rf electron linac

7/29/2019 RF Electron LINAC

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RF ELECTRON LINAC

SHU-HONG WANG ()

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 Linac1.1. Properties of the RF Electron LinacElectrons 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.


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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

zktjc erkJEtzrz

E'

)(),,( 00= ,

zktjc

crr erkJjEtzrE

'

)(])(1[),,( 12/12

0=

,

0=E , (1)

zktjc erkJEjtzrB

'

)(),,( 100= ,

0== zr BB .

where J 0 and J 1 are ze ro-order and f i r s t -order Besse l f unc t ions , r e spec t ive ly ;

/crck = is the wave number, its frequency is the waveguide's cutoff frequency 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 E z(R) = 0, i. e the first root of 0)(0 =RkJ c :

405.2== Rc

RK crc

(2)


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0

'jkk += , where is the field attenuation factor due to the rf loss on a resistive wall;

pvk

=0 is

a wave number with frequency and phase velocity vp. Let us first consider the case of no power loss

(ideal conductor, = 0), then its propagation property (dispersive relation) is as follows:

222220 )()()(

cck

ck

crc

== . (3)

It describes the relations among kc, and k0 in)(

00)(

zktj

c erkJ

, as shown in Figure 2.

Figure 2 . Dispersion curve for a uniform waveguide

For TM01-mode to exist in the waveguide, 0k 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

cc

kv

cr

p

==20 )/(1

(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 cvp , we must modify the structure to slow down the pv , for


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instance, by introducing a periodic disk-loaded structure, as shown in Figure 3. Then the wave

amplitude is periodically modulated:

)( 0),(),,(zktj

cLzezrEtzrE

= (5)

where ),( zrEcL

is a periodic function with period d = cL . 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 cLjk

e 0 . We can express ),( zrEcL

as a Fourier series in z :

z

L

nj

nn

nLccerkJEzrE

2

0 )(),(

== (6)

where the coefficients )(0 rkJE nn are the solutions of the wave motion equation with cylindrical

boundary condition, so that

=

=n

zktjnnz

nerkJEtzrE)(

0 )(),,(

(7)

where cn Lnkk /20 += is the wave number of theth

n space harmonic, which has the phase

velocity of

cLknkk

vcn

np +==

)/21( 00

(8)

With above expressions we find that :

A traveling wave consists of infinite space harmonic waves, as shown in Figure 4.

Harmonic waves with 0>n , propagating in the z+ direction, are forward waves; those with

0


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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 0 andmoves along the axis has an energy gain per period ( cL , cell length) of

W = e E0Lccos0 (9)

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

= cL zc

dzzEL

E 00 ),0(1

. (10)

Figure 4 also shows a second upper branch, which is one of an infinity of such high-order modes(HOMs), and intercepts the cvp = 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

dzdP

EZ

w

as

/

2

= (M/m). (11)

It expresses that, given the rf power loss per unit length, how high an electric field aE can be

established on the axis. Since 2aw EP , therefore sZ is independent of aE and the power loss,

depends only on the structure itself: its configuration, dimension, material and operating mode.


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2.2.2 Quality Factor Q

The unloaded quality factor of an accelerating structure is defined as

dzdP

UQ

w /=

(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 aE (since

2

aEU ).

2.2.3 Zs/QWith the definitions of sZ and Q , we have

Zs/Q = Ea2/U. (13)

This defines that for establishing a required electric field aE , the minimum stored energy required.

Obviously QZs / 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,

UPv wg /= (14)

where wP is the power flow, defined by integrating the Poynting vector over a transverse plane

within the inner diameter of the dick. For TM01-mode, rdrHEPa

rw 20= here a is the iris radius,

and for this mode, rEr and rH , so that4

avg .

2.2.4 Attenuation ConstantWe define

= sL dzz00 )( (15)


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as the attenuation constant, where )(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

02= ePP inout and )1( 02= e

L

P

dz

dP

s

inw (16)

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 smaller0 gives a larger group velocity of the structure and thus a largerinner radius of the disk ( 4avg ) and a larger transverse acceptance. Finally, 0 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:

Shunt-impedance 2/10fZs , Quality Factor2/1

0 fQ ,

Total rf peak power 2/10 fPtot , Minimum energy stored 0/ fQZs ,

RF energy stored 20 fU , Power filling time 2/30

ftF ,

Transverse dimension of structure a and 10 fb .

The final choice of 0f 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 ( 10.5 cm) for the SLAC linac and many others.

