Ion and proton linacs Øystein Midttun – University of Agder and European Spallation Source (ESS) FYS4550/FYS9550, University of Oslo, Autumn 2015 1.

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Ion and proton linacsØystein Midttun– University of Agder and European Spallation Source (ESS)

FYS4550/FYS9550, University of Oslo, Autumn 2015

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Literature

• Books– Theory and Design of Charged Particle Beams – Martin Reiser

– RF linear accelerators – Thomas Wangler

• Lectures and proceedings from– CERN Accelerator School – http://cas.web.cern.ch/cas/

• M. Vretenar – 2009

– Joint Universities Accelerator School – https://espace.cern.ch/juas/SitePages/Home.aspx • A. Lombardi – 2014

• J.-B. Lallement – 2015

3

Linear acceleratorsWhat, how, why?

z Fundamental of RF cavities

Commonly used accelerating structures

Beam dynamics

Content of this lecture

4

€

r a=

qm( r E+

r v×

r B)

A LINear ACcelerator (linac) is a device where chargedparticles acquire energy moving on a linear path

Type of accelerated particles• Electrons• Protons and light ions• Heavy ions

Type of accelerating structure• Electric field for acceleration• Magnetic field for focusing / steering

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This lecture focuses on RF linear accelerators

Linear accelerators

Electrostatic Time varying

Induction Radio frequency

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Electrostatic linacs are limited by the high voltage

750 kV Cockcroft-Walton Linac2 injector at CERN from 1978 to 1992

Constant potential difference (electric field)

Acceleration limited to few MeV (electric field breakdown)

Still used in very first stage of acceleration. Range: 10-100 keV

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Induction linacs are suited for medium energy, high current, short pulse applications

€

r E⋅dl∫ =−

d r B

dt∫∫ ⋅dS

Faraday’s law: A time varying magnetic field generates an electric field

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The first radio frequency linac was designed by Rolf Widerøe in 1928

1 MHz, 25 kV rf source to accelerate potassium ions up to 50 keVOptimum gap distance d = βλ/2 = βc/2f

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Problems of the Widerøe linac are long gap distancesat low frequencies, and power loss at high frequencies

0 2 4 6 8 10 12 14 16 18 200

0.51

1.52

2.53

3.5

Proton energy [MeV]

Gap

dis

tanc

e [m

]

10 MHz RF source

Above 10 MHz the drift tubes basically become antennas

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Solution: enclose the gap between drift tubes in a cavity to store the energy in the form of a magnetic field

The Alvarez drift tube linac (DTL) – 1955 – is the basis of modern linear accelerator technology

Used RF amplifiers developed during WWII

Each cell is the equivalent of a resonant cavityf = 1/(2πLC)L: shape of cavityC: shape and distance between drift tubes

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RF linear accelerators are mainly used for:

1. Low energy accelerators for protons and ions– particles are synchronized with the RF field in the region where velocity

increases with energy. When velocity is almost constant with increased energy, synchrotrons are more efficient (multiple crossings)

2. Production of high intensity proton beams– compared with synchrotrons, linacs have higher repetition rate, and are

less affected by resonances

3. High energy electron colliders– no synchrotron radiation

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Difference in ion and electron velocity

0 100 200 300 400 500 600 700 800 900 1000-0.2

-1.66533453693773E-16

0.2

0.4

0.6

0.8

1

1.2

1.4

ElectronsProtons"Newton" protons

Kinetic energy [MeV]

β (v

/c)

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Synchronism condition between accelerated particleand RF-wave

d

t (travel between two cells) = T (RF period)d: distance between two consecutive cells

d = vt = βc/f = βλ

The ion linac cell length has to increase as β increases, and the linac will be made of a sequence of different accelerating structures matched to the ion velocity

An electron linac (β≈1) will be made of an injector and a series of identical accelerating structures

Linacs adapt the gap distance to the velocity, whereas circular accelerator have a fixed gap distance

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d

d = βc/f = βλ

d = 2πR = constant

d

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Travelling wave cavities are essentially used for acceleration of ultra-relativistic particles, i.e. electrons

