Linear Collider Damping Rings

Document Sample
Linear Collider Damping Rings Powered By Docstoc
					Linear Collider Damping Rings

            Andy Wolski
Lawrence Berkeley National Laboratory

   USPAS Santa Barbara, June 2003
What do they look like?
                                                    30 m Wiggler




             Circumference       300 m Main Damping Ring
             Correction and        3 Trains of 192 bunches
                 Extraction         1.4 ns bunch spacing


                                             Injection and RF
                  90 m
             Extraction
                   Line

                                 30 m Wiggler

                                                         110 m
                   Spin                                Transfer
               Rotation                                    Line




                        110 m
                     Injection
                          Line
                                               231 m
                                          Predamping Ring
     NLC                                2 Trains of 192 bunches


  Positron
    Rings
                                                                   2
Operating Cycle in NLC/JLC MDRs
• Each bunch train is stored for three machine cycles
    – 25 ms or 25,000 turns in NLC
• Transverse damping time  4 ms
• Horizontal emittance ×1/50, vertical ×1/7500

               160 m                                         Spin
           Extraction                                    Rotation
                 Line


                                 30 m Wiggler

                                                           103 m
         Circumference                                     Injection
         Correction and                                    Line
         Extraction


         300 m Main Damping Ring
              3 Trains of 192 bunches
               1.4 ns bunch spacing
                                                Injection and RF



     30 m Wiggler

                                                                       3
What do they look like?




TESLA Damping Rings
                          4
Performance Specifications
                           NLC MDR         TESLA e+
Injected γε                150 µm rad     10 000 µm rad
Extracted Horizontal γε     3 µm rad        8 µm rad
Extracted Vertical γε     0.02 µm rad      0.02 µm rad
Injected Energy Spread    1% full width   1% full width
Extracted Energy Spread    0.1% rms          0.13%
Extracted Bunch Length       4 mm            6 mm
Bunch Spacing                1.4 ns           20 ns
Bunches per Train             192             2820
Repetition Rate              120 Hz           5 Hz

                                                          5
Radiation Damping…
• Longitudinal phase space
   – Particles perform synchrotron oscillations in RF focusing potential
   – Higher energy particles radiate energy more quickly in bends
   – At the equilibrium energy, the revolution period is an integer times the
     RF period (the synchrotron principle…)
• Transverse phase space
   – Particles perform betatron oscillations around the closed orbit
   – Radiation is emitted in a narrow cone centered on the instantaneous
     direction of motion
   – Energy is restored by the RF cavities longitudinally
   – Combined effect of radiation and RF is a loss in transverse momentum
• Damping time in all planes is given by:                  E0
                                                     J  2 T0
                                                           U0
                                                                                6
…and Quantum Excitation
• Radiation is emitted in discrete quanta
• Number and energy distribution etc. of photons obey
  statistical laws
• Radiation process can be modeled as a series of “kicks” that
  excite longitudinal and transverse oscillations




                                                                 7
Synchrotron Oscillations
                                                         Equilibrium orbit
Dispersive orbit


                                d
                                     p
                                dt
                   d eVRF                     1      dU 
                          sin  s   RF   U 0    
                   dt E0T0                     T0     dE 

                                     
                            d 2 2 d
                                         s2  0
                            dt 2  E dt




                                                                         8
Longitudinal Damping
           ˆe t  cos s t   s 
                       E



              
           s ˆe t  sin  s t   s 
                            E

             p

                    eVRF  RF
     s2                     p cos s 
                     E0 T0


                            E0
              J E E  2       T0
                            U0
                                I4
                    JE  2 
                                I2
                                                      Problem 1
                                                      Show that:
         1                            1                               1 
I2                       I4        2  2k1 ds              1
                                                                         C0  
               ds
            2
                                     
                                      
                                                
                                                
                                                          p 
                                                                 C0
                                                                    I1         ds

