Design of a high energy, high duty cycle, racetrack microtron

NUCLEAR

INSTRUMENTS

AND

METHODS

56

(I967)

I97-2o8; © NORTH-HOLLAND

PUBLISHING

CO.

DESIGN OF A H I G H ENERGY, H I G H DUTY CYCLE, RACETRACK M I C R O T R O N B. H. WIIK and P. B. WILSON*

High Energy Physics Laboratory, Stanford University, Stanford, California, U.S.A. Received 12 June 1966 The design of a split-magnet racetrack microtron with a maximum final energy between 400 and 600 MeV is considered. Detailed calculations of the beam optics are presented. It is shown that adequate focusing can be provided by quadrupoles placed on the individual orbits. The phase stability of the device is investigated. The phase stable region is sufficiently large so that the tolerances on the homogeneity of the magnetic field and stability of the accelerating voltage are reasonable. As the result of phase focusing, the energy resolution of the microtron is on the order of 10-3. Design parameters for a conventional (non-supercon-

ducting) rf system are given. Depending on the choice of an rf power source, the proposed microtron can provide an average current at maximum energy of 50 to I00/~A at a duty cycle of 3 to 6%. At 200 MeV the average current is 75 to 200/~A and the duty cycle is 10 to 30%. In comparison with an electron linac using a comparable amount of average rf power, the energy homogeneity and duty cycle of the microtron are superior by an order of magnitude. In addition the microtron is considerably more compact and is less expensive to build.

1. Introduction A t the present there is considerable interest in a high current, high duty cycle electron accelerator with an energy of several h u n d r e d MeV for nuclear research. The racetrack microtron*, using s u p e r c o n d u c t i n g cavities as proposed earlierS), meets these requirements. However, the efficient use of rf power in the m i c r o t r o n also makes it feasible to achieve high duty cycle with c o n v e n t i o n a l n o n - s u p e r c o n d u c t i n g rf cavities. It is in fact possible by using two commercially available klystrons to reach unity duty cycle at a b o u t 100 MeV and 13°,o duty cycle at 400 MeV with an average current of 100 pA. F u r t h e r m o r e , the energy resolution is of the order of 10 -3 , and the injection and extraction of the beam pose n o serious problems. We therefore feel it worthwhile to consider this device in greater detail. In the next section a general description of the microtron is given, including a discussion of the methods chosen for the injection and extraction of the beam. A detailed investigation of beam optics a n d phase stability using measured fringing fields from a model m a g n e t is given in sections 3 a n d 4. The design of the rf system is presented in section 5, together with a discussion of problems due to beam loading, beam break-up, a n d electron capture. I n the c o n c l u d i n g section a rough cost estimate for the device is given.

2. Description of the microtron A schematic drawing of the proposed m i c r o t r o n is shown in fig. 1, a n d the m a i n design parameters are listed in table 1. A configuration has been chosen for the m i c r o t r o n which consists of two 180 ° b e n d i n g magnets with u n i f o r m fields, an accelerating linac section providing an energy gain W per t u r n located

TABLE 1 Parameters for the proposed microtron.

Maximum energy Average current at max. energy Duty cycle at max. energy Orbit parameter v Orbit parameter/~ Injection energy (total) Energy gain per turn Total number of orbits Magnetic field Operating wavelength Spacing between orbits Diameter of last orbit Magnet gap Distance between magnets

Rfsystem A

Rfsystem B

595 MeV 100/tA 6% 1 39 44.5 MeV 19.7 MeV 28 17.3 kG 23.g cm 7.6 cm 2.3 m 2.5 cm 4.2 m

415 MeV 45/zA 3% 1 39 31 MeV 13.7 MeV 28 12.0 kG 23.8 cm 7.6 cm 2.3 m 2.5 cm 4.2 m

in the field-free region between the magnets on the path c o m m o n to all the orbits, and an external injector linac, it is a d v a n t a g e o u s to use a long accelerating linac section in order to reach a high duty cycle. This requires a large separation between the magnets, which in t u r n makes it difficult to insure the horizontal and vertical stability of the beam by using only wedge focusing. However, the large separation between subsequent orbits makes it possible to put q u a d r u p o l e s o n each individual orbit if necessary. Thus to a large extent in this m a c h i n e the guiding a n d focusing properties of the magnet system are separated. * Work supported in part by the U. S. Office of Naval Research, Contract [Nonr 225(67)]. t The racetrack microtron was first suggested by Schwinger. In recent years calculations for several magnet configurations have been publishedx 3). Brannen and Froelich 4) have built a 12 MeV racetrack microtron utilizing four sector magnets. 197

198

1~. H . W I I K

AND

P. B. W I L S O N

....fHIGH VOLTAGE ELECTRON SOURCE

~INJECTION

/X INJECTION SECTIONS

',,MAGNET

~-~-'~- 7~ . . . . . . . . .

" \~ ~

L PREBUNCHER OR PHASE CHOPPER GUIDING 'x MAGNET . . . .

~E~_g:rfR6fi-6~Bl-f~

~-~-~--~

SIDE VIEW

GUIDING' MAGNET

, QUADRUPOLES

....... g:~ .... --:~b __l-~- . . . . . . . . . . .

/)} ~j//

-t-- E

TOP VIEW Fig. 1. Schematic drawing of the microtron.

Electrons with a total energy W0 are injected into the microtron from the external injector linac. The injection is accomplished by bending the electrons into the median plane of the microtron with a small 180 ° magnet. Upon entering the field of the microtron they are deflected once more through 180 ° into the common path. To extract the electrons a small magnet is placed in the region between the bending magnets as indicated in fig. 1. To miss the microtron magnet proper, the electrons will be bent about 15 ° and subsequently refocused. In order to satisfy the resonance conditions, the revolution time for the first orbit and the time lag between subsequent orbits must be integral numbers/~ and v of the rf period. Using the effective magnet approximation 6) these conditions lead to W0/(mo c2) = (B2/I0.7 [-kG.cm]) { # - v - ( S l 2 ) } ,

