Let ρ be the mean radius of the beam pipe in a basic magnet configuration of a betatron. If the total magnetic flux enclosed by the beam circumference is ramped up by a time-dependent magnetic flux density, the induced electric field along the beam axis and the particle momentum are
E ·ds= 2πρE=πρ2B˙av, E=1
2B˙avρ, p˙=eE=1 2eB˙avρ, p= 1
2eBavρ=eBgρ, or Bg=1
2Bav, (1.4)
whereE is the induced electric field,Bav is the average magnetic flux density inside the circumference of the beam radius. We obtain the betatron principle: the guide fieldBgis equal to 1/2 of the average fieldBav, first stated by R. Wieder¨oe in 1928.11
9See e.g., J.W. Beal, N.C. Christofilos and R.E. Hester,IEEE Trans. Nucl. Sci. NS16, 294 (1958) and references therein; Simon Yu, Review of new developments in the field of induction accelerators, inProc. LINAC96(1996).
10See e.g., R.B. Miller, inProc. NATO ASI on High Brightness Transport in Linear Induction Accelerators, A.K. Hyder, M.F. Rose, and A.H. Guenther, Eds. (Plenum Press, 1988); R.J. Briggs, Phys. Rev. Lett. 54, 2588 (1985); D.S. Prono,IEEE Trans. Nucl. Sci. NS32, 3144 (1985); G.J.
Caporaso,et al., Phys. Rev. Lett. 57, 1591 (1986); R.B. Miller,IEEE Trans. Nucl. Sci. NS32, 3149 (1985); G.J. Caporaso, W.A. Barletta, and V.K. Neil,Part. Accel.11, 71 (1980).
11In 1922, Joseph Slepian patented the principle of applying induction electric field for electron beam acceleration in the U.S. patent 1645304.
Figure 1.3 is a schematic drawing of a betatron, where particles circulate in the vacuum chamber with a guide field Bg, which is equal to half of the average flux densityBav enclosed by the orbiting particle.
Figure 1.3: Schematic drawing of a betatron. The guide field for beam parti-cles Bg must equal to the average flux densityBav en-closed by the orbiting path.
It took many years to understand the stability of transverse motion. This problem was solved in 1941 by D. Kerst and R. Serber.12 We design a magnet so that the magnetic field is
Bz=B0
R r
n
, with n=−R B0
dBz dr
r=R
, (1.5)
whereRis the orbit radius of a reference particle,ris a radius with small deviation fromR, andnis thefocusing index. Letx=r−Randzbe small radial and vertical displacements from a reference orbit, then the equations of motion become
d2z
dt2 +ω2nz= 0, d2x
dt2 +ω2(1−n)x= 0. (1.6) The motion is stable and simple harmonic if 0 ≤ n ≤ 1 (see Exercise 1.14). The resulting frequencies of harmonic oscillations are fx = f0√
1−n and fz = f0√n, wheref0 = ω/2π =v/(2πR) is the revolution frequency, andv is the speed of the particle.
In 1940 D. Kerst built and operated the first betatron achieving 2.3 MeV at University of Illinois. In 1949 he constructed a 315-MeV betatron13at the University of Chicago with parametersρ= 1.22 m,Bg= 9.2 kG,Einj= 80−135 keV,Iinj= 1−3 A. The magnet weighed about 275 tons and the repetition rate was about 6 Hz. The limitations of the betatron principle are (1) synchrotron radiation loss (see Chapter 4) and (2) the transverse beam size limit due to its intrinsic weak-focusing force.
I.4 Radio-Frequency (RF) Accelerators
Since the high-voltage source can induce arcs and corona discharges, it is difficult to attain very high voltage in a single acceleration gap. It would be more economical to
12D. Kerst and R. Serber,Phys. Rev. 60, 53 (1941). See also Exercise 1.14. Since then, the transverse particle motion in all types of accelerators has been calledbetatron motion.
13D.W. Kerstet. al., Phys. Rev. 78, 297 (1950).
I. HISTORICAL DEVELOPMENTS 9 make the charged particles pass through the acceleration gap many times. This con-cept leads to many different rf accelerators,14which can be classified as linear (RFQ, linac) and cyclic (cyclotron, microtron, and synchrotron). An important milestone in rf acceleration is the discovery of the phase-focusing principle by E. M. McMillan and V. Veksler in 1945 (see Ref. [21] and Chap. 2, Sec. IV.3). Accelerators using an rf field for particle acceleration are described in the following subsections.
