1. Nonlinear Integrable Optics with Space Charge in the IOTA Ring
C. Hall& N. Cook& D.L. Bruhwiler& S.D. Webb&
A. Romanov# E. Prebys# A. Valishev#
& #
presented at the HSC Section Meeting, CERN
(Hadron Synchrotron Collective effects)
24 April – Geneva
This work is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC

2. Motivation Hadron accelerators must advance the intensity frontier
High Energy Physics
neutrino production; creation of exotic nuclei
Basic Energy Sciences
spallation neutron sources
Nuclear Energy
subcritical nuclear reactors; transmutation of nuclear waste
Orders-of-magnitude reduction in beam loss is required
existing systems are intensity limited
require losses to be limited at ~W/m scales
new ideas are required to limit instabilities & beam halo
space charge, wakefields
IOTA – the Integrable Optics Test Accelerator
part of the Fermilab Accelerator Science and Technology (FAST) facility
proving ground for new ideas and advanced concepts w/ hadrons
Danilov & Nagaitsev, “Nonlinear accelerator lattices with one and two analytic invariants,” PRSTAB 13, (2010)

3. Diagnostic plots at location of bmin
Early Work ~2012
Worked with PyORBIT
2D dynamics with space charge
Linear, symmetric FoFo lattice
arcs are designed to have the transfer matrix of a thin, radially-focusing lens
thin, radially-focusing lens placed at entrance
phase advance across the drift (i.e. fractional tune) is 0.3
Primary result
core oscillations generate beam halo in the linear lattice
strong tune-spread suppresses oscillations; prevents halo
we call this “nonlinear decoherence”
Diagnostic plots at location of bmin
Webb, Bruhwiler, Nagaitsev, Danilov, Valishev, Abell, Shishlo, Danilov & Cary, “Effects of Nonlinear Decoherence on Halo Formation” (2013), arXiv:

4. For a linear lattice, core mismatch oscillations quickly drive test-particles into the halo

5. For integrable nonlinear magnetic fields (D&N 2010), nonlinear decoherence suppresses halo formation

6. Conclusions from early work (~2012)
Nonlinear decoherence suppresses SC effects
space charge tune shift of 0.1 or larger is tolerated
no concern about numerical noise
all oscillations rapidly suppressed in a few passes
distribution rapidly finds quasi-stationary Vlasov distribution
stability observed for 100,000 passes
However, the ring was strongly approximated
modeled as a thin, symmetric, linear focusing lens
Relation to recent work (2015 – 2017):
presence of a real ring is fundamentally different
single-turn phase advance is crucial
zero tune (i.e. multiple of p phase advance) in the ‘arc’
small deviations (i.e. tune shift errors ~0.01) damage invariance
incoherent space charge tune shift is a fundamental problem

7. A. Valishev, “IOTA – A Brief Parametric Profile,” presented at Focused Workshop on Scientific Opportunities in IOTA (Batavia, April, 2015);

8. A Hierarchy of Lattices – definition
Tier 0
drift of specified length (1.8 m for IOTA)
called the ‘insertion drift’
thin, symmetric, linear focusing lens with specified phase advance (0.3 for IOTA)
horiz. & vert. beta functions are equal, with waist in the center
Toy Lattice
Tier 0 lattice, plus the ‘nonlinear insert’ (aka the ‘elliptic magnet’)
completely fills the drift
magnet strength varies inversely with beta function of Tier 0 lattice
quadrupole component (horiz. focusing) breaks symmetry of the beta functions
Tier 1
thin lens is replaced by the quads and dipoles of IOTA (called ‘the arc’)
dynamics is assumed to be linear
phase advance in ‘the arc’ is an integer multiple of p (emulates thin lens)
dispersion in the ‘insertion drift’ is zero
Tier 2
sextupoles are added to Tier 1 for chromaticity correction
Tier 3
the ‘nonlinear insert’ is added to Tier 2
sextupoles may be turned off
S.D. Webb, D.L. Bruhwiler, A. Valishev, S. Nagaitsev and V. Danilov, “Chromatic and Dispersive Effects in Nonlinear Integrable Optics” (2015), arXiv:
discussed below

