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1. Introduction

1.1 Essential Properties of the RF Linacs

Charged particles can be resonantly accelerated, along an almost linear orbit, by an rf electric field. This accelerating facility is called rf linac. The rf accelerating field is either a traveling wave provided by the loaded waveguides, or a standing wave by the loaded cavities. RF linac has the following features, compared with other types of accelerator: ? RF linacs have no difficulties with the beam injection (into the linac) and ejection (from the linac ), compared with the circle/ring-type accelerators. ? RF linac can accelerate the charged particles from low energy (a few keV) to very high energy, 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 strong transverse focusing and adiabatic longitudinal bunching. ? It can work at a pulsed mode with any duty factor in principle, and 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/maintained with high stability/reliability, and its construction/operation costs per unit beam power are expensive compared with circle/ring accelerators.

1.2 RF Linac Applications

? To be as injectors for synchrotrons, SR light sources and colliders. ? Medical uses, such as radio-therapy and production of medical isotopes. ? Industrial-irradiation and ion implantation. ? RF linac-based Free Electron Laser (FEL). ? Electron-positron linear colliders. ? High intensity proton linac for the Spallation Neutron Sources (SNS) and for other uses related to the nuclear energy.

-1-

2. Electron Linacs

2.1 Acceleration with the RF Linac

Assuming an electric-magnetic 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 Fig.1. Their distributions are analytically described in the following expressions:

Figure 1. EM field pattern of TM01 mode

E (r , z , t ) ? E 0 J 0 (k c r )e z

j? t ? k z

,

J 1 (k c r )e

j? t ? k z

E r ( r , z , t ) ? jE 0 [1 ? (

? cr ?

) ]

2 1/ 2

, (1)

E? ? 0 ,

B? ( r , z , t ) ? j ? 0 E 0 J 1 ( k c r ) e

j? t ? k z

,

Br ? B z ? 0 .

where J0 and J1 are zero-order and first-order Bessel functions, respectively; k c ? ? cr / ? is the wave number, its frequency is the waveguide cutoff frequency, and its phase velocity is the velocity of light. It is obtained from the first root of J 0 ( k c r ) ? 0 , as shown in Fig. 2:

Figure 2.

Function

J 0 (k c r )

-2-

kcr ?

k ? ? ? jk 0

? cr

c

? r ? 2 . 405

(2)

, where ? is the field attenuation factor due to the rf loss on a resistive wall,

k0 ?

?

vp

is a wave number with frequency ? and phase velocity vp.

Let us first consider the case of no power loss (ideal conductor, ? = 0) first, then its propagation property (dispersive relation) is as follows:

k0

2

?(

?

c

) ? kc

2

2

?(

?

c

) ?(

2

? cr

c

)

2

.

j(? t ? k

0

(3)

z)

It describes the relations among kc, ? and k0 in

J 0 ( kc r) e

, as shown in Fig.3.

Figure 3 .

Dispersion curve for a uniform waveguide

For TM01 to exist in the waveguide, k 0 should be a real number, so that ? ? ? cr .This means that only the waves with ? ? ? cr can be propagated in the waveguide. But their phase velocity is

vp ?

?

k0

?

c 1 ? (? cr / ? )

2

?c

(4)

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 instance, by introducing a periodic disk-loaded structure, as shown in Fig.4. Then the wave amplitude is periodically modulated:

Figure 4. Disk-loaded TW structure

-3-

E z (r, z, t) ? E L ( r, z)e

j (? t ? k 0 z )

(5)

where EL (r, z) is a periodic function with period Lc . This is the Floquet theorem: at the same places in different periods, the amplitudes of the propagated field are the same but their phases differs by a factor of e jk 0 L . We can express EL (r, z) as a Fourier series in z :

? ? j 2? n L z

E L ( r,z ) ?

? E n J 0 ( k n r )e

n ? ??

where the coefficients En J 0 (k n r) are the solutions of the wave motion equation with cylindrical boundary condition, so that

E z ( r , z ,t ) ?

? E n J 0 ( k n r )e

n ? ??

?

j( ? t ? k n z )

(6)

where kn is the wave number of the which has the phase velocity of

n

th

space harmonic wave,

?

kn ?

k n ? k 0 ? 2? n / L

v np ?

?

k 0 (1 ? 2? n / k 0 L )

?c

.

With the above expressions we find that: ? A traveling wave consists of infinite space harmonic waves, as shown in Fig. 5.

Figure 5. Brillouin diagram for a periodically loaded structure

? 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 a standing wave and a traveling wave. ? Because 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-

? When a TW is used to accelerate particles, a particle that “ rides ” on the wave at phase ?0, as shown in Fig.6, and moves along the axis has an energy gain per period ( Lc , cell length) of

Fig. 6 A particle “ rides ” on the wave at phase ?s,

?W = e E0Lccos?s

where E0 is the field on axis, averaged over a period

E0 ?

