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Unit 2 Magnet specifications in circular accelerators

Unit 2 Magnet specifications in circular accelerators. Soren Prestemon and Paolo Ferracin Lawrence Berkeley National Laboratory (LBNL) Ezio Todesco European Organization for Nuclear Research (CERN). QUESTIONS. Order of magnitudes of the size of our objects: why ?

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Unit 2 Magnet specifications in circular accelerators

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  1. Unit 2Magnet specifications in circular accelerators Soren Prestemon and Paolo Ferracin Lawrence Berkeley National Laboratory (LBNL) Ezio Todesco European Organization for Nuclear Research (CERN)

  2. QUESTIONS • Order of magnitudes of the size of our objects: why ? High energy circular accelerators • Length of an accelerator: Km 46° 14’ 15” N – 6 ° 02’ 51” E 15 m 40° 53’ 02” N – 72 ° 52’ 32” W 41° 49’ 55” N – 88 ° 15’ 07” W 1 Km 1.9 Km RHIC ring at BNL, Long Island, US Main ring at Fermilab, Chicago, US

  3. QUESTIONS • Order of magnitudes of the size of our objects: why ? High energy linear accelerators • Length of a linear accelerator: Km - but we will not deal with them 46° 14’ 15” N – 6 ° 02’ 51” E 15 m 37° 24’ 52” N – 122° 13’ 07” W 3.5 Km Linear accelerator at Stanford, US

  4. QUESTIONS • Order of magnitudes of the size of our objects: why ? High energy circular accelerators • Length of an accelerator magnet: 10 m • Diameter of an accelerator magnet: m • Beam pipe size of an accelerator magnet: cm 46° 14’ 15” N – 6 ° 02’ 51” E 15 m 6 cm 0.6 m A stack of LHC dipoles, CERN, Geneva, CH Dipole in the LHC tunnel, Geneva, CH

  5. CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit • Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing • Aperture requirements for arc quads and dipoles (size of the beam) • Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters • Example: the LHC 5. The interaction regions: low-beta magnet specifications • How to squeeze the beam • Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

  6. 1. PRINCIPLES OF A SYNCHROTRON • Electro-magnetic field accelerates particles • Magnetic field steers the particles in a closed (circular) orbit • To drive particles through the same accelerating structure several times • As the particle is accelerated, its energy increases and the magnetic field is increased (“synchro”) to keep the particles on the same orbit • Limits to the increase in energy • The maximum field of the dipoles (proton machines) • The synchrotron radiation due to bending trajectories (electron machines) • Colliders: two beams with opposite momentum collide • This doubles the energy ! • One pipe if particles collide their antiparticles (LEP, Tevatron) • Otherwise, two pipes (ISR, RHIC, HERA, LHC)

  7. 1. PRINCIPLES OF A SYNCHROTRON • The arcs: region where the beam is bent • Dipoles for bending • Quadrupoles for focusing • Correctors • Long straight sections (LSS) • Interaction regions (IR) where the experiments are housed • Quadrupoles for strong focusing in interaction point • Dipoles for beam crossing in two-ring machines • Regions for other services • Beam injection (dipole kickers) • Accelerating structure (RF cavities) • Beam dump (dipole kickers) • Beam cleaning (collimators) A schematic view of a synchrotron The lay-out of the LHC

  8. CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit • Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing • Aperture requirements for arc quads and dipoles (size of the beam) • Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters • Example: the LHC 5. The interaction regions: low-betamagnet specifications • How to squeeze the beam • Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

  9. 2. THE ARC:HOW TO KEEP PARTICLES ON A CIRCLE • Kinematics of circular motion • Relativistic dynamics • Lorentz (?) force Putting all together • Hyp. 1 - longitudinal acceleration<<transverse acceleration Hendrik Antoon Lorentz, Dutch (18 July 1853 – 4 February 1928), painted by Menso Kamerlingh Onnes, brother of Heinke

  10. 2. THE ARC:HOW TO KEEP PARTICLES ON A CIRCLE • Relation momentum-magnetic field-orbit radius • Preservation of 4-momentum • Hyp. 2 Ultra-relativistic regime • Using practical units for particle with charge 1, one has magnetic field in Tesla … • Remember 1 eV=1.60210-19 J • Remember 1 e= 1.60210-19 C

