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Radio-frequency linear accelerators for ion implanters
218
Nuclear Instruments and Methods in Physics Research B21 (1987) 218-223 North-Holland, Amsterdam
Section IV. N e w equipment and systems
R A D I O - F R E Q U E N C Y LINEAR ACCELERATORS FOR ION I M P L A N T E R S H.F. GLAVISH GMW Associates, 1060 Lakeview Way, Redwood City, California 94062, USA
The principles of radio-frequency acceleration are reviewed including a discussion of the flexibility derived from independent phase and amplitude control, and the use of radio-frequency quadrupoles to accelerate beams of very high current.
1. Introduction
2. Principle of the d linear accelerator
In recent years high energy, deep implant studies on silicon have uncovered a number of promising new applications [1-4], including the elimination of epitaxial layers in the manufacturing process, late modification of device characteristics, and improved device performance in high density situations. Implanters with substantial beam currents are needed in order to implement these new processes economically in semiconductor production. Hitherto, the applications have been addressed only with high voltage dc implanters having very low output beam currents. Radio-frequency linear acceleration is an alternate technique, capable of much higher beam currents. The radio-frequency (rf) linear accelerator, unlike a dc machine, is a resonance device in which the electric component of a time varying rf electromagnetic field acts on a charged particle to produce acceleration. Although proposed in the 1920s [5-7], technical difficulties impeded development until the new radar technologies emerged in the post-war years. In 1946-47 a 32 MeV proton linear accelerator was built at Berkeley, California under the supervision of Alvarez [8]. Other machines quickly followed, experience grew rapidly, and the techniques and understanding are now quite mature. Two recent developments are significant in the application to high energy, high current ion implanters. One is the idea of using independent phase and amplitude control (see section 4) in a sequence of rf accelerating gaps, to achieve universality in accelerating a wide variety of different particles over a broad, continuously variable energy range. In this regard, an rf accelerator can offer at least as much flexibility as a dc machine. The other new development is the radio-frequency quadrnpole accelerator (see section 5) capable of accelerating beams with currents even as high as 100 mA.
Perhaps the easiest representative linac to understand is the Sloan and Lawrence machine [7] developed in 1931 and shown in fig. 1. Barrett [9] reported on a pilot proton machine of this type at the 1982 Ion Implantation Conference. The accelerating electric field is localized in the gaps between a series of drift tubes. Alternate drift tubes are linked together, one set being driven by an rf voltage of tens of kilovolts, and the other set tied to ground. If a positive charged particle arrives at the first gap when the rf voltage is negative, it is accelerated across the gap and passes into the field free region of the first drift tube. The length of this drift tube is such that the rf phase advances by 180 ° in the time it takes for the particle to cross the first gap and drift through the tube to the second gap. The tube shields the particle from the electric field as the electric field is changing in phase. At the second gap the first drift tube has assumed a positive voltage, and acceleration again occurs, this time to the second, grounded drift tube. The process is repeated at each gap thus producing a resonant acceleration. Since the particle velocity increases along the accelerator path, there is a corresponding increase in the length of the drift tubes, as shown schematically in fig. 1. A linac such as the one just described has a fixed velocity profile for a given operating frequency. Thus for a specific particle charge to mass ratio, there is a unique injection energy, a unique final energy, and a unique rf field amplitude. Other possibilities do not exist except at much lower velocity profiles corresponding to 1/3, 1/5, 1 / 7 etc. harmonic timing conditions, generally of little interest in a practical linac. When a dc beam is injected into a linac continuously, particles which do not have the correct phase relation relative to the rf accelerating field are quickly lost. Particles that do have the correct phase relation
0168-583X/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
219
H.F. Glavish / R f linear accelerators Vcol~>t
phase stability [12,13] but with a more limited phase acceptance.
x/x°
3. Beam dynamics Fig. 1. The Sloan and Lawrence drift tube linac. become grouped or bunched and emerge as a continuous stream of micropulses separated by a distance d = o / f , where o is the final particle velocity and f is the frequency of the applied voltage. For a linac to work at all, the correct timing for resonant acceleration is a necessary condition. For it to work well: (1) The phase acceptance must be large enough to obtain satisfactory beam currents and prevent the beam pulses from expanding longitudinally and debunching as they travel along the linac axis; (2) The transverse (radial) acceptance must be large enough to obtain satisfactory beam currents, with sufficient radial focusing to keep the beam pulses collimated as they travel along the linac. The first condition, known as phase stability, occurs if particles cross an accelerating gap when the electric field is increasing in magnitude, for then late particles receive more energy and early particles less energy, enabling the late particles to catch up with the early ones. In axial symmetric structures phase stability simultaneously leads to radial defocusing, a result known as the incompatibility or MacMillan theorem [10]. As shown in fig. 2, the electric field produces radial focusing in the first half of the gap and radial defocusing in the second half. An increasing electric field therefore generates a defocusing impulse which is larger than the focusing impulse with the net result of radial defocusing. Radial focusing may be restored by including discrete focusing elements along the beam path such as electrostatic quadrupoles, magnetic quadrupoles or solenoids; or by using nonaxial symmetric gaps as reviewed by Boussard [11] and as in the rf quadrupole (see section 5). Finally, phase changes may be introduced so that some gaps are focusing and some are defocusing. This can produce simultaneous radial and
A typical beam micropulse contains many particles in motion relative to each other under the influence of forces associated with the time varying electromagnetic field in the accelerator gaps and the particle charges (space charge). Phase stability implies particles in the bunch must experience a restoring force acting more or less towards the center of the bunch as illustrated in fig. 3. The motion may be viewed as generated by a potential energy well, just as the harmonic oscillator well produces simple harmonic motion (slam). A more detailed, quantitative description of the principles of rf linear acceleration is given by Vlasov [14]. 3.1. Simple harmonic motion
Simple harmonic motion (shin) is characterized by a linear restoring force F acting towards a central reference point: (1)
F = mYc = - k x ,
where k is the force constant and x is the displacement from the center. The motion may be thought of as generated from an oscillator potential energy well (see fig. 4): (2)
V = ½kx 2,
and F = - d U / d x = - k x as in eq. (1). The particle velocity o is found by integrating eq. (1). On a o versus x diagram the particle motions describe a family of ellipses as shown in fig. 4: /,/,IV 2
X2
÷
= 1.