(7)

? Lc 0

1

Lc

E z ( 0 , z ) dz

? Figure 5 also shows a second upper branch, which is one of an infinity of such highorder 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 1) Shunt-Impedance Zs

The shunt-impedance per unit length of the structure is defined as

Zs ? Ea 2 ? dPw / dz

(M?/m)

(8)

It expresses, given the rf power loss per unit length, how high an electric field E a can be established on the axis. Since Pw ? E a2 , 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.

2) Quality Factor Q

The unloaded quality factor of an accelerating structure is defined as

Q ?

?U

? dPw / dz

(9)

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 2 U ? E a ).

-5-

3) Zs/Q

With the definitions of

Zs

and

Q

, we have (10)

Zs/Q = Ea2/?U

This defines, for establishing a required electric field E a in a structure, the minimum stored energy required. Obviously Z s / Q is independent of power loss in the structure.

4) Group Velocity vg

Group velocity is the velocity at which the field energy travels along the waveguide,

v g ? Pw / U

(11)

where Pw is the power flow, defined by integrating the Poynting vector over a transverse plane. For TM01 mode,

Pw ?

?0

a

E r H ? 2 ? rdr

(12) and

H? ? r ,

here a is the iris radius, and for this mode,

Er ? r

so that

vg ? a

4

.

5) Attenuation Constant ?0

We define

?0 ?

?0

Ls

? ( z ) dz

(13)

where ? (z) is the attenuation per unit length of the structure. This is one of the most important parameters for a TW structure, 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

? 2? 0

(14)

? 2? 0

dPw dz

?

Pin Ls

(1 ? e

).

(15)

It is clear that, the larger the ?0 , the smaller the output power, and hence the higher the rate

of power use . On the other hand, a smaller ?0 gives a larger group velocity of the structure and thus a larger iris radius ( v g ? a 4 ) and a larger transverse acceptance. This is important in the positron injector design, since the positron beam usually has a large transverse emittance at the beginning of acceleration. Finally, ?0 should be chosen by a compromise between these two effects. The output power is usually absorbed by a load installed at the end of the section, as shown in Fig. 7.

-6-

(a) Disk-loaded TW Structure

(b) SW Structure

Figure 7. Power absorber at the end of TW section

6) Working Frequency f0

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: ? Shunt-impedance ? Total rf peak power ? RF energy stored

Z s ? f0

Ptot ?

1/ 2

, ,

? Quality Factor

Q ? f0

?1 / 2

,

Z s / Q ? f0

1

?1 / 2 f0

? Minimum energy stored ? Power filling time

b?

?1 f0

,

U ?

?2 f0

,

tF ?

?3 / 2 f0

,

? Transverse dimension of structure, a and

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 linacs work at a frequency of about 3000 MHz (S-band), e.g. 2856 MHz (? ? 10.5 cm) for the SLAC linac and many others.

7) Operation Mode

Here we define the operation mode, which is specified by the rf phase difference between two adjacent accelerating cells. For instance 0-mode, ?/2-mode, 2?/3 -mode and ?mode are the operation modes that have the phase differences 0, ? /2, 2?/3 and ? respectively between two adjacent cells, as shown in Fig. 8.

Figure 8. Operation modes

For a SW structure, 0-mode or ?-mode has the highest shunt-impedance but the lowest group velocity, thus it has high accelerating efficiency, but may not be stable in operation. On the other hand, the ? /2-mode has the lowest shunt-impedance but the largest group velocity, and thus has lower accelerating efficiency but high operation stability. For a structure with both high efficiency and high stability, the solution is to use a so-called bi-

-7-

periodic structure, e.g. a side-coupled cavity structure, which combines the advantages of ?mode and ?/2-mode. We will talk about the operation modes for SW structure in the later. For a disk-loaded TW structure the optimum operation mode is the 2?/3-mode, that has the highest shunt-impedance, as shown in Fig.9.

Figure 9. Shunt-impedances vs. operation modes.

2.3 Constant Impedance Structure

With the definitions of structure parameters, the rf power distribution along the linac section is

dPw dz ? ?

? Pw

Qv g

? ? 2 ? 0 Pw

(16)

?0 ? ?

2Qv g

where ?0 is the attenuation per unit length of structure,

.

If the structure is uniform along the z axis, from the above equations, we have

Ea 2 ?

? Zs

Qv g

Pw

and

dE a dz

??

? Ea

2Qv g

? ? ? 0 Ea

.

(17)

For a uniform structure, ?0 = constant,

E a ( z) ? E 0 e

?? 0 z

, and

Pw ( z) ? P0 e

? 2? 0 z

.