  11. 2. THE ARC:HOW TO KEEP PARTICLES ON A CIRCLE Nikolai Tesla (10 July 1856 - 7 January 1943) • Born at midnight during an electrical storm in Smiljan near Gospić (now Croatia) • Son of an orthodox priest • A national hero in Serbia Career • Polytechnic in Gratz (Austria) and Prague • Emigrated in the States in 1884 • Electrical engineer • Inventor of the alternating current induction motor (1887) • Author of 250 patents Miscellaneous • Strongly against marriage [brochure of Nikolai Tesla Museum in Belgrade (2000)] • Considered sex as a waste of vital energy [guardian of Nikolai Tesla Museum in Belgrade, private communication (2002)] Tesla, man of the year

  12. 2. THE ARC:HOW TO KEEP PARTICLES ON A CIRCLE • Relation momentum-magnetic field-orbit radius

  13. 2. THE ARC:HOW TO KEEP PARTICLES ON A CIRCLE • The magnet that we need should provide a constant (over the space) magnetic field, to be varied with time to follow the particle acceleration • This is done by dipoles • As the particle can deviate from the orbit, one needs a linear force to bring it back • We will show in the next section that this is given by quadrupoles

  14. CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit • Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing • Aperture requirements for arc quads and dipoles (size of the beam) • Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters • Example: the LHC 5. The interaction regions: low-betamagnet specifications • How to squeeze the beam • Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

  15. 3. THE ARC:SIZE OF THE BEAM AND FOCUSING • The force necessary to stabilize linear motion is provided by the quadrupoles • Quadrupoles provide a field which is proportional to the transverse deviation from the orbit, acting like a spring • One can prove that the motion equation in transverse space (with some approximations) is where

  16. 3. THE ARC:SIZE OF THE BEAM AND FOCUSING • A sequence of focusing and defocusing quadrupoles with the same (opposite) strength and spaced by L is a providing linear stability to the beam – this is called a FODO cell • Let L be the distance between two consecutive quadrupoles • The equations of transverse motion are • Where the term K is zero in dipoles, and in focusing quadrupoles, in defocusing quadrupoles

  17. 3. THE ARC:SIZE OF THE BEAM AND FOCUSING • The motion equation in the transverse space is similar to a harmonic oscillator where the force depends on time … • Solution: a oscillator whose amplitude and frequency are modulated •  and  give the beam size • xy are the invariants (emittances) [m rad] • x and y are the beta functions [m] •  is the phase advance, related to the beta function • The beta functions oscillate along the ring, reaching maxima and minima in the quadrupoles

  18. 3. THE ARC:SIZE OF THE BEAM AND FOCUSING • Relations for a FODO cell: beam size vs cell length • Let 2L be the cell length – we consider it for the moment as an independent variable • We define (2L) as the phase advance per cell A typical cell has (2L)=/2 (90° phase advance) – for this cell one has Beta functions in a FODO cell with L=50 m Beta functions in a FODO cell with L=100 m

  19. 3. THE ARC:SIZE OF THE BEAM AND FOCUSING • Example of the LHC: L=50 m, f=170 m, d=30 m • The beta functions are in meters they are related, but not equal to the beam size • Pay attention ! f=170 m does not mean that the beam size is 170 m !! • It is not easy to “feel” the dimension of a beta function • Radius of the beam in the arc (1 sigma) • LHC: n=3.75 10-6 m rad • High field E=7 TeV, =7460 - =0.29 mm • Injection E=450 GeV, =480 - =1.2 mm • Beam size depends on cell length, energy and normalized emittance

  20. 3. THE ARC:SIZE OF THE BEAM AND FOCUSING • Focusing in a FODO cell • Thin lens approximation: focusing strength in a 90° FODO cell is • The focusing strength is related to K1 and to the quadrupole length ℓq and the quadrupole gradient is • LHC: at high field B=8.33 T, =2801 m, L=50 m, G ℓq=660 T • For a 60° phase advance the same linear dependence on L, with different constants • It looks worse: same beam size, 50% more focusing required

  21. CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit • Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing • Aperture requirements for arc quads and dipoles (size of the beam) • Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters • Example: the LHC 5. The interaction regions: low-betamagnet specifications • How to squeeze the beam • Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

  22. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS

  23. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS

  24. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS • Input 1. Collision energy Ec • Gives a relation between the dipole magnetic field B and the total length of the dipoles Ld • Technology constraint 1. Dipole magnetic field B • Does not depend on magnet aperture • Bt <2 T for iron magnets • Bt <13 T for Nb-Ti superconducting magnets (10 T in practice) • Bt <25 T for Nb3Sn superconducting magnets (16-17 T in practice) • Output 1. Length of the dipole part Length in m, B in T, energy in GeV

