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General introductionF. Gerigk (CERN/BE/RF)

LINEAR ACCELERATORS

OVERVIEW

• Historical developments,

• Fundamental concepts,

• Characteristics of accelerating cavities,

• Electron/hadron accelerators

COCKROFT - WALTON (1932)voltage multiplier + proton accelerator (< 1 MeV)

typically used up to 750 kV

crucial technology: voltage multiplier

the original machine(200 keV)

CERN Linac2 pre-injectoruntil 1993 (750 keV)

VAN DER GRAAFF GENERATOR (1931)

• a DC voltage is connected to the lower electrode (7),

• charges are transported (4) to the dome (1), where they are collected by the upper electrode (2)

• until a spark equalises the potentials

• 1 MV for 90 $!

(< 25 MV, tandem operation)

crucial technology: charge separation and accumulation

5 MV generator in 1933 (MIT, Round Hill, USA)

20 MeV accelerator in 1981 (NSF, Daresbury, UK)

• one sphere contains an ion source, the other one a target,

• beam through the air or later through vacuum,

From DC to RF acceleration

THE WIDERÖE LINAC (1927)

period length increases with

velocity:

energy gain:

the RF phase changes by 180 deg, while the particles travel from one tube to the next

E-field

particles

crucial technology: RF oscillators & synchronism

The use of RF enables to have ground potential on both sides of

the accelerator. This allows a limitless cascade of accelerating gaps!!

• the Wideröe linac was only efficient for low-energy heavy ions,

• higher frequencies (> 10 MHz) were not practical, because then the drift tubes would act more like antennas,

• when using low frequencies, the length of the drift tubes becomes prohibitive for high-energy protons:

BUT:

e.g. 10 MHz proton acceleration

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THE ALVAREZ LINAC (1946)

after WW2 high-power high-frequency RF sources became available (radar technology):

most old linacs operate at 200 MHz!

the RF field was enclosed in a box: RF resonator

While the electric fields point in the

“wrong direction” the particles are shielded

by the drift tubes.crucial technology: high-freq. RF sources & RF resonators

inside a drift tube linac

Linac2 at CERN, 50 MeV

DIFFERENCES BETWEEN HADRON AND ELECTRON

ACCELERATION

Newton: Einstein:

rest energy:

total energy: 0

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energy [MeV]

v/c - electrons (Einstein)v/c - protons (Einstein)v/c - protons (Newton)

relativistic factor:

PROTON VS. ELECTRON ACCELERATION

• protons change their velocity up to the GeV range (β=0.95 at W=2 GeV),

➡accelerating structures (distance between gaps) need to be adapted to the changing velocity,

• electrons are almost immediately relativistic (β=0.95 at W=1.1 MeV),

➡basically from the source onwards one can use the same accelerating structure (optimised for β=1.0) for the rest of the linac,

Example of a 2/3 π-mode travelling wave structure for electrons

synchronism condition:- explanations on 2/3 π-mode in appendix

FUNDAMENTAL CAVITY CHARACTERISTICS

BASICS OF RF ACCELERATION I

gap

energy gain of a particle with charge q:

-L/2 -L/2

passing a gap with the electric field E:

RF phase

this can be written as:

average electric field on axis

cavity or cell length

transit time factor

synchronous phase

BASICS OF RF ACCELERATION I

gap

energy gain of a particle with charge q:

-L/2 -L/2

passing a gap with the electric field E:

RF phase

this can be written as:

average electric field on axis

cavity or cell length

transit time factor

synchronous phase

BASICS OF RF ACCELERATION I

gap

energy gain of a particle with charge q:

-L/2 -L/2

passing a gap with the electric field E:

RF phase

this can be written as:

average electric field on axis

cavity or cell length

transit time factor

synchronous phase

BASICS OF RF ACCELERATION IIaverage electric field:

transit time factor:

ignoring the velocity change in the cavity and assuming a constant field between -g/2 and g/2, T simplifies to:

assuming:

FUNDAMENTAL CAVITY CHARACTERISTICS: SHUNT IMPEDANCE

shunt impedance (linac definition):

maximising ZT2: maximising energy gain per length for a given power loss

be careful: shunt impedance (synchrotron definition):

FUNDAMENTAL CAVITY CHARACTERISTICS: (R/Q)

quality factor of a resonator: Q= f(surface resistance, geometry)

acceleration efficiency per unit stored energy: (r/Q)= f(geometry)

surface losses

(independent of surface losses!)

DESIGNERS OF NORMAL CONDUCTING CAVITIES ARE OPTIMISING FOR:

• maximum effective shunt impedance ZT2(high electric efficiency), different structures are efficient for different particle velocities,

• peak fields below a certain threshold (avoid sparking and breakdowns),

• maintain synchronism between the cells and the particles,

• choose a number of coupled cells so that: i) structure can still have a flat field (stabilisation), ii) power consumption is compatible with existing power sources, iii) there is enough space for transverse focusing (quadrupoles between multi-cell cavities)

SUPERCONDUCTIVITY

• In 1965 the High-Energy Physics Lab (HEPL) at Stanford University accelerated electrons in a lead plated cavity.

• In 1977 HEPL operated the first superconducting linac (with niobium cavities), providing 50 MeV with a 27 m long linac.

• In 1996, 246 metres of SC (Nb sputtered on Cu) cavities are used in LEP with an installed voltage (per turn) of 1320 MV (electrons).

• In 2005 SNS commissions a SC proton linac providing 950 MeV in 230 m (incl. transverse focusing).