9. A Hierarchy of Lattices – variations
dp/p can be set to zero
we frequently zero the momentum spread to eliminate chromatic effects
the ‘small ring nonlinearities’ of IOTA can be included
these are predominantly 2nd-order in each element
in a sequence of elements, the effects can feed up to higher order
this breaks the attempted emulation of a thin lens
e.g. we observe nonlinear chromaticity and nonlinear dispersion
by default, these effects are ignored in our hierarchy of lattices
this is necessary for making theoretical progress
it is a good assumption for large rings
however, these nonlinearities must be included to fully understand IOTA
a ‘compensated’ arc design is required for space charge
space charge introduces a tune shift (reduction)
the phase advance of the arc must be increased to accommodate
a lattice redesign is required for each value of tune shift
incoherent tune shift due to nonlinear space charge is a problem
this difficulty is not present in the ‘Toy lattice’

10. Simulation Tools
Results presented here are based primarily on Synergia
developed by Fermilab’s accelerator simulation group
important contributions through code modifications and technical support
J. Amundsen, Q. Lu, A. Macridin, L. Michelotti, C.S. Park, E. Stern, T. Zolkin
Synergia has unique features that have enabled our success
control of linear, 2nd-order, … or fully nonlinear dynamics in each element
2D space charge algorithms: both gridded Poisson and ‘frozen’ models
We also use the code sixdsimulation
envelope design code with linear space charge
used to create the compensated lattice designs for use in Synergia
A. Romanov, G. Kafka, S. Nagaitsev, A. Valishev, “Lattice Correction Modeling for Fermilab IOTA Ring,” TUPRO058 IPAC 2014.

11. IOTA – no space charge – 1 million turns
Coasting proton bunch in IOTA w/ single NL ‘elliptic’ magnet
Generalized KV distribution with H0 = 9.47x10-6
ex = 0.03 mm-mrad (rms normalized), dp/p = 0
Resulting variation is consistent over 100k turns
𝜎H = 1.07%, 𝜎I = 2.56% for the particle shown below
Periodic variation which appears bounded 1M

12. The Potential of the Nonlinear ‘Elliptic’ Magnet
The integrable potential of D&N (2010) has the form:
Where the ‘elliptic’ coordinates are:

13. Matching to the Potential
e is a generalized emittance; ~H
A generalized KV distribution would be ideal
nearly linear space charge
mostly coherent tune depression
incoherent spread is small
Problems with KV Distribution:
it is prone to numerical instabilities, which manifest as rapid emittance increase
not a realistic distribution

14. Matching to the Potential
e is a generalized emittance; ~H
A generalized KV distribution would be ideal
nearly linear space charge
mostly coherent tune depression
incoherent spread is small
Problems with KV Distribution:
it is prone to numerical instabilities, which manifest as rapid emittance increase
not a realistic distribution
Solution: Use a waterbag distribution
Uniform distribution out to the max value: 0 < e < e0
Not prone to numerical instability
More ‘realistic’ distribution
Nonlinear space charge forces
incoherent tune spread

15. Matching to the Potential – Generating Particles
Best stability achieved if bunch distribution is matched into the elliptic potential:
Bunch Generation Routine:
Generate a trial particle inside some bounds
Check that tU ≤ e
Assign remaining momentum based on e - tU

16. Matching to the Potential – Waterbag in H
Recall that the generalized emittance, e ~ H

17. Matching to the Potential – X/Y Distribution
The x/y distributions do not assume the usually expected parabolic forms due to the fact that we are matching to the elliptic potential.

18. A theoretical point design for IOTA with finite current
Lattice:
IOTA v8.2 lattice with a single nonlinear insert
space charge tune depression is 0.03
phase advance increased by 0.03 to recover np requirement
nonlinear strength parameter t = (max is t = 0.5)
the Tier 1 lattice approximation is assumed
not consistent with IOTA, but pretty good for large proton RCS
no sextupoles; no RF
RFQ bunches stream longitudinally to fill the entire ring
injection needs to be modeled in 3D
could be done with Warp
2.5 MeV proton beam:
normalized emittance = 8 mm-mrad RMS
distribution assumed to be ‘generalized waterbag’
scraping of Gaussian tails will be required at injection
even so, the ‘real’ IOTA beam distribution will be different
dp/p < 0.1 RMS (to avoid issues with chromaticity)
electron cooling may be required
A practical point design for IOTA is needed  more work
this is a near-term priority, assuming funding is continued

