Paul Derwent14 Dec 001

Stochastic Cooling

Paul Derwent

14 Dec 00

http://cosmo.fnal.gov/organizationalchart/derwent/cdf_accelerator.htm

Paul Derwent14 Dec 002

Idea Behind Stochastic Cooling

Phase Space compressionDynamic Aperture: Areawhere particles can orbit

Liouville’s Theorem:Local Phase Space

Densityfor conservative systemis conserved

Continuous MediaDiscrete Particles

Swap Particles and Empty

Area -- lessen physicalarea occupied by beam

x

x’

x

x’

Paul Derwent14 Dec 003

Idea Behind Stochastic Cooling

Principle of Stochastic cooling Applied to horizontal tron oscillation

A little more difficult in practice. Used in Debuncher and Accumulator to cool

horizontal, vertical, and momentum distributions

COOLING? Temperature ~ <Kinetic Energy>minimize transverse KE minimize E longitudinally

Kicker

Particle Trajectory

Paul Derwent14 Dec 004

Stochastic Coolingin the Pbar Source

Standard Debuncher operation: 108 pbars, uniformly distributed ~600 kHz revolution frequency

To individually sample particles Resolve 10-14 seconds…100 THz bandwidth

Don’t have good pickups, kickers, amplifiers in the 100 THz range Sample Ns particles -> Stochastic process

» Ns = N/2TW where T is revolution time and W bandwidth

» Measure <x> deviations for Ns particles

Higher bandwidth the better the cooling

Paul Derwent14 Dec 005

Betatron Cooling

With correction ~ g<x>, where g is gain of system New position: x - g<x>

Emittance Reduction: RMS of kth particle

Add noise (characterized by U = Noise/Signal) Add MIXING

Randomization effects M = number of turns to completely randomize sample

xk −g⟨x⟩( )2 =xk2 −2gxk + g2 ⟨x⟩2

⟨x⟩ = 1Ns

xi =1Ns

xk +1Nsi

∑ xii≠k∑

Average over all particles and do lots of algebra

d⟨x⟩2

dn=−2g⟨x2 ⟩

Ns+ g2

Ns⟨x2 ⟩, where n is 'sample'

⇒ Cooling Time1τ=2W

N2g−g2( )

⇒ Cooling Time 1

τ=

2W

N2g − g2 M +U[ ]( )

Paul Derwent14 Dec 006

Momentum Cooling

Momentum Cooling explained in context of Fokker Planck Equation

Case 1: Flux = 0 Restoring Force (E-E0)Diffusion = D0

Cooling of momentum distribution (as in Debuncher)

‘Small’ group with Ei-E0 >> D0

Forced into main distribution MOMENTUM STACKING

∂ψ∂t

= −∂

∂EC E( )ψ − D E( )

∂ψ

∂E ⎛ ⎝

⎞ ⎠

where ψ = density function ∂N

∂EC E( ) is energy gain function

D E( ) represent diffusion terms (noise, mixing, feedback)

ψ =ψ 0 exp−α E −E0( )

2

2D0

⎛

⎝ ⎜

⎞

⎠ ⎟

Paul Derwent14 Dec 007

Stochastic Stacking

Gaussian Distribution CORE

Injected Beam (tail) Stacked

E0

‘Stacked’

C(E)

D(E)

Paul Derwent14 Dec 008

Pbar Storage Rings

Two Storage Rings in Same Tunnel Debuncher

» Larger Radius

» ~few x 107 stored for cycle length• 2.4 sec for MR, 1.5 sec for MI

» ~few x 10-7 torr

» RF Debunch beam

» Cool in H, V, p

Accumulator» ~1012 stored for hours to days

» ~few x 10-10 torr

» Stochastic stacking

» Cool in H, V, p

Both Rings are ~triangular with six fold symmetry

Paul Derwent14 Dec 009

Debuncher Ring

ßtron cooling in both horizontal and vertical planes

Momentum cooling using notch filters to define gain shape

4-8 GHz using slot coupled wave guides in multiple bands

All pickups at 10 K for signal/noise purposes

Paul Derwent14 Dec 0010Accumulator Ring

Not possible to continually inject beam Violates Phase Space Conservation Need another method to accumulate beam

Inject beam, move to different orbit (different place in phase space), stochastically stack

RF Stack Injected beam Bunch with RF (2 buckets) Change RF frequency (but not B field)

» ENERGY CHANGE

Decelerates ~ 30 MeV Stochastically cool beam to core

Decelerates ~60 MeV

Injected Pulse

Core

Stacktail

Frequency(~Energy)

Power(dB)

Paul Derwent14 Dec 0011Stochastic Stacking

Simon van Der Meer solution: Constant Flux:

Solution:

Exponential Density Distribution generated by Exponential Gain Distribution

Max Flux = (W2||Ed)/(f0p ln(2))

∂ψ∂t

= constant

∂ψ∂E

=ψ

Ed, where Ed characteristic of design

ψ =ψ 0 expE − Ei( )

Ed

⎡

⎣ ⎢ ⎤

⎦ ⎥

Gain

Energy

Density

Energy

StacktailCore

Stacktail

Core

Using log scales on vertical axis

Paul Derwent14 Dec 0012

Implementation in Accumulator

Stacktail and Core systems How do we build an exponential gain

distribution? Beam Pickups:

Charged Particles: E & B fields generate image currents in beam pipe

Pickup disrupts image currents, inducing a voltage signal

Octave Bandwidth (1-2, 2-4,4-8 GHz) Output is combined using binary combiner

boards to make a phased antenna array

Paul Derwent14 Dec 0013Beam Pickups

Pickup disrupts image currents, inducing a voltage signal

3D Loops Planar Loops

Paul Derwent14 Dec 0014Beam Pickups

At A:

Current induced by voltage across junction splits in two, 1/2 goes out, 1/2 travels with image current

AI

Paul Derwent14 Dec 0015Beam Pickups

At B:

Current splits in two paths, now with OPPOSITE sign Into load resistor ~ 0 current Two current pulses out signal line

B

I

T = L/ c

Paul Derwent14 Dec 0016Beam Pickups

Current intercepted by pickup:

Use method of images

In areas of momentum dispersion D

Placement of pickups to give proper gain distribution

+w/2-w/2

y

x

x

d

Current Distribution

I =Ibeamπ

tan−1 sinhπd

x+w2

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎡ ⎣ ⎢

⎤ ⎦ ⎥−tan−1 sinh

πd

x−w2

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

≈Ibeamπ

exp−πxd

⎛ ⎝

⎞ ⎠ for largex

Δx = Dβ2

ΔEE

Paul Derwent14 Dec 0017Accumulator Pickups

Placement, number of pickups, amplification are used to build gain shape

StacktailCore = A - B

Energy

Gain

Energy

StacktailCore

Paul Derwent14 Dec 0018

AntiProton Source

Shorter Cycle Time in Main Injector Target Station Upgrades Debuncher Cooling Upgrades Accumulator Cooling Upgrades

GOAL: >20 mA/hour