1. Circular Machines Moohyun Yoon Department of Physics
Pohang University of Science and technology (POSTECH)
Pohang, S. Korea
SLRI - CERN ASSEAN Accelerator School
Synchrotron Light Research Institute
Nakhorn Ratshasima, Thailand
August 27 – September 1, 2017

2. Reading…
H. Wiedemann, Particle Accelerator Physics 4th ed. (Springer 2015)
J.J. Livingood, Principles of Cyclic Particle Accelerators (Van Nostrand 1961)
M. Sands, Physics of Electron Storage Rings, SLAC-121 (1970)
CERN 96-02, Cyclotrons, Linacs, and Their Applications

3. Outline Cyclotron FFAG Betatron Microtron
Synchrotron, storage ring, collider
Synchrotron radiation source etc.

4. Livingston diagram
History of circular accelerators began with the (classical) cyclotron. Limitation in classical cyclotron led to the development of modern circular particle accelerators.
Kinetic energy in log scale
Classical cyclotrons
Year in linear scale
“Per every ten-year period, approximately one order of magnitude increase in beam energy has been achieved.”

5. Cyclotron Ion source “Dees” for acceleration
Ernest Lawrence
A constant vertical B field provides the force to maintain the ion’s circular orbit
Particles pass dee gaps repeatedly, always gaining energy
As the energy of the particles increases, the radius of the orbit increases and the motion becomes spiral
Used for ion acceleration (protons, deuterons, -particles)

6. Fall 1930 UC Berkeley
The first cyclotron by Lawrence, 4-inch diameter, 80 keV proton (Jan 1931), 1 kV gap voltage

7. A few basics
In the presence of magnetic field (B=ẑB) only, the centripetal force is
Br : magnetic rigidity
(r, q, z) = (r, q, z)
In practical units,
Br[T•m] = p[GeV/c] for proton or electron
Cyclotron frequency (revolution frequency of a particle)
m0 : rest mass
Electric field
h : harmonic number (= a positive integer)
f0 : revolution frequency of a particle
h=1 : first harmonic or push-pull mode of operation
h=2 : second harmonic or push-push mode of operation

8. Isochronous condition
The cyclotron frequency
must be constant for a particle to see always the accelerating phase at the dee gap (i.e., synchronism between the particle revolution period and the RF period).
This means that B must increase with the radius in accordance with the rate of increase in g.
But this requirement contradicts the condition for field variation to achieve vertical focusing!! (In cyclotron, achieving proper vertical focusing is more important than radial focusing because of the limitation in the vertical space)

9. Orbital stability (vertical motion)
focusing
defocusing
These figures show that the magnetic field must decrease with radius in order to achieve stable vertical motion (i.e. focusing).
Contradicts the isochronous condition!

10. Vertical motion
From the Lorentz equation in the cylindrical coordinate system (r, q, z) where z is the coordinate specifying the component perpendicular to the orbital plane, the vertical equation of motion of a particle neglecting the effect of the electric field and expanding the radial component of the magnetic field to first order in z we can derive the Kerst-Serber equation for vertical motion:
vertical (or axial) focusing frequency or vertical tune
where
: field index
R: equilibrium orbit radius
Note: positive (negative) n means the vertical field decreases (increases) with radius.
Horizontal motion
Kerst-Serber equation for horizontal motion
horizontal (or radial) focusing frequency or horizontal tune

11. To achieve both vertical and horizontal stability we must have 0 < n < 1
The greater the focusing frequency (or tune) is, the stronger is the focusing of a particle.
When nz,r > 1, we call “strong focusing”.
When nz,r ≤ 1, we call “weak focusing”.
Cyclotron is a weak-focusing accelerator whereas synchrotron in which the focusing is based on quadrupoles is a strong-focusing accelerator.
Summary
To achieve isochronism, the magnetic field must increase with radius.
To achieve vertical orbit stability, nz must be real meaning the field index n must be > 0. So the field must decrease in radius.
To achieve both vertical and horizontal stability we must have 0 < n < 1
1 and 2 are contradictory! This is the limitation of the classical cyclotron so that the maximum achievable energy in classical cyclotron is limited to ~20 MeV. See the Livingston diagram.

12. How can we resolve these contradictory requirements?
Recall
Three choices
Set B to constant (or slightly decrease) with radius, and make wrf vary (in pulsed mode) → synchrocyclotron (or equivalently FM cyclotron)
Emax : MeV (limited by the magnet size)
e.g. Berkeley 184” cyclotron (1946)
B ~1.5 T, 350 MeV later 460 MeV, r ~ 2.3 m
Biggest one: Gatchina, 1000 MeV, 7800 tons, 6.85 m pole diameter
Vary wrf (or in a certain case wrf =constant) and increase B with time such that the orbit radius r = constant. The transverse focusing is achieved by quadrupoles (strong focusing) → synchrotron
e.g. First for electron, Berkeley 300 MeV (1949)
First for proton, Brookhaven 3 GeV Cosmotron (1952)
Biggest one: weak focusing synchrotron : Dubna, 10 GeV, tons

