LHC Project Note 111

15 September 1997

Bernard.Jeanneret@cern.ch

Geometrical acceptance in LHC Version 5.0

J.B. Jeanneret and R.Ostojic

Keywords: aperture, injection, optics, collimation, crossing angle

Summary

The geometrical acceptance of LHC Version 5.0 is presented for the regular arc and the machine

insertions. The acceptance and beam separation in the low-� triplets are determined for various

crossing conditions at injection.

1 Introduction

The algorithms used to compute the geometrical acceptance of the LHC regular cell have beenexpanded to allow element by element evaluation of the acceptance in the complete LHC ring.The trajectory of the beam relative to the magnet axis is now taken exactly into account,allowing the analysis of the crossing beams in the experimental insertions, as well as optimisingthe acceptance in the separation dipoles. In the injection insertions, besides the acceptancefor the circulating beam, the acceptance for the injected and miskicked beams is calculated.Finally, to complete the procedures, a simple database containing the mechanical and alignmenttolerances, and the geometrical de�nition of the vacuum chambers associated with magnetclasses has been developed.

The procedure is applied to the calculation of the geometrical acceptance of the Version 5.0 ofthe LHC ring at injection, and attention is drawn to possible bottlenecks. The beam separationand the geometrical acceptance of the low-� triplets is presented for various crossing conditionsat injection. Finally, on the basis of the nominal collision parameters, a thicker cold bore ofthe low-� triplets in the high luminosity insertions is proposed. The cold bore adds to theprotection of the triplets without limiting the acceptance of the insertions.

2 De�nitions and computational methods

We give here a summary of the de�nitions and of the computational method for determiningthe geometrical acceptance presented in detail in [1]. Normalised transverse space coordinatesare used, de�ned by:

nx =x

k��xny =

y

k��y(1)

1

nr

n1

ny

nxna

na

Figure 1: The geometrical edge of the secondary halo as given by the cleaning system. For the

ratio of primary and secondary apertures of n2=n1 = 7=6, the parameters of the secondary halo are

nr=n1 = 1:4 and na=n1 = 1:22. For the nominal primary aperture of n1 = 7:0, nr = 9:8 and na = 8:5.

where: x, y are the transverse coordinates, k� is the square root of the �-beating factor, and�x;y the r.m.s beam size.

The primary collimators of the machine are at the distance n1 in the nx � ny plane, andde�ne the primary aperture, Figure 1. The secondary collimators are at n2, retracted by afactor n2=n1 = 7=6. For these parameters of the cleaning system it has been shown that thesecondary halo is de�ned by nr=n1 = 1:4 and na=n1 = 1:22, so that for the primary apertureof n1 = 7:0, nr = 9:8 and na = 8:5 [2].

The minimum required primary aperture is

nmin1

= 5:5 (2)

However, to allow for an operational margin, the primary aperture of the LHC is speci�edas [1, 3]:

nspec1 = 7:0 (3)

This value of n1 is the criterion for the geometrical acceptance of the ring. Since the machineis cold almost everywhere, the secondary halo escaping the collimation system could inducea quench at a location where the n1(s) < n

spec1 . The geometrical edge of the secondary halo

coincides quite well with a detailed simulation of the secondary halo generated by the collimationsystem. The Version 5.0 of the collimation system, located in IR7, is now quite well de�ned,and will be reported in the near future.

The geometrical acceptance in any part of the ring is calculated on the basis of the largestsecondary halo that can be inscribed in the vacuum chamber, taking into account the displace-ment of the beam at a particular point, Figure 2. The maximum beam displacement is givenas a linear sum:

�x;y(s) = COpeakx;y + [�mech

x;y (s) + �alx;y(s)] + k� �Dx;y(s) � �p + [dsepx;y(s) + dinjy (s) + daxisx (s)] (4)

where COpeak is the peak closed orbit excursion, and �mech and �al aggregate mechanical andalignment tolerances of the vacuum chamber. The dispersion term contains the linear term and

