INTRODUCTION TO RF LINEAR ACCELERATORS (LINACS)1)P.M. Lapostolle

AbstractThe history of linear accelerators is now more than 50 years old andafter a difficult start presented many remarkable successes. Theory andtechnology encounter difficult problems briefly presented here with thesolutions so far developed on orbit stability, focusing, RF structure andbeam dynamics computations. Questions still remain however abouthigh intensity limits which, when answered, might still open a morebrilliant future for these types of machine.

1 . BRIEF HISTORY OF LINACS

The first accelerators built for nuclear physics were of electrostatic type; such machineswere efficient but limited in voltage due to electrical breakdown.

In 1924 a proposal was made by Ising to add several accelerations without havinganywhere the total voltage. The method was based on the use of drift tubes and time varying fields, as sketched in Fig. 1.

Fig. 1 Drift tube accelerator of Ising

Along the axis of metallic tubes, charged particles can drift without being subject to anyelectric field except at their ends, in the gaps between two consecutive drift tubes, according totheir respective voltage V. If an accelerating voltage is applied initially to all the tubes and isswitched off on each of them between the time of entrance and exit of a charged particle, theparticle will receive in each gap a succession of accelerations. In practice, voltage pulses aremore appropriate, as used in present day induction linacs; needless to say, however, that at thetime of the proposal shape and timing of high voltage pulses were not good enough to producea useful operation.

In 1928, Widere proposed to replace voltage pulses by an RF voltage (with a constantfrequency, the drift tube lengths have to increase with the b of the particles). The method wastested on a single drift tube (input + output) and with 25 kV RF peak voltage, it was possible toobserve an acceleration of 50 keV for singly-charged Na and K ions.

In 1931 Sloan and Lawrence built a real linac of 30 drift tubes giving to Hg ions anenergy of 1.25 MeV; lengthening it later in 1934 to 36 drift tubes and increasing their voltagethey reached 2.8 MeV. the intensity was of course very low and the beam quality not specified:R.F. voltage differed from Ising pulses, having no real flat top, phase stability (see Section 2)was not yet discovered and focusing was not ensured, except maybe by ES lens effect (seeSection 5).

No further development occurred until the war, due to the lack of proper high power RFtechnology (limited then to 10 MHz) and to the discovery of the cyclotron. The length of theSloan and Lawrence machine approached 2 m with a b max of only a few thousandths; with the

1) More details on the subject can be found in a Los Alamos report LA 11601 MS, Proton Linear Accelerators,

or in a CERN Yellow Report (in French) CERN 87-09

same RF wavelength of 30 m acceleration of protons would have led to a prohibitive length. Incyclotrons, on the contrary, the spiralling of the trajectories together with some focusing effectallowed the succession of many accelerations over a limited extent, leading to the possibility ofproducing 10 MeV protons.

The development of radars offered, after the war, pulsed high voltage equipment in themetric and centimetric wave ranges; science and technology of electromagnetism and beamdynamics of already high level were then available.

The parallel development of circular accelerators (synchrocyclotrons and synchrotrons)was also helpful with the discovery of phase stability.

Ginzton, Hansen, Chodorow, Slater, Walkinshaw used 3 GHz to accelerate electrons.

Alvarez and Panofsky used 200 MHz for protons.

2 . LONGITUDINAL MOTION. PHASE STABILITY. ACCELERATION BYA TRAVELLING WAVE

For a given geometry of the drift tubes which corresponds to a certain rate of acceleration,particles must receive at each gap an exact energy gain and the voltage must have an exact value.The RF voltage V applied is larger and there are two phases per RF period for which thevoltage has the right value Vs (see Fig. 2). When the field is rising the phase is stable since aparticle arriving too early will be less accelerated and slip slightly in phase until the next gap;vice versa for a late particle. The other phase is unstable. The stable phase is called thesynchronous phase f s and one has

Vs = Vcos s

with f = 0 corresponding to the crest (proton case).

Fig. 2 Phase instabilityIn a first approach one can replace the discrete gap configuration by an equivalent

continuous interaction (in Section 5 will appear a justification). One may say that the fielddistribution along the axis in the successive gaps presents an analogy with a standing wavepattern, sum of a forward and a backward wave. One would then consider the interaction of theforward synchronous wave with the particles.

Introducing relativity symbols and letting s , s , s refer to the synchronous particle,one has

= 1 + Wm0c

2

and putting = s

one can write, following the path of a particle, along the axis

m0c2

w

d ( )ds

=qET

wcos coss( )

dds

=

c

1

1s

=

c

s3 s3

whence

m0c3 d

dss3 s3

dds

= qET cos cos s( )

Forgetting about the change in g s one then obtains

2 + 2qETs3 s3

m0csin cos s( ) = Ct

where ET is the amplitude of the synchronous wave.

Such a motion derives from the Hamiltonian

H = c

22s3 s3

qETm0c

2 sin cos s( )

Neglecting the change in g s is only valid for heavy particles (protons and ions) for whichit is slow enough. It corresponds to an acceleration fighting against a constant breaking force(or to a forced pendulum model as often presented for circular machines).

One gets from it the usual stability bucket of Figs. 3 and 4; Fig. 5 shows the phase spaceplot relative to an operation with s = 0 (fixed point as used by Sloan and Lawrence).

Taking into account the change in g s opens up the bucket and gives the so-called golfclub, see Fig. 6 (notice that the coordinates used are not conjugate).

Fig. 3 Classical stability bucket

Fig. 4 Non-accelerating bucket

Fig. 5 Fixed-point operation

Fig. 6 Golf club

For electrons g increases very rapidly and b becomes close to 1 over a short distance.Most of the accelerating structure (apart from a buncher) is a constant velocity b = 1 structure.

The previous derivation must be modified; keeping g instead of dg , it leads to:

1 1 + =

21 = eET

m0ccos cos 0( )

where refers now to a 0 of the field expressed as ET sin (electron linac convention).Figure 7 shows a corresponding phase space plot. One may notice that the 0 of a curvecorresponds to its position for .

Introducing, as is the custom

=eETm0c

2 = 2pieET

m0c

one can see that for 2pi the curve corresponding to 0 = p /2 which provides the maximumacceleration can start from W = 0. For < pi , however, there can be low energy electrons justslipping in phase and not accelerated (Fig. 7).

Fig. 7 Electron acceleration phase plot for < pi

3 . TRANSVERSE MOTION. DEFOCUSING ACTION OF THE ACCELERATING FIELD

The fields at the entrance and exit of a gap produce focusing and defocusing actions (seeFig. 8). If the field is rising the overall effect is defocusing.

Fig. 8 Focusing and defocusing actions in a gap

Calling on the travelling wave accelerator concept one may transform the EM field of thesynchronous wave in a frame moving at the same velocity (supposed constant). A TMaccelerating field becomes electrostatic. Then

Ezz =

Exx

Eyy = 2

Err

If there is a longitudinal stability in z, there is instability in the transverse direction.

This has been a strong handicap for ion linacs until AG focusing was invented andapplied (thin foils were destroyed by breakdown and thick wire grids, if transparent enough,introduced bad aberrations). New methods exist now, in particular the RFQ principle for lowb .

For electrons, which become very quickly relativistic, the transverse effects are veryweak.

4 . LINAC ACCELERATING STRUCTURES

(Only a general review is given here; details can be found in more specialized lectures, inparticular on ion linacs.)

In a circular waveguide, the phase velocity of the waves ph is always larger than 1. It isthen necessary to slow them down. Periodic loading is used and/or field concentration orreorientation via drift tubes.

The structure must also provide space to install quadrupole focusing (at least at low b forions and protons).

Stability of the field distribution is also a concern.b

Sloan and Lawrence structure (present design, see Fig. 9a); the large drift tubes canhouse quadrupoles (frequency from 10 to 100 MHz).

H-type structure (see Fig. 9b); the field of the transverse electric mode is madeaccelerating through the drift tube configuration; since there is not much space for focusing, thefixed point operation (see Fig. 5) is used, with focusing and rebunching from place to place(frequency up to 100 MHz).

Fig. 9a Fig. 9b

0.02 < b < 0.1

Quasi Alvarez structure (see Fig. 10); the large drift tubes (almost 2 b l long) can housequadrupoles; the rate of acceleration is lower than with the H-type structure (frequency up to200 MHz); high intensities can be accelerated.

Fig. 10

0.03 < b < 0.4

Alvarez structure (most common for protons, see Fig. 11); a quadrupole is put in eachdrift tube. Such a structure allows very large intensities (frequency between 100 and400 MHz). For b approaching 0.5 the RF losses increase, the drift tubes becoming resonantlike l /2 antenna.

Fig. 11b > 0.4 (protons)

Side-coupled cavities (Fig. 12). At this b the focusing is installed in between sections ofcavities (frequency from 600 to 1200 MHz).

Fig. 12b = 1 (electrons)

Iris-loaded cavity (Fig. 13) with three or four irises per wavelength l (frequency usually3 GHz).

Fig. 13

Other types of structures

Superconducting (or normal temperature) independent cavities for heavy-ion boosters: afew gaps or even a helix. Very flexible for various particles and different b : for each velocityand particle (charge and mass) the phase of the independent cavities is correspondinglyadjusted.

RFQ (low- b protons and ions): cavity excited on a quadrupolar EM mode with profiledvanes or rods to provide acceleration.5 . DETAILED PARTICLE DYNAMICS COMPUTATION

Acceleration through a gapTransit time factorSimulation codes

The initial approach (Panofsky equations) was developed from the experience gained onelectron beam tubes with grids (see Fig. 14). In the case of grids the field in a gap can beuniform of value E0. With open holes on the contrary it penetrates inside and can have, on theaxis, a distribution as shown on the figure.

