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CERN-ACC-SLIDES-2014-0066

HiLumi LHC

FP7 High Luminosity Large Hadron Collider Design Study

Presentation DELPHI: an Analytic Vlasov Solver for Impedance-Driven Modes

Mounet, N (CERN)

07 May 2014

The HiLumi LHC Design Study is included in the High Luminosity LHC project and is partly funded by the European Commission within the Framework Programme 7 Capacities Specific Programme, Grant Agreement 284404. This work is part of HiLumi LHC Work Package 2: Accelerator Physics & Performance.

The electronic version of this HiLumi LHC Publication is available via the HiLumi LHC web site <http://hilumilhc.web.cern.ch> or on the CERN Document Server at the following URL: <http://cds.cern.ch/search?p=CERN-ACC-SLIDES-2014-0066>

CERN-ACC-SLIDES-2014-0066

DELPHI: an analytic Vlasov solver for impedance-driven modes

• N. Mounet

Acknowledgments: X. Buffat, A. Burov, K. Li, E. Métral, G. Rumolo,

• B. Salvant, S. White

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 2

 Introduction & motivation  Getting to Sacherer integral equation  How to solve Sacherer equation: Discrete Expansion over

Laguerre Polynomials and HeadtaIl modes

 Landau damping  Some benchmarks  What else could be done ?

Outline

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 3

 For given machine and beam parameters, we often need to

evaluate transverse beam stability w.r.t. impedance effects.

 Also important to assess the efficiency of stabilization

techniques: damper, non-linearities (Landau damping).

 Time domain macroparticles tracking is often a very good tool

and can give a complete vision, BUT:

➢ Too slow for certain large scale problems, e.g. typical LHC

problem (~1400/2800 bunches with transverse damper → need fine modeling of intrabunch motion and more than 100000 turns to see an instability...),

➢ Too slow to perform large parameter space scans (chromaticity,

non-linearities, intensity, damper gain, etc.),

➢ Very difficult to be sure the beam is stable in a certain

configuration.

Introduction & motivation

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 4



Another possibility: Vlasov solver in ”mode domain”

→ solver that looks for all modes that can develop, among which one can easily spot the most critical (i.e. unstable).

 Idea is not new (non exhaustive list): ➢ Laclare formalism [J. L. Laclare, CERN-87-03-V-1, p. 264], ➢ MOSES [Y. Chin, CERN/SPS/85-2 & CERN/LEP-TH/88-05], ➢ NHTVS [A. Burov, Phys. Rev. ST AB 17, 021007 (2014)].



Another idea: write transfer map with azimuthal and radial mesh of the bunch(es) (no macroparticles) [V. V. Danilov & E. A. Perevedentsev,

• Nucl. Instr. Meth. in Phys. Res. A 391 (1997) pp. 77-92], recently extended &

improved by S. White and X. Buffat. → all linear collective dynamics represented by a matrix; then eigenvalues = modes.

Introduction & motivation

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 5

 All current Vlasov solvers have limitations:

➢ Laclare cannot solve problems that involve too many betatron

sidebands (give size of matrix to diagonalize),

➢ MOSES limited to single-bunch, resonator models, w/o damper,

➢ NHTVS does not automatically check convergence, relies

• n airbag rings for radial discretization & treats Landau

damping in the framework of stability diagram theory (approximation). → Can we do better ?

Introduction & motivation

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 6

 Outline: ➢ Vlasov equation ➢ Hamiltonian ➢ Perturbative approach adopted ➢ Impedance term

→ Sacherer integral equation for transverse modes.

 We follow here an approach largely inspired from A. W. Chao,

Physics of Collective Beam Instabilities in High Energy Accelerators, John Wiley & Sons (1993), chap. 6.

 Note: here, unlike Chao we use ”engineer” convention for the

Fourier transform → e jωt (unstable modes have imag. part<0). Also, SI units (c.g.s in Chao), and notations often different.

Getting to Sacherer integral equation

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 7

 Vlasov equation expresses that the local phase space density

does not change when one follows the flow of particles.

 In other words: local phase space area is conserved in time:  Assumptions:

➢ conservative & deterministic system (governed by Hamiltonian) – no

damping or diffusion from external sources (no synchrotron radiation),

➢ external forces (no discrete internal force or collision).

→ impedance seen as a collective field from ensemble of particles.

