# Presentation: 274532

## Stochastic Cooling in the Fermilab AntiProton Source

- Paul Derwent
- Beams Division/Pbar/CDF

## Stochastic Cooling

- Main Entry: **sto·chas·tic**
-                Pronunciation: st&-'kas-tik, stO-
-                Function:* adjective*
-                Etymology: Greek *stochastikos* skillful in aiming, from
-                *stochazestha*i to aim at, guess at, from *stocho*s 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: 2**cool**
-                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**_

## Why an Antiproton source?

- p pbar physics with one ring
- Dense, intense beams for high luminosity

## Luminosity HistoryCollider Run I

- It’s all in the pbars!

## Making 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%)

## Why 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

## Pbar Longitudinal Distribution

## Overview 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 MHz

## Idea Behind Stochastic Cooling

- Phase Space Compression:
- Dynamic Aperture: Area	where 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!

## 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

## Why 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

## Simple 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

## Stochastic 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

## Cooling 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!

## Momentum 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

## Simple Example

- Linear Restoring Force with Constant Diffusive Term (Electronic noise)
- Gaussian Distribution
- Inject at E> E0
- Coherent force dominates  --- collected into core!

- Simulation!

## Types of Momentum Cooling

- Filter Cooling:
- Use Momentum - Frequency map
- Notch Filters for Gain Shaping
- Debuncher
- Recycler
- Stack tail (as correction)

## Types 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)

## Momentum Stacking

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

## Van der Meer’s Solution

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

- Exponential Density Distribution generated by Exponential Gain Distribution
- Max Flux = (W2||*E**d*)/(f0p ln(2))

## Implementation 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

## Beam Pickups

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

## Beam Pickups

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

## Current Intercepted by Pickup

- In areas of momentum dispersion D
- Placement of pickups to give proper gain distribution

- Use Method of Images

## Accumulator Pickups

- Placement
- number of pickups
- amplification
- used to build gain shape
- Also use Notch filters to zero signal at core

## Accumulator Stacktail

- Not quite as simple:
- -Real part of gain cools beam
- frequency dependson momentumf/f = -p/p  (higher f at lower p)
- Position depends on momentumx = Dp/p
- Particles at different positions have different flight times
- Cooling system delay constant
**OUT OF PHASE WITH COOLING SYSTEM AS MOMENTUM CHANGES**

## Accumulator Stacktail

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

- Stacktail Design Goal For Run II
- 	Ed ~ 7 MeV
- 	Flux ~ 35 mA/hour
- Show simulation!

## Performance Measurements

- 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

## Performance 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

## Performance Measurements

- Best Performance:
- 39.9 mA in 4 hours
- Restricted by core-stacktail couplings

## 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

- Beam

- Since beam does not decohere,
- Carry information back to pickup
- 	Feedback!

- Schottky

- Stacktail

- Core

## Stacktail Schottky Signals

- Core

- Freshly injected
-  beam

- Later in cycle

- Stacktail Leg1

## Core 2-4 Schottky Signals

- Core

- Freshly injected
-  beam

- Later in cycle

- Stacktail Leg1

## Pbar Longitudinal Distribution

## Antiprotons & 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