Travelling wave structures can not be used for protons or ions with v<c: 1. constant cell length does not allow synchronism 2. structures are long without space for transverse focusing

(from previous lecture: Fr = e(1-β2)Er)

Particle velocity must be close to the phase velocity of the travelling wave (vph)Disc-loaded waveguide for vph=c at a given frequency

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Linear acceleratorsWhat, how, why?

z Fundamental of RF cavities

Commonly used accelerating structures

Beam dynamics

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Fundamental cavity characteristics:Electric field (V/m)

€

E0 =1

LE(0,0,z)dz

0

L

∫-L/2 L/2

L= cavity length

E-field

z

Cavity

Average electric field when E(t) is maximum

€

ΔW =q Ez(0,0,z)e− j(ωt+φ )dz

−L/ q

L/ 2

∫

Time varying field:Energy gain of a particle with charge q and phase φ

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Fundamental cavity characteristics:Transient time factor (dimensionless)

€

T = Ez(z)e−j

ωzβc

⎛

⎝ ⎜

⎞

⎠ ⎟

dz−L/ 2

L/ 2

∫ / Ez(z)dz−L/ 2

L/ 2

∫

-L/2 L/2

L= cavity length

E-field

z

Cavity

with the transient time factor

We assume constant velocity (z=cβt) through the gap, and write the energy gain as

€

ΔW =qE0LTcos(φ)

If we assume constant field and velocity in the gap, T simplifies to

€

T =sin

πLβλ

⎛

⎝ ⎜

⎞

⎠ ⎟

πLβλ

0 0.5 1 1.5 2 2.5

-0.4

-0.2

-1.11022302462516E-16

0.2

0.4

0.6

0.8

1

L/βλ

T

If we don’t get our gap length right, we could end up decelerating the beam! ratio of energy gain with E(t) to Emax(t)

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Fundamental cavity characteristics:Quality factor (dimensionless)

€

Q=2πf

PU -L/2 L/2

L= cavity length

E-field

z

Cavity

Defines the ratio of the stored energy (U) to the power lost on the wall (P) in one RF cycle (f = frequency).

Q is a function of the geometry and of the surface resistance of the cavity material.

Examples at 700 MHz

Superconducting (niobium): Q=1010 (depends on temperature)Normal conducting (copper): Q=104 (depends on cavity mode)

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Characteristics of RF cavities and linacsShunt impedance (Ω/m)

€

Z=E02 L

P

€

ZTT=(E0T)2 L

P

-L/2 L/2

L= cavity length

E-field

z

Cavity

The shunt impedance measures how well we concentrate the RF power in the useful region.

The effective shunt impedance measures if the structure is optimized and adapted to the velocity of the particle to be accelerated

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RF proton and ion linacs use standing wavesfor particle acceleration

Mode Cell length

0 βλ

π/2 βλ/4

2π/3 βλ/3

π βλ/2

Named from the phase difference between adjacent cells

Particle must be in phase with the E-field, and the cell length matched with β

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Transverse electric and transverse magnetic resonance modes

TE mode (transverse electric): TEmn

The electric field is perpendicular to the direction of propagation in a cylindrical cavity.

TM mode (transverse magnetic): TMmn

The magnetic field is perpendicular to the direction of propagation in a cylindrical cavity.

m: azimuthaln: radial

In a bounded medium the electric and magnetic field must obey the boundary conditions:E∥=0B⊥=0

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The two Transverse Electric modes for accelerating structures are

TE11: dipole mode TE11: quadrupole mode

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The most common Transverse Magnetic mode for accelerating structures is TM01

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Linear acceleratorsWhat, how, why?

z Fundamental of RF cavities

Commonly used accelerating structures

Beam dynamics

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The radiofrequency quadrupole (RFQ)

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The RFQ has a four vane structure, where each quadrant is a resonator

The RFQ focuses, accelerates, and bunches the beam

+ +

- - -

+ +

+ +

βλ

+

+

- -

Opposite vanes (180°) Adjacent vanes (90°)

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Quadrupole focusing

Acceleration

Bunching

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The interdigital H-mode structure uses a TE11 field for acceleration

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The interdigital H-mode structure uses a TE11 field for acceleration

Very good shunt impedance in the low beta region (β from 0.02 to 0.08 ) and low frequency (up to 200MHz)

Ideal for low beta heavy ion acceleration

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The drift tube linac (DTL) accelerates particles with a TM11 field

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DTLs operate with a standing RF-waves in the 0-mode

π-mode 0-mode (2π)

DTL prototype for CERN Linac4 (352 MHz).

