Cyclotron

Cyclotron Stewart Stewart Clelland, Clelland, 64901 6490111 11

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(Dated: November 28, 2014) The cyclotron is the third iteration of particle accelerator, the successor to the linear accelerator (lineac ) and the Van Van De Graaf generator. generator. It was developed developed in the early early 20th cen century tury to analyze atomic structure and is still used today as a producer of medical radiation and radioisotopes. This paper will detail the history, development, theory and uses of the cyclotron. I. A.

HIST HISTOR ORY Y

B.

Previous Previous Accelerat Accelerators ors

The Cyclotron was developed in the wake of the successful cessful linear accelerator accelerator ( linac ), ), a particle particle accelerator accelerator still used today due to its advantages over circular accelera celerator torss in the realm realm of heavy heavy ion acceler accelerati ation. on. The motivation of the  linac  was to accelerate charged particles using a lower voltage than a Van De Graff generator. It succeeded by using a series of charged tubes fed by an RF source to vary the charge to reuse the same voltage differential many times, thereby accelerating a charged particle[6] through multiple electrode gaps. See figure 1.

Cyclotro Cyclotron n Develop Developmen mentt

The conception of the cyclotron is attributed to Ernest O. Lawrence beginning in 1929 when he discovered a paper written by Rolf Widerøe describing a drift tube  linac  [5]. [5]. Rolf, Rolf, a German German electri electrical cal enginee engineerr who had studied both electrical engineering and physics, conceived of  a method of accelerating charged particles with an RF electrical electrical field(cite field(cite german). german). The issue with Widerøe’s proposal was the need for the accelerated ion to spend equal time in each subsequent subsequent drift tube. his limitation limitation naturally leeds to the increase in length of the subsequent quent drift tubes in Widerøe’s Widerøe’s proposal (proportional (proportional to the square roots of integers)[5].Due to the relatively low frequency circuits available at the time, and the scaling of  drift tubes, acceleration of heavy ions was deemed practical due to the lower velocities. However, attempting to accelerate light ions to high energy leads to impractically long tubes. To overcome the issue with lighter ions, Lawrence conceived of a circular accelerator with two ”D” shaped electrodes, trodes, of oscillating oscillating opposite charged used to accelerate accelerate ions to high energies (see figure 2).

Figure 1. Overview of a Linear Accelerator

Modern  linacs  such   such as the Stanford Linear Accelerator Center Center (SLAC ) can produce produce electr electrons ons of 25G 25GeV. eV. The drawba drawback ck is, that that to reach reach these energi energies, es, the   SLAC  needs needs 3.2km and costs costs 315 million million dollar dollars. s. The size, size, complexit complexity y and costs involv involved ed motiva motivated ted the developdevelopment of the first circular accelerator, the cyclotron.[8] Figure 2. View of cyclotron from above

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The undertaking of transforming this concept into reality was seized by a graduate student named M. Stanley Livingston. Livingston. He produced the first working working model of the

2 cyclotron at 4.5in in diameter with the ability to accelerate ions up to 80 000 eV [7].

II.

HARDWARE

The construction of a primitive cyclotron is a product of relatively few pieces. To construct a cyclotron we will refer to original patent filings[5] to describe necessary parts and their roles.

(b) A rectifier circuit to convert utilities AC to DC (24). (c) An high frequency oscillatory electrical circuit to provided voltage to the two hollow semihemispherical electrodes   oscillator . 5. A means to inject ions into the center of the apparatus  30 . 6. An exit for the beamline  55 . III. A.

THEORY

Cyclotronic Frequency

The basic theory of operation of a cyclotron revolves around a equation relaying the frequency of the electric circuit, the mass of the particle, the force of the magnetic field, and the charge. The cyclotron frequency (f cyclotron ) or the equivalent angular frequency (ωcyclotron ) can be derived by looking at Newtons equation [5]. F magnetic = F centripetal

qυB  =

mυ 2 r

Where m is the mass of the particle,  υ  is its velocity and r is its gyroradius. r =

Figure 3. A technical drawing of the view from figure 2

1. Two semi-hemispherical hollow electrodes denoted 6  and 7 . In between these ”dees” are where the particle is accelerated by the potential difference of  the two electrodes 2. A sealed metal pressure vessel to contain the moving particle  8 . 3. A vacuum pump attached to  15 , along with hydrogen gas to flush the system through  16 . 4. Electrical (a) A powerful electromagnet producing a magnetic field normal to the electric field between the electrodes 11. These serve a dual purpose, they keep the path of the charged particle curved within the apparatus aswell as keep the beam focused in the middle of the upper and lower surface of the electrode.

T  =

mυ qB

2πr 2πmυ 2πm = = υ qBυ qB

ωresonance =

qB m

or equivalently f resonance =

qB 2πm

This equation holds only for non-relativistic conditions. Due to the high velocities we can obtain through cyclotrons, it is necessary to modify this characteristic equation to account for relativistic mass. However, it should be noted that due to constraints at the time, cyclotrons had a constant frequency. Therefore, as a particle was accelerated and its mass increased, the frequency of the cyclotron would be too high for the relativistic mass unless the magnetic field was varied. β  =

υ c

3 is the relativistic velocity, while γ  = is the Lorentz factor.

where ∆α   is the deflection at distance z   of the central plane by an ion of charge  e, average energy E  with velocity υ, crossing a sinusoidal electric field of  ω/2π at phase θ relative to maximum potential 2V 0   [3]. The electronic focusing is dominant for about the first third of the radius, with magnetic focusing dominating thereafter.