13. wrf : constant (as in classical cyclotron)
3. Relativistic cyclotron (sector-focusing cyclotron, AVF cyclotron, Isochronous cyclotron)
AVF: Azimuthally Varying Field
L.H. Thomas 1938
‘Radial ridge cyclotron’
wrf : constant (as in classical cyclotron)
B(r) : increase with radius to satisfy isochronism
Vertical focusing is obtained from azimuthally varying field (AVF)
Fz = q(v×B)z = q(vrBq – vqBr) 13

14. AVF is based on alternating gradient focusing (F-D-F… or D-F-D…)
AVF cyclotron
-f0 f0
-f0 L
AVF is based on alternating gradient focusing (F-D-F… or D-F-D…)
f : focal length
For f1 = -f2 = f0, we find
: always focusing

15. focusing in sector cyclotrons
Spiral-ridge cyclotron
Flutter factor:
With flutter and additional spiral angle of bending field we find (derivation omitted)
for AVF
e: spiral angle
For a 35º spiral, the effect of the flutter is doubled and for 45º trebled. To a good approx. we find the spiral has no effect on the radial tune.
[illustration of focusing at edges]
By making magnet sectors spiral ridged, further vertical focusing based on the “edge focusing” can be achieved. Spiraling is employed for most isochronous cyclotrons over 40 MeV.
Edge focusing

16. Various sector focused cyclotrons

17. FFAG (Fixed Field Alternating Gradient accelerator)
Similar to a synchrotron
Strong focussing (‘Alternating Gradient’)
Dipole field increases with particle energy through path variation (unlike synchrotron magnets do not cycle so operation is simple and cheap)
Orbit changes with energy
RF changes slightly (pulsed but high rep rate ~ kHz)
Rapid acceleration ~ms so suitable for acceleration of particles with short lifetime
Large transverse and momentum acceptances
(good for muon acceleration and carbon acceleration
and transportation for cancer therapy)
Ref. K.R. Symon, D.W. Kerst, L.J. Laslett, K.M. Terwilliger, Phys. Rev. 103, 6, 1837 (1956)
Radial-sector FFAG magnets and orbits

18. Scaling FFAG: constant orbit shape, constant betatron tune
Nonscaling FFAG: tunes vary to avoid resonances, greater momentum compaction
Scaling FFAG: constant orbit shape, constant betatron tune
Nonscaling FFAG, EMMA
10-20 MeV electron
KEK FFAG, 1 MeV
Kyoto U, 150 MeV FFAG

19. Betatron
Cyclotron is not suitable for acceleration of electrons because they quickly become relativistic. For electron acceleration, the betatron idea was invented in which the acceleration field is provided by induction.
1922 Slepian, US patent
1928 Wideroe, 2:1 principle
1940 Kerst and Serber, first successful operation of 2.3 MeV electron
Electrons are accelerated with the electric fields produced by changing magnetic fields (Faraday’s law) linking the electron orbit
The magnetic field B0 makes the electrons moving in a circle
The magnetic field also serves to guide the particles and its gradients provided focusing
Early betatron, U Illinois

20. Force acting on electron
Cross section of a Betatron
Bguide= <B>/2
p=erBguide
Bguide = 1/2 Baverage
Coil
Steel
Vacuum chamber Ba Bg r
Let
Ba : uniform time-varying magnetic field applied in the central gap
Bg : nonuniform field applied in the vacuum chamber (nonuniformity is achieved by shaping the pole piece)
Let
Variation of the magnetic flux in the vacuum chamber Ya = pr2Ba induces the electric field (i.e. Faraday law)
Force acting on electron
On the other hand, Bg must satisfy the condition for circular trajectory, i.e p = eBg r
Thus we find
Betatron 2:1 rule

21. Betatron is an electron accelerator (5 – 300 MeV)
Not used for ion acceleration because the required radius is too large for ion
Maximum achievable beam energy is limited by synchrotron radiation
Principle of orbit stability is discovered by Kerst and Serber, hence the name betatron oscillations
Typical parameters for 315 MeV University of Chicago betatron (Kerst 1949)
r = 1.22 m
(Bg)max = 9.2 kG
magnet wight = 215 ton
injection energy = 80 – 135 keV
injected current = 1 – 3 A
repetition rate = 6 Hz

22. Microtron
In 1944 V.I. Veksler proposed a microtron concept for electron acceleration based on the principle of phase stability
Ref. Veksler V.I., Dokl. Akad. Nawk USSR 43 (1944) 329
The energy of the particle at each passage through the cavity increases by an increment of particle’s rest energy
RF Cavity
Extraction B
n: harmonic number

23. At each revolution, particles remain in resonance with higher and higher harmonics of the accelerating frequency frf
Maximum energy of a particle is limited by the size of the magnet
Microtron is still used for low energy electron acceleration (~100 MeV) and (although not popular) as an injector for high energy electron synchrotron (e.g. Alladin, Univ. Wisconsin)

24. Synchrotron
RF cavities
Focusing magnets
Deflecting magnets
Extraction magnets
Injection magnets
The accelerating particle beam travels around a fixed closed-loop path
The magnetic field which bends the particle beam into its closed path increases with time during the accelerating process, being synchronized to the increasing kinetic energy of the particles
Transverse focusing is strong and provided by a number of quadrupoles
A storage ring is a special type of synchrotron in which the kinetic energy of the particles is kept constant

25. Fixed target vs. Colliding beam
Fixed Target accelerator
Much of the energy is lost in the fixed target and only part is used to produce secondary particles
Collider
All energy will be available for particle production
The energy of a moving particle interacting with a fixed target must be ~2gcm times greater than the energy in the center-of-mass (CM) system (exercise)
e.g. 1 GeV electron for fixed target accelerator is equivalent to 16 MeV counter rotating beams in the collider

26. Large Hadron Collider (LHC) CERN, 6
Large Hadron Collider (LHC) CERN, 6.75 TeV p-p collider, 27 km, L ~ 1034 cm-2s-1

27. Synchrotron Radiation Sources
Storage ring for synchrotron radiation (SR)
First generation: parasitic operation
Second generation: dedicated to the generation for SR
Third generation: magnet lattice optimized to undulators and wigglers (insertion devices (ID))
Injector linac – booster synchrotron – storage ring

28. Figure of merit parameters of the SR source
Spectral brightness (nth harmonic) or brilliance
Angle-integrated photon spectral flux in the forward direction
Photon beam emittance
Lu: length of an undulator
Photon beam b function
* Spectral flux: number of photons emitted per second within Dw/w bandwidth (usually 0.1%)

29. The phase space area (2pSxS’x) (2pSyS’y) is at minimum if the b functions of the electron beam are matched to those of the photon beam, bx,y = bl. Then the
spectral brightness is maximized to
Spectral brightness increases as the electron beam emittance reduces
If we include the energy spread and the local dispersion in the ID section then the spectral brightness is reduced
The electron beam’s vertical emittance is determined by a coupling between H and V motion and can be controlled by skew quadrupoles (i.e. rotated quads)
As the emittances approach the diffraction limit, i.e. ex,y ≈ ln/4p, the radiation beam becomes coherent in the transverse plane = 1 Å, el ≈ 8 pm)
An electron storage ring in which the beam emittance approaches the diffraction limit is called “diffraction-limited storage ring” or “ultimate storage ring”.

30. In an electron storage ring, the equilibrium (or natural) emittance is determined by balancing between the radiation damping and the quantum excitation, which can be expressed by (ref. M. Sands, SLAC-121, 1970) for isomagnetic lattice
: bending angle
Jx : horizontal damping partition number
F : a constant depending on the structure of the magnet arrangement
The emittance is determined by the behaviour of the dispersion and the horizontal betatron functions within bending magnets
The vertical emittance is determined by a coupling between H and V motion and can be controlled by skew quadrupoles
≪ 1% in 3rd generation light sources
e.g. SPEAR3: ~0.05%, ALS: ~0.1%

31. Various magnetic lattices for synchrotron radiation sources
Double Bend Achromat (DBA) lattice (or Chasman-Green lattice) hx
DBA: Two bends per cell
This is the theoretical minimum emittance of a DBA lattice with achromatic condition. In reality however ex is a factor of 3 to 4 higher than the theoretical value

32. hx Theoretical minimum emittance (TME) lattice
bx and hx are minimum in the middle of the bend
The emittance is by a factor of 3 smaller than the emittance in the DBA structure

33. Multi-Bend Achromat (MBA) lattice
The emittance can be made lower by employing multiple bends
matching dipoles
MAX-IV lattice

34. Multi-Bend Achromat (MBA) lattice with longitudinal variation of the magnetic field
In 1992, A. Wrulich proposed that the emittance can be lowered by introducing longitudinal variation of dipole field
ref. Nagaoka and Wrulich, NIMA 575 (2007) 292
emittance for non-isomagnetic lattice

37. Upgrade of SLS (Unit Cell)
E=2.4 GeV
ε=134 pm*rad Sf Sd Sd Sf
B= T
k= 3.87 m-2
q =-0.78 deg
B= T
k= m-2
q =1.09 deg
There are 5 bendings in a unit cell

38. Achromat of SLS- II E=2.4 GeV, 134 pm rad, C=288 m, N=12 Unit Matching
Cell
Matching

39. Emittance and energy for various light source storage rings•
PLS-II
R. Bartlini, EPAC2008

40. Summary
Today we have reviewed briefly the history of the development of circular particle accelerators
Limitation of the classical cyclotron led to ideas of various modern circular accelerators, synchrocyclotron, synchrotron, relativistic cyclotrons, etc.
We have also introduced operating principles of betatron and microtron
We have introduced various magnet structures popular for 3rd generation synchrotron radiation storage ring

41. Thank you!

42. Circular accelerators
Bending radius
Bending field with time
Bending field with radius
RF frequency with time
Operation mode (pulsed/CW)
Betatron
induction
Microtron
varying h
Classical cyclotron
simple, but limited Ek
Isochronous cyclotron
high power
Synchro- cyclotron
higher Ek, but low P
FFAG
strong focusing, large acceptance
Synchrotron
high Ek, strong focusing
from M. Seidl, CERN accelerator school – introductory course, Prague, Czech Republic, Sep 12, 2014