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Proceedings of the CERN–Accelerator–School course: Introduction to Accelerator Physics
Available online at https://cas.web.cern.ch/previous-schools
1
LINAC
D. Alesini
Laboratori Nazionali di Frascati dell’INFN, Frascati, Italy
Abstract
A linac (linear accelerator) is a system that allows to accelerate charged particles through
a linear trajectory by electromagnetic fields. This kind of accelerator finds several
applications in fundamental research and industry. The main devices used to accelerate
the particle beam will be introduced in the first part of the paper, while in the second
part, the fundamentals of the longitudinal and transverse beam dynamics will be
highlighted. A short paragraph is finally dedicated to radiofrequency quadrupoles
(RFQ).
Keywords:
Radio Frequency; particle accelerators; cavity; electromagnetic field; superconductivity;
linac.
1 Introduction
A linear accelerator (linac) is a device that allows accelerating charged particles (electrons, protons, ions) along
a straight trajectory [1,2]. The main advantage of linacs is their capability to produce high-energy, high-intensity
charged particle beams of excellent quality in terms of beam emittance and energy spread. These devices find
applications in different fields such as research, healthcare and industry [3-8].
The acceleration can be obtained with constant (DC) or time-varying electric fields. In the second case, we
can have two types of linacs: radio frequency (rf) and induction. In this proceeding we will refer to rf linacs in
which the particle acceleration is obtained by electromagnetic fields confined in resonant cavities fed by sinusoidal
time-varying power sources. The design and the structure of a linac depend on the type of accelerated particles
(e.g. electrons, protons or ions) and on the required final beam parameters in terms of energy, energy spread,
emittance and current. In the design and in the choice of the technology, several constraints have also to be taken
into account such as cost, available footprint and power consumption.
The main advantages [9] of linacs with respect to other possible accelerators (as synchrotrons or cyclotrons)
are the fact that they can handle high peak current beams, they can run with high duty-cycle and they exhibit low
synchrotron radiation losses (in the case of light particle acceleration as electron or positrons). On the contrary,
the main drawback is the fact that they require a large number of cavities to reach a desired energy, since the beam
passes only once in the accelerating structures. Moreover, synchrotron radiation damping (in the case of light
particles) cannot be used to reduce the beam emittance unless to adopt complicated schemes.
Linacs are mainly used in fundamental physics research as injectors for synchrotrons and storage rings, free
electron lasers and injectors for colliders. They also find a huge number of medical and industrial applications for
cancer therapy, x rays generation, material treatment, food irradiation and ion implantation.
A linac conceptual scheme, where its technology complexity can be also assessed, is shown in Fig. 1.
Particles are generated and pre-accelerated in the injector, and then accelerated by accelerating structures and
focused by magnetic elements (quadrupoles and solenoids). Beam trajectory and dimensions along the linac are
measured by different types of diagnostic devices (striplines, cavity beam position monitors and beam screens)
2
depending on the particle beam and energy. In the figure are reported also the control, cooling and vacuum systems,
the rf distribution and rf power sources.
Fig.1: Schematic layout of a linac
2 Acceleration process: particle velocity and energy
The first historical linear accelerator was conceived by the Nobel prize Wilhelm Conrad Röntgen (1901). It
consisted in a vacuum tube containing a cathode connected to the negative pole of a DC voltage generator of
few kV. Electrons emitted by the heated cathode (by thermionic emission) were accelerated while flowing to
another electrode connected to the positive generator pole (anode). Collisions between the energetic electrons and
the anode produced x-rays and gave the possibility to do the first radiography.
In modern electrostatic accelerators high voltage is shared between a set of electrodes creating
an accelerating field between them. This type of accelerator is better known as the Cockcroft-Walton accelerator
[10]. Its main limitation in term of achievable energy, is due to the fact that all partial accelerating voltages add
up at some point and insulation problems or discharges occur, thus limiting the maximum achievable voltage to
a few tens of MV.
Electrostatic accelerators are still used for several applications as x ray generation, material analysis, or ion
implantation. The simple scheme of a DC acceleration process between two electrodes is reported in Fig. 2(a).
Considering the energy-momentum relation and the Lorentz force we can easily write the following
relations:
vdpdEdp
E
mc
vdEpdpcEdEcpEE ===+= 2
2222
0
222
(1)
====
=zzz
vtz
zqE
dz
dW
qE
dz
dE
Eq
dz
dp
vEq
dt
dp also and
, (2)
3
where E0 (=m0c2) is the particle rest energy, E is the total energy, m0 is the rest mass, m is the relativistic mass, v is
the particle velocity, p (=mv) is the particle momentum, W=E-E0 is the kinetic energy and Ez is the accelerating
field. In Eq. (2) dE/dz is the rate of energy gain per unit length and it is proportional to the accelerating field Ez.
Integrating Eq. (2) on the accelerating gap we obtain the energy gain per electrode pair
E:
VqEdzqEdz
dz
dE
E
gap z
gap
===
. (3)
Fig.2: Scheme of acceleration processes between two electrodes: (a) electrostatic acceleration; (b) rf acceleration.
If now we consider the relativistic factors
=v/c and
=E/E0 we can write:
2
0
0
2
0
211
1
1
+
−=
−=−= WE E
E
E
. (4)
The behavior of
and
as a function of the kinetic energy is given in Fig. 3 for an electron and a proton.
From the plot it is evident that light particles (electrons) are practically fully relativistic (1, >>1) at relatively
low energies (above 10 MeV) and reach a constant velocity (
c). Thus, for this type of particles, the acceleration
process occurs at constant velocity. On the contrary heavy particles (protons and ions) are typically weakly
relativistic and reach a constant velocity only at very high energies. This means that their velocity changes during
the acceleration process. As illustrated in the following, the possible velocity change during acceleration generates
important differences in the accelerating structures design and beam dynamics between light and heavy particles.
Fig. 3:
(top) and
(bottom) as a function of the kinetic energy for an electron (blue) and a proton (red).
4
3 Radiofrequency accelerators: the Wideröe linac
In the late 1920's propositions were made by Ising (1924) and implemented by Wideröe (1927) to overcome
the limitation of electrostatic devices in terms of reachable energy [11-13]. The proposed scheme is illustrated in
Fig. 4. An AC voltage generator feeds, alternately, a series of electrodes in such a way that particles do not
experience any force while travelling inside the tubes (Drift Tubes) while are accelerated across the gaps. In
particular, this last statement is true if the drift tube length (or, equivalently, the distances between the centers of
the accelerating gaps Ln) increases with the particle velocity during the acceleration, such that the time of flight
between gaps is kept constant and equal to half of the rf period. If this condition is satisfied, the particles are
subject to a synchronous accelerating voltage and experience an energy gain of ∆E=q∆V at each gap.
In this type of structures, called Drift Tube Linac (DTL), a single rf generator can be used, in principle, to
indefinitely accelerate the beam, avoiding the breakdown limitations that affect the electrostatic accelerators.
Fig. 4: Conceptual scheme of a Wideröe linac.
The Wideröe linac is the first rf linac. As illustrated in Fig. 1(b), a rf acceleration process does not allow to
accelerate a continuous particle beam but the beam needs to be “bunched” with a distance between bunches exactly
equal to the rf period.