A. LINAC
In 1925 G. Ising pointed out that particle acceleration can be achieved by using an alternating radio-frequency field. In 1928 R. Wieder¨oe reported the first working rf accelerator, using a 1-MHz, 25-kV oscillator to produce 50-kV potassium ions shown in the top plot of Fig. 1.4. In 1931 D.H. Sloan and E.O. Lawrence built a linear accelerator using a 10-MHz, 45 kV oscillator to produce 1.26 MV Hg+ion.15
Figure 1.4: Top: schematic drawing of the Wieder¨oe rf LINAC, where drift tubes shield particles from the decelerating rf electric field.
Wider¨oe used a 1-MHz, 25-kV oscillator to pro-duce 50-kV potassium ions. Bottom: enclos-ing the drift tubes in a metallic cylinder, the capacitance of the gap can be coupled to the inductance for a resonance cavity to achieve a higher efficiency in acceleration gradient. This cavity invented by Alvarez is called Alvarez linac or drift-tube linac (DTL).
Since the distance between adjacent drift tubes isβλ/2 =βc/(2frf), it would save space by employing higher frequency rf sources. However, the problem associated with a high frequency structure is that it radiates rf energy at a rate ofP =ωrfCVrf2, whereωrf is the rf frequency,C is the gap capacitance, andVrf is the rf voltage. The rf radiation power loss increases with the rf frequency. To eliminate rf power loss, the drift tube can be placed in a cavity so that the electromagnetic energy is stored in the form of a magnetic field (inductive load). At the same time, the resonant frequency of the cavity can be tuned to coincide with that of the accelerating field.16
In 1948 Louis Alvarez and W.K.H. Panofsky constructed the first 32-MV drift-tube proton linac (DTL or Alvarez linac) shown schematically in the bottom plot of Fig. 1.4. Drift-tube linacs have been used as injectors for high energy accelerators at
14The rf sources are classified into VHF, UHF, microwave, and millimeter waves bands. The microwave bands are classified as follows: L band, 1.12-1.7 GHz; S band, 2.6-3.95 GHz; C band, 3.95-5.85 GHz; X band, 8.2-12.4 GHz; K band, 18.0-26.5 GHz; millimeter wave band, 30-300 GHz.
See also Exercise 1.2.
15D.H. Sloan and E.O. Lawrence,Phys. Rev.38, 2021 (1931).
16L. Alvarez,Phys. Rev.70, 799 (1946).
BNL, KEK, Fermilab, SNS, and CERN. In the 1970’s Los Alamos constructed the first side-coupled cavity linac (CCL), reaching 800 MeV. Fermilab upgraded part of its linac with the CCL to reach 400 MeV kinetic energy in 1995.
After World War II, rf technology had advanced far enough to make magnetron and klystron amplifiers that could provide MW rf power at 3 GHz (S band).17 Today, the highest energy linac has achieved 50-GeV electron energy operating at S band (around 2.856 GHz) at SLAC, and has achieved an acceleration gradient of about 20 MV/m, fed by klystrons with a peak power higher than 40 MW in a 1-μs pulse length. To achieve 100 MV/m, about 25 times the rf power would be needed. The peak power is further enhanced by pulse compression schemes.
Superconducting cavities have substantially advanced in recent years. At the Continuous Electron Beam Accelerator Facility (CEBAF) at the Thomas Jefferson National Accelerator Laboratory (JLAB) in Virginia, about 160 m of superconduct-ing cavity was installed for attainsuperconduct-ing a beam energy up to 6 GeV in 5 paths ussuperconduct-ing 338 five-kW CW klystrons. During the LEP-II upgrade more than 300 m of super-conducting rf cavity was installed for attaining more than 100-GeV beam energy.
Many accelerator laboratories, such as KEK in Japan, Cornell and Fermilab in the U.S. and DESY in Germany, are collaborating in the effort to achieve a high-gradient superconducting cavity for a linear collider design called the International Linear Collider (ILC). Normally, a superconducting cavity operates at about 5–10 MV/m.
After extensive cavity wall conditioning, single-cell cavities have reached far beyond 25 MV/m.18
B: RFQ
In 1970, I.M. Kapchinskij and V.A. Teplyakov invented a low energy radio-frequency quadrupole (RFQ) accelerator – a new type of low energy accelerator. Applying an rf electric field to the four-vane quadrupole-like longitudinally modulated structure, a longitudinal rf electric field for particle acceleration and a transverse quadrupole field for focusing can be generated simultaneously. Thus the RFQs are especially
17The klystron, invented by Varian brothers in 1937, is a narrow-band high-gain rf amplifier. The operation of a high power klystron is as follows. A beam of electrons is drawn by the induced voltage across the cathode and anode by a modulator. The electrons are accelerated to about 400 kV with a current of about 500 A. As the beam enters the input cavity, a small amount of rf power (< 1 kW) is applied to modulate the beam. The subsequent gain cavities resonantly excite and induce micro-bunching of the electron beam. The subsequent drift region and penultimate cavity are designed to produce highly bunched electrons. The rf energy is then extracted at the output cavity, which is designed to decelerate the beam. The rf power is then transported by rf waveguides.
