Stochastic CoolingStochastic Cooling • • • • • •

(1)

Aim of Cooling :

⇒

^Reduction ^of ^emittances

(transverse and longitudinal).

⇒

^Increase ^of ^phase ^space ^density^.

•

^Basic ^p^rinciple ^of

stochastic cooling

•

^Histo^ry

•

^Simple ^theory ⁱⁿ ^time ^domain

•

^Sto^chastic ^coôling ât ÂC/AA

•

^Sto^chastic ^coôling ât ÂD

•

^Hints ^fo^r ^op^eration

(2)

(transverse) stochastic cooling

Measure position and apply a correction.

PickUp

Kicker

x x’

One particle passing with maximal position at PU

PickUp

Kicker

x x’

Particle crossing the reference orbit at PU

⇒

^Phase ^from ^PU ^to ^Kicker ^:

90 ^o

^plus ^a ^multiple ^of

180 ^o

(3)

Pick Up

Kicker

x x’

: mean position in phase space

A sample of particles passing the cooling system -

correction of the mean position.

Cooling is hampered by ... noise :

•

^from ^other ^particles ^'in ^the ^sample ^slice'.

•

^noise ^from ^pick-up ^and electronics.

Longitudinal cooling :

•

^Palmer ^cooling ^(not ^used ^for ^AD) ^: ^Measure

mean energy with a PU at a location with dis-

persion and correct for it.

•

^Filter ^method (quantitative description dicult in time domain).

(4)

1968

^Idea ^of ^sto^chastic ^cooling ^by ^van ^der ^Meer.

1972

^First observation of Schottky noise at ISR.

1972

^Theo^ry ^of ^transverse ^sto^chastic ^co^oling.

1975

^First ^exp^erimental demonstration at ISR.

1975 ¯p

accumulation schemes for ISR and SPS.

1975

^Filter ^method ^for ^momentum ^co^oling.

1977 & 1978

^Renement ôf ^theo^ry ând ^detailed êx-

perimental verication at ICE.

1981 & 1982

Accumulation of several

10 ¹¹ ¯p

^'s ⁱⁿ

AA from batches of several

10 ⁶ ¯p

^'s.

1986 & 1987

Construction of AC.

1983

Observation of W and Z bosons (carriers of the electro-weak force) in Sp

¯p

^S

1995

Observation of top-quark at Fermilab.

(5)

Schottky scan for

2 × 10 ⁸

^protons ^at ^2.0 ^Ge^V/c ⁱⁿ ^AD

ξ ’

ξ

( ε

σ

)

^1/2

= ( <ξ

²

> )

^1/2

( ε

σ

)

^1/2

= ( <ξ

^{’ 2}

> )

^1/2

Particles in normalized phase space

(6)

Bandwidth and Kicker response

•

^Transmission in frequency range

f ₁

^-

f ₂

i.e. bandwidth :

W := f 2 − f 1

nominal for AD : 900 - 1600 MHz (ampliers)

realistic : 1200 - 1500 MHz (entire system)

•

Ârrival ât ^the ^kicker ^with â ^delay

t _d

⁽

.315µs

ⁱⁿ

AD at 3.57 GeV/c)

•

^Response ⁱⁿ ^time ^domain

Band from 0 MHz to 700

MHz

frequency / MHz

Transmission

0 500 1000 1500 0.312 0.314 0.316

time/

µ

s

Signal

0 1/2W

Band from 900 MHz to

1600 MHz

frequency / MHz

Transmission

0 500 1000 1500

^0.312 ^0.314 ^0.316

time/µs Signal 0

1/2W

(7)

•

^One ^sample ^contains

N _S = N(1/2W )/t rev

^out

of

N

^ions ⁽

t _rev

^is ^the ^revolution ^period).

•

Measurement (normalized phase space

⇒

< ξ ² >=< ξ ² >= ε σ

⁾ ^gives

¯ξ+ ξ _n

^, ^where

ξ _n

^is

due to noise.