Let us now consider the acceleration between a pair of two electrodes. The behavior of the accelerating
electric field and voltage are given by the following relations:
( ) ( ) ( )
cos, tzEtzE RFRFz
=
( ) ( )
2
2 cos
==== gap RFRF
RF
RFRFRFRF dzzEV
T
ftVV
. (5)
The energy gain per electrode (supposing a symmetric accelerating field with respect to z=0) is given by:
( ) ( ) ( )
( )
( )
( )
( )
acc
inj
acc
RFinj
gap RF
gap RFRF
RF
gap RF
L
LinjRFRF
gap particle
by
seen
z
V
V
TVq
T
dzzE
dz
v
z
zE
V
dzzEqdz
v
z
zEqdztzEqE
cos
ˆ
cos
cos
cos, 2
2
=
=
+==
+
−
(6)
where
inj is the injection phase of a generic particle. T is called the “transit time factor” and is always less than 1.
It takes into account the fact that the rf voltage is oscillating in time while the beam is traversing the gap and that
the effective peak accelerating voltage (
acc
V
ˆ
) is in fact equal to the rf voltage (VRF) multiplied by this factor.
mVLVE accacc /=
is the average accelerating field seen by the particle.
5
If we consider a Wideröe structure, the energy gain, at each gap, is equal to
En=qVacc while the particle
velocity increases accordingly to Eq. (4). In order to be synchronous with the accelerating field at the center of
each gap, the time of flight (tn) between gaps is given by:
RF
RFnRFnn
RF
n
n
ncTTvL
T
v
L
t
2
1
2
1
2====
, (7)
where TRF is the generator rf period,
n
v
is the average particle velocity between the gap n and n+1 and
RF is the rf
wavelength. Then, the distance between the centers of two consecutive gaps has to be increased as follows:
RFnn
L
2
1
=
. (8)
The energy gain per unit length is then given by:
nRF
acc
n
acc qV
L
qV
L
E
2
==
. (9)
4 Rf cavities
There are two important consequences of Eq. (9) obtained in the previous paragraph. First of all, the condition
Ln<<
RF (necessary to model the tube as an equipotential region) requires
1
n
. This implies that the Wideröe
acceleration technique cannot be applied to relativistic particles. Moreover, when particles velocity increases
the drift tube gets longer, reducing the acceleration efficiency (energy gain per unit length
LE
). The average
accelerating gradient (defined as
naccacc LVE =
) increase pushes towards the use of small RF (high frequencies),
but the concept of equipotential drift tubes cannot be applied at small RF and the power lost by free space radiation
increases proportionally to the rf frequency.
All previous considerations combined with the fact that high frequency, high power rf sources became
available only after the 2nd world war (thanks to the radar technology applied to military purposes), pushed to
develop more efficient accelerating structures than simple drift tubes. In order to avoid electromagnetic (e.m.)
radiation processes and to allow the use of high frequency sources, the accelerating system had to be enclosed in
a metallic volume: a rf cavity. In a rf cavity the e.m. field has a particular spatial configuration (resonant mode)
whose components, including the accelerating field Ez, oscillates at a specific frequency (called resonant
frequency) characteristic of the mode. The mode is excited by the rf generator that is coupled to the cavity through
waveguides, coaxial cable antennas or loops. The resonant modes are called Standing Wave (SW) modes since
they have a spatial fixed configuration that oscillates in time. The spatial and temporal field profiles in a cavity
have to be computed (analytically or numerically) by solving the Maxwell equations with the proper boundary
conditions [14,15].
For a SW cavity the accelerating field on the z-axis can be written as:
( ) ( ) ( )
tj
ezEtfzEtzE RF
z
RF
RFRFz~
Real 2cos,
=
+=
, (10)
6
where fRF is the excitation frequency of the generator equal to (or close to) the resonant frequency of the cavity,
RF is the angular excitation frequency,
( )
zERF
(or the phasor
z
~
E
) is a real (complex) function related to the spatial
configuration of the mode.
Rf linacs use different type of cavities depending on the species and energy range of particles to be
accelerated, as described in the following paragraphs.
5 Alvarez structures
The Alvarez structures [16], reported in Fig. 5, can be described as special DTLs in which the electrodes are part
of a resonant macrostructure. They work in the so-called “0-mode”, since the accelerating field at a given time has
the same phase in each gap. The Wideroe structures, on the contrary, work in the “-mode”, i.e. the accelerating
field in consecutive gaps has opposite sign. They are used for protons and ions in the range of
=0.05-0.5. They
typically operate at fRF=50-400 MHz,
RF=6-0.7 m in the 1-100 MeV energy range. Usually, they are simply called
DTL (instead of the Wideroe structures that are not used anymore).
As for the Wideroe structure, also in the Alvarez linac the accelerating field is concentrated between gaps
and the beam crosses the drift tubes when the electric field is decelerating. The drift tubes are suspended by stems
and quadrupole magnets (for transverse focusing) can fit inside them.
To maintain the synchronism between the beam and the accelerating field at each gap, the distance between
the accelerating gaps Ln has to be varied according to the formula:
RFnn
L
=
. (11)
To maintain this synchronism, in principle, both the gap and the drift tube length can be varied. A generally
applied criterion is to maintain a constant transit time factor according to Eq. (6).
Some examples of Alvarez structures can be found at CERN linac 4 [17] operating at 352 MHz frequency
with 3 resonators tanks of 500 mm diameter, 19 m long, 120 Drift Tubes, that accelerate protons from 3 MeV to
50 MeV (=0.08 to 0.31). In this case the distance between gaps varies from 68mm to 264mm.
Fig. 5: Schematic views of the Alvarez structure (Drift Tube Linac).
7
6 High cavities: cylindrical structures
When the
of the particles increases (>0.5) one has to use higher rf frequencies (>400-500 MHz) to increase
the accelerating gradient per unit length. It is possible to demonstrate that DTL structures become less efficient
(effective accelerating voltage per unit length for a given input power) for such values of
.
In this range of frequencies, cylindrical single or multiple cavities working in the TM010-like mode are used.
For a pure cylindrical structure (usually called a “pillbox cavity”) the first accelerating mode (i.e. the mode with
non-zero longitudinal electric field on axis) is the TM010 mode. It has a well-known analytical solution (obtained
solving Maxwell’s equations), and its spatial configuration in the case of a pure cylindrical cavity is given in Fig.
6(a). For this mode the electric field has only a longitudinal component, while the magnetic one is purely
azimuthal. The corresponding complex phasors are given by [14]:
−=
=
a
r
pJ
Z
jAH
a
r
pAJEz
01
'
0
0
010
1
~
~
, (12)
where a is the cavity radius, A is the mode amplitude and p01 (= 2.405) is the first zero of the Bessel function J0.
The resonant frequency of this mode is given by:
a
cp
f
201
res =
.
Fig. 6: (a) Ideal pillbox cavity and e.m. field configuration; (b) Sketch of a real cavity operating in the TM010-like mode with
two type of coaxial couplers (antenna and loop).
The geometry of real cylindrical cavities is somewhat different to that of a pillbox. In fact, one has to also
consider the perturbation introduced by the beam pipe, the power couplers to rf generators and any antenna or
pick-up used to monitor the accelerating field inside the cavity. For this reason, the actual accelerating mode is
called the TM010-like mode. Real cavities and their couplers to the rf generators are designed using numerical
codes that solve the Maxwell equations with the proper boundary conditions. A sketch of a cavity fed through
a loop or a coaxial probe to an external generator is given in Fig. 6(b), where there are also reported the electric
and magnetic field lines and the longitudinal electric field profile on axis. Details of the coupler design can be
found in [18].
