Download Electron linac

RF ELECTRON LINAC
SHU-HONG WANG (王书鸿)
Institute of High Energy Physics (IHEP)
Yuquan Road 19, Beijing 100049, China
E-mail: [email protected]
RF Electron Linac is introduced in this lecture note, including the principles of acceleration, their basic features,
main structures, pre-injector, longitudinal and transverse motions, and beam physics for a high current and high
energy injector linac.
1.
Introduction to RF Electron Linac
1.1. Properties of the RF Electron Linac
Electrons can be resonantly accelerated, along an almost linear orbit, by an rf electric field. This
accelerating facility is called RF Electron Linac. The rf accelerating field is either a traveling wave in
loaded waveguides, or a standing wave in loaded cavities.
RF electron linac has the following features, compared with other types of accelerator:
♦ It has no difficulties with the beam injection (into the linac) and ejection (from the linac ),
compared with the circle / ring-type accelerators.
♦ It can accelerate electrons from low energy (a few tens keV) to very high energy (~ TeV), does
not like dc high-voltage accelerator which has the dc voltage breakdown limitation, and does not like
electron ring-type accelerator which has a beam energy loss limitation caused by the synchrotron
radiation.
♦ It can provide a high current (or high intensity) beam with transverse focusing and longitudinal
bunching.
♦ It can work at a pulsed mode with any duty factor, and / or at a CW mode.
♦ It can be designed, installed and commissioned section by section.
♦ It is mostly equipped by rf accelerating structures, not easy to be operated and maintained with
high stability and reliability, and its construction and operation costs per unit beam power are
expensive compared with circle / ring accelerators.
1.2. Applications of RF Electron Linac
♦ To be as injectors for synchrotrons, synchrotron radiation light sources and
electron-positron colliders.
♦ Medical uses, such as radiotherapy and production of medical isotopes.
♦ Industrial irradiation for various materials and products.
1
♦ Linac-based Free Electron Laser (FEL).
♦ Electron-positron linear colliders.
2. Elementary Principles of RF Electron Linac
2.1 Acceleration with RF Linac
Assuming an rf electric-magnetic (EM) field travels in a uniform cylindrical waveguide, its
fundamental mode TM01 has the EM components of longitudinal electric field E and azimuthal
magnetic field B, as shown in Figure.1. Their distributions are analytically described with following
expressions;
Figure 1.
EM field pattern of TM01 mode
'
E z (r , z, t ) = E 0 J 0 (k c r )e jωt − k z ,
E r (r , z , t ) = jE 0 [1 − (
'
ω cr 2 1 / 2
) ] J 1 (k c r )e jωt − k z ,
ω
Eθ = 0 ,
(1)
'
Bθ (r , z, t ) = jμ 0 E 0 J 1 (k c r )e jωt −k z ,
Br = B z = 0 .
where J0 and J1 are zero-order and first-order Bessel functions, respectively;
k c = ω cr / ω is the wave number, its frequency is the waveguide's cutoff frequency ω cr , and its
p h a s e v e l o c i t y i s t h e v e l o c i t y o f l i g h t . T h e c u t o f f
frequency for a given radius R of the waveguide can be obtained from the boundary condition of Ez
(R) = 0, i. e the first root of J 0 (k c R) = 0 :
Kc R =
ω cr
c
2
R = 2.405
(2)
k ' = α + jk 0 , where α is the field attenuation factor due to the rf loss on a resistive wall; k 0 =
ω
vp
is
a wave number with frequency ω and phase velocity vp. Let us first consider the case of no power loss
(ideal conductor, α = 0), then its propagation property (dispersive relation) is as follows:
ω
ω
ω
k 0 2 = ( ) 2 − k c 2 = ( ) 2 − ( cr ) 2 .
c
c
c
It describes the relations among kc, ω and k0 in J 0 (k c r )e
j (ωt − k 0 z )
(3)
, as shown in Figure 2.
Figure 2 . Dispersion curve for a uniform waveguide
For TM01-mode to exist in the waveguide, k 0 should be a real number, so that ω ≥ ω cr . This means
that only the waves with ω ≥ ω cr can be propagated in the waveguide. But their phase velocity is
vp =
ω
k0
=
c
1 − (ω cr / ω ) 2
≥c
(4)
Figure 3. Disk-loaded TW structure
Obviously, these waves can not resonantly accelerate electrons. To have an accelerating structure in
which the propagated waves have v p ≤ c , we must modify the structure to slow down the v p , for
3
instance, by introducing a periodic disk-loaded structure, as shown in Figure 3. Then the wave
amplitude is periodically modulated:
E z (r , z , t ) = E L c (r , z )e j (ωt − k0 z )
(5)
where E L c (r , z ) is a periodic function with period d = Lc . This is Floquet theorem: at the same
locations in different periods, the amplitudes of the propagated field are the same but their phases
differ by a factor of e jk0 Lc . We can express E L c (r , z ) as a Fourier series in z :
E Lc (r , z ) =
∞
∑
n = −∞
E n J 0 ( k n r )e
−j
2πn
z
Lc
(6)
where the coefficients E n J 0 (k n r ) are the solutions of the wave motion equation with cylindrical
boundary condition, so that
E z (r , z , t ) =
∞
∑
n = −∞
E n J 0 (k n r )e j (ωt − kn z )
(7)
where k n = k 0 + 2πn / Lc is the wave number of the n th space harmonic, which has the phase
velocity of
vnp =
ω
kn
=
ω
k 0 (1 + 2πn / k 0 Lc )
≤c
(8)
With above expressions we find that :
♦ A traveling wave consists of infinite space harmonic waves, as shown in Figure 4.
♦ Harmonic waves with n > 0 , propagating in the + z direction, are forward waves; those with
n < 0 , propagating in the − z direction, are back waves.
♦ If each forward wave has the same amplitude and phase velocity as a back wave, then they form
a standing wave. Therefore a method of analysis using space harmonic waves can be used to describe
both standing waves and traveling waves.
♦ Because of the various space harmonic waves have different phase velocities, only one of the
harmonic waves can be used to resonantly accelerate particles. The fundamental mode (n = 0)
generally has the largest amplitude and hence is used for acceleration.
4
Figure 4. Brillouin diagram for a periodically loaded structure [5]
♦ When a TW is used to accelerate particles, a particle that “ rides ” on the wave at phase ϕ0 and
moves along the axis has an energy gain per period ( Lc , cell length) of
ΔW = e E0Lccosϕ0
where E0 is the field on axis, averaged over a period
1
E0 =
Lc
L
∫0 c
E z (0, z )dz .
(9)
(10)
♦ Figure 4 also shows a second upper branch, which is one of an infinity of such high-order modes
(HOMs), and intercepts the v p = c line. These modes are so-called wake fields, which can be excited
by the transversely offset beam.
2.2 Essential Parameters of a TW Accelerating Structure
2.2.1 Shunt-Impedance Zs
The shunt-impedance per unit length of the structure is defined as
Zs =
Ea 2
− dPw / dz
(MΩ/m).
(11)
It expresses that, given the rf power loss per unit length, how high an electric field E a can be
established on the axis. Since Pw ∝ Ea2 , therefore Z s is independent of E a and the power loss,
depends only on the structure itself: its configuration, dimension, material and operating mode.
