Thin-lens tracking through the Combined-Function Magnets in the PS - - PowerPoint PPT Presentation

Thin-lens tracking through the Combined-Function Magnets in the PS

Malte Titze

April 27, 2016

Introduction Symplecticity Results & Conclusion References

Outline

1 Introduction 2 Symplecticity 3 Results & Conclusion

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 2 / 15

Introduction Symplecticity Results & Conclusion References

Introduction

• Topic: Accelerator

physics; Space Charge (SC) effects (= interaction between charged particles).

• LHC Injector Upgrade

(LIU): Deliver twice higher brightness beams compared to today’s nominal values. SC effects play an increasing role [Bartosik et al. (2016)].

• SC effects are larger at low energies (time dilation @ high energies)
• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 3 / 15

Introduction Symplecticity Results & Conclusion References

Current Situation

• Usually the tracking of the individual particles and the simulation
• f their interaction goes in an alternating fashion (i.e. tracking,

interaction, tracking, interaction, ...).

• In particular, there are so-called ’Analytical’ or ’Frozen’ SC models

(using analytical solutions of the Poisson equation) and ’Self Consistent’ PIC Codes (solving the Poisson equations on a grid; PIC = Particle In Cell).

• The Frozen SC model we currently use at CERN is based on the

widely used tracking program MAD-X.

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 4 / 15

Introduction Symplecticity Results & Conclusion References

Current Situation p. 2

• Recall: Simulate SC effects at CERN in with the Frozen SC model

requies in particular tracking with MAD-X through the entire complex of preaccelerators, especially the PSB, PS and the SPS.

• The PS is somewhat special in that it includes 100 so-called

Combined-Function Magnets (CFMs) which consist of the superposition of bending magnets with multipoles of higher order.

Figure 1: CFM in the PS, courtesy by Schoerling (2014)

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 5 / 15

Introduction Symplecticity Results & Conclusion References

Purpose of these magnets?

• Bending magnets and multipoles are used to keep particles on

track.

• Like marbles in a larger

tracking pipe, the particles generally perform oscillations around the reference trajectory.

Figure 2: Marble in a pipe with offset

• Tune: phase advance of these oscillations during one revolution in

the ring.

• Chromaticity: dependency of tune w.r.t. the longitudinal

momentum relative to the reference particle.

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 6 / 15

Introduction Symplecticity Results & Conclusion References

Purpose of these Magnets? p. 2

• It turns out that many aspects of beam physics are similar to
• ptics. In particular, one can describe most focusing / defocussing

elements as thin lenses.

• A thin lens description usually leads to fast tracking algorithms.
• For ’thick’ elements like

bending magnets, one is looking for a corresponding decomposition into several thin slices.

Figure 3: Schematic view of the bending situation on the dark side of the moon ...

• CFM = all in one = save space for other instruments. However

that makes a thin-lens description not easy. The answer requires a small excursion.

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 7 / 15

Introduction Symplecticity Results & Conclusion References

Excursion: Symplecticity

Hamilton Equations ˙ z = J(dH(z, s))tr (1) with z := (q, p)tr and J =

• 1

−1

• 2n×2n

. If z0 is a solution to (1) and M is a differentiable map on phase space, leaving H invariant, then z1 = M(z0) is another solution if M′(z0) ˙ z0 = ˙ z1

!

= J(dH(z1, s))tr = J(dH(z0, s)(M′(z0))−1)tr = J[(M′(z0))−1]tr(dH(z0, s))tr = J[(M′(z0))tr]−1(−J) ˙ z0, consequently1: Symplecticity Condition M′(z0)trJM′(z0) = J

1with a grain of salt ...

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 8 / 15

Introduction Symplecticity Results & Conclusion References

Symplectic vs. non-symplectic integrators

Simulating “zi = M(zi−1)”

(a) explicit euler

(b) implicit euler

(c) midpoint

(d) gauss-legendre2 Figure 4: Arnold cats under the effect of various Runge-Kutta solvers for the harmonic

• scillator with same step sizes.

