NON-INTERLEAVED SEXTIJPOLF.: SCHEMES FOR LEP AND RELATED EFFECTS OF SYSTEMATIC MULTIPOLE COMPONE~rs IN THE LATTIG~ MAGNETS

A. Verdier L~P Division, CERN, CH-1211 Geneva 23

Summary

Non-interleaved sextupole arrangements (i.e. sex­tupole families free from geometric aberrations) have been used for the chromaticity correction o f LEP. It is shown here that care had to be taken for optimizing the sextupole arrangement. Under this condition, the per­formance obtained was substantially better than that of the standard interleaved scheme. On the other hand, the perturbation of the non-interleaved scheme by undesired multipoles helped to specify tolerances on the field quality of the LEP cell magnets.

Introduction

Non-interleaved sextupole schemes have been pro­posed1 for the chromaticity correction of large elec­tron machines such RS LEP. In such schemes, the sextu­poles are grouped into pairs of identical e lements s o that firstly only linear components sit between the sextupoles of each pair a nd secondly the betatron phas e advances in a pair are TI. Under these conditions, the non-linear motion is exactly localized inside the pairs and the complete machine behaves like a linear machine, i.e. betatron oscillations with any amplitude are stable .

In an actual electron machin e , the synchrotron os­cillations produce an uninterrupted change of the par­ticle momentum so that the phase advance between the sextupoles of each pair is practically never TI and the above nice property of the non-interleaved scheme breaks down, i.e. the betatron amplitudes associated with synchrotron motion are limited.

A method has recently been proposed2 for determin­ing the influence of this limitation on the luminosity of the machine. In the light of this analysis, which will be shortly recalled below, it is shown in this pa­per that different non-interleaved schemes may lead to considerably different performance. On the other hand, the reduction of performance associated with the intro­duction of perturbing multipole components can be es­timated, which is useful for tolerance specifications.

I . Analysis2 of performance limitation due to the stability limits

The stability limit of the non-linear betatron mo­tion is a geometrical property of a given machine. It can be conveniently expressed as a maximum stable emit­tance £(0) which is a function of the amplitude of the synchrotron oscillation 0 • For our purpose, this is re­stricted to the case where the vertical emittance is equal to half the horizontal emittance, which will be referred to as the emittance in what follows. The maxi­mum stable emittance is that of a particle whose trans­verse coordinates remain bounded over a certain number of turns. In our case, the traiectories are tracked for 400 turns with the program PATRICIA3.

As the actual standard deviations of the trans­v:rse and longitudinal particle distributions are func­tions of the beam energy and the damping partition num­bers, the latter can be adjusted as a function of the energy E so that an arbitrary number F1 of standard transverse emittance Ex is linked to an arbitrarv num­ber F2 of relative standard energy deviation a /Evia the function £( 0) by: e

(1)

It is thus possible to det e rmine th e emittance at each energy, but the analytical computRtion is onlv possible when the function E(Jx) can be obtained expli­citly2 (this is the case when the function E(o) is linearized). If we impose a beam-beam tune shift ( ~~ v = 0.03 in what follows) and an optimum vertic­al e~ittanc e , this enables to compute the luminosity. Below

4 this computation is made with the program REAM­

PARAM It is important to note that the energy defined by eq. (1) has a maximum value for a certain Jxm• and that Jx must be in the range 0.5 ~ Jx ~ 2.5. A certain range of energy standard deviations corresponds to the interval (0.5, Jxm) which in turn defines a useful range of the function £(0) on the stability limit curves. For energies corresponding to .Jx values smaller than 0.5, Jx is in fact maintained equal to 0.5 and the lu~inosit~ is scaled as E2, which implies a constant emittance •

2. Two non-int e rleaved schemes for LEP 11

(Version 115, 90° per cell, B~ 1,2 = 0.1 - 0.2,

B~ 1,2 = 1.6 - 3.2)

2.1 Scheme A

This scheme was so designed6 as to fill all pos­sibl e sextupole places starting from the first place clos e to the dispersion suppr essor. A half-octant is shown in Fig. 1 (LEP has four superoeriods, each being divided int o two symme trical octants).

