Hereward - CERN Document Server...H.G. Hereward 1. General 2o Luminosity 3. Limitine cases 4. Application to tolerances 5. Acimowledgement s 1 2 3 5 7 Appendix 1. The integrations

CERN/MPS/DL 69-15 5 November 1969

HOW GOOD IS THE R.M.S. AS A MEAS;JRE OF BEAM SIZE ?

H.G. Hereward

1. General

2o Luminosity

3. Limitine cases

4. Application to tolerances

5. Acimowledgement s

1

2

3

5

7

Appendix 1. The integrations 8

Appendix 2. Extrer:ia 13

Appendix 3. Mismatching an:. re-matc~1ing 15

PS/8038

CERN/MPS/DL 69-15 Erratum 18 November 1969

HOW GOOD IS THE R.M.S. AS A MEASURE OF BEAM SIZE?

B.de Raad has pointed out a mistake in the "Uniform density" entry on page 6 of the above report. Please replace the table and the line following it by:

factor according to

<Xf > i<d2> <x 2 >cc ( 17) H+K

Uniform density*) --

0.2500 0.1250 0.3750 0.8165 0.8201

Parabolic density 0.1667 0.1250 0.2917 0.7559 0.7668

Gaussian beam 0.5000 0.1250 0.6250 0.8944 0.8907

The biggest discrepancy is in the parabolic density case, and is 1.4%.

H.G. Hereward

1. General

The mean sq·.iare displace:nent <y 2 > , and its square

root (the r.m.s. displacement) are very convenient quantities to

~se as measures of beam width, especially in the case that the

shape of the particle distribution is not lmow~1. This ..L.s because

the way they increase in the presence of a statistical disturbance,

li~e random kicks or misalignments or gas scattering, is independent

of the distribution shape.

It is worth insisting on this point, as it is sor:ietimes

supposed that tl~ property is limited to Gaussians. Let y 1 be a

particle's initial position, let y 2 be its position after receiving

an additional displace~ent d :

Y2 = Y1 + d

Norr look at

Take the ave rage

ses where y 1 and d are uncorrelated - for

, ~f the kick that a particle receives is independent

of its y1 - then we have

2 <Y 1. d> = 2 <Y 1 > < d >

( 1)

( 2)

( 3)

(4)

- 2 -

ani it is s:1fficient to ha ·.;e <. y 1> = 0 or <d.> = 0 to give

<y~> = 2 2 <y1> + <d. > ( 5)

Thus the well-known process of add.ing together "q'..ladratically"

the r.m.s. initial displacement and. the r.rn.s. extra displacement

is valid if they are uncorrelated and at least one has zero average.,

2 .. Luminosity

The main purpose of this note is to consider

a measure of' beam height in intersecting storage rings,. We are

thinldng of rings in w1dch the beams intersect at an angle in the

horizonta:'.. plane, an:l take the case where the two beams have the

same z-d.istrib~tion. Call it p(z) , normalised to unity :

+oo

as

/_00

p( z) dz = (7)

By defini tio~'l

+oo

< z 2 > = r z 2 o c z) il:~ ( 8) - 00

And for given currents tl!e interaction rate is proportional to*)

L = }' +oo

*) Looking at ( 9) a:'.1.d the

< f> =

general iefinition

r +oo f(z) p( z;) dz

,! -oo

one sees that L = <p >

( 9)

of an average

and it is correct to think of L as the p of the target beam,

as seen on average by the bombardin8 beamo

- 3 -

For a distribution of given for;n, just scaled, 1 is

inversely proportional to its width, so is then constant.

How constant it is for differen~ distributions is shown below (more

deta~ls in Appendix 1)

Rectangle, 1 <z2> ~ = o.2887

Parabola, " = 0.2683

Cosine, " = 0.2685 ( 10)

Tria11c~.0, " = 0 .. 2722 Ga-.issian, II = 0.2821

The range of the values listed is ~ 3.?fo.

The same considerations can be presented in terms of the

effective beam heisht as defined by Darriulat and Rubbia 1). Whe,-i the

two beams have the sa11e nor:nalised distribution p( z), t!lis definition

becomes

( 11)

-oo

so it is just 1/1 o Thus the tabulation (10) says that

( 12)

within a range of ~ 3.7% for the distributions listed ..

3. Limiting cases

It wo-..ild be even more in Uc.Li.d prove 1.

that L<z 2 > 2 falls within a reasonably narrow range for all

possible distributions, and one can show (Appendix 2) that it is

never less than 0.,2683 for any p ?: 0 distribu~ion .. Unfortunately

it does not have an upper bound: even if we restrict ourselves to

symmetrical distributions with p ?: O, only one peak, an:'!. a finite

- 4 -

range of z, we can inve21t distributions like that of Figo1 which .1.

can have L <Z 2> 2 as large as we like

Fig.1

(a certa:'.. n fraction of the particles are spread out over a co:cstan-t

width, an:i cm:tribute a coGstant; a:::o:.mt to <Z 2 > , w!:iile the remainint;

fraction are in a peak that contributes increasingly to 1 as it is

made narrJcner).

However, it seerns reasonable to assu:ne that the bea11s put

into the ISR will be so:nething like Gaussian, with the ta.ils clipped

o:f'f; or, if they have been clipped more severely, like one of the

distributio.1.s that I have listed. If it is reaso·1ably narrow, gas

scatterifig w:'..11 make it more and more Gaussiai1 '.mtll the aperture

begins to affect the tails significar:tly. The va:'..ues of L <Z 2 > t for a trcmcated Gaussia21, fro11 the formulae in Appendix 1, are:

<Z2> .l,.

L

-I> 0 -I> 0.2887 highest

1,.63 0.2694 minimum ( 14)

~ 00 n,2821

As an alter 1.,!' :1 old beau, whose shape

is the result of scattering within an aperture that imposes a

p = 0 boundary co:iditio:i at :!:: Zo, I have considered the family

p == A cos 1TZ 2z 0

- 5 -

+ B cos~ 2z 0 '

-Zo<Z<Zo ( 1 5)

with B decreasing from A/3 (it ca-"lYl.ot be more, because p ~ o)

to zero. One finds

B/A L <Z 2 >~

1/3 0.2734

0.27 0.2747 maximum ( 16)

0 0.2685 lowest

Thus the luminosity can be calc~lated from

( 17)

or the effective beaD height from ( 12), with reas· 1nable confidence

of being within ± 3.7%, and with the certainty that they are not

more than + 3.7% too optimistic.

4. Application to tolera_~ces

Consider a bean with a know:.i ?

<Y-> u~dergoing some

injection, ejection, or deflection processo Let it suffer some

unwanted tra...~sverse displacement do The immediate effect on

<Y 2 > is given by (5) .. However, for the primary beam in ma.chines

and transport channels we are usually interested in the effect

much further downbea.o, w:1ere filamentation2) has randomised the

phase. This *) phase-randomised y has the property

*) It is referred to the plane wl1ere is defined and where

( 19)

a. occurs.

- 6 -

A co:nparison between the predictions of (19), (17) ani the precise

computation of Hanney anl Keil 2) gives for the factor of luminosity

reauction at d = 0.5 factor according to

<Xf > 1 2 <X2> ( 17) H+K ~ <d >

00

Uniform density*) 0.3333 001250 0.,4583 o.8528 0.8201

Parabolic density 0.1667 0.1250 0.2917 0.7559 0.7668

Gaussian beam 0.5000 0.1250 o.625 o.8944 o.8907

The biggest discrepancy is in the uniform beam case, and is just over 3%.

The effect of a deflection-error ¢ is analogous to (19):

( 20)

Both (19) and (20) make the assumptions that (i) there are

no matching errors, (ii) the mat chins is calcuh.ted for the beam

without the dipole perturbations d or ¢ . There is more about

these matc!iing questions in Appendix 3 b·1t it is worth pu"'cting down 2 here for co:npleteness the effect on <Y > of an independent mis-

**) 1 match., Take a mismatch characterised by the factors >.., ~ - -. Acting on any matched distribJtion it gives:

1 2

( 21)

This shows that sizeable mismatches have rather little

effect; for example ~ = 1.4 deforms a circle into a 2:1 ellipse,

but increases <y 2 >i by only 12%.