2.2.7 Operation Mode


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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 of 0, /2, 2/3 and , 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 2/3-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 StructureThere 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

wg

ww PQv

P

dz

dP02

== (17)

where 0 is the attenuation per unit length of structure,gQv2

0

= . If the structure is uniform along

the z axis, from the above equations, we have

wg

sa P

Qv

ZE

=2 and a

g

aa EQv

E

dz

dE0

2

== (18)

For a uniform structure, 0=constant,z

a eEzE0

0)(= , and zw ePzP 0

20)(

= .

Thus in a constant-impedance structure, )(zEa and )(zPw are decrease along the z axis in a section.

At the end of a section with length sL :

00)(

= eELE sa and 02

0)(= ePLP sw (19)

where g

s

s Qv

L

L 200

== 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

00 0

01)(

==

eLeEdzzEeW sL sa . Using ins PZE 0

20 2 = ( inP =input power), then

)1

(2

0

0

=

eLPZeW sins (20)

For an optimized design of a constant impedance structure, we should maximize W . Given inP

and sL we make

sZ maximum and 26.1)1

( 0max0

0

=

e

(21)

Given sL and Q , we can obtain the optimized group velocity gv . Obviously the smaller gv , the

bigger 0, and the bigger gv , the lower 0E . An effective way to control gv 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

/2/ 0== gsF vLt . To decrease Ft , then, 0 should be < 1.26.


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3.2 Constant Gradient Structure

To keep 0EEa = = 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 , tovary gv and to keep frequency constant along the section, also to keep the variations of Q and sZ

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

wW PzdzdP )(2/ 0= and 02

0= ePP

sL(22)

where = sL dzz0 00 )( is a section attenuation. Sincedz

dPZE wsa =2 , to keep aE = constant, we

need dzdPw / = const., so that

=

+=

zL

ePz

L

PPPzP

ss

L

ws

02

0

0

0

11)(

(23)

Thus in a constant gradient structure, wP should be linearly decreased along the structure. With

WW PzdzdP )(2/ 0= and )(2/)( 0 zQzvg = , we have

)1(1

1

2

1)(

0

0

2

2

0

=

eLz

e

L

z

s

s

(24)

and0

0

2

2

1

)1(1

)(

=e

eL

z

Q

Lzv ssg (25)

Thus in a constant gradient structure, the )(zvg also decreases along the structure in the same way as

)(zPW . The energy gain for an on-crest particle is

== sL sa LeEdzzEeW 0 0)( Since )1(0202

0== e

L

PZ

dz

dPZE

s

sL

ss , then

)1( 020 = eLPZeW ss (26)

To have maxW , we should have Zs maximum and 0 maximum, all power should be lost in

the structure. On the other hand we should also consider the filling time, /2QtF = , and 0 should

be chosen by a compromise among some effects. An example of a SLAC constant gradient structure


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is shown in Figure 8. Each section is designed to be a tapered structure: 2b 8.4 to 8.2 cm, 2a 2.6to 1.9 cm, vg / c 0.021 to 0.007,

sL = 3.05 m, and sZ 57 M/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 02/ kd = , with pvk /0 = . 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.


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From Figure.4 again, one finds that SW accelerators operate either at the lowest or highest

frequency of the pass band, where NLkn = , and 1,0 =N . That means the operation modes in SW

accelerators is either 0 or .

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, /2, -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 inthe 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 n th solution of these equations can be

nnn nAX cos= , n = 0, 1, 2,...N (27)

where An is a constant and n is the phase shift between adjacent cells. For a SW structure, at the two

ends of the chain we have X0 = XN, so that

N = q , q = 0, 1, 2,....N. (28)

Then the coupled equations have their solutions of Eigen function (field amplitude)

)cos(0N

qnAX

qn

= (29)


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and Eigen value (frequency)

)/cos(1 Nqkc

aq

+= (30)

where a 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 q corresponds a mode which has a phase shift of = q/N between adjacent cells.

The number q for the modes of 0, /2 and -modes are 0, N/2 and N, respectively, and only whenN is even, there is a /2-mode in the chain.

The band width of the chain is b=-0, as shown in Figure.9. Usually the coupling constant

kc


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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.

-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 /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

-modes /2-mode

Effective shunt impedance 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 -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. 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-mode and /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 /2 and -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 -mode there is also a faster group velocity and abigger 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)


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The CCL is operated at /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 on-

axis-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 2/ -mode operation of a cavity resonator chain.[3]


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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 side-

coupled 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 LinacAs 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.


