Pichhoff Linac Lecture

8/4/2019 Pichhoff Linac Lecture

1/15

RF Linac Structures

Nicolas Pichoff

CEA/DIF/DPTA/SP2A

[email protected]

Outline

q Basis

o Advantage/disadvantage of linacs

o RF Cavities

q Beam dynamics though linacs

o Energy gain

o Linear motion

o RMS matching


2/15

Why RF linacs ?

Goal of an accelerator : Accelerate a wanted beam within the lower cost

wanted : particle, energy, emittance, intensity, time structure

cost : construction, operation

Main competitors : RF linacs, Synchrotrons, Cyclotrons...

Advantages : High current, high duty-cycle, low synchrotron radiation losses.

Drawbacks : High room & cavities consumption, no synchrotron radiation

damping

RF linacs : Particles accelerated on a linear path with RF cavities.

Main use of linacs : Low energy injectors, high intensity protons beam,high energy lepton colliders.

Linacs main applications

Synchrotron injectors : High intensity, high duty-cycle

Neutron sources : High Power. Material study, transmutation, nuclear fuelproduction, irradiation tools, exotic nucleus production

Protons

High energy collider :No synchrotron losses

Medical/Industrial irradiation : Low energy

High-quality e- beam for FEL : Strong focusing

Electrons

Neutron sources : Material study

Heavy ions

Nuclear physics research : High intensity, high duty-cycle

Implantation : Semi-conductors

Driver for inertial-confinement fusion


3/15

RF resonant cavity

Goal : Give kinetic energy to the beam

Basic principle

- The wanted (accelerating) mode is excited at

the good frequency and position from a RF

power supply through a power coupler,

RF power supply

Wave guide

Power coupler

Cavity

- Conductor enclosing a close volume,- Maxwell equations +Boundary conditions

allow possible electromagnetic fieldEn/Bnconfigurations each oscillating with a given

frequencyfn : a resonnant mode. The field is

a weighted superposition of these modes.

- The phase of the electric field is adjusted to

accelerate the beam.

Elements of mode calculation

0//

rr

=EBoundary conditions :

0rr

=^

Bclose to the surface

( ) ( ) ( ) = rEtetrE nnr

r

r

r

,Electric field :

02

22

rrrr

=+ nn

n Ec

Ew

Mode calculation : nnf= pw 2

c : speed of light

TM011 : f=548 MHz TM020 : f=952 MHzTM010 : f=352.2 MHz

Ex : Drift Tube Linac (DTL) tank


4/15

Field time variation

?edt

edn

2

n2

n

2

=w+

1

Joule losses in conductor1) S: conductor surfacenn

RF eQ

&-=

0

w

( ) em

w-

S

n

2

n dSnHEr

rr

3

V: enclosed volume3) Energy exchange with beam : Beam loading( )tIkn =

( ) ( )

e-

V

n dVrEt,rJdt

d1 rrrr

2

S: open surface2) Energy exchange with outside

( ) ( )0j++-= tjnnexn

RF RFetSeQ

ww

&

losses feed

( ) e+ Sn SdnEHdt

d1 rrr

Field time variation (cont)

( ) ( )tIkeSedt

de

Qdt

edn

tjnnn

n

n

RFn RF+=++

j+ 02

2

2w

w

w

Which is a damped harmonic oscillator in a forced regime

With :

The last equation can be modelled by :

exnnn QQQ

111

0

+= the quality factor of the cavity

RF

nQ

w

t = 2 is the filling time of the cavity

( )0j+

tjn

RFeSw is the RF source

( )tIkn is the beam loading


5/15

RF definitions and properties

Cavity voltage V0 : ( ) = dzzEV z0

Shunt impedance R :dP

VR

=

2

20

Dissipated powerPd

: Mean power dissipated in conductor over one RF period

Cavity length : L

Transit time factorT (calculated latter) : TqVW =D 0max

DWmax : Maximum energy that can be gained by a particle in the cavity

Effective shunt impedance :dP

WRT

2

2max2 D

=

Per cavity

20

2

1VRPd =

L

RF definitions and properties

Effective shunt impedance per unit length :

Cavity mean electric field E0

: ( ) == dzzELLV

E z10

0

Shunt impedance per unit length Z :

L

R

P

EZ

d

=

=

2

20

Dissipated power per unit length Pd

over one RF period

TqEW =D 0maxMaximum energy that can be gained per unit

length by a particle with charge q in the cavity:

Per unit length

dP

WZT

D=

2

2max2

20

2

1EZPd =


6/15

Example of use of effective shunt impedance ZT2

The effective shunt impedance of the structures has been chosen to set the

transition energy between sections for TRISPAL project (C. Bourra, Thomson).

