT/ET~C~ which satisfies y I = JTT/2, y is
zero and so the potential of the vane surface is constant along the
length of the cavity, that is, the acceleration bore is excited with
the TE210 mode. On condition of resonance that y I < jir/2, y is ima-
ginary and the potential distribution is tilted. With y I > JTT/2, y
is real and the distribution is bowed, (see Appendix A)
The above model describes only the longitudinal field distribution
in the ideal cavity. The symmetry of the azimuthal field distribution
is important to get desired particle motions in the cavity. Practically
the cavity has four chambers with slightly different dimensions, that is,
it is equivalent to four resonant circuits which are coupled each other.
Therefore, the end tuners are also useful to regulate the azimuthal field
distribution. In the cold model test, a slight regulation has changed
largely the azimuthal distribution with little effect on the longitudinal
one.
2.3. Vane extender
A shorter distance between the end wall and vane is desired to
suppress a effect of the fringing field at both ends of the vane. The
distance can be shortened by an extension of the vanes with the end
space for the magnetic flux kept constant (Fig. 2.1). This end shape
partially blocks the return path of the magnetic flux for the TElln family.
- 9 -
End wallVane extender
"Side wall
Fig. 2.1. RFQ cavity of a four vane type.
End tuners
»Lo
1T
c
IT
L
C'T-
i
1
l:T:1i
L.—1—iitr—|—vrn^-
XjCq_5Lo
°
L.
T T -T T
•z2
Fig. 2.3. Equivalent circuit for the RFQ cavity of
a four vane type.
- 1 0 -
a (= 0.4, 0.6, 0.8, 1.0, 1.4)cm: • : : ] • • ••:• I
20 30 40 50
Fig. 2.2. Resonant frequency vs. cavity dimension.
R (cm)
3. Cold Model I
3. 1. rf measurement
The first cavity with straight vanes was manufactured to grasp basic
characteristics of the four-vane type. It is made of aluminum, 1 m long
and 18.8 cm in inner diameter (Fig. 3.1). The vane has thickness of
10 mm. The vane tip is a circular arc of 5 mm radius in the transverse
cross section and machined with a planer. The bore radius is 10 mm.
End tuners of screws with 10 mm diameter are attached to the end walls.
An rf power is fed to the cavity with a coupling loop.
The measured resonant frequency for the TE210 mode is 452.8 MHz,
while the ones given by SUPERFISH and eq.(2.1) are 453.9 and 457.5 MHz,
respectively. The capacitance C between the vanes is given by eq.(2.2).
—3 2From S = 6.22xio m , C is calculated to be 15.8 pF/m, and so the
o
total capacitance is 63.2 pF/m. The dispersion curves of the TE21n and
tht TElln families are shown in Fig. 3.2. The frequency difference
between the TE210 and the TE110 modes is 22.7 MHz, and the overlapping
of their resonance curves is not observed. The measured Q-value is 2400.
The relative magnetic field strengths in the four chambers are
measured by comparing the frequency perturbation by a brass rod inserted „
through holes on the side of the cylindrical cavity (Fig. 3.3). The
cavity is excited with a weak coupling. Figures 3.4(a)^(c) show the
field distributions along the length of the cavity in the four chambers.
The results showed that the azimuthal symmetry and longitudinal uniformity
of the field distribution are obtained independently of the position
and the number of coupling loops. The observed azimuthal asymmetry is
caused by mechanical errors.
- 12 -
3.2. Relation between the end space of the cavity and the longitudinal
field distribution.
The geometry of the end space has an influence on the longitudinal
field distribution. The end capacitance C has different values by
shapes of the end space even if its area is kept constant. With unapp-
ropriate value of the product of the end capacitance and inductance,
the uniformity of the longitudinal field distribution is not produced.
To measure the electric field in the acceleration bore along the beam
axis, perturbation method is used. The perturbing disk of bakelite is
2 mm thick and 18 mm in diameter.
The first shape of the vane extender is illustrated in Fig. 3.5.
The end space area is 88 % of a half cross section of one chamber. As
C is too large in this shape, the longitudinal field distribution is
bowed as shown in Fig. 3.6, therefore shortening the end tuner gap more
and more bows the field distribution. To flatten the longitudinal field
distribution, the vane extenders have been cut (Fig. 3.7). Since C
decreased in a larger ratio than increase of L by this change of the
end space, the field distribution was tilted. Regulation with the end
tuners, however, produces the uniformity of the field distribution (
Fig. 3.8).
An equivalent circuit analysis of the cavity describes thses phe-
nomena, (see Appendix A) The end space of the cold model II was designed
on the basis of these experimental results.
- 13-
Fig. 3 .1 . Cold model cavity I.
- 1 4 -
F i g . 3 . 2 .
1 2 3 ^
Harmonic no.
Dispersion curves
SZ.
120 ,l20.Ll40
1 2 3 4 5 6 7
Hole no.
Fig. 3.3. Positions of holes for the magnetic field measurement.
- 1 5 -
oHX
10
8
6
4
2
0
7
H 4 - _
H 3 - - |Feeder
g = =
•
Pick
= 1 =
./
=>\/
up
P ;"1
Feeder
H2
H4
I3
Hl
3 4 5
Hole no.
(a)
I
Fig. 3.4(a) (c). Magnetic field distributions with
the different positions and the
number of coupling loops.
The end tuners are kept at the
same positions at each measurement-
10
8
6
4
2
0
I \H H
2 3 4 5
Hole no.
(b)
FeedersH . \**T**S Hi
^Feeders
-/ la// \-•\ -y r
FeedersPick up
H2 »3
7
3 4 5
Hole no.
(c)
80 —
Beam axis
Sej end space area
Fig. 3.5. Shape of the end space before
the vane extenders are cut.
Size*: are in mm.
1 . 0 •
0.8 -
0.6 -
0.4
0.2 -
10 20 30 40 50 60 (mm)
Fig. 3.6. Electric field distribution along the
beam axis before the vane extenders
are cut.
- 17 -
Fig. 3.7. End space after the vane
extendeis are cut.
Sizes are in mm.
Pick up
iO 20 30 ' 0 50 50 70Z
Feeder
3
0) °
> 2
0)
H+1 0id
Ha 3
•5
1
0j
, . — — ^ —
f= 449 MHz
f= 453 MHz
f= 456 MHz
1
\ ,
.0 SO BO
End tuners
Fig. 3.8. Electric field distributions along
the beam axis with the end tuners
closer (upper) and farther (bottom)
from the vanes.
- 18 -
4. Cold Model II
4.1, Structure of the second model cavity
The other model cavity and two sets of vanes, modulated and straight,
were constructed to study the following subjects;
(1) Relation of the resonant frequencies for the straight and
modulated vanes with the ones calculated by SUPERFISH.
(2) Setting and contact method of the vanes to the tank.
(3) Effects of mechanical errors on the field distribution.
(4) Tuning method to get the required field distribution and
resonant frequency.
(5) Relation of the electric field distribution in the acceleration
bore with the theoretical one, and with the magnetic field
distributions in the four chambers.
(6) Feasibility of an rf feed with a single loop couplar.
Photos of the cavity are given in Figs. 4.1 and 4.2.
The inner diameter and length of the cavity are 258 mm and 1000 mm,
respectively. The tank is made of aluminum alloy whose electric
conductivity is 23.4 % of copper. The vanes are made of copper. Its
length and thickness are respectively 960 mm and 35 mm for the both sets.