2

O’ A

C

B

D

∆x

x

y

∆y

H=18 mm

R=22 mm

Figure 2: Fit of the largest secondary halo in the vacuum chamber with the beam displaced by �x;y

with respect to the ideal centre of the chamber.

a correction due to parasitic coupling:

Dx;y(s) = Dlinearx;y (s) + kD

vuut�x;y(s)

�QF;xDQF (5)

Here, kD is the coupling coe�cient, andDQF is the peak linear dispersion in the arc (2.1 m in thefocusing quadrupole). The displacement due to the dispersion is proportional to the momentumo�set �p (di�erent for injection and collision energies), and is corrected for �-beating.

In the crossing points the beams travel in a common beam tube and are separated to avoidparasitic collisions and collective beam-beam e�ects. The separation is generated by kickerslocated on either side of the crossing points. The separation is de�ned by the crossing angleand the transverse separation of the beams at the interaction point. For a given set of crossingparameters, the orbit is displaced with respect to the ring axis by dsep, not only in the commonpart of the ring (from D1R-D1L), but also within all elements which are located between theseparation kickers. In the machine section common to the circulating and injected beams,i.e. between Q5 and D2 in IR2 and IR8, the displacement dinj describes the e�ect of theinjection kicker MKI. Besides the case where the beam is injected onto the circulating orbit,two other possibilities, related to MKI malfunction, are considered: an injected beam which isnot diverted onto the circulating orbit, and a circulating beam which is accidentally miskickedout of the ring. Since MKI acts in the vertical plane, dinj has a vertical component only.Finally, in the dipole separators in the insertions (D1-D2 and D3-D4 pairs), the magnet axisdoes not necessarily coincide with the ring axis, and the ideal orbit is displaced by daxis. Thisvalue may be modi�ed by selecting the separation of the two apertures so as to improve thegeometrical acceptance of the separators (particularly useful for D3-D4 in IR4, and for D2 inIR2/8). Depending on the location in the ring, e.g. in the arc cells or straight sections withoutcrossing, some or all of the d terms may be identical to zero.

3

Table 1: Coe�cients entering the calculation of beam displacement �x;y

Coe�cient Name Value

Peak Close orbit excursion COpeakx;y 4.0 mm

Beta beating (root) k� 1.1Relative parasitic dispersion kD 0.273Momentum o�set,injection �p 10�3

Momentum o�set,collision �p 2 � 10�4

The displacements �x;y are calculated with PERL scripts under UNIX. The linear opticalfunctions and separations are retrieved by running MAD using the LHC V5.0 database, andare calculated exactly inside the magnets, with a step of one metre. The additional parame-ters entering the calculation for Version 5.0 of LHC are given in Table 1. For each location,the largest secondary halo which �ts inside the vacuum chamber of a given magnet class iscalculated. This in turn gives the equivalent primary collimator aperture n1, which serves asthe de�nition of the local acceptance. The results are presented as tables and �gures, and arecompared with the required global acceptance of nspec1 = 7. The mechanical and alignmenttolerances for magnet classes presently used in LHC Version 5.0 are stored in a PERL moduleas a simple database, and are given in Table 2. The database also contains the de�nition ofthe vacuum chamber geometry and magnet axis o�set relative to the average ring axes. It canbe easily extended for new properties and magnet classes.

4

Table 2: Database of aperture parameters for the LHC magnet classes. The database contains

mechanical and alignment tolerances, axis o�set relative to the average ring axis, and the vacuum

chamber type, described by its shape and dimensions.