Fig. 14 Accelerating gap

With the initial approach (with grids) if one assumes a particle crosses the gap with aconstant velocity v with the phase in the middle, the energy gain is:

W = q E0 cos z

v+

g /2

+g /2

dz = qVT cos with

V = E0g

voltage across the gap, and

T = sin / 2 / 2

where

= g / v

q is the transit time (in phase) through the gap and T is called the transit time factor (always< 1): it is the reduction in acceleration with respect to what the voltage V would give.

In the real case (without grids) it is possible to express the amplitude of the field Ez (z) onthe axis (as obtained from measurements or rather from a computer) with the help of a Fourierintegral:

Ez z,r = 0( ) = V2pi T kz( )

+

cos kzz dzwith the inverse relation

VT kz( ) = Ez z,0( )

+

cos kzz dzfrom which

VdT kz( )

dkz= zEz

+

z,0( ) sin kzz dzIn such an expression the field is represented as the sum of an infinite set of travelling

waves of amplitude proportional to T(kz) (in a standing wave configuration only two waves arepresent). Each wave T(kz) is an EM wave the complete distribution of which can be knownfrom Maxwell's equation (assuming circular symmetry) in such a way that the complete fielddistributions can be derived throughout the gap inside of a cylinder of radius a, the hole radius:

Ez z,r,t( ) = V2pi T kz( )

+

I0 krr( ) cos kzz cos t + ( )dkzEr z,r,t( ) = V2pi T kz( )

kzkr

+

I1 krr( ) sin kzz cos t + ( )dkzcB z,r,t( ) = V2pi T kz( )

w

ckr

+

I1 krr( ) cos kzz sin t + ( )dkzwith

kr2 = kz2 2 / c2

With such expressions, making use of inverse Fourier relations, one can easily computethe energy gain. Along a parallel to the axis, at the distance r, one has:

W = qVT kz( )I0 krr( ) cos where kz = w /v f and T(kz) is the amplitude factor relative to the synchronous wave; the factthat only the synchronous wave interacts justifies the approach used in Section 2. It alsoexplains a correction often introduced in the expression of the transit time, as follows:

T = sin / 2 / 2

I0 krr( )I0 kra( )

with, for q , a slight correction to take into account the effect of the chamfer of the hole.

With the field equations above one can compute the phase change across the gap,accounting for the change in velocity, and the transverse motion. It is also possible to considera trajectory with a slope, such that

r = r0 + r0' z

Such integrals lead to expressions involving T(kz) and its first and second derivatives withrespect to kz, according to the relation given above (see the report LA-11601-MS, page 80).

These expressions are used in common codes like PARMILA and MAPRO, validhowever only for protons and ions.

A more recent code, DYNAC, valid for all particles (including electrons), with a betteraccuracy, introduces instead of the radius r the reduced radius

R = r z with R' = dRdz

leading, in the paraxial approximation, to:

R"R q2m0c

31

3 3Ezt + R

q2m0c

2

2 2 + 2 4 4 Ez

2= 0

such that R and R' are conjugate (which was not the case for r and r').This equation shows, in addition to the phase dependent term usually defocusing (see

Section 3), a focusing term of the electron lens type, mainly important for electrons but also forvery low velocity particles (explaining probably the weak focusing observed by Sloan andLawrence, see Section 1).

Such a term for a uniform field just leads by integration to the classical solution and therelation between r and R; it can however in the case of localized fields (gaps) exhibit anappreciably enhanced effect, not at all in contradiction with the considerations developed in amoving frame at constant velocity.

The derivation of the equation in R, not given in the report LA-16601-MS, is developedin the Appendix.

6 . SPACE CHARGE EFFECTS. INTENSITY LIMITS

Particles of the same charge repel each other. Space charge field tends to increase thebeam size and hence entails a risk of loss.

According to Poisson's law a linear space charge field requires r = Cte.

The only solution for a uniform density, only possible for a continuous beam is the so-called Kajchinsky-Vladimirsky (K.V.) distribution, which is a surface distribution in the 4Dspace. Such a distribution is, of course, not very physical but fortunately it happens, fromenergy consideration, that the K.V. beam equations are satisfied approximately (over shortenough distances) by r.m.s. dimensions (size and emittances) for all distributions (for a K.V.distribution r.m.s. dimensions are half the real ones and r.m.s. emittances one quarter of thereal ones). Such a property is extremely useful, in simulation studies for instance, for findinggood matching conditions.

When the intensity increases, various phenomena occur, due to the non-linear character ofspace charge forces.

Beam emittance can transfer from one coordinate to another and total emittance tends toincrease.

Around the beam core a "halo" develops; when trying to scrape it, it reappears. Even ifsuch a halo contains only one or a few per cent of the particles or even less this can be a veryserious problem for large intensity machines.

For small average intensity on the other hand, the emittance (beam quality) is the mainconcern. A practical recipe which leads to good operation is as follows:

AG focusing is usually specified by the phase advance per focusing period of theincoherent oscillations; calling s 0 the value for zero intensity and s for full beamone must have s 0 < 90 and usually s 0 in the range of 60 or 70 with s / s 0 > 0.4.

In addition focusing and longitudinal stability should be adjusted, as far as possible,in such a way that transverse kinetic energy and longitudinal oscillation energyremain as close as possible to avoid emittance transfer.

Several simulation codes are available to study numerically the space charge effects. Thebeam is represented by a few thousand of macroparticles and several methods are used tocompute their space charge field:

A fast Fourier transform (FFT) routine to solve Poisson's law when one canassume a circular symmetry for the bunch (PIC, particle in cell code)A numerical integration using analytical expressions valid for an ellipsoidal bunchshape (Ellipsoidal profile code)A point-to-point computation which does not make any assumption on the geometrybut requires some care to avoid "collisional effects" (Particle-to-particle interactioncode); this last code is much slower when increasing the number of macroparticles.Present theoretical approaches to study the details and analyze the phenomena strongly

related to highly non-linear space charge fields are making use of the modern theories ofstochasticity with resonance overlap effects and Arnold's diffusion. They are still underdevelopment.

BIBLIOGRAPHY

G. Ising, Ark. Mat. Astron. Fyz. 18 (1924) 1.R. Widere, Archiv fr Electrotechnik 21 (1928) 387.D.H. Sloan and E.O. Lawrence, Phys. Rev. 38 (1931) 2021.D.H. Sloan and W.M. Coates, Phys. Rev. 46 (1934) 539.L. Alvarez, The design of a proton linear accelerator, Phys. Rev. 70 (1946) 799.J.P. Blewett, Phys. Rev. 88 (1952) 1197.L. Alvarez et al., Berkelez proton linear accelerator, Rev. Sci. Instrum. 26 (1955) 111 and 210.V.I. Veksler, A new method of acceleration, J. Phys. USSR 9 (1945) 153.I. Kapchinskij and V. Vladimirskij, Proc. Int. Conf. on High Energy Accelerators, Geneva,1959 (CERN, Geneva, 1959), p. 274.E.M. MacMillan, The synchrotron, Phys. Rev. 68 (1945) 143.L. Smith, Linear accelerators, Handbuch der Physik, Band XLIV (Springer, Berlin, 1959), p.341-389.

H. Bruck, Acclrateurs circulaires de particules (Presses Univ.de France, Paris, 1966).[English translation by Los Alamos: Circular particle accelerators, LASL.LA TR 72-10 Rev.(1972).]A. Lichtenberg, Phase space dynamics of particles (J. Wiely and Sons, New York, 1969).E. McMillan, The relation between phase stability and first order focusing, Phys. Rev. 80(1950) 493.L. Teng, AG focusing for linacs, Rev. Sci. Instrum. 25 (1954) 264.L. Smith et al., Focusing in linacs, Rev. Sci. Instrum. 26 (1955) 220.E. Nolte et al., The Munich heavy ion post accelerator, IEEE Trans. Nucl. Sci. NS-26 (1979)3724.

T. Fukushima et al., Measurement of interdigital H type linac, Proc. Linear Accelerator Conf.,Santa Fe, 1981 (LA 9234C, Los Alamos, 1982), p. 296.E. Knapp et al., Standing-wave high-energy linacs, Rev. Sci. Instrum. 39 (1968) 979-991.D. Bhne, The UNILAC, Proc. Linear Accelerator Conf., Chalk River, 1976 (AECL 5677,Chalk River, 1976), p. 2.L. Smith, Beam dynamics in induction linacs, Proc. Linear Accelerator Conf., Santa Fe, 1981(LA 9234C, Los Alamos, 1982), p. 111.T. Fessenden, Induction linacs for HIF, Proc. Linear Accelerator Conf., Seeheim, 1984 (GSI84-11, Darmstadt, 1984), p. 485.A. Schempp et al., A heavy ion post accelerator, Proc. Linear Accelerator Conf., Montauk,1979 (BNL 51134, Brookhaven, 1980), p. 159.

L. Bollinger et al., Concept of a superconducting linac, Proc. Linear Accelerator Conf.,Seeheim, 1984 (GSI 84-11, Darmstadt, 1984), p. 217.I. Kapchinskij et al., The linac with space uniform quadrupole focusing, IEEE Trans. Nucl.Sci. NS-26 (1979) 3462.J. Potter et al., RFQ research at Los Alamos, IEEE Trans. Nucl. Sci. NS-26 (1979) 3745.R. Stokes et al., RFQ dynamics, IEEE Trans. Nucl. Sci. NS-26 (1979) 3469.K. Halbach et al., Properties of the code SUPERFISH, Proc. Linear Accelerator Conf., ChalkRiver, 1976 (AECL 5677, Chalk River, 1976), p. 122, and SUPERFISH, Part. Accel. 7,(1976) 213.M. Bell et al., Numerical computation of field distribution, Proc. Linear Accelerator Conf.,Batavia, 1970 (NAL, Batavia, 1971), p. 329.W. Panofsky et al., Berkeley proton linac, Rev. Sci. Instrum. 26 (1955) 119.P. Lapostolle, Equations de la dynamique des particules, rapport CERN AR/Int. SG5/11(1965).A. Carne et al., Design equations in a linac, Proc. Linear Accelerator Conf., Los Alamos, 1966(LA 3609, Los Alamos, 1966), p. 201.M. Prom and M. Martini, Computer studies of beam dynamics, Part. Accel. 2 (1971)289-299.