End parts of each vane are removable and the areas of the end space can
be enlarged by re-machining of the end parts. The vane geometry and
modulation parameters are given in Fig. 4.3. The modulated vane has a
constant cell length of 30 mm and average aperture radius of 14.28 mm,
same as the aperture of the straight vanes. The minimum aperture radius
is 10 mm. The modulation factor m is 2. The modulation and aperture
- 19 -
are exaggerated for the convenience of field distribution measurement
in the acceleration bore.
The theoretical field is generated with the vane shape expressed
as
2
x 2 + y2 = r2cos2i)/= -|- {1 - AI (kr)coskz} . (4.1)A O
The symbols are explained in Appendix D. The cross section is hyperbolic
at kz = IT/2, where it has a guadrupole symmetry. The vane tip is,
however, approximated to a circular arc and tangential line in the trans-
verse plane, because higher vane voltage is applicable with the circular
tip owing to the larger intervane distance and the calculated field is
practically quadrupolar one in the acceleration bore. The straight vane
has the same cross section as that of the modulated vane at kz = ir/2.
Each vane is fixed with screws on four base flanges which are mounted
on the tank side. With the vanes set temporarily, the distances between
the guide pins fixed on the each vane end were measured. Then the surfaces
of the base flanges were cut slightly to give the desired inter pin dis-
tances. The final errors of the vane setting are within 0.1 mm in the
inter pin distances. The vanes and tank are contacted with copper braids
with rubber cords in it.
The vane tip has been machined by the use of a ball cutter of 25 mm
diameter at 2500 rpm with an NC milling machine. The cutting step along
the vane length is 1.0 mm. It took about an hour to cut 50 mm vane
length. The NC machined surface has cusps of 10 um height. They have
been polished away with sand paper, and the surface has been finished
- 2 0 -
with a buff polisher. The final surface, roughness is within 1 ym. The
machining error of the modulation is within 70 ym (Figs. 4.4 and 4.5).
Opposed to each vane end, eight end tuners are mounted to the end
walls. The tuner is a copper rod of 25 mm diameter.
The gap between the tuner and vane is variable from 0 to 20 mm with a
micrometer head. The side wall has lb holes of 40 mm diameter and 20
holes of 14 mm for r.ne field measurement and the rf coupling.
4.2 Field distribution with the straight vanes
In an RFQ linac with a small acceleration bore radius, a precise
measurement of an electric field in the bore is difficult. For example,
the minimum aperture radius is designed to be 2.5 mm at the LITL. The
electric field distribution in the bore will be appraised according to
the measurements of the magnetic field distributions in the four cham-
bers. This model cavity has a bore big enough to measure the electric
field. The relative electric and magnetic field distributions are
measured with perturbation method. The magnetic field distributions in
the four chambers are obtained by measurement of frequency shifts with
a brass rod inserted through a hole on the side wall (Fig. 4.6). A
perturbator used for the electric field measurements is an aluminum bead
of 6 mm diameter. The measure system is controlled by a mini-computer
HP-1000 and the measured frequency is precise to within 300 Hz. (see
Appendix B)
- 21 -
An rf power is fed to the cavity with a single coupling loop. To
avoid an external disturbance, the measurements are done with a weak
coupling. Figure 4.7 shows the magnetic field distribution with the end
tuner heads on the same plane of the end wall. The resonant frequency
is 299.5 MHz. The field has an asymmetry of ± 25 % azimuthally and tilts
longitudinally. After the end tuning, the azimuthal symmetry and
longitudinal uniformity of the field distributions are obtained, as
shown in Fig. 4.8. The azimuthal distribution is uniform to within 2 %.
The resonant frequency is 296.0 MHz, nearly equal to 296.5 MHz predicted
by SUPERFISH. A calculated one using eq.(2.1) is 336.9 MHz. The total
equivalent capacitance between the vanes is calculated to be 108 pF/m.
F and B in the figures mean the gaps between the end tuners and
the vanes on both ends of the cavity.
An output impedance of the rf power supply and a characteristic
impedance of a power feeder line are usually chosen at 50 ft. A larger
coupling loop can match an input impedance of the cavity to a feeder
line of a 50 U characteristic impedance, (see Sec. 4.7) In Figs. 4.9
(a) and (b) are shown the magnetic field distributions when a coupling
loop is replaced to the one for 50 f2 matching. The end tuners are kept
at the positions adjusted with a weak coupling. The field is strongest
in the chamber with the coupling loop and weakest in the opposite chamber.
However, the azimuthal symmetry has been recovered by a slight end tuning,
as shown in Fig. 4.10.
The magnetic field strength in the chamber is proportional to the
electric one in the vane gap. Figure 4.11 shows the magnetic field
- 2 2 -
distribution along the length of the cavity in one chamber and the
electric one in the vane gap it has. It is found that their field
distributions are almost similar. The symmetry of the magnetic fields
in the four chambers produces the quadrupole symmetry of the electric
field in the acceleration bore. The electric field in the bore is
measured as described below and the quadrupole symmetry is obtained.
The pole-tip shape is approximated with a circular arc, as described
in Sec. 4.1. The measured electric field distributions in the bore are
shown in Figs. 4.12 and 13. A solid line in Fig. 4.13 indicates the
expected values, obtained by calculating the square root of a relative
electric energy included in a spherical region of 3 mm radius in an
exact quadrupole field. In the region within the bore radius, the
sufficient quadrupole field is obtained with the pole tip approximated
with the circular arc. Quadrupole fields predicted by SUPERFISH are
shown in Fig. 4.14. A broken line expresses a quadrupole field with a
pole-tip shape of a hyperbola and a solid line with one of a circular
4.3. Field distribution with the modulated vanes.
The pole-tip modulation produces a longitudinal accelerating field
in addition to a transverse focusing field. The desired electric field
components are given by Appendix D.I. Only axial field exists on the
beam axis, and its strength changes sinusoidally in a unit cell.
- 2 3 -
Figures 4.15 and 16 show the magnetic field distributions without and
with the end tuning, respectively. An asymmetry of the fields after the
end tuning is within 2.5 %. The resonant frequency reduces to 293.5 MHz,
which means that the capacitance between the vanes is 1.7 % larger than
that with the straight vanes.
The electric field distribution on the beam axis is shown in Fig.
4.17. Their values are measured along the axis at the positions, kz = nir/2,
where the pole tips have an exact quadrupole symmetry and the field on
the axis is strongest. The detailed field distributions in the cell are
shown in Figs. 4.18 and 19. Solid lines in Fig. 4.19 indicate the
expected value with a computer calculation and a sine curve giving an
axial field distribution on the beam axis, respectively. At kz = nir/2
the electric energy perturbed by the used bead includes not only the
axial component but the transverse of 17 % for this vane geometry. It
is found that the desired electric field exist near the beam axis. In
Fig. 4.20 is shown the measured electric field distribution at kz = nir
along the radius of the cavity. The measured values agree with the
expected ones within the minimum aperture radius.
4.4. End tuning to obtain the desired field distribution
The field distributions described in Sees. 4.2 and 4.3 are obtained
in the cavity having the end space illustrated in Fig. 4.21. The end
space area is 86 % of a half cross section of one chamber. The end tuners
are set about 3 mm from the vanes to produce the azimuthal symmetry and
- 2 4 -
longitudinal uniformity of the field distribution. It is possible to
increase the end inductance t without a wide change of the end capacitance
Cg by a suitable cutting of the vane extenders. An increase of L results
in expansion of the gap between the end tuner and the vane.