DTLs are ideal for for low β (0.04-0.5), high current beams of light or heavy ions

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A Coupled Cavity DTL (CCDTL) consists of a series of DTL-like tanks (0-mode), coupled by coupling cells (π/2 mode)

Coupling cell

DTL-like tank2 drift tubes

Quadrupoles are placed between tanks for longer focusing lengths, and easier access and alignment. The CCDTL has lower cost than a standard DTL

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The π/2-mode is used to stabilize long chains of coupled resonators

ω

kππ

0 6.67 13.34 20.01 26.68 33.35

Dispersion curve for a 7-cell coupled resonator chain.1. the modes allowed will be equally

spaced in k2. The number of modes will be

identical to the number of cells3. k represents the phase difference

between the field in adjacent cells

Perturbations from the π/2 mode will cancel each other

Operating mode

Perturbingmode

Perturbingmode

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In side coupled linacs (SCL), the cells that are not excited are removed form the beam axis

From the wave point of view: π/2-modeFrom the beam point of view: π-mode

Frequency range 800-3000 MHzProton β = 0.5-1 (ideal value is 1)

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π-mode structures (PIMS) is a standing wave linac structurefor protons with β > 0.4

Simple structure with identical cell length (βλ/2) within a module

Since β is large, the phase slippage is small

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Every proton linac structure has a characteristic curve of shunt impedance (=acceleration efficiency) as function of energy, which depends on the mode of operation.

The choice of the best accelerating structure for a certain energy range depends on shunt impedance, but also on beam dynamics and construction cost

CERN’s Linac4

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Superconducting cavities have less power losses, thus (much) higher quality factor and shunt impedance

Multi gap cavities (elliptical) for proton or electron accelerationOperates in the π-mode (cell-length βλ/2)High β (0.5-1)Proton frequency: 350-700 MHzElectron frequency: 0.35-3 GHz

Spoke cavityOperates in a TEM mode (coaxial resonator)Low β (0.1-0.75)Proton frequency: 100-400 MHz

But, they require a cryogenic system!

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Comparison of some different RF accelerating structures (not exhaustive list)

Cavity Type Beta Range Frequency Particles

RFQ Low! – 0.1 40-500 MHz Protons, Ions

IH 0.02 – 0.08 40-100 MHz Ions (Protons)

DTL 0.05 – 0.5 100-400 MHz Protons, Ions

SCL 0.5 – 1 (ideal is 1) 600-3000 MHz Protons, Electrons

Spokes 0.1-0.75 100-400 MHz Protons, Ions

Elliptical > 0.5 350 – 3000 MHz Protons, Electrons

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The European Spallation Source (ESS) super conducting linacvs. CERN’s normal conducting Linac4

ESS 2 GeV proton linac (0-90 MeV NC, 0.09-2 GeV SC)Total length: 600 m -> average gradient 3.3 MeV/m

Linac4160 MeV H- linac (NC)Total length: 80 m -> average gradient 2 MeV/m

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Linear acceleratorsWhat, how, why?

z Fundamental of RF cavities

Commonly used accelerating structures

Beam dynamics

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Longitudinal dynamics – energy gain is maximum when φ=0

-90 -45 0 45 90 135 180 225 270

RF phase

RF si

gnal

E0

cos

(φ)

Acceleration

Deceleration

€

ΔW =qE0LTcos(φ)

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With an RF linac it is only possible to transfer energy to a bunched beam

€

ΔW =qE0LTcos(φ)

-90 0 90 180 270 360 450 540 630 720 810 900 990 1080

RF phase

RF si

gnal

– E

0 co

s(φ

)

Unbuched

Bunched

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In longitudinal dynamics: phase, time and longitudinal position of a particle are used to describe the same thing

φ-W z-z’ t-W

Referring to the synchronous particle: an imaginary particle whose velocity is used to determine the synchronicity with the electric field

Distance from bunch to bunch: βλ corresponds to 360° and 1 RF period in time 1/f.352.2 MHz, 50 MeV protons:β=0.314, λ=c/f, βλ=267 mm, T=2.84 ns.