5
2.2.2 Quality Factor Q
The unloaded quality factor of an accelerating structure is defined as
Q=
ωU
− dPw / dz
(12)
where U is the stored energy per unit length of structure. The Q also describes the efficiency of the
structure. With this definition one can see that, given the stored energy, the higher the Q , the less the
rf loss; or given rf loss, the higher the Q , the higher the E a (since U ∝ E a2 ).
2.2.3
Zs/Q
With the definitions of Z s and Q , we have
Zs/Q = Ea2/ωU .
(13)
This defines that for establishing a required electric field E a , the minimum stored energy required.
Obviously Z s / Q is independent of power loss in the structure.
2.2.4 Group Velocity vg
It is the velocity of the field energy traveling along the waveguide,
v g = Pw / U
(14)
where Pw is the power flow, defined by integrating the Poynting vector over a transverse plane
within the inner diameter of the dick. For TM01-mode, Pw = ∫0a E r H ϑ 2πrdr here a is the iris radius,
and for this mode, E r ∝ r and H θ ∝ r , so that v g ∝ a 4 .
2.2.4
Attenuation Constant
We define
τ 0 = ∫0Ls α ( z )dz
6
(15)
as the attenuation constant, where α ( z ) is the attenuation per unit length of the structure, as
mentioned in section 2.1. This is one of the most important parameters for TW structures, since it
defines the ratio of output power to input power for an accelerating section (of length Ls ), and
determines the power loss per unit length
Pout = Pin e −2τ 0
and
dPw Pin
=
(1 − e − 2τ 0 )
dz
Ls
(16)
It is clear that, the larger the τ0, the smaller the output power, and hence the higher the rate of power
use. On the other hand, a smaller τ0 gives a larger group velocity of the structure and thus a larger
inner radius of the disk ( v g ∝ a 4 ) and a larger transverse acceptance. Finally, τ0 should be chosen by
a compromise between these two effects. The residual output power is absorbed by a load installed at
the end of the section, as shown in Figure. 5.
(a) Disk-loaded TW Structure
(b) SW Structure
Figure 5. Power absorber at the end of a TW section
2.2.6 Working Frequency
The working frequency is one of the basic parameters of the structure, since it affects on most of
the other parameters according to the following scaling laws:
Q ∝ f 0−1 / 2 ,
♦ Shunt-impedance Z s ∝ f 01 / 2 , ♦ Quality Factor
♦ Total rf peak power Ptot ∝ f 0−1 / 2 , ♦ Minimum energy stored Z s / Q ∝ f 0 ,
♦ RF energy stored
U ∝ f 0−2 ,
♦ Power filling time
t F ∝ f 0−3 / 2 ,
♦ Transverse dimension of structure a and b ∝ f 0−1 .
The final choice of f 0 is usually made by adjusting all of the above factors and by considering the
available rf source as well. Most electron linac work at a frequency of about 3000 MHz (S-band), e.g.
2856 MHz (λ ≈ 10.5 cm) for the SLAC linac and many others.
2.2.7 Operation Mode
7
Here we define the operation mode, which is specified by the rf phase difference between two
adjacent accelerating cells. For instance 0-mode, π/2-mode, 2π/3 -mode and π-mode are the operation
modes that have the phase differences of 0, π /2, 2π/3 and π, respectively, between two adjacent cells,
as shown in Figure. 6.
Figure 6. Operation modes
For a disk-loaded TW structure the optimum operation mode is the 2π/3-mode, that has the highest
shunt-impedance, as indicated in Figure. 7.
Figure 7 Shunt-impedance vs. operation modes in TW structure [2]
3.
Traveling Wave Accelerating Structure
There is no firm rule with which to decide whether a traveling wave or a standing wave structure is to
be chosen. However, traveling wave structure is usually used when dealing with short beam pulses
and when particle velocities approach the velocity of light, as is the case with electrons.
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3.1 Constant Impedance Structure
With the definitions of structure parameters, the rf power distribution along the linac section is
dPw
ωP
= − w = −2α 0 Pw
dz
Qv g
where α0 is the attenuation per unit length of structure, α 0 =
the
(17)
ω
. If the structure is uniform along
2Qv g
z axis, from the above equations, we have
Ea 2 =
ωZ s
Qv g
Pw
dE a
ωE a
=−
= −α 0 E a
2Qv g
dz
and
For a uniform structure, α0=constant, E a ( z ) = E 0 e −α 0 z , and Pw ( z ) = P0 e −2α 0 z .
Thus in a constant-impedance structure, E a (z ) and Pw (z ) are decrease along the
At the end of a section with length L s :
E a ( Ls ) = E 0 e −τ 0
where τ 0 = α 0 L s =
ωL s
2Qv g
z axis in a section.
Pw ( Ls ) = P0 e −2τ 0
and
(18)
(19)
is the section attenuation. The energy gain of an electron that "rides" on
the crest of the accelerating wave and moves to the end of section is
1 − e −τ 0
. Using E 02 = 2Z sα 0 Pin ( Pin =input power), then
ΔW = e ∫0Ls E a ( z )dz = eE0 Ls
τ0
ΔW = e 2 Z s Pin Ls ⋅ (
1 − e −τ 0
τ0
)
(20)
For an optimized design of a constant impedance structure, we should maximize ΔW . Given Pin
and L s we make
Z s ⇒ maximum and (
1 − e −τ 0
τ0
) max ⇒ τ 0 = 1.26
(21)
Given L s and Q , we can obtain the optimized group velocity v g . Obviously the smaller v g , the
bigger τ0, and the bigger v g , the lower E 0 . An effective way to control v g is to adjust the inner
radius a of the disk along the section. On the other hand, the power filling time of a waveguide is
t F = Ls / v g = 2πτ 0 / ω . To decrease t F , then, τ0 should be < 1.26.
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3.2 Constant Gradient Structure
To keep E a = E 0 = constant along the structure, the structure is not made uniform, so that α0 =
α0( z ). The question is how to determine α0( z ). Let us change the radii of the structure, a and b , to
vary v g and to keep frequency constant along the section, also to keep the variations of Q and Z s
along z so small that they can be neglected, then we have
dPW / dz = −2α 0 ( z ) Pw and PLs = P0 e −2τ 0
where τ 0 = ∫0Ls α 0 ( z )dz is a section attenuation. Since E a2 = − Z s
(22)
dPw
, to keep E a = constant, we
dz
need dPw / dz = const., so that
PLs − P0
Pw ( z ) = P0 +
Ls
⎡ 1 − e −2τ 0
z = P0 ⎢1 −
Ls
⎣⎢
⎤
z⎥
⎦⎥
(23)
Thus in a constant gradient structure, Pw should be linearly decreased along the structure. With
dPW / dz = −2α 0 ( z ) PW and v g ( z ) = ω / 2Qα 0 ( z ) , we have
α 0 ( z) =
and
v g ( z) =
1
⋅
2 Ls
ωL s
1 − e −2τ 0
z
1−
(1 − e − 2τ 0 )
Ls
1−
⋅
(24)
z
(1 − e − 2τ 0 )
Ls
(25)
1 − e − 2τ 0
Thus in a constant gradient structure, the v g (z ) also decreases along the structure in the same way as
Q
PW (z ) . The energy gain for an on-crest particle is
ΔW = e ∫0Ls E a ( z )dz = eE 0 Ls Since E 0 2 = − Z s
dPLs
dz
=
Z s P0
(1 − e − 2τ 0 ) , then
Ls
ΔW = e Z s P0 Ls (1 − e −2τ 0 )
(26)
To have ΔWmax , we should have Z s ⇒ maximum and τ0⇒ maximum, all power should be lost in
the structure. On the other hand we should also consider the filling time, t F = 2Qπ / ω , and τ0 should
be chosen by a compromise among some effects. An example of a SLAC constant gradient structure
10
is shown in Figure 8. Each section is designed to be a tapered structure: 2b ≈ 8.4 to 8.2 cm, 2a ≈ 2.6
to 1.9 cm, v g / c ≈ 0.021 to 0.007,
L s = 3.05 m, and Z s ≈ 57 MΩ/m.
The advantages of the constant gradient structure are its uniform power loss and lower average
peak surface field, thus most TW electron linacs are designed as constant gradient structures.
Figure 8. Parameters of a SLAC constant gradient Structure [2]
4.
Standing Wave Accelerating Structure
4.1 Standing wave for acceleration
A direct and a reflected sinusoidal varying waves, traveling with the same velocity but in opposite
directions, combine to create a standing wave (SW) pattern. If the amplitudes of the direct and
reflected wave are A and B, respectively, the SW pattern has maximum A+B and minimum A-B,
distant from each other by d = π / 2k 0 , with k 0 = ω / v p . The average amplitude of the SW pattern is
A, hence the same as the direct traveling wave (TW). Such a SW pattern is not useful since the
reflected wave only dissipates power traveling backwards and does not contribute to the acceleration
of particles.
However, SW accelerators use both the direct and reflected waves to accelerate particles. It can be
understood from Figure 4: at the points where the direct and reflected space harmonics join, they have
the same phase velocity, and if this velocity synchronous with the particle, both harmonics contribute
to the acceleration.
11
From Figure.4 again, one finds that SW accelerators operate either at the lowest or highest
frequency of the pass band, where k n L = Nπ , and N = 0,±1 . That means the operation modes in SW
accelerators is either 0 or π .
4.2 Stabilized SW accelerating structure
In addition to the high accelerating efficiency, the structure should have a high stability as well in
the operation.
4.2.1 Properties of a structure with single-periodic chain
A linac structure usually consists of a series period to provide a space harmonic wave that has
phase velocity to be equal to the particle velocity for accelerating particles resonantly. In this case the
structure has only one kind of period, or so-called single-periodic chain. It is known that in a cavity
many "separated modes" can be excited, such as TM010, TM011,..TM01n and so on. They have different
frequencies and amplitudes. On the other hand, in a single-periodic chain, many "operating mode" can
be excited, that are defined by their phase shift between adjacent periods, such as 0, π/2, π-modes.
These two kinds of modes are related to and different from each other. To simply study the properties
of a single-periodic structure, an equivalent circuit is usually used. In this analytical way each period
of the structure is described by an equivalent circuit with lumped parameter (L,C,R) and all periods in
the structure are coupled each other with those lumped parameters.
Assuming that a structure consists of N periods (cells) in the chain, and is terminated by an half
cell at both ends. Each mode's frequency, amplitude and phase can be obtained by solving N+1
coupled equations. With the periodic property of the chain, the nth solution of these equations can be
X n = An cos nϕ n ,
n = 0, 1, 2,...N
(27)
where An is a constant and ϕn is the phase shift between adjacent cells. For a SW structure, at the two
ends of the chain we have X0 = XN, so that
Nϕ = qπ ,
q = 0, 1, 2,....N.
(28)
Then the coupled equations have their solutions of Eigen function (field amplitude)
X n q = A0 cos(
12
qnπ
)
N
(29)
and Eigen value (frequency)
ωq =
ωa
(30)
1 + k c cos(qπ / N )
where ωa is the resonant frequency in each cell. These equations describe the dispersion relation, by
that one can see:
♦ If a chain consists of N+1 cells, then only N+1 modes can be excited in the chain.
♦ Each ωq corresponds a mode which has a phase shift of ϕ = qπ/N between adjacent cells.
♦ The number q for the modes of 0, π/2 and π-modes are 0, N/2 and N, respectively, and only when
N is even, there is a π/2-mode in the chain.
♦ The band width of the chain is ωb=ωπ-ω0, as shown in Figure.9. Usually the coupling constant
kc<< 1 (e.g 5%), then ωb ≈ kcωa, is proportional to kc.
Figure 9. Band width and modes
♦ The mode separation between two adjacent modes is varied with mode's number. From (30), tt is
easy to have the following expressions of the mode separations at 0-, π/2- and π - modes:
(
Δω
ωa
) 0,π =
k cπ 2
4N
2
,
and (
Δω
ωa
)π / 2 =
k cπ
2N
(31)
Thus the biggest mode separation is at π/2-mode and it is proportional to kc/N2, while the smallest one
is at 0- and π-modes
dω
≈ 0 for 0 and
♦ The group velocity of each mode can be also easily obtained, for instance v g =
dk
dω
π-modes, and v g =
= ω a k c Lc / 2 for π/2-mode. Thus π/2 mode has the biggest group velocity
dk
13
and hence the fastest energy propagation in the structure. This is very helpful to overcome the beam
loading effect, particularly for high current beam acceleration.
♦ π-mode is located at the edge of the pass band. Both of the forward wave of n = 0 and its
backward wave's harmonic wave of n = 1 make the contributions to accelerate particles, see Figure. 4,
hence it has the highest shunt impedance compared with all other modes. On the contrary π/2-mode
has the lowest shunt impedance, since all of its backward waves have the phase velocities in the
opposite direction to particle's motion, so they do not make any contribution to the acceleration and
just to be lost on the cavity wall. A brief summary of mode's comparison is listed in Table 1.
Table 1. A brief summary of mode's comparison
Effective shunt impedance
π-modes
π/2-mode
maximum
minimum
Mode's separation
minimum
maximum
Group velocity
minimum
maximum
Field distortion by perturbation
maximum
minimum
By Table 1, one can see that for a SW structure, 0-mode or π-mode has the highest shuntimpedance but the lowest group velocity, thus it has high accelerating efficiency but may not be stable
in operation. On the other hand, the π /2-mode has the lowest shunt-impedance but the largest group
velocity, and thus has lower accelerating efficiency but high operation stability. To have a structure
with both high efficiency and high stability, the solution is to use a so-called biperiodic structure,
which combines the advantages of π-mode and π/2-mode.
4.2.2 Properties of a biperiodic chain
The purpose of introducing a biperiodic chain is to have an operation mode which can combine all
advantages of π/2 and π-modes, and hence to form a very effective and very stable accelerating
structure. Let us introduce a coupling periodic chain to the accelerating periodic chain and to form a
biperiodic chain. By adjusting the coupling chain to make its pass band to be coupled resonantly with
the accelerating pass band, so that at 0-mode or π-mode there is also a faster group velocity and a
bigger mode separation, as shown in Figure. 10. The example of this kind of structure for electron
linac is so called coupled-cavity linac (CCL).