*) In this table "uniform" and "parabolic'' -refer to the phase-plane density as a function of amplitude. Th~s is in contrast to the terms used on page 3 •

**) One of them is the mismatch param3ter called z in reference 3) c

- 7 -

Note that deflection errors are (quadratically) additive,

while mismatches ~re multiplicative; if we have a mixture, the order

must be respected. Early deflection errors get multiplied by any sub­

eequent mismatches.

5. AcknowlAdgements

I should like to thank Bovet and Keil for useful comme:::its o

- 8 -

Appendix

THE INTEGRATIONS

The normalised rectangular distribution is

p = 2zo

Hence <Z2> -- J +Zo

- 2zo -zo

And 1 1 l +Zo

= 4z5 .I

-Zo

(The effective height

this distribution).

-1 1

So =

in

z 2dz =

dz =

+ Zo *)

1 z~ 3

1 2zo

is equal to the full width

= /J. 6

2zo for

The distribution expressed in terms of its r.m.s. width is someti~es

01seful. It can be o-btained by s·,1bsti tuting

into p •

*Ywe ta1rn

Zo = =

* * * * *

1.7321 z rms

The normalised parabolic distribution is

...l (1 z2 ) p = 4-zo - -ze in -zo ~ z ~ Zo

---------------p = 0 where not otherwise state do

Hence

And 1 9 = 16~-

So 1 <Z 2 >.;. =

- 9 -

I +zo ( z 2_ ~ ) dz =

-zo

J +zo (1 2z 2

z 4) -~ + -r dz

Zo Zo -zo

~ = :d2. 25

1 zJ 5

3 = 5zo

The distribution expressed in terms of its r.m.s.

can be obtained by substi bting into p :

Hence

Zo = J5 z rms = 2.2361

* * * * *

z rrns 0

The norrna::.ised cosine distrib·.ition is

p = 1T cos !!...E. 4-zo 2z 0

in -zo ~ z ~ zo o

-zo

= 2~s [ 2x cos x +

,0 ( 1 - ~ )

1TZ dz 2z 0

- 10 -

+Zo

And L 11'2 f cos 2

1T'Z dz = ~ 2zo -zo

11'2

= 16zo

So = (, 8 )~ 1 - ?

The distribution in terms of its r.m.s. can be obtained

by s'.lbsti tu ting into p :

Hence

And

Zo = (1 8 )-~ - -~ z \ 1T' rms

* * * * *

= 2.2976 z rms

The normalised triangular distribution

1 p = Zo

<Z2>= 2 Zo

L 2

= ;g

(1 - /~ 0 /,)in

Zo

r (z2_ z3

J Zo 0

z

f 0 (1 0

2 /-1 3~6

) dz

2z Zo

-zo ~ Z ~ Zo

1 z~ = 7

0

z2 ) dz

2 + '"""':! = 3 Zo

is

1 Zo

The distribution in terms of its r.m.s. can be obtained

by substi~uting into p

Zo = J6 z = rms 2.41+95

* * * * *

z rms

- 11 -

The full Gaussian :listribution is a limiting case of

the truncated Gaussian, so we consider them together.

Put

p = A

Fexp -z2 2 a in -zo ~ z ~ Zo •

At z 0 ~ oo this is alreaiy normalised if' A= 1, and bas <z 2> =a,

but for other z0 the normalisation requires

where a is the "ncirc.ial probability integral"

a(x) 1 r +x -t2

= $ exp 2 dt I . -x

We have

<Z2> A r +zo z2 -z2 =

J2rra exp dz

2a . -zo

Using

f -t2 I exp 2 dt = ~~.ex( t)

-t2 - t exp 2

We get

- 12 -

This clParly tends to a as z0 tends to infinity. Wit:1 a little

more work one can chec~ that it tends to z~/3, the rectangular­

distribution value, if z 0/a~ is s~all.

Now

1 =---2 1T ~

-z2 dz exp

a

The integral can be expressed in terms of a, but it is more

convenient to use the other forill of error function 4)

to give

1 Erf(x)= J;r

A2

+x

r -x

1 = 2./rr a

tends to

= 0.28209

exp(-t 2 ) dt

and at the other limit again the rectangular value (o.28863) can

be verified., Between these two there is a minimum which can be

found. by tabulatio!l. a:'ld inverse interpolation.

