accelerator basics or things you wish you knew while at ir-2 and talking to pep-ii folks martin...
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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
SASS
September 10, 2008
Outline
Introduction Strong focusing, lattice design Perturbations due to field errors Chromatic effects Longitudinal motion
How to design a storage ring? Uniform magnetic field B0 →
circular trajectory
Cyclotron frequency:
00 qBP
m
qB
00
Why not electric bends?
c
v
mMVE
TB
B
E ]/[
][300
What about slight deviations?
6D phase-space stable in 5
dimensions beam will leak out in
y-direction
Let’s introduce a field gradient
magnetic field component Bx ~ -y will focus y-motion
Magnet acquires dipole and quadrupole components
combined function magnet
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
Equation of motion
)ˆˆ(ˆ0 yxxyGyBB
0)(2
2
usKds
udu
2
11
x
B
BK yx
x
B
BK yy
1
0)(2
2
usKds
udu
x
BG y
},{ yxu Hill’s equation:
Equation of motion
)ˆˆ(ˆ0 yxxyGyBB
0)(2
2
usKds
udu
2
11
x
B
BK yx
x
B
BK yy
1
0)(2
2
usKds
udu
x
BG y
},{ yxu Hill’s equation:
natural dipol focusing
Weak focusing ring K ≠ K(s) define uniform field index n by:
Stability condition: 0 < n < 1
x
B
B
n y
1
2
01
''2
x
nx
0''
2 y
ny
natural focusing in x is shared between x- and y-coordinates
Strong focusing K(s) piecewise constant Matrix formalism:
Stability criterion: eigenvalues λi of one-turn map M(s+L|s) satisfy
1D-system:
)('
)()(
su
susU )()|()|()( 001122 sUssMssMsU
10
1 l
11
01
fKl
f
1
drift space, sector dipole with small bend angle
quadrupole in thin-lens approximation
2|)(| , yxMTr
1|| i ni 21
Alternating gradients
quadrupole doublet separated by distance d:
if f2 = -f1, net focusing effect in both planes:
2121
111
ff
d
fff
d
ff
21
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
s
dss
0 )'(
')(
)('2
1)( ss
)(
)(1)(
2
s
ss
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
s
dss
0 )'(
')(
)('2
1)( ss
)(
)(1)(
2
s
ss
properties of lattice design
properties of particle (beam)
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
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
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
Beam field and space-charge effectsuniform beam distribution: beam fields:
• E-force is repulsive and defocusing
• B-force is attractive and focusing
rLa
NqFr 22
0
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
b
x
LR
LR
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:
sincossin
sinsincos)|(
sLsM
sin)(
)2
(cos
12
22111
ms
mm
sin)(
sin2)(
21
2211
ms
mms
)|()|()|()|( 121
111222 ssMsLsMssMsLsM
betatron tune
)'(
'
2
1
2 s
ds
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
''
BB
yKy xy
''
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
Tune shift due to quadrupole field error
0)()('' usksKu
0
0
)(')'()(' 00
s
s
squdssksuu
1
01)|()|(
~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’
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 ss
q
β-beat:
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 ss
q
β-beat:
twice the betatron frequency
half-integer resonances
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
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!
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)(''
sDsKD x
)()()( sDsxsx
)1( onPP
)()( sDsxCOD
Calculation of dispersion function
1
)0('
)0(
1001
)('
)(
232221
131211
D
D
mmm
mmm
sD
sD
100
102
1
ll
transfer map of betatron motioninhomogeneous driving term
Sector dipole, bending angle θ = l/ρ << 1
quadrupole FODO cell
0'2
sin
)2
sin21
1(
,
2,
DF
DF
D
LD
…Φ = horizontal betatron phase advance per cell
x
Dispersion suppressors
100
sincossin
1)cos1(sincos
FFF
FF
FODO D
D
M
1
0F
F
D
D
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(2
1
x
(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
RF cavity
)(),( 00 rc
JEtrEz
)(),( 10 r
cJ
c
iEtrB
Simple pill box cavity of length L and radius R
Bessel functions: tie R
c405.2
Transit time factor T < 1:
Ohmic heating due to imperfect conductors:
u
uT
sin
v
Lu
2
T
v
LqEPz
0
c
skin
2
skin
cdissP
Cavity design3 figures of merit: (ωrf, R/L, δskin) ↔ (ωrf, Q, Rs)
Quality factor Q = stored field energy / ohmic loss per RF oscillation
A
V
LR
RL
P
UQ
skinskindiss 2
)(
volume
surface area
Shunt impedence Rs = (voltage gain per particle)2 / ohmic loss
cavitysizeP
LTER skin
disss
1)( 2
0
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 a
specific phase pattern
between adjacent cavities drive only one eigenmode
)/cos(10)(
Nqmq
, m = coupling coefficient
relative phase between adjacent cavities
large frequency spacing → stable mode
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
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
v
C
C
T
TT
T
h = integer
C = circumference
v = velocity
Synchrotron equation of motion rf 2
1
sc
c
trans 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
Phase space topology sss
ss E
qVhH
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