Superconducting Electron Linacs

1
Superconducting Electron Linacs

• Nick Walker
• DESY
• CAS Zeuthen 15-16 Sept. 2003

2
Whats in Store

• Brief history of superconducting RF
• Choice of frequency (SCRF for pedestrians)
• RF Cavity Basics (efficiency issues)
• Wakefields and Beam Dynamics
• Emittance preservation in electron linacs

• Will generally consider only high-power
  high-gradient linacs
• sc ee- linear collider
• sc X-Ray FEL

TESLA technology
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Status 1992 Before start of TESLA RD(and 30
years after the start)
L Lilje
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S.C. RF Livingston Plot
courtesy Hasan Padamsee, Cornell
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TESLA RD
2003 9 cell EP cavities
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TESLA RD
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The Linear Accelerator (LINAC)
standing wave cavity bunch sees fieldEz E0
sin(wtf )sin(kz) E0 sin(kzf )sin(kz)
c
c

• For electrons, life is easy since
• We will only consider relativistic electrons (v?
  c)we assume they have accelerated from the
  source by somebody else!
• Thus there is no longitudinal dynamics (e do not
  move long. relative to the other electrons)
• No space charge effects

8
SC RF
Unlike the DC case (superconducting magnets), the
surface resistance of a superconducting RF cavity
is not zero

• Two important parameters
• residual resistivity
• thermal conductivity

9
SC RF
Unlike the DC case (superconducting magnets), the
surface resistance of a superconducting RF cavity
is not zero

• Two important parameters
• residual resistivity Rres
• thermal conductivity

? surface area ? f-1
losses
Rs ? f 2 when RBCS gt Rres
f gt 3 GHz
fTESLA 1.3 GHz
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SC RF
Unlike the DC case (superconducting magnets), the
surface resistance of a superconducting RF cavity
is not zero
Higher the better!

• Two important parameters
• residual resistivity
• thermal conductivity

LiHe
heat flow
Nb
I
skin depth
RRR Residual Resistivity Ratio
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RF Cavity BasicsFigures of Merit

• RF power Pcav
• Shunt impedance rs
• Quality factor Q0
• R-over-Q

rs/Q0 is a constant for a given cavity
geometry independent of surface resistance
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Frequency Scaling
normal superconducting
normal superconducting
normal superconducting
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RF Cavity BasicsFill Time
From definition of Q0
Allow ringing cavity to decay(stored energy
dissipated in walls)
Combining gives eq. for Ucav
Assuming exponential solution(and that Q0 and w0
are constant)
Since
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RF Cavity BasicsFill Time
Characteristic charging time
time required to (dis)charge cavity voltage to
1/e of peak value. Often referred to as the
cavity fill time. True fill time for a pulsed
linac is defined slightly differently as we will
see.
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RF Cavity BasicsSome Numbers
fRF 1.3 GHz S.C. Nb (2K) Cu
Q0 5?109 2?104
R/Q 1 kW 1 kW
R0 5?1012 W 2?107 W
Pcav (5 MV) cw! 5 W 1.25 MW
Pcav (25 MV) cw! 125 W 31 MW
tfill 1.2 s 5 ms
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RF Cavity BasicsSome Numbers
fRF 1.3 GHz S.C. Nb (2K) Cu
Q0 5?109 2?104
R/Q 1 kW 1 kW
R0 5?1012 W 2?107 W
Pcav (5 MV) cw! 5 W 1.25 MW
Pcav (25 MV) cw! 125 W 31 MW
tfill 1.2 s 5 ms
Very high Q0 the great advantage of s.c. RF
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RF Cavity BasicsSome Numbers

• very small power loss in cavity walls
• all supplied power goes into accelerating the
  beam
• very high RF-to-beam transfer efficiency
• for AC power, must include cooling power

fRF 1.3 GHz S.C. Nb (2K) Cu
Q0 5?109 2?104
R/Q 1 kW 1 kW
R0 5?1012 W 2?107 W
Pcav (5 MV) cw! 5 W 1.25 MW
Pcav (25 MV) cw! 125 W 31 MW
tfill 1.2 s 5 ms
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RF Cavity BasicsSome Numbers

• for high-energy higher gradient linacs(X-FEL,
  LC), cw operation not an option due to load on
  cryogenics
• pulsed operation generally required
• numbers now represent peak power
• Pcav Ppkduty cycle
• (Cu linacs generally use very short pulses!)

fRF 1.3 GHz S.C. Nb (2K) Cu
Q0 5?109 2?104
R/Q 1 kW 1 kW
R0 5?1012 W 2?107 W
Pcav (5 MV) cw! 5 W 1.25 MW
Pcav (25 MV) cw! 125 W 31 MW
tfill 1.2 s 5 ms
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Cryogenic Power Requirements
Basic Thermodynamics Carnot Efficiency (Tcav
2.2K)
System efficiency typically 0.2-0.3 (latter for
large systems)
Thus total cooling efficiency is 0.14-0.2
Note this represents dynamic load, and depends
on Q0 and V Static load must also be included
(i.e. load at V 0).
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RF Cavity BasicsPower Coupling

• calculated fill time was1.2 seconds!
• this is time needed for field to decay to V/e for
  a closed cavity (i.e. only power loss to s.c.
  walls).

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RF Cavity BasicsPower Coupling

• calculated fill time was1.2 seconds!
• this is time needed for field to decay to V/e for
  a closed cavity (i.e. only power loss to s.c.
  walls).
• however, we need a hole (coupler) in the cavity
  to get the power in, and

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RF Cavity BasicsPower Coupling

• calculated fill time was1.2 seconds!
• this is time needed for field to decay to V/e for
  a closed cavity (i.e. only power loss to s.c.
  walls).
• however, we need a hole (coupler) in the cavity
  to get the power in, and
• this hole allows the energy in the cavity to leak
  out!

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RF Cavity Basics
cavity
coupler
circulator
Generator (Klystron)
matched load
Z0 characteristic impedance of transmission
line (waveguide)
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RF Cavity Basics
impedance mismatch
Generator (Klystron)
Klystron power Pfor sees matched impedance Z0
Reflected power Pref from coupler/cavity is
dumped in load
Conservation of energy
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Equivalent Circuit
Generator (Klystron)
cavity
coupler
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Equivalent Circuit
Only consider on resonance
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Equivalent Circuit
Only consider on resonance
We can transform the matched load impedance Z0
into the cavity circuit.
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Equivalent Circuit
define external Q
coupling constant
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Reflected and TransmittedRF Power
reflection coefficient(seen from generator)
from energy conservation
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Transient Behaviour
steady state cavity voltage
from before
think in terms of (travelling) microwaves
remember
steady-state result!
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Reflected Transient Power
time-dependent reflection coefficient
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Reflected Transient Power
after RF turned off
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RF On
note No beam!
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Reflected Power in Pulsed Operation
Example of square RF pulse with
critically coupled
under coupled
over coupled
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Accelerating Electrons

• Assume bunches are very short
• model current as a series of d functions
• Fourier component at w0 is 2I0
• assume on-crest acceleration (i.e. Ib is
  in-phase with Vcav)

t
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Accelerating Electrons
consider first steady state
whats Ig?
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Accelerating Electrons
steady state!
Consider power in cavity load R with Ib0
From equivalent circuit model (with Ib0)
NB Ig is actually twice the true generator
current
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Accelerating Electrons
substituting for Ig
introducing
beam loading parameter
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Accelerating Electrons
Now lets calculate the RF?beam efficiency
power fed to beam
hence
reflected power
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Accelerating Electrons
Note that if beam is off (K0)
previous result
For zero-beam loading case, we needed b 1 for
maximum power transfer (i.e. Pref 0) Now we
require
Hence for a fixed coupler (b), zero reflection
only achieved at one specific beam current.
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A Useful Expression for b
efficiency
voltage
can show
optimum
where
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Example TESLA
beam current
cavity parameters (at T 2K)
For optimal efficiency, Pref 0
From previous results
cw!
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Unloaded Voltage
matched condition
hence
for
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Pulsed Operation
From previous discussions
Allow cavity to charge for tfill such that
For TESLA example
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Pulse Operation
generatorvoltage
RF on

• After tfill, beam is introduced
• exponentials cancel and beam sees constant
  accelerating voltage Vacc 25 MV
• Power is reflected before and after pulse

cavity voltage
beam on
beaminduced voltage
t/ms
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Pulsed Efficiency
total efficiency must include tfill
for TESLA
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Quick Summary
cw efficiency for s.c. cavity
efficiency for pulsed linac
fill time

• Increase efficiency (reduce fill time)
• go to high I0 for given Vacc
• longer bunch trains (tbeam)

• some other constraints
• cyrogenic load
• modulator/klystron

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Lorentz-Force Detuning
In high gradient structures, E and B fields exert
stress on the cavity, causing it to deform.
detuning of cavity

• As a result
• cavity off resonance by relative amount D
  dw/w0
• equivalent circuit is now complex
• voltage phase shift wrt generator (and beam) by
• power is reflected

require
few Hz for TESLA
For TESLA 9 cell at 25 MV, Df 900 Hz !! (loaded
BW 500Hz) note causes transient behaviour
during RF pulse
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Lorentz Force Detuning cont.
recent tests on TESLA high-gradient cavity
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Lorentz Force Detuning cont.

• Three fixes
• mechanically stiffen cavity
• feed-forwarded (increase RF power during pulse)
• fast piezo tuners feedback

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Lorentz Force Detuning cont.

• Three fixes
• mechanically stiffen cavity
• feed-forwarded (increase RF power during pulse)
• fast piezo tuners feedback

stiffening ring
reduces effect by 1/2
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Lorentz Force Detuning cont.

• Three fixes
• mechanically stiffen cavity
• feed-forwarded (increase RF power during pulse)
• fast piezo tuners feedback

Low Level RF (LLRF) compensates. Mostly
feedforward (behaviour is repetitive) For TESLA,
1 klystron drives 36 cavities, thus vector sum
is corrected.
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Lorentz Force Detuning cont.

• Three fixes
• mechanically stiffen cavity
• feed-forwarded (increase RF power during pulse)
• fast piezo tuners feedback

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Lorentz Force Detuning cont.

• Three fixes
• mechanically stiffen cavity
• feed-forwarded (increase RF power during pulse)
• fast piezo tuners feedback

recent tests on TESLA high-gradient cavity
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Wakefields and Beam Dynamics

• bunches traversing cavities generate many RF
  modes.
• Excitation of fundamental (w0) mode we have
  already discussed (beam loading)
• higher-order (higher-frequency) modes (HOMs) can
  act back on the beam and adversely affect it.
• Separate into two time (frequency) domains
• long-range, bunch-to-bunch
• short-range, single bunch effects (head-tail
  effects)

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Wakefields the (other) SC RF Advantage

• the strength of the wakefield potential (W) is a
  strong function of the iris aperture a.
• Shunt impedance (rs) is also a function of a.
• To increase efficiency, Cu cavities tend to move
  towards smaller irises (higher rs).
• For S.C. cavities, since rs is extremely high
  anyway, we can make a larger without loosing
  efficiency.

significantly smaller wakefields
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Long Range Wakefields
Bunch current generates wake that decelerates
trailing bunches. Bunch current generates
transverse deflecting modes when bunches are not
on cavity axis Fields build up resonantly latter
bunches are kicked transversely ? multi- and
single-bunch beam break-up (MBBU, SBBU)
wakefield is the time-domain description of
impedance
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Transverse HOMs
wake is sum over modes
kn is the loss parameter (units V/pC/m2) for the
nth mode
Transverse kick of jth bunch after traversing one
cavity
where yi, qi, and Ei, are the offset wrt the
cavity axis, the charge and the energy of the ith
bunch respectively.
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Detuning
next bunch
no detuning
HOMs cane be randomly detuned by a small
amount. Over several cavities, wake decoheres.
abs. wake (V/pC/m)
with detuning
Effect of random 0.1 detuning(averaged over 36
cavities).
abs. wake (V/pC/m)
Still require HOM dampers
time (ns)
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Effect of Emittance
vertical beam offset along bunch train(nb 2920)
Multibunch emittance growth for cavities with
500mm RMS misalignment
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Single Bunch Effects

• Completely analogous to low-range wakes
• wake over a single bunch
• causality (relativistic bunch) head of bunch
  affects the tail
• Again must consider
• longitudinal effects energy spread along bunch
• transverse the emittance killer!
• For short-range wakes, tend to consider wake
  potentials (Greens functions) rather than modes

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Longitudinal Wake
Consider the TESLA wake potential
(r(z) long. charge dist.)
wake over bunch given by convolution
average energy loss
V/pC/m
head
tail
For TESLA LC
z/sz
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RMS Energy Spread
RF
accelerating field along bunch
DE/E (ppm)
wakeRF
Minimum energy spread along bunch achieved when
bunch rides ahead of crest on RF. Negative slope
of RF compensates wakefield. For TESLA LC,
minimum at about f 6º
z/sz
rms DE/E (ppm)
f (deg)
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RMS Energy Spread
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Transverse Single-Bunch Wakes
When bunch is offset wrt cavity axis, transverse
(dipole) wake is excited.
V/pC/m2
kick along bunch
tail
head
Note y(s z) describes a free betatron oscillatio
n along linac (FODO) lattice (as a function of s)
z/sz
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2 particle model
Effect of coherent betatron oscillation - head
resonantly drives the tail
head eom (Hills equation)
solution
tail eom
resonantly driven oscillator
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BNS Damping
If both macroparticles have an initial offset y0
then particle 1 undergoes a sinusoidal
oscillation, y1y0cos(kßs). What happens to
particle 2?
Qualitatively an additional oscillation
out-of-phase with the betatron term which grows
monotonically with s. How do we beat it? Higher
beam energy, stronger focusing, lower charge,
shorter bunches, or a damping technique
recommended by Balakin, Novokhatski, and Smirnov
(BNS Damping)
curtesy P. Tenenbaum (SLAC)
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BNS Damping
Imagine that the two macroparticles have
different betatron frequencies, represented by
different focusing constants kß1 and kß2
The second particle now acts like an undamped
oscillator driven off its resonant frequency by
the wakefield of the first. The difference in
trajectory between the two macroparticles is
given by
curtesy P. Tenenbaum (SLAC)
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BNS Damping
The wakefield can be locally cancelled (ie,
cancelled at all points down the linac) if
This condition is often known as autophasing.
It can be achieved by introducing an energy
difference between the head and tail of the
bunch. When the requirements of discrete
focusing (ie, FODO lattices) are included, the
autophasing RMS energy spread is given by
curtesy P. Tenenbaum (SLAC)
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Wakefields (alignment tolerances)
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Cavity Misalignments
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Wakefields and Beam Dynamics
The preservation of (RMS) Emittance!
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Emittance tuning in the Linac
Consider linear collider parameters

• DR produces tiny vertical emittances (gey
  20nm)
• LINAC must preserve this emittance!
• strong wakefields (structure misalignment)
• dispersion effects (quadrupole misalignment)
• Tolerances too tight to be achieved by surveyor
  during installation

