LONGITUDINAL DYNAMICS IN PARTICLE ACCELERATORS

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LONGITUDINAL DYNAMICSIN PARTICLE ACCELERATORS
by Joël Le DuFF (retired from LAL-IN2P3-CNRS)
Cockroft Institute, Spring 2006
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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)
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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
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Types of accelerators
Total energy Rest energy Kinetic energy
electron E00,511 MeV protons E0938 MeV
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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.
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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
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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.

BENDING is generated by a magnetic field
perpendicular to the plane of the particle
trajectory. The bending radius ? obeys to the
relation
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.
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Acceleration Curvature
Within the assumption
z
x, r
s
the Newton-Lorentz force
?
?
o
becomes
leading to
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Energy Gain
In relativistic dynamics, energy and momentum
satisfy the relation
Hence
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
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Methods of Acceleration
1_ Electrostatic Field Energy gain
Wn.e(V2-V1) limitation Vgenerator S
Vi
Electrostatic accelerator
2_ Radio-frequency Field Synchronism
LvT/2
vparticle velocity
T RF period
Wideroe structure
also
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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
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Electrostatic accelerator
Accélérateur colonne
d.c. high voltage generator
Accelerating column
HV system used by Cockcroft Walton to break the
lithium nucleus
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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
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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.
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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
The corresponding angular frequency is
Synchronism if
vVsin?t
m m0 (constant) if W ltlt E0
If so the cyclotron is isochronous
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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).
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Cyclotron (3)
Cyclotron SPIRAL at GANIL
Here below is an artist view of the spiral shaped
poles and the radio-frequency system.
Here above is the magnet and its coils
SPIRAL accelerates radio-active ions
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Cyclotron (4)
Cyclotrons at GANIL, Caen
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Cyclotron (5)
Energy-phase equation
Energy gain at each gap transit
Particle RF phase versus time
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
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Cyclotron (6)
Separating
Integrating
with
Rest energy
Injection phase
Starting revolution frequency
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Microtron (Veksler, 1954)
The expression
shows that if the mass increases, the frequency
decreases m ?r
Synchronism condition
If the first turn is synchronous
Since required energy gains are large the concept
is essentially valid for electrons.
electrons
0.511 MeV Energy gain per turn
protons 0.938 GeV !!!
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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
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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
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The Pill Box Cavity
From Maxwells equations one can derive the wave
equations
Solutions for E and H are oscillating modes, at
discrete frequencies, of types TM ou TE. For llt2a
the most simple mode, TM010, has the lowest
frequency ,and has only two field components
Ez
H?
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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.

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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.
Neglecting the transit time in the gap.
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Transit Time Factor
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 t0
The total energy gain is
( 0 lt T lt 1 )
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Transit Time Factor (2)
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
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Important Parameters of Accelerating Cavities
Shunt Impedance
Relationship between gap voltage and wall losses.
Quality Factor
Relationship between stored energy in the volume
and dissipated power on the walls.
Filling Time
Exponential decay of the stored energy due to
losses.
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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
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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
leading to an exponential decay of the stored
energy
avec
(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
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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, WseWsm
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
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Input Impedance of a Cavity
The circuit impedance as seen from the input is
Within the approximation ???0 the impedance
becomes
When ?? satisfies the relation Q?0/2?? one has
Ze 0,707 Zemax , with Zemax R. The quantity
2??/?0 is called the bandwidth (BW)
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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, QeRL/?0L, one gets
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Principle of Phase Stability
Lets consider a succession of accelerating gaps,
operating in the 2p mode, for which the
synchronism condition is fulfilled for a phase ?s
.
For a 2p mode, the electric field is the same in
all gaps at any given time.
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).
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A Consequence of Phase Stability
Transverse Instability
Longitudinal phase stability means
defocusing RF force
The divergence of the field is zero according to
Maxwell
External focusing (solenoid, quadrupole) is then
necessary
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Focusing
Accelerating section, of an electron linac,
equipped with quadrupoles
Accelerating section, of an electron linac,
equipped with solenoids
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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.
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The Traveling Wave Case
The particle travels along with the wave, and k
represents the wave propagation factor.
If synchronism satisfied
where ?0 is the RF phase seen by the particle.
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Multi-gaps Accelerating StructuresA- Low
Kinetic Energy Linac (protons,ions)
Mode 2p L vT bl
Mode p L vT/2
In  WIDEROE  structure radiated power ? w 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