SUMMARY OF THE FIRST PARTELECTROSTATIC ACCELERATORS WIDERÖE DTL
injaccinjRF VqTqVE cosˆcos RFnnL
RF CYLINDRICAL CAVITIES (PILLBOX-LIKE)
ΩP
VR
diss
acc ˆ 2
dissacc QP
Q
RV
ˆ
RF
F
Q
2/2 MODE
ALVAREZ
=0.05-0.5, fRF=50-400 MHz
nn VqE
Transit time factor
>0.5, fRF=0.3-3 GHz (or above)
nR
F
t
accacc eVtV
1ˆ
2
RFd
d
MODE
TW CAVITIES: IRIS LOADED STRUCTURES(electrons, β=1)
g
Fv
L
z
INPUTacc eEzE
z
INPUTF ePzP 2
L
INPUTacc
eEV
1
L
INPUTOUT ePP 2
2
ˆ 2
w
Pv
P
p
p
Er
Fg
F
diss
diss
acc
LINAC TECHNOLOGY
The cavities (and the related LINAC technology) can be of different material: copper for normal conducting (NC, both SW than TW) cavities; Niobium for superconducting cavities (SC, SW);
We can choose between NC or the SC technology depending on the required performancesin term of: accelerating gradient (MV/m); RF pulse length (how many bunches we can contemporary accelerate); Duty cycle (see next slide): pulsed operation (i.e. 10-100 Hz) or continuous wave (CW)
operation; Average beam current. …
ACCELERATING CAVITY TECHNOLOGY
Dissipated power intothe cavity walls isrelated to the surfacecurrents
Frequency [GHz]
NIOBIUM
Between copper and Niobium there is a factor 105-106
wallcavity
tansdiss dSHRP
densitypower
2
2
1
COPPER
RF STRUCTURE AND BEAM STRUCTURE: NC vs SCThe “beam structure” in a LINAC is directly related to the “RF structure”. There are two possible type of operations:
• CW (Continuous Wave) operation allow, in principle, to operate with a continuous (bunched) beam• PULSED operation there are RF pulses at a certain repetition rate (Duty Cycle (DC)=pulsed width/period)
SC structures allow operation at very high Duty Cycle (>1%) up to a CWoperation (DC=100%) (because of the extremely low dissipated power) withrelatively high gradient (>20 MV/m). This means that a continuous (bunched)beam can be accelerated.
NC structures can operate in pulsed mode at very low DC (10-2-10-1 %)(because of the higher dissipated power) with, in principle, larger peakaccelerating gradient(>30 MV/m). This means that one or few tens of bunchescan be, in general, accelerated. NB: NC structures can also operate in CW but atvery low gradient because of the dissipated power.
Bunch spacing
RF pulsesRF power
t
Amplitude 103-108 RF periods
time
PERFORMANCES NC vs SC: MAXIMUM Eacc
If properly cleaned and fabricated, NCcavities can quite easily reach relativelyhigh gradients (>40 MV/m) at rep. rate upto 100-200 Hz and pulse length of few µ.
Longer pulses or higher rep. rate can bereached but in this case the gradient has tobe reduced accordingly (MV/m).
The main limitation comes frombreakdown phenomena, whose physicsinterpretation and modelling is still understudy and is not yet completelyunderstood.
Conditioning is necessary to go in fulloperation
SC cavities also need to beconditioned
Their performance is usually analyzedby plotting the behavior of thequality factor as a function of theaccelerating field.
The ultimate gradient ( 50 MV/m) isgiven by the limitation due to thecritical magnetic field (150-180 mT) .
NC
SC
PARAMETERS SCALING WITH FREQUENCYWe can analyze how all parameters (r, Q) scale with frequency and what are the advantages or disadvantages in accelerate with low or high frequencies cavities.
parameter NC SC
Rs f1/2 f2
Q f-1/2 f-2
r f1/2 f-1
r/Q f
w// f2
w┴ f3
Wakefield intensity: related to BD issues
r/Q increases at high frequency
for NC structures also r increases and this push toadopt higher frequencies
for SC structures the power losses increases with f2
and, as a consequence, r scales with 1/f this push toadopt lower frequencies
On the other hand at very high frequencies (>10 GHz)power sources are less available
Beam interaction (wakefield) became more critical athigh frequency
Cavity fabrication at very high frequency requireshigher precision but, on the other hand, at lowfrequencies one needs more material and largermachines
short bunches are easier with higher f
SW SC: 500 MHz-1500 MHzTW NC: 3 GHz-6 GHzSW NC: 0.5 GHz-3 GHz
Compromisebetween several requirements
EXAMPLES: EUROPEAN XFEL
EXAMPLE: SWISSFEL LINAC (PSI)
Courtesy T. Garvey
LINAC: BASIC DEFINITION AND MAIN COMPONENTSLINAC (linear accelerator) is a system that allows to accelerate charged particles through a linear trajectoryby electromagnetic fields.
