Skin depth




No perfect conductors exist, but certain metals are very good
conductors.
Copper, with 1/ = 1.7 x 10-8 m, where  is the conductivity, is
the most commonly used metal for accelerator applications.
For a good but not perfect conductor, fields and currents are not
exactly zero inside the conductor but are confined to within a
small finite layer at the surface called the skin depth.
In a real conductor, the electric and magnetic fields and current
decay exponentially with distance from the surface of the
conductor, a phenomenon known as the skin effect.
Value or R




Physically, the skin effect is explained by the fact that RF
electric and magnetic fields applied at the surface of a
conductor induce a current, which shields the interior of
the conductor from those fields.
For frequencies in the 100-MHz range and for a good
conductor such as copper, the skin depth  is on the
order of 10-6 m, and R, is in the milliohm range
The use of superconducting materials reduces the
surface resistance dramatically.
The superconducting surface resistance can be roughly
10-5 times that of copper
Resonance and phase



Q=10
Note that the /2 between the resonator and the external driving force
For an high Q value small differences in the driving frequency with respect to
the resonant frequency resolves in a remarkable phase shift
Limit of a single resonator
For a single resonator


Each cavity has its own power source, all at the same frequency
and amplitude
We also adjusted all the phases so that the ultrarelativistic particle
sees always the field at its maximum
Travelling wave
0 is the phase
in the first cell

In the center of every cell (z=nd) the field is equal
to the field of a travelling wave with constant
phase velocity vf=/k
Transverse Travelling Wave
t=0
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t=T/8
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t=T/4
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t = 3T / 8
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t=T/2
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t = 5T / 8
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t = 3T / 4
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t = 7T / 8
Transverse Standing Wave
0.5 
Transverse Travelling Wave
t=T
Transverse Standing Wave
0.5 
Advantages of acceleration with E.M. wave

Using a travelling wave has several benefit:


Transit time factor = 1
Easier power system. Remember that small
differences in the driving frequency resolve in strong
differences in the phase
But…


Unfortunately the Lawson-Woodward theorem states that the
energy gain of a charged particle, interacting with e.m. field in the
vacuum without any static or magnetic field or boundaries, is zero
if the average is performed over a long distance (ideally infinity)
Also if we consider a e.m. propagating with an angle with respect
to the charge motion the phase velocity>c
0
0/cos
Group velocity



There are no truly
monochromatic waves in
nature.
A real wave exists in the form
of a wave group, which
consists of a superposition of
waves of different frequencies
and wave numbers.
If the spread in the phase
velocities of the individual
waves is small, the envelope of
the wave pattern will tend to
maintain its shape as it moves
with a velocity that is called
the group velocity.
Phase velocity vs group velocity


The exponential factor describes a traveling wave with a mean
frequency and mean wave number, and the first factor
represents a slowly varying modulation of the wave amplitude.
The mean phase velocity is
Starting equations
Equation for Ez

In the case of a plane wave with only z component
different from zero
Equation for k
Spatial solution
Frequency cutoff


If the wavenumber kz is complex, then the amplitude
of the wave travelling through the waveguide falls
off exponentially, i.e. loss-free wave propagation is
not possible.
Loss-free propagation occurs only when kz is real.
Dispersion relation


In the regime of loss-free wave propagation in the
waveguide, the wavelength z is always larger than
the wavelength in free space.
This means that the phase velocity of the wave within
the waveguide is greater the speed of light
Example cylindrical waveguide

In a lossless uniform waveguide with azimuthal
symmetry, the axial electric field for the lowest
transverse magnetic mode is
We need to modify the structure

Because the phase velocity is always larger than c, the uniform
guide is unsuitable for synchronous particle acceleration, and we
must modify the structure to obtain a lower phase velocity.
Periodic loaded structure

d

One might expect
that converting the
uniform guide to a
periodic structure
might perturb the
field distribution by
introducing a zperiodic modulation
of the amplitude of
the wave.
Ed(r, z) is a periodic function with the same period
d as the structure
Brillouin plot
TRAVELING-WAVE LlNAC STRUCTURES





The linac consists of a sequence of identical tanks, each consisting of an array of
accelerating cells separated by the irises.
The electromagnetic wave is launched at the input cell of each tank; the wave propagates
along the beam direction, and beam bunches are injected along the axis for acceleration
by the wave.
The electromagnetic energy is absorbed by the conductor walls and by the beam, and the
field amplitude attenuates along each tank.
At the end of each tank the remaining energy is delivered to an external resistive load.
It is desirable to maximize the energy gain of the beam over a given distance and to
minimize the power lost to the walls and to the external load.
More details




