- 1 Introduction
- 2 Booster
- 2.1 Linear Optics
- 2.2 Tune correction
- 2.3 Chromaticity Correction
- 2.4 Error Tolerance Specifications
- 2.5 Closed orbit correction system
- 2.6 Aperture requirements
- 2.7 Dynamic aperture
- 2.8 Booster injection
- 2.9 Booster extraction
- 2.10 Booster RF system
- 2.11 Booster straight sections allocation
- 3 Linac
- 4 Linac to booster transport line (LTB)
- 5 Booster to storage ring transport line (BTS)
- 6 Injection into the Storage Ring
- 7 Pulsed Magnets Parameters
The Sirius injection system is composed of a 150 MeV Linac, a full energy 3 GeV synchrotron booster operating in top-up mode and 2 transport lines, one from the Linac to the Booster (LTB) and another from the Booster to the Storage Ring (BTS). The booster and the storage ring are concentric and share the same tunnel. Their injection sections are placed close together to restrict the 'dirty' area in terms of radiation. A general layout of the Sirius injection system is shown in Figure 1.
The booster lattice consists of 50 modified FODO cells with combined-function magnets suitable to provide low emittance. The main difference from other booster designs is that the Sirius booster contains only arc sections: no long dispersion-free straight sections are provided. This means that injection and extraction, as well as the Petra 5-cell RF cavity, are located in these dispersive straight sections. With this concept a very high lattice symmetry for the booster is obtained and only a few magnet families are necessary. A high lattice symmetry is not only beneficial for beam dynamics but also to maximize the radial distance between the storage ring and the booster beam centers, minimizing the variations in the size of the corridor between the two concentric rings.
The lattice is composed of three main families of magnets: combined function dipoles (BD), with defocusing quadrupole and sextupole field components, focusing quadrupoles (QF) and focusing sextupoles (SF). Together, they define the working point and set the chromaticities to +0.5 in both planes. The achieved natural emittance at the extraction energy of 3 GeV is very low, which is essential to provide a clean injection process into the storage ring with the nonlinear kicker injection process.
The families of focusing quadrupoles QF and the defocusing quadrupolar component in the dipoles BD set the nominal horizontal and vertical tunes. In order to allow for flexibility in tune adjustment, an extra family of weak corrector quadrupoles QD is added to the lattice. A QD quadrupole is placed every two dipoles, right next to it. In the same way, the chromaticities are set to +0.5 in both planes by the family of sextupoles SF and the defocusing sextupolar component in the dipoles BD; and to allow flexibility for chromaticity adjustment, an extra family of corrector sextupoles SD is introduced. A sextupole corrector SD is placed every five dipoles in the lattice. There are 2 kinds of straight sections connecting the dipoles: odd sections contain orbit correctors and even sections contain quadrupoles QD and sextupoles SF. All sections contain quadrupoles QF and BPMs. Ten sextupoles SD are distributed in odd and even sectors. A schematic diagram of the Booster straight sections is shown in Figure 3
Tune correction in the Sirius Booster will be performed with QF and QD families:
where is the tune and is the integrated quadrupole strength.
Chromaticity correction in the Sirius Booster will be performed with SF and SD families.
where is the chromaticity and is the integrated sextupole strength. From the above equation, we notice the chromaticity is corrected to 0.5 in both planes when and are equal to 1.189 and 0.742, respectively.
To specify the maximum SD integrated strength, we considered, among others, the effect of a systematic error of 12% in the sextupolar component of the dipoles and a range of variation of chromaticity of [0.5,4.0] in both planes.