* * * * *

.. 13 -

Appendix 2

The extrema of can rnost easily be investigated

by abandociine:; the restriction that p is normalised. Then we define

.I = J +oo p( z) dz ( 2, 1)

-oo

and we have

+oo

<z2> = I-11 z 2p dz, = I-1 J, say

-oo

and +oo

1 = I- 2 ( p 2 dz, = I- 2 K, say

We shall look first fo:' the minimum of L 2<z 2> , so of

F = J ( 2 ,2)

Cons:.der a sr:iall chaii3e op of p over an interval dz located

at z g The consequental changes in the integrals are

oI = op dz

oK = 2pop dz

ani so

- 14 -

z2 +-J

+ !±E. K ) Foo dz

Physical distributions have p ~ O for all z, so we can assume

I, J, K, F are all positive. If F is minimal, 8F is zero

or positive for any permitted change, so we must have:

-5 + z 2 + !±£. _ I J K - 0

for all z where p > O

and -5 + z 2 + !±E. ~ I J K 0

for all z where p = O ,

The parabolic distribution

K ( 5 z 2) • P = 4 1 - ~ in ' z 'j 5J/I

p = 0 outside

satisfies these conditions, and it is straightforward to verify

that it is consistent with its own I, J, and K, provided they

satisfy F = 32/53 •

A maximum of F would have to satisfy (2,4), and

( 2 '3)

( 2 ,4)

( 2 '5)

(2,5) with the inequality sign reversed, aYJ.d there is no p ~ 0 solutiono

- 15 -

Appendix 3

We suppose there exists a transformed phase-plane in

which the equi-density curves are circles. Take all the particles on

the circle of radius R,

= R cos 0

= R sin )6

with 0 running through an in'.:erval of 27T. Averaging over these

particles o~e has

2 2 R2 <Xt + P1> =

Now take a mismatch matrix 3)

( 0

The matrix of any equivalent mismatch oan be expressed as th:l,s

matrix together with rotations, and rotations do not change

x 2 + p 2 • The circle becomes

and averaging over 27T in 0 one has

- 16 -

Since the factor by which <X 2 + p2 > increases is independent of

R, it applies to the whole distribution, independently of its kind.

* * * * *

If a bea~ suffers a rand.om displacement (or deflection)

whose r.m. s .. is know:i and is s'.lbstantial compared. with the beam width,

it becomes interesting to allow for it in the matching process .. So

far we have explicitly excluded this trick (page 6 ), but here we

write down what it giveso

Again ta~<:e a phase-pla.1e in which the equi-d.ensi ty

curves are circles

=

Consider a random displacement d

= +

< 2> <X12> P2 =

1 2 <Rf>

Ordinarily ~ < R~> gives the phase-randomised. effect of this,

yielding (19) but let us instead apply a re-matching transfor­

mation

- 17 -

( "' 0 ) 0 /\. -1

so that

<X~> = /\. 2<Xf> + "'2<d2>

2 /\.""'2 <Xf> <p3> =

and choose A to make this circular:

The result is

Taken as far as the <d2> term in the Taylor expansion this is the

san1e as ( 19) that is, re-matching has only a secon:i order effect.

But increasing <d 2 > ma~es a gradual transition to "matching the

tolerances" instead of "matching the beam" • We give below 3. few

comparisons between the blow-up factors (for <x 2 > ; ) witho~t 00

re-matching

and with re-matching

... 18 -

ana the estimated gain of luminosity due to re-JIBtching

d /x without with gain rms rms

0.5 1.061 1.057 0.3%

1 1.225 1.189 3.a% 2 1. 732 1 .4-95 16 % 4- 2u236 10 732 29 %

10 7.141 30170 A factor 2.2

These numbers are independent of the kind of distributiono

... 19 -

References

1) Darriulat and Rubbiao CERN internal document 68/340/5 SIS/si

See also S.Va.n der Meer ISR-P0/68-31.

2) Hanney and Keil. "How do angle and position errors •••••• ?"

CERN/ISR-TH/69-44.

3) HaCJ.ney and Keil. "Ho71' does betatron mismatching affect ••••••• ?"

CERN/ISR-TH/69-32.

4) Tables of Probability Functions Vol.I and II. A.N.Lowru1

New York 1941/42.

Distribution: (open)

Scientific Staff MPS, ISR, SI Divisions. P.Darriulat B.D.Hyams C .Rubbia A.Wetherell K.Winter

•