Description Class �mechx �alx �mech

y �aly Axis Vacuumname [mm] [mm] [mm] [mm] [mm] chamber

Main dipole MB 2.5 1.6 1.1 1.6 0.0 r22-v18Warm separator D1 MBXW 0.0 1.0 0.0 1.0 0.0 r65-v27.5Cold separator D1 MBS 0.6 1.6 0.6 1.6 0.0 r37Separator D2 (IR1) MBT+ 0.6 1.6 0.6 1.6 90.0 r37Separator D2 (IR2) MBT- 0.6 1.6 0.6 1.6 -90.0 r37Separator IR4 D4B MBR4B 1.5 1.6 1.1 1.6 97.0 r34-v30Separator IR4 D4A MBR4A 1.5 1.6 1.1 1.6 117.0 r34-v30Separator IR4 D3B MBR3B 1.5 1.6 1.1 1.6 191.0 r34-v30Separator IR4 D3A MBR3A 1.5 1.6 1.1 1.6 200.0 r34-v30Warm separator IR3 MBW3 0.6 1.0 0.6 1.0 104.5 r31.5-v23Warm separator IR7 MBW7 0.6 1.0 0.6 1.0 -104.5 r31.5-v23

Main quad MQ 1.0 1.6 1.0 1.6 0.0 r22-v18Main quad (no screen) MQ NOBS 0.6 1.6 0.6 1.6 0.0 r25Wide aper. quad MQY 0.6 1.0 0.6 1.0 0.0 r31.5Low-� quad (IR2) MQX 0.6 1.0 0.6 1.0 0.0 r31.5Low-� quad (IR1, Q1) MQX1 0.6 1.0 0.6 1.0 0.0 r23Low-� quad (IR1, Q2) MQX2 0.6 1.0 0.6 1.0 0.0 r30Low-� quad (IR1, Q3) MQX3 0.6 1.0 0.6 1.0 0.0 r30Warm quad MQW 0.6 1.0 0.6 1.0 0.0 hyp-r20.5-h24

Injection kicker MKI 0.0 1.0 0.0 1.0 0.0 r19Injection septum MSI 1.0 1.0 1.0 1.0 0.0 r20Dump kicker MKD 1.0 1.0 1.0 1.0 0.0 r29Dump septum MSD 1.0 1.0 1.0 1.0 0.0 r23

Orbit corrector MCBY 0.6 1.6 0.6 1.6 0.0 r33.5TAS absorber ABS 0.2 0.5 0.2 0.5 0.0 r17

5

3 Arc aperture at injection

The geometrical acceptance of the regular cell is presented in Figure 3. As in other �guresshown in later sections, the magnet layout is displayed at the bottom of the �gure. Thecell shown in Figure 3 begins with a defocusing quadrupole QD16.R1, contains the focusingquadrupole QF17.R1 and two strings of three main dipoles, and ends with the trim quadrupoleQT.QD18.R1 associated with the defocusing quadrupole QD18.R1. The local geometrical ac-ceptance is calculated inside each magnet as described above.

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100

n1

s

n1(s)layout schem

Figure 3: The geometrical aperture n1(s) in the LHC cell. The element at the centre of the �gure

is the focusing quadrupole QF17.R1. The minimum acceptance occurs in the main dipoles on either

side of this quadrupole, and for the present table of tolerances is n1 = 5:95.

The geometrical acceptance of the regular cell shown in Figure 3 corresponds to the setof parameters and tolerances given in Table 1 and 2. The geometrical acceptance of the cellis n1 = 5:9, 15% below the speci�cation n1 = 7:0. This situation was discussed a year ago,and it was decided [3] that the maximum displacements �x(s);�y(s) would be reduced fromthe present value of 11 and 7 mm, by � 1:4 mm in both planes, to meet the speci�cation ofn1 = 7:0. The improvement could be obtained by revising

1. the thickness and mechanics of the cold bore

2. the thickness and the tolerances of the beam screen

3. the alignment tools and procedures

4. a reduction of the peak closed orbit excursion

The geometrical acceptance with these modi�cations is given in Figure 4.

6

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100

n1

s

n1(s)layout schem

Figure 4: The geometrical acceptance n1(s) in the LHC cell with the displacements �x;y reduced

by 1.4 mm. The minimum acceptance is n1 = 7:0.

