Diapositive 1
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