Boosters for synchrotrons - Aarhus Universitet · Chapter 1 gives an introduction to synchrotrons...

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Transcript of Boosters for synchrotrons - Aarhus Universitet · Chapter 1 gives an introduction to synchrotrons...

Page 1: Boosters for synchrotrons - Aarhus Universitet · Chapter 1 gives an introduction to synchrotrons and in particular the design goals and so- lutions for the Australian Synchrotron

Boosters for synchrotrons exemplied by the Australian Synchrotron

Booster

Thesis submitted for the degree ofPh.D. in accelerator physics

by

Søren Vinter Weber

December 18, 2007

Danfysik A/SJyllinge, Denmark

Institute for Storage Ring FacilitiesUniversity of Aarhus, Denmark

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Copyright c©2007 Søren Vinter Weber. This thesis may be freely copied, distributed and transmit-

ted. It is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of

this license, visit http://creativecommons.org/licenses/by/3.0/ or send a letter to Creative Commons,

171 Second Street, Suite 300, San Francisco, California, 94105, USA. An electronic copy of this

thesis can be obtained from the ISA webpage http://www.isa.au.dk, or by contacting the author at

[email protected].

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About this thesis

This thesis is submitted to the Faculty of Science at Aarhus University, Denmark to fulllthe requirements of the Ph.D. degree. It is the product of a collaboration between thecompany Danfysik A/S and the Institute for Storage Ring Facilities at Aarhus Univer-sity (ISA) on an Industrial Ph.D. concerning the booster synchrotron accelerator for theAustralian Synchrotron Project in Melbourne, Australia.

The contract to build a full energy turn-key injection system for the Australian Syn-chrotron Project (ASP) storage ring was awarded Danfysik A/S in December 2003, andthe completed and fully operational system was accepted in October 2006.

Danfysik A/S is a supplier of accelerator components as well as complete acceleratorsystems. The larger part of the Angströmquelle Karlsruhe (ANKA) injector was suppliedby Danfysik in 1999 [PBE+01]. In 2002 the injector for the Canadian Light Source (CLS)was completed [GPMD04]. In the past the design, construction, and commissioning of anaccelerator facility was done by the scientic community itself. Construction of certaincomponents was often subcontracted to industrial partners. However, the constructionof larger, international accelerator facilities together with a maturing industry initiated atrend to transfer more and more of the overall construction from the scientic communityto industry. This has also been the case for Danfysik A/S. The fraction of the whole systembeing delivered by Danfysik A/S increased from ANKA to CLS. With the ASP injectionsystem Danfysik A/S committed itself to delivering a complete turn-key system, includingperformance demonstration. In construction of the injection system, Danfysik A/S madeuse of sub-contractors for larger parts for which it had insucient expertise. The Germancompany ACCEL Instruments delivered the 100 MeV pre-injector linac, the booster cavityand low-level RF, including the necessary software for their operation [PDM+06], and theSlovenian company Cosylab delivered the control system software [ZVP04].

To be able to independently supply and commission particle accelerators, DanfysikA/S had to acquire knowledge on accelerator physics and beam dynamics. Starting withthe ANKA project, Danfysik A/S made an arrangement with ISA to supply expertise inaccelerator research and development and commissioning. To transfer knowledge of thedesign, building, and characterization of circular accelerators from ISA to Danfysik A/S,the collaboration on the ASP injection system includes an industrial Ph.D. project on thebooster synchrotron.

This thesis is in large part based on the work done in connection with the commissioningand performance demonstration of the ASP injection system, with focus on the boostersynchrotron.

Outline

Chapter 1 gives an introduction to synchrotrons and in particular the design goals and so-lutions for the Australian Synchrotron booster. Chapter 2 describes the combined-function

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magnets and their inuence on the beam. Theoretical beam dynamics studies in the formof numerical simulations and analytical investigations are presented in chapter 3. Chapter4 contains the experimental ndings from the commissioning and performance vericationof the injection system, and a comparison to other booster synchrotrons are made. Chapter5 contains the summary and recommendations.

Involvements

The magnet lattice of the booster was designed by Søren Pape Møller (ISA) inspired by theSwiss Light Source booster lattice [GIJ+98], for which Danfysik A/S built the combined-function magnets. The detailed design of the injection system was done by Danfysik A/Sin collaboration with Søren Pape Møller.

The measurements of the combined-function magnets was done by Danfysik A/S, withSøren Pape Møller and the author evaluating the accelerator physics aspects of the mea-surement results as part of the quality assurance process.

The analytical calculations (section 3.2) were done solely by the author. The numericalinvestigation (section 3.1) were done by the author, with practical assistance with thesoftware from primarily Etienne Forest (KEK, Japan) in relation to MAD-XP [SSF07] andAndreas Streun (SLS, Switzerland) in relation to Tracy2 [Ben97].

The commissioning of the ASP booster was done by a collaboration consisting of SørenPape Møller, Jørgen S. Nielsen, and Niels Hertel (ISA), the author (Danfysik A/S andISA), and the Major Projects Victoria (MPV) Delivery Team: Greg LeBlanc, Marc Boland,Rohan Dowd, Eugene Tan, and Martin Spencer. The MPV physicists were present andactively participating during the entire commissioning period, although their presence wasreduced in the later part of the injection system commissioning as they needed to preparefor the storage ring commissioning. The ISA personnel were present in shifts of 35 weeks,with overlaps of up to a week. The author was present during the entire commissioningperiod, except the last two-week shift in August 2006.

Acknowledgements

Several people have provided support and feedback during this Ph.D. project. Søren Pape-Møller has been an excellent, committed and present supervisor whose large knowledge ofaccelerator physics has been invaluable for the academic part of the project. He hasalso been good to work with during the commissioning of the injection system. For thecommercial part of the project, Managing Director of Danfysik A/S Bjarne Roger Nielsenhas been an extremely helpful supervisor, both concerning information on Danfysik A/Sand the writing of the business report.

During the commissioning, Niels Hertel and Jørgen S. Nielsen were pleasant workingpartners, and I would also like to thank Jørgen for his proofreading of this thesis. DanfysikRF and diagnostics specialist Povl Abrahamsen was also very helpful prior to and duringcommissioning with advice and help in these areas.

The ASP commissioning team: Greg LeBlanc, Mark Boland, Rohan Dowd, EugeneTan, and Martin Spencer have been great to work with before and during the commission-ing, and they have also been helpful in providing me with information and supplementalmeasurements after the commissioning was nished.

Last, but not least, I would like to thank my wife Mette Vinter Weber for her supportduring the course of this project, and particularly her patience during the extended periodI spent in Australia.

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I have made contributions to the following scientic papers1 during the project [FNBB+06,FNM05, MBB+05, WBB+07].

1In some papers I am credited with my previous surname Friis-Nielsen.

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Contents

1 Overview and design of the ASP injection system 1

1.1 Introduction to synchrotrons . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Design goals and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Linac and transfer lines overview . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Booster synchrotron overview . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.1 Injection and extraction . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.2 The RF system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Diagnostics equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6.1 Wall Current Monitor and Faraday Cup . . . . . . . . . . . . . . . . 131.6.2 Energy Dening Slits . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.3 Beam viewers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.4 Synchrotron Light Monitors . . . . . . . . . . . . . . . . . . . . . . . 141.6.5 Fast Current Transformers . . . . . . . . . . . . . . . . . . . . . . . . 151.6.6 DC Current Transformer . . . . . . . . . . . . . . . . . . . . . . . . . 151.6.7 Strip lines and tune measurement . . . . . . . . . . . . . . . . . . . . 151.6.8 Beam Position Monitors . . . . . . . . . . . . . . . . . . . . . . . . . 151.6.9 Beam Loss Monitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 The combined-function magnets 18

2.1 Basic formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Induced eddy currents in the vacuum chamber . . . . . . . . . . . . . . . . . 182.3 O-axis elds in a combined-function magnet . . . . . . . . . . . . . . . . . 192.4 Tune-shifts due to o-axis beam and variable dipole current . . . . . . . . . 212.5 Measurement of the combined-function magnets . . . . . . . . . . . . . . . . 24

2.5.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.2 Measuring the multipole components . . . . . . . . . . . . . . . . . . 242.5.3 Multipole measurement results . . . . . . . . . . . . . . . . . . . . . 252.5.4 Integrated dipole measurement results . . . . . . . . . . . . . . . . . 27

3 Theoretical beam dynamics studies 31

3.1 Transverse dynamic aperture . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.2 The transverse apertures and investigated area . . . . . . . . . . . . 323.1.3 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.4 Tune maps for the error-free lattice . . . . . . . . . . . . . . . . . . . 333.1.5 The inuence of misalignments and eld errors . . . . . . . . . . . . 35

3.1.5.1 Closed-orbit excursion eects . . . . . . . . . . . . . . . . . 393.1.6 Long-term stability and losses . . . . . . . . . . . . . . . . . . . . . . 40

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3.1.7 Summary, further investigations, and optimizations . . . . . . . . . . 423.2 Longitudinal acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Experimental results 46

4.1 Linac emittance and momentum-spread . . . . . . . . . . . . . . . . . . . . 464.2 Closed-orbit and beam position . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Tunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.1 Tune sensitivity to BF/BD ratio . . . . . . . . . . . . . . . . . . . . 534.3.2 Average beta function at the QD positions . . . . . . . . . . . . . . . 58

4.4 Chromaticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.5 Dispersion function and transverse momentum acceptance . . . . . . . . . . 594.6 Emittance measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.7 Position stability of the extracted beam . . . . . . . . . . . . . . . . . . . . 664.8 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8.1 Multi-bunch mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.8.2 Single-bunch mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.8.3 Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.8.3.1 Momentum acceptance . . . . . . . . . . . . . . . . . . . . 704.8.3.2 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.9 Performance and improvement suggestions . . . . . . . . . . . . . . . . . . . 724.9.1 Comparison to other booster synchrotrons . . . . . . . . . . . . . . . 724.9.2 Summary and recommendations . . . . . . . . . . . . . . . . . . . . 73

5 Summary 86

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List of Figures

1.1 Simplied synchrotron and ASP overview . . . . . . . . . . . . . . . . . . . 21.2 Phase-space ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Linac multi-bunch beam signal . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 LTB Twiss parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 BTS Twiss parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Booster lattice functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Injection kicker pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.8 Extraction septum and kicker pulses . . . . . . . . . . . . . . . . . . . . . . 121.9 View of the BPM button layout. . . . . . . . . . . . . . . . . . . . . . . . . 161.10 BPM cable attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Induced sextupole strengths from eddy currents . . . . . . . . . . . . . . . . 192.2 Change in eective focusing with BF/BD ratio . . . . . . . . . . . . . . . . 232.3 Measurement of a BF magnet . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 CF quadrupole and sextupole components . . . . . . . . . . . . . . . . . . . 262.5 3D calculation of BF magnet saturation . . . . . . . . . . . . . . . . . . . . 272.6 Tunes and chromaticities calculated from the CF multipole measurements . 282.7 Dipole kicks from CF magnet variations . . . . . . . . . . . . . . . . . . . . 292.8 Distribution of dipole kick angles . . . . . . . . . . . . . . . . . . . . . . . . 292.9 Closed-orbits excursion from dipole kicks . . . . . . . . . . . . . . . . . . . . 30

3.1 Maximum beam envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Aperture sizes at injection point . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Tune map for on-momentum particles . . . . . . . . . . . . . . . . . . . . . 353.4 Tune maps for o-momentum particles . . . . . . . . . . . . . . . . . . . . . 363.5 Dynamic aperture for selected tunes . . . . . . . . . . . . . . . . . . . . . . 373.6 Usable aperture when including errors . . . . . . . . . . . . . . . . . . . . . 383.7 Dynamic aperture for best and worst case . . . . . . . . . . . . . . . . . . . 393.8 Dynamic aperture as function of closed-orbit RMS excursions . . . . . . . . 403.9 Simulated longterm transverse stability and losses without damping . . . . . 413.10 Alternate tracking grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.11 The RF eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.12 Longitudinal phase-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Linac momentum-spread estimate . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Closed-orbit during ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Tune measurement with spectrum analyzer . . . . . . . . . . . . . . . . . . 504.4 The ramping curves for the main magnets . . . . . . . . . . . . . . . . . . . 514.5 Tunes as a function of time . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Tune variation during ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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4.7 Dynamic aperture close to injection . . . . . . . . . . . . . . . . . . . . . . . 534.8 Horizontal dierence orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.9 The ratio of BF, QD, and QF to BD during ramping . . . . . . . . . . . . . 554.10 Calculated focusing and tunes with changing BF/BD ratio . . . . . . . . . . 564.11 Measured tune shift with RF frequency . . . . . . . . . . . . . . . . . . . . . 604.12 The dispersion function at 100 MeV and 3 GeV . . . . . . . . . . . . . . . . 614.13 The dispersion function by quadrant at 100 MeV and 3 GeV . . . . . . . . . 624.14 Current as a function of momentum-spread at 100 MeV . . . . . . . . . . . 624.15 Emittance measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . 634.16 Transverse beam size on CCD chip with camera position . . . . . . . . . . . 644.17 Synchrotron light images at 3 GeV . . . . . . . . . . . . . . . . . . . . . . . 754.18 SLM captures of the beam during ramp . . . . . . . . . . . . . . . . . . . . 764.19 The emittances during ramping . . . . . . . . . . . . . . . . . . . . . . . . . 764.20 Horizontal beam proles at 550 ms . . . . . . . . . . . . . . . . . . . . . . . 774.21 Position stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.22 Extraction septum stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.23 Kicker inuence on extraction septum stability . . . . . . . . . . . . . . . . 794.24 First hundreds of turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.25 Multi-bunch current during ramp . . . . . . . . . . . . . . . . . . . . . . . . 814.26 Single turn extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.27 The multi-bunch trace through the injection system . . . . . . . . . . . . . . 834.28 Single-bunch current during ramp . . . . . . . . . . . . . . . . . . . . . . . . 844.29 Single-bunch purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.30 Vacuum gauge readings for May and August 2006 . . . . . . . . . . . . . . . 85

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List of Tables

1.1 Linac specications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Design parameters of the combined-function magnets . . . . . . . . . . . . . 101.3 Design parameters of the tuning quadrupole and sextupole magnets . . . . . 101.4 Kickers and septa parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Main parameters for the booster synchrotron . . . . . . . . . . . . . . . . . 13

2.1 Average multipole values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Closed-orbit excursions at injection due to magnet variations . . . . . . . . 30

3.1 Misalignments and eld errors . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Total ∆p/p for particles with dierent phase θ . . . . . . . . . . . . . . . . . 44

4.1 Chromaticities at 0.1, 2, and 3 GeV . . . . . . . . . . . . . . . . . . . . . . . 594.2 Emittances up the ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 Beam losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4 Comparison to other booster synchrotrons . . . . . . . . . . . . . . . . . . . 72

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Chapter 1

Overview and design of the ASP

injection system

This chapter gives a short introduction to synchrotrons and some of the elementary beamdynamics of synchrotrons. After the introduction an overview of the entire injection systemis given.

1.1 Introduction to synchrotrons

Synchrotrons are a class of circular, charged particle accelerators. Figure 1.1(a) shows thebasic layout of an injection system consisting of a linac, a booster synchrotron, and transferlines. Inside the vacuum chamber, the beam of charged particles circulate. They are bendand focussed by the magnets, and accelerated by the RF cavity. Typical synchrotroncircumferences are in tens of meters to hundreds of meters.

There are several uses for a synchrotron. Three common uses are as a collider, as asynchrotron light source, and as an energy booster.

In a collider counter circulating beams are brought to collide at chosen interactionpoints around the synchrotron, or a single beam is directed at a xed target. Examplesof modern high energy colliders are the Relativistic Heavy Ion Collider (RHIC) [HFF+03],and the soon to be completed Large Hadron Collider (LHC) [Eva07].

Synchrotron light sources uses the synchrotron light created by the bending of electrons.A sketch of the Australian synchrotron light facility is shown in gure 1.1(b). The radiatedsynchrotron light power is given by [Jac01, eq. (14.46)]

P(t′)

=23

e2 |v|2

c3γ4

rel (1.1)

where e is the elementary charge, c is the speed of light, γrel is the relativistic gamma factor,and v is the rate of change of the particle velocity vector. The reason for using light, chargedparticles such as electrons is evident in the fourth power of the relativistic gamma factor.The light is emitted over a broad spectrum from far infrared to hard x-rays, and depends onthe beam energy and bending radius. Because the beam is highly relativistic (γrel >∼ 103),the synchrotron light is emitted in a narrow forward cone with an RMS opening angle of∼ γ−1

rel . The smaller the beam size in phase-space, called the emittance, and the greaterthe beam current, the larger the brightness of the synchrotron light. So-called insertion

devices: wigglers and undulators, are incorporated into the storage ring to increase thebrightness of the synchrotron light. The synchrotron light is used primarily in life andmaterial sciences.

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2 Overview and design of the ASP injection system

(a) Schematic view of an injection system based on a linac and booster synchrotron[Møl]

(b) Schematic view of ASP [ASP]

Figure 1.1: A schematic view of an injection system with a linac pre-accelerator, transfer lines,and a booster synchrotron is shown in (a), and in (b) is shown an overview of ASP. The greenline is the beam path, the yellow arrows indicated the beam direction, and the blue arrows showthe synchrotron light. From the electron source the beam is accelerated to 100 MeV in the linac,injected into the booster synchrotron and accelerated to 3 GeV, extracted, and injected into thestorage ring where it circulates giving o synchrotron radiation. The synchrotron light is usedat the experimental stations distributed around and outside the storage ring.

A booster synchrotron is used to boost the energy of the beam. Because increasing theenergy means increasing the magnetic elds that keep the beam bending radius constant,the magnets in a booster must be able to give a homogeneous eld over a large range in eldstrength. For the ASP booster this is from a beam energy of 0.1 to 3.0 GeV, i.e. a factorof 30. For modern storage rings the essential features of a booster, aside from stabilityand reproducibility, are a small beam size in phase-space (emittance) and a large beam

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1.2 Basic equations of motion 3

current. The large beam current reduces the time it takes for the booster to ll the storagering, while the small emittance is necessary as the storage ring itself has a small emittanceto achieve a high synchrotron light brightness. The smaller the emittance, the smaller themagnet gaps can be made which reduces the size and cost of the magnets. Also, a smallbooster emittance reduces the beam losses at injection and allow for top-up operation,where additional current is added to the existing beam in the storage ring, continuallyreplenishing that which has been lost due to gas-scattering and other eects. This allowsfor a more constant current in the storage ring, and thus a more constant brightness of thesynchrotron light used at the experimental stations.

1.2 Basic equations of motion

The force governing the particle motion is the Lorentz force

F = qe (E + v ×B) (1.2)

where qe is the particle charge, E is the electric eld vector, v is the particle velocity vector,and B is the magnetic ux density vector. The magnetic ux density is also referred toas the magnetic eld. The magnets responsible for the bending of the particle trajectoriesare the dipole magnets, while the quadrupole magnets focus the beam. Sextupole magnetsare used to correct the focusing of particles that have a slightly dierent energy from theaverage beam energy.

The bending of the beam of charged particles is dependent on the momentum of theparticles and the magnetic eld in the dipole magnet. The magnetic rigidity determinesthe relationship between the magnetic eld B, the bending radius ρ, and the beam particlemomentum p and charge qe

Bρ =p

|q| e(1.3)

If the particle momentum is either too great or too small, the particle will deviate fromthe desired orbit, hit the vacuum chamber wall and be lost.

The acceleration is done in the RF cavity by an oscillating electric eld, the frequencyof which is such that a particle arriving at the correct time, i.e. phase angle, alwaysexperiences the same electric eld at each full turn in the synchrotron. The eld frequencyin the cavity is usually in the radio frequency region, hence the name RF cavity . Because ofthe constant phase requirement, the beam particles will naturally be grouped into bunches,each bunch always arriving at the RF cavity at approximately the nominal phase angle.For synchrotrons the nominal phase angle is normally such that the synchronous particlearrives at the cavity when the electrical eld is zero, thus recieving no energy increment. Ifthe particle arrives sooner, it means that its momentum is smaller than the nominal and itwill recieve a small positive energy increment, and vice versa. Particles that arrive slightlyo the nominal phase angle will therefore oscillate around the nominal, stable phase angle.These oscillations are called synchtron oscillations. Acceleration is done by increasing themagnetic eld in the dipoles causing the beam to arrive earlier at the RF cavity thusreceiving an energy increment.

A synchrotron, unlike a betatron or a microtron, keeps the beam orbit radius constantduring acceleration by increasing the magnetic elds.

Apart from the longitudinal force of the RF, the beam is also aected by transverseforces that focuses the beam. In most cases the transverse forces can be described ac-curately by linear approximations. Designating the horizontal coordinate x (s), and the

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4 Overview and design of the ASP injection system

vertical coordinate y (s), where s is the arc length along the particle trajectory, the basicequations for the particle trajectory are

∂2x

∂s2−(

k (s)− 1ρ2

)x =

∆p

p0(1.4a)

∂2y

∂s2+ k (s) y = 0 (1.4b)

where ∆p/p0 is the momentum deviation of the particle from the nominal momentum p0,and k (s) is the strong focusing caused by a transverse gradient in the magnetic eld atposition s

k (s) = −g (s)Bρ

(1.5)

where g (s) is the gradient at s. There is no general agreement on the sign of k (s). Thegradient is often created by a quadrupole magnet, or by increasing/decreasing the pole gapin a dipole with x. For ∆p/p0 = 0, and using that the synchrotron is periodic, the solutionto (1.4) is an oscillating trajectory given by

xβ (s) =√

εxβx (s) cos (φx (s)− φ0x) (1.6a)

yβ (s) =√

εyβy (s) cos (φy (s)− φ0y) (1.6b)

where ε is the emittance, β (s) is the beta-function, φ0 is an arbitrary constant phase, and

φx,y (s) =∫ s

0

ds

βx,y (s)(1.7)

are the betatron phases. The numbers of betatron oscillations per revolution are called thebetatron tunes, or simply the tunes

Qx,y =12π

∮ds

βx,y (s)(1.8)

where∮denotes integration around the full circumference of the booster.

The value of the beta-function at any point in the synchrotron is determined uniquelyby the magnet lattice, while the emittance is a property of the beam although at highenergy it is aected by the lattice design. In phase-space a stable particle moves along anellipse, the so-called Courant Snyder invariant , given by

γz2 + 2αzz′ + βz′2 =εz

π(1.9)

where z = x, y, z′ = ∂z/∂s, and

α (s) = −β′ (s)2

(1.10a)

γ (s) =1 + α2 (s)

β (s)(1.10b)

where β′ = ∂β/∂s. The area of the phase-space ellipse is therefore πε. Geometric propertiesof the phase ellipse is shown in gure 1.2.

A beam bunch consists of ∼ 1010 particles, and each particle has its own emittance.However, for light particles, such as electrons the emittances from the electron source are

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1.2 Basic equations of motion 5

z′

z

z′ =

γzεz

πz′

int =

εz

πβz

z =

βzεz

π

zint =

εz

πγz

Figure 1.2: Geometric properties of the (z, z′) phase ellipse of area πεz.

to very good approximation Gaussian in distribution, and one can refer to the emittance ofthe beam as one standard deviation of the phase-space density distribution function. Theemittance is often given in units of π mm mrad, i.e. 50 π mm mrad. However, both theπ and mrad are often left out, e.g. 30 π nm rad and 30 nm are equivalent. As long asthe energy of the beam is conserved, i.e. no beamgas scattering, synchrotron radiation,and acceleration, the emittance is conserved. This is a consequence of Liouville's theorem,stating that in the absense of collissions and dissipation the phase-space density is constant[Zot06].