(31
general particle F synchronousreference
RADIAL FOCUSI
IAL DEFOCUSING
Fig. 2. Focusing effect of an electric field in an accelerating gap.
Fig. 3. In a beam micropulse particles are in motion relative to each other as if they experienced a restoring force F(x). IV. NEW EQUIPMENT AND SYSTEMS
220
H.F. Glavish / Rf linear accelerators
PotentialEnerQy u -- ~ k x 2 ,
//f
--x... Xo X
~
~Velocity v f~~ v~/~ x. s
Fig. 5. Potential energy well for a linac, showing a stable region between A and B.
FJi.
Fig. 4. An oscillator potential energy well (upper) generates simple harmonic motion described as ellipses (lower) in a velocity versus displacement diagram.
3.2. Phase motion in a linac
If the gaps along the linac are spaced at a distance L, the force on the particle is the gradient of the differential kinetic energy ( d W - d W ~ ) / L . If the motion is viewed as resulting from a potential U, then aU ax
It is customary when considering a linac to describe the motion relative to a so-called synchronous particle, more or less at the center of the bunch (see fig. 3), perfectly synchronized to the rf field at each gap, and crossing the center of a gap when the electric rf field is at a phase q~s: E=E0cos@s.
(4)
If the particle charge is q, the applied voltage on the gap V, and the effective length of the gap g, the energy increment imparted to the synchronous particle is d W s = q E o g T c o s Os = qVTcos ~ ,
(5)
where T is the transit time factor introduced by Panofsky [15]. It is a number that is always _< 1 and accounts for the phase change occurring in the field in the finite time it takes for the particle to cross the gap. If the angular frequency of the rf field is ~, then a particle at a distance x from the synchronous particle, crosses the center of the gap at a phase q~= q~ + q,,
O(W-W~) -
(dW
dW~)
-L
az
.
(9)
The explicit form of U is found by substituting eqs. (5), (7) and (8) into eq. (9) and integrating: U=
q VTvs Lto ( ~ c o s O ~ - s i n ( ¢ ~ + ~ ) + s i n ¢ ~ } .
(10)
As shown in fig. 5: (1) A potential well exists around the synchronous particle at ~ = 0 if and only if the synchronous phase angle ~s lies between 0 ° and 90°; (2) At the right the well extends to -= ~#s and at the left to - 2q~s. The total phase width of the well is thus 3q~; (3) Unlike slam, particles can only have a certain maximum kinetic energy relative to the synchronous particle, and still be bound. By analogy to the previous discussion on shm the motion may be described in terms of a velocity versus displacement diagram, or as is customary for a linac, a relative kinetic energy W - IV, versus phase diagram, using the linear relationship Ox Oz
v - v~ _= v~
W-W~
(11)
2W~
(6) and eq. (7). Thus
where
q, = -x. v~
(7)
The energy increment imparted to this general particle is: d W = qVTcos (q~s + ¢).
(8)
a( w - w~) aq,
a( w - w~) az a~ az
0x 0¢
au 2w~ a¢ w - w ~
(12)
Integrating the last result produces the curves of fig. 6, called separatrices [14]. For small ¢ a separatrix is a
H.F. Glavish/ Rf linearaccelerators
Energy Deviation (W -W s )
JJ /Ph .
r U /without space charge ~/ //~with spacecl'mrge
slightly distorted ellipse, while at the limit of stability it is the shape of a fish. During acceleration in a linac a particle moves along a separatrix. An initial phase spread is transformed into an energy spread: the larger the phase acceptance the larger the energy spread. When the synchronous phase angle ~ increases towards 90 ° the phase stability and phase acceptance increase but the energy gain decreases, as shown by eq. (5). Nevertheless, a linac may simultaneously have a large phase acceptance and a high energy gain. For example, when e~s = 30 ° the phase acceptance is A~ = 90 °, or 25% of a dc beam. But since cos ¢s = 0.866 this fraction of the peak electric field is still available for acceleration. The overall transmission through a linac is further improved by bunching (¢s = 90 °) a dc beam prior to injection into the acceleration cavities. Also, the intrinsic energy spread in a linac may be reduced by debunching (~,~ = - 90 °) after acceleration.