W/(mo c2) = (B2/10.7 [ k G . cm])v,

(1) (2)

in the limit in which the electron velocity approaches the velocity of light. In the preceding relations ½S is the distance between the effective field boundaries of the microtron magnets, 2 is the free space wavelength at the operating frequency, B is the magnetic field strength in kG, and moc2 is the electron rest energy. The separation of the magnets in a racetrack microtron results in several advantages as compared to a conventional microtron. First, the final energy can be changed continuously for a fixed extraction point by changing Wo, W and B according to eqs. (1) and (2). Second, the product B2 is no longer limited to approximately 10 4 G - c m , as in the case of the conventional microtron and thus a much larger value of B can be

chosen. In fact, the field is now only limited by the saturation of the magnets. Third, the flexibility in choosing B2 also makes it possible to operate at longer wavelengths. As the separation between subsequent orbits is equal to v2/n, a long wavelength makes it possible to put quadrupoles and other correcting devices on each individual orbit, even for v = 1. Also, the phase error caused by the fringing field and velocity changes of the electrons during the first few orbits is reduced as 2 is increased. An important feature of the design presented in this paper is the injection of the electrons at high energy. Since in the racetrack microtron the injection energy is independent of the energy gain per turn, it might at first sight seem advantageous to inject at low energies. However, a closer examination shows there are several reasons to choose a high energy injection. The more important are: 1. At the lowest output energies of the microtron the voltage gradient in the accelerating section is low, and hence the injection energy must be rather high in order to capture the electrons. For an output energy of 100 MeV, corresponding to an energy gradient of about 104 V/cm, a minimum kinetic injection energy of about 1.4 MeV is needed for capture. Now to raise the output energy without changing the mode of operation or the distance between the magnets, the total injection energy must be raised proportionally, thus leading to an injection energy on the order of 10 MeV at the highest output energy of the device. 2. The injection phase is determined uniquely by the resonant phase and the phase slip due to the fringing field and the straight sections. This phase angle is in general different from the phase angle determined by

A HIGH

ENERGY, HIGH

DUTY

optimum capture. However, if we inject a relativistic beam from a high energy, external injector this problem vanishes. 3. Detailed calculations show the acceptance of the microtron to be severely limited by vertical defocusing produced by the fringing fields of the magnets. However, the defocusing decreases rapidly with increasing beam energy, and injection at high energy will result in a larger acceptance and smaller beam losses. To achieve a similar result using low energy injection, the microtron can be operated in a higher mode (v > 2). In this case the available phase-space is greatly reduced, and in addition the duty cycle of the microtron will be lower compared to operation at v-- 1. 3. B e a m o p t i c s

3.1. BEAM OPTICS IN THE ACCELERATING SECTION A number of transverse forces act to produce focusing, defocusing, or deflection of the electron beam in the accelerating linac. The effect of these forces will be most severe on the first passage of the electrons through the accelerator section. We will estimate the divergence angle 30 introduced by each of the effects for the orbit with lowest energy. The effects which will be considered are space charge, the effect of transverse rf fields off the axis of the structure, the lens effect due to the fringing fields at the entrance and exit of the accelerating section, the focusing effect due to nonuniformities in the amplitude of the axial rf field within the accelerating structure, and coupler asymmetry. The first four effects act only on off axis electrons, while the last produces a deflection of the beam as a whole. The divergence angle produced by space charge forces acting on an electron at the edge of a beam of radius r with current I 0 can be conveniently written as (60)s c = (loZo/Mo) {L/(2~r)}/(?afl~).

(3)

Here L is the length of the accelerator section, Zo = 377 ~, M o = mocZ/e = 0.511 MV, fl~ = G/c where G is the electron velocity, and ? = (1 -fl~) 2 -~ . An impulse approximation is used in deriving this and the following relations in the sense that the transverse momentum 6p imparted to an electron is calculated assuming no change in radial position during passage of the electron through the section. The greatest spreading of the beam due to space charge forces will occur for cw operation when ? is the lowest. It will be shown later that the injection energy is about 6 MeV and the circulating current about 7 mA for this case. Using L = 330 cm and r = 0.1 cm in eq. (3), a value of (60)sc ~ 1 0 - 6 is obtained.

CYCLE, RACETRACK

MICROTRON

199

The transverse electromagnetic forces acting on an off-axis electron can be derived following the work of Loew and Helm7). Assuming that the electron and wave velocities are nearly equal so that the phase angle of the electrons with respect to a wave traveling with phase velocity Cflw is nearly constant, the divergence angle can be written (60)E M~ (nr/2) (67/7) {(1 - fl¢flw)/(fl~flw)} tg A.

(4)

Here A is the phase angle measured in space with respect to the position of maximum acceleration, and 67,~, (EoL cos A ) / M o is the energy gain measured in units of the electron rest energy, where E 0 is the peak axial field strength in the accelerating section. In the microtron the equilibrium phase angle is behind the crest in space (although ahead of the crest in time), A is therefore negative and hence the effect of this force is focusing for fleflw < 1. Such would be the case if, for example, heating due to rf power dissipation caused an appreciable expansion of the accelerating structure. The change in flw can then be calculated as

6fiw = - fl2(C/Vg)6t/t, where 6Ill is the relative expansion of the structure. If 200 kW of average power is dissipated in a section 3 m in length, an average temperature rise of about 2°C and a value of 61/l~ 3 x 10-5 can be expected for a copper structure with a wall thickness of 1 cm. If Vg/C is taken to be 0.003, then 6flw ~ - 0.01 from the preceding relation. Substituting flo ~ 1, flw = 0.99, A = 20 °, r = 0 . 5 cm and 2 = 2 4 cm into eq. (4), together with 67/y ~ ½ for the first orbit, we obtain (60)EM~ 1 x 10 -4.

The fringing fields at the entrance and exit of the accelerator section can be thought of as weak lenses with focal length 2yMo/(E o cos A), converging at the entrance and diverging at the exit7). The effect of both lenses produces a net convergence angle given by

(60)v r = ( -- ¼r/L) (3?/7) 2.

(5)

Assuming for the first orbit that 67/7 = ½, r = 0.5 cm, L = 330 cm, and cos A ~ 1, then (60)v v ~ - 1 x 10 -4 . The effect of ripple in the amplitude of the axial field can be considered as a series of weak converging doublets. Using a model where the axial field is given by E = E {1 + a sin (bz)}, the net convergence angle is that given by eq. (5) multiplied by ½a2. In typical traveling wave structures the amplitude of the nonuniformities due to higher order space harmonics is on the order of a ~ ½. Thus the convergence produced by this effect is an order of magnitude less than that given by eq. (5).