If the beam is accelerated the emittance shrinks, decreasing inversely proportional tothe momentum. This eect is known as adiabatic damping , and it can be intuitivelyunderstood by observing that the RF cavity increases only the longitudinal momentum,leaving the transverse momentum unchanged, thus the divergence of the beam is reduced.The normalized emittance is dened such that it is invariant during acceleration

εN = βrelγrelε (1.11)

where βrel and γrel are the beta and gamma functions from special relativity, respectively.The solutions (1.6) were for ∆p/p = 0. With momentum deviation, (1.3) shows that

the particle has depending on the sign of the deviation either a larger or smallerbending radius ρ + xD, and that if the magnetic eld remains constant it scales linearlywith the momentum deviation

xD = D (s)∆p

p(1.12)

where D (s) is the lattice dependent dispersion function. Thus, the trajectories in (1.6)become

x (s) = D (s)∆p

p+√

εxβx (s) cos (φx (s)− φ0x) (1.13)

y (s) =√

εyβy (s) cos (φy (s)− φ0y) (1.14)

Since there is no bending in the vertical plane, there is no dispersion in that plane. [RS94,Win94]

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6 Overview and design of the ASP injection system

1.3 Design goals and solutions

The primary goal of the ASP booster was to design a synchrotron which could deliver asmall emittance, while at the same time have a small circumference that would keep theprice down by having fewer magnets, and also allow it to t into the existing buildingdesign. The small emittance was desired to give a low loss in the injection into the storagering and to allow for top-up operation.

A small emittance can be achived by having

1. a large bending radius of the dipoles,

2. many cells,

3. a smaller dispersion in the bending magnets

From (1.4a) it is seen that the momentum-spread drives the perturbations. However, if thebending radius is very large it reduces the impact. A large bending radius means having alarge booster circumference and many magnets, since the sum of the bending angles mustadd up to 2π. This is counter of what was desired in the ASP booster and cannot be usedto reduce the emittance. However, other facilities does use this approach by installing theirbooster in the same tunnel as the storage ring, reducing the emittance but also increasingthe monetary cost of the booster.

A cell is a sequence of magnets where the lattice functions, i.e. βx,y (s) and D (s), andtheir derivaties are the same at the start and end of the cell, thus allowing the cells tobe placed in succession. A large number of cells is benecial: the horizontal emittance isinversely proportional to the cube of the number of cells. It is also proportional to thesquare of the beam energy [Win94]

εx = kLE2

N3c

(1.15)

where kL is a lattice type dependent constant, and Nc the number of identical cells. To haveboth many cells and a short circumference entails that the cells must be short. The latticetype chosen for the ASP booster is the so-called FODO lattice, consisting of a Focusingquadrupole magnet, a drift (dipole magnet), a Defocusing quadrupole, and a second drift(dipole magnet). It is a simple, eective, and well understood lattice type.

To have a small dispersion in the bending magnets requires the use of strong focus-ing quadrupoles. However, strong quadrupoles introduce large focusing variations for o-momentum particles, i.e. a large tune spread, increasing the risk of beam loss on tuneresonances. Strong sextupole magnets are needed to compensate for this eect which iscalled chromaticity. Consequently, each cell must include six magnets: the two dipole mag-nets, two quadrupoles of opposite polarity, and two sextupoles also of opposite polarity.If the three magnet types are combined into one magnet, which is possible because themagnetic eld is a vector eld, it reduces the number of magnets needed from six to two.

By combining the dipole, quadrupole, and sextupole magnets into one combined-function magnet, the length of the cell can be kept small, and more cells can be ttedinto the same space. The number of magnets and consequently also power supplies, canalso be reduced, reducing the over-all cost of the machine. The disadvantage of combiningthe magnets are that the freedom to individually adjust the magnets is lost. However, thisis a exibility that is rarely used to a large extend, and by including a few, smaller tuningquadrupoles and sextupoles in the lattice some tunability can be accomodated.

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1.4 Linac and transfer lines overview 7

In 2003 Danfysik A/S submitted two tender quotations for the ASP injection system:one based on separate-function (SF) dipole magnets, and a second based on combined-function (CF) dipole magnets [Møl03]. The reason behind submitting two quotationswas based on a caution principle: in case the CF lattice was considered to risky by thecustomer a SF lattice had also been submitted before the deadline. Both lattices had thesame circumference and the same price. The SF lattice had 20 focusing and 24 defocusingquadrupole magnets, which also included sextupole elds for chromaticity correction, whilethe CF lattice had 28 focusing and 32 defocusing magnets. In addition to this both latticeshad additional, weaker quadrupole magnets for tune correction. The higher number ofcells, N cf

c /N sfc ∼ 1.4, of the cf lattice resulted in an emittance that was 2.5 times smaller

as given by (1.15): the CF lattice had an emittance of 31 nm rad, while the SF latticehad an emittance of 77 nm rad. Due to the lower emittance of the CF lattice design, andDanfysik's prior experience in producing CF dipole magnets with sextupole elds [GIJ+98],the CF lattice was chosen for the ASP booster.

1.4 Linac and transfer lines overview

The ASP injector system (see full drawing at the back of this thesis) consists of a linac anda booster synchrotron. In addition there are the linac-to-booster (LTB) and the booster-to-storage ring (BTS) transfer lines.

The linac is a turn-key solution delivered to specications by the sub-contractor ACCELto Danfysik A/S [PDM+06]. The specications are detailed in table 1.1.

Table 1.1: The linac specications. [Møl04]

Energy Elinac ≥ 100 MeVPulse to pulse energy variation ≤ 0.25% (RMS)Relative energy spread ≤ 0.5% (RMS), ±1.5% (full width)Repetition rate ≤ 5 HzNormalized emittance εN ≤ 50 π mm mradEnd charge in single-bunch mode > 0.31 nCEnd charge in multi-bunch mode > 3.1 nC

The electrons are extracted from the electron source using a grid modulated at 500MHz. They are accelerated to 90 keV in a 4-cell 500 MHz pre-buncher. Before enteringthe two main accelerating structures, they are bunched at 3 GHz in a 16-cell nal buncher.Each 500 MHz bunch will on average consist of three 3 GHz bunches with 85% of theelectrons in the second bucket and 7% in the rst and third bucket. Each of the two 3GHz main accelerating structures provides the beam with 50 MeV. A quadrupole tripletis placed between the two structures, and 15 solenoids are used after the pre-buncher andpartway through the rst main accelerating structure to provide focusing of the beam. Thelinac can give either a single bunch of 2 ns or a bunch train with a length determined bythe trigger gate size. A length of about 150 ns (75 bunches) has been chosen.

When a highly charged bunch train enters the linac, the interaction of the front buncheswith the linac structures can create an electromagnetic eld of a strength comparable tothe accelerating RF elds. This is called beam loading and aects the trailing bunches.Figure 1.3 shows the multi-bunch beam signal from the FCT located at the start of theLTB. It is evident that the charge per bunch varies. It increases almost linearly in timein the beginning of the bunch train and drops o quickly at the end with a trailing tail of

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8 Overview and design of the ASP injection system

several nanoseconds. The reason is not known, and although the manufacturer believes itis due to beam loading, it would not explain why the leading bunches would be aected.

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0 20 40 60 80 100 120 140 160 180

Sig

nal

[V

]

Time [s]

"LTB FCT1"

Figure 1.3: The multi-bunch beam signal (negative sign) as observed by the rst FCT in theLTB.

The LTB provides an optically matched transport of the beam from the linac to thebooster: the horizontal and vertical beta functions and their derivatives, βH , β′H , βV , β′V ,together with the dispersion function and its derivative, D and D′, are all matched in thebeginning to the linac and at the end to the booster (see gure 1.4). Large parts of theLTB also has vanishing D and D′ to allow more accurate emittance measurements usinga so-called quadrupole scan.

Horizontal beam envelope [m] versus distance [m]

Vertical beam envelope [m] versus distance [m]

Note: Linear optics

0.000

0.000

16.361

16.361

0.003500

0.003500

-0.003500

-0.003500

Betatron envelope Momentum envelope

Figure 1.4: The beta functions (top left) and dispersion function (bottom left) of the LTB, andthe horizontal (top right) and vertical (bottom right) 1σ beam envelopes (ε = 0.25 mm mradand ∆p/p = 0.5%).

Once transported into the booster, the beam will be accelerated from 100 MeV to 3.0GeV, extracted into the BTS and injected into the storage ring. Like the LTB, the BTSprovides matched transport between the booster and the storage ring.

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1.5 Booster synchrotron overview 9

Horizontal beam envelope [m] versus distance [m]

Vertical beam envelope [m] versus distance [m]

Note: Linear optics

0.000

0.000

39.865

39.865

0.001500

0.001500

-0.001500

-0.001500

Betatron envelope Momentum envelope

Figure 1.5: The beta functions (top left) and dispersion function (bottom left) of the BTS, andthe horizontal (top right) and vertical (bottom right) 1σ beam envelopes (ε = 0.030 mm mradand ∆p/p = 0.094%).

1.5 Booster synchrotron overview

The full energy ASP booster was designed to have a small circumference of 130.2 m as itshould t within the designed shielding walls. By using strong focusing and many cellsthe booster is able to deliver a beam with a small emittance of ∼ 33 nm rad. The strongfocusing gives rise to large negative chromaticities that, if uncorrected, would cause a largetune spread. This would increase the probability of loosing particles on a resonance; inparticular at low energies where the energy-spread is high. There is also a risk of head-tailinstabilities. To correct these problems strong sextupoles are used.

Including most of the needed quadrupole elds in the bending magnets allows for ahigher number of cells with the same booster circumference. The sextupoles needed tocorrect the chromaticity to +1 in both planes should be placed close to the quadrupoles.This is achieved by also integrating the sextupole elds in the bending magnets. Including aquadrupole component in the bending magnets is common in synchrotrons, while includinga sextupole component is rare.

The booster lattice is four-fold super symmetric, each of the four arcs consisting of 8horizontally defocusing (BD) and 7 horizontally focusing (BF) CF magnets (see table 1.2).The BF magnets have a larger radius of curvature to ensure stability of the synchrotronoscillations [Wal94]. There are also 2 horizontally defocusing (QD), and 2 horizontallyfocusing (QF) quadrupole magnets in each of the four arcs. The QD and QF allows the tuneto be varied freely around the nominal tunes (QH , QV ) = (9.2, 3.25) in the intervals (9.05 9.45, 3.05 3.45). To allow additional chromaticity correction small tuning sextupoles havebeen included in equal numbers to the quadrupoles, and are able to adjust the chromaticityby ±1 unit at 3 GeV (see table 1.3). A total of 24 horizontal and 12 vertical correctordipole magnets are also present with maximum bending angles at 3 GeV of ±1.2 and ±0.6mrad, respectively.

There are four straight sections used for injection, extraction, RF, and diagnostics.A lattice that also makes use of combined-function magnets with sextupole elds hasbeen used successfully for the Swiss Light Source (SLS) booster. Their combined-function

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10 Overview and design of the ASP injection system

Table 1.2: Design parameters of the combined-function magnets.

Combined-function magnets BD BF

Number of magnets 32 28Magnetic eld (100 MeV 3 GeV) [T] 0.0418 1.2529 0.0148 0.4436Bend angle [] 8.250 3.429Arc length [m] 1.150 1.350Radius of curvature [m] 7.9867 22.5602

Gradient [T/m] (−dBz/dx) 6.6977 -8.2559at max. eld [T/m] (6.6662)a (−8.2681)a

Sextupole component [T/m2] 49.2477 -35.4062at max. eld [T/m] (50.7493)a (34.6162)a

Vacuum chamber inner dimension [mm2] 40× 24aThe measured average values.

Table 1.3: Design parameters of the tuning quadrupole and sextupole magnets.

Quadrupole magnet QF QD

Number of quadrupoles 8 83 GeV gradient g [T/m]

Nominal −23.51 4.00Maximum −27.00 6.50

Magnetic length [cm] 25 15

Sextupole magnet XH XV

Number of sextupoles 8 83 GeV gradient G [T/m2]

Nominal 0 0Maximum −313 481

Magnetic length [cm] 18 20

magnets were build and eld mapped by Danfysik. Figure 1.6 shows the ASP boosterlattice functions for a single arc.

1.5.1 Injection and extraction

The injection into the booster is through a thin septum. The septum blade is o-set 20 mmfrom the nominal booster orbit. Once through the septum, the beam crosses the boosteraxis one meter downstream where a fast kicker deects it by 18 mrad onto the boosteraxis. The injection kick delivers a virtually square pulse (see gure 1.7) which falls tozero within the time it takes for the head of the complete injected beam to return to theinjection point. The parameters for the kickers and septa are given in table 1.4.

The extraction is done in a single turn by a fast kicker deecting the beam into a thinseptum magnet. A slow four-magnet bump osetting the beam a maximum of 14.5 mmtowards the septum blade positioned 18 mm from the nominal orbit relaxes the requirementon the extraction kick angle from about 16 to 4.9 mrad.

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1.5 Booster synchrotron overview 11

Figure 1.6: The booster lattice functions for a quarter of the booster. XH and XV are thehorizontal and vertical tuning sextupoles, respectively.

Table 1.4: Parameters for the kickers and septa.

Septa Kickers

Injection Extraction Injection Extraction

Angle [mrad] 139 120 18 4.9Beam size at exit entrance

(σx, σy) [mm] (1.1, 0.9) (0.38, 0.1) Aperture

[mm2

]14× 14 12× 12 45× 30 40× 24

Length [mm] 1020 1520 1000 1000Thickness [mm] 1.8 1.5 Field [gauss] 455 7880 59 500

1.5.2 The RF system

The acceleration is done by the RF system consisting of a 5-cell cavity of type PETRA fedby a 75 kW Inductive Output Tube (IOT). The cavity and the low-level RF was deliveredand commissioned by sub-contractor ACCEL [PDM+06], while the IOT was bought fromthe company Thales Radio Broadcast Inc. (Virginia, USA) [Møl04] and commissioned by

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12 Overview and design of the ASP injection system

Figure 1.7: A trace of the injection kicker pulse. The division is 100 ns. The beam is injectedat t = 0 ns, and the pulse is constant for ∼170 ns after which the pulse falls to zero before a fullrevolution time (434 ns) has passed.

(a) Extraction kicker pulse (100 ns division) (b) Extraction septum pulse (100 µs division)

Figure 1.8: The traces from the extraction kicker (a) and septum (b). The kicker pulse risesfrom zero to the extraction eld strength within the (434− 150) ns = 284 ns it takes from thetail of the multi-bunch beam passes the kicker until the head arrives back at the kicker location.The ringing seen in the kicker pulse trace is due to reections in the monitor cable. The septumpulse has a sine-like shape, and the beam is extracted at the top.

Povl Abrahamsen from Danfysik A/S.The RF frequency is 499.654 MHz given by the same Master Oscillator (MO) that

supplies the linac (3 GHz) and storage ring RF. Using the same MO for all RF cavitiessimplies top-up operation, where a single bucket in the storage ring can be selected andlled by adjusting the trigger delay for the injection system as a whole, and keeping alldelays internal to the injection system constant. It also reduces the losses occurring dueto mismatch between the RF frequency and the bunch train time structure when goingbetween the dierent RF cavities (linac, booster, storage ring). The dis-advantage is thatit makes it dicult to change the RF frequency by very much in order to accommodate aparticular RF frequency preference (e.g. in the booster).

The maximum accelerating RF voltage is 1.2 MV. To reduce the RF induced energyspread at injection, which will add to the existing energy spread from the linac, the RFvoltage is ramped together with the booster magnets from around 120 kV at 100 MeV, seealso section 3.2. [Møl04, MBB+05]

1.6 Diagnostics equipment

The diagnostics equipment is used to measure beam and machine parameters and is es-sential during commissioning and fault-nding. Choosing the right kind of diagnostics isimportant. With the injection system being a commercial product the amount, and price,

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1.6 Diagnostics equipment 13

Table 1.5: The main parameters for the booster synchrotron.

General parameters

Energy E [GeV] 0.1 3.0Single-bunch mode current I [mA] > 0.5Multi-bunch mode current I [mA] > 5.0Bunch charge (single-/multi-bunch) q [nC] 0.22 / 0.03Circumference L [m] 130.2Revolution time T [ns] 434Injected pulse length [ns] 2 150Injected emittance εinj [nmrad] < 250Injected energy spread (RMS) ≤ 0.5%Injected bunch length [mm] ≤ 33

Lattice parameters

Tunes QH , QV 9.2, 3.25Natural chromaticities ξH , ξV -8.83, -11.50Momentum compaction factor αp 0.0098

Synchrotron radiation parameters

Natural emittance εH [nmrad] 0.04 33Energy loss per turn U0 [keV] 9× 10−4 743Synchrotron radiation power P [kW] 0 3.7Characteristic wavelength λc [Å] 44632 1.65Damping time, horizontal τH [ms] 72000 2.7Damping time, vertical τV [ms] 93000 3.5Damping time, longitudinal τE [ms] 54000 2.0

RF parameters

Damped energy spread σE/E 3.1× 10−5 94× 10−5

Damped bunch length σL [mm] 0.3 19RF frequency fRF [MHz] 499.654Harmonic number H 217RF voltage V [MV] 0.12 1.2Over-voltage factor Q inf 1.6Quantum lifetime [s] inf 205Synchrotron frequency (ν = Ω/2π) [kHz] 184 100

of the diagnostics equipment is kept to an absolute minimum. Every diagnostics equip-ment desired in the injection system is weighed as need to have or nice to have, withthe latter not being included unless it can be argued to save more money than it coststhrough a faster commissioning.

1.6.1 Wall Current Monitor and Faraday Cup

The linac is equipped with a Wall Current Monitor (WCM) and a Faraday Cup (FCP), bothinstalled immediately behind the electron source. The WCM allows for non-destructive

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14 Overview and design of the ASP injection system

current measurement, and is equipped with a ferrite ring to reduce higher order modesinduced in the stainless steel housing by the beam's passing. It has a bandwidth largerthan 2 GHz. The FCP uses stainless steel tubing and copper for the beam stop. It isinserted into the beam, thereby destroying it, using pneumatic drive, and has a bandwidthgreater than 2 GHz.

1.6.2 Energy Dening Slits

An Energy Dening Slit (EDS) is located in the LTB between the two rst bending magnetsand shortly before the rst beam viewer. It consists of two slit blades in the vacuumchamber. One of them is slightly displaced along the beam axis relative to the other,allowing them to overlap when the slits are fully closed. The movements of the slits arecontrolled by stepper motors. The position of each slit can be observed remotely on aPC and directly on a position indicator on the slits, and from this the slit opening andtransverse displacement relative to the center of the beam pipe can be inferred. When theslits are fully retracted the the opening inside the beam pipe is 50 mm.

1.6.3 Beam viewers

The LTB is equipped with three beam viewing screens, the booster with four, and the BTSwith three. The LTB screens, the BTS screens, and two of the booster screens (diagnosticsand RF sections) are operated by a single stroke actuator. The remaining two in thebooster (injection and extraction sections) are operated by a dual stroke actuator. Thescreens have a size of 50× 50 mm2 and consist of Cr-doped alumina (Al2O3). The screensare mounted at an angle of 45 degrees to the beam and can be inserted into the beam by apressurised-air actuator. All screens except the third LTB screen are mounted and insertedhorizontally into the beam. The third LTB screen, due to space restriction, is mountedand inserted into the beam vertically. Each screen is observed by a JAI A11 CCD camera[JAIb] via a quartz window. A cross hair is mounted 3 mm above the screen. The vieweris mounted and aligned so that the beam is observed centred on the cross hair when itis centred in the vacuum chamber. Inserting a viewer into the beam eectively stops thebeam at that point.

1.6.4 Synchrotron Light Monitors

Two Synchrotron Light Monitors (SLM) are mounted in the booster, one placed after theBD dipole (BD-2-03) and another after the BF dipole (BF-2-06). Synchrotron light emittedin a bending magnet is reected by a 15× 15 mm2 Ni:P coated at copper mirror angledat 46.4 (rst SLM) and 46.0 (second SLM) horizontally relative to the light path. Themirror can be inserted into the beam by a horizontally mounted manual operated actuatorwith a stroke from 0 to 25 mm. The nominal position is 13 mm into the chamber.

The reected light exits through a quartz window viewer port and is observed by atriggerable JAI A1 high resolution CCD camera [JAIa] mounted in the horizontal planeon the inside of the booster ring. The lens is mounted in a xed postion on the camerasupport and has a focal length of 200 mm. The position of the camera can be adjustedin the horizontal plane. Four intensity lters in standard 30 mm Barrel C-mounts withtransmissions T = 50%, 32%, 10% and 5%, respectively, can be mounted in front of thecamera. A blue colour lter can also be mounted in front of the camera allowing anincreased emittance measurement resolution (see section 4.6).

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1.6 Diagnostics equipment 15

1.6.5 Fast Current Transformers

There are four Bergoz in-ange Fast Current Transformers (FCTs) installed in the injectionsystem. Two are installed in the LTB: one at the very beginning just after a horizontalsteerer, and another at the very end just before the septum. The third FCT is installed inthe booster diagnostics section, and the fourth in the BTS just after SEP-A-3. The FCTshave a rise time down to 175 ps and up to 2 GHz bandwidth, and a nominal sensitivity of1.25 V/A.

1.6.6 DC Current Transformer

The Direct Current Transformer (DCCT) is a Bergoz New Parametric Current Transformer(NPCT-075-C000-H) with an optional Radiation Hard sensor head. The DCCT is placedin the diagnostics section of the booster and measures the DC current component of thecirculating beam. The voltage signal from the DCCT is analysed by a Bergoz electronicsmodule. The module gives a voltage proportional to the beam current, and it has anoutput range of ±10 V, corresponding linearly to ±20 mA beam current.

1.6.7 Strip lines and tune measurement

The tune measurement system consists of two identical strip line systems of each fourstrip lines shifted 90 relative to each other. The strip lines are oset 45 relative to thehorizontal plane to minimize the amount of synchrotron radiation irradiating the electrodes.Each strip has a length of 150 mm, equal to 1/4 of the distance between the booster bunches.One strip line system functions as an exciter and the other as a detector. The measurementis done using an Agilent spectrum analyser. The swept exciter signal from the spectrumanalyser is amplied by a 20 W RF amplier with a constant gain of 63 dB and a bandwidtharound 0.5 525 MHz. During commissioning it was chosen to drive only one strip by theamplier, thereby at the same time exciting equally vertical/horizontal oscillations due tothe 45 oset. RF combiners/splitters are used to combine/split the signals from the striplines as desired, though all tune measurements were carried out using only one pick-upmeasuring simultaniously both tunes.

The detector is placed a quarter of the booster circumference downstream of the ex-citer. This corresponds to ∼ 2.3 horizontal and ∼ 0.81 vertical betatron oscillations. Themaximum signal is achieved if the fractional part of the number of betatron oscillationsare either 1/4 or 3/4.

1.6.8 Beam Position Monitors

The Beam Position Monitor (BPM) system consists of 32 in-ange BPM units, each withfour button pickups placed symmetrically about the vertical and horizontal plane, and withand angle of ±30 with the horizontal plane (see gure 1.9). The button pickups are 50SMA FB from Metaceram, France. They are constructed with circular copper electrodescoupling capacitively to the booster bunches, and the button pickup signal is available onan SMA female coaxial connector. The buttons are welded in a separate BPM block withConat ange interface on the two sides connecting to the booster beam line. The inducedvoltage from each of the electrodes is read by Bergoz MX-BPM modules mounted in aBergoz BPM-RFC 19" crate. The voltage on each of the four pickups is given by

V (ω) = Zt (ω) Ib (ω) (1.16)

Page 26: Boosters for synchrotrons - Aarhus Universitet · Chapter 1 gives an introduction to synchrotrons and in particular the design goals and so- lutions for the Australian Synchrotron

16 Overview and design of the ASP injection system

Figure 1.9: View of the BPM button layout.

where Ib is the beam current, and Zt is the beam coupling impedance. For a circular beampipe the beam coupling impedance is given by

Zt =a2

2r

ω

βc

R

1 + iωRC(1.17)

where a is the radius of the button pickup, r is the radius of the vacuum chamber, βc isthe velocity of the beam, R is the system impedance (50 Ω), i =

√−1, and C is the button

pickup capacitance [Hin00]. For small variations around the center, the beam position isgiven by

x =(VA (ωf ) + VD (ωf ))− (VB (ωf ) + VC (ωf ))

VA (ωf ) + VB (ωf ) + VC (ωf ) + VD (ωf )1Sx

(1.18a)

y =(VA (ωf ) + VB (ωf ))− (VC (ωf ) + VD (ωf ))

VA (ωf ) + VB (ωf ) + VC (ωf ) + VD (ωf )1Sy

(1.18b)

where VA (ωf ), VB (ωf ), VC (ωf ), and VD (ωf ) are the respective button pickup signalamplitudes at the chosen fundamental frequency ωf , and Sx and Sy are the sensitivityconstants (in units of %/mm). The bunch structure gives rise to a 500 MHz signal. Forthe geometrically simpler cases of 90 between each button, the sensitivity constants are1/r and

√2/r for the cases of the buttons on the x and y axes or rotated 45, respectively.