Error Tolerance Specifications
Booster Magnets Multipole Errors
@r = 17.5 mm
|B2/B0 (sextupole)||--||5.5×10-4 *||1.0×10-4|
- * This spec is replicated in the dipole alignment and rotation errors table
- Actual designed dipole model shows numbers are that in accordance with these specs. (see fieldmap analysis)
@r = 17.5 mm
1 Multipoles of prototype magnets measured with radial rotating coils
@x = 17.5 mm
1Relative multipoles calculated around the Runge-Kutta trajectory for the latest sextupole model-03 fieldmap at 3 GeV
|Transverse position, ,||160||160||160||300||μm|
|Rotation around longitudinal axis||0.8||0.8||0.8||--||mrad|
|Excitation error (static or low frequency)||0.15||0.3||0.3||--||%|
|Dipole Gradient Error||2.4||--||--||--||%|
|Girder vibration rms amplitude||500||nm|
|Power supply ripple|
Closed orbit correction system
The specification for the closed orbit correction system for the Sirius booster is in Table 8 and its distribution along the ring is shown in Figure 4. To calculate corrector kicks and estimate residual orbit displacements, we have simulated static orbit distortions due to random alignment and excitation errors in all magnets in the lattice (see Table 6). These are relaxed error tolerances, we expect better results for the real machine. The implicit safety margin that is assumed should account for the dynamic effects during energy ramping that were not taken into account in these calculations. The closed orbit before correction is shown in Figure 5 and the residual closed orbit distortion after correction is shown in Figure 6. The corrector strengths are listed in Table 9
|Number of BPMs||50|
|Number of horizontal dipole correctors||25|
|Number of vertical dipole correctors||25|
|Maximum horizontal dipole corrector strength||310||μrad|
|Maximum vertical dipole corrector strength||310||μrad|
|COD rms @ BPMs||247||284||µm|
|Rms corrector strength||60||89||µrad|
|Max. corrector strength||186||253||µrad|
The physical aperture requirements for the booster are defined by the injected Linac beam size and energy variation from pulse to pulse, by the closed orbit distortions and by a tolerance for beam oscillations after injection. During acceleration the beam size shrinks as well as the oscillations due to mismatched energy, position and angle of the injected beam.
The beam stay clear (half-aperture) was defined by the following formulae:
The resulting requirements are illustrated in Figure 7.
The booster vacuum chambers will be made of 1 mm thick stainless steel cylindrical tubes with inner diameter of 23.4 mm at the dipoles and 36 mm at the straight sections.
The chromaticity correction sextupoles, multipole components in dipoles and quadrupoles, and alignment errors reduce the dynamic aperture to about ±12 mm in the horizontal and ±4 mm in the vertical plane at the injection point. This is sufficiently large for an efficient on-axis injection and for beam lifetime. Figure 8, Figure 9 and Figure 10 show the dynamic and momentum apertures for 20 machines with random multipole, alignment and excitation errors in all magnets.
The 150 MeV electron beam from the Linac is injected on axis into the booster. The injection septum and the on-axis injection kicker are shown in Figure 43, the horizontal trajectory of the injected beam in Figure 12 and a cross section at the injection point in Figure 13. The injection point at the booster is defined at the physical end of the injection septum.
Two kickers are used to extract the beam from the booster. They deflect the beam towards the extraction septum, passing through a defocusing dipole and a tune corrector quadrupole along the way. The nominal strength for this quadrupole is zero, it will only be used for small tune corrections. The arrangement of the booster extraction system elements is shown in Figure 44.
The extracted beam horizontal trajectory is shown in Figure 15. Due to the large amplitude of the extracted beam, the horizontal size of the dipole vacuum chamber next to the septum will have to be increased. The beam arrives at the septum with an amplitude of x=22 mm and angle of x'=5.0 mrad. This amplitude considers:
|half-aperture for stored beam||18.0||mm|
|vacuum chamber thickness||1.0||mm|
|half-aperture for extracted beam (4σ)||1.0||mm|
Booster RF system
Table:Booster RF parameters shows the design parameters of the RF system for the booster. The value of the Peak RF voltage presented in the table was calculated assuming the energy spread of the beam could be 90% larger than the expected value, also present in the table, and the momentum compaction could be 30% higher than the value in the table. These are very pessimistic scenarios for the system design, being the nominal
|Cavity type||PETRA 5-cell|
|Peak RF voltage||1.05||MV|
|Total scaled shunt impedance||15||MÎ©|
|Maximum RF power needed||45.0||kW|
|Momentum compation factor||7.19×10-4|
|Natural bunch length||11.25||mm|
|Natural bunch length||37.53||ps|
|Linac to Booster phase stability @500MHz||5.0||°|
QS = Skew Quadrupole
GSL = Generic Stripline
TunePkup = Tune Pickup
TuneShkr = Tune Shaker
InjKckr = Injection Kicker
InjSept = Injection Septum
EjeKckr = Ejection Kicker
Scrn = Fluorescent Screen
|Frequency (e-gun and SHB)||499.664||499.664||MHz|
|Frequency (accelerating structures)||2997.948||2997.948||MHz|
|Normalized emittance||≤ 50||≤ 50||π·mm·mrad|
|Relative energy spread (rms)||≤ 0.5||≤ 0.5||%|
|Pulse to pulse energy variation||≤ 0.25||≤ 0.25||%|
|Pulse to pulse beam position variation||≤ 0.20||≤ 0.20||mm|
|Pulse to pulse jitter||≤ 100||≤ 100||ps|
|Pulse charge||≥ 3||≥ 1||nC|
|Pulse duration||150 to 300||≤ 1||ns|
Linac pictures during FAT at SSRF
Linac to booster transport line (LTB)
The Linac-to-Booster transport line (LTB - naming initials TB), transports the 150 MeV electron beam from the end of the Linac (by definition, just after the last accelerating section) to the booster injection point, just after the injection septum. A general view of the LTB transport line is shown in Figure 24. A schematic diagram of the LTB elements with their names is shown in Figure 25. The main parameters are shown in Table 15.