Vlasov equation

[A. A. Vlasov, J. Phys. USSR 9, 25 (1945)]

Courtesy A. W. Chao

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 8

 Simplest expression: with ψ the general 6D phase space

distribution density (and t the time),

 In our case: ➢ independent variable chosen as s=v t (longitudinal position

along accelerator orbit),

➢ particle coordinates (4D – no x/y coupling):

• transverse: (y, py ) ⇔ (Jy , θy ) (action/angle)
• longitudinal: (z, δ)

→

Vlasov equation

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 9

 Simplest expression: with ψ the general 6D phase space

distribution density (and t the time),

 In our case: ➢ independent variable chosen as s=v t (longitudinal position

along accelerator orbit),

➢ particle coordinates (4D – no x/y coupling):

• transverse: (y, py ) ⇔ (Jy , θy ) (action/angle)
• longitudinal: (z, δ)

→

Vlasov equation

How to write them ?

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 10

 In the presence of a dipolar vertical impedance resulting in a

force Fy(z,s): with and

Hamiltonian

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 11

 In the presence of a dipolar vertical impedance resulting in a

force Fy(z,s): with and

Hamiltonian

unperturbed tune chromaticity machine radius slippage factor synchrotron freq. total energy velocity=βc

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 12

 In the presence of a dipolar vertical impedance resulting in a

force Fy(z,s): with and

Hamiltonian

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 13

 In the presence of a dipolar vertical impedance resulting in a

force Fy(z,s): with and

Hamiltonian

transverse part longitudinal part (linear) dipolar wake fields

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 14

 In the presence of a dipolar vertical impedance resulting in a

force Fy(z,s): with and

Hamiltonian

transverse part longitudinal part (linear) dipolar wake fields unperturbed !

→ important assumption : invariant (and action-angle variables) stay as in linear case...

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 15

 … will then give derivatives of Jy, θy, z and δ w.r.t. s:

Hamilton's equations...

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 16

 … will then give derivatives of Jy, θy, z and δ w.r.t. s:

Hamilton's equations...

Neglected (not even mentioned in Chao's book) Neglected (from

Chao: OK when far from synchro-betatron resonances & small transverse beam size)

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 17

 Equation remains quite complicated: partial differential eq. for

distribution function ψ (s, Jy, θy, z, δ):

 To simplify the problem: ➢ Assume a mode is developping in the bunch along the

revolutions, with a certain (complex) frequency Ω=Qcω0,

➢ Assume we stay close to the stationary unperturbed

distribution ψ0, function of invariants Jy and → perturbation formalism:

How to solve Vlasov equation ?

r=√ z

2+ η 2v 2δ 2

ωs

2

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 18

 Equation remains quite complicated: partial differential eq. for

distribution function ψ (s, Jy, θy, z, δ):

 To simplify the problem: ➢ Assume a mode is developping in the bunch along the

revolutions, with a certain (complex) frequency Ω=Qcω0,

➢ Assume we stay close to the stationary unperturbed

distribution ψ0, function of invariants Jy and → perturbation formalism:

How to solve Vlasov equation ?

r= z

2 2v 2 2

s

2

∆ψ1: self- consistent perturbation to be found

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 19

 Use polar coordinates in longitudinal:  After some algebra, neglecting second order terms

proportional to ∆ψ1 Fy (wake field force assumed to be small):

Rewriting Vlasov equation

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 20

 Use polar coordinates in longitudinal:  After some algebra, neglecting second order terms

proportional to ∆ψ1 Fy (wake field force assumed to be small):

Rewriting Vlasov equation

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 21

 Writing f1 as a Fourier series

we can show that all f1

k are zero except for k=-1 (this is exact

except for k=1 for which it relies on |Qc-Qy|<<|Qc+Qy| )

→

 For g1 decomposition is more subtle:

The trick is to find appropriate decompositions...

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 22

 Writing f1 as a Fourier series

we can show that all f1

k are zero except for k=-1 (this is exact

except for k=1 for which it relies on |Qc-Qy|<<|Qc+Qy| )

→

 For g1 decomposition is more subtle:

The trick is to find appropriate decompositions...

→ azimuthal modes decomposition

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 23

 Writing f1 as a Fourier series

we can show that all f1

k are zero except for k=-1 (this is exact

except for k=1 for which it relies on |Qc-Qy|<<|Qc+Qy| )

→

 For g1 decomposition is more subtle:

The trick is to find appropriate decompositions...

→ azimuthal modes decomposition

to cancel some term in Vlasov eq.