The wasted electrons are collected at a water-cooled collector. If the efficiency were 50%, a klystron with the above parameters would produce 100 MW of rf power. See also E.L. Ginzton, “The $100 idea”,IEEE Spectrum,12, 30 (1975).
18See e.g., J. Garber,Proc. PAC95, p. 1478 (IEEE, New York 1996). Single-cell cavities routinely reach 30 MV/m and beyond.
I. HISTORICAL DEVELOPMENTS 11 useful for accelerating high-current low-energy beams. Since then many laboratories, particularly Los Alamos National Laboratory (LANL), Lawrence Berkeley National Laboratory (LBNL), and CERN, have perfected the design and construction of RFQ’s, which are replacing Cockcroft-Walton accelerators as injectors to linac and cyclic accelerators.19
C: Cyclotron
Thesynchrotron frequencyfor a non-relativistic particle in a constant magnetic field is nearly independent of the particle velocity, i.e.,
ωsyn=eB0
γm
γ ≈1
−−−−→ ωcyc=eB0
m , (1.7)
whereB0 is the magnetic field, andmis the particle mass. In 1929 E.O. Lawrence combined the idea of a nearly constant revolution frequency and Ising’s idea of the rf accelerator (see Sec. I.4A of Wieder¨oe linac), he invented thecyclotron.20 Historical remarks in E.O. Lawrence’s Nobel lecture are reproduced below:
One evening early in 1929 as I was glancing over current periodicals in the University library, I came across an article in a German electrical engineering journal by Wider¨oe on the multiple acceleration of positive ions. . . .This new idea immediately impressed me as the real answer which I had been looking for to the technical problem of accelerating positive ions, . . . Again a little analysis of the problem showed that a uniform magnetic field had just the right properties – that the angular velocity of the ion circulating in the field would be independent of their energy so that they would circulate back and forth between suitable hollow electrodes in resonance with an oscillating electric field of a certain frequency which has come to be known as thecyclotron frequency.
If two D plates (dees) in a constant magnetic field are connected to an rf electric voltage generator, particles can be accelerated by repeated passage through the rf gap, provided that the rf frequency is an integer multiple of the cyclotron frequency, ωrf=hω0. On January 2, 1931 M.S. Livingston demonstrated the cyclotron principle by accelerating protons to 80 keV in a 4.5-inch cyclotron, where the rf potential applied across the the accelerating gap was only 1000 V. In 1932 Lawrence’s 11-inch cyclotron reached 1.25-MeV proton kinetic energy that was used to split atoms, just a few months after this was accomplished by the Cockcroft-Walton electrostatic
19See e.g. A. Pisent,HIGH POWER RFQS, Proc. of PAC09, 75 (2009)
20E.O. Lawrence and N.E. Edlefsen, Science, 72, 376 (1930). See e.g. E.M. McMillan,Early Days in the Lawrence Laboratory (1931-1940), in New directions in physics, eds. N. Metropolis, D.M. Kerr, Gian-Carlo Rota, (Academic Press, Inc., New York, 1987). Thecyclotronwas coined by Malcolm Henderson, popularized by newspaper reporters; see M.S. Livingston,Particle Accelerators:
A Brief History, (Harvard, 1969).
accelerator. Since then, many cyclotrons were designed and built in Universities.21 Figure 1.5 shows a schematic drawing of a classical cyclotron.
Figure 1.5: Schematic drawing of a classical cyclotron. Note that the radial distance be-tween adjacent revolutions becomes smaller as the turn number increases [see Eq. (1.9)]. A septum is a device that can kick the beam into an external beam transport line.
The momentump and kinetic energyT of the extracted particle arep =mγβc andT =mc2(γ−1) =p2/[(γ+ 1)m]. Using Eq. (1.2), we obtain the kinetic energy per amu as
T
A= e2B02R20 (γ+ 1)mu
Z A
2
≡K Z
A 2
, (1.8)
whereB0R0=Bρis the magnetic rigidity, ZandAare the charge and atomic mass numbers of the particle,mu is the atomic mass unit, andK is called theK-value or bending limitof a cyclotron. In the non-relativistic limit, theK-value is equal to the proton kinetic energyT in MeV, e.g. K200 cyclotron can deliver protons with 200 MeV kinetic energy.
