1. Evaluation of fringe ﬁeld effect of the Booster bump ... · effect of the Booster bump magnets to the extracted beam trajectory in a scheme ... 155.86 mm 130.46 mm 4.2 mm 19.76

1. Evaluation of fringe field effect of theBooster bump magnets to the extracted beamtrajectory in a scheme with increased distance

between bump magnets.2. Lattice functions perturbation at extraction

in a scheme with pulsed bump-septummagnets.

A. Drozhdin, J. Lackey, J.-F. Ostiguy

August 29, 2003

1 Introductions

It was recently found that linear optics of the Booster is significantly perturbedby the edge focusing of the injection and extraction orbit bumps. The maximumhorizontal βx is increased with respect to linear optics by 36% from 33.7 m to 46.0m and dispersion by 94% from 3.2 m to 6.2 m; in the vertical plane, the maximumβy is increased by 25% from 20.5m to 25.6 m. The edge focusing strength of abending magnet is a function of bending angle θ and magnet length L:

K = −θ2

L. (1)

Both sector and rectangular bending magnets produce focusing effect in onlyone plane, and sign of focusing does not depend on the sign of magnetic field. Forthe Booster doglegs, L is small (0.247 m) and θ is large (60 mrad). There are twodoglegs (at Long03 and Long13 straight sections), each with 4 bending magnets.

1

The focusing effects are additive, giving rise to a significant amount of extra fo-cusing (0.115 m−1, close to one main magnet which has 1/f = 0.157 m−1) andleading to a big perturbation to the linear lattice.

The most effective way to mitigate edge focusing effect is based on the dis-tance increasing between magnets, that permits to reduce bending angle. Thisallows to decrease horizontal βx by 29% and dispersion by 74% in the Booster ifspace between magnets in both Long03 and Long13 straight sections is increasedby 0.56 m. If this is done in the Long03 straight section only it reduces horizontalβ by 15% (to 40.5 m) and dispersion by 46% (to 4.65 m). The last scheme willbe realized during the summer shutdown as a short-term solution to mitigate theproblem.

As a long term solution the new design of injection and extraction was pro-posed with increased distance between magnets in the injection bump and withextraction bump used only at the accelerator top energy with extraction septumlocated behind the envelope of the beam at injection. The lattice function per-turbations are reasonably small in this case. Unfortunately this solution requiressufficient amount of design and construction work and is pretty expensive.

Maximum value of βx,y and dispersion at injection for different solutions ofinjection and extraction in the Booster are presented in Table 1. The fringe fieldeffect of the Booster bump magnets to the extracted beam trajectory in a schemewith increased distance between bump magnets and lattice function perturbationat extraction in a scheme with pulsed bump-septum magnets are evaluated in thisnote.

2 Fringe field effect of the Booster bump magnets tothe extracted beam trajectory

The proposed layout and strength of the new extraction system are presented inFig. 1 and Table 2. An extracted beam passes pretty close to the dogleg magnetsgap in a strong fringe fields (Fig. 2) that may effect extracted beam trajectorydisplacement, and possibly may require realignment of sufficient part of extractedbeam line.

The circulating and extracted beam trajectories, β functions, dispersion andbeam half-size at extraction in the Booster Long-02 and Long-03 straight sectionsfor the new scheme are shown at Figs. 3 and 4. Vertical phase space and beampopulation at the center of the first vertical magnet of extracted beam line are

2

∆βx ∆βy ∆D ∆νx ∆νy

% % %

without injection and extraction bumps 100 100 100 0.000 0.000

with existing injection and extraction bumps 136 125 195 0.069 0.011

without extraction bump at Long13 118 124 144 0.032 0.027

distance between magnets increased 107 124 120 0.012 0.015by 0.56 m at Long03 and Long133-magnet extraction bumps with distance between 104 126 114 0.006 0.032magn. increased by 0.56 m at Long03 and Long13distance between magnets increased 121 124 148 0.037 0.013by 0.56 m at Long033-magnet extraction bumps with distance between 118 125 143 0.034 0.019magnets increased by 0.56 m at Long03

new injection and extraction schemes 104 117 111 0.003 0.024

Table 1: Maximum of βx, βy, dispersion and betatron tune deviation at injectionfor different solutions of injection and extraction schemes in the Booster. Betatrontune without injection and extraction bumps is νx = 6.70, νy = 6.80.

shown in Figures 5 and 7 for calculated fringe field (Fig. 6) at the extractedbeam trajectory in the dogleg magnets number 3 and 4. As shown in Figs. 8 acompensation of this effect can be obtained by displacement of magnet number 3down and magnet number 4 up by (155.73mm-124.44mm)/2=15.6 mm from thedesigned position (see Fig. 2). Fringe field gradient is calculated from Fig. 6,and is equal to G = −3.862 T/m in the magnet number 3, G = +2.219 T/m inthe magnet number 4, and G = ∓3.329 T/m in the magnet number 3 and 4 withmagnets displacement for field effect compensation.

The sextupole component of magnetic field in the Booster bump magnets ispretty big. But as the sextupole components are approximately equivalent in thesemagnets, and sign of current is different, this displacement will compensate (infirst approximation) sextupole component effect of these magnets to the circulat-ing beam stability.

3

Table 2: Strength of extraction magnets.element location length Magnetic field bending angle

m kG mradMKS02-1,2,3,4 Long-02 1.08 0.08388 1.222, (4 magnets)

IBEX02 Long-02 0.265 0.0903 0.0807DOG03-1,2,3,4 Long-03 0.24722 3.018 2.5164, (1 magnet)

SEPTUM03 Long-03 1.524 10.85752 55.807

3 Evaluation of lattice functions perturbation at ex-traction in a scheme with pulsed bump-septum mag-nets

Table 3: Booster lattice functions at the center of Long03 with new extractionbump with increased distance between magnets, and with pulsed septum-bumpmagnets. The edge focusing strength of one bump magnet for different schemesis shown in the bottom of this table.

increased distance pulsed bump-septumbetween magnets magnets

βx, m 6.162 6.070αx 0.074 0.073

βy, m 20.002 20.003αy 0.021 0.021

dispersion D, m 1.847 1.870dispersion D’ 0.000 -0.001

K=θ2/L at injection 0.000500 -K=θ2/L at extraction 0.000006 0.001500K=θ2/L at injection 0.009000for existing schemeK=θ2/L at extraction 0.000100for existing scheme

The new proposed extraction bump at Long-03 straight section with pulsedbump-septum magnets used only at the top energy and septa located outside ofthe main magnet aperture is shown in Fig. 9.

4

septum-magnet

2.755866 m 1.427354 m 1.02085 0.79593

155.73 mm199.94 mm

0.4572 0.79375 m

199.94 mm

Booster center line

Booster center line

dog-3

dog-4

130.46+25.4=155.86mm

155.86+25.4=181.26mm

99.04+25.4=124.44mm

155.73mm

new schemeschemeexisting

respect to DogLeg centerextracted beam position with

Bo

ost

er d

efo

c. m

agn

et

Bo

ost

er d

efo

c. m

agn

etB

oo

ster

def

oc.

mag

net

dog-4

New Scheme

Existing Scheme

25.4

mm

Bump at injection

25.4

mm

dog-3 dog-4

dog-3

Bump at injection

55.54 mrad

99.04 mm

do

g-2

55.54 mrad

155.86 mm130.46 mm

4.2

mm

19.76 mm

Bump+Bex at extraction

Bump+Bex at extraction

dog-1

23.0 mm

Figure 1: Extracted beam trajectories in the Booster Long-03 straight section forthe new (top) and existing (bottom) schemes.

The horizontal and vertical β functions and horizontal dispersion at the accel-erator top energy in the extraction region (Long03) are shown in Fig. 10 for thenew extraction bump with increased distance between bump-magnets and for thescheme with pulsed bump-septum magnets used only at the top energy. A com-parison of the Booster lattice functions for these two cases at the center of Long03is presented in Table 3. The edge focusing strength of one bump magnet shown inthis table is a factor of 6 less at the top energy for the scheme with pulsed magnetscompared to this strength at injection in the existing Booster. Along with a factorof 3 less circulating beam size at the top energy this should not effect a problemat extraction.

5

��

��

��

��

DogLeg-4

DogLeg-3

63.5 mm

114.

3 m

m

25.4

124.

44 m

m

155.

73 m

m

Magnet center

DogLeg-3

DogLeg-4

4 m

m

Booster center linefor DogLeg-3

for DogLeg-4Booster center line

62.611 mm

38.9

89 m

m

Figure 2: Extracted and circulating beam positions in the Booster DoLeg-3 andDoLeg-4 for new scheme.

6

-5

0

5

10

15

20

25

0 5 10 15 20 25 30

Y, m

m

Path length, m

SEPTkm1,2 km3,4

bex

bumpbump+bex

bump+bex+kickbump+bex+kick_septum

0

20

40

60

80

100

120

140

160

18 20 22 24 26 28 30

Y, m

m

Path length, m

SEPTUM

bump+bex+kick_septumVert. aperture

Figure 3: Circulating (top) and extracted (bottom) beam trajectories in the BoosterLong-02 and Long-03 straight sections for the new scheme.

7

-5

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30

Bet

a fu

nctio

ns, m

Path length, m

KM1,2 KM1,2bex

SEPT

horizontal beta, mvertical beta, m

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30

Dis

pers

ion,

m

Path length, m

KM1,2 KM1,2

bex

SEPT

dispersion

-1

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30

Bea

m h

alf-

size

, mm

Path length, m

KM1,2 KM1,2

bex

SEPT

horizontalvertical

Figure 4: β functions (top), dispersion (middle) and beam half-size at extractionin the Booster Long-02 and Long-03 straight sections. Normalized emittance at95% is 12 mm.mrad.

8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-5 -4 -3 -2 -1 0 1 2 3 4 5

X’,

mra

d

X, mm

horizontal phase space at VBC1 entrance

55.2

55.3

55.4

55.5

55.6

55.7

55.8

55.9

282 284 286 288 290 292 294 296 298

Y’,

mra

d

Y, mm

vertical phase space at VBC1 center

282

284

286

288

290

292

294

296

-4 -3 -2 -1 0 1 2 3 4

Y, m

m

X, mm

beam size at VBC1 center

Figure 5: Horizontal (top), vertical (middle) phase space and beam population(bottom) at VBC1 center without fringe fields in the DogLeg magnets. Normal-ized emittance at 95% is 12 mm.mrad.

9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Rel

ativ

e fie

ld s

tren

gth

Distance from center [inch]

Booster Dogleg Magnet

Figure6:

Calculated

magnetic

fieldin

them

ediumplane

ofthe

Booster

DogL

egm

agnet.10

55.5

56

56.5

57

282 284 286 288 290 292 294 296 298 300

Y’,

mra

d

Y, mm

no fieldDogLeg-3 - 62%, DogLeg-4 - 32%, no gradient

DogLeg-3 - 62%, DogLeg-4 - 32%, positive gradientDogLeg-3 - 62%, DogLeg-4 - 32%, negative gradient

55.2

55.4

55.6

55.8

56

56.2

56.4

56.6

56.8

282 284 286 288 290 292 294 296 298 300

Y’,

mra

d

Y, mm

no fieldDogLeg-3 - 62%, DogLeg-4 - 32%

285

290

295

300

-4 -3 -2 -1 0 1 2 3 4

Y, m

m

X, mm


Figure 7: Vertical phase space (top and middle) and beam population (bottom) atVBC1 center for magnetic field of 62% and 32% of the nominal gap field in theDogLeg magnets number 3 and 4 at the extracted beam trajectory. With and with-out gradient of fringe field in magnets number 3 and 4. The beam displacement atVBC1 center is ∆Y = 3.423 mm, ∆Y ′ = 0.758 mrad.

11

55.2

55.3

55.4

55.5

55.6

55.7

55.8

55.9

56

56.1

282 284 286 288 290 292 294 296 298

Y’,

mra

d

Y, mm

no fieldDogLeg-3 - 62%, DogLeg-4 - 32%, no gradient

DogLeg-3 - 62%, DogLeg-4 - 32%, negative gradientDogLeg-3 - 62%, DogLeg-4 - 32%, positive gradient

55.2

55.3

55.4

55.5

55.6

55.7

55.8

55.9

56

282 284 286 288 290 292 294 296 298

Y’,

mra

d

Y, mm


282

284

286

288

290

292

294

296

298

300

-4 -3 -2 -1 0 1 2 3 4

Y, m

m

X, mm


Figure 8: Vertical phase space (top and middle) and beam population (bottom) atVBC1 center for magnetic field of 47% and 47% of the nominal gap field in theDogLeg magnets number 3 and 4 at the extracted beam trajectory. With and with-out gradient of fringe field in magnets number 3 and 4. The beam displacementat VBC1 center is ∆Y = 1.22 mm, ∆Y ′ = 0.0032 mrad. An equal field at theextracted beam trajectory can be obtained by displacement of magnet number 3down and magnet number 4 up by (155.73mm-124.44mm)/2 from the designedposition (see Fig. 2).

12

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��

��

��

50m

m

magnet halfaperture 30 mm

circulating beam 0.4 GeV

circulating beam 8 GeV

=43 mradα

aperture 30 mmmagnet half

aperture 30 mmmagnet half

magnet halfaperture 30 mm

aperture 60 X 105

20m

m36m

m

kick

10 mm

15 mm

kick=36 mm= 2 X 7.6 mm + 7 mm + 2 X 6.9 mm

A(inj)=23 mm, A(extr)=7.6 mm

emitt(95%)=18 mm-mrad

aperture 60 X 105 aperture 60 X 105

aperture 60 X 105

Figure 9: Proposed extraction bump at Long-03 straight section with pulsedbump-septum magnets used only at the top energy. The septa is outside of themain magnet aperture.

13

0

5

10

15

20

25

30

44 46 48 50 52 54

Hor

izon

tal b

eta

func

tions

, m

Path length, m

pulsed bump-septum magnets scheme

scheme with increased distance between bump-magnets

increased distance between bump-magnets at Long03pulsed bump-septum magnets at Long03

0

5

10

15

20

25

30

44 46 48 50 52 54

Ver

tical

bet

a fu

nctio

ns, m

Path length, m




1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

44 46 48 50 52 54

Dis

pers

ion,

m

Path length, m




Figure 10: Horizontal and vertical β functions (top) and horizontal dispersion(bottom) at the top energy in the extraction region (Long03) for new extractionbump with increased distance between bump-magnets and for the scheme withpulsed bump-septum magnets used only at the top energy.14

Documents

1. Evaluation of fringe ﬁeld effect of the Booster bump ... · effect of the Booster bump magnets to the extracted beam trajectory in a scheme ... 155.86 mm 130.46 mm 4.2 mm 19.76