4 Straight sections without crossing

4.1 Cleaning Insertions in IR3, IR7

The geometrical acceptance in the betatron cleaning insertion in IR7 is shown in Figure 5. Itshould be noted that the large part of the insertion, starting from D4L to D4R, contains onlywarm dipoles and quadrupoles (MBW and MQW). The smallest value of the acceptance isn1 = 6:73 and occurs in Q6R. Although it is below the speci�cation of n1 = 7:0, this limitationis acceptable in a warm area, and would result in a slightly larger loss rate if the collimatorswere not a considerably deeper aperture limitation. The situation in the momentum cleaninginsertion in IR3, which has identical layout and very similar optics, is almost identical to thatshown in Figure 5.

4.2 RF Insertion in IR4

The geometrical acceptance in the RF insertion in IR4 is well above the speci�cation, as shownin Figure 6. The acceptance of the separator dipoles D3 and D4 has been maximized by thechoice of the progressively larger axis separation, and is above n1 = 8:0 even though the magnetsare equipped with a beam screen. The minimum acceptance occurs in Q5R and is n1 = 7:9.

7

0

2

4

6

8

10

12

14

16

18

20

200 300 400 500 600

n1

s

n1(s)layout schem

Figure 5: The geometrical acceptance n1(s) in the betatron cleaning insertion in IR7. The element

limiting the aperture is a twin warm quadrupole Q6R where n1 = 6:73.

0

2

4

6

8

10

12

14

16

18

20

9800 9900 10000 10100 10200

n1

s

n1(s)layout schem

Figure 6: The geometrical aperture n1(s) in the RF insertion in IR4.

8

0

2

4

6

8

10

12

14

16

18

20

16500 16600 16700 16800 16900

n1

s

n1(s)layout schem

Figure 7: The geometrical acceptance n1(s) in IR6. The elements limiting the acceptance are warm

septum MSD and the cold quadrupoles Q6 and Q7.

4.3 Dump Insertion in IR6

The geometrical acceptance in the dump insertion in IR6 is shown in Figure 7. Several di�cultpoints may be noted. The minimum acceptance occurs in the warm septum MSD and isn1 = 5:6 for a round vacuum chamber with a radius of 23 mm. Several serious limitationsoccur in the cold quadrupoles, in Q6L where n1 = 6:3, and similarly in Q6R (n1 = 6:4) andin Q7L (n1 = 6:6). It must be envisaged to either change the optics in order to reduce the�-functions by � 25%, or to use large aperture quadrupoles and to modify the vacuum chamberin the MSD.

9

5 High Luminosity Insertions

5.1 Injection

As mentioned in section 2, the beams cross in the four LHC experimental insertions, and areseparated in the common part of the ring. The separation is de�ned in terms of the crossingangle and the transverse separation of the beams at the interaction point, and is generatedby kickers located on either side of the crossing points. As long as the separation kickers arebeyond Q3, the displacement in the inner triplet does not depend on their exact location, andis to a large degree determined only by the crossing angle.

For the purposes of the present study the separation kickers are assumed to be located nextto Q4 and Q5 facing the IP, on both sides of the experimental insertions. This position of theseparation kickers is the natural choice if independent control of the two beams is required,since it limits to Q4 and D2 the region of the matching section where the beams are displaced.Moving the kickers closer to the dispersion suppressor would increase the number of magnetsa�ected by displaced orbits. However, in terms of kicker force, this scheme is far from optimal,since it requires kicker strengths about two times higher than that of the MCBY correctors.Typical orbit displacement in IR1 is shown for the injection optics in Figure 8. The parametersof the crossing are: crossing angle per beam 175 �rad (full crossing angle of 350 �rad), 2.5 mmhalf beam separation (full separation between the two beams 5 mm), and 90 deg. crossingplane rotation angle (vertical crossing). As can be seen, the orbit is displaced vertically by asmuch as 8 mm in Q4R, and horizontally by about 12 mm in D2R.