•

^With ^a ^correction

g(¯ξ+ ξ n )

⁽

g

^is ^the ^a ^factor)

the relative emittance change is :

∆ε σ

ε _σ = < (ξ − g(¯ξ+ ξ n )) ² > − < ξ ² >

< ξ ² > + < ξ ² >

= −2g ¯ξ ² − 2¯ξξ _n + g ² (¯ξ+ ξ _n ) ²

< ξ ² > + < ξ ² >

•

^Cooling ^rate ^is ^the ^mean ^relative ^emittance ^de-

crease per turn divided by

t _rev

^:

1 τ = 1 t _rev

−∆ε _σ

ε _σ = 1 t _rev

g

N _S − g ²

2N S (1 + U)

with

U :=< ξ _n ² > / < ¯ ξ ² >

^the ^noise ^power ^to

signal power ratio. Note

< ¯ ξ ² >=< ξ ² > /N _S

•

^Optimum

g = (1 + U) ⁻¹

(neglecting mixing)

gives :

1 τ =

W N

1 1 + U

(8)

•

^Correct ^dependancy ôn ^number ôf îons

N

^and

bandwidth

W

^.

•

^Impo^rtance (qualitative) of mixing :

–

^Good ^(wanted) ^mixing

M

^Kick^er

→

^PU.

–

^Bad ^(unwanted) ^mixing

M ˜

^PU

→

^Kicker.

•

^Mixing ^dicult ^to ^evaluate numerically.

•

^Filter ^method ^fo^r longitudinal stochastic cooling dicult to explain.

A more precise theory in frequency domain :

•

^Includes ^lter ^metho^d ^for longitudinal cooling.

•

^Explains attenuation of Schottky sidebands when the cooling loop is closed.

•

^T^ransverse ^co^oling ^time ⁽

M

^and

M > 1 ˜

⁾ ^:

1 τ = W

N

2g(1 − ˜ M ⁻² ) − g ² (M + U)

•

^Optimum ^:

_τ ¹ = ^W _N ^{(1− ˜} _M+U ^M

⁻²

⁾

²^.

(9)

using the simple formula without mixing

- Bandwidth

W

⁼ ³⁰⁰ ^MHz ^(more ^realistic ^than

the nominal 700 MHz).

- Number of

¯p

^:

N = 5 × 10 ⁷

- Noise to signal ratio :

U =

¹ ^to ³⁰ ^(at ^large

and small emittances respectively).

- Expect to cool from

200 π µ

^m ^to

5 π µ

^m ^:

ln

200π µ

^m

5π µ

^m

5 × 10 ⁷

300 × 10 ⁶ s ⁻¹ (1 + 10)

=τ=1.8

^s

= 7

^s.

Optimistic Value not including eect of mixing.

- Gives correct order of magnitude - design re-

port states 20 s. But we were lucky to obtain

such a good result with the simple theory.

- Be careful applying this formula !!!

- Eect of mixing can lead to signicantly dier-

ent results !!!

(10)

History of rst large scale application

•

^Increase ^of (6-dimensional) phase space density by a factor

10 ⁹

^!

• ¯p

^collection ⁱⁿ ^AC ^:

–

^Large acceptances in all 3 planes.

–

^Fast ^co^oling ^of

¯p

^beam.

⇒

^Large ^bandwidth overlapping 3 bands with distinct hardware.

–

Ôperated ât ônly ône ênergy^.

•

Accumulation in AA :

–

Accumulation of batches injected from AC.

–

^Separation ⁱⁿ ^momentum ^of ^stack ⁽ⁱⁿ ^AA)

and batch coming from the AC.

–

^Coôling ôf ^newly înjected

¯p

^into ^stack.

–

Deceleration (with RF) to bring new batch close to stack.

–

^Stack-tail ^system ^co^ols ^new ^batch ^into ^stack

–

^Stack-core ^system ^k^eeps ^stack ^small

(11)

(12)

(13)

•

^Slower ^Coôling ^w.r.t. ÂC ^can ^be âccepted

⇒

^Band ¹ ôut ôf ^three ⁱⁿ ÂC îs ^sucient.

•

^Three ^coôling ^systems ûsing ^two ^PU ând ^two

kickers (horizontal and vertical). Both PU's

and both kickers for longitudinal system.

•

^Cooling ^at ^two ^distinct ^momenta ^(and ^thus ^ion

velocities) implies :

–

^T^wo ^distinct ^delays ^necessary ^(for ^all ^three

system)

⇒

^(pa^rtly) ^distinct transmission paths.

