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Accelerator Magnet Design

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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  jddq
                                                                                                             w

    Taking into account of current signs
                                                                                                    -            a
                                                                                                                     +
                      a r w
           j 0                cosq                   2 j 0                                             r
   B1  4
           2         
                      0   r
                                 
                                      ddq  
                                                          
                                                               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 rw
                                       j 0                 cos2 q             j 0
                               B1  4
                                       2         
                                                 0    r
                                                               
                                                                     ddq  
                                                                                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 rw
                                     exp(inq )                         j 0 Rref 1
                                                                              n      a                     r w
    C n  2
                   2        a 
                                r        n
                                                     ddq  
                                                                                      exp(inq )dq
                                                                                      a
                                                                                                             r
                                                                                                                1 n d

     for n=2 one has
                                                    j 0 Rref                w
                                         B2                   sin(2a ) log1  
                                                                            r
     and for n>2
                                               j 0 R ref 1 2 sin(an) (r  w) 2  n  r 2  n
                                                      n

                                        Bn  
                                                               n           2n
•   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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