# Introduction to Cyclotrons

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```					Introduction to Cyclotrons

Dong Hyun An
Lab. of Accelerator Development
KIRAMS
(Korea Institute of Radiological & Medical Sciences)
Seoul, Korea

AESJ-                   2008- 08-
The 4th AESJ-KNS summer school, 2008-08-07, Kyushu Univ.
Contents
■   Beam & Lorentz Force
■   Classification of accelerators
■   Cyclotron
- Composition
- Electromagnet
: magnetic rigidity & extraction radius
: beam stability in axisymmetric field
: Isochronous cyclotron
: Additional Axial focusing structure (AVF, SSC, SCC)
- RF system
: harmonic number & Dee angular width
: Axial focusing with RF electric field
- Spiral Inflector
- Central Region
- Extraction
: turn separation & transit time factor
■   KIRAMS cyclotrons

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The Motion of a charged particle
Define a charged particle
type of ions : assemble of proton, neutron, and electron -> m0(or E0), q
12C6+ : proton(6), neutron(6), electron(0), Charge state = proton-electron=6

Energy : E = E + E = m 2 c 4 + p 2 c 2
total    0      k        0

g = 1 + Ek / E0
100 MeV proton means that its kinetic energy, Ek, is 100 MeV.
position : (x,y,z) or (r, θ, z),...
momentum : (px, py, pz), b = 1 - 1 / g 2                 (x,x’), (y.y’), (z,z’)
x: displacement from the reference orbit
x’: dx/dz : direction difference from the reference orbit

Describe the motion of a charged particle
x’
x      oscillation

reference orbit                         F : restoring force
(straight line or
equilibrium orbit)   real space (x, x’)                                                                 x[m]
trace space (x, x’)

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Define a Beam
Beam : a group of charged particles
: distribution of a group of charged particles
: group properties

Described to a very good approximation by Gaussian distribution
- Energy spread : Del Ek/Average Ek
- spatial distribution : beam size in transverse plane

1        æ x2 ö                 rms=2sigma : 68% occupation
r ( x) =        expç -    ÷
ç 2s 2 ÷
2p s x    è   x ø
1.65 rms : 90%
2 rms : 95%
s x : rms half of sizes               2.58 rms : 99%
Energy, size, diverge direction            4 rms : 100%

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Describe the motion of beam
Let’s describe the behavior of an entire composite beam.
Assume that 1 / R = 0, Dp/p = 0 .
æ 1              ö         1 Dp
x( s )' '+ç
ç R(s) 2 - k (s) ÷ x(s) =
÷                               Þ         x( s )' '- k ( s ) x ( s ) = 0
è                ø        R ( s ) p0
We introduce beta function b (s ) also known as the amplitude function.
,

x ( s) = e b ( s ) cos(Y ( s ) + f )                  ds
s
with Y ( s) = ò
= E ( s ) cos(Y ( s ) + f )                 0
b (s )

e : emittance, E(s) : envelope
e
x' ( s ) = -         [a ( s) cos(Y ( s) + f ) + sin (Y ( s) + f )]
b (s)
b ' ( s)                1 + a 2 ( s)
with a ( s) º -                g ( s) º
2                       b ( s)
g ( s ) x 2 ( s) + 2a ( s) x( s ) x' ( s ) + b ( s) x'2 ( s) = e
a , b , g : courant - snyder parameters
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Emittance & CS parameters
The area of phase ellipse : A = p     e
Liouville’s theorem :
every element of a volume of phase
space is constant with respect to
time if the particles obey the
canonical equations of motion.
→
Focusing
The area of the phase ellipse and
the beam emittance are
invariants of the particle motion.

