10.94 SCMULT—Tracks through a zero length multipole to simulate space charge effects

Tracks through a zero length multipole to simulate space charge effects
Parallel capable? : yes
GPU capable? : no
Back-tracking capable? : no






Parameter NameUnitsType Default

Description






GROUP stringNULL

Optionally used to assign an element to a group, with a user-defined name. Group names will appear in the parameter output file in the column ElementGroup






Important notes:

This element simulates transverse space charge (SC) kicks using K.Y. Ng’s formula [24].

The linear SC force is given by:

              2   2
   ′  KscLe--z-∕(2σz)-----x-----
Δx  =     √2-πσz    σx(σx + σy)

            -z2∕(2σ2)
Δy ′ = KscLe√------z------y-----
            2πσz    σy(σx + σy)
(153)

where Ksc = 2Nre
γ3β2, L is the integrating length, σx,y,z are rms beam size.

The non-linear SC force is given by:

                       ⌊   (            )               (             ) ⌋
      K   Le-z2∕(2σ2z)            x+ iy         --x2--y2     xσσy + iyσσx
Δx ′ =--sc∘---------Im ⌈w  ( ∘----------)  - e 2σ2x 2σ2yw ( ∘--x------y-) ⌉
       2σz  σ2x - σ2y            2(σ2x - σ2y)                   2(σ2x - σ2y)

                       ⌊  (             )               (             ) ⌋
       K  Le- z2∕(2σ2z)           x + iy         - x22--y22    xσσy + iy σσx
Δy ′ = -sc∘---------Re ⌈w ( ∘-----------)  - e 2σx 2σyw ( ∘--x------y-) ⌉
       2σz  σ2x - σ2y          2(σ2x - σ2y)                    2(σ2x - σ2y)
(154)

where w(z) is the complex error function

           ⌊                ⌋
                    ∫z
w (z) = e-z2⌈1 + 2√i--  eζ2dζ ⌉
                  π
                    0
(155)

Equation 154 appear to diverge when σx = σy. In fact, this is not true, because the expressions inside the square brackets will provide zero too at σx = σy to cancel the poles outside. In our code, we calculate this equation at 1.01σx and 0.99σx, and average the total effects.

To invoke the calculation, one must use set up command “insert_sceffects” proceed “run_setup” and “Twiss_output” command proceed “track”.

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