|Total length including septum||21.2475||m|
The LTB lattice is composed of two chromatic sections separated by an achromatic section. In the first chromatic section just after the Linac the dispersion function is negative, created by two negative deflection dipoles. The first dipole is also a beam energy spectrometer for the Linac. An energy slit is placed in this section at the position where the energy resolution is maximum, i.e., where the relation ηx/βx is maximum. In the achromatic section, the LTB line goes through a shielding wall separating the Linac from the storage ring tunnel. In the last chromatic section the dispersion is positive and the optical functions are matched to the Booster's functions. The beam is injected on-axis into the Booster.
Since the optical functions at the Linac exit are not precisely known, we have considered 6 different initial condition modes (M1, ... , M6) for the transport line and looked for solutions that are flexible enough to accommodate all modes. The initial Twiss functions for the modes are shown in Table 16, where several different combinations of initial αx and αy are covered. The corresponding optical functions are shown in Table 17.
|Linac exit||Booster injection|
LTB Beam Size and Aperture Requirement
The figures below show the horizontal and vertical beam sizes along LTB for the different modes M1 to M6. The beam parameters at the Linac end are: emittance: εx = εy = 170 nm.rad; rms energy spread: δ=0.5%.
The aperture requirements and BSC for the LTB transport line are calculated considering a clearance for residual orbit distortion and trajectory variations due to pulse to pulse differences in beam energy and launching conditions (position and angle) as described in Table 18.
|Max. orbit distortion along LTB (H and V)||± 1||mm|
|Linac pulse to pulse energy variation||0.4||%|
|Linac pulse to pulse beam centroid position/angle variation (H and V)||40||nm.rad|
|LTB vacuum chamber inner aperture (full) at quadrupoles (H / V)||36.1 / 36.1||mm|
|LTB vacuum chamber inner aperture (full) at dipoles (H / V)||23.4 / 23.4||mm|
|Number of beam sigmas that fit in the aperture (H / V)||± 3.3 / ± 4.6|
The calculated horizontal and vertical BSC for LTB transport line modes are shown in the figures below.
LTB Orbit Correction
The LTB orbit correction parameters are shown in Table 19 and the magnet alignment and excitation error tolerances assumed for orbit correction studies in the LTB transport line are shown in Table 20. The trajectory correction statistics are shown in Table 21 and the orbits before and after correction for all modes are shown in Table 22. The correction system uses a beam position measurement station with integrated fluorescent screens and striplines. The injection septum is used as corrector to adjust horizontal position and angle at the booster injection point.
|Number of beam position measurement stations||6|
|Number of horizontal correctors||6|
|Number of vertical correctors||6|
|Maximum corrector strength||± 2.5||mrad|
LTB Diagnostic Elements
|Bunch waveform (micro & macro bunch)||Stripline BPM||1% peak value|
|Bunch charge||ICT||2.0 %|
|Position||Stripline BPM||0.2 mm|
|Fluorescent screen||0.5 mm|
|Profile||Fluorescent screen||0.5 mm|
|Energy slit / collimation||Beam scraper||0.1 mm|
LTB Horizontal/Vertical Scrapers
A horizontal scraper with a pair of blades will be used in the LTB line as energy slit, to select the portion of the beam coming from the Linac that is within the Booster energy acceptance at injection energy. For this purpose it is installed at a high dispersion and low horizontal betatron function place. An identical scraper will also be installed in the vertical plane, at a high vertical betatron function place, to reduce particle losses at the septum or in the Booster.
|Quantity of scrapers||2||(1 horizontal + 1 vertical)|
|Number of blades per scraper *||2|
|Blade position resolution (transverse)||10||μm|
|Motion range per blade **||20||mm|
|* Blades with independent movement|
|** Vacuum chamber diameter ∅=36 mm|
LTB Beam Position Measurement Stations
Beam position measurement stations combining a stripline BPM and a fluorescent screen will be used in the LTB line to measure the beam position. The specifications for the stripline BPMs are shown in Table 25 and for the fluorescent screens in Table 26.