Eight vane extenders are cut, as indicated by a broken line in Fig.
4.21. The end spaces increase by 12 %. An azimuthal and longitudinal
uniformity of the field distribution has been obtained with only two end
tuners set 6.5 mm to a vane (Fig. 4.22). Relation between the resonant
frequency and the end gap is shown in Fig. 4.23. Eight end gaps have the
same distances. With the end gaps larger than about 8 mm, the resonant
frequency or gap capacitance depends less sharply on the gap distance.
A change of the end capacitance gives the same effect as that of the
end inductance to the field distribution. With disks attached on the end
walls, the. end capacitance can be increased with less decrease of the in-
ductance (Fig. 4.24). The resonant frequency vs. the gap distance with
disks is shown in Fig. 4.25.
It is possible to generate various field distributions with end
tuning. A shorter end tuner gap on only one side raises the vane potential
on the' side. Figure 4.26 shows the magnetic field distribution measured
with such condition. The electric field distribution along the beam axis
is shown in Fig. 4.27. The notches represent the variation of field
strength owing to the vane modulation. In Fig. 4.28 is shown the
magnetic field distribution when the end tuners on both sides are placed
nearer to the vanes. The propagation constant y becomes real and the
-25 -
longitudinal field distribution is bowed.
4.5. Field distribution with a displaced vane
Setting errors of the vanes occurs in the process of a fabricaion.
A displacements of the vanes from their proper positions introduce mul-
tipoles, which is specified in ref. 4. On the other hand, the gap errors
between the vanes produce a large asymmetry of the azimuthal field distri-
bution. The gap error within a certain extent can be compensated by the
end tuners.
A vane gap is shortest at a position where the pole tips have an
exact quadrupole symmetry. The minimum gap for this model cavity is 11.83
mm. The setting errors of the modulated vanes are listed in Table 1
together with those of the straight vanes. The magnetic field distribution
in the cavity having the modulated vanes without end tuning was shown in
Fig. 4.15. The asymmetry was within 12 %.
One vertical modulated vane is horizontally displaced to examine a
field disturbance due to a larger vane gap error. The gap errors are
lested in Table 1. The maximum gap error averaged over the vane length is
-0.18 mm. The measured magnetic field is strongest in the chamber with
the largest vane gap and weakest with the smallest one (Fig. 4.29). The
asymmetry is about 2 times larger than that with smaller errors shown in
Fig. 4.15. The field distribution after end tuning are shown in Fig. 4.30.
The field asymmetry due to the vane gap errors was also calculated
by SUPERFISH. The dimension of the straight vane described in Sec. 4.1
- 2 6 -
is used in the calculation. One vertical vane is horizontally displaced.
The vane gap error is ± 0.21 mm. The magnetic field in the cavity has
about the same distribution with the measured one, however, the asymmetry
is ± 5 % (Fig. A.31). While in the actual cavity the four chambers are
coupled both capacitively and inductively, in the SUPERFISH calculation
the chambers are coupled only capacitively, because the RFQ cavity is
approximated to a part of a large torus and magnetic fluxes do not run
round the vane. Therefore, the calculated asymmetry is smaller than the
measured one.
4.6 Mode separation
The four vane cavity has various resonance modes besides the TE210 mode
which is used for the focusing and acceleration in an RFQ. In Figs. 4.32 a nd
33 are shown the resonant frequencies of various modes for the straight and
modulated vanes. The modes have been identified by comparing the phases
azicnuuhally and the amplitudes longitudinally as to the magnetic fields in
the four chambers with pick up loops. As for a TElln mode the magnetic flux
is dominant in a pair of diagonally opposing chambers and faint in the other
pair. The TElln modes have two resonant frequencies corresponding to two
choices of the pair. The end tuners have been set to give a longitudinally
and azimuthally uniform field for the TE210 mode.
For both sets of the vanes TE110 modes have the nearest resonant fre-
quencies to those of the TE210 modes. In the case of the straight vane
the measured resonant frequencies are 296.0 MHz for TE210, 294.0 and 293.2
- 2 7 -
MHz for TE110 modes. The predicted values by SUPERFISH are 296.5 MHz for
TE210 and 285.7 MHz for TE110 modes (see Chap.5). The measured f-pequencies
for the TE110 are 8 MHz higher than the calculated one, and the measured
difference of 2 MHz between the TE210 and TE110 is considerably smaller
than the calculated difference of 10 MHz. This is explained as follows:
The phase advance for TE11 mode in the end space, which is tuned to give
a phase advance of ir/2 for TE21 mode with a higher cut off frequency,
is smaller than ir/2. Then the resonant frequency for TE110 mode becomes
higher than the cut off frequency so that the phase advance in the vane
compensates the shortage of the advance, in the end space.
In the case of the modulated vane the measured frequencies are 293.5
MHz for TE210, 289.6 and 288.7 MHz for TE110 modes. The difference with
the modulated vanes is larger than that for the straight vane. On this
model cavity the resonant frequency difference is big enough to separate
easily. In the case of an actual RFQ linac, however, the mode separation
will be closer due to its smaller aperture. It is desirable to have a
larger mode separation to get a stable accelerating condition.
In order to get the larger mode separation, eight plates have been
placed trially on each zero potential surface for TE210 mode on the end
walls, so that they give little effect on the TE210 mode and interrupt
the magnetic flux for the TE110 modes. The resonant frequencies have been
measured with the mode separation plates for the modulated vane. The
differences between the TE210 and 110 are shown in Fig. 4.34. With the
closer plate position to the axis, the bigger frequency difference has
been obtained. The electric field distribution near the vane end is shown
- 2 8 -
in Fig. 4.35 with the plates placed closest to the axis. No significant
perturbation to the field has not been observed.
4.7. The Q value and rf coupling
The Q value has been measured for the cavity with the modulated vanes.
A block diagram of the measurement is shown in Fig. 4.36. From a signal
generator an rf signal is fed through a circulator to the cavity in pulse
with a single loop coupler. The reflected power is dissipated in a 50 fi
resistor. The decay time constants of the stored energy have been measured
for various rotation angles of the coupler, and the loaded Q value have
been obtained as
Q = UT ,
where u is the angular frequency and x is the interval during which the
stored energy decreases to 1/e. In Fig. 4.37 are shown the loaded Q
values versus effective loop area A. cos8. . The measurement has been done
2 2
with two loop areas of 25 and 1 cm . With a loop of 1 cm the decay time
constant scarcely depends on the rotation angle. The unloaded Q value of
the cavity, Q , has been determined from the decay constant x for an
infinitesimal coupling. The obtained Q values agree with ones obtained
from resonance curves.
The measured Q is 3400, which is 44 % of the ideal value.
By the use of a model shown in Fig. 4.38 the Q value is calculated as follows:
- 2 9 -
Qcal
where p. and p9 are specific resistivities of the materials of the vane
and tank respectively, D is the inner diameter, w is the vane thickness,
M is the permeability for vacuum and f is the operating frequency. The
-8following values of the specific resistivities are used: p^ = 1.72 x 10 ft'i
_Q
(copper), Py = 7.08 x 10 fi'm (aluminum alloy).
