Alessandra Lombardi version 28/07/2017 18:24
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3.3
Accelerator design
3.3.1
Choice of geometrica lcavity beta
Use material from
F. Gerigk, M. Eshraqi, M. Schuh, Choice of the Optimum Beta for the SPL Cavities, CERNsLHC-Project-Note-0001.
.
3.3.2
Beam dynamics in SC section
Two different layouts have been considered for the SC section. : one based on doublet focusing and
one based on FODO focusing. The difference from the strict beam dynamics point of view (beam
quality, sensitivity to errors) is negligible [1], but their implication on the magnets design is very
different. A FODO type layout is incompatible, at least for the lower beta section, with a fully
segmented cryomodule and would rule out the use of warm quadrupoles. On the other hand a FODO
focusing system, which requires higher magnetic field for the same average beam size, limits to about
450 mm the minimum length of the quadrupole at high energy, where magnetic stripping is an issue.
In order to use only one family of warm quadrupoles, a mixed layout is adopted [2] consisting of a
doublet focusing lattice till 2.5 GeV and a long FODO cells from 2.5 GeV to the final energy. This
layout has the advantages of both, it is highly segmented and uses warm quadrupoles – which are
much easier to align – and lower field in the quadrupoles which reduces the probability of Lorentz
stripping. As an additional minor advantage the mixed layout uses 12 less quadrupoles with respect to
nominal doublet layout. The quadrupoles used in this study have a magnetic length of 350 mm vs. the
450 mm used in the previous studies.
In the following the beam dynamics of the mixed layout is reported and -where interesting- it is
compared with the fully doublet or fully FODO system.
The magnetic length of quadrupoles, 350 mm, is such that for a chosen transverse focusing the
maximum gradient is below the threshold to keep losses below 0.1 W/m [6], such a length would not
be advised for any of the previous designs. In this architecture the distance between two adjacent
cavities, a quadrupole and a cavity, and cavity to the end wall of the cryo-module are based in a highly
segmented SPL design, i.e. warm quadrupoles, and the values are quoted in Table 2. All quadrupoles
are equipped with a steerer, assuming nested coils. In this compact design, the resulting sextupole
component has to be controlled to few units to avoid exciting an envelope oscillation. This design
uses just two types of cryo-modules for the accelerator.
In the FD & F0D0 mixed architecture of the highly segmented SPL there are:
•
3 cavities (low β cryo-module) with a NC doublet per period in the low energy part.
•
8 cavities (high β cryo-module) and a NC doublet in the high energy region before the 2.5
GeV.
•
2x one high β cryo-module and one single NC quadrupole after 2.5 GeV. This makes a F0D0
lattice twice the length of the F0D0 previously discussed
2
F D
low β - FD
F D
12.9 m
high β - FD
F D
13.3 m
F
high β
F0D0
D
28.2 m
Figure 1: Building blocks for the SPL mixed solution.
Table 2: Distance of the quads and cavities in low and high beta mixed SPL cryo modules
From
Quadrupole
Cavity
Quadrupole
Cryo-wall
Cavity
Cryo-wall
To
Cavity
Cavity
Quadrupole
Cavity
Cryo-wall
Quadrupole
Length (mm)
350
580 (low β) / 494 (high β)*
500
800
645 (low β) / 490 (high β)*
350
1
2
3
4
5
6
* The distance from end of last cell to the beginning of first cell
In the doublets upstream of 2.5 GeV branching, afterwards there is just one quad
Sketch showing the lengths in Table 2
In the SPL layout there is provision to deliver a beam for ISOLDE, the Isotope Separation On-Line
Experiment, and EURISOL, European Isotope Separation On-Line at energies of 1.5 and 2.5 GeV
respectively. Magnetic stripping of the extra loosely bound electron in H- ions is limiting the
maximum magnetic field in bending magnet, resulting in long dipoles in the extraction regions. To
insure lossless extraction to ISOLDE one period, 15.1 m, (equal to SPL high beta period) is added
between cryo-modules No. 5 and 6, leaving a drift space of 12.8 m. Lossless extraction region for
EURISOL at 2.5 GeV requires that a longer drift length of 22.5 m is added between high beta cryomodules No. 11 and 12, leaving an empty space of 21.05 m.
All three layouts respect the three general beam dynamics criterions:
a- The phase advance per period for zero current does not exceed 90 degrees
b- The phase advance per meter changes smoothly
3
c- Longitudinal to transverse phase advance ratio should not fall on peaks of the Hofmann plots
[3] to avoid resonances which cause emittance exchange), Fig. 2.
Figure 2. The red circles show the working points of mixed layout on a Hofmann plot
for a peak current of 60 mA.
Table 3: Comparison of layouts. The respective values for the low beta and the high beta regions are
separated by a “&” sign.
Doublets
L (m)
501
E (MeV)
786 & 4989
Periods
20 & 23
Cav/period
3&8
Total Quads (PS)
86 + 4 warm (54)
Total Cavities
244
FoDo
510
710 & 5020
24 & 24
2&8
96 + 4 warm (59)
240
Mixed
505*
753 & 5005
20 & 23
3 & 8 & 16
78 warm
244
* With longer drift at 2.5 GeV extraction region
The average energy gain per unit length is comparable for the three layouts. There are 10 low beta
cryo modules as in the doublet case and in the high beta region there are 23 cryo-modules as a sum of
11 of them using a doublet focusing and the next 12 long FoDo focusing. The total number of cavities
is comparable while the total number of quadrupoles and power supplies is less. The doublet option
has the advantage of being more flexible for segmentation plus its cryo-module design and
construction is simpler. To achieve the same phase advance per period using the FoDo interlacing
needs quadrupoles with half the gradients of the doublet interlacing , Fig. 3. By keeping the integrated
gradient constant FoDo interlacing would allow reducing the length of quadrupoles by two in the
FoDo with respect to the doublet without causing any magnetic stripping. A long FoDo architecture,
possible at high energies, would allow segmentation while it reduces the Lorentz losses significantly
with respect to the nominal doublet layout.
4
Fig. 3: Left: Gradient of the quadrupoles along the line in both layouts. Low beta region extends to
120 m, and the jump in gradients happens after the transition, Right: Maximum gradient which gives
less than 0.1 W/m of loss for Gaussian and double Gaussian distribution with two rms sizes.
Fig. 4: Energy of beam along the LINAC In all the three layouts
In the mixed architecture of the highly segmented SPL, there are three cavities per period in the low
energy region being housed in a cryo-module preceded by a pair of normal conducting quadrupoles.
The optimum number of cavity per period is given by the necessity to avoid resonances. In our case
and without compromising on the gradient, three cavities per period turns out to be the optimum
number to limit the longitudinal phase advance to 73 degrees per period. Falling on a resonance will
cause emittance exchange between longitudinal and transverse planes resulting in an unacceptable
emittance increase in the transverse planes.
5
In the high-energy region, before the 2.5 GeV branching a normal conducting pair of quadrupoles is
followed by eight cavities, one cryo-module, while after the 2.5 GeV branching each single
quadrupole will be followed by one cryo-module, making a long FoDo focusing optics, e.g. F—
Cryo—D—Cryo—, Fig. 5. The length of the low energy cryo modules is 4.68 m and the period
length is 6.48 m, length of the high energy cryo-modules is 13.26 m, and the period length is 15.06 m
before branching and 28.22 m after the branching.
As there are less quadrupoles after the transition at 2.5 GeV, the mixed LINAC can reach the same
energy within the same length while the branching length is longer than the other two designs.
Maybe move up
The beam dynamics simulations are performed for a 60 mA beam from LINAC4. In absence of any
manufacturing and assembly errors and any RF amplitude and phase errors, SPL transports and
accelerates the H- beam from LINAC4, preserving the beam quality, i.e. with full transmission and
minimum emittance growth.
.
Beam dynamics design, simulation, and calculations are done using the multi-particle code TraceWin
[4], which includes routines for 2D and 3D space-charge calculations using 50,000 macro particles.
The phase advances in each section are chosen to have equal phase advance per meter, average
focusing force, at the interconnection of two regions, as a result the quadrupoles next to the matching
region have a negligible difference from their nominal value, specially at the transition from low beta
region to high beta region where the two structures are connected seamlessly.
Fig. 6: Left: Emittance evolution in the SPL, Right: Halo evolution for gaussian beam.
6
Fig. 7: Beam envelopes along the LINAC. Top: X, Middle: Y, Down: Phase at 352.2 MHz
The matching must be done very carefully at the transition from low energy to high energy and also
across the extraction branches; however, long extraction drift spaces have an impact on the beam halo,
visible on the graphs, Fig. 6. The longitudinal halo is enhanced at 1.4 GeV, 200 m from the beginning
of SPL, where the acceleration is interrupted for several hundred βλs, and the transverse halo is mainly
increased where there is a change of focusing scheme from doublet to FoDo, at 310 m. The major
emittance increase in the transverse plane happens at the 2.5 GeV extraction region where the long
drift is followed by a change of focusing scheme. At the same place the maximum beam size occurs,
having an rms value of 4 mm in horizontal plane, while in the rest of the machine this value stays
below 3 mm, Fig. 7, resulting in the minimum aperture to ratio of 17.
The outer most particles stay confined within 10 mm, almost all along the LINAC, Fig. 8L. In
Longitudinal plane the beam is well within the acceptance, Fig. 8R, although the bucket size has not
been kept constant across the frequency jump, to shorten the overall length of LINAC. Still if a higher
margin is needed the synchronous phase as well as the accelerating voltage in superconducting
cavities should be decreased, this would result to a longer LINAC.
7
Fig. 8: Left, Beam density along the LINAC, Right, longitudinal acceptance at 165 MeV, the phase
on abscissa is in degrees at 352.2 MHz and the ordinate is energy spread in MeV.
.
Table 4: RMS Normalized Emittance comparison in layouts
Initial (PIMSin)
Final
π.mm.mrad
Δε/εin [%]
Layout
X
Y
Z
Both
0.328
0.334
0.458
Mixed
0.387
0.384
0.515
Doublet
0.369
0.365
0.486
FoDo
0.359
0.356
0.546
Mixed
18.0
15.0
12.4
Doublet
12.5
9.4
3.8
FoDo
9.5
6.5
16.6
The performance of the mixed architecture is comparable to the doublet and the FoDo interlacing
architecture from the beam dynamics point of view. It is worth mentioning that the mixed architecture
has a longer drift space, for lossless extraction at 2.5 GeV. The extra few-percent emittance increase
is mostly due to this increased free space.
Need to add the effect of errors and correction system.
Reference for this session:
[1] M. Eshraqi, P. A. Posocco, Doublet vs. FoDo Structure: Beam dynamics and Layout,
CERN-BE-Note-2010-007
[2] M. Eshraqi, A. M. Lombardi, P. A. Posocco, Highly Segmented SPL as a Mixture of
Doublet and FoDo, CERN-ATS-2010-228
8
[3] I. Hofmann, G. Franchetti, O. Boine-Frankenheim, J. Qiang, and R. D. Ryne, Phys. Rev.
ST-AB 6, 024202 (2003).
[4] R. Duperrier, N. Pichoff and D. Uriot, CEA Saclay codes review, Proc. International
Conf. on Computational Science, Amsterdam, The Netherlands, 2002.
3.3.3
H- stripping
When an H- ion moves in a magnetic field B, it experiences a Lorentz force that bends its trajectory
and also tends to strip its electrons. For a particle travelling at a reduced velocity β, a transverse
magnetic field in the laboratory frame produces a transverse electric field in the rest frame of the
particle. According to the Lorentz transformation of the fields,
.
This electric field, if strong enough, can remove the extra electron of an H- ion; the H- lifetime, in its
rest frame, has been expressed by Scherk (Scherk, 1979):
,
with
and
.