                                                                                     9
Quantum Excitation (Longitudinal)
          δ
                                   p ˆ             
                            1        1 cos1   p ˆ cos 
                                   s               s

                            1  ˆ1 sin 1   ˆ sin   
                                                                 u
                                                                 E0
                       τ
                                          2
                             ˆ 2   2  u  2 u  sin  
                            1      ˆ             ˆ
                                          2
                                         E0    E0


                           Including damping:
                              ˆ
                            d 2    1             ˆ 2
                                  2  N u ds  2
                                          2

                             dt   E0 C0           E




                                                                      10
Equilibrium Longitudinal Emittance
• We have found that:
                         ˆ
                       d 2    1             ˆ 2
                             2  N u ds  2
                                     2

                        dt   E0 C0           E

• From synchrotron radiation theory:
            1                  2 E02 I 3                 1
                N u ds  4Cq J E E  I 2       I3  
                    2
                                                                  ds
                                                          
                                                              3
            C0


Problem 2
Find an expression for the equilibrium energy spread,
and show that:

                     0et      ,equ 1  et   

                                                                       11
Betatron Oscillations: Action-Angle Variables
• It is often more convenient to describe betatron oscillations
  using action-angle variables:
                      2 J  x 2  2xx  x2
                                     x
                   tan     
                                     x

• The old variables are related to the new ones by:
                   x  2 J cos 

                   x  
                            2J
                                 sin     cos 
                            

• The equations of motion take the simple form:
                     d 1                dJ
                                           0
                     ds                 ds
                                                                  12
Damping of Vertical Oscillations
• Radiation is emitted in a narrow cone (angle ~1/γ) around
  instantaneous direction of motion, so vertical co-ordinate and
  momentum are not changed by photon emission
• RF cavity changes longitudinal momentum, and hence the
  vertical direction of motion:
                            py          p y  p         p 
          y1  y     
                    y1                    1      y1  
                           p  p       p     p
                                                 
                                                         
                                                            p
                                                              

• Averaging over all betatron phase angles gives (per turn):
                                 U0
                       J         J                  Problem 3
                                 E0
                                                       Show this!
• Hence the equation of motion is:
                      dJ   U
                          0 J
                      dt   E0T0
                                                                    13
Damping of Horizontal Oscillations
• When a photon is emitted at a point where there is some
  dispersion, the co-ordinates with respect to the closed orbit
  change:
                          u                   u
               x1  x            
                                  x1  x       
                          E0                  E0


• Taking the energy loss to first order and averaging around the
  ring, we find after some work:
                dJ     I  U             U
                    1  4  0 J   J x 0 J
                       I ET
                dt        2  0 0        E0T0




                                                                   14
Quantum Excitation of Betatron Motion
• Let us now consider the second order effects. It is easy to
  show that the change in the action depends to second order on
  the photon energy as follows:
                     2
               1 u 
           J    H              H  γη2  2αηη  βη2
               2  E0 
                  


• Averaging over the photon spectrum and around the ring, and
  including the radiation damping gives:
              dJ x      1                 2
                     2 E02C0 
                              N u 2 Hds  J x
               dt                         x
                               2 I5 2
                   C q 2              Jx                          H
                             J x x I 2  x                 I5             ds
                                                                     
                                                                         3


                                                                                  15
Summary of Dynamics with Radiation
     d                                                                                    E0
     dt
                                
          inje 2t    equ 1  e 2t              J x x  J y y  J E E  2
                                                                                           U0
                                                                                              T0

                                                                         I4                                   I4
                                                           J x  1                    Jy 1       JE  2 
                                                                         I2                                   I2
         
I1       ds                                                       C
                                                         U0            E04 I 2
          1                                                         2
I2              ds                                                                             p
            2
                                                           2  Cq 2
                                                                               I3
                                                                                                 
          1                                                                   JE I2              s
I3              ds
         
              3
                                                                    I1
                                                          p 
          1                           1 B y                     C0
I4       2  2k1 ds          k1 
                                    B x                        eVRF  RF
                                                         s2                p cos s 
         H                                                            E0 T0
I5              ds   H   2  2      2
         
