1. USPAS June 2007, Superconducting accelerator magnets Unit 2 Magnet specifications in circular accelerators Soren Prestemon and Paolo Ferracin Lawrence Berkeley National Laboratory (LBNL) Ezio Todesco European Organization for Nuclear Research (CERN)

2. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.2 QUESTIONS Order of magnitudes of the size of our objects: why ? High energy circular accelerators Length of an accelerator:  Km 15 m 1.9 Km Main ring at Fermilab, Chicago, US 41° 49’ 55” N – 88 ° 15’ 07” W 1 Km 40° 53’ 02” N – 72 ° 52’ 32” W RHIC ring at BNL, Long Island, US 46° 14’ 15” N – 6 ° 02’ 51” E

3. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.3 QUESTIONS Order of magnitudes of the size of our objects: why ? High energy linear accelerators Length of a linear accelerator:  Km - but we will not deal with them 15 m Linear accelerator at Stanford, US 46° 14’ 15” N – 6 ° 02’ 51” E 3.5 Km 37° 24’ 52” N – 122° 13’ 07” W

4. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.4 QUESTIONS Order of magnitudes of the size of our objects: why ? High energy circular accelerators Length of an accelerator magnet:  10 m Diameter of an accelerator magnet:  m Beam pipe size of an accelerator magnet:  cm 15 m A stack of LHC dipoles, CERN, Geneva, CH 46° 14’ 15” N – 6 ° 02’ 51” E Dipole in the LHC tunnel, Geneva, CH 0.6 m 6 cm

5. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.5 CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing Aperture requirements for arc quads and dipoles (size of the beam) Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters Example: the LHC 5. The interaction regions: low-beta magnet specifications How to squeeze the beam Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

6. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.6 1. PRINCIPLES OF A SYNCHROTRON Electro-magnetic field accelerates particles Magnetic field steers the particles in a closed (  circular) orbit To drive particles through the same accelerating structure several times As the particle is accelerated, its energy increases and the magnetic field is increased (“synchro”) to keep the particles on the same orbit Limits to the increase in energy The maximum field of the dipoles (proton machines) The synchrotron radiation due to bending trajectories (electron machines) Colliders: two beams with opposite momentum collide This doubles the energy ! One pipe if particles collide their antiparticles (LEP, Tevatron) Otherwise, two pipes (ISR, RHIC, HERA, LHC)

7. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.7 1. PRINCIPLES OF A SYNCHROTRON The arcs: region where the beam is bent Dipoles for bending Quadrupoles for focusing Correctors Long straight sections (LSS) Interaction regions (IR) where the experiments are housed Quadrupoles for strong focusing in interaction point Dipoles for beam crossing in two-ring machines Regions for other services Beam injection (dipole kickers) Accelerating structure (RF cavities) Beam dump (dipole kickers) Beam cleaning (collimators) A schematic view of a synchrotron The lay-out of the LHC

8. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.8 CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing Aperture requirements for arc quads and dipoles (size of the beam) Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters Example: the LHC 5. The interaction regions: low-beta magnet specifications How to squeeze the beam Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

9. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.9 2. THE ARC: HOW TO KEEP PARTICLES ON A CIRCLE Kinematics of circular motion Relativistic dynamics Lorentz (?) force Putting all together Hyp. 1 - longitudinal acceleration<<transverse acceleration Hendrik Antoon Lorentz, Dutch (18 July 1853 – 4 February 1928), painted by Menso Kamerlingh Onnes, brother of Heinke

10. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.10 2. THE ARC: HOW TO KEEP PARTICLES ON A CIRCLE Relation momentum-magnetic field-orbit radius Preservation of 4-momentum Hyp. 2 Ultra-relativistic regime Using practical units for particle with charge 1, one has magnetic field in Tesla … Remember 1 eV=1.602  10 -19 J Remember 1 e= 1.602  10 -19 C

11. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.11 2. THE ARC: HOW TO KEEP PARTICLES ON A CIRCLE Nikolai Tesla (10 July 1856 - 7 January 1943) Born at midnight during an electrical storm in Smiljan near Gospić (now Croatia) Son of an orthodox priest A national hero in Serbia Career Polytechnic in Gratz (Austria) and Prague Emigrated in the States in 1884 Electrical engineer Inventor of the alternating current induction motor (1887) Author of 250 patents Miscellaneous Strongly against marriage [brochure of Nikolai Tesla Museum in Belgrade (2000)] Considered sex as a waste of vital energy [guardian of Nikolai Tesla Museum in Belgrade, private communication (2002)] Tesla, man of the year

12. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.12 2. THE ARC: HOW TO KEEP PARTICLES ON A CIRCLE Relation momentum-magnetic field-orbit radius

13. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.13 2. THE ARC: HOW TO KEEP PARTICLES ON A CIRCLE The magnet that we need should provide a constant (over the space) magnetic field, to be varied with time to follow the particle acceleration This is done by dipoles As the particle can deviate from the orbit, one needs a linear force to bring it back We will show in the next section that this is given by quadrupoles

14. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.14 CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing Aperture requirements for arc quads and dipoles (size of the beam) Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters Example: the LHC 5. The interaction regions: low-beta magnet specifications How to squeeze the beam Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

15. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.15 3. THE ARC: SIZE OF THE BEAM AND FOCUSING The force necessary to stabilize linear motion is provided by the quadrupoles Quadrupoles provide a field which is proportional to the transverse deviation from the orbit, acting like a spring One can prove that the motion equation in transverse space (with some approximations) is where

16. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.16 3. THE ARC: SIZE OF THE BEAM AND FOCUSING A sequence of focusing and defocusing quadrupoles with the same (opposite) strength and spaced by L is a providing linear stability to the beam – this is called a FODO cell Let L be the distance between two consecutive quadrupoles The equations of transverse motion are Where the term K is zero in dipoles, and in focusing quadrupoles, in defocusing quadrupoles

17. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.17 3. THE ARC: SIZE OF THE BEAM AND FOCUSING The motion equation in the transverse space is similar to a harmonic oscillator where the force depends on time … Solution: a oscillator whose amplitude and frequency are modulated  and  give the beam size  x  y are the invariants (emittances) [m rad]  x and  y are the beta functions [m]  is the phase advance, related to the beta function The beta functions oscillate along the ring, reaching maxima and minima in the quadrupoles

18. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.18 3. THE ARC: SIZE OF THE BEAM AND FOCUSING Relations for a FODO cell: beam size vs cell length Let 2L be the cell length – we consider it for the moment as an independent variable We define  (2L) as the phase advance per cell A typical cell has  (2L)=  /2 (90° phase advance) – for this cell one has Beta functions in a FODO cell with L=50 m Beta functions in a FODO cell with L=100 m

19. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.19 3. THE ARC: SIZE OF THE BEAM AND FOCUSING Example of the LHC: L =50 m,  f =170 m,  d =30 m The beta functions are in meters they are related, but not equal to the beam size Pay attention !  f =170 m does not mean that the beam size is 170 m !! It is not easy to “feel” the dimension of a beta function Radius of the beam in the arc (1 sigma) LHC:  n =3.75 10 -6 m rad High field E =7 TeV,  =7460 -  =0.29 mm Injection E =450 GeV,  =480 -  =1.2 mm Beam size depends on cell length, energy and normalized emittance

20. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.20 3. THE ARC: SIZE OF THE BEAM AND FOCUSING Focusing in a FODO cell Thin lens approximation: focusing strength in a 90° FODO cell is The focusing strength is related to K 1 and to the quadrupole length ℓ q and the quadrupole gradient is LHC: at high field B =8.33 T,  =2801 m, L =50 m, G ℓ q =660 T For a 60° phase advance the same linear dependence on L, with different constants It looks worse: same beam size, 50% more focusing required

21. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.21 CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing Aperture requirements for arc quads and dipoles (size of the beam) Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters Example: the LHC 5. The interaction regions: low-beta magnet specifications How to squeeze the beam Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

22. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.22 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS

23. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.23 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS

24. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.24 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS Input 1. Collision energy E c Gives a relation between the dipole magnetic field B and the total length of the dipoles L d Technology constraint 1. Dipole magnetic field B Does not depend on magnet aperture B t <2 T for iron magnets B t <13 T for Nb-Ti superconducting magnets (10 T in practice) B t <25 T for Nb 3 Sn superconducting magnets (16-17 T in practice) Output 1. Length of the dipole part Length in m, B in T, energy in GeV

25. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.25 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS

26. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.26 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS Input 2. Injection energy E i Determines the relativistic factor, that affect the beam size Constraint 2. Normalized beam emittance  n Determined by the beam properties of the injectors Semi-cell length L This is a free parameter that can be used to optimize Determines the beta functions Output 2. Aperture of the arc magnets (also determined by field errors and beam stability) Size of the beam at injection Magnet aperture

27. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.27 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS

28. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.28 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS Technology constraint 1. Quadrupole magnetic field vs aperture Output 3. Gradient of the quadrupoles Semi-cell length L Also determines the focusing, i.e. the integrated gradient Output 4. Length of the quadrupoles

29. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.29 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS Output 5. Number of semi-cells and arc length Equal to the number of quadrupoles

30. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.30 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS Example: Large Hadron Collider E =7000 GeV Nb-Ti magnets, dipole field B =8.3 T L d =17600 m Cell length L =50 m  f =170 m  n =3.75  10 -6 m rad Injection energy 450 GeV,  =480 Beam size  =0.0012 m (at injection) 2*10  =0.024 m, i.e., much less than the available aperture of 0.056 m Aperture is larger then needed to have the beam at injection in the zone of “good field”

31. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.31 4. THE ARC: FLOWCHART FOR MAGNET PARAMETERS Example: Large Hadron Collider Arc magnets aperture and technology constraint determine quadrupole gradient: 8.3 T at 28 mm radius gives  300 T/m for Nb-Ti at 1.9 K – large safety margin taken, operational gradient chosen at 220 T/m Cell length determines focusing strength, i.e. quadrupole length Quadrupole length → length in the cell available for dipoles together with total length of dipoles → number of quadrupoles  400 is the space for correctors, instrumentation, interconnections

32. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.32 CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing Aperture requirements for arc quads and dipoles (size of the beam) Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters Example: the LHC 5. The interaction regions: low-beta magnet specifications How to squeeze the beam Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

33. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.33 5. THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS We are now in the straight sections of the machine There are no dipoles Only quadrupoles to keep the beam focused In the middle of the straight section one has a free space for the experiment, with the interaction point (IP) where beams collide Around the experiment the optics must keep two distinct aims Keep the beam focused Reduce the size of the beam in the interaction point (IP) to increase the rate of collisions (luminosity) Beam size proportional to  (  ) – but  is invariant, so act on 

34. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.34 5. THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS A system of quadrupoles is used to reach a very low beta function, called  *, in the IP (LHC: 0.55 m instead of the 30-200 m in the arcs) Physical constraint: empty space around the IP – distance of the first magnet to the IP, called l *, (LHC: 23 m) – needed for the detectors ! The lay-out of quadrupoles close to the interaction point in the LHC, and the beta functions

35. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.35 5. THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS Drawback: beta function gets huge in the quadrupoles ! But this happens only in collision, where the beam is smaller In free space around IP ( s =0), one has At the entrance of the triplet one has In reality, the situation is even worse: the maximum beta function in the LHC triplet is much larger than at the entrance at the entrance we have whereas in the triplet we have  m =4400 m

36. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.36 5. THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS Aperture requirement:  a+c/  * and dependent on l *, l t Given the aperture, the technology limits the maximal gradient At first order, G  1/  We will show the limits of the approximation, and a more precise estimate, in Unit 8 The triplet has to focus the beam in the interaction point The focusing strength is a function of l *, l t, and is not related to  * This gives a requirement on the integrated gradient … … that together with the maximum gradient gives the triplet length The 4 equations are coupled

37. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.37 5. THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS The 4 equations are coupled For the LHC, one has  * =0.55 m  m =4400 m With respect to the arc,  m is ~22 times larger, but the  is ~16 times larger in collision  the aperture is not so different from the cell magnets  = 0.070 m instead of  = 0.056 m in the arcs With a triplet length of 24 m the required integrated gradient of 4800 T This requires a quadrupole gradient of 200 T/m With Nb-Ti one can get up to 300 T/m quadrupoles of  = 0.070 m – one has a good safety margin

38. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.38 5. THE INTERACTION REGIONS: LOW-BETA MAGNET SPECIFICATIONS Example: the LHC interaction regions Baseline: Nb-Ti quadrupoles, 200 T/m, 70 mm aperture,  * =0.55 m LARP : Nb 3 Sn quadrupoles, 200 T/m, 90 mm aperture,  * =0.25 m

39. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.39 CONTENTS 1. Principles of synchrotron 2. The arc: how to keep particles on a circular orbit Relation between energy, dipolar field, machine length 3. The arc: size of the beam and focusing Aperture requirements for arc quads and dipoles (size of the beam) Gradient requirements (focusing force) for arc quadrupoles 4. The arc: a flow chart for computing magnet parameters Example: the LHC 5. The interaction regions: low-beta magnet specifications How to squeeze the beam Gradient and aperture requirements for low-beta quadrupoles 6. The interaction regions: detector specifications

40. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.40 6. THE INTERACTION REGIONS: DETECTOR SPECIFICATIONS Detector magnets provide a field to bend the particles The measurement of the bending radius gives an estimate of the charge and energy of the particle Different lay-outs A solenoid providing a field parallel to the beam direction (example: LHC CMS, LEP ALEPH, Tevatron CDF) Field lines perpendicular to (x,y) A series of toroidal coils to provide a circular field around the beam (example: LHC ATLAS) Field lines of circular shape in the (x,y) plane Sketch of a detector based on a solenoid Sketch of the CMS detector in the LHC

41. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.41 6. THE INTERACTION REGIONS: DETECTOR SPECIFICATIONS Detector transverse size The particle is bent with a curvature radius B is the field in the detector magnet R t is the transverse radius of the detector magnet The precision in the measurements is related to the parameter b A bit of trigonometry gives The magnetic field is limited by the technology If we double the energy of the machine, keeping the same magnetic field, we must make a 1.4 times larger detector …

42. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.42 6. THE INTERACTION REGIONS: DETECTOR SPECIFICATIONS Detector transverse size B is the field in the detector magnet R t is the transverse radius of the detector magnet The precision in the measurements is  1/ b Examples LHC CMS: E =2300 GeV, B =4 T, R l =12.9 m, R t =5.9 m, b =9 mm LEP ALEPH: E =100 GeV, B =1.5 T, R l =6.5 m, R t =2.65 m, b =16 mm that’s why we need sizes of meters and not centimeters ! The magnetic field is limited by technology But fields are not so high as for accelerator dipoles (4T instead of 8 T) Note that the precision with BR t 2 – better large than high field … Detector longitudinal size several issues are involved – not easy to give simple scaling laws

43. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.43 SUMMARY We gave the principles of a synchrotron The problem is not only accelerating …but also keeping on a circle ! Magnets are needed for keeping particle on the orbit Arcs: dipoles for bending and quadrupoles for focusing How to determine apertures, fields and gradients Input: machine energy and beam emittance (injectors) Free parameter: cell length Output: dipole field, quadrupole gradient, magnet lengths and numbers (i.e. machine length, excluding IR regions) Interaction regions How to squeeze the beam size Determination of the aperture, gradient and length of the IR quads

44. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.44 COMING SOON During the next days: How these technological limits are determined ? What is the physics and the engineering behind?

45. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.45 REFERENCES Beam dynamics - arcs P. Schmuser, et al, Ch. 9. F. Asner, Ch. 8. K. Steffen, “Basic course of accelerator optics”, CERN 85-19, pg 25-63. J. Rossbach, P. Schmuser, “Basic course of accelerator optics”, CERN 94- 01, pg 17-79. Beam dynamics - insertions P. Bryant, “Insertions”, CERN 94-01, pg 159-187. Beam dynamics - detectors T. Taylor, “Detector magnet design”, CERN 2004-08, pg 152-165.

46. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.46 ACKNOWLEDGEMENTS J. P. Kouthcouk, M. Giovannozzi, W. Scandale for discussions on beam dynamics and optics www.wikipedia.orgwww.wikipedia.org for most of the pictures The Nikolai Tesla museum of Belgrade, for brochures, images, and information, and the anonymous guard I met in August 2002 F. Borgnolutti for listening all my dry talks B. Bellesia for providing the slides template

47. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.47 APPENDIX A: DEPENDENCE ON THE CELL LENGTH Example: Large Hadron Collider Larger L → larger beta function → larger beam size → larger magnet aperture, but Larger L → small number of cells → smaller focusing strength → smaller number of quadrupoles

48. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.48 APPENDIX A: DEPENDENCE ON THE CELL LENGTH Example: Large Hadron Collider Dipoles contribute for around 17.5 Km With a cell length of 30 m quads are 3.5 Km long (20%), with 70 m quads are 1 Km long (6%) – baseline is 50 m, giving 1.3 Km

49. USPAS June 2007, Superconducting accelerator magnets Unit 2 – Magnet specifications in particle accelerators 2.49 APPENDIX A: DEPENDENCE ON THE CELL LENGTH Example: Large Hadron Collider The amount of the cable needed for dipoles and quadrupoles can also be estimated – equations will be derived in Unit 8 The quantity of cable is roughly independent of the cell length, with a minimum around 50 m (but this was not the criteria used to select L !)