The fraction f of the stripped ions per unit path length can be expressed as (Jason, Hudgings, & Dyck,
1981):
,
which leads to the stripping probability
.
For 1 GeV H- ions, a magnetic field of few tens of Tesla is sufficient to strip the first electron, due to
its low binding energy (0.755 eV).
To estimate the probability of stripping in a quadrupole we have to consider that the magnetic field B
increases linearly with the radius. For a given distribution the probability is very low for particles
travelling around the magnetic centre of the quadrupole and it increases with the distance from the
axis. In general the problem of magnetic stripping concerns only the halo particles. However, if the
beam is off-centred, the high density part becomes closer to the higher B region and the total number
of stripped particles will increase. The stripping probability as function of energy for the two SPL
linac layouts is shown in Figure 2 and reaches its maximum at 5 GeV. At this energy the probability as
function of the quadrupole length is plotted in Figure 3.
9
F0D0 lattice
FD lattice
1
1
at 25% of the bore
at 50% of the bore
at 75% of the bore
at the pole-tip
0.1
0.01
110
110
110
110
110
110
110
110
110
3
0.01
110
Stripping probability (%)
Stripping probability (%)
110
0.1
4
5
6
7
8
110
110
110
110
110
9
110
 10
110
 11
110
 12
1
2
3
4
5
110
3
4
5
6
7
8
9
 10
 11
 12
1
2
E (GeV)
3
4
5
E (GeV)
Figure 2: Stripping probability for the nominal SPL quadrupole (50 mm bore radius, 450 mm
long) as function of the beam energy.
FD lattice
F0D0 lattice
1
1
0.1
0.1
0.01
0.01
110
110
110
110
110
110
110
110
110
3
110
Stripping probability (%)
Stripping probability (%)
110
4
5
6
7
8
110
110
110
110
110
9
110
 10
110
 11
110
 12
10
20
30
40
50
110
3
4
5
6
7
8
9
 10
 11
at 25% of the bore
at 50% of the bore
at 75% of the bore
at the pole-tip
 12
10
20
30
40
50
Quad. length (cm)
Quad. length (cm)
Figure 3: Stripping probability at 5 GeV as function of the quadrupole length (50 mm bore
radius).
10
Assuming a magnet with a bore of 50 mm and 450 mm in length the probability of stripping is
extremely low for both FD and F0D0 layouts if the beam is contained within a radius of 12.5 mm.
However to quantify the amount of stripped particles we have to combine the stripping probability
with the probability of a particle being at a certain distance from the center. For these reasons we
express B as function of r (
) and we derive the amount of losses in the quadrupole
integrating over the particle distribution
and the length d, obtaining:
.
The result will depend on the type of the distribution (Gaussian, double Exponential, etc…), on the
width σ of the distribution itself and on the displacement from the center.
Since the distribution at the ion source output is not yet experimentally known, two types of
distribution are used to test the effect of the magnetic stripping probability on a beam with various
radial extent, the Gaussian distribution
and the Double Exponential (Dex)
.
Both of them are normalized to 1 with the same standard deviation σ. The kurtosis parameter k1 is 3
for the Gaussian distribution and 6 for the Dex and it can be directly converted into the geometrical
halo parameter (
) (Allen & Wangler, 2002). Both distributions will be cut at 5 σ and the
results in the terms of losses will be renormalized.
We will analyze two σ value scenarios (Figure 4):
a. σ = 1.7 mm corresponding to the nominal case2 with full steerers correction ( < 1 mm).
b. σ = 2.5 mm corresponding to the nominal case without correction (
1
< 10 mm).
.
Input beam jitter: ±0.2 mm & ±0.2 mrad; quadrupole gradient error: ±0.5 %; quadrupole
misalignment: ±0.2 mm.
2
11
Figure 4: Beam radial density over 500 simulations for the nominal case (F0D0) without and
with steering correction.

= 2.5 mm, r.0 = 10 mm
dist. (log scale to 10-4)
0.01
4
110
6
110
8
110
 10
110
Stripping Probability
1
Gauss.
Dex
F0D0
FD
 12
 50  40  30  20  10 0
10
20
30
40
110
50
radius (mm)

= 1.7 mm, r.0 = 1 mm
0.01
4
110
6
110
8
110
 10
110
Stripping Probability
dist. (log scale to 10-4)
1
Gauss.
Dex
F0D0
FD
 12
 50  40  30  20  10 0
10
20
30
40
110
50
radius (mm)
Figure 5: Beam distributions at 5 GeV (in log scale to 10-4). The distributions radial extent is
compared to the stripping probability profile for the nominal SPL quadrupole (50 mm bore
radius, 450 mm long) for both layouts.
12
The plots in Figure 5 show the radial extent the distribution in the worst case compared to the
stripping probability at 5 GeV for both layouts and for the nominal SPL quadrupole. The probability
of stripping is sufficiently low in the FODO layout for any of the case considered whereas for the
Doublet layout, at high energy, problem could arise if the beam centre position error is not fully
compensated by the steerers.
The maximum power losses acceptable in SPL are 1 W/m which represents about 1 particle in 107 at
the highest energy. The losses per unit length are calculated assuming that all the particles are lost
inside the magnet, i.e. the power of the beam losses obtained via the stripping probability is divided by
the length of the magnet.
In Figure 6 the dependence of the losses on the beam displacement
is shown: for the same
integrated gradient it is possible to reduce the quadrupole length until the loss-limit is reached. With a
quadrupole length of 450 mm there is no risk of going beyond the limit even for the FD layout without
beam steering correction.