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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 s to several s and a beam energy of 50 keV to 200 keV. If the gunuses 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 electronsexperience 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 pv < 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


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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 = v/c < 1 for protons and =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 (=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 fieldeffect 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:

scs TLeEW cos0= , and cos0 cTLeEW= (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 sWWW = is small that

)cos(cos0 sTeEdz

Wd

=

(33)

where = sscmW32

0 , and s = .

Simultaneously, their phase difference and velocity difference per cell are given by follows,

respectively:

2s

ss

zz == , and

dz

ds

=

2

2

(34)


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Combining W and expressions, one can get longitudinal motion equation for non-synchronous

particles:

0)cos(cos2

)(1

3320

033

33=+

s

ss

ss

ss cm

TeE

dz

d

dz

d

(35)

Since2

0

33

2 cm

W

dz

dsss

=

==

So the longitudinal motion equation describes the non-synchronous particles motion in ( W , )

phase space.

6.3 Stability of the Longitudinal MotionAssuming that the accelerating rate (per unit length) is so small that one can neglect its damping

termdz

d ss )( , then the longitudinal motion equation can be simplified as:

0)cos(cos2

3320

0

2

2

=+ ssscm

TeE

dz

d

(36)

or dzcm

TeE

dz

dd s

ss

)cos(cos2

)(332

0

0

= (37) using

dzd

ddz

/

=

then

d

cm

TeE

dz

dd

dz

ds

ss

)cos(cos2

)(332

0

0 =

by integrating above equation both sides, and considering2

033

2

cm

W

dz

d

ss

=

we have

HcWc

sk =+ )cos(sin

22

21 (38)

where

331

2

ss

c = ,2

0

02

cm

TeEc = ,

20cm

WWk

== .

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 H in right side is an integration constant, or say Hamitonian. Hence Equation

(40) describes the energy conservation of motion. That is Liouvilles theorem: in a motion systemwhich 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.


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Figure 15 Potential well and stable region of the longitudinal motion.By these figures, one can see that:

z Within 02


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In above description we have ignored the damping termdz

d ss )( . If we now take it account into

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-particles, if

the variations ofs and s in a cell are so large that can not be neglected, then this stable phase area is

increased and hence the accelerators acceptance is improved. In this case the buckets boundary is

changed from a fish to a golf club, as shown in figure 16.

Figure 16 Stable boundaries

(a) fish (ignoring acceleration) (b) golf club (including acceleration)

6.5 Small Angle Oscillation

Expending( )

ss

sss

cos2

sin)(coscos

2 ,

and just taking its linear term for small angle oscillation, then the longitudinal motion is linearized as

0)()( 2

2

2

=+

lkdz

d(40)

where

3320

02 sin2

ss

sl

cm

TeEk = is the phase oscillation wave number.

The related phase oscillation period and frequency are

102=

ll

kL and rfl ff %10 . As 1 ,

the phase oscillation is going to be disappeared. For a small angle oscillation, one can easily get the

phase damping law:

4/3)( s , and

4/3)( sW (41)

also as shown in Fig.17.


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Figure 17. Phase damping in small angle oscillation.

7. Transverse Motion

7.1 Field Distribution Near Axis of StructureIn 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 ExpressionLet us start with wave motion equation in cylindrical coordinate system:

0)()( 22 =+ rEkrE ,

0)()( 22 =+ rHkrH

and consider the axis-symmtry of TM01 and/or TM010 modes 0=

E, and 0

2

2

=

E, ( also for H )

so that

0)(1 2

2

2

=+

+

zzz Ek

z

E

r

Er

rr

moving the 2nd and 3rd terms to the right side, and integrating over r, we have

drEk

z

Er

rr

Ez

zz

+

=

22

21

Assuming the radius of the cylinder R, and expanding the field near axis, r


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23

+

=

=

=

02

0

2

2

)(2

rz

r

zz Ekz

Er

r

E, and

+

=

=

=

02

0

2

2

2

2

)(2

1rz

r

zz Ekz

E

r

E

so that

+

= =

=

02

0

2

22

)(4

),,0(),,( rzr

zzz Ek

z

ErtzEtzrE (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

)cos(),0(),,0( 0 += tttzE zz (43)

where 0 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:

)2

cos(),0( zL

nAz

cnz

= (44)

Put (44) into (43) then

)2

cos(),,0( 0

+=

z

L

ntBtzE

c

nz (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:

c

nL

nk

2= , and

n

pnk

v= (46)

