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J-M Lagniel ESS-Lund Jan 29, 2014Zero-current longitudinal beam dynamics(in linacs)

Jean-Michel Lagniel (CEA/GANIL)

Longitudinal focalization = nonlinear forces => separatrix, tune spreadAcceleration => damping of the phase oscillations.

The longitudinal beam dynamics is complex,even when the nonlinear space-charge forces are ignored.

The three different ways to study and understandthis zero-current longitudinal beam dynamicswill be presented and compared.

Page NI- EoM integration in field maps Mappings(Transit-Time-Factor)

II- Mappings 2nd order differential EoM(Smooth approximation)

III- Longitudinal beam dynamics without damping(Phase-space portraits @ Poincar surface of section)Smooth approximation EoM / Mapping / Integration in field map

IV- Longitudinal beam dynamics with damping (Basin of attraction, bifurcation diagrams)

V- Concluding remarks

ProgramJ-M Lagniel ESS-Lund Jan 29, 2014Page NNumerical integration (dz) of the EoM in field maps (, W) phase-spaceI- From EoM integration in field maps to mappings

J-M Lagniel ESS-Lund Jan 29, 2014Page NMapping i+1 = i + Wi+1 = Wi + W

=> Integration over one accelerating cell - cavityI- From EoM integration in field maps to mappings

(Panofsky 1951) The Transit-Time-Factor contains all the information on thefield map and speed + radial evolution over the accelerating cell / cavity

Without approximation

(z = 0)Ez mean valueMore complicated than the original only useful with approximationsJ-M Lagniel ESS-Lund Jan 29, 2014Page NI- From EoM integration in field maps to mappingsOdd function of z

Ez(r,z) = even function

+ constant speedover the cell

Constant speed and radius over the cell

Under this form, the TTF is the an component of the Ez(z) Fourier transformwith n = Lc / => n = h ( loss of information on the shape of Ez(z) ) The TTF of each particle is a functionof the particle mean radius and velocities (input values in practice)(not function of the particle radius and speed evolution over the cell)J-M Lagniel ESS-Lund Jan 29, 2014Page NI- From EoM integration in field maps to mappings

Allows to find analytical expressions of the TTF for particular field distributions

J-M Lagniel ESS-Lund Jan 29, 2014Page NI- From EoM integration in field maps to mappings

Using the approximated formula to evaluate the particle TTF we have found a practical way to build a mappingJ-M Lagniel ESS-Lund Jan 29, 2014Page NI- From EoM integration in field maps to mappings

Using an approximated formula to evaluate the particle TTF we have found a practical way to build a mappingNO ! This mapping is not (by far) symplectic (area preserving) when the TTF is calculated taking into account the particle mean speed and radius

A phase correctionmust be added to obtain a symplectic mapping (1st order)

Pierre Lapostolle et al1965 1975 (B.C. age).

J-M Lagniel ESS-Lund Jan 29, 2014Page NI- From EoM integration in field maps to mappings

The only way to produce a simple symplectic mappingis to consider the synchronous particle TTF for every particle

TTF analytical expression => neglect the evolution of the velocity in the cell

Simple symplectic mapping => neglect the effect of the particle velocity spread on the TTF(Phase and energy evolution with respect to the synchronous particle)(Mapping used for the comparison with the other methods)J-M Lagniel ESS-Lund Jan 29, 2014Page NII- From mappings to 2nd order differential EoM Smooth approximation considering the mapping without phase correction

Low amplitude oscillationsLarge amplitude oscillationsLong term behaviorJ-M Lagniel ESS-Lund Jan 29, 2014Page NII- From mappings to 2nd order differential EoM Error on the longitudinal phase advance per focusing periodinduced by the smooth approximation

Twiss matrix

MappingSmooth approximation

J-M Lagniel ESS-Lund Jan 29, 2014Page NThe 3 ways to study the longitudinal beam dynamics