The iron saturates at a field of about 1.8 T (depending slightly on the quality of iron and magnet design). The total volume of iron-core is proportional to the cubic power of the beam rigidityBρ. Thus the weight of iron-core increases rapidly with itsK-value: Weight of iron = W ∼(Bρ)3 ∼ K1.5, whereBρ is the beam rigidity.
Typically, the magnet for a K-100 cyclotron weighs about 160 tons. The weight problem can be alleviated by using superconducting cyclotrons.22
Beam extraction systems in cyclotrons is challenging. LetV0 be the energy gain per revolution. The kinetic energy at N revolutions is KN = eNV0 =e2B2r2/2m, whereeis the charge,mis the mass,Bis the magnetic field, andris the beam radius at theN-th revolution. The radiusr of the beam at theN-th revolution becomes
r= 1 B
2mV0 e
1/2
N1/2, (1.9)
21M.S. Livingston, J. Appl. Phys, 15, 2 (1944); 15, 128 (1944); W.B. Mann, The Cyclotron, (Wiley, 1953); M.E. Rose,Phys. Rev.,53, 392 (1938); R.R. Wilson,Phys. Rev.,53, 408 (1938);
Am. J. Phys.,11, 781 (1940); B.L. Cohen,Rev. Sci. Instr.,25, 562 (1954).
22See H. Blosser, inProc. 9th Int. Conf. on Cyclotrons and Applications, p. 147 (1985).
I. HISTORICAL DEVELOPMENTS 13 i.e. the orbiting radius increases with the square root of the revolution numberN. The beam orbit separation in successive revolutions may becomes small, and thus the septum thickness becomes a challenging design problem.
Two key difficulties associated with classical cyclotrons are the orbit stability and the relativistic mass effect. The orbit stability problem was partially solved in 1945 by D. Kerst and R. Serber (see Exercise 1.14). The maximum kinetic energy was limited by the kinetic mass effect. Because the relativistic mass effect can destroy particle synchronism [see Eq. (1.7)], the upper limit of proton kinetic energy attainable in a cyclotron is about 12 MeV (See Exercise 1.4).23 Two ideas proposed to solve the dilemma are the isochronous cyclotron and the synchrocyclotron.
Isochronous cyclotron
The radius of an orbiting particle and the magnetic field that maintain isochronism with a constantωare
ρ= p eB = p
γmω = c ω
1−E02
E2 1/2
, Bz=γmω
e = ω
ec2E(ρ) = ωE0
ec2
1−ωρ c
2−1/2
. (1.10)
whereE0=mc2is the rest energy,ωis the angular revolution frequency, andBzorB is the guide field. When the magnetic flux density is shaped according to Eq. (1.10), the focusing index becomesn <0, and the vertical motion is unstable.
In 1938 L.H. Thomas pointed out that, by using an azimuthal varying field, the orbit stability can be retained while maintaining the isochronism. The isochronous cyclotron is also called the azimuthal varying field (AVF) cyclotron. Orbit stability can be restored by shaping the magnetic pole plates with hills and valleys.24 The success of sector-focused cyclotron constructed by J.R. Richardsonet al. led to the proliferation of the separate sector cyclotron, or ring cyclotron in the 1960’s.25 It gives stronger “edge” focusing for attaining vertical orbit stability. Ring cyclotrons are composed of three, four, or many sectors. Many universities and laboratories built ring cyclotrons in the 1960’s.
Synchrocyclotron
Alternatively, synchronization between cyclotron frequency and rf frequency can be achieved by using rf frequency modulation (FM). FM cyclotrons can reach 1-GeV
23H. Bethe and M. Rose,Phys. Rev.52, 1254 (1937).
24L.H. Thomas,Phys. Rev.54, 580 (1938).
25H.A. Willax, Proc. Int. Cyclotron Conf. 386 (1963); Design and operation aspects of a 1.3 MW high power proton ring cyclotron at the PSI by M. Seidel et. al. is available at http://accelconf.web.cern.ch/AccelConf/IPAC10/papers/tuyra03.pdf
proton kinetic energy.26 The synchrocyclotron uses the same magnet geometry as the weak-focusing cyclotrons. Synchronism between the particle and the rf accelerating voltage is achieved by ramping the rf frequency. Because the rf field is cycled, i.e.
the rf frequency synchronizes with the revolution frequency as the energy is varied, synchrocyclotrons generate pulsed beam bunches. Thus the average intensity is low.