In the actual model cavity the power loss increases owing to imperfect
contacts between the vane and the tank, the holes on the side wall, the
current in the end parts, surface oxidation, and so on, which are neglected
in the calculation. Considering these conditions the obtained Q value is
satisfactory.2
With an effective coupler area of 23 cm , the loaded Q value is half
the unloaded value, QQ and the reflected power is the minimum. It has been
observed with a network analyzer that the input impedance of the cavity is
a 50 Q at the coupling. An equivalent circuit analysis gives the coupler
area which matches the cavity to a feeder line with a characteristic
impedance R
2A 2__1_ Lj l+(Rc/a%)
S^ ~Q (L /4) (R,,/wLi) , (4.3)o o *-
where A and S are the area of the coupling loop and the one of a quarter
cross section of the cavity, L is the inductance of the one chamber and
- 3 0 -
L. is the self inductance of the loop (see Appendix C). With Q = 3400,
2 8H 7S = 85.5cm , L. = y b (Jin T-) where the wire radius a = 0.5 nun
and the loop radius b= 28 mm, the effective loop area A is calculated to
2be 27 cm . The experimental loop area agrees with the calculated value.
4.8 Field penetration into a hole and the slots on the cavity wall.
On the wall of RFQ tank the various holes and slots are made for RF
coupler, rf monitor pumping port, and viewing port. The penetration of RF
field into the holes and slots have not a little effects on the resonant
frequency and the Q-value of the accelerating cavity. The dimensions of
these holes and slots must be determined not only according to their
purposes but also considering the effect due to field penetration. In
this section, the field penetration has been investigated, in order to
estimate the change of the resonant frequency by the hole or slot.
The field penetration into a hole and two slots which are made on the
cylindrical wall of the RFQ tank as shown in Fig. 4.39, is measured by
perturbation method. The measurement method is explained in Fig. 4.40.
The perturbing plunger is moved through a series of positions at which the
resonant frequencies are measured. The experimental results are shown in
Fig. 4.41.
The experimental results are explained as follows. A resonant fre-
quency shift Af is given by the general expression.
y-= |j/(yH 2- £E2)d5 (4.4)
-31 -
where F is a constant which depends upon the shape of the perturbing
volume, u and e are the permeability and the dielectric constant of the
vacuum , C is the volume of the perturbing object and U is the average
energy stored in the cavity. In the rectangular wave guide and the
circular wave guide, the strength of an rf with a lower frequency than the
cut-off one of dominant mode decreases exponentially along the wave guide.
Equation (4.4) is rewrited as follows;
f- = JL / (pH2-£E2)dSdz= ^f (Ae-kz)2dz, (4.5)
where A is obtained by integration over the cross section S of the
plunger. The resonant frequency f at z= 0 is given by
fl " £o _ I^.i_ (4 6)
f 4U 2k ' ( }
o
and f„ at z = d is given by
FA 2e~ 2 k d
4u—a-• (4<7)
where f is the resonant frequency at z = <». When f. is considered to
be the proper resonant frequency, the resonant frequency changes by
pulling out the plunger as follows,
fl - £2 „ £1 - £2 _ FA2 1f f, 4U 2kvo 1
Therefore, the equation which explaines the experimental result, is given by
- 3 2 -
(4 9)
Since the dominant mode is TE11 in the case of hole, the propagation
constant is given by
(4.10)
where the radius of hole a is 2.0 cm, the frequency f is 293 MHz, the
light velocity c i, 3 x l O 1 0 cm/sec. The coefficient of eq.(4.9) is
_3obtained from the experimental value, Af/f=10.4xl0 at d = °°. The
experimental result is explained by equation
/Af= 10.4xl0-3(l _
In the cases of the slots, .which can be approximated by rectangular
waveguides, the experimental results have been calculated using domenant
mode TE10.
As described above, the experimental results are well explained
by the s mple theoretical consideration. Therefore, t. e resonant frequency
shift due to the field penetration can be estimated, if the direction
and relative strength of the field on the wall are known.
- 3 3 -
Fig. 4.1. Cold model cavity II.
Fig. 4.2. Inside view of the cavity.
- 3 4 -
f/,
(UNIT CELL)
0A = 60 mm
a = 10 mm
m = 2
8 = 30°
r Q = 1 4 . 2 8 mm
Fig. 4.3. Vane geometry.
. 9 -l-ft .U40
VJLH-
S i d e F 1 2 3 4 5 6 7 8 9 S i d e B
H o l e n o .
Fig. 4.6. Positions of holes for the magnetic field measurement.
- 3 5 -
Fig. 4.4. Machining with NC-milling machine.
Fig. 4.5. After polishing with sand paper
and buff. Surface roughness <Q.8 pm
- 3 6 -
ICO
Feeder Pick up
iError
End tuners1-2 2-3 3-4 4-1
F 20 20 20 20 mmgap
B a 20 20 20 20 mm
f = 299.49 MHz
Pick up
Feeder
Side F 1 4 5 6
Hola no.
9 Side B
Fig. 4.7. Magnetic field distribution in the cavity
with the straight vanes before the end tuning,
(with weak coupling).
8.8
7.6
Feeder Pick up
I Error
End tuners1-2 2-3 3-4 4-1
F 2.7 2.7 3.2 3.5 mm
B 2.7 2.7 3.0 3.5 mmgap
f = 296.02 MHz
Pick up
Feeder
Side F 1 4 5 6
Hole no.
9 Side B
Fig. 4.8. Magnetic field distribution after
th end tuning, (with weak coupling)
7 h
Side F 1
Feeder Pick up
(~4~l"3/
I Error
-
i l i
Fgap
Bgap
X
Pick up
1-2
2.7
2.7
o
X
—o—
fFeeder
I I i
End2-3
2.72.7
296
1
tuners3-4 4-1
3.2 3.5 ran
3.0 3.5 irni
11 MHz
•
X
• - «
i i
3 4 5 6
Hole no.
9 Side B
Feeder Pick up End tuners1-2 2-3 3-4 4-1
Fg a p 2.7 2.7 3.2 3.5 ranBgap 2 - ? 2*7 3 - 0 3 - 5 n i n
fQ= 296.09 MHz
Feeder
(a) The rf power i s fed to chamber 1.
Side F 1 2 3 4 5 6 7 8 9 Side B
Hole no.
(b) The rf power is fed to chamber 4.
Fig. 4.9. Magnetic f ield dis t r ibut ions in the cavity,
with the s t ra ight vanes, matched to the
feeder l i ne .
8.8
Feeder Pick up
End tuners1-2 2-3 3-4 4-1
F 2.2 2.6 3.1 2.85 mm
B 2.4 2.6 2.9 3.05 mm
fQ= 295.87 MHz
Pick up
Feeder
Side F 1 4 5 6
Hole no.
8 9 Side B
Fig. 4.10. Magnetic field distribution in the cavity
with the straight vanes matched to the
feeder line after the end tuning.
8 -
Perturbator for mag,
J Error
Mag. field distribution
_ o oo — —
Elec. field distribution
Perturbatorfor elec.
r9 18 27 36 45 54 63 72 81 90
z (cm)
Fig. 4.11. Magnetic and electric field
distributions along the length
of the cavity.
H
?•H+Jid
20 mm
Perturbator
Fig. 4.12. Electric field distribution near the center of
the cavity with the streight vanes.
- 4 0 -
relative value, E
wo(D
cr t3 *
r t3 *(D
COr t
01H-
0Q3"r t
^
3n>Ul
>—*(DO
11
O
H iH-(DH*O.
p .
COr ti-lH-CJ*Cr tH-o3H-
r t3*(D
oO M-16
-8
-i
--> o33
CO
en
•
- F"
o o o -t » <35 CD O
-
; Expected
X. '
oula
r tn>o.SIH-r t3"
cy>
naw
cn
notri
oHiH-(D
O.