4.3 Coupled-Cavity Linac (CCL)
14
The CCL is operated at π/2-mode of the biperiodic chain. In this structure,
the couple element is also cavity-type. There are two types of CCL for the electron linac. One is onaxis-coupled structure, where each couple cavity is located on axis and between two accelerating
cavities, as indicated in Figure 11 (b). Another one is so called side-coupled structure, as shown in
Figure 11 (c)
Figure 10. Resonantly coupling of accelerating passband with coupling passband
Figure 11
π / 2 -mode operation of a cavity resonator chain.[3]
15
and in Figure 12, where each couple cavity is alternately located at the side of the accelerating cavity.
The coupling factors for both on-axis-coupled and sidecoupled structures are adjusted by modifying the slot size between accelerating cavity and coupling
cavity. In the side-coupled structure, the traveling ways for beam and for rf field are separated, so that
both accelerating cavity and coupling cavity can be optimized independently from each other.
Figure 12 Side-Coupled Cavity structure
5.
Electron Pre-injector Linac
As we have mentioned at the beginning of this lecture note that electron linacs are widely used as the
injectors of synchrotrons, SR light sources and ring-type electron-positron colliders, linac based FEL,
radiotherapy machines, and electron-positron linear colliders. All of these electron linacs should have
the pre-injectors at the beginning, even though these pre-injectors are some different from each other
for the various uses. However, they consist of most basic and common components of the electron
linac.
Figure 13 shows a schematic layout of an injector linac. Two types of electron pre-injector are
commonly used: a dc high voltage gun with a bunching (velocity modulation) system (as shown in
Figure 13), and an rf gun followed by a short accelerating structure.
16
Figure. 13 Schematic layout of an electron injector linac [5]
The dc electron gun has a cathode (thermionic or photo-cathode) and an anode. It produces
electrons with pulse lengths of 1 μs to several μs and a beam energy of 50 keV to 200 keV. If the gun
uses a thermionic cathode, then a wire-mesh control grid is needed to form a beam pulse, which
normally works at a voltage of about minus 50V with respect to the cathode. The cathode consists of
some oxide and is heated to reduce the work function of electron emission. Beam dynamics in the gun
is usually simulated with the EGUN code.
Since the electrons from the gun have the velocities of v < c (e.g. ~ 0.5 c), the electron bunch can
be shortened by using a bunching system that modulates the electron velocity with an rf field in the
cavity or in the waveguide bunchers, followed by a drift space. The rf is phased with respect to
electron beam such that the front electrons experience a decrease in energy and the back electrons
experience an increase in energy. Beams can be bunched to about 100 of the fundamental frequency.
The bunched beam is accelerated to about 20~50 MeV before the space charge effects can be
neglected in these system.
The first stage of bunching the beam from the dc high voltage gun is commonly accomplished by
using SW single-cavity bunchers, followed by some TW bunchers for further bunching and
acceleration. Usually the first few cells of the TW buncher have v p < c in order to synchronize with
the beam.
If the gun uses a photo-cathode, then the electrons are produced by the photo-electric effect, using
a laser pulse incident on the cathode, and no grid in the gun.
The rf gun is followed by an accelerating structure, since the electrons from the cathode are soon
bunched by the rf field. The rf gun consists of one or more SW cavities with the cathode installed in
the upstream wall of the first cavity. Compared with the dc gun, the rf gun has the advantage of
quickly accelerating electrons to relativistic velocity (about 5 to 10 MeV), which avoids collective
effects such as the space-charge effect and provides a shorter bunch length and lower beam emittance
at the cathode. However, the rf gun has some time-dependent effects due to its time-dependent rf field,
which may dilute the performance of electron bunches.
6. Longitudinal Motion
17
The principles of the longitudinal motion (described in this section) and the transverse motion
(described in the next section) are almost the same both for electron linac and proton linac. However,
since the static mass of proton is much heavier then electron (by a factor of about 1830), so that in the
RF linacs, mostly β= v/c < 1 for protons and β=1 for electrons, and hence there are some differences
between proton and electron in these motion., that will be mentioned somewhere in these sections.
6. 1 Stable Synchronous phase in Linacs
In the linacs, the stable synchronous phase should be selected in a phase region where the electric
field is rising with time, as shown in Figure 14. For high energy electrons (β=1), the longitudinal
phase oscillation (any particle in a bunch to the reference particle or synchronous particle) is almost
disappeared. One can put the beam bunch just on crest to have the maximum acceleration. However if
electron bunch charge is high (e.g. > 1 nC) and bunch length is small, then the longitudinal wake field
effect will induce an additional energy spread of the bunch (single bunch beam loading effect). To
cure this effect, we have to select the bunch center off crest.
Figure 14. Stable synchronous phase for linacs
6.2 Longitudinal Motion Equation
The energy gain per cell for synchronous and non-synchronous particles are given as follows,
respectively:
δWs = eE 0TLc cosϕ s , and δW = eE 0TLc cosϕ
(32)
where T is so called time transit factor which will be talked about in the Proton Linac Chapter. For the
TW accelerating structure, T = 1, and for the SW structure, T< 1.
Assuming ΔW = δW − δWs is small that
dΔW
= eE 0T (cos ϕ − cos ϕ s )
dz
(33)
where ΔW = m0 c 2γ s3 β s Δβ , and Δβ = β − β s .
Simultaneously, their phase difference and velocity difference per cell are given by follows,
respectively:
z − zs
β 2 λ dΔϕ
Δϕ = ϕ − ϕ s = −
2π , and Δβ = − s
(34)
β sλ
2π dz
18
Combining ΔW and Δϕ expressions, one can get longitudinal motion equation for non-synchronous
particles:
2πeE0T
1 d
dΔϕ
( β s3γ s3
)+
(cos ϕ − cos ϕ s ) = 0
(35)
3 3 dz
dz
m0 c 2 β s3γ s3 λ
βsγ s
ΔW
dΔϕ
=
dz
m0 c 2
So the longitudinal motion equation describes the non-synchronous particle’s motion in ( ΔW , Δϕ )
phase space.
Since Δγ = γ − γ s = −
λ
2π
β s3γ s3
6.3 Stability of the Longitudinal Motion
Assuming that the accelerating rate (per unit length) is so small that one can neglect its damping
d (β sγ s )
, then the longitudinal motion equation can be simplified as:
term
dz
d 2ϕ
dz
d(
or
then
2
+
2πeE 0T
m0 c 2 β s3γ s3 λ
(cos ϕ − cos ϕ s ) = 0
(36)
2πeE0T
dϕ
)=−
(cos ϕ − cos ϕ s )dz
dz
m0 c 2 β s3γ s3 λ
(37) using dz =
dϕ
dϕ / dz
2πeE 0T
dϕ dϕ
d( ) = −
(cos ϕ − cos ϕ s )dϕ
dz
dz
m0 c 2 β s3γ s3 λ
by integrating above equation both sides, and considering
dϕ
2π
ΔW
=− 3 3
dz
β s γ s λ m0 c 2
we have
c1Wk2
+ c 2 (sin ϕ − ϕ cos ϕ s ) = H ϕ
2
eE T
2π
ΔW
.
where c1 = 3 3 , c 2 = 0 2 , Wk = δγ =
βsγ s λ
m0 c
m0 c 2
(38)
The 1st term in left side of Equation (40) is a “kinetic energy”, and the 2nd is a “potential energy” of
the motion, and the Hϕ in right side is an integration constant, or say Hamitonian. Hence Equation
(40) describes the energy conservation of motion. That is Liouville’s theorem: in a motion system
which can be described by a Hamitonian, the area surrounded by a phase trajectory is constant. The
potential well and phase trajectory are shown in Figure 15.