200

B.H.

WIlK

AND

P. B. W I L S O N

Here dis the length of the coupler in the axial direction, 2a is the diameter of the beam aperture, and 6E and 6q~ are the variations in amplitude and phase of the axial electric field component over the aperture. Couplers have been developed 7) in which the asymmetry has been reduced to 3E/E ~ 10- 3, 6q~ ~ 2 x 10 z. Assuming cos A ~ 1 and ld/a ~ 1, we find from the preceding relation that (60)c a ~ 1 x 10 -4. For a bunch of negligible phase spread the transverse deflection introduced by couple asymmetry on a given orbit can be cancelled out by a steering dipole. However, if the spread in the bunch is appreciable, electrons at different phases will receive differing deflections. The net effect is to couple the phase spread into a spread in the transverse momentum. The spread in 0 which is coupled into the transverse phase space from this cause is estimated to be about 10% of the total deflection, or 6 0 ~ 1 x 10 -s.

a gap width of 3 cm. Pole pieces rounded off with a radius of 1.25 cm, using also a clamp on the fringing field, gave K = 0 . 7 . The clamp was an iron window frame 13 cm high, 6 0 c m wide, and 7.5 cm long, with an aperture 2.5 cm high and 40 cm wide. The frame was spaced 3 cm (one gap width) away from the pole faces. The proposed microtron magnet has a gap width of 2.5 cm and a radius for the injection orbit of 9.2 cm. Using these values, together with 6 y = 0 . 2 cm and K = 0 . 7 in eq. (7), we obtain a divergence angle of 2 x 10 -3 tad. This divergence angle is an order of magnitude larger than the effects caused by the linac section. Therefore, in the turn by turn calculation of the beam extent and divergence, given in the following section, only the fringing field effects were taken into account. The preceding values of K were computed for an electron energy of 40 MeV and a magnetic field of 12 kG. If the electron energy is varied between 30 and 100 MeV, the same numerical value of K is obtained. Thus in the energy range of interest for this microtron, the defocusing introduced by the fringing field is correctly described, even on the first few turns, by the impulse approximation using the numerically determined value for K.

3.2. DEFOCUSING IN THE FRINGING FIELD OF THE

3.3. ORBIT BY ORBIT COMPUTATION OF THE BEAM OPTICS

Finally we consider the effects of coupler asymmetry. The main effect is to deflect the beam as a whole, in the direction of the maximum gradient of E= with respect to the transverse coordinate, by an angle v)

(riO)CA~ 'e~ '(,ST/~'){d;./(2aL)} [(,SE/E)sin a + a,bcosA]. (6)

BENDING MAGNETS

An electron entering or leaving normal to the pole faces of a homogeneous magnet experiences to first order no focusing or defocusing effects. To second order, however, the passage through the fringing field acts in the vertical plane as a defocusing lens. An electron traveling parallel to the median plane at a distance 6y above it will have a divergence angle (30)My after passing through the first fringing field. It is convenient to write this divergence angle in the form 8) (60)MV = (I 6yt/R)tg(½ KD/R).

(7)

Here D is the gap width, R the bending radius and K is a constant determined by the magnet geometry. To get a reliable estimate of the defocusing effect, the fringing field of a scaled-down model of the microtron magnet was measured for different shapes of the pole pieces. Using the measured field, the paths of electrons through the fringing field were computed step by step and the transfer matrix determined. This matrix was then equated with the transfer matrix as given by the impulse approximation. From this equation K can be determined. Pole pieces having a squareedge profile gave K = 1.0 for this magnet, which had

The small number of orbits in the microtron makes it feasible to calculate the beam properties through the whole accelerating cycle. We have done this turn by turn using the SLAC Beam Transport ProgramO). In this program the (6 dimensional) phase volume occupied by the beam is computed as the electrons pass through different beam elements. After each element the extent, divergence, and momentum spread, as well as the length of the electron bunch, is printed out. The assumed width + x, height _+y, and divergences _+ 0x and + Oy of the electron bunch at the entrance to the injection magnet are listed in the first line of table 2. The spread in momentum was assumed to be 0.5O/o. The extent and divergence of this bunch was then computed to second order through all orbits. Table 2 lists the computed beam properties at the exit of the linac section for each orbit. With the exception of the vertical defocusing, the correction due to second order effects is small. Fifteen quadrupole pairs were used, and their strength determined to keep the extent of the beam within 1 cm and as parallel as possible. In the computation the quadrupoles are assumed to be 20 cm long spaced 10 cm apart. The radius of the aperture is 2 cm. In table 2 the magnetic field strength at the pole face is listed. A positive sign implies focusing

A HIGH

ENI='RGY, H I G H

DUTY CYCLE, RACETRACK

201

MICROTRON

TABLE2 Properties of the electron beam in the microtron. Orbit nr.

E (MeV)

± x (cm)

_+0x (mrad)

+ y (cm)

_+0u (mrad)

Q1 (G)

Injected

33

0.10

0.10

0.10

0.10

154

-

0.21 0.18 0.20 0.23 0.24 0.24 0.32 0.30 0.32 0.35 0.46 0.27 0.20 0.30 0.35 0.38 0.47 0.43 0.34 0.26 0.23 0.24 0.21 0.18 0.16 0,17 0.15 0.13

0.37 0.45 0.11 0.41 0.59 0.23 0.31 0.15 0.14 0.18 0.17 0.26 0.25 0.29 0,16 0.15 0.17 0.14 0.13 0.13 0.24 0.05 0.05 0.04 0,04 0.10 0.11 0.10

0.07 0.I1 0.18 0.03 0.08 0.18 0.08 0.05 0.32 0.19 0.06 0.12 0.18 0.34 0.08 0.19 0.46 0.24 0.14 0.18 0.25 0.14 0.10 0.15 0.29 0.39 0.24 0.15

0.31 0.35 0.10 0.31 0.27 0.08 0.16 0.13 0.37 0.33 0.18 0.06 0.11 0.10 0.22 0.22 0.16 0.15 0.04 0.06 0.02 0.07 0.04 0.06 0.13 0.06 0.10 0.04

