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

AC

BEA

M D

YN

AM

ICS

LIN

AC

CO

MP

ON

ENTS

AN

D

TEC

HN

OLO

GY

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

LIN

AC

BEA

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AC

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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

LIN

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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: