accelerator basics or things you wish you knew while at ir-2 and talking to pep-ii folks
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Accelerator Basics or things you wish you knew while at IR-2 and talking to PEP-II folks. Martin Nagel University of Colorado SASS September 10, 2008. Outline. Introduction Strong focusing, lattice design Perturbations due to field errors Chromatic effects Longitudinal motion. - PowerPoint PPT PresentationTRANSCRIPT
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Accelerator Basicsor things you wish you knew while at IR-2 and talking to PEP-II folksMartin Nagel
University of Colorado
SASSSeptember 10, 2008
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Outline Introduction Strong focusing, lattice design Perturbations due to field errors Chromatic effects Longitudinal motion
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How to design a storage ring? Uniform magnetic field B0 →
circular trajectory
Cyclotron frequency:
00 qBP
mqB
00
Why not electric bends?
cv
mMVETB
B
E ]/[
][300
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What about slight deviations?
6D phase-space stable in 5
dimensions beam will leak out in
y-direction
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Let’s introduce a field gradient
magnetic field component Bx ~ -y will focus y-motion
Magnet acquires dipole and quadrupole components
combined function magnet
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Let’s introduce a field gradient
magnetic field component Bx ~ -y will focus y-motion
Magnet acquires dipole and quadrupole components
Problem! Maxwell demands By ~ -x
focusing in y and defocusing in xcombined function magnet
)ˆˆ(ˆ0 yxxyGyBB
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Equation of motion
)ˆˆ(ˆ0 yxxyGyBB
0)(2
2
usKdsud
u
2
11
xB
BK yx x
BB
K yy
1
0)(2
2
usKdsud
u
xB
G y
},{ yxu Hill’s equation:
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Equation of motion
)ˆˆ(ˆ0 yxxyGyBB
0)(2
2
usKdsud
u
2
11
xB
BK yx x
BB
K yy
1
0)(2
2
usKdsud
u
xB
G y
},{ yxu Hill’s equation:
natural dipol focusing
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Weak focusing ring K ≠ K(s) define uniform field index n by:
Stability condition: 0 < n < 1
xB
Bn y
1
2
01'' 2
xnx
0'' 2 yny
natural focusing in x is shared between x- and y-coordinates
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Strong focusing K(s) piecewise constant Matrix formalism:
Stability criterion: eigenvalues λi of one-turn map M(s+L|s) satisfy
1D-system:
)(')(
)(susu
sU )()|()|()( 001122 sUssMssMsU
10
1 l
11
01
fKl
f
1
drift space, sector dipole with small bend angle
quadrupole in thin-lens approximation
2|)(| , yxMTr1|| i ni 21
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Alternating gradients
quadrupole doublet separated by distance d:
if f2 = -f1, net focusing effect in both planes:2121
111ffd
fff
dff21
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FODO cell
stable for |f| > L/2
2
2
2
2
2
21)
21(
2
)2
1(22
1
fL
fL
fL
fLL
fL
M x
)( ffMM xy
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Courant-Snyder formalism Remember: K(s) periodic in s Ansatz: ε = emittance, β(s) > 0 and periodic in s Initial conditions phase function ψ determined by β: define: β ψ α γ = Courant-Snyder functions or Twiss-parameters
0)('' usKu
0)(cos)()( sssu
),()',( 000 uu
s
sdss
0 )'(')(
)('21)( ss
)()(1)(
2
sss
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Courant-Snyder formalism Remember: K(s) periodic in s Ansatz: ε = emittance, β(s) > 0 and periodic in s Initial conditions phase function ψ determined by β: define: β ψ α γ = Courant-Snyder functions or Twiss-parameters
0)('' usKu
0)(cos)()( sssu
),()',( 000 uu
s
sdss
0 )'(')(
)('21)( ss
)()(1)(
2
sss
properties of lattice design
properties of particle (beam)
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ellipse with constant area πε shape of ellipse evolves as particle propagates particle rotates clockwise on evolving ellipse after one period, ellipse returns to original shape, but particle moves
on ellipse by a certain phase angle
trace out ellipse (discontinuously) at given point by recording particle coordinates turn after turn
Phase-space ellipse 22 ''2 uuuu
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Adiabatic damping – radiation dampingWith acceleration, phase space
area is not a constant of motion
Normalized emittance is invariant: N
• energy loss due to synchrotron radiation
• SR along instantaneous direction of motion
• RF accelerartion is longitudinal
• ‘true’ damping
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particle → beam different particles have different values of ε and ψ0
assume Gaussian distribution in u and u’ Second moments of beam distribution:
rms
rms
rms
u
uu
u
2
2
'
'
beam size (s) =
beam divergence (s) =
)(s
)(/ s
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Beam field and space-charge effectsuniform beam distribution: beam fields:
• E-force is repulsive and defocusing
• B-force is attractive and focusing
rLa
NqFr 220
2
2
relativistic cancellation
beam-beam interaction at IP: no cancellation, but focusing or defocusing!