I. Hofmann, Generalized three-dimensional equations for the emittance and field energy ofhigh-current beams in periodic focusing structures, Part. Accel. 21 (1987) 69.P. Lapostolle, Relations nergtiques dans les faisceaux continus, rapport CERN ISR DI/71-6(1971). [English translation by Los Alamos: LA TR 80-8 (1980.]F. Sacherer, r.m.s. envelope equations with space charge, IEEE Trans. Nucl. Sci. NS-18(1971) 1105.T. Wangler, Relation between field energy and r.m.s. emittance, IEEE Trans. Nucl. Sci. NS-32 (1985) 2196.D. Warner, Accelerating structure of CERN Linac, Proc. Linear Accelerator Conf., ChalkRiver, 1976 (AECL 5677, Chalk River, 1976), p. 49.M. Weiss, The new CERN Linac, Proc. Linear Accelerator Conf., Montauk, 1979 (BNL51134, Brookhaven, 1980), p. 67.R. Chasman, Numerical calculation of transverse emittance growth, IEEE Trans. Nucl. Sci.NS-16 (1969) 202.R. Jameson, Equipartitioning in linacs, Proc. Linear Accelerator Conf., Santa Fe, 1981 (LA9234C, Los Alamos, 1982), p. 125.P. Lapostolle, Etude numrique de charge d'espace, rapport CERN ISR/78-13 (1978).

APPENDIXEQUATION OF THE TRANSVERSE MOTION WITH THE REDUCEDRADIUS IN THE PARAXIAL APPROXIMATION

The equation

d mvr( )dt

= q Er vzB( ) (A.1)can be written

d r' ( ) = qm0c

Er cB( )dt = k r, z,t( )dt (A.2)Putting

R = r = r 2 1( )1/4 and R' = dR / dz (A.3)one has

r' = R' 2 1( )1/4 R2 2 1( )5/4 ddz (A.4)and

ddz

r' ( ) = R" 2 1( )1/4+

R2

2 1( )3/4 + 3R4 2 2 1( )7/4 ddz 2

R2

2 1( )3/4 d2dz2(A.5)

Now, from Eq. (A.2), in the paraxial approximation

k r, z,t( ) = k z,t( )r = qm0c

Err c

Br

r (A.6)with, according to Maxwell's equations

Err =

12

Ezz and

Br =

12c2

Ezt (A.7)

But one has from Eq. (A.2)

ddr

r' ( ) = k r, z,t( )c (A.8)and, for the longitudinal motion

m0c2 d

dz= qEz (A.9)

so that one also has

m0c2 d2

dz2= q dEz

dz= q Ezz +

qc

Ezt (A.10)

Eventually one obtains

d2Rdz2

R q2m0c

31

3 3Ezt + R

q2m0c

2

2 2 + 2 4 4 Ez

2= 0

FUNDAMENTALS OF ION LINACSM. WeissCERN, Geneva, Switzerland

AbstractThe operating principles of ion RF linear accelerators are described,based on the concept of 'slow waves' generated in 'loaded cavities'.Fundamental cavity parameters are presented, and equationsdescribing the action of electromagnetic fields on the particles aregiven. Space charge influences are touched upon only briefly. Varioustypes of linear accelerators are introduced according to their use atdifferent particle energies and frequencies.

1. INTRODUCTIONLinear accelerators (linacs) accelerate particles on a linear path. Usually linacs operate

with sinusoidally varying electromagnetic fields and are called RF linear accelerators. RFfields are created in a bounded volume, known as cavity, and the cavity operates either as awaveguide (the electromagnetic energy enters the cavity at one end, is dissipated partly alongthe cavity walls which are of finite conductivity, and exits at the other end to be dissipated ina matched load), or as a resonator (the electromagnetic energy is reflected back and forth onthe end walls). Accelerators working on these principles are called travelling- and standing-wave linacs, respectively. RF linacs operate in the frequency range of a few MHz up toseveral GHz. They are suitable for the acceleration of light particles, like electrons (at10 MeV electrons already have the relativistic velocity b = 0.999), as well as of heavierparticles, ions (protons at 10 MeV have b = 0.145). Ions are slow particles and ion linacsusually operate in the MHz region whilst electron linacs operate in the GHz region. Thestructure of the cavity differs when used for slow or rapid particles. However, the basicprinciples of operation are always the same. This paper deals with ion linacs which have avast field of applications, such as physics research, nuclear waste transmutation, breeding ofnuclear fuel, materials study (neutron spallation sources) and medical applications (cancertreatment).

Sections 2 and 3 are essential for understanding the principles of operation of linearaccelerators with RF electromagnetic fields. The accelerating structures have been treated aslossless for better clarity; lossy structures would be treated in a similar way. Section 4outlines the basic cavity parameters, whilst Sections 5 and 6 deal with some more commonaccelerator types. Section 7 shows how the action of electromagnetic fields on particles iscomputed, and Section 8 touches the problem of space charge influences and the beamloading of cavities. Section 9 treats a particular accelerator called the radio-frequencyquadrupole (RFQ), which is in wide use all over the world. Section 10 gives some concludingremarks.

2. EMPTY CAVITIES: WAVE TYPES AND MODESIn free space, the electromagnetic wave has electric and magnetic field vectors

perpendicular to the direction of propagation. Such a wave type is called the transverseelectromagnetic wave (TEM). In bounded media, where the boundary is a perfect conductor,such a wave type is not possible because the boundary conditions cannot be satisfied. On theboundary, the tangential component of the electric field Et, as well as the normal componentof the magnetic field Bn, have to be zero. These conditions are satisfied if the wave has one ofthe field components in the direction of propagation (the longitudinal or z-axis of thecoordinate system). If it is the electric field component Ez, the wave type is called transversemagnetic (TM); if it is the magnetic field component Bz the wave type is called transverseelectric (TE) , [1].

Wave propagation in cavities can be explained in terms of reflections from wall to wall;in this way one has a field component in the direction of propagation. Not all the angles of

wave incidence and reflection are allowed; only a discrete number of them satisfy theconditions. For each wave type, one can have certain wave modes, which are indicated bysubscripts. For example, in a rectangular cavity one can have cavity modes

TEmn or TMmn ,

m and n indicating the number of half waves in the x and y direction, respectively (see Fig. 1).If there is a third subscript (in resonators), it indicates the number of half waves in the zdirection.

Fig. 1 TE01 wave in a rectangular cavityIn linacs one distinguishes between two wave velocities: the phase velocity n ph and the

group velocity n g. The former is the velocity of the wave phenomenon, like that of rapid wavepropagation along the shore, resulting from an obliquely incident sea wave. Electromagneticwaves, moving with the velocity c and falling obliquely on the side walls of a cavity, wouldresult in a longitudinal (along the cavity) wave phenomenon moving with a phase velocitybigger than the velocity of light c. Of course, the wave energy does not move in thelongitudinal direction with the phase velocity, but with the much slower group velocity.However, for the acceleration of particles in accelerators it is the phase velocity which counts.To see this in more detail, we shall write the wave equation in cylindrical coordinates(accelerators usually have a circular cross section) for a rotationally symmetricelectromagnetic field like TM01, where the subscripts indicate the number of field variationsazimuthally, and the number of zeros of the axial field radially (0 < r Rmax), respectively:

2Ezz2

+1r

r r

Ezr

1c2

2Ezt2

= 0 .

The solution is usually given in the form of a product of functions of one variable:

Ez (z,r,t) = Z(z) R(r) T(t) ,where the functions Z(z) and T(t) are of the form:

Z(z) exp (jkz) ; T(t) exp (jw t) .The factor k in the exponent represents the phase advance of the wave per unit length. Thefactor w represents the phase change per unit time or the angular frequency. The exponentj(w t kz) is typical for travelling waves. Counting z from the position of the crest of the waveat time t = 0, one stays on the crest if during motion the condition

w t kz = 0 ,

remains satisfied. One has

z

t=

k= ph > c ,

as we already know. Inserting the assumed solution for E z into the wave equation, oneobtains an equation for R(r) :

d2Rdr2

+1r

dRdr

+ 2

c2 k2

Kr21 24 34

R= 0 .

This is the Bessel equation of zero order (due to the assumed field symmetry), and thesolution is given by the Bessel function of first kind and zero order:

R(r) = AJ0 (Krr) .At the boundary (cylinder of radius Rmax = a), Ez (z,a,t) must be zero, hence:

J0 (Kra) = 0 Kra = 2.405 (first zero of the Bessel function).To satisfy the boundary conditions, the value of Kr is fixed. From its definition:

c

2

= Kr2 + k2 .

This is the very important dispersion relation for empty cavities and for a given wave typeand mode. Plotting w = f (k), we obtain the dispersion or Brillouin diagram presented inFig. 2.

ph =

k

g =ddk

Fig. 2 Dispersion (Brillouin) diagram for empty cavitiesThe plotted curve is a hyperbola and the slope of the radius vector from the origin to a

point on the curve gives the phase velocity n ph = w /k ; all the points above the asymptote, forwhich n ph = c , have n ph > c . The slope of a point on the hyperbola gives the group velocity[2]

g =ddk

.

The lowest frequency for which the boundary conditions are satisfied is w c. At this frequencythe phase advance k, as well as the group velocity, are zero, whilst the phase velocity isinfinite. To accelerate particles we need a longitudinal electric field but as the electromagneticwave is a time varying field we must maintain a synchronism between the wave and theparticle, which means that the phase velocity n ph of the wave and the velocity of the particlen p must be equal. One sees immediately that in order to use electromagnetic waves foracceleration one must find methods to slow the waves down.