Particle source
Accelerated beam
Accelerating structures Focusing elements: quadrupoles and solenoids
Transverse dynamics of accelerated particles
Longitudinal dynamics of accelerated particles
LIN
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BvEqdt
pd chargeq
velocityv
massm
momentump
ACCELERATIONBENDING AND FOCUSING
Transverse DynamicsLongitudinal Dynamics
LORENTZ FORCE: ACCELERATION AND FOCUSINGParticles are accelerated through electric field and are bended and focalized through magnetic field.The basic equation that describe the acceleration/bending /focusing processes is the Lorentz Force.
To accelerate, we need a force in thedirection of motion
2nd term always perpendicular to motion => no energy gain
MAGNETIC QUADRUPOLEQuadrupoles are used to focalize the beam in the transverseplane. It is a 4 poles magnet:
B=0 in the center of the quadrupole
The B intensity increases linearly with the off-axisdisplacement.
If the quadrupole is focusing in one plane is defocusing in theother plane
N
N S
S
m
TG
xqvGF
yqvGF
xGB
yGB
x
y
y
x
gradient quadrupole
z
x
beam
y
x
y
Electromagnetic quadrupoles G <50-100 T/m
01.0@31
1@3001
m
MVFTF
m
MVFTF
vF
F
EB
EB
E
B
LINAC: BASIC DEFINITION AND MAIN COMPONENTSLINAC (linear accelerator) is a system that allows to accelerate charged particles through a linear trajectoryby electromagnetic fields.
Particle source
Accelerated beam
Accelerating structures Focusing elements: quadrupoles and solenoids
Transverse dynamics of accelerated particles
Longitudinal dynamics of accelerated particles
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SYNCHRONOUS PARTICLE/PHASE
Let’s assume that the “perfect” synchronism
condition is fulfilled for a phase s (called
synchronous phase). This means that a particle
(called synchronous particle) entering in a gap with
a phase s (s=RFts) with respect to the RF voltage
receive an energy gain (and a consequent change in
velocity) that allow entering in the subsequent gap
with the same phase s and so on.
for this particle the energy gain in each gap is:
Let us consider a SW linac structure made by acceleratinggaps (like in DTL) or cavities.
In each gap we have an accelerating field oscillating in timeand an integrated accelerating voltage (Vacc) still oscillating intime than can be expressed as:
TRF
sacc
sacc
sacc qV
V
VqE _
_
cosˆ
RF
RFRFT
f
2
2 Vacc
ts*s
Vacc_s
obviously both s and s* are synchronous phases.
cosˆ tVV RFaccacc
accV
PRINCIPLE OF PHASE STABILITY
Vacc
ts*s
Let us consider now the first synchronous phases (on the positive slope of the RF voltage). If weconsider another particle “near” to thesynchronous one that arrives later in the gap(t1>ts, 1>s), it will see an higher voltage, it willgain an higher energy and an higher velocity withrespect to the synchronous one. As a consequenceits time of flight to next gap will be shorter, partiallycompensating its initial delay.
Similarly if we consider another particle “near”to the synchronous one that arrives before in thegap (t1<ts, 1<s), it will see a smaller voltage, it willgain a smaller energy and a smaller velocity withrespect to the synchronous one. As a consequenceits time of flight to next gap will be longer,compensating the initial advantage.
On the contrary if we consider now thesynchronous particle at phase s
* and anotherparticle “near” to the synchronous one that arriveslater or before in the gap, it will receive an energygain that will increase further its distance form thesynchronous one
The choice of the synchronous phase in the positiveslope of the RF voltage provides longitudinal focusingof the beam: phase stability principle.
The synchronous phase on the negative slope ofthe RF voltage is, on the contrary, unstable
Relying on particle velocity variations, longitudinalfocusing does not work for fully relativistic beams(electrons). In this case acceleration “on crest” is moreconvenient.
12
(protons and ions or electrons at extremely low energy)
ENERGY-PHASE EQUATIONS (1/2)
In order to study the longitudinal dynamics in a LINAC, the following variables are used, which describe the generic particlephase (time of arrival) and energy with respect to the synchronous particle:
s
sRFs
EEw
tt
The energy gain per cell (one gap + tube in case of a DTL) of a generic particleand of a synchronous particle are:
saccacc
saccs
VqVqE
VqE
cosˆcosˆ
cosˆ
Energy of the synchronous particle at a certain position along the linac
Energy of a generic particle at a certain position along the linac
Arrival time (phase) of the synchronous particle at a certain gap (or cavity)
Arrival time (phase) of a generic particle at a certain gap (or cavity)
cosˆ tVV RFaccacc
ssaccs VqEEw coscosˆ
L (accelerating cell length)
ssaccEqL
w coscosˆ
z
accacc EL
V ˆˆ
subtracting
Dividing by theaccelerating cell lengthL and assuming that:
Average acceleratingfield over the cell (i.e.average acceleratinggradient)
Approximating
dz
dw
L
w
ssaccEqdz
dw coscosˆ
(protons and ions or electrons at extremely low energy)
sRF tt
On the other hand we have that the phase variation per cell of a generic particle and of a synchronous particle are:
t
t
RF
sRFs
wcE
MATvvL
t
L
t
L ss
RF
s
RFs
RF 33
0
11
v, vs are the average particles velocities
This system of coupled (non linear)differential equations describe the motionof a non synchronous particles in thelongitudinal plane with respect to thesynchronous one.