3.3. Space charge effects Difficulty is always encountered when space charge effects are analyzed because the motion of a particle is no longer independent of the presence of other particles. The description is simplified and still quite realistic if the particle distribution within a bunch is assumed to be uniform and the bunch itself of ellipsoidal shape. The effect of the aggregate space charge forces on the longitudinal or phase motion can then be represented by a simple potential energy function of the form: =
,I
[',
9~
Fig. 6. Separatrices describing the longitudinal motion of a particle in a linac beam micropulse. The solid curves correspond to stable motion in the re#on A to B of the linac well of fig. 5. The broken curve represents unstable motion.
U~ -~kcx-,
221
(13)
where k c is dependent on the physical dimensions of the ellipsoid, but independent of the individual particle coordinates. In the case of shin, provided k > k c, the particles are still bound but have a larger amplitude of oscillation. In the case of a linac the effect of space charge is more devastating since the well extends a long
\\
Fig. 7. Space charge forces reduce the size of the well (upper) and the area of the limiting stable separatrix (lower).
way in the direction of negative ~,. Both the phase acceptance and the energy spread are greatly reduced as shown in fig. 7. Diabolically, as the number of particles increases with increasing beam current, the size of the phase space they are allowed to occupy decreases. At some limiting beam current instability ensues. This simplified model predicts a space charge limited beam current of
3rTsinths[Ws(keV)] 1/2 Im~x (mA) ---
4L
A (ainu-------)-
× [ ~ ] 2 V
(kV)
(14)
corresponding to a limiting stable separatrix that is 2/3 the size of the case of no space charge. In the above equation, r is the geometric mean of the transverse semiaxes of the ellipsoid. The current is most seriously limited at low energies, i.e. at the beginning of the linac. It varies with the square root of the particle mass.
4. Independent phase and amplitude control As discussed in the introduction a great deal of flexibility is possible if the sequence of accelerating gaps are independently controlled in phase and amplitude. Phase control always enables the synchronous timing condition to be satisfied from one cavity to the next regardless of particle velocity. Amplitude control, including turning the rf power off in a cavity, or block of cavities, enables the energy to be varied continuously. Operation in a variety of novel ways is possible including: alternating the sign of the synchronous phase angle to provide simultaneous phase and radial focusing (see IV. NEW EQUIPMENT AND SYSTEMS
222
H.F. Glavish / Rf linear accelerators
section 2); tailored bunching at the beginning of the linac to maximize the capture effidiency of an injected dc beam; tailored debunching at the end of the linac to minimize the final energy spread. High rf power dissipation has in the past rendered independent phase and amplitude control somewhat impractical. However, this is not a limitation in the case of superconducting cavities where the scheme was first used [16]. Nor is it a limitation in ion implanters, where the energies of a few MeV, and the frequencies of a few MHz, are extremely low compared with the much higher energies and frequencies typical of linac applications in nuclear physics. For instance, the Eaton NV1000 [17] produces 1 MeV singly charged beams with a total rf power dissipation in the entire ten cavities of only 20 kW. Class C solid state power amplifiers provide efficient drivers and account for only a further 18 kW of power dissipation. The Eaton series of high energy implanters [17] illustrates the utility of independent phase and amplitude control. A single machine accelerates all particles from boron to antimony, singly or doubly charged, through any energy from zero to a maximum value dependent only on the number of rf cavities used. High beam currents are possible. For example, in the case of phosphorus, eq. (14) suggests a space charge limited current of 2.8 mA. In practice a beam current of 1.5 mA has been achieved without observing significant degrading effects from space charge.
5. Rf quadrupole linacs Even now, the need for high energy very high beam currents is forseeable. One example is oxygen ion implantation [18]. Another is metals implantation. Great flexibility with regard to particle species and energy variability is not always necessary in these applications, in which case an immediate possibility is the radiofrequency accelerator (rfq) proposed by Kapchinskii and Teplyakov [19] in 1970, and first implemented in this country at Los Alamos Scientific Laboratory in 1980 [20-22]. The rfq is capable of capturing a very high percentage of an injected dc beam (at least 80%) and accelerating very high beam currents, even as much as 100 mA. The principle of an rfq is shown in fig. 8. An rf rather than a dc voltage is applied to a quadrupole comprised of 4 symmetrically arranged electrodes. At an instant in time, at a certain position along the beam path, the voltage distribution may be such as to generate radial focusing in the vertical direction and radial defocusing in the horizontal direction. After a time equal to one half of an rf period, the voltages become reversed. The drifting beam thus sees the structure as a
V = Vo c o s w t
D-I-G D Fig. 8. An rf quadrupole generates strong radial forces (upper) and also axial electric fields if the electrodes are corrugated (lower).