3. LOADED CAVITIES: SPACE HARMONICS, TRAVELLING AND STANDINGWAVE STRUCTURESOne method to slow down the waves is to 'load' the cavity by introducing some periodic

obstacles into it [3], as shown in Fig. 3. The study of the solution of the wave equation insuch a case is based on two essential points:

Fig. 3 Disc-loaded cavity (schematic)i) Floquet's theorem for periodic structures, which states that in a given mode of oscillation

and at a given frequency, the wave function is multiplied by a constant such as ejkL , whenmoving from one period to the next. A simple function that satisfies Floquet's theorem isejkz.

ii) The complicated boundary conditions cannot be satisfied by a single mode, as was thecase with an empty cavity, but by a whole spectrum of space harmonics , which is in fact aFourier series applied to a periodic case. We now have:

Ez (r, z,t) = F(r,z)e j(tkz)F (r, z + L) = F(r,z)

F(r, z) = an(r)e j(2pin/L)z .n

Introducing these expressions in the wave equation for rotationally symmetric waves weget:

e jt e j(k+2pin / L)z d2an(r)dr2

+1r

dan(r)dr

+ Krn2 an(r)

n

= 0 ,

with

Krn2

=

c

2

k + 2pinL

2,

and

ph =

k + 2pin / L=

kn.

For each n we have a travelling wave with its own velocity n ph, which is slowed downcompared to the case of an empty cavity. In principle we can find an n such that n ph = n p ,which is what we wanted. For n ph < c , one has Krn2 < 0 and so Krn = jkr. The solution of theBessel equation is now expressed by the Bessel functions of imaginary argument, which arecalled modified Bessel functions:

an (r) = An I0 (krr)where An is a constant.

It is not easy to find the functional relationship between w and kn for the dispersiondiagram of a periodic structure. However, this diagram must reflect the periodicity of thestructure and it will be drastically different from that of an empty cavity, even in the region ofn = 0. Figure 4 shows the dispersion diagram of loaded cavities.

Fig. 4 Dispersion diagram of a periodic structure (loaded cavity); unloaded cavitycase shown for comparison

Interesting features can be observed:

i) for a given wave type and mode, there is a limited passband of possible frequencies fromw c to w p ; at both ends of the passband, the group velocity n g is zero;

ii) for a given frequency, one has an infinite series of space harmonics , from n = to n =+ . All space harmonics have the same group velocity, but different phase velocities;

iii) when the electromagnetic energy propagates only in one direction (full curves ondiagram), we have a travelling wave accelerator (TW); when the energy is reflected backand forth at both ends of the cavity (full and dotted curves), we have a standing waveaccelerator (SW).

vi) the group and phase velocities do not have to be in the same direction; if they are, one hasa forward wave and if they are in opposite directions, one has a backward wave.

4. FUNDAMENTAL CAVITY PARAMETERSIn the design of linear accelerators one is concerned with a certain number of

parameters, of which the fundamental ones are the following [4,5]:i) Transit-time factor T

T =Ez (z,0)e

j(z / p ) dz0L

Ez0L (z,0) dz .

The transit-time factor T indicates how well a field Ez accelerates particles. The integral in thedenominator is equal to E0 L , where E0 is the average field in a cell, and L the cell length. Thesignificance of the transit-time factor will became clearer in Section 7.

ii) Effective shunt impedance per unit length Zseff

Zseff =E0T( )2

dP / dz[Mm1] .

The effective shunt impedance indicates how efficient the acceleration for a given dissipatedRF power dP/dz is, where P is the power flowing in the structure and E0T is the amplitude ofthe relevant space harmonic.

iii) Quality factor Q

Q = wdP / dz

,

where w is the stored energy per unit length.The ratio Zseff/Q = (E0 T)2/w w is also an important parameter as it depends only on thegeometry of the structure and not on the quality (conductivity) of the walls; it indicates howmuch acceleration one has for a given stored energy.

iv) Group velocity n g

g =Pw

.

P is the power flowing in the structure; as w E2, a small n g is usually preferred for betteracceleration efficiency.

v) Frequency fThe choice of the frequency of an accelerator depends on many factors. In general, withhigher frequencies one has a better Zseff (Zseff w 1/2), but the aperture for the beam getssmaller.

vi) Filling time t

tF =dz

g(z)0L ,

where here L is the length of the cavity.The factor tF indicates the time needed for the electromagnetic energy to fill the cavity; theabove formula is valid for TW accelerators. For SW accelerators, where the electromagneticenergy builds up progressively in time one has

tF Q

.

vii) Duty factor hThe duty factor indicates the fraction of time during which the accelerator is in active mode.Linear accelerators usually operate in a pulsed regime where the pulse is repeated a certainnumber of times per second. The duty factor is given by the product of the pulse length andrepetition rate.

5. SOME TYPICAL ION LINACSBefore describing some ion linear accelerators, it seems appropriate to analyse briefly

the TW and SW modes of operation. The TW mode is usually used for particles whichapproach the velocity of light; the SW mode is used for slower particles. Both modes can beunderstood by observing the dispersion diagram of Fig. 5:

Fig. 5 Dispersion diagram of periodic structures i) the working point of the TW mode is usually around point A (phase advance per period of

the structure @ p /2; the phase velocity @ c, the group velocity corresponds to the tangent tothe curve at point A);

ii) the working points of the SW mode must have phase advances of 0 or p , modulo 2 p (as forexample points B and C); it is only at these points that the direct (full line) and reflectedwave (dotted line) have the same phase velocity, so they can both be used to accelerateparticles. A disadvantage is that at these points the group velocity is equal to zero; we shallcome to this problem later.

Figure 6 shows a disc-loaded ion linac which operates in the SW mode with a phaseadvance of p . The electric field Ez has opposite directions in successive cells. The 'noses'around the beam aperture are present in order to increase the transit-time factor T (or theamplitude of the space harmonic which is relevant for the acceleration). Figure 7 shows theAlvarez or drift-tube linac (DTL), where the field E z is in the same direction in all cells.Hence the DTL operates in the 0 mode (in fact, for the beam, the mode is 2 p ). As theazimuthal magnetic field is also in the same direction in all cells, the walls separating the cellscould have been left out. The DTL therefore has a higher shunt impedance and is better thanthe disc-loaded structure at relativistic velocities of 0.04 < b < 0.4. Another advantage is thatthe drift tubes can house focusing elements (like magnetic quadrupoles), very important atlow particle velocities. Figure 8 shows a photo of drift tubes taken out of CERN Linac 2 (forprotons, output energy 50 MeV, frequency 200 MHz).

Fig. 6 Disc-loaded linac (schematic)

Fig. 7 DTL linac (schematic)

Fig. 8 Linac 2: girders with drift tubesAt lower velocities, b @ 0.01, a very convenient accelerator is the radio-frequency

quadrupole, RFQ . This widespread accelerator will be treated in some detail in Section 9. Forhigher ion velocities, b > 0.4, a convenient structure is the side-coupled linac, SCL [6],shown schematically in Fig. 9. Its particularity is that in addition to accelerating cells it alsocontains coupling cells, placed sidewise. The field vector Ez is in phase opposition in adjacentaccelerating cells. The coupling cells form another chain of oscillators and the SCL istherefore a 'biperiodic structure'. Biperiodic structures have interesting properties: if theaccelerating and coupling cells are properly tuned, their passbands in the dispersion diagramare brought to a confluence and form a curve as shown in Fig. 10. The original operatingpoint of accelerating cells, with the group velocity zero (A), becomes a point with a finalgroup velocity (B). Hence, energy can flow and the structure is said to be compensated.(Energy in non-compensated SW structures must also flow to cover RF losses in the structurewalls; to render this possible, the electromagnetic field pattern gets slightly distorted.Compensated structures avoid this field pattern distorsion.)

Fig. 9 Side-coupled linac, SCL (schematic)

Fig. 10 Confluence of two passbandsThe DTL can also be compensated, and a second chain of oscillators can be brought

into the cavity in the form of post couplers (see Fig. 11). The inductance of the post couplerand its capacity towards the drift tube are tuned so as to bring the two passbands toconfluence. CERN Linac 2 is compensated in this way.

Fig. 11 DTL cavity compensated with post couplers (schematic)

6. STRUCTURE COMPUTATIONS: DRIFT TUBE LINACLinear accelerators are periodic structures with complicated boundary conditions, and to

design them (geometry corresponding to a given frequency and field pattern) one usescomputer programs. To illustrate this procedure we take as an example the computation of theDTL, which operates with a TM010 field type and mode. The DTL is composed of one orseveral tanks (cavities), each tank containing a certain number of cells, coupled together, andall resonating at the same frequency. Each cell is composed of a gap and two half drift tubes,as seen in Fig. 7. The length of the cells increases according to the velocity of the particles,which have to cross a cell in one RF period. One computes only a certain number of cells, forvarious velocities, and then interpolates the results. Owing to symmetry it suffices to solvethe wave equation, with proper boundary conditions, in only a quarter of a cell, as shown inFig. 12. (In fact a cell is not symmetric because of the stem which holds the drift tube.However, one first neglects the stem and introduces it only later as a perturbation.) Toincrease the precision of calculations, the mesh size can be diminished in regions where thefield vectors have to be determined more accurately (see Fig. 13). In addition to fields, 2-Dprograms like SUPERFISH [7] or URMEL [8] compute also other cavity parameters, such asZseff, Q , power loss in the walls PL, average accelerating field E0, stored energy w, transit-time factor T, etc. Some of these parameters are used in beam dynamics calculations.

Fig. 12 Computation of DTL cells by SUPERFISH

Fig. 13 SUPERFISH calculations with a variable mesh size

7. ACTION OF ELECTROMAGNETIC FIELDS ON PARTICLESIn Section 6 we discussed how one computes an accelerator cell from the

electrodynamic point of view. Now we shall analyse how the electromagnetic field acts onparticles. As an example we consider the situation in a gap of a DTL accelerator (see Fig. 14).The field pattern is TM010 and thus we have the field components Ez, Er, and B [9].