O-H-COrtri
Cr tH*
rt3*(0o*on(H
Electric field strength (arb.)H M W
W 1 1
H- H- \rt rr \
y- p- v
^ a \t) r( • \(DO .o o>
•
. ;C
ircu
lar
po
le tip
/
\ ; H
yp
erbo
la jr
10
9
8
7
Feeder\ ^
\ 4
V a
-
-
1 1
Pick up
13 J Fo y * a a DB
gap
^ — * — *
==— fPick
I i r
1-2
20
20
End2-3
20
20
tuners3-4 4-1
20
20
fo= 296.85
upT
Feeder
i I
20
20
MHZ
i
mm
mm
l
Side F 1 9 Side B
Hole no.
Fig. 4.15. Magnetic field distribution in the cavity with
the modulated vanes before the end tuning,
(with weak coupling)
8.8 t-
X 8.4O
•H 8.0
7.6
Side F 1
Feeder\
AT\Ti"\ a 1 o
iError
-A a'-» T"
1 I
Pick
2r\3~7y
1
up
FgapBgap
Pick
End
1-2 2-3
3.0 3.7
3.0 3.8
s ^up |
Feeder
1 i I
tuners
3-4
2.8
3.0
293
4-1
4.0
4.0
.53
I I
. mm
mm
MHz
^ *
I9 Side B
Hole no.
Fig. 4.16. Magnetic field distribution In the cavity witli
the modulated vanes after the end tuning,
(with weak couping)
vane
,Perturbator-••beam axis
7/ vane
Fig. 4.18. Electric field distribution on the beam axis in cells.
- 4 3 -
3 -
H 2 -
l< i -
o L.
Hill
II r '-• I "
UfHl
r ' i
mil
i i
Hill
' I I ,0 10.. 20 30 40 50 60 70 80 90 100
z (cm)
Fig. 4.17. Electric field distribution on the beam axis.
1.0
5•H
0.5
Transeversal component
.of electric energy
,17 X
Expectedvalue
JPerturbator
Beam axis (kz)
Fig. 4.19. Electric field distribution on the beam
axis in a cell.
- 44 -
lue,
>
>
rela
ti1.0
0.8
0.6
0.4
0.2
n
-
•
-16
; expected value
\ /
-8 0 8r (mm)
-
-
_
-
16
Fig. 4.20. Electric field distribution along the radius
of the cavity at kz=mr.
- 4 5 -
Fig. 4.21. End space of the cavity II.
7.6
Feeder Pick up
Error
End tuners
1-2 2-3 3-4 4-1
F 20 20 20 6.5 mmgapBgap 20 20 20 6.5 mm
fQ= 293.46 MHz
1 I I I I I I I IS i d e F l 2 3 4 5 6 7 8 9 S i d e B
H o l e n o .
Fig. 4.22. Magnetic field distribution after the vane
extenders are cut. (with weak couping)
- 4 6 -
Disk
Fig. 4.24. Disk attached on the end wall
to increase the capacitance
between the vanes and end wall.
294
uc;u
I 290H
m5» 236
-
• /
/
I End tuner
.1J
/
— .
\
29D -
0 5 10 15 20
End tuner gap x (mm)
Fig. A.23. Resonant frequency vs. the
end tuner gap.
5 10 15 20
End gap x (ram)
Fig. 4.25. Resonant frequency vs. the end
disk gap.
-47 -
10
Feeder Pick up
Side P i 2 3
End tuners
FgapBgap 2-4 3-3
f = 289.92 MHz
1 1 I I I4 5 6
Hole no.7 8 9 Side B
Fig. 4.26. Magnetic field distribution with a
shorter end tuner gap on one side.
i i i i i . . . . ] . i
40 60 80 100
Z(cm)
Fig. 4.27. Electric field distribution on the
beam axis corresponding to the magnetic
field distribution shown in Fig. 4.26.
9
8
7
Feeder
V 4 1\ A 1
-
i i
Pick up
" ^
JjS Fgapgapfo=
1-2
1.1
1.1
287
_ — — •
o ^
*
End
2-3
1.1
1.1
buners
3-4
1.0
1.0
18 MHz
1 TPick up
l i
i
Feeder
1 1
€-
i
4-1
1.01.0
/
1
mm
mm
s
S i d e F 1 2 3 4 5 6 7 8 9 S i d e B
H o l e n o .
Fig. 4.28. Magnetic field distribution with shorter
end tuner gaps on both sides.
Relative field strength
Hl H2 H3 H41.08 1.00 0.99 1.07
Fig. 4.31- Calculated magnetic field strengths in
four chambers with a displaced vane.
- 4 9 -
11
10
Feeder Pick up
End tuners1-2 2-3 3-4 4-1
F 20 20 20 20 mm
B 20 20 20 20 ran
f = 296.9 MHZ
Pick up
Feeder
I I l i i8 9 Side BSide F 1 2 3 4 5 6
Hole no.
Fig. 4.29. Magnetic field distribution when a
vane is displaced. Gap error of
chamber 1 is -0.18 mm. (with weak
coupling)
10
9
8
7
6
Feeder
(\ -»
V 4 1
-
— ¥ — - •
-
Pick up_ /
VP
0
-• 1 : —
TPick up
1-2
3.0
3.0
V-~.
t
End2-3
3.0
3.0
tuners3-4 4-1
3.0 2.0
3.0 1.8
293.0
-dfe
Feeder
1 1 I I I
_ - — • — •
MHZ
, -
mn
nfn
a.
5T"
1Side F 1 4 5 6
Hole no.
9 Side B
Fig. 4.30. Magnetic field distribution after
the end tuning. (with weak coupling)
- t s -
00
•E-
r t
(-f* j *ID
COr tH
00*
r t
<
3IDDl
IDCOO31119r t
ht
(t
(DPOH1
(DCO
OH i
r t
ID
r>oI"1
a.i(D
Resonant frequency (MHz)IOUlO
UlOO
U>Ul
ooo
g
o ©
io to tokD VO <£>Ul il^ cri
M
3BO
1-3
ai ^Mto
0)H.H-
rt
OHi
rt
g"Hi
O U> Crt
Ul O lo
Resonant frequency (MHz)
ID
Ulu>
to(Dto
o
§OH-IDCO
OHi
rt
o
soa<D
roLno
U)oo
P>
o
u>
o
.6.oo
I
K) tOtO
00 VO U)
• J CT\ Ul
-
(
Ul Ul Ulto to uiCT\ 00 O
Ul Ul to
I
Q
TEllnB
TEllnA
1
^©
1. .
w
TE21n
o
c<
0
01
to Hi
the flua
o oUl -JM 00
\
7J 294
s
u0)D
+1
O
292
290
2S8
286
284
TE210
TEllOA
TEllOB
I
61 41 21r (mm)
Without i n t . vanes
Interstitial vane
Fig. 4,34. Mode separation with interst i t ial vanes.
- 5 2 -
X
I ok
20
; With int. vanes
; Without
Perturbator
40 60 z(mm)
r
r= 21 mm
Interstitial "vane
Vane
Fig. 4.35.. Electric field distribution near
the end of the cavity with the
interstitial vanes.
-S3 -
50 a
Fig. 4.36. Measurement system of the Q value. The
pictures show a decay of the stored energy
of the TE210 mode (293.5 MHz) with the
modulated vanes.
