Download higher order mode studies

ELECTROMAGNETIC DESIGN OPTIMIZATION STUDIES FOR
g=0.61, 650 MHz ELLIPTIC SUPERCONDUCTING RADIO
FREQUENCY CAVITY*
Arup Ratan Jana#, Vinit Kumar, and Rahul Gaur, Accelerator and Beam Physics Laboratory,
Materials and Accelerator Science Division,
Raja Ramanna Centre for Advanced Technology, Indore, India
calculations of wakefield and its effect on the beam.
Abstract
For the proposed Indian Spallation Neutron Source
(ISNS), an injector linac is being designed at RRCAT.
The injector linac will comprise of multi-cell
superconducting elliptic cavities to accelerate the H- beam
in the range 200 MeV – 1 GeV. Two families of such
cavities are proposed to be used – g=0.61 cavities to
accelerate the beam in the range 200 – 500 MeV, and
g=0.9 cavities to accelerate the beam in the range 500
MeV – 1 GeV. For the optimization of the geometry of
such cavities, we have recently developed a generalized
approach and used it for the optimization of g=0.9, 5-cell
cavities [1]. In this paper, we use the approach described
in Ref. 1 to optimize the geometry of g= 0.61, 5-cell
cavities to maximize the achievable accelerating gradient.
The geometry of the end cells of the cavity have been
especially optimized to facilitate the extraction of trapped
monopole higher order modes. For the dipole higher order
modes, we have performed a calculation to estimate the
threshold current for excitation of regenerative beam
break up instability. We discuss the details of these design
calculations in the paper.
CELL GEOMETRY OPTIMIZATION
For the cavity geometry, we have followed the typical
TESLA-type cavity shape [2], which has evolved over
several years to minimize the multipacting problems[2-3].
Fig. 1 shows the schematic of the half-cell where, the
geometry can be described by seven independent
parameters - the half-cell length L, iris ellipse radii a and
b, equator ellipse radii A and B, iris radius Riris and
equator radius Req. The wall angle α can be calculated as a
derived parameter from these seven independent
parameters. For the mid cells, we choose L = g/2, and
Req is tuned to achieve the resonant frequency as 650
MHz[1].
INTRODUCTION
To facilitate neutron based multidisciplinary research
in India, there is a plan to build an Indian Spallation
Neutron Source(ISNS)[1] where, for medium and high
energy range of the1GeV H- linear accelerator, two
classes of 650 MHz, 5-cell elliptical superconducting
radiofrequency(SRF) cavities will be used. In this paper,
we present the physics design of the multicell g = 0.61
650-MHz SRF cavity, which can be used to accelerate
particles in the medium energy section of the linear
accelerator. Here, first we present the electromagnetic
design studies for the optimization of the mid-cell
geometry, for which we follow a step by step one
dimensional optimization technique developed earlier for
the g = 0.9 cavities[1]. Next, we present the calculation
of R/Q for monopole higher order modes(HOMs) and
optimize the end cell geometry to ensure that there is no
trapped mode with significant value of R/Q. Then we
present an analysis of the dipole HOMs to estimate the
threshold limit for the beam current responsible for
regenerative the beam break-up(BBU) instability. Before
we conclude, briefly we present a discussion on the
Figure 1: Schematic of the half cell of an elliptic cavity.
In this electromagnetic design optimization, our target
is to achieve the maximum possible accelerating field Eacc
in the cavity. Here the two constraints i.e. the peak
surface magnetic field Bpk that leads to the breakdown in
the superconducting properties and the peak surface
electric field Epk that leads to field emission, limit the
maximum achievable value of Eacc[3-4.] Hence, the aim is
to minimize the value of Bpk/Eacc, keeping a satisfactory
value for Epk/Eacc. The peak magnetic field Bpk, developed
on the inner surface of the cavity wall must be less than
the critical magnetic field and for an SRF cavity operating
at 650 MHz, 70 mT can be taken as the value for the Bpk
as a safe margin[1]. Similarly, from the field emission
point of view, the maximum tolerable value of peak
electric field Epk on the inner surface of the cavity wall,
must be ≤ 40MV/m[1]. For our design, the target value of
Epk/Eacc≤ 2.355and Bpk/Eacc≤ 4.56 mT(MV/m) will ensure
an accelerating gradient ~ 15.5 MV/m.
Optimized Mid Cell Geometry
____________________________________________ ____________________________________________________________________
*Work supported by the Department of Atomic Energy,
#E-mail: [email protected]
To optimize the electromagnetic design of the mid cell,
electric field amplitude i.e. Eaxial along the cavity length
as shown in Fig. 2(a) and discover that the mode is
trapped inside the cavity. To solve this problem we do the
fine-tuning of the end cell geometry to make that
individually end cell resonates at 1653.2 MHz. In that
case, the end cell will act as a resonator for that particular
HOM, and, as shown in Fig. 2(b) the location of
maximum field amplitude will shift to the end cell. We
achieve this by changing the semi major axis A of that end
cell from 52.64 mm to 52.25 mm. At the same time, end
cell length is so tuned that the fundamental mode
frequency is fixed at 650 MHz. We finalize the end
halfcell length at 71.24 mm.