13
110
110
Losses
110
4
5
6
8
110
9
110
110
FD (300mm)
4
110
Gauss. (1.7mm)
Dex (1.7mm)
Gauss. (2.5mm)
Dex (2.5mm)
1 W/m
5
110
6
110
7
110
110
FD (200mm)
Gauss. (1.7mm)
Dex (1.7mm)
Gauss. (2.5mm)
Dex (2.5mm)
1 W/m
7
110
Losses
110
8
110
9
110
 10
 10
110
 11
 11
110
 12
 12
0.1
110
10
1
0.1
Beam Displacement (mm)
4
1
10
Beam Displacement (mm)
FD (450mm)
110
5
110
6
110
Gauss. (1.7mm)
Dex (1.7mm)
Gauss. (2.5mm)
Dex (2.5mm)
1 W/m
7
Losses
110
8
110
9
110
 10
110
 11
110
 12
110
0.1
1
10
Beam Displacement (mm)
Figure 6: Beam losses in a SPL quadrupole at 5 GeV for different quadrupole compared to the
radio protection limit of 1 W/m. The F0D0 scenario is not shown as for none of the distributions
the limit for displacement less than 10 mm is reached.
For low losses, like the ones requested by the radioprotection limit, the stripping probability is
proportional to the length of the magnet. This means that the maximum gradient as function of beam
energy we can accept is independent on the length of the magnet itself and it is univocally determined.
This value is calculated using the distributions with the maximum displacement, 1 mm for σ = 1.7 mm
and 10 mm for σ = 2.5 mm and is reported in Figure 7 for the 1 W/m limit and the more conservative
0.1 W/m. Since the slope of these curves is always greater than the one of derived from Error!
14
Reference source not found. for the nominal 450 mm quadrupole, the bottleneck is represented by
the values at 5 GeV, which are reported in Table 1.
Given the maximum quadrupole gradient as function of the beam energy (Figure 7) and the integrated
gradient along the machine for the FD and F0D0 scenarios, it is possible to calculate the minimum
magnetic length required as function of energy (Figure 8 and Figure 9). Again the bottleneck is found
for the values obtained at 5 GeV, reported in Table 2.
0.1 W/m
1 W/m
55
55
Gauss (1.7mm)
Dex (1.7mm)
Gauss (2.5mm)
Dex (2.5mm)
45
40
35
30
25
20
15
45
40
35
30
25
20
15
10
10
5
5
0
1
1.5
2
2.5
3
3.5
4
4.5
Gauss (1.7mm)
Dex (1.7mm)
Gauss (2.5mm)
Dex (2.5mm)
50
Max. quad. gradient (T/m)
Max. quad. gradient (T/m)
50
0
5
E (GeV)
1
1.5
2
2.5
3
3.5
4
4.5
5
E (GeV)
Figure 7: Maximum quadrupole gradient as function of beam energy and distribution.
Table 1: Maximum quadrupole Gradient at 5 GeV for Gaussian and Double Exponential
Distribution
Limit
0.1 W/m
1 W/m
Gaussian
σ = 1.7 mm
σ = 2.5 mm
14.0 T/m
5.5 T/m
15.6 T/m
6.1 T/m
15
Double Exponential
σ = 1.7 mm
σ = 2.5 mm
10.8 T/m
4.9 T/m
12.0 T/m
5.4 T/m
0.1 W/m - F0D0
1 W/m - F0D0
250
Gauss (1.7mm)
Dex (1.7mm)
Gauss (2.5mm)
Dex (2.5mm)
225
200
Minimum quadrupole magnetic length (mm)
Minimum quadrupole magnetic length (mm)
250
175
150
125
100
75
50
25
0
Gauss (1.7mm)
Dex (1.7mm)
Gauss (2.5mm)
Dex (2.5mm)
225
200
175
150
125
100
75
50
25
1
1.5
2
2.5
3
3.5
4
4.5
0
5
1
1.5
2
2.5
E (GeV)
3
3.5
4
4.5
5
4
4.5
5
E (GeV)
Figure 8: Minimum quadrupole magnetic length for the F0D0 layout.
0.1 W/m - FD
1 W/m - FD
450
Gauss (1.7mm)
Dex (1.7mm)
Gauss (2.5mm)
Dex (2.5mm)
400
350
Minimum quadrupole magnetic length (mm)
Minimum quadrupole magnetic length (mm)
450
300
250
200
150
100
50
0
1
1.5
2
2.5
3
3.5
4
4.5
350
300
250
200
150
100
50
0
5
Gauss (1.7mm)
Dex (1.7mm)
Gauss (2.5mm)
Dex (2.5mm)
400
1
1.5
2
E (GeV)
2.5
3
3.5
E (GeV)
Figure 9: Minimum quadrupole magnetic length for the FD layout.
16
Table 2: Minimum Quadrupole magnetic Length at 5 GeV for Gaussian and Double
Exponential Distribution
limit
0.1 W/m
1 W/m
Scenario
F0D0
FD
F0D0
FD
Gaussian
σ = 1.7 mm σ = 2.5 mm
64 mm
163 mm
129 mm
325 mm
58 mm
148 mm
115 mm
295 mm
Double Exponential
σ = 1.7 mm
σ = 2.5 mm
83 mm
185 mm
167 mm
369 mm
75 mm
167 mm
149 mm
334 mm
The considerations on the H- stripping are an important input for the decision of the focusing lattice of
the SPL. If an FD lattice with constant quadrupole length was adopted for the SPL then the minimum
magnetic length should be 450 mm (5GeV limit). This implies that the quadrupoles at low energy are
over specified. As already detailed above, the choice for SPL is to keep one family of quadrupoles for
the whole linac, and to use a FD lattice at low-energy, preferable for the cryostats segmentation, and a
F0D0 lattice at high energy. The transition between the two focusing systems happens at the 2.5 Gev
branching
Another phenomenon which causes unwanted H- stripping is the so called intra-beam stripping [6].
This phenomenon occurs when two H- with very different velocities come very close to each other.
The relative velocity (β) has to be more than 2×10-4. If a Gaussian distribution for both spatial and
velocity distribution is assumed and the three planes are decoupled, the resulting fractional loss can be
written as
,
is the rms spatial width of the bunch,
is the rms velocity width
and
is a form factor which is =2/√3 (max) when all 3 velocity spreads are equal. The
first evidence of this phenomenon is reported in [7] and its cross section reviewed in [8]. It is
important to underline that the fractional loss depends proportionally to the peak current: keeping the
product of the peak current and the pulse length constant (i.e. maintaining the beam power), the power
loss is proportional to the peak current itself.
