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Accelerator Magnet Design Soren Prestemon Lawrence Berkeley National Laboratory Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 References - Acknowledgments • USPAS course “Superconducting Accelerator Magnets”, Ezio Todesco, Paolo Ferracin, Soren Prestemon • USPAS Course “Magnetic Systems: Insertion Devices”, Ross Schlueter Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Outline • The magnets of an accelerator • Some magnetics fundamentals • Review of magnetic multipoles – Definition: Taylor series – Inverse problem: how to create multipole fields • Iron-dominated (scalar potential) • Biot-Savart • Design and fabrication issues with real accelerator magnets Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Layout of an accelerator • Magnets play key role: – Kick beam into accelerator during injection: Kicker magnets – Align injected beam with stored beam: Septum magnets – Bend beam in circle: bend magnets (dipoles) – Focus beam to allow storage (quadrupoles) – Compensate for electron energy variation (sextupoles) n Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Additional magnet systems • Correctors – Dipoles for field trajectory correction – Can be “slow”: compensate static or slow-varying drifts – Can be fast: allow fast feedback for beam control • Chicanes – Versions of corrector magnets (not used for beam feedback) – Used to provide mild steering of beam, e.g. in straights, or for dispersion • Light-source Wigglers and undulators – Used in to produce synchrotron radiation of particular quality – Ideally are transparent to beam storage Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Examples of Accelerator Magnets • Dipoles – Steering • Quadrupoles – focusing • Sextupoles – chromaticity • These components are analogous to optical elements, e.g. mirrors – Charged-particle optics Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Simplest dipole: windowframe • Assume μ=∞; field in center is uniform • Field across coil is linear • What happens if μ is finite? • What happens at the ends? Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Another “simple” geometry • Constant J in ellipse => J=0 in intersecting zone – Field in center is uniform => perfect dipole – This is the motivation for standard “Cos(θ)” dipoles Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Transitioning from theory to practice • Coil is made from a wire/cable => J~constant – Discretize Cos(θ) distribution using wedges – Ends must allow beam-passage – These “details” introduce errors in the form of harmonic content Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Some classic configurations • These configurations are often mentioned in the literature – Combined-function magnets can take a variety of forms • Scalar potential can define combination of fields • Scalar potential can be defined for “dominant” multipole of interest – other multipoles are then added via additional energization Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Review: Maxwell Ampere Faraday Continuity across interfaces implies: Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Allowed multipoles Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Multipole fields • Potential isosurfaces, m=1, 2, 3, 4 Normal Skew Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Some comments • The series expansion is only valid out to the minimum radius rs of any potential surface rs • Non-dimensionalization by rs is often replaced by Rref, a convenient measurement radius • The coefficients beyond Bm (the dominant mode)are then often normalized by 10-4Bm – resulting terms are said to be in “units” Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 FIELD HARMONICS OF A CURRENT LINE • Field given by a current line (Biot-Savart law) Félix Savart, Or, in terms of multipoles: French (June 30, 1791-March 16, 1841) Jean-Baptiste Biot, French (April 21, 1774 – February 3, 1862) 15 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 FIELD HARMONICS OF A CURRENT LINE • The multipoles of a line current then scale like 1/n – The details of the decay depend on the line current position – Adding multiple line currents judiciously positioned can result in a multipole field of order m with fairly small multipoles n≠m iy -z* z x -z z* The line currents can be connected so as to create a dipole, quadrupole, etc 16 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 HOW TO GENERATE A PERFECT FIELD • Perfect dipoles – Cos theta: proof – homework from last Monday j (q ) j 0 cos(mq ) The vector potential reads n 0 j 1 Az ( , ) n 2 n 1 0 cos[n( q )] 60 and substituting one has j = j0 cos q n - + j 1 2 q Az ( , ) 0 0 n cos(mq ) cos[n( q )]dq 0 -0 4 0 40 2 - + 0 0 n 1 using the orthogonality of Fourier series m 0 j0 -0 6 Az ( , ) cos(mq ) 2m 0 17 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Basic features of “sector” coils (Ezio Todesco) • We compute the central field given by a sector dipole with uniform current density j I jddq w Taking into account of current signs - a + a r w j 0 cosq 2 j 0 r B1 4 2 0 r ddq w sin a - + This simple computation is full of consequences – B1 current density (obvious) – B1 coil width w (less obvious) – B1 is independent of the aperture r (much less obvious) • For a cosq, / 2 rw j 0 cos2 q j 0 B1 4 2 0 r ddq 2 w 18 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 SECTOR COILS FOR DIPOLES • Multipoles of a sector coil j 0 Rref 1 n a rw exp(inq ) j 0 Rref 1 n a r w C n 2 2 a r n ddq exp(inq )dq a r 1 n d for n=2 one has j 0 Rref w B2 sin(2a ) log1 r and for n>2 j 0 R ref 1 2 sin(an) (r w) 2 n r 2 n n Bn n 2n • Main features of these equations – Multipoles n are proportional to sin ( n angle of the sector) • They can be made equal to zero ! – Proportional to the inverse of sector distance to power n • High order multipoles are not affected by coil parts far from the centre 19 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Using free parameters • First allowed multipole B3 (sextupole) 0 jRref sin(3a ) 1 2 1 B3 3 r r w w - a + for a=/3 (i.e. a 60° sector coil) one has B3=0 r - + • Second allowed multipole B5 (decapole) 0 jRref sin(5a ) 1 4 1 B5 3 5 r 3 r w 50.0 for a=/5 (i.e. a 36° sector coil) or for a=2/5 (i.e. a 72° wedge 45.0 40.0 sector coil) 35.0 one has B5=0 30.0 25.0 a3 a 2 20.0 a1 • With one sector one cannot set to zero both multipoles 15.0 10.0 … but it can be done with more sectors! 