293 304 358 467 394 377 400

- 297 -314 -354 - 479 -410 -399 - 400

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

46.7 60.4 74.0 87.7 101.4 115.1 128.7 142.4 156.1 169.8 183.4 197.1 210.8 224.5 238.2 251.8 265.5 279.2 292.9 306.5 320.2 333.9 347.6 361.3 374.9 388.6 402.2 416.0

356

-

495

-539

-625

625

-697

722

-600

600

-668

660

Extracted

416.0

0.12

0.29

0.04

0.14

672

-636

i

2 3 4 5 6 7 8

in the horizontal plane. On a few orbits wedge focusing* is used to provide additional vertical focusing at the expense o f some horizontal defocusing. The calculated beam properties 10 m from the exit o f the m i c r o t r o n are also listed in the last line o f table 2. F u r t h e r c o m p u t a t i o n s show that a beam with a h o r i z o n t a l and vertical extent o f 0.5 cm, a divergence angle o f 0.5 mrad, and a m o m e n t u m spread o f 1°/ /o will be accepted, assuming some additional q u a d r u p o l e pairs are used. It is possible to obtain a beam meeting these requirements on phase space and m o m e n t u m spread f r o m an injector linac*°). If the b e a m passes t h r o u g h a q u a d r u p o l e pair off axis, in a d d i t i o n to being focused it will be slightly deflected. The bending angle is to first order equal to r / f where r is the a m o u n t by which the beam is off axis and f the focal length o f the q u a d r u p o l e pair. As the focal length typically is on the o r d e r o f several meters, the bending angle is quite small and can be corrected for by a steering coil located near the q u a d r u p o l e pair. A n o t h e r possibility would be to use q u a d r u p o l e s on

Q2 (G) 175

429

each orbit. The beam is then a u t o m a t i c a l l y steered toward the optical axis o f the q u a d r u p o l e system. This scheme is feasible if the focal lengths o f the q u a d r u p o l e s are c o m p a r a b l e to or longer than the distances between the quadrupoles. We conclude that a d e q u a t e focusing can be provided by the use o f q u a d r u p o l e s in the region between the magnets. Errors due to steering o f the q u a d r u p o l e s and inhomogeneities in the field o f the bending magnets can be corrected for each orbit individually if necessary. 4. Phase

stability

The resonant electron enters the linac section with a phase t ~00 and a total energy Wo. After c o m p l e t i n g n orbits, an electron injected with a phase and energy different f r o m the resonant co n d i t i o n will have a phase * In practice, in order to keep the bending magnet simple, it may be advantageous to use a single quadrupole located at the entrance or exit to the magnets instead of wedge focusing. -t The phase convention is such that the energy gain is maximum (Win) for a relativistic electron crossing the reference plane at the entrance to the linac section (z = 0) at phase q) = 0.

202

B. H. WIIK AND P. B. WILSON

deviation 69(n) and an energy deviation 6W(n) from the r e s o n a n t conditions. In m a k i n g the c o m p u t a t i o n it is c o n v e n i e n t to consider the phase and energy errors between two successive orbits of the same electron. These relations can immediately be written down to yield

J

% 8o td FZ

--

>-

,~9(,,) = ,~o(n - 1 ) + {2~,' /(WmCOS~O~)},~W(")+ + {2rc(S--L)/2} {(1 --,q,)//~,} +

70

kLl

z eo I,

(S)

o [{1

~ 5o

=

WmCOS 0J,

(9)

A~: -z,ld3

~8

~=0

Br ~

8r =2×

id3

40

g

w(,,) = (WIn/C) (10)

~ 30 w

Wm cos 9~ is the r e s o n a n t increase in energy per orbit, and c~q, is the electron velocity on the n TM orbit. The phase errors introduced by the straight sections are p r o p o r t i o n a l to S ( I - / ~ , ) / / ~ , . The q u a n t i t y bS~/[~, is a correction to the path length caused by the fringing field of the magnet, and was determined numerically from the measured profile of the fringing field. The phase slip resulting from s y n c h r o t r o n radiation from the electrons in the b e n d i n g magnets is negligible. In order to estimate the phase stable region as a f u n c t i o n of q)~ we rearrange the equations in the limit [], = 1, and assume also that the phase slip due to the fringing field is small. The result is

69(n + 1 ) -

AB

2 6 9 0 0 + 690, - 1) =

---- 2/tV{COS[q~r+6tp(n)]/COS¢Pr--l}.

(ll)

This equation is identical to the phase equation for a conventional microtron. The solution t~) for small oscillations is given by cos(Zg/A) = 1 -7~vtgq),,

(12)

where A is the period of one phase oscillation expressed in n u m b e r of turns. F r o m this expression it is seen that the range in r e s o n a n t phase over which stable phase oscillations are possible* is given by 0 < 9r < arctg {2/(gv)}.

(13)

F r o m eq. (13) we see that in order to insure the largest phase space v should be chosen equal to 1. I n this case we have stable orbits when the r e s o n a n t phase is chosen between 0 ° and 32.5 °. Thus we would expect the tolerances on the magnetic field to be o n the order of

ABIB ,~ 32.5°/{(//2)x

360°},

\\

~ 20 z~ ~o

-I.2 - I . 0 - 0 . 8 - G 6 - 0 . 4 - 0 . 2

0

0.2 0.4 0.6 0.8

1.0 1.2

DEVIATION FROM NOMINAL ENERGY IN MeV

Fig. 2. The energy spectrum after 28 orbits showing the effect of deviations of the magnetic field from the resonant value. The final energy is 416 MeV, the nominal resonant phase is 20° and v--1. Electrons are injected with a phase spread of 12°, an energy spread of 0.5~o and a total energy of 33 MeV. where l is the total path length in the magnets on the last orbit a n d 2 is the rf wavelength. In our case l/2~ 30 a n d thus AB/B should be on the order of _+2 × 10 -3 . The size of the phase stable region also indicates that the peak voltage in the linac section can vary by several percent without loss of electrons. To check this, taking into a c c o u n t the phase slip due to the fringing field and the straight sections, electrons were traced through all 28 orbits for different values of the injection parameters using eqs. (8)-(10). I n fig. 2 the energy spectrum after 28 orbits is given for electrons with a final energy of 416 MeV. The operating mode was v - - 1 and the magnetic field was varied by _ 2 x 10 -3 from the r e s o n a n t value. F o r this variation * At the upper limit of the phase stable region the period of the phase oscillation is two turns. The condition A = 2 can aso be expressed as 9(n + 1) = - 9(n) = 9(n - 1). This condition represents a phase oscillation of constant amplitude, but it is also apparent that it represents a limit of stability; a slight increase in the phase difference from turn to turn results in an oscillation of increasing amplitude. By substituting the preceding phase relation into eq. (l l), the upper limit on the phase stable region is obtained directly.