Fig. 14 Cell of a DTL accelerator (schematic)In Section 3 we expressed the Ez field by a Fourier series. In the same way, but to be

more general, we now express this field by a Fourier integral

Ez (r, z) = Ak I0

(krr) cos kz dk ,where cos kz has replaced the exponential function because Ez is an even function of z and

kr = k2

c

2

1/2

.

By inversion of the Fourier integral one gets

Ak I0(krr) =1

2piEz (r, z) cos kz dz .

To determine the constant Ak, we shall assume that at the drift tube bore radius a, the Ezfield has a constant value in the gap, and zero within the drift tube:

Ez (a, z) = const = E , g2

z g2

.

Solving the integral we get

Ak =Eg2pi

sin kg / 2( )kg / 2

1I0(kra)

,

and for the field

Ez (r, z) =Eg2pi

sin kg / 2( )kg / 2

I0(krr)I0(kra) cos kz dk .

The gain in energy of a particle q, when passing through the i-th cell of the DTL, is computedas follows:

Wi Wi1 = q Ez Li /2Li /2 (r, z,t)dz = q Ez

Li /2Li /2 (r, z) cos (t + )dz ,

where Li is the cell length and j is the RF phase (counted from the crest of the wave) whenthe particle crosses the mid gap (t = 0). At a given time t, the particle will be at the position zgiven by

z = u pt ,

where u p is the velocity of the particle, considered here as constant. Substituting t = z/u p inthe integral one gets

Wi Wi1 = q Ez Li /2Li /2 (r, z) cos z

pcos sin z

psin

dz .

We recognize that the first term under the integral gives the inverse Fourier integralmultiplied by cos j , and the second term is zero, as the even function Ez is multiplied by theodd function sin w z/u p. As the inverse Fourier integral has already been computed (note that k= w /u p), we have

Wi Wi1 = q Egsin kg / 2( )

kg / 2I0(krr)I0(kra)

cos .

The result is interesting: q Eg cos j would be the energy gain due to an alternatingvoltage Eg traversed at a phase j from the peak; the factor

sin kg / 2( )kg / 2

,

indicates the reduction in the energy gain due to the finite time the particle travels across thegap; the factor

1I0(kra)

,

is the additional reduction due to the fact that the field Ez on the axis penetrates into the drifttube aperture and is weaker than E, the field at the drift tube bore radius. Of course

Ez (0, z)dz = Eg . L /2L /2

The field off axis is always stronger than the field on axis [factor I0(krr)].Substituting Eg with E0L (E 0 is the average longitudinal field on axis in a cell), we

finally get

Wi Wi1 = q E0L T(k,r) cos j ,with

T(k,r) = sin kg / 2( )kg / 2

I0(krr)I0(kra)

= T(k) I0(krr) .

T(k,r) is the transit-time factor off axis (on axis it is T (k)). The transit-time factor makes itpossible to compute the energy gain in the whole gap, without the necessity of integrating themotion of particles point by point.

Knowing Ez, one can compute the other electromagnetic field components: Er with thedivergence theorem

divr

E = 1r

r rEr( )+

Ezz ,

giving

Er r( )= 1r

Ezz r d r ,0

rand B via

rotv

B = 1c2

v

Et ,

giving

Bz =

1c2

Ert .

One also has to make use of a property of Bessel functions:

x I0(x)dx = xI1(x) .

The formulae for all the TM010 field components are written below:

Ez (r, z,t) =E0L2pi

T(k)I0

(krr) cos kz cos(t + )dk

Er (r, z,t) =E0L2pi

T(k) kkr

I1

(krr) sin kz cos(t + )dk

B (r, z,t) =E0L2pi

T(k) 1kr

I1

(krr) cos kz sin(t + )dk .

The components Er and B produce a radial force on the moving particle. The changeof the radial momentum pr of the particle in a cell is

pr,i pr,i1 = q Er (r, z) cosz

p+

pB (r, z) sin

z

p+

Li /2

Li /2 dzp

.

The computation proceeds in an analogous way as with Wi Wi1.

8. BEAM LOADING OF CAVITIES: SPACE CHARGEIn Section 7 we have seen how the electromagnetic field in the cavity acts on particles.

Particles also act on the fields, and this interaction between beam and cavity is called beamloading of the cavity [10]. For reasons of phase stability [11,12], the beam bunch (particles inthe linac are grouped into bunches) and the RF field are not in phase: the bunch crosses thegap before the crest of the electric RF field. Therefore beam loading has a resistive as well asa capacitive component. The RF power supplied by the RF generator must, in addition tostructure losses, cover also the 'beam loading losses'. During the passage of the beam in thecavity, an RF amplitude and RF phase control is in action, and additional RF power iscorrectly supplied to keep the original pattern of the electromagnetic field in the cavity. Afeedback system acts on the tuners to keep the operating frequency stable.

The space charge of the beam which loads the cavity also produces a repulsive actionbetween particles. As the charge distribution in the beam is not uniform, the space chargeforces are non-linear. The question arises: how can one match a beam with non-uniformforces to a structure where the forces (as for example of focusing quadrupoles) arepredominantly linear? The answer is that it can be done provided one considers the root-mean-square (r.m.s) values of the particle distribution in phase space. Without going intomuch detail here, we just mention that Sacherer [13] has demonstrated that the evolution ofr.m.s values depends only on the linearized part of self forces, and that all beams having thesame intensity and the same r.m.s values behave in the same way. Therefore, once we knowthe r.m.s values of a real beam it can be replaced by an equivalent one having the same r.m.svalues, but a uniform density distribution. Such a beam can be matched.

The r.m.s beam sizes are computed via the second momenta of a normalizeddistribution function f(x,x,y,y,z,z)

x = x2( )1/2 = ... (x x)2 f (x, x , y, y , z, z )dx dy dz d x d y d z ,where x is the first moment (average value), usually zero. The same is valid for other phasespace coordinates. The r.m.s beam emittance in the phase plane (x,x) is defined as

rms = x2

x2

(x x )2[ ]1/2 .9. THE RFQ LINEAR ACCELERATOR

The RFQ accelerator is a very particular ion linac because it accepts an unbunched(continuous) beam, bunches it, focuses it and accelerates it [14,15]. All these actions areperformed with RF fields: no external focusing is needed and the bunching efficiencyapproaches 100%. The RFQ is a Russian invention (Kapchinskij and Teplyakov), but itbecame popular worldwide only after the Los Alamos National Laboratory, LANL,established a design method and built a prototype in 1980 [16]. This ion accelerator is the bestone in the low energy range, from a few keV to a few MeV per atomic mass unit.

The operation of the RFQ can best be understood by considering a long electricquadrupole with an alternating voltage on it, as shown in Figs. 15 and 16. Particles movingalong the z-axis and staying inside the RFQ for several periods of the alternating voltage,would be exposed to an alternating gradient focusing (Fig. 15). If the tips of the quadrupoleelectrodes are not flat but 'modulated' in an appropriate way, a part of the electric field is'deviated' into the longitudinal direction and this field can be used to bunch and accelerateparticles (Fig. 16).

Fig. 15 Long electric quadrupole with alternating voltage

Fig. 16 Modulation of RFQ electrodes to create longitudinal fieldsIt is very instructive to analyse the RFQ from the beam dynamics and electrodynamics

point of view.

2.1 Beam dynamicsThe particles move along the z-axis of the RFQ, and thus the beam dynamics considers

only the small central region around this axis. The electrodes form a well defined boundaryand, due to symmetry, the magnetic field on the axis is zero and nearly zero in the closeneighbourhood. Therefore, in this region one can replace the wave equation by the muchsimpler Laplace equation for the electric field potential U(r, ,z):

1r

r r

Ur

+1r2

2U 2

+ 2Uz2

= 0 .

The general solution of this equation has the form:

U(r, , z) = V2

[ Aon r2n cos2n + Aln I2n (lkr) cos2n cos lkz] ,l

n

n

wherel + n = 2p + 1, p = 0,1,2,... V/2 is the electrode potential with respect to the axisI2n (x) is the modified Bessel function of order 2nk = 2 p/bl b is the relativistic factor and l the wavelength

The general solution contains all the harmonics in an infinite series, but an RFQ is usuallywell described with only a few harmonics. The lowest-order solution has only two terms, oneof each infinite series:

U(r, , z) = V2

A01 r2 cos2 + A10I0 (kr) cos kz[ ] .

The first term is the potential of an electric quadrupole (focusing term); the second,containing cos kz , is linked with the acceleration. Constants A01 and A10 are determined byimposing that the potential U along the electrode stays constant and equal to V/2:

A10 =m2 1

m2I0(ka) + I0(mka),

A01 =1a2

[1 A10I0(ka)] =a2

.

The constants A10 and A01 are expressed by geometric parameters, a, the minimum distance ofthe electrode from the axis, and m , the modulation factor. Increasing m one gets moreacceleration; decreasing a one gets more focusing. Multiplying the equation for A01 by V onegets:

V + A10I0(ka)V = V .This equation tells us that part of the electrode voltage V is required for focusing ( c V), andanother part for acceleration [A10 I0 (ka)V]. As the electrode modulation can be produced in avery precise manner (e.g. by a numerically-controlled milling machine), we can obtain fieldswhich act on the beam as we want. The field components, derived from the two-termpotential function, are:

Er = Ur =

V2

2A01 r cos2 + kA10I1(kr) cos kz[ ]E =

1r

U = V A01r sin2

Ez = Uz =

V2

kA10I0(kr) sin kz .

In addition, all these fields have to be multiplied by a time factor, such as sin ( w t + j ), to takecare of their a.c. character. The computation of the energy gain per cell is done as with theDTL linac. The RFQ operates with a phase advance per cell of p , hence the cell length isbl /2, i. e. the particles cross it in half of the RF period.

