1. PHY862 Accelerator Systems Hadron linacs (protons, H-minus, ions)
Peter N. Ostroumov Professor of Physics
Michigan State University

2. Content LINAC layout, LINAC systems Proton/H-minus linacs RFQ
Pulsed and CW linacs
Linacs for light ions (protons, H-minus, deuterons)
Linacs for heavy ions up to uranium
Proton/H-minus linacs
RF system
Focusing
Accelerating structures
RFQ
RF resonators
Beam dynamics
Heavy ion linacs
Multi-charge acceleration
Literature:
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

3. Conditions for RF acceleration
Two conditions for RF acceleration should be fulfilled:
The wave has electric field component in the direction of particle motion
The wave phase velocity is equal to the particle velocity: synchronism
Two periodic systems can be used for the RF acceleration
Periodically loaded waveguides: acceleration by a traveling wave. Periodic reflections from the conducting walls reduces phase velocity of the wave below the speed of light
So far traveling wave structures are use in electron linacs only
Periodic coupled resonator structure: acceleration by a standing wave. Standing wave is composed of a sum of two traveling waves prorogating in opposite directions
Room temperature linacs
Single resonators structures
Mainly superconducting cavities
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

4. Transit time factor
Electric field distribution in the accelerating gap in resonator
Note: sometimes q is the charge and sometimes q is a number of electrons removed from an atom
General expression for the energy gain
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

5. Energy Gain  is the synchronous phase
Note : q or qe in ion accelerator
Transit time factor ( T or TTF)
In linear hadron accelerators the phase is referenced to the crest
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

6. Transit time factor for even function Ez
If the electrical center coincides with geometrical center
If the velocity change in the gap is small
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

7. Transit time factor in DTL gap, simplified
Some approximations g Ez z
g/2
-g/2
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

8. Transit time factor for a multi-gap SC resonators
Ez is the odd function Ez is the even function
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

9. Typical RF Linac structure
Hadron RF Linac Layout
Typical RF Linac structure
Front end
Medium-energy
section
High-energy section
Drift Tube Linac (DTL)
Separated DTL (SDTL)
IH-structure
SC cavities
Ion source
Radio Frequency
Quadrupole
Coupled Cavity Linac
(Side coupled structure
Disk-and Washer Structure
Annular Coupled Structure)
SC Cavities
(Elliptical
Spoke-loaded TEM-class)
Frequency jump
Lattice transition
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

10. Hadron Linear Accelerator Systems
Ion source
Accelerating structures
Standing wave structures (resonators)
To reduce the cost of the linac, radio frequency can change as velocity increases
Focusing structure
Pulsed or DC power supplies
RF power amplifiers
Based on vacuum tubes, klystrons, solid state
HV modulators, cooling system
Beam diagnostics
Sensors and electronics
Control system
Cryogenic system
Vacuum system
Provide residual pressure below 10-7 Torr
In SC resonators, vacuum is ~10-9 Torr
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

11. RF linacs RF Linacs CW NC* SC Pulsed NC ATLAS ISAC-I (RFQ, IH) ISAC-II
RIKEN inj.
SARAF RFQ
ANL RFQ
FRIB RFQ
RISP RFQ
GANIL RFQ
ATLAS
ISAC-II
INFN
ReA3
SARAF
ADS-IMP
FRIB
ADS front end
SPIRAL-2
PIP-II
EURISOL
LANSCE
Synchrotron
Injectors (FNAL,KEK, CERN, IHEP….)
MMF (Moscow)
SNS
CSNS
SNS
ESS
CSNS
CERN SPL
*Low-energy,
several MeV/u
Heavy-ions
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