5.0 0.0 20 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Examples of real magnets • Number of sectors is chosen based on: – Multipole content that can be tolerated – Fabrication issues Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Example geometries for real superconducting accelerator magnets Paolo Ferracin, USPAS 2009 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Real superconducting magnets: Basic design / fabrication Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Overview of Nb3Sn coil fabrication stages After winding After reaction After impregnation Cured with matrix Reacted Epoxy impregnated 24 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Overview of accelerator dipole magnets Tevatron HERA SSC RHIC LHC 25 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Design issues • Superconducting magnets store energy in the magnetic field – Results in significant mechanical stresses via Lorentz forces acting on the conductors; these forces must be controlled by structures – Conductor stability concerns the ability of a conductor in a magnet to withstand small thermal disturbances, e.g. conductor motion or epoxy cracking, fluxoid motion, etc. – The stored energy can be extracted either in a controlled manner or through sudden loss of superconductivity, e.g. via an irreversible instability – a quench • In the case of a quench, the stored energy will be converted to heat; magnet protection concerns the design of the system to appropriately distribute the heat to avoid damage to the magnet 26 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Lorentz force - Dipole magnets • The Lorentz forces in a dipole magnet tend to push the coil – Towards the mid plane in the vertical-azimuthal direction (Fy, Fq < 0) – Outwards in the radial-horizontal direction (Fx, Fr > 0) Tevatron dipole HD2 27 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Lorentz force - Quadrupole magnets • The Lorentz forces in a quadrupole magnet tend to push the coil – Towards the mid plane in the vertical-azimuthal direction (Fy, Fq < 0) – Outwards in the radial-horizontal direction (Fx, Fr > 0) TQ HQ 28 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Stress and strain Mechanical design principles LHC dipole at 0 T LHC dipole at 9 T Displacement scaling = 50 • Usually, in a dipole or quadrupole magnet, the highest stresses are reached at the mid-plane, where all the azimuthal Lorentz forces accumulate (over a small area). 29 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Lorentz force - Solenoids • The Lorentz forces in a solenoid tend to push the coil – Outwards in the radial-direction (Fr > 0) – Towards the mid plane in the vertical direction (Fy, < 0) 30 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Concept of stability • The concept of superconductor stability concerns the interplay between the following elements: – The addition of a (small) thermal fluctuation local in time and space – The heat capacities of the neighboring materials, determining the local temperature rise – The thermal conductivity of the materials, dictating the effective thermal response of the system – The critical current dependence on temperature, impacting the current flow path – The current path taken by the current and any additional resistive heating sources stemming from the initial disturbance Input of spurious energy Q Depending on new state Qnew Qnew ( J non sc , T T ) Local temperature rise T T (C p ) Superconducting state is impacted J c J c ( B, T T ) Heat is conducted to neighboring material 31 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Calculation of the bifurcation point for superconductor instabilities Thanks to Matteo Allesandrini, Texas Heat Balance Equation in 1D, without coolant: [W/m3] Center for Superconductivity, for these calculations and slides d dT dT k (T ) (T ) J 2 Qinitial _ pulse C(T )volume 0 dx dx dt Heat conduction Joule effect Quench trigger Heat stored in the material Ex. RECOVERY of a potential Quench t=0 T(x,t) t=1 t=2 I0 Wire Heater x 32 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Example of quench initiation QUENCH with 1 [mJ] Linear Scale Quench Temperature [K] Time [s] 33 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Analysis of SQ02 – quench propagation QUENCH with 1 [mJ] Tcritical Tsharing Flux Jumps and Motion in Superconductors 6.34 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Analysis of SQ02 – quench propagation QUENCH with 1 [mJ] Hot Spot temp. profile Tcritical Tsharing 35 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Magnet protection • The quench propagation aids in distributing stored energy to the rest of the magnet • Often we accelerate the process by actively heating the magnet once a quench initiation has been detected (“Active protection”) • If possible, much of the energy is also absorbed by a dump resistor • The energy can also be absorbed by inductive coupling to a secondary Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010 Permeability and field-lines Problem 1: find the functional relationship α1=f(μ1,μ2,α2). Plot the function for μ1=1, μ2=1 and μ2=10 α2 B2, H2 μ2 μ1 α1 Fundamental Accelerator Theory, Simulations and Measurement Lab – UC Santa Cruz, San Francisco, January 18-29, 2010

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