Message: Numerical errors introduce artificial effects, like blow up of phase space in single particle tracking, which leads to wrong predictions over long periods.

2Symplectic; for a proof see Hairer et al. (2006).

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 9 / 15

Introduction Symplecticity Results & Conclusion References

... now back to CFMs

• Without going into the details, the Hamiltonian of CFMs in

comoving coordinates looks as follows H = pσ −(1+Kxx +Kyy)

• (1 + ˆ

η)2 − p2

x − p2 y + e

p0 As

• . (2)
• The mixture of x and px etc. makes life more complicated:

Hamiltons equations (1) can be written as ˙ z = −{H, z} = − : H : z and its (symplectic) solution is therefore z = exp(−s : H :)z0.

• (2) ⇒ H = D + K, where D is the drift term in curvilinear

coordinates and K the kick term involving the As potential. Since D and K do not commute → BCH formula necessary for a drift-kick decomposition (remember: ’Beam optics’ picture).

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 10 / 15

Introduction Symplecticity Results & Conclusion References

Solution

• Fortunately, there is an alternative way: One can still write down

an implicit symplectic solution for even the most general Hamiltonians [Titze (2015)] ¯ q = q −

K

• µ=0

1 (µ + 1)!(−∆s)µ+1(∂p(−H. + ∂s)µH)(q, ¯ p; sf ), ¯ p = p +

K

• µ=0

1 (µ + 1)!(−∆s)µ+1(∂q(−H. + ∂s)µH)(q, ¯ p; sf ).

• This was utilized by me in first order to obtain a suitable thin-lens

tracking map and implemented in MAD-X.

• Important: The vector potential of the CFMs in comoving

coordinates had to be determined as well.

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 11 / 15

Introduction Symplecticity Results & Conclusion References

Results

20 40 60 80 100 120 Slice number 0.005 0.004 0.003 0.002 0.001

Qx

6.075 20 40 60 80 100 120 Slice number 6.995 6.990 6.985 6.980 6.975 6.970 6.965

Qy

Preliminary

' '

Figure 5: Chromaticity convergence in the PS against analytic Lie-Transformation result.

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 12 / 15

Introduction Symplecticity Results & Conclusion References

Results p. 2

(x, px )

(y, py)

(x, px )

(y, py)

P r e l i m i n a r y

Figure 6: Tracking results of the unmodified code (Top left/right) and the corrected (symplectic) code (Bottom left/right).

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 13 / 15

Introduction Symplecticity Results & Conclusion References

Conclusion

• We have now a correct thin-lens description of CFMs available.

Moreover, we have a method at hand in which we can basically slice every finite-dimensional problem which can be described by a Hamiltonian in terms of implicit symplectic steps (For example: many body problems in astronomy, resonance driven bunch envelope equations etc.).

• This model is already implemented in a test version of MAD-X and

currently running first Frozen SC tracking simulations in the PS.

• A future official release of MAD-X is planned with includes the

corrected CFM map. Many thanks for your attention!

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 14 / 15

Introduction Symplecticity Results & Conclusion References

Further Reading I

• H. Bartosik, A. Oeftiger, F. Schmidt, and M. Titze. Space charge

studies with high intensity single bunch beams in the CERN SPS. In Proceedings of the IPAC, 2016. To be published.

• E. Hairer, C. Lubich, and G. Wanner. Geometric Numerical Integration:

Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics. Springer Berlin Heidelberg, 2006. ISBN 9783540306665.

• D. Schoerling. Prediction of the field distribution in CERN-PS
• magnets. In Proceedings of the IPAC, 2014.
• M. Titze. Symplectic maps for general hamiltonians. CERN HSC

Group Meeting, November 2015. URL https://espace.cern.ch/ be-dep/ABP/HSC/Meetings/slides_MT_02-11-15.pdf.

• H. Yoshida. Construction of higher order symplectic integrators.

Physical Letters A, 150(5), 1990.

• M. Titze (CERN / HU Berlin)

Gentner Day 27.04.2016 15 / 15