t is basically the time offlight between twoaccelerating cells
subtracting
Dividing by theaccelerating celllength L
33
0
332
32
22
2
11 that grememberin 11
ss
RF
ss
RF
s
RF
s
RF
s
RF
s
ss
s
sRF
s
RFE
w
E
cccdd-β
cv
v
vvvvvv
vv
vv
vv
MAT
dz
d
L
Approximating
wcEdz
d
ss
RF
33
0
ssaccEqdz
dw coscosˆ
ENERGY-PHASE EQUATIONS (2/2)(protons and ions or electrons at extremely low energy)
SMALL AMPLITUDE ENERGY-PHASE OSCILLATIONS
wcEdz
d
Eqdz
dw
ss
RF
ssacc
33
0
coscosˆ
Assuming small oscillations around the synchronousparticle that allow to approximate
Deriving both terms with respect to z andassuming an adiabatic accelerationprocess i.e. a particle energy and speedvariations that allow to consider
sss sincoscos
dz
dw
cEw
dz
cEd
ss
RFss
RF
33
0
33
0
dz
dw
cEdz
d
ss
RF
33
0
2
2
0
sinˆ
2
33
0
2
2
s
ss
saccRF
cE
Eq
dz
d
harmonic oscillator equationThe condition to have stable longitudinaloscillations and acceleration at the same time is:
0
20cos0
0sin02
s
sacc
ss
V
Vacc
ts
Vacc_s
if we accelerate on the risingpart of the positive RF wave wehave a longitudinal forcekeeping the beam bunchedaround the synchronous phase.
zww
z
s
s
sinˆ
cos
(protons and ions or electrons at extremely low energy)
accV
The angular frequencyis simply: T= ssc;
The angular frequencyscale with 1/3/2 thatmeans that for ultrarelativistic electronsshrinks to 0 (the beamis frozen)
The energy-phase oscillations can be drawn in the longitudinal phase space:
zww
z
s
s
sinˆ
cosw
w
The trajectory of a generic particle in thelongitudinal phase space is an ellipse.
The maximum energy deviation is reached at=0 while the maximum phase excursioncorresponds to w=0.
the bunch occupies an area in the longitudinalphase space called longitudinal emittance andthe projections of the bunch in the energy andphase planes give the energy spread and thebunch length.
Gain energy withrespect to thesynchronous one
Loose energy
move backward
move forward
w
0
Number of particles
Bunch length
Energy spread
0
Longitudinalemittance is thephase space areaincluding all theparticles
w
ENERGY-PHASE OSCILLATIONS IN PHASE SPACE
Synchronous particle
(protons and ions or electrons at extremely low energy)
s -s
saccEq cosˆ
To study the longitudinal dynamics at large oscillations, we have to consider the non linear system of differentialequations without approximations. In the adiabatic acceleration case it is possible to easily obtain the following relationbetween w and that is the Hamiltonian of the system related to the total particle energy:
APPENDIX: LARGE OSCILLATIONS AND SEPARATRIX
Hconstsincossinˆ
2
133
0
2
2
33
0
sss
ss
accRF
ss
RF
cE
Eqw
cE
For each H we have different trajectories in thelongitudinal phase space
the oscillations are stable within a region bounded by aspecial curve called separatrix: its equation is:
the region inside the separatrix is called RF bucket. Thedimensions of the bucket shrinks to zero if s=0.
trajectories outside the RF buckets are unstable.
we can define the RF acceptance as the maximumextension in phase and energy that we can accept in anaccelerator:
0sincos2sinˆ2
1 2
33
0
ssssacc
ss
RF EqwcE
2
1
33 )sincos(ˆ2
RF
sssaccsso
MAX
EqcEw
sMAX 3
MAX
MAXw
Small amplitude oscillations
RF bucket
From previous formulae it is clear that thereis no motion in the longitudinal phaseplane for ultrarelativistic particles (>>1).
It is interesting to analyzewhat happen if we injectan electron beamproduced by a cathode (atlow energy) directly in aTW structure (with vph=c)and the conditions thatallow to capture the beam(this is equivalent toconsider instead of a TWstructure a SW designed toaccelerate ultrarelativisticparticles at v=c).
Particles enter the structure withvelocity v<c and, initially, they arenot synchronous with theaccelerating field and there is a socalled slippage.