19. Space Charge breaks Integrability
Singe-particle invariants are broken
Ensemble average is better behaved
With space charge:
‘Time independence’ of Danilov & Nagaitsev theory is broken
Both zero-current invariants now fluctuate significantly at 2 frequencies
Some ensemble properties still appear to be approximately maintained
we don’t yet understand how meaningful this may be
Bounded motion + nonlinear decoherence may be all that is required

20. Adjusting for Space Charge Tune Depression
Need an nπ phase advance around the ring between the exit and entrance of the nonlinear magnet with space charge.
nπ phase advance
NL insert
One then selects an optimal current to approximately zero the x,y phase advances.
Iterative matching is used to correct the lattice for any tune depression and preserve the phase advance.
Results shown here are optimized for a modest depression of dQSC= 0.03

21. Results with the Compensated Lattice
Phase advance around the ring is now corrected with the new compensated lattice.
This leads to much better behavior of the first invariant (the Hamiltonian).

22. Simulating Nonlinear Decoherence
We simulate a disturbance to the bunch as follows:
Create matched bunch as described above
Displace the bunch centroid horizontally
Tune spread from the nonlinear magnet should restore centroid.

23. Simulating Nonlinear Decoherence
Simulation without space charge:
Centroid motion rapidly damps.
Horizontal bound now set by new equipotential line.
No change in vertical bound

24. Simulating Nonlinear Decoherence
Simulation without space charge:
Centroid motion rapidly damps.
Horizontal bound now set by new equipotential line.
No change in vertical bound
What happens to the nonlinear decoherence when space charge is included?
See next slide…

25. Nonlinear Decoherence – with Space Charge
Space charge acts as a driving term against the nonlinear decoherence.

26. Decoherence for nonlinear insert at ‘half’ strength
For t=0.2 (max value is t=0.5), decoherence is too weak
some improvement seen at large emittance

27. Decoherence for nonlinear insert at ‘full’ strength
For t=0.4 (close to max of 0.5), decoherence is effective
phase space mixing speeds up as emittance is increased

28. Comparison with simple Octupole
Nonlinear Magnet
It is possible to achieve comparable rates of damping with an octupole
Octupole produces larger fluctuations
Requires high pole-tip magnetic fields

29. Tune Spreads – Octupole vs D&L Nonlinear Insert
Centroid motion removed from tune calculation

30. Summary & Future Work (1)
Theoretical understanding of IOTA is improving
Hamiltonian perturbation theory – Lie operator approach
simulations with Synergia & sixdsimulation
detailed understanding of phase advance(s), nonlinearities, space charge
other codes can be used: MAD-X, IMPACT-Z, MaryLIE/IMPACT
We’ve established a working point with space charge
the community can build on this result – much work to do!
J. Eldred et al. (Fermilab) are designing an RCS for PIP-II
Integrability lost due to space charge – what are the implications?
zero-current integrability may still be important, but if not:
carefully shaped octupoles (or combined quad./oct.) may be enough
symmetry of Danilov & Nagaitsev ring may be enough?
use a simple octupole?
dynamic aperture is the merit function.

31. Summary & Future Work (2)
Symplectic space charge solvers are essential
nonsymplectic noise is injected into the dynamics at every step
simple ‘frozen’ models (e.g. linear) are typically symplectic
the force must come from the gradient of a potential, to machine precision
S. Webb derived a general approach:
O. Boine-Frankenheim tested a prototype tracking code implementation, in consultation with S. Webb (discussion begins on slide #12):
Impedance-driven instabilities may be a better target
insights from S. Nagaitsev:
they have less effect on phase advance in the arcs
incoherent tune shifts are small
hence, the dynamics may remain integrable
A. Macridin et al. (Fermilab) are simulating impedance effects in IOTA
S.D. Webb, “A spectral canonical electrostatic algorithm,” Plasma Phys. Contr. Fusion 58, (2016);
O. Boine-Frankenheim, “PIC solvers for Intense Beams in Synchrotrons,” presented at High Brightness (2016);

32. The IOTA Collaboration (beam physics):
Acknowledgments
The IOTA Collaboration (beam physics):
G. Stancari# J. Eldred# S. Nagaitsev#
C. Mitchell% R. Ryne% D.T. Abell& R. Kishek
S. Chattopadhyay$,# S. Szustkowski$ F. Blampuy$
Supporting members of the Synergia development team:
E. Stern# J. Amundson# A. Macridin# C.S. Park# L. Michelotti# % $ 
& #
This work is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC