Cyclotron Magnets
William Beeckman
ECPM37 - Groningen
October 28 2009
1
Presentation overview (1)
Mechanics
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Orbit in a uniform magnetic field – cyclotron frequency
Consequence of mass increase
Isochronous cyclotron
Stability and Focusing
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The need for extra focusing
Edge focusing
Spiral focusing
The alternating gradient principle
Betatronic oscillations
No 2 sectors cyclotron
Resonances
2
Presentation overview (1)
The field shaping zoo
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steel modifications
coils
Types of steel
Magnet design
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Basic parameters : C235 example
Using 2d and 3d codes
Coils … well, 1 hour is short and I needed a victim
3
TRIUMF 520 MeV H-
18m diameter, 4000 tons, 0.46T
4
PSI 590 MeV H+ separated sectors
5
Cyclone 10/5 and cyclone 3D
6
Mechanics
7
Acceleration - The Lorentz force (1)
Force generated by an electric field E and a magnetic field on a
particle of charge q and velocity v
r
r r r
F = q (E + v × B)
r
r
Felec = q E
r
r r
Fmagn = q (v × B)
Electric field along the
trajectory supplies energy
Force orthogonal
to velocity and field
Velocity tangent to
trajectory
No work done
No energy supplied
Accelerating gaps
Electric field orthogonal
to the trajectory does
NOT supply energy
Extraction deflector
8
Acceleration - The Lorentz force (2)
Not all forces supply energy
BUT
All forces give acceleration
Change in the velocity as a vector
Force tangent to
trajectory
Tangential
acceleration
Force orthogonal to
trajectory
Centripetal (normal)
acceleration
r
v2 r
Fcentripetal = m n
R
Change in the modulus
of velocity
Change in the direction
of velocity
9
Orbit in a uniform magnetic field
The motion in the magnetic field is defined by the balance between
the magnetic part of the Lorentz force and the centrifugal force
v2
qvB=m
r
Bz
V
FL
The magnetic rigidity Br is
proportional to the momentum p=mv
H- FC
v p
Br = m =
q q
ωr
This can also be written Br = m
q
thus defining the cyclotron frequency
qB
ω=
m
(in radians/s)
10
Rigidity
Br =
1
T 2 + 2E 0 T
qc
B(T), r(m), q(C), c(m/s), T(J), E0(J)
From this formula, we can derive the 2 extreme cases
Classical limit: T << 2E0 : T = B2r2q2/2m0
Ultrarelativistic limit: T >> E0 : T = Brqc
11
Consequences of the mass increase
B
qB q B
= Cst = B0
then B = B0 γ
ω
=
Cst
if
=
ω=
γ
m m0 γ
ω ⋅m 0 2π f rev E 0
B(T), m(kg), f(Hz), ω(radians/s), E(J)
B0 =
=
2
q
qc
Due to the increase of mass, the revolution frequency decreases as
acceleration proceeds.
For protons at rest:
1.6 10 -19
6
B(T)
15
.
24
10
B(T)
f(Hz) =
=
- 27
2π 1.627 10
2 solutions
• the RF frequency varies to follow the decreasing revolution
frequency during acceleration : synchrocyclotron
• the RF frequency is fixed throughout the whole energy range and
the mass increase is compensated by increasing the magnetic field
in the same proportion: isochronous cyclotron
• The classical cyclotron can be viewed as limit of one of these 2
12
Isochronism (1)
Isochronism (Iso=same, chronos=time) is achieved when the
particle's revolution frequency is the same for all energies. The
accelerating system frequency (or a multiple of it) is then made
identical to this constant revolution frequency and, turn after turn,
every time the particle shows up at the entrance of the accelerating
gap, the voltage value is the same and acceleration takes place.
f RF = h.f rev
h is the harmonic number.
If this condition is not met, the particle will show up in the gap too
soon or too late, and will, under certain circumstances, finally
experience a voltage of the wrong polarity (deceleration).
Once the machine is isochronous, a particle that once was
accelerated with a given dee voltage will keep receiving the same
voltage throughout the whole acceleration. In other words, the
particle doesn’t change its phase wrt the RF system.
13
Isochronism (2)
If we compare that particle to one that is at another phase, the
difference is in the energy they receive in each gap. The one that
gets the largest acceleration will remain such all the time and thus
make less turns in the machine. Its phase is kept constant by the
fact that, having gained more energy, its path becomes longer,
keeping the transit time between gaps constant.
All accelerating phases (-89.99° to +89.99°) could in principle be
used … with an infinite number of turns at the boundaries. In
practice, a phase span of +/-30° about the maximum is used. All
particles will reach the final energy but those at or close to +-30°
will make twice more turns than the central particle.
14
Stability and focusing
15
The need for extra focusing
Stability in the vertical (also called axial) plane requires that the
field decreases with radius.
This condition is naturally fulfilled in flat pole magnets since the
field leaks outside the magnet. However, in the inside of such a
magnet, the field index is very small and hardly ensure axial
stability (weak focusing).
Synchrocyclotrons are purposedely designed with decreasing fields
along radius
But, as the particle's mass increases during the acceleration process,
one would like to increase the field with radius, in complete
opposition with stability requirements.
Extra axial stability must be provided in a different way
16
Edge focusing (1)
Edge focusing (Thomas focusing) comes from the alternance of
high field (hills or sectors) and low field (valleys) regions.
Instead of being circular, like in a homogeneous field, the particle
path is distorted in a polygon-like shape that depends on the
number of hills encountered in 1 turn.
The figure below shows an extreme deformation of the orbit going
from circular to triangle (3 hills) and to square (4 hills). Of course
real orbits are somewhere in between the circle and the polygon
17
Edge focusing (2)
On such a polygon-like orbit, the velocity is no longer purely
azimuthal but gets a radial component.
If this radial velocity interacts with an azimuthal component of the
magnetic field, we get a vertical component of the Lorentz force.
Such an azimuthal component of the magnetic field exists in the
fringing field produced at the interface between a hill and a valley,
out of the median plane … which is precisely the place we need it
Since both the radial velocity AND the azimuthal field change sign
when going from the entrance to the exit edge, both edges have the
same focusing effect.
18
Edge focusing (3)
19
Edge focusing (4)
20
Spiral focusing
Spiral focusing plays on the other pair (velocity, field), the
azimuthal velocity and the radial field.
The azimuthal velocity is the main component of the velocity. The
radial field is created by giving the hills a spiral shape.
BUT, contrary to edge focusing
that produces the same effect on
both edges, the spiral focusing
produces an alternance of
focusing and defocusing effects
since the azimuthal velocity
doesn’t change sign from one
edge to the other.
21
Alternating-gradient principle
Optics states that an alternating series of focusing and defocusing
lenses leads to an overall focusing provided the distance between
the lenses is not too large and the focusing strenghs are not too
different. This is the alternating-gradient principle.
α
β
β
focusing lens
defocusing lens
defocusing lens
focusing lens
α
22
Betatronic oscillations (1)
The conditions for stability require a restoring force proportional to
the displacement. This is the situation of the harmonic oscillator.
The particles will perform oscillations around their equilibrium
orbit. These oscillations are called betatronic oscillations because
they were first observed in an electron accelerator, the betatron.
There are 2 such oscillations, one in the horizontal plane and one in
the vertical plane. As all harmonic motions, they have a frequency
and an amplitude. While the horizontal amplitude may be rather
large, the vertical one is restricted by the size of the magnetic gap.
The notations for the frequencies are νr (QH) for the horizontal
(radial) motion and νz (QV) for the vertical (axial) motion.
They are not time inverse values but indicate the number of full
oscillation(s) performed by the particle during one revolution.
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Betatronic oscillations (2)
24
Betatronic oscillations (3)
From the study of the different terms, we can for sure say that the
focusing force varies with
• the difference between hill and valley fields
• the number of hills in one turn
• the spiral angle
These effects are taken into account in the following formula for
the vertical betatronic frequency
2


N
2
ν z = n +  2  F (1 + 2 tan 2 (ξ ))
 N −1
 N 2  3 
2
2
 2
ν r = 1 − n +  2
 F (1 + 2 tan (ξ ))
 N − 1  N − 4 
2
F=
B − B
B
2
2
is called the flutter and represents the hill to
valley field difference
25
Betatronic oscillations (4)
 N2 
 2 
 N −1
in which N is the number of hills per turn, represents the
fact that a triangular shape is more favourable to the
existence of a radial velocity that a square shape, etc.
(1 + 2 tan 2 (ξ ))
ξ
Takes into account
• edge focusing
• spiral focusing
Remark that they are both multiplied by F
is the spiral angle, i.e. the angle between the orbit and
the tangent to the pole edge. A mean spiral takes into
account the effect from both edges
dB r
n=−
dr B
is the field index and
n = 1 - γ2
26
No 2 sectors cyclotron
From the expression of the horizontal betatronic frequency, we see
that if N<2, the flutter dependent term becomes very large and
negative, making
 N 2  3 
2
 2
ν = 1 − n +  2
 F (1 + 2 tan (ξ ))
 N − 1  N − 4 
2
r
negative !! Physically, this translates the change from bounded,
oscillatory-type solutions to unlimited exponential ones. This is N
called the π stop-band and it is reached every time we have ν r =
2
It implies that
• N must be larger than 2 (lower limit of the π stop-band)
N
• there is an energy limit for every N value T = ( − 1)E 0
2
Protons can for example be accelerated to a maximum of 0.5*938 =
469 MeV for N=3 and to 938 MeV for N=4
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Number of sectors
N small
low maximum energy
distorted orbits – spaced lenses good focusing
N large
high maximum energy
F small and quasi circular orbits -> spiral compulsory
• 1 or 2 No way !!
• 3 - large gap between poles, 3 dees, only harmonic
modes=3, 6 if resonators coupled, more if independent
• 4 - two valleys with RF cavities, all harmonic modes
possible, two valleys for other devices
• >4 - better for SSC
28
Resonances
During the acceleration, QH and QV change. The plot of QV vs QH
is the working point diagram.
Like any oscillatory phenomenon, the amplitude of a betatronic
motion can grow uncontrolled whenever an external source excites
it with its own frequency. This resonance occurs as the betatronic
frequency is a multiple of the "geometrical frequency" of the
cyclotron. In this case, any kick given to the particle because of its
particular position will be experienced again and again. These are
known as systematic resonances.
Under proper circumstances and frequency ratios, the 2 oscillators
can be coupled and the energy stored in one motion, transferred to
the other. These are coupling resonances. They are indicated by
lines in the working point diagram, which the particle’s working
point curve should avoid or cross as fast as possible.
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Betatronic oscillations (6)
Systematic resonances
Working point curve
Coupling resonances
30
The field shaping zoo
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Shaping the field to the goal
• Average field value <B(r)> is important
• The requested field shape can be obtained by different
methods or, more frequently, combination of methods
• Warm coils machines: Iron dominated
• SC machines: Coil dominated
• … and everything in between
• Constant gaps : B(r) naturally decreasing
The larger the gap, the stronger the decrease
Mid-60' synchrocyclotrons e.g. Orsay SC
• Coil field : B(r) naturally increasing
Important only when iron becomes saturated
e.g. AGOR SC cyclotron
32
Shaping the iron contribution
Pros
• Very effective, even in superconducting machine
• Simple and cheap
• Very reliable
Cons
• Highly non linear
• Demands accurate computation and machining
• Fixed correction: no flexibility
Model it before implementing it to avoid unexpected effects
33
The iron shaping methods zoo
• Change the ratio of hill/valley spanned angle with radius
e.g. "horns" in C10/5, spanned angle in C235
• Prevent field decrease at large radii
e.g. Pole end chamfer in C18/9, valley inserts in C235
• Decrease the gap along radius
e.g. Elliptical gap in C235
• Lateral edges milling
e.g. C18/9 – C235
• Iron inserts
e.g. flaps in C10/5
• Change local saturation
e.g. trim rods in Chalk River synchrocyclotron
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1. Hill spanned angle, C10/5
Horns
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1. Hill spanned angle, C235
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2. Pole end chamfer, C18/9
Chamfer
37
2. Valley inserts, C235
Valley steps
38
3. Elliptical gap between poles C235
39
4. Lateral edge milling, C18/9
Shimmed pole edge
Fixed pole edge
40
4. Lateral edge milling C235
3
2
1
41
5. Movable iron inserts, C10/5
Flaps
42
5. Movable iron inserts, C18/9
Flaps
43
5. Movable iron inserts, C18/9
Flaps
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5. Movable iron inserts, C18/9
H– measured magnetic field
D
Two measured
magnetic field in C18/9
cyclotron
Difference between
magnetic fields
provided by iron insert
(flap) movement.
Flap effect
45
6. Trimming rods, Chalk River SC
46
Shaping the coil contribution
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•
•
•
•
•
•
•
•
Pros
Great flexibility
Multiparticles, multienergy machines
Reliable to the level of the power supply
Linear … but iron makes it not so linear
Cons
Very weak except in superconducting machines
Trim coils increase the gap
Demands accurate computation
Insulation in high vacuum (H-), connection outputs
Complex and expensive
Model it before implementing it to avoid unexpected effects
47
The coil shaping methods zoo
• Choose the coil position at nominal current
e.g. all SC cyclotron
• Coil subdivision
e.g. VECC
• Trim coils around poles
e.g. C70, AGOR
• Trim coils on poles
e.g. MC40
• Harmonic coils for beam centering
e.g. C235
48
2. Coil subdivision, VECC
β coils
α coils
49
3. Trim coils around poles, C70
<B>
B0
B0γ(H-)
B0γ(D-)
<r>
50
3. Trim coils around poles, C70
51
3. Trim coils around poles, AGOR
52
4. Trim coil on poles, MC40
53
4. Trim coil on poles, MC40
54
5. Harmonic Coils, C235
55
Construction
56
Parameters for choice
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•
•
•
•
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Cast
Large parts, low deflection, poor mechanical properties
Chemical composition and thermal treatment paramount
Risk of defects, porosities, sand inclusions, etc
Accuracy from machining, no accuracy loss from assembly
Laminated
Limited thickness : 300 mm max, usual 200 mm
Good magnetic and mechanical properties
Slight anisotropy
Forged
Very good magnetic and mechanical properties
Expensive (casting + forging)
Large parts possible
57
Assembly Many possibilities
.
From Annual Report of China Institute of Atomic Energy 2005 CYCIAE 100 MeV
58
Fully cast
59
Fully cast
60
Cast and assembled
61
Designing the magnet
62
Strategic decisions of design
To decide
•
•
•
•
•
magnetic field level on poles, Bhill
pole mechanical radius
gap between poles
valley gap and field, Bvalley
number of sectors, N
63
Magnetic field on poles
Determined by the type of accelerated ions
negative ions, H- and D• low field then large pole radius (EM stripping)
• easy, 100% extraction by stripping
fully stripped positive ions
• high field, small pole radius
• complicated extraction system or self-extraction
64
Gap between poles
Compromise between
small gap
• reduced number of At of coils
• pole radius reduced, orbits close to outer edge
• no space
• very sensitive to errors : vertical losses
large gap
• large space: injection, extraction, probes
• easier vacuum pumping
• lower field
65
Valley gap
Compromise between
small gap
• low flutter
• axial focusing problems
• spiralization of poles necessary
• more field
large gap
• deep valley design
66
Well designed cyclotron magnet
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•
•
•
•
•
•
Isochronous magnetic field during acceleration
Axial (vertical) and radial focusing of beam
Operation point far from resonances
Fast passage through resonance(s) zone(s)
Possibility to install all subsystems
Efficient use of power
Good cooling
67
Design steps
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•
•
•
•
step0: Squeeze requirements and extract juice
step1: Get starting numbers from hand calculation
step2: 2d global model
step3: 3d global model
step4: 2d cuts for detailed local objects
Steps 2, 3 and 4 are combined in an iterative process and
2d calculations must be preferred.
68
Fixed energy
235 MeV
Non superconducting coils
Isochronous
Machine Dimensioning
Magnetic rigidity
Bρ = 2.35 T.m
Maximum field limited
by coil power efficiency
B(r) = B0*γ(r)
High B obtained using ELLIPTIC gap
Bmax ~ 3T close to coil
B min ~ B coil
∆Β = saturation magnetization 2.1 T
Small gap and little axial
focusing possible thanks to
the small current (300 nAmps)
required for treatment
Azimuthal field modulation (flutter)
Spiral shape of poles
2


2
N

ν z = n +  2 −1 ⋅ F ⋅ (1+ 2 tan(ξ ))
N

2
2
 d B  r 
n = −
= 1−γ



dr B 

For best RF efficiency, k=0.5
BUT
to decrease machine dimensions
k >0.5 (more hill, thus more field)
CHOICE : k=0.67 (60° hills)
<B> near coil is thus
0.67*3 + 0.33*(3-2.1) = 2.31 T
2
F=
B
−
B
γ(235 MeV) = 1.25
thus central field B0
~2.31/1.25 =1.8 T
Coil dimension, position and current to match these values,
taking into account power consumption and cooling
Return yoke dimensions function of
Flux to transport
Allowed iron saturation
Radially focusing
Axially defocusing
Extra axial focusing needed
At high radii,
<B> = k*Bhill+ (1-k)*Bvalley
k = Staking factor = hill angle/90°
Radius close to
coil is R~
2.35/2.31 = 1.02m
B increases with radius
B
2
2
CHOICE : ν z = 0.2
Flutter and spiral not too large
Field gradient can be strong
Spiral angle of pole completely
determined since n, N, F and
ν z are known
69
Computing a dipole field (1)
How can we compute the field in the air gap between the poles ?