In Figure 10 the Fractional Loss is calculated by means of a program supplied by FNAL [9] for all the
SPL beam dynamics previously described: since the transverse and longitudinal phase advances are
almost the same for the 3 cases, so are the beam sizes and the velocity distributions. This means that
the 3 cases are indistinguishable. The resulting Power Loss exceeds the 0.1 W/m limit in many areas
and in particular in the achromatic bend transfer line from Linac4, where a tight waist in both
transverse and longitudinal plane is achieved. The only way to reduce these losses is to reduce the
peak current: if the Low Current scenario is chosen instead of the High Current, the power loss would
be reduced by a factor 2, preserving the above limit
17
1E-05
FD
FODO
mixed
Fractional Loss
1E-06
1E-07
1E-08
1E-09
0.20 0
0.18
200
300
400
500
FD
FODO
mixed
600
400
500
600
z position (m)
Linac4 – PIMS
& TL
0.16
Power Loss (W/m)
100
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
100
200
300
z position (m)
Figure 10: Fractional Loss and Power Loss calculated as function of the position along the linac
for the High Current scenario. The data were generated by a program supplied by FNAL [9].
Refernce for this session
[7] M. Chanel, et al., Physics Letters B, Volume 192, Issues 3-4, 2 July 1987, Pages 475-477.
[8] D. Raparia, Contribution at HB2010 Conference.
[9] J. F. Ostiguy, Private Communication.
3.3.4
HOM effects
The effect of the HOM on beam quality is analysed in detail for the SPL in order to define damping
requirements as function of the operation parameters. For this purpose a dedicated beam dynamics
simulation code, Simulation of higher order Mode Dynamics (SMD), focusing on beam-HOM
interaction, has been developed in the frame of this project. SMD allows analysing the beam
behaviour under the effect of HOMs, taking into account many important effects, such as for example
the HOM frequency spread, beam input jitter, different chopping patterns, as well as klystron and
alignment errors. SMD performs bunch tracking simulations of point-like bunches taking into account
the effects on the time of flight and energy gain induced by an initial energy error. Space charge
effects are neglected in all simulations.
In [1-2] it was shown, that HOMs are only a concern in the longitudinal plane, the deflection forces of
non-monopole modes being too weak to cause transverse Beam Break Up.
18
3.3.4.1
Simulation Input Parameters
The simulation input parameters can be split up into machine specific and cavity specific parameters.
All machine parameters used in the simulations are listed in Table HOM simulation machine
parameters and are based on Linac4 [Linac4] simulations and experience at SNS [HOMws]. A
Gaussian distribution is assumed, unless otherwise stated.
Table HOM simulation machine parameters
Beam Intensity
40-400 mA
Intensity bunch-to-bunch jitter
3%
Injection Energy
160± 0.078 MeV
Phase at 704 MHz
-15 ±0.4 deg
RF phase and amplitude jitter
0.5 deg and 0.5% (uniformly distributed)
Bunch frequency
352.2MHz
Pulse Length
1 ms
Repetition rate
50 Hz
The cavity specific simulation parameters are the mode frequencies and its R/Q(β) values. In Figure
RQ_beta_max the first three monopole bands of the SPL baseline cavities and their maximum R/Q
value are shown. The R/Q(β) dependency of some modes which can vary over several orders of
magnitude is shown in Figure RQ_beta. A Gaussian HOM frequency spread of 1MHz is assumed for
all non TM010 modes based on the experience at DESY [privJ] and Jefferson Laboratory [privM]. The
properties of the modes above the beam pipe cut-off frequency depends on the inter cavity sections
and is under investigation [HOMcryo].
Figure RQ_beta_max: The first three monopole bands of the SPL baseline cavities and their
maximum R/Q value in the used energy range are shown. The beampipe cut-off frequency is
indicated by the dashed line.
Figure RQ_beta: R/Q(β) dependency of some HOMs in the SPL baseline cavities. (1) medium
beta cavity; (2) high beta cavity
3.3.4.2
Reference simulation
In all simulations one or more pulses consisting of 350,000 point-like bunches (≈1ms) are tracked
through the linac and then the phase space area created by the last pulse is recorded. Then an effective
phase space of the pulse in the longitudinal plane is calculated by
Tracking one pulse with a Gaussian distributed injection energy and phase error and no RF errors
through the linac results in the phase space distribution shown in Figure InitialSimulation.
Figure InitialSimulation: 2D phase space histogram of one pulse at the end of the linac, which is
not disturbed by HOMs. This distribution will be used as reference for HOM simulation.
3.3.4.3
RF-Errors
The amplitude and phase jitter in the RF system lead to an increase in the average energy and phase
error of the bunches along the linac, which can also be represented as phase space increase. The result
of 1.000 end to end simulation for one pulse is shown in the histogram of Figure RFerror
Figure RFerrors: 2D phase space histogram at the exit of 1,000 linacs with RF errors, but no
HOM is present and the phase space distribution of the single pulses. The phase space area is
increased significantly compared to Figure InitialSim. On average ε is ∼ 3.8 times higher.
19
The phase space increase due to RF errors can be used as upper limit for the tolerable impact of
HOMs.
3.3.4.4
Higher Order Monopole Modes
The modes with the highest R/Q(β) values are chosen and the beam current Ib as well as the damping
are varied. Figure IbQex shows the resulting phase space increase. Each set is simulated for 100
linacs, where the HOM frequency pattern is varied. For all currents and damping the impact of HOMs
can be neglected compared to the influence of RF errors. The beam noise is not strong enough to drive
HOMs.
Figure IbQex: Average longitudinal phase space increase and deviation of 100 linacs against the
damping Qex for different Ib. The increase goes quadratic with Ib. The plateau around Qex ≈
4·107 is due to the pulse period structure.
3.3.4.5
Resonances
Resonances created by the pulse-substructure as well as the principal machine lines created by bunch
spacing can drive HOM significantly as shown in Figure MachineLines. Three different pulsesubstructures are shown and the mean HOM frequency is set to the resonance frequencies between the
3rd and 4th fundamental machine line. The notation n/m means that n out of m bunches stay and m-n
bunches in between are chopped. In order to keep the total charge per pulse constant, the charge per
bunch is increased. Most critical are the principal machine lines, but also chopping with a highest
repetition rate – small m – can cause a significant HOM impact. Hence, no mode with an artificial
high R/Q(β) value should be close to such a resonance line. The fundamental machine lines can be
considered during the cavity design. Nevertheless in order to keep full flexibility for the chopping
pattern, the case that a HOM falls on a high frequent chopping machine line cannot be a priori
excluded. Therefore to guarantee a stable beam for any chopping pattern a damping in the order of
Qex=105 is required for modes with high (>10Ω) R/Q values.
Figure MachineLines: Longitudinal phase space increase against the HOM frequency where a
HOM falls on a chopping machine line at nominal current and Q ex = 107 for different chopping
patterns. The RF induced growth (black dashed line) is indicated together with the identified
monopole modes (violet dashed lines) in the investigated frequency range.
3.3.4.6
TM010 Passband modes
The modes beside the accelerating π-mode in the TM010 band have a small frequency spread and can
have an artificially high R/Q value for velocities different from the geometrical beta (β≠βg). HOM
dampers are generally insensitive to these modes due to the rejection filter for the acceleration mode.
Hence, only the fundamental power coupler can provide a damping for these modes. The modes and
the damping calculated with different EM simulation tools are listed in Table TM010damping. The
most critical mode is the TM010,4/5π mode. Its influence as function of the damping for different
currents is shown in Figure TM010. For a beam current of 400mA a significant longitudinal emittance
increase is visible, but which remains below the growth caused by RF-error at the expected damping.
Table: TM010 modes and the expected damping by the Fundamental power coupler.
Superfish HFSS
MWS
f [MHz]
f [MHz] Qex [106]
f [MHz] Qex [106]
(R/Q)(βg)[
(R/Q)max[
Ω]
Ω]
βg=0.65
TM010,1/5π 695.4
696
6.46
695.2
5.73
0.003
0.8
TM010,2/5π 697.9
698.5
1.78
697.8
1.6
0.203
0.5
TM010,3/5π 701
701.6
0.91
700.9
0.81
0.198
52
20
TM010,4/5π
TM010,π
βg=1
TM010,1/5π
TM010,2/5π
TM010,3/5π
TM010,4/5π
TM010,π
703.5
704.4
704.1
705
0.61
1.1
703.4
704.4
0.57
1.03
0.475
299
268
324
692.5
695.7
699.8
703.1
704.4
692.5
695.7
699.7
703.1
704.4
6.32
1.73
0.91
0.65
1.17
692.3
695.5
699.6
703
704.3
6.07
1.67
0.88
0.63
1.15
0.001
0.037
0.011
0.1
565
0.1
0.4
25
167
547
Figure TM010: Average longitudinal phase space growth versus Qex caused by the TM010,4/5π
mode and different currents; RF error caused growth (dashed black line); simulated Qex (shaded
area).
The chopping pattern can cause a further longitudinal emittance increase, especially if a chopping
machine line matches exactly a TM010 mode as illustrated in [MSIPAC10].
3.3.4.7
HOM Power Dissipation
The additional load for the cryogenic system is estimated for the SPL in [Schuh:1323892]. At
moderate damping (Qex=107) the dissipated HOM power at a chopping resonance line can easily
exceed 10 W/Ω. This is an enormous load for the cryogenic system, if the power is dissipated at 2 k. A
Qex of 107 is expected when using slightly lossy material between cavities as the only HOM damping
mechanism. Instead, a dedicated HOM coupler can reduce the Qex further and decreases the dissipated
HOM power on resonance. On the other hand it increases the width of the resonance and the power
dissipation close to a resonance. Nevertheless, this situation is preferable, because it enables the
operation with HOMs at chopping machine lines and does not limit the operation to low frequency
chopping patterns.
3.3.4.8
Damping Requirements
Based on the discussed scenarios a moderate damping in the order of Qex=107 is sufficient as long as
no high frequency chopping pattern is introduced. Allowing any chopping pattern a damping of
Qex=105 or even stronger is required to reduced the heat load caused by a resonant excitation to a
tolerable level and cause no beam degradation.
Sources:
[1] M. Schuh et al., Influence of Higher Order Modes on the Beam Stability in the High Power
Superconducting Proton Linac (SPL), PRST-AB
[2] M. Schuh, Study of Higher Order Modes in Superconducting Accelerating Structures for Linac
Applications, PhD Thesis, 2011
[Schuh:1323892] M. Schuh & W. Weingarten, Power dissipation by Higher Order Modes, CERNsLHC-Project-Note-0027, 2010
[Linac4] G. Bellodi, et al. End to End Beam Dynamics and RF Error Studies for Linac4, Proceedings
of
LINAC08,
Victoria,
BC,
Canada,
MOP086
(2008).
http://accelconf.web.cern.ch/AccelConf/LINAC08/papers/mop086. pdf
[HOMws] A. Lombardi, S.-H. Kim, and W. Weingarten: Summary of the work- shop on HOM in
SPL,
Tech.
Rep.
sLHC-Project-Note-0003,
CERN,
Geneva
(2009).
http://cdsweb.cern.ch/record/1189889
[privJ] J. Sekutowicz, private communication
21
[privM] F. Marhauser, private communication
[HOMcryo] S. Molloy and R. Calaga: Simulations of Proposed Accelerating Cavities for the CERN
SPL,
Proceedings
of
IPAC’10,
Kyoto,
Japan,
WEPEC055
(2010).
http://accelconf.web.cern.ch/AccelConf/IPAC10/papers/wepec055. Pdf
[MSIPAC10] M Schuh et al., Higher Order Mode Analysis of the SPL Cavities, Proceedings of
IPAC'10,
Kyoto,
Japan,
2010
http://accelconf.web.cern.ch/AccelConf/IPAC10/papers/thpe082.pdf
3.4
Tables
All tables must be referred to in the text in consecutive numerical order. Tables should be referred to
as ‘Table 1, Tables 2–7’. (The word ‘Table’ should never be abbreviated.)