Take a harmonic wave of n = 1, which has the phase velocity vp1 as the same as particles velocity v,

then

)2

cos(),,0( 01

+= zL

ntBtzE

cz (47)

For a synchronous acceleration with 0-mode, for instance, the phase velocity and the length of each

accelerating period should be kept as:

vT

L

Lv

rf

c

c

p ===/2

1

or == rfc vTL (48)


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24

where the particles velocity variation within a period is assumed to be omitable.Above equation is so

called resonant / synchronous accelerating condition. From ),,0( tzEz expression mentioned above,

one has

),,0()2

(),,0( 2

2

2

tzEz

tzEz

z

=

Put this equation into(44), and consider 222 )2

()(

==

ck , then we have

+ 2)(1),,0(),,(

rtzEtzrE zz (49)

To reduce the beam energy spread, one should take r, , .

2) Er Expression

Similar with deduction of Ez expression, the Erexpression can be deduced as

)sin(),0(),,( 0

+= ttrtzrE zr (50)

It shows that ),,( tzrEr is increased linearly with r, but decreased with .

3) H Expression

Similarly, we can have

)sin(),0(),,( 00

+= tz

crtzrH z (51)

Hand Ercontribute the transverse forces to particles motion, their forces ratio is

20 =

r

z

eE

Hve

Since these two forces have different signs (+/-) from each other, hence the total transverse force is

)sin(),0()1( 02

0

+== tz

erHveeEF zzrr (52)

Obviously, when 1, then Fr 0.

For a synchronous particle we take st =+ 0 , and 02

0 , defocusing force.

if 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.


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25

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

produce a constant transverse quadrupole gradientx

B

y

BG

yx

=

= .

Figure 18 A quadrupole magnet cross section

For a pole tipe with radius r0, and pole-tip field B0, then the gradient is0

0

r

BG = .

For a particle moving along the z direction with velocity v and with transverse coordinates (x, y),theLorentz force components are

xx evGF = , and yy evGF =

If eG is positive, the quadrpole lens focuses in x and defocuses in y.

The transverse motion equations (Hills Equations) of this particle in this lens are

0)(22

2

=+ xzkdz

xd, and 0)(2

2

2

= yzkdz

yd(53)

wherecm

zeGzk

0

2)(

)( = 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.


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26

Figure 19. FODO quadrupole lattice with accelerating gaps (or tubes)

7.3 Matrix Solution of Hills Equation

A quadupole-transport (drift) channel can be described by the motion equation:

0)('' =+ xzKx (54)

where 22''

dzxdx = , and )()( 2 zkzK = .

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:

=

'0

0

2221

1211' x

x

mm

mm

x

x(55)

(a) Drift space ( K = 0 ):

=

10

1 lM

where =l drift length

(b) Focusing quadrupole ( K=eG/m0> 0)

=

lKlKK

K

lKlK

M

cossin

sincos

(c ) Defocusing quadrupole ( K=eG/m0< 0)

=

lKlKK

K

lKlK

M

coshsinh

sinhcosh

(d) Thin lens:


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27

A quadupole lens approaches its thin-lens approximation when 0lK ,

while lK remains finite. Then we have

= 11

01

f

M

where f is the focal length, "+"---defocusing; "" ---focusing and

cm

eGllK

f 0

1

==

The total transfer matrix through a sequence of constant elements is obtained by forming the product

of the individual M matrices, such as:

1231...., MMMMMM nn = .

7.5 Courant-Snyder invariant

As we have discussed that the transverse motion equations (Mathieu-Hill Equation) are:

0)('' =+ xzKx x

0m

eGkx =

0)('' =+ yzKy y 0m

eGky =

Their solutions ina phase-amplitude form could be

))(cos()()( += zzAzx

[ ]))(sin()(cos()()(

)('

+++= zzzz

Azx

where A and are the constants of integration, and2

)()(

'z

z

The amplitude function, (or- function), satisfies

44'2 22'' =+ xK , (with phase advanceof

1=

dz

d)

Let us introduce the Courant-Snyder invariant

2''22 2 xxxxA ++= = constant. (58)


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28

with

21+=

7.6 Transfer Matrix Through One Focusing Period

The transfer matrix M through one full period can be written:

+=+

sincossin

sinsincos)( pLzzM (59)

1) Here 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 2


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29

maxmax Arx a == . (60)

Here ar is the inner radius of accelerating tube, and A is the acceptance value of this accelerator:

max

2

arA = , and 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:

2''2 2 xxxx bbb ++= (61)Beam Matching

If the injected beams ellipse is well matched to the focusing system, that means at the injection

location every particles 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 beams 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 isvery 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:

axbx = , ayby = , axbx = , ayby =

hence one can adjust 4 quadrupoles gradient prior its injection into the accelerators periodic

focusing system.