375 MeV85 MeV45 MeV19 MeV 234 MeV

Designing a cavity consists in :

Rejecting the unwanted modes frequencies far from the RF frequency,

Calculating the tuning system,

Increasing the Q0 of the accelerating mode,

Calculating the energy deposition geometry to define the cooling system (~20W/cm2 max),

the temperature increase and the associated frequency shift,

Matching the coupler to the accelerating mode,

Damping the High Order Modes (HOM) considered as dangerous (having a frequency

close to a multiple of the RF frequency), mainly excited by the beam itself and responsible of

power losses or beam dynamics perturbations,

Increasing the beam aperture to reduce beam losses,

Reducing the peak electric field to reduce electron field emission,

Reducing the peak magnetic field to avoid quenches (th. Nb max at 2K : 200 mT),

Adjusting the cavity geometry to reduce the multipactor probabilities,

Calculating and minimising the cavity deformation and the associated frequency shift

through the actions of electromagnetic forces pressure (Lorentz forces),

Introducing RF peak-up necessary for the field phase and amplitude control,

Transmitting all these data to the guy calculating the low level RF control,

...

Fitting the accelerating mode frequency with the RF frequency,

Maximise the shunt impedance of the cavity


7/15

Various types of cavity : Tank

DTL (medium energy ~5-100 MeV)

RFQ (low energy ~50 keV-7 MeV)

Various types of cavity : Coupled cavity

CCDTL (medium energy ~5-100 MeV)

CCL (high energy ~80 MeV-2 GeV)


8/15

Various types of cavity : Superconducting

Elliptical (high energy ~100MeV- 2 GeV)

Spoke (medium energy ~20-100 MeV)

One word on travelling wave cavity

These cavities are essentially used for acceleration of ultra-relativistic particles

( ) ( ) ( )-

=

n

zktj

nznerEt,z,rE

w

The longitudinal field component is :

( ) ( )zktjn nerE-

w

is a space harmonic of the field, given by the cavity periodicity

Particle whose velocity is close to the phase

velocity of the space harmonic exchanges energy

with it. Otherwise, the mean effect is null. nnp

kvv

w

=j


9/15

The transit time factor and the particle phase

( ) ( )( ) =D dsssqEzW fcos

Energy gained by a particle in a cavity of length L :

( )( )b

w+f=w+f=f

s

s z

00

0s

ds

ctswith :

with :

( ) pTqVW fb cos0 =DAssuming a constant velocity : b

( ) ( )

-+=D dsss

ccossqEzW 00

b

wf

( ) = dssEzV0 Cavity Voltage( ) ( )( )

( ) ( )( )

=

dsssEz

dsssEz

pf

ff

cos

sinarctan Synchronous phase

-0 .2

0

0 .2

0 .4

0 .6

0 .8

1

El

ti

fild

t

ti

l

iti

F ieldA mpl itudefs

=-3 0

Df=Df=Df=Df=

120Df= 160

Df=f =D 3 60

L

- 250

- 200

- 150

- 100

-50

0

50

100

150

200

P

il

RF

h

-0 .2

0

0 .2

0 .4

0 .6

0 .8

1

N

li

d

E

G

i

( ) ( )( ) -= dsssEzV

T pffcos1

0

( ) ( ) = dsesEzVsjf

0

1

Transit-time factor : 0 < T< 1

Example 1 : The transit time factor in a one-cell cavity

Field in the cavity with time

Field amplitude

Field seen by a non

synchronous particle

Energy gain

Energygain

ElectricField

( )bTqVW =D 0

Fast particle : T@ 1


10/15



Field amplitude

Field seen by a non


Energy gain

Energygain

ElectricF

ield

( )bTqVW =D 0

Medium fast particle : T@ 0.85



Field amplitude

Field seen by a non


Energy gain

Energygain

ElectricField

( )bTqVW =D 0

Slow particle : T@ 0.3


11/15

The synchronous particle - Linac design

fsi fsi+1fsi-1

DiDi-1

fi fi+1fi-1

bsi-1

Cavity number i-1 i i+1

Synchronous phase

Particle velocity

RF phase

Distances

bsi

The linac is designed with a hypotheticalsynchronous particle. Its phase in a

cavity is called here thesynchronous phase :

The absolute phase ji and the velocity bi-1 of this particle being known at the entrance of

cavity i, its RF phase fi is calculated to get the wanted synchronous phase fsi,

the new velocitybi of the particle can be calculated from,

if the phase difference between cavities i and i+1 is given, the distanceDibetweenthem is adjusted to get the wanted synchronous phase fsi+1 in cavity i+1.

if the distanceDibetween cavities i and i+1 is set, the RF phase fi of cavity i+1 is

calculated to get the wanted synchronous phase fsi+1 in it.

nc

Dii

is

isisi pff

bwff 211 +-+=- ++Synchronism condition :

siii ff -j=

sii TqVW fcos0 =D

Linac with coupled cavities (DTL)

Field in cavities

Particle synchronous with the field Its energy gain

Particle not synchronous with the field Its energy gain

Gaps have the same phase. Distances between them are adjusted for synchronism.


