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Longitudinal Dynamics in Particle Accelerators - Leduff
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1
LONGITUDINAL DYNAMICSIN PARTICLE ACCELERATORS
by
Joël Le DuFF
(retired from LAL-IN2P3-CNRS)
Cockroft Institute, Spring 2006
2
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)
3
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
4
Types of accelerators
Kinetic energy W
Electrons Protons/ ions
Electrostatic Van de Graaf &Tandems
20- 35 MeV (Vivitron)
Betraton
10- 300 MeV Microtron 25- 150 MeV
Cyclotron
10- 100 MeV
Synchro- cyclotron
100- 750 MeV
Synchrotron
1- 10 GeV 1- 1000 GeV
Storage ring
1- 7 GeV (ESRF)
Collider ring
10- 100 GeV (LEP) 1- 7 TeV (LHC)
Linacs
20 MeV- 50 GeV (SLC) 50- 800 MeV(LAMPF) MeV(LAMPF) (LAMPF) Linear collider
50- 1000 GeV (TESLA)
Total energy = Rest energy + Kinetic energy
E0 = m0c2
= E0+ W electron E0=0,511 MeVprotons E0=938 MeV
5
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.
6
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
7
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 , preferably along the direction of the initial momentum.
eEdtdp
BENDING is generated by a magnetic field perpendicular to the plane of the particle trajectory. The bending radius obeys to the relation :
Be
p
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.
E
8
Acceleration & Curvature
o
zx, r
s
Within the assumption:
EE
zBB
the Newton-Lorentz force:
BveEedtpd
becomes:
rzr uBevueEuv
mudtmvd
2
leading to:
zBep
eEdtdp
9
Energy Gain
2220
2 cpEE
zeEdtdp
dzdpv
dzdE
eVdzEeWdzeEdEdW zz
In relativistic dynamics, energy and momentum satisfy the relation:
Hence: vdpdE
The rate of energy gain per unit length of acceleration (along z) is then:
and the kinetic energy gained from the field along the z path is:
where V is just a potential
WEE 0
10
Methods of Acceleration
1_ Electrostatic Field
Energy gain : W=n.e(V2-V1)
limitation : Vgenerator = Vi
2_ Radio-frequency Field
Synchronism : L=vT/2
v=particle velocity T= RF periodWideroe structure
Electrostatic accelerator
220
TvLalso :
11
3_ Acceleration by induction
From MAXWELL EQUATIONS :
AHB
tAE
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
Methods of Acceleration (2)
12
Accélérateur colonne
d.c. high voltage generator
HV system used by Cockcroft & Walton to break the lithium nucleus
Electrostatic accelerator
Accelerating column
13
Van de Graaf type electrostatic accelerator
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.
14
Betatron
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.
A constant trajectory also requires :
dtBd
Rdtd
ER z22
Induction law
Newton-Lorentz force
dt
BdReeE
dt
dp z
2
1
dt
dBRe
dt
dp
BRep
0
0
zo BB2
1
15
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:
eB
mT
eB
mv
eB
pr
2
The corresponding angular frequency is :
m
eB
Trfr
2
2
Synchronism if : rRF
m = m0 (constant) if W <<
E0
v=Vsint
Cyclotron
If so the cyclotron is isochronous
16
Cyclotron (2)
Cyclotron of M.S.Livingstone (1931)On the left the 4-inch vacuum chamberUsed to validate the concept.On the right the 11-inch vacuum chamberof the Berkeley cyclotron that produced1,2 MeV protons.In both cases one single electrode (dee).
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.
17
Cyclotron (3)
Cyclotron SPIRAL at GANIL
Here above is the magnet and its coils
Here below is an artist view of the spiral shaped poles and the radio-frequency system.
SPIRAL accelerates radio-active ions
18
Cyclotron (4)
Cyclotrons at GANIL, Caen
19
Cyclotron (5)
Energy-phase equation:
Energy gain at each gap transit:Particle RF phase versus time:
sinV̂eE
tRF
where is the azimuthal angle of trajectory
Differentiating with respect to time gives: EBecRFrRF
2
Smooth approximation allows:
r
rT 2/
12Bec
ERF
r
Relative phase change at ½
revolution
1sinˆ 2Bec
E
VeEdEd RF
And smooth approximation
again:
20
Cyclotron (6)
2000
00
0 ˆ21ˆcoscos EE
EVeEE
Ve r
RF
r
RF
with :
dEBecE
Ved RF
1ˆcos 2
Integrating:
Separating:
0
0
0
r
E
Rest energy
Injection phase
Starting revolution frequency
21
The expression shows that if the mass
increases, the frequency decreases :
m r
Synchronism condition:
m
eBr
mTr
If the first turn is synchronous :
electrons 0.511 MeV
Energy gain per turn protons 0.938 GeV !!!
Since required energy gains are large the concept is essentially valid for electrons.
)1(intint 0 egeregerT
T turnRF
r
Microtron (Veksler, 1954)
22
Microtron « Racetrack »
Synchronism is obtained when the energy gain per turn is a multiple of the rest energy:
()/turn = integer
Allows to increase the energy gain per turn by using several accelerating cavities (ex : linac section)
Carefull !!!! This is not a « recirculating » linac
23
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.
-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)
RF
Ez
JouH
24
The Pill Box Cavity
02
2
002
tAA )( HouEA
krJEz 0
krJZj
H 10
37762,220Za
ck
e tj
Ez H
From Maxwell’s equations one can derive the wave equations :
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:
25
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.
26
Energy Gain with RF field
RF acceleration
cosˆ
cosˆcosˆˆˆ
VeW
tzEtzEzEVdzzE RF
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.
Neglecting the transit time in the gap.
27
Transit Time Factor
tgVtEEz coscos0
vtz
Oscillating field at frequency and which amplitude is assumed to be constant all along the gap:
Consider a particle passing through the middle of the gap at time t=0 :
The total energy gain is: dzvz
geVW
g
g
2/
2/
cos
eVTeVW 2/
2/sin
angletransitvg
factortimetransitT( 0 < T < 1 )
28
Transit Time Factor (2)
dzezEeE tjgze
0
dzezEeeE v
zjgz
j
ep
0
g v
zj
zjj
e dzezEeeeE ip
0
cos0 g vzj
z dzezEeE
g
z
g tjz
dzzE
dzezET
0
0
pvzt
ip
Consider the most general case and make use of complex notations:
p is the phase of the particle entering the gap with respect to the RF.
Introducing:
and considering the phase which yields the maximum energy gain:
29
Important Parameters of Accelerating Cavities
Shunt Impedance
Filling Time
RVPd
2
PWQ
d
sW
VQR
s2
QWQdtWd
P ss
d
Quality Factor
Relationship between gap voltage and wall losses.
Relationship between stored energy in the volume and dissipated power on the walls.
Exponential decay of the stored energy due to losses.
30
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).
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
PWQ
d
s
WV
QR
s2
RVPd
2
31
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:
WQdtWd
P ss
d
leading to an exponential decay of the stored energy:
eWWt
ss 0 avec Q
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:
eWWt
ss 212
0
(filling time)
32
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
dVHdVE VV 2020
22
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:
*
0*
21
0*
21
4141
CVVWWW
LIVavecIILW
LCCVVW
smses
lLlsm
se
Leading to:
LRRCQ
RVV
Pd0
0
*
21
33
Input Impedance of a Cavity
The circuit impedance as seen from the input is:
0
111 avecCj
LjRZ e
0
20
20
212
Qj
RRjL
RLZ e
Within the approximation «0 the impedance becomes:
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) :
BWQ 1
34
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 ):
Defining an external Q as, Qe=RL/0L, one gets:
L
Lt
tl RR
RRRavec
LRQ 0
QQQ el
1110
35
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.
sVeseV sinˆ 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.
Principle of Phase Stability
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).
36
Transverse Instability00
z
zE
t
VLongitudinal phase stability means :
The divergence of the field is zero according to Maxwell : 000.
x
E
z
E
x
EE xzx
defocusing RF force
External focusing (solenoid, quadrupole) is then necessary
A Consequence of Phase Stability
37
Focusing
Accelerating section, of an electron linac, equipped with
solenoids
Accelerating section, of an electron linac, equipped with
quadrupoles
38
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.
39
The Traveling Wave Case
0
0 cos
ttvz
vk
kztEE
RF
RFz
velocityparticlev
velocityphasev
The particle travels along with the wave, and k represents the wave propagation factor.
00 cos
tvvtEE RFRFz
If synchronism satisfied:
00 cos EEandvv z
where 0 is the RF phase seen by the particle.
40
Multi-gaps Accelerating Structures:A- Low Kinetic Energy Linac (protons,ions)
Mode L= vT/2 Mode 2 L= vT =
In « WIDEROE » structure radiated power 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. ALVAREZ structure