A HIGH ENERGY, HIGH DUTY CYCLE, RACETRACK MICROTRON :I O~u~LJI.O

2:

o 2~ D2 ud

b~ o (12

0.5

---- z

:

i t

l 0.96 0.98 1.00 1.02 1.04 1.06 1.08 RELATIVE PEAK ACCELERATING VOLTAGE V/V n

I,lO

Fig. 3. Number of electrons (normalized with respect to the number of injected electrons) with an energy deviation less than _+ 1.5 MeV from the mean energy as a function of voltage variations in the accelerating section. The energy spread in the injected electrons is 0.5%, and the phase spread is 13° centered around a resonant phase angle of 15°. The final energy is 416 MeV and v = 1. in the magnetic field none of the electrons are lost. The main effect of the small variation in the magnetic field is to shift the nominal resonant angle to compensate for the change in the field. Thus an increase in field corresponds to a nominal resonant angle closer to 0 ° and a correspondingly narrower spectrum. Since in practice B can be adjusted to give the right average value, local variations of B several times higher than 4-2 x 10 -3 , or on the order of one percent, can be tolerated. This result agrees with the calculations of Robinson12). Fig. 3 shows the n u m b e r of electrons lost during the accelerating cycle as a function of the relative peak accelerating voltage V/V., where V, is the nominal accelerating voltage. The magnetic field was chosen one part in a t h o u s a n d too high in order to provide the m a x i m u m tolerance against the energy variations occurring during the transient beam loading period. We see that the peak voltage, as anticipated, can vary considerably before any losses occur. In making the computation, the injection conditions were kept fixed. By varying the injection phase, the region o f m a x i m u m acceptance can be made considerably wider. The average time for an electron in the microtron to complete a full phase oscillation is a b o u t 0.3/~sec. As the rf filling time o f the structure is o f the order of 3 /~sec, any change in the peak voltage will be adiabatic as seen by the electron. Therefore, if the instantaneous peak voltage varies, the electrons will adjust their resonant phase angle to achieve on the average the energy gain as determined by the magnetic field. Hence we would

203

not expect variations of the peak voltage on the order of several percent to produce large changes in the shape of the spectrum nor to shift the mean energy. Fig. 4 shows the energy spectrum for values of the peak voltage near the limits of m a x i m u m acceptance as shown in fig. 3. To achieve higher resolution the resonant phase angle must be chosen closer to 0 °. In this case the tolerances on the magnetic field and on the allowable high voltage ripple becomes more severe. Fig. 5 shows the energy spectrum after 28 orbits for ~0r = 5 °. We see that 50% of the electrons are in an energy bin AE/E= +_10 -4. Finally we want to consider the behavior of the microtron at low final energies. To produce a low energy beam the injection energy, energy gain per turn and the magnetic field must be decreased according to eqs. (i) and (2). The injection phase and energy must also be changed from the resonant conditions in order to compensate for the large phase slip caused by the straight sections. The proper injection phase and energy are given by the condition that the electrons must approach the resonant value after many orbits. This is illustrated in fig. 6, where the phase deviations from the resonant value as a function of the orbit number is plotted for two electrons with a final energy of 83.4 MeV. The phase slip on the first few orbits due to the

J .~ ~ ~ >~ ~ > 3 ~

70 60 V ~nn = 1.06

50 40 50 20 io

o~ ~ ' ~ 60 ~ 50 ~ 40 ~ 30

-2.0-I.6-I.2-0.8-0.4

0

0.4 0.8

1.2 1.6 2.0

V

Vn = 1.00

O3

~ 2o io -2.0 - I . 6 - I . 2 - 0 . 8 - 0 . 4 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 DEVIATION FROM NOMINAL ENERGY IN MeV

Fig. 4. The energy spectrum after 28 orbits for two values of the ratio of the peak accelerating voltage to the nominal peak voltage. Other conditions are the same as for fig. 3.

204

B. H. W I I K

AND

P. B. W I L S O N

cycle are well separated in energy from the main peak of the spectrum. 5. R f system 5 . 1 . G E N E R A L DESCRIPTION >

=: 40 LLI

E:

LIJ

z >

g

3O

,5

,=, {L

=o u.I u.I

m z

I0

I

L

[

-0.6 -0,4 -0.2 DEVIATION

FROM

0

NOMINAL

0.2

I

I

0.4

0.6

ENERGY

IN

MeV

Fig. 5. T h e e n e r g y s p e c t r u m a f t e r 28 o r b i t s s h o w i n g the h i g h r e s o l u t i o n a t t a i n e d by c h o o s i n g a s m a l l r e s o n a n t p h a s e (~r = 5°). O t h e r c o n d i t i o n s a r e the s a m e as for the c a s e AB/Br = 0 in fig. 2.

straight sections is clearly seen. The width of the spectrum, however, is the same as for higher energies. The period of the phase oscillation obtained from the curves in fig. 6 is about 4.5 turns. This can be compared with a calculated value of A = 4.45 turns obtained from eq. (12) using v = 1 and ¢pr= 15 ° . The agreement is reasonable, even for the case of curve ll where the amplitude of the oscillation is relatively large. F r o m the phase calculations we can draw the following conclusions : 1. If the cavity voltage is stable to about 3% and the magnetic field integrated along any orbit deviates +_ 2 x 10 -3 or less from the resonant value, few electrons will be lost during the accelerating cycle. These requirements should easily be met in practice. 2. As the result of the phase focusing, the energy spectrum of the electrons extracted from the microtron is inherently narrow. For 28 orbits the width is of the order of 10 -3 or better for v = 1. Furthermore, those electrons which do get lost during the accelerating