19
Figure 15． Potential well and stable region of the longitudinal motion.
By these figures, one can see that:
π
z Within − < ϕ s < 0 , there is a potential well, all particles in the well are steady oscillated
2
around the synchronous particle ( Ws , ϕ s ) and are steady accelerated along the structure.
z The boundary of stable region is ϕ 2 < ϕ < −ϕ s where, ϕ 2 is a solution of H ϕ (ϕ 2 ) = H ϕ (−ϕ s ) .
For a small angle oscillation, ϕ 2 ≈ 2 ϕ s , so that the width of stable region ≈ 3 ϕ s .
z
The phase boundary of stable region (so called “Separatrix”, “fish” or “bucket”) is described by
c 2 [sin( −ϕ s ) − (−ϕ s ) cos ϕ s ] = H ϕ .
z
The energy boundary of stable region can be obtained by taking
Wk ,max =
ΔWmax
m0 c 2
⎡ 2eE 0Tβ s3γ s3 λ
⎤
=⎢
(ϕ s cos ϕ s − sin ϕ s )⎥
2
⎢⎣ πm0 c
⎥⎦
6.4 Acceleration Effects
20
ϕ = ϕs ,
1/ 2
(39)
d ( β sγ s )
. If we now take it account into
dz
the motion equation, then one can find that the related phase trajectory is changed a little, but the
phase area surround by this trajectory is still a constant. For the acceleration of very low-βparticles, if
the variations of βs and γs in a cell are so large that can not be neglected, then this stable phase area is
increased and hence the accelerator’s acceptance is improved. In this case the bucket’s boundary is
changed from a “fish” to a “golf club”, as shown in figure 16.
In above description we have ignored the damping term
Figure 16 Stable boundaries
(a) fish (ignoring acceleration)
(b) golf club (including acceleration)
6.5 Small Angle Oscillation
Expending
cos ϕ ≈ cos ϕ s − (ϕ − ϕ s ) sin ϕ s −
(ϕ − ϕ s )2 cos ϕ
s,
2
and just taking its linear term for small angle oscillation, then the longitudinal motion is linearized as
d 2 (Δϕ )
dz
where k l2 =
2
+ k l2 (Δϕ ) = 0
2πeE0T sin ϕ s
m0 c 2 β s3γ s3 λ
(40)
is the phase oscillation wave number.
2π
≥ 10λ and f l ≤ 10% f rf . As β → 1 ,
kl
the phase oscillation is going to be disappeared. For a small angle oscillation, one can easily get the
phase damping law:
The related phase oscillation period and frequency are Ll =
Δϕ ∝ ( βγ ) −s 3 / 4 , and ΔW ∝ ( βγ ) 3s / 4
also as shown in Fig.17.
21
(41)
Figure 17. Phase damping in small angle oscillation.
7. Transverse Motion
7.1 Field Distribution Near Axis of Structure
In this section we discuss the transverse motion both for electron linac and proton linac with TW
(waveguide, TM01-like mode) and SW (cavity, TM010-like mode), respectively. Since in the linacs,
beam is accelerated along / near axis, hence the RF field distribution near the axis is the most
interested.
1) Ez Expression
Let us start with wave motion equation in cylindrical coordinate system:
∇ 2 E (r ) + k 2 E (r ) = 0 ,
∇ 2 H (r ) + k 2 H (r ) = 0
and consider the axis-symmtry of TM01 and/or TM010 modes
∂E
∂2E
= 0 , and
= 0 , ( also for H )
∂ϑ
∂ϑ 2
so that
∂ 2 Ez
1 ∂ ∂E z
(r
)+
+ k 2 Ez = 0
r ∂r
∂r
∂z 2
moving the 2nd and 3rd terms to the right side, and integrating over r, we have
⎞
∂E z
1 ⎛ ∂ 2 Ez
+ k 2 E z ⎟dr
= − ∫ r⎜
2
⎜
⎟
∂r
r ⎝ ∂z
⎠
Assuming the radius of the cylinder R, and expanding the field near axis, r << R, by
r 2 ∂ 2 Ez
E z (r , z , t ) = E z (0, z , t ) +
(
) r =0 + ........
2 ∂r 2
∂E z
over r, and omit the terms above r3, we have
then integrate
∂r
22
∂E z
r ⎡⎛ ∂ 2 E z
= − ⎢⎜
∂r
2 ⎢⎜⎝ ∂z 2
⎣
⎤
⎞
∂2Ez
1 ⎡⎛⎜ ∂ 2 E z
⎟
⎢
=
−
+ (k 2 E z ) r =0 ⎥ , and
⎟
2 ⎢⎜⎝ ∂z 2
⎥⎦
∂r 2
⎠ r =0
⎣
⎤
⎞
⎟
+ (k 2 E z ) r =0 ⎥
⎟
⎥⎦
⎠ r =0
so that
E z (r , z , t ) = E z (0, z , t ) −
r2
4
⎡⎛ ∂ 2 E
z
⎢⎜
⎢⎣⎜⎝ ∂z 2
⎤
⎞
⎟
+ ( k 2 E z ) r =0 ⎥
⎟
⎥⎦
⎠ r =0
(42)
For a SW structure, the field distribution does not change with time except its amplitude, so one can
separate the varieties of r, z, and t as
E z (0, z , t ) = Ε z (0, t ) cos(ωt + ϕ 0 )
(43)
where
ϕ0
is the phase at t = 0. Assuming the structure is periodic with period of Lc, then according
to the Flouqeut theorem, the field can be expressed by a periodic function:
Ε z (0, z ) =
∞
∑
−∞
An cos(
2πn
z)
Lc
(44)
Put (44) into (43) then
E z (0, z , t ) =
∞
∑
−∞
Bn cos(ωt −
2πn
z + ϕ0 )
Lc
(45)
It means that the Ez field consists of a series of space harmonic waves, which have wave numbers and
phase velocities, respectively, as follows:
kn =
2πn
ω
, and v pn =
Lc
kn
(46)
Take a harmonic wave of n = 1, which has the phase velocity vp1 as the same as particle’s velocity v,
then
E z (0, z , t ) = B1 cos(ωt −
2πn
z + ϕ0 )
Lc
(47)
For a synchronous acceleration with 0-mode, for instance, the phase velocity and the length of each
accelerating period should be kept as:
v p1 =
ω
2π / Lc
=
Lc
= v or Lc = vTrf = βλ
Trf
23
(48)
where the particle’s velocity variation within a period is assumed to be omitable.Above equation is so
called resonant / synchronous accelerating condition. From E z (0, z , t ) expression mentioned above,
one has
∂ 2 E z (0, z , t )
∂z 2
= −(
2π 2
) E z (0, z , t )
βλ
ω
2π
Put this equation into(44), and consider k 2 = ( ) 2 = ( ) 2 , then we have
c
λ
⎡
πr 2 ⎤
E z (r , z , t ) ≈ E z (0, z , t ) ⎢1 + (
)
(49)
βγλ ⎥⎦
⎣
To reduce the beam energy spread, one should take r ↓, λ↑, βγ↑.