lo,, . n insertion

1nteract1on

point

F D

Standard 90° cells ---'- :1 J:1 SF1 501 SD1 SF2 SF 2 502 502 SF1 SF1 501 501 I 1

dispersion

suppressor 2 cells

horizontal focusing vertical focusing

one cell without sextupole

I mid

octanf

Fig. 1 Non-Interleaved Sextupole Scheme A

The same situation occurs in the other half-octant, which has a different low-B* insertion. The sextupole strengths are computed with the program HARMON? by half-octant in order to have minimum derivatives of B with respect to momentum at the middle of the octant and at the crossings. It should be noted that the ver­tical chromaticity is the most critical one because of the small value of B~ at the interaction point. As a consequence, the D-sextupoles are at least twice as strong as the F-sextupoles and influence to a great extent the non-linear motion.

for scheme A, the strength of the pair S02 is large: 1.1 m-3 (length 0.76 m) compared to the value of 0.562 m-3 required for having o~ = 0 with two sextu-pole families in total. ,v

This large strength results in a quick decrease of the stable emittance when increasing the amplitud e of synchrotron oscillat~on (Fig. 3). Following the analy-

-173-

sis of Chapter l, the luminosity curve shown in Fig. 4 is obtained. The stability limits an<l the associated luminosity curve of the standard interleaved scheme 2

are given for comparison in Figs. 3 and 4. It should be noted that for this case, the linearization of the function E(o) is restricted to its useful range as de­fined in Chapter 1.

2, 2 Scheme B

A second non-interleaved sextupole scheme was made8 by moving the SDl pair close to the middle of the octant by one cell so that it becomes a SD2 pair (Fig. 2).

t Standard 90 ° cells

,. ~SF1 SF1 501 SD1 SF2 SF2 SD2 SD2 SF1 SF1 ISD2 SD21

\ow - n 1n sert1 on

F D

d1spers1on 12 .._cells suppressor

horizontal focusing vertical focusing

one cell without sextupole

mid

oct a nt

Fig. 2 Non-Interleaved Sextupole Scheme B

Thus, the phase advance between this new SD2 pair and the SD2 pair in the middle of the half-octant is 4n, The SF-arrangement has not been modified. The strength of the SD2's becomes 0.63 m-3 which results in stabili­ty limits larger than those of scheme A (Fig. 3). The consequence of this increase on the maximum operating energy and on the luminosity is large as can be seen in Fig. 4. Indeed, the maximum energy (Chapter l) scales with the abscissa for which E( o) = 0 if E( o) is repre­sented as a straight line with a constant slope, and the luminosity scales with E4 for a constant Jx, which is almost the case at these oarticular maximum ener­gies.

Sqor e root of the l

stable emm d tance x 10 over 400 turn s

0 0 0

0 0

"

I see sect 21 I}

I see sect 2.2 I

non- interleaved schemes

Standard interleaved scheme The cir cles represent t he stable emittance

Relative amplitude of the synchro tron oscillation ( 0/o)

-----~-+-~--''---~-+--Cl-I---~~-~--~--~

Fig. 3 Stable Emittance for Three Sextupole Schemes

3. Effects of Parasitic ~ultipole Components9 in the Non-Interleaved Scheme

Since the LEP structures used in the beam dynamics programs contain one sextupole close to each quadru­pole, it is very easy to use the sextupoles which are not excited in the non-interleaved scheme, for oerturb­ing the non-interleaved scheme by the introduction of multipole components inside the pairs. Figure 5 shows an example of an SD-pair with the non-excited sextu-poles.

40

30

20

10

I (mA)

interleaved

~ .... _.., \ -- \

-- 15

-- 10

B Intensity curves

I 5 Al interleaved

c:=----::_::-:::-:::~- - -"'"""'

50 60 70 BO 90 E( GeV)

Fig, 4 Luminosity vs. Energy for Three Sextupole Schemes

§l~I BENDINGI ~ [1J JBENDINGJ §] [~ JBENDINGJ ~ ~J JBENDINGj §] ['.]

SD missing SF missing SD missing SF SD

Fig. 5 A Non-Interleaved SD Pair

Of course, this is not a perfect simulation of multipole components in the lattice magnets, but this is a simple way to simulate these components in order to get the order of magnitude of the effect for a subse­quent tolerance estimate. The computations were made for scheme B which is described in Section 2.2. Keeping in mind that "integrated" means "integrated over two LEP cells", the results are as follows:

3.1 Sextupoles

If a horizontally focusing sextupole component of integrated value 0.006 m-2 is introduced inside all the sextupole pairs, the maximum energy is reduced by 3 % and the luminosity by 10 % in the overlapping energy range (Fig. 6).