8
6.1 SW cavity parameters: shunt impedance and quality factor
For a SW cavity the first figure of merit is the shunt impedance defined by [1,2,14,19]:
Ω
P
V
Racc
ˆ
diss
2
=
, (13)
where
acc
V
ˆ
is the accelerating voltage for a given dissipated power into the cavity (Pdiss). This parameter qualifies
the efficiency of the cavity: the higher its value, the larger is the achievable accelerating voltage for a given
dissipated power. As an example, at 1 GHz for a normal conducting (NC) copper cavity a typical shunt impedance
of the order of 2 M can be obtained, while a superconducting (SC) cavity, at the same frequency, can reach
values of the order of 1 T, due to the extremely lower dissipated power. It is also useful to refer to the following
quantity called shunt impedance per unit length:
( )
mΩ
p
E
LP LV
L
R
r
diss
acc
diss
acc
ˆˆ 2
2
===
, (14)
where L is the cavity length,
acc
E
ˆ
.the average accelerating field and pdiss the dissipated power per unit length.
The quality factor of the accelerating mode is then defined by the ratio of the cavity stored energy (W) and
the dissipated power on the cavity walls:
diss
RF P
W
Q
=
0
. (15)
For a NC cavity operating at 1 GHz the quality factor is of the order of 104 while, for an SC cavity, values
of the order 109 to 1010 can be achieved.
It can be easily demonstrated that the ratio R/Q is a pure geometric factor and it does not depend upon
the cavity wall conductivity or operating frequency. This is the reason why it is always taken as a geometric design
qualification parameter. The R/Q of a single cell is of the order of 100.
6.2 Multi-cell SW cavities
In linacs, rf cavities are used in systems of multi-cavities. In a multi-cell structure, there is one input coupler that
feeds a system of coupled cavities as sketched in Fig. 8. The field of adjacent cells is coupled through the cell
irises (and/or through properly designed coupling slots). It is quite easy to demonstrate that, for such structures
the shunt impedance is N times the impedance of a single cavity. Moreover, with one source, it is possible to feed
a set of cavities with a simplification of the power distribution system layout. On the other hand, the fabrication
of multi-cell structures is more complicated than single-cell cavities.
6.3 -mode structures
The N-cell structure behaves like a system of N coupled oscillators with N coupled multi-cell resonant modes. As
an example, the field configuration of a two-cell resonator is shown in Fig. 7. The mode in which the two cells
oscillate with the same phase is called 0 mode, while the one with 180° phase shift is called -mode. It is quite
easy to verify that the most efficient configuration (generally used for acceleration) is the -mode, which is shown
in Fig. 8 for a system of five cells. In this system, in order to have a synchronous acceleration in each cell,
the distance (d) between the centre of two adjacent cells has to be d=
v/(2fRF) where v is the particle velocity. As
sketched in Fig. 8 this allows to synchronize the beam passage in each cell with the accelerating field, granting
a continuous acceleration process.
9
To maintain the synchronism, for ions and protons the cell length has to be increased and the linac will be
made of a sequence of different accelerating structures matched to the ion/proton velocity. For electrons, after
the injector, d=
RF/2 and the linac is made by a series of identical accelerating structures.
Examples of this type of system are the LINAC 4 (CERN) PIMS (PI Mode Structure) that operates at
fRF=352 MHz with
>0.4 [17] while, for electrons, the superconducting cavities of the European X-FEL that
operate with modules of nine coupled cells at 1.3 GHz [20-21].
Fig. 7: Resonant modes in a system of two coupled cavities. The mode typically used for acceleration is the -mode.
Fig. 8: Acceleration in multi-cell SW cavity operating on the mode: the cell length is equal to
RF/2 and the bunch in
each cell is always synchronous with the electric field positive half-wave.
6.4 /2 mode structures
It is possible to demonstrate that in a multi-cell system the different resonant modes are distributed in a curve
called “dispersion curve”. As an example, the case of a 9-cell structure is reported in Fig. 9. Each mode has
a bandwidth proportional to the quality factor [19] and, over a certain number of coupled cavities, the overlap of
the tails of adjacent modes can introduce problems for field equalization and structure operability. This limits
the maximum number of multi-cell structures to around 10-15. Since the criticality of a working mode depend on
the frequency separation between the working mode and the adjacent modes, the /2 mode, from this point of
view, is the most “stable” one. Unfortunately, for this mode it is possible to demonstrate that the accelerating field
is zero every two cells. One possible solution to use this mode is to put off-axis the empty cells (called coupling
cells) like in the system schematically represented in Fig. 10(a). In spite of this mechanical complication with
respect to a -mode system (see as example the mechanical drawing of Fig. 10(b)), this allows to increase
the number of cells to more than 20 without problems.
10
Fig. 9: Resonant modes distributed in the “dispersion curve” for a 9-cell multi-cell structure.
Fig. 10: (a) Sketch of a multi cell cavity of 5 cell operating on the /2 mode; (b) Mechanical sketch of the cavity.
These types of structures are used both for electrons than protons. As example in the Spallation Neutron
Source [22] there is the Coupled Cavity Linac (CCL) section with 4 modules, each containing 12 accelerator
segments. It operates at 805 MHz and accelerates the proton beam from 87 to 186 MeV in a length of about
55 meters.
7 Travelling wave structures
There is another possibility to accelerate particles: using travelling wave (TW) structures. In TW structures
an e.m. wave travels together with the beam in a special guide in which the phase velocity of the wave matches
the particle velocity. In this case the beam absorbs energy from the wave and it is continuously accelerated.
Typically, these structures are used for electrons because for such particles the velocity can be assumed
constant all along the structure and equal to c, while it would be difficult to modulate the phase velocity for heavy
particles that change their velocity during acceleration.
In the simple case of an e.m. wave propagating into a constant cross-section waveguide, the phase velocity
is always larger than the speed of light and thus the e.m. wave will never be synchronous with a particle beam. As
example in a circular waveguide (Fig. 11(a)) the first propagating mode with Ez 0 is the TM01 mode, whose
longitudinal electric field (neglecting the attenuation) can be expressed by the well know formula [23]:
( )
2
cut
2
RF
**
zRF
01
00
TM
z1
cos
01
−=−
=c
kzktr
a
p
JEE z
, (16)
where a is the radius of the waveguide,
*
z
k
is the propagation constant,
cut is the cut-off angular frequency of
the waveguide equal to
acp01cut =
. The corresponding phase velocity is given by:
11
2
RF
2
cut
*
RF
ph 1
−
== c
k
v
z
, (17)
which is always larger than c.
The behaviour of the propagation constant as a function of frequency is the well-known dispersion curve
and is sketched in Fig. 11(a). It is important to remark that the phase velocity is not the velocity of the energy
propagation into the structure, which, instead, is given by the group velocity (vg):
2
RF
2
cutg 1
−==
=
c
dk
d
v
RF
z
(18)
and is always smaller than c.
Fig. 11: (a) Circular waveguide example (top) and its dispersion curve (bottom); (b) Iris loaded waveguide model (top) and
its typical dispersion curve (bottom); (c) TW structure with input and output couplers: the e.m. field travels together with
the bunch, continuously transferring its energy to the particles.
In order to slow down the wave phase velocity, the structure through which the wave is travelling, is
periodically loaded with irises. A sketch of an iris-loaded structure is given in Fig 11(b) and it can be designed to
have the phase velocity equal to the speed of the particles allowing a net acceleration over large distances.
The propagating field, in this case, is that of a special wave travelling within a spatial periodic profile (TM01-like
mode) and, according to the Floquet’s theorem [1, 2, 23], can be expressed as:
( )
( )
z
zP
TM
zezktzrEE
−
−=
−
*
RF
D periodwith func tion periodic
cos,
like01
. (19)
In Eq. (19) the propagating constant does not have an analytical expression as in the case of constant cross-
section waveguides and the dispersion curve for this type of structures is given in Fig. 11(b) and shows that, at
a given frequency, the phase velocity can be equal to (or even slower than) c. In Eq. (19) we have also included
the attenuation constant
[1/m] of the accelerating field due to the rf losses in the metallic walls. Typical values
of are in the range 0.2-0.3 1/m.