|Resolution @ 1 nC||100||μm|
|Measurement range (radius)||± 5||mm|
|Spatial resolution (transverse)||0.2||mm|
|Mechanical positioning accuracy||0.2||mm|
|Mechanical positioning repeatability||0.1||mm|
Two Integrating Current Transformers (ICTs) will be installed at the LTB extremities allowing measurements of its transmission efficiency. The first ICT is being purchased as part of the Linac. The specifications for the LTB ICTs are shown in Table 27.
|Quantity||3||(1 from Linac + 2)|
|Model||ID 34.9 - Bergoz CF4.5"-34.9-40/1.1|
Booster to storage ring transport line (BTS)
The main function of the booster-to-storage ring transport line (BTS - with naming initials TS), is to transport the 3.0 GeV electron beam from the booster synchrotron to the storage ring (SR). The geometric requirements for this line are determined by the lattice of the two accelerators located in the same machine tunnel. Since the BTS line traverses the machine tunnel, a long element-free drift section is required to facilitate the passage of people and equipment through this region. To save costs, the BTS line uses the same quadrupoles as the storage ring and same dipoles and correctors as the booster. The booster dipole is scaled to a lower value as compared to the booster peak value, so that it can be set to DC operation without overheating the coils.
|Total length including septa||26.89||m|
|Number of dipoles||3|
|Number of quadrupoles||8|
|Number of horizontal correctors (CH + septa)||4 + 2|
|Number of vertical correctors||6|
The BTS lattice is designed with the same magnetic elements of the Booster (dipoles and correctors) and of the Storage Ring (quadrupoles Q14 and Q20). We define the BTS transport line from the Booster extraction septum to the Storage Ring thin injection septum, with both septa included.
The BTS initial optical conditions are matched to the Booster parameters at the extraction point, but the final conditions are mismatched to optimize the injection efficiency into the storage ring with a nonlinear kicker (NLK). The 2 modes M1 and M2 are optimized for different positions of the injected beam at the NLK. Mode M1 assumes the beam is injected at x=-8.0 mm, close to the NLK field maximum, and mode M2 assumes the beam is injected at x=-5.3 mm, where the NLK field has a slope. We expect to inject the beam close to the NLK field maximum, but, for safety, in case the dynamic aperture is not sufficiently large in the beginning, we have studied the possibility of injecting the beam closer to the storage ring axis.
In addition to usual requirements for beam transmission, the BTS transport line optics also requires a free space to allow people and equipment to pass by bending below the beam pipe.
|Booster extraction point||Storage Ring injection point||NLK|
The considerations for the extracted beam amplitude at the booster extraction septum are described in Table 10, and the considerations for the injected beam aperture at the storage ring injection point (end of thin septum), in Table 31.
BTS Beam Size and Aperture Requirement
The figures below show the horizontal and vertical beam sizes along BTS for the modes M1 and M2.
The aperture requirements and beam stay clear (BSC) for the BTS transport line are calculated considering a clearance for residual orbit distortion and trajectory variations due to pulse to pulse differences in beam energy and launching conditions (position and angle) as described in Table 32.
|Max. orbit distortion along LTB (H and V)||± 1||mm|
|Booster pulse to pulse energy variation||0.1||%|
|Booster pulse to pulse beam centroid position/angle variation (H and V)||0.4||nm.rad|
|BTS vacuum chamber inner aperture (full) at quadrupoles (H / V)||23.4 / 23.4||mm|
|BTS vacuum chamber inner aperture (full) at dipoles (H / V)||23.4 / 23.4||mm|
The calculated horizontal and vertical BSC for BTS transport line modes are shown in the figures below.
BTS Orbit Correction
The BTS transport line orbit correction main parameters are shown in Table 33. The specification of error tolerances takes into consideration the maximum Booster corrector strength at 3 GeV, since the same correctors are used. The tolerances are shown in Table 34 and the correction statistics in Table 35. The orbits before and after correction for all modes are shown in Table 36. The correction system uses the same beam position measurement station with integrated fluorescent screens and striplines. The Booster extraction septum is used as a horizontal corrector as well as the storage ring injection septa, that are used to adjust horizontal position and angle at the storage ring injection point.
|Number of beam position measurement stations||5|
|Number of horizontal correctors||4|
|Number of septa used as horizontal corrector||2|
|Number of vertical correctors||6|
|Maximum corrector strength||±0.35||mrad|
BTS Diagnostics Elements
Injection into the Storage Ring
The injection point in the storage ring is, by definition, the physical end of the thin septum.