- 5 4 -
4000,
3000
a)2000
lOOO
Coupl ing l o o po ; A-i = 1 cm2
With a matching to
the 50n feeder line
1 110 15
A. cos 8.
20
(cm2)
25
Fig. 4.37. Apparent Q value vs . coupling loop area
with the straight vanes.
Fig. 4.38. Model for calculationof the Q value.
R=0.25 an
3.75 cm
TANK AXIS 0.5 an
Fig. 4.39. A hole and two slots used in the
measurement of field penetration
RFQ TANK WALL
PERTURBING PLUNGER
Fig. 4.40. Cross section of RFQ tank,hole and
plunger which illustrates perturbation method
-56 -
10 15 20 25 30
POSITION OF PERTURBING PLUNGER d (mm)
35
Fig. 4.41. Experimental results of the fieldpenetration
into the hole and the two slots. The results
A, B and C are corresponding to the cases of
A, B and C shown in Figs. 4.39. The solid lines
represent the calculation curves.
-57 -
Table 1. Vane gap errors
Pin
PI -
P5 -
PI -
P5 -
P3 -
P7 -
P2 -
P6 -
P4 -
P8 -
no.
P3
P7
P5
F3
P7
PI
P6
P4
P8
P2
Straight
Side F
+ 0
+ 0
- 0
- 0
+ 0
+ 0
+ 0
0
+ 0
- 0
.02
.06
.03
.02
.09
.02
.06
.00
.10
.06
Gap
. vanes
Side B
- 0
+ 0
+ 0
- 0
- 0
+ 0
- 0
- 0
+ 0
+ 0
.09
.05
.06
.09
.03
.02
.04
.10
.05
.04
errors [mm)
Modulated vanes
Side F
+ 0
- 0
- 0
- 0
+ 0
- 0
- 0
- 0
- 0
+ 0
.07
.05
.03
.05
.02
.03
.01
.05
.03
.04
Side B
+ 0.08
- 0.04
+ 0.02
+ 0.01
- 0.04
+ 0.05
- 0.02
- 0.02
- 0.01
+ 0.10
With
Side
+ 0.
- 0.
+ 0.
- 0.
+ 0.
0.
—
—
—
—
displaced vane
F
07
03
05
27
24
00
Side B
+ 0.04
- 0.02
0.00
+ 0.07
- 0.09
+ 0.04
—
— -
—
Side F Side B
- 58 -
5. Calculation with SUPERFISH
Field distribution in an RFQ cavity has been estimated with a well
known computer code SUPERFISH. ^ The code can be applied only to solve
axisymmetric fields, whereas TE modes which are excited in an RFQ cavity
have no such symmetry. To apply SUPERFISH to those modes, we regard the
straight cavity as a small part of a large torus, where a magnetic flux
makes a closed loop around the symmetry axis and an axisymmetric field is
generated. As SUPERFISH requires cylindrical symmetry, it cannot be
applied to a cavity with modulated vanes.
The relation between resonant frequency f and the radius of the
large torus L is shown in Fig. 5.1 in the case of the cold model I, The
figure shows the convergence of f on accordance with increasing cf the
rsL'->, L/R, where the radius of the cavity R is 9.5 cm. In the calcu-
lations, the ratio L/R is set at 1000, which gives a sufficiently azi-
muthal symmetry of a field distribution.
In Fig. 5.2 and 5.3 are shown electric field lines for the TE210
and TE110 modes calculated on the full cross section of the cold model
II. For the TE210 mode it is enough to calculate on one eighth of the
cross section owing to its field symmetry. On the other hand it should
be calculated at least on a quarter cross section for the TE110 mode.
The mode appeared in Fig. 5.3 as the combination of the two TE110 modes
shown in Figs. 5.4(a) and (b) and there is no difference in f among
these three modes. The calculated resonant frequencies are 285.7 MHz
for the TE110 and 296.5 MHz for the TE210. The difference is 10.8 MHz.
The calculations have beenexecuted under the following boundary conditions.
- 5 9 -
The fine lines in the figures represent a surface of ideal conductor.
Electric field lines are normal to the fine lines and entirely tangien-
tial to the bold lines.
Figure 5.5 shows f is inversely proportional to the cavity radius
R with the other parameters kept constant as the bore radius r = 0.5 cm,
taper angle 6 = 15° and vane thickness W= h cm. The geometry of the vane
is similar to that in Fig. 4.3. With a cavity radius of 12.9 cm and
a taper angle of 30° fixed, f has been computed for various bore radii
r .(Fig. 5.6) Then the capacitance C between the vanes has been cal-
culated with obtained f and eq. 2.2. Calculated C is the total of the
four vane gaps. In Fig. 5.7 is shown the dependence of C on the taper
angle 8 of the vane. As taper angle goes close to zero, the inductance
increases more than the capacitance decreases, then the resonant frequency
becomes lower.
An ideal quadrupole field is generated with a vane cross section
of a hyperbola, whereas an adequate electric quadrupole field is obtained
in the useful aperture by approximating the vane tip to a circle with
the radius of the curvature at the vane top as in Fig. 4.3. Figure
5.8 shows the calculated electric field strength along the x axis defined
in the figure and the relative deviation of the electric field gradient
in the case of circular approximation. The field gradient can be ex-
panded in a series of the powers of (x/r ) as;
- 6 0 -
By fitting the calculated gradients to the series,
the values of A. listed in Table 2 have been obtained.
With modulated vanes are installed, the distance
between adjacent vanes takes the minimum value at
kz=ir/2, where the cross section has a quadrupole
symmetry discussed above. The surface field in the
transverse plane becomes strongest at the point a
little near the vane tip than the point of minimum
intervane distance and the value is 1.36 times larger
than that on the tip, as is represented in Fig. 5.9 for
less than 30°.
Table 2
A± -0.00-30
A2 0.05995
A3 -0.02437
A 4 -0.74629
A5 0.17953
A, 4.66634b
A ? -0.37027
Ao -9.40483o
Ag 0.22121
A 1 Q 6.78471
the taper angle
500
I450 •'
400
Fig. 5.1. Dependence of the resonant
frequency on the radius of
the torus.
-61 -
Fig. 5.2. Electric field lines Fig. 5.3.
for the TE210 mode.
fQ = 296.5 MHz.
Electric field lines
for the TE110 mode.
f = 285.7 MHz.o
(b) fQ=285.7 MHz (a) fQ=285.7 MHz
Fig. 5.4. Electric field lines for
the two TE110 modes calculated
on the half cross section.
- 6 2 -
R(cm) 100
Fig. 5.5. Dependence of the resonant frequency
or the free space wave length on the
radius of the cavity.
(rQ= 0.5 cm, e= 15°.)
ro(cm)
100
1.5
Fig. 5.6. Dependence of the resonant
frequency and the capacitance
on r .
(R= 12.9 cm, 9= 30°)
280
Fig. 5.7.
15 9 (Degree)3090
Dependence of the resonant
frequency and capacitance
on the taper angle.
.(R=12.9 cm, r = 1.43 cm)
Fig. 5.8. Calculated electric field strength
and the relative deviation of the
field gradient of the cold model II
along the x axis at kz = 90°.
2.0
1.5Q.
1U
UJ1.0
0.5
1 1
9 = 15°0
0=30- 0=45°
' 1
/ ,fl^ if/*
" 0
^ • • - -
Fig.
10I (mm)
i5 20
5.9. Vane surface field of the cold model II
in the transverse plane at kz = 90°.
The chain line represents the point of
minimum intervane distance. The mesh
density along the vane surface is about
1 mesh/mm.
