USPAS January 2012, Superconducting accelerator magnets Unit 2 Magnet specifications in circular accelerators Helene Felice, Soren Prestemon, Lawrence Berkeley National Laboratory (LBNL) Paolo Ferracin and Ezio Todesco European Organization for Nuclear Research (CERN)

USPAS January 2012, Superconducting accelerator magnets E. Todesco - Superconducting magnets 2 FOREWORD The science (or the art …) of superconducting magnets is a exciting, fancy and dirty mixture of physics, engineering, and chemistry Chemistry and material science: the quest for superconducting materials with better performances Quantum physics: the key mechanisms of superconductivity Classical electrodynamics: magnet design Mechanical engineering: support structures Electrical engineering: powering of the magnets and their protection Cryogenics: keep them cool … The cost optimization also plays a relevant role Keep them cheap …

USPAS January 2012, Superconducting accelerator magnets E. Todesco - Superconducting magnets 3 FOREWORD An example of the variety of the issues to be taken into account The field of the LHC dipoles (8.3 T) is related to the critical field of Niobium-Titanium (Nb-Ti), which is determined by the microscopic quantum properties of the material The length of the LHC dipoles (15 m) has been determined by the maximal dimensions of (regular) trucks allowed on European roads This makes the subject complex, challenging and complete for any physicist or engineer A 15m truck unloading a 27 tons LHC dipole Quantized fluxoids penetrating a superconductor used in accelerator magnets

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.4 QUESTIONS Order of magnitudes of the size of our objects: why ? High energy circular accelerators Length of an accelerator:  Km 15 m 1.9 Km Main ring at Fermilab, Chicago, US 41° 49’ 55” N – 88 ° 15’ 07” W 1 Km 40° 53’ 02” N – 72 ° 52’ 32” W RHIC ring at BNL, Long Island, US 46° 14’ 15” N – 6 ° 02’ 51” E

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.5 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 15 m A stack of LHC dipoles, CERN, Geneva, CH 46° 14’ 15” N – 6 ° 02’ 51” E Dipole in the LHC tunnel, Geneva, CH 0.6 m 6 cm

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.6 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators PRINCIPLES OF A SYNCHROTRON Electro-magnetic field accelerates particles Magnetic field steers the particles in a  circular orbit Driving particles in the same accelerating structure several times Particle accelerated  energy increased  magnetic field increased (“synchro”) to keep the particles on the same orbit of curvature  Limits to the increase in energy The maximum field of the dipoles (proton machines) This is why high field magnets are important to get high energies! The synchrotron radiation due to bending trajectories (electron machines) Constant

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators PRINCIPLES OF A SYNCHROTRON 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)

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.10 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators A DIGRESSION ON “RELATIVISTIC MASS” Note: in several books The mass is called the rest mass A new relativistic mass is defined So that the momentum is still defined as This is why you see in several texts that accelerating corresponds to increase the mass (?) This is a rather controversial intepretation of the formalism Einstein was not in favour of it – I will follow his advice “It is not good to introduce the concept of the mass of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion.” — Albert Einstein in letter to Lincoln Barnett, 19 June 1948 (quote from L. B. Okun (1989), p. 42 [1]Lincoln Barnett [1] Other authors against this concept " The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself. " [18] [18] E. F. Taylor, J. A. Wheeler (1992), Spacetime Physics, second edition, New York: W.H. Freeman and Company, ISBN Spacetime Physics, second editionW.H. Freeman and CompanyISBN Albert Einstein, German (14 March 1879 – 18 April 1955)

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: RELATION BETWEEN GAMMA, VELOCITY AND ENERGY What are the ranges of velocity obtained in particle accelerators? From to from the speed of light ! Please note e - in LEP (100 GeV) go faster than p + in LHC (7 TeV)

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: HOW TO KEEP PARTICLES ON A CIRCLE Preservation of 4-momentum Rest energy Hyp. 2 Ultra-relativistic regime Using practical units for particle with charge as electron, one has magnetic field in Tesla … Remember 1 eV=1.602  J Remember 1 e=  C

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators TESLA DIGRESSION Nikolai Tesla (10 July 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: HOW TO KEEP PARTICLES ON A CIRCLE Relation momentum-magnetic field-orbit radius

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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 increased 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 (like a spring) We will show in the next section that this is given by quadrupoles

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.18 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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 Then, with some approximations one gets where

