ELENA Tracking studies
P.Belochitskii, O.Berrig
With thanks to: C.Carli, L.Varming Jørgensen, G.Tranquille
BMAD – an alternative tracking program
BMAD is a program from Cornell by
D.Sagan. It is symplectic ( which basically
means energy conservation:
http://en.wikipedia.org/wiki/Symplectic
_integrator), and has the added
capability of tracking through user
defined fields. Since the electron cooler
is described by a field map (and the
space charge from the electron beam
also) it would have been a fine candidate
for ELENA.
However there were some issues with
the running of the program (that has
been solved later) which at the time
discouraged us from using BMAD.
Plot of the ELENA ring; made by BMAD
MADX does not calculate correct tunes for off-momentum particles
Sextupoles ON; Solenoids OFF; EBEAM OFF; Tracking with MADX
Qhor
MADX TRACK
MADX TWISS
0.280
MADX
0.278
0.276
0.274
0.010
0.005
0.005
0.010
PP
Sextupoles ON; Solenoids OFF; EBEAM OFF
Qhor
0.280
PTC_TRACK
PTC_TWISS
0.279
PTC
0.278
Even though tunes, chromaticities,
etc. are not calculated correctly in
MADX, the tracking in MADX is the
same as the tracking in PTC up to
second order (but only for onmomentum particles; for offmomentum particles there are
errors in MADX)
See: http://cern-acceleratorsoptics.web.cern.ch/cernacceleratorsoptics/OtherInfo/MADX_PTC_Bend
ingMagnet_difference_14Nov.doc
0.277
All simulations were therefore
done with PTC
0.276
0.275
0.010
0.005
0.005
0.010
PP
How to simulate the space charge generated by the electron beam?
The electron beam makes de-focusing of the antiprotons in both planes; This process
cannot be simulated by PTC. MADX however, can do simulations with an arbitrary
MATRIX element – but only for thin lens optics.
We made a model of ELENA with thin lens, however the beta-functions were not perfect.
Comparizon Thick and Thin lens optics
Because of the following arguments:
• In order not to spend too much time
optimizing the thin-lens beta-functions
(which would in any case always represent
an approximation).
• The MATRIX element of MADX only simulate
the effect on the protons that are inside the
radius of the electron beam, but not the
protons that are outside.
• MADX is only precise to second order.
beta function
14
12
betax thin
10
betx thick
8
6
4
2
0
0
5
10
15
20
25
30
s
Comparizon Thick and Thin lens optics
beta function
14
12
betay thin
10
bety thick
8
We decided to simulate the electron cooler
with Mathematica and the rest of the ring
with PTC
6
4
2
0
0
5
10
15
20
25
30
s
It was decided to run without the main sextupoles
magenta=10mm, green=20mm, red=30mm, blue=40mm, orange=50mm
Without sextupoles
With sextupoles
The simulations on this page were
done with the electron cooler
simulated with MADX solenoids
(the electron beam was not
included).
pxn
pxn
Horizontal
0.04
0.04
0.02
0.02
0.04
0.02
0.02
0.04
xn
0.04
0.02
0.02
0.02
0.04
0.04
pyn
Vertical
xn
The strength of the sextupoles was
nominal, i.e. set to make the
chromaticity equal to zero, so that
off-momentum particles will not
touch neighboring resonances
0.04
0.02
0.02
0.04
pyn
0.04
0.04
0.02
0.02
0.02
0.04
yn
0.04
0.02
0.02
0.02
0.02
0.04
0.04
0.04
yn
Since the sextupoles brought too
much non-linearity (leading to loss
of particles with 50mm amplitude)
we decided to temporarily switch
off the main sextupoles in order to
concentrate on resonances from
the rest of the machine (especially
the electron cooler)
Philosophical interlude (1)
There are no linear magnets, not even quadrupoles and bending magnets.