A constant gradient traveling wave linac section is proposed for use as the accelerating cavity in the microtron. Four similar accelerator sections, with certain refinements to be discussed later, are to be used as an injector. Thus two high duty cycle, high power rf sources are required, one for the accelerating section between the magnets and one for the four injector sections. The design of the injector will be discussed in detail in a later section. Although a standing wave structure of the type proposed for the meson factory at Los Alamos 13) would have a somewhat higher shunt impedance, a traveling wave structure has several important advantages in this application. The filling time for the traveling wave structure (based on the time required to reach 95% of the final value of the field) is about a factor of three shorter. In addition, the traveling wave structure does not have a high initial reflected power during the filling time, and the reflections do not change as the beam current is changed. Because the power levels in the proposed rf system are very high, such reflections could be a serious problem if a standing wave cavity were to be used. In considering the design of a suitable traveling wave structure, several features are important. A constant gradient design is desirable since the power dissipation along the length of the section is then nearly uniform. A fairly low value of the attenuation parameter should be chosen for several reasons. First, beam S4~¢n)

;;

o

& I

-i(r

'-g 8 E ( o ) = - 0.195 MeV F[

_is ~

~E(o)=

-0.243MeV

INJECTION

Fig. 6. T h e d e v i a t i o n in p h a s e f r o m t h e r e s o n a n t v a l u e p l o t t e d as a f u n c t i o n o f o r b i t n u m b e r . T h e final e n e r g y is 83.4 M e V , t h e i n j e c t i o n e n e r g y is 6.6 M e V , t h e r e s o n a n t p h a s e is 15 ° a n d v = 1.

A HIGH ENERGY~ HIGH DUTY CYCLE, RACETRACK MICROTRON TABLE 3

(V o = Wo - 0 . 5 1 MeV) and V~ is the r e s o n a n t energy gain per turn. The energy gain per t u r n for a c o n s t a n t gradient structure of length L can be written as

Design of the accelerating sections. Length of sections Guide wavelength Shunt impedance per unit length Q factor Attenuation parameter Filling time Group velocity at z = 0 Group velocity at z = L Minimum disk hole diameter

L = 2 = r = Q = r = TF = vg = v¢ = 3.2 cm

3.3 m 23.9 cm 42 M.Q/m 20 000 0.57 2.9/tsec 0.0064 0.0020

Vr =

VmCOS(lgr-(~V

Vm =

( r L P o ) + (1 -

(15)

,

e- 2~)~,

6V = ½icrL[1-{2re-2~/(l

(16)

-e-2~)}],

(17)

where r is the s h u n t impedance per unit length, Po is the peak input power, i¢ is the peak circulating beam current, 6 V is the energy sag due to beam loading, and Vm is the energy gain for ic = 0, q~r = 0, and fie = 1. The peak power and peak circulating current in t u r n are related to the average power P~, average current i,, a n d duty cycle D by ic = N i a / D , (18)

loading effects are reduced considerably for a relatively small loss in energy gain. Second, lower z implies a higher group velocity and hence a less critical dependence of phase velocity on t u n i n g errors a n d on temperature variations d u r i n g operation. Third, a substantial fraction of the i n p u t power is dissipated in an external t e r m i n a t i o n rather than in the structure itself. Other advantages of a low r c o n s t a n t gradient design have been pointed out in c o n n e c t i o n with the design of the SLAC accelerator structure~4). When all of these factors are taken into account, a choice of = 0.57 provides a reasonable compromise. Once the value of the a t t e n u a t i o n parameter has been chosen, the other properties of the accelerating sections can be computed. Typical properties for a ~Tr-mode structure with r = 0.57 are given in table 3. The final o u t p u t energy of a microtron with N turns is given by

(19)

Po = q e ~ / o .

Here q is a factor which takes into account the fact that the efficiency of a pulsed klystron decreases as the duty cycle is increased. It is taken to be unity at maxim u m peak power and m i n i m u m duty cycle, and typically falls to a b o u t 0.5 when D is increased by a factor of 5. The detailed variation of the efficiency factor r/with duty cycle will depend on the particular choice which is made for the high power klystron. For this reason two specific klystrons have been chosen, from a m o n g m a n y available tubes, in order to present a more quantitative design for two representative rf systems. Using the k n o w n properties of these tubes, together with eqs. (14)-(19) and the parameters of the linac section listed in table 3, the results in table 4 are obtained. The design

(14)

Vf = V o + N V r

205

Here Vo is the kinetic energy of the injected electrons

TABLE 4

Design parameters for two rfsystems. Vt (MeV)

D (~)or

ia (~A)

Vr (MeV)

V0 (MeV)

2Vm (MeV)

P0 (MW)

Pa (kW)

Ph

(kW)

ie

6 I1"/(Vm " •COSq~r) (~)

(kG)

47 34 19 7

7 8 8 8

17.3 12.0 6.2 2.4

0 41 21 9

0 9 9 10

13.2 12.0 5.8 2.4

i (mA)

B

i System A 595 414 213 84

6 12 30 100

100 150 200 250

19.7 13.7 7.0 2.8

44.0 30.5 15.5

3 3 10 30

0 45 75 100

15.0 13.7 6.8 2.7

34.1 30.5 15.0 5.6

43.8 30.5 15.8

[

5.0 2.4 0.65

6L_L

300 285 195 100

60 60 40 20

75 75 62 30

0 18 15 8

i i

System B 454 414 206 81

31 31 15.5 6.2

2.5 2.5 0.6 0.1

206

B. H. W I I K AND P. B. WILSON

of the first system shown in the table is based on the largest available tube at this frequency, the Litton L-3401 klystron. This tube has a peak power output of 5.0 MW, an average power output of 300 kW, and a maximum pulse width of 550 #sec at this power output level. Using this system, the microtron would have a maximum final energy of about 600 MeV with a duty cycle of 6%, an average beam current of 100/~A, and an average beam power, Pb, of 60 kW. At 400 MeV the duty cycle increases to 13% with a somewhat higher average current. The L-3401 is also capable of being operated cw at a power output of 100 kW. At this power level the energy of the microtron would be about 85 MeV. It is seen that the efficiency of the klystron decreases by a factor of three in going from 6% duty cycle to cw operation. This decrease in efficiency with increasing duty cycle has been taken into account in computing the values listed in table 4. For operation at the maximum power level for rf system A in table 4, about 200 kW of average power would be dissipated in the accelerating section, or about 10 W/cm z over the outer cylindrical surface area of the structure. Admittedly there are engineering problems to be solved in designing the accelerating section to handle a heat flux of this magnitude. However, the design of the remainder of the rf system should be relatively straightforward, since systems incorporating the L-3401 klystron have been built and are being operated routinely. The design of an rf system with a more conservative average power and duty cycle is also given in table 3. This system provides a maximum output energy of somewhat over 400 MeV at 3% duty cycle, with an average current of about 50 #A. The required 2.5 M W of peak power and 75 kW of average power could be provided by, for example, an Eimac X780 klystron. Using this klystron, the maximum pulse length would be 2000/Jsec, and the maximum practical duty cycle about 30% at an output energy of 80 MeV. 5.2. BEAM LOADING