Injection with Nonlinear Kicker (InjNLKckr)
On-axis injection (InjDpKckr)
Pulsed Magnets Parameters
|Injection||Extraction Thin||Extraction Thick||Injection Thick||Injection Thin|
|Nominal magnetic field||0.380||-1.089||-1.089||1.089||1.089||T|
|Nominal beam trajectory radius||1.317||-9.188||-9.188||9.188||9.188||m|
|Magnet shape radius||1.350||-12.500||12.500||12.500||12.500||m|
|Horizontal Beam Stay Clear (full)||≥ 22||≥ 9||≥ 9||≥ 9||≥ 9||mm|
|Vertical Beam Stay Clear (full)||≥ 16||≥ 8||≥ 8||≥ 7||≥ 7||mm|
|Amplitude reproducibility (rms)||≤ 0.07||≤ 0.006||≤ 0.006||≤ 0.017||≤ 0.017||%|
|Flat top (rms)||≤ 0.07||≤ 0.006||≤ 0.006||≤ 0.017||≤ 0.017||%|
|Flat top width||150||150||150||150||150||ns|
|Minimum half-sine pulse duration to satisfy flat-top||≥ 6.2||≥ 21||≥ 21||≥ 12.6||≥ 12.6||μs|
|Integrated leak field||≤ 50||≤ 200||≤ 200||≤ 3.7||≤ 3.7||G.cm|
|Nominal deflection||19.34||2.52||6.1||2.9 (@x=-8 mm)||mrad|
|Maximum deflection||23.4||2.9||6.7||3.4 (@x=-6 mm)||mrad|
|Nominal integrated field||0.010||0.025||0.061||0.029 (@x=-8 mm)||T.m|
|Maximum integrated field||0.012||0.029||0.067||0.034 (@x=-6 mm)||T.m|
|Horizontal Beam Stay Clear (full)||≥ 36||≥ 36||≥ 24||≥ 24||mm|
|Vertical Beam Stay Clear (full)||≥ 15||≥ 15||≥ 9||≥ 9||mm|
|Amplitude reproducibility (rms)||≤ 0.31||≤ 0.37||≤ 0.33||≤ 0.7||%|
|Flat top (peak-to-peak)||≤ 0.62||≤ 0.74||≤ 0.66||≤ 1.4||%|
|Flat top width||150||150||150||150||ns|
|Rise time||-||≤ 1.5||-||-||μs|
|Fall time||≤ 1.5||-||≤ 1.5||≤ 1.5||μs|
|Minimum half-sine pulse duration to satisfy flat-top||≥ 3.0||≥ 2.7||≥ 2.9||≥ 2.0||μs|