 1 1− β 

2

m = γm0 is the corrected mass Therefore, with relativistic consideration, the cyclotron resonance frequency would be f  =

f 0 γ 

However, if we may vary the magnetic field as previously stated, something that was possible during the 1930s. We can simply set B = γB0 which allows our γ   factors to cancel, and returns our gyroradius to m0 υ r = qB0

Figure 4. Electric focusing in between the Dees [3]

D. B.

Cyclotron Kinetic Energy

When discussing particle accelerators, the characteristic most often mentioned is the energy of the apparatus. This energy is the kinetic energy  KE  of an outgoing particle. In a cyclotron, the KE   varies according to charge and the mass of the particle involved. We begin with the maximum gyroradius r max a particle can have within the cyclotron. mυ rmax  = qB

KE  =

C.

Due to the finite nature of the electromagnets, on the outside bounds of the magnets we encounter fringing (see figure 5). This fringing acts on the ions pushing them back towards the central plane as illustrated by the solid arrows. The amplitude of the oscillations about the central axis near the rim decay over time as a function of  this restoring force. Stanley writes it as 2

−

 π d ln H  

z = −

z

1 4

d ln R

Where H z   is the field component producing motion in the z   direction at r. The resulting movement of the charged particle will be a function of both the electrical and magnetic field. The focusing will be adequate if  the maximum amplitude of oscillation is less than half  the internal height of the electrodes.

We then solve for  υ υ =

Magnetic Focusing

qBrmax m

mυ 2 (qBr )2 = 2 2m

Electric Focusing

During the path of a charged particle about the cyclotron, it is necessary to ensure the particle stays in the center plane. This is done in the acceleration regions between the ”dees”. This focusing is done by two cylindrical electric lenses that run along the top and the bottom of gap between the Dees. The equation governing the magnitude of electrical focusing is described by Stanley Livingston as 2

  λz

eV 0 ωz  1 eV 0 ∆α = −   sinθ − E  υ 2 E 

k

 cosθ 2

Figure 5. Magnetic focusing[3]

4 IV. A.

USES

would be collected. After many subsequent runs through calutrons , the U-235 would be weapons grade.

Medical

The primary use of cyclotrons in the medical field are isotope production and a positron emission source [2]. Due to the short half life of some medical isotopes, the ability to have on site production is indispensable [10]. The nature of the radioisotopes produced is also very different from that of a reactor.

Figure 6. The difference between radioisotpes of reactors and cyclotrons [10]

The difference, as you can see, is that reactor radioisotopes generally lie above the line of stability. Therefore their decay mode is often β  whereas accelerator isotopes tend to have  β + emission or electron capture. This decay mode leads to applications in molecular imaging due to its high specific activity. Today there are close to 20 radioisotopes with medical applications that can be produced by cyclotrons, amongst which more than a third have half lives less than an hour and a half [10]. These short lived isotopes must be manufactured on site so that research may be carried out prior to total decay. The advent of ”low cost” medical cyclotrons has the ability to revolutionize isotope research due to its new found viability. Furthermore, as seen in the past 4 years with the Chalk River reactor, relying on a single large source for isotopes such as Technetium 99 can be disastrous for the medical community[4]. −

Figure 7. The operation of a calutron [9]

V.

NOTABLE EXAMPLES[1]

1. UC Berkley 27-inch was used to produce 6MeV deuterons in 1937, it was the first to probe neutrondeuteron interactions. 2. In 1942, Berkley Rad Lab produced a 184-inch cyclotron whose energy exceeded 100 MeV. It was used to develope the  calutrons   as well as bombard U238 to add a proton, becoming U239, which decays to Plutonium-239. In the end, the X-10 reactor was used in the Manhattan project, but the decay path remains the same.

VI.

CONCLUSION

During WWII, Lawrence developed a device for the Manhattan project that was derivative of his cyclotron. This device, named a   calutron   functioned as a hybrid mass spectrometer/cyclotron. The  calutron  was used to separate isotopes of Uranium-235 from Uranium-238 by function of accelerating them through a magnetic field, the lighter Uranium-235 would have a tighter arc and

The cyclotron, while being a simple device by modern standards, is arguably the first generation of modern particle accelerators. The energies obtained allowed the us to examine the nucleus using particle bombardment, allowed us to create new isotopes of medical importance and create nuclear weapons. The fact that cyclotrons are still used today is a testament to their simple and versatile nature. While outclassed in modern physics by CERN scaled accelerators, with the ability for more research hospitals to experiment with previously exotic radioisotopes, it is likely we will see many more advancements from the cyclotron in the applied domain.

[1]   ELECTROMAGNETIC AND NUCLEAR INTERACTIONS , chapter 3, pages 205–276. [2] Leonard G. Gomella and Steven A. Haist.   Chapter 15. Imaging Studies . The McGraw-Hill Companies, New York, NY, 2007.

[3] M. Stanley Livingston. The cyclotron. i.  Journal of Applied Physics , 15(1), 1944. [4] Tim Lougheed. Cyclotron production of medical isotopes scales up.   Canadian Medical Association Journal , 947, 2013.

B.

Manhattan

5 [5] L.E. O. Method and apparatus for the acceleration of  ions, February 20 1934. US Patent 1,948,384. [6] American Institute of Physics. Early particle accelerators, 2014. [7] APS Physics. Ernest lawrence and m. stanley livingston, 2014.

[8] Stanford Physics. The stanford linear accelerator center, 2014. [9] AlfredL. Yergey and A.Karl Yergey. Preparative scale mass spectrometry: A brief history of the calutron. Journal of the American Society for Mass Spectrometry , 8(9):943–953, 1997. ´ [10] M. A. AvilaRodr´ ıguez, A. Z´arateMorales, and A. FloresMoreno. Cyclotron production of medical radioisotopes. AIP Conference Proceedings , 1265(1), 2010.