An important parameter in the operation of the microtron is the decrease in energy gain per turn in going from zero current to full current. The relative change in energy expressed as a percentage of the unloaded energy gain per turn is given in table 4. In compiling the table, the average current has been chosen such that the energy change due to beam loading lies in the range 7 to 10%. In the earlier discussion of phase stability, it was shown that by proper adjustment of the resonant phase angle and magnetic field the microtron can accommodate changes on the order of

6% in the energy gain provided by the accelerating section. By adjusting the timing of the injected current so that injection begins about a half a microsecond before filling of the rf accelerator section is complete, additional current can be accommodated corresponding to several percent more in the value of the beam loading parameter. Hence the currents shown in table 4 can probably be accelerated with no shaping of the rf input pulse. By proper shaping of the amplitude of the input rf pulse it should be possible to achieve appreciably higher currents. 5.3. BEAM BREAK-UP

The possibility of beam break-up in the proposed microtron should be considered. The type of beam break-up observed in multisection linacs [-e.g. at SLAC16)] should not be of concern here. Although the microtron has 28 turns, it is not equivalent to a 28 section machine because the transverse modulation at the break-up frequency is not coherent from turn to turn if the break-up frequency and operating frequency are not harmonically related. The regenerative type of beam break-up17), in which the section can act as a backward-wave oscillator, is of more concern. The largest circulating currents occur at the highest energies when the duty cycle is the lowest. For the cases listed in table 4, the largest peak circulating current is about 50 mA. This current can be compared with measured break-up currents for constant gradient sections having about the same length. These currents are typically several hundred milliampere or higher for long pulses. We conclude that there is an adequate safety factor against this type of beam break-up at the highest peak circulating currents for the microtron design considered here. 5.4. INJECTION

From eqs. (1) and (2) it is seen that the required kinetic energy for the electrons injected into the microtron is V0 = kVr-0.51 MeV, where k is a constant determined by the mode of operation and the magnet spacing. Values for 11Io, normalized to an injection energy Vo = 30.5 MeV at a resonant energy gain Vr = 13.67 MeV, are listed in table 4. If the power from the klystron supplying the injector were to be split equally among the four injector linac sections, then the maximum injection energy would be 2Vm, where Vm is given by eq. (16). In obtaining this result the effect of phase slip during capture and any

A HIGH ENERGY, HIGH DUTY CYCLE, RACETRACK MICROTRON additional voltage due to the electron source are neglected. Values of 2 Vm are listed in table 4. It is seen that 2V m is approximately equal to Vo, and therefore the capability of the rf system is well matched to the requirement on injection energy for the value of k which has been chosen in computing the examples. As the output energy of the microtron is decreased, it is highly desirable to take advantage of the lower peak power requirement by increasing the duty cycle. To do this most effectively the mode of operation must remain at v = 1, and this is assumed in the examples given in table 4. However, as the peak power input to the injector is decreased, the field strength in the first injector linac section may fall below the minimum value required to capture the electrons from the electron source. The field strength required for capture decreases with increasing source voltage and is lower for capture into a section with a phase velocity less than the velocity of light. Assuming for the moment that a capture section having fiw = I is used, then the minimum electric field strength, Ec, required for capture is given by 15)

eE~2/(mo c2) = n{(l -flo)/(1 +flo)} ½.

(20)

Here Cflo is the velocity of the electrons from the electron source. The field strength is given in terms of the power flow P and the geometrical properties of the structure, in particular the group velocity Vg, by

E = [2nP(r/Q)/{)t(Vg/C)}] -~.

(21)

If we first assume that the capture section is identical to the other accelerator sections, then eq. (21) indicates that two steps can be taken to increase the field strength in the capture region. First, the power flow in the initial section can be increased by putting half of the power from the injector klystron into the first

207

section, and splitting the remaining half equally among the other three sections. This increases the field in the capture section by a factor of~/2, while decreasing the total energy produced by the injector by only 3.5%. Second, the orientation of the first section can be reversed so as to put the low group velocity end at the front. This increases the field at the front end by a factor of e ~ (the ratio of group velocities at the input and output of a constant gradient section is e 2~) over the field for the same input power when the section has the normal orientation for constant gradient operation. The energy output of the reversed section is also slightly reduced (by 5% for r = 0.57), leading to a reduction in the energy of the whole injector by 1.8%. By combining both of these steps the field in the capture region is enhanced by a factor (~/2) e °'57, o r by a factor of 2.5, while the total loss in energy from the injector is only about 5%. The voltage required under these conditions from the electron source can be computed using eqs. (20) and (21), together with the parameters of the accelerator section listed in table 3. The results are given in the last column of table 5. In computing the required voltages, it has been assumed that the field strength in the capture section is 1.25 times the minimum value for capture. For this field strength the energy loss due to phase slip in the capture section will be about 7% of the total injection energy. It is seen that a 1.4 MeV electron source is required for capture at the lowest final energy where the duty cycle is greatest. Also shown in table 5 are the peak currents required at each duty cycle, and currents designated as six times the peak current. These last figures are the currents required from the electron source assuming that no prebunching is done and that only electrons in a 60 ° phase spread are accepted by the capture section. Elec-

TABLE 5

Voltage and current requirements for the electron source. Final energY (MeV)

T /

Duty cycle (%)

Average current (/~A)

Peak injected current Io (mA)

(mA)

Electron source voltage (kV)