sequence of converging and diverging ion optical lenses. Such alternating forces can provide strong simultaneous focusing in both the vertical and horizontal directions [23]. Furthermore, if the electrodes are appropriately contoured along the beam path, then in addition to strong radial focusing, axial acceleration also occurs. The contours have the form of corrugations in the electrode surface nearest the beam axis. At certain points along the beam path the tips of the horizontal pair of electrodes are closer to the axis than the tips of the vertical pair of electrodes. At in-between points the opposite is true, the tips of the vertical electrodes now being closer to the axis. Such a structure produces an axial component of electric field alternating in direction. If the spatial pitch of the corrugations corresponds to the distance the synchronous particle moves in an rf period, resonant acceleration results. The ability of an rfq to capture and accelerate large beam currents is immediately evident from eq. (14). Compared with a drift tube linac, an rfq typically operates at a higher frequency, and therefore the cell length L is smaller. Also, the initial capture and acceleration occurs adiabatically with large synchronous phase angles near 90 ° . Both these features, as shown by eq. (14), greatly increase the space charge limited beam current. Simultaneously, by the very nature of the structure, there are very strong radial focusing forces to keep the beam well collimated under the action of the strong transverse space charge forces. At this conference, DiBitonto et al. [24] report on an rfq designed to deliver beams of 35 mA for metals
H.F. Glavish / Rf linear accelerators
implantation. V a r i a n / E x t r i o n are investigating [25] the rfq structure for a high energy semiconductor implanter. Hitachi Ltd. [26] have also proposed such a machine, b u t of limited flexibility. Operating parameters for the two last mentioned machines are not in the public d o m a i n at this time. The rfq is similar to a conventional linac in the sense that the velocity profile is fixed (see section 2). Universality with respect to particle type and energy variability requires a convenient method to be developed for varying the resonant frequency over a wide range, perhaps 3-1. However, the rfq is a natural choice in a dedicated application such as an oxygen implanter where the particl~ species is fixed and the final energy range is narrow. We wish to acknowledge that much of the work on rf accelerators carried out by G M W Associates has been made possible through support from Eaton Corporation.
[10] [11] [12] [13] [14]
[15] [16] [17]
[18]
[19] [20] References [21] [1] J.F. Ziegler, Nucl. Instr. and Meth. B6 (1985) 270. [2] N.W. Cheung, Proc. SPIE 530 (1985) 2. [3] M.I. Current, R.A. Martin, K. Doganis and R.H. Bruce, Semiconductor International (June 1985) 106. [4] C. McKenna, C. Russo, B. Pedersen, D. Downey and R. Liebert, Semiconductor International (April 1986). [5] G. Ising, Arkiv Mat. Astron. och Fys. 18 (1925) 1. [6] R. Wideroe, Arch. Electrotechuik 21 (1928) 387. [7] E.O. Lawrence and'D.H. Sloan, Proc. Nat. Acad. Sci. 17 (1931) 64. [8] L.W. Alvarez et al., Rev. Sci. Instr. 26 (1955) 111. [9] N.J. Barrett, Ion Implantation: 4th Int. Conf. on
[22] [23] [24]
[25] [26]
223
Equipment and Techniques, eds., H. Ryssel and H. Glawischnig (Springer, Berlin, 1982) p. 45. E.M. McMillan, Phys. Rev. 80 (1950) 493. D. Boussard, Focusing of Charged Particles, Vol. II, ed., A. Septier (Academic Press, New York, 1967) p. 327. I.B. Fainberg, Proc. CERN Symp. High Energy Accelerator Pion Phys., Geneva (1956). L.B. Mullet, Atomic Energy Res. Estab. Harwell GP/M147, GP/M149 (1953). A.D. Vlasov, Theory of Linear Accelerators, Translation by Z. Lerman (1968) U.S. Department of Commerce, AEC-tr-6718. W.K.H. Panofsky (Univ. Calif. Rad. Lab. Berkeley 1951) UCRL 1216. P.H. Ceperley, J.S. Sokolowski, I. Ben-Zvi, H.F. Glavish and S.S. Hanna, Nucl. Instr. and Meth. 36 (1976) 421. H.F. Glavish, D. Bernhardt, P. Boisseau, B. Libby, G. Simcox and A.S. Denholm, these Proceedings (Ion Implantation Technology, Berkeley, 1986) Nucl. Instr. and Meth. B21 (1987) 264. J.P. Ruffell, D.H. Douglas-Hamilton, R.E. Kaim and K. Izumi, these Proceedings (Implantation Technology, Berkeley, 1986) Nucl. Instr. and Meth. B21 (1987) 229. I.M. Kapchinskii and V.A. Teplyakov, Prib. Tekh. E.ksp. 2 (1970) 19. R.H. Stokes, K.R. Crandall, J.E. Stovall and D.A. Swenson, IEEE Trans. Nucl. Sci. NS-26 (1979) 3469. J.M. Potter, S.W. Williams, F.J. Humphry and G.W. Rodenz, IEEE Trans. Nucl. Sci. NS-26 (1979) 3745. R.W. Hamm et al., Proc. Int. Conf. on Low Energy Ion Beams (University of Bath, April 1980) p. 2. E.D. Courant, M.S. Livingston and H.S. Snyder, Phys. Rev. 88 (1952) 1190. D. DiBitonto, R. Huson, P. Mclntyre, A. Nassiri, D. Raparia and C. Swenson, these Proceedings (Ion Implantation Technology, Berkeley, 1986) Nucl. Instr. and Meth. B21 (1987) 155. C. McKenna, private communication and ref. [4]. N. Sakudo et al., Hitachi Ltd. European Patent Application No. 84904176.9, Publication No. 0163745.