Table 1: RF parameters of the optimized mid-cell
geometry for the -mode operation.
On the other hand, for the dipole modes, the beam gets
a transverse kick and exchange energy with that dipole
and under suitable circumstances, an instability, known as
regenerative beam break up instability may build up if
the beam current is higher than a threshold current Ith. It is
therefore necessary to estimate the threshold current
corresponding to prominent dipole modes. Strength of a
dipole mode is given in terms of R⊥/Q, which has the
following expression[4]
RF Parameter
Frequency
Transit-time factor(T)
Qo
R/Qo
Epk
Bpk
Magnitude
650
0.710
> 1×1010
327.910
33.66
70.83
Unit
MHz.
Ω.
MV/m.
mT
HIGHER ORDER MODE STUDIES
In this analysis, we restrict ourselves to transverse
magnetic(TM) type monopole and dipole modes only. We
look at the HOMs having resonant frequency up to the
upper cut off frequency of ~2.0 GHz, which is decided by
the beam pipe diameter. The cavity shunt impedance R
has square dependency on the energy gain, hence has
strong dependence on We have calculated R/Q for
different monopole modes, as a function of β using
SLANS[6]. Only for the π mode at 649.99 MHz i.e. the
operating mode frequency and 4π/5 mode at 649.44 MHz,
the R/Q values are significant. R/Q for the π mode
dominates and shows a maximum of ~ 354  at β ~ 0.65.
However, the 4π/5 mode has higher value of R/Q
compared to the  mode, near the two end points of the
operating β range. This restricts the use of g = 0.61
cavities to β ranging from ~0.51 to ~0.76.
We notice that except the mode at 1653.20 MHz, all the
modes have their R/Q values less than 10  and, for the
mode at 1653.2 MHz, R/Q gradually increases with β and
approaches ~ 20  at β = 0.76. We check the axial
1.0
1.0
0.5
0.5
Eaxial (MV/m)
Eaxial (MV/m)
first we fix the half-cell length L equal to βgλ/2 and for
our case this is equal to 70.336 mm. We choose Riris = 44
mm, which basically is governed by the beam dynamics
considerations as well as by the requirement of the cellto-cell coupling kc. For the mid cell geometry, the ratio
a/b is optimized to obtain the minimum value of Epk/Eacc
in order to achieve the maximum Eacc, whereas the other
parameter A/B is mainly decided by mechanical
requirement like stiffness, rigidity etc. Following the step
by step optimization procedure[1], finally for the mid cell
geometry we obtain that Bpk/Eacc is minimum for A =
52.64 mm and B = 55.55 mm, and a/b = 0.53 respectively,
which corresponds to a = 15.28 mm and b = 28.83 mm
for = 880. Table-1 summarizes the corresponding RF
parameters for the TM010 like π mode. We have obtained
these values using the 2D code SUPERFISH[5].
The end half-cell needs to be tuned such that the cavity
with end cells resonate at the same frequency as that of
mid cells and the field flatness is maximized. Therefore,
we vary the end half-cell length Le, keeping all other
parameters fixed and find that for Le = 71.5495 mm, the
target is achieved. This needs to be further optimized to
ensure that there are no trapped higher order modes with
significant strength in the cavity, which we discuss in
next the section.
0.0
0.0
-0.5
-0.5
-1.0
-1.0
(a)
-60
-40
-20
0
20
40
(b)
-60
60
-40
-20
0
20
40
60
Lcavity (mm)
Lcavity (mm)
Figure 2: Axial electric field of the mode at 1653.20
MH. (a)Here, the field is trapped inside. (b) and in this
modified geometry there is no more field trapping.
n z
2
 E z  i c
zs  r   e dz
.
2
The above expression is same as 𝑘𝑛 times the expression
described in Ref [4] and is in agreement with the
commonly followed convention. Here, 𝑘𝑛 , 𝜔𝑛 and
𝑈𝑛 denote the wave vector, the angular frequency and the
stored energy respectively of the nth dipole mode. Here
𝜕𝐸𝑧 ⁄𝜕𝑟 is the transverse gradient of the axial electric
field. The dipole mode resonating at a frequency 961.982
MHz shows the maximum value of R⊥/Q which is
approximately 44.3×103 Ω/m2. Figure 3 shows the plot of
R⊥/Q for the five modes of the first dipole pass band as a
function of β.
Using the R⊥/Q data for these prominent dipole
modes, we now estimate the threshold current Ith given by
the following expression [1,6]
 3 cp k n
I th 
.