12
In a TW structure, the rf power enters into the cavity through an input coupler and flows through the cavity
in the same beam direction. An output coupler, at the end of the structure, connected to a matched load, absorbs
the residual power not transferred to the beam or dissipated in the cavity wall, avoiding reflections, as sketched in
Fig. 11(c). If there is no beam, the input power simply dissipates on the cavity walls and the remainder is finally
dissipated into the power load. In the presence of a beam current a fraction of this power is, indeed, transferred to
the beam itself. TW structures allow acceleration over large distances (few meters, hundreds of cells) with just
an input coupler and a relatively simple geometry.
For example, the SLAC electron Linac [24] is composed by more than one hundred 3 m long structures
operating at 2.856 GHz while the SwissFEL linac by 96 structures, 2 m long operating at 5.712 GHz [25-27].
Similarly to what has been done for SW cavities, it is possible to define some figures of merit for TW
structures as well. A complete description of these quantities is given, as example, in [1,2,19,28]. Here we want
just to mention three quantities that are generally considered in the definition of the TW parameters. The first one
is the attenuation constant
, already mentioned, and defined as:
F
F
FPdzdP
P
p2
2diss −==
, (20)
where pdiss is the dissipated power per unit length and PF is the power flowing into the structure at a given section.
The second one is the shunt impedance per unit length r defined as:
=mp
E
r
diss
2
acc
, (21)
where Eacc is the average accelerating field. The higher the value of r, the higher the available accelerating field
for a given rf power. Typical values for a 3 GHz structure are 60 M/m. The last quantity is the filling time F
defined as the time necessary to the rf wavefront to propagate from the input coupler to the end of a section of
length L, and given by:
s
g
Fv
L
=
. (22)
Typical values of the group velocity vg are 1-2% the speed of light and, as a consequence, the filling times
are of the order of few hundreds on ns up to 1 s. After one filling time the structure is completely filled of e.m.
energy and the beam can be injected and efficiently accelerated. It can be demonstrated that the average
accelerating field in a simple TW structure, can be expressed as
z
INacc erPE
−
=2
being PIN the input power.
In Appendix I it is also shown how the SW field of a multi-cell structure can be written as the sum of two
counter propagating travelling waves.
8 Linac technology
An extensive dissertation on linac technology is given in [19]. Here we limit to point out that the linac structures
can be realized with normal conducting or superconducting technology depending on the required linac
performances in term of average accelerating field, type of particles, rf pulse length (i.e. how many bunches we
can contemporary accelerate), duty cycle (i.e. pulsed operation or continuous wave operation) and average beam
current.
The materials used for linac structure fabrication are: oxygen-free high conductivity (OFHC) copper for
normal conducting structures (both SW and TW) and niobium for superconducting cavities (SW only). OFHC
copper is the most common material used for NC structures because it has a very good electrical (and thermal)
conductivity, it has a low Secondary Emission Yield (SEY), that allows to minimize multipacting phenomena [29]
13
during structure power up, it shows good performance at high accelerating fields, it is easy to machine with very
good roughness (up to the level of a few nm), it can be brazed or welded.
The most common material for the fabrication of SC cavities is niobium for several reasons [31-37]: it has
a relatively high transition temperature (Tc=9.25 K), very low surface resistance when exposed to rf fields, it has
a relatively high critical magnetic field, Hc=170-180 mT, it is chemically inert, can be machined and deep drawn,
it is available either as bulk or sheet in any size, fabricated by forging and rolling, it can also be used as a coating
(e.g. by sputtering) on NC materials like Cu; it has good thermal stability and has a relatively low cost.
Fig. 12: (a) Sketch of the input rf power into accelerating structures and (b) related beam structure.
9 Beam structure and rf structure
Rf structures are fed, in general, by pulses with a certain repetition rate and duty cycle (DC), which is defined as
the ratio between the pulse width (Trf_pulse) and the period (Trep) (DC=Trf_pulse/Trep). Each pulse includes from
thousands up to several million rf periods as schematically represented in Fig. 12(a). The “beam structure” in
a linac is correlated to the “rf structure” since a fraction of the rf pulse is used for beam acceleration (as represented
in Fig. 12(b)). SC structures allow operation at very high DC (>1%) up to a continuous wave (CW) operation
(DC=100%), because of the extremely low dissipated power, with relatively high accelerating field (>20 MV/m).
This means that a continuous (bunched) beam can be accelerated. NC structures can operate in pulsed mode at
lower DC (<0.1 %), because of the higher dissipated power with, in principle, larger peak accelerating field
(>30 MV/m). In low DC linacs, depending on the application, from one up to few hundreds bunches can be, in
general, accelerated.
10 Longitudinal beam dynamics of accelerated particles
In this second part of the paper the basic principles of the longitudinal and transverse beam dynamics will be
illustrated. A complete dissertation is given as example in [1,2].
10.1 Synchronous phase
Let us consider a SW linac structure made by accelerating gaps (like a DTL) or cavities as shown in Fig. 13(a). In
each gap the accelerating field oscillates in time and the accelerating voltage (Vacc) sketched in Fig. 13(b) can be
expressed as:
( )
cos
ˆtVV RFaccacc
=
. (23)
14
We can assume that the “perfect” synchronism condition is fulfilled for an ideal particle that crosses each
gap with a phase
s with respect to the accelerating voltage. By definition this phase is called synchronous phase
and the particle, synchronous particle. Then, the synchronous particle enters in each gap with a phase
s (
s=
RFts)
with respect to the rf voltage, has an energy gain (and a consequent change in velocity) that allows to enter in
the subsequent gap with the same phase
s with respect to the accelerating voltage and so on. For this particle
the energy gain in each gap is simply given by:
( )
sacc
sacc
sacc qV
V
VqE _
_
cos
ˆ==
. (24)
Looking at the plot of Fig. 13 one case easily understand that both
s and -
s are synchronous phases. Let
us consider the first synchronous phase
s (on the positive slope of the rf voltage). If we consider another “non-
synchronous” particle “near” to the synchronous one, that arrives later in the gap (t1>ts, 1>s), this particle will
experience a higher voltage (i.e. gaining a slightly larger amount of energy) and thus will have an higher velocity
than the synchronous one. Its time of flight to next gap will be reduced, partially compensating its initial delay
with respect to the synchronous particle. Similarly, if we consider a particle that enters the gap before
the synchronous one (t1<ts, 1<s), it will experience a smaller accelerating voltage, gaining a smaller amount of
energy and its time of flight to next gap will increase, compensating the initial advantage with respect to
the synchronous particle. On the contrary, if we consider the synchronous particle at phase -
s and other particles
“near” to it that arrive later or before in the gap, they will receive an energy gain that will increase further their
distance with respect to the synchronous one. In conclusion, the synchronous phase on the positive slope of the rf
voltage provides a longitudinal focusing of the beam allowing to have a stable beam acceleration. This mechanism
is called “phase stability principle”. On the contrary, the synchronous phase on the negative slope of the rf voltage
is an unstable position.
Since it relies on particle velocity variations, longitudinal focusing does not work for fully relativistic beams
(electrons). In this case acceleration “on crest” is more convenient.
Fig 13: (a) Sketch of accelerating gaps; (b) accelerating voltage in each gap.
10.2 Energy-phase equations
The previous intuitive approach can find a more rigorous mathematical formalism. To this purpose, the following
variables,
and w, are used to describe a generic particle longitudinal position with respect to the synchronous
one. They are defined as follows:
( )
−=
−=−=
s
sRFs
WWw
tt
, (25)
15
where t (
) is the arrival time (phase) of a generic particle at a certain gap and W is the kinetic energy of the same
particle at a certain position along the linac. ts,
s and Ws are the same quantities referred to the synchronous
particle.
The energy gain per “accelerating cell” (considering as “accelerating cell” one gap plus the two half drift
tubes in case of a DTL) of a generic particle and of the synchronous one are respectively:
( )
+==
=
saccacc
saccs
VqVqW
VqW
cos
ˆ
cos
ˆ
cos
ˆ
. (26)
The delta energy gain is then given by:
( )
ssaccs VqWWw
coscos
ˆ−+=−=
. (27)
Dividing by the accelerating cell length L and assuming that
LVE accacc = ˆˆ
is the average accelerating field
over the cell (i.e. average accelerating gradient) we obtain:
( )
( )
ssaccssacc Eq
dz
dw
Eq
L
w
coscos
ˆ
coscos
ˆ−+=−+=
, (28)
where we have approximated
dzdwLw
.
On the other hand, the phase variation per cell of a generic particle and of a synchronous particle are:
=
=
t
t
RF
sRFs
, (29)
where
t is basically the time of flight between two accelerating cells. Considering the variation of
between two
accelerating gaps,
( )
sRF tt −=
, and dividing by
L we have that (
1
):
w
cEdz
d
w
cE
MAT
vvL
t
L
t
Lss
RF
dz
d
L
ss
RF
s
RF
s
RF 33
0
33
0
11
−=−
−=
−
=
. (30)
The system of coupled (non-linear) differential equations represented by Eqs. (28) and (30), and reported in
the following, describes the motion of a non-synchronous particle in the longitudinal plane with respect to
the synchronous one:
1
In the MAT passage we have considered the following approximations:
33
0
332
32
22
2
11 ;
11
ss
RF
ss
RF
s
RF
s
RF
s
RF
s
ss
s
s
RF
s
RF E
w
E
ccc
dd-β
c
v
v
vvv vvv
vv vv
vv
−=
−
−==
−=−
−
−
=
−
16
( )
−=
−+=
w
cEdz
d
Eq
dz
dw
ss
RF
ssacc
33
0
coscos
ˆ
. (31)
10.3 Small amplitude energy-phase oscillations
Deriving the second equation of (31) with respect to z and assuming an adiabatic acceleration process (i.e.
( )
( )
dzdwcEwdzcEdssRFssRF 33
0
33
0
) we obtain:
dz
dw
cEdz
d
ss
RF 33
0
2
2
−=
. (32)
Assuming in the first equation of (31) small oscillations around the synchronous particle that allow to
approximate
( )
sss
sincoscos −+
and substituting this equation in (32) we finally obtain:
( )
0
sin
ˆ
2
33
0
2
2=
−
+
s
ss
saccRF cE
E
q
dz
d
. (33)
This is the equation of a harmonic oscillator with angular spatial frequency
s. The conditions to have stable
longitudinal oscillations and acceleration at the same time are then:
( )
0
2
0cos0
0sin0
2
−
−
s
sacc
ss
V
. (34)
Equation (33) and the condition (34) summarizes what we have intuitively described with the “phase
stability principle”: if we accelerate on the rising part of the positive rf wave we have a longitudinal force keeping
the beam “focused” around the synchronous phase and oscillating during acceleration with spatial angular
frequency
s. The angular frequency can be simply obtained substituting in Eq. (33) z with
sct (and then dz=
scdt)
and it is related to the angular spatial frequency by:
T=
s
sc.
Each particle during acceleration describes in the longitudinal plane, an ellipse around the synchronous
particle, as schematically represented in Fig. 14.
The maximum energy deviation is reached at
=0 while the maximum phase excursion corresponds to w=0.
All particles in the bunch occupy an area in the longitudinal phase space called longitudinal emittance and
the projections of the bunch in the energy and phase planes give the energy spread and the bunch length as
schematically represented in Fig. 14.
From Eq. (33) it is also evident that the angular frequency scale with 1/3/2 that means that for ultra-
relativistic electrons shrinks to zero and the beam is “frozen” and does not oscillate in the longitudinal plane.
17
Fig 14: oscillating particles in the longitudinal phase space around the synchronous one.
11 Longitudinal dynamics of low energy electrons
From previous formulae it is clear that there is no motion in the longitudinal phase space for ultra-relativistic
particles with >>1. This is the case of electrons, whose velocity is always close to the speed of light even at low
energies. For electrons, accelerating structures are designed to provide an accelerating field synchronous with
particles moving at v=c, as in the case of TW structures with phase velocity equal to c.
It is interesting to analyze what happens if we inject an electron beam generated in an electron gun (at low
energy) directly in a TW structure (with vph=c) and the conditions that allow to capture the beam (
2
). The sketch
is given in Fig. 15. Particles enter in the TW structure with velocity v<c and, initially, they are not synchronous
with the accelerating field. In this part of the accelerator there is a so-called “phase slippage”. After a certain
distance they can reach enough energy (and velocity) to become synchronous with the accelerating wave
(Fig. 15(a)). This means that they are captured by the accelerator and, from this point on, they are permanently
accelerated. If this does not happen (i.e. the energy increase is not enough to reach the velocity of the wave) they
are lost.
Fig. 15: basic scheme of a low energy electron gun coupled with a TW accelerating structure: capture process (a) and
phase slippage (b).
11.1 Phase slippage
We will now describe with a more precise mathematical approach this phenomenon. The accelerating field of
a TW structure can be expressed by:
2
This is equivalent to consider instead of a TW structure a SW designed to accelerate ultra-relativistic particles at v=c
18
( )
( )
tz
kztEE RFaccacc
,
cos
ˆ
−=
, (35)
where
represents the phase of the particle with respect to the accelerating wave.
The equation of motion of a particle with a position z at time t accelerated by the TW is then, from
the Lorentz force, given by:
( ) ( ) ( )
cos
ˆ
,cos
ˆ3
00 accacc Eq
dt
d
cm
dt
d
cmtzEqmv
dt
d===
. (36)
Integrating both terms of Eq. (36) between an initial and a final state [1,38] we can find which is the relation
between
and
from an initial condition to a final one:
+
−
−
+
−
+=
fin
fin
in
in
accRF
infin Eq
E
1
1
1
1
ˆ
2
sinsin 0
. (37)
Supposing that the particle reaches, asymptotically, the value
fin=1 we have:
in
in
accRF
infin Eq
cm
+
−
+= 1
1
ˆ
2
sinsin 2
0
. (38)
Equation (38) (or the more general Eq. (37)) gives several information on the physics of the low energy
acceleration process. First of all, in order to have a solution for the final phase
fin, the second term of the Eq. (38)
should be in the interval [-1,1]. This means that, for a given initial injection phase, we always have that
infin
sinsin
that means
infin
. This is the mathematical description of the phase slippage phenomenon as
represented in Fig. 15 (b). Another important consequence of Eq. (38) is that, for a given accelerating field and rf
frequency, there are only some possible injection phases for which we can capture the beam. Conversely, for
a given injection energy (
in) and phase
in we can find which is the accelerating peak field (
acc
E
ˆ
) that is necessary,
to have a relativistic beam at phase
fin (i.e. that is necessary to capture the beam at phase
fin).
11.2 Bunch compression
Equation (38) is useful to describe the bunch compression during the capture process. As the injected beam moves
up to the crest, in fact, it experiences also a longitudinal bunching, which is caused by velocity modulation (hence
the name “velocity bunching”). This is evident plotting the final phases as a function of the injection phases for
different accelerating field
acc
E
ˆ
as reported, for example, in Fig. 16(a). Differentiating Eq. (37) it is
straightforward to derive the first order compression factor:
fin
in
infin
cos
cos
=
. (39)
This mechanism can be used to compress the electron bunches in the first stages of acceleration [39].
19
Fig. 16: (a) Final phase as a function of the injection phase for different accelerating field, assuming fRF=3 GHz with two
sketched bunches before and after the capture process; (b) Scheme of two bunches with different length captured by a TW
structure coupled to a thermionic gun (the tails of the longer one are not captured by the low accelerating field).
11.3 Capture efficiency and buncher
For a given Eacc we can easily calculate the range of the injection phases
in actually accepted in the capture process
(i.e. particles whose injection phases are within this range can be captured while the other are lost). Fig. 16(b)
illustrate this concept. Assuming an accelerating voltage of Eacc=17 MV/m (and fRF=3 GHz), only the electrons
that enter into the TW structure with a phase between -120 and -60 deg are captured, the other are lost. To reduce
the number of lost particles there are, in principle two possibilities. The first one is to increase the accelerating
voltage, thus increasing the range of phases. The other is to pre-shape the beam in order to increase the number of
electrons that occupy the right phase interval. While the first solution requires more rf power to increase
the accelerating field, this second approach can be pursued with a simple scheme as reported in Fig. 17, where
a typical injector scheme is reported. The scheme foresees the use of a buncher. It is a SW cavity aimed at pre-
forming the particle bunch gathering particles continuously emitted by a source by modulating the energy (and
therefore the velocity) of the continuous emitted beam, using the longitudinal E-field of the SW cavity itself. After
a certain drift space, the velocity modulation is “converted” in a density charge modulation. The density
modulation depletes the regions corresponding to injection phase values incompatible with the capture process.
The TW accelerating structure (capture section) is placed at an optimal distance from the buncher, to capture
a large fraction of the charge and accelerate it up to relativistic energies. The amount of charge lost using this
scheme is drastically reduced, while the capture section provides also further beam bunching.
Fig. 17: Scheme of a typical electron injector using a buncher.
20
12 Transverse beam dynamics of accelerated particles
We will now describe more in detail the transverse motion of particles that are accelerated in a linac. The transverse
beam dynamics is determined by the effect of the rf fields, by the magnetic elements (quadrupoles or solenoids)
and by the collective effects such as space charge forces and wakefields.
12.1 Rf transverse forces: defocusing term
The rf fields that accelerate particles act also on the transverse beam dynamics, because of the off axis transverse
components of the electric and magnetic field, as schematically represented in Fig. 18(a), where the electric field
lines in a generic accelerating gap are reported. More precisely, considering the Maxwell’s equations in vacuum
and assuming an accelerating SW field of the type reported in Eq. (10), we can calculate the transverse force, as
reported in the following:
( )
( )
+
+
+
−
=−=
=
−=
=
=
oncontributi B
oncontributi E
Force
Lorentz
2
scoordinate
lcy lindrica
2sin
2
cos
2
2
2
1
0
c
z
zE
c
r
c
z
zzE
r
vBE
q
F
t
E
c
r
B
z
E
r
E
E
c
B
E
RFRFRF
RF
RF
r
r
z
z
r
. (40)
As an example, the transverse forces as a function of the longitudinal coordinate are reported in Fig. 18(b)
for an accelerating gap of L= 3 cm, working at fRF=350 MHz, for a =0.1 and for two different injection phases.
From the plot it is evident that the transverse forces are equal to zero in the center of the gap (except Fr|B in the case
0), while they have an opposite sign at the entrance and at the exit of the gap itself, as also visible in
the schematic picture of Fig. 18(a). It is also evident that the electric field contribution is dominant and that, if
the injection phase is negative (as required for longitudinal focusing) the two in/out contributions do not
compensate, resulting in an integrated defocusing force.
Fig. 18: (a) Electric field lines in a generic accelerating gap; (b) transverse forces as a function of the longitudinal coordinate
in traversing the accelerating gap.
More precisely, from the previous formulae it is possible to calculate the transverse momentum increase
due to the rf transverse forces. Assuming that the velocity and particle position do not change across the gap, we
obtain to the first order:
r
cLEq
c
dz
Fp
RF
acc
L
Lrr
22
2/
2/
sin
ˆ
−==
+
−
. (41)
21
The formula highlights the defocusing nature of such term (since sin
<0), that scales as 1/2. As
a consequence, it disappears at relativistic regime (i.e. for electrons
3
). For a correct evaluation of the defocusing
effect in the non-relativistic regime we have also to take into account the velocity change across the accelerating
gap, the transverse beam dimensions changes across the gap (with a general reduction of the transverse beam
dimensions due to the focusing in the first part). Both contributions give a reduction of the defocusing force but
the resulting one is still defocusing.
12.2 Rf focusing in electron linacs
We have pointed out that the rf defocusing term is negligible in electron linacs. It is important to mention that, for
this type of linacs, there is a second order effect due to the non-synchronous harmonics of the accelerating field
that give a net focusing contribution [40]. These harmonics generate a ponderomotive force i.e., a force in
an inhomogeneous oscillating electromagnetic field. As discussed in [40] it is possible to demonstrate that we have
an average focusing force given by this expression:
( )
ecm
E
rqFacc
r2
0
2
8
ˆ
−=
, (42)
where the term () is a factor that depends on the harmonic content of the accelerating field and is of the order
of 0.1 [40]. This harmonic content is larger in SW cavities because of the presence of the wave that propagate in
the opposite direction with respect to the beam (as pointed out in Appendix I). With accelerating gradients of few
tens of MV/m it is quite easy to verify that this transverse gradient can easily reach the level of MV/m2.
12.3 Collective effects: space charge forces
Collective effects are phenomena related to the number of particles in a bunch, and, in linacs, they can play
a crucial role in the longitudinal and transverse dynamics of intense beams. They are typically related to space
charge effects and wakefield and here we will focus only on the former. This effect is generated by the Coulomb
repulsion between particles. If we consider a uniform and infinite cylinder of charge with radius Rb moving along
the longitudinal axis z with an average current I, it is quite easy to verify that the force experienced by a generic
particle inside the cylinder at a radius rq is given by:
rr
cR
I
qF q
b
SC ˆ
222
0
=
. (43)
This simple example gives us a couple of interesting information: the effect of space charge is of particular
concern for low energy particles and high currents, because the space charge forces scale as 1/2 and linearly with
the current I. In this particular example the force is linear with the displacement rq, but in general space charge
forces are non-linear and depend on the beam particle distribution.
12.4 Magnetic focusing, transverse oscillations and
-function
In the previous paragraphs we pointed out that the rf and space charge forces (or the natural divergence of the beam
given by a generic source) give a defocusing effect and need to be compensated, in order to take under control,
the transverse beam dynamics in a linac. For this reason, quadrupoles are used to focus the beam and, at low
energies, also solenoids. We will consider, in particular, the first type of elements. As done in all accelerator
machines with strong focusing magnets, quadrupoles are used in an alternating configuration, since they are
focusing in one plane and defocusing into the other. In a linac they are interleaved by either accelerating gaps or
accelerating structures. The type of magnetic configuration and the magnets distance depend on the type of
3
In the case of electrons, moreover, we already pointed out that, in general, =0 for maximum acceleration and this also cancel the defocusing effect.