IV. NEW EQUIPMENT AND SYSTEMS
Nuclear Instruments and Methods in Physics Research B21 (1987) 218-223 North-Holland, Amsterdam
Section IV. N e w equipment and systems
R A D I O - F R E Q U E N C Y LINEAR ACCELERATORS FOR ION I M P L A N T E R S H.F. GLAVISH GMW Associates, 1060 Lakeview Way, Redwood City, California 94062, USA
The principles of radio-frequency acceleration are reviewed including a discussion of the flexibility derived from independent phase and amplitude control, and the use of radio-frequency quadrupoles to accelerate beams of very high current.
1. Introduction
2. Principle of the d linear accelerator
In recent years high energy, deep implant studies on silicon have uncovered a number of promising new applications [1-4], including the elimination of epitaxial layers in the manufacturing process, late modification of device characteristics, and improved device performance in high density situations. Implanters with substantial beam currents are needed in order to implement these new processes economically in semiconductor production. Hitherto, the applications have been addressed only with high voltage dc implanters having very low output beam currents. Radio-frequency linear acceleration is an alternate technique, capable of much higher beam currents. The radio-frequency (rf) linear accelerator, unlike a dc machine, is a resonance device in which the electric component of a time varying rf electromagnetic field acts on a charged particle to produce acceleration. Although proposed in the 1920s [5-7], technical difficulties impeded development until the new radar technologies emerged in the post-war years. In 1946-47 a 32 MeV proton linear accelerator was built at Berkeley, California under the supervision of Alvarez [8]. Other machines quickly followed, experience grew rapidly, and the techniques and understanding are now quite mature. Two recent developments are significant in the application to high energy, high current ion implanters. One is the idea of using independent phase and amplitude control (see section 4) in a sequence of rf accelerating gaps, to achieve universality in accelerating a wide variety of different particles over a broad, continuously variable energy range. In this regard, an rf accelerator can offer at least as much flexibility as a dc machine. The other new development is the radio-frequency quadrnpole accelerator (see section 5) capable of accelerating beams with currents even as high as 100 mA.
Perhaps the easiest representative linac to understand is the Sloan and Lawrence machine [7] developed in 1931 and shown in fig. 1. Barrett [9] reported on a pilot proton machine of this type at the 1982 Ion Implantation Conference. The accelerating electric field is localized in the gaps between a series of drift tubes. Alternate drift tubes are linked together, one set being driven by an rf voltage of tens of kilovolts, and the other set tied to ground. If a positive charged particle arrives at the first gap when the rf voltage is negative, it is accelerated across the gap and passes into the field free region of the first drift tube. The length of this drift tube is such that the rf phase advances by 180 ° in the time it takes for the particle to cross the first gap and drift through the tube to the second gap. The tube shields the particle from the electric field as the electric field is changing in phase. At the second gap the first drift tube has assumed a positive voltage, and acceleration again occurs, this time to the second, grounded drift tube. The process is repeated at each gap thus producing a resonant acceleration. Since the particle velocity increases along the accelerator path, there is a corresponding increase in the length of the drift tubes, as shown schematically in fig. 1. A linac such as the one just described has a fixed velocity profile for a given operating frequency. Thus for a specific particle charge to mass ratio, there is a unique injection energy, a unique final energy, and a unique rf field amplitude. Other possibilities do not exist except at much lower velocity profiles corresponding to 1/3, 1/5, 1 / 7 etc. harmonic timing conditions, generally of little interest in a practical linac. When a dc beam is injected into a linac continuously, particles which do not have the correct phase relation relative to the rf accelerating field are quickly lost. Particles that do have the correct phase relation
0168-583X/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
219
H.F. Glavish / R f linear accelerators Vcol~>t
phase stability [12,13] but with a more limited phase acceptance.
x/x°
3. Beam dynamics Fig. 1. The Sloan and Lawrence drift tube linac. become grouped or bunched and emerge as a continuous stream of micropulses separated by a distance d = o / f , where o is the final particle velocity and f is the frequency of the applied voltage. For a linac to work at all, the correct timing for resonant acceleration is a necessary condition. For it to work well: (1) The phase acceptance must be large enough to obtain satisfactory beam currents and prevent the beam pulses from expanding longitudinally and debunching as they travel along the linac axis; (2) The transverse (radial) acceptance must be large enough to obtain satisfactory beam currents, with sufficient radial focusing to keep the beam pulses collimated as they travel along the linac. The first condition, known as phase stability, occurs if particles cross an accelerating gap when the electric field is increasing in magnitude, for then late particles receive more energy and early particles less energy, enabling the late particles to catch up with the early ones. In axial symmetric structures phase stability simultaneously leads to radial defocusing, a result known as the incompatibility or MacMillan theorem [10]. As shown in fig. 2, the electric field produces radial focusing in the first half of the gap and radial defocusing in the second half. An increasing electric field therefore generates a defocusing impulse which is larger than the focusing impulse with the net result of radial defocusing. Radial focusing may be restored by including discrete focusing elements along the beam path such as electrostatic quadrupoles, magnetic quadrupoles or solenoids; or by using nonaxial symmetric gaps as reviewed by Boussard [11] and as in the rf quadrupole (see section 5). Finally, phase changes may be introduced so that some gaps are focusing and some are defocusing. This can produce simultaneous radial and
A typical beam micropulse contains many particles in motion relative to each other under the influence of forces associated with the time varying electromagnetic field in the accelerator gaps and the particle charges (space charge). Phase stability implies particles in the bunch must experience a restoring force acting more or less towards the center of the bunch as illustrated in fig. 3. The motion may be viewed as generated by a potential energy well, just as the harmonic oscillator well produces simple harmonic motion (slam). A more detailed, quantitative description of the principles of rf linear acceleration is given by Vlasov [14]. 3.1. Simple harmonic motion
Simple harmonic motion (shin) is characterized by a linear restoring force F acting towards a central reference point: (1)
F = mYc = - k x ,
where k is the force constant and x is the displacement from the center. The motion may be thought of as generated from an oscillator potential energy well (see fig. 4): (2)
V = ½kx 2,
and F = - d U / d x = - k x as in eq. (1). The particle velocity o is found by integrating eq. (1). On a o versus x diagram the particle motions describe a family of ellipses as shown in fig. 4: /,/,IV 2
X2
÷
= 1.