This is the case of electrons whose velocity is always close tospeed of light c even at low energies.
Accelerating structures are designed to provide an acceleratingfield synchronous with particles moving at v=c. like TW structureswith phase velocity equal to c.
LONGITUDINAL DYNAMICS OF LOW ENERGY ELECTRONS
After a certain distance they can reach enoughenergy (and velocity) to become synchronouswith the accelerating wave. This means thatthey are captured by the accelerator and fromthis point they are stably accelerated.
z
If this does not happen (theenergy increase is not enough toreach the velocity of the wave)they are lost
Eacc
cosˆ,cosˆ 3
00 accacc Eqdt
dcm
dt
dcmtzEqmv
dt
d
LONGITUDINAL DYNAMICS OF LOW ENERGY ELECTRONS: PHASE SPLIPPAGE
The accelerating field of aTW structure can beexpressed by
tz
kztEE RFaccacc
,
cosˆ
The equation of motion of a particle with a position z at time t accelerated by theTW is then
It is useful to find which is the relation between and from an initial condition (in) to a finalone (fin)
fin
fin
in
in
accRF
infinEq
E
1
1
1
1
ˆ
2sinsin 0
Suppose that the particle reachasymptotically the value fin=1 we have:
in
in
accRF
infinEq
cm
1
1
ˆ
2sinsin
2
0
infininfin sinsin
Eacc
in
in
fin
fin=1
z
Eacc
Should be in the interval [-1,1] to have a solution for fin
This quantity is >0This limits the possible injection phases (i.e. thephase of the electrons that is possible to capture)
We always have that
in
in
infinRF
accq
EE
1
1
sinsin
2ˆ 0
For a given injection energy (in) and phase (in)we can find which is the accelerating field (Eacc) thatis necessary to have the completely relativistic beamat phase fin (that is necessary to capture the beamat phase fin)
Example: Ein = 50 keV, (kinetic energy), in =-/2,
fin =0 in≈ 1.1; in≈ 0.41fRF= 2856 MHzRF≈ 10.5 cm
We obtain Eacc 20MV/m;
in
in
RF
MINaccq
EE
1
1ˆ 0_
The minimum value of the electric field (Eacc) thatallow to capture a beam. Obviously this correspond toan injection phasein =-/2 and fin=/2.
Example: For the previous case we obtain: Eacc_min 10MV/m;
LONGITUDINAL DYNAMICS OF LOW ENERGY ELECTRONS: CAPTURE ACCELERATING FIELD
LONGITUDINAL DYNAMICS OF LOW ENERGY ELECTRONS: CAPTURE EFFICIENCY AND BUNCH COMPRESSION
fin
ininfin
cos
cos
Ezin<1
fin=1
During the capture process, as the injected beam moves up to the crest, the beam is also bunched, which is caused byvelocity modulation (velocity bunching). This mechanism can be used to compress the electron bunches (FEL applications).
Bunch length variation
in=0.01fin1
These particle are lostduring the capture process
BUNCH COMPRESSION CAPTURE EFFICIENCY
All particlesare captured
in=0.01fin1
In order to increase the capture efficiency of atraveling wave section, pre-bunchers are oftenused. They are SW cavities aimed at pre-formingparticle bunches gathering particlescontinuously emitted by a source.
I
t
I
tTRF
I
tTRF
Continuous beamwith velocity modulation
Bunching is obtained by modulating the energy (and therefore the velocity) of a continuousbeam using the longitudinal E-field of a SW cavity. After a certain drift space the velocitymodulation is converted in a density charge modulation. The density modulation depletesthe regions corresponding to injection phase values incompatible with the capture process
A TW accelerating structure (capture section) is placed at an optimal distance from the pre-buncher, to capture a large fraction of the charge and accelerate it till relativistic energies. Theamount of charge lost is drastically reduced, while the capture section provide also furtherbeam bunching.
Once the capture condition ERF>ERF_MIN is fulfilled thefundamental equation of previous slide sets the rangesof the injection phases in actually accepted. Particleswhose injection phases are within this range can becaptured the other are lost.
BUNCHER AND CAPTURE SECTIONS
Thermionic
Gun
Continuous beam
Buncher
(SW Cavity)
Drift L
Bunched beam
Capture section
(SW or TW)
Bunched and captured beam
(electrons)
v
t
vE
t
TRF
LINAC: BASIC DEFINITION AND MAIN COMPONENTSLINAC (linear accelerator) is a system that allows to accelerate charged particles through a linear trajectoryby electromagnetic fields.