-20

-15

-10

-5

0

5

10

15

20

13100 13200 13300 13400 13500

Sep

arat

ion

[mm

]

s [m]

layout schemx_sepy_sep

x_sep_toty_sep_tot

Figure 8: Beam trajectory in Ring 1 of IR1 corresponding to the crossing angle of 175 �rad per

beam, 2.5 mm half beam separation at the IP, and crossing plane rotation of 90 deg. The insertion is

tuned to the injection �� of 18 m.

10

0

5

10

15

20

13260 13280 13300 13320 13340 13360 13380 13400

Sep

arat

ion

s [m]

layout schemseparation in sigma

separation in mm

Figure 9: Physical and normalized beam separations (in mm and units of �, resp.) in IR1 region

from D1R-D1L. (Vertical crossing, 175 �rad per beam, 2.5 mm half beam separation at the IP). The

insertion is tuned to the injection �� of 18 m.

The separation of the two beams in the common part of the insertion is shown for theabove crossing parameters in Figure 9. Also shown is the normalized separation n�(s), i.e. thephysical separation normalized with respect to the larger of �x or �y. In this particular case,the 2.5 mm half beam separation at the IP was chosen so that the minimum beam separation nboccurs inside the inner triplet, and is 12.2�. The discontinuity of the normalized separation inQ3 is due to the fact that �x and �y change magnitude, and the maximum of the two transversebeam sizes is used for normalization.

The geometrical acceptance in IR1 at injection is shown in Figure 10. The smallest value ofthe acceptance is n1 = 3:5 in Q4R (cold bore 50 mm), which is clearly unacceptable, and resultsfrom the combination of large vertical orbit displacement and a �y of 230 m in this quadrupole.Elsewhere in the matching section the acceptance is above the speci�cation. The acceptance inQ4 is around n1 = 3:0 irrespective of the crossing plane rotation angle, and could be improvedeither by reducing the injection �� (to 12 m as in IR2, section 6), by displacing the separationkickers, or by introducing large aperture quadrupoles at this location.

From Figure 10, the acceptance of the low-� triplets is n1 = 8:9. This result assumesreduced cold bore apertures of the low-� quadrupoles corresponding, as explained in the nextsection, to the requirements of radiation protection of the triplets in collision. An interestingsituation occurs in the front absorber TAS (shown in Figures 8-10 as the �rst element of thetriplet looking from the IP), whose aperture has been increased from the \Yellow Book" valueof 28 mm to 34 mm in order to improve its acceptance at injection to n1 = 6:5. In collision,Figure 13, the absorber is not limiting the acceptance of the inner triplet.

11

0

5

10

15

20

13100 13200 13300 13400 13500

n1

s

n1(s)layout schem

Figure 10: The geometrical acceptance n1(s) in IR1 for injection optics. (Vertical crossing, 175 �rad

per beam, 2.5 mm half beam separation at the IP).

Figure 11: The minimum beam separation nb and the geometrical acceptance n1 in the IR1 low-�

triplet at injection as function of the crossing angle. (Vertical crossing, 2.5 mm half beam separation

at the IP).

12

In order to determine the range of crossing angles available at injection, the geometricalacceptance of the low-� quadrupoles in IR1 is plotted in Figure 11 as function of the crossingangle. Also shown in this �gure is the minimum beam separation nb in the inner triplet. Withrespect to the speci�cation of n1 = 7:0, the maximum crossing angle in IR1 is very close to500 �rad. The corresponding value of the minimum beam separation is nb = 16:6�. On theother hand, in order for nb to be larger than 10�, a crossing angle greater than 260 �rad isrequired. Based on these criteria, the allowed range of crossing angles in IR1 extends from 260to 500 �rad.

5.2 Collision

The nominal �� in the high luminosity insertions at collision is 0.5 m, and the beams collidewith a crossing angle of 100 �rad per beam. Obviously, the acceptance limit in IR1 occurs inthe low-� triplet, where the � function reaches 4500 m. The beam separation in the commonpart of the insertion is shown in Figure 12, and the geometrical acceptance in the insertion inFigure 13.