(31
general particle F synchronousreference
RADIAL FOCUSI
IAL DEFOCUSING
Fig. 2. Focusing effect of an electric field in an accelerating gap.
Fig. 3. In a beam micropulse particles are in motion relative to each other as if they experienced a restoring force F(x). IV. NEW EQUIPMENT AND SYSTEMS
220
H.F. Glavish / Rf linear accelerators
PotentialEnerQy u -- ~ k x 2 ,
//f
--x... Xo X
~
~Velocity v f~~ v~/~ x. s
Fig. 5. Potential energy well for a linac, showing a stable region between A and B.
FJi.
Fig. 4. An oscillator potential energy well (upper) generates simple harmonic motion described as ellipses (lower) in a velocity versus displacement diagram.
3.2. Phase motion in a linac
If the gaps along the linac are spaced at a distance L, the force on the particle is the gradient of the differential kinetic energy ( d W - d W ~ ) / L . If the motion is viewed as resulting from a potential U, then aU ax
It is customary when considering a linac to describe the motion relative to a so-called synchronous particle, more or less at the center of the bunch (see fig. 3), perfectly synchronized to the rf field at each gap, and crossing the center of a gap when the electric rf field is at a phase q~s: E=E0cos@s.
(4)
If the particle charge is q, the applied voltage on the gap V, and the effective length of the gap g, the energy increment imparted to the synchronous particle is d W s = q E o g T c o s Os = qVTcos ~ ,
(5)
where T is the transit time factor introduced by Panofsky [15]. It is a number that is always _< 1 and accounts for the phase change occurring in the field in the finite time it takes for the particle to cross the gap. If the angular frequency of the rf field is ~, then a particle at a distance x from the synchronous particle, crosses the center of the gap at a phase q~= q~ + q,,
O(W-W~) -
(dW
dW~)
-L
az
.
(9)
The explicit form of U is found by substituting eqs. (5), (7) and (8) into eq. (9) and integrating: U=
q VTvs Lto ( ~ c o s O ~ - s i n ( ¢ ~ + ~ ) + s i n ¢ ~ } .
(10)
As shown in fig. 5: (1) A potential well exists around the synchronous particle at ~ = 0 if and only if the synchronous phase angle ~s lies between 0 ° and 90°; (2) At the right the well extends to -= ~#s and at the left to - 2q~s. The total phase width of the well is thus 3q~; (3) Unlike slam, particles can only have a certain maximum kinetic energy relative to the synchronous particle, and still be bound. By analogy to the previous discussion on shm the motion may be described in terms of a velocity versus displacement diagram, or as is customary for a linac, a relative kinetic energy W - IV, versus phase diagram, using the linear relationship Ox Oz
v - v~ _= v~
W-W~
(11)
2W~
(6) and eq. (7). Thus
where
q, = -x. v~
(7)
The energy increment imparted to this general particle is: d W = qVTcos (q~s + ¢).
(8)
a( w - w~) aq,
a( w - w~) az a~ az
0x 0¢
au 2w~ a¢ w - w ~
(12)
Integrating the last result produces the curves of fig. 6, called separatrices [14]. For small ¢ a separatrix is a
H.F. Glavish/ Rf linearaccelerators
Energy Deviation (W -W s )
JJ /Ph .
r U /without space charge ~/ //~with spacecl'mrge
slightly distorted ellipse, while at the limit of stability it is the shape of a fish. During acceleration in a linac a particle moves along a separatrix. An initial phase spread is transformed into an energy spread: the larger the phase acceptance the larger the energy spread. When the synchronous phase angle ~ increases towards 90 ° the phase stability and phase acceptance increase but the energy gain decreases, as shown by eq. (5). Nevertheless, a linac may simultaneously have a large phase acceptance and a high energy gain. For example, when e~s = 30 ° the phase acceptance is A~ = 90 °, or 25% of a dc beam. But since cos ¢s = 0.866 this fraction of the peak electric field is still available for acceleration. The overall transmission through a linac is further improved by bunching (¢s = 90 °) a dc beam prior to injection into the acceleration cavities. Also, the intrinsic energy spread in a linac may be reduced by debunching (~,~ = - 90 °) after acceleration.