|Tolerance for integrated dipole field at center 1||-||-||-||< 3.7||G.cm|
|Tolerance for integrated quadrupole gradient at center 2||-||-||-||< 0.12||T|
1 For horizontal stored beam centroid oscillation < 10% of beam size.
2 For horizontal stored beam size oscillation < 10%.
Assumptions for aperture requirements and tolerance calculations
|Septum||Booster Injection||Booster Extraction||Storage Ring Injection|
|beam size, ±3σ||mm||±6.3||±4.0||±1.0||±1.0||±1.0||±0.5|
|tolerance for vacuum chamber||mm||±1.0||±1.0||±1.0||±1.0||±1.0||±1.0|
|Total Full Size||mm||22||16.0||9.0||8.0||9.0||7.0|
Flat-top and pulse-to-pulse reproducibility
The requirements for the 3 GeV kickers and septa flat-top and pulse-to-pulse reproducibility were determined based on the NLK injection efficiency in the storage ring. The injected beam position and angle stability at the NLK are required to be within Δx < 150 μm and Δx' < 50 μrad. Random variations in the injected beam position and angle at the NLK are supposed to be caused by uncorrelated random variations in:
1) Booster extraction kickers
2) Booster extraction septa
3) Storage ring thick injection septum
4) Storage ring thin injection septum
5) Storage ring NLKckr
6) BTS transport line magnet vibrations and ripple
If we suppose these effects add in quadrature and have the same weight, the tolerance for each contribution becomes:
Δx < 60 μm and Δx' < 20 μrad
1) Booster extraction kickers (bek): A variation in the bek kick (same kick in both kickers) affects the position and angle at the NLK according to:
Δx [mm] = 11.1 Δθbek[mrad] and Δx' = 3.4 Δθbek , thus Δθbek < 5.4 μrad or Δθbek/θbek < 0.3%
2) Booster extraction septa (bes): A variation in the bes kick (same kick in both septa) affects the position and angle at the NLK according to:
Δx [mm] = 13.9 Δθbes[mrad] and Δx' = 0.5 Δθbes , thus Δθbes < 4.3 μrad or Δθbes/θbes < 0.006%
3) Storage ring thick injection septum (stk): A variation in the stk kick affects the position and angle at the NLK according to:
Δx [mm] = 5.3 Δθstk[mrad] and Δx' = Δθstk , thus Δθstk < 11.4 μrad or Δθstk/θstk < 0.01%
4) Storage ring thin injection septum (stn): A variation in the stn kick affects the position and angle at the NLK according to:
Δx [mm] = 3.7 Δθstn[mrad] and Δx' = Δθstn , thus Δθstn < 16.2 μrad or Δθstn/θstn < 0.019%
5) Storage ring NLKckr (pmm):
Δθpmm < 20 μrad or Δθpmm/θpmm < 0.7%
For the other pulsed magnets:
1) Storage ring on-axis injection kicker (sik): The tolerance is calculated by requiring that the injected beam oscillation amplitude be limited to Δxmax < 0.5 mm along the storage ring horizontal plane. The trajectory of the electron beam after a residual kick at sik is:
Δx(s) = Δθsik [βx,sik βx(s)]1/2 sin(Δφ)
Δxmax is limited if
Δθsik < Δxmax / [βx,sik βx,max]1/2
Using βx,sik=18.6 m and βx,max=19.3 m, we have
Δθsik < 26.4 μrad or Δθsik/θsik < 0.33 %
2) Booster on-axis injection kicker (bik): The tolerance can be calculated in a similar way, by requiring that the injected beam oscillation amplitude be limited to Δxmax < 1.5 mm along the booster horizontal plane. When specifying the booster aperture, an allowance of 4.5 mm was considered for beam oscillations after injection. For the booster βx,bik=17.9 m and βx,max=23.2 m, so we have
Δθbik < 0.074 mrad or Δθbik/θbik < 0.31 %
3) Booster injection septum (bis): A variation in the bis kick affects the position and angle at the injection kicker (bik) according to:
Δx [mm] = 1.85 Δθbis[mrad] and Δx' = 0.2 Δθbis ,
if we require Δx < 0.5 mm at the booster injection kicker, we have:
Δθbis < 0.27 mrad or Δθbis/θbis < 0.07%
To calculate the allowed storage ring injection septa leakage field at the stored beam position in the storage ring, we have assumed that the oscillations caused by the perturbation be limited to 10% of the beam size. The beam trajectory after a kick is given by:
Δx(s) = Δθ0 [β0β(s)]1/2 sin[φ(s)-φ0] < 0.1 σx(s) = 0.1 [εx β(s)]1/2
and as a worst-case estimate we take sin[φ(s)-φ0]=1, and the residual kick at the stored beam position should satisfy:
Δθ0 < 0.1 [εx/β0]1/2 for εx=0.27 nm.rad and β0=20 m, we have Δθ0 < 0.37 μrad, or ∫B.dl < 3.7 G.cm at E = 3 GeV.
For the booster extraction septum, we set the allowed leakage field so that the orbit distortion amplitude is limited to 0.3 mm.
Δx(s) = Δθ0 [β0β(s)]1/2 sin[φ(s)-φ0] < 0.3 mm , then for βx,max=23.2 m and β0=9.2 m, we have Δθ0 < 20 μrad, or ∫B.dl < 200 G.cm at E = 3 GeV.
For the booster injection septum, we set the allowed leakage field so that the orbit distortion amplitude is limited to 1.5 mm.
Δx(s) = Δθ0 [β0β(s)]1/2 sin[φ(s)-φ0] < 1.5 mm , then for βx,max=23.2 m and β0=9.2 m, we have Δθ0 < 0.1 mrad, or ∫B.dl < 50 G.cm at E = 150 MeV.
Half-sine pulse duration
If we define the pulse amplitude by
A = A0 cos(π t/T)
then the half-sine pulse duration T required for a flat-top dA/A0 in the time range 2t0 << T is given by
T = π t0 * (2 dA/A0)-1/2