Lecture 1 Introduction to RF for Accelerators

Dr G Burt Lancaster University

Engineering

Electrostatic Acceleration

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Van-de Graaff - 1930s A standard electrostatic accelerator is a Van de Graaf

These devices are limited to about 30 MV by the voltage hold off across ceramic insulators used to generate the high voltages (dielectric breakdown).

RF Acceleration

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By switching the charge on the plates in phase with the particle motion we can cause the particles to always see an

acceleration

You only need to hold off the voltage between two plates not the full accelerating voltage of the accelerator.

We cannot use smooth wall waveguide to contain rf in order to accelerate a beam as the phase velocity is faster than the speed of light, hence we cannot keep a bunch in phase with the wave.

Early Linear Accelerators (Drift Tube)

• Proposed by Ising (1925) • First built by Wideröe (1928) • Alvarez version (1955)

Replace static fields by time-varying fields by only exposing the bunch to the wave at certain selected points. Long drift tubes shield

the electric field for at least half the RF cycle. The gaps increase length with distance.

Cavity Linacs

• These devices store large amounts of energy at a specific frequency allowing low power sources to reach high fields.

Cavity Quality Factor • An important definition is the cavity Q factor, given

by

Where U is the stored energy given by, The Q factor is 2π times the number of rf cycles it

takes to dissipate the energy stored in the cavity. • The Q factor determines the maximum energy the

cavity can fill to with a given input power.

cPUQ ω=0

dVHU ∫=2

021 µ

00

exp tU UQ

ω= −

Cavities • If we place metal walls at

each end of the waveguide we create a cavity.

• The waves are reflected at both walls creating a standing wave.

• If we superimpose a number of plane waves by reflection inside a cavities surface we can get cancellation of E|| and BT at the cavity walls.

• The boundary conditions must also be met on these walls. These are met at discrete frequencies only when there is an integer number of half wavelengths in all directions.

(ω/c)2=(mπ/a)2+ (nπ/b)2+ (pπ/L)2

L The resonant frequency of a rectangular cavity can be given by

Where a, b and L are the width, height and length of the cavity and m, n and p are integers

a

Pillbox Cavities

• Transverse Electric (TE) modes

• Transverse Magnetic (TM) modes ( ) ϕςϕ imnmmz ea

rJArE ±

= ,1, zt

nm

zt E

aikE ∇= 2,

2

ς( )zt

nmt Ez

aiH ∇×= ˆ2,

2

ςεω

( ) ϕςϕ imnmmz ear

JArH ±

= ,1

', zt

nm

zt H

aikH ∇= 2,

2

'ς( )zt

nmt Hz

aiE ∇×−= ˆ'2 ,

2

ςµω

011 2222 =

−+

∂∂

+

∂∂

∂∂ ψµεω

ϕ zk

rrr

rr

ϕψ imtm erkJA±= )(1

Wave equation in cylindrical co-ordinates

Solution to the wave equation

Bessel Function • Ez (TM) and Hz (TE)

vary as Bessel functions in pill box cavities.

• All functions have zero at the centre except the 0th order Bessel functions.

-0.5

-0.3

0.0

0.3

0.5

0.8

1.0

0 2 4 6 8 10

m=0

m=1

m=2 m=3

Jm(kTr)

kTr

First four Bessel functions.

One of the transverse fields varies with the differential of the Bessel function J’

All J’ are zero in the centre except the 1st order Bessel functions

Cavity Modes

TE1,1 TE0,1 TM0,1

TE2,1 TEr,θ

r θ

Cylindrical (or pillbox) cavities are more common than rectangular cavities.

The indices here are

m = number of full wave variations around theta

n = number of half wave variations along the diameter

P = number of half wave variations along the length

The frequencies of these cavities are given by f = c/(2π) * (ζ/r)

Where ζ is the nth root of the mth bessel function for TE modes or the nth root of the derivative of the mth bessel function for TE modes or

TM010 Accelerating mode Electric Fields

Magnetic Fields Almost every RF cavity operates using the TM010 accelerating mode.

This mode has a longitudinal electric field in the centre of the cavity which accelerates the electrons.

The magnetic field loops around this and caused ohmic heating.

TM010 Monopole Mode

0 0

0 10

2.405

00

2.405

00

i tz

z

r

i t

r

rE E J eR

HH

i rH E J eZ R

EE

ω

ωϕ

ϕ

−

−

=

==

− =

=

=

E

H

Beam

Z0=377 Ohms

A standing wave cavity

Accelerating Voltage

Position, z

Ez, at t=0 Normally voltage is the potential difference between

two points but an electron can never “see” this voltage as it

has a finite velocity (ie the field varies in the time it takes the electron to cross the cavity

Position, z

Ez, at t=z/v

The voltage now depends on what phase the electron

enters the cavity at.