The synchrocyclotron is limited by the rf frequency detuning range, the strength of the magnet flux density, etc. Currently two synchrocyclotrons are in operation, at CERN and at LBL.
D: Microtron
The accelerating rf cavities are expensive, it would be economical to use the rf struc-ture repetitively. The microtron, originally proposed by V. Veksler in 1944, is designed to do this. Repetitive use requires synchronization between the orbiting and the rf periods. For example, if the energy gain per turn is exactly equal to the rest mass of the electron, the energy at then−1 passage isγm0c2 =nm0c2. the orbit period is an integral multiple of the fundamental cyclotron period: T =n2πmeB .Thus, if the rf frequency ωrf is an integral multiple of the fundamental cyclotron frequency, the particle acceleration will be synchronized. Such a scheme or its variation was invented by V. Veksler in 1945.
The synchronization concept can be generalized to include many variations of magnet layout, e.g. the race track microtron (RTM), the bicyclotron, and the hexa-tron. The resonance condition for the RTM with electrons traveling at the speed of light is
nλrf=βcΔt= 2πβΔE
ecB, (1.11)
where ΔEis the energy gain per passage through the rf cavity,Bis the bending dipole field,λrf is the rf wavelength, andn is an integer. This resonance condition simply states that the increase in path length is an integral multiple of the rf wavelength.
Some operational microtrons are the three-stage MAMI microtron at Mainz, Ger-many,27and the 175-MeV microtron at Moscow State University. Several commercial models have been designed and built by DanFisik.The weight of the microtron also increases with the cubic power of beam energy.
E: Synchrotrons, weak and strong focusing
After E.M. McMillan and V. Veksler discovered the phase focusing principle of the rf acceleration field in 1945, a natural evolution of the cyclotron was to confine the particle orbit in a well-defined path while tuning the rf system and magnetic field
26For a review, see R. Richardson,Proc. 10th Int. Conf. on Cyclotrons and Their Applications, IEEE CH-1996-3, p. 617 (1984).
27See e.g., H. Herminghaus, inProc. 1992 EPAC, p. 247 (Editions Fronti`eres, 1992).
I. HISTORICAL DEVELOPMENTS 15 to synchronize particle revolution frequency.28 The first weak-focusing proton syn-chrotron, with focusing index 0< n <1, was the 3-GeV Cosmotron in 1952 at BNL.
A 6-GeV Bevatron constructed at LBNL in 1954, led to the discovery of antiprotons in 1955.
An important breakthrough in the design of synchrotron came in 1952 with the discovery of the strong-focusing or the alternating-gradient (AG) focusing principle by E.D. Courant, H.S. Snyder and M.S. Livingston.29 Immediately, J. Blewett in-vented the electric quadrupole and applied the alternating-gradient-focusing concept to linac30 solving difficult beam focusing problems in early day rf linacs. Here is
“Some Recollection on the Early History of Strong Focusing” in the publication BNL 51377 (1980) by E.D. Courant:
Came the summer of 1952. We have succeeded in building the Cosmotron, the world’s first accelerator above one billion volts. We heard that a group of European countries were contemplating a new high-energy physics lab with a Cosmotron-like accelerator (only bigger) as its centerpiece, and that some physicists would come to visit us to learn more about the Cosmotron. . . . Stan (Livingston) suggested one particular improvement: In the Cosmotron, the magnets all faced outward. . . .Why not have some magnets face inward so that positive secondaries could have a clear path to experimental apparatus inside the ring?
. . .I did the calculation and found to my surprise that the focusing would be strengthened simultaneously for both vertical and horizontal motion.. . .Soon we tried to make the gradients stronger and saw that there was no theoretical limit – provided the alterations were made more frequent as the gradient went up. Thus it seemed that aperture could be made as small as one or two inches – against 8×24 inches in the Cosmotron, 12×48 in the Bevatron, and even bigger energy machines as we then imagined them. With these slimmer magnets, it seemed one could now afford to string them out over a much bigger circles, and thus go to 30 or even 100 billion volts.
. . .I did the calculation and found to my surprise that the focusing would be strengthened simultaneously for both vertical and horizontal motion.. . .Soon we tried to make the gradients stronger and saw that there was no theoretical limit – provided the alterations were made more frequent as the gradient went up. Thus it seemed that aperture could be made as small as one or two inches – against 8×24 inches in the Cosmotron, 12×48 in the Bevatron, and even bigger energy machines as we then imagined them. With these slimmer magnets, it seemed one could now afford to string them out over a much bigger circles, and thus go to 30 or even 100 billion volts.