–

^Dierent ^revolution ^harmonics

⇒

^two ^dis-

tinct notch lters for long. cooling.

–

^PU constructed for injection momentum :

⇒

^at ^2.0 ^Ge^V/c ^: ^lower sensitivity and thus reduction of signal/noise

–

^Fo^r ² ^Ge^V/c ^delay ^correction ^power ^am-

pliers grouped in 3 blocks

•

Programmable PU and Kicker movement as in the AC. (Only PU movement used now)

•

Programmable phase invariant attenuators (static and dyn.) to optimize gain.

•

Programmable delays (gain invariant) to optimize phase.

(14)

(15)

Setting of following example such that :

⇒

^Cooling ⁱⁿ âll ^three ^planes ât ^3.5 ^Ge^V/c ând ât

2.0 GeV/c.

•

^Check ôptics ôf ÂD, ê.g. ô^rbits ând ^wo^rking

point (tunes).

•

Âre âmpliers ô.k. ^(DC ^current) ^? ^They ^should

be used in adjacent pairs.

•

^Check ^system ^switches ^on ^stochastic ^co^oling

control display.

•

^Pick-up ^movement. ^Closes ^slo^wly ^during ^3.5 ^Ge^V/c

attop, closed at 2.0 GeV/c and open otherwise.

Position reading possible via display ("knob").

•

^Dynamic ^attenuato^rs ^: ^function ^increases ^with

time to decrease gain. No way to check exe-

cution of programmation from ACR.

•

^Check ^timings ^and ^make ^sure ^that ^the ^system

works only during the plateaus required.

Stochastic CoolingStochastic Cooling • • • • • •

⇒

⇒

•

•

•

•

•

•

⇒

90 o

180 o

Pick Up

Kicker

•

•

•

•

1968

1972

1972

1975

1975 ¯p

1975

1977 & 1978

1981 & 1982

10 11 ¯p

10 6 ¯p

1986 & 1987

1983

¯p

1995

2 × 10 8

ξ ’

ξ

( ε

)

= ( <ξ

> )

( ε

)

= ( <ξ

> )

•

f 1

f 2

W := f 2 − f 1

•

t d

.315µs

•

frequency / MHz

0 500 1000 1500 0.312 0.314 0.316

time/

s

0

1/2W

frequency / MHz

0 500 1000 1500

•

N S = N(1/2W )/t rev

N

t rev

•

⇒

< ξ 2 >=< ξ 2 >= ε σ

¯ξ+ ξ n

ξ n

•

g(¯ξ+ ξ n )

g

∆ε σ

ε σ = < (ξ − g(¯ξ+ ξ n )) 2 > − < ξ 2 >

< ξ 2 > + < ξ 2 >

= −2g ¯ξ 2 − 2¯ξξ n + g 2 (¯ξ+ ξ n ) 2

< ξ 2 > + < ξ 2 >

•

t rev

1

τ = 1 t rev

90 ^o

180 ^o

10 ¹¹ ¯p

10 ⁶ ¯p

2 × 10 ⁸

f ₁

f ₂

t _d

N _S = N(1/2W )/t rev

t _rev

< ξ ² >=< ξ ² >= ε σ

¯ξ+ ξ _n

ξ _n

ε _σ = < (ξ − g(¯ξ+ ξ n )) ² > − < ξ ² >

< ξ ² > + < ξ ² >

= −2g ¯ξ ² − 2¯ξξ _n + g ² (¯ξ+ ξ _n ) ²

< ξ ² > + < ξ ² >

t _rev

τ = 1 t _rev

−∆ε _σ

ε _σ = 1 t _rev

N _S − g ²

U :=< ξ _n ² > / < ¯ ξ ² >

< ¯ ξ ² >=< ξ ² > /N _S

g = (1 + U) ⁻¹

2g(1 − ˜ M ⁻² ) − g ² (M + U)

_τ ¹ = ^W _N ^{(1− ˜} _M+U ^M

⁾

N = 5 × 10 ⁷

ln

5 × 10 ⁷

300 × 10 ⁶ s ⁻¹ (1 + 10)

10 ⁹