If we calculate the voltage at two phases 90 degrees apart we get real and

imaginary components

Accelerating voltage • An electron travelling close to the speed of light traverses through a

cavity. During its transit it sees a time varying electric field. If we use the voltage as complex, the maximum possible energy gain is given by the magnitude,

• To receive the maximum kick with multiple cells the particle should traverse the cavity in a half RF period (see end of lecture).

2cLf

=

( )/ 2

/

/ 2

,L

i z cb z

L

E eV e E z t e dzω+

−

∆ = = ∫

Transit time factor • An electron travelling close to the

speed of light traverses through a cavity. During its transit it sees a time varying electric field. If we use the voltage as complex, the maximum possible energy gain is given by the magnitude,

• Where T is the transit time factor given by

• For a gap length, g. • For a given Voltage (=E0L) it is clear that we get maximum energy gain for a

small gap.

( )/ 2

/0

/ 2

,L

i z cz

L

E eV e E z t e dz E LTω+

−

∆ = = =∫

( )

( )

( )/ 2

/

/ 2/ 2

/ 2

, sin

,

Li z c

zL

L

zL

gE z t e dzT g

E z t dz

ω πβλ

πβλ

+

−+

−

= =∫

∫

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

Tran

sit t

ime

fact

or, T

g/βλ

φ

V

φs

Vp

Phase stability is given by off-crest acceleration

Stable region

Overvoltage • To provide a stable bunch you often will accelerate off

crest. This means the particles do not experience the maximum beam energy.

• Vb=Vc cos(φs) = Vc q • Where Vc is the cavity voltage and Vb is the voltage

experienced by the particle, φ is the phase shift and q is known as the overvoltage.

For TM010 mode

( )

( )

( )

( )

/ 2/

/ 2

/ 2

0/ 2

/ 2

0/ 2

0

,

cos /

sin //

2sin / 2/

Li z c

zL

L

LL

L

V E z t e dz

E z c dz

z cE

c

L cE

c

ω

ω

ωω

ωω

+

−

+

−

+

−

= ℜ

=

=

=

∫

∫

( )

( )

0

0

cos2cos

zV E LT

V E L

ϕ

ϕπ

=

=

This is often approximated as

Where L=c/2f, T=2/π

Hence voltage is maximised when L=c/2f

Position, z

Ez, at t=z/v

Gas Breakdown • If we apply a high

voltage across a gap we can ionise the molecules in the intervening gas.

• At high pressure the mean free path is too low to gain enough momentum

• At low pressure there are not enough molecules to ionise.

Does this mean we don’t get breakdown in vacuum?

Field Emission • High electric fields can lead to electrons quantum

tunnelling out of the structure creating a field emitted current.

Once emitted this field emitted current can interact with the cavity fields.

Although initially low energy, the electrons can potentially be accelerated to close to the speed of light with the main electron beam, if the fields are high enough.

This is known as dark current trapping.

Field Enhancement • The surface of an accelerating

structure will have a number of imperfections at the surface caused by grain boundaries, scratches, bumps etc.

• As the surface is an equipotential the electric fields at these small imperfections can be greatly enhanced.

• In some cases the field can be increase by a factor of several hundred.

1

10

100

1000

10000

100000

1 10 100 1000h/b

Beta

h

2b

Elocal=β E0

Vacuum Breakdown • Breakdown occurs when a

plasma discharge is generated in the cavity.

• This is almost always associated with some of the cavity walls being heated until it vaporises and the gas is then ionised by field emission. The exact mechanisms are still not well understood.

• When this occurs all the incoming RF is reflected back up the coupler.

• This is the major limitation to gradient in most pulsed RF cavities and can permanently damage the structure.

Kilpatrick Limits • A rough empirical formula for the peak surface electric

field is

• It is not clear why the field strength decreases with frequency.

• It is also noted that breakdown is mitigated slightly by going to lower group velocity structures.

• The maximum field strength also varies with pulse length as t-0.25 (only true for a limited number of pulse lengths)

• As a SCRF cavity would quench long before breakdown, we only see breakdown in normal conducting structures.

Dark Current Trapping • When we looked at beam dynamics we saw

that we could inject a low energy bunch in a beta=v/c=1 structure and it could be accelerated to the speed of light and arrive on crest.

• If we have field emitted electrons in the structure these could also be capture and can travel with the main beam.

• The gradient at which this occurs is given by

Surface Resistance

δ

As we have seen when a time varying magnetic field impinges on a conducting surface current flows in the conductor to shield the fields inside the conductor.

However if the conductivity is finite the fields will not be completely shielded at the surface and the field will penetrate into the surface.

2δσωµ

=This causes currents to flow and hence power is absorbed in the surface which is converted to heat.

Skin depth is the distance in the surface that the current has reduced to 1/e of the value at the surface, denoted by

Current Density, J.

x

.