? Need beam-based alignment
mma!
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Basics (linear optics)
thin-lens quad approximation Dy-KY
gij
Yj
Yi
yj
Ki
linear system just superimpose oscillations
caused by quad kicks.
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Introduce matrix notation
Original Equation
Defining Response Matrix Q
Hence beam offset becomes
G is lower diagonal
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Dispersive Emittance Growth
Consider effects of finite energy spread in beam
dRMS
chromatic response matrix
dispersivekicks
latticechromaticity
dispersive orbit
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What do we measure?
BPM readings contain additional
errors boffset static offsets of monitors wrt
quad centres bnoise one-shot measurement noise
(resolution sRES)
random(can be averagedto zero)
fixed fromshot to shot
launch condition
In principle all BBA algorithms deal with boffset
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Scenario 1 Quad offsets, but BPMs aligned
quad mover dipole corrector

• Assuming
• a BPM adjacent to each quad
• a steerer at each quad

steerer
simply apply one to one steering to orbit
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Scenario 2 Quads aligned, BPMs offset
one-to-one correction BAD! Resulting orbit not
Dispersion Free ? emittance growth
Need to find a steering algorithm which
effectively puts BPMs on (some) reference line
real world scenario some mix of scenarios 1 and 2
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BBA

• Dispersion Free Steering (DFS)
• Find a set of steerer settings which minimise the
  dispersive orbit
• in practise, find solution that minimises
  difference orbit when energy is changed
• Energy change
• true energy change (adjust linac phase)
• scale quadrupole strengths
• Ballistic Alignment
• Turn off accelerator components in a given
  section, and use ballistic beam to define
  reference line
• measured BPM orbit immediately gives boffset wrt
  to this line

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DFS
Problem
Note taking difference orbit Dy removes boffset
Solution (trivial)
Unfortunately, not that easy because of noise
sources
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DFS example
mm
300mm randomquadrupole errors 20 DE/E No BPM
noise No beam jitter
mm
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DFS example
Simple solve
In the absence of errors, works exactly
original quad errors fitter quad errors

• Resulting orbit is flat
• Dispersion Free
• (perfect BBA)

Now add 1mm random BPM noise to measured
difference orbit
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DFS example
Simple solve
original quad errors fitter quad errors
Fit is ill-conditioned!
85
DFS example
mm
Solution is still Dispersion Free but several mm
off axis!
mm
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DFS Problems

• Fit is ill-conditioned
• with BPM noise DF orbits have very large
  unrealistic amplitudes.
• Need to constrain the absolute orbit

minimise

• Sensitive to initial launch conditions (steering,
  beam jitter)
• need to be fitted out or averaged away

87
DFS example
Minimise
absolute orbit now constrained
remember sres 1mm soffset 300mm
original quad errors fitter quad errors
88
DFS example
mm
Solutions much better behaved! ! Wakefields !
mm
Orbit not quite Dispersion Free, but very close
89
DFS practicalities

• Need to align linac in sections (bins), generally
  overlapping.
• Changing energy by 20
• quad scaling only measures dispersive kicks from
  quads. Other sources ignored (not measured)
• Changing energy upstream of section using RF
  better, but beware of RF steering (see initial
  launch)
• dealing with energy mismatched beam may cause
  problems in practise (apertures)
• Initial launch conditions still a problem
• coherent b-oscillation looks like dispersion to
  algorithm.
• can be random jitter, or RF steering when energy
  is changed.
• need good resolution BPMs to fit out the initial
  conditions.
• Sensitive to model errors (M)

90
Ballistic Alignment

• Turn of all components in section to be aligned
  magnets, and RF
• use ballistic beam to define straight reference
  line (BPM offsets)
• Linearly adjust BPM readings to arbitrarily zero
  last BPM
• restore components, steer beam to adjusted
  ballistic line

62
91
Ballistic Alignment
62
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Ballistic Alignment Problems

• Controlling the downstream beam during the
  ballistic measurement
• large beta-beat
• large coherent oscillation
• Need to maintain energy match
• scale downstream lattice while RF in ballistic
  section is off
• use feedback to keep downstream orbit under
  control

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TESLA LLRF