22
particles, energy, beam intensity and beam dimensions requirements. Due to the alternating quadrupole focusing
system, each particle (as in synchrotrons) performs transverse oscillations and the equation of motion in
the transverse plane is of the type:
( ) ( )
0
2
defocusing
RF
2
focusing
magnetic
2
2
2=−
−+
SCRF Fx
sK
sks
dsxd
. (44)
The K(s) term takes into account the magnetic configuration and the rf defocusing term that exhibits a linear
behavior with the particle displacement, while the FSC term is the non-linear space charge term.
If we neglect the space charge forces and we assume K2>0, the solution of the single particle trajectory
described by the Eq. (44) is a pseudo-sinusoid described by the equation (
4
):
( ) ( )
+= 0
0
cos)(
s
s
os
ds
ssx
, (45)
where the characteristic
-function depend on the magnetic and rf configuration along the linac and the constants
0 and
0 depend on the initial conditions of the particle at the entrance of the linac (i.e. position and angle).
Fig. 19(a) shows, as an example, different particle trajectories corresponding to different initial conditions. It is
quite easy to verify that, because of Eq. (45), all particles oscillations are contained within an envelope that scales
with the square root of the
-function, sketched, as an example in Fig 19(a) and also the final transverse beam
dimensions (
x,y(s)) vary along the linac within the same envelope.
Because of the regular magnetic and accelerating structure configuration, the
-function is “locally”
periodic, and it is useful to refer to the focusing period (Lp) that is the length after which the focusing structure is
repeated (usually equal to N
). The transverse phase advance per period is defined as:
( )
p
L
L
s
ds
p
=
(46)
And, for transverse oscillation stability, should be in the range 0<
<
[1]. The quantity
/Lp is called phase
advance per unit length.
Fig. 19: (a) Sketch of the
-function and transverse particle trajectories along the linac; (b)
-function and transverse
particle trajectories in the case of “smooth approximation”.
4
Unfortunately, the Twiss function here has the same notation than the relativistic factor . They are, obviously, two completely different quantities.
23
12.5 Smooth approximation of the transverse oscillations
In case of “smooth approximation” of the linac, we consider an average effect of the quadrupoles and rf and, as
a consequence, a constant
-function. Assuming a focusing structure of the type sketched in Fig 20, which
alternates focusing quadrupoles with accelerating gaps (also called FODO lattice) we obtain a simple harmonic
motion along s coordinate of the type:
( )
000 cos1)(
+= sKKsx o
. (47)
In particular for this simple case, we obtain that the phase advance per unit length K0 is equal to [1]:
( )
( )
termdefoc using
RF
3
2
0
termfoc using m agnetic
2
0
0sin
ˆ
2
RF
acc
cm
Eq
cm
qGl
K−
−
=
, (48)
where G [T/m] is the quadrupole gradient and l [m] is the quadrupole length, and all other quantities have been
already defined in the previous sections (
5
). In the previous formula it is possible to recognize the two contributions
of the magnetic focusing and of the rf defocusing (with opposite sign with respect to the previous one). This
simplified approach allows to make several considerations. First of all, the rf defocusing term scales with fRF
(=1/RF), and this sets a higher limit to the working frequency of the cavities (i.e. at lower particle energies it is
better to operate at lower frequency). Moreover, as already pointed out, the rf defocusing term plays a crucial role
at low energy, since the defocusing term scale as 1/(
)3.
If we consider also the space charge contribution and the simple case of an ellipsoidal beam of charge Q
(that gives linear space charge forces) we obtain for K0 the following expression [1]:
( )
( ) ( )
termdefocusing cha rge spa ce
322
0
0
termdefocusing
RF
3
2
0
termfocusing m agnetic
2
0
0813sin
ˆ
2zyx
RF
RF
acc rrrcm fqIZ
cm
Eq
cm
qGl
K
−
−
−
−
=
, (49)
where I is the average beam current (=Q/TRF), rx,y,z are the ellipsoid semi-axis, f is a form factor (0<f<1) and Z0 is
the free space impedance (equal to 377 Ω). For ultra-relativistic particles (e.g., electrons of several MeV) both
the rf defocusing and the space charge terms disappear and the external focusing is only required to control
the emittance and beam dimensions and to stabilize the beam against instabilities.
Fig 20: Focusing structure considered in the smooth approximation (FODO lattice).
5
Please note that in this formula is the relativistic factor.
24
12.6 General considerations on linac optics design for protons and ions
According to what illustrated, the beam dynamics is completely dominated by space charge and rf defocusing
forces. Focusing is usually provided by quadrupoles that are also integrated in the drift tubes of DTL structures.
The phase advance per period () should be, in general, in the range 30-80 deg [1,2]. This means that, at low
energy, we need a strong focusing term (short quadrupole distance and high quadrupole gradient) to compensate
for the rf defocusing, but the limited space in the drift tubes (proportional to ) limits this achievable integrated
gradient and, as a consequence, the beam current. As
increases, the distance between focusing elements can
increase:
in the DTL goes, as example, from 70 mm at 3 MeV, fRF=350 MHz to 250 mm at 40 MeV and can
be increased to 4-10 at higher energy (>40 MeV). As already pointed out, the overall linac is made of a sequence
of structures, matched to the beam velocity, and where the length of the focusing period increases with energy. As
increases, longitudinal phase error between cells of identical length becomes small, and we can have short
sequences of identical cells (instead of cells all with different dimensions) with a reduction of the overall
construction costs. From beam dynamics simulations it is possible to calculate the beam radius along the structures
that allows to calculate the margin between beam radius and physical apertures.
12.7 General considerations on linac optics design for electrons
For electrons, space charge forces act only at low energy or high peak current. Below 10-20 MeV (injector)
the beam dynamics optimization has to include emittance compensation schemes using, typically, solenoids. At
higher energies, no space charge and no rf defocusing effects occur, but we have rf focusing due to
the ponderomotive force in accelerating structures. In this part of the linac focusing periods up to several meters
can occur. Nevertheless, the optics design has to take into account longitudinal and transverse wakefields (due to
the higher frequencies used for acceleration) that can cause energy spread increase, head-tail oscillations and multi-
bunch instabilities. The longitudinal bunch compression schemes based on magnets and chicanes have also to take
into account, for short bunches, the interaction between the beam and the emitted synchrotron radiation in bending
magnets (Coherent Synchrotron Radiation effects-CSR) [41,42]. All these effects are important especially in linacs
for Free Electron Lasers (FEL) that require extremely good beam qualities, short bunches and high peak current.
13 Radio Frequency quadrupoles (RFQ)
At low proton (or ion) energies (0.01), space charge defocusing is strong and quadrupole focusing is not very
effective (because of the low
). Moreover, cell lengths become small and conventional accelerating structures
(DTL) are very inefficient. At these energies, it is used another type of structure, called Radio Frequency
Quadrupole (RFQ), invented in the ’60 by Kapchinskiy and Tepliakov [43]. These structures allow to
simultaneously provide transverse focusing, acceleration and bunching of the beam. The sketch of the structure
and the picture of a fabricated one are given in Fig. 21.
Fig. 21: Sketch of a RFQ structure and picture of a fabricated one.
25
The transverse focusing effect is due to the fact that the resonating mode of the cavity (between the four
electrodes) is the quadrupole mode (TE210) whose electric field lines are sketched in Fig. 22(a). The working
principle of a quadrupole electric mode is similar to that of a magnetic quadrupole with no field in the center of
the structure and a transverse electric field that increases linearly with beam off-axis and that is focusing in one
plane and defocusing in the other. The rf alternating voltage on the electrodes produces an alternating focusing
channel with a period rf.