Synchronous particle and oscillations around the synchronous particle

J-M Lagniel ESS-Lund Jan 29, 2014Page NIII- Longitudinal beam dynamics without dampingSmooth approximation Choice of s Beam size vs longitudinal aperture The temptation is high to increase the synchronous phase

High-power LINAC designers (and managers) must bringas much attention to the longitudinal beam size / longitudinal aperture ratioas they bring to the radial beam size / radial aperture ratio J-M Lagniel ESS-Lund Jan 29, 2014

-20 -15 Long. Acc. / 2Page NIII- Longitudinal beam dynamics without dampingSmooth approximation vs Mapping s = -90

J-M Lagniel ESS-Lund Jan 29, 2014Page NIII- Longitudinal beam dynamics without dampingMapping s = -90

0l* > 50

More and more resonances => resonance overlaps=> larger choatic area

82, 86, 90 / lattice => real phase advance value higher than the one given by the smooth approximation

J-M Lagniel ESS-Lund Jan 29, 2014Page NIII- Longitudinal beam dynamics without dampingAs 0l* increases the phase-space portraits plotted using the mapping show

more and more resonancesmore and more resonance overlapslarger and larger choatic areas longitudinal acceptance reduction[P. Bertrand, EPAC04]

Is it true or is it a spurious effect of the mapping ?

( periodic error = excitation of the resonances ? )

... If yes, why ?

Check making a direct integration of the EoM J-M Lagniel ESS-Lund Jan 29, 2014Page NIII- Longitudinal beam dynamics without dampingLongitudinal toy Direct integration of the EoM s = -90

TTFField map = First-harmonic-model

0l* EpicJ-M Lagniel ESS-Lund Jan 29, 2014h frfPage NIII- Longitudinal beam dynamics without dampingPhase-space portraits plotted using the Longitudinal ToyLc = L h = 1

Ez(z) = pure sinusoid(first-harmonic)

1/4 resonance not excited !!!!

Stable oscillations aroundthe inverted pendulum position

0l* = 80

0l* = 90

0l* = 95J-M Lagniel ESS-Lund Jan 29, 2014Page N

III- Longitudinal beam dynamics without dampingPhase-space portraits plotted using the Longitudinal ToyLc = L/4 h = 4

Ez(z) with harmonics > 1

The resonances are excited

Mapping 0l* = 500l* = 700l* = 90J-M Lagniel ESS-Lund Jan 29, 2014Page N

III- Longitudinal beam dynamics without dampingsummaryEoM @ smooth approximation Ez(z) = Constant => Resonances not excited ... but essential to understand the longitudinal beam dynamics PhysicsMapping Ez(z) = Dirac comb (period L) => FT[Ez(z)] = Dirac comb (1/L) All resonances excitedThe longitudinal acceptance is significantly reduced at high 0l* EoM using field maps Ez(z) = Field map => FT[Ez(z)] = some harmonics (1/L) Some resonances excited (need more work !)J-M Lagniel ESS-Lund Jan 29, 2014Page NIV- Longitudinal beam dynamics with damping

(, d/ds) planeAttractor = 0d/ds = 0Damped linear harmonic oscillator

Linac = under-damped regime (RFQ ?) Longitudinal phase advance J-M Lagniel ESS-Lund Jan 29, 2014Page NIV- Longitudinal beam dynamics with damping

Attractor = W axis W Damping = energy spread (, W) phase-plane

Phase-space area preservationif adiabatic evolution

J-M Lagniel ESS-Lund Jan 29, 2014Smooth approximationSmall amplitude oscillationsPage NIV- Longitudinal beam dynamics with damping

Smooth approximation (, d/ds) plane Attractor = (0, 0) s = -30

Acceptances+ separatrix K = 0Basin of attractionJ-M Lagniel ESS-Lund Jan 29, 2014Page NIV- Longitudinal beam dynamics with damping

Mapping

(, d/ds) planeAttractors = (0, 0) and the 1/4 resonance islands J-M Lagniel ESS-Lund Jan 29, 20140l* = 82Page NIV- Longitudinal beam dynamics with dampingMapping Basin of attraction (, d/ds) plane