3.3. Space charge effects Difficulty is always encountered when space charge effects are analyzed because the motion of a particle is no longer independent of the presence of other particles. The description is simplified and still quite realistic if the particle distribution within a bunch is assumed to be uniform and the bunch itself of ellipsoidal shape. The effect of the aggregate space charge forces on the longitudinal or phase motion can then be represented by a simple potential energy function of the form: =
,I
[',
9~
Fig. 6. Separatrices describing the longitudinal motion of a particle in a linac beam micropulse. The solid curves correspond to stable motion in the re#on A to B of the linac well of fig. 5. The broken curve represents unstable motion.
U~ -~kcx-,
221
(13)
where k c is dependent on the physical dimensions of the ellipsoid, but independent of the individual particle coordinates. In the case of shin, provided k > k c, the particles are still bound but have a larger amplitude of oscillation. In the case of a linac the effect of space charge is more devastating since the well extends a long
\\
Fig. 7. Space charge forces reduce the size of the well (upper) and the area of the limiting stable separatrix (lower).
way in the direction of negative ~,. Both the phase acceptance and the energy spread are greatly reduced as shown in fig. 7. Diabolically, as the number of particles increases with increasing beam current, the size of the phase space they are allowed to occupy decreases. At some limiting beam current instability ensues. This simplified model predicts a space charge limited beam current of
3rTsinths[Ws(keV)] 1/2 Im~x (mA) ---
4L
A (ainu-------)-
× [ ~ ] 2 V
(kV)
(14)
corresponding to a limiting stable separatrix that is 2/3 the size of the case of no space charge. In the above equation, r is the geometric mean of the transverse semiaxes of the ellipsoid. The current is most seriously limited at low energies, i.e. at the beginning of the linac. It varies with the square root of the particle mass.
4. Independent phase and amplitude control As discussed in the introduction a great deal of flexibility is possible if the sequence of accelerating gaps are independently controlled in phase and amplitude. Phase control always enables the synchronous timing condition to be satisfied from one cavity to the next regardless of particle velocity. Amplitude control, including turning the rf power off in a cavity, or block of cavities, enables the energy to be varied continuously. Operation in a variety of novel ways is possible including: alternating the sign of the synchronous phase angle to provide simultaneous phase and radial focusing (see IV. NEW EQUIPMENT AND SYSTEMS
222
H.F. Glavish / Rf linear accelerators
section 2); tailored bunching at the beginning of the linac to maximize the capture effidiency of an injected dc beam; tailored debunching at the end of the linac to minimize the final energy spread. High rf power dissipation has in the past rendered independent phase and amplitude control somewhat impractical. However, this is not a limitation in the case of superconducting cavities where the scheme was first used [16]. Nor is it a limitation in ion implanters, where the energies of a few MeV, and the frequencies of a few MHz, are extremely low compared with the much higher energies and frequencies typical of linac applications in nuclear physics. For instance, the Eaton NV1000 [17] produces 1 MeV singly charged beams with a total rf power dissipation in the entire ten cavities of only 20 kW. Class C solid state power amplifiers provide efficient drivers and account for only a further 18 kW of power dissipation. The Eaton series of high energy implanters [17] illustrates the utility of independent phase and amplitude control. A single machine accelerates all particles from boron to antimony, singly or doubly charged, through any energy from zero to a maximum value dependent only on the number of rf cavities used. High beam currents are possible. For example, in the case of phosphorus, eq. (14) suggests a space charge limited current of 2.8 mA. In practice a beam current of 1.5 mA has been achieved without observing significant degrading effects from space charge.
5. Rf quadrupole linacs Even now, the need for high energy very high beam currents is forseeable. One example is oxygen ion implantation [18]. Another is metals implantation. Great flexibility with regard to particle species and energy variability is not always necessary in these applications, in which case an immediate possibility is the radiofrequency accelerator (rfq) proposed by Kapchinskii and Teplyakov [19] in 1970, and first implemented in this country at Los Alamos Scientific Laboratory in 1980 [20-22]. The rfq is capable of capturing a very high percentage of an injected dc beam (at least 80%) and accelerating very high beam currents, even as much as 100 mA. The principle of an rfq is shown in fig. 8. An rf rather than a dc voltage is applied to a quadrupole comprised of 4 symmetrically arranged electrodes. At an instant in time, at a certain position along the beam path, the voltage distribution may be such as to generate radial focusing in the vertical direction and radial defocusing in the horizontal direction. After a time equal to one half of an rf period, the voltages become reversed. The drifting beam thus sees the structure as a
V = Vo c o s w t
D-I-G D Fig. 8. An rf quadrupole generates strong radial forces (upper) and also axial electric fields if the electrodes are corrugated (lower).