(18) are decrease along the z axis in a

Thus in a constant-impedance structure, Ea (z) and section. At the end of a section with length Ls ,

E a ( Ls ) ? E 0 e

?? 0

P (z) w

and

Pw ( L s ) ? P0 e

?2 ? 0

.

(19)

where

? 0 ? ? 0 Ls ?

? Ls

2Qv g

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

-8-

?W ? e?

Ls

0

E a ( z ) dz ? eE 0 L s

1?e

?? 0

?0

(20)

Using

E 0 ? 2 Z s ? 0 Pin

2

( Pin = input power), then

?W ? e 2Z s Pin L s ? ( 1? e

?? 0

?0

).

(21)

?W .

For an optimized design of a constant impedance structure, we should maximize Given Pin and Ls we make

Zs

maximum

and

(

1? e

?? 0

?0

) max ? ? 0 ? 1. 26 .

(22)

Given Ls and Q , we can obtain the optimized group velocity v g . Obviously, the smaller v g , the bigger ?0 ; and the bigger v g , the lower E 0 . An effective way to control v g is to adjust the iris radius a along the section. On the other hand, the power filling time of a waveguide is

t F ? L s / v g ? 2Q ? 0 / ?

.

(23)

To decrease

tF

, then, ?0 should be adjusted to be < 1.26.

2.4 Constant Gradient Structure

To keep Ea ? E 0 = constant along the structure, the structure is not made uniform, so that ?0 = ?0( z ). The question is how to determine ?0( z ). Let us change the radii of the structure, a and b , to vary v g along the section, and keep the variations of Q and Z s along

z so small that they can be neglected, then we have

dPW / dz ? ?2? 0 (z)P w

Ls

and

PL s ? P0 e

? 2? 0

(24)

dPw dz

where

?0 ?

?0

? 0 ( z ) dz

is a section attenuation. Since

Ea ? ? Z s

2

, to keep

Ea =

constant,

we need

dPw / dz =

constant, so that

PL s ? P0 Ls

? 2? 0 ?? ?? 1? e z ? P0 ?? ? 1 z ??. Ls ?? ??

Pw ( z) ? P0 ?

(25)

Thus in a constant gradient structure, With

dPW / dz ? ?2? 0 (z)P W

Pw

should be linearly decreased along the structure. , we have

1? ? z Ls 1? e

?2 ? 0

and

v g ( z ) ? ? / 2 Q ? 0 ( z)

? 0 (z ) ?

1 2 Ls

1? e ? z ? 2? 1? (1 ? e 0 ) Ls

? 2? 0

and

vg(z ) ?

? Ls

Q

(1 ? e

?2 ? 0

)

.

(26)

-9-

In a constant gradient structure, the v g ( z ) also decreases along the structure in the same way as PW (z ) . The energy gain for an on-crest particle is

?W ? e?

dPL s dz

Ls

0

E a ( z ) dz ? eE 0 L s

.

(27)

Since

E 0 2 ? ? Zs

?

Z s P0 Ls

(1 ? e

?2? 0

),

then

?2 ? 0

? W ? e Z s P0 Ls (1 ? e

).

(28)

To have

Zs

? Wmax

, we should have: ?0 maximum ? all power should be lost in the structure.

tF ?

maximum and

On the other hand, we should also consider the filling time,

?0

Ls

1 v g (z)

dz

, and ?0 should

be chosen by a compromise among some effects. An example, SLAC constant gradient structure ( see Fig.10), each section is designed to be a tapered structure: 2 b ≈ 8.4 to 8.2 cm, 2 a ≈ 2.6 to 1.9 cm, v g / c ? 0.021 to 0.007, Ls = 3.05 m, Z s ? 57 M?/m, tF = 0.83μ s.

Figure 10. Parameters of a SLAC constant gradient Structure

The advantages of the constant gradient structure are its uniform power loss and lower average peak surface field, thus most electron linacs are designed as constant gradient structures.

2.5 Standing 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, while the standing wave structures is usually used for proton and heavy ion linacs. However, due to the high shunt impedance and high power use rate of a standing wave structure, it is also used for the electron linacs. A direct and a reflected sinusoidally varying wave, 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

- 10 -

and minimum A-B, distant from each other by d ? ? / 2 k 0 , with k 0 ? ? / 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 5: 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. From Figure.5 again, one finds that SW accelerators operate either at the lowest or at the highest frequency of the pass band, where k n L ? N ? , and N ? 0 , ? 1 . That means the operation modes in SW accelerators is either 0 or ? . The details for a standing wave structure will be introduced in the chapter of proton linacs.

2.6 Electron Pre-injector Linac

As we have mentioned in the section 1.2 of this lecture, 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 use. However, they consist of most basic and common components of the electron linac. Figure 11 shows a schematic layout of an injector linac. Two types of electron preinjector are commonly used: a dc high voltage gun with a bunching (velocity modulation) system and an rf gun followed by a short accelerating structure.