In the present case where buttons are placed ±30 o the axis, Sx ≈ 8.78%/mm andSy ≈ 5.25%/mm close to the center are found from numerical 2D calculations.

It is important that the four signals are neither attenuated too much by the cablesfrom the pickup to the module, nor attenuated dierently. Under the assumption that theattenuation was directly proportional to the cable length, all four buttons on the sameBPM was connected with equal length cable. Unfortunately the attenuation varied greatlywith length, and the attenuation (in dB) varied by up to 20% (5.7 dB) and on averagewith an RMS of 8.6% (2.3 dB) between the individual pickups on the same BPM (seegure 1.10). These variations would give false closed-orbit excursions of over 5 mm, andalthough this could in theory be compensated in software it would have a negative impacton linearity and sensitivity. Rather than installing new and better cables a costly andtime consuming task it was decided to add additional attenuators to the least attenuatedpickups in a group, thereby reducing the variation in attenation between the pickups of agiven BPM.

The additional attenuators brought the variation down to 4.7% (0.48 dB) and 1.2%(0.2 dB) maximum and RMS, respectively. Connecting the cables at the BPM end to apower splitter fed from a 500 MHz RF power generator simulated the signals induced by acentered beam. The BPM software was calibrated, adjusting for acceptably small o-sets

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1.6 Diagnostics equipment 17

0

10

20

30

40

50

Cable

att

enuati

on

[dB

]C

able

att

enuati

on

[dB

]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

BPMBPM

Pickup A

Pickup B

Pickup C

Pickup D

Figure 1.10: The BPM button pickup cable attenuations.

caused by the dierences in attenuation of up to 0.6 mm, and horizontal and vertical RMSvalues of 0.07 and 0.15 mm, respectively.

The MX-BPM modules have a dynamic range of 75 dB and amplify the input voltages(minimum −70 dBm) before calculating the beam position from the relative strength ofthe voltages using (1.18). The modules calculate the position from the signals and give theresults as (x, y) position pairs in units of Volts. The 32 modules' (x, y) position voltagesare read by four National Instruments NI-DAQ ADC units, which in turn are read by aPC via a USB connection. The reading of the Bergoz electronics modules by the ADCs isgated by a DG535.

Two Libera BPM Electronics modules from Instrumentation Technologies were pro-vided to obtain First Turn Capability and 1000 Turn to Turn readings. Two specialBergoz Interface boards were delivered, and can replace any two MX-BPM modules, thuspermitting to guide the corresponding BPM button to the Libera modules. These Liberamodules proved tricky to get working and were not used during the commissioning.

1.6.9 Beam Loss Monitors

The Beam Loss Monitors (BLMs) measure the beam loss at the location they are placed.They have a counting rate of up to 10 MHz with a dynamic range of 108. It takes 100 nsfor a BLM to recover from a hit. Each BLM is connected a CNT-66 timer/counter.

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Chapter 2

The combined-function magnets

This chapter details the design of the combined-function magnets and how they inuenceon the beam. The method of measurement at the Factory Acceptance Test (FAT) as wellas the measurement results are also shown.

2.1 Basic formulas

Dening the magnetic length as

L =∫

B (s) ds

B0(2.1)

where B0 is the magnetic eld in the center of the magnet, the total bending angle can bewritten as

θ =BL |q| e

p(2.2)

where (1.3) has been used.If p is given in units of eV/c, (1.3) can be written as

BT ρT =peV/c

q cm/s

(2.3)

The focusing strength of a quadrupole is the normalized gradient given by

k = − 1Bρ

dBy

dx(2.4)

There is no general agreement on the sign. The strength of a sextupole is

κ = − 1Bρ

d2By

dx2(2.5)

where the same sign convention as for the quadrupole is used.

2.2 Induced eddy currents in the vacuum chamber

When the combined-function magnets are ramped, the changes in magnetic elds give riseto eddy currents according to Farady's law of induction

∇×E = −∂B∂t

(2.6)

18

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2.3 O-axis elds in a combined-function magnet 19

where E is the electric eld. The eddy currents in the laminated magnets are neglible,whereas the eddy currents in the vacuum chambers must be considered. They give rise toa sextupole component in the center of the vacuum chamber with a strength given by

κVC = 2Fµ0σδ

∂B/∂t

B(2.7)

where µ0 is the vacuum permability, σ is the conductivity of the vacuum chamber, andδ and h are the thickness and height of the vacuum chamber, respectively. The factor Fdepends on the geometry of the chamber and is given by

F = 2∫ 1

0

√x2 +

(h

w

)2

(1− x2)dx (2.8)

where w is the width of the chamber. For the elliptical vacuum chambers of height 23.5mm and width 40 mm, F = 1.48 [ES93]. The induced sextupole strengths by the BDand BF are calculated from the nal ramping curves used in commissioning (gure 4.4 onpage 51) and shown in gure 2.1, where (2.7) has been used with σ = 1.35× 106 (Ωm)−1

and δ = 0.7 mm.

0

0.1

0.2

0.3

0.4

0.5

Induced

sextu

pole

κ[

m−

2]

Induced

sextu

pole

κ[

m−

2]

0 0.2 0.4 0.6 0.8 1

Time [s]Time [s]

0

0.5

1

1.5

2

2.5

3

Beam

energ

y[G

eV

]B

eam

energ

y[G

eV

]

Energy ramp

BD

BF

Figure 2.1: Sextupole strengths induced by the eddy currents in the vacuum chambers causedby the ramping of the magnets.

The smaller bending radius and larger eld increase of the BD family results in a largerinduced sextupole strength, which is largest at low energy where the magnetic eld issmall. The induced sextupole strengths are κVC,BD ≤ 0.40 m−2 and κVC,BF ≤ 0.05 m−2

and they have the same sign as the sextupole component in the BD family. Using Winagile[Bry99] it is found that the induced sextupole strengths change the chromaticities from thenominal (ξH , ξV ) = (1, 1) to (0.55, 3.61) at the beginning of the ramp where the inducedsextupoles are largest. Changes that are easily corrected, if desired, using the trimmingsextupoles by setting them to (κSF, κSD) = (−13.7, 2.2) m−2 at the beginning of the ramp.

2.3 O-axis elds in a combined-function magnet

To understand the eects on the o-axis beam inside an accelerator magnet it is benecialto investigate the two-dimensional multipole expansion of the transverse magnetic eld.

Page 30: Boosters for synchrotrons - Aarhus Universitet · Chapter 1 gives an introduction to synchrotrons and in particular the design goals and so- lutions for the Australian Synchrotron

20 The combined-function magnets

Using Amperes Law the transverse components Bx and By of the magnetic eld in thepole gap of an accelerator magnet can be written in terms of the scalar eld φ:

Bx = −∂φ

∂xBy = −∂φ

∂y(2.9)

where

φ = −∞∑

m=0

[−AmRe (x + iy)m + BmIm (x + iy)m] (2.10)

in Cartesian coordinates. The terms Bm designate normal multipoles, and the terms Am

designate skew multipoles. A skew multipole magnet is a normal multipole magnet rotatedabout the beam axis. A pure normal multipole magnet that is rotated 90

m becomes a pureskew multipole magnet. In the booster are no skew multipole magnets by design. However,small skew multipole contributions will always be present due to magnet alignment errors,and errors in the eld.

Inserting (2.10) into (2.9), neglecting the skew terms, and setting the dipole eld B0 ≡−B1, the quadrupole gradient g ≡ −2B2, and the sextupole gradient g′ ≡ −6B3, the rstthree multipole terms are [Bry92]

Dipole m = 1

φ = B0y (2.11a)

Bx = 0 (2.11b)

By = B0 (2.11c)

Quadrupole m = 2

φ = gxy (2.12a)

Bx = gy (2.12b)

By = gx (2.12c)

Sextupole m = 3

φ =g′

6(3yx2 − y3

)(2.13a)

Bx = g′yx (2.13b)

By =g′

2(x2 − y2

)(2.13c)

Combining the elds in (2.11) (2.13) and ignoring higher order multipoles gives

Bx = gy + g′yx (2.14a)

By = B0 + gx +g′

2(x2 − y2

)(2.14b)

If the beam is o-axis, (x, y) 6= (0, 0), it is evident from (2.14) that the higher order eldswill give contributions to the lower order elds. The eective quadrupole gradient becomes

∂Bx

∂y=

∂By

∂x= g + g′x (2.15)

The sextupole gradient remains unchanged

∂2Bx

∂y2= 0 (2.16a)

∂2By

∂x2= g′ (2.16b)

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2.4 Tune-shifts due to o-axis beam and variable dipole current 21

Octupole and higher order terms have been neglected here, since they are expected to berelatively weak.

If the beam on average is horizontally oset (x 6= 0) in a combined-function mag-net family, the change in eective quadrupole strength can inuence the transverse tunes(Qx, Qy). Assuming all magnets' multipole gradients have been accurately measured, thisx oset can be calculated from the measured value of the tune-shift where assumptionson the average oset in the tuning sextupoles have been made.

2.4 Tune-shifts due to o-axis beam and variable dipole cur-

rent

The additional focusing on an o-axis beam feed-down from the sextupole component canbe calculated in a simple way for small deviations from the nominal orbit by means oflinear algebra.

The tune shift vector ∆Q =(∆Qx,∆Qy

)can be obtained from the focusing shift

vector ∆k = (∆kBF,∆kBD) using the matrix T

∆Q = T∆k (2.17)

From (2.15) and (2.5) the change in focusing strength ∆k can be written as

∆k = (κBFxBF, κBDxBD) (2.18)

The measured average values of the quadrupole and sextupole strengths are given in table1.3. Using these values together with lattice design software, in this case Winagile [Bry99],the relation between the horizontal o-set and tune-shifts is found to be(

∆Qx

∆Qy

)=(

0.0417 −0.0198−0.0547 0.179

)(xBF

xBD

)(2.19)

where the o-set x is measured in mm. The vertical tune can be seen to depend verystrongly on the horizontal position inside the BD magnets. If the average displacementsinside the BD and BF families are of opposite sign the sensitivity is even stronger.

Nominally the ratio between the magnetic elds of the BF and BD magnets should bexed. However, during commissioning it was found that the beam could not be acceleratedunless this ratio was varied.

The actual relative integrated magnetic eld ratio of the BF and BD magnets relativeto their nominal ratio is dened as

R =(ΘBF/ΘBD)actual

(ΘBF/ΘBD)nominal

(2.20)

where Θ = nΦ, and n is the number of magnets of that family, and Φ is the integratedmagnetic eld of one of the magnets. The beam momentum depends on the value of R.Increasing R so that R becomes greater than one, the magnetic eld in the BF magnetsincreases. This will shift the beam inward towards the booster center in the BF magnets,due to the dispersion a larger magnetic eld is bending wise equivalent to a smallermomentum. This shift will result in a slightly shorter circumference of the orbit, equivalentto a higher revolution frequency, which will cause the RF to swiftly increase the momentumof the beam, thereby decreasing the revolution frequency, so that the harmonic numberremains an integer. Since the BD magnet eld is unchanged, the orbit inside the BD

Page 32: Boosters for synchrotrons - Aarhus Universitet · Chapter 1 gives an introduction to synchrotrons and in particular the design goals and so- lutions for the Australian Synchrotron

22 The combined-function magnets

magnets will shift outward as given by the local value of the dispersion. The end resultwill be that the closed-orbit is shifted inward in the BF magnets and outward in the BDmagnets for increasing R. Also the gradients of the magnets change, with the BF gradientsincreasing, and the BD gradients decreasing due to the beam momentum change.

From (1.3) is found that∆B

B=

∆p

p− ∆ρ

ρ(2.21)

A change in bending radius corresponds to a change in the booster circumference, thusthe RF will change the momentum to preserve the harmonic number, and the change inbending radius will be negated. Because there are two dipole families, this can be writtenas

∆ρ

ρ= NBDLBD

(∆ρ

ρ

)BD

+ NBFLBF

(∆ρ

ρ

)BF

= 0

⇔(∆ρ/ρ)BF

(∆ρ/ρ)BD

= −NBDLBD

NBFLBF= −184

189(2.22)

where N is the number of magnet of length L. Thus, on average an o-set in the horizontalin a BF magnet can be countered by an opposite o-set in a BD magnet.

The total relative integrated magnetic elds are 4/15 and 11/15 for the BF and BDfamilies, respectively. From (2.20) and (2.21) the change in momentum becomes

∆p

p=

∆B

B=

4R + 1115

− 1 (2.23)

Since ∆ρ/ρ = x/D, where x is the o-set and D the dispersion, the o-set in the BDs canbe written as

xBD = DBD

(∆p

p−(

∆B

B

)BD

)= DBD

(4R + 11

15− 1)

(2.24a)

where the last equality uses that the BD strength is not changed, i.e. (∆B/B)BD = 0.Using (2.22) the o-set in the BF becomes

xBF = − 184DBF

189DBDxBD =

184189

DBF

(1− 4R + 11

15

)(2.24b)

Increasing R, i.e. increasing the BF strength, will increase the beam momentum givenby (2.23) causing o-sets in the BD and BF magnets given by (2.24a) and (2.24b), respec-tively. The increased momentum, the o-sets, and the changed BF magnet strength willgive a new, eective focusing strength given by

k =p0

p(k0 + κ0x) (2.25)

where (2.15) has been used, and subscript zero denotes the value for R = 1. Using (2.23) (2.25) the new focusing for the BD and BF becomes

kBD =15

4R + 11k0BD + κ0BDDBD

(1− 15

4R + 11

)(2.26a)

kBF =15

4R + 11Rk0BF +

184189

Rκ0BFDBF

(15

4R + 11− 1)

(2.26b)

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2.4 Tune-shifts due to o-axis beam and variable dipole current 23

while the new sextupole strengths without the presence of higher order multipoles simplyscales with p0/p

κBD =15

4R + 11κ0BD (2.26c)

κBF =15

4R + 11Rκ0BF (2.26d)

and the bending angles of the magnets become

θBD =15

4R + 11θ0BD (2.26e)

θBF =15

4R + 11Rθ0BF (2.26f)

Figure 2.2 shows a plot of (2.26a) and (2.26b) using the measured 3 GeV values givenin table 2.1 on page 26. The BD focusing changes very little, which was to be expectedsince only the momentum of the beam is changed. If these dependencies on R can be

0.66

0.6625

0.665

0.6675

0.67

0.6725

0.675

kB

D

[

m−

2]

kB

D

[

m−

2]

1 1.005 1.01 1.015 1.02

BF/BD ratio RBF/BD ratio R

−0.84

−0.8375

−0.835

−0.8325

−0.83

−0.8275

−0.825

kB

F

[m−

2]

kB

F

[m−

2]

BD

BF

Figure 2.2: The change in focusing strength of the BD and BF as a funtion of the BF/BD ratioR from (2.26a) and (2.26b) using the measured 3 GeV values from table 2.1.

incorporated into the lattice model software with any precission, it can be used to give amore accurate prediction of the lattice when R 6= 1. Assuming that the dispersion changesinsignicantly, it should be possible to implement this in the lattice when using MAD-XP[SSF07].

Normally one can derive the focusing strengths of the quadrupoles from the tunes bytting the model quadrupoles to give the measured tunes. However, since changing Ralso changes the energy, the chromaticities as well as the change in focusing strengthswill change the tunes. The chromaticities depend on the sextupoles, to which there arecontributions from the combined-function magnets (which change with R), the tuningsextupoles, and the eddy currents in the vacuum chamber. To separate these contributionsis complicated, however, since these eects are all linear for small changes in R, one canincorporate them in the calculated focusing strengths and use them to calculate the changein tunes for identical chromaticities.

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24 The combined-function magnets

The equivalent focusing strengths calculated from the tunes which includes thechromaticity eect can be approximated as

kc,BF = ∆kBFR + k0BF (2.27a)

kc,BD = ∆kBDR−1 + k0BD (2.27b)

The focusing of the magnet at R = 1 will be ∆k + k0. This is a sum of the nominalfocusing strength and a factor κBFx0BF which can be interpreted as either a quadrupolegradient induced by an o-axis orbit, or an error in measurement of either the tune or ofthe quadrupole component of the magnet. Since there is an uncertainty in the positioningof the booster elements, and thus of the circumference, it would very likely that x0 6= 0 forthe chosen RF frequency.

2.5 Measurement of the combined-function magnets

2.5.1 Measurement setup

The measurement setup of a combined-function magnet during the FAT is shown in gure2.3.

Figure 2.3: The integrated eld in a BF magnet is measured using a long integrating coil witha curvature matching the magnets bending radius. The magnet is ramped to full excitation andthe voltage induced in the coil measured at four dierent currents. This procedure is repeatedat dierent radial displacements of the coil. The multipole eld components are extracted fromthe data, taking into account the nite width of the coil.

The measurements was done for dierent points in the horizontal plane on the verticalsymmetry axis. By ramping the magnetic eld a voltage E is introduced in the coil givenby Faraday's law of induction

E = −NdΦB

dt(2.28)

where N is the number of coil windings, and ΦB =∫S B · dA is the total magnetic eld

ux through the surface area S contained within the coil loop.

2.5.2 Measuring the multipole components

The measurement was carried out at discrete positions in the x direction resulting anintegrated eld magnetic eld along the path of the beam s. The eld outside the magnetfalls o very quickly, and the length of the coil was made such as to include practically all

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2.5 Measurement of the combined-function magnets 25

of the fringe-eld at both ends of the magnet. Approximating the surface area of the coilloop S = L×w where w is the width and L the length of the coil, the measured integratedeld at the i'th position becomes

ΦB (xi) =∫ +L/2

−L/2ds

∫ xi+w/2

xi−w/2Bz (x, s) dx (2.29)

where s = 0 is in the center of the magnet. Since for a properly aligned coil only the z-component of the magnetic eld will contribute, the subscript z is dropped in the following.Likewise, the coil is taken to be innitely long.

On the vertical symmetry axis the magnetic eld can be written as

B (x, s) =∞∑

j=0

Bj (s)j!

xj (2.30)

where Bj (s) is the strength of the 2 (j + 1)-pole eld component at a given position alongthe beam axis. Inserting (2.30) into (2.29) and changing the summation gives

ΦB (xi) =∞∑

j=1

∫B (s)j−1 ds

j!

[(xi +

w

2

)j−(xi −

w

2

)j]

= b0w︸︷︷︸dipole

+ b1wxi︸ ︷︷ ︸quadrupole

+ b2

(12wx2

i +w3

24

)︸ ︷︷ ︸

sextupole

+ . . . (2.31)

where bj =∫Bjds. To establish whether the quadrupole and sextupole strengths are

correct, the eld is normalized to the measured dipole

ΦB (xi)b0w

= 1 +b1

b0xi +

b2

b0

(12x2

i +w2

24

)+ . . . (2.32)

The normalization cancels out the dependence on the coil width for the quadrupole part.For the sextupole part there is a position independent constant. In the data analysiscarried out at Danfysik, the measurement was tted with a polynomial to extract thequadrupole and sextupole strengths, and the later value was then corrected by performing

the coordinate substitution xi →√

x2i −

w2

12 .

2.5.3 Multipole measurement results

The measured multipole strengths for dierent excitations were normalized to their dipoleexcitation and then multiplied by the nominal dipole strength at 3 GeV. The results ofthe multipole measurements of the BD and BF families are shown in gure 2.4 and theaverage values are given in table 2.1.

The sextupole strength is dicult to measure very precisely with the chosen method,and the large variation in sextupole strength relative to the variation in quadrupole strengthis suspected to be primarily measurement uncertainties. The size of the uncertainties in themeasurements were not investigated due to time issues, and because the variation in themeasured quadrupole strengths were considered acceptable. It is therefore unknown howlarge a part of the spread is caused by magnet-to-magnet variation, and how much is causedby measurement errors such as variations in measurement coil alignment, temperaturevariations, time-jitter, etc.

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26 The combined-function magnets

Quadrupole component

0

2.5

5

7.5

10

Num

ber

ofm

agnets

Num

ber

ofm

agnets

−0.75−0.5−0.250

Deviation from design [%]Deviation from design [%]

−6.7 −6.68 −6.66

Normalized integrated gradientNormalized integrated gradient

Sextupole component

01234

Deviation from design [%]Deviation from design [%]

−25.5 −25 −24.5

Normalized integrated gradient [m−1]Normalized integrated gradient [m−1]

26A (3%)

92A (10%)

495A (60%)

830A (100%)

26A average

92A average

495A average

830A average

BD

(a) BD quadrupole and sextupole components

Quadrupole component

0

5

10

Num

ber

ofm

agnets

Num

ber

ofm

agnets

−0.25 0 0.25 0.5

Deviation from design [%]Deviation from design [%]

8.225 8.25 8.275 8.3

Normalized integrated gradientNormalized integrated gradient

Sextupole component

−6 −4 −2 0

Deviation from design [%]Deviation from design [%]

16.75 17 17.25 17.5 17.75

Normalized integrated gradient [m−1]Normalized integrated gradient [m−1]

26A (3%)

92A (10%)

495A (60%)

830A (100%)

26A average

92A average

495A average

830A average

BF

(b) BF quadrupole and sextupole components

Figure 2.4: CF quadrupole and sextupole components normalized to the integrated center(x = 0) eld and multiplied with the nominal 3 GeV dipole eld.

Table 2.1: Average CF quadrupole and sextupole components normalized to the integratedcenter (x = 0) eld and multiplied with the nominal 3 GeV dipole eld.

Normalized quadrupole sextupole[m−1

]Family BD BF BD BF

26 A (3%) -6.686425 8.245918 -25.081167 16.80909192 A (10%) -6.688750 8.255327 -24.836750 17.052091495 A (60%) -6.685969 8.267989 -24.828063 17.331071830 A (100%) -6.666219 8.268082 -25.374645 17.308107

The shift in the multipole components for the BD family at full excitation is caused bysaturation which reduces the eld strength at the edges of the pole as illustrated in gure2.5.

The variations in the multipole elds between the magnets were translated into vari-ations in the tunes and chromaticities, in order to estimate their eects in the booster,which are of primary concern.

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2.5 Measurement of the combined-function magnets 27

Figure 2.5: The magnetic eld density in the BF magnet as calculated by Opera 3D [Fie]. Thered areas are the rst to saturate.

In order to get an approximate measure for the inuence of the quadrupole and sex-tupole variations from magnet to magnet, which could be used in the quality acceptancecriteria, the variations in multipole strengths were translated into variations in tunes andchromaticities. For each magnet family the matrix T from (2.17) giving the relation be-tween a change in quadrupole strengths and a change in tunes was calculated usingWinagileby changing the focusing strength by 1% relative to the average focusing strength. Usingthe average value of the measurements of the BF family, the resulting tunes were foundfor each BD magnet by assuming that all the BD magnets were identical, and vice versathe average of the BD values was used for each BF magnet. The matrix T was similarlyobtained for the chromaticities by varying the sextupole strengths. The results are shownin gure 2.6. In the real world the actual tunes and chromaticities are the result of theindividual magnet contributions but are expected not to deviate signicantly from thevalues found from this procedure.

