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LINACS
Alessandra Lombardi and Jean-Baptiste Lallement
CERN
http://lombarda.home.cern.ch/lombarda/juasLINAC/
1
Contents
• Lesson 1 : introduction, RF cavity
• Lesson 2 : from RF cavity to accelerator
• Lesson3 : single particle dynamics in a
linear accelerator
• Lesson 4 : multi-particle dynamics in a
linear accelerator and –time permittingbeam dynamics tour of linac2
• Guided study : Drift Tube Linac Design
• Visit around PS : linac2 and linac3 ??
2
credits
• much of the material is taken directly from
Thomas Wangler USPAS course
(http://uspas.fnal.gov/materials/SNS_Front-End.ppt.pdf) and
Mario Weiss and Pierre Lapostolle report
(Formulae and procedures useful for the design of
linear accelerators, from CERN doc server)
• from previous linac courses at CAS and JUAS
by Erk Jensen, Nicolas Pichoff, Andrea
Pisent,Maurizio Vretenar ,
(http://cas.web.cern.ch/cas)
3
LECTURE 1
• what is a LINAC
• historical introduction
• parameters of a Radio Frequency (RF)
cavity
4
motivation
• particle beams accelerated in controlled
condition are the probe for studying/acting
the structure of matter and of the nucleus
• controlled condition means generating a
high flux of particles at a precise energy
and confined in a small volume in space
5
what is a linac
• LINear ACcelerator : single pass device that
increases the energy of a charged particle by
means of an electric field
• Motion equation of a charged particle in an

electromagnetic field
x  positionvector


d x q   dx  


E


B


2
dt
m 
dt

2
q, m  ch arg e, mass
 
E , B  electric , magneticfield
t  time
6
what is a linac-cont’ed

2
d x q   dx  


E


B


2
dt
m 
dt

type of particle :
charge couples with the
field, mass slows the
type of structure
acceleration
7
Rate of change of energy


d x q   dx  


E


B


2
dt
m 
dt

2


dp
  dx  
 q  E   B
dt
dt


can be written as :

p  momentum
W  energy
 

 