9.2 ElectrodynamicsFor the beam dynamics it was sufficient to consider a small region around the axis and

deal with the Laplace equation. For the electrodynamics one has to consider the whole cavityand solve the wave equation. The RFQ operates with the TE210 field configuration, which isschematically shown in Fig. 17, for an empty cavity as well as for a loaded cavity (electrodesof such a shape are called vanes). The modulation of the vanes is neglected as it has littleeffect on the design of the cavity. The electric field is transverse, and the magnetic field islongitudinal and of opposite sign in adjacent quadrants. An electric equivalent scheme for thefour-vane RFQ is shown in Fig. 18. To compute the cavity with a 2-D program likeSUPERFISH, it suffices to consider, due to symmetry, only one half of a quadrant (seeFig. 19). In the region where the fields have to be known with a greater precision, the mesh isfiner.

Fig. 17 TE210 field pattern in empty and loaded cavities

Fig. 18 Equivalent scheme for a four-vane RFQ

Fig. 19 SUPERFISH calculations of a half quadrant of a four-vane RFQSo far we have left out one question: what happens to the magnetic field at both ends of

the RFQ? It cannot be perpendicular to the end covers of the cavity (perfect conductors), itmust be parallel to them. In fact the magnetic field of a quadrant satisfies this condition byturning around the vane as shown in Fig. 20. These end regions of the RFQ are computedwith 3-D programs like MAFIA [17].

Apart from the four-vane RFQ, one very often also uses a four-rod RFQ [18],developed by the University of Frankfurt and shown in Fig. 21. This RFQ resembles atransmission line and, indeed, the cavity which encloses it affects very little the resonantfrequency. This can be an advantage when operating at lower frequencies ( 100 MHz),where a four-vane RFQ would require too large a cavity. Figure 22 shows a design byMAFIA of a four-rod type RFQ with symmetric supports, built by the Laboratori Nazionali diLegnaro, LNL, Italy, for the CERN lead-ion project [19].

Fig. 20 End region of a four-vane RFQ and its equivalent circuit

Fig. 21 A four-rod type RFQ developed by the University of Frankfurt

Fig. 22 The lead-ion RFQ designed with MAFIA

10. CONCLUSIONThe basic features of RF ion linear accelerators have been described in this paper. In

order to accelerate particles, a synchronism between the electromagnetic wave and thevelocity of particles was required. The concept of the slowing down of waves linked withloaded cavities was essential for understanding how linacs function. The presentation ofspace harmonics in the dispersion diagram for periodic structures made clear the choice ofthe working point for either the TW or the SW structure. The fact that SW structures operateonly with phase advances per period of 0 or p became clear when considering that both thedirect and the reflected wave have to contribute to the acceleration of particles.

A few examples of ion linear accelerators operating in the SW mode have beenpresented. All of them, with the exception of the SCL, usually operate in the frequency rangeof one hundred to a few hundreds of MHz. Structure computations, as well as thecomputation of the action of the electromagnetic field on particles, have been presented withthe aim to show the methods applied. The notion of space charge and beam loading completethis introduction to RF ion linacs.

REFERENCES

[1] C. Ramo, J.R. Whinnery and T. van Duzer, in Fields and waves in communicationelectronics (John Wiley, New York, 1965) p. 374.

[2] J.C. Slater, in Microwave electronics (D. van Nostrand, New York, 1951) p. 395.[3] Ibid., pp. 169177.[4] G.A. Loew and R.B. Neal, Accelerating structures, in Linear Accelerators, eds.

P.M. Lapostolle and A.L. Septier (North Holland, Amsterdam, 1970), pp. 4752.[5] G. Dme, Review and survey of accelerating structures, in Linear Accelerators (see

Ref. [4]).[6] E.A. Knapp, High energy structures, in Linear Accelerators (see Ref. [4]), pp. 601616.[7] K. Halbach and R.F. Holsinger, Superfish A computer program, Part. Accel. 7 (4)

(1976) pp. 213222.[8] T. Weiland, On the computation of resonant modes in cylindrically symmetric cavities,

Nucl. Instrum. Methods 216 (1983) pp. 229248.[9] P. Lapostolle, Introduction la thorie des acclrateurs linaires, CERN 8709 (1987).[10] D. Boussard, Beam loading, Proc. CAS, Queens College, Oxford, 1985; CERN 8703

(1987).[11] P.M. Lapostolle, Introduction to RF linear accelerators (Linacs), these proceedings.[12] M. Weiss, Introduction to RF linear accelerators, Proc. CAS, University of Jyvskyl,

Finland, 1992; CERN 9401 (1994).[13] F. Sacherer, RMS envelope equations with space charge, CERN/SI/Int. DL/7012.[14] I.M. Kapchinskiy, Strahldynamik in den Sektionen mit rumlich uniformer

Fokussierung, GSItr14/76 (Translation from Russian).[15] M. Weiss, Radio frequency quadrupole, Proc. CAS, Aarhus, Denmark,1986; CERN 87

10 (1987).[16] R.H. Stokes et al., The radio frequency quadrupole, Proc. 11th Int. Conf. on High

Energy Accelerators, Geneva, 1980 (Birkhauser, Basle, 1980).[17] R. Klatt et al., Mafia A three dimensional electromagnetic CAD system, Proc. Linear

Accelerator Conf., 1986, SLAC report 303, pp. 276278.

[18] A. Schempp et al., Zero-mode RFQ development in Frankfurt, Proc. Linear AcceleratorConf., Seeheim, Germany, 1984, ed. N. Angert (ISSN: 01714546, GSI8411, 1984)p. 338.

[19] A. Lombardi et al., Comparison study of RFQ structures for the lead ion Linac atCERN, Proc. EPAC, Berlin, 1992, eds. H. Henke, H. Honeyer and C. Petit-Jean-Genaz(Editions Frontires, Gif-sur-Yvette, 1992) pp. 557559.

RF SYSTEMS FOR LINACSW. PirklCERN, Geneva, Switzerland

AbstractAn overview of RF systems for linacs is given, problems associatedwith beam loading are discussed and examples of RF systeminstallations presented.

1 . RF POWER REQUIREMENTSIn linear accelerators the beam has to gain its energy during a single pass through the

structure. This leads to high requirements on accelerating fields and on overall RF power. Atypical linac has therefore installed RF power in the range of tens to hundreds of Megawatts. Itis then justified to consider linacs as RF-oriented machines, in contrast to circular acceleratorswhich are rather magnet-oriented.

In the present context the RF system proper starts at the RF connection to the structure.The latter can be of the standing- or travelling-wave type, but whatever the mechanisms ofpower build-up and energy transfer to the beam may be there are two distinct requirements to becovered: delivering power to create and maintain a well defined field pattern in the structure(wall power Pwall), and replacing the power transmitted to the particle beam (beam powerPbeam), i.e.

PRF = Pwall + Pbeam

All wall losses can be thought of as concentrated in a single resistor, the "shunt impedance" rdefined as r = Uge

2 / 2 * Pwall( )where Uge is the effective structure voltage seen by a beam of given velocity, i.e. taking intoaccount the transit time factor T that expresses the voltage reduction due to the sinusoidal fieldvariation during beam-passage time. This is the "equivalent circuit" definition of the shuntimpedance hence the factor of 2 in the denominator which stems from the fact that the powergenerated by a sinusoidal RF voltage is a factor of two smaller than the power generated byDC.

On the other hand, the beam power is given by

Pbeam = Uacc * I1 / 2Uacc = Uge * cos

where Uacc is the average energy gain of the beam expressed in eV, the stable phase angleand I1 the spectral component of the beam current at the RF frequency. For short bunches thiscomponent is close to twice the DC current I0 so that the beam power is approximately Uacc*I0.

Note that there is no third component representing additional losses caused bytransporting the beam power through the structure. This is due to the fact that it rides on top ofthe very large reactive energy flows related with the fields so that the additional transport lossescan be neglected for all practical purposes.

2 . BEAM LOADING

To study the effects of beam-structure interaction from the external RF system point ofview the equivalent circuit of Fig. 1 below is often used, idealizing the structure as a resonator

with a single accelerating gap across which the full effective gap voltage Uge is applied. Thisresonator is driven by two sources, namely the RF system and the beam, which are supposedto act all directly on the gap plane. While this model is a fairly accurate representation of asimple buncher or even a complete standing wave tank it is not directly applicable to standing-wave structures.

Fig. 1 Equivalent circuit of the cavity (single-gap resonator)The corresponding phasor diagram is given in Fig. 2 for a data set corresponding

approximately to the conditions of Tank 1 at the CERN linac 2. As phase reference theeffective gap voltage has been chosen and positioned at 0 in the first quadrant. For phasestability reasons the center of the bunch crosses the midplane of the accelerating gap(s) not atthe instant of maximum voltage Uge but at some time before. This time corresponds to the"stable-phase angle" of the RF, whose value according to the "linac" definition which countsthis phase from the top amounts to -30 in the example. (The beam current is situated in thethird quadrant due to the sign convention in the diagram defining it as a generator rather than aload.)

According to Kirchhoff's first law, the current to be fed by the RF amplifier is

Iamp = Icav Ibeam = Iloss + Ireact Ibeam

where Iloss stands for the loss current (in phase with the voltage causing driving wall losses)and Ireact the reactive current of the cavity (orthogonal to voltage causing lossless reactivepower exchange). This is represented by the phasor triangle in the fourth quadrant. Since theamplifier requires a resistive load for optimal working conditions, the cavity is intentionallydetuned to provide a reactive component that just cancels the reactive component of the beamcurrent. This sets the necessary angle of detuning (47.3 in the example, i.e. slightly higherthan one half-bandwidth).

Now that the complex impedance of the detuned resonator is determined one can also, asan alternative, analyze the circuit by superposition of the voltages. Responses of the resonatorto the amplifier current alone (Uamp) and to the beam current alone (Ubeam) are combined togive again the effective gap voltage Uge as shown by the phasors in the first quadrant. (It isassumed here for simplicity that the generator is of infinite output impedance, i.e. that it acts asan ideal current source.)