0

5

10

15

20

13260 13280 13300 13320 13340 13360 13380 13400

Sep

arat

ion

s [m]

layout schemseparation in sigma

separation in mm

Figure 12: Beam separation in IR1 from D1R-D1L for beams colliding in the vertical plane with an

angle of 100 �rad per beam. The collision �� is 0.5 m.

In view of improving the protection of the triplet against high secondary beam ux, thecold bore aperture assumed in these calculations has been reduced from the nominal value of63 mm to 46 mm for Q1, and to 60 mm for Q2 and Q3 (MQX1, MQX2 and MQX3 in Table 2).As a result, the geometrical acceptance of Q1-Q3 in collision has become almost identicalin all magnets, but reduced with respect to the maximal acceptance of the 70 mm low-�quadrupoles. It is nevertheless su�cient for the nominal collision parameters since n1 = 8:0.

13

Calculations of the energy deposition in the quadrupole coils assuming these values of the coldbore apertures [4], indicate that the peak azimuthal energy density in the quadrupoles is withinthe required safety margin, even though the aperture of the front absorber has been increasedto 34 mm, as necessary for injection in IR1.

0

5

10

15

20

13100 13200 13300 13400 13500

n1

s

n1(s)layout schem

Figure 13: The geometrical acceptance n1 in IR1 for collision optics. (Vertical crossing, 100 �rad

per beam). The collision �� is 0.5 m.

The geometrical acceptance of the inner triplet and the minimum beam separation are shownas function of the crossing angle in Figure 14. With respect to the acceptance speci�cation ofn1 = 7:0, the maximum allowed crossing angle in collision is 300 �rad. For these conditions,the beam separation inside the triplet is nb=7.2�, and increases to 8.4� for a crossing angle of350 �rad, for which the acceptance of the triplet is still acceptable. If the separation of thecolliding beams in the triplet would need to be still larger, the issue of the thickness of the coldbore wall would need to be reviewed.

14

Figure 14: The minimum beam separation nb and the geometrical acceptance n1 in the low-� triplet

of IR1 at collision as function of the crossing angle.

6 Experimental Insertions Combined with Injection

The LHC injection insertions in IR2 and IR8 feature similar layout of the inner triplet as inIR1. The matching section, however, is modi�ed due to the presence of injection elements. Inparticular, the injection kicker MKI is located in between Q4 and Q5 quadrupoles, which impliesthat both magnets must have wider apertures than elsewhere. For this reason, 70 mm aperturequadrupoles are foreseen, but the acceptance of these quadrupoles may not be su�cient if thecrossing orbit is not closed before Q5. Therefore, we consider the same layout of the separationkickers as in IR1, i.e. a set of horizontal/vertical kickers next to Q4 and Q5. In addition, thepresence of the injection kicker close to the separation dipole D2 requires that the acceptanceof the Q4-D2 assembly be considered also for the case of eventual kicker mis�re. Two situationsare identi�ed: a) premature �ring of the kicker which deviates the circulating orbit (standardsecondary halo of Figure 1) out of the ring, and b) absence of kicker action on the injectedbeam (round beam halo) which may scrape the neighbouring elements.

The beam separation in the common part of IR2 is shown in Figure 15 for the injectionoptics (��=12 m), and for the following set of crossing parameters: crossing angle 175 �rad perbeam, 2 mm half beam separation, and vertical crossing plane. With respect to beam separationin IR1, Figure 9, it can be observed that the optical functions, and therefore beam separation,are not symmetric around the IP. This is due to the fact that the requirement of round beamsat the IP has been relaxed in order to maximize the phase advance between the kicker and theinjection dump TDI. As a consequence, the minimum beam separation nb=8.7 occurs in theinner triplet on the injection side, and is substantially less than what could be expected on thebasis of a relatively large injection �� of 12 m. Similarly, the low-� quadrupoles on the MKI

15

0

5

10

15

20

16600 16620 16640 16660 16680 16700 16720

Sep

arat

ion

s [m]

layout schemseparation in sigma

separation in mm

Figure 15: Beam separation in IR2 from D1R-D1L. (Vertical crossing, 175 �rad per beam, 2 mm

half beam separation at the IP). The insertion is tuned to an injection �� of 12 m

side determine the geometrical acceptance of the triplet, as can be seen in Figure 16, where thegeometrical acceptance of IR2 is shown at injection.