12. Typical parameters of hadron linacs
Normal conducting (usually pulsed machines due to limitations of thermal issues )
Beam current: up to 200 mA
Beam energy: up to1 GeV for protons or H-minus
Uranium (UNILAC - GSI): up to 11 MeV/u
Duty cycle up to 12% (LANSCE)
Superconducting ion accelerators (CW)
ATLAS
50 SC cavities, ~70 MV total voltage
238U up to 10 MeV/u
PIAVE-ALPI (INFN, Legnaro, Italy)
40 SC cavities, ~50 MV total voltage
132Xe up to ~7 MeV/u
ISAC (TRIUMF) –about 40 MV total voltage
FRIB: 200 MeV/u Uranium, 400 MeV/u light ions, 400 kW beam power, CW
SNS (NC up to 187 MeV and SC from 187 MeV to 1 GeV)
1.4 MW proton beam on target
Projects
FNAL PIP-II: 0.8 GeV, 1 mA CW
GANIL: 5 mA, 40 MeV, q/A=1/3
SARAF: 5 mA 40 MeV, q/A=1/2
IFMIF: 125 mA, 40 MeV deuterons
ESS: pulsed, 2 GeV, 5 MW
MYRRHA, CW, 600 MeV, 4mA
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

13. Standing wave structure: DTL (Alvarez)- drift tube linac
Long cylinder resonator
Loaded with drift tubes
Electromagnetic field is
TM010 like
Drift tubes are usually
used to house focusing devices: magnetic quadrupoles
Traveling wave structures are not efficient for low velocities due to high RF losses (heat in the walls)
Proton and ion accelerators use standing wave structures
Decrease d to reduce 0
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

14. DTL Protons, f=198.2 MHz, resonator diameter is ~1 meter
From 0.75 to 20 MeV from 90 MeV to 100 MeV
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

15. Drift Tube Linac (DTL) Fall 2018
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

16. Wideroe or interdigital structure
Wideroe or Sloan–Lawrence coaxial-line
structure in a π−3π configuration
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

17. IH and CH structures Focusing triplet Fall 2018
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

18. IH and CH structures Very high shunt impedance up to  ~ 0.5 Fall 2018
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

19. JPARC H-minus linac Ion Species H- Energy 400 MeV Peak intensity
50 mA*
Duty
1.25 %
Average beam power
133 kW
Cavity type
RT DTL, RT CCL
Frequency
324/972 MHz
Status
Under operation
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

20. J-PARC Linac Layout DTL SDTL
　　 Drift-Tube LInac　　　　　　　 　Separated DTL　　　 Annular-Ring Coupled Structure (ACS)
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

21. J-PARC DTL F=325 MHz, H-minus, 3 MeV to 50 MeV
H-minus accelerators are popular as injectors to synchrotrons
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

22. Side coupled structure, LANSCE, SNS section
Energy range: MeV
High shunt impedance ~50 M/m
See Lecture 7-10, p
Room temperature
Known as CCL – coupled cavity linac
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

23. Annular coupled structure at JPARC
972 MHz
Axially symmetric coupling cell
High shunt impedance
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

24. High power amplifiers for DTL
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

25. Klystron
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

26. 805 MHz Klystron
Klystron in its solenoid mounted on its pulse transformer
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

27. JPARC klystron gallery
Klystron gallery of the J-PARC linac
972MHz
330m
324MHz
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
P.N. Ostroumov Lecture PHY862 "Aceclerator Systems"
Fall 2018

28. Fixed velocity and variable velocity accelerating structures
Normal Conducting
Beam
β/2
Normal or Super Conducting
Beam
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
βOPT/2
P.N. Ostroumov Linac Overview - Introduction
Fall 2018 28
June 15, 2014

29. CW Linacs 200 MeV/u FRIB linac Fall 2018
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

30. FRIB Linac configuration
Cavity Type
QWR
HWR 0
0.041
0.085
0.285
0.53
f [MHz]
80.5
322
Va [MV]
0.810
1.80
2.09
3.70
Eacc [MV/m]
5.29
5.68
7.89
7.51
Ep /Eacc
5.82
5.89
4.22
3.53
Bp /Eacc [mT/(MV/m)]
10.3
12.1
7.55
8.41
R/Q [Ω]
402
455
224
230
G [Ω]
15.3
22.3
77.9
107
Aperture [m]
0.036
0.040
Leff ≡  [m]
0.153
0.317
0.265
0.493
Lorenz detuning [Hz/(MV/m)2]
< 4
Specific
1.4E+9
2.0E+9
5.5e+9
9.2E+9 QL
6.3E+6
1.9E+6
5.6E+6
9.7E+6 N=
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