-65 -
6. Concluding remarks
The model study shows that a sufficient mode separation and field
uniformity can be obtained with a single loop coupler. The vane assembly
and electric contact method used at the second model cavity has given
a satisfactory mechanical reliability and reasonable Qvvalue. On the
basis of the model study, the cavity of the RFQ test linac 'LITL1 has
been designed and is under construction.
Acknowledgement
The authors would like to thank Prof. Y. Hirao for helpfu Jis-
cussions. They also thank Dr. N. Tokuda who read and criticized the
manuscript. They are much indebted to the members of the machine shop
at INS who manufactured the first model cavity. The second model cavity
was manufactured at Toshiba Corporation, Tsurumi Works.
References
1) Y. Hirao et al., "NUMATRON, PART II", INS-NUMA-5, 1977.
2) Y. Hirao, "NUMATRON Project", Proc. of the Int. Conf. on Nuclear
Structure, p. 594 (1977).
3) I. M. Kapchinskij and V. A. Teplyakov, Prib. Tekh. Eksp., No. 2,19
(1970).
4) K. R. Crandall et al., Proceedings of the 1979 Linear Accelerator
Conference, Montank, N. Y., 1979, p. 205.
- 66 -
5) N. Ueda et al., "An RFQ Linac for Heavy Ion Acceleration , Proceedings
of the 1981 Linear Accelerator Conference, Santa Fe, USA, 1981.
6) S. Yamada, "Buncher Section Optimization of Heavy lea RFQ Linacs",ibid.
7) N. Tokuda and S. Yamada, "New Formulation of the RFQ Radial Matching
Section",ibid..
8) J. M. Potter, et al., "Radio Frequency Quadrupole Acceleraing
Structure Research at Los Alamos", IEEE Trans, on Nucl. Sci., Vol.
NS-26, No. 3, 1979.
9) H. Lancaster and K. J. Kim, "An Analytical Solution for the Resonant
Frequency of a Simple Radio-Frequency Quadrupole Structure",
LBL-Medical Accelerator Note 13, 14 July 1981.
10) E. L. Ginzton, "Microwave Measurements", McGRAW-HILL BOOK COMPANY,
INC. NEW YORK, TORONTO, LONDON (1957).
11) K. Halbach and R. F. Holsinger, "SUPERFISH-A Computer Program for
Evaluation of RF Cavities with Cylindrical Symmetry", Particle
Accelerator vol. 7, p. 213, 1976.
- 67 -
Appendix A: Equivalent circuit analysis of a four vane cavity
The longitudinal field distribution in a four vane cavity is bowed
or tilted with unsuitable shapes of the end space, that is, the vane-
tip potential is changed along the beam axis. These phenomena are
explained by an equivalent circuit analysis.
We consider the distributed constants of an ideal four vane cavity
as shown in Fig. 2.3. As the ideal cavity has four identical chambers,
we consider a resonant condition of one chamber. The schematic of an
equivalent transmission line is shown in Fig. 2.3(bellow). This circuit
is resonator short-circuited at both ends. The equations for the voltage
and current on the transmission line are
V U ) = V. cosh yl + Z I. sinh yl, (A.I)
and
1(1) = I. cosh yl + (V./Z ) sinh yl, (A.2)
where V., I.; initial voltage and current,
Y ; propagation constant,
Z ; characteristic impedance.
The propagation constant Y and the characteristic impedance Z in the
end space are given by;
Y = Jw/L C , (A.3)e e e
and
Z = /L /C . (A.4)e e e
-68 -
Those in the vane gap are
Y v = /(L./Lo)(l-<AoCo) , (A.5)
and
Zy = (1)/(-LiLo)/(l-u)2LoCo) . (A. 6)
This circuit is divided into two sections to obtain the resonant condition.
When the input impedances of two sections are respectively expressed by
Z^ and Z^, resonance occurs when the following relation is satisfied;
(A.7)
The input impedances Z^ and Z. shown in Fig. 2.3 are obtained from eqs.
(A.I) (A.6). The initial voltage V of a line short-circuited at the
far end is zero, therefore,
Zl = V ( V / 1 ( V = Ze tanh
To obtain Z^ we use V.=VUJ and I.=I(2,e> ;
z2 =
) coshY H + Z I(£ ) sinh y i.e v ~v v e v v
/Z ) sinh Y
+ Z tanh y I2 ^ ( A . 9 )
1 + (Z,/Z ) tanh y I1 v 'v v
-69 -
The vane-tip potential has a constant value along the beam axis when
the distributed constants and the resonant frequency satisfies the
following equations ;
2irf = 1//L C , (A.10)
2irf/L C -Si = TT/2 . (A.11)e e e
The distributed constants in the cold model II
We estimate the distributed constants in the cold model II with
the modulated vanes. The measured resonant frequencies f of the TE21n
modes are as follows ;
£1 = 293.5 (MHz) (or n=0 on the vane) ,
f2 = 326.5 (MHz).
For C , we use eq.(2.2);
C Q = 27.5 (pF/m).
L satisfies eq.(A.lO);
L =10.7 (nH-m)^o
As the calculations of L and C are difficulc, we give a qualitative
analysis here. To estimate roughly L , we express it as a ratio of an
end space area S to a vane thickness w,
L = -e w a
e
- 70 -
For this cavity, S =4.13xlO~3 m2, W=3.50x10 2 m and I =0.135 m.
Then,
Lg = 1.10 (viH/m) .
C satisfies eq.(A.ll), then
Ce = 36.2 (pF/m) .
We can obtain^L. using the relation of eq.(A.7) with the above values
and f=326.5 MHz ;
L± = 0.34 (uH/m) .
Those calculated for the straight vanes and the cold model I are 0.32
and 0.42 pH/m, respectively.
We can calculate the valtage distribution on the resonant circuit
using eqs.(A.2) and (A.2) with the above distributed constants, and
three examples are shown in Fig. A.l(aH(c). The resonant frequency is
given as a function of C or L in Fig. A.2.
-71 -
-ZL-
Relative value
H i0h
(11
3"0>H3o3H-noo3•ao3(D
r t
, .
o•
o3*(U3
iQCD
0H i
o
„—,
(1)J3p j
o3r t3"(D
hato03(li3r t
0H-HOCH-r f
33H-r t3*
n0)
oc£)r t(D
a
or t
ft)
CLH-Ulr th(H-trcr tH-03(n
Ul ±_
nn4.oCO
n(0
— '
toV£)- J
Relative valueO O O O H
KJ jk m oo o
Ul
O
Ul
^^ a^ ^ ^ to
<Z °
— - — - ^ ~
Hi0
IItotv>O\
SsN
Relative valueO - O O O
to **. on oo
COin ;
ui
n0
i(—•w
(1
_ _ — •
H iOIIto0000
S \
a \N
\
CO
I
N03
-310
-305
-300
-295
0.5 0.6 0.7 0.8 0.9 292
287
282
277
VLe
oo i o»T
kL'kCJ I
1.2 1.3 1.4 1.5 1.6
Fig. A.2. Calculated resonant frequency as a function of Cg or
Appendix B: The field measurement system of the RFQ cavity
The field measurement system for the RFQ cavity consists of a mini-
computer system (HP-1000), HP-IB* devices, and a perturbator shifter,
as shown in Fig. B.I.