Particle source
Accelerated beam
Accelerating structures Focusing elements: quadrupoles and solenoids
Transverse dynamics of accelerated particles
Longitudinal dynamics of accelerated particles
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RF TRANSVERSE FORCESThe RF fields act on the transverse beam dynamics because of the transverse components of the E and Bfield
z
According to Maxwell equations thedivergence of the field is zero and this impliesthat in traversing one accelerating gap there is afocusing/defocusing term
t
E
c
rB
z
ErE
Ec
B
E
z
zr
22
2
210
t
E
cz
ErqvBEqF zz
rr
2
tzEtzE RFRFz cos,
injRF
RF
Erc
z
z
zErqF
cos
2
injRFRFRFBr
c
zzE
c
rqF
sin
2
fRF=350 MHz=0.1L=3cm
r
RF DEFOCUSING/FOCUSING
ts
rc
LEq
c
dzFp
RF
acc
L
L
rr
22
2/
2/
sinˆ
Transversemomentumincrease
Gaplength
Defocusingforce sincesin<0
Defocusing effect
transverse defocusing scales as 1/2 and disappears at relativistic regime (electrons)
At relativistic regime (electrons), moreover, we have, in general, =0 for maximum acceleration and thiscompletely cancel the defocusing effect
Also in the non relativistic regime for a correct evaluation of the defocusing effect we have to:
take into account the velocity change across the accelerating gap
the transverse beam dimensions changes across the gap (with a general reduction ofthe transverse beam dimensions due to the focusing in the first part)
Both effects give a reduction of the defocusing force
From previous formulae it is possible to calculate the transverse momentum increase due to the RFtransverse forces. Assuming that the velocity and position changes over the gap are small we obtain to thefirst order:
RF FOCUSING IN ELECTRON LINACS-RF defocusing is negligible in electron linacs.-There is a second order effect due to thenon-synchronous harmonics of theaccelerating field that give a focusing effect.These harmonics generate a ponderomotiveforce i.e. a force in an inhomogeneousoscillating electromagnetic field.
Ez
z
Direction of propagation
P RF inIs equivalent to thesuperposition oftwocounterpropagatingTW waves
Ez
z
Direction of propagation
The forward wave only contribute tothe acceleration (and does not givetransverse effect).The backward wave does notcontribute to the acceleration butgenerates an oscillating transverseforce (ponderomotive force)
ecm
ErqF acc
r 2
0
2
8
ˆ
With accelerating gradients offew tens of MV/m can easilyreach the level of MV/m2
t
E
cz
ErqF zz
r
2
The Lorentz force is linear withthe particle dispacemement
MECHANISM
tnarm
rn
n
cos1
0
Transverse equation of motion
RF harmonics
z
rThis generate aglobal focusing force
NON-SYNCHRONOUS RF HARMONICS: SIMPLE CASE
OF SW STRUCTURE
Average focusing force
Particle trajectory
APPENDIX: PONDEROMOTIVE FORCELet us consider a particle under the action of a non-uniform force oscillating at frequency in the radial direction. Theequation of motion in the transverse direction is given by:
trgr cos
We are searching for a solution of the type:
trtrr fs
Where rs represents a slow drift motion and rf a fastoscillation. Assuming rf<<rs we can proceed to a Taylorexpansion of the function g(r) writing the equation as:
tdr
dgrrgrr
srr
fsfs cos
trgr sf cos
srr
fs
fs
dr
dgrrg
rr
On the time scale on which rf oscillatesrs is essentially constant, thus, theequation can be integrated to get:
trg
r sf
cos
2
Substituting in the main equationand averaging over the one period
ss rrrr
ss
dr
rdg
dr
dgrgr
2
22 4
1
2
Thus, we have obtained an expressionfor the drift motion of a charged particleunder the effect of a non-uniformoscillating field
assuming
In the case of the Lorentz force due to the harmonics of the acceleratingfield g(r)=Ar and we obtain:
02 2
ss rA
r
We have then an exponential decay of the amplitude due to a constant focusing force
APPENDIX: RF NON-SYNCRONOUS HARMONICS
tkzEE RFRFz coscosˆ
kztE
kztE
E RFRF
RFRF
z cos2
ˆcos
2
ˆ
RFRFRF
RF
cTc
k
,
2
kzEE
kzEc
zkzEE RFRF
RFRFRF
ctzparticlebyseen
z 2cos2
ˆ
2
ˆcosˆcoscosˆ 2
The SW can be written as the sum of two TWs in the form:
kzEE
kzc
zEkz
c
zEE RFRF
RFRF
RFRF
ctzparticlebyseen
z 2cos2
ˆ
2
ˆcos
2
ˆcos
2
ˆ
Let us consider the case of a multi-cell SWcavity working on the -mode. TheAccelerating field can be expressed as:
In order to have synchronism between the acceleratingfield and and ultrarelativistic particle we have to satisfy thefollowing relation:
The accelerating field seen by the particle is given by:
Synchronous waveco-propagatingwith beam
NON-Synchronous wave (called RFharmonic) counter-propagatingwith beam (opposite direction)
The accelerating field seen by the particle is given by:
tEE
kzEE
E RFRFRFRFRF
ctzparticlebyseen
z 2cos2
ˆ
2
ˆ2cos
2
ˆ
2
ˆ
Oscillating field that does not contributeto acceleration but that gives RF focusing
Ez
z
Direction of propagation
P RF in
Synchronouswave:acceleration
COLLECTIVE EFFECTS: SPACE CHARGE AND WAKEFIELDS
Effect of Coulomb repulsionbetween particles (space charge).