0l* = 60 K = 0.02K = 0.100l* = 70 K = 0.01K = 0.10Attractors : (0, 0) (1/6 resonance) J-M Lagniel ESS-Lund Jan 29, 2014Page NIV- Longitudinal beam dynamics with dampingMapping (fractal) Basin of attraction (, d/ds) plane

K = 0.01K = 0.050l* = 82 K = 0.10K = 0.20Attractors(0, 0)(1/4 resonance) J-M Lagniel ESS-Lund Jan 29, 2014Page NIV- Longitudinal beam dynamics with dampingESS linac (2012)

K = 0.36 0.10 0.04 DTL 0.015 0.005 high energyJ-M Lagniel ESS-Lund Jan 29, 2014Page NIV- Longitudinal beam dynamics with damping

SPIRAL 2 superconducting linacK = 0.05 0.08 0.12 0.19 0.16 0.08 0.05J-M Lagniel ESS-Lund Jan 29, 2014Page NIV- Longitudinal beam dynamics with dampingJ-M Lagniel ESS-Lund Jan 29, 2014summaryDamping induced by the acceleration

Phase-width reduction and energy-spread growth The stable fix points of the resonance islandsact as main attractors at low damping rates The damping can annihilate the effect of the resonances Page NV- Five points to keep in mindWhen the Transit-Time-Factor is used(linac designs and optimizations, understanding of the basic physics) (i.e. as soon as a direct numerical integration of the EoM is abandoned !)the longitudinal beam dynamics must be computed in such a waythat the longitudinal motion in the (, W) phase planeremains symplectic (area preserving)

Option #1 = Use the synchronous-particle TTF for all the particles(but keep in mind the consequences of this approximation) Option #2 = Follow the work done at the B.C. age(A.C. age prefer numerical integrations !)J-M Lagniel ESS-Lund Jan 29, 2014

Page NV- Five points to keep in mindJ-M Lagniel ESS-Lund Jan 29, 2014The nonlinear character of the accelerating field induces aphase-advance spread (tune shift) which must be considered when the phase width is important (or halo)

This nonlinear character makes thelongitudinal beam dynamics much more complicated than the radial one !Space-charge induced nonlinearitieswill obviously complicate the situation !

Think phase advance evolution with amplitude

Page NV- Five points to keep in mind J-M Lagniel ESS-Lund Jan 29, 2014

The results obtained using the classical approximations(TTF & smooth approximation)are very useful to understand the longitudinal beam dynamics(including the large-amplitude motions and damping)

BUT

For longitudinal phase advances greater than 60/period,these approximations induce errors on the values of the parameters calculated using them (e.g. 0l * )

AND

Hide the resonances excited by the nonlinear accelerating field localized in the cavities = acceptance reduction

Page NV- Five points to keep in mindTo understand the longitudinal beam dynamics in linacsit is essential to take into account the damping induced by the accelerationwhen the damping rate is significant with respect to the period of the longitudinal oscillations.J-M Lagniel ESS-Lund Jan 29, 2014

k should be considered as an important parameterto analyze a linac design and understand its longitudinal beam dynamicsPage NV- Five points to keep in minds = ???J-M Lagniel ESS-Lund Jan 29, 2014

-20 -15 Longitudinal Acceptance / 2A systematic and well defined ruleto choose the synchronous phaseshould be defined taking into accountboth risk and project economy

Page NConcluding remarksJ-M Lagniel ESS-Lund Jan 29, 2014Hope you are now convinced thatthe zero-current longitudinal beam dynamics is complex !

At least more complex than what is taught inclassical Accelerator Books and Accelerator Schools

Several questions still open

Why no (nearly no) excitation of the resonances for Lc = L and h = 1 ?(numerical integration of the EoM)

How different Fourier spectra of the longitudinal repartition of Ez(z)act on the beam dynamics ? (series of multi-cell cavities)

Which second order differential equation can be used as modelto study, understand and predict the effect (resonances) of this longitudinal repartition ?

What happen when the space-charge forces are added ?Page N