Spatial amplitude : b
focusing status : a (+ : defocusing, - : focusing)              Defocusing

focusing or defocusing amplitude : g
Normalized emittance : By considering beam’s energy variation,
the normalized emittance is constant.
en = b g e
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beam current & time structure
■   beam current
1 Ampere= 1Coulomb per 1sec ( 1 |e| = 1.6 x 10-19 Coulomb),
Number of electrons in 1 Coulomb = 6.25 * 1018
ex) 1 mA Proton = 6.25*1015 proton
ex) 1 microA or 1 e microA C6+= 1.04*1012 carbon ion
ex) 1 p microA C6+= 6.25*1012 carbon ion

■   Time structure of beam

macrostructrue

group of bunches                   group of bunches
microstructure

Tpulse

1 cycle, T sec
Repetition rate [Hz] = 1/T
Duty factor = Tpulse/T
MicroDutyFactor=RFacceptance[deg]/360
Average current : Ipeak*DutyFactor*MicroDutyFactor

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Lorentz Force
■   Lorentz force by Electromagnetic field
■   The equation of motion is
r dpr    r    r v
F=    = qE + qv ´ B
dt
■   Energy variation in electromagnetic field
r                 r
r2                r2
r r           r r r r
DE = ò F × dr = q ò (v ´ B + E ) × dr
r                r
r1               r1

■   By Magnetic Field : No energy gains
r r        r r        r    r r r
v // dr ® (v ´ B ) ^ dr Þ (v ´ B ) × dr = 0
: The force is always perpendicular to the moving direction of
paticle ( steering, bending, focusing & defocusing )
■   By Electric Field : Energy variation
r
r2
r r
DE = q ò E × dr = qU
r
r1

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Classification of accelerators
■ Linear accelerator : beam trajectory is linear
- electron Linac
- ion or proton linac
■ Circular accelerator : beam trajectory is circular
Separated
AVF                       Synchro-
-sector                  Synchrotron               FFAG
Cyclotron                   cyclotron
Cyclotron
RF
fixed         fixed          varied           varied          varied
frequency

Magnet
field         fixed         fixed           fixed           varied          fixed
Distribution

beam          CW             CW            pulse            pulse           pulse

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Cyclotrons

AESJ-                   2008- 08-
The 4th AESJ-KNS summer school, 2008-08-07, Kyushu Univ.
Composition of a Cyclotron

1930, Ernest. O. Lawrence
proposed the first circular accelerator, cyclotron.
1932, Lawrence and Livingston together
built a cyclotron for 1.2MeV proton.
•   Ion source to generate ions
•   Electromagnet to circulate ions
•   RF Dee to accelerate ions
•   Deflector or stripper foil to extract ions from the cyclotron

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Block-diagram of cyclotrons
Cyclotron
Central region(4)
External                Cyclotron
Injection
Ion                     Spiral
Source                 inflector(3)              Cyclotron                   Cyclotron
- Electromagnet(1)              Extraction(5)
stripper foil or
- RF structure(2)                 deflector
Internal
Ion source
Ion generation & Injection            acceleration & circulation            Extraction

Beam Transport
Targets for                Beam output                     - energy modulation
experiments &              - type of ions                - bending, steering, focusing
RI production        - Energy & Energy spread                   - collimator
- beam currents & time structure     - beam diagnostics (FC,BPM,..)
- beam emittance

- power supply (High voltage, High current, RF Amplifier, ... )
- vacuum system, cooling system, control system & program
- in superconducting cyclotron, sc coil and ultra low temperature cryostat system

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Lorentz Force & Equation of motion

• RF Dee gap                • Electromagnet
r
r dP r r v
F = = qE + qv ´ B
dt
• Spiral Inflector
• Central Region
• Deflector

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Electromagnet of a cyclotron

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Lorentz force = centrifugal force
mv2
qvB =
r
2
Ek + 2 E0 Ek
Magnetic rigidity : Br =
q *c
v qB        q B qB0                B : average magnetic field [T],
revolution frequency : wrev = =        =    =                   r or R ext : average radius [m]
r gm0 m0 g       m0                Ek : kinetic energy [eV], E0 : rest energy [eV]