These effects cannot beneglected especially at low energyand at high current because thespace charge forces scales as 1/2
and with the current I.
EXAMPLE: Uniform and infinite cylinder of charge moving along z
z
rEr
B
rq
rrcR
IqF q
b
SCˆ
2 22
0
Rb
IIn this particular case it islinear but in general it is anon-linear force
Collective effects are all effects related to the number of particles and they can play a crucial role in thelongitudinal and transverse beam dynamics of intense beam LINACs
SPACE CHARGE
WAKEFIELDS
The other effects are due to thewakefield. The passage of bunchesthrough accelerating structures exciteselectromagnetic field. This field canhave longitudinal and transversecomponents and, interacting withsubsequent bunches (long rangewakefield), can affect the longitudinaland the transverse beam dynamics. Inparticular the transverse wakefields, candrive an instability along the train calledmultibunch beam break up (BBU).
z
wTSeveral approaches are used toabsorb these field from thestructures like loops couplers,waveguides, Beam pipe absorbers
MAGNETIC FOCUSING AND CONTROL OF THE TRANSVERSE DYNAMICS
Defocusing RF forces, space charge or thenatural divergence (emittance) of the beam needto be compensated and controlled by focusingforces.
This is provided by quadrupoles alongthe beam line.At low energies also solenoids can beused
In a linac one alternates accelerating structureswith focusing sections.
Quadrupoles are focusing in one plane anddefocusing on the other. A global focalization isprovides by alternating quadrupoles with oppositesigns
F D F D F
s
x
The type of magnetic configuration and magnetstype/distance depend on the type ofparticles/energies/beam parameters we want toachieve.
Due to the alternating quadrupole focusing system eachparticle perform transverse oscillations along the LINAC.
The equation of motion in the transverseplane is of the type:
0
0
cos)(
s
so
s
dsssx
The single particle trajectory is a pseudo-sinusoiddescribed by the equation:
0
2
22
2
2
SCRF Fx
sK
sksds
xd
Term dependingon the magneticconfiguration
RF defocusing/focusingterm
Characteristic function(Twiss -function [m]) thatdepend on the magneticand RF configuration
Depend on the initialconditions of the particle
The final transverse beam dimensions (x,y(s)) varyalong the linac and are contained within an envelope
TRANSVERSE OSCILLATIONS AND BEAM ENVELOPE
Space charge term
p
L
L
s
ds
p
Transverse phaseadvance per period Lp.For stability should be0<<
Focusing period(Lp)= length afterwhich thestructure isrepeated (usuallyas N).
Distance [mm or m]
SMOOTH APPROXIMATION OF TRANSVERSE OSCILLATIONS
In case of “smooth approximation” of theLINAC (we consider an average effect of thequadrupoles and RF) we obtain a simpleharmonic motion along s of the type ( isconstant):
000 cos1)( sKKsx o
32
0
2
0
0
sinˆ
2
RF
acc
cm
Eq
cm
qGlK
x Phase advance per unit length (/Lp)
s
Magnetic focusing elements (for a FODO)
RF defocusing term
If we consider also the Space Charge contribution in the simple case of an ellipsoidal beam (linear spacecharges) we obtain:
zyx
RF
RF
acc
rrrcm
fqIZ
cm
Eq
cm
qGlK
322
0
0
32
0
2
0
08
13sinˆ
2
I= average beam current (Q/TRF)rx,y,z=ellipsoid semi-axisf= form factor (0<f<1)Z0=free space impedance (377 Ω)
Space charge term
For ultrarelativistic electrons RF defocusing and space charge disappear and the external focusing is required to controlthe emittance and to stabilize the beam against instabilities.
G=quadrupole gradient [T/m] l=quadrupole length
NB: the RF defocusing term f sets a higher limit to theworking frequency
Lp
l
GENERAL CONSIDERATIONS ON LINAC OPTICS DESIGN (1/2)
Beam dynamics dominated by space charge and RF defocusing forces
Focusing is usually provided by quadrupoles
Phase advance per period () should be, in general, in the range 30-80 deg, this means that, at lowenergy, we need a strong focusing term (short quadrupole distance and high quadrupole gradient) tocompensate for the rf defocusing, but the limited space () limits the achievable G and beam current
As increases, the distance between focusing elements can increase ( in the DTL goes from 70mm(3 MeV, 352 MHz) to 250mm (40 MeV), and can be increased to 4-10 at higher energy (>40 MeV).
