IUH =

BNL‑18395

CRISP 73‑19

NG‑269

BROOKHAVEN NATIONAL LABORATORY

Associated Universities, Inc.

Upton, L.I., N.Y.

ACCELERATOR DEPARTMENT

Informal Report

STOICHASTIC COOLING IN GENERAL AND ITS APPLICATION TO

p STORAGE IN PARTICULAR

R. B. Palmer

October 15, 1973

[N 0 T I C E]{.underline}

> This report was prepared as an account of work sponsored by the United
> States
>
> Government. Neither the United States nor the United States Atomic
> Energy Com-
>
> mission, nor any of their employees, nor any of their contractors,
> subcontractors
>
> or their employees, makes any warranty, express or implied, or assumes
> any legal
>
> liability or responsibility for the accuracy, completeness or
> usefulness of any
>
> information, apparatus, product or process disclosed, or represents
> that its use
>
> would not infringe privately owned rights.

CRISP 73‑19

R. B. Palmer

> [Stoichastic cooling in general and its]{.underline}
>
> [application to p storage in particular]{.underline}
>
> I present here some basic formulae for stoichastic cooling. The

formulae are not exact but suffice to get order of magnitude results.

> [1. Simplest Calculation]{.underline}
>
> Consider a ring of some large circumference S containing N~T~
> particles traveling near the velocity of light. Now suppose that we
> have some scheme that detects the mean lateral position (Y) of an
> element of the beam at a **"**pick up station\" of length . The number
> of particles in that element we call N.
>
> N = (1)

If the signals corresponding to such displacements (y) are amplified and
used to correct these displacements at a \"corrector station\" at a
point further round the ring, then the mean squared displacement a 2
will be slowly reduced. First we assume:

> 1\) negligible noise in the amplifier;
>
> 2\) adequate band width, ie band width F
>
> F 2c/ cycles/~sec~
>
> Where c is the velocity of the particles;
>
> 2\.

3)  particle transit time between pickup and

> corrector station constant to a time small
>
> compared with /c.

4)  transit time for a complete rotation around

> the machine must have random fluctuations large
>
> compared with /c.
>
> Given these conditions we can now calculate what might be called the
>
> ideal stoichastic Fooling rate. Let y~i~ be the displacement of the
> ith par‑
>
> ticle. The mean square displacement will be
>
> \(3\)
>
> The mean positions detected
>
> \(4\)
>
> will themselves have a distribution and will have a mean square value
>
> \(5\)
>
> If the signals corresponding to these mean positions are amplified by
> a factor
>
> \(a\) and used to correct the trajectories, then the y position of
> each particle
>
> after correction will be y~i~ ‑ a y and the resultant mean square
> displacement '
>
> will be given by:
>
> 3\.
>
> On average
>
> ie on average the mean square deviation is reduced by the fraction
>
> \(6\)
>
> Clearly maximum cooling is obtained with a = 1 and
>
> \(7\)
>
> If, as at the ISR, NT = 4 10^14^, = 3 cm, F = 1GHz, then N
> [\~]{.underline} 10 ^10^ and f =
>
> 10^‑10.^ The time τ reduce a2 to about 1/e is then 10^10^ x 2 sec
> [\~]{.underline} 6 hours.
>
> The CERN expected time T is 24 hours but it turns out that at CERN our
> assump-­
>
> tion (3) is not satisfied, i.e. there is significant transit time
> spread be­-
>
> tween pick‑up and corrector stations.
>
> [2. Consider transit time spread between pick‑up and
> corrector]{.underline}
>
> If on average a fraction ε of the particles in the length at the
>
> pick‑up station remain in a length at the correction station then
> equation(6)
>
> must be modified (see Appendix I):
>
> \(8\)
>
> 4\.
>
> Maximum reduction is obtained for a=ε and for this value
>
> \(9\)
>
> At CERN ε [\~]{.underline} 1/2 and thus the cooling time τ is
> increased to 24 hours. This
>
> agrees with the CERN calculated number as reported in Months foreign
> travel
>
> report.
>
> At CERN the transit time fluctuations for a complete cycle are very
>
> large compared with /c and thus no correction is needed for this. Only
> in
>
> a machine near transition or a machine with very small momentum spread
> would this become a problem. If then the fluctuations in transit time
> are t and we define = then we find that the [maximum]{.underline}
> cooling fraction is given by
>
> Since is in general negligible we will not consider this effect
> further.
>
> [3. Consideration of Noise]{.underline}
>
> At CERN the noise is less than 10% of the pick‑up signal and has a
>
> negligible effect. The situation is very different, however, in the
> case of
>
> a p storage ring. In a p storage ring the total number of particles
> N~T~ is
>
> unlikely to be more . than 10^12^ finally and not more than 10^7^
> would be made per
>
> AGS pulse. If these were in a storage ring like the ISR the number N
> in a length
>
> of 3 cm would be only 200. Under such circumstances the cooling time
> is only
>
> 400 sec. Noise in the pick‑up system however makes this unattainable.
>
> 5\.
>
> Let b be the noise quoted in terms of the apparent RMS y position of
> one particle in cm\'s. At CERN b = 10^4^ cm, ie unless we have more
> than 10^4^ particles one could not find where they were to better than
> 1 cm!
>
> Given this definition of b we get instead of equation (8)
>
> \(10\)
>
> The maximum cooling is obtained for
>
> \(11\)
>
> If b is large then ti‑e reduction factor is given by:
>
> \(12\)
>
> and this is not a function of N!