In one RF period, a 50 MeV proton travels over 267 mm during 2.87 ns. On the plot -> φ = 4.5° -> z = -3.3 mm -> t = 3.55e-11 s

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-90 -45 0 45 90 135 180 225 270 RF phase

RF si

gnal

E0

cos

(φ)

Bunching is achieved and maintained by kicking the later arriving particle with a higher electric field

Synchronous particle

Early arriving

Late arriving

– The synchronous particle gets the correct kick by definition

– The late arriving particle gains slightly more energy

– The early arriving particle gains slightly less energy

The particles oscillate around the synchronous particle

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-90 -45 0 45 90 135 180 225 270 RF phase

RF si

gnal

E0

cos

(φ)

Bunching is achieved and maintained by kicking the later arriving particle with a higher electric field

Accelerationand bunching

Accelerationand de-bunching

Decelerationand bunching

Decelerationand de-bunching

– The synchronous particle gets the correct kick by definition

– The late arriving particle gains slightly more energy

– The early arriving particle gains slightly less energy

The particles oscillate around the synchronous particle

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At the same time as we accelerate our beam, we must keep it in focus transversally

x’

x

x’

x

x’

x

Defocused beam Apply a force towards the axis proportional to the distance from the axis:F(x) = -kx

Focused beam

€

r F=

r v×

r B

€

r F=q

r E

Magnetic focusing

Electric focusing

Proportional to the particle velocity

Independent of the particle velocity

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x envelopey envelope

In a magnetic quadrupole, the B-field gradientis proportional to the distance from the beam axis

Quadrupole gradient: G [T/m]

N

S N

S

Magnetic field Magnetic force

€

Bx =−GyBy =Gx

€

Fx =−qvGyFy =qvGxx

y

– A quadrupole focuses in one plane and defocuses in the other– Alternation between focusing and defocusing along the beam line

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Space charge force in a uniform cylindrical beam

Gauss’ law

€

E⋅dS∫∫ =1

ε 0ρdV∫∫∫

€

B⋅dl∫ =μ0 J⋅dS∫∫

€

2πrlEr =ρ

ε 0πr2l

€

Er =ρr2ε 0

€

2πrBθ =μ 0Jπr2

€

Bθ =μ0Jr2

€

J =ρv

€

r F= (q

r E+

r v×

r B)

Ampere’s law

beam

Fr

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Space charge is compensated by induced B-field at high β, no space charge issues in electron linacs

€

Fr =e(Er −vBθ )= (1e - β)Er =eEr

γ2

€

Bθ =βc

Er

€

J =ρv

€

Er =ρr2ε 0

€

Bθ =μ0Jr2

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RF defocusing

z

The fields vary in time as the particles cross the gap.

The fields acting on the particle depend on the radial particle displacement, which varies across the gap.

The particle velocity increases while the particle crosses the gap, so that the particle does not spend equal times in each half of the gap.

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Transverse focusing equilibrium

Ion linac:Phase advance = External focusing - RF defocusing - space charge - instabilities

Electron linac:Phase advance = External focusing - RF defocusing - space charge - instabilities

The equilibrium between external focusing force and internal defocusing forces defines the frequency of beam oscillations in the transverse plane.

We characterize these oscillations in terms of – phase advance per focusing period– phase advance per unit length

x’

x

Beam dynamics design needs to minimise emittance growth and halo development to:1. avoid uncontrolled beam loss (activation of machine parts)2. preserve small emittance (high luminosity in the following accelerators)

53

Linear acceleratorsWhat, how, why?

z Fundamental of RF cavities

Commonly used accelerating structures

Beam dynamics

Ion and proton linacs