6 12 30 100

1130 150 200 250

1.7 1.2 0.7 0.25

10 7.5 4.0 1.5

o 40 300 14oo

3 10 30

45 75 100

1.6 0.7 0.3

10 4.5 2.0

40 300 1400

6 Ip

System A 595 414 213 84 System B 414 206 81

208

B. H. W I I K

AND

trons injected over this phase range will be c a p t u r e d into a b o u t a l0 ° o u t p u t phase spread. A l t h o u g h a p r e b u n c h e r cavity could be used at injection voltages o f less than a b o u t 200 kV, a b o v e this voltage level the p o w e r r e q u i r e m e n t rapidly becomes prohibitive because o f the relativistic longitudinal stiffness o f the beam. If it is desirable to remove the electrons outside o f the 60 ° phase spread, this can be done by using a transverse deflection cavity to p h a s e - c h o p the beam. Such a cavity would require only a few h u n d r e d watt, even at the highest injection voltage. In s u m m a r y , from these c o n s i d e r a t i o n s it is seen that if the m i c r o t r o n is to be o p e r a t e d at high duty cycle in the 100 MeV energy range a high voltage electron source is required which has a cw o u t p u t current o f a b o u t 2 m A at 1.5 MeV, increasing to 10 m A (pulsed) at lower voltages. Van de G r a a f f generators, which can be a d a p t e d to meet these requirements, are c o m m e r cially available. An alternative solution to the c a p t u r e p r o b l e m for final energies below 200 MeV would be to accelerate the electrons to the 1.5 to 2 MeV energy range using a specially designed bunching and c a p t u r e section powered by a cw klystron delivering 50-75 kW. By tapering the phase velocity in several steps, a conventional 80 kV gun and p r e b u n c h e r a r r a n g e m e n t can be utilized. This alternative m e t h o d for accelerating to 1.5 MeV or a b o v e would p r o b a b l y be s o m e w h a t m o r e expensive than o b t a i n i n g a beam o f this energy directly from a high voltage generator.

6. Conclusion The calculated properties o f the split-magnet racetrack m i c r o t r o n c o m p a r e very f a v o r a b l y with the properties of other types o f electron accelerators in the 100 600 MeV energy range. In c o m p a r i s o n with an electron linac using a c o m p a r a b l e a m o u n t o f average rf power, the energy h o m o g e n e i t y and duty cycle o f the m i c r o t r o n are superior by an o r d e r o f magnitude, while the average current is c o m p a r a b l e . In addition, the m i c r o t r o n is c o n s i d e r a b l y m o r e c o m p a c t . C o m p a r e d with other circular accelerators, the m i c r o t r o n has the a d v a n t a g e o f high average c u r r e n t and the ease o f b e a m extraction associated with a linac. We have m a d e a rough estimate o f the cost o f the m a j o r c o m p o n e n t s for the p r o p o s e d m i c r o t r o n . Included in the estimate is the cost of magnets, quadrupoles, klystrons, m o d u l a t o r s , rf c o m p o n e n t s ,

P. B. W I L S O N

accelerating structures, v a c u u m c h a m b e r , injector, and an allowance for miscellaneous items. The cost o f these c o m p o n e n t s for the 400 M e V machine, using the lower p o w e r 3% duty cycle rf system, is estimated to be less than 1½ million dollars. F o r the 600 MeV microtron, using the higher duty cycle rf system, the cost of the bare machine is estimated to be a b o u t 2 million dollars. The cost o f housing, installation, and experimental facilities must be a d d e d to the preceding figures. We estimate that the cost for a c o m p l e t e d machine o f the design p r o p o s e d here would be a b o u t 20 to 30% o f the cost for an electron linac with c o m p a r a b l e energy and d u t y cycle. The a u t h o r s wish to t h a n k Professor W. C. Barber for c o n t i n u e d s u p p o r t and e n c o u r a g e m e n t d u r i n g this study.

References 1) A. Roberts, Ann. Phys. 4 (I 958) 115. 2) E. M. Moroz, Sov. J. At. Energy 4 (1958) 323. a) A. Paulin, Nucl. Instr. and Meth. 12 (1961) 155. 4) E. Brannen and H. Froelich, J. App[. Phys. 32 (1961) 1179. 5) B. H. Wiik, H. A. Schwettman and P. B. Wilson, Proc. Fifth Int. Conf. High Energy Accelerators (Frascati, Italy, Sept. 1965) Nat. Comm. for Nucl. Energy (Rome, 1966) 686. 6) K. T. Bainbridge, Experimental Nuclear Physics 1 (ed. E. Segre; Wiley, New York, 1953) p. 582. 7) G. A. Loew and R. H. Helm, IEEE Trans. Nucl. Sci. NS-12, no. 3 (1965) 580. 8) K. L. Brown, private communication. 9) C. H. Moore, S. K. Howry and H. S. Butler, SLAC Beam Transport Program, Stanford Lin. Accel. Center, Stanford, California. 10) R. H. Miller, R. F. Koontz and D. D. Tsang, IEEE Trans. Nucl. Sci. NS-12, no. 3 (1965) 804. 11) C. Henderson, F. F. Heymann and R. E. Jennings, Proc. Phys. Soc. (London) B66 (1953) 41. 12) C. S. Robinson, Technical Report no. 156, Physics Research Laboratory, Univ. of Illinois, Urbana, Ill. (1966). 13) H. C. Hoyt, D. D. Simmonds and W. F. Rich, Rev. Sci. Instr. 37 (1966) 755. 14) R. P. Borghi, A. L. Eldredge, G. A. Loew and R. B. Neal, Advances in Microwaves 1 (ed. L. Young; Academic Press, New York, 1966) p. 29. 15) M. Chodorow, E. L. Ginzton, W. W. Hansen, R. L. Kyhl, R. B. Neal and W. K. H. Panofsky, Rev. Sci. Instr. 26 (1955) 181. 16) O. H. Altenmueller, E. V. Farinholt, Z. D. Farkas, W. B. Herrmannsfeldt, H. A. Hogg, R. F. Koontz, C. J. Kruse, G. A. Loew and R. H. Miller, Proc. 1966 Linear Accel. Conf., Report no. LA-3609, Los Alamos Sci. Lab., Los Alamos, New Mexico (Dec. 1966) 267. 17) T. R. Jarvis, G. Saxon and M. C. Crowley-Milling, Proc. IEE 112 (1965) 1795.