Image current: beam position monitor:
)2/()2/sin(2
e
e
bx
LRLR
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How to calculate Courant-Snyder functions? can express transfer matrix from s1 to s2 in terms of α1,2 β1,2 γ1,2 ψ1,2
then one-turn map from s to s+L with α=α1=α2, β=β1=β2, γ=γ1=γ2, Φ=ψ1-ψ2 = phase advance per turn, is given by:
obtain one-turn map at s by multiplying all elements
can get α, β, γ at different location by:
sincossinsinsincos
)|(
sLsM
sin)(
)2
(cos
12
22111
ms
mm
sin)(
sin2)(
21
2211
ms
mms
)|()|()|()|( 121
111222 ssMsLsMssMsLsM
betatron tune
)'('
21
2 sds
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Example 1: beta-function in drift space
*
2** )()(
sss
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Example 2: beta-function in FODO cell
QDQF/2 QF/2
discontinuity in slope by -2β/f
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Perturbations due to imperfect beamline elements Equation of motion becomes inhomogeneous:
Multipole expansion of magnetic field errors: Dipole errors in x(y) → orbit distortions in y(x) Quadrupole errors → betatron tune shifts
→ beta-function distortions Higher order errors → nonlinear dynamics
BB
xKx yx
''
BByKy x
y''
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Closed orbit distortion due to dipole errorConsider dipole field error at s0 producing an angular kick θ
|)()(|cossin2
)()( 0
0 sss
su
integer resonances
ν = integer
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Tune shift due to quadrupole field error
0)()('' usksKu
0
0
)(')'()(' 00
s
s
squdssksuu
101
)|()|(~ 0000 qsLsMsLsM
2sin2cos22cos2 0q 4
0qtune shift
can be used to measure beta-functions (at quadrupole locations):
• vary quadrupole strength by Δkl
• measure tune shift
klyx
yx
,, 4
q = integrated field error strength
quadrupole field error k(s) leads to kick Δu’
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beta-beat and half-integer resonances
])[22cos(2cos2
2sin2sin 00 q
quadrupole error at s0 causes distortion of β-function at s: Δβ(s)
(1,2)-element of one-turn map M(s+L|s)
|)()(|22cos2sin2 00 ssq
β-beat:
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beta-beat and half-integer resonances
])[22cos(2cos2
2sin2sin 00 q
quadrupole error at s0 causes distortion of β-function at s: Δβ(s)
(1,2)-element of one-turn map M(s+L|s)
|)()(|22cos2sin2 00 ssq
β-beat:
twice the betatron frequency
half-integer resonances
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Linear coupling and resonances So far, x- and y-motion were decoupled Coupling due to skew quadrupole fields
νx + νy = n sum resonance: unstable νx - νy = n difference resonance: stable
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Linear coupling and resonances So far, x- and y-motion were decoupled Coupling due to skew quadrupole fields
νx + νy = n sum resonance: unstable νx - νy = n difference resonance: stablemymx
mymx
nonlinear resonances
ν = irrational!