              3

                                                     sin  s  
                                                                     U0
                                                                    eVRF
                                                                               I5
                                                            0  C q 2
                                                                              J x I2
                                                                                                                   16
The NLC TME Cell

                           High field in dipole




                                                  Sextupoles at high dispersion
                                                  points, with separated betas


        Vertical focusing in the dipole




Low dispersion and horizontal
beta function in the dipole
                                                   Cell length ≈ 5 m              17
H Function in the NLC TME Cell




                                 18
The TESLA TME Cell

                                 Low field in dipole




           No vertical focusing in the dipole



                                      Sextupoles at high dispersion points




   Larger dispersion and horizontal
   beta function in the dipole
                                                      Cell length ≈ 15 m

                                                                             19
NLC and TESLA TME Cells Compared
• NLC
   –   Compact cell to keep circumference as short as possible
   –   High dipole field for greater energy loss, reducing wiggler length
   –   Short dipole requires very low values for dispersion and beta function
   –   Gradient in dipole field to improve transverse dynamics
• TESLA
   – Circumference fixed by bunch train and kicker rise/fall time
   – Long dipole for larger momentum compaction, longer bunch
• Optimum lattice functions at center of dipole:
                  L            L                              3
           0            0              min  Cq   2

                2 15           24                            J x12 15

   – Obtained by minimizing I5 for a ring without a wiggler
   – It is not usually possible to control the dispersion and beta function
     independently
                                                                                20
Two Simple Scaling Relationships
Problem 4
Show that for an isomagnetic ring with the lattice functions tuned
for minimum emittance:
                                                          2

                               8 2 Lcell     Cq        3
                  N cellB0                         
                             C eme c  
                                  2 6      12 15 
                                                   0 
                                                          1

                              8 2 Lcell     Cq         3
                   2 B0                         
                            C eme c  
                                 2 6      12 15 
                                                  0 




                                                                     21
Scaling Relationships Applied to the NLC

                            1 bunch train   γε0 = 3 μm
                                            Lcell = 6 m
                                            τ = Ntrain 1.6 ms




           6 bunch trains




                                                          22
Scaling Relationships Applied to the NLC

                            2 bunch trains   γε0 = 1 μm
                                             Lcell = 6 m
                                             τ = Ntrain 1.6 ms

           6 bunch trains




                                                           23
Damping Wiggler
• A wiggler reduces the damping time by increasing the energy
  loss per turn:

                         C           e 2 c 2C
                  U0         E I 
                               4
                                                  E0  B 2 ds
                                                   2

                         2             2
                               0 2




• Wiggler must be located where nominal dispersion is zero,
  otherwise there can be a large increase in the natural emittance
• If horizontal beta function is reasonably small, wiggler can
  significantly reduce the natural emittance (through reduced
  damping time)
• Drawbacks include possible detrimental effect on beam
  dynamics
                                                                 24
Types of Wiggler
• A wiggler is simply a periodic array of magnets, such that the
  field is approximately sinusoidal
• Different technologies are possible:
   – Electromagnetic
   – Permanent magnet
   – Hybrid (permanent magnets driving flux through steel poles)
• Choice of technology comes down to cost optimization for
  given requirements on field strength and quality
• Both TESLA and NLC damping rings have opted for hybrid
  technology




                                                                   25
Modeling the Dynamics in the Wiggler
• Magnet design is produced using a standard modeling code
• Field representation must be obtained in a form convenient for
  fast symplectic tracking

                                         sin mk x x sinh k y ,mn y cosnk z z 
                                 mk x
              Bx   cmn
                                 k y ,mn
             By   cmn cosmk x x  coshk y ,mn y cosnk z z 

                                         cosmk x x sinh k y ,mn y sin nk z z 
                                  nk z
              Bz   cmn
                                 k y ,mn
           k y ,mn  m 2 k x  n 2 k z2
             2             2




