Tracking at resonance tune for linear lattice--> emittances grow

[Youssef]

Hi,

for tracking at resonance tune, i.e. Qx=Qy and for a example lattice just with two Quadrupoles and drifts, I dont expect any emittance changes...
this is my lattice:

"mqd: MULT, KnL =-0.2169824517, order = 1
mqf: MULT, KnL = 0.2169824517, order = 1
drift1:DRIF, L = 0.65
quad1:Line=(mqd)
drift2 : DRIF, L =2.6
quad2:line =(mqf)
drift3 : DRIF, L = 9.65

fodo1cell : line = (drift1,quad1,drift2,quad2,drift3)
use,fodo1cell
return
"

After running tracking

"
&run_control
n_passes = 5000000
&end

&bunched_beam
n_particles_per_bunch = 1000
bunch=..\bunch\track_UNIELLIPSE_Emittances_Div_by_4.bun
use_twiss_command_values = 1
emit_x = 8.75e-06
emit_y = 3.75e-06
distribution_cutoff[0] = 1, 1
distribution_type[0] = "uniform-ellipse", "uniform-ellipse"
enforce_rms_values[0] = 1, 1
limit_in_4d = 1
&end

&track
&end
"

I found the there is some changes in the emittances, emi_x increases and emi_y decreases.
I checked it already for kquad and edrift, some thing happened.
Do you have any idea where this emittance changes come?

Regards,
Youssef

[michael_borland]

Youssef,

The problem seems to be that when using the exact Hamiltonian (the default in elegant), there is "hidden" coupling between x and y motion. In particular, to compute the changes in x and y, we have to compute the slopes x' and y' from the momenta qx and qy. As you can see in equation 6 of LS-356, the transformations involve coordinates from both planes. When using the expanded Hamiltonian, the transformations involve coordinates from a single plane only.

You can see this by using KQUAD elements and setting EXPAND_HAMILTONIAN=1. (MULT has the same parameter, but there's a bug in its implementation.)

The bottom line is that x and y emittances are not separately preserved when using the exact Hamiltonian. Presumably this is made worse by sitting on the coupling resonance, though I didn't check that.

--Michael