It can be seen that the amplifier with unchanged drive would induce, in the absence of thebeam, a resonator voltage Uamp that is grossly out of amplitude and out of phase. After thebeam has arrived the resonator voltage would only gradually approach the wanted voltage Ugein a (spiral) trajectory that takes several cavity time constants to complete. During this transientthe beam would not receive the proper energy gain and be useless.

Fig. 2 Phasor diagram for the cavity

What is needed is some circuitry that would:

change the amplifier drive in amplitude and in phase such that it results in Icav before thebeam arrives and in Iamp thereafter,

set a cavity tuning angle that compensates the reactive part of the beam current.

These tasks are normally covered by three different regulation systems or servos, namely foramplitude, phase and tuning. Combining amplitude and phase servos by a single overall RFfeedback is in principle possible (and applied in circular accelerators) but has not yet foundwide application in linacs.

3 . ANATOMY OF A FULLY REGULATED RF CHAIN

Figure 3 shows a block diagram of the Tank 1 RF chain in CERN linac 2 as anillustration of how the three servos mentioned above can be integrated into the system.

Fig. 3 Block diagram of a fully regulated RF chain

3 . 1 Forward RF path (bold lines in Fig. 3)A low-level RF signal of about 2 W arrives at the input plane with a well-defined phase

from the phase reference line. The signal then travels through the detector box, 6-bit phaseshifter, phase and amplitude modulator, is gradually amplified in the transistor amplifier, poweramplifier, drive amplifier and final amplifier before arriving at the feeder loop of the resonator atsufficiently high power to cover tank requirements. Somewhere in the forward RF path therehave to be means to adjust output phase and amplitude at high speed in response to the electricalcorrecting signals out of the servos. This is straightforward for the phase which is in generalwell transmitted through almost any amplifier chain so that a low-level phase shifter can be usedat the input. However, the amplitude modulation is quite critical since most high poweramplifiers exhibit some kind of saturation and over saturation, where the RF output stalls oreven decreases for increased drive. In the present case these adverse effects are avoided by aspecial current stabilisation in the final amplifiers so that the amplitude modulator can beimplemented as a low-level device at the input of the chain. In other machines the amplitudelinearity problem is solved by controlling the RF output amplitude via the plate supply voltageusing a high-voltage floating modulator with one or more series tubes.

3 . 2 Amplitude and phase servo

A sample of the tank RF field is taken from the tank via a coupling loop, cleaned in abandpass filter and routed to the detector box ( lines in Fig. 3). It carries information ontank amplitude as well as tank phase. Both are demodulated in the detector box and appear atthe output in the form of video signals (D A (amplitude) and D (phase with respect to the inputplane)). In the feedback control module they are compared to the command value ATANK fortank amplitude generated in the control system, and to 0 as implied command value for the tankphase. The resulting error signals are amplified and fed to the amplitude and phase modulator,thus closing the loop.

For constant beam current the amplitudes of the error signals are predictable so thatcomputer-generated "feedforward" signals can be injected for open-loop correction. This canalso reduce the transient at the start of the beam. Inputs for such feed-forward signals are

provided at the feedback control, but not used in normal operation. The absolute phase of thetank with respect to the input plane is stabilized to some value that depends on the phasingbetween cavity and detector box. Phase adjustment could be implemented by an adjustablephase shifter in this line, but the unavoidable mismatches would also produce unwantedamplitude variation at the same time. It is therefore preferable to set the operating phase by #1in the reference signal phase feeding the chain. A motor-driven line stretcher "trombone" with aresolution of 0.25 degrees is foreseen for this purpose.

The 6-bit digital phase shifter #2 is provided to center the initial phase of the main signalpath within the limited operating range of the fast phase modulator.

3 . 3 Tuning servo

To sense the tuning state of the resonator the phase of the voltage across its gap(s) has tobe compared with the phase of the forward wave in the feeder line. This information isextracted by a directional coupler and fed via a low-pass filter and 6-bit phase shifter #2 to thedetector box where information on resonator phase is already available. An error signal isgenerated and fed to the tuning control, which in turn actuates the tuner motors if the tuningstate is outside an adjustable error band. Phase shifter #3 allows setting of the tuning angle in5.6 steps to optimize the power transfer between amplifier and cavity; it has virtually NOinfluence on the phase seen by the beam due to the action of the phase servo.

4 . OVERALL STRUCTURE OF THE RF SYSTEM

A linac RF system consists in general of many independent RF chains for a number ofreasons: accelerating tanks cannot exceed a certain length to avoid beam-breakup; poweramplifiers cannot exceed practical power limits; different functions like bunching, debunchingand diagnostics require independent settings; different harmonic numbers in the structurerequire amplifiers of different frequency, etc.

The individual RF chains have to be linked together with constant phase-relationship thatshould be unaffected by manipulations on any of its members. This is achieved by a phasereference line whose outputs are strongly decoupled from each other, preferably throughdirectional devices to keep the power loss in reasonable limits. Figure 4 shows the overallstructure of CERN linac 2. Each chain receives its input signal via a directional coupler of20 dB coupling and 40 dB directivity, such that the effective separation is as high as 60 dB.The phase reference line is running in along the tanks, with couplers mounted in closeproximity to the outputs of the return loops. The two cables carrying respectively the referenceand the return signals are of the same type, tied and laid together so that a common variation inlength due to temperature differences will not influence the phasing of the resonator.

The power stages for Tanks 2 and 3 deserve special mention since there are two identicalamplifiers with separate feeder lines and coupling loops to work into a single tank. Thisconfiguration has been chosen in order to provide two feed points into the relatively long tanksand to improve this way the inter-tank transient conditions under heavy beam loading. From apurely RF-plumbing point of view the two amplifiers could also have been connected to thetank by a dedicated combiner, as explained in section 5.4 below.

Figure 5 gives an impression of the physical layout of such a chain:

Fig. 4 RF system of CERN linac 2

Fig. 5 Physical layout of the RF chain for tank 2 in CERN linac 2

The four racks at the left contain respectively: the computer interface and a HV supply forthe pre-predriver; low-level circuitry, servos and the pre-prediver amplifier itself (5 kW);the HV power supply for the prediver; the prediver (50 kW).

the three cylinders to the right are the covers of identical triode power amplifiers used asdriver and composite final stage for tank 2. The height of these amplifiers is over threemeters, their lower part extends into a service gallery that also contains the infrastructure(water supplies, blowers etc.) as well as the RF feeder lines and high-power phaseshifters (trombones).

the enclosure against the right wall contains the HV modulator for the final stages whichis implemented as a pulse-forming network discharged by an ignition.

The layouts for tanks 1 and 3 are similar. Buncher and debuncher chains consist only ofthe four standard racks since the power delivered by the "predriver" is sufficient to power thecorresponding resonators.

5 . SPECIAL TOPICS

Now that a general picture has been obtained some special topics can be discussed thatinfluence the performance of the RF system.

5 . 1 Limits of the regulation systems

5.1.1 Loop delay

We consider as approximation a servo system with a single time constant, together withan overall time delay t delay (caused by amplifiers, RF feeder lines, structure, return cable etc.).The only limit on servo performance is then the overall loop delay t delay. It is well known fromthe theory of servomechanisms that a certain "phase margin" is needed at the unity-gainfrequency to assure satisfactory transient behaviour and to avoid instabilities. Assuming apermissible phase margin of 45, the total phase at unity gain frequency must not exceed(180 - 45) = 135 from which 90 are already reserved for the 6db/octave "rolloff" from theleading time constant (the cavity in most cases). In other words the unity-gain frequency fug islimited to a value where the delay contributes no more than about 45, i.e. since

f ug * delay * 360 = 45f ug < 1 / 8 * delay( ) .

Knowing fug one can derive t servo, the servo time constant after which a step perturbation isreduced by a factor of e = 2.718. Theory of the simple model then predicts

servo = 1 / 2 * pi * f ug( ) = 8 * delay / 2 * pi( ) 1.3* delay .Hence the servo time constant is always somewhat longer than the overall delay in the loop, aresult that is also intuitively evident.

5.1.2 Cross coupling of servos

Phase and amplitude servos are not independent as can be seen in the phasor diagramFig. 6a (left). The two components of the gap voltage, induced by amplifier and beam, arepresented in the same form as earlier in Fig. 2. Suppose that a test signal is injected in theamplitude loop so that the voltage Uamp generated by the amplifier is (slowly) varied with

Fig. 6 Cross-coupling of servos

constant phase. The resulting gap voltage Uge will however not only have a component inamplitude D A' but also in phase D '; an unwanted return signal will therefore be seen in thephase servo.

Similarly, a pure phase variation of the RF drive as in Fig. 5b (right) will lead to wantedphase component D " together with unwanted cross-coupling amplitude component D A".

Another source of cross coupling of amplitude and phase is inherent in most high poweramplifiers due to non-linearities.

Since both servos have to react equally quickly they need similar cut-off frequencies.Cross-coupling introduces additional phase shifts at these critical spots with the result that theachievable frequency limit for both servos combined is much lower than for one servo alone.The tuning servo, as the third element in the regulation system can fortunately be designed forsignificantly slower response since it has only to cope with temperature drifts or air pressurevariations. Its crossover frequency is therefore much lower and will not interfere with the fastsystems.

5.1.3 Non-linearities in the RF chain

All power amplifiers have a tendency to saturate at the upper end of their operating rangei.e. their differential amplification is reduced or may even change sign. The loop amplificationof the amplitude servo varies accordingly. Despite linearization circuits in the feedbackcontroller there remains a residual effect that sets an upper limit to the applicable loop gain.

5.1.4 Overall performance of the servomechanisms

Due to the effects described above, the practical performance of the servomechanisms isan order of magnitude lower than the theoretical limit set by the loop delay alone. Typical loopgains are in the range of 20 40 dB, together with time constants of the order of microseconds.This leads to operating precision of about 1% in amplitude and 1 in phase even under heavybeam loading.