2) Er Expression
Similar with deduction of Ez expression, the Er expression can be deduced as
π
(50)
Ε z (0, t ) sin(ωt + ϕ 0 )
βλ
It shows that E r (r , z , t ) is increased linearly with r, but decreased with βλ.
E r (r , z, t ) = −r
3) Hθ Expression
Similarly, we can have
πcε 0
Ε z (0, z ) sin(ωt + ϕ 0 )
(51)
λ
Hθand Er contribute the transverse forces to particle’s motion, their force’s ratio is
−eμ 0 v z H ϑ
= −β 2
H ϑ (r , z, t ) = −r
eE r
Since these two forces have different signs (+/-) from each other, hence the total transverse force is
eπ
(52)
Fr = eE r − eμ 0 v z H ϑ = −(1 − β 2 )r
Ε z (0, z ) sin(ωt + ϕ 0 )
βλ
Obviously, when β→ 1, then Fr → 0.
For a synchronous particle we take ωt + ϕ 0 = ϕ s , and −
π
2
≤ ϕ s < 0 for a stable longitudinal motion.
Then we find that
if β< 1, Fr> 0 , defocusing force.
if β→1, Fr → 0.
Therefore we have to introduce a focusing force into the linac, e.g. a series of quardrupole magnets to
keep transverse motion to be stable.
24
7.2 Focusing Element and Transverse Motion Equation
A typical transverse focusing element is a quadrupole (magnetic or electric). It produces the
transverse forces (Lorentz forces) when a particle is in a quadrupole and offseted from longitudinal
axis, as shown in Fig 18. In an ideal quadrupole field the pole tips have hyperbolic profiles and
∂B y
∂B
.
produce a constant transverse quadrupole gradient G = x =
∂x
∂y
Figure 18 A quadrupole magnet cross section
B0
.
r0
For a particle moving along the z direction with velocity v and with transverse coordinates (x, y),the
Lorentz force components are
For a pole tipe with radius r0, and pole-tip field B0, then the gradient is G =
Fx = −evG x , and F y = evG y
If eG is positive, the quadrpole lens focuses in x and defocuses in y.
The transverse motion equations (Hill’s Equations) of this particle in this lens are
d 2x
dz
2
+ k 2 ( z ) x = 0 , and
where k 2 ( z ) =
eG ( z )
m0 cβγ
d2y
dz
2
− k 2 ( z) y = 0
(53)
is so called focusing strength.
Although individual lenses focus in only one plan ( say in x plan ), they can be combined (so called
lattice) with both polarities to give overall strong focusing (Alternating Gradient Focusing , Periodic
Focusing System) in both x and y plans, as shown in Figure 19.
25
Figure 19. FODO quadrupole lattice with accelerating gaps (or tubes)
7.3 Matrix Solution of Hill’s Equation
A quadupole-transport (drift) channel can be described by the motion equation:
x '' + K ( z ) x = 0
(54)
2
where x '' = d x
, and K ( z ) = k 2 ( z ) .
dz 2
if K = 0, drift motion ; K > 0, focusing; K < 0, defocuing.
Above equation is a linear second-order differential equation, its solution can be given in matrix form:
⎡ x ⎤ ⎡ m11
⎢ x ' ⎥ = ⎢m
⎣ ⎦ ⎣ 21
m12 ⎤ ⎡ x 0 ⎤
m 22 ⎥⎦ ⎢⎣ x 0' ⎥⎦
(55)
(a) Drift space ( K = 0 ):
⎡1 l ⎤
M =⎢
⎥
⎣0 1⎦
where l = drift length
(b) Focusing quadrupole ( K=eG/m0βγ> 0)
⎡
⎢ cos K l
M =⎢
⎢⎣− K sin K l
sin K l ⎤
⎥
K ⎥
cos K l ⎥⎦
(c ) Defocusing quadrupole ( K=eG/m0βγ< 0)
⎡
sinh K l ⎤
⎢ cosh K l
⎥
M =⎢
K ⎥
⎢⎣ K sinh K l cosh K l ⎥⎦
(d) Thin lens:
26
A quadupole lens approaches its thin-lens approximation when
Kl → 0 ,
while K l remains finite. Then we have
⎡ 1
M = ⎢± 1
⎢ f
⎣
0⎤
⎥
1⎥
⎦
where f is the focal length, "+"---defocusing; "–" ---focusing and
eGl
1
= Kl=
f
m0 cβγ
The total transfer matrix through a sequence of constant elements is obtained by forming the product
of the individual M matrices, such as:
M = M n M n −1...., M 3 M 2 M 1 .
7.5 Courant-Snyder invariant
As we have discussed that the transverse motion equations (Mathieu-Hill Equation) are:
eG
kx =
x '' + K x ( z ) x = 0
m0 βγ
eG
y '' + K y ( z ) y = 0
ky = −
m0 βγ
Their solutions in a phase-amplitude form could be
x( z ) = A β ( z ) cos(ψ ( z ) + δ )
x' ( z ) = −
A
β ( z)
[α ( z ) cos(ψ ( z ) + δ + sin(ψ ( z ) + δ )]
where A and δ are the constants of integration, and α ( z ) ≡ −
The amplitude function, (or
β - function), satisfies
2 ββ '' − β ' 2 +4 β 2 K x = 4 , (with phase advance of
β ' ( z)
2
dψ
1
= )
dz
β
Let us introduce the Courant-Snyder invariant
A 2 = γx 2 + 2αxx ' + βx '2 = constant.
(58)
27
with γ =
1+α 2
β
7.6 Transfer Matrix Through One Focusing Period
The transfer matrix M through one full period can be written:
⎡cos σ + α sin σ
M (z → z + Lp ) = ⎢
⎣ − γ sin σ
β sin σ
⎤
cos σ − α sin σ ⎥⎦
(59)
1) Here σ is the phase advance per period of every particle, each lying on its own trajectory ellipse.
If the focusing is either too strong or not enough, then the solution is unstable. The required condition
for stability of the transverse motion is trM < 2 . Here “tr” is the trace of the M matrix. From above
equation we find that the stability requirement corresponds to cos σ < 1 , or say 0 < σ < π .
2) Each particle’s motion in the linear system (only with linear forces) has an ellipse trajectory in
phase space of ( x, x’ ) or ( y, y’ ), as shown in figure 25. This ellipse is described with three
1+α 2
independent parameters:ε(ellipse area), α and β(ellipse shape); with γ =
. These parametersε, α,
β
and β(or γ ) are involved in above matrix M, and be called as Courant-Snyder parameters or Twiss
parameters. The unit of ε: m.rad (or mm.mrad), The unit of β: m (or cm, mm ). The unit of γ: 1/m. α
is dimensionless.
Figure 25. An ellipse trajectory in phase space
7.7 Acceptance, Emittance and Beam Matching
Accelerator’s Acceptance
From (60) in (x, x’) plan, the maximum outer boundary of the phase space (ellipse, determined by
the M matrix) is called as Acceptance, with its
28
x max = ra =
Aβ max .
(60)
Here ra is the inner radius of accelerating tube, and A is the acceptance value of this accelerator:
A=
ra2
β max
, and
β max
is determined by the focusing strength.