If we again use the CERN signal to noise ratio we can then derive this
maximum cooling rate.

Two ways present themselves for further improving this rate:

> \(1\) lower the noise.

\(2\) increase the rate of corrections per second.

The former can possibly be achieved when it is noted that the band

width is not specified and need not be nearly as large as that used at
CERN.

Theoretically this does not lower the noise ratio since the plate
lengths have

to be lengthened as the band width is lowered. Despite this, the freedom
of

choosing the band width could probably allow lower noise amplifiers.

The second solution suggests the use of a [small]{.underline} ring.
Although the

number of particles per unit length will be larger this does not affect

the rate. The increased number of corrections per sec, however, will be
directly

useful. For instance, if we capture p's at 3 GeV we can have a ring 1/10
the

size of the ISR and get a cooling time of only 20 seconds. This
consideration

must, however, be weighed against the smaller total phase space capacity
of

the smaller ring and thus the smaller number of p\'s that can be
captured prior

to cooling.

We have shown that the cooling for a p ring is noise limited yet, when

figures taken from CERN are used and the best gain selected, then
cooling

times as short as 20 sec are still possible.

I conclude then that Stoichastic Cooling would allow the collection,
cooling and stacking of p\'s from some quite large number of AGS pulses.

> I 7.
>
> APPENDIX
>
> [Consideration of Transit Time Fluctuations between pick up and
> Corrector]{.underline}
>
> [Stations.]{.underline}
>
> We will consider the sample of particles (N 0 between the pick up
> elec­-
>
> trodes, and the sample of particles (N~4~) that lie between the
> corrector plates
>
> at a later time when the correction signal derived from the first
> sample is
>
> applied to the corrector plates. Ideally these two samples are
> identical and
>
> the time delay is obviously set to maximize the number of particles
> that are
>
> present in both samples. Let us now subdivide each of the two samples
> derived
>
> above (N~0~ and N~4~ )into subsets:
>
> Let N~1~ be the number of particles found in both samples N~0~ and
> N~4~.
>
> Let N~2~ be those in the sample N~0~ that are not in N~1~: N~2~ = N~0~
> ‑ N ~1~
>
> Let N~3~ be those in the sample N~4~ that are not in N~1~: N~3~ = N~4~
> ‑ N~1~
>
> We will denote sums over the samples N~k~ by ε~k~ and define S~k~ =
> ε~k~ y~i­~
>
> The correction signal is now given by the amplification factor(a)
> times
>
> the mean position of the particles between the pick up electrodes;
> i.e. correction
>
> signal = a S~0~ /N~o~ = a (S~1~ + S~2~ )/ N~O~. This signal is then
> used to correct the
>
> positions of all the particles between the corrector plates. Positions
> (y~i~)
>
> of the particles i in the sample N~4~ are then changed to y~i~ ‑
> a(S~1~ + S~2~ )/ N~0~.
>
> The mean square deviation after correction (')^2^ is then given by:
>
> 8\.
>
> Now
>
> Now we must consider what the average or expectation values of the
>
> Sums S are when averaged over some number of cycles. The mean square
> expectation value of any sum S~k~ is given by
>
> However the expectation value of the product of two uncorrelated
> samples,
>
> e.g. N~1~ and N~2~ is zero
>
> \< S ~1~ S~2~ \> = 0
>
> Thus
>
> 9\.
>
> We now note that N~4~ [\~]{.underline} N~o~, and define ε by N~1~ = ε
> N~0~ . In the
>
> ideal situation of no transit time fluctuations ε = 1. As the
> fluctuations
>
> become significant so ε is reduced. At the CERN ISR ε [\~]{.underline}
> 0.5.
>
> So using this definition of ε we obtain.
>
> or if