Both the tune variations for each excitation and the drift in tunes with excitation wereconsidered acceptable, with the latter, if needed, being easily correctable using the tuningquadrupoles. The large change in vertical tune comes from the saturation of the BD at fullexcitation: gure 2.4 shows virtually no change in BF's quadrupole component when goingfrom 495 A to 830 A excitation, while the BD quadrupole component shows a signicantchange of 0.3%.

2.5.4 Integrated dipole measurement results

The integrated magnetic eld as a function of excitation current was also measured. Theresults were used in the generation of the ramping curves for the magnets, thereby incorpo-rating saturation eects directly. Small variations in the dipole eld as a function of currentfrom magnet to magnet lead to kicks because many magnets are coupled in series, and thushave the same current. The 28 BF magnets are all in series, while the 32 BD magnets aresplit into two identically sized groups. To nd how the variations between the magnetswould aect the beam at dierent energies, all magnets in a series connected group was setto the same current the current was chosen so that the integrated eld of each magneton average would give the nominal eld strength for that particular energy. The smallvariations in magnetic eld from magnet to magnet for that current was translated intovariations in bending angle. The results are shown in gure 2.7, where the magnets have

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28 The combined-function magnets

0

2.5

5

7.5

10

12.5

Num

ber

ofm

agnets

Num

ber

ofm

agnets

9.15 9.175 9.2 9.225 9.25

Horizontal tune QHHorizontal tune QH

3.1 3.15 3.2 3.25 3.3

Vertical tune QVVertical tune QV

0

2.5

5

7.5

10

12.5

Num

ber

ofm

agnets

Num

ber

ofm

agnets

−0.5 −0.25 0 0.25 0.5

Deviation from design [%]Deviation from design [%]

−4 −3 −2 −1 0 1 2

Deviation from design [%]Deviation from design [%]

26A (3%)

92A (10%)

495A (60%)

830A (100%)

26A average

92A average

495A average

830A average

BD

BF

(a) Tunes

0

10

20

30

Num

ber

ofm

agnets

Num

ber

ofm

agnets

−0.25 0 0.25 0.5 0.75 1

Horizontal chromaticity ξHHorizontal chromaticity ξH

1.25 1.5 1.75 2 2.25 2.5

Vertical chromaticity ξVVertical chromaticity ξV

0

5

10

15

Num

ber

ofm

agnets

Num

ber

ofm

agnets

−125 −100 −75 −50 −25 0

Deviation from design [%]Deviation from design [%]

25 50 75 100 125 150

Deviation from design [%]Deviation from design [%]

26A (3%)

92A (10%)

495A (60%)

830A (100%)

26A average

92A average

495A average

830A average

BD

BF

(b) Chromaticities

Figure 2.6: Histograms of the equivalent tunes and chromaticities for the measured quadrupoleand sextupole component values of the combined-function magnets. The average values for thefamilies are the same because of the method used to derive the values.

been placed in the same sequence as they are installed in the booster. The kick angles areseen not to decrease proportionally to the energy but rather staying somewhat constant.This could be explained if the variations are due to mechanical variations, i.e. dierencesin the size of the pole gaps. Unfortunately, it could also be articially introduced if the

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2.5 Measurement of the combined-function magnets 29

−0.2

−0.1

0

0.1

0.2

kic

kangle

[mra

d]

kic

kangle

[mra

d]

5013

5049

5007

5060

5005

5039

5017

5058

5021

5037

5002

5033

5015

5045

5027

5011

5050

5019

5038

5009

5056

5018

5035

5031

5034

5008

5057

5025

5047

5030

5022

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5016

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5024

5036

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5040

5001

5048

5029

5054

5014

5032

5051

5004

5055

5012

5041

5006

5044

5026

5042

5023

5053

5010

5043

5020

Dipole magnet serial numberDipole magnet serial number

100 MeV0.5 GeV1 GeV1.5 GeV2 GeV2.5 GeV3 GeV

Figure 2.7: The dipole kick angles from variations in the integrated dipole eld for dierentenergies along the booster lattice starting from injection.

measuring coil is precisely not centered, due to the presence of the strong multipoles.A histogram of the kick angle distribution for the injection energy is shown in gure 2.8

and is seen to be approximately Gaussian centered on µ = 0.046 mrad and with a standarddeviation of σ = 0.12±0.02 mrad. The injection energy is the more interesting one, becauseat the low energy the beam is larger and more sensitive to closed-orbit excursions. Entering

0

5

10

15

Num

ber

ofm

agnets

Num

ber

ofm

agnets

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

Kick angle [mrad]Kick angle [mrad]

BD

BF

BD and BF

Gaussian fit

Figure 2.8: Distribution of the integrated dipole kick angles for each CF species and the total.The Gaussian t of the total distribution has σ = 0.12± 0.02 mrad and is centered on µ = 0.046mrad.

the kick angle values at injection energy into the model, the resulting closed-orbit is foundand corrected using Winagile [Bry99]. The uncorrected and corrected closed-orbits are

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30 The combined-function magnets

shown in gure 2.9, and the statistics for them in table 2.2.

Horizontal closed orbit [m] versus distance [m]

Vertical closed orbit [m] versus distance [m]

Note: Linear optics

0.000

0.000

130.20

130.20

0.004500

0.004500

-0.004500

-0.004500

Figure 2.9: The uncorrected (blue) and corrected (red) closed-orbit created by the kicks ingure 2.7.

Table 2.2: The uncorrected and corrected closed-orbit excursions due to variations in dipoleeld from magnet to magnet at injection energy.

Uncorrected Corrected

Max. peak-to-peak [mm] 8.751 1.327Average excursion [mm] −0.079 −0.038RMS excursion [mm] 2.217 0.230

The uncorrected orbit has large excursions, and to these must be added those createdby misalignment. Depending on the dynamic aperture at the chosen injection settings, onecould expect it would be dicult to achieve rst turn without resorting to the correctors.Indeed, during commissioning it did prove necessary to manually adjust, by trial and error,horizontal and vertical correctors to get rst turn.

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Chapter 3

Theoretical beam dynamics studies

This chapter describes the theoretical beam dynamics studies, where investigation of thetransverse dynamic aperture was done using simulation software, as well as analyticalcalculations on the induced longitudinal momentum-spread caused by the RF. A partialstudy of the dynamic aperture was done prior to commissioning of the injection system.However, the work presented here was carried out after the booster commissioning had n-ished and the experimental investigations were concluded (chapter 4). Part of the ndingspresented here were also presented with a poster at the particle accelerator conference inAlbuquerque, New Mexico [WBB+07].

3.1 Transverse dynamic aperture

3.1.1 Introduction

The inclusion of sextupole elds in the lattice were necessary for chromaticity correction.However, non-linear elds have perturbing eects on the beam. Beam particles that aretoo far from the nominal orbit risk being lost due to chaotic motion caused by the non-linear elds. The dynamic aperture is the limit within which the beam is stable for asucient time typically of the order of the damping time or the required lifetime of thebeam. The dynamic aperture should always be larger than the physical aperture set bythe vacuum chamber. The size of the domain in phase-space wherein the beam particletrajectories are stable is referred to as the dynamical acceptance.

If certain criterias are met, the total six-dimensional phase-space can be subdivided intothe longitudinal phase-space and the four-dimensional transverse phase-space, and thesecan then be treated separately. The main criterias are that intra-beam particle scatteringsare suciently few, and that the dispersion at the RF cavity is not too large. Both ofthese criteria are met in the ASP booster. For the purpose of this study, the transversephase-space is not split into a horizontal and vertical due to the presence of the sextupoleswhich couple the horizontal and vertical particle motion (see (2.13)).

Calculations of the dynamic aperture that incorporate the non-linearities are typicallydone with two complimentary approaches: using numerical tools or perturbative analysis.The numerical approach makes use of tracking codes where approximate solutions of theequations of motion are calculated. It can handle strong non-linearities but being numericalcan give less insight into the underlying behavior. Perturbative analysis is less precise thanthe numerical approach but has the advantage that one can derive analytical informationon some dynamical quantities such as detuning, phase-space deformations, resonances, etc.[Tod99, Smi02]

31

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32 Theoretical beam dynamics studies

Presently, the dynamic aperture was investigated with single particle tracking, i.e.intra-beam interactions were neglected. The primary software tools used were MAD-XPand PTC [SSF07] while additional investigations into the longterm transverse stability ofthe beam was made with Tracy2 [Ben97]. Common for these tools is that they provide moreaccurate tracking results for thick lens elements through the use of symplectic integration

which conserves the phase-space volume by preserving the Hamiltonian structure of theindividual ring elements [Sch05]. Earlier investigations into the dynamic aperture doneusing the software tool Winagile [Bry99] used thick lens tracking as well, but did not usesymplectic integration resulting in a growing emittance of the test particle beam [FNM05].

3.1.2 The transverse apertures and investigated area

The transverse size of the beam at any point in the orbit is determined by the emittanceand the values of the optical functions at that point. The largest emittance that can beaccomodated is called the admittance. The physical aperture is the limit on the transversebeam size imposed by the dimensions of the vacuum chamber.

In the main dipole magnets, the chamber is approximately an ellipsoid with an innerwidth of 40 mm and inner height of 23.5 mm, everywhere else the chamber is circular withan inner diameter of 40.5 mm. The usable physical aperture is limited horizontally by theaperture of the rst and last QF magnet and vertically by the aperture of the rst and lastBD magnet in each quadrant. Using the optical funtions to calculate the beam size aroundthe ring, the maximum possible beam size at the injection point is found to be 35.0 mmhorizontally and 10.8 mm vertically. The beam envelope in a quadrant is shown in gure3.1. Figure 3.2 shows the investigated area of dynamic aperture, the vacuum chamber, thephysical aperture, and the 100 MeV beam size at the injection point.

As the beam is accelerated, its transverse size shrinks due to adiabatic damping. Con-sequently, the apertures are more important at the low energies of injection. Indeed, thebeam must be injected on-axis within only ±3 mm vertically and less if the dynamicaperture is smaller than the physical.

In the dynamic aperture investigations, the size of the investigated area was chosensuch that it was reasonably larger than the limits imposed by the physical aperture. Theinvestigated 50× 50 mm2 area was lled with test particles positioned in a grid formation,with a spacing of 1 mm horizontally and vertically between neighbouring test particles.These test particles were then tracked through each element in the lattice for a givennumber of turns.

3.1.3 Resonances

Resonances are areas in tune-space where small perturbations of the closed-orbit causesthe amplitude of the beam particle oscillations to grow rapidly, increasing the beam sizeand eventually leading to beam loss. The resonances are given by

mQH + nQV = p (3.1)

where m, n, and p are integers. The order of the resonance is m + n. The lower theorder, the stronger the resonance. Additionally the beam is particularly sensitive to theso-called systematic resonances, i.e. where p is a multiple of the machine symmetry, in thepresent case four due to the four-fold symmetry of the booster. There are two systematicresonances within the adjustable tune range:

3QH + 0QV = 28 (3.2a)

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3.1 Transverse dynamic aperture 33

Horizontal beam envelope [m] versus distance [m]

Vertical beam envelope [m] versus distance [m]

Note: Linear optics

0.000

0.000

32.550

32.550

0.025000

0.025000

-0.025000

-0.025000

Betatron envelope Momentum envelope

Figure 3.1: Maximum beam envelopes (violet) in a quadrant with vacuum chamber walls(cyan).

and1QH + 2QV = 16 (3.2b)

Resonances have a nite width in tune-space, and the presence of magnetic eld errors andmisalignment of magnetic elements greatly increases the width of the resonances. At anygiven time the beam occupies a nite area in tune-space, the size of which depends on themomentum-spread and chromaticities.

3.1.4 Tune maps for the error-free lattice

The dynamic aperture was investigated in the area of tune-space spanned by (9.0, 3.0) <(QH , QV ) < (9.5, 3.5) by tracking the square grid of test particles described in section 3.1.2for 1000 turns, corresponding to 0.434 ms. Only the part of the grid above the horizontalsymmetry plane was tracked, since the booster is mirror symmetric in the horizontal plane.The resulting plot of the dynamic aperture as a function of tunes is commonly referred toas a tune map.

Figure 3.3 shows a contour map of the remaining number of test particles as a functionof tunes for the on-momentum case ∆p/p = 0. The small grey crosses indicate the positionsof the investigated tune-points. The rst through fourth order resonances are shown withred, gold, black, and grey solid, straight lines, respectively. The third order systematicresonances from (3.2) are shown with red-black dashed lines. The black circles show theposition of nine tune-points that later are investigated further. The numbers in the plot are

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34 Theoretical beam dynamics studies

−30

−20

−10

0

10

20

30

Heig

ht

[mm

]H

eig

ht

[mm

]

−30 −20 −10 0 10 20 30

Width [mm]Width [mm]

Investigated area

Injection vacuum chamber

Physical aperture limit

Injected beam envelope

3 GeV beam envelope

Figure 3.2: The investigated area of the dynamic aperture (±25 mm) shown with the size ofthe vacuum chamber and physical aperture overlayed. The 99% beam envelope of the nominal100 MeV beam is shown in the center.

the total remaining number of particles, since these are spaced 1 mm apart this correspondsroughly to the area of the dynamic aperture in units of mm2 above the horizontal symmetryplane.

It can be seen from gure 3.3 that the systematic resonances, particularly (3.2b) havea strong and far reaching inuence on the dynamic aperture in the quadrant. So muchin fact that the nominal tune point (9.2, 3.25) is far less favorable than something closerto (9.08, 3.15). The non-systematic third order and the fourth order resonances appear tohave no signicant inuence, apart from the fourth order resonance 2QH + 2QV = 25 thatgoes through (9.5, 3.0). This resonance is close to being a systematic resonance, 25 ≈ 24,which would be expected to increase its eect.

It is clear that the largest dynamic aperture is in the lower left corner of the map. Thelower right corner also has two smaller areas of large dynamic aperture, but being smallerthey have less room to accomodate the tune-spread.

To include the inuence of beam momentum-spread, tune maps were made with parti-cles having ∆p/p = ±1% without using the tuning sextupoles to correct the chromaticity.For a Gaussian density distribution in momentum with σp/p = 0.5% this would encompass95% of the particles in the beam. The tune maps are shown in gure 3.4. Again the lowerleft corner has the largest dynamic aperture, while the lower right corner, as expected, hasmuch less than in the on-momentum case.

Since any part of the beam that is outside the dynamic aperture is lost, not only the areabut also the shape of the dynamic aperture is important. Approximate envelope plots weregenerated from the surviving particle's distribution. Because the envelope encompassesthe outerlying particles, minor islands of instability may exist just inside the edge of theenvelope. The size of these islands is suggested by gure 3.7 on page 39. Figure 3.5 showsapproximate envelope plots of the on- and o-momentum dynamic apertures at three

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3.1 Transverse dynamic aperture 35

3.5

3.4

3.3

3.2

3.1

3.0

QV

9.59.49.39.29.19.0QH

900

850

800

800

750

750

700

700

700

650

650

650

650 600

600 600

600

550

550

550

500

500

500

500 450

450

400

400

350

350 300 300

250

250

250

250

200

200

200

200

200

150

150

150

100

100

100

100 50

50

50

50

Figure 3.3: Tune map for on-momentum particles. The numbers in the plot refer to the numberof remaining test particles, and the straight lines are the resonances.

dierent tune points with the physical aperture and the injected beam envelope overlayed.It is seen in gure 3.5 that the available aperture at the tune point (9.2070, 3.2295)

chosen because it is close to the nominal tune point (9.2, 3.25) is reduced for particleswith a negative momentum deviation, unless the chromaticity is adjusted to (1, 1). Forthe tune point (9.0608, 3.1264) the stable region is much larger than the physical aperture,and appears to extend beyond the area investigated. The stable region inside the physicalaperture for the tune point (9.3803, 3.1202) is severely limited, and with larger variationsbetween the on- and o-momentum dynamic apertures than the previous two tune points.However, adjusting the chromaticy to (1, 1) has a positive eect and can be explained bythe tune point's closer proximity to resonances. The break in left-right mirror symmetryseen in all cases comes from (2.13c).

3.1.5 The inuence of misalignments and eld errors

The presence of errors in the physical positioning of the magnets as well as errors inthe magnetic elds inuence the beam and cause reduction in the dynamic aperture. As

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36 Theoretical beam dynamics studies

3.5

3.4

3.3

3.2

3.1

3.0

QV

9.59.49.39.29.19.0QH

1150

1150

1100 1050

1050

1000

1000

950 900

850 800

800

750

750

700

700

650 650

650

600

600

600

550

550

550

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350

300

300

300

250

250

250

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200

150

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100

100

100

50

50

50

(a) ∆pp

= +1%

3.5

3.4

3.3

3.2

3.1

3.0

QV

9.59.49.39.29.19.0QH

1050 1000

950

900 850 800

750

700

650

600

600

550

550

500

500

450

450

400

400

400

350

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300

300

250

250

250

200

200

150

150

150

100

100

50

50

50

50

50

50

(b) ∆pp

= −1%

Figure 3.4: Tune maps for the o-momentum particles without chromaticity correction.

discussed in section 2.3, o-axis quadrupoles and sextupoles give dipole elds, i.e. kicksthe beam. This results in closed-orbit excursions, thus reducing the eective physical anddynamic apertures. The sextupoles also give a quadrupole eld and thereby a changein tune. Being random, the errors partially breaks the four-fold booster symmetry. Thecorrection dipoles distributed around the booster can be used to adjust the closed-orbit towithin 1 mm RMS in both planes of the nominal center. However, this is only meassuredat the BPMs' positions, and the beam may have larger excursions elsewhere, such as insidethe combined-function magnets. In generel the eects of misalignment and eld errors areequivalent to a widening of the tune resonances.

In the simulation, the misalignments are generated on a per-magnet basis rather thana per-girder basis and this gives an overestimate of the random errors. On the girder, themagnets can be positioned with a higher acuracy, e.g. σy ∼ 0.01 mm due to machiningof the magnet surfaces, than the girders themselves can. Apart from the BD magnetsat the start and end of each quadrant, the BD and a BF magnets are placed as pairs oncommon girders, thus making their relative position error much smaller. It is principally thequadrupole component that is responsible for the dipole kick of the beam, and with the BDand BF magnets having opposite polarities, the total kick angle from the magnets as a pairwill be reduced. Taking a 0.15 mm displacement as an example, the BD quadrupole willgive a kick of 0.115 mrad while the BF quadrupole gives a 0.167 mrad kick in the oppositedirection. With a betatron wavelength being much larger than the distance between thetwo magnets, the kick angles can be added linearly making their combined kick angleas a pair 0.05 mrad. The simulation code does not take this into account, hence theoverestimate.

To quantify the eects of the errors, nine tune sets were selected. Gaussian distributedrandom errors were added to every magnetic element in the ring, except the dipole cor-rectors. This was done one hundred times for each tune set, giving one hundred dierentvirtual boosters for each tune set, each with its own set of errors. The closed-orbit ex-cursions resulting from these errors were minimized using the correction dipoles, and thetuning quadrupoles QD and QF adjusted to give the desired tunes. Both the eld andalignment errors were assumed to be random and Gaussian in distribution, with the align-

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3.1 Transverse dynamic aperture 37

(QH , QV ) = (9.2070, 3.2295)

0

5

10

15

20

25

y[m

m]

y[m

m]

−20 −10 0 10 20

∆p

p= +1%

∆p

p= 0%

∆p

p= −1%

∆p

p= +1% (1, 1)

∆p

p= −1% (1, 1)

Physical aperture

Injected beam

(QH , QV ) = (9.0680, 3.1264)

0

5

10

15

20

25

y[m

m]

y[m

m]

−20 −10 0 10 20

(QH , QV ) = (9.3803, 3.1202)

0

5

10

15

20

25

y[m

m]

y[m

m]

−20 −10 0 10 20

x [mm]x [mm]

Figure 3.5: Approximate envelopes of the dynamic apertures for ∆p/p = −1, 0, 1 at threedierent tune points. The cases where the tuning sextupoles have been used to set the chro-maticities (ξH , ξV ) to one in both planes, are indicated with (1, 1). The tuning sextupoles wereleft o for ∆p/p = 0. The horizontal axis x points away from the booster ring center.

ment error distribution truncated at 2σ. The limits on the errors are given in table 3.1. Foreach of these virtual boosters the dynamic aperture for on-momentum test particles wasinvestigated at the nine tune sets in the same way as in the error-free cases. The fractionof the area inside the physical aperture at the injection point into the booster (see gure

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38 Theoretical beam dynamics studies

Table 3.1: The misalignments and eld errors used in the simulation. The alignment errordistributions are truncated at 2σ.

Errors Alignment [mm] Relative eld

σx σy σs Normal Skew

BD and BF 0.15 0.15 0.5 3.0× 10−4 1.0× 10−3

QD and QF 0.20 0.15 0.5

3.2) found to be dynamically stable was recorded, and the parts of the dynamic aperturethat lay beyond the physical aperture was disregarded. Figure 3.6 shows the number ofoutcomes, i.e. the number of virtual booster, where the size of the dynamic aperture rela-tive to the physical aperture is within the given ranges. Those outcomes without a stableclosed orbit, i.e. 0% dynamic aperture, were not included. Next to the graph keys forthe tunes in the upper right corner of the gure is given the size of the dynamic aperturerelative to the physical when no misalignments nor eld errors are present.

Corrected closed orbits

Num

ber

ofoutc

om

es

≤ 75% 75%− 95% > 95%

Usable physical apertureUsable physical aperture

0

20

40

60

80

100

(9.0558, 3.051) 99%(9.0558, 3.151) 100%(9.0558, 3.251) 100%(9.1558, 3.051) 100%(9.1558, 3.151) 100%(9.1558, 3.251) 96%(9.2558, 3.051) 100%(9.2558, 3.151) 95%(9.2558, 3.251) 70%

Figure 3.6: Histogram of the fraction of usable, i.e. dynamically stable, physical aperture for100 random error scenarios for nine selected tune sets. Next to the graph key is shown thefraction of usable physical aperture in the case of no errors.

Two main conclusions can be drawn from this gure. Firstly, in the large majority ofthe outcomes, the dynamic aperture is markedly reduced compared to the error-free case,and is more restrictive of the beam size than the physical aperture. Secondly, the tune setsclosest to the integer resonances suer more strongly. The tune point (9.1558, 3.1510) hasthe largest number of outcomes with the largest usable aperture. This is unsurprising as ithas both a large dynamic aperture in the error-free case and is located far from any strongresonances. In the error-free case the tune point (9.0558, 3.1510) had the largest dynamicaperture of the nine, yet being relatively close to the integer resonance QH = 9 there areonly a few outcomes where the usable aperture is greater than 95%.

The dynamic aperture for the tune set (9.1558, 3.1510) in the case of largest dynamicaperture (best) and smallest, non-zero, dynamic aperture (worst) is shown in gure 3.7.Even in the best case, the dynamic aperture is close to cutting into the physical aperture,

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3.1 Transverse dynamic aperture 39

Misaligned (QH , QV ) = (9.1558, 3.1510)

−20

−10

0

10

20

y[m

m]

y[m

m]

−20 −10 0 10 20

x [mm]x [mm]

Worst caseBest casePhysical apertureInjected beam

Figure 3.7: The dynamic apertures at (QH , QV ) = (9.1558, 3.1510) for the misaligned andclosed-orbit corrected best case (green) and worst case (red). The latter is the worst case wherea stable closed-orbit existed. Islands of stability are seen detached from the bulk.

and in the worst case the beam would have to be o-axis at the injection point to avoidlosses. The introduced errors also clearly break the mirror symmetry in the horizontaly = 0 plane.

The results found here are neither qualitatively unexpected nor surprising. However,the extend to which a given choice of limits on the tolerances on the magnets' eld qualityand alignment quantitatively aects the available aperture, even when the orbit iscorrected is not triviel to estimate. A study such as this can be used to check if a givenset of tolerances are acceptable. Although the errors are overestimated, as discussed atthe beginning of this section, the results indicate that the tolerances on the ASP boostercannot be relaxed.