r
r r ∂D r
4th Maxwell's equation ∇ × H −
=J
∂t
(Ampere's law)
here: no charge => no displacement
current.
r
r
r r
r r
B
.dl = ∫ J .ds
In integral form ∫ H .dl = ∫
µ0 µ r r
S
r
r
r r
B
B r
∫iron µ 0 µ r .dl + air∫ µ 0 .1 .dl = ∫S J .ds
In iron, µr >>1 and the whole contribution of the path in iron is
negligible compared to the path in air (µr =1)
B
.h = nI where h is the gap height, I the current,
µ0
and n the number of turns in the solenoid
70
Computing a dipole field (2)
From this formula, we see that the field B =
•
•
•
µ 0 nI
h
is proportional to the total current in the solenoid,
is inversely proportional to the magnetic gap and
is independent on the pole surface, a rather counter-intuitive
fact to most people.
We also see that it is very important that the magnetic circuit is
made out of very high relative permeability material. Since the
total current is shared between the path in the air and the path in
iron, it is important to reduce the contribution from the latter to
make the best possible use of the solenoid current.
71
Permeability of ferromagnetics
Material
Initial relative
permeability
0.002 Wb/m²
Maximum
relative
permeability
Steel (0.9% C)
Iron (99.8%)
Iron (99.95%)
78 Permalloy (78.5% Ni, 21.5% Fe)
Superpermalloy
Cobalt (99%)
Nickel (99%)
Mumetal (75% Ni, 2% Cr, 5% Cu , 18% Fe)
50
150
10,000
8,000
100,000
70
110
20,000
1000
5000
200,000
100,000
1,000,000
250
600
100,000
72
Permeability for a good steel (Poisson)
Permeability (log scale) as a function of Induction
permeability ( µ )
10000.0
1000.0
100.0
10.0
1.0
0
0.5
1
1.5
2
2.5
B [T]
73
2D Modelling in early study
The 3D geometry is modelled with a 2D code in axisymmetry using pseudo-materials
The stacking factor is the proprtion of the circle occupied by the real material.
Each pseudo-material is defined by a modified B-H curve
B
pseudo
= µ H + k ⋅ (B − µ H )
0
0
74
3D Modelling (1)
75
3D Modelling (1)
76
Field Vs Radius for Different Azimuths
77
Play every possible trick to
use 2d instead of 3d
78
3D Modelling (2)
79
2D Modelling for details study
Strong iron saturation allows to use axi-symmetrical models to
compute the field along a radial cut. The 2d (axi) model is simply
given the shape of the radial cut, with all its features. It allows to
study details in a much more quick and efficient way.
The technique is remarquably accurate but it requires an initial
"calibration" of the models with angles to cope for different offsets.
It was used for instance in the study of the gradient corrector in
C230. This is a tiny slab of iron, a few mm thick with thickness
variations of 0.1 mm in an overall structure with typical dimensions
of a couple of meters
80
2D Modelling : Gradient Corrector
81
3d tips
You all know this but it is
worth repeating it
anyway!
82
Meshing
•
Codes compute potentials and we want 1st and 2nd order
derivatives for fields and tunes
Prepare your model to have a proper mesh where you
compute values, the rest of the model is subsidiary
YOU
must be in control of the mesh, not the code
•
A limited amount of quadratic elements is much more
effective to accuracy than many linear elements
83
Betatron Frequencies
84
Boundaries
• Is the rest of the universe far enough ? TEST IT!
Don't be happy with default far field boundary conditions.
Solution must be stable with normal far field conditions
•
Use symmetry boundary conditions where possible
85
Coils and steel
•
The farther from saturation you are, the more you have to
care about correctly describing your steel (BH curve)
•
Correction coils and saturated steel won't necessarily
behave the way you expect they would
•
Codes are very good in the computation of small changes
between 2 models but less good at absolute values.
86
Thank you
87