                                                                                      26
Fitting the Wiggler Field




                            27
Tracking Through the Field
• Using an appropriate field representation (that satisfies
  Maxwell’s equations), one can construct a symplectic
  integrator:
                                  
                          xnew  m xold 

• M is an explicit function of the phase-space co-ordinates, and
  satisfies the symplectic condition (so the dynamics obey
  Hamilton’s Equations):
                          M S M T  S
                                      m
                                M ij  i
                                      x j
                                      0 1
                                      1 0
                                  S      
                                          
                                                                   28
Dynamics in the NLC Wiggler
         Horizontal Kicks and Phase Space




           Vertical Kicks and Phase Space




                                            29
Chromaticity
• Chromaticity is the tune variation with energy
• Quadrupole focusing strength gets smaller as particle energy
  increases
• It can easily be shown that:
                               x     1
                                      4 
                          x              x k1ds
                              
                                   1
                          y  y 
                                  4    y k1ds

• Since beta functions peak at the focusing quadrupoles in the
  appropriate plane, the natural chromaticity is always negative
• Chromaticity is connected to beam instabilities
   – particles with large energy deviation cross resonance lines
   – some collective effects (e.g. head-tail instability) are sensitive to the
     chromaticity
                                                                                 30
Correcting Chromaticity with Sextupoles




                 sextupole
                  k1= x k2
                                          31
Dynamics with Sextupoles
• Sextupoles can be used to correct chromatic aberrations…
                                1
                        x  
                               4   x k1-ηx  x k2ds
                              1
                             4 
                        y        y k1   x  y k 2ds

• …but introduce geometric aberrations and coupling:
                  x  1 k 2l  x 2
                        2                y  k 2lY  x


• It is important to keep the required strengths to a minimum by
  designing the linear lattice functions for effective sextupole
  location



                                                               32
Dynamic Aperture
• Geometric aberrations from sextupoles (and other sources)
  distort the transverse phase space, and limit the amplitude
  range of stable betatron oscillations


Horizontal phase space of NLC TME cell




                                         Vertical phase space of NLC TME cell



                                                                         33
Transverse and Longitudinal Aperture
• Damping rings require a “large” dynamic aperture
   – Injected beam power ~ 50 kW average, and radiation load from any
     significant injection losses will destroy the ring
   – Nonlinear distortion of the phase space may lead to transient emittance
     growth from inability properly to match injected beam to the ring
   – For NLC Main Damping Rings, the target dynamic aperture is 15 times
     the injected rms beam size
• We also need a large momentum acceptance
   – Injected beam has a large energy spread
   – Particles may be lost from insufficient physical aperture in dispersive
     regions, or through poor off-momentum dynamics
   – Particles within a bunch can scatter off each other, leading to a
     significant change in energy deviation (Touschek Effect)
• It is important to perform tracking studies with full dynamic
  model and physical apertures
                                                                               34
NLC Main Damping Ring Dynamic Aperture
         δ= -0.005




         δ= +0.005



                       Dynamic Aperture On-Momentum

                                 15× Injected Beam Size



                                                          35
Longitudinal Acceptance
• The longitudinal acceptance has three major limitations:
   – Poor off-momentum dynamics
   – Physical aperture in dispersive regions
   – RF bucket height
• Off-momentum dynamics can be difficult to quantify
   – see previous slides
• Physical aperture can be a significant limitation
   – 1% momentum deviation in 1 m dispersion is a 1 cm orbit offset
• RF bucket height comes from non-linearity of the longitudinal
  focusing
   – Previous study of longitudinal dynamics assumed a linear slope of RF
     voltage around the synchronous phase
   – Valid for small oscillations with synchronous phase close to zero-
     crossing

                                                                            36
RF Bucket Height
• The “proper” equations of longitudinal motion (without
  damping) are:
                      d
                           p
                      dt
                      d eVRF
                                sin s   RF   sin s 
                      dt E0T0