2e  g ( )  R Lcav  L2cav
Here, p is the momentum of the charged particle and Lcav
~0.710 mm is the length of the cavity, e denotes the
magnitude of the charge of the particle being accelerated.
R
1

Q nU n
ze
Here, the function g(α) is a measure of synchronization
between the particle velocity βc and the phase velocity
𝑣𝑝 of the particular mode. The prominent dipole modes,
their corresponding R Q and Ith are given in Table-2.
frequency of 325 MHz. Thus charge q per micropulse is
~12.3 pC. Hence, for a PRR of 325 MHz, q = 12.31 pC
and k|| = 0.531 V/pC, we get power dissipation within the
macropulse as 26.18 mW(CW average will be 2.62 mW).
3.0
4
4.50x10
st
k|| for 1 pass-band
nd
2.5
rd
Mode at 961.982 MHz.
Mode at 965.852 MHz.
Mode at 971.217 MHz.
Mode at 976.495 MHz.
Mode at 980.104 MHz.
4
2.0
2
 k|| (V/pC)
RT/Q (/m )
3.00x10
k|| for 2 pass-band
k|| for 3 pass-band
k|| for ~200 modes
1.5
1.0
4
1.50x10
0.5
0.00
0.50
0.55
0.60
0.65
0.70
0.75
0.80

Figure 3: R Q of the dipoles plotted as a function of β.
0.0
0.00
0.25
0.50
0.75
1.00

Fig. 4. Integrated loss factor Σ k|| is plotted with β.
Table 2: Details of the prominent dipoles and the Ith.
Frequency
(in MHz)
961.98
965.85
971.22
976. 50
980.10
p
0.75967
0.65890
0.58792
0.53191
0.49359
Q
9
2.17×10
2.23×109
2.34×109
2.47×109
2.57×109
R⊥/Q
(Ω/m2)
44313.7
28279.6
16987.0
8868.9
4606.5
Ith
(in mA)
1.01
0.70
0.88
1.34
2.19
WAKEFIELD ANALYSIS
The standard software, e.g., ABCI[8] calculates the
wake and the loss factor for the ultra-relativistic cases
only, assuming β ~ 1. In our case, the beam is not ultrarelativistic, and we have therefore calculated kn for the
entire range of β by evaluating R Q and using the
following formula[4]
1  n 
2
  R    
k n   n n   e 2  c 
 4Q 
where kn is the loss factor for the nth mode and σ is the
rms length of the Gaussian beam bunch, which is
assumed around 5 mm in our calculations. The integrated
loss factor k|| is obtained by summing the value for all n.
The plot of the loss factor with β is shown in Fig. 4. Using
this figure we get the integrated loss factor for ~ 0.61 as
~0.531 V/pC. Using this data we have calculated the
parasitic heat loss on the cavity walls due to the excitation
of wakefields by the beam. The power P0 appearing as
parasitic heat loss is related to k|| by the following
equation [4]
P0  PRR  q 2  k||
where PRR is the pulse repetition rate. In our case, the
pulse length is 2 ms, and macropulse current is 4 mA, and
within a macropulse, the micropulses repeat at a
DISCUSSIONS AND CONCLUSIONS
In this paper, we have presented the design of a βg =
0.61 650-MHz SRF cavity for the proposed ISNS project.
Particles traversing through βg=0.61 class of cavities, will
have their β varying from ~0.51 to ~0.76. Analysis of the
monopole showed the existence of a mode trapped inside.
We discussed a procedure in order to make extraction of
the trapped mode possible. In this paper we presented a
procedure to estimate the threshold current for the
regenerative beam breakup instability. We also present a
procedure to calculate the wake loss parameter based on
the R/Q of the individual monopole.
ACKNOWLEDGEMENT
The authors would like to thank Dr. P. D. Gupta for the
constant encouragement and Dr. S. B. Roy for the helpful
discussions and suggestions.
REFERENCES
[1] A. R. Jana et.al., IEEE Trans. Applied Superconductivity.
IEEE-TAS (4), 2013, pp. 3500816
[2] B. Aune et al, Phys. Rev. ST Accel. Beams, vol. 3, 092001,
Sep. 2000.
[3] H. Padamse, J. Knobloch, and T. Hays, RF
Superconductivity for Accelerators. NY, Wiley, 1998.
[4] T. P. Wangler, Principles of RF Linear Accelerators, Wiley
Series in Beam Physics and Accelerator Technology ISBN
0-471-16814-9, 1998.
[5] K. Halbach et al, in Particle Accelerators, vol. 7, 1976, pp.
213-222,
[6] D. Myakishev and V. Yakovlev, in Proc. of the 1995 PAC.
Dallas, TX, 1995, pp. 2348-50,
[7] J. J. Bisognano, in Proc. 3rd Workshop RF Supercond.,
1987, pp. 237–248.
[8] Y. H. Chin, User's guide for ABCI version 8.8, LBL- 5258.