31. High power heavy ion linac is required for production of radioactive beams
Beam power is limited by available current from the ECR
ECR LINAC Target
St1
St1
St1
Ions
Uranium
Energy
400 MeV/u
Power
400 kW
q is the ion charge state
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

32. Stripper effect on beam parameter
Stripping lowers intensity in each charge
state
Main issue is the stripper damage due to
(a) heating
(b) radiation
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

33. Effective shunt impedance of accelerating structure
From Yakovlev’s lecture, p. 45. V is the effective voltage which includes transit time factor, R is the shunt impedance of the cavity
In Linacs we also use effective shunt impedance per unit length, L is the length of the accelerating structure
The maximum accelerating field in resonators is limited with breakdown field EPEAK
Depending on specific type of the resonator, E0 is lower than EPEAK by factor of 2-6
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

34. Peak fields in accelerating cavities
Normal conducting structures made from copper
Kilpatrick limit was introduced in 1950s, it is an empirical formula
In modern structures, electric field exceeds Kilpatrick limit by a factor of
1.5 – 2.0
Superconducting structures
Peak magnetic field is limited by quench, theoretical value is ~200 mT at 2K
Peak electric field is limited by the surface quality. ~120 MV/m can be achieved. Operational values are lower
Peak fields can not be measured
These ratios are known from the simulations of the resonator design
EACC can be obtained experimentally from the stored energy
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

35. Linac economics The linac cost is sum of capital and operation cost
The capital cost is the cost of accelerating structure
The operational cost is the electric bill and maintenance effort
The cost of a linac depends from the choice of an average accelerating gradient
Total cost, L is the length of linac
Capital cost per meter CL
Capital cost per watt of power CP
Energy gain
Power loss in the resonators’ walls, E includes TTF
Beam power
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

36. Linac cost Total cost Structure power cost Beam power cost
Structure length cost
Cost
E (MV/m)
Accelerating field can be limited by breakdowns in resonators
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

37. Continuous Wave Linac (100% duty cycle): NC or SC ?
Required wall plug power to create accelerating field
where  is the efficiency of the RF generator
Typical example: 1 GeV CW linac
Superconducting CW linac is much more economic than NC
Both pulsed or CW SC linacs require NC front end for
~0.1 to 10 MeV/u depending on q/A and duty factor
Transition energy is higher for pulsed linacs: SNS – 187 MeV
ESS MeV
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

38. Notations Axial distance s z Phase
Circular Accelerators
Hadron Linear Accelerators
Axial distance s z
Phase
With respect to zero crossing (sin-like)
With respect to maximum (cos-like)
Particle kinetic energy U W
Energy gain U W
Energy deviation E W
Phase advance of transverse oscillations 
 (sometimes, )
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

39. Radial dependence of the accelerating field
See Lectures 7-10, p 19,20
The radial dependence of the accelerating field is notable only for hadron accelerating structures due to <c
The Bessel functions appear as a result of wave equation solution in axially-symmetric structures
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

40. Longitudinal motion Compare to Lecture 5,
now we have radial dependence
of the accelerating field
In most of accelerators particles perform radial oscillation close to axis in the
way that << 1 and the value of the modified Bessel function is close to 1.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

41. Hamiltonian 𝑊
is generalized momentum which is canonically conjugate to the generalized motion coordinate  Hamiltonian describes particle oscillations around synchronous particle. If we assume that particle energy and velocity are changing slowly during particle oscillations then the Hamiltonian does not depend on time and it is a constant of motion.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

42. Phase space trajectories
Potential energy
Phase trajectory equation for each value of H
Separatrix extension in phase is ~3s
Relief of potential function and a family of phase trajectories 42
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

43. Separatrix
For stability condition, the synchronous phase must be negative The value of Hamiltonian, corresponding to separatrix, is
New variable, more commonly used in Linacs
Separatrix equation
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

44. Acceptance .
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

45. Accelerating field and phase trajectories
This separatrix is plotted for a conservative approximation (“fish”)
These trajectories include acceleration, non-conservative approximation (“golf club” shape) 4.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

46. Stable and unstable phases in standing wave
P.N. Ostroumov Lecture PHY862 "Accelerator Systems" 1 46
Fall 2018

47. Small longitudinal oscillations
Matched Mismatched
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

48. Adiabatic invariant theorem for small oscillations
The phase trajectory of small oscillations is an ellipse
The amplitude and frequency of oscillations are slowly varying functions of time – apply adiabatic invariant theorem for harmonic oscillations, Sa=const
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

49. Theory of multi-q beam acceleration
Energy gain per nucleon
q is the ion charge state,
A is the mass number
Fixed velocity profile
(RFQ, RT DTL), energy gain per nucleon
will be the same for any q/A if
Velocity is defined from energy gain per nucleon
Variable velocity profile
(SC Linac)
E0=const,
Tune phases of individual cavities
Multi-q heavy-ion beam
acceleration
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

50. Synchronous phase as a function of uranium ion charge state
Synchronous phase as a function of uranium ion charge state. The designed synchronous phase is –30 for q0 =75.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

51. Synchronous phase of multi-q beam
Single accelerating gap E
q=75 g
(z)
-0.15
-0.1
-0.05
0.05
0.1
0.15
Distance, m w t
q=77
q=73
Earlier arrival
Later arrival S
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

52. Separatrix and small longitudinal oscillations
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

53. Uranium beam stripping and total voltage, 400 kW, 400 MeV/u
Bunching efficiency =80% q
V, MV
Efficiency (%)
St.
IECR (pA) 29
3790
100 no
5.25
29-82
1677= 25 1 21
1352=
6.3 2 83
1292=
1.33 3
395
Multiple charge state acceleration
St.
From
ECR
After St1
After St2
After St3
Efficiency
(%)
IECR (pA) 2
28-29
69-73
88-91 -
147
3.6 3
58-61
79-83 97
5.4
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

54. RF Defocusing in Particle Accelerator
Electric field lines between the ends drift tubes. If accelerating, the field is focusing at input and defocusing at output. While field level is increasing while particles cross the gap to provide longitudinal beam bunching, the defocusing effect is larger.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

55. Defocusing due to accelerating field
Cylindrical coordinate system
Apply Maxwell’s equation, Lectures 7-10, p.12
Vacuum, =1
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

56. Field Distribution Accelerating gap of an resonator
Accelerating field distribution
Radial field distribution
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

57. Earnshaw's Theorem
Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw in It is usually referenced to magnetic fields, but originally applied to electrostatic fields. It applies to the classical inverse-square law forces (electric and gravitational) and also to the magnetic forces of permanent magnets and paramagnetic materials or any combination (but not diamagnetic materials).
Effective potential created by static field
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

58. Equations of motion in x - and y - directions (Hill’s Equations) (see S. Lund’s slide 23 and 24):
The dissipative term
is negligible in Linacs
Focusing function for simple FD or FODO structure
No accelerating field With the accelerating field
Lp is the length of focusing period
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

59. Grid or Foil Focusing of Charged Particles
Foil or grid focusing—the defocusing effect is suppressed
RF Defocusing effect is suppressed by closing the the drift-tube hole at the exit of the gap with a foil thin enough to be crossed by particles.
First test: 1947, Alvarez linac
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

60. Quadrupole Focusing Lorentz Force
Arrows indicate direction of Lorentz force acting on positively charge particle moving from the screen. Field is proportional to distance from axis, G- gradient of quadrupole field.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

61. Various Types of Focusing Periods
FODO D F D F D F
FOD (Doublets) D F F D D F
FOF-DOD
F/2 D
F/2
F/2 D
F/2
Triplets
Fall 2018
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"

62. Radio Frequency Quadrupoles
Short history
RF quadrupole focusing properties
Creating longitudinal electric fields
Electric potential in the RFQ accelerating cell
Vane tip shaping
Transit time factor
Synchronous motion
Focusing
Potential expansion for arbitrary vane tip shape
Examples of beam physics design
Multi-harmonic buncher upstream of the RFQ
Different resonator structures for RFQs
4-vane structure: advantages and disadvantages
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

63. Quadrupole mass-spectroscopy
Different focusing effect for ions with different q/A
Can transport selected q/A, all other ions will be lost inside the structure
For slow ion beams, frequencies <several MHz
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

64. Stability of different ions in the mass-separator
Phase advance as a function of mass-to-charge ratio, A/q for different values of focusing field
Ion motion for 0 above 180 is unstable
Transfer matrix for periodic structure (Yue Hao’s lecture)
If , the transverse motion is unstable and ions are lost
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

65. Radio frequency quadrupole
RF voltage is applied to 4 electrodes (vanes) a
Potential function in cylindrical coordinates
Horizontal vanes
Vertical vanes
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

66. Axial accelerating field
To produce the axial field suppose we modulate the vane tips along the axial direction.
If this is done with x and y modulations that are 180 degrees out of phase, the on-axis potential will follow the potential variations of the vane tips and a sinusoidal on-axis electric field is produced.
Longitudinal electric field is
between A and B
Point B
Point B
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Point A
Fall 2018

67. Short history
4 electrodes with RF voltage have been used for mass separation since 1950s
I.M. Kapchinski and V.A. Teplyakov invented RFQ. The first publication in 1969 in the Proc. of HEP Accelerator Conference
First operational RFQ – 1972, Protvino, Russia
Linear Accelerator Structures with Space-Uniform Quadrupole Focusing
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

68. Significant contribution by Los Alamos group
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

69. CW Radio Frequency Quadrupoles
ANL
FRIB
GANIL
RAON
IMP
f, MHz
60.625
80.5
88.0
81.25
Type
4-vane with windows, brazed
4-vane, brazed
4-vane, bolted
A/q 7 3
6.8
V, kV 70
60-115
50 – 138.5 Kp
1.42 (1.6)
1.63
1.65
1.7
1.55
P, kW 60 90
177 94 57
L, m 4 5 6
W, keV/u
300
500
750
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

70. Vane tips
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

71. Potential distribution
The accelerating gap dimensions are much smaller than RF wavelength
Electrostatic approximation can be applied
Generalized solution can be found as a solution for Laplace equation with quadrupolar boundary conditions transversely and periodic conditions in z
Infinite Fourier-Bessel series
For “ideal” vane tip electrodes the potential is described as
This formula for quasi-electrostatic potential was introduced by
I.M. Kapchinskiy
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

72. Electrode surface Fall 2018
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

73. Physical interpretation of A
On axis, r=0
Potential difference across the bsl/2 cell length
A equals the fraction of the accelerating voltage
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

74. Transit time factor and energy gain
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

75. Synchronous particle in the RFQ
As in a DTL, the synchronous particle “sees” the accelerating field at phase s when the particle is located in the cell center
Cell center
the cross-section has exact quadrupole symmetry
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

76. Electric field Cylindrical coordinate system Time dependence Fall 2018
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

77. Transverse focusing This is Mathieu-Hill equation Simplifications
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

78. Apply “smooth” approximation
Valid for 0 <90
Typical value of 0 40, in high intensity RFQs, 40 <0 <90
Focusing DOES NOT depend from ion velocity
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

79. RFQ focusing
Use of electric rather than magnetic fields is superior for low velocity particles.
Use of RF focusing fields rather than DC fields allows higher peak fields
About twice of Kilpatrick limit
More conservative in CW RFQs, ~1.5
The focusing alternates in time but is spatially uniform. When the fields are focusing they focus everywhere in the RFQ. When the fields are defocusing they defocus everywhere in the RFQ.
Spatially uniform quadrupole focusing in the RFQ increases the fraction of space used for focusing to 100%.
The short focusing period (beta lambda) keeps the phase advance per focusing period small which helps keep the beam away from the unstable limit at =π.
Very efficient focusing for space charge dominated beams
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

80. Longitudinal motion The same as in DTL
Frequency of small longitudinal oscillations
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

81. Beam physics design of RFQs
Define vane tip modulation for machining
Design the geometry of the RF structure
Kapchinski –Teplyakov beam dynamics design
Keep bunch physical length constant along the RFQ to minimize space charge effects
Los Alamos introduced the beam dynamics design for proton/H-minus RFQs
Use radial matcher to match static phase space to dynamic phase space
Shaper
Gentle buncher
Acceleration
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

82. Vane tip modulation S1 S2 S3 S4 S5 S6 S7 S8 S9 x, y z L vanes LRFQ
Lf,in
Lf,out R0
S S S S S S6 S S S9
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