Extraction radius :    Rext = Br / Bext                         q* : charge state, q = q* | e |, c : speed of light [m/s]

proton              proton              C6+
r or Rext
Ek [MeV/u]        30                  250                 400
Real
equilibrium orbit
Br [m.T]          0.7977              2.4321              6.3680
frev [MHz]        16.006              36.113              18.696

Bext[T]           1.083               3.0                 3.5
B0 [T]            1.050               2.3688              2.4544
Rext[m]           0.736               0.811               1.819

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The motion in Axisymmetric field magnet
r
Bq = 0 for an axisymmetric field Þ B = Br r + Bz z        ˆ       ˆ
r
dP       r r
= qv ´ B, (dg / dt = 0, Q no electric field )
dt
& ˆ       &     & & ˆ zˆ                           ˆ
gm0 [(&& - rq 2 )r + (rq& + 2rq )q + &&z ] = q (vq Bz r - vr Bzq - vq Br z )
r                                               ˆ                  ˆ
&
rq = v = -v, r = R + x = R (1 + x / R ), ( x << R )
q                                   0           0            0            0

Radial Motion & Stability                                                 Axial Motion & Stability
¶Bz                 æ     x ö                                    ¶Br                           ¶Bz
Bz (r ) » Bz ( R0 ) + x                  = B0 ç1 - n ÷
ç
Br = 0 + z                + O( z 2 ) » z                (¬ Ñ ´ B = 0)
¶r   r = R0        è     R0 ÷
ø                                    ¶z   z =0                     ¶r   z =0

v2
&& + 2 (1 - n) x = && + wrev 2 (1 - n) x = && + w r 2 x = 0
x                  x                       x                                   && + wrev 2 nz = && + w z 2 z = 0
z                z
R0
- radial betatron frequency :                           w r = wrev 1 - n       - axial betatron frequency : w z                               = wrev n
wr                                                                                wz
- radial betatron tune : n r º                               = 1- n            - axial betatron tune :                      nz º               = n
wrev                                                                              wrev
- radial orbit stability                                                       - axial orbit stability
wr 2 > 0 or n < 1 : stable                                                    w z > 0 or n > 0 : stable
2

2
wr < 0 or n > 1 : unstable                                                    w z < 0 or n < 0 : unstable
2

Magnetic field with r
æ                    B ¶B                                             ö
stable condition : 0 < n < 1 , ç field index, n = -                                                  ÷       should be decreased
ç                    R0 ¶r                                            ÷         for beam stability
è                                                              r = R0 ø     in axisymmetric magnet.

AESJ-                   2008- 08-
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Problems in classic cyclotrons
r
Classical cyclotron B = constant or axisymmetr ic
γincreases with particle’s energy but RF frequency is fixed.
→ ωrev decreases
→ The RF phase at the acceleration gap is shifted,
finally particles are decelerated.
→ Limitation for particle’s acceleration, < 22 MeV/e

q
wrev   =     Bz
gm0

r
Synchrocyclotron B = constant or axisymmetr ic

ωrf at each energy is synchronized as the same with ωrev.
→ The output beam is pulsed.
→ It makes the output beam current be lower.

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Isochronous Cyclotron

Isochronous cyclotron
: the revolution time is the same.

wrev = constant
Bz = Bz (r ( E )) = g ( E ) Br =1
Bz q      q
wrev =        = B0    = const., (¬ B0 = Bg =1 )
g m0      m0

→ RF frequency is constant.             wrf º hwrev (h : harmonic number)
→ But, the isochronous magnet field distribution with r
makes the axial motion be unstable due to the field index, n<0.

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dpr                        dpq
= q(vq Bz - v z Bq ),     = q (vz Br - vr Bz )
dt                         dt
dp z
= q (vr Bq - vq Br )
dt
The B-field variation along the             The radial B-field
azimuthal direction can be                  can be constructed by
constructed by Hill-Valley Structure.       Spiral Hill Edge Structure.

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AVF Cyclotron
Bz ( R, q ) = B0 ( R )(1 + f ( R ) g (q ) ) = B0 ( R )Y ( R, q )
f ( R ) : flutter amplitude , Y ( R, q ) : modulation function
g(q ) : function of max. value = 1 and average value = 0

Flutter function
2              2p
é B ( R, q ) - B0 ( R) ù                    1
ò [F( R,q ) - 1] dq
2
F ( R) = ê z                    ú                 =
ë      B0 ( R)         û                   2p    0
q

g (q ) = step function Þ F ( R) = f ( R) 2
1
g (q ) = sine function Þ F ( R) = f ( R ) 2
2

N2
axial tune : n z                    F ( R )(1 + 2 tan 2 z ) + L
2
= n+ 2
N -1
N : No. of sectors, n : Field Index( º -rdB/Bdr )
2                  3N 2
radial tune : n r       = 1- n + 2                   F ( R )(1 + 2 tan z ) + L F ( R) : flutter function, z : inclinatio n angle
2

( N - 1)( N 2 - 4)

normal AVF cyclotron : Bvalley < Bhill
Separated-sector Ring cyclotron : Bvalley~0

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Electromagnet of 30MeV AVF Cyclotron

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Tunes & resonance lines
Magnet properties of 30MeV cyclotron
Isochronous field                          Integrated Phase Error

Horizontal & Vertal tunes                  Tunes & resonance lines

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Separated-Sector Cyclotron
Eliminating all the iron in the valley region,
The flutter amplitude f(R) is maximized.
It makes the axial tune be higher.

This cyclotron is also an isochronous one.

PSI RING cyclotron

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Superconducting Cyclotron
This Superconducting isochronous cyclotron
is adequate for the high-energy and low-current application,
such as particle therapy, etc.

The higher the magnetic field,
250MeV proton
RED: Bext
Blue,dashed : B0

The flutter amplitude remains constant,
but only the average magnetic field is increased.
This makes the Flutter function be lower.
The axial tune is lower by amount of the ratio of
average magnetic field to that of normal conducting
cyclotron.

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RF system for a cyclotron
Signal Generator

FineTuner
Controller       RF Duty controller

8)
Driver Amp.
3)              3)
5)                                    5)
4)    1)         2)   1)   4)                   Intermediate power Amp.

6)
7)          Power Amp.

1)   RF Dee
2)   Center Dee
Dee gap                                    3)   Liner
4)   Dee stem
5)   RF cavity
6)   RF coupler
7)   Transmission Line
Dee    Liner                                 8)   RF pickup coupler

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Harmonic number & Dee angular width

Harmonic Number, h

wrf º hwrev

Dee Angular Width for the maximum acceleration

180
q Dee   =
h
θdee : the angle between gap centers

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Axial Focusing with RF electric field

■   E-field variation
Da
eV0 sin q w
( Da ) f .v.   =-             z
E      v
independent of gap geometry,

θ > 0 (after peak voltage) axially focusing
θ < 0 (before peak voltage) axially defocusing

■   Energy variation through a gap                           Vt ( x, z, t ) = V ( x, z ) cos(w t + q )
t = 0 at the gap center
eV0 cos 2 q
(Da ) e.v. /(Da ) f .v.   =b                                    q : RF phase at the gap center

E sin q
b is normally around 1, a constant depending on the
cos q                 sin q
geometry. The smaller the gap width, The larger b.

After several turns from cyclotron center, the                              cos2 q
particle energy E at the gap center is much                                  sin q
larger than the dee voltage V0. The term of field
variation is dominant in focusing effect.