Beam emittance
Beam emittance is a beam projection in the phase space (say x and x’). In a linear system, the
emittance has a shape of ellipse usually. The beam emittance (its area and shape) at each location
along the linac axis is determined by both initial beam parameters and accelerator structures. The
outer ellipse of the projection in the phase space is so called Beam Emittance, or say the beam
emittance is defined to be the phase space (x, x') occupied by the beam:
ε = γ b x 2 + 2α b xx ' + β b x ' 2
(61)
Beam Matching
If the injected beam’s ellipse is well matched to the focusing system, that means at the injection
location every particle’s trajectory has the same orientation and same shape of the ellipse as the ones
of acceptance (determined by the accelerator).
If the injected beam is not matched with the accelerator acceptance, then the beam’s trajectories in
the phase space will be changed from its original one to the new one which is determined by the
accelerator. In this case the beam projection in the phase space or say its emittance will be enlarged,
and may induces beam loss. Therefore, a beam matching to the accelerator at the injection location is
very important to keep the beam quality.
It is clear that a beam matching in the 4-dimentional transverse phase spaces (x, x', y, y') requires
following 4 conditions:
α bx = α ax , α by = α ay , β bx = β ax , β by = β ay
hence one can adjust 4 quadrupoles’ gradient prior its injection into the accelerator’s periodic
focusing system.
If the beam is not matched, then the beam emittance will be enlarged by a factor of Bmag :
ε 2 = Bmag ε 1
Bmag =
α
α ⎤
1 ⎡ β1 β b
+
+ β b β1 ( b + 1 ) 2 ⎥
⎢
2 ⎣ β b β1
β b β1 ⎦
(62)
where α1 , β 1 designed lattice (acceptance) parameters at the injection position and α b , β b
unmatched beam parameters at the injection position.
It is convenient to express the equation of beam emittance in a matrix form as
29
X T σ −1 X = 1
[
⎡x⎤
where X = ⎢ ' ⎥ , and X T = x
⎣x ⎦
σ −1 =
(63)
]
x ' is the transpose matrix;
1 ⎡γ α ⎤
⎡ β −α⎤
and its inverse matrix is σ = ε ⎢
⎥
⎢
⎥
ε ⎣α β ⎦
⎣− α γ ⎦
By this expression of the emittance, for a well matched beam’s motion from location 1 to location 2
with transfer matrix of R, then its emittance at location 2 can be obtained by
σ 2 = Rσ 1 R T .
(64)
It is worth to mention that for the beam transformation in a periodic focusing lattice, the phase
advance σ per period Lp is related to the βfunction by
Lp
σ =
∫
0
dz
β( z )
(65)
and the average βover the period is approximately β ≈ σ / L p .
The βmax is usually located in the middle point of focusing quadrupole, and the βmin in the middle point
of defocusing quadrupole.
Normalized Emittanc
If the emittance is defined in the phase space of ( x, Px ), where Px is the momentum of the beam
in x-plan, then in the linear transformation of beam motion(including acceleration), the emittance (the
phase space area occupied by the beam) is constant according to Liouville’s theorem. We call this
emittance as normalized emittance.
If the emittance is defined in the phase space of (x, x’ ), where x’= dx/dz then even in the linear
transformation of beam motion(including acceleration), the emittance is not constant during
acceleration, since x' ∝ Px / βγ . We call this emittance as un-normalized emittance, or simply
emittance.
8. Beam physics of high energy / current electron injector linac
Both factory type of e+e- colliders and the 3rd generation light sources need their injector linacs to
have high energy (for full energy injection) and high current (for having high beam injection rate into
rings) with small beam emittance and small energy energy spread. To have those high quality beams
from the injector linacs, one has to consider some beam physics issues to overcome the possible beam
performance dilutions caused by the high current.
8.1 Space charge effects on the low energy part of the injector
30
Let us consider two of many electrons in a bunch going on the same direction. There are two
kinds of forces acting on each electron and caused by all other electrons in the bunch . One is the
statistic electric force eE r according to the Coulomb law. Another one is the Lorenz force with the
magnetic field Bθ caused by the moving electrons according to the Ampere law
z
F1 = eEr =
e2 r
∫ n( r )dr
ε 0r 0
e
e 2 v 2 μ0 r
F2 = − v × Bθ = −
∫ n( r )dr
c
cr 0
where n(r ) is the charge density distribution function. The total space charge force is
z
Fsc = e( Er −
v
Bθ ) = eEr (1 − β 2 ) = eEr / γ 2
c
(66)
So that if β < 1 ， Fsc ≠ 0 ； and if β = 1 ， Fsc = 0 .
The longitudinal space charge effects will cause an additional beam energy spread and the
transverse space charge force will cause a normalized emittance growth. For the beam coming from
thermionic cathode electron gun which is commonly used for the injector linac, the beam energy
(depending on the extract voltage between anode and cathode) is usually 80 keV ~ 200 keV, the
related β = 0.413 ~ 0.659 , hence the space charge effects may be strong for the high current beam,
e.g. for 10 A with 1 ns beam pulse (the pulse charge 10 nC/pulse). To partially cure these effects, a
higher extract voltage of the gun is preferable. Here are some examples of simulated beam
performances with different extract voltages by the EGUN and PARMELA codes.
Figure 26 shows the normalized beam emittance variations along the pre-injector (consisting of an
electron gun, an bunching system and an pre-accelerator up to about 30 MeV with optimized solenoid
focusing field) for gun extract voltages of 80 kV, 120 kV and 200 kV. Some drupes of the emittance
just downstream of the buncher are due to the beam loss in the bunching processes.
From this figure one can see that the final normalized emittance at the exit of the pre-accelerator is
about 3.5 πcm-mrad for 80 keV beam and less than 2.0πcm-mrad for 150 keV beam. Figure 28 shows
the longitudinal bunch distribution in a beam pulse with different gun extract voltages. One can see by
this figure that, if the gun extract voltage is high than 150 kV then a clean three bunches’ pattern can
be obtained by the buncher (2856 MHz). It means that with this beam energy from the gun the
longitudinal space charge effects can be well cured in the bunching processes.
31
Normalized emittance
（cm.mrad)
4
3.5
3
2.5
2
1.5
1
0.5
0
120KV, Xrms
80KV,Xrms
150KV, Xrms
Z
82
100
120
169
274
365
386
Location along the pre-injector（cm)
Figure 26. Normalized beam emittance variation along the pre-injector
a) 80 kV
b) 120 kV
c) 150 kV
Figure 27. Bunched particle distribution in energy-phase space with different gun voltage
8.2 Wakefield effects
As the beam current increasing the head particles in a bunch (or the upstream bunch in a bunch train)
may produce the electric-magnetic fields in the structure, which travel in the structure following the
head particles (or the upstream bunch). These fields are called to be single bunch wakefield (or multibunch wakefields) which act on the tail particles in a bunch (or on the downstream bunches)
longitudinally and transversely and make the beam performance dilutions. Figure 28 shows the
transverse wakefield effects when the bunch offset from the axis transversely.
32
Figure 28 Transverse wakefield effects
Upper is single bunch effect and lower is the multi-bunch effect
8.2.1
Single bunch longitudinal effect
The single bunch wake effect can be well described by a two-macroparticle model. The energy
variation due to the single bunch longitudinal wake for head and tail macroparticles (each having
charge of Ne/2 and separated by a distance d) respectively are
dEh
Ne2
=−
W// (0) and
4
dz
dEt
Ne2
Ne2
=−
W// (0) −
W// ( d )
4
2
dz
(67)
For the SLAC type accelerating structure (2856 MHz) and if with a bunch length of 3 mm, the
calculated wake functions are W// (0) = 225V / pC / m and W// (3mm) = 57.4V / pC / m . Hence one can
estimate the averaged bunch energy loss (beam loading) in the accelerating structures and the energy
difference between head and tail macroparticles leading to the additional beam energy spread.