5 . 2 Choice of RF power source

It has been seen that the RF system of a linac is comprised of a number of individual RFchains of very different output power requirements. It is then useful to keep the same basicstructure (like interlocks, computer controls, servomechanisms, etc.) for all these individualchains and provide a set of modular amplifier blocks that are combined to provide the necessaryoutput power. A short overview of industrial power amplifier elements is presented in Fig. 7.Legal power limits for different TV and broadcast services are also indicated since industry

Fig. 7 Approximate RF average power limits of different active devices (source: VALVOcatalogue)

designs the main line of readily available off-the-shelf components within these limits. Lettercodes for main radar bands have been added since related equipment is and was thecrystallisation point for many linac RF systems. It may be worth remembering that the verywidespread operating frequency of 202.56 MHz had initially been chosen because of theavailability of surplus World War II radar components.

Some comments on the different amplifier categories:

5.2.1 Semiconductors

The "Transistor amplifier limit" is a quickly moving target since semiconductor amplifierswith bipolar or FET-devices are in constant progress. They have replaced most of the tubecircuitry in all preamplifier stages and are gaining acceptance as final stages for certainapplications such as bunchers and debunchers. From the point of view of circuit design theyare low-voltage, low-impedance, low-Q devices.

Semiconductor components are inherently unforgiving concerning overvoltages; sincethey are also quite sensitive to load conditions, they need adequate protection elements likecirculators which, however, make the amplifier an ideally-matched source.

Advantages: compact and reliable packages with virtually no maintenance; very widebandwidths possible. Disadvantages: Tend to have high electrical delay times; are radiation sensitive (less importantin linacs than in other accelerators).

5.2.2 Tubes (Triodes and Tetrodes)

Tubes use a control grid to modulate the density of an internal electron beam which, afteracceleration, transmits its RF energy to the external load circuit. Transit-time effects reduce theeffectiveness of grid control at high frequencies and mandate small system dimensions oflimited power carrying capability. Tubes are the workhorse for linacs up to about 300 MHz.Gains are moderate, about 10 15 dB per stage for triodes and up to 20 dB for tetrodes. Theyare generally run in "grounded-grid" configuration which reduces the gain but is the onlypractical possibility to assure stability above a few MHz for triodes and above about 100 MHzfor tetrodes. From the point of view of circuit design, tubes are high voltage, high impedance,high-Q devices.

A typical tube circuit provides coaxial input and output resonators to tune out the tubecapacitances over a relatively small bandwidth. The two resonators are often implemented in"reentrant" configuration, i.e. they share one cylindrical metallic wall with inner/outer facesassigned to inner/outer resonator (Fig. 8). It is of course the skin effect that separates the twocircuits at RF frequencies although they are at DC, galvanically connected by the conductingwall.

Advantage: Relatively robust devices.

Disadvantage: Limited lifetime of the active elements, need regular maintenance.

Fig. 8 Structure of a triode amplifier

5.2.3 Klystrons

Klystrons are the active elements of choice for the higher linac frequency bands. Theyovercome the limits of gridded tubes by the use of velocity modulation (rather than densitymodulation). The internal electron beam is initially directed through a pillbox-type cavity whereits velocity is modulated with the frequency of the input signal. After a certain drift space thefaster electrons overtake the slower ones and create regions of higher density. Passive RFcavities provided at these regions further enhance the beam structure by the self-induced electricfield. The RF beam power is finally transferred to an internal RF cavity from where it isextracted to the external load.

Klystrons are very high voltage, high-Q devices. The long internal drift space of thebeam needs transverse focusing by solenoids or permanent magnets. As the length of this drift

space increases with the RF wavelength there is a practical lower limit for the operatingfrequency. Figure 9 shows the "Klystron Skyline", the pictorial representation of the unitsfrom one large European manufacturer. Also shown is the largest commercially availableklystron, almost 6 m in length for operation at 200 MHz. The large dimension has so farhindered its use in low-frequency linacs.

Advantages of klystrons: High gain of 40 50 dB (about 10 dB per internal cavity).High lifetime.

Disadvantages: Supply voltages of 100 200 kV necessitate special technology; X-ray hazardexists if not properly shielded; pulsed high-power klystrons tend to be noisy electrically andacoustically. In general only moderate DC-to-RF conversion efficiency ~ 50%.

5.2.4 Other RF power sources.

The upper power limit in klystrons is ultimately determined by the output resonator. Itsdimensions scale directly with the wavelength for a given cavity mode; at higher frequenciessome critical part (usually the coupling slot) becomes breakdown-limited due to the smalldimensions involved. Gyroklystrons, gyratrons and similar construction principles push thislimit upwards by the use of transverse rather than longitudinal extraction mechanisms whichallow larger cavity dimensions; their development is in full swing.

Two-beam accelerators first generate an auxiliary beam which runs in parallel with thephysics beam and serves as a distributed power source. A prominent example for extremelyhigh operating frequency is the CERN CLIC proposal which is presented in section 6.3.

5 . 3 Amplifier matching and resonator transients

According to a well known theorem in circuit theory, maximum power transfer betweensource and load occurs if their impedances are complex conjugate to each other. This isperfectly valid for low-level, linear circuitry but additional constraints of dynamic range have tobe considered at high power. Take for example an amplifier for P = 100 kW whose tube asactive element presents 15 k W internal (source) impedance and supports a maximum RFvoltage U = 10 kV. 15 kW is the optimum load at low level, but only up to the output voltagelimit of 10 kV corresponding to 5 kW. Beyond this point the load impedance has to be reducedto avoid saturation and sparking; full power can only be achieved at a load impedance ofU2/(2*P)= 500 W with drastically reduced amplification.

In virtually all active elements the optimal load resistance for maximum output powerdiffers by one or two orders of magnitude from the internal source resistance. Looking backfrom the feeder line into the amplifier, this internal source impedance appears at a voltage-standing-wave ratio (VSWR) of 10 to 100. It is assumed that the tube-to-feeder coupling isarranged such that the optimal load for maximum output power corresponds to the characteristicimpedance of the line.

The load impedance of cavities and tanks is not constant but exhibits a sharp resonance inthe frequency domain. It is customary to refer to cavity impedances at the "Detuned Short"

(DS) plane of the feeder line. This has been implicitly been done in Fig. 1. The inputimpedance begins and ends at 0 W for the fully detuned cavity, and describes the well known

Fig. 9 The 'klystron skyline' of one European manufacturer

circle going through the center of the Smith-diagram for perfect match at the resonancefrequency (Fig. 10).

Fig. 10 Influence of the line length between amplifier and cavity: maxi Q/mini QFourier transform allows one to derive the behaviour in the time domain for transient

conditions, i.e. for abrupt switching-on of the cavity by a gated sinusoid at the resonancefrequency. The apparent instantaneous cavity impedance is zero at the beginning, follows thereal axis, and finally stops at the steady-state resistance after several cavity time constants.

If the transformed source impedance is situated on this low-impedance part of the realaxis (Fig. 10 middle), then there is perfect match at least at one moment of the transient. Theoverall average power transfer during the filling phase is optimal and the fastest possiblerisetime results. In the frequency domain, the bandwidth is widest due to the strongestdamping of the resonator by the amplifier. The term "Mini-Q" has been coined for this regime

Inversely, if the transformed source impedance is on the high-impedance part of the realaxis, there is never a perfect match; the average power transfer is worst; the risetime is slowest;the bandwidth smallest. This is designated as "Maxi-Q". These two extreme cases representthe only two points where the overall system is at "resonance", i.e. where the amplituderesponse in the frequency domain peaks at the operating frequency and where there is nobeating in time domain. These conditions should be set for best overall performance. The"Mini-Q" point is the preferred mode of operation, but is there is a tendency for amplifiersparking in case of cavity breakdowns. Therefore the more forgiving "Maxi-Q" mode hassometimes to be set during conditioning.

It should be pointed out that the above qualitative description is strictly valid only for low-level conditions. Saturation effects modify more or less the behaviour at nominal operatinglevels but the basic effects remain valid. As a practical consequence, a high-power phaseshifter (trombone) is in general necessary between amplifier and cavity for the optimisation oftransient behaviour (and not to adjust the overall phase at an ill-conditioned spot as issometimes suspected).

Some generators, especially semiconductors and klystrons, have stringent requirementsfor the load impedance that preclude the direct connection of a cavity in pulsed regime. Non-reciprocal ferrite devices such as circulators or insulators have to be provided in these cases toabsorb all load reflections. The transient behaviour becomes then independent of feeder linelength.

5 . 4 RF plumbing

The RF feeder line between the power amplifier and the structure transports in generalconsiderable RF power and has to be designed with some precaution. Except for very highduty cycle it is the peak-pulse power rather than the average power that sets the limits.

In coaxial systems the support of the inner conductor is often the weakest spot.Accumulation of dust and metallic particles, e.g. stemming from contact wear on phase shifters(trombones), may substantially reduce the flashover rating in particular on horizontally orienteddisks. The use of metallic l /4 stubs eliminates this problem altogether since there are noinsulating surfaces to be contaminated. It is of course prudent to choose the largest linediameter possible. This is more an economic than a technical problem, since coaxial linediameters of up to 300 mm having peak-power capabilities of tens of Megawatts can be usedaround 200 MHz without problems from higher-order modes. Contact erosion from RFcurrents has been a problem at some machines, this is again alleviated by large diametersresulting in lower current densities. Pressurisation by air or SF6 is sometimes used to increasethe peak power capability, at least permanent venting with dry air is advisable as a preventivemeasure. This complicates the mechanical design since continuity of the air channel has to beassured and excessive leaks avoided.

In waveguide systems the dimension of the feeder is in general predetermined by theoperating frequency. Pressurisation or evacuation are the means to achieve high peak-powercapability; sometimes bo