3.4.1
Positioning and layout of the table
Each table should be centred on the page width, with a table number and brief caption in point size
10 typed above it.
Tables should be open, drawn with a double thin horizontal line (3/4 pt) at the top and bottom,
and a single horizontal line (3/4 pt) separating column headings from data.
3.4.2
Formatting and layout within the table
Write the column headings in sentence case but do not use full stops. Units should be entered in
parentheses on a separate line below the column heading. (If the same unit is used throughout the
table, it should be written in parentheses on a separate line below the caption.)
Unsimilar items should be aligned on the left, whereas similar items should be aligned on the
operator or decimal point. All decimal points must be preceded by a digit.
3.4.2.1
Table captions
Table captions should be brief and placed centrally above the table, e.g., ‘Table 1: A short title’. No
full stop is necessary unless the caption is more than one sentence long, in which case full punctuation
should be used. Captions should be typed in point size 10.
3.4.2.2
Notes in the table
Notes in tables should be designated by superscript lower-case letters, and begun anew for each table.
The superscript letter should be placed in alphabetical order moving from left to right across the first
row and down to the last. The notes should then be listed directly under the table.
References cited in tables should appear in consecutive numerical order, following the order
explained above for notes (left to right, top to bottom). They should appear exactly as references cited
in the main body of the document (and be treated as part of the text in terms of numbering, so be
careful when moving tables containing references).
3.5
Bibliographical references
References should be cited in the text using numbers within square brackets: ‘example [1], example
[1, 2], example [1–5]’, or ‘see Ref. [1], Refs. [1–5]’. The word ‘Reference’ should be written in full if
this word occurs at the beginning of a sentence. References should appear in consecutive numerical
order in the text and should be listed in numerical order at the end of the text. Punctuation can be used
22
either within or outside the brackets, but please ensure that one method is used consistently throughout
the contribution.
3.5.1
List of references
Unless you are near the bottom of the last page of text, do not start a new page for the list of
references, but continue on the same page. The list of references should appear as an unnumbered
subsection, entitled ‘References’. The references themselves are numbered automatically. Note that in
the list of references it is unnecessary to state the title of an article or chapter in proceedings or in a
collection of papers unless a page number cannot be quoted, e.g., for forthcoming publications.
For abbreviations of names of journals quoted in the references, see the Journal abbreviations
entry available from the DTP Web page at the URL http://cern.ch/DTP/dtpgrammar.htm3. The entry
Citation of references on the same Web page shows more details on how to present references.
If you need to provide a bibliography, this should come after the list of references.
If you have appendices with references cited therein, the list of references must follow the
appendices. If no references are cited in the appendices, the list of references precedes the appendices.
3.6
Footnotes
Footnotes are to be avoided. If absolutely necessary, they should be brief, and placed at the bottom of
the page on which they are referred to, in point size 9. The footnote marker should be place inside any
punctuation mark. Take care when citing references in the footnotes to ensure that these are correctly
numbered.
3.7
Appendices
Each appendix should be laid out as the sections in the text. Appendices should be labelled
alphabetically and be referred to as Appendix A, Appendices A–C, etc. Equations, figures and tables
should be quoted as Eq. (A.1) and Fig. A.1, etc.
3.8
Acknowledgements
If required, the acknowledgements should appear as an unnumbered subsection immediately before
the references section.
4
Spelling and grammar
For more information on English grammar rules and commonly misused words and expressions
(including a guide to avoiding ‘franglais’), please see the files available from the DTP Web pages
(http://cern.ch/DTP).
4.1
Spelling
CERN uses British English spelling, and ‘-ize’ rather than ‘-ise’. Here we provide a few examples for
guidance:
-il
fulfil (not fulfill)
-re
centre (not center), metre (not meter for length)
-our
colour (not color)
3
The Reviews of Modern Physics site: http://rmp.aps.org/info/manprep.html also has a list. (footnote style)
23
-gue
catalogue (not catalog, but analog is used in electronics)
-mme programme (not program, unless referring to a computer program)
-ell-
labelled (not labeled)
-ce/-se licence (noun), license (verb), practice (noun), practise (verb)
-ize
organization, authorize.
Exceptions to the ‘-ize’ rule include advise, comprise, compromise, concise, demise, devise,
enterprise, exercise, expertise, improvise, incise, precise, revise, supervise, surmise, surprise, televise.
4.2
4.2.1
Punctuation
Hyphen
Hyphens are used to avoid ambiguity, i.e., in attributive compound adjectives (compare ‘a little used
car’ and ‘a little-used car’), to distinguish between words such as ‘reform’ (change for the better) and
‘re-form’ (form again), and to separate double letters to aid comprehension and pronunciation (e.g.,
co-operate).
Hyphens are also used if a prefix or suffix is added to a proper noun, symbol, or numeral, and in
fractions: e.g., non-Fermi, 12-fold, three-quarters.
4.2.2
En dash
En dashes are used to mean ‘and’ (e.g., space–time, Sourian–Lagrange) or ‘to’ (e.g., 2003–2004,
input–output ratio).
4.2.3
Em dash
An em dash is used as a parenthetical pause. Simply type with no space on either side (Word will
automatically insert a thin space), e.g., ‘the experiment—due to begin in 2007—represents a major
advance...’.
4.2.4
Quotation marks
Double for true quotations, single for anything else. Single within double for a quotation within a
quotation. Our preferred method of punctuation around quotation marks is to place punctuation marks
(except an exclamation mark, question mark, dash or parenthesis belonging only to the quotation)
outside the quotation marks, to avoid any ambiguity: Oxford has been called a ‘Home of lost causes’.