Figure 11. Schematic layout of an electron injector linac The dc electron gun has a cathode (thermionic or photo-cathode) and an anode. It produces electrons with pulse lengths of 1 ns to several ?s 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

- 11 -

of electron emission. Beam dynamics in the gun is usually simulated with the EGUN code. Fig.12 shows the simulation results of an thermionic electron gun with EGUN code, giving the gun optics at 160 kV and showing equal-potential surfaces and the beam-focusing pattern.

Figure 12. A simulation result of a thermionic electron gun with 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 linac 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 continuous beam (un-bunched) has the parameters of W0, β0 and γ0. When it goes into a buncher to have an energy modulation , as shown in Fig.13,

? W ? W? ? W 0 ? eE 0TL b sin ?

(29)

where Lb is the buncher length, and E0 the averaged Ez field over the length.

Figure 13. Energy modultion with a buncher

- 12 -

Its velocity modulation

?? ? ?? ? ? 0 ? ?W m0c ? 0 ? 0

2 3

, and

? ? ? ?max

so that

1 ) ?

?

eE 0 TL b m0c ? 0 ? 0

2 3

(30)

At downstream buncher,

d?? ds ? 2? ( 1

?? ? ?? ? ? ? ? 0 ,

1 2? 1

?

?0

?

??

) ?

?

(

?0

?

2?

eE 0 TL b

? 0 ? ??

3 3 ? m 0 c 2? 0 ? 0

?

1 Lf

(31)

here Lf is the longitudinal focusing length. Choosing a drift length Ld between buncher and accelerator, so that ? ?

?

?

2

(i.e. at Ld, ? ? , max

? m0c ? 0 ? 0

2 3 3

? ?0

), as shown in figure 14. Then

Ld ?

(32)

4

eE 0 TL b

Figure 14. Beam phase distribution at a distance downstream the buncher

Fig. 15 shows a SLC preinjector system. It consists of a dc high voltage gun (80 kV), two sub-harmonic SW bunchers (178.5MHz), a four-cell TW fundamental frequency buncher (2856 MHz, v p = 0.75 c), and a fundamental frequency accelerator (2856 MHz, 3 m long, 10 v p = c). This preinjector can bunch a beam of 7 x 10 electrons with bunch length 2.5 ns (FWHM) and energy 120 keV to a bunch of 5 x 1010 electrons in 20 ps at 40 MeV.

Figure 15. SLC preinjector system with dc HV gun and prebunchers

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.

- 13 -

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 timedependent effects due to its time-dependent rf field, which may dilute the performance of electron bunches. Like the dc gun, the rf gun can have a thermionic cathode or a photo-cathode. As an example, Figure 16 shows an L-band photo-cathode rf gun at LANL, which consists of a single half-cell cavity at L-band, a Cs3Sb cathode, and a laser pulse train with wavelength 522 nm. The electron pulse pattern produced from the cathode is a bunch train of 6 ?s, with micro-bunches of 53 ps (FWHM), separated by 9.2 ns. A high bunch charge of 27 nC was produced in each micropulse. At the bunch charge of 10 nC, the measured emittance is about 3x10-5 m.

Figure 16. Photo-cathode rf gun at LANL

2.7 RF Pulse Compressor

Another essential component of the electron linac is high power rf source which in most machines are klystrons with its associated high voltage modulators. Magnetrons are used in low energy electron linacs with single sections. Since the high peak power required (say 10 - 80 MW), these klystrons work at low duty cycles, for example with the pulse repetition rate of about 100 Hz and pulse lengths about a few μs. To increase the peak power and hence to increase the accelerating gradient, some linas use rf pulse compression system, such as so called SLED (SLAC Energy Development). The rf pulse compressor is used to compress a longer pulse at a lower power into a shorter pulse with higher power.

- 14 -

Figure 17. SLAC rf system (a) without SLED, (b) with SLED

The principle of SLED can be seen in Figure 17, where part (a) shows non-SLED operation, with the rf power from klystron directly transmitted to the linac. Part (b) shows the SLED system, which has two major components: a ?-phase shifter on the drive side of the klystron, and two high-Q (Q0= 100,000) cavities on the output side of the klystron, operated at TE015-mode and connected to a 3-db coupler (hybrid). During the first part of the pulse(4.2 μs), the phase of the rf drive signal is positive, the rf cavities fill up with energy at that phase, and emits the field ( E e ). During the second part of the pulse (0.8μs), the rf drive signal is reversed. In this case the fields emitted by the cavities ( E e ) add to the field reflected by the cavity coupling irises ( RE in ):

E out ? E e ? RE in

(33)

and the power flows toward the accelerator sections.

- 15 -

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