In reality they contain non-linear elements like sextupoles and octupoles and up to any higher
order mode. This is not only because the fringe fields at the end of the magnets are non-linear,
but even the magnets themselves. See: http://zwe.home.cern.ch/zwe/talks/cern_nonlinear.pdf
A bending magnet contains e.g. sextupolar components (the larger the bending angle, the
larger the sextupolar component). This was already discovered by Karl Brown in 1982:
http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-r-075.pdf :
X2-y2 is a sextupolar component:
HORIZONTAL motion:
D[P[x,
y], x]
VERTICAL motion:
D[P[x,
y], x]
x·y is a sextupolar component, see J.Jowett:
\\cern.ch\dfs\Projects\ .. \MultipoleFields.nb
For a bending magnet,h is equal to 1 divided by the bending radius: h=1/r
Compensation of sextupolar component in the main magnets
NO compensation
sextupole
Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF No Mathematica
Qx 2.22
Qy 1.205 Momentum offset 0.000
HOR phase space diagram
With compensation
Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF No Mathematica
Qx 2.22
Qy 1.205 Momentum offset 0.000
HOR phase space diagram
pxn
0.03
pxn
0.03
1. mm
1. mm
6. mm
6. mm
11. mm
11. mm
16. mm
16. mm
21. mm
21. mm
26. mm
0.02
26. mm
0.02
31. mm
0.01
0.03
0.02
0.01
31. mm
0.01
0.01
0.02
xn
0.03 0.03
0.02
0.01
0.01
0.01
0.01
0.02
0.02
0.03
0.03
0.02
xn
0.03
Philosophical interlude (2)
Any machine with non-linear elements will couple the horizontal and vertical planes and
lead to lines in the tune diagram that should be avoided:
Avoid tunes that fulfill the equation:
where: l,m and r are integers
See page 69 in http://cds.cern.ch/record/212880/files/CERN-92-01.pdf
See also: http://cern.ch/bruening/CAS/Driven_Resonances.ppt
and: http://cds.cern.ch/record/1694484/files/CERN-2014-002.pdf
Qx=2.22 Qy=1.205
Qx=2.26 Qy=1.29
Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Real field in solenoids
Qx 2.22
Qy 1.205 Momentum offset 0.000
VER phase space diagram
Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF
Qx 2.26
Qy 1.29 Momentum offset
VER phase space diagram
pyn
0.03
Qx=2.28 Qy=1.30
Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF
Qx 2.28
Qy 1.30 Momentum offset
VER phase space diagram
Real field in solenoids
0.000
Real field in solenoids
0.000
pyn
0.03
pyn
0.03
1. mm
1. mm
1. mm
6. mm
6. mm
11. mm
11. mm
0.03
0.02
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
0.02
yn
0.03
0.03
0.02
0.01
0.01
0.02
yn
0.03
0.03
0.02
0.01
0.01
0.01
0.01
0.01
0.02
0.02
0.02
0.03
0.03
0.03
0.02
yn
0.03
Biggest problem. Vertical emittance is growing – why?
No shielding – No correctors
With shielding – With correctors
Observe first that there is
coupling between the
horizontal and vertical
planes (Qx-Qy = 1). The
signatures of this resonance
are seen in the FFT plots
(next slide) and in the
variations of Wx and Wy
versus turn; when one of
them is on the crest, the
other is in the trough - but
amplitude of variations is
different for x and y-planes.
On top of that, in the
vertical plane Wy is growing
with number of turns!
Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Real field in solenoids
Qx 2.26 Qy 1.29 Momentum offset 0.000
Wxn xn^2 pxn^2 Versus turn no.
Sextupoles OFF; Solenoids ON 100 Gauss No Shielding Scaled ; EBEAM OFF Real field in solenoids
Qx 2.26 Qy 1.29 Momentum offset 0.000
Wxn xn^2 pxn^2 Versus turn no.