4.2.5
Apostrophe
Do not use in plural acronyms (e.g., JFETs), decades (1990s). Do use in plural Greek letters and
symbols (e.g., π’s).
4.2.6
Colon, semicolon, exclamation mark, question mark
Please note that in English these punctuation marks do not require a space before them.
4.3
Lists
In a series of three or more terms, use a comma (sometimes called the serial comma) before the final
‘and’ or ‘or’ (e.g., gold, silver, or copper coating). In a run-on list, do not introduce a punctuation
mark between the main verb and the rest of the sentence. Avoid the use of bullet points.
For a displayed list there are two options:
24
i)
finish the introductory sentence with a colon, start the first item of the list with a lower-case
letter, finish it with a semicolon, and do the the same for all items until the last, where a full
stop is placed at the end of the text (as here); (item list style)
ii) finish the introductory sentence with a full stop, start the first item with a capital letter and
finish it with a full stop, and do the same for the remaining items.
4.4
Capitalization
Capitalize adjectives and nouns formed from proper names, e.g., Gaussian. Exceptions to this rule
include units of measure (ampere), particles (fermion), elements (einsteinium), and minerals (fosterite)
derived from names. Capitalize only the name in Avogadro’s number, Debye temperature, Ohm’s law,
Bohr radius.
Never capitalize lower-case symbols or abbreviations. When referring to article, paper, or
report, column, sample, counter, curve, or type, do not capitalize.
Do capitalize Theorem I, Lemma 2, Corollary 3, etc.
4.4.1
Acronyms
In the first instance, spell out the acronym using capital letters for each letter used in the acronym, and
provide the acronym in parentheses, e.g., Quark–Gluon Plasma (QGP).
4.5
Numbers
Spell out numbers 1 to 9 unless they are followed by a unit or are part of a series containing the
number 10 or higher (as here); numbers are always in roman type. Numbers should always be written
out at the beginning of a sentence.
4.6
Symbols
Names of particles, chemicals, waves or states, covariant couplings and monopoles, and mathematical
abbreviations are all written in roman type,
Symbols of variables (i.e. anything that can be replaced by a number) should be typed in italics.
Take care that this is consistent throughout the contribution.
4.7
Units
Symbols for units are printed in roman type. Symbols for units derived from proper names are written
with capital letters (e.g., coulomb, 6 C). Insert a non-breaking space between the number and the unit,
unless it is % or a superscript (e.g., 10 cm, 100 GeV, 20%, 27°C). Write the unit out in full in cases
such as ‘a few centimetres’.
Please see the file on symbols and units, available from the DTP Web pages, for a list of
abbreviations for the most commonly used units4.
4
NIST also has a useful summary on the subject, see http://physics.nist.gov/cuu/Units/.
25
Appendix A: Styles for Microsoft Word (Appendix heading 1 style)
We accept contributions prepared on a Macintosh or PC using Word (version 2001 or later).
Table A.1 lists the styles to be used in formatting each part of your contribution.
Table A.1: List of styles to be used in preparing contributions in Word (Table caption style)
Style name
Part of contribution (Cell heading style)
Title
Title of the contribution
Author
Names of the authors
Affiliation
Affiliation of the authors
Abstract heading
Abstract heading
Abstract body
Text of abstract
1 Heading 1
Section heading (level 1 heading) numbered
1.1 Heading 2
Subsection heading (level 2 heading) numbered
1.1.1 Heading 3
Subsubsection heading (level 3 heading) numbered
1.1.1.1 Heading 4
Subsubsubsection heading (level 4 heading) numbered
Section
Section heading (level 1 heading) unnumbered
Subsection
Subsection heading (level 2 heading) unnumbered
First paragraph
Text following all numbered and unnumbered heading
Paragraph
Other text
Item list
List (to be used for itemized list)
Figure caption
Figure number and caption
Figure
Figure (to be used when inserting figures in the text)
Table caption
Table caption
Cell heading
Cell heading in table
Cell body
Cell text in table
Equation
Numbered equation
References
List of references
Footnote
Footnote
A Appendix heading 1
Appendix section heading (level 1 heading) numbered manually
A.1 Appendix heading 2
Appendix subsection heading (level 2 heading) numbered manually
A.1.1 Appendix heading 3
Appendix subsubsection heading (level 3 heading) numbered manually
A.1.1.1 Appendix heading 4
Appendix subsubsubsection heading (level 4 heading) numbered manually
26
A.1
Obtaining the template file and making your contribution available (Appendix heading 2
style)
You can find the instructions as a Word template file (cernrep.dot) and an example (cernrepexa.dot) at
the URL http://cern.ch/DTP/dtpcernreport.htm.
Please contact the editors to determine the best way to submit your contribution.
Appendix B: Guidelines for Microsoft Word files
While articles may be submitted in Word format, we have found the use of Word files in production to
be problematic. To minimize possible problems, please observe the following guidelines.
1. Use Word 2001 or later.
2. Use MathType for both displayed (stand-alone) equations and mathematics run into the text,
including individual mathematical symbols, Greek letters, and other special characters not
found on the keyboard.
3. Do not embed graphics in the text. We require separate electronic figure files (EPS, PICT,
TIFF, PNG, PDF, and JPEG). A ‘place holder’ may be indicated for placement purposes.
4. Please, keep it simple. Do not change fonts or try to achieve a ‘typeset’ look. Final page
layout is done by the DTP. Use exclusively the styles described in the present paper and
summarized in Table A.1.
5. Do the references ‘by hand’, without using an external program. The entry Citation of
references on the Web page at the URL http://cern.ch/DTP/dtpgrammar.htm shows how to
present references. The entry Journal abbreviations on the same Web page gives the correct
abbreviations for names of journals quoted in the references.
6. Do not hide text or insert comments into the text.
7. For tables, use the Word table editor. Do not create ‘by hand’ using tab stop or spaces.
8. Do not use Rich Text Format.
27