A linac is made of a sequence of structures, matched to the beam velocity, and where the length ofthe focusing period increases with energy. As increases, longitudinal phase error between cells ofidentical length becomes small and we can have short sequences of identical cells (lower constructioncosts).
Keep sufficient safety margin between beam radius and aperture
PROTONS AND IONS
Courtesy A. Lombardi
Space charge only at low energy and/or high peak current: below 10-20 MeV (injector) the beamdynamics optimization has to include emittance compensation schemes with, typically solenoids;
At higher energies no space charge and no RF defocusing effects occur but we have RF focusing due tothe ponderomotive force: focusing periods up to several meters
Optics design has to take into account longitudinal and transverse wakefields (due to the higherfrequencies used for acceleration) that can cause energy spread increase, head-tail oscillations, multi-bunch instabilities,…
Longitudinal bunch compressors schemes based on magnets and chicanes have to take into account,for short bunches, the interaction between the beam and the emitted synchrotron radiation (CoherentSynchrotron Radiation effects)
All these effects are important especially in LINACs for FEL that requires extremely good beamqualities
GENERAL CONSIDERATIONS ON LINAC OPTICS DESIGN (2/2)ELECTRONS
Courtesy C. Vaccarezza
Courtesy M. Ferrario
RADIO FREQUENCY QUADRUPOLES (RFQ)At low proton (or ion) energies (0.01), space charge defocusing ishigh and quadrupole focusing is not very effective. Moreover celllength becomes small and conventional accelerating structures (DTL)are very inefficient. At this energies it is used a (relatively) newstructure, the Radio Frequency Quadrupole (1970).
These structures allow to simultaneously provide:
AccelerationTransverse Focusing
Bunching of the beam
Electrodes
Courtesy M. Vretenar
1-FocusingThe resonating mode of the cavity (between the fourelectrodes) is a focusing mode: Quadrupole mode (TE210).The alternating voltage on the electrodes produces analternating focusing channel with the period of the RF(electric focusing does not depend on the velocity and is idealat low )
RFQ: PROPERTIES
2-AccelerationThe vanes have a longitudinal modulation with period = RF thiscreates a longitudinal component of the electric field that acceleratethe beam (the modulation corresponds exactly to a series of RF gaps).
3-BunchingThe modulation period (distance between maxima) can beslightly adjusted to change the phase of the beam inside the RFQcells, and the amplitude of the modulation can be changed tochange the accelerating gradient. One can start at -90 phase(linac) with some bunching cells, progressively bunch the beam(adiabatic bunching channel), and only in the last cells switch onthe acceleration.
The RFQ is the only linear accelerator that can accept a low energy continuous beam.
Courtesy A. Lombardi
Courtesy M. Vretenarand A. Lombardi
RFQ: EXAMPLES The 1st 4-vane RFQ, Los Alamos1980: 100 KeV - 650 KeV, 30 mA , 425 MHz
TRASCO @ INFN LegnaroEnergy In: 80 keVEnergy Out: 5 MeVFrequency 352.2 MHzProton Current (CW) 30 mA
The CERN Linac4 RFQ45 keV – 3 MeV, 3 m80 mA H-, max. 10% duty cycle
In general the choice of the acceleratingstructure depends on:
Particle type: mass, charge, energy Beam current Duty cycle (pulsed, CW) Frequency Cost of fabrication and of operation
Cavity Type Range Frequency Particles
RFQ 0.01– 0.1 40-500 MHz Protons, Ions
DTL 0.05 – 0.5 100-400 MHz Protons, Ions
SCL 0.5 – 1 600 MHz-3 GHz Protons, Electrons
SC Elliptical > 0.5-0.7 350 MHz-3 GHz Protons, Electrons
TW 1 3-12 GHz Electrons
THE CHOICE OF THE ACCELERATING STRUCTURE
Moreover a given accelerating structure hasalso a curve of efficiency (shunt impedance)with respect to the particle energies and thechoice of one structure with respect to anotherone depends also on this.
As example a very general scheme is given in theTable (absolutely not exhaustive).
LINAC: BASIC DEFINITION AND MAIN COMPONENTSLINAC (linear accelerator) is a system that allows to accelerate charged particles through a linear trajectoryby electromagnetic fields.
Particle source
Accelerated beam
Accelerating structures Focusing elements: quadrupoles and solenoids
Transverse dynamics of accelerated particles
Longitudinal dynamics of accelerated particles
LIN
AC
BEA
M D
YN
AM
ICS
LIN
AC
CO
MP
ON
ENTS
AN
D
TEC
HN
OLO
GY
ELECTRON SOURCES: RF PHOTO-GUNSRF guns are used in the first stage of electron beam generation in FEL andacceleration.
