Download Accelerator 1 Ted Wilson

Accelerators - Lectures I and II by Ted Wilson
(John Adams Institute for Accelerator Science)
‹The
history of accelerators
‹Linear accelerator
‹Dispersion in a waveguide
‹Slowing down the wave
‹Quality factor and Shunt Impedance
‹Cyclotron and Magnetic Rigidity
‹The Synchrotron
‹Magnet types and their multipole field shapes
‹Transverse coordinates
‹Quadrupoles and AG focusing
‹Equation of motion in transverse co-ordinates
‹The lattice and Beam sections
‹Emittance , Beam Size, Q and Beta
‹Phase stability, and Closed orbit
‹Dispersion, and Synchrotron motion
‹Chromaticity
‹Luminosity
‹Final focus optics
‹Beam Beam Force and other LC limits
‹Instability and the impedance of the wall
‹How to organize a Design Study
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
1
Reference Material
This talk
http://acceleratorinstitute.web.cern.ch/acceleratorinstitute/Ambleside%20Wilson.pdf
A text book
“An Introduction to Particle Accelerators” – E. Wilson - OUP
http://ukcatalogue.oup.com/product/9780198508298.do
A general introduction –
“Engines of Discovery” A. Sessler and E. Wilson - WSP
http://www.worldscibooks.com/physics/6272.html
Lattice design
Rossbach ,J. and Schmüser, P. (1992). Basic course on accelerator optics.
Proceedings of the 1986 CERN Accelerator School, Jyvaskyla, Finland, CERN 87-1
http://doc.cern.ch/yellowrep/2005/2005-012/p55.pdf
Errors and Corrections
http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=95-06_v1
Magnet and power Supply
http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=92-05
Radio Frequency System and Cavities
http://preprints.cern.ch/cernrep/2005/2005-003/2005-003.html
Instabilities
http://doc.cern.ch/yellowrep/2005/2005-012/p139.pdf
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
2
The history of accelerators
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
3
Linear accelerator
‹ Particle
gains energy at each gap
‹ Lengths of drift tubes follow increasing
velocity
‹ Spacing becomes regular as v approaches c
‹ Wideroe’s first linac:
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Inside the Fermilab linac
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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The International Linear Collider
(ILC)
TESLA technology: these
superconducting accelerator structures
are built of niobium, and are the crucial
components of the International Linear
Collider.
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Transverse magnetic (E) modes
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Dispersion in a waveguide
c /ω = λ
k2 =
c / ω c = λc
2
ω
⎛ ⎞
⎝ c⎠
v ph = λ f = ω / k
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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−
k = 2π / λ g
2
ω
⎛ c⎞
⎝ c⎠
dω
vg =
dk
Two travelling waves in a guide.
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Slowing down the wave
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Cavity resonators
‹•
n x E= 0 – because the E field should be
normal to the perfectly conducting walls.
‹ Assume we can separate out a time
dependent solutions
aM = e
‹ leaving
−
ωM
2Q
t
{A1 cos Ω M t + A2 sin Ω M t}
a spatial solution:
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Quality factor
Energy stored and dissipated per cycle
Us
Us
Q = 2π
=ω
Ud
W
1 ˆ2
Us = ∫ εE dv
2
where W is the power dissipated
2
V0
1 Isurf
W = Rs = ∫
dA
2
2 σδ
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Cyclotron
ev × B =
mv 2
ρ
ρ
mv p
Bρ =
=
e
e
‹
LIV1_4(Cyclotron_Side).PCT, LIV1_4(Cyclotron_Top).PCT,force,gi, NEWCLASSCYC.AD5
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Magnetic Rigidity
‹
from resolution of momenta that:
‹
the magnitude of the force may be written:
‹
Equating the right hand sides of the two expressions above,
we find we can define a quantity known as magnetic rigidity:
A common convention in charged particle dynamics is to
quote pc in units of electron–volts. Whereupon:
‹
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Components of a synchrotron
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Magnet types
By
x
‹ Dipoles
bend the beam
B
y
x
‹ Quadrupoles
focus it
0
0
‹ Sextupoles
correct chromaticity
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Multipole field expansion
Scalar potential
obeys Laplace
1 ∂ 2φ 1 ∂ ⎛ ∂φ ⎞
∂ 2φ ∂ 2φ
+ 2 = 0 or 2 2 +
⎜r ⎟ = 0
2
r ∂θ r ∂r ⎝ ∂r ⎠
∂x ∂y
whose solution is
∞
φ = ∑φn r n sin nθ
n=1
Example of an octupole whose potential
oscillates like sin 4θ around the circle
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
17
Taylor series expansion
Field in polar coordinates:
To get vertical field
Taylor series of multipoles
Bz = φ0 + φ2 .2 x + φ3 .3 x 2 + φ4 .4 x 3 + ......
1 ∂Bz
∂ 2 Bz 2 1 ∂ 3 Bz 3
= B0 +
x+2 2 x +
x + ...
3
1! ∂x
3! ∂x
∂x
Dip. Quad Sext
Octupole
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Fig. cas 1.2c
Multipole field shapes
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Multipoles in the Hamiltonian
‹ We
said
contains x (and y) dependence
‹ We
find out how by comparing the two
expressions:
‹
We find a series of multipoles:
‹ For
a quadrupole n=2 and:
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Q diagram
nQ = p ,
l QH + mQV = p ,
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Transverse coordinates
ρ
s
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Gutter
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Fields and force in a quadrupole
No field on the axis
Field strongest here
(hence is linear)
Force restores
∂B y
Gradient
Normalized:
∂x
1 ∂By
.
k =−
( B ρ ) ∂x
POWER OF LENS
Defocuses in
vertical plane
l ∂By 1
lk = −
.
=
( B ρ ) ∂x f
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Fig. cas 10.8
Alternating gradients
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Equation of motion in transverse coordinates
‹ Hill’s equation (linear-periodic coefficients)
where
at quadrupoles
like restoring constant in harmonic motion
‹ Solution (e.g. Horizontal plane)
y = β (s) ε sin[φ ( s) + φ0 ]
‹ Condition
‹ Property of machine
‹ Property of the particle (beam) ε
‹ Physical meaning (H or V planes)
Envelope
Maximum excursions
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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The lattice
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Beam sections
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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after,pct
Example of Beam Size Calculation
‹ Emittance
at 10 GeV/c
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Physical meaning of Q and β
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Betatron phase space at various
points in a lattice
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Closed orbit of an ideal machine
F
F
F
In general particles executing betatron oscillations
have a finite amplitude
One particle will have zero amplitude and follows an
orbit which closes on itself
In an ideal machine this passes down the axis
x′
x
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Closed orbit
Zero betatron
amplitude
Phase stability
V = V 0 sin(2 π f a + φ s )
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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PHS.AD5
Dispersion
F
Low momentum particle is bent more
It should spiral inwards but:
There is a displaced (inwards) closed orbit
Closer to axis in the D’s
Extra (outward) force balances extra bends
F
D(s) is the “dispersion function”
F
F
F
F
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Fig. cas 1.7-7.1C
Synchrotron motion
F
This is a biased rigid pendulum
2
2
π
η
V
h
f
φ&& = − 0 2 (sinφ − sinφs )
E0 β γ
F
For small amplitudes
2
V
h
f
π
η
2
0
φ&& +
φ =0
2
E0 β γ
F
Synchrotron frequency
η hV0 cos φs
fs =
f .
2
2πE0β γ
F
Synchrotron “tune”
η hV0 cos φs
fs
Qs = =
.
2
f
2πE0β γ
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Bucket and pendulum
•
θ
θ
F The
“bucket” of synchrotron motion is just
that of the rigid pendulum
F Linear motion at small amplitude
F Metastable fixed point at the top
F Continuous rotation outside
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
36
Measurement of Chromaticity
‹ We
can steer the beam to a different mean
radius and a different momentum by
changing the rf frequency and measure Q
‹ Since
‹ Hence
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Correction of Chromaticity
‹ Parabolic
field of a 6 pole is really a gradient
which rises linearly with x
‹ If x is the product of momentum error and
B" D Δp
dispersion
Δk =
.
( B ρ) p
‹ The
effect of all this extra focusing cancels
chromaticity
⎡ 1 B"(s )β ( s )D( s)ds ⎤ dp
ΔQ = ⎢ ∫
.
⎥
( Bρ)
⎣ 4π
⎦ p
‹ Because
gradient is opposite in v plane we
must have two sets of opposite polarity at F
and D quads where betas are different
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Luminosity
‹
‹
Imagine a blue particle colliding with a beam of cross
section area - A
σ
Probability of collision is
⋅N
A
‹
For N particles in both beams
σ
A
‹
Suppose they meet f times per second at the revolution
frequency
f rev =
‹
⋅ N2
Event rate
2
f rev N
⋅σ
A
βc
2πR
Make big
e.g. 10 −25
Make small
LUMINOSITY
≈ 1030 to 1034 [cm-2 s-1 ]
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
39
Hour-glass effect
β (s) = β * +
s2
β*
Luminosity has to be calculated in slices
desirable to have σz ≤ βy ⇒ short bunch
length
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
40
Final focus optics
final
doublet (FD)
IP
f
1
f (=L*)
2
f1
f2
f2
‹ Need
large demagnification of the (mainly
vertical) beam size
‹ β y*
of the order of the bunch length σz (hourglass effect)
‹ Need free space around the IP for physics
detector
‹ Assume f2 = 2 m ⇒ f1 ¡ 600 m
‹ Can make shorter design but this roughly
sets the length scale
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Example of a final focus
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Field around a moving cylinder of
charge
‹ Take
unit length of the bunch
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
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Beam Beam Force
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
44
Beam-beam Q shift limit
‹
This is the (linear) defocusing effect of one beam on
the other and since it is comparable to the non–linear
effects, we take it as a practical limit for beam stability
r p Nβ *
dQ =
≤ 0.003
γA
‹
‹
Make the β as small as possible
The luminosity formula is similar, and to win
luminosity for a fixed Q shift we only gain linearly
with energy
*
γ
f rev N 2
L=
A
‹
‹
We increase the value of N up to the limit imposed by
instabilities
It also helps to split into many bunches
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
45
Beam-Beam for linear colliders
‹ Luminosity
‹ where
enhancement (pinch)
D is “disruption”
‹ Beamstrahlung
‹ Make
bunch length as large as possible but
must be less than Courant’s β which is the
length of the waist (hourglass effect)
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
46
Linear collider total power limit
‹
All the N particles accelerated f times per second must
each be given energy U
Prf =
‹
Pbeam
η
η
With an efficiency, η, the power limits f and the
luminosity – can be hundreds of MW
fN
L=
A
‹
=
NfeU
2
While N is limited by instabilities and A by focusing
techniques
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
47
Impedance of the wall
Wall current Iw due to circulating bunch
Vacuum pipe not smooth, Iw sees an
IMPEDANCE :
Resistive = in phase.
capacitive lags,
inductive leads
Impedance Z = Zr + iZi
Induced voltage V ~ Iw Z = –IB Z
V acts back on the beam
Ö INSTABILITIES
INTENSITY DEPENDENT
Test: If Initial Small Perturbation is :
INCREASED? INSTABILITY
DECREASED? STABILITY
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
48
Stability diagram
‹ Keil
Schnell stability criterion:
Z Fm0 c 2 β 2γη ⎛ Δp ⎞
2
≤
⎜ ⎟
n
Io
⎝ p ⎠ FWHH
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
49
Accelerators - Lectures I and II by Ted Wilson
(John Adams Institute for Accelerator Science)
‹The
history of accelerators
‹Linear accelerator
‹Dispersion in a waveguide
‹Slowing down the wave
‹Quality factor and Shunt Impedance
‹Cyclotron and Magnetic Rigidity
‹The Synchrotron
‹Magnet types and their multipole field shapes
‹Transverse coordinates
‹Quadrupoles and AG focusing
‹Equation of motion in transverse co-ordinates
‹The lattice and Beam sections
‹Emittance , Beam Size, Q and Beta
‹Phase stability, and Closed orbit
‹Dispersion, and Synchrotron motion
‹Chromaticity
‹Luminosity
‹Final focus optics
‹Beam Beam Force and other LC limits
‹Instability and the impedance of the wall
‹How to organize a Design Study
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
50
Reference Material
This talk
http://acceleratorinstitute.web.cern.ch/acceleratorinstitute/Ambleside%20Wilson.pdf
A text book
“An Introduction to Particle Accelerators” – E. Wilson - OUP
http://ukcatalogue.oup.com/product/9780198508298.do
A general introduction –
“Engines of Discovery” A. Sessler and E. Wilson - WSP
http://www.worldscibooks.com/physics/6272.html
Lattice design
Rossbach ,J. and Schmüser, P. (1992). Basic course on accelerator optics.
Proceedings of the 1986 CERN Accelerator School, Jyvaskyla, Finland, CERN 87-1
http://doc.cern.ch/yellowrep/2005/2005-012/p55.pdf
Errors and Corrections
http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=95-06_v1
Magnet and power Supply
http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=92-05
Radio Frequency System and Cavities
http://preprints.cern.ch/cernrep/2005/2005-003/2005-003.html
Instabilities
http://doc.cern.ch/yellowrep/2005/2005-012/p139.pdf
Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide
51