Transcript Particle Accelerators - Lancaster University

LONGITUDINAL DYNAMICS
IN PARTICLE ACCELERATORS
by
Joël Le DuFF
(retired from LAL-IN2P3-CNRS)
Cockroft Institute, Spring 2006
1
Bibliography : Old Books
M. Stanley Livingston
High Energy Accelerators
(Interscience Publishers, 1954)
J.J. Livingood
Principles of cyclic Particle Accelerators
(D. Van Nostrand Co Ltd , 1961)
M. Stanley Livingston and J. B. Blewett Particle Accelerators
(Mc Graw Hill Book Company, Inc 1962)
K.G. Steffen
High Energy optics
(Interscience Publisher, J. Wiley & sons, 1965)
H. Bruck
Accelerateurs circulaires de particules
(PUF, Paris 1966)
M. Stanley Livingston (editor) The development of High Energy Accelerators
(Dover Publications, Inc, N. Y. 1966)
A.A. Kolomensky & A.W. Lebedev Theory of cyclic Accelerators
(North Holland Publihers Company, Amst. 1966)
E. Persico, E. Ferrari, S.E. Segre Principles of Particles Accelerators
(W.A. Benjamin, Inc. 1968)
P.M. Lapostolle & A.L. Septier
Linear Accelerators
(North Holland Publihers Company, Amst. 1970)
A.D. Vlasov
Theory of Linear Accelerators
(Programm for scientific translations, Jerusalem 1968)
2
Bibliography : New Books
M. Conte, W.W. Mac Kay
An Introduction to the Physics of particle Accelerators
(World Scientific, 1991)
P. J. Bryant and K. Johnsen The Principles of Circular Accelerators and Storage Rings
(Cambridge University Press, 1993)
D. A. Edwards, M. J. Syphers An Introduction to the Physics of High Energy Accelerators
(J. Wiley & sons, Inc, 1993)
H. Wiedemann
Particle Accelerator Physics
(Springer-Verlag, Berlin, 1993)
M. Reiser
Theory and Design of Charged Particles Beams
(J. Wiley & sons, 1994)
A. Chao, M. Tigner
Handbook of Accelerator Physics and Engineering
(World Scientific 1998)
K. Wille
The Physics of Particle Accelerators: An Introduction
(Oxford University Press, 2000)
E.J.N. Wilson
An introduction to Particle Accelerators
(Oxford University Press, 2001)
And CERN Accelerator Schools (CAS) Proceedings
3
Types of accelerators
Kinetic energy W
Electrons
Protons/ions
Electrostatic
Van de Graaf &Tandems
Betraton
Microtron
20-35 MeV
(Vivitron)
10-300 MeV
25-150 MeV
Cyclotron
10-100 MeV
Synchro-cyclotron
Synchrotron
Storage ring
Collider ring
Linacs
Linear collider
100-750 MeV
1-10 GeV
1-7 GeV (ESRF)
1-1000 GeV
10-100 GeV (LEP)
20 MeV-50 GeV (SLC)
50-1000 GeV (TESLA)
1-7 TeV (LHC)
50-800 MeV(LAMPF)
MeV(LAMPF) (LAMPF)
Total energy = Rest energy + Kinetic energy
= E0+ W
E0 = m0c2
electron E0=0,511 MeV
protons E0=938 MeV
4
Brief history of accelerators
1919 Rutherford gets the first nuclear reactions using natural alpha rays
(radio activity) of some MeV).
« He notes already that he will need many MeV to study the atomic nucleus »
1932 Cockcroft & Walton build a 700 KV electrostatic generator and break
Lithium nucleus with 400 KeV protons.
(Nobel Price in 1951)
1924 Ising proposes the acceleration using a variable electric field between drift tubes
( the father of the Linac).
1928 Wideroe uses Ising principle with an RF generator, 1MHz, 25 kV
and accelerate potassium ions up to 50 keV.
1929 Lauwrence driven by Wideroe & Ising ideas invents the cyclotron.
1931 Livingston demonstrates the cyclotron principle by accelerating hydrogen ions
up to 80 KeV.
5
Brief history of accelerators (2)
1932 The cyclotron of Lawrence produces protons at 1.25 MeV and « breaks atoms »
a few weeks after Cockcroft & Walton
(Nobel Prize in 1939)
1923 Wideroe invents the concept of betatron
1927 Wideroe builds a model of betatron but fails
1940 Kerst re-invents the betatron which produces 2.2 MeV electrons
1950 Kerst builds a 300 MeV betatron
6
Main Characteristics of an Accelerator
ACCELERATION is the main job of an accelerator.
•The accelerator provides kinetic energy to charged particles, hence increasing their
momentum.

•In order to do so, it is necessary to have an electric field E , preferably along the
direction of the initial momentum.
dp
eE
dt
BENDING is generated by a magnetic field perpendicular to the plane of the
particle trajectory. The bending radius  obeys to the relation :
p
 B
e
FOCUSING is a second way of using a magnetic field, in which the bending
effect is used to bring the particles trajectory closer to the axis, hence
to increase the beam density.
7
Acceleration & Curvature
x, r
Within the assumption:

E  E
z
s


o
becomes:
leading to:

B  Bz
the Newton-Lorentz force:

  
dp
 eE  ev  B
dt


dmv  
v2 
u  m ur  eE u  ev Bzur
dt

dp
 eE
dt
p
 Bz 
e
8
Energy Gain
In relativistic dynamics, energy and momentum satisfy the relation:
E  E0  W 
E2  E02  p2c2
Hence:
dE  vdp
The rate of energy gain per unit length of acceleration (along z) is then:
dE  v dp  dp  eE
z
dz
dz dt
and the kinetic energy gained from the field along the z path is:
dW  dE  eEzdz

W  e Ezdz  eV
where V is just a potential
9
Methods of Acceleration
1_ Electrostatic Field
Energy gain : W=n.e(V2-V1)
limitation
: Vgenerator =S Vi
Electrostatic accelerator
2_ Radio-frequency Field
L=vT/2
Synchronism :
v=particle velocity
also :
T= RF period
0
T
Lv 
2
2
Wideroe structure
10
Methods of Acceleration (2)
3_ Acceleration by induction
From MAXWELL EQUATIONS :
The electric field is derived from a scalar potential  and a vector potential A
The time variation of the magnetic field H generates an electric field E



E     A
t




B  H    A
11
Electrostatic accelerator
d.c. high voltage generator
A
c
c
é
l
é
r
a
t
e
u
r
c
o
l
o
n
n
e
Accelerating column
HV system used by Cockcroft & Walton to break the lithium nucleus
12
Electrostatic accelerator (2)
An insulated belt is used to
transport electric charges to a
HV terminal .
The charges are generated by
field effect from a comb on the
belt . At the terminal they are
extracted in a similar way.
The HV is distributed along the
column through a resistor.
Van de Graaf type electrostatic accelerator
13
Betatron
Induction law
d
2 d Bz
2  R E  
  R
dt
dt
Newton-Lorentz force
dp
dt
 eE  
1
2
eR
d Bz
dt
A constant trajectory also requires :
p   e R B0
dp
dt
eR
dB0
dt
Bo 
1
2
Bz
The betatron uses a variable
magnetic field with time. The
pole shaping gives a magnetic
field Bo at the location of the
trajectory, smaller than the
average magnetic field.
14
Cyclotron
At each radius r corresponds a velocity v for the
accelerated particle. The half circle corresponds to
half a revolution period T/2 and B is constant:
r
p
eB

mv
T
eB
2

m
eB
The corresponding angular frequency is :

2
eB
 2  fr 

r
T
m
Synchronism if :
 RF   r
v=Vsint
m = m0 (constant) if
W << E0
If so the cyclotron is isochronous
15
Cyclotron (2)
Here below the 27-inch cyclotron,
Berkeley (1932). The magnet was
originally part of the resonant
circuit of an RF current generator
used in telecommunications.
Cyclotron of M.S.Livingstone (1931)
On the left the 4-inch vacuum chamber
Used to validate the concept.
On the right the 11-inch vacuum chamber
of the Berkeley cyclotron that produced
1,2 MeV protons.
In both cases one single electrode (dee).
16
Cyclotron (3)
Cyclotron SPIRAL at GANIL
Here below is an artist view of the
spiral shaped poles and the radiofrequency system.
Here above is the magnet and its coils
SPIRAL accelerates radio-active ions
17
Cyclotron (4)
Cyclotrons at GANIL, Caen
18
Cyclotron (5)
Energy-phase equation:
Energy gain at each gap transit:
E  eV̂ sin 
Particle RF phase versus time:
  RF t 
where  is the azimuthal angle of trajectory
Differentiating with respect to time gives:
Smooth approximation allows:
 
Relative phase change at ½ revolution
And smooth approximation again:
  RF  r  RF  ec2 B
E
 r
 
Tr / 2 
 E
       RF2  1
r
 ec B 
d 
 RF E  1




dE E eVˆ sin   ec2B

19
Cyclotron (6)
Separating:
 E
dcos      RF2  1dE
eVˆ  ec B 
Integrating:

RF
2

cos  cos0   1  RF E  E0   
E  E0 
eVˆ  r0 
2eVˆE0 r0
with :
E0 
 0
r 0 
Rest energy
Injection phase
Starting revolution
frequency
20
Microtron
(Veksler, 1954)
The expression
eB
r 
m
shows that if the mass
increases, the frequency decreases :
m
r
Synchronism condition:
If the first turn is synchronous
:
electrons
Tr int eger turnint eger ( 01)
TRF
0.511 MeV
Energy gain per turn
protons
Tr  m  
0.938 GeV !!!
Since required energy
gains are large the
concept is essentially
valid for electrons.
21
Microtron
« Racetrack »
Allows to increase the energy gain per turn by using several
accelerating cavities (ex : linac section)
Synchronism is obtained
when the energy gain per
turn is a multiple of the
rest energy:
()/turn = integer
Carefull !!!! This is not a « recirculating » linac
22
The advantage of Resonant Cavities
- Considering RF acceleration, it is obvious that when particles get high
velocities the drift spaces get longer and one loses on the efficiency. The
solution consists of using a higher operating frequency.
- The power lost by radiation, due to circulating currents on the electrodes,
is proportional to the RF frequency. The solution consists of enclosing the
system in a cavity which resonant frequency matches the RF generator
frequency.


H ou J

Ez
RF
-Each such cavity can be independently
powered from the RF generator.
- The electromagnetic power is now
constrained in the resonant volume.
- Note however that joule losses will
occur in the cavity walls (unless made
of superconducting materials)
23
The Pill Box Cavity
From Maxwell’s equations one can derive
the wave equations :
A
 A  0 0  2  0
t
2
Ez
H
2
( A  E ou H )
Solutions for E and H are oscillating modes,
at discrete frequencies, of types TM ou TE.
For l<2a the most simple mode, TM010, has
the lowest frequency ,and has only two field
components:
E z  J 0 kr 
j


J 1 kr 
H
Z0
e
j t
k  2   2,62 a Z 0  377
 c
24
The Pill Box Cavity (2)
The design of a pill-box cavity can
be sophisticated in order to
improve its performances:
-A nose cone can be introduced in
order to concentrate the electric
field around the axis,
-Round shaping of the corners
allows a better distribution of the
magnetic field on the surface and a
reduction of the Joule losses. It
also prevent from multipactoring
effects.
A good cavity is a cavity which
efficiently transforms the RF
power into accelerating voltage.
25
Energy Gain with RF field
RF acceleration
In this case the electric field is oscillating. So it is for the potential.
The energy gain will depend on the RF phase experienced by the
particle.
 Eˆ z dz  Vˆ
W  e Vˆ cos 
E z  Eˆ z cos  RF t

 Eˆ z cos  t
Neglecting the transit
time in the gap.
26
Transit Time Factor
Oscillating field at frequency  and which amplitude
is assumed to be constant all along the gap:
V cos t

cos

t

E z E0
g
Consider a particle passing through the middle of
the gap at time t=0 :
zvt
The total energy gain is:
sin  / 2
W eV
 eVT
 /2

T
g
v
g/2
eV
W 
cos z dz

g g / 2
v
transit angle
transit time factor
(0<T<1)
27
Transit Time Factor (2)
Consider the most general case and make use of complex notations:
E  ee 0 Ez z e dz
j t
g
 t   z  p
v
p is the phase of the particle entering the gap with respect to the RF.
j z
  j p g

v
E  ee e 0 Ez z e dz 


j z
  j j g

v
E  eee e 0 Ez z e dz 


   p  i
p
Introducing:
E  e  Ez z e
g
0
i
j z
v
dz cos 
and considering the phase which yields the
maximum energy gain:
T
g
0

j t


Ez z e dz
g
0 Ez z dz
28
Important Parameters of Accelerating Cavities
Shunt Impedance
Relationship between gap
voltage and wall losses.
2
Pd  VR
Quality Factor

Q  Ws
Pd
Relationship between
stored energy in the
volume and dissipated
power on the walls.
R  V2
Q Ws
Filling Time
dWs 
Pd   dt  Q W s
Exponential decay of the
stored energy due to losses.

Q

29
Shunt Impedance and Q Factor
The shunt impedance R is defined
as the parameter which relates the
accelerating voltage V in the gap to
the power dissipated in the cavity
walls (Joule losses).

Q  Ws
Pd
R  V2
Q Ws
2
Pd  VR
The Q factor is the parameter
which compares the stored
energy, Ws, inside the cavity to
the energy dissipated in the walls
during an RF period (2/). A
high Q is a measure of a good RF
efficiency
30
Filling Time of a SW Cavity
From the definition of the Q factor one can see that the energy is
dissipated at a rate which is directly proportional to the stored
energy:
dW 
Pd   dt s  Q W s
leading to an exponential decay of the stored energy:
t
W s  W s0 e

avec

Q
(filling time)

Since the stored energy is proportional to the square of the
electric field, the latter decay with a time constant 2 .
If the cavity is fed from an RF power source, the stored energy
increases as follows:


2
 t
2
W s  W s0 1e
31
Equivalent Circuit of a Cavity
RF cavity: on the average, the stored energy in the magnetic field equal
the stored energy in the electric field, Wse=Wsm
0

0
E
dV

H dV


V
V
2
2
2
2
RLC circuit: the previous statement is true for this circuit, where the
electric energy is stored in C and the magnetic energy is stored in L:
Leading to:
1
1
*
Wse  VV C
0   LC  2
4
Wsm  1 L I l I *L avec V  0 LIl
4
Ws  Wse  Wsm  1 CVV *
2
*
1
VV
Pd  2 R
Q  0RC  R
0L
32
Input Impedance of a Cavity
The circuit impedance as seen from the input is:
1
 1  1  jC 

Z e  R jL



avec    0  
Within the approximation «0 the impedance becomes:
Ze 
 02 RL
 L  j2R
2
0

R
1  j2Q 
0
When  satisfies the relation Q=0/2 one has Ze= 0,707 |Ze|max
, with |Ze|max= R. The quantity 2/0 is called the bandwidth (BW) :
Q 1
BW
33
Loaded Q
If R represents the losses of the equivalent resonant circuit of the
cavity, then the Q factor is generally called Q0.
Introducing additional losses, for instance through a coupling loop
connected to an external load, corresponding to a parallel resistor
RL , then the total Q factor becomes Ql ( loaded Q ):
Q l  Rt
0L
avec
Rt 
RR L
R  RL
Defining an external Q as, Qe=RL/0L, one gets:
1  1  1
Ql Q0 Qe
34
Principle of Phase Stability
Let’s consider a succession of accelerating gaps, operating in the 2π mode,
for which the synchronism condition is fulfilled for a phase s .
For a 2π mode,
the electric field
is the same in all
gaps at any given
time.
eVs  eVˆ sin s
is the energy gain in one gap for the particle to reach the next
gap with the same RF phase: P1 ,P2, …… are fixed points.
If an increase in energy is transferred into an increase in velocity, M1 & N1
will move towards P1(stable), while M2 & N2 will go away from P2 (unstable).
35
A Consequence of Phase Stability
Transverse Instability
Longitudinal phase stability means :
V
t
0
E z
z
0
defocusing
RF force
The divergence of the field is
zero according to Maxwell :
.E  0 
E x E z

0 
x
z
E x
0
x
External focusing (solenoid, quadrupole) is then necessary
36
Focusing
Accelerating section, of an electron
linac, equipped with quadrupoles
Accelerating section, of an electron
linac, equipped with solenoids
37
Focusing (2)
For protons & ions linacs,
small quadrupoles are
generally placed inside the
drift tubes. Those
quadrupoles can be either
electro-magnets or
permanent magnets.
38
The Traveling Wave Case
Ez  E0 cosRFt  kz 
k
RF
v
z  vt  t0 
The particle travels along with the wave, and
k represents the wave propagation factor.
v  phase velocity
v  particle velocity


v
Ez  E0 cos RFt  RF t   0 
v


v  v and Ez  E0 cos 0
If synchronism satisfied:
where 0 is the RF phase seen by the particle.
39
Multi-gaps Accelerating Structures:
A- Low Kinetic Energy Linac (protons,ions)
Mode  L= vT/2
In « WIDEROE » structure radiated power
ALVAREZ structure
Mode 2 L= vT = 
  CV
In order to reduce the
radiated power the gap is
enclosed in a resonant
volume at the operating
frequency. A common wall
can be suppressed if no
circulating current in it for
the chosen mode.
40