If a vertically focusing sextupole of the same va­lue is introduced, the oerformance is almost not af­fected (Fig. 6). This can be explained by the reductio i;i of the maximum Sn excitation which mainly determines I the stability limit, as well as a partial compensation of the decrease of the vertical phase advance with mo­mentum in the SD-pairs. However, such a compensation cannot be oushed too far: a further increase of the perturbing sextupole strength leads to a decrease of the performance. Furthermore , the introduction of F­and D-sextupoles close to each F- and n- quadrupole for the exact correction of the linear variation of their gradients with momentum leads to a performance similar to that of the interleaved schemeR.

-174-

0

bO

l I a rbitrar y un its I

lF' n - 1nl r1 i e a ve~ sc-x tu po les . ,., ,--. r 1:-r 1 :1 r~r. t1 , 'n 1r1~1d e the pairs

Per t urb ir'J 0- <;f'xtupole

K'1i ! t i:= - 0003m-~

Fr. r fl.ir ~ rng F - sex t upo lp

Kt 1, l' t =-G003m 1

"(•

Marh1ne para meters

n; , c 1 r: , 1 6

Q, ' 94 2 Q' ' 90 35 Q, : 0 07

: .r ' 2 7 10 ; rod m

o If ' 1 6 10

E I Ge V I

-l-- ---90

Fig . 6 Luminosity of LEP vs. e nergy with the non-interleaved Scheme B perturbed with sextupoles. The integration symbol means that the perturbing component is inte­grated over two cells.

3.2 Octupoles

If a horizontally focusing octupole component of integrated value 0.12 m-3 is introduced inside all the sextupol e pairs, the maximum energy is r educ ed by l % and th e luminosity by 4 % in the overlapping energy range (Fig. 7).

L (arbitrary unit s l

+--60

·1on-1nter!raverl st:xlupcdes,

no perturbn1i on inside th e roirs

Per turb ing f - octupole

K'r' [eel(= 006rri l

Perturbinq 0- ocrupole

Kt'. lcrll = 006m-~

+-70

Scrne machine pararne ters

r.s 1r Fig 6

[ !Ge Vl

+-- - ----- 1-- --~

~o 90

Fig. 7 Luminosity of LEP vs. energy with the non-interleaved Scheme B perturbed with octupoles. The integration symbol means that the perturbing component is inte­grated over two cells.

If a ver tically focusing octupole component of the same value is introduced, the maximum e ne rgy is reduced by 5.5 % and the luminosity by 13 % in the overlapping energy rang e (Fig. 7).

Conclusion

Non-int e rleaved sextupole schemes are attractive for circular LEP-type machines, especially when consid­ering small momentum deviations of some permilles.

The rec ent possibility of analysing the influence of the stability limits on the ma chine performance2 al­lows a quantitative comparison between various schemes. It was thus possible to show th e importance of well choosing th e sextupole famili e s fo r the case of LEP 11.

The clear advantage of the best non-inter\eaved scheme over the standard interleaved scheme implied that the LEP sextupoles had to be designed so that they meet the requirements for the non-interleaved sextupole scheme.

Moreove r, the consequence of introducing perturb­ing multipol e components in the non-int erl eaved scheme could be quantified, which was extremely useful for specifying tolerances on the field quality of the LEP cell magnetslO,

I .

2 .

3 .

4 .

5 .

6 .

7 .

8 .

9 .

10 .

-175-

References

K. Br own and R.V. Servranckx, Chromatic correc­tions for large storage rings, IEEE Trans. on Nucl. Sci. NS-26, No. 3, p. 3598 (1979).

E. Ke il, High-energy performanc e of LEP, LEP note 337 (1981) - unpublished.

H. Wi edeman, Chromaticity correction in large storage rings, PEP-220 (1976).

M. Hann ey and E. Keil, BEAMPARAM - A program for computing beam dynamics and performance of e+e­storage rings, Div. Report CERN ISR-TH/79-29 (1979).

Compiled by A. Hutton, Parame t e r list for LEP ver­sion 11, LEP note 289 (1981) - unpublished.

E. Keil, Chromaticity correction in some variants of the LEP lattice, LEP note 350 (1981) - un­published.

M. Donald, Chromaticity correction in large storage rings, IEEE Trans. on Nucl. Sci., Vol. NS-24, No . 3, p . 1200 (June 1977).

A. Verdi er, La luminosite ~ haut e energie pour le LEP 11 avec a* nominaux suivant l es arrangements hexapolaires, LEP note 396 (1982) - unpublished.

A. Verdier, Effect of some systematic multipole imperfections in the LEP lattice, LEP note 429 (1983) - unpublished.

G. Guignard, Tolerances for the magnetic elements of the LEP lattice, Paper presented at this Con­ference.

sis of Chapter l, the luminosity curve shown in Fig. 4 is obtained. The stability limits an<l the associated luminosity curve of the standard interleaved scheme 2

are given for comparison in Figs. 3 and 4. It should be noted that for this case, the linearization of the function E(o) is restricted to its useful range as de­fined in Chapter 1.

2, 2 Scheme B

A second non-interleaved sextupole scheme was made8 by moving the SDl pair close to the middle of the octant by one cell so that it becomes a SD2 pair (Fig. 2).

t Standard 90 ° cells

,. ~SF1 SF1 501 SD1 SF2 SF2 SD2 SD2 SF1 SF1 ISD2 SD21

\ow - n 1n sert1 on

F D

d1spers1on 12 .._cells suppressor

horizontal focusing vertical focusing

one cell without sextupole

mid

oct a nt

Fig. 2 Non-Interleaved Sextupole Scheme B

Thus, the phase advance between this new SD2 pair and the SD2 pair in the middle of the half-octant is 4n, The SF-arrangement has not been modified. The strength of the SD2's becomes 0.63 m-3 which results in stabili­ty limits larger than those of scheme A (Fig. 3). The consequence of this increase on the maximum operating energy and on the luminosity is large as can be seen in Fig. 4. Indeed, the maximum energy (Chapter l) scales with the abscissa for which E( o) = 0 if E( o) is repre­sented as a straight line with a constant slope, and the luminosity scales with E4 for a constant Jx, which is almost the case at these oarticular maximum ener­gies.

Sqor e root of the l

stable emm d tance x 10 over 400 turn s

0 0 0

0 0

"

I see sect 21 I}

I see sect 2.2 I

non- interleaved schemes

Standard interleaved scheme The cir cles represent t he stable emittance

Relative amplitude of the synchro tron oscillation ( 0/o)

-----~-+-~--''---~-+--Cl-I---~~-~--~--~

Fig. 3 Stable Emittance for Three Sextupole Schemes

3. Effects of Parasitic ~ultipole Components9 in the Non-Interleaved Scheme

Since the LEP structures used in the beam dynamics programs contain one sextupole close to each quadru­pole, it is very easy to use the sextupoles which are not excited in the non-interleaved scheme, for oerturb­ing the non-interleaved scheme by the introduction of multipole components inside the pairs. Figure 5 shows an example of an SD-pair with the non-excited sextu-poles.

40

30

20

10

I (mA)

interleaved

~ .... _.., \ -- \

-- 15

-- 10

B Intensity curves

I 5 Al interleaved

c:=----::_::-:::-:::~- - -"'"""'

50 60 70 BO 90 E( GeV)

Fig, 4 Luminosity vs. Energy for Three Sextupole Schemes

§l~I BENDINGI ~ [1J JBENDINGJ §] [~ JBENDINGJ ~ ~J JBENDINGj §] ['.]

SD missing SF missing SD missing SF SD

Fig. 5 A Non-Interleaved SD Pair

Of course, this is not a perfect simulation of multipole components in the lattice magnets, but this is a simple way to simulate these components in order to get the order of magnitude of the effect for a subse­quent tolerance estimate. The computations were made for scheme B which is described in Section 2.2. Keeping in mind that "integrated" means "integrated over two LEP cells", the results are as follows:

3.1 Sextupoles

If a horizontally focusing sextupole component of integrated value 0.006 m-2 is introduced inside all the sextupole pairs, the maximum energy is reduced by 3 % and the luminosity by 10 % in the overlapping energy range (Fig. 6).

If a vertically focusing sextupole of the same va­lue is introduced, the oerformance is almost not af­fected (Fig. 6). This can be explained by the reductio i;i of the maximum Sn excitation which mainly determines I the stability limit, as well as a partial compensation of the decrease of the vertical phase advance with mo­mentum in the SD-pairs. However, such a compensation cannot be oushed too far: a further increase of the perturbing sextupole strength leads to a decrease of the performance. Furthermore , the introduction of F­and D-sextupoles close to each F- and n- quadrupole for the exact correction of the linear variation of their gradients with momentum leads to a performance similar to that of the interleaved schemeR.