AESJ-                   2008- 08-
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Spiral Inflector
- External Ion source
- Spiral Inflector bends axially injected beam                         Spiral Inflector Parameters
onto cyclotron’s median plane                             Beam Energy                 24 keV (±5%)

Inflector Height            1.872 cm

Fringe Constant             0.147
tilt parameter k’           0.0

electrode potentials        ±9.02 kV

electrode gap               8 x 16 mm
Central ray trajectories

Beam acceptance of spiral inflector
ü x-x’ trace space : 581 πmm.mrad
ü y-y’ trace space : 300 πmm.mrad
ü Shifted in y-z plane by difference in fringing      ü beam center at Initial Beam : (-0.25, -0.98)mm
fields at the entrance of inflector

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Central Region 1/3

Drawing and Picture of Central Region of KIRAMS-30 Cyclotron

TOSCA, OPERA3D

Dee Voltage
Using electric field distribution and                                   Beam tracking with Dee Voltages
Q obtained from MWS                                                     Resultant Dee Voltage : 52 kV

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Central Region 2/3

RF Acceptance                                    Beam Acceptance
ü Beam Train Simulation                           ü   Initial Beam : 20x20mm, ±100mrad in trace spaces
ü length : βλ=33.5mm                              ü   Energy spread : 5% →24±1.2 keV
ü Center position : (0,0,52mm)                    ü   Center position : (0,0,52mm)
ü Energy with respect to initial RF phase         ü   The initial distribution of the final survivals
After acceleration(2 turns) in CR              ü   x-x’ (y-y’) trace space : 293(139) πmm.mrad
ü 5% energy spread → RF Accep. 60 deg

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Central Region 3/3

Horizontal and Axial Beam Trajectory
(The average positions of beam at each time)

Electric fields           Magnetic fields

Beam Envelop evolution      Beam Energy with turns

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Extraction by carbon stripper
When cyclotron accelerates negative ions,
it easily extract positive ions by stripping            stripper position and
ion’s electron with thin carbon foil.                   beam trajectory after stripping

Extraction Efficiency is almost 100 %.

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Turn separation
Energy Gain per turn
qVdee
g /2
æw          ö
DEk = 2 N dee
g      ò/ 2
-g
dz sin ç rf z + fc ÷
ç v
è
÷
ø
= 2 N dee qVdeeT sin fc
Vdee : Dee voltage                                                    sin[ h g / 2r ]
Φc: RF phase at the gap center                                   T=
h : harmonic number,                                                     h g / 2r
g : dee gap width
T : transit time factor

Turn Separation
2
Ek + 2 Ek E0   dr    E + E0           dB
Br =                  ®    = 2k          dEk -
qc           r  Ek + 2 Ek E0        B

dr        ( E k + E0 )
=                         dEk
r ( Ek 2 + 2 Ek E0 )(1 - n)
n: field index

Turn separation of 30Mev cyclotron
Vdee=50kV, T~1, Φc=78 deg from θdee=39 and h=4

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Extraction by Electrostatic deflector
The internal beam is extracted by
an electrostatic deflector and several magnet or electromagnet channels.

The deflector is composed of
a septum(thin plate, electrically grounded) and
an electrode(antiseptum) for high voltage.

In order to minimize the beam loss by hitting the septum,
: beam is extracted at n r =1 radial position where
the beam is forced coherently off-center.

After extraction by deflector,
the negative field gradient of the fringing field               K250 SC cyclotron, NSCL, MSU
makes beam defocus in the horizontal direction.
The axial motion of beam is oscillatory by the strong
focusing by n<0.

In order to prevent the radial beam blow-up from the negative field gradient
and bring the beam out of cyclotron,
usually several magnet channels as a field corrector and reducer are used.

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KIRAMS CYCLOTRONs
Internal Ion Source (KIRAMS-13)             External Ion Source(KIRAMS-30)

KIRAMS-13     KIRAMS-30

B0                  1.27 T        1.05 T

h                     4             4

Dee Angular         39 deg        39 deg
Width

RF freq.           77.3 MHz      63.96 MHz

Max. axial tune      0.40          0.75

Extraction         0.407 m        0.736 m
proton         proton

No. of sectors        4             4

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