As the cavity is about 30 m far from the computer, a serial data
link with the HP3070A real time application terminal is used between
HP-IB network and the computer. Power given to a coupler is supplied
by a signal generator 8640B into which is put the attenuated voltage
( O'vl V ) from a D/A power supply programmer 59501A for frequency
modulation. The frequency modulation rate is 100 Hz per digit of the
power programmer, and the frequency is measured by a universal counter
5328A in seven significant figures. Output signal from a pick up is
rectified with a crystal detector and digitized by a digital voltmeter
3455A in six significant figures. An interface for the pulse motor
controller is a microcircuit 12566B which is 16 bits parallel DO/DI.
This link is 4 m long.
The field distribution of the cavity is measured as follows. The
perturbator is set at a position. The output voltage from the pick up
is digitized at each time when the frequency is changed by a 200 Hz step
over a 20 kHz range. The resonance curve is illustrated on the graphic
display as shown in Fig. B.2. The signal includes noise of ^0.01 % due
to instability of the measuring instruments. The resonant frequency is
determined by fitting the peak region of the curve to a parabola. Then
* Hewlett Packard Interface Bus
- 74-
the perturbator is shifted to the next position. This is one loop.
After the repetition of the above procedure, the field distributic 's
plotted on the display as shown in Fig. B.3. A loop takes about 100
seconds.
The measurement error including the fitting error of a loop was
decided as follows. The measurement of the resonant frequency was re-
peated 10 times in a short times under the same measurement condition.
The standard deviation was about 300 Hz for a resonant frequency of
300 MHz.
CPU
IIP-1000
/
Serial Data Link (30 m)
/i -/Microcircui
GraphicCOT ,
// Key Doard /
Motor /"c
Pulse riotor
. Driver
t {4 m)
Real Time Appli-
cation Terminal
D/A Power SupplyProgrannior
V
Signal
GeneratorI
IiP-IB
Digital
Voltmeter
1
Universal
Counter
A1
>, 1 Coupler Pick up
c
(£
Fig. B.I. Field mesurement system for the RFQ cavity.
- 75
Fig. B.2. Resonance curve on the display.
The frequency modulation rate is
10 kHz/V (horizontal axis).
Fig. B.3. Field distribution along the
beam axis on the disply.
- 7 6 -
Appendix C: An rf coupling loop for a four vane cavity
An equivalent circuit analysis gives the dimension of a coupling
loop which matches a four vane cavity to a feeder line in the following
way.
Using an equivalent circuit and symbols given in Fig. C.I, we have
i i I i + M-I), (C.I)
E = j(0(MvIi + L-I). (C.2)
Using Z=-E/I and eliminating E and I from eqs.(C.l) and (C.2), we obtain
Z. - E./I. = L . + J 2 V _ (c.3)
Upon putting
Z 1 = R± + jX± , (C.4)
and using
Z = r + 1/jcoC , (C.5)
then we obtain
4 2 2
R. = ^ H P j . (C.6)1 (1-WLC) +((orC)
1 (1 - a) LC) + (urC)
The circuit is matched to a feeder line with a characteristic impedance
- 7 7 -
R , when the following relations are satisfied.
(C8)
X ± - 0 . (C.9)
Equations (C.7) and (C.9) give
Q'k 4 Q k
2 \1 -k +-=-
2 X/2where Q = uL/r is the Q value of the resonant circuit and k= (M /L.L)
2 2is the coupling factor. Under the condition 1» k » 1/Q , which is satisfied
usually, eq.(C.lO) leads to
where
f = 1/2TT/LC" . (C.12)o
In Fig. C.2 is shown the shift of the resonant frequency vs. the coupling
factor. Upon considering that the measured resonant frequency becomes
higher with increasing the coupling, the branch with minus sign of the
sqare root is discarded. From eqs.(C6) and (CIO) we obtain
± = O)Li(Qk2/2)(l - A- (2/Qk2)2) . (C.13)
- 78 -
Equations (C.8) and (C.13) give the matching condition as follows;
* L = k2 = J 1 + ( R 2k
L.L k Qo (Rc /O)L.) • ( C - 1
The magnetic flux $ in a chamber generated by the current I on
the chamber wall i s given as
$ = Lo- IQ . (C.15)
Assume that the flux density is uniform in the chamber, then the flux
$ penetrating the coupling loop with an area A is obtained as
Then we obtain the mutual inductance as
M = LQA/AS . (C.17)
Since L=L Ik, eq.(C14) lead to
{S) " Q Q (Lo/4) (Rc/UL.) • ( C- 1 8 )
2In the case of the RFQ cold model II with S=85.5 cm and 1= 100 cm,
we have
L = u S/X, = 10.7 nH .o o
The used loop is an oval one made of copper wire of 1 mm thickness.
Let the loop be approximated to a circle which has the same area, then
the self inductance L. of the loop is obtained as
- 79 -
where a is the radius of the wire and b is the radius of the circle.
Put a =0.5 mm and b = 28 mm; then
L. = 153 nH .
For the resonant frequency of 296 MHz, 0)L. = 285 fi
The characteristic impedance of the feeder line is 50 Q, then
R /wL. = 0.175 .c l
Using the measured Q value 3400, from eq.(C18) we obtain
2A = 26.9 cm
The calculated area agrees satisfactorily with the experimental value
2of 23 era , considering the errors due to the approximations employed
in the calculation.
- 80 -
r I
Fig. C.I. Equivalent circuit for a four vane cavity.
L = u S/l, I; cavity length. L = L /4. C = 4C .
Z=-E/I= r + l/jioC. Z. = E /!.. I = 41 .
xlO
I"4-1
2Q,
k2= K2/(U)
Fig. C.2. Shift of the resonant frequency vs.
the coupling factor k.
-81 -
Appendix D: Formulae of RFQ Beam Dynamics*' **)
1. Electr ic Field Components
kAVEz = —T— Io(kr) sinkz ,
Er = - f j r c o s 2 f - ^
XVE i 2^
c o s k z
each multiplied by sin (iot +
k = 2 IT / gA
A y m2 - 1rn^Io (ka) + I o (mka)
X = 1 - AI0 (ka)
2. Energy Gain a Cell
AW = qE0S,T costj>s
= j qAV cos<jis
E = 2AV/BAo
3A/2
TT/4
: space-average longitudinal field
: unit c e l l length
: t rans i t time factor
3 . Focusing and Defocusing
_ TT2qVA s i n t|)A
A2 XV
ro= a/
: defocusing strength
: focusing strength
: characteristic average radius
* K. R. Crandall et al., Proceedings of the 1979 Linear Accelerator
Conference, Montank, N. Y., p. 205.
** S. W. Williems et al., ibid., p. 144.
- 82 -
4. Longitudinal Motion
o ir2qVAf2 sin U J n
a^ = —^ H 2 ^ — : angular frequency
sin $ — $tan 4- = —= —° 1 - cos $
$ - 3 I $ I : phase length
Z b = ,— : spatial length
5. Transversal Motion
flr2 = f2(A + -,j—7 ) : angular frequency
B2
E n : beam envelope
E n : normalized emittance/ir
A_ = flrya2/c : normalized acceptance/ir
6. Stability Criterion
7. Kilpatrick's Criterion
f = 1.643 x 104 E 2 e - ° - 0 8 5 / E
f : MHz
E : MV/cm
E s = KV/r0 : highest surface field that occurs
K = 1.36 at the point of pure qundrupole
symmetry.
3. Limiting Currents
Il= —™—b— : longitudinal limit_L<£(JA
rh : beam radius
f = r b / 3 b for 0.8 < b / r b < 5
b = ISX |4>s | / 2 f
- 8 3 -