If the beam is not matched, then the beam emittance will be enlarged by a factor of Bmag :

12 magB=

+++= 2

1

11

1

1 )(2

1

b

bb

b

b

magB (62)

where 1 , 1 designed lattice (acceptance) parameters at the injection position and b , b

unmatched beam parameters at the injection position.

It is convenient to express the equation of beam emittance in a matrix form as


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30

11 = XXT (63)

where

= '

x

xX , and

'xxX

T = is the transpose matrix;

=

11and its inverse matrix is

=

By this expression of the emittance, for a well matched beams motion from location 1 to location 2

with transfer matrix of R, then its emittance at location 2 can be obtained by

TRR 12 = . (64)It is worth to mention that for the beam transformation in a periodic focusing lattice, the phase

advance per period Lp is related to the function by

=pL

0)z(

dz

(65)

and the average over the period is approximately pL/ .

The max is usually located in the middle point of focusing quadrupole, and the min 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 Liouvilles 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 /' xPx . 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 tohave 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


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31

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 reE according to the Coulomb law. Another one is the Lorenz force with the

magnetic field B caused by the moving electrons according to the Ampere law

z == rr drrnr

eeEF

00

2

1 )(

z == r drrncr

veBv

c

eF

0

022

2 )(

where )(rn is the charge density distribution function. The total space charge force is

22 /)1()( rrrsc eEeEBc

vEeF === (66)

So that if 1


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32

0

0.5

1

1.5

2

2.5

3

3.5

4

Z 82 100 120 169 274 365 386

Location along the pre-injectorcm)

Normalizedemittance

cm.mrad)

120KV, Xrms

80KV,Xrms

150KV, Xrms

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 effectsAs 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 multi-

bunch 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 thetransverse wakefield effects when the bunch offset from the axis transversely.


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33

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

)0(4 //

2

WNe

dz

dEh = and )(2

)0(4 //

2

//

2

dWNe

WNe

dz

dEt = (67)

For the SLAC type accelerating structure (2856 MHz) and if with a bunch length of 3 mm, the

calculated wake functions are mpCVW //225)0(// = and mpCVmmW //4.57)3(// = . 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 effectFor multi-bunch dynamics only the fundamental accelerating mode (beam loading) is important,

and for a constant gradient structure, the loaded accelerating gradient is

)2

1(

1

)( 22

2//0

= e

e

e

QsW

dz

dE

dz

dE b (68)


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34

where bQ is the bunch charge, the attenuation of the structure and )(// sW 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 57.0= , nCQb 5.2= and there

are 3 bunches in 1 ns beam pulse with bunch spacing 10.5 cm, then we have

mpCVcmW //11.53)5.10(// = and mpCVcmW //47.40)21(// =

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 2

ndand the 3

rdbunches

enter the structures at 0.70 ns, 0.35 ns and 0 ns respectively, before the filling time of 0.83s, so thatthe 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 bunchs 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 0x at 0=z , then at sz = , the tail

particles further offset caused by the wake )(dW of head particle is

skE

dWNe

x

x=

4

)(2

max0

(69)

where k is the quadrupole focusing strength.r

B

E

Lk

q

, with quadrupole effective length qL ,

magnetic gradientr

B

and particle energy E. For the SLAC type structure, if the bunch length is 3

mm, then we have

2//4.3)3( mpCkVmmW = .

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 2/0


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35

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 Break-

UP) effect. Different from the single bunch BBU, its wake function )(dW is dominated by one or a

few resonators having large shunt-impedance nr ,

)c

d

sin(eQ

r

)d(WncQ2

d

n n

nn 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

2//064.2)5.10( mpCkVcmW = and2//548.0)21( mpCkVcmW =

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 particles energy

r

B

Ek

1

andr

B

E

Ek

2

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.

8.4 Dispersive effectIf 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

particles trajectory further oscillated along the linac axis


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36

E

Ex ii

=

where 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.

References

1. Lapostolle and A. Septier, Eds., Linear Accelerators,North Holland andWiley, 1970.2. 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 andEngineering, 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 pre-

injector, p. 99.

7.3.5 G.A. Loew, Normal Conducting p = 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.

rf electron linac

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