83. RFQ for heavy ions
Must operate for wide range of q/A – charge to mass ratio
Requires different level of RF power in the RFQ resonator
Beam space charge is not significant in heavy ion RFQs
Requires low frequency of the RFQ resonator: beam velocity after ion source is low
Heavy ion RFQs are used in the front end of CW SC linacs: must operate in CW regime
Focusing by RFQ
To provide we need to increase  (decrease frequency)
Operational frequencies of heavy ion RFQs ~ MHz
This results to long RFQs
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

84. RFQ potential for arbitrary shape of vane tips
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

85. RFQ resonators
4-vane structures are used in the high frequency range above about 200 MHz.
Most common structure for light ions especially protons.
Built with two specially configured end cells to produce a longitudinally uniform fields throughout interior of cavity.
Transverse electric field is localized near vane tips.
Magnetic field is longitudinal localized in four outer quadrants.
Efficiency is high because vane charging currents are uniform along the length of the vanes.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

86. 2.5 MeV proton RFQ
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

87. Several methods have been devised to suppress the effects of unwanted modes.
Vane coupling rings that electrically connect opposite vanes ensuring the same vane potentials. Shifts the dipole frequencies upwards eliminating their effect
Tuning rods that shift the dipole mode frequencies upwards. The simplest approach is rods attached to the end plates that extend into the midplane of each quadrant.
IH resonator (-type)
Adjustable slug tuners in all four quadrants along the outer walls. These also allow us to adjust the longitudinal vane voltage profile and compensate for nearby longitudinal modes.
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

88. 6.7-MeV RFQ at LANL
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

89. 4-rod structures Mode mixing is not an issue
Suitable for low frequencies 200 MHz
Good for pulsed operation
Shunt impedance is lower by factor of 3
compared to 4-vane structures
These structures have proven
in operation
Have not seen detailed 3D simulations
Some asymmetric distribution
especially for magnetic field
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

90. ReA3 RFQ
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

91. Beam matching in FODO channel
If =0, the ellipse is upright: easy to define matching condition to the focusing channel
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

92. Smooth approximation, beam matching, zero current
Mismatch beam envelope oscillation, linearize the envelope equation
Beam envelope oscillates with twice of betatron frequency x’
Matched ellipse
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

93. Unmatched Beam in Periodic Structure
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

94. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

95. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

96. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

97. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

98. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

99. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

100. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

101. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

102. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

103. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

104. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

105. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

106. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

107. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"
Fall 2018

108. Zero current matched beam in periodic focusing channel
0=43
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

109. Zero current mismatched beam in periodic focusing channel
0=43
One betatron oscillation takes place over 8.4 periods
One oscillation of the envelope takes place over 4.2 periods
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

110. BACKUP SLIDES
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

111. Real linac for ion beam acceleration
Pillbox or coaxial /4 (Quarter Wave Resonator) and /2 (Half Wave Resonator) cavities can accelerate particles
A string of pillbox (or /4 and /2) cavities powered individually can be tuned to accelerate particles synchronously
Continuous wave (100% duty cycle) superconducting accelerators
The cavity array can be constructed in one RF structure
DTL
RFQ
Coupled structures
Very economic for pulsed (low duty cycle) accelerators
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

112. Advanced EM Optimization of SC resonators
Advanced EM optimization : outer conductor:
form cylinder to conical shape
Drift tubes are highly optimized to reduce EPEAK
2.5 deg drift tube face tilt to compensate beam
steering effect
109 MHz
Frequency
72.75
MHz
beta
0.14
0.077
U0 at 1 MV/m
0.4
0.39 J
bl/2 39 32 cm
EPEAK at 1 MV/m
5.0
4.6
MV/m
BPEAK at 1 MV/m 92 76 Gs G 40 26
Ohm
Rsh/Q
548
575
Voltage per cavity
2.1
2.5 MV
Dynamic LHe load at 4K 6 11 W
72.75 MHz
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018

113. Radial dependence of the accelerating field
Traveling waves in the infinite periodic structures
Only one harmonic effects on beam acceleration for which 0=s=/k0c
Accelerating cavity is an resonator therefore the field is standing wave
P.N. Ostroumov Lecture PHY862 "Accelerator Systems"
Fall 2018