• 2010 DESY is constructing XFEL (electrons), which will provide 20 GeV of acceleration (electrons) within 1.6 km.

• European Spallation Source (ESS) is funded and will be constructed in Lund (Sweden).

SPALLATION NEUTRON SOURCE, OAKRIDGE

1 GeV, 1-1.4 MW on target, 60 Hz, linac pulse length 1 ms

WHEN ARE SC CAVITIES ATTRACTIVE?

Instead of Q values in the range of ~104, we can now reach 109 - 1010, which drastically reduces the surface losses (basically down to ~0) ➜ high gradients with low surface losses

However, due to the large stored energy, also the filling time for the cavity increases (often into the range of the beam pulse length):

(only valid for SC cavities)

PULSED OPERATION & DUTY CYCLES FOR RF, CRYO, BEAM DYNAMICS

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beam duty cycle

cryogenics duty cycle

RF duty cycle

• beam duty cycle: covers only the beam-on time,

• RF duty cycle: RF system is on and needs power (modulators, klystrons)

• cryo-duty cycle: cryo-system needs to provide cooling (cryo-plant, cryo-modules, RF coupler, RF loads)

Depending on the electric gradient, beam current, particle velocity, and pulse rate, SC cavities can actually be less cost efficient than NC cavities!

Nevertheless, one can generally get higher gradients (for high beta) than with NC standing-wave cavities! (E.g. XFEL cavities: ~23.6 MeV/m in a 9-cell 1300 MHz cavity, vs 3-4 MeV/m in traditional NC standing wave cavities.)

LEP Nb on Cu cavity

XFEL 9-cell cavity

ANL triple spoke cavity

THANK YOU!!

MATERIAL USED FROM:

• M. Vretenar : Introduction to RF Linear Accelerators (CAS lecture 2008)

• T. Wangler : Principles of RF Linear Accelerators (Wiley & Sons)

• H. Braun: Particle Beams, Tools for Modern Science (5th PP Workshop, Islamabad)

• D.J. Warner: Fundamentals of Electron Linacs (CAS lecture 1994, Belgium, CERN 96-02)

• Padamsee, Knobloch, Hays: RF Superconductivity for Accelerators (Wiley-VCH).

• F. Gerigk: Formulae to Calculate the Power Consumption of the SPL SC Cavities, CERN-AB-2005-055.

APPENDIX:Basics of Accelerating Cavities

WAVE PROPAGATION IN A CYLINDRICAL PIPE

Maxwells equationssolved in cylindrical coordinates for

the simplest mode with E-field on axis: TM01

propagation constant:

cut-off wave number:

wave number:

TM01 field configuration

E-fieldB-field

λp

+ boundary conditions on a metallic cylindrical pipe: Etangential=0

cut-off wavelength in a cylindrical wave-guide (TM01 mode)

TM01 waves propagate for :

the phase velocity is:

dispersion relation

a

Brioullin diagram (dispersion relation)

no waves propagate below the cut-off frequency, which depends on the radius of the cylinder,

each frequency corresponds to a certain phase velocity,

the phase velocity is always larger

than c! (at ω=ωc: kz=0 and vph=∞),

energy (and therefore information) travels at the group velocity vgr<c,

synchronism with RF (necessary for acceleration) is impossible because a particle would have to travel at v=vph>c!

group velocity:

We need to slow down the phase velocity!

put some obstacles into the wave-guide: e.g: discs

2b

L

2a

h

Dispersion relation for disc loaded travelling wave structures:

disc loaded structure:

structure with: vph=c at kz= 2π/3 (SLAC/LEP injector)

Brioullin diagram

damping:

Example of a 2/3 travelling wave structure

synchronism condition:

TRAVELLING WAVE STRUCTURES

• The wave is damped along the structure and can be designed as “constant-impedance” structure or as “constant-gradient” structure.

• Travelling wave structures are very efficient for very short (us) pulses, and can reach high efficiencies (close to 100% for CLIC), and high accelerating gradients (up to 100 MeV/m, CLIC).

• are used for electrons at β≈1,

• cannot be used for ions with β<1: i) constant cell length does not allow for synchronism, ii) long structures do not allow for sufficient transverse focusing,

STANDING WAVE CAVITIES• Closing of the walls on both sides

of the waveguide or disc-loaded structure yields multiple reflections of the waves.

• After a certain time (the filling time of the cavity) a standing wave pattern is established.

• Due to the boundary conditions only certain modes with distinct frequencies are possible in this resonator:

Brioullin diagram

dispersion relation

STANDING WAVE CAVITIES• for n cells the fundamental

pass-band has n modes from 0 to (n-1)π/(n-1),

• the frequency difference between 0 and π-mode is given by the cell-to-cell coupling k,

• usually the 0, π/2, or π-mode is used for acceleration,

• cell length can be matched to any particle velocity!

• the mode names correspond to the phase difference from one cell to the next,

0-MODE CAVITIES: ALVAREZ DTL• most common structure

for protons and ions with β<0.3-0.4 (< 50 - 100 MeV for protons),

• one gap per βλ,

• optimum for gap/cell length ≈0.2 - 0.3,

• at higher energies the drift tubes become very long and increase the losses,

Electrical efficiency depends on the electric field (P∼E2) and beam current (50 MeV DTL with 3.2 MV/m, Pbeam ≈ Pcopper ≈ 4.7 MW ⇒ ηDTL ≈ 50%)

Pumping port

Power coupler

Quadrupoles

Drift Tubes