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Chromatic effects off-momentum particle: equation of motion:
to linear order, no vertical dispersion effect similar to dipole kick of angle define dispersion function by
general solution:
)()(''
sxsKx x
/l
)(1)(''s
DsKD x
)()()( sDsxsx
)1( onPP
)()( sDsxCOD
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Calculation of dispersion function
1)0(')0(
1001)(')(
232221
131211
DD
mmmmmm
sDsD
10010
21
ll
transfer map of betatron motioninhomogeneous driving term
Sector dipole, bending angle θ = l/ρ << 1
quadrupole FODO cell
0'2
sin
)2
sin211(
,
2,
DF
DF
D
LD
…Φ = horizontal betatron phase advance per cell
x
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Dispersion suppressors
100
sincossin1)cos1(sincos
FFF
FF
FODO D
D
M
10F
F
DD
at entrance and exit:
after string of FODO cells, insert two more cells with same quadrupole and bending magnet length, but reduced bending magnet strength:
QF/2 (1-x)B QD (1-x)B QF xB QD xB QF/2
)cos1(21
x
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(z, z’) → (z, δ = ΔP/P) → (Φ = ω/v·z, δ) allow for RF acceleration
synchroton motion very slow
ignore s-dependent effects along storage ring avoid Courant-Snyder analysis and consider one
revolution as a single “small time step”
syx 1.
Longitudinal motion
Synchroton motion
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RF cavity
)(),( 00 rc
JEtrEz
)(),( 10 r
cJ
ciEtrB
Simple pill box cavity of length L and radius R
Bessel functions: tie Rc405.2
Transit time factor T < 1:
Ohmic heating due to imperfect conductors:
uuT sin
vLu2T
vLqEPz 0
c
skin2
skin
cdissP
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Cavity design3 figures of merit: (ωrf, R/L, δskin) ↔ (ωrf, Q, Rs)
Quality factor Q = stored field energy / ohmic loss per RF oscillation
AV
LRRL
PUQ
skinskindiss 2
)(
volume
surface area
Shunt impedence Rs = (voltage gain per particle)2 / ohmic loss
cavitysizePLTER skindiss
s
1)( 20
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Cavity array cavities are often grouped into an array
and driven by a single RF source
N coupled cavities → N eigenmode frequencies
each eigenmode has aspecific phase patternbetween adjacent cavities
drive only one eigenmode
)/cos(10)(
Nqmq
, m = coupling coefficient
relative phase between adjacent cavities
large frequency spacing → stable mode
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cavity array field pattern:
pipe geometry such that RF below cut-off (long and narrow)
side-coupled structure in π/2-mode behaves as π-mode as seen by the beam
coupling
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Synchrotron equation of motion
)sin(0 srfrf tVV
synchronous particle moves along design orbit with exactly the design momentum
0 hrf
Principle of phase stability:
• pick ωrf → beam chooses synchronous particle which satisfies ωrf = hω0
• other particles will oscillate around synchronous particle
synchronous particle, turn after turn, sees ss VV sin0
RF phase of other particles at cavity location: srf t
)sin(sin2 2
00 s
ss EqV
ssrf
srf v
vCC
TTT
T
h = integer
C = circumference
v = velocity
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Synchrotron equation of motion rf 2
1
sc
ctrans 1
η = phase slippage factor
αc = momentum compaction factor
transition energy: …beam unstable at transition crossing
linearize equation of motion:
• stability condition
• synchrotron tune:
0cos s
sss
ss E
hqV
cos
2 20
0
“negative mass” effect
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Phase space topology sss
ss EqVhH sin)(coscos22
1),( 2002
0 Hamiltonian:
• SFP = stable fixed point
• UFP = unstable fixed point
• contours ↔ constant H(Φ, δ)
• separatrix = contour passing through UFP,
separating stable and unstable regions
bucket = stable region inside separatrix
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RF bucketParticles must cluster around θs and stay away from (π – θs)
(remember: Φ ↔ z)
Beams in a synchrotron with RF acceleration are
necessarily bunched!
bucket area = bucket area(Φs=0)·α(Φs)s
ss
sin1sin1)(