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 24

 After such decompositions, Vlasov eq. now looks like

The trick is to find appropriate decompositions...

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 25

 After such decompositions, Vlasov eq. now looks like

The trick is to find appropriate decompositions...

must be constant w.r.t Jy → → dipole mode

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 26

 After such decompositions, Vlasov eq. now looks like

The trick is to find appropriate decompositions...

must be constant w.r.t Jy → → dipole mode

 Next step is to evaluate Fy: ➢ written initially as a 4D integral (convolution of the wake in

z, weighted by total distribution function):

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 27

 After such decompositions, Vlasov eq. now looks like

The trick is to find appropriate decompositions...

must be constant w.r.t Jy → → dipole mode

 Next step is to evaluate Fy: ➢ written initially as a 4D integral (convolution of the wake in

z, weighted by total distribution function):

multiturn sum (importance of self-consistency)

distribution wake fo

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 28

 Using the fact that the unperturbed distribution has no dipole

moment, and the previous decompositions:

How to write the wake fields force

S(z)

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 29

 Using the fact that the unperturbed distribution has no dipole

moment, and the previous decompositions:

How to write the wake fields force

 After some tricks we get for an impedance (details in Chao):

and for an ideal damper (constant imag. wake, no multiturn):

S(z)

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 30

 Using the fact that the unperturbed distribution has no dipole

moment, and the previous decompositions:

How to write the wake fields force

 After some tricks we get for an impedance (details in Chao):

and for an ideal damper (constant imag. wake, no multiturn):

S(z)

Q'y / (η R)

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 31

 Using the fact that the unperturbed distribution has no dipole

moment, and the previous decompositions:

How to write the wake fields force

S(z)

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 32

 Using the fact that the unperturbed distribution has no dipole

moment, and the previous decompositions:

How to write the wake fields force

 After some tricks we get for an impedance (details in Chao):

and for an ideal damper (constant imag. wake, no multiturn):

S(z)

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 33

 Using the fact that the unperturbed distribution has no dipole

moment, and the previous decompositions:

How to write the wake fields force

 After some tricks we get for an impedance (details in Chao):

and for an ideal damper (constant imag. wake, no multiturn):

S(z)

Q'y ω0 / η

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 34

 Finally, combining the 2 previous slides, integrating over φ,

defining τ = r/v (maximum long. amplitude in seconds), neglecting Qc Q ‑

y0 in the impedance and Bessel functions, and generalizing

to M equidistant bunches of intensity per bunch N with the usual assumption they all oscillate in the same way:

Sacherer integral equation – with damper

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 35

 Finally, combining the 2 previous slides, integrating over φ,

defining τ = r/v (maximum long. amplitude in seconds), neglecting Qc Q ‑

y0 in the impedance and Bessel functions, and generalizing

to M equidistant bunches of intensity per bunch N with the usual assumption they all oscillate in the same way:

Sacherer integral equation – with damper

no damping turns tune fractional part

coupled-bunch mode

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 36

 Finally, combining the 2 previous slides, integrating over φ,

defining τ = r/v (maximum long. amplitude in seconds), neglecting Qc Q ‑

y0 in the impedance and Bessel functions, and generalizing

to M equidistant bunches of intensity per bunch N with the usual assumption they all oscillate in the same way:

Sacherer integral equation – with damper

no damping turns

damper part

tune fractional part

coupled-bunch mode

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 37

 Finally, combining the 2 previous slides, integrating over φ,

defining τ = r/v (maximum long. amplitude in seconds), neglecting Qc Q ‑

y0 in the impedance and Bessel functions, and generalizing

to M equidistant bunches of intensity per bunch N with the usual assumption they all oscillate in the same way:

Sacherer integral equation – with damper

no damping turns

damper part impedance part

tune fractional part

coupled-bunch mode

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 38



Using a decomposition over Laguerre polynomials of the radial functions (idea from Besnier 1974, used then by Y. Chin in code MOSES – 1985):

How are we going to solve this ?

→ in principle any long. distribution can be put in.

Laguerre polynomial

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 39



Using a decomposition over Laguerre polynomials of the radial functions (idea from Besnier 1974, used then by Y. Chin in code MOSES – 1985):

How are we going to solve this ?



Then the following integrals can be computed analytically: → in principle any long. distribution can be put in. → can also play with parameters a & b → exponentials make impedance sum convergence easy.