The geometrical acceptance in IR2 for the circulating beam, Figure 16, satis�es the speci-�cation everywhere. The minimum occurs in Q4R, n1=7.15, in marked contrast to IR1, whichhas an identical separation kicker layout. The di�erence can be traced to the lower ��, whichresults in smaller orbit displacements in the Q4-D2 section for equal crossing parameters, andto the lower �-functions in the matching section, which in IR2 are below 100 m in both planes.The acceptance of MSI is n1=7.26, and of MKI n1=7.22 for the tolerance and vacuum chamberparameters of Table 2.

The minimum beam separation and the geometrical acceptance of the inner triplets in IR2are shown as function of the crossing angle in Figure 17. As mentioned above, due to theasymmetric �-functions the di�erence between the acceptance and beam separation in the leftand right triplet is of the order of one sigma. As a consequence, in order to have nb=10,a crossing angle of 400 �rad is required, while the acceptance of the inner triplet (injectionside) ful�lls the n1=7 speci�cation at 440 �rad. The range of allowed crossing angles in IR2 istherefore substantially smaller than in IR1, largely due to the asymmetric optical functions inthe inner triplet.

The geometrical acceptance for the two cases of kicker mis�ring mentioned above are shownin Figures 18 and 19. In Figure 18, the injected round beam is assumed to pass through Q5Land MKI, and to continue without deviation through Q4L and D2L. Note also the position ofthe separation kickers MCBY, located next to Q5L and Q4L, with a slightly larger apertureof 72 mm. The minimum acceptance for the round injected beam is 6.8 � at the Q5 end of

16

0

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16400 16500 16600 16700 16800 16900

n1

s

n1(s)layout schem

Figure 16: The geometrical acceptance n1(s) for the circulating beam in IR2 at injection. (Vertical

crossing plane, 175 �rad per beam, half beam separation at the IP of 2 mm.)

Figure 17: The minimum beam separation nb and the geometrical acceptance n1 in IR2 the low-�

triplet at injection as function of the crossing angle. (Vertical crossing, 2 mm half beam separation at

the IP.

17

0

5

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20

16490 16500 16510 16520 16530 16540

n1

s

n1(s)layout schem

Figure 18: The geometrical acceptance n1(s) in IR2 from Q5-D2 for the case of injected beam (round

beam halo). (Vertical crossing plane, 175 �rad per beam, half beam separation at the IP of 2 mm.)

MKI. The acceptance for the miskicked circulating beam, Figure 19, is smaller in all magnetsand is minimum at the IP end of MKI, n1=4.0, while it is 7.52 in Q4L and 7.35 in D2L. Thissituation indicates that it would be highly advisable to close the orbit before the kicker, sincein that case the acceptance of the MKI would satisfy the speci�cation for all three situations.

The vertical crossing (-90 deg. rotation angle) assumed in the previous �gures gives thelargest acceptance in IR2 for all cases considered. This is a consequence of the vertical injec-tion plane, with a negative MKI kick. For crossing plane rotations larger than -90 deg., theacceptance in the Q4L and D2L (and even more so in the MKI) is reduced considerably bothfor the injected and miskicked beams, as shown in Figure 20, where the acceptance in Q4L,which is slightly smaller than in D2L, is shown. Besides the rotation angle, the position ofthe beam at the IP (above or below the crossing plane, as indicated by �2 mm half beamseparation) plays a very important role. For the crossing angle of 350 �rad, and the assumedposition of the separation kickers, the horizontal crossing in IR2 is not feasible because of verylow acceptance in the Q4L and D2L, both for injected and miskicked beams.