sequence of converging and diverging ion optical lenses. Such alternating forces can provide strong simultaneous focusing in both the vertical and horizontal directions [23]. Furthermore, if the electrodes are appropriately contoured along the beam path, then in addition to strong radial focusing, axial acceleration also occurs. The contours have the form of corrugations in the electrode surface nearest the beam axis. At certain points along the beam path the tips of the horizontal pair of electrodes are closer to the axis than the tips of the vertical pair of electrodes. At in-between points the opposite is true, the tips of the vertical electrodes now being closer to the axis. Such a structure produces an axial component of electric field alternating in direction. If the spatial pitch of the corrugations corresponds to the distance the synchronous particle moves in an rf period, resonant acceleration results. The ability of an rfq to capture and accelerate large beam currents is immediately evident from eq. (14). Compared with a drift tube linac, an rfq typically operates at a higher frequency, and therefore the cell length L is smaller. Also, the initial capture and acceleration occurs adiabatically with large synchronous phase angles near 90 ° . Both these features, as shown by eq. (14), greatly increase the space charge limited beam current. Simultaneously, by the very nature of the structure, there are very strong radial focusing forces to keep the beam well collimated under the action of the strong transverse space charge forces. At this conference, DiBitonto et al. [24] report on an rfq designed to deliver beams of 35 mA for metals
H.F. Glavish / Rf linear accelerators
implantation. V a r i a n / E x t r i o n are investigating [25] the rfq structure for a high energy semiconductor implanter. Hitachi Ltd. [26] have also proposed such a machine, b u t of limited flexibility. Operating parameters for the two last mentioned machines are not in the public d o m a i n at this time. The rfq is similar to a conventional linac in the sense that the velocity profile is fixed (see section 2). Universality with respect to particle type and energy variability requires a convenient method to be developed for varying the resonant frequency over a wide range, perhaps 3-1. However, the rfq is a natural choice in a dedicated application such as an oxygen implanter where the particl~ species is fixed and the final energy range is narrow. We wish to acknowledge that much of the work on rf accelerators carried out by G M W Associates has been made possible through support from Eaton Corporation.
[10] [11] [12] [13] [14]
[15] [16] [17]
[18]
[19] [20] References [21] [1] J.F. Ziegler, Nucl. Instr. and Meth. B6 (1985) 270. [2] N.W. Cheung, Proc. SPIE 530 (1985) 2. [3] M.I. Current, R.A. Martin, K. Doganis and R.H. Bruce, Semiconductor International (June 1985) 106. [4] C. McKenna, C. Russo, B. Pedersen, D. Downey and R. Liebert, Semiconductor International (April 1986). [5] G. Ising, Arkiv Mat. Astron. och Fys. 18 (1925) 1. [6] R. Wideroe, Arch. Electrotechuik 21 (1928) 387. [7] E.O. Lawrence and'D.H. Sloan, Proc. Nat. Acad. Sci. 17 (1931) 64. [8] L.W. Alvarez et al., Rev. Sci. Instr. 26 (1955) 111. [9] N.J. Barrett, Ion Implantation: 4th Int. Conf. on
[22] [23] [24]
[25] [26]
223
Equipment and Techniques, eds., H. Ryssel and H. Glawischnig (Springer, Berlin, 1982) p. 45. E.M. McMillan, Phys. Rev. 80 (1950) 493. D. Boussard, Focusing of Charged Particles, Vol. II, ed., A. Septier (Academic Press, New York, 1967) p. 327. I.B. Fainberg, Proc. CERN Symp. High Energy Accelerator Pion Phys., Geneva (1956). L.B. Mullet, Atomic Energy Res. Estab. Harwell GP/M147, GP/M149 (1953). A.D. Vlasov, Theory of Linear Accelerators, Translation by Z. Lerman (1968) U.S. Department of Commerce, AEC-tr-6718. W.K.H. Panofsky (Univ. Calif. Rad. Lab. Berkeley 1951) UCRL 1216. P.H. Ceperley, J.S. Sokolowski, I. Ben-Zvi, H.F. Glavish and S.S. Hanna, Nucl. Instr. and Meth. 36 (1976) 421. H.F. Glavish, D. Bernhardt, P. Boisseau, B. Libby, G. Simcox and A.S. Denholm, these Proceedings (Ion Implantation Technology, Berkeley, 1986) Nucl. Instr. and Meth. B21 (1987) 264. J.P. Ruffell, D.H. Douglas-Hamilton, R.E. Kaim and K. Izumi, these Proceedings (Implantation Technology, Berkeley, 1986) Nucl. Instr. and Meth. B21 (1987) 229. I.M. Kapchinskii and V.A. Teplyakov, Prib. Tekh. E.ksp. 2 (1970) 19. R.H. Stokes, K.R. Crandall, J.E. Stovall and D.A. Swenson, IEEE Trans. Nucl. Sci. NS-26 (1979) 3469. J.M. Potter, S.W. Williams, F.J. Humphry and G.W. Rodenz, IEEE Trans. Nucl. Sci. NS-26 (1979) 3745. R.W. Hamm et al., Proc. Int. Conf. on Low Energy Ion Beams (University of Bath, April 1980) p. 2. E.D. Courant, M.S. Livingston and H.S. Snyder, Phys. Rev. 88 (1952) 1190. D. DiBitonto, R. Huson, P. Mclntyre, A. Nassiri, D. Raparia and C. Swenson, these Proceedings (Ion Implantation Technology, Berkeley, 1986) Nucl. Instr. and Meth. B21 (1987) 155. C. McKenna, private communication and ref. [4]. N. Sakudo et al., Hitachi Ltd. European Patent Application No. 84904176.9, Publication No. 0163745.
IV. NEW EQUIPMENT AND SYSTEMS