  25. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS

  26. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS • Input 2. Injection energy Ei • Determines the relativistic factor, that affect the beam size • Constraint 2. Normalized beam emittance n • Determined by the beam properties of the injectors • Semi-cell length L • This is a free parameter that can be used to optimize • Determines the beta functions • Output 2. Aperture of the arc magnets(also determined by field errors and beam stability) • Size of the beam at injection • Magnet aperture

  27. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS

  28. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS • Technology constraint 1. Quadrupole magnetic field vs aperture • Output 3. Gradient of the quadrupoles • Semi-cell length L • Also determines the focusing, i.e. the integrated gradient • Output 4. Length of the quadrupoles

  29. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS • Output 5. Number of semi-cells and arc length • Equal to the number of quadrupoles

  30. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS • Example: Large Hadron Collider • E=7000 GeV • Nb-Ti magnets, dipole field B=8.3 T • Ld=17600 m • Cell length L=50 m • f=170 m • n=3.7510-6m rad • Injection energy 450 GeV, =480 • Beam size =0.0012 m (at injection) • 2*10=0.024 m, i.e., much less than the available aperture of 0.056 m • Aperture is larger then needed to have the beam at injection in the zone of “good field”

  31. 4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS • Example: Large Hadron Collider • Arc magnets aperture and technology constraint determine quadrupole gradient: 8.3 T at 28 mm radius gives 300 T/m for Nb-Ti at 1.9 K – large safety margin taken, operational gradient chosen at 220 T/m • Cell length determines focusing strength, i.e. quadrupole length • Quadrupole length → length in the cell available for dipoles together with total length of dipoles → number of quadrupoles400 is the space for correctors, instrumentation, interconnections

  32. CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit • Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing • Aperture requirements for arc quads and dipoles (size of the beam) • Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters • Example: the LHC 5. The interaction regions: low-beta magnet specifications • How to squeeze the beam • Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

  33. 5. THE INTERACTION REGIONS:LOW-BETA MAGNET SPECIFICATIONS • We are now in the straight sections of the machine • There are no dipoles • Only quadrupoles to keep the beam focused • In the middle of the straight section one has a free space for the experiment, with the interaction point (IP) where beams collide • Around the experiment the optics must keep two distinct aims • Keep the beam focused • Reduce the size of the beam in the interaction point (IP) to increase the rate of collisions (luminosity) • Beam size proportional to () – but  is invariant, so act on 

  34. 5. THE INTERACTION REGIONS:LOW-BETA MAGNET SPECIFICATIONS • A system of quadrupoles is used to reach a very low beta function, called b*, in the IP (LHC: 0.55 m instead of the 30-200 m in the arcs) • Physical constraint: empty space around the IP – distance of the first magnet to the IP, called l*, (LHC: 23 m) – needed for the detectors ! The lay-out of quadrupoles close to the interaction point in the LHC, and the beta functions

  35. 5. THE INTERACTION REGIONS:LOW-BETA MAGNET SPECIFICATIONS • Drawback: beta function gets huge in the quadrupoles ! • But this happens only in collision, where the beam is smaller • In free space around IP (s=0), one has • At the entrance of the triplet one has • In reality, the situation is even worse: the maximum beta function in the LHC triplet is much larger than at the entrance at the entrance we have whereas in the triplet we have m =4400 m

  36. 5. THE INTERACTION REGIONS:LOW-BETA MAGNET SPECIFICATIONS • Aperture requirement:  a+c/* and dependent on l*, lt • Given the aperture, the technology limits the maximal gradient • At first order, G1/ • We will show the limits of the approximation, and a more precise estimate, in Unit 8 • The triplet has to focus the beam in the interaction point • The focusing strength is a function of l*, lt, and is not related to * • This gives a requirement on the integrated gradient … • … that together with the maximum gradient gives the triplet length • The 4 equations are coupled

  37. 5. THE INTERACTION REGIONS:LOW-BETA MAGNET SPECIFICATIONS • The 4 equations are coupled • For the LHC, one has • *=0.55 m • m=4400 m • With respect to the arc, m is ~22 times larger, but the  is ~16 times larger in collision the aperture is not so different from the cell magnets •  = 0.070 m instead of  = 0.056 m in the arcs • With a triplet length of 24 m the required integrated gradient of 4800 T • This requires a quadrupole gradient of 200 T/m • With Nb-Ti one can get up to 300 T/m quadrupoles of  = 0.070 m – one has a good safety margin

  38. 5. THE INTERACTION REGIONS:LOW-BETA MAGNET SPECIFICATIONS • Example: the LHC interaction regions • Baseline: Nb-Ti quadrupoles, 200 T/m, 70 mm aperture, * =0.55 m • LARP :Nb3Sn quadrupoles, 200 T/m, 90 mm aperture, * =0.25 m