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: SIZE OF THE BEAM In this way 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 We write the amplitude as a s -dependent part (beta function) plus an invariant along the ring The invariant is inverse proportional to the energy – so one can define a new invariant, also independent of the energy

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: SIZE OF THE BEAM Size of the beam is given by  emittance invariant along the ring, invariant from ring to ring This is a property of the injectors! Beam energy: the larger the energy, the smaller the beam  is the beta function [m], giving the envelope of the beam along the ring (the so-called optics)

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: SIZE OF THE BEAM A bit more insight on beta functions A typical structure is the FODO cell: alternating quadrupoles spaced by length L of similar gradient One can prove that this gives positive focusing in both transverse planes The phase advance usually is set to  integer have a periodicity LHC: 90° phase advance, i.e. one transverse oscillations after s =8 L One can prove that for 90° cells The larger L, the less quads you need  The larger L, the larger the beam Cell

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: SIZE OF THE BEAM 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 = m rad High field E =7 TeV,  =7460,  =0.29 mm Injection E =450 GeV,  =480,  =1.2 mm LHC requirement: beam is cut around 6  = 7 mm – radius of arc magnets aperture is 28 mm, i.e. about a factor four larger

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FOCUSING Focusing in a FODO cell Thin lens approximation: focusing strength in a 90° FODO cell is The focusing strength is related to K 1 and to the quadrupole length ℓ q and the quadrupole gradient is LHC: at high field B =8.33 T,  =2801 m, L =50 m, G ℓ q =660 T Case of 60° phase advance: linear dependence on L, different constants It looks worse: same beam size, 50% more focusing required

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.25 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS Input 1. Collision energy E c Gives a relation between the dipole magnetic field B and the total length of the dipoles L d Technology constraint 1. Dipole magnetic field B Does not depend on magnet aperture B t <2 T for iron magnets B t <13 T for Nb-Ti superconducting magnets (10 T in practice) B t <25 T for Nb 3 Sn superconducting magnets (16-17 T in practice) Output 1. Length of the dipole part Length in m, B in T, energy in GeV

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS Input 2. Injection energy E i Determines the relativistic factor, that affect the beam size Constraint 2. Normalized beam emittance  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 (diameter)

USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS Output 5. Number of semi-cells and arc length Number of semi-cells = number of quadrupoles = n q Length of the arc L a

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE ARC: FLOWCHART FOR MAGNET PARAMETERS Example: Large Hadron Collider E =7000 GeV Nb-Ti magnets, dipole field B =8.3 T Total length of dipoles L d =17.6 km Cell length L =50 m Beta function in the focusing quad  f =170 m Emittance  =3.75  m rad Injection energy 450 GeV,  =480 Beam size  = m (at injection) Coil aperture of m (coil-to-coil) Reduced size available for the beam Usually beam does not take more then 2/3 of space

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.36 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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)  reduce 

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS A system of quadrupoles is used to reach a very low beta function, called  *, in the IP (LHC: 0.55 m instead of the 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS Aperture requirement:  a+c/  * and dependent on l *, l t 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 *, l t, and is not related to  * This gives a requirement on the integrated gradient … … that together with the maximum gradient gives the triplet length

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS The 4 equations are coupled For the LHC, one has  * =0.55 m  max =4400 m With respect to the arc,  max is ~22 times larger, but the  is ~16 times larger in collision  the aperture is not so different from the cell magnets  = 70 mm instead of  = 56 mmm 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  = 70 m – one has a good safety margin

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 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 First target: Nb 3 Sn quadrupoles, 200 T/m, 90 mm aperture,  * =0.25 m Present target: Nb 3 Sn quadrupoles, 150 T/m, 140 mm aperture,  * =0.15 m LHC Nb-Ti baseline 90 mm aperture Nb 3 Sn TQ 120 mm aperture Nb 3 Sn HQ

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.43 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, gradient and length of the IR quads

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.44 COMING SOON During the next days: How these technological limits are determined ? What is the physics and the engineering behind?

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.45 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 J. Rossbach, P. Schmuser, “Basic course of accelerator optics”, CERN , pg Beam dynamics - insertions P. Bryant, “Insertions”, CERN 94-01, pg Beam dynamics - detectors T. Taylor, “Detector magnet design”, CERN , pg

USPAS January 2012, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.46 ACKNOWLEDGEMENTS J. P. Kouthcouk, M. Giovannozzi, W. Scandale for discussions on beam dynamics and optics 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