Laguerre polynomial

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 40

 In the end Sacherer integral equation can be set as an

eigenvalue problem:

 In DELPHI, convergence is automatically checked with respect

to the matrix size (no radial & azimuthal modes) for every single calculation.

 Matrix can be computed only once for a full set of intensities,

damper gain or phase, and/or Qs. → such parameter scans can be done quite fast.

Final eigenvalue problem

impedance and damper matrix

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 41

 To include Landau damping, we simply need to replace the

tune by

 Then, assuming the transverse invariant stays ~ the same, the

transverse part of the perturbation becomes:

 And the expression of the force from dipolar wake fields

Landau damping

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 42

 We define the dispersion integral as

which can be computed analytically for many transverse distributions (Gaussian, parabolic, and others).

 Then the equation becomes:

This is a non-linear equation of the coherent (complex) tune shift Qc, which can be solved numerically.

Landau damping

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 43

Benchmarks



DELPHI vs MOSES, for single-bunch TMCI without damper (LEP RF cavities modelled as a broadband resonator):

Real part, Q'=0

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 44

Benchmarks



DELPHI vs MOSES, for single-bunch TMCI without damper (LEP RF cavities modelled as a broadband resonator):

Imag. part, Q'=0

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 45

Benchmarks



DELPHI vs MOSES, single-bunch without damper (LEP RF cavities modeled as a broadband resonator):

• Imag. part,

Q'=22

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 46

Benchmarks



DELPHI vs Karliner-Popov, single-bunch with damper (VEPP-4, broadband resonator):

Real, part, Q'=0

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 47

Benchmarks



DELPHI vs Karliner-Popov, single-bunch with damper (VEPP-4, broadband resonator):

• Imag. part,

Q'=0

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 48

Benchmarks



DELPHI vs Karliner-Popov and HEADTAIL (macroparticle simulation code –

• G. Rumolo et al), single-bunch with damper (VEPP-4, broadband resonator):

Imag, part, Q'=-7.5 DELPHI is closer to HEADTAIL. Karliner-Popov is more stable → due to their non flat damper ?

(we cannot check because Karliner-P damper parameters are not provided).

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 49



Transverse feedback:

➢

First idea: reactive feedback (prevent mode 0 to shift down and couple with mode -1) → not more than 5-10 % increase in threshold, despite several attemps and models developped [Danilov-Perevedentsev 1993, Sabbi 1996, Brandt et al 1995],

➢

Another idea: resistive feedback, first found ineffective [Ruth 1983], tried at LEP but never used in operation. Recently (2005) thought to be a good option by Karliner- Popov with a possible increase by factor ~5 of TMCI threshold → can we confirm ?

Re-analysing LEP TMCI



Impedance model: two broad-band resonators (RF cavities + bellows), the rest is relatively small (<10%)

[G. Sabbi, 1995].

→ experimental tune shifts and TMCI threshold (from simple formula) well reproduced, → threshold slightly less than 1mA.

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 50

LEP



LEP without damper (typical LEP2 parameters)

• Imag. part, Q'=0

Note: we had to change the bunch length (1.3cm instead

• f 1.8cm) to match

Karliner-Popov's result.

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 51

LEP



LEP with resistive damper (typical LEP2 parameters)

• Imag. part,

Q'=-22 Again, we see that Karliner- Popov model gives more stability than DELPHI → we cannot

reproduce their result.

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 52

LEP: stability analysis with resistive damper



Instability threshold vs. Q' and damper gain (up to 10 turns) with DELPHI: Essentially, one cannot do better than the natural (i.e. without damper) TMCI threshold.

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 53

LEP: stability analysis with reactive damper



Instability threshold vs. Q' and damper gain (up to 10 turns) with DELPHI: We can do a little better than the ”natural” TMCI. → seems to match (qualitatively) LEP

• bservations.

The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014 54

Conclusions and future possible work



Developped a new code (DELPHI) to study stability in mode-coupling conditions, with transverse damper, in multibunch, for any longitudinal distribution, and including transverse Landau damping (beyond stability diagram approximation). Benchmarks done (vs. MOSES, Karliner & Popov, HEADTAIL), many more to be done.



As an example, LEP experimental results (relative ineffectiveness of transverse flat damper – being reactive or resistive) qualitatively obtained.



In the future, all kinds of longitudinal non-linearities could be included, but with probably some difficulties:

➢ non-linear bucket, ➢ quadrupolar wakes, ➢ second order chromaticity.