The acceleration process is achieved by means of a longitudinal modulation of the vanes with period RF.
This creates a longitudinal component of the electric field (as given in Fig. 22(b)) that accelerates the beam
(the modulation corresponds exactly to a series of rf gaps).
The third effect, i.e. the bunching, is generated by changing the modulation period (distance between electric
field maxima), since it is designed to change the phase of the beam with respect to the rf field during beam
acceleration, while the amplitude of the modulation can be varied to change the accelerating gradient.
The continuous beam enters the first cell and it is bunched around the -90 deg phase (bunching cells), progressively
the beam is bunched and accelerated (adiabatic bunching channel) and, only in the last cells we have a switch on
a pure acceleration process. The process is illustrated in Fig. 22(c).
The mathematical description of this process would require a dedicated course and is out of the scope of
the present proceeding. Details can be found in [1,44].
Fig. 22: (a) Quadrupole mode (TE210) of the RFQ four vane geometry; (b) Longitudinal modulation of the electrodes with
sketched electric field lines; (c) Bunching mechanism in RFQ.
14 Choice of the frequency of operation and accelerating structures
The choice of the frequency and type of accelerating structures depends on several factors such as: type of particles
to be accelerated, average beam current and duty cycle, available space and compactness, cost.
Table I schematically illustrates how the accelerating structure parameters scale with frequency [19]. For
NC structures r increases with frequency as f 1/2 and this pushes to adopt higher frequencies. Nevertheless, at high
frequencies the beam-cavity interaction due to wakefields becomes more critical (wz f 2, w⊥ f 3). On the other
hand, for SC structures it can be demonstrated that the power losses increase with f 2 and, as a consequence, r scales
with 1/f, this pushes to adopt lower frequencies.
Higher frequencies allow to achieve, in general, higher accelerating gradients in NC structures but require
higher mechanical precision in the realization of the structures. Moreover, at very high frequencies (>10-12 GHz)
power sources are either commercially not available or very expensive. On the other hand, at low frequencies one
needs more bulk material and machines have generally a larger footprint.
In DTL, for protons and ions, the accelerating cell dimensions (that scale as c/fRF) become not practical
at high frequency (as example at 3 GHz and beta=0.1 the accelerating cells are 10 mm) and also the insertion of
magnetic elements in the drift tubes for transverse focusing is not possible. Also, the rf defocusing effects scale as
1/f and they do not allow a stable acceleration of the beam.
26
In general, a given accelerating structure has a curve of efficiency (shunt impedance per unit length) with
respect to the particle energy and, if it works at given , the same structure, becomes not efficient at larger and
the transition to another type of structure is necessary.
As a general consideration normal conducting linacs require high peak power from power sources and also
high average power in case of operation at relatively high DC (0.1-1%). For high duty cycle linacs (>1%) the use
of superconducting structures is the only possible solution.
For all previous considerations, it is possible to make some general considerations schematically reported
in Table II. In low duty cycle (<10-3) electron linacs higher frequencies (up to X band) can be used for both SW
than TW structures. Proton and ion linacs use low frequencies (40-500 MHz) up to 0.5 with RFQ or DTL
structures. At higher energies (>0.5) SW multi cell structures at higher operating frequencies are used (from
500 MHz up to few GHz).
Superconducting multi cell cavities working on the -mode, are used in high duty cycle linacs starting from
larger than 0.5 and, in case of protons or ions, combined with low frequency structures in the first stages of
acceleration. A complete review of superconducting accelerating cavities is given in [45].
Table I: Scaling laws for cavity parameters with frequency
Parameter
NC
SC
Surface resistance (Rs)
f 1/2
f 2
Quality factor (Q)
f -1/2
f -2
Shunt impedance per unit length (r)
f 1/2
f -1
r/Q
f
Longitudinal wakefield (w//)
f 2
Transverse wakefield (w┴)
f 3
Table II: typical use of different accelerating structures (not exhaustive)
Acknowledgements
I would like to thank Luca Piersanti for the careful reading, corrections and helpful suggestions. Several pictures,
schemes, images and plots have been taken from papers and presentations reported in the References: I would like
to acknowledge all the authors.
Cavity Type
Range
Frequency
Particles type
RFQ
0.01– 0.1
40-500 MHz
Protons, Ions
DTL
0.05 – 0.5
100-500 MHz
Protons, Ions
Multi cell ( or /2 cavities)
0.5 – 1
600 MHz-3 GHz
Protons, Electrons
SC multi cell -mode
> 0.5-1
350 MHz-3 GHz
Protons, Electrons
TW
1
3-12 GHz
Electrons
27
Appendix I: SW field as a sum of two counter-propagating TW waves.
In a multi cell SW structure working on the -mode, the accelerating field can be expressed in a simplified form,
as:
( ) ( ) ( )
( )
( )
t
zE
zkEEtzEE RF
RF
zRFzRFRFz
coscos
ˆ
cos
===
, (A1)
where we have considered a simple expression for ERF(z). In order to have synchronism between the accelerating
field and the particle of velocity v, and supposing that the velocity variation of the particle while traversing
the structure is negligible, kz has to satisfy the following relation:
vvT
kRF
RFRF
z
22 ===
. (A2)
The accelerating field seen by the particle is then given by (t=z/v):
( ) ( ) ( )
zk
EE
zkE
v
z
zkEE z
RFRF
zRFRFzRF
vtz
particle
by
seen
z2cos
2
ˆ
2
ˆ
cos
ˆ
coscos
ˆ2+==
=
=
. (A3)
On the other hand, the SW can be written, with some math, as the sum of two TWs in the form:
( ) ( ) ( ) ( )
zkt
E
zkt
E
tzkEE zRF
RF
zRF
RF
RFzRFz++−==
cos
2
ˆ
cos
2
ˆ
coscos
ˆ
. (A4)
Substituting in this last expression t=z/v we obtain the same expression (A3). In conclusion the SW field
can be “seen” as the superposition of two counter propagating TW waves. The co-propagating one is those that
gives the net acceleration
2
ˆRFzEE =
, while the other one (back propagating) does not contribute to
the acceleration but generates an oscillating term with no net acceleration effect.
Appendix II: Large longitudinal oscillations and separatrix
To study the longitudinal dynamics at large oscillations, we have to consider the non-linear system of differential
equations without approximations as given in Eq. (31). In the adiabatic acceleration case it is possible to obtain
the following relation between w and [1]:
( ) ( )
Hconstsincossin
ˆ
2
133
0
2
2
33
0
==−−++
sss
ss
accRF
ss
RF cE Eq
w
cE
. (A5)
For each H we have different trajectories in the longitudinal phase space as schematically reported in Fig. A1.
The oscillations are stable within a region bounded by a special curve called separatrix whose equation is:
( ) ( ) ( )
0sincos2sin
ˆ
2
12
33
0
=++−++ ssssacc
ss
RF Eqw
cE
. (A6)
The region inside the separatrix is called rf bucket and the dimensions of the bucket shrink to zero if
s=0.
28
Trajectories outside the rf bucket are unstable and the rf acceptance is defined as the maximum extension
in phase and energy that we can accept in an accelerator. They are given by:
s
MAX
3
2
1
33 )sincos(
ˆ
2
−
=
RF
sssaccsso
MAX
EqcE
w
. (A7)
Fig. A1: Different trajectories in the longitudinal phase space corresponding to different values of H (the red one is
the separatrix).
29
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