12/15

Linac with independently phased cavities (SCL)

Field in cavities

Particle synchronous with the field Its energy gain

Particle not synchronous with the field Its energy gain

The distance between the cavities is given. Cavities are phased to accelerate a

given particle.

Choice of the synchronous phase

[ ]-> 0,90:00 pqV f

[ ]< 180,90:00 pqV f

Stability condition : Late particles should gain more energy that early ones

00 0 D

p

p

sinTqVd

Wdf

f[ ]-> 0,180:00 pqV f

[ ]< 180,0:00 pqV f

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-90 -60 -30 0 30 60 90

f ()

qV0(normalised)

Acceleration condition : The field should accelerate the particle

0

0

0 >

>D

pcosTqV

W

f

[ ]-> 90,90:00 pqV f

[ ]< 270,90:00 pqV f

-1

-0.5

0

0.5

1

-360 -270 -180 -90 0 90 180 270 360


13/15

General equations of motion

=

==

==

z

zz

z

z

yy

z

z

xx

mcF

dsd

ds

yd

mc

F

ds

d

ds

xd

mc

F

ds

d

bgb

gb

b

gb

gb

b

gb

2

2

2

bwc is the particle velocity along w direction

g is the particle reduced energy,

q and m its charge and rest mass.

x andy are transverse directions,

s is the abscissa along longitudinal directionz,

xandyare called the particle slopes.

( ) FEBvqdt

pd rrrrr

=+=

mcp ww = gb+

cdsdt zb=+

These equation are non linear, coupled and damped.

Each element (cavity, quadrupole ) contributes to the force.

22z

x

z

z

mc

Fx

dsdx

gbgb

gb=+

Linear force

Thephase advance of the particle in a lattice is then : ( ) ( )sSs mms -+=

Giving : ( ) ( ) ( )( )00 cos mmbe += sssw wm

with : m0 and e0 constant, ( ) ( )+=

s

s wms

dss

0

0b

mm , and : ( ) ( )sSs wmwm bb =+

( ) 02

2

=+ wskds

wdw

In the highest simplification level, the external force along direction w (x,y orj)

can be considered periodic, linear, uncoupled and undamped over one period :

( ) ( )skSsk ww =+Hill equation :

In the (w, w) phase-space, the particle is moving on an ellipse of equation :

( ) ( ) ( ) 022 2 ebag =++ wswwsws wmwmwm

( )ds

ds wmwm

ba

2

1-= ( )

( )( )ss

swm

wmwm

b

ag

21+

=with : and Courant-Snyder parameters.


14/15

Particle motion in a FODO Lattice

Particle trajectory

Particle

Particle ellipse maximum size

Particle ellipse

Phase-space trajectory Phase-space periodic looks

Foc. Quad. middle Foc. Drift. middle

Defoc. Drift. middleDefoc. Quad. middle

RMS dimensions and Beam Twiss parameters

The rms dimensions of the beam are defined statistically as followed :

( )2~ www -=rms size :

rms slope : ( )2~ www -=

rms emittance : ( ) ( )222 ~~~ wwwwwww ---=e

The beam Twiss parameters are then :

w

w

w

eb ~

~2=

w

w

w

eg ~

~ 2=

( ) ( )

w

w

wwww

e

a ~

---=

wwww wwww ebag =++ 5222


15/15

RMS matched beam

50% mismatched beam

The beam is matched when :

wmw bb =

wmw aa =

wmw gg =

Phase-space trajectory Phase-space periodic looks

Matched beam

Bigger input beam

Smaller input beam

Phase-space scanned by

the mismatched beams

Summary

Linacs are competitive for low energy, high current, high duty cycle beams or

very high energy light-particles (e+-e-) colliders.

Acceleration is generally done with RF resonant cavities, confinement with

quadrupoles (except at very low energy).

Cavities are pieces of metal (Cu or Nb) whose shape is optimised to accelerate

the particles at the RF frequency with the higher efficiency (ZT2

as high aspossible) and the lower cost. The choice of the accelerating electric fieldEis a

compromise between the linac length reduction (E) and the power dissipation

(E) .

RF phases in cavities are adjusted with respect to a synchronous particle to

accelerate the beam and keep it bunched (synchronous phase choice).

Forces are linearised to calculate the beam matching to the structure.

Documents

Pichhoff Linac Lecture