The surface resistance is defined as 1

surfR δσ=

For copper 1/σ = 1.7 x 10-8 Ωm

Power Dissipation • The power lost in the cavity walls due to ohmic heating is given by,

Rsurface is the surface resistance • This is important as all power lost in the cavity must be replaced by

an rf source. • A significant amount of power is dissipated in cavity walls and hence

the cavities are heated, this must be water cooled in warm cavities and cooled by liquid helium in superconducting cavities.

212c surface

P R H dS= ∫

Pulsed Heating

Pulsed RF however has problems due to heat diffusion effects.

Over short timescales (

Peak Surface Fields • The accelerating gradient is the average gradient seen by an

electron bunch,

• The limit to the energy in the cavity is often given by the peak surface electric and magnetic fields. Thus, it is useful to introduce the ratio between the peak surface electric field and the accelerating gradient, and the ratio between the peak surface magnetic field and the accelerating gradient.

max

2acc

EE

π=

cacc

VE

L=

max /2430/acc

H A mE MV m

=

Electric Field Magnitude

For a pillbox

Maximum Gradient Limits • All the limiting

factors scale differently with frequency.

• They also mostly vary with pulse length.

• The limiting factor tends to be different from cavity to cavity.

For a CW machine the gradient is limited by average heating instead. Also need to think about the electricity bill as 1 MW is £200 per day.

Average Heating • In normal conducting cavities, the RF deposits large

amounts of power as heat in the cavity walls. • This heat is removed by flushing cooling water through

special copper cooling channels in the cavity. The faster the water flows (and the cooler), the more heat is removed.

• For CW cavities, the cavity temperature reaches steady state when the water cooling removes as much power as is deposited in the RF structure. (Limit is ~ 1 MW but 500 kW is safer)

• This usually is required to be calculated in a Finite Element code to determine temperature rises.

• Temperature rises can cause surface deformation, surface cracking, outgassing or even melting.

• By pulsing the RF we can reach much higher gradients as the average power flow is much less than the peak power flow.

Q factor Pillbox

( )2

220 01 2.4052

EU R L Jπε=

( ) ( )2

2012

0

2.405c surfaceEP R R L R JZ

π= +

( ) ( )

( )

00

453 /2 1 /

453 / 2601 /

surface surface

RL L RQR L R R L RL RGL R

ωµ= =

+ +

= =+

Shunt Impedance

• Another useful definition is the shunt impedance,

• This quantity is useful for equivalent circuits as it relates the voltage in the circuit (cavity) to the power dissipated in the resistor (cavity walls).

• Shunt Impedance is also important as it is related to the power induced in the mode by the beam (important for unwanted cavity modes)

212

cs

c

VR

P=

TM010 Shunt Impedance 0

0 10

2

2.405

cE LV

i rH E JZ Rϕ

π=

− =

( )

( ) ( )

2

220

, 120

220

, 120

220

120

12

2.4052

2.405

2.405

c surface

c ends surface

c walls surface

c surface

P R H dS

E rP R r J drZ R

EP RL R JZ

EP R R L R JZ

π

π

π

=

=

=

= +

∫

∫

( )( ) ( )

2 40

231

2 5 10

2.405s

surfacesurface

Z L xRRR R L R Jπ

= =+

Cavity geometry • The shunt impedance is

strongly dependant on aperture

Figures borrowed from Sami Tantawi

Similarly larger apertures lead to higher peak fields. Using thicker walls has a similar effect. Higher frequencies need smaller apertures as well

Multicell • It takes x4 power to double the voltage in one cavity but only x2 to

use two cavities/cells to achieve the same voltage (Rs ~number of cells).

• To make it more efficient we can add either more cavities or more cells. This unfortunately makes it worse for wakefields (see later lectures) and you get less gradient per unit power.

• In order to make our accelerator more compact and cheaper we can add more cells. We have lots of cavities coupled together so that we only need one coupler. For N cells the shunt impedance is given by

This however adds

complexity in tuning, wakefields and the gradient of all cells is limited by the worst cell.

sintotal gleR NR=

Lecture 1 Introduction to RF for Accelerators Electrostatic AccelerationVan-de Graaff - 1930sRF AccelerationEarly Linear Accelerators (Drift Tube)Cavity LinacsCavity Quality FactorCavitiesPillbox CavitiesBessel FunctionCavity ModesTM010 Accelerating modeTM010 Monopole ModeA standing wave cavityAccelerating VoltageAccelerating voltageTransit time factorOvervoltageFor TM010 modeGas BreakdownField EmissionField EnhancementVacuum BreakdownKilpatrick LimitsDark Current TrappingSurface ResistancePower DissipationPulsed HeatingPeak Surface FieldsMaximum Gradient LimitsAverage HeatingQ factor PillboxShunt ImpedanceTM010 Shunt ImpedanceCavity geometryMulticell