dW dx dp
dx 
dx



 q   E   B
dt
dt dt
dt 
dt

Energy change via the electric field
8
type of particles – 4 groups
• Electrons :
–
–
–
–
High energy collider
High-quality e- beam for FEL;
Medical/Industrial irradiation;
Neutron sources
• protons and light ions :
– Synchrotron injectors : High intensity, high duty-cycle
– Neutron sources : High Power. Material study, transmutation, nuclear fuel
production, irradiation tools, exotic nucleus production
• heavy ions
– Nuclear physics research : High intensity, high duty-cycle
– Implantation : Semi-conductors
– Driver for inertial-confinement fusion
• short lived particles (e.g. muons)
9
type of linacs
electric field
static
time varying
induction
Radio Frequency Linac
10
static linac
• device which provides a constant potential
difference (and consequently electric field).
Definition of the Volt, measure of the energy in
eV.
• acceleration is limited to few MeV. Limitation
comes from electric field breakdown
• still used in the very first stage of acceleration
when ions are extracted from a source.
11
induction linac
12
Radio Frequency Linac
• acceleration by time varying
electromagnetic field overcomes the
limitation of static acceleration
• First experiment towards an RF linac :
Wideroe linac 1928
• First realization of a linac : 1931 by Sloan
and Lawrence at Berkeley laboratory
13
Wideroe linac
potassium ionBUNCHED
lenght of the tube
proportional to the
velocity
25 kV 1 MHz
• the energy gained by the beam (50 keV) is
twice the applied voltage (25 keV at 1 MHz)
14
from Wideroe to Alvarez linac
• to proceed to higher energies it was necessary to
increase by order of magnitude the frequency and to
enclose the drift tubes in a cavity (resonator)
• this concept was proposed and realized by Luis Alvarez
at University of California in 1955 : A 200 MHz 12 m
long Drift Tube Linac accelerated protons from 4 to 32
MeV.
• the realization of the first linac was made possible by the
availability of high-frequency power generators
developed for radar application during World War II
15
principle of an RF linac
RF power supply
1) RF power source:
generator of electromagnetic wave of
a specified frequency. It feeds a
2) Cavity : space enclosed in a
metallic boundary which resonates
with the frequency of the wave and
tailors the field pattern to the
3) Beam : flux of particles that we
Wave guide
Power coupler
push through the cavity when the
field is maximized as to increase its
4) Energy.
Cavity
16
A (Linac) RF System:
transforms mains power
into beam power
RF line
Mains
220 V
DC Power
supply
RF Amplifiers
Beam in
W, i
Cavity
Beam out
W+DW, i
Transforms mains power
into DC power
Transforms DC power
Transforms RF power
into RF power
into beam energy
[klystron/tube efficiency] [efficiency=shunt impedance]
17
designing an RF LINAC
• cavity design : 1) control the field pattern inside
the cavity; 2) minimise the ohmic losses on the
walls/maximise the stored energy.
• beam dynamics design : 1) control the timing
between the field and the particle, 2) insure that
the beam is kept in the smallest possible volume
during acceleration
18
electric field in a cavity
•
assume that the solution of the wave equation in a bounded
medium can be written as
E( x, y, z, t )  E ( x, y, z )  e
function of space
 jt
function of time oscillatin
at freq = ω/2π
• cavity design step 1 : concentrating the RF power
from the generator in the area traversed by the beam in the
most efficient way. i.e. tailor E(x,y,z) to our needs by choosing
the appropriate cavity geometry.
19
cavity geometry and related
parameters definition
field
y
beam
x
Cavity
L=cavity length
z
1-Average electric field
2-Shunt impedance
3-Quality factor
4-Filling time
5-Transit time factor
6-Effective shunt impedance
20
standing vs. traveling wave
Standing Wave cavity : cavity where the forward
and backward traveling wave have positive
interference at any point
21
cavity parameters-1
• Average electric field ( E0 measured in V/m) is the
space average of the electric field along the direction
of propagation of the beam in a given moment in time
E( x, y, z, t )  E ( x, y, z )  e  jt
when F(t) is maximum.
L
1
E0   Ez ( x  0, y  0, z )dz
L0
• physically it gives a measure how much field is
available for acceleration
• it depends on the cavity shape, on the resonating
mode and on the frequency
22
cavity parameters-2
•
Shunt impedance ( Z measured in Ω/m) is defined as the ratio of the
average electric field squared (E0 ) to the power (P) per unit length (L)
dissipated on the wall surface.
Z
•
•
•
E
2
0
L

P
or
Z
2
0
E
dL
for TW

dP
Physically it is a measure of well we concentrate the RF power in the
useful region .
NOTICE that it is independent of the field level and cavity length, it
depends on the cavity mode and geometry.
beware definition of shunt impedance !!! some people use a factor 2 at
the denominator ; some (other) people use a definition dependent on the
cavity length.
23
cavity parameters-2
optimized (from the ZTT point of view) cavity offers
minimum surface for the max volume : spherical cavity.
y
x
z
24
cavity parameters-2
But a more realistic shape includes –at least- an iris for the beam to pass
through!
y
x
z
25
cavity parameters-3
• Quality factor ( Q dimension-less) is defined as the ratio
between the stored energy (U) and the power lost on the
wall (P) in one RF cycle (f=frequency)
2   f
Q
U
P
• Q is a function of the geometry and of the surface
resistance of the material :
superconducting (niobium) : Q= 1010
normal conducting (copper) : Q=104
example at 700MHz
26
cavity parameters-3
• SUPERCONDUCTING Q depends on
temperature :
– 8*109 for 350 MHz at 4.5K
– 2*1010 for 700 MHz at 2K.
• NORMAL CONDUCTING Q depends on
the mode :
– 104 for a TM mode (Linac2=40000)
– 103 for a TE mode (RFQ2=8000).
27
cavity parameters-4
• filling time ( τ measured in sec) has different definition on
the case of traveling or standing wave.
• TW : the time needed for the electromagnetic energy to
fill the cavity of length L
L
dz
tF  
v z
0 g 
velocity at which the energy
propagates through the cavity
• SW : the time it takes for the field to decrease by 1/e
after the cavity has been filled
tF 
2Q

measure of how fast the stored
energy is dissipated on the
28
wall
cavity parameters-5
• transit time factor ( T, dimensionless) is defined as the maximum
energy gain per charge of a particles traversing a cavity over the
average voltage of the cavity.
• Write the field as
Ez( x, y, z, t )  Ez ( x, y, z )e i (t )
• The energy gain of a particle entering the cavity on axis at phase
φ is
L
•
DW   qEz (o, o, z )e i (t  ) dz
0
29
cavity parameters-5
• assume constant velocity through the cavity
(APPROXIMATION!!) we can relate position
and time via
z  v  t  ct
• we can write the energy gain as
DW  qE0 LT cos 
• and define transit time factor as
L
 E  z e

 j


z
c
z
T 
0
L
 E  z dz
z
0




dz
T depends on the
particle velocity and on
the gap length. IT
DOESN”T depend on
the field
30
cavity parameters-5
• NB : Transit time factor depends on x,y (the
distance from the axis in cylindrical symmetry).
By default it is meant the transit ime factor on
axis
• Exercise!!! If Ez= E0 then
 L 
L=gap lenght
β=relativistic parametre
λ=RF wavelenght
sin 

 

T
 L 


  
31
cavity parameter-6
ttf for 100 keV protons, 200 MHz., parabolic distribution
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
-0.2
lfield (cm)
if we don’t get the length right we can end
up decelerating!!!
32
effective shunt impedance
• It is more practical, for accelerator designers
to define cavity parameters taking into
account the effect on the beam
• Effective shunt impedance ZTT
Z
E
2
0
L

P
measure if the structure
design is optimized

E T
ZTT 
2
0
L

P
measure if the structure is optimized
and adapted to the velocity of the
particle to be accelerated
33
limit to the field in a cavity
• normal conducting :
– heating
– Electric peak field on the cavity
surface (sparking)
• super conducting :
– quenching
– Magnetic peak field on the surface (in
Niobium max 200mT)
34
Kilpatrick sparking criterion
(in the frequency dependent formula)
f = 1.64 E2 exp (-8.5/E)
Kilpatrick field
40
electric field [MV/m]
35
30
25
20
15
10
5
0
0
100
200
300
400
500
600
700
800
900
1000
frequency [MHz]
GUIDELINE
nowadays : peak
surface field up to
2*kilpatrick field
Quality factor for normal
conducting cavity is
Epeak/EoT
35
summary of lesson 1
• first step to accelerating is to fill a cavity with
electromagnetic energy to build a resonant field.
in order to be most efficient one should :
1) concentrate the field in the beam area
2) minimise the losses of RF power
3) control the limiting factors to putting energy into
the cavity
• This is achieved by shaping the cavity in the
appropriate way
36
Photo gallery
• 352 MHz cavities for 3 MeV protons
• 80 MHz cavity for muons
37
352 MHz cavity
38
88 MHz test cavity
(made from an 114 MHz structure)
88 MHz
cavity
2 MW
amplifier
Nose
Cone
(closed gap)
39
LECTURE 2
• modes in a resonant cavity
• TM vs TE modes
• types of structures
• from a cavity to an accelerator
40
wave equation -recap
• Maxwell equation for E and B field:
 2
2
2
1 2  
 2  2  2  2 2 E  0
 x
y
z
c t 
•
•
In free space the electromagnetic fields are of the transverse electro
magnetic,TEM, type: the electric and magnetic field vectors are  to each
other and to the direction of propagation.
In a bounded medium (cavity) the solution of the equation must satisfy the
boundary conditions :


E//  0


B  0
41
TE or TM modes
• TE (=transverse electric) : the electric field is
perpendicular to the direction of propagation.
in a cylindrical cavity
n : azimuthal,
TE nml
m : radial
l longitudinal component
• TM (=transverse magnetic) : the magnetic
field is perpendicular to the direction of
propagation
n : azimuthal,
TM nml
m : radial
l longitudinal component
42
wave equation-recap
•
•
in cylindrical coordinates the solution for a TM wave can be expressed as
Ez  ARr  e j t k z z 
The function () is a trigonometric function with m azimuthal periods the function
R(r), is given with the Bessel function of first kind, of order m and argument sqrt
(2/c2-kz2)r:
1 d 2
 m2  0
2
 d
d 2 R 1 dR   2
m2 
2



k


R  0
z
dr 2 r dr  c 2
r2 
 2

2
  AJ K r 
Rr   AJ m 

k

r
z
m
r
 c2



•
,At the boundary (a cylinder of radius a), the condition for TM waves is Ez = 0, i.
e. Jm(Krr) = 0 , and the first solution (lowest frequency) is for the TM 01 wave,
with Kra = 2.405 or Kr = 2.405/a
•
dispersion relation links, for a given wave type and mode, the frequency of
oscillation  to the phase advance per unit length k
2
c2
 K r2  k z2
fixed by boundary
conditions
43
dispersion diagram
Backward wave
Forward wave Kz>0
cut off frequency
44
wave equation- recap
•
consider one component of the wave equation and express the solution as
a product of functions like (travelling wave case)
Ez  AR(r )( )e
j t  k z z 
z 

 v ph
t kz
 being the angular frequency and kz the phase advance per unit length;
•
the phase velocity must be matched to the velocity of the particle that
needs to be accelerated. In empty cavities Vph  c so the waves must be
slowed down by loading the cavity with periodic obstacles.
disc loaded cavity. The
obstacles delimit cells, and
each cell is a resonator,
coupled to its neighbours
through the central aperture
45
phase velocity /group velocity
Ez  AR(r )( )e
j t  k z z 
moving with the wave one can put (t - kz z) = 0
z 

 v ph
t kz
velocity of the wave
phenomenon > c to satisfy
boundary condition
the electromagnetic energy propagates with the a smaller
velocity, the group velocity, given by :
d
vg 
dk z
46
wave equation-recap
•
•
in a periodic structure of period l the solution of the wave equation is such
that
--- for a given frequency and mode of oscillation, and in absence of losses,
the solution may differ from a period to the next only by a factor like
 jk z l
(Floquet theorem)
e
•
--- the complicated boundary conditions can be satisfied only by a whole
spectrum of modes called space harmonics.
E r , z   e j t k z z   an r e  j 2n / l z
n
•
 d 2 an r  1 dan r 


 K rnan r   0

2
r dr
 dr

2n 


K      kz 

 c

l 
2
2
2
rn
For each n one has a travelling wave with its own, slowed down velocity
vph. The an are modified Bessel functions:
an(r) = An I0(krr)
dispersion relation for
a chain of coupled
oscillators
47
wave equation-recap
•
•
•
•
•
•
for a given wave type and mode, there is a limited passband between o and ; at these points, vg = 0;
the bigger the coupling between cells, the bigger the passband;
for a given frequency, there is an infinite series of space harmonics, with the same vg, but with different
vph;
when the electromagnetic energy propagates only in one direction (solid curves on the diagram) 
travelling wave accelerator, TW; when the energy is reflected back and forth at both ends of the cavity
(solid and dotted curves)  standing wave accelerator, SW;
if vph and vg in the same direction  forward wave; if in opposite directions  backward wave;
travelling wave accelerators operate near the middle of the passband, where vg is maximum and the
mode spacing biggest, see point A
standing wave accelerators operate on the lower or upper end of the passband, point B or C , because
only there the direct and reflected wave have the same vph, and they both accelerate particles. At these
points the group velocity is zero.
48
TE modes
dipole mode
quadrupole mode used in
Radio Frequency Quadrupole
49
TM modes
TM010 mode , most commonly used
accelerating mode
50
cavity modes
•
• 0-mode Zero-degree phase shift
from cell to cell, so fields adjacent cells
are in phase. Best example is DTL.
•
• π-mode 180-degree phase shift from
cell to cell, so fields in adjacent cells
are out of phase. Best example is
multicell superconducting cavities.
•
• π/2 mode 90-degree phase shift
from cell to cell. In practice these are
biperiodic structures with two kinds of
cells, accelerating cavities and
coupling cavities. The CCL operates in
a π/2structure mode. This is the
preferred mode for very long multicell
cavities, because of very good field
stability.
51
basic accelerating structures
• Radio Frequency Quadrupole
• Interdigital-H structure
• Drift Tube Linac
• Cell Coupled Linac
• Side Coupled Linac
52
derived/mixed structure
• RFQ-DTL
• SC-DTL
• CH structure
53
Radio Frequency Quadrupole
54
Radio Frequency Quadrupole
55
Radio Frequency Quadrupole
cavity loaded with 4 electrodes
TE210 mode
56
RFQ Structures
57
four vane-structure
1. capacitance between
vanetips, inductance in the
intervane space
2. each vane is a resonator
3. frequency depends on
cylinder dimensions (good
at freq. of the order of
200MHz, at lower
frequency the diameter of
the tank becomes too big)
4. vane tip are machined by a
computer controlled milling
machine.
5. need stabilization (problem
of mixing with dipole
modeTE110)
58
four rod-structure
•
capacitance between rods, inductance with
holding bars
•
each cell is a resonator
•
cavity dimensions are independent from the
frequency,
•
easy to machine (lathe)
•
problems with end cells, less efficient than 4vane due to strong current in the holding bars59
transverse field in an RFQ
+
-
alternating gradient
focussing structure with
period length 
(in half RF period the
particles have travelled a
length /2 )
+
-
-
+
+
-
60
transverse field in an RFQ
DT1
DT3
t0
t1
DT2
t2
DT5
t3
DT4
t4
time
RF signal
-
ion beam
+
-
+
-
+
animation!!!!!
-
+
electrodes
DT1,DT3......
t0,t1,t2........
....
DT2,DT4.....
61
acceleration in RFQ
longitudinal modulation on the electrodes creates a longitudinal
component in the TE mode
62
acceleration in an RFQ

D
(1 
)
2
2
longitudinal
radius of
curvature
modulation X aperture
aperture
beam axis
63
important parameters of the RFQ
 q  V  1  1  I o ka  I o mka 

B     2   2
 m0  a  f  a  m I o ka  I o mka 
type of
particle
limited by
sparking
Transverse field distortion due to
modulation (=1 for un-modulated
electrodes)
m 1
2 
E0T  2
V
m I o (ka)  I o (mka)    4
2
Accelerating efficiency : fraction of the field
deviated in the longitudinal direction
(=0 for un-modulated electrodes)
cell
length
transit
time factor
64
.....and their relation
2
 I o ka  I o mka 
m 1


 I 0 (ka)  1
2
2
 m I ka  I mka  m I (ka)  I (mka)
o
o
o
o


focusing
efficiency
accelerating
efficiency
a=bore radius, ,=relativistic parameters, c=speed of light, f= rf frequency,
I0,1=zero,first order Bessel function, k=wave number, =wavelength,
m=electrode modulation, m0=rest q=charge, r= average transverse beam
dimension, r0=average bore, V=vane voltage
65
RFQ
•
•
•
•
•
•
The resonating mode of the cavity is a focusing mode
Alternating the voltage on the electrodes produces an alternating
focusing channel
A longitudinal modulation of the electrodes produces a field in the
direction of propagation of the beam which bunches and
accelerates the beam
Both the focusing as well as the bunching and acceleration are
performed by the RF field
The RFQ is the only linear accelerator that can accept a low
energy CONTINOUS beam of particles
1970
Kapchinskij and Teplyakov propose the idea of the
radiofrequency quadrupole ( I. M. Kapchinskii and V. A.
Teplvakov, Prib.Tekh. Eksp. No. 2, 19 (1970))
66
Interdigital H structure
67
Interdigital H structure
the mode is the TE110
68
Interdigital H structure
•stem on alternating side of the drift tube force a longitudinal field between the
drift tubes
•focalisation is provided by quadrupole triplets places OUTSIDE the drift
tubes or OUTSIDE the tank
69
IH use
• very good shunt impedance in the low
beta region ((  0.02 to 0.08 ) and low
frequency (up to 200MHz)
• not for high intensity beam due to long
focusing period
• ideal for low beta heavy ion acceleration
70
Drift Tube Linac
71
DTL – drift tubes
Quadrupole
lens
Drift
tube
Tuning
plunger
Post coupler
Cavity shell
72
Drift Tube Linac
73
DTL : electric field
Mode is TM010
DTL
The DTL operates in 0 mode
for protons and heavy ions in the
range =0.04-0.5 (750 keV - 150 MeV)
1.5
E
1.5
1
1
0.5
0.5
0
Synchronism condition (0 mode):
0
0
20
40
60
0
-0.5
-0.5
-1
-1
-1.5
-1.5
80
20
100
40
120
60
140
80
l=
100
120
z
140
l
c
f
 
The beam is inside the “drift tubes” when the
electric field is decelerating
The fields of the 0-mode are such that if we
eliminate the walls between cells the fields are
not affected, but we have less RF currents
and higher shunt impedance
75
Drift Tube Linac
1. There is space to insert
quadrupoles in the drift
tubes to provide the strong
transverse focusing needed
at low energy or high intensity
2. The cell length ()
can increase to
account for the
increase in beta
 the DTL is the ideal
structure for the
low  - low W range
76
RFQ vs. DTL
DTL can't accept low velocity particles, there
is a minimum injection energy in a DTL due to
mechanical constraints
77
Side Coupled Linac
Chain of cells, coupled via slots and
off-axis coupling cells.
Invented at Los Alamos in the 60’s.
Operates in the /2 mode (stability).
CERN SCL design:
Each klystron feeds 5
tanks of 11 accelerating
cells each, connected by
3-cell bridge couplers.
Quadrupoles are placed
between tanks.
The Side Coupled Linac
multi-cell Standing Wave
structure in /2 mode
frequency 800 - 3000 MHz
for protons (=0.5 - 1)
Rationale: high beta  cells are longer  advantage for high frequencies
• at high f, high power (> 1 MW) klystrons available  long chains (many cells)
• long chains  high sensitivity to perturbations  operation in /2 mode
Side Coupled Structure:
- from the wave point of view, /2 mode
- from the beam point of view,  mode
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Room Temperature SW structure:
The LEP1 cavity
5-cell Standing Wave
structure in  mode
frequency 352 MHz
for electrons (=1)
To increase shunt impedance :
1. “noses” concentrate E-field in “gaps”
2. curved walls reduce the path for RF currents
“noses”
BUT: to close the hole between
cells would “flatten” the dispersion
curve  introduce coupling slots to
provide magnetic coupling
80
example of a mixed structure : the
cell coupled drift tube linac
linac with a reasonable shunt impedance in the range of 0.2 <  < 0.5, i. e.
at energies which are between an optimum use of a DTL and an SCL
81
accelerator
example of a mixed structure : the
cavity coupled drift tube linac
Single Accelerating CCDTL tank
1 Power coupler
/ klystron
ABP group seminar 16 march 06
Module
CCDTL – cont’ed
• In the energy range 40-90 MeV the velocity of the
particle is high enough to allow long drifts between
focusing elements so that…
• …we can put the quadrupoles lenses outside the drift
tubes with some advantage for the shunt impedance but
with great advantage for the installation and the
alignment of the quadrupoles…
• the final structure becomes easier to build and hence
cheaper than a DTL.
• The resonating mode is the p/2 which is intrinsically
stable
83
Coupling cell
1st Half-tank
(accelerating)
2nd Half-tank
(accelerating)
ABP group seminar 16 march 06
overview
Ideal range of frequency
beta
Particles
RFQ
Low!!! - 0.05
40-400 MHz Ions / protons
IH
0.02 to 0.08
40-100 MHz Ions and also
protons
DTL
0.04-0.5
100-400
MHz
Ions / protons
SCL
Ideal Beta=1
But as low as
beta 0.5
800 - 3000
MHz
protons / electrons
take with
CAUTION!
85
Summary of lesson 2
• wave equation in a cavity
• loaded cavity
• TM and TE mode
• some example of accelerating structures
ad their range of use
86