Waveguide, cavities, linear accelerators and RFQs Lars Hjorth Præstegaard Aarhus University Hospital

Outline • Waveguides • Cavities • Linear accelerators (linacs) – Accelerating structures – Traveling and standing wave acceleration

– Longitudinal dynamics – Beam loading

• RFQ linear accelerator Aarhus University Hospital, Århus Sygehus

1

Waveguides

Electromagnetic wave in free space Planar wave: E(z,t)=E0Exp(i(t+kzz))

Electromagnetic wave in free space: Transverse electromagnetic wave (TEM wave)

 Both electric and magnetic field perpendicular to direction of wave propagation (z axis)

 No electric field in direction of wave propagation: No acceleration Dispersion relation for TEM wave (propagation along z axis) :  ≡ 2f = 2c/z ≡ kzc

: Angular frequency (2f). z: Wavelength kz : Wave number (2/). c: Speed of light

Dispersion relation: Relation between  and kz(z)

 Phase velocity ≡ /kz = c

Group velocity ≡ d/dkz = c

wave in -z dir.

wave in +z dir.

Aarhus University Hospital, Århus Sygehus

2

Waveguide: Boundary conditions Wave propagation in +z direction

z z

n

Conductor boundary conditions: Conductor = equipotential: Ez=0 Conductor skin depth c

Goal: Transfer of energy from microwave source to the particle beam

 Wave crest desynchronizes with the particle

 No net transfer of energy to particles Aarhus University Hospital, Århus Sygehus

Disk-loaded waveguide Disk-loaded waveguide: Addition of discs to the waveguide:

Disk-loaded waveguide with period d

 Reflections from discs  Positive interference of reflections from neighbor disks if Phase advance = kz*2d  2  kzd=  Standing wave behavior for kzd= (no energy transport)  Group velocity close to zero for kzd    Large perturbation of waveguide dispersion relation for kzd≈ Aarhus University Hospital, Århus Sygehus

14

Disk-loaded waveguide: Dispersion Dispersion relation for disk-loaded waveguide:

Electric coupling via beam iris

Passband: Large for large cellto-cell coupling

wave in -z dir.

wave in +z dir.



kzd

Disks  Frequency exist for which = c  Acceleration of relativistic electrons (v=c) Aarhus University Hospital, Århus Sygehus

Disk-loaded waveguide: Modes

p=2

Small holes in discs  Scattering of forward wave (hole=source of wave propagation)  Positive interference: phase advance = kz*pd = 2  Loss-free propagation: kzd = 2/p

p=1,2,3,...

Modes used for particle acceleration: 0 mode: kzd=2 (p=1)  mode: kzd= (p=2) 2/3 mode: kzd=2/3 (p=3) /2 mode: kzd=/2 (p=4) Aarhus University Hospital, Århus Sygehus

15

Linacs: Traveling wave (TW) acceleration

Traveling wave (TW) acceleration • Disk-loaded waveguide (slowed-down wave)

• TM wave and beam travels synchronous • Input of RF power at first cell • Output of RF power at last cell • Injection of beam along axis of waveguide RF load

RF power in

Resistive loss in walls + Energy transfer to beam  Reduction of microwave power along waveguide

Beam source

Aarhus University Hospital, Århus Sygehus

16

TW acceleration: Power dissipation in walls Change of power along dz:

dP 

dP: Change of power along dz

2  E z 0dz  

2

E   z 0 dz dz Zs 2Rs l E z20 Zs   dP dz



TW power: Q factor:

Ez0: Axial electric field amplitude l: Length of TW structure Zs: 2Rs/l: Shunt impedance per unit length

w: Stored energy per unit length g: Group velocity

P z    gw

Q  

Energy stored wdz w     Power loss  dP  dP dz

Attenuation of electric field:

dEz 0 z     z Ez 0 z  dz

(z): Field attenuation per unit length

Aarhus University Hospital, Århus Sygehus

TW acceleration: Power dissipation in walls P z  



Definition of Zs, Q, and P

Q g 2 E z 0 z  Z s

Definition of Zs, P, Q and 

Attenuation of TW power:

Q g dP z  dE z   2 E z 0 z    2 z P z  dz Z s dz

dP z  dz   z    2P z  Equation for P(z) above



Definition of Q and P



E z 0 z   2

 2Q g z  Z s P z   2Zs z P z  Q g

Aarhus University Hospital, Århus Sygehus

17

TW acceleration: Constant impedance E z 0 z  

Constant impedance accelerating structure:

E z 0 0  e  z

Uniform cell geometry  Q, Zs, g and  do not depend on z

Energy gain for synchronous particle at wave crest: l

W  q  E z 0 0  e Equation for Ez02



 z

0

1  e  l dz  qE z 0 0  l l

1  e  l 2 q Zs P 0  l l

l: Length of TW structure

Low l: High power loss in load High l: High power dissipation to walls Tuning of K by changing g (disk aperture)

K (typically K≈0.8)

Importance of shunt impedance

Maximum of K (l=1.26): K = 0.903

Ez0(l) = 0.28Ez0(0) , P(l) = 0.08 P(0) (remaining power in load)

Aarhus University Hospital, Århus Sygehus

TW acceleration: Constant gradient Constant gradient accelerating structure: Ez0 does not depend on z (change of structure geometry): • g and  depends on z (sensitive to structure geometry) • Q and Zs=Ez02/(-dP/dz) do not depend on z (approximately correct)

 dP(z)/dz = constant

 P z   P 0 

P l   P 0 z l

Use of equation for attenuation of TW power: dP z    2 z P z   dz

 P l   P 0 e 2

dP z    2   z dz  P z  P 0  0 P l 

,  

l

l

  z dz 0

Aarhus University Hospital, Århus Sygehus

18

TW acceleration: Constant gradient



 dP z    P 0 1  e 2 dz l





z  P z   P 0 1  1  e  2  l   Expression for Zs



E z 0 z    Zs 2

Z P 0  dP z   s 1  e  2 dz l







Energy gain for synchronous particle at wave crest: l

W  q  E z 0 0 dz  qEz 0 0 l 0

 q 1  e 2 Zs P 0 l

Tuning of K by changing g (disk aperture) Importance of shunt impedance

K (typically K≈0.8)

Maximum of K (=): K=1

 Ez0(l) = P(l) = 0 (infinite filling time as g=0)

Aarhus University Hospital, Århus Sygehus

TW acceleration: Stanford linear accelerator 50 GeV electrons 932 disc-loaded sections of 3.05 m

RF input Water cooling

Discs

Aarhus University Hospital, Århus Sygehus

19

TW acceleration: Stanford linear accelerator

Energy: 50 GeV electrons (3 km linac with 932 linac sections) Accelerating structure: • • • • • Constant gradient:

2/3 mode (large group velocity  short fill time for pulsed operation) Period (d): 35 mm Structure length: 3.05 m Iris (hole) tapering: 26 mm to 20 mm Cavity radius tapering: 84 mm to 82 mm

• Uniform power dissipation • Lower peak surface electric field

=0.57 (K=0.82): Compromise between high energy gain and short filling time Aarhus University Hospital, Århus Sygehus

Linacs: Standing wave (SW) acceleration

20

SW acceleration Disk-loaded waveguide with reduced apertures at ends:

d

1. Full reflection of traveling waves at structure ends 2. Low field attenuation (l