Accelerator Basicsor things you wish you knew while at IR-2 and talking to PEP-II folksMartin Nagel

University of Colorado

SASSSeptember 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

mqB

00

Why not electric bends?

cv

mMVETB

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

usKdsud

u

2

11

xB

BK yx x

BB

K yy

1

0)(2

2

usKdsud

u

xB

G y

},{ yxu Hill’s equation:

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

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

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

Alternating gradients

quadrupole doublet separated by distance d:

if f2 = -f1, net focusing effect in both planes:2121

111ffd

fff

dff21

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

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

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)

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 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

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

Example 1: beta-function in drift space

*

2** )()(

sss

Example 2: beta-function in FODO cell

QDQF/2 QF/2

discontinuity in slope by -2β/f

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''

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

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’

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:

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

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)(''s

DsKD x

)()()( sDsxsx

)1( onPP

)()( sDsxCOD

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

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

(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

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

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

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

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

vCC

TTT

T

h = integer

C = circumference

v = velocity

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

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

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)(