RF power is introduced via the input coupler.
Part of RF power is dissipated in the structure, part is
taken by the beam (beam loading) and the rest is
absorbed in a matched load at the end of the structure.
Usually, structure length is such that ~30% of power goes
to the load.
The “traveling wave”structure is the standard linac for
electrons from β~1.
Choice of the frequency

Some basic considerations for the choice include:




higher power efficiency at higher frequencies because
of the 1/2 dependence of shunt impedance
tighter beam-positioning tolerances at higher
frequencies because of smaller apertures.
Other considerations are often equally
important.
For applications requiring acceleration of very
short, intense bunches of electrons, it is desirable
to provide large stored energy per unit length,
which scales as -2 favoring lower frequencies
Comparison
Long pulses. Gradients 2-5 MeV/m
Used for Ions and electrons at all
energies
Short pulses, High frequency.
Gradients 10-20 MeV/m
Used for Electrons at v~c
At the beginning




Many particle sources, be it for electrons, protons, or
ions, produce a continuous stream of particles at
modest energies limited by electrostatic acceleration
between two electrodes.
Not all particles of such a beam will be accelerated
because of the oscillatory nature of the accelerating
field.
For this reason and also in the case short bunches or
a small energy spread at the end of the linac is
desired, the particles should be concentrated at a
particular phase.
This concentration of particles in the vicinity of an
optimum phase maximizes the particle intensity in
the bunch
Prebuncher





A bunched beam can be
obtained from a
continuous stream of
nonrelativistic particles by
the use of a prebuncher.
The basic components of a
prebuncher is an rf-cavity
followed by a drift space.
As a continuous stream of
particles passes through
the prebuncher, some
particles get accelerated
and some are decelerated.
Particles before
acceleration (a) and right
after (b).
A distance L downstream of
the buncher cavity (c) the
phase space distribution
shows strong bunching
Chopper



There are still particles between the bunches which could
either be eliminated by an rf-chopper or let go to be lost in the
linear accelerator because they are mainly distributed over the
decelerating field period in the linac
This is a chopper which consists of an rf-cavity excited similar to the
prebuncher cavity but with the beam port offset by a distance r from the
cavity axis. In this case the same rf source as for the prebuncher or main
accelerator can be used and the deflection of particles is effected by the
azimuthal magnetic field in the cavity
To produce a single pulse, the chopper system may consist of a
permanent magnet and a fast pulsed magnet. The permanent magnet
deflects the beam into an absorber while the pulsed magnet deflects the
beam away from the absorber across a small slit
Klystron


Cavities installed in circular accelerators and linac structures
require RF power of at least a few tens of kW and as much as
several MW in large high-energy accelerators or linacs.
The klystron has proven to be the most effective power
generator for accelerator applications
How the klystron works








Electrons are emitted from a round cathode
with large surface area, and are accelerated by
a voltage of a few tens of kV.
This yields a round beam, with a current
ranging from a few amperes up to more than
10 A.
Suitably shaped electrodes near the cathode
are used to focus the beam
Sometimes several solenoids are added along
the tube to ensure good collimation of the
beam.
The beam comes out of the cathode with a very
well-defined particle velocity and passes
through a first cylindrical cavity.
A wave is excited in this resonator by an
external preamplifier with a power output of a
few tens of watts, which, depending on the
phase, will accelerate, brake, or simply not
influence the particles in the beam.
The velocity of the particles through the cavity
is thus modulated, with a frequency exactly
equal to the resonant frequency.
In the zero-field drift section which follows, the
faster particles move ahead, while the slower
ones lag behind.
Cont.




After a certain distance bunches of particles are
formed, with a separation given by the wavelength
of the driving wave.
The continuous current from the cathode thus
becomes a pulsed current, with a frequency equal to
the frequency of the coupled driving supply
A second cavity mounted at this point is resonantly
excited by the pulsed current, and the RF wave
generated in this cavity can then be coupled out.
In an optimal design the beam is almost completely
stopped by the RF field it excites in the second
cavity, leaving just a small residual energy, i.e. the
kinetic energy stored in the beam is transformed
into RF energy
Klystron efficiency




Here U0 is the supply voltage of the klystron, Ibeam the beam
current and  the efficiency of the klystron, which nowadays
ranges from 45% and 65%.
In particular, klystrons used to drive linac structures operating in
the S-band at the commonly-used frequency of 2.998 GHz require
voltages between 250 and 300 kV, traversed by beam currents of
around 250 A.
For an efficiency of  = 45% this yields a power output of Pkl = 3035 MW.
Naturally such high power can only be handled in pulsed
operation. The pulse length in linacs is a few s with a repetition
rate of a few hundred Hz. This yields an average power output of
a few tens of kW, which is relatively easy to manage