3.1.5.1 Closed-orbit excursion eects

Closed-orbit excursions always lead to reduced physical aperture, and the larger the ex-cursions the smaller the physical aperture. However, their eects on the dynamic apertureis slightly more complicated. The misalignments and eld-errors will act like higher or-der multipoles on the beam, and the exact eects of these will depend on their strengthsaround the closed-orbit's position. The result will be seen as islands of stability in the(xRMS, yRMS) position space. This is shown in gure 3.8 for three of the nine tune setsinvestigated. Generally there is an inclination towards larger dynamic aperture for smallerexcursions. An interpretation of this result is that the RMS position corresponds to aneective integrated quadrupole eld along that particular closed-orbit, and thereby to atune set. If that tune set has a large dynamic aperture in the error-free case (gure 3.3), itcould have a positive eect on the dynamic aperture for that particular closed-orbit. If thisintepretation is correct, the RMS position could be an additional parameter to optimizeon in a case where the dynamic aperture of a machine is severely limited by misalignmentsand eld errors, although it will not be easy to realize, particularly for a ramping machine.

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40 Theoretical beam dynamics studies

7

6

5

4

3

2

1

y rms

[mm

]

3.02.52.01.51.00.5xrms [mm]

(9.0558, 3.0510)

120

100

80

60

40

20

0

survivors

(a)

1.4

1.2

1.0

0.8

0.6

0.4

y rms

[mm

]

1.21.00.80.60.4xrms [mm]

(9.1558, 3.2510)

400

350

300

250

200

survivors

(b)

2.0

1.5

1.0

0.5

y rms

[mm

]

1.00.80.60.4xrms [mm]

(9.1558, 3.1510)

1000

800

600

400

200

0

survivors

(c)

Figure 3.8: Contour plot of the dynamic aperture as a function of closed-orbit RMS excursions.Islands of stability and instability are seen with a tendency towards higher stability for smallerexcursions.

3.1.6 Long-term stability and losses

When determining the dynamic aperture using single particle tracking, the maximum num-ber of full turns in the synchrotron to track a test particle before declaring its orbit stablemust be chosen. At high energy, the beam particles emit synchrotron radiation therebyloosing energy on their way around the booster. After each full turn they are acceleratedin the RF cavity. This has a net damping eect on their transverse motion, thus reducingperturbations in the beam over time. Within a few damping times any perturbation willpractically have died out. The common approach is therefore to track the number of turnscorresponding to a few times the damping time of the synchrotron. In cases where damp-ing times are very large, the number of turns to track also becomes very large, makingthe tracking very time consuming. It is therefore common practise to limit the number ofturns and assume the result is representative.

As stated, the dynamic apertures of the previous sections were based on tracking of

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3.1 Transverse dynamic aperture 41

only 1000 turns in the booster. This was approximately an order of magnitude lowerthan the damping time at 3 GeV, and more than ve orders of magnitude lower than thedamping time at injection energy. To determine whether the ndings obtained from so fewturns could be meaningful, longer trackings of 1.4 million turns were done at nine dierenttunes, indicated by black circles in gure (3.3). The number of turns correspond to 607.6ms, during which time the beam has been injected, ramped to full energy, and extracted,i.e. the complete lifetime of the beam in the booster. The assumption in this case is thatthe behaviour at the nine tunes are representative of all other tunes. The damping eectsof the RF cavity was not included. The square grid of test particles as described in section3.1.2 was injected into the booster. The fraction of test particles remaining as a functionof the number turns for the nine sets of tunes are shown in gure 3.9.

0

0.2

0.4

0.6

0.8

1

Fra

cti

on

ofre

main

ing

test

part

icle

sFra

cti

on

ofre

main

ing

test

part

icle

s

0.001 0.01 0.1 1 10 100

time [ms]time [ms]

100 101 102 103 104 105 106

Number of turnsNumber of turns

Tracking limit(9.0558, 3.0510)(9.0558, 3.1510)(9.0558, 3.2510)(9.1558, 3.0510)(9.1558, 3.1510)(9.1558, 3.2510)(9.2558, 3.0510)(9.2558, 3.1510)(9.2558, 3.2510)

Figure 3.9: The simulated fraction of particles remaining after a given number of turns fordierent sets of tunes. No damping eects have been included.

Seven of the nine tune sets show a quick decline, with little particle loss occuring after100 turns. However, the two sets (9.0558, 3.1510) and (9.1558, 3.1510) behave dierentlyand show a slower decay that continues even after 1000 turns. The (9.1558, 3.1510) tunepoint stabilizes around 30,000 turns while the (9.0558, 3.1510) point continually showssmall signs of beam loss.

Figure 3.9 suggests that the qualitative behaviour, i.e. the relative sizes of the dynamicapertures for dierent tunes, is suciently well established after 1001000 turns to give agood insight into the dynamics, and to select the areas in tune-space where the dynamicaperture is large. However, the areas of largest dynamic aperture appear also to be theslowest to converge (if ever), and so the more precise quantitative behaviour can only beextracted by using the full number of turns. In this context it is important to note thatthe adiabatic shrinking of the beam during acceleration makes the dynamic aperture lessinuential over time.

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42 Theoretical beam dynamics studies

3.1.7 Summary, further investigations, and optimizations

Neglecting the misalignments and eld errors, the dynamic aperture of the lattice is largerthan the physical aperture for a large part of the tune-space, and it is also possible to acco-modate momentum-spreads of ∆p/p ≤ ±1%, particularly if the chromaticity is correctedto (ξH , ξV ) = (1, 1). The errors result in a markedly reduction in available aperture and ahigh probability of beam loss from this for the majority of the nine investigated tune sets.However, as was mentioned in section 3.1.5 the random misalignments are overestimated.A more detailed analysis based on per-girder position errors was not possible within thegiven time-frame, and would possibly require changes in the source code of the trackingsoftware. However, considering the considerable impact that errors have shown, it wouldbe prudent to include such a study in the design of a future booster synchrotron of similarlattice characteristics.

A further improvement of the study would be to also include errors in the generatedtune map, i.e. generate a tune map similar to gure 3.3 for each of a number of virtualmachines. A third dimension could be added to the gure by color-coding the grey crosses(white→black) to indicate the fraction of the outcomes that are above a certain acceptablethreshold in physical aperture usability (e.g. 95%). Alternatively some sort of quality factorrepresenting the weighted dynamic aperture of the virtual machines could be assigned toeach tune point. Such a study would require many weeks of computing time if done in detailfor a similar number of tunes. To improve eciency while still obtaining representativeresults several things could be done. Firstly, one could limit the number of turns trackedto around 100 as this was seen in gure 3.9 to be an adequate number of turns to getqualitatively useful results. Secondly, the number of tracked particles could be reduced.This could be accomplished by a larger spacing between the particles or by changing howthe tracking is done: rather than indiscrimenately tracking all particles in a grid patternone could distribute particles on a circle with xed radius (see gure 3.10).

If a particle with a given φi does not

0

10

20

30

Heig

ht

[mm

]H

eig

ht

[mm

]

−20 0 20

Width [mm]Width [mm]

φi

r

Figure 3.10: Alternate tracking distribu-tion.

survive, the length of the radius vector r forthat particle is reduced by a xed amount,and a new particle is tracked from thatposition. If on the other hand the par-ticle survives, it is kept, and no trackingis done at that φi for shorter r. Trackingin this way eliminates tracking the major-ity of stable particles, and thus the major-ity of time needed to track, as the track-ing time is proportional to the number ofturns× particles.

A dynamic aperture obtained in thisway will include islands of instability as was seen in gure 3.7 islands of stability existand would falsely be taken to belong to the bulk, thus including the unstable areas in thedynamic aperture result but since this is a qualitative survey it would most likely beacceptable. Of course one could choose to limit this eect at the cost of computing timeby requiring that a certain number of consecutive particles, say three, with the same φi allsurvives before declaring all particles with equal φi, but shorter r, stable. Once a smallerregion of tune-space has been identied to have a large dynamic aperture, one could do amore thorough investigation for selected tunes in that region.

In section 3.1.5.1 it was seen that the cases of smaller closed-orbit excursions hadlarger dynamic aperture, as one would expect. It was suggested that the islands of high

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3.2 Longitudinal acceptance 43

dynamic aperture seen in gure 3.8 was related to the eective tunes of the particles forthat particular closed-orbit. A way to investigate if this was the case would be to note thetransverse position of each particle for every full turn and Fourier transform them to getthe fractional parts of the tunes. From this one should then be able to conclude whetherit is the eective tunes of the test particles that determines their survival, or if some othereect may be at work.

3.2 Longitudinal acceptance

The RF cavity is responsible for the acceleration of the beam, and it sets the limits on thelongitudinal acceptance of the booster.

−Emax

0

Emax

EE

−π φ0 01

2π π

φφ

Figure 3.11: The RF eld.

The synchronous angle for the booster is θ0 = −π2 (see gure 3.11). Every time the

synchronous particle has made a full revolution it arrives at θ0. If the particle's energyis slightly higher, dispersion will result in a longer orbit and the particle will arrive at alater time. This is similar to a smaller θ which means the particle will be decelerated bya negative voltage, and vice versa. Particles that are slightly o-momentum will thereforeoscillate around θ = θ0. These oscillations are called synchrotron oscillations, and thenumber of oscillations per turn is call the synchrotron tune. For small ∆p/p these oscilla-

tions are ellipses in the(θ, ∆p

p

)phase-space plane. If the particle's momentum deviation

is too large, it will arrive too much out of phase and be lost. The envelope in(θ, ∆p

p

)phase-space wherein the particles are stable is called the separatrix , and the area insidethe separatrix is the acceptance, sometimes also reered to as the bucket . For the caseθ0 = −π

2 the expression for the separatrix in(θ, ∆p

p

)phase-space simplies to

∆p

p= ±

√2 |e|U

πH |η|m0β2relγrel

cos(

θ + θ0

2

)(3.3)

where |e| = 1 is the elementary unit charge, U is the RF amplitude in V, H = 217 is theharmonic number, η is the fractional change in revolution frequency in units of ∆p/p, m0

is the electron mass in eV, βrel and γrel are the relativistic β and γ, respectively. [BJ93]

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44 Theoretical beam dynamics studies

The choice of cavity amplitude for a given particle energy must be suciently largeto compensate for the energy lost to synchrotron radiation, and to accelerate the beam.However, additional voltage, the so-called over-voltage, is needed to increase the lifetimeof the beam.

When a beam electron emits a photon, its reduction in energy may bring it outsidethe bucket, and it will be lost. The quantum lifetime is the time it takes for the numberof particles in the beam to be reduced to a factor 1/e due to this eect. Intra-beamcoulomb scattering where transverse momentum is converted into longitudinal momentumcan also cause beam loss if the acquired longitudinal momentum lands the particle(s)outside the bucket. This is called the Touschek eect. The Touschek lifetime dependsstrongly on the beam energy and is important for small beams of energies < 1 GeV.Though these two eects call for a large over-voltage, a too large over-voltage also hasnegative eects. The most important for the ASP booster being that it introduces a largemomentum-spread in the beam, which, through the dispersion, will enlarge the beam inthe transverse direction. Of lesser importance for the ASP booster, is that a large over-voltage increases the synchrotron tune, which in case of synchrotronbetatron coupling,from non-zero dispersion at the RF cavity, will enlarge the beam in the betatron tunespace. This in turn can lead to losses from resonances (see chapter 3.1).

When particles are injected into the booster at a phase angle slightly dierent from θ0,the momentum deviation will be increased. The distribution of particles in the bunchesfrom the linac is according to the manufacturer such that 85% is in the central 500 MHzbucket within ±10, and the remainder of the particles are in the two other 3 GHz buncheslocated ±2π/3 away from the center of the bucket, i.e. at θ = π/6 and −7π/6. Themomentum deviation ∆p/p for a particle as a function of the phase angle θ is given by(

∆p

p

)2

=

(∆p

p

)2

− |e|UπH |η|m0β2

relγrel(∆θ cos θ0 − sin θ + sin θ0) (3.4)

where ∆p/p is the maximum momentum deviation found at θ = θ0, and can be calculatedfor a particle with a given ∆p/p and θ. Table 3.2 shows the total momentum deviation as afunction of injected phase angle for the central bunch's outer envelope, and the inner edgeof the side bunches. The trajectories and the separatrices for U = 1.2 MV and U = 120kV are shown in gure 3.12. The transverse booster momentum acceptance shown in thegure was found during commissioning for a 100 MeV coasting beam (see section 4.5).

Table 3.2: The total momentum deviation for particles injected o the synchronous phase atan angle θ with an existing momentum deviation ∆p/p.

Maximum momentum deviation ∆p/p

Cavity voltage 120 kV 1.2 MV

∆p/p at injection 0% ±1.5% 0% ±1.5%

Center bunch: θ = θ0 ± 10 ±0.17% ±1.51% ±0.52% ±1.59%with 300 ps jitter: θ = θ0 ± 37 ±0.60% ±1.62% ±1.90% ±2.42%

Side bunches: θ = θ0 ±(

2π3 − 10

)±1.55% ±2.16% ±4.91% ±5.14%

As can be seen in the gure, the RF cavity voltage increases the momentum deviation ofall particles in the 3 GHz side bunches to more than the transverse momentum acceptancefor both cavity voltages. Any particles in these bunches will therefore be quickly lost afterinjection. Although the increase in the momentum-spread for the center bunch is small

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3.2 Longitudinal acceptance 45

−5

−2.5

0

2.5

5

∆p

p[%

]∆

p

p[%

]−1 −0.5 0 0.5 1

Time [ns]Time [ns]

−3

2π −π −

1

2π 0

1

θθ

Separatrix for U = 1.2 MV

Separatrix for U = 120 kV

Center bunch

Side bunches

Center bunch envelope

Center bunch envelope

Envelope with 300 ps jitter

Envelope with 300 ps jitter

Booster acceptance

Booster acceptance limit

Booster acceptance limit

Figure 3.12: The longitudinal phase-space. The separatrices for two dierent RF cavity volt-ages with the three 3GHz bunches having ∆p/p ≤ ±1.5% ((∆p/p)rms = 0.5%). The centralbunch has according to the manufacturer at least 85% of the total charge. The width of thecenter bunch is ±10, and the envelope of its longitudinal phase trajectories are given for 1.2MV (solid blue line) and 120 kV (dashed blue line) cavity voltages. The measured peak-to-peakjitter is 300 ps corresponding to ∆θ = ±27. The envelopes in the cases of jitter are given inred. The orange lines indicate the full width of the transverse booster momentum acceptancefound during commissioning for a 100 MeV coasting beam (see section 4.5). From the transverseacceptance, a limit on the longitudinal acceptance can be made (in yellow).

for both voltages, a large part of the center bunch tail is already outside the transverseacceptance.

The electron source was triggered by an electrical signal from a DG535 that was con-verted to an optical signal and then back again to an electrical signal. This setup had alarge jitter of 300 ps peak-to-peak. Including this jitter in the calculations, the inducedmomentum deviation can be seen to increase drastically for the high voltage while havingless impact for the low voltage.

The limited transverse momentum acceptance, and in particular its asymmetry, sig-nicantly reduces the available longitudinal phase-space as shown by the yellow solid anddashed lines. Assuming a Gaussian current density distribution in momentum-spread anda constant charge density in phase angle, about 18% of the center bunch charge will belost for U = 120 kV. This is without including the jitter.

By injecting at low voltage and then increasing the voltage during energy ramp, thetotal momentum-spread can be limited. More importantly, the choice of a lower cavityvoltage signicantly helps to reduce the losses by allowing capture from a larger intervalin phase angle, thereby limiting the eects of the jitter. Complete beam loss can occurat U = 1.2 MV if jitter shifts the bunch outside the acceptance. It was found duringcommissioning that having the maximum cavity voltage at injection would completelydestroy the beam for any number of shots.

The main conclusion from these calculations is that the asymmetry of the trans-verse momentum acceptance will place a strong limit on the injection eciency. If themomentum-spread from the linac should be reduced from the nominal ±0.5% RMS, thiswould have signicant impact on the injection eciency. Alternatively, the transversemomentum acceptance of the booster should be increased.

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Chapter 4

Experimental results

In this chapter some of the parameters that dene the performance of the booster are pre-sented. These are experimental results obtained mainly during the booster commissioningfrom February to May in 2006, but some were also obtained in August the same year.They have been presented with a poster at the European particle accelerator conferencein Edinburgh, Scotland in 2006 [FNBB+06], and the particle accelerator conference inAlbuquerque, New Mexico in 2007 [WBB+07].

4.1 Linac emittance and momentum-spread

The extracted beam current from the booster depends on the injected beam from the linac.For this reason investigation of the linac beam emittance and momentum-spread may benecessary in the case where the booster performance is unsatisfactory, or when increasedperformance is wanted.

A quadrupole scan was used to measure the beam emittance by recording the beam sizeon a screen as a function of quadrupole strength. If the screen saturates, the measurementwill give a too high emittance. Indeed a suspiciously large emittance was measured, and itwas decided to temporarily replace the Cr-doped alumina screens in the LTB with OpticalTransition Radiation (OTR) screens. Although OTR screens give less light they do notsaturate. They use the eect that when a charge particle crosses between two mediums withdiering dielectric constants it emits radiation [Str06, sec. 8.6.4]. Using the OTR screensan emittance measurement in the dispersion-free section found the normalized emittanceto be within specication (εN < 50 π mm mrad).

The energy dening slits were used to estimate the momentum-spread. Using the FCTin the LTB and the DCCT in the booster, the transmission was measured as a functionof slit opening. The second LTB FCT was unavailable at the time of the measurement,so when using the booster DCCT it is assumed that the beam is not scraped in theinjection septum1. Figure 4.1 shows the current transmission as a function of slit open-ing overlaid the calculated transmission for three 6.8 mA beams with Gaussian currentdistribution, normalized emittances of εN = 50 π mm mrad, and momentum-spreads of∆p/p = 0.6%, 0.5%, 0.7%.

There is good agreement between the measured data and the overlaid calculated beam,suggesting that the calculated beam is a resonably representation of the actual beam. The∆p/p = 0.6% case is the best least-squares t to the data. From this, the momentum-

1There may be some scraping as will be seen in gure 4.27 on page 83, in which case the momentum-spread found here is slightly overestimated.

46

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4.2 Closed-orbit and beam position 47

0

2

4

6

8

Curr

ent

inboost

er

[mA

]C

urr

ent

inboost

er

[mA

]

0 1 2 3 4 5

Energy defining slit opening [±mm]Energy defining slit opening [±mm]

Booster DCCT

∆p/p = 0.6%

∆p/p = 0.5%

∆p/p = 0.7%

Figure 4.1: The current measured by the DCCT in the booster as a function of EDS opening(black sqaures). Overlaid are the predicted currents for εN = 50 π mm mrad beams with threedierent ∆p/p. The ∆p/p = 0.6% case is the best (least-squares) t to the measurement.

spread must be concluded to be around 20% higher than the specied value of 0.5%, whichwill be seen to have a negative impact on the injection eciency.

4.2 Closed-orbit and beam position

The BPMs measure the closed-orbit position at 32 points around the booster and aredescribed in section 1.6.8. During the rst part of the commissioning, when large beamlosses occured shortly after injection, there were problems with roughly half the BPMsgiving bogus positions: large shot-to-shot variations were observed, and no observableeect from steering the beam with a corrector magnet were seen. These problems occuredfor those BPMs that were least attenuated, and also for BPM 31 and 32 (see gure 1.10).Later in the commissioning, when the initial losses were greatly reduced, it was only duringthe rst 100 ms after injection that BPMs other than 31 and 32 gave implausible values.Although not checked in detail, it is believed that the problem is caused by the automaticgain controllers (AGC) in the MX-BPM units, which read the pickup voltages. The AGCtries to keep the sum of the pickup signals constant, and during the time after extractionuntil a new beam is injected there is no signal, and the gain is increased. When the beamis injected, a strong signal is induced, which then decays as the beam suers initial losses.The large gain of the large signal causes those MX-BPMs connected with small attenuationcables to saturate until the AGC has adjusted to the new signal. With the MX-BPM unitsdesigned for storage ring use, the AGC is suspected to have a long response time. If thebeam losses are large, by the time the AGC has adjusted it may have reduced the gain toomuch, and the MX-BPMs with large attenuation cables get too little signal. This lattereect was observed during the early times of commissioning: during the rst part of theramp one half of the BPMs would function while the other half would not, and vice versaduring the last part of the ramp.

The beam position measurements are used to correct the closed-orbit by reducing itsexcursions. This is done using dipole corrector magnet of which there are 24 horizontal and12 vertical. The kick angle applied by a dipole corrector magnet to the beam is independent

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48 Experimental results

of the beam position and angle inside the corrector, and for small kick angles there is alinear relation between the kick and the amplitude of the resulting closed-orbit change.Thus, a so-called response matrix A can be constructed, which relates the changes in thecorrector kick angles to the change in beam position at the BPMs

∆x = A∆θ (4.1)

where ∆x is a vector of the change in the 32 horizontal beam position values, and ∆θ is avector of the change in the 24 horizontal corrector kick angles, making A a 32×24 matrix.The vertical case is similar with A being a 32 × 12 matrix. The matrix elements Aij canbe calculated from the model, as they can be written in terms of the beta-function valuesat BPM position i, and corrector position j

Aij =

√βiβj

2 sinπQcos Q (π − |φj − φi|) (4.2)

where Q is the betatron tune and φ is the betatron phase. Alternatively A can be measureddirectly by successively changing the corrector magnets and measuring the position change.Once A has been constructed, it can be used to calculate the kick angles that will minimizethe measured orbit excursions. Several methods exist for this purpose and it is oftennecessary to correct the orbit a few times before it is acceptably minimized. [Win94]

During the rst part of commissioning, it was found that the beam could not make a fullturn in the booster without resorting to the corrector magnets. As many turns are neededfor the MX-BPMs to calculate a position, the correction had to be done using the fourbeam viewers and adjusting the corrector manually. With separate function magnets, thequadrupoles can be used to align the beam in their center, where a change in quadrupolestrength changes the focus in the two transverse planes but does not move the beamcenter. In a combined function magnet, the quadrupole component cannot be changedwithout changing the dipole component as well. Thus, without the First Turn capabilitiesof the Libera units, it was necessary to use trial and error combined with intuitive guessesto arrive at a useful set of corrector values. However, once a set of corrector values hadbeen found that would allow many thousands of turns with low beam loss, the MX-BPMscould be used to ne tune the orbit.

Figure 4.2 shows the closed-orbit during the ramp. Apart from the last measurementdone in August 2006, the others were done in May the same year. The large excursionsseen for the last BPMs in August are suspected to be caused by a too low signal-to-noiseratio (see gure 1.10), as they did not change when the beam was kicked using a corrector.The integer parts of the horizontal and vertical tunes, 9 and 3, respectively, can also beseen.

The original correction procedure was based on entering the 64 measured orbit excur-sions by hand into Winagile [Bry99] and use its model-based minimisation routine to derivea set of kick angles to be added to the existing corrector settings in the spreadsheet usedfor the ramps. This was later improved by adapting the measuring software to write thebeam positions to a le that could be read directly by Winagile. However, the derivedkick angles still needed to be added by hand to the spreadsheet. This process was te-dious and time consuming, and since the ramp was constantly being changed to minimizelosses, it was only done if the closed-orbit was suspected of being a large cause of beamloss. However, later in the commissioning the storage ring orbit correction software wasadapted to the booster and used for orbit correction, allowing the orbit to be corrected inincremental steps of 10 seconds rather than 10 minutes, and with no more than 3 stepsto converge. As discussed in section 2.3, if the beam is o-axis in the combined-function

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4.3 Tunes 49

−10

−5

0

5

10

Hori

zonta

lexcurs

ion

[mm

]H

ori

zonta

lexcurs

ion

[mm

]

5 10 15 20 25 30

BPMBPM

50 ms (110 MeV)

100 ms (230 MeV)

200 ms (800 MeV)

300 ms (1.4 GeV)

400 ms (2.0 GeV)

500 ms (2.6 GeV)

600 ms (3.0 GeV)

August (3.0 GeV)

−10

−5

0

5

10

Vert

icalexcurs

ion

[mm

]V

ert

icalexcurs

ion

[mm

]

50 ms (110 MeV)

100 ms (230 MeV)

200 ms (800 MeV)

300 ms (1.4 GeV)

400 ms (2.0 GeV)

500 ms (2.6 GeV)

600 ms (3.0 GeV)

August (3.0 GeV)

Figure 4.2: The closed-orbit during ramp. The measurements were done in May, 2006, exceptthe last (thick, black line) which was done in August, 2006.

magnets it will change the tune, and if this brings the beam closer to a resonance it canincrease the emittance. This was observed at one point during commissioning when theextracted beam had a large vertical emittance. By correcting the orbit at high energy, thevertical emittance was reduced.

