Paul DerwentApr 20, 20231

Stochastic Cooling in the Fermilab AntiProton Source

Paul Derwent

Beams Division/Pbar/CDFThursday, April 20, 2023

Paul DerwentApr 20, 20232Stochastic Cooling

Main Entry: sto·chas·tic

Pronunciation: st&-'kas-tik, stO-

Function: adjective

Etymology: Greek stochastikos skillful in aiming, from

stochazesthai to aim at, guess at, from stochos target,

aim, guess -- more at STING

Date: 1923

1 : RANDOM; specifically : involving a random variable

<a stochastic process>

2 : involving chance or probability : PROBABILISTIC <a

stochastic model of radiation-induced mutation>

- sto·chas·ti·cal·ly /-ti-k(&-)lE/ adverb

Main Entry: 2cool

Date: before 12th century

intransitive senses

1 : to become cool : lose heat or warmth <placed the pie

in the window to cool> -- sometimes used with off or

down

2 : to lose ardor or passion <his anger cooled>

transitive senses

1 : to make cool : impart a feeling of coolness to

<cooled the room with a fan> -- often used with off or

down <a swim cooled us off a little>

2 a : to moderate the heat, excitement, or force of :

CALM <cooled her growing anger> b : to slow or lessen

the growth or activity of -- usually used with off or

down <wants to cool off the economy without freezing

it -- Newsweek>

- cool it : to calm down : go easy <the word went out to

the young to cool it -- W. M. Young>

- cool one's heels : to wait or be kept waiting for a long

time especially from or as if from disdain or discourtesy

From Webster’s Collegiate Dictionary

Paul DerwentApr 20, 20233Why an Antiproton source?

p pbar physics with one ring Dense, intense

beams for high luminosity

L =f0N

pN

p_

4πσ 2 F(β,σz)

Paul DerwentApr 20, 20234

Luminosity HistoryCollider Run I

0

5

10

15

20

25

30

0 100 200 300 400 500 600

Number of Antiprotons (/109)

It’s all in the pbars!

Paul DerwentApr 20, 20235Making Anti-protons

120 GeV protons off metal target Collect some fraction of anti-protons which are created

Within collection lens aperture Momentum ~8 GeV (±2%)

Paul DerwentApr 20, 20236Why an Anti-proton source?

~11,000 cycles

Store and cool in the process!

Collect ~2 x 10-5 pbars/proton on target ~5e12 protons on target ~1e8 pbars per cycle 0.67 Hz Large Energy Spread & Emittance

Run II Goals 36 bunches of 3 x 1010 pbars Small energy spread Small transverse dimensions

Paul DerwentApr 20, 20237Pbar Longitudinal Distribution

Paul DerwentApr 20, 20238Overview Information

Frequency Spectrum Time Domain:

(t+nT0) at pickup

Frequency Domain:harmonics of revolution frequency f0 = 1/T0

Accumulator:T0~1.6 sec (1e10 pbar = 1 mA)f0 (core) 628890 Hz

127th Harmonic ~79 MHzη=

1γt

2 −1γ2

Δff

=−ηΔpp

Paul DerwentApr 20, 20239

Idea Behind Stochastic Cooling

Phase Space Compression:

Dynamic Aperture: Areawhere particles can

orbit

Liouville’s Theorem*:

Local Phase Space Density for conservative system is conserved

*J. Liouville, “Sur la Théorie de la Variation des Constantes arbitraires”, Journal de Mathematiques Pures et Appliquées”, p. 342, 3 (1838)

WANT TO INCREASE PHASE SPACE DENSITY!

x

x’

x

x’

Paul DerwentApr 20, 202310

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 DerwentApr 20, 202311Why more difficult in practice?

Standard Debuncher Operation: 108 particles, ~uniformly distributed Central revolution frequency 590035 Hz

» Resolve 10-14 seconds to see individual particles!

» 100 THz antennas = 3 µm!

pickups, kickers, electronics in this frequency 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 DerwentApr 20, 202312Simple Betatron Cooling

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

Emittance Reduction: RMS of kth particle

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

=2WN

2g−g2( )

Paul DerwentApr 20, 202313Stochastic Nature?