Wxn mm mrad
Wxn mm mrad
120
80
1. mm
100
1. mm
6. mm
6. mm
11. mm
80
60
16. mm
60
11. mm
40
40
20
20
200
400
600
n
1000
800
200
400
600
800
1000
n
Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Real field in solenoids
Qx 2.26 Qy 1.29 Momentum offset 0.000
Wyn yn^2 pyn^2 Versus turn no.
Sextupoles OFF; Solenoids ON 100 Gauss No Shielding Scaled ; EBEAM OFF Real field in solenoids
Qx 2.26 Qy 1.29 Momentum offset 0.000
Wyn yn^2 pyn^2 Versus turn no.
Wyn mm mrad
70
Wyn mm mrad
100
60
1. mm
1. mm
6. mm
80
6. mm
50
11. mm
11. mm
16. mm
40
60
30
40
20
20
10
200
400
600
800
n
1000
200
400
600
800
1000
n
Biggest problem. Vertical emittance is growing – why?
No shielding – No correctors
With shielding – With correctors
No essential difference
in the tune spectra,
except for the presence
of the linear coupling
resonance Qx-Qy = 1.
But coupling cannot
increase the emittance
in one plane and not in
the other!
Biggest problem. Vertical emittance is growing – why?
Other observations:
1. The electron cooler field without shielding and without correctors is a theoretical calculation
followed by a Mathematica interpolation at a given particle position. The electron cooler
field with shielding and with correctors is calculated by OPERA with a given mesh; for
tracking we again use Mathematica with interpolation at a given particle position.
2. The curl (=rotB) of the electron cooler field without shielding and without correctors is much
smaller than for the electron cooler field with shielding and with correctors
Summary:
solenoid
solenoid
solenoid
solenoid
solenoid
(without shielding and without correctors)
(without shielding and without correctors)
(without shielding and without correctors)
(with shielding and with correctors)
(with shielding and with correctors)
with
with
with
with
with
2.5
5
10
3
5
mm
mm
mm
mm
mm
spacing;
spacing;
spacing;
spacing;
spacing;
the
the
the
the
the
rot
rot
rot
rot
rot
B
B
B
B
B
amplitude
amplitude
amplitude
amplitude
amplitude
is
is
is
is
is
0.0000022
0.0000230
0.0001600
0.0090000
0.0650000
3. The vertical emittance growth depends on the working point. The vertical emittance grows
slower for WP (2.26 ; 1.29) than for WP (2.28 ; 1.30).
Conclusion (1)
A. We found that the main bending magnets have strong sextupolar
components. We compensated for these by placing sextupoles (with
opposite polarity) around the main bending magnets.
B. We simulated several working points
C. We simulated off-momentum particles
Problems:
1. The vertical emittance is growing. Hypothesis: Numerical imprecision in the
electron cooler field (simulated with OPERA; which is a standard program in
CERN for simulating magnets).
2. Comparison of MADX solenoid field with a Mathematica solenoid (with hard
edges). Big differences were observed – is it because of fringe fields that are
automatically added in MADX. But in that case I would assume that the
fringe fields depends on the aperture of the solenoid and since the aperture
is not part of the MADX model, it seems there is an error ?
Conclusion (2)
Things to do:
1. Understand the problem with the vertical emittance growth
2. Possibly solve the problem with the vertical emittance growth by making an
extremely fine mesh in OPERA
3. Do the simulations with space charge, in order to see the effect of the electron
beam.
4. Documentation – how to use Mathematica tracking inside PTC. It is a fairly
simple and modular setup.
Nice things to do:
1. Fully compensate the main bending magnets. The compensation with the
sextupoles significantly reduced the coupling in the machine; however quite
some coupling remains, especially at large amplitudes. Is the remainder of the
coupling linear ?
2. Do more investigations on why a hard edge solenoid does not give the same
result as a MADX solenoid
3. Run ELENA as a transfer line. See what compensation gives the best closed orbit
solution
4. Decomposition of the magnetic field in the electron-cooler field into thin lenses