Insucient shielding of the C-magnet referred to as the thick septum placedin the BTS just downstream of the extraction septum gave a leak eld that aected theclosed-orbit in the booster. Full energy in the booster was achieved before the BTS wasready to accept the extracted beam, and so for safety reasons, the extraction septa hadbeen powered o. When the thick septum was turned on, its leak eld was strong enoughto prevent the beam from making a single turn in the booster. However, this eect could becompletely negated by using the booster corrector magnets. Later, additional shielding wasadded around the booster vacuum chamber to reduce the need for the corrector magnets.

4.3 Tunes

The betatron tune Q is dened as the number of betatron oscillations per revolution. Thebetatron frequency is fβ = Q×f0, where f0 is the revolution frequency. The fractional partof the tune, Qf , is detected using a spectrum analyzer. The spectrum analyzer performsa step-wise sweep over a given frequency range, exciting the beam at each frequency in

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50 Experimental results

turn while measuring the response signal (power) induced by the beam position shifts inthe stripline system. Strong responses are seen when the frequency matches resonancefrequencies of the beam, the strongest of which are the revolution and betatron frequenciesand their integer multiples. Qf appears as sidebands to the revolution frequency and itshigher harmonics, as shown in gure 4.3. The integer part can be inferred from the closed-orbit by applying a kick to the beam using a corrector magnet (see gure 4.8 on page 54).

-110

-90

-70

-50

-30

-10

518.0 518.4 518.8 519.2 519.6 520.0

Frequency [MHz]

Power [dBm]

Figure 4.3: Tune measurement with spectrum analyzer. The large left-most spike is 225th har-monic of the revolution frequency. The two smaller spikes around 518.5 MHz are the horizontaland vertical tunes.

The BD, BF, QD, and QF magnets are ramped virtually identically, and with analmost linear ramp. Figure 4.4 shows the ramping curves on a beam momentum scale: themagnetic elds that for nominal bending angles and tunes would correspond to a beamof the given momentum. During commissioning the relative strengths were optimizedwith the sole purpose of reducing the beam losses as much as possible (see section 4.8).Investigation of the resulting tunes were done afterwards.

The tunes were measured up the ramp around the 225th harmonic, 518.074 MHz, sincethe signals here were among the strongest found. The sweep time was set to 100 ms.Because of the non-zero sweep time, the horizontal and vertical tunes were measured atslightly dierent times ( <∼ 10 ms). The horizontal and vertical tunes were identied byhow they moved when either of the tuning quadrupoles were changed. The matrix T from(2.17) was calculated using WinAgile and used to this end.

Figure 4.5 shows the tunes as a function of time with the ramping curve for the BDmagnets overlaid. The change of working point during the ramp overlaid the dynamicaperture tune maps is shown in gure 4.6.

From gure 4.6(a) it is seen that the injection tune point is apparently in an area ofvery low dynamic aperture for both the May and August ramps. Overlaying the tuneson the o-momentum dynamic aperture maps (gures 4.6(b), 4.6(c)) shows a somewhatlarger dynamic aperture at injection. This is a peculiar result, insofar that a lot of timewas spent on nding the optimum injection point, which would give the smallest injection

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4.3 Tunes 51

0

500

1000

1500

2000

2500

3000

0 0,2 0,4 0,6 0,8 1Time [s]

Equi

vale

nt p

[MeV

/ c]

BD BF QD QF

Figure 4.4: The main magnets' ramping curves on a scale corresponding to the beam momen-tum. The BD magnets are divided into two groups, each group powered by a single power supply.The two supplies are set to the same current. For nominal current settings as dened by thedesign tune and magnet measurements all curves will overlap and top at 3 GeV. As can beenseen there are dierences in the settings. These settings were found to be optimal, resulting invirtually no losses above ∼ 500 MeV. The actual beam momentum is dened by the combinedbending strength of the BD and BF magnets (see (2.23)).

Tunes

9.05

9.10

9.15

9.20

9.25

9.30

0 100 200 300 400 500 600

Time [ms]

QH

3.00

3.05

3.10

3.15

3.20

3.25

QV

Figure 4.5: The tunes as a function of time with the ramping curve from 100 MeV to 3 GeVoverlaid. This measurement was done in May 2006.

losses. It was done by varying the individual magnet strengths and observing the beam

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52 Experimental results

3.5

3.4

3.3

3.2

3.1

3.0

QV

9.59.49.39.29.19.0QH

(a) ∆pp

= 0%

3.35

3.30

3.25

3.20

3.15

3.10

3.05

3.00

QV

9.359.309.259.209.159.109.059.00QH

(b) ∆pp

= −1%

3.35

3.30

3.25

3.20

3.15

3.10

3.05

3.00

QV

9.359.309.259.209.159.109.059.00QH

(c) ∆pp

= +1%

Figure 4.6: The variation in tune during the energy ramp overlaid the dynamic aperture plotfor ∆p/p = 0 detailed in section 3.1.4. The injection tune is the right-most tune point, andthe extraction tune the left-most. For the red curve measured in May 2006, the rst ve points(circles) are 100 500 MeV, the last ve are 1 3 GeV, all equidistantly spaced in energy. Forthe black curve measured in August 2006, the points (squares) are at 100 MeV, 2 GeV, and 3GeV, respectively. The 3rd and 4th order resonances are shown as thick black and thin grey lines,respectively.

current. It is highly unlikely that these optimizations would converge on the least favorabletune point in the vicinity, as gure 4.6(a) indicates.

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4.3 Tunes 53

There may be two obvious reasons for this result. Firstly, the dynamic aperture mayalready be suciently large (see gure (4.7)), and the matching between the LTB and thebooster favours this injection tune. Secondly, the dynamic aperture plots are for the ideal

(QH , QV ) = (9.2458, 3.1827)

0

5

10

15

20

25

y[m

m]

y[m

m]

−20 −10 0 10 20

x [mm]x [mm]

∆p

p= +1%

∆p

p= 0%

∆p

p= −1%

Physical aperture

Injected beam

Figure 4.7: Dynamic aperture envelope plot for a tune point very close to the measured injectiontune point of (9.253, 3.182). No chromaticity correction has been done for the o-momentumparticles. The dynamic aperture limits the o-momentum parts of the beam somewhat.

case, and it is possible that the actual booster has a set of misalignment and eld errors,that changes the tune map in such a way as to give the injection point larger dynamicaperture relative to the immidiate surrounding area.

To obtain the integer part of the tune, the corrector magnets were used. A horizontaland a vertical corrector were used to apply a small kick to the beam at 3 GeV. Thedierence between the orbit before and after the kick is shown in gure 4.8. Overlaidare the predictions of the lattice model, and also of the so-called adapted model, whereinthe quadrupole components of the BF and BD magnets have been changed to match themeasured tunes. This is elaborated further in section 4.3.1. For the horizontal tune thereis good agreement between the models and the measurements suggesting that the integerpart of the tune is indeed 9. For the vertical tune, only the adapted model is a reasonablyt, although there are somewhat large discrepancies. However, the integer part of the tuneis clearly 3.

4.3.1 Tune sensitivity to BF/BD ratio

When the tunes were rst measured during commissioning, the vertical tune was found tobe below 3. The tunes were found to be around (QH , QV ) ≈ (9.3, 2.85) at 3 GeV ratherthan the designed (9.2, 3.25) for the nominal settings of the tuning quadrupoles, QD andQF, of kQD = 0.4 m−2 and kQF = −2.35 m−2. The dierence of (∆QH ,∆QV ) ≈ (0.1,−0.4)between the observed tunes and the designed tunes can be explained if the beam experiencesintegrated quadrupole elds in the BD and BF magnets that are dierent from the valuesmeasured during the FAT. A change of −0.94% and +0.53% of the respective BD and BFquadrupole focusing strengths relative to the measured can account for the dierent tuneat 3 GeV. At 2.7 GeV the corresponding change of strengths are −0.83% and 0.58%, whileat 1.8 GeV they are −0.86% and 0.84%.

During commissioning it was found that the beam was extremely sensitive to the ratioof the magnetic eld strengths between the BD and BF magnet families. In constructing

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54 Experimental results

Horizontal difference orbit

-8

-6

-4

-2

0

2

4

6

8

0 130.2

Position [m]

Beam excursion [mm]

Measured

Model

Adapted model

(a) Horizontal dierence orbit

Vertical difference orbit

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 130.2

Position [m]

Beam excursion [mm]

Measured

Model

Adapted model

(b) Vertical dierence orbit

Figure 4.8: The measured horizontal (a) and vertical (b) dierence orbits originating from asmall kick from a horizontal/vertical corrector magnet. The measured points are overlaid withthe predictions of the untouched lattice model and of the adapted model where the focusingstrengths of the BD and BF magnets have been changed to match the measured fractional tune.

the ramping curves for the magnets excitation, measurements of∫

B (I) ds were used totransparently compensate for saturation. Taking the ramping curves in gure 4.4 for the

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4.3 Tunes 55

BF, QD, and QF magnets and dividing with that of the BD magnet, the ratio for eachmagnet during the ramp is obtained and shown in gure 4.9 as a function of the beammomentum rather than time. It is seen that in the beginning of the ramp, the ratio BF/BDis below one, but quickly shoots up to 1.02 during the ramp to 500 MeV, after which itdecreases slightly. The ratio QD/BD starts being very high, but is approximately 1 after500 MeV with a small dip around 2.3 GeV. The ratio QF/BD starts very low, becomesslightly greater than 1 around 500 MeV, after which is dips below 1 and stays constantfrom around 1 GeV.

0,95

1

1,05

1,1

1,15

0 500 1000 1500 2000 2500 3000p [MeV/c ]

Rel

ativ

e st

reng

th

BF QD QF

Figure 4.9: The ratios of BF, QD, and QF to BD during ramping. The design ratio is 1.

During commissioning it was attempted to keep the ratio BF/BD constant. It wasbelieved that the change in ratio was necessary only because of a tune resonance, thus theQD and QF were used in an attempt to stay clear of this resonance. These attempts wereunsuccessful, in that acceleration could not be achived without changing the BF/BD ratio.The reason for the necessity to vary the ratio is not understood, and the fact that manythings are changed simultaneously when the BF/BD ratio is changed complicates matters(see section (2.4)). However, at SLS where the booster also operate with two dierentcombined-function dipole families, they have included a trim coil in the BF magnet tocompensate for possibly dierent residual elds between the BD and BF magnets [GIJ+98] at SLS they use a single power supply to drive both families, at ASP the two familiesare on separate supplies so there is no need for a trim coil. It is therefore possible that thevariation in the BF/BD ratio is prompted by dierent residual elds between the BD andBF magnets.

The change in tunes with the change in the BF/BD ratio R given in (2.20) was in-vestigated at 3 GeV. Here the beam is geometrically smallest and the beam current thusleast sensitive to changes in R. The ramp was repeatedly modied to give dierent R at3 GeV. Since this changed the tunes and Twiss parameters it was sometimes necessary tomodify the values of QD and QF as well as perform closed-orbit corrections in order toavoid resonances which would give beam loss. For this reason it was not straight forwardto obtain the tunes as a function of R, and the following steps were instead taken.

The settings of the QD and QF magnets were entered into the lattice model, and the

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56 Experimental results

0.664

0.662

0.660

0.658

k BD

[m

-2]

1.0151.0101.0051.000R

-0.832

-0.830

-0.828

-0.826

-0.824

kB

F [m-2]

(a) Calculated BD (blue circles) and BF (red triangles) focusing as afunction of BF/BD ratio R. The t functions are given in (2.27).

3.5

3.4

3.3

3.2

3.1

3.0

QV

9.59.49.39.29.19.0QH

(b) Changes in tune for equidistant steps in R from 1.00 (upperleft) to 1.02 (lower right). The green lines with circle markersare for (kQD, kQF) = (0.5946,−2.294) m−2, and the red (blue)line with triangles (squares) are for kQD (kQF) increased by10% (2%).

Figure 4.10: The inuence on focusing and tunes of a variation in the BF/BD ratio R (see(2.20)). From tune measurements the equivalent focusing strengths (kBD, kBF) were calculatedand shown in (a) these also include the chromaticity eect, and are therefore dierent fromthose shown in gure 2.2 on page 23. From these the changes in tunes for xed tuning quadrupolestrengths were obtained (b).

values of the focusing strengths kBD and kBF were calculated to give the measured tunes.

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4.3 Tunes 57

It is important to note that the focusing strengths found this way include the eect of thechromaticity due to the change of beam momentum (see (2.23)), and are therefore dierentfrom those given in (2.26a) and (2.26b) and shown in gure 2.2 on page 23. Figure 4.10(a)shows the calculated focusing strengths as a function of R tted with (2.27). The resultsof the ts are

∆kBF = −0.41016± 0.00828 m−2 (4.3a)

kBF0 = −0.41349± 0.00835 m−2 (4.3b)

⇒ kBFR=1 = ∆kBF + kBF

0 = −0.82365± 0.01663 m−2 (4.3c)

⇒kBF

R=1

kBFFAT

− 1 = −0.38%± 2.0%

∆kBD = 0.30041± 0.00658 m−2 (4.3d)

kBD0 = 0.36464± 0.00653 m−2 (4.3e)

⇒ kBDR=1 = ∆kBD + kBD

0 = 0.66505± 0.01311 m−2 (4.3f)

⇒kBD

R=1

kBDFAT

− 1 = −0.24%± 2.0%

where kFAT are the average values measured during FAT (see gure 2.4). Inserting thetted values in (2.27), and using that the tunes change approximately linearly with thefocusing strengths, the changes in tunes as a function R can be calculated. Figure 4.10(b)shows these changes. The high sensitivity of the vertical tune to the quadrupole strengthsis clearly seen. The change in tunes from R = 1 to 1.02 is (∆QH ,∆QV ) = (0.12,−0.37).For the nominal QD and QF settings, the BF/BD ratio of R ≈ 1.02, which was used duringthe larger part of acceleration, would result in a vertical tune below 3, as was also observed.Mainly the QD was used to correct this, moving the vertical tune to above 3. These tuneestimates depend on the chromaticity: by comparing gure 4.10(a) with gure 2.2 onpage 23 it is seen that the chromaticity has a signicant impact on the calculated focusingstrengths, particularly the kBD.

It is also interesting to calculate what the focusing strengths are from the tunes mea-sured at R = 1, since these should be identical to those measured during FAT and couldtherefore serve as an independent check of the FAT measurements. The results of the tin (4.3) are in agreement with the FAT measurements. However, the large uncertainties aswell as the dependence on the assumptions of the relation between k and R means that theyshould not be used to estimate the R = 1 strengths. Instead a single tune measurementcarried out at R = 1 is used, and from this the BF and BD focussing strengths were foundto dier from the FAT measurements by 0.01%±0.05% and −0.61%±0.04%, respectively.The uncertainties are from the resolution limit of the tune measurement alone. Translatingthis into an average horizontal o-set inside the magnets yields xBF = (0.4± 1.4) mm andxBD = (−21± 1.3) mm, where the average sextupole strengths from the FAT measure-ments and (2.15) have been used. The calculated oset in BD is unrealisticly large andwould place the beam just outside the vacuum chamber. From gure 2.4 it appears thatthe variation in quadrupole strength is around ±0.1%. If this is dominated by randommeasurement errors, there should be a systematic error of the order ±0.6% in the BD mea-surements. This is a rather large error, but it cannot be ruled out. During the prototypemeasurements, the quadrupole gradient was found to be 0.7% lower than designed at fullexcitation. This was caused by saturation. It was believed that by increasing the edgeangles by 0.25, the change in focus could be improved at high excitations with accept-able impact at low excitions. For a combined-function magnet, measurement of the edge

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58 Experimental results

angles is inherently dicult due to the gradients. Even if either the measurement or thecalculation was awed and the magnet altered needlessly, it could still not account for theerror: to achieve the measured tunes would require inconsistent changes in edge angles,with the horizontal tune requiring an edge angle of 5.80 and the vertical requiring 4.33.Also, the measurement procedures were identical for the BD and BF magnets, and the BFfocusing shows no deviation. Other than systematic errors in the BD magnet the mostlikely other options are (possibly systematic) misalignments including misalignment ofthe BD magnet on the girder or damage to one or more of the magnetic elements.

In case of the reason for R 6= 1, the residual eld-problem discussed earlier in thissection is a possibility. However, there may also be something blocking the beam path,which can be gotten around by changing the individual bending strengths of the dipoles.Unfortunately, since the tunes are global properties of the booster they cannot be usedto nd such localized errors. Instead the corrector magnets can be used to create localbumps that could steer the beam around the obstacle. However, this was not investi-gated during commissioning, because the functionality for easily making local bumps wasnot implemented in the control system, and it would have been very cumbersome to domanually.

4.3.2 Average beta function at the QD positions

A model of the lattice was used to calculate tune-shifts as a function of changes in thequadrupole strengths of QD and QF, which in turn was used to distinguish between hor-izontal and vertical tunes. By measuring the tune shift ∆Q as a function of change inquadrupole strength ∆k, the average beta function for the serial connected quadrupolescan be found from [Zim98]

〈βquad〉H,V = ± 2n∆kLquad

[cot (2πQH,V ) (1− cos (2π∆QH,V )) + sin (2π∆QH,V )] (4.4)

where the ± refers to the horizontal (H) and vertical (V ) plane, respectively, and the n tothe number of serially connected quadrupoles. Increasing the QD 5% yielded 〈βQD〉H =1.85 m and 〈βQD〉V = 11.0 m at p = 113 and 105 MeV, respectively. At 3 GeV therespective values are 1.92 m and 8.2 m. The model gives 1.94 m and 13.1 m. There is afairly good agreement between the horizontal values, while the vertical values are somewhatdierent. Changing the model's BD and BF quadrupole strengths in order to match themeasured tunes at injection time found the horizontal beta function changed by less than5% while the vertical beta function increased by 25% to 16.3 m.

The corresponding measurement for the QF family was not made.

4.4 Chromaticities

Particles that have a small momentum deviation also have a slightly dierent focusinglength. The chromaticity

ξ =∆Q

∆p/p(4.5)

is the change in tune resulting from a change in momentum, and is linear for small changesin momentum. Sometimes the chromaticity is also dened as the relative change in tunewith momentum, in which case (4.5) should be divided with Q.

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4.5 Dispersion function and transverse momentum acceptance 59

The momentum-compaction factor αp is dened as the relative change in orbit lengthwith the relative change in momentum. Thus, for a constant magnetic eld

∆p

p= −α−1

p

∆frev

frev= −α−1

p

∆fRF

fRF(4.6)

where frev is the revolution frequency and fRF is the RF frequency. Inserting (4.6) into(4.5) results in

ξ = −∆QαpfRF

∆fRF(4.7)

Knowing αp and varying the RF frequency while measuring the changes in tunes, thechromaticities can be obtained.

In the following the design value of the momentum compaction, αp = 0.00985465,has been used since two attempts to measure it directly yielded conicting results, andfurther attempts could not be completed within the given time. The measurement of αp isdone by scaling the magnetic elds of all magnets and observing the change in revolutionfrequency. This must be done while the booster is not ramping and the RF is turnedo. It is suspected that the reason for the failed measurements was due to an insucientdetuning of the RF cavity, allowing the beam to excite the cavity, which in turn acted backon the beam, changing its momentum.

With the entire facility using the same RF frequency, changing the RF in the boosterwould also change the linac RF. Changing the linac RF changes the energy of the linacbeam, giving a mismatch between the injection energy and the booster magnet injectionsetting, resulting in large injection losses. Thus, the spare frequency generator, the MO,was used for the booster RF, allowing the booster RF to be changed without aecting thelinac RF.

The chromaticities were measured at three dierent times corresponding to three dif-ferent energies: 100 MeV, 2 GeV, and 3 GeV. The changes in tunes are shown in gure4.11 as a function of momentum deviation and RF frequency. The relation is resonablylinear within the investigated range, and the results of the linear ts, the chromaticities,are given in table 4.1. The chromaticities are postive and low in all cases.

Table 4.1: The chromaticities at injection, extraction, and 2 GeV.

ξH ξV

10 ms (100 MeV) 1.45± 0.10 0.900± 0.100400 ms (2 GeV) 0.483± 0.045 1.23± 0.06600 ms (3 GeV) 0.442± 0.099 2.09± 0.04

During commissioning measurements of the chromaticities were not done. Rather thetuning sextupoles were manually adjusted to maximize the beam current at injection. Thetuning sextupoles were not ramped, as this showed no eect on the beam current afterinjection. Instead they were set at a constant value, thus having a larger inuence at lowenergy than at higher energy.

4.5 Dispersion function and transverse momentum acceptance

The displacement xD of a particle with a momentum deviation ∆p/p is given by (1.12).The bending occurs in the horizontal plane, thus the dispersion is that of the horizontalplane. There is also a slight bending in the vertical plane originating from the vertical

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60 Experimental results

−0.02

−0.01

0

0.01

∆Q

H∆

QH

−1% −0.5% 0% 0.5% 1%

∆p/p∆p/p

10 ms (100 MeV)

400 ms (2 GeV)

600 ms (3 GeV)

−60 −40 −20 0 20 40

∆fRF [kHz]∆fRF [kHz]

−0.02

−0.01

0

0.01

∆Q

V∆

QV

10 ms (100 MeV)

400 ms (2 GeV)

600 ms (3 GeV)

Figure 4.11: The change in tune as a function momentum deviation for three dierent times(energies).

betatron oscillations. However, the amplitude is very small making the vertical dispersioncompletely negligible compared with the horizontal. Using (4.6), the dispersion can begiven as a function of RF frequency

xD (s) = −D (s)αp

∆fRF

fRF(4.8)

where αp is the momentum compaction factor. As was discussed in section 4.4, the designvalue of the momentum compaction, αp = 0.00985465, will be used instead of a measuredvalue.

The dispersion function was measured both at 100 MeV without ramping the booster,referred to as DC, and at 3 GeV during a normal ramping cycle. Figure 4.12 showsresults overlayed the untouched model's and the adapted model's predictions (see section4.3). There is over-all a very good agreement, though there are discrepancies in the straightsections.

Since the booster lattice is four-fold symmetric, the dispersion function measured shouldbe the same for each quadrant. Figure 4.13 shows the dispersion function as a function ofquadrant. Variations in the dispersion function are clearly seen, and are most likely causedby the closed-orbit excursions as well as variations in the magnets and their alignment.

Simultaneously with the position measurements at 100 MeV, the current remaining inthe booster after a full second was recorded. Due to variations in the injection septum

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4.5 Dispersion function and transverse momentum acceptance 61

100 MeV (DC)

0

0.1

0.2

0.3

0.4

0.5

D[m

]D

[m]

0 20 40 60 80 100 120

Position [m]Position [m]

3 GeV

0

0.1

0.2

0.3

0.4

0.5

D[m

]D

[m]

Model

Adapted model

Measured

Figure 4.12: The measured values of the dispersion function at 100 MeV (DC) and 3 GeVoverlayed the model's and the adapted model's predictions.

strength and the RF phase mismatch, the shot-to-shot current varied by more than 50%.Thus, the current recorded is the maximum current from ten consecutive shots. Using (4.8),the current in the booster as a function of the momentum-deviation ∆p/p was obtained.The results are shown in gure 4.14.