Result depends upon independence of the measured centroid <x> in each sample In case where have no frequency spread in beam, cannot cool

with this technique!

Some number of turns M to completely generate independent sample

But… Where is randomization occurring?

» WANT: kicker to pickup GOOD MIXING

» ALSO HAVE: pickup to kicker BAD MIXING

Δff

=−ηΔpp

Paul DerwentApr 20, 202314Cooling Time

Electronic Noise: Random correction applied to each sample More likely to heat than cool Noise/Signal Ratio U

High Bandwidth Low Noise

Optimum Gain (in correction g) goes down as N goes up!

⇒ Cooling Time 1

τ=

2W

N2g − g2 M +U[ ]( )

Paul DerwentApr 20, 202315Momentum Cooling

Time evolution of the particle density function, (E) = ∂N/∂E

Fokker-Planck Equation -- c. 1914 first used to describe Brownian motion

Two Pieces: Coherent self force through pickup, amplifier, kicker

» Directed motion of the particle

Random kicks from other particles and electronic noise» Diffusive flux from high density to low density

∂ψ∂t

=−∂∂E

F E( )ψ −D E( )∂ψ∂E

⎛ ⎝

⎞ ⎠

F E( ) is gain function

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

Paul DerwentApr 20, 202316Simple Example

Linear Restoring Force with Constant Diffusive Term (Electronic noise) Gaussian Distribution

Inject at E> E0

Coherent force dominates --- collected into core!

F(E) =−α(E −E0)

D(E) =D0

⇒ Ψ(E) =Ψ0e−α

2D(E−E0 )2

E0

‘Stacked’

F(E)

D(E)

Simulation!

Paul DerwentApr 20, 202317Types of Momentum Cooling

Filter Cooling: Use Momentum - Frequency map Notch Filters for Gain Shaping

» Debuncher

» Recycler

» Stack tail (as correction)

Splitter Combiner

Adjustable DelayNotch Filter

0.01

0.1

1

10

300 360 420 480 540 600 660 720 780 840 900

Frequency (KHz)

Paul DerwentApr 20, 202318Types of Momentum Cooling

Palmer Cooling Use Momentum - Position Map in regions of Dispersion Pickup Response vs Position to do Gain Shaping

» Accumulator Core: Signal(A) - Signal(B)

» Accumulator Stacktail (described in coming slides)

A B

Beam Distribution

Top View

Paul DerwentApr 20, 202319Momentum Stacking

Van der Meer’s solution: desire constant flux past energy point static solution !

∂Ψ∂t

=∂∂E

(F(E)Ψ(E)−D(E)∂Ψ∂E

)=∂∂E

φ=0

V(E) volts per turn applied at kicker

diffusive term depends upon particle density and mean square voltage applied

(ignoring amplifier noise for the moment)

φ0 constant flux

F(E) =eV/T where T is period

D(E) =AV(E)2Ψ(E)

⇒ φ0 =eVT

Ψ −AV2∂Ψ∂E

Paul DerwentApr 20, 202320Van der Meer’s Solution

∂Ψ∂E

=−φ0

AV2Ψ+

eAVT

&Maximize Gradient term

V =2φ0TeΨ

Substitute and Integrate

⇒ Ψ =Ψ0e(E−E i ) Ed , where Ed =4Aφ0T

2 e2

V =2φ0TeΨ0

e−(E−E i )Ed

To build constant flux, build voltage profile which is exponential in shape and results in density distribution which is exponential in

shape!

Paul DerwentApr 20, 202321

Exponential Density Distribution generated by Exponential Gain Distribution

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

Gain

Energy

Density

Energy

StacktailCore

Stacktail

Core

Using log scales on vertical axis

Paul DerwentApr 20, 202322Implementation in Accumulator

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 DerwentApr 20, 202323Beam Pickups

At A:

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

AI

Paul DerwentApr 20, 202324Beam 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 DerwentApr 20, 202325Current Intercepted by Pickup

In areas of momentum dispersion D

Placement of pickups to give proper gain distribution

+w/2-w/2

y

x

x

d

Current DistributionI =

Ibeamπ

tan−1 sinhπd

Δx+w2

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎡

⎣ ⎢ ⎤

⎦ ⎥ −tan−1 sinhπd

Δx−w2

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎡

⎣ ⎢ ⎤

⎦ ⎥ ⎧ ⎨ ⎩

⎫ ⎬ ⎭

≈Ibeamπ

exp −πΔxd

⎛ ⎝

⎞ ⎠ for large Δx Δx≥w( )

