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					               MUON LINAC
              Solenoids Fringe Fields

                  M.Aslaninejad
                     J.Pasternak
                      J.Pozimski




21.04.2009,           M. Aslaninejad
Lancaster
                Fast acceleration




            Linear Pre-accelerator (244 MeV to 900 MeV)
        RLA I - 4.5 pass, 0.6 GeV/pass, (0.9 GeV to 3.6 GeV )
        RLA II - 4.5 pass, 2 GeV/pass (3.6 GeV to 12.6 GeV )
        Non scaling FFAG - 8 revolutions (12.6 GeV to 25 GeV )


21.04.2009,                  M. Aslaninejad
Lancaster
                                     SC Linac - 244 to 909 MeV. 25 cavities: First 6 cavities
                                     each 0.744826 meters. Second 8 cavities each 1.489652.
                                       Third 11 cavities each 2* 1.489652. Frequency 201 MHz
                     Tue Feb 12 12:47:13 2008    OptiM - MAIN: - M:\casa\acc_phys\bogacz\IDS \PreLinac\Linac_sol.opt
10




                                                                                                                                                     5
BE TA_ X&Y [m]




                                                                                                                                                     DISP _X&Y [m]
                                         8 one meter solenoids.
                                                 ks=1                                              11 one meter solenoids. ks=0.83
                 6,one meter
                 solenoids.
                  ks=1.4
0




                                                                                                                                                     0
                 0        BE TA_X      BE TA_Y       DISP_X      DISP_Y                                                                        146




                 6 short cryos                    8 medium cryos                                                       11 long cryos

                      15 MV/m                         17 MV/m                                                            17 MV/m




1.1 Tesla solenoid                                         1.4 Tesla solenoid                                             2.4 Tesla solenoid

                            21.04.2009,                                                   M. Aslaninejad
                            Lancaster
              Initial MADX




21.04.2009,     M. Aslaninejad
Lancaster
              MADXP, with Alex initial condition.




21.04.2009,                M. Aslaninejad
Lancaster
                           Concern
Difference of beta functions between OptiM code and MADX

Source of the inconsistency:
1-Fringe fields for solenoids, not included in MADX yet.
2- Mismatch in optics:
 linac focusing without a periodic boundary condition. Small
mismatch at the beginning results in large beta beating.




21.04.2009,                 M. Aslaninejad
Lancaster
        Ideal linear solenoid( zero aperture )transfer matrix:
                      1 + cos(kL)      sin(kL)          sin(kL)       1 - cos(kL) 
                           2               k                 2              k     
                                                                                  
                      - k sin(kL)   1 + cos(kL)
                                                      -k
                                                         1 - cos(kL)     sin(kL) 
                            4              2                   4            2             k = eB0 /pc
                    =
                                                                         sin(kL) 
             Msol
                      - sin(kL)     -
                                       1 - cos(kL)     1 + cos(kL)                 
                            2               k                2              k     
                      1 - cos(kL)       sin(kL)         k sin(kL)     1 + cos(kL) 
                     k                -               -                           
                            4               2                 4             2     

Non-zero aperture: correction due to the finite length of the edge.
It decreases the solenoid total focusing – via the effective length of:
                                                                                  1          0      0     0
                                                                                  -                      0
                                  1                    
                                      
                                                              k 2a                            1      0
                                                                                =                          
   1
L=       B (s) ds         edge =   Bz (s) ds - B0 L   -
                                         2          2
                                                                        Medge
                                                                                     edge

   B0   -
              z
                                  2  -                      8                  0          0      1     0
                                                                                                           
                                                                                  0
                                                                                             0   -edge   1
                                                                                                            


                                       Msoft sol = Medge Msol Medge

        21.04.2009,                                  M. Aslaninejad
        Lancaster
   The first 6 solenoids, with the Alex initial conditions.
      The strength of the 6th solenoids a bit smaller




21.04.2009,               M. Aslaninejad
Lancaster
 All cavities off. Different initial values. Different KS for matching solenoids




21.04.2009,                       M. Aslaninejad
Lancaster
                  Solenoid Fringe field

Thin solenoids defined in MADX can not resolve the
problem as they produce focusing in both planes.
Fringe fields for solenoids, not yet included in MADX.
But, it is ongoing through PTC.

Define an “arbitrary element matrix” in MADX
  according to the fringe field defocusing matrix.
50 matrices have been defined. Two for each solenoids
according to the corresponding solenoids strength.
Radius of each solenoids=25 cm



  21.04.2009,             M. Aslaninejad
  Lancaster
21.04.2009,   M. Aslaninejad
Lancaster
21.04.2009,   M. Aslaninejad
Lancaster
21.04.2009,   M. Aslaninejad
Lancaster
              Summary and future plans

• Main source of inconsistency was
  identified – fringe fields of solenoids
• Effects of Cavities Focusing.
• Tracking in Field Maps For Solenoids
  and RF.



21.04.2009,            M. Aslaninejad
Lancaster

				
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