Increasing the frequency decreases the orbit diameter and vice versa. Thus, the currentas a function of RF frequency is dependent on the set of orbit corrections. Also, since limit isset by the dynamic aperture rather than the physical aperture at least for positive ∆p/pas seen by the presence of an island of stability around ∆p/p ≈ 3% the measurementclearly depends on the tunes and chromaticities as well.

At the chosen RF frequency of 499.654 MHz corresponding to ∆p/p = 0, the current isat its maximum. The current decreases faster for lower momenta than for higher, suggest-ing that on average the beam is closer to the inner (dynamic or physical) aperture than theouter. The FWHM of the peak, corresponding the the full peak-to-peak transverse momen-tum acceptance, is 2.1%. There is an asymmetry around zero, with the acceptance beingin the interval −0.675% < ∆p/p < 1.42%. As discussed in section 3.2, it is the numericallylower of the limits that sets the maximum momentum acceptance, and in this respect the

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62 Experimental results

100 MeV (DC)0

0.2

0.4

D[m

]D

[m]

1 2 3 4

QuadrantQuadrant

3 GeV

0

0.2

0.4

D[m

]D

[m] BPM 1

BPM 1 averageBPM 2BPM 2 averageBPM 3BPM 3 averageBPM 4BPM 4 average

1 2 3 4

QuadrantQuadrant

BPM 5BPM 5 averageBPM 6BPM 6 averageBPM 7BPM 7 averageBPM 8BPM 8 average

Figure 4.13: The dispersion function variation between the four quadrants for each of the eightbeam position monitors at 100 MeV (DC) and 3 GeV.

0

0.25

0.5

0.75

1

Im

ax

[mA

]aft

er

1s

Im

ax

[mA

]aft

er

1s

499.5499.55499.6499.65499.7

fRF [MHz]fRF [MHz]

−1% 0% 1% 2% 3%

∆p/p∆p/p

2.1%

Figure 4.14: The current remaining in the booster after a full second as a function of themomentum-deviation calculated using equation (4.8) from the RF frequency at 100 MeV (DC).The red line indicates the half maximum of the current. At the highest momentum shown,the current is seen to increase again, which could mean that the beam has crossed a betatronresonance. Lower and higher RF frequencies could not be investigated, as these would cause thebooster low-level RF to trip.

large asymmetry leads to an unnecessarily small momentum acceptance. To change thecenter from ∆p/p = 0 to ∆p/p = 1

2 (1.42%− 0.675%) = 0.37% ⇒ fRF = 499.636 MHz a small fractional change in RF frequency corresponding to the booster circumferencebeing 4.8 mm larger than 130.2 m. Unfortunately, because the RF frequency is used bythe linac and storage ring as well, it may not be over-all optimal to change it.

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4.6 Emittance measurements 63

4.6 Emittance measurements

The transverse beam emittance was measured non-destructively at 3 GeV using a trigger-able frame-grabbing CCD synchrotron light monitor (SLM). The synchrotron light emit-ted in a bending magnet gives information on the electron beam prole from which thetransverse emittances can be found. Since the power of the emitted synchrotron lightis proportional to the inverse square of the bending radius [Jac01], the SLM was placeddownstream of the 3rd BD magnet after injection.

The synchrotron light emitted in a BD bending magnet was reected by a 15×15 mm2

Ni:P coated at copper mirror angled 46 to the light path. The light exited the vacuumchamber via a quartz window. It was then focused by a lens and passed through a holein an aluminum foil to remove scattered light. To have a well dened optical wavelengthand low diraction, the CCD camera was tted with a blue color lter. Intensity lterswere also tted onto the camera to avoid saturation of the CCD chip. Figure 4.15(a)shows the synchrotron light monitor system setup with a red line indicating the path ofthe synchrotron light. A schematic view of the setup is given in gure 4.15(b).

(a)

(b)

Figure 4.15: The synchrotron light imaging system setup. A top-view photo with a red lineindicating the synchrotron light path from the source (bending magnet) to the CCD camera (a),and a schematic view (b).

The transverse electron beam density distribution is assumed to be Gaussian. The 1σelectron beam size can then be found from the synchrotron light prole on the CCD

σbeam = M × σCCD (4.9)

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64 Experimental results

where σCCD is the standard deviation obtained by tting the CCD intensity prole witha Gaussian, and M is the magnication by the lens given by

M =d0

df(4.10)

where d0 is the distance from the source to the lens, and df is the distance from the lensto the CCD as seen in gure 4.15(b). There is some uncertainty on d0 caused by theextended source region. However, d0 was estimated using the 3D drawings of the boosterby extending a straight line back from the mirror to the tangential point of the centeredbeam.

The camera was triggered with a delay of 600 ms from injection, corresponding to abeam energy of 3 GeV. The longitudinal camera position was varied to nd the focus whichis where the horizontal and vertical image sizes are minimal. At each camera position an 8bit gray scale picture was grabbed with the CCD camera. The horizontal and vertical pixelsums were taken to obtain horizontal and vertical intensity distributions. The intensitydistributions were both tted with Gaussians, and transverse image sizes were found foreach camera position. These sizes were then plotted as a function of camera position andtted with a second order polynomial to estimate the minimal horizontal and vertical beamspot sizes on the CCD, σCCD,H and σCCD,V , respectively (see gure 4.16). The obtained

y = 0.0483x2 - 10.027x + 580.38

R2 = 0.9936

y = 0.0105x2 - 2.4575x + 180.25

R2 = 0.5604

0

20

40

60

80

100

120

140

160

180

80 100 120 140 160

Relative camera position [mm]

Bea

m s

pot s

ize

on C

CD

[ µm

]

x [µm]y [µm]Poly. (x [µm])Poly. (y [µm])

Figure 4.16: The 1σ transverse beam size on the CCD chip as a function of camera position.

estimates found were σCCD,H = 60 µm and σCCD,V = 36 µm. Because the synchrotronlight from the horizontal plane comes from an extended source region due to the horizontalbending, a deviation from ideal imaging can occur in this plane. This, together with a

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4.6 Emittance measurements 65

large uncertainty in the exact position of the vertical minimum, is most likely the causeof the dierent positions of the horizontal and vertical minima. Using equation (4.9) andthe estimated spot sizes, the transverse electron beam sizes are σbeam,H = 243 µm andσbeam,V = 148 µm. Figure 4.17 on page 75 shows synchrotron light images for the 3 GeVmulti-bunch and single-bunch beam.

The horizontal and vertical emittances are given by

εH =σ2

beam,H −(

∆pp D

)2

βH(4.11a)

εV =σ2

beam,V

βV(4.11b)

where D is the dispersion function, ∆p/p the momentum spread, and β the beta-function.The momentum-spread has not been measured, thus using equation (4.11a) only an upperlimit on the horizontal emittance can be estimated. Figure 4.8 suggests that the horizontalbeta-function is in reasonable agreement with the adapted model and that the verticalbeta-function is fairly close to the adapted model as well.

Using the beta-function values at the center of the BD magnet from the adapted latticemodel, βH = 1.21 m and βV = 12.9 m, the emittances are found to be εH < 49.2 nm radand εV = 1.7 nm rad. The horizontal value is much larger than the expected 33 nm rad,which would suggest that the beam is close to a tune resonance and later measurements,shown in table 4.19, did indeed yield lower values.

Figure 4.12 shows the dispersion function measured at 100 MeV without ramping thebooster. It was not measured directly at the source of the synchrotron light, which ismidway between the 3rd and 4th BPM. However, the measured dispersion function at theBPMs agrees well with the model in that region. Thus, using the dispersion value fromthe model, D = 0.2024 m, an upper limit on the damped momentum-spread can be set

∆p

p<

σbeam,H

DH= 0.12% (4.12)

This limit is in agreement with the one calculated from the lattice model where a dampedenergy-spread of 0.094% was found. From the obtained emittances, the coupling factor is3% < εV /εH < 19%, where the lower limit is assuming zero momentum-spread and theupper limit uses the model's damped energy-spread.

Leaving the camera position, such that df = 247 mm, the emittances in single-bunchmode at 3 GeV were found to be εH < 31.3 nm rad and εV < 6.6 nm rad. The single- andmulti-bunch results should be the same, however the measurements were obtained morethan three months apart with dierent magnet settings and therefore cannot be directlycompared.

The beam prole was also measured during ramping in multi-bunch mode, starting at150 ms after injection and proceeding in steps of 50 ms. Due to the lack of synchronizationbetween the gun trigger and the camera, 30 screenshots were taken at each step to ensurea reasonable number of screenshots with sucient luminosity. A single screenshot fromeach step is shown in gure 4.18. Assuming zero momentum-spread (4.11) was used tocalculate the emittances. The results are shown in table 4.2 and gure 4.19 with theuncertainties dominated by the shot-to-shot variations in the beam size. Also shown isthe equilibrium emittance, εeq., calculated from the lattice model based on the equilibriumbetween synchrotron radiation damping and excitation.

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66 Experimental results

Table 4.2: The beam size, and upper limits on the emittances and momentum-spread duringramping.

Beam size Maximum emittance Momentum-spread

t [ms] p [GeV] σH [µm] σV [µm] εH [nm] εV [nm](

∆pp

)max

[10−5]

150 0.51 786 ± 10 477 ± 9 512 ± 13 17.6 ± 0.7 388 ± 5200 0.80 533 ± 11 317 ± 20 236 ± 9 7.8 ± 1.0 263 ± 5250 1.10 364 ± 9 165 ± 188 110 ± 6 2.1 ± 5.0 180 ± 5300 1.39 206 ± 5 215 ± 7 35 ± 2 3.6 ± 0.2 102 ± 3350 1.68 129 ± 5 175 ± 11 14 ± 1 2.4 ± 0.3 64 ± 3400 1.96 145 ± 20 158 ± 25 17 ± 5 1.9 ± 0.6 72 ± 10450 2.27 126 ± 14 174 ± 19 13 ± 3 2.4 ± 0.5 62 ± 7500 2.59 154 ± 7 228 ± 9 20 ± 2 4.0 ± 0.3 76 ± 4550 2.90 236 ± 25 223 ± 17 46 ± 10 3.8 ± 0.6 117 ± 13600 3.02 187 ± 7 272 ± 11 29 ± 2 5.7 ± 0.5 92 ± 4

The emittance values found in this section are in reasonable agreement with the val-ues from the adapted lattice model, although they tend to be consistently smaller thanpredicted insofar that the momentum-spread contribution to the beam size has not beensubtracted. Although it is possible that the beta-function values used in calculating theemittances are too small, it was seen in section 4.3.2 that the average value of the hori-zontal beta-function measured at the location of the QD magnets in good agreement withthe lattice model. Rather, it is more likely that the distance d0 between the source pointand the lens is underestimated, and since the calculated emittances depend on the squareof d0, this has a large impact on the calculation of the emittances from the CCD beamproles.

At 550 ms the horizontal emittance increases, and also varies relatively more from shotto shot. As shown in gure 4.20 this is not due to a bad Gaussian t to the beam prolebut rather a real eect. The most likely explanation is that the beam is close to a tuneresonance at this time.

The emittance was also attempted measured in the BTS using a quadrupole scan inthe same way as the linac emittance was measured in the LTB. The rst quadrupole inthe BTS was varied while keeping the second quadrupole xed. This was done for threedierent values of the second quadrupole, observing the beam size on the screen placed justdownstream of the quadrupole. The screens in the BTS are of the same type as those inthe LTB which had saturation problems, whereby the width of the Gaussian beam prolet became to wide (see discussion in section 4.1). Thus, a single-bunch beam was usedwith a current ∼ 1.6 mA to minimize saturation on the screen. However, the emittancesobtained were larger than the ones measured in the booster by a factor of 3 5, and thehorizontal emittance varied from 117 to 150 nm rad, depending on the strength of thesecond quadrupole. There was a tendency for smaller beam sizes calculated using theBTS lattice model to give larger measured emittances. It is therefore suspected that,like in the LTB, the screen indeed saturated.

4.7 Position stability of the extracted beam

The energy and position stability of the extracted beam is important to ensure minimalloss at injection into the storage ring. If the beam is injected into the storage ring with an

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4.7 Position stability of the extracted beam 67

oset relative to the nominal orbit it corresponds to an emittance increase.A measurement of the shot-to-shot energy stability is done most reliably in the stor-

age ring and has not been done in these investigations. The position stability can beinvestigated in the BTS by use of a uorescent screen and a triggerable frame grabber.Any shot-to-shot energy variation will to rst order contribute linearly to the measuredmovement of the beam, unless measured in the dispersion-free part of the transfer line (seegure 1.5).

Though any element in the injection system that can vary in time can aect the positionstability of the extracted beam it is most often caused by one of the fast-pulsed magnetsystems, such as the septum, kicker, or bumper magnets and their respective power sup-plies.

The position stability was investigated by inserting a 50× 50 mm2 Cr-doped alumina(Al2O3) beam viewer into the beam shortly after the second quadrupole magnet in theBTS. The light emitted by the screen was detected by a non-triggerable2 JAI A11 CCDcamera [JAIb]. A sequence of 60 images were taken free running at 30 fps at one minuteintervals for an hour. From each sequence one non-saturated picture with the beam spoton was selected. For each picture the horizontal pixel value sum and the vertical ditto wasmade and a Gaussian t was performed using Matlab[Mat] on both of the distributions.The measurements were done in May 2006 for the multi-bunch mode and again in Augustthe same year for both multi-bunch and single-bunch mode. The results for the horizontaland vertical sum Gaussian ts of the data from May are shown in gure 4.21 on page 78.The histogram of the horizontal positions was tted with a Gaussian and the standarddeviation found. The middle, left sub-gure gives the variation in position in the form ofa 6σ value. The variation in units of the average sigma are then

σpos,H

σbeamsize,H=

0.0048/69× 10−4

= 0.89 (4.13a)

σpos,V

σbeamsize,V=

0.0008/61.54× 10−3

= 0.09 (4.13b)

A maximum position variation of 0.1 was requested to ensure an acceptable injection intothe storage ring. The horizontal variation found was 9 times higher than this.

As gure 4.21(a) shows, the beam moves roughly within a span of ±1.5 mm, which canbe explained by a kick angle at the extraction septum of ±0.25 mrad, corresponding to aseptum excitation current variation of about ±20 A. Disregarding the extreme variationsin current seen in gure 4.22, the current is seen to vary in good accordance with thisprediction. It should be noted that the gure was made after the measurement. However,it is representative of the behavior and amplitudes observed. These ndings strongly sug-gest that the extraction septum current variation alone could explain the large horizontalposition variation. Consequently, the septum current variation woudl have to be reducedby at least a factor of ten for the beam position stability to be acceptable.

Between May and August 2006, work was done on the septum stability. Particularly,the large spikes (∼ 200 A) in septum current were eliminated. The measurement wasre-done in August for both multi-bunch and single-bunch mode and yielded better results[WMN06]. In multi-bunch the variations were now found to be

σpos,H

σbeamsize,H=

0.18× 10−3

1.16× 10−3= 0.16 (4.14a)

2At the time of measurement a triggerable camera could not be made available at the given location.

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68 Experimental results

σpos,V

σbeamsize,V=

0.25× 10−3

1.40× 10−3= 0.18 (4.14b)

and in single-bunchσpos,H

σbeamsize,H=

0.167× 10−3

0.96× 10−3= 0.17 (4.15a)

σpos,V

σbeamsize,V=

0.133× 10−3

1.36× 10−3= 0.10 (4.15b)

Although the position variation is still larger than the target value of 0.1, the horizontalstability has been vastly improved. Additional work on the septum stability should yieldfurther improvement to the beam position stability.

The septum pulse is generated by discharging a capacitor bank. The septum currentis measured with a current transformer and the peak current is detected and used in aregulation loop to adjust the charge of the capacitor bank to achieve the desired septumpeak current. Disabling the kicker, which is located close to the septum, reduced the currentvariations drastically as seen in gure 4.23 from October 2007. Consequently, the causefor the septum instability is concluded to be noise from the kicker: when the kicker res,it gives a signal that spills over into the septum current measurements, thereby creating aspike which the septum peak detection loop locks on to. The induced spike varies in sizefrom shot to shot, thus perturbing the septum loop calculation.

The cable connecting the kicker supply to the kicker is run in a shielded cable tray,which, if made properly, should prevent the signal from spilling over into other cables.Although the kicker and septum supplies are placed in the same rack, this is not cause forthe interference. This was seen when the kicker supply was taken out and placed next tothe rack causing an increase in the septum variations by a factor of four.

Presently, it is not yet known how to prevent the kicker noise from aecting the septum,but it is under investigation by the Australian Synchrotron sta. A possible next step inthe investigation could be to use an antenna to try and pin-point a leak location: thekicker pulse is roughly 800 ns long and should therefore show up around 625 kHz onceevery second.

4.8 Current

The current is, together with the emittance, considered the most important parameter ina booster, since it determines how fast the storage ring can be lled for a given repetitionfrequency. During the commissioning the current was the primary parameter that wasoptimized on; not only because of its importance but also because the amount of currentin the booster is dependent on other important parameters such as the emittance, energy-spread, closed-orbit position, tunes, etc.

First turn was achieved February 15th 2006, with hundreds of turns 4 days later (seegure 4.24 on page 80). The turns were measured using the rst strip line system (describedin section 1.6.7).

The electron source can operate in two modes: single-bunch and multi-bunch mode.In single-bunch mode a single bunch is emitted through the 500 MHz grid in the electronsource. The trigger for the source is independent on the RF frequency. Thus, any jitter inthe trigger will move the bunch in the RF bucket, possibly splitting it into two buckets.

In multi-bunch mode a gated trigger signal determines the length of the bunch train,and the grid is modulated using the RF frequency, thus avoiding the eects of triggerjitter. The train can be from ∼ 20 ns up to at least several microseconds, but has been

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4.8 Current 69

xed in the timing system to 150 ns. During commissioning the multi-bunch mode wasused almost exclusively, as it induced the strongest signal in diagnostics equipment, suchas the BPMs, due to its higher current.

4.8.1 Multi-bunch mode

The circulating current in the booster was measured with a DCCT, a Bergoz New Para-metric Current Transformer [NPC], placed at the end of the rst quadrant. It measuresthe DC component of the circulating beam current using two transformers in a commonfeedback loop.

Figure 4.25 shows the circulating current in the booster during the energy ramp. Thisis the ramping curve from August 2006 that was also used for the tune measurements insection 4.3. It is a 1st shot, subsequent shots have slightly lower currents due to outgassingcaused by the synchrotron radiation.

The extraction is done in a single turn using a fast kicker magnet. This is veried withthe booster FCT and shown in gure 4.26, where no current is evident in the booster afterthe last turn.

Using the WCM in the linac section, and the FCTs in the LTB, booster, and BTS,the multi-bunch traces through the injection system can be observed. The result is shownin gure 4.27. The traces do not all go to zero which is due to limited bandwidth of theFCT. However, the integral which is proportional to the beam charge is conservedand can be used as a measurement of the total charge [Abr06]. The current in the tail ofthe beam suers a larger loss than the head after a quarter of a turn in the booster. Thiscould be due to beam loading in the linac where the head of the beam induces a voltage inthe accelerating structure, and thereby reduces the accelerating voltage for the tail of thebeam, resulting in a smaller kinetic energy of the tail. When the beam is injected into thebooster, the lower energy tail is bend more and scraped, most likely in the septum. Thebeam loading can be compensated for by adjusting the timing and/or modulation of thelinac RF pulse [TY06].

4.8.2 Single-bunch mode

Using single-bunch mode a single bucket in the storage ring can be lled. The circulatingsingle-bunch current is shown in gure 4.28. The waves in the current signal are causedby ringing in the signal cable where the signal is reected multiple times in the cable dueto imprecise matching of the terminations.

The single bunch purity target was that 99% of the charge should be contained inthe central 500 MHz bunch. Using the FCT in the BTS it was observed that too muchcurrent was contained in a trailing bunch. Figure 4.29 shows the single bunch and a secondbunch 2 ns behind it. It is suspected to be caused by a phase dierence between the twoaccelerating structures of the linac. Presently, it gives a 3:1 ratio of current in two adjacentbunches in the storage ring. Since this is undesirable when doing top-up, the trailing bunchis removed by re-phasing the booster RF when in single-bunch mode.

4.8.3 Losses

The beam losses at dierent times in the cycle were measured by comparing the FCP signalfrom the linac, the FCT signals from the LTB and the booster at injection, the boosterFCT at 300 MeV and again at extraction, and the BTS FCT. The FCTs are not idealto use as a measure of the absolute current, due to the nite rise time of 0.175 ns and

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70 Experimental results

that the signal has to be numerically integrated by the oscilloscope. The uncertainty onthe measured charge obtained from the FCT signals are estimated to be around ±20%,while the uncertainty on the DCCT signal from the booster is believed to be less than 5%.Calibration was done against the WCM. The results are summarized in table 4.3. Over-all

Table 4.3: The beam losses in the injection system.

Multi-bunch mode Single-bunch mode

Charge [nC] Remain Charge [nC] Remain

FCP 7.02 100% 0.81 100%

LTB FCT1 7.52 107% 0.96 119%

LTB FCT2 6.76 96% 0.82 101%

Booster 100 MeV 3.47 (8.0 mA) 49% 0.43 (1.0 mA) 53%

Booster 300 MeV 1.95 (4.5 mA) 28% 0.26 (0.6 mA) 32%

Booster 3 GeV 1.95 (4.5 mA) 28% 0.26 (0.6 mA) 32%

BTS FCT 1.76† 25% 0.23† 28%†Measured as 90% of booster FCT signal.

about 75% of the beam charge is lost, with the majority, 50%, lost in the injection intothe booster, and the last 25% lost at low energy. Although these are large losses, it is anadvantage that they occur at low energy, thus reducing the total radiation doses created.

Observing gure 4.25 it is seen that virtually all the current loss occurs shortly afterinjection, and the large majority before the beginning of the ramp. The greatest loss occurduring the rst 40 ms, and only a minor fraction is lost after the rst 75 ms (154 MeV).After 188 ms (708 MeV) there is no measurable loss. The range on the oscilloscope onlywent to 10 mA, and the rst 0.5 ms are saturated, suggesting that the current injected ishigher. From table 4.3 a current at LTB FCT2 of roughly 15 mA ± 3 mA is found. If nocurrent is lost in the septum it would imply that between 25% and 87% (2.5 8.7 mA) ofthe beam current is lost in the rst 1000 turns in the booster.

4.8.3.1 Momentum acceptance

The losses can be partially explained by insucient transverse momentum acceptance. Insection 4.5 the transverse momentum acceptance was found to be −0.675% < ∆p/p <1.42%, where, as was discussed in section 3.2, it is the numerically lower of the limitsthat sets the upper limit on the momentum acceptance, ∆p/p < 0.675%. In section 3.2 itwas also concluded that for a beam with a Gaussian momentum-spread distribution with∆p/p = 0.5%, the transverse momentum acceptance would result in an 18% loss. Withthe injected momentum-spread most likely being ∆p/p ≈ 0.6% (see section 4.1) this comesto around 26%. The transverse momentum acceptance is dependent on the tunes andchromaticities, and the acceptance for the injection settings used here were not measured.Thus, they may dier somewhat. The abrupt losses seen in 4.25 at 2.6 and 14.4 ms suggestthat the beam is crossing tune resonances at these times.