To compensate the averaged bunch energy loss, one can apply a litter more RF power from the
power source. While to compensate the bunch energy spread, one can put the bunch center off crest of
the accelerating wave, so that the particles in the tail and head parts having higher and lower energy
gain, respectively.
8.2.2 Multi-bunch longitudinal wake effect
For multi-bunch dynamics only the fundamental accelerating mode (beam loading) is important,
and for a constant gradient structure, the loaded accelerating gradient is
dE dE0 W// ( s )Qbτ 1 − e −2τ
(
− e − 2τ )
=
−
− 2τ
2
τ
dz
dz
1− e
33
(68)
where Qb is the bunch charge, τ the attenuation of the structure and W// ( s ) the wake function at a
distance of s. The long range wakefields can be specified in a simply form in which wakefields from
one bunch to the next are presented by a point-like wake kick V/pC/m plus its first derivative in
position s along the bunch. For the SLAC type S-band structure, if τ = 0.57 , Qb = 2.5nC and there
are 3 bunches in 1 ns beam pulse with bunch spacing 10.5 cm, then we have
W// (10.5cm) = 53.11V / pC / m and W// ( 21cm ) = 40.47V / pC / m
To compensate the multi-bunch longitudinal wake effect, the simplest scheme is that the
amplitude of the input rf field is linearly ramped during the bunch train injection in to the structure.
One could adjust the timing of the bunch train, for example, let the 1st, the 2nd and the 3rd bunches
enter the structures at 0.70 ns, 0.35 ns and 0 ns respectively, before the filling time of 0.83μs, so that
the input RF field in the structure is ramped during the beam pulse and hence the most bunch to bunch
energy variation in a short bunch may be compensated. The best timing can be defined by measuring
each bunch’s energy with beam position monitor (BPM) installed at a downstream position where the
dispersion is larger
8.2.3 Single bunch transverse wake effects
By the two macroparticle model, if the initial bunch offset x0 at z = 0 , then at z = s , the tail
particle’s further offset caused by the wake W⊥ (d ) of head particle is
⎡ Δx ⎤
Ne2W⊥ ( d )
=
×s
⎢ ⎥
4kE
⎣ x0 ⎦ max
where k is the quadrupole focusing strength. k ∝
magnetic gradient
(69)
Lq ∂B
, with quadrupole effective length Lq ,
E ∂r
∂B
and particle energy E. For the SLAC type structure, if the bunch length is 3
∂r
mm, then we have
W⊥ (3mm ) = 3.4kV / pC / m 2 .
To cure this effect the well known BNS damping can be employed. In this scheme, the
accelerating phase of the bunch center is selected in the range of 0 < ϕ < π / 2 , so that the tail particle
in the bunch will have its energy of less than the head particle. Since the focusing strength is inversely
proportional to the beam energy, hence the tail particle will meet its stronger focusing than the head
particle. The cost of using this scheme is a dilution of bunch energy spread. As we have seen from the
formula (71), the single bunch transverse effect is inversely proportional to the bunch energy. Thus in
the low energy part of the injector linac, one can employ the BNS damping scheme and then in the
high energy part one can shift the accelerating phase of the bunch center to the normal range of
−π / 2 < ϕ s < 0 so that the bunch energy spread can be damped soon after.
34
For the most injector linacs which are not long enough to employ the BNS damping scheme
(only SLC-linac used), then an orbit correction scheme may be adopted to cure the single bunch
transverse wakefield effect, together with controlling the misalignment of the accelerating structures.
This scheme will be discussed in the later.
8.2.4 Multi-bunch transverse wake effects
As it is well known, the multi-bunch transverse wake causes the cumulative BBU (Beam BreakUP) effect. Different from the single bunch BBU, its wake function W⊥ (d ) is dominated by one or a
few resonators having large shunt-impedance r⊥n ,
ωnd
r ω −
ω d
W⊥ ( d ) = ∑ ⊥ n n e 2 cQn sin( n ) .
Q
c
n
n
(70)
As the same as the long range longitudinal wake, the long range transverse wakefields from one
bunch to the next can also be represented by a point-like wake kick V/pC/m2 plus its first derivative in
position d along the bunch. By this simplification, for SLAC-type (2856 MHz, 10.5 cm wavelength)
structure and if 3 bunches ina bunch train, one has
W⊥ (10.5cm) = 2.064kV / pC / m 2 and W⊥ ( 21cm) = 0.548kV / pC / m 2
To cure this BBU effect except controlling the misalignment of the accelerating structures, an orbit
correction scheme can be adopted.
8.3 Chromatic effect
If a high current beam has a large energy spread in the low energy part of
the linac, then a chromatic effect may be appeared in the quadrupole focusing system. Since the
quadrupole focusing strength is inversely proportional to the particle’s energy
ΔE ∂B
1 ∂B
and Δk ∝ 2
E ∂r
E ∂r
giving the magnetic field gradient, hence the particles with different energies will meet the different
focusing strength leading to an normalized emittance growth. To cure this effect, one may need to
optimize the bunching processes to make the bunch energy spread as small as possible giving a high
bunching efficiency, and may employ a high accelerating gradient to make the bunch energy
increasing as rapid as possible.
k∝
8.4 Dispersive effect
If the high current beam has an offset with respective to the quadrupole
center, then the beam may be affected by a dipole component in the quadrupole system and make the
particle’s trajectory further oscillated along the linac axis
35
ΔE i
E
where α is the dispersive function caused by the dipole component. In the low energy part of the
linac usually the beam energy spread is not so small that the oscillation trajectories of particles with
different energy could be separated leading to an effective emittance growth with respect to the linac
axis. To cure this effect, one has to control the quadrupole misalignment and employ the beam orbit
correction scheme.
xi = α
References
1.
2.
Lapostolle and A. Septier, Eds., Linear Accelerators, North Holland and Wiley, 1970.
G. A. Loew and R. Talman, Elementary principles of linear accelerators,
AIP Conf. Proc. 105, 1983.
3. T. P. Wangler, Principles of RF Linear Accelerators, Jone Wiley & Sons,
Inc. 1998.
4. D.H. Whittum, Introduction to Electrodynamics for Microwave Linear
Accelerators, SLAC-PBU-7802, April 1998.
5. A. W. Chao and M. Tigner, Eds., Handbook of Accelerator Physics and
Engineering, World Scientific Publishing Co., Inc., 1998.
The relevant Sections in this Handbook are as follows:
1.6.10 G.A. Loew, Linear accelerator for electrons , p. 26.
2.4
H.G. Kirk, R. Miller and D. Yeremian, Electron gun and preinjector, p. 99.
7.3.5 G.A. Loew, Normal Conducting υp = c Linac Structures , p. 516.
6.7
Z.D. Farkas, RF Pulse Compression, p. 374.
7.1.1 A.D. Yeremian , R.H. Miller, Electron gun and preinjector, p. 419.
2.5.1 K. Thompson , K. Yokoya, Collective effect in high energy electron
linacs, p. 103.
36