• These may be derived from the Hamiltonian:

             H  - 1  p 2 
                                 eVRF
                                         coss   RF   sin s  RF 
                                E0T0 RF
                   2




                                      d H
                                         
                                      dt 
                                      d    H
                                         
                                      dt    
                                                                                37
Longitudinal Phase Space
• The Hamiltonian is a constant of the motion, which allows us
  to draw a phase-space portrait
                                                   Stable fixed point
                                                   Unstable fixed point
                                                 S Separatrix
     RF
                                                         VRF
                        S

                                                                           




                                                                               
                                                       coss   s   sin s 
                                          4eVRF
                              RF  
                               2
                                                      
                                        E0T0 RF p                  2          

                                                                                  38
Alignment Issues
• The final luminosity of the collider is critically dependent on
  the vertical emittance extracted from the damping rings
• In a perfectly flat lattice, the lower limit on the vertical
  emittance comes from the opening angle of the radiation
   – Gives about 10% of the specified values for NLC and TESLA
• Magnet misalignments give the dominant contribution to the
  vertical emittance
   – Quadrupole vertical misalignments
       • Vertical dispersion
       • Vertical beam offset in sextupoles
   – Quadrupole rotations and sextupole vertical misalignments
       • Couple horizontal dispersion into the vertical plane
       • Couple horizontal betatron oscillations into the vertical plane

                                                                           39
Betatron Coupling
• In a damping ring, the dominant sources of betatron coupling
  are skew quadrupole fields
   – Normal quadrupoles have some “roll” about the beam axis
   – Sextupoles have some vertical offset with respect to the closed orbit
• Particles with a horizontal offset get a vertical kick

                                                               Particle on
                                                               closed orbit
                                                               Particle with
                                                               horizontal
                                                               amplitude

                                                               Vertical kick
                                                               depends on
                                                               horizontal
                                                               amplitude
                                                                               40
Effects of Betatron Coupling
• In action-angle variables, the “averaged Hamiltonian” for a
  coupled storage ring can be written:

                         H   x J x   y J y   n J x J y cos x   y 
                      C0                         ~
                      2


• The equations of motion are:

                                                   d x 2        ~
                                                                                                
            J x J y sin  x   y                                             cos x   y 
      dJ x ~                                                                   Jy
                                                            x 
       ds                                           ds   C0 
                                                                  2           Jx                
                                                                                                 


                                                   d y  2          ~
                                                                                                
                J x J y sin  x   y                                        cos x   y 
      dJ y       ~                                                             Jx
                                                               y 
      ds                                            ds   C0    
                                                                     2        Jy                
                                                                                                 

                                                                                                     41
Solutions to the Coupled Hamiltonian
• The sum of the horizontal and vertical actions is conserved:
                         J0  J x  J y
                       dJ 0
                            0
                        ds

• There are fixed points at:

                                         
                        1               
                   J x  J 0 1 
                        2             ~2 
                                  2   
                                         
                                                 x  y
                                         
                        1               
                   J y  J 0 1 
                        2             ~2 
                                  2   
                                         


• With radiation, the actions will damp to the fixed points
                                                                 42
The Difference Coupling Resonance
• The equilibrium emittance ratio is given by:
                        Jy         ~
                                   
                                       2

                             
                        Jx       ~
                                   42
                                  2




• The measured tunes are given by:
               C0 d 1              ~2
                      2   1 2           x  y
               2 ds
                             2




                                                          43
What is the Coupling Strength?
• We add up all the skew fields around the ring with an
  appropriate phase factor:
              C0

                                              n s    x   y   2  x   y  n 
          1                                                                                   s
    ~
    n            x  y k s ei s ds
                                 n

         2   0                                                                              C0


• ks is the skew quadrupole k-value.
• For a rotated quadrupole or vertically misaligned sextupole,
  the equivalent skew fields are given by:

                        k s  k1 sin 2           k s  k 2 y




                                                                                                  44
Vertical Dispersion
• In an electron storage ring, the vertical dispersion is typically
  dominated by betatron coupling
   – Emittance ratios of 1% are typical
• For very low values of the vertical emittance, vertical
  dispersion starts to make a significant contribution
• Vertical dispersion is generated by:
   – Vertical steering
       • vertically misaligned quadrupoles
   – Coupling of horizontal dispersion into the vertical plane
       • quadrupole rotations
       • vertical sextupole misalignments




                                                                      45
Vertical Steering: Closed Orbit Distortion
• A quadrupole misalignment can be represented by a kick that
  leads to a “cusp” in the closed orbit

                                             




• We can write a condition for the closed orbit in the presence of
  the kick:
         y0   y 0                      cos x    x sin  x        x sin  x        
         y   y   
     M                            M
                                                                                                
         0  0                                x sin  x        cos x    x sin  x 
                                                                                                 
• We can solve to find the distortion resulting from many kicks:
                             y s1          y s 
              ys1  
                         2 sin  y        s 
                                                                                    
                                                        cos  y   y s1    y s  ds

                                                                                                     46
Vertical Steering: Vertical Dispersion
• The vertical dispersion obeys the same equation of motion as
  the vertical orbit, but with a modified driving term:

      y  k1 y 
                         1               differentiate wrt               k1 y  k1 y 
                                                                                                1
                     1                                              y
                                                                                                


• We can immediately write down the vertical dispersion arising
  from a set of steering errors:
                    y s1 
                                y s  k1 y   cos y   y s1    y s   ds
                                              1
      y s1              
                        2 sin  y           
                                                         

• Including the effect of dispersion coupling:
                        y s1 
     y s1   
                    2 sin  y  
                                                                        1
                                                                                                          
                                       y s  k1  k 2 x  y  k s x   cos  y   y s1    y s  ds
                                                                           
                                               
                                                                                                                   47
Effects of Uncorrelated Alignment Errors
• Closed orbit distortion from quadrupole misalignments:
                             y2            Yq2
                                                         y k1l 2
                            y        8 sin 2  y 


• Vertical dispersion from quadrupole rotation and sextupole
  misalignment:
      y
       2
                q   2
                                                        y
                                                         2
                                                                  Ys    2

                            y k1l x 2                                   y k 2 l x 2
      y   2 sin 2  y                               y   8 sin 2  y 


• Vertical emittance generated by vertical dispersion:
                                         JE y
                                              2

                                  y  2         2
                                         J y y
                                                                                                  48
Examples of Alignment Sensitivities
                                       KEK-
                          APS    SLS          ALS   NLC MDR   TESLA DR
                                       ATF
Energy [GeV]               7     2.4   1.3    1.9     1.98       5
Circumference [m]         1000   288   140    200     300      17,000
γεx [µm]                   34    23    2.8    24       3         8

γεy [nm]                  140    70     28    20      19         14
Sextupole vertical [µm]    74    71     87    30      53         11
Quadrupole roll [µrad]    240    374   1475   200     511        38
Quadrupole jitter [nm]    280    230   320    230     264        76

• Note:
  Sensitivity values give the random misalignments that will
  generate a specified vertical emittance. In practice, coupling
  correction schemes mean that significantly larger
  misalignments can be tolerated.
                                                                        49
Collective Effects
• Issues of damping, acceptance, coupling are all single particle
  effects - they are independent of the beam current
• Particles in a storage ring interact with each other (directly or
  via some intermediary e.g. the vacuum chamber)
• A wide variety of collective effects limit the achievable beam
  quality, depending on the bunch charge or total current
• The consequences of collective effects are
   – Phase space distortion and/or emittance growth
   – Particle loss
• Damping rings have high bunch charges, moderate energies
  and small emittance
   – Vulnerable to a wide range of collective effects
• Too wide a subject to enter into here!
                                                                  50

				
DOCUMENT INFO