Use Method of Images

Δx = Dβ2

ΔEE

Paul DerwentApr 20, 202326Accumulator Pickups

Placement number of pickups amplification used to build gain

shape

Also use Notch filters to zero signal at core

StacktailCore = A - B

Energy

Gain

Energy

StacktailCore

Paul DerwentApr 20, 202327Accumulator Stacktail

Not quite as simple: -Real part of gain cools beam frequency dependson momentum

f/f = -p/p (higher f at lower p) Position depends on momentum

x = Dp/p

Particles at different positions have different flight times

Cooling system delay constant

» OUT OF PHASE WITH COOLING SYSTEM AS MOMENTUM CHANGES

Paul DerwentApr 20, 202328Accumulator Stacktail

105

106

107

108

109

1010

1011

1012

-100-50050

Leg 1Leg 2CoreTotal

Abs(Real)

Energy

0

60

120

180

240

300

360

-100-50050

Leg 1Leg 2CoreTotal

Phase (degrees)

Energy

Use two sets of pickups at different Energies to create exponential Distribution with desired phase Characteristics

Stacktail Design Goal For Run IIEd ~ 7 MeVFlux ~ 35 mA/hour

Show simulation!

Paul DerwentApr 20, 202329Performance Measurements

-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

720 740 760 780 800 820 840 860 880 900 920

Frequency - 628000 Hz

A:IBEAM 44.684498Line Fit

0

5

10

15

20

25

30

35

40

0 8 16 24 32 40 48 56 64 72 80Beam Current

11-Oct31-Jul

Max Flux

Stacking Rate

Fit to exponential in region of stacktail (845-875 in these units)

Calculate Maximum Flux for fitted gain shape

Different beam currents

Independent of Stack Size

Max Flux ~30 mA/hour

Paul DerwentApr 20, 202330Performance Measurements

Engineering Run Iia Best Achieved

Run Goal

Protons on Target 3.8e12 5e12 5e12

Cycle Time (sec) 3.2 1.5 2.2

Production Efficiency 10 20 15(pbars/106protons)

Stacking Rate 4 18 10.3(1e10 per hour)

Stacking rate limited by input flux and cycle time » Which we limit because of core-stacktail coupling problems

Paul DerwentApr 20, 202331Performance Measurements

Best Performance: 39.9 mA in 4 hours

Restricted by core-stacktail couplings

Paul DerwentApr 20, 202332

Stacktail - Core Coupling

Coupling in regions where frequency bands overlap 2-4 GHz ! much larger than previous overlap

Two phenomena Coherent beam feedback

» Stacktail kicks beam and coherent motion is seen at core

Misalignment gives transverse - longitudinal coupling» Try to correct with kickers

Pickup Kicker

BeamSince beam does not decohere,Carry information back to pickup

Feedback!Schottky

Pickup

Stacktail

Core

Paul DerwentApr 20, 202333Stacktail Schottky Signals

-90

-85

-80

-75

-70

-65

-60

2.999287 2.999387 2.999487 2.999587 2.999687 2.999787 2.999887

Frequency (GHz)

2.12 after $90

1.07 after $90

No Beam

Core

Freshly injected beam

Later in cycle

Stacktail Leg1

Paul DerwentApr 20, 202334Core 2-4 Schottky Signals

-85.00000

-80.00000

-75.00000

-70.00000

-65.00000

-60.00000

2.99929 2.99939 2.99949 2.99959 2.99969 2.99979 2.99989

Frequency (GHz)

2.12 after $90

1.07 after $90

No Beam

Core

Freshly injected beam

Later in cycle

Stacktail Leg1

Paul DerwentApr 20, 202335Pbar Longitudinal Distribution

Paul DerwentApr 20, 202336Antiprotons & the Collider

From the H- source, Linac, booster, Main Injector 120 GeV protons

on the target From the target:

8 GeV antiprotons through the Debuncher & Accumulator

Send them off to the Tevatron & D0 & CDF