• Multi cell: typically 2-3 cells• SW mode cavities• operate in the range of 60-120 MV/m cathode peak accelerating field
with up to 10 MW input power.• Typically in L-band- S-band (1-3 GHz) at 10-100 Hz.• Single or multi bunch (L-band)• Different type of cathodes (copper,…)
The electrons are emitted onthe cathode through a laserthat hit the surface. They arethen accelerated trough theelectric field that has alongitudinal component onaxis TM010.
cathode
RF PHOTO-GUNS: EXAMPLES
LCLS Frequency = 2,856 MHz Gradient = 120 MV/m Exit energy = 6 MeVCopper photocathodeRF pulse length 2 sBunch repetition rate = 120 Hz Norm. rms emittance0.4 mmmrad at 250 pC
PITZ L-band Gun Frequency = 1,300 MHz Gradient = up to 60 MV/m Exit energy = 6.5 MeV Rep. rate 10 HzCs2Te photocathodeRF pulse length 1 ms800 bunches per macropulseNormalized rms emittance1 nC 0.70 mmmrad0.1 nC 0.21 mmmrad
Solenoids field are used tocompensate the space chargeeffects in low energy guns. Theconfiguration is shown in thepicture
Basic principle: create a plasma and optimize its conditions (heating, confinement and loss mechanisms) toproduce the desired ion type. Remove ions from the plasma via an aperture and a strong electric field.
ION SOURCES
CERN Duoplasmatron proton Source Electron Cyclotron Resonance (ECR) ECR
Medical applications
Security: Cargo scans
Industrial applications:Ion implantation for semiconductors
Injectors for colliders and synchrotron light sources
Neutron spallation sources
FEL
THANK YOU FOR YOUR ATTENTION AND….
REFERENCES1. Reference Books:
T. Wangler, Principles of RF Linear Accelerators (Wiley, New York, 1998).P. Lapostolle, A. Septier (editors), Linear Accelerators (Amsterdam, North Holland, 1970).
2. General Introduction to linear accelerators
M. Puglisi, The Linear Accelerator, in E. Persico, E. Ferrari, S.E. Segré, Principles of Particle Accelerators (W.A. Benjamin, NewYork, 1968).P. Lapostolle, Proton Linear Accelerators: A theoretical and Historical Introduction, LA-11601-MS, 1989.P. Lapostolle, M. Weiss, Formulae and Procedures useful for the Design of Linear Accelerators, CERNPS-2000-001 (DR), 2000.P. Lapostolle, R. Jameson, Linear Accelerators, in Encyclopaedia of Applied Physics (VCH Publishers,New York, 1991).
3. CAS Schools
S. Turner (ed.), CAS School: Cyclotrons, Linacs and their applications, CERN 96-02 (1996).M. Weiss, Introduction to RF Linear Accelerators, in CAS School: Fifth General Accelerator Physics Course, CERN-94-01(1994), p. 913.N. Pichoff, Introduction to RF Linear Accelerators, in CAS School: Basic Course on General Accelerator Physics, CERN-2005-04(2005).M. Vretenar, Differences between electron and ion linacs, in CAS School: Small Accelerators, CERN-2006-012.M. Vretenar, Low-beta Structures, in CAS RF School, Ebeltoft 2010J. Le Duff, High-field electron linacs, CAS School: Fifth Advanced Accelerator Physics Course, CERN 95-06, 1995.J. Le Duff, Dynamics and acceleration in linear structures, CAS School: Fifth General Accelerator Physics Course, CERN 94-01,1994.D. Alesini, “Linear Accelerator Technology”, CERN Yellow Reports: School Proceedings, v. 1, p. 79, CERN-2018-001-SP (CERN,Geneva, 2018).
ACKNOWLEDGEMENTSSeveral pictures, schemes, images and plots have been taken from papers and previous presentations ofthe following authors that I would like to thank.
I would like to acknowledge in particular the following authors:
Hans Weise, Sergey Belomestnykh, Dinh Nguyen, John Lewellen, Leanne Duffy, Gianluigi Ciovati, Jean Delayen, S. Di Mitri, R.Carter, G. Bisoffi, B. Aune, J. Sekutowicz, H. Safa, D. Proch, H. Padamsee, R. Parodi, Paolo Michelato, Terry Garvey, YujongKim, S. Saitiniyazi, M. Mayierjiang, M. Titberidze, T. Andrews, C. Eckman, Roger M. Jones, T. Inagaki, T. Shintake, F. Löhl, J.Alex, H. Blumer, M. Bopp, H. Braun, A. Citterio, U. Ellenberger, H. Fitze, H. Joehri, T. Kleeb, L. Paly, J.Y. Raguin, L. Schulz, R.Zennaro, C. Zumbach. Detlef Reschke, David Dowell, K. Smolenski, I. Bazarov, B. Dunham, H. Li, Y. Li, X. Liu, D. Ouzounov, C.Sinclair
A. Gallo, E. Jensen, A. Lombardi, F. Tecker and M. Vretenar
I would like also to acknowledge the following authors:
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