4.8.3.2 Vacuum

The residual gas in the vacuum chamber interacts with the electron beam causing the beamcurrent to decay exponentially over time. The primary eects are elastic scattering of theelectron beam on the gas nuclei, and bremsstrahlung from the electromagnetic interaction

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4.8 Current 71

between the beam electrons and the gas nuclei. The scattering decay constant is

1τs

=2πr2

eZ2ρc

γ2rel

(〈βH〉βH,max

a2+〈βV 〉βV ,max

b2

)(4.16a)

and the bremsstrahlung decay constant is

1τb

=16r2

eZ2ρc

411ln(

183Z1/3

)(− ln εRF −

58

)(4.16b)

where

ρ = 7.24× 1024nZ

Pmbar

TK(4.16c)

is the nuclei charge density, nZ is the number of nuclei per gas molecule, P is the residualpressure, T is the temperature, Z is the nuclei charge, γrel is the relativistic gamma factor, cis the speed of light, re is the classical electron radius, 〈β〉 is the average beta-function, a andb are the horizontal and vertical limiting apertures, respectively, βmax is the beta-functionvalue at the limiting aperture, and εRF = 0.0189 is the RF acceptance [Mur96, Præ01].The residual gas is mostly water vapor and molecular nitrogen, of which the nitrogen hasthe strongest inuence on the beam. At injection energy the scattering eect completelydominates bremsstrahlung, e.g. for nitrogen (nZ = 2, Z = 7) at P = 10−7 mbar, τs = 10seconds while τb = 49 minutes. As seen from (4.16a) and (4.16c), the scattering decayconstant is proportional to the pressure, and inversely proportional to the square of theenergy. The vacuum gauges are placed at the same locations as the vacuum pumps. Thisreduces cost but also means that the gauges report only on the pressure at the most idealplaces in the booster. According to the vacuum calculations carried out during the detaileddesign, the pressure farthest away from the pumps should be at most a factor of 10 higherthan reported at the gauges [Ped04].

During commissioning the inuence of the vacuum quality on the beam lifetime wasmost clearly seen when comparing the rst shot with subsequent shots, where the syn-chrotron light from the previous shot had heated the vacuum chamber causing an increasein pressure due to outgassing. This would be seen as a reduction in accelerated current.In May 2006, this was of the order 35% 45%, going from 4.5 mA to about 2.7 mA. Thepressure at the gauges would on average be around 10−7 mbar due to outgassing (see gure4.30). Assuming an average pressure of ∼ 3.5 × 10−6 mbar, this would give a scatteringdecay constant of τs = 0.28 seconds, which in the 54.3 ms between injection and start ofacceleration, where the decay constant swiftly increases, would give signicant beam lossesof about 17%. After three months of using the booster for the storage ring commissioning,the losses in August 2006 were of the order 15% 20%, going from 6 mA to about 5 mAof accelerated current where some optimizations on the ramp had also been done.

Improvements in the vacuum from May to August 2006 are suspected to have beena signicant reason for the ability to go from 4.5 to 6.1 mA (rst shots). In August thepressure at the gauges were around 5 × 10−8 mbar, assuming the average pressure wouldthen be ∼ 2× 10−7 this would give a scattering decay constant of τs = 5 seconds. Duringthe time between injection and acceleration, the beam loss would be roughly 1.1%, which isinsignicant. Even in the worst case with an average pressure ten times the gauge reading,the losses would only be 2.7%. Consequently, the losses seen in gure 4.25(b) are highlyunlikely to be dominated by an insucient vacuum.

The losses virtually stops above 130 MeV (after 75 ms), which suggests that the beamis too large, and the adiabatic shrinking of the beam prevents it from being scraped furtherby the dynamic, or physical, aperture. Also, when stopping the ramping, the losses became

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72 Experimental results

very small after just two seconds. Although the synchrotron light causes an outgassing andgives poorer vacuum, it took several tens of seconds rather than two seconds for the pumpsto reach a signicantly smaller pressure. Thus, the momentum-spread of the beam orthe limited transverse momentum acceptance and not the vacuum level appears to beprimarily responsible for the injection losses.

4.9 Performance and improvement suggestions

4.9.1 Comparison to other booster synchrotrons

In table 4.4 the main parameters of the ASP booster are compared to those of otherrecently build booster synchrotrons for light sources with storage rings of comparableenergy. ASP, SLS, and the Spanish ALBAmake use of combined-function dipole magnets intheir boosters. SLS includes quadrupole and sextupole components in their dipoles similarto ASP. ALBA includes vertical focusing and a sextupole component in their dipoles;horizontal focusing is done by separate quadrupole magnets which also include a sextupolecomponent.

Table 4.4: Comparison of the ASP booster to other modern booster synchrotrons.

Magnets Combined-function Separated-function

Facility ASP SLSa ALBAab CLS Diamond Soleil

Energy [GeV] 3.0 2.7 3.0 2.9 3.0 2.75Emittance [nm] 30 9c 9 552 135 130Circumference [m] 130.2 270.0 249.6 102.5 158.4 156.6Current [mA] 4.56.1 12 5 11 6 12Full energy chargeper second [nA] 2.02.7 32 12 3.8 6.3 16 19Repetition rate [Hz] 1 3 3 1 25 3Top-up Yes Yes Yes No Yes Yes

SLS [JMS06], ALBA [MEDG+06], CLS [GPMD04],Diamond [Dia07], Soleil [Lou06].

aPlaced in storage ring tunnel bDesign values cAt 2.4 GeV

Top-up operation has become a must-have for all modern facilities. A low injectionemittance into the storage ring is a necessity for successful top-up operation. Both theSLS and ALBA boosters have lower a emittance than the ASP booster. However, in bothcases this is achieved with twice the circumference. The boosters of the English DiamondLight Source and the French Soleil both have comparable circumferences, and here theemittance of the ASP is much smaller by at least a factor of four.

Another common parameter to look at is the charge that can be accelerated to fullenergy per second as this determines the lling time of the storage ring. Here the ASPbooster delivers the lowest performance. In this respect one should consider the importanceof this parameter: using top-up operation, the smaller charge per second means that thestorage ring should be topped up more frequently. However, since users of the storage ringsynchrotron light prefers as small a variation in brightness, i.e. storage ring current, aspossible, the storage ring would in any case need to be lled frequently with small amountsof charge, e.g. Diamond plans to do top-up every other minute [CK07]. A complete re-llof the storage ring from scratch would also take longer, yet the time for ASP is around oneminute assuming 200 mA storage ring current and eectively 3.3 mA per shot muchshorter than the lifetime of the beam of several hours.

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4.9 Performance and improvement suggestions 73

Lastly, the ASP booster is a commercial turn-key solution. Price, therefore, is animportant parameter, and although no direct citations can be given for commercial reasons,the general consensus from potential new buyers of an ASP booster synchrotron model isthat it is remarkably inexpensive when considering the performance.

4.9.2 Summary and recommendations

The experimental investigations showed that the booster was able to achieve both theemittance and the current that were the main goals of the design.

The relative beam current losses at injection were higher than expected while at higherenergy, they were lower. The total relative current loss, from electron source to the ex-tracted beam was as expected, only with a dierent loss pattern a more advantageouspattern from a radiation safety point of view. The cause of the larger injection losses are fora large part suspected to be due to the booster having a too small momentum acceptancefor the linac beam. Figure 4.14 suggests that the RF frequency may not be optimal forthe booster. However, that measurement was not done with the present injection settings,and since at least the upper limit on the momentum-acceptance is set by the dynamicaperture, it is likely that the gure will look somewhat dierent with the present injectionsettings. Regardless, it suggests, as was also seen in section 3.1, that the dynamic apertureis small and can be limiting, i.e. smaller than the physical aperture, and also somewhatsensitive to momentum deviations. However, the full width of the momentum acceptancewas 2.1%, and for a perfectly injected beam with an RMS momentum-spread of 0.5% thatwould be enough to accommodate 95% of the beam, if not for the asymmetry in accep-tance. Since the RF frequency is determined by what is optimal for the storage ring, itcannot be changed to better suit the booster. Thus, it would only make sense to repeat themomentum acceptance measurement with the present settings, and ramping, to see if it isworth doing further optimisations on the energy-spread from the linac, or if the boosteracceptance is fairly symmetric.

The closed-orbit correction scheme worked suciently well, and it was possible tocorrect the closed-orbit up the ramp to within 1 2 mm RMS. The large attenuation inthe BPM cables (see gure 1.10) caused problems during commissioning, because it madeit dicult to measure and correct the orbit. All cables were terminated in the same rack.However, at the small expense of an additional low-end PC, the MX-BPMs could easilyhave been placed in two racks at opposite sides of the booster ring, thereby eliminatingthe need for the longest cables. The corrector magnets were also able to reduce the eectof the BTS thick septum leak eld on the beam in the booster.

The need to vary the strength of the BF magnet family relative to the BD is puzzling,and because, due to their combined-function nature, it changes many things at once: beammomentum, closed-orbit, tunes, and chromaticities, it has not been possible to locate acause with the diagnostics equipment available. It may be dierent residual elds in theBD and BF families, which is compensated by changing their relative strengths. If thereare errors in any of the magnets, a resent article [FTS07] outlines a method to measure themagnet strength in a live machine using a transverse kick and turn-by-turn beam positionmeasurements. Unfortunately, the BPM system for the booster, unlike for the storage ring,does not allow turn-by-turn beam position measurements.

The chromaticities were measured at injection, 2 GeV, and extraction, and found tobe small and positive as desired with the settings of the tuning sextupoles used duringcommissioning. The chromaticities were not used as an optimisation parameter duringcommissioning, rather the tuning sextupoles were manually adjusted simply to reduce thebeam loss as much as possible.

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74 Experimental results

As mentioned earlier, the emittances measured in the booster using the synchrotronlight are close to the predictions of the lattice model, although they tend to be lower. Thiscould be explained if the distance from the source point to the camera lens, d0, is largerthan assumed. The emittance was attempted measured in the BTS using a quadrupolescan. However, due to saturation on the screens, the found emittances were both muchlarger than measured in the booster and diered between measurements depending on thebeam size on the screen (quadrupole settings).

There were problems obtaining a satisfactory position stability after extracting thebeam in the BTS. The noise from the extraction kicker caused the extraction septumcurrent to vary from shot to shot. The large septum angle of 120 mrad, together with thesmall beam emittance, meant that the RMS beam movements were up to 0.18 times theRMS beam size. The beam movements give a larger eective emittance when injectinginto the storage ring, reducing the benet of the small booster emittance. In a futureinjection system design, the large septum angle should be reduced as much as possible,say a factor of ten or more, thereby reducing the position jitter proportionally. A strongerstatic septum, which can be made signicantly more stable than a pulsed septum, shouldbe placed downstream to give the necessary total extraction angle.

The injection eciency was found to be very dependent on the injection settings of themagnets due to the limited dynamic aperture. The trimming of the injection settings wastedious to perform, as the ramp would have to be stopped, a new ramp loaded, and theramp started again, which would easily take a minute. This could be avoided if a sumregister was added to the magnet power supplies. Presently, the current output from apower supply during ramping is set using a single register that cannot be change on-the-y.If an additional register was added that could be changed on-the-y, and the output currentset to the sum of the ramping value and the value in the additional register, it would allowthe injection settings to be easily ne tuned, while having little impact at high energy. Themaximum value in the additional register should be no more than 3% 5% of the injectioncurrent value. This type of functionality is successfully used at ISA's ASTRID (AarhusSTorage RIng in Denmark) [Møl92]. This change needs only be implemented in the largepower supplies driving the main magnets, and it should be neither dicult nor expensiveto do.

Injection and extraction on-the-y was considered during commissioning. Injection on-the-y has the advantages that it allows the beam to swiftly go to higher energies wherevacuum lifetime is higher and dynamic aperture is less important factors that are be-lieved to signicantly inuence the injection eciency in the booster. Also, either less timecan be used, or the magnet ramp be made less steep, i.e. smaller ∂B/∂t, when injectingand extracting on-the-y. However, it requires a very small degree of shot to shot varia-tions for all elements. Injection on-the-y was attempted shortly during commissioning,but the inability to easily make small adjustments to the magnet injection settings madeit impossible to nd an acceptable injection setting, and it was quickly abandoned. If theproposal for the additional sum register is implemented it may be possible to do injectionon-the-y in the future.

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4.9 Performance and improvement suggestions 75

025

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Figure 4.17: Synchrotron light images at 3 GeV for multi-bunch and single-bunch mode. TheGaussian tted intensity proles are also shown. The width of the CCD images corresponds to3.28 mm at the synchrotron light source.

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76 Experimental results

(a) 0.51 GeV (b) 0.80 GeV (c) 1.1 GeV (d) 1.4 GeV (e) 1.7 GeV

(f) 2.0 GeV (g) 2.3 GeV (h) 2.6 GeV (i) 2.9 GeV (j) 3.0 GeV

Figure 4.18: The transverse electron density prole as observed by the SLM during ramping.The proles were taken with 50 ms spacing starting at 150 ms. The full width and height of animage corresponds to 4.06 mm (215 pixels) at the synchrotron light source.

0

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Figure 4.19: The emittances during ramping, asuming a zero momentum-spread. The equilib-rium emittance εeq. is calculated from the lattice model.

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4.9 Performance and improvement suggestions 77

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Figure 4.20: The horizontal beam proles at 550 ms for two dierent shots (numbered 0 and13). It is seen that the dierence in widths for the shots are real and not due to a bad t of theGaussian to the prole.

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78 Experimental results

0.0050.01

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4.9 Performance and improvement suggestions 79

Figure 4.22: The output current of the extraction septum as a function of time over two hours.A span of ±20 A is marked in gray. This data represents the behavior of the septum during theposition measurement but was not taken at that time.

Figure 4.23: The output current of the extraction septum as a function of time over threehours taken in October 2007. Erratic behaviour is seen during the two periods where the kickeris active. The spikes at the start and end of each period is due to the kicker being switched onand o.

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80 Experimental results

Figure 4.24: The strip line signal (yellow) showing the rst hundreds of turns achieved on

February 19th 2006. The downward slope is caused by the injection septum signal adding to thebeam induced signal in the strip line.

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4.9 Performance and improvement suggestions 81

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Figure 4.25: Multi-bunch current during the nal, optimized energy ramp from August 25th

2006. The blue line is the beam current (negative), and the dotted red line is the output currentfrom the rst BD power supply converted to equivalent beam energy. The area within the dashedgreen line in (a) is shown in (b). The dashed grey line in (b) indicates the start of the ramp at54.3 ms, while the dotted grey lines indicate times just after a steep loss. They are at 2.6 ms,14.4 ms, and 39.4 ms.

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82 Experimental results

Figure 4.26: Traces from the oscilloscope showing the circulating current in the booster (yellowtrace). No current can be seen a revolution time (434 ns) after the last bunch train, demonstratingthat the current was removed from the booster in a single turn.

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4.9 Performance and improvement suggestions 83

−1

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]B

TS

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gnal[V

]

Figure 4.27: The multi-bunch traces through the injection system starting with the WCM inthe linac section, then the second FCT in the LTB, in the booster after a quarter turn, againin the booster a quarter turn before extraction, and ending with the rst FCT in the BTS(November 2007).

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84 Experimental results

Figure 4.28: Single-bunch current (blue line and left scale) during the nal, optimized energyramp (red line and right scale) from August 2006. Due to the zoom-in on the rst half of thecurve, the current for energies above 2.1 GeV is not visible, though as was the case with themulti-bunch mode, there is no loss higher up the ramp.

Figure 4.29: BTS FCT trace of a single bunch. A trailing second bunch is evident 2 ns afterthe primary bunch.

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4.9 Performance and improvement suggestions 85

(a) Vacuum gauge readings for May 2006

(b) Vacuum gauge readings for August 2006

Figure 4.30: The vacuum gauge readings for May (a) and August (b) 2006. The horizontal scaleis in days, and the vertical scale is in mbar. The pressure is seen to increase due to outgassingcaused by the synchrotron radiation. The increase is much less in August compared to May.

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Chapter 5

Summary

Danfysik A/S has delivered a full energy 3 GeV booster synchrotron as part of a completeturn-key injection system for the Australian Synchrotron Project (ASP) storage ring. Theinjection system consists of a 100 MeV linac, sub-contracted to the German company AC-CEL [PDM+06], a linac-to-booster (LTB) transfer line, a full energy booster synchrotron,and a booster-to-storage ring (BTS) transfer line. The contract for the delivery of a turn-key booster was awarded Danfysik A/S in competition with other, larger manufacturers ofaccelerators based on price, performance specication, and Danfysik's history of successfuldelivery of injection systems to the Canadian Light Source [GPMD04] and Forschungszen-trum Karlsruhe [PBE+01].

The subject of this thesis has been the full energy booster synchrotron. Based on theexperimental ndings presented here, the Australian Synchrotron booster must be deemeda very successful commercial 3rd generation light source booster, delivering a reliable andreproducible1, very low emittance beam. The use of combined-function dipole magnetswith integrated quadrupole and sextupole elds has enabled the booster to be smaller andcheaper than if separate-function magnets had been used to achieve the same low emit-tance. The reduced exibility inherent in the combined-function lattice has not hamperedthe booster's ability to deliver a beam that fullls the specications of the injection system.The target current of more than 5 mA in multi-bunch mode and 0.5 mA in single-bunchmode was achieved, with the injection system delivering 5.5 6.1 mA and 0.6 mA, respec-tively. The target horizontal emittance at 3 GeV of 33 nm rad was also achieved, withthe emittance being less than 31 nm rad. Compared to other modern 3rd generation lightsources, the ASP booster is the most compact booster to deliver such a low emittance.

However, the advantage of the small booster emittance is reduced by the positionstability issue of the extracted beam. The issue is ascribed to uctuations in the currentof the fast extraction septum, caused by its controller being disturbed by noise from theextraction kicker. The remedy for this is to discover where the noise penetrates the septumsystem's shielding, and to improve the shielding at this/these locations. In future designs itwould be better to minimize the impact of such noise on the position stability by reducingthe fast septum angle thereby reducing the position variation proportionally andusing a DC septum further downstream in the BTS.

The DC septum, the so-called thick septum, in the BTS had a signicant leak eld,which prevented the injected beam from making a full turn in the booster. However, thebooster correctors could be used to correct the orbit, and additional magnetic shieldingwas later added around the booster vacuum chamber closest to the septum, negating the

1According to lead accelerator physicist Greg LeBlanc of the Australian Synchrotron there is no needfor any day to day adjustment or optimization of the injection system.

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87

eect of the leak eld. Still, in the future it would be better to build a better shielding onthe septum itself.

Tune measurements in section 4.3.1 showed the combined-function dipole magnetsworking very well, with only a small (in the case of the defocusing BD magnet family)or no discrepancy (in the case of the focusing BF magnet family) between the factoryacceptance test measurements of their quadrupole components, and the values calculatedfrom a tune measurement done at 3 GeV with the relative BF and BD strength ratio beingnominal. The reason for the need to vary the BF/BD strength ratio to achieve acceler-ation is not fully understood, primarily because varying their ratio changes many thingssimultaneously: closed-orbit, energy, tunes, and chromaticities. Possible reasons may bea dierent residual eld at injection energy, or, less likely, a localized error in one of themagnets including misalignments.

The total losses observed were close to the expected, however, the loss pattern wasdierent, with the vast majority of the losses occurring at injection and low energy in thebooster. From a radiation safety point of view this is better than losses at high energywhere more radiation is produced. The low observed injection eciency of ∼ 50% issuspected to be dominated by a limited momentum acceptance of the booster, made worseby the asymmetry of the transverse acceptance of −0.675% < ∆p/p < 1.42% shown ingure 4.14, but the variation in the BF and BD strength ratio may also play a role. Boththe injection system and the storage ring use the same Master Oscillator to drive the RFunits, which means that the booster RF frequency cannot be changed in an attempt toreduce the acceptance asymmetry. Rather, the injection settings of the booster, i.e. thedipole eld, tunes and chromaticities, should be adjusted to nd an injection setting withlarger dynamic aperture for o-momentum particles. The linac should also be trimmed toprovide a beam of lower momentum-spread, insofar it does not reduce the total amount ofcurrent within the momentum acceptance span.

The tweaking of the injection settings would be made very much simpler by includinga sum register in the magnet power supplies of the combined-function magnets and thetrimming quadrupoles, which could be adjusted independently of the running ramp andproduce a small additional current to be added to the ramping current in real time. It islikely that such a change to the power supplies would allow for on-the-y injection, andpossibly extraction as well, if desired.

The dynamic aperture of the booster was seen in the simulations to be very sensitiveto misalignments and eld errors. Although the ndings overestimate the consequences ofmisalignments somewhat because they are simulated on a per-magnet basis, rather thanthe more realistic per-girder basis, gure 3.6 indicates clearly that the tolerances shouldnot be relaxed. This was also observed during commissioning, where the corrector magnetswere essential to get rst turn.

The large attenuation in the cables connecting the beam position monitor (BPM) but-tons to the electronics module (MX-BPM) system made, in particular, the rst part of thecommissioning more cumbersome, as it was either dicult or impossible to calculate a setof corrector settings to correct the closed-orbit. Not much can presently be done aboutthis in the ASP booster, as pulling new cables or rearranging the existing will be far tooexpensive to be worthwhile the eort. However, in a possible future booster, better cablesshould be used, and the MX-BPM units should perhaps be placed in two racks, rather thanone. These racks should be located at opposite sides of the booster, allowing the longestcable lengths to be reduced. This could easily be done at the expense of a single low-endPC.

All-in-all the available diagnostics equipment proved capable for the commissioning

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88 Summary

of the injection system, although using Optical Transition Radiation (OTR) screens forthe beam viewers rather than Cr-doped alumina (Al2O3) screens would have allowed forreliable emittance measurements in the transfer lines. Also, less radiation sensitive andtriggerable cameras for the screens would have been an advantage for a minimal additionalcost. The beam loss monitors were not useful during commissioning and could be omittedin the future.

If the above mentioned recommendations are taken into consideration it would makethe ASP booster an even more attractive commercial booster for new 3rd generation lightsources.

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References

[Abr06] P. Abrahamsen. Personal correspondance between julien bergoz and povlabrahamsen, referenced by p. abrahamsen in an e-mail dated 10-05-200617:40. Unpublished, May 2006.

[ASP] The australian synchrotron homepage. Available from World Wide Web:http://www.synchrotron.vic.gov.au.

[Ben97] J. Bengtsson. Tracy-2 user's manual. Internal SLS document, PSI, Villingen,1997.

[BJ93] Phillip J. Bryant and Kjell Johnsen. The Principles of Circular Accelerators

and Storage Rings. Cambridge University Press, 1993.

[Bry92] P.J. Bryant. Basic theory for magnetic measurements. In Stuart Turner,editor, CERN 92-05, Yellow Report. CERN, 1992. Available from WorldWide Web: http://doc.cern.ch/cgi-bin/tiff2pdf?/archive/cernrep/

1992/92-05/p52.tif.

[Bry99] Phillip J. Bryant. Winagile: Windows alternating gradient interactive latticedesign software for synchrotrons & transfer lines, Februrary 1999. Availablefrom World Wide Web: http://bryant.home.cern.ch/bryant/.

[CK07] C. Christou and V. C. Kempson. Operation of the diamond light source injec-tor. In Proceedings of PAC07, Albuquerque, New Mexico, USA, pages 11121114, 2007. Available from World Wide Web: http://cern.ch/AccelConf/p07/PAPERS/TUPMN086.PDF.

[Dia07] Diamond light source website, 2007. Available from World Wide Web: http://www.diamond.ac.uk/Technology/InjectionSystem.

[ES93] D. A. Edwards and M. J. Syphers. An Introduction to the Physics of High

Energy Accelerators. John Wiley and Sons, 1993.

[Eva07] L. Evans. Lhc: Construction and commissioning status. In Proceedings of

PAC07, Albuquerque, New Mexico, USA, pages 15, 2007. Available fromWorld Wide Web: http://cern.ch/AccelConf/p07/PAPERS/MOXKI01.PDF.

[Fie] Vector Fields. Opera-3d. Available from World Wide Web: http://www.

vectorfields.com/content/view/26/49/.

[FNBB+06] Søren Weber Friis-Nielsen, Henning Bach, Franz Bødker, Alexander Elkjær,Nils Hauge, Jesper Kristensen, Lars Kruse, Stig Madsen, Søren Pape Møller,Mark James Boland, Rohan Dowd, Gregory Scott LeBlanc, Martin John

89

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