In LHC Version 5.0, IR2 is tuned in collision to a high-� optics with a �� of 250 m. Thepeak value of the �-functions do not occur in the inner triplet, but rather in Q5, where theyare around 3700 m. The geometrical acceptance of the insertion is shown for collision crossingangle of 200 �rad in Figure 21. The minimum n1 is 8.64 in the MKI. The acceptance of thetriplets in the low-� optics will be studied when it becomes available.

18

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16500 16505 16510 16515 16520 16525 16530 16535 16540

n1

s

n1(s)layout schem

Figure 19: The geometrical acceptance n1(s) in IR2 from MKI-D2L for the case of miskicked

circulating beam. (Vertical crossing plane, 175 �rad per beam, half beam separation at the IP of

2 mm.) The minimim acceptence (n1=6.70) occurs in the IP side of MKI.

Figure 20: The geometrical acceptance n1 in Q4L of IP2 as function of crossing plane rotation for

the injected and miskicked beams and half beam separation of �2 mm at the IP. (Crossing angle

175 �rad per beam).

19

0

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16400 16500 16600 16700 16800 16900

n1

s

n1(s)layout schem

Figure 21: The geometrical acceptance n1 in IR2 for collision optics. (Vertical crossing, 100 �rad

per beam). The collision �� is 250 m.

7 Conclusions

In this report, we have presented the geometrical acceptance in LHC Version 5.0. The al-gorithms previously used to compute the geometrical acceptance of the LHC arc have beenimproved, and extended to treat all the magnets in the ring.

The previously noted acceptance limitations in the LHC arcs remain in Version 5.0, andthe improvements in the tolerance budget and reduction of the peak closed orbit error havealready been identi�ed as possible remedies. In the machine insertions, serious di�cultiesare encountered in the dump insertion in IR6, where both the extraction kicker and the coldquadrupoles have unacceptably small acceptances. In the cleaning insertions, the acceptanceof the warm magnets is slightly below the speci�cation, but nevertheless acceptable. Theacceptance in the RF insertion is very comfortable, well above the rest of the ring.

A study of the crossing beams in the low-� triplets in the experimental insertions indicatesthat there exists a range of crossing angles at injection which satis�es both the beam separationrequirements and the speci�cation for the geometrical acceptance in the common section of themachine. The range is from 260 �rad to 500 �rad in IR1, and from 400 �rad to 440 �rad inIR2, reduced by the fact that in the injection insertions the �-functions are asymmetric aroundthe IP. In IR1, it is proposed that the cold bore aperture of the low-� quadrupoles is reduced by3 mm with respect to the standard cold bore (63 mm) of a 70 mm low-� quadrupole in order toimprove the inner triplet protection, while preserving its acceptance within speci�cation. Therange of crossing angles at collision is then from 200 �rad to 350 �rad.

In the injection insertions, the acceptance of the matching section needs to be determinedalso in cases of injection kicker failure. It has been shown that a suitable solution in IR2 exists

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in case of a vertical crossing plane. For other crossing plane rotations, the acceptance is reducedeither for the injected beam or for the miskicked circulating beam, and is unacceptably low forless than 45 degrees.

No attempt has been made to optimise the parameters of the separation kickers. We havenoted, however, that the range of crossing parameters that are compatible with the acceptanceof the experimental insertions depends strongly on the injection optics and the exact positionof the kickers, and requires extensive studies if full exibility is required.

Acknowledgements

We thank O.Bruening, W.Chou, J.Miles, N. Mokhov, J.P.Koutchouk and E.Weisse for theircollaboration.

References

[1] J.B. Jeanneret and T.Risselada, LHC project note 66, September 1996.

[2] D. Kaltchev et al., PAC'97 Vancouver and LHC Project Report, to be issued.

[3] Minutes of the 9th meeting of the Parameter & Layout Committee, P.Lef�evre, revisedversion, 4th September 1996.

[4] N. Mokhov, private communication, July 1997.

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