1. Ion and proton linacs Øystein Midttun
– University of Agder and European Spallation Source (ESS)
FYS4550/FYS9550, University of Oslo, Autumn 2015
CERN’s Linac2 - proton injector for the pre-accelerators of the LHC.

2. Literature Books Lectures and proceedings from
Theory and Design of Charged Particle Beams – Martin Reiser
RF linear accelerators – Thomas Wangler
Lectures and proceedings from
CERN Accelerator School –
M. Vretenar – 2009
Joint Universities Accelerator School –
A. Lombardi – 2014
J.-B. Lallement – 2015

3. Content of this lecture
Linear accelerators
What, how, why? z
Fundamental of RF cavities
Commonly used accelerating structures
Beam dynamics

4. A LINear ACcelerator (linac) is a device where charged
particles 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

5. This lecture focuses on RF linear accelerators
Electrostatic
Time varying
Induction
Radio frequency

6. Electrostatic linacs are limited by the high voltage
Constant potential difference (electric field)
Acceleration limited to few MeV (electric field breakdown)
Still used in very first stage of acceleration. Range: keV
750 kV Cockcroft-Walton
Linac2 injector at CERN from 1978 to 1992

7. Induction linacs are suited for medium energy,
high current, short pulse applications
Faraday’s law:
A time varying magnetic field
generates an electric field

8. The first radio frequency linac was designed by Rolf Widerøe in 1928
Rolf Widerøe used a 1 MHz, 25 kV RF source to accelerate potassium ions up to 50 keV.
The optimal choice of the distance between acceleration gaps is d = βλ/2 = βc/2f, where d is the distance between drift tube gaps, βc is the velocity of the particle, and λ and f are the wavelength and frequency of the RF wave.
1 MHz, 25 kV rf source to accelerate potassium ions up to 50 keV
Optimum gap distance d = βλ/2 = βc/2f

9. Problems of the Widerøe linac are long gap distances
at low frequencies, and power loss at high frequencies
Above 10 MHz the drift tubes basically become antennas

10. 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
The Alvarez drift tube linac became possible because by the development of high-power, high-frequency RF amplifiers. This technology was invented during World War II to power radar systems. The typical frequency of these systems was 200 MHz.
Each cell is the equivalent of a resonant cavity
f = 1/(2πLC)
L: shape of cavity
C: shape and distance between drift tubes

11. RF linear accelerators are mainly used for:
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)
Production of high intensity proton beams
compared with synchrotrons, linacs have higher repetition rate, and are less affected by resonances
High energy electron colliders
no synchrotron radiation

12. Difference in ion and electron velocity
Protons have a rest mass of MeV/c2: They follow Newton mechanics up to some tens of MeV, and then slowly become relativistic. In the GeV range the velocity is almost constant: v≈0.95c at Ek=2.1 GeV.
Proton and ion linacs have a structure adapted to the increased particle velocity, whereas synchrotrons cover the range where the velocity is nearly constant.
Electrons have a rest mass of 511 keV/c2 (proton mass/1836): Relativistic in the keV range, and increase the velocity to the MeV range: v≈0.95c at Ek=1.1 MeV. This velocity is reached after a few meters in a linac.
Electron linacs consist of a series of identical accelerating structures.

13. Synchronism condition between accelerated particle and RF-wave
t (travel between two cells) = T (RF period)
d: distance between two consecutive cells
d = vt = βc/f = βλ d
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

14. Linacs adapt the gap distance to the velocity, whereas
circular accelerator have a fixed gap distance d d
d = βc/f = βλ
Linear accelerator:
Particles accelerated by a sequence of gaps (all at the same RF phase).
Distance between gaps increases proportionally to the particle velocity, to keep synchronicity.
Used in the range where β increases. “Newton” machine
Circular accelerator:
Particles accelerated by one (or more) gaps at given positions in the ring.
Distance between gaps is fixed. Synchronicity only for β~const, or varying (in a limited range!) the RF frequency.
Used in the range where β is nearly constant. “Einstein” machine
d = 2πR = constant

15. Travelling wave cavities are essentially used for acceleration
of ultra-relativistic particles, i.e. electrons
Particle velocity must be close to the phase velocity of the travelling wave (vph)
Disc-loaded waveguide for vph=c at a given frequency
Travelling wave structures can not be used for protons or ions with v<c:
constant cell length does not allow synchronism
structures are long without space for transverse focusing
(from previous lecture: Fr = e(1-β2)Er)

16. Fundamental of RF cavities
Linear accelerators
What, how, why? z
Fundamental of RF cavities
Commonly used accelerating structures
Beam dynamics

17. Fundamental cavity characteristics: Electric field (V/m)
E-field z
Average electric field when E(t) is maximum
-L/2
L/2
L= cavity length
Time varying field:
Energy gain of a particle with charge q and phase φ
Average electric field: E0 measured in V/m.
Average electric field on beam axis in the direction of the beam propagation at a given moment in time when E(t) is maximum.
Measure how much field is available for acceleration
Depends on the cavity shape, resonating mode and frequence

18. Fundamental cavity characteristics:
Transient time factor (dimensionless)
Cavity
E-field
We assume constant velocity (z=cβt) through the gap, and write the energy gain as z
-L/2
L/2
with the transient time factor
L= cavity length
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)
Transit time factor: T dimension-less.
Defines the ratio of the energy gained in the time varying RF field to that in a DC field.
T is a measure of the reduction in energy gain caused by the sinusoidal time variation of the field in the gap.
T depends on the particle velocity and on the gap length. It does not depend on the field.
If we assume constant field and velocity in the gap, T simplifies to

19. Fundamental cavity characteristics: Quality factor (dimensionless)
E-field z
-L/2
L/2
L= cavity length
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)

20. Characteristics of RF cavities and linacs Shunt impedance (Ω/m)
Cavity
E-field z
-L/2
L/2
L= cavity length
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
Shunt impedance (per unit of length): Z measured in Ω/m.
Defines the ratio of the average electric field squared (E02) to the power (P) per unit of length (L) dissipated on the walls surface.
Independent on the field level and cavity length. Depends on cavity mode and geometry.
Effective shunt impedance: ZT2.
More practical for accelerator designers who want to maximize the particle energy gain per unit power dissipation.
While the shunt impedance measures if the structure design is optimized, the effective shunt impedance measures if the structure is optimized and adapted to the velocity of the particle to be accelerated.

21. RF proton and ion linacs use standing waves for particle acceleration
Particle must be in phase with the E-field, and the cell length matched with β
Mode
Cell length βλ
π/2
βλ/4
2π/3
βλ/3 π
βλ/2
To obtain an accelerating structure we close our disc-loaded structure at both ends with metallic walls -> multiple reflections of the waves.
Boundary condition at both ends is that electric field must be perpendicular to the cover -> only some modes on the disc-loaded dispersion curve are allowed.
Named from the phase difference between adjacent cells

22. Transverse electric and transverse magnetic resonance modes
In a bounded medium the electric and magnetic field must obey the boundary conditions:
E∥=0
B⊥=0
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: azimuthal
n: radial

23. The two Transverse Electric modes for accelerating structures are
TE11: dipole mode
TE11: quadrupole mode

24. The most common Transverse Magnetic mode
for accelerating structures is TM01

25. Fundamental of RF cavities
Linear accelerators
What, how, why? z
Fundamental of RF cavities
Commonly used accelerating structures
Beam dynamics

26. The radiofrequency quadrupole (RFQ)

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

28. The RFQ focuses, accelerates, and bunches the beam+ -
Quadrupole focusing βλ + + + +
Acceleration + + - - -
Opposite vanes (180°)
Adjacent vanes (90°)
The RFQ consists of four electrodes (vanes) between which we excite an RF Quadrupole mode (TE210). This electric focusing channel has an alternating gradient with the period of the RF. Since the electric focusing does not depend on the velocity, this structure is ideal for low β values.
The vanes have a longitudinal modulation with a period of βλ. This structure creates a longitudinal component of the electric field. The modulation corresponds exactly to a series of RF gaps and provides acceleration.
The alternating longitudinal field has also a bunching effect on the beam. The RFQ is the only linear accelerator that can accept a low energy continuous beam of particles.
Bunching 28

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

30. The interdigital H-mode structure uses a TE11 field for acceleration
The stems are located on alternating sides of the drift tubes, and this configuration forces a longitudinal field between the drift tubes. The structure uses the π-mode resonance.
The beam focalisation can be provided by quadrupole triplets placed outside the drift tubes or outside the tank
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

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

32. DTLs operate with a standing RF-waves in the 0-mode
Drift tube linacs operate in the 0-mode. Since the wall current then becomes zero, the walls are useless. The RF-currents flow on the external tank, and on the tubes.
The drift tubes are held by stems, and they shield the particles during the half RF period when the electric field on axis is decelerating.
The drift tubes can house focusing magnetic quadrupoles. Cell length should increase with increasing β.
DTLs are ideal for for low β ( ), high current beams of light or heavy ions
DTL prototype for CERN Linac4 (352 MHz).

33. A Coupled Cavity DTL (CCDTL) consists of a series of DTL-like
tanks (0-mode), coupled by coupling cells (π/2 mode)
DTL-like tank
2 drift tubes
Coupling cell
Quadrupoles are placed between tanks for longer focusing lengths, and easier access and alignment. The CCDTL has lower cost than a standard DTL

34. The π/2-mode is used to stabilize long chains of coupled resonators
Dispersion curve for a 7-cell coupled resonator chain.
the modes allowed will be equally spaced in k
The number of modes will be identical to the number of cells
k represents the phase difference between the field in adjacent cells
When β increases, and the cells become longer, it is advantageous to use higher frequencies to reduce the overall length. To reduce the RF cost, linacs use high-power RF sources which feed a large number of coupled cells. Longer chains have higher sensitivity to perturbation, and the π/2-mode is therefore preferred to eliminate these perturbations.
Even though, the π/2 mode has lower acceleration efficiency than the 0 or π modes, is it interesting to use because of the reduced cost of RF. Long chains of coupled cells that can
be fed by single high-power RF sources are less expensive than using many smaller power units.
Perturbing
mode
Perturbing
mode
Operating
mode
Perturbations from the π/2 mode will cancel each other

35. 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-mode
From the beam point of view: π-mode
Frequency range MHz
Proton β = (ideal value is 1)

36. π-mode structures (PIMS) is a standing wave linac structure
for 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

37. Every proton linac structure has a characteristic curve
of shunt impedance (=acceleration efficiency) as function
of energy, which depends on the mode of operation.
CERN’s Linac4
Even though the shunt impedance is around 20% lower for a π-mode structure (PIMS) than for a Side-Coupled Linac (SCL) operating at 704 MHz, the PIMS has the advantage of using the same RF frequency (352 MHz) as all the other accelerating structures in Linac4, thus simplifying and standardising the linac RF system.
The choice of the best accelerating structure for a certain energy range depends on shunt impedance, but also on beam dynamics and construction cost

38. Superconducting cavities have less power losses, thus
(much) higher quality factor and shunt impedance
Spoke cavity
Operates in a TEM mode (coaxial resonator)
Low β ( )
Proton frequency: MHz
Multi gap cavities (elliptical) for proton or electron acceleration
Operates in the π-mode (cell-length βλ/2)
High β (0.5-1)
Proton frequency: MHz
Electron frequency: GHz
In a spoke cavity, the electric field across the gaps is generated by a magnetic field turning around some supports, the spokes. The spokes represent sections of short-circuited coaxial resonators. The field distribution resembles the interdigital H-mode structure.
But, they require a cryogenic system!

39. Comparison of some different RF accelerating structures
(not exhaustive list)
Cavity Type
Beta Range
Frequency
Particles
RFQ
Low! – 0.1
MHz
Protons, Ions IH
0.02 – 0.08
MHz
Ions (Protons)
DTL
0.05 – 0.5
MHz
SCL
0.5 – 1 (ideal is 1)
MHz
Protons, Electrons
Spokes
Elliptical
> 0.5
350 – 3000 MHz
Higher frequencies are economically convenient (shorter, less RF power, higher gradients possible) but limitation comes from mechanical precision in construction (tight tolerances are expensive!) and beam dynamics for ion linacs at low energy.
Electron linacs tend to use higher frequencies ( GHz) than ion linacs. Standard frequency 3 GHz (10 cm wavelength). No limitations from beam dynamics, travelling wave structures require less accurate machining than standing wave structures.
Proton linacs use lower frequencies ( MHz), increasing with energy (ex.: MHz): compromise between focusing, cost and size.
Heavy ion linacs tend to use even lower frequencies ( MHz), dominated by the low beta in the first sections.

40. The European Spallation Source (ESS) super conducting linac
vs. CERN’s normal conducting Linac4
ESS
2 GeV proton linac (0-90 MeV NC, GeV SC)
Total length: 600 m -> average gradient 3.3 MeV/m
The ESS accelerator is a 2 GeV proton linac. At this energy, superconducting cavities are required, and the accelerator can then use spokes cavities to accelerate the beam from the DTL to medium β.
Linac4 is a 160 MeV H- linear accelerator that will be used as injector to the PS booster at CERN. This energy does not require superconducting cavities, and the acceleration from the DTLs is made by CCDTLs and PIMS. This avoids using a cryogenic system.
Linac4
160 MeV H- linac (NC)
Total length: 80 m -> average gradient 2 MeV/m

41. Fundamental of RF cavities
Linear accelerators
What, how, why? z
Fundamental of RF cavities
Commonly used accelerating structures
Beam dynamics

42. Longitudinal dynamics – energy gain is maximum when φ=0
Acceleration
Deceleration

43. With an RF linac it is only possible to transfer energy
to a bunched beam
Unbuched
Bunched

44. In longitudinal dynamics: phase, time and longitudinal
position of a particle are used to describe the same thing
Referring to the synchronous particle: an imaginary particle whose velocity is used to determine the synchronicity with the electric field
φ-W
z-z’
t-W
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

45. Bunching is achieved and maintained by kicking
the later arriving particle with a higher electric field
– 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
Late arriving
Synchronous particle
Early arriving
The particles oscillate around the synchronous particle

46. Bunching is achieved and maintained by kicking
the later arriving particle with a higher electric field
– 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
Acceleration
and bunching
Acceleration
and de-bunching
We prepare (bunch) our beam for acceleration by:
- generating a velocity spread inside the beam
letting the beam distribute itself around the particle with the average velocity
Discrete bunching, DTL.
Adiabatic bunching, RFQ. We generate the velocity spread continuously with small longitudinal fields, and bunch the beam over several oscillation in the phase space (up to 100!). This method allows a better capture around the stable phase : 95% capture vs. 50% for discrete bunching. With an RFQ, we can start at -90º phase with some bunching cells, then progressively bunch the beam, and only in the last cells switch on the acceleration.
Deceleration
and bunching
Deceleration
and de-bunching
The particles oscillate around the synchronous particle

47. 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
Magnetic focusing
Proportional to the particle velocity
Electric focusing
Independent of the particle velocity

48. In a magnetic quadrupole, the B-field gradient
is proportional to the distance from the beam axis y N S
Quadrupole gradient: G [T/m]
Magnetic field
Magnetic force x
– A quadrupole focuses in one plane and defocuses in the other
– Alternation between focusing and defocusing along the beam line
x envelope
y envelope

49. Space charge force in a uniform cylindrical beamFr
Gauss’ law
Ampere’s law

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

51. RF defocusing The fields vary in time as the particles cross the gap.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.

52. Transverse focusing equilibriumx’ x
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
Ion linac:
Phase advance = External focusing - RF defocusing - space charge - instabilities
If the phase advance is too large, we run into instabilities because of resonances. If it is too small, the beam becomes too large in size.
The transverse RF defocusing is proportional to 1/γ2, which means that is disappears at relativistic velocity (transverse magnetic force cancels the transverse RF electric force).
The transverse defocusing from space charge disappears at high velocity due to an induced magnetic field.
External focusing is required only to control the emittance and to stabilize the beam against instabilities (as wakefields and beam breakup).
Electron linac:
Phase advance = External focusing - RF defocusing - space charge - instabilities
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. Ion and proton linacs Linear accelerators What, how, why?z
Fundamental of RF cavities
Commonly used accelerating structures
Beam dynamics