-174-

0

bO

l I a rbitrar y un its I

lF' n - 1nl r1 i e a ve~ sc-x tu po les . ,., ,--. r 1:-r 1 :1 r~r. t1 , 'n 1r1~1d e the pairs

Per t urb ir'J 0- <;f'xtupole

K'1i ! t i:= - 0003m-~

Fr. r fl.ir ~ rng F - sex t upo lp

Kt 1, l' t =-G003m 1

"(•

Marh1ne para meters

n; , c 1 r: , 1 6

Q, ' 94 2 Q' ' 90 35 Q, : 0 07

: .r ' 2 7 10 ; rod m

o If ' 1 6 10

E I Ge V I

-l-- ---90

Fig . 6 Luminosity of LEP vs. e nergy with the non-interleaved Scheme B perturbed with sextupoles. The integration symbol means that the perturbing component is inte­grated over two cells.

3.2 Octupoles

If a horizontally focusing octupole component of integrated value 0.12 m-3 is introduced inside all the sextupol e pairs, the maximum energy is r educ ed by l % and th e luminosity by 4 % in the overlapping energy range (Fig. 7).

L (arbitrary unit s l

+--60

·1on-1nter!raverl st:xlupcdes,

no perturbn1i on inside th e roirs

Per turb ing f - octupole

K'r' [eel(= 006rri l

Perturbinq 0- ocrupole

Kt'. lcrll = 006m-~

+-70

Scrne machine pararne ters

r.s 1r Fig 6

[ !Ge Vl

+-- - ----- 1-- --~

~o 90

Fig. 7 Luminosity of LEP vs. energy with the non-interleaved Scheme B perturbed with octupoles. The integration symbol means that the perturbing component is inte­grated over two cells.

If a ver tically focusing octupole component of the same value is introduced, the maximum e ne rgy is reduced by 5.5 % and the luminosity by 13 % in the overlapping energy rang e (Fig. 7).

Conclusion

Non-int e rleaved sextupole schemes are attractive for circular LEP-type machines, especially when consid­ering small momentum deviations of some permilles.

The rec ent possibility of analysing the influence of the stability limits on the ma chine performance2 al­lows a quantitative comparison between various schemes. It was thus possible to show th e importance of well choosing th e sextupole famili e s fo r the case of LEP 11.

The clear advantage of the best non-inter\eaved scheme over the standard interleaved scheme implied that the LEP sextupoles had to be designed so that they meet the requirements for the non-interleaved sextupole scheme.

Moreove r, the consequence of introducing perturb­ing multipol e components in the non-int erl eaved scheme could be quantified, which was extremely useful for specifying tolerances on the field quality of the LEP cell magnetslO,

I .

2 .

3 .

4 .

5 .

6 .

7 .

8 .

9 .

10 .

-175-

References

K. Br own and R.V. Servranckx, Chromatic correc­tions for large storage rings, IEEE Trans. on Nucl. Sci. NS-26, No. 3, p. 3598 (1979).

E. Ke il, High-energy performanc e of LEP, LEP note 337 (1981) - unpublished.

H. Wi edeman, Chromaticity correction in large storage rings, PEP-220 (1976).

M. Hann ey and E. Keil, BEAMPARAM - A program for computing beam dynamics and performance of e+e­storage rings, Div. Report CERN ISR-TH/79-29 (1979).

Compiled by A. Hutton, Parame t e r list for LEP ver­sion 11, LEP note 289 (1981) - unpublished.

E. Keil, Chromaticity correction in some variants of the LEP lattice, LEP note 350 (1981) - un­published.

M. Donald, Chromaticity correction in large storage rings, IEEE Trans. on Nucl. Sci., Vol. NS-24, No . 3, p . 1200 (June 1977).

A. Verdi er, La luminosite ~ haut e energie pour le LEP 11 avec a* nominaux suivant l es arrangements hexapolaires, LEP note 396 (1982) - unpublished.

A. Verdier, Effect of some systematic multipole imperfections in the LEP lattice, LEP note 429 (1983) - unpublished.

G. Guignard, Tolerances for the magnetic elements of the LEP lattice, Paper presented at this Con­ference.