  39. CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit • Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing • Aperture requirements for arc quads and dipoles (size of the beam) • Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters • Example: the LHC 5. The interaction regions: low-betamagnet specifications • How to squeeze the beam • Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

  40. 6. THE INTERACTION REGIONS:DETECTOR SPECIFICATIONS • Detector magnets provide a field to bend the particles • The measurement of the bending radius gives an estimate of the charge and energy of the particle • Different lay-outs • A solenoid providing a field parallel to the beam direction (example: LHC CMS, LEP ALEPH, Tevatron CDF) • Field lines perpendicular to (x,y) • A series of toroidal coils to provide a circular field around the beam (example: LHC ATLAS) • Field lines of circular shape in the (x,y) plane Sketch of a detector based on a solenoid Sketch of the CMS detector in the LHC

  41. 6. THE INTERACTION REGIONS:DETECTOR SPECIFICATIONS • Detector transverse size • The particle is bent with a curvature radius • B is the field in the detector magnet • Rt is the transverse radius of the detector magnet • The precision in the measurements is related to the parameter b • A bit of trigonometry gives • The magnetic field is limited by the technology • If we double the energy of the machine, keeping the same magnetic field, we must make a 1.4 times larger detector …

  42. 6. THE INTERACTION REGIONS:DETECTOR SPECIFICATIONS • Detector transverse size • B is the field in the detector magnet • Rt is the transverse radius of the detector magnet • The precision in the measurements is  1/b • Examples • LHC CMS: E=2300 GeV, B=4 T, Rl=12.9 m, Rt=5.9 m, b=9 mm • LEP ALEPH: E=100 GeV, B=1.5 T, Rl=6.5 m, Rt=2.65 m, b=16 mm • that’s why we need sizes of meters and not centimeters ! • The magnetic field is limited by technology • But fields are not so high as for accelerator dipoles (4T instead of 8 T) • Note that the precision with BRt2 – better large than high field … • Detector longitudinal size • several issues are involved – not easy to give simple scaling laws

  43. SUMMARY • We gave the principles of a synchrotron • The problem is not only accelerating …but also keeping on a circle ! • Magnets are needed for keeping particle on the orbit • Arcs: dipoles for bending and quadrupoles for focusing • How to determine apertures, fields and gradients • Input: machine energy and beam emittance (injectors) • Free parameter: cell length • Output: dipole field, quadrupole gradient, magnet lengths and numbers (i.e. machine length, excluding IR regions) • Interaction regions • How to squeeze the beam size • Determination of the aperture, gradientand length of the IR quads

  44. COMING SOON During the next days: How these technological limits are determined ? What is the physics and the engineering behind?

  45. REFERENCES • Beam dynamics - arcs • P. Schmuser, et al, Ch. 9. • F. Asner, Ch. 8. • K. Steffen, “Basic course of accelerator optics”, CERN 85-19, pg 25-63. • J. Rossbach, P. Schmuser, “Basic course of accelerator optics”, CERN 94-01, pg 17-79. • Beam dynamics - insertions • P. Bryant, “Insertions”, CERN 94-01, pg 159-187. • Beam dynamics - detectors • T. Taylor, “Detector magnet design”, CERN 2004-08, pg 152-165.

  46. ACKNOWLEDGEMENTS • J. P. Kouthcouk, M. Giovannozzi, W. Scandale for discussions on beam dynamics and optics • www.wikipedia.org for most of the pictures • The Nikolai Tesla museum of Belgrade, for brochures, images, and information, and the anonymous guard I met in August 2002 • F. Borgnolutti for listening all my dry talks • B. Bellesia for providing the slides template

  47. APPENDIX A: DEPENDENCE ON THE CELL LENGTH • Example: Large Hadron Collider • Larger L → larger beta function → larger beam size → larger magnet aperture, but • Larger L → small number of cells → smaller focusing strength → smaller number of quadrupoles

  48. APPENDIX A: DEPENDENCE ON THE CELL LENGTH • Example: Large Hadron Collider • Dipoles contribute for around 17.5 Km • With a cell length of 30 m quads are 3.5 Km long (20%), with 70 m quads are 1 Km long (6%) – baseline is 50 m, giving 1.3 Km

  49. APPENDIX A: DEPENDENCE ON THE CELL LENGTH • Example: Large Hadron Collider • The amount of the cable needed for dipoles and quadrupoles can also be estimated – equations will be derived in Unit 8 • The quantity of cable is roughly independent of the cell length, with a minimum around 50 m (but this was not the criteria used to select L!)

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