Practical Lattice Design

Transcription

1 Practical Lattice Design Dario Pellegrini (CERN) USPAS January, 15-19, /17 D. Pellegrini - Practical Lattice Design

2 Lecture 5. Low Beta Insertions 2/17 D. Pellegrini - Practical Lattice Design

3 Insertions Insertions are interruptions of the normal periodicity of the lattice, which allow to accomplish specific optics requirements for special elements and instruments. We can have insertions for all kinds of scopes, acting on both the machine geometry and the optics. We saw already the dispersion suppressor, an insertion that creates a region of vanishing (or smaller) dispersion useful for: RF cavities (why?) Reducing the beam size, remember: σ 2 = βε + (Dδ) 2 Undulator magnets (see tomorrow) Today: low beta insertions! 3/17 D. Pellegrini - Practical Lattice Design

4 Why Low Beta? When delivering a particle beam to a target, the number of events per second can be written as: dr dt = Lσ p were σ p is the cross section of a specific process and L is the luminosity. If we have two colliding beams we can write the luminosity as: L = N 1N 2 f rev N b 4πσ x σ y For a given machine size and energy, f rev, the revolution frequency, is fixed. If we want to push the luminosity we can take: 1. the intensity route, acting on the number of bunches N b and on the number of particles in the two colliding bunches, N 1 N the beam size route, making σ x σ y as small as possible by having low emittances and low beta. 4/17 D. Pellegrini - Practical Lattice Design

5 How big can an insertion be? At the LHC there are 46 m of empty space! 5/17 D. Pellegrini - Practical Lattice Design

6 How hard can it be? These 46 m cost more than 1 km of optics manipulations! 6/17 D. Pellegrini - Practical Lattice Design

7 The most problematic insertion: the drift space Let s see what happens to the Twiss parameters α, β, and γ if we stop focusing for a while β α γ = C 2 CC C 2 2SC SC + S C 2S C S 2 SS S 2 β α γ s 0 for a drift: M drift = ( C S C S ) = ( 1 s 0 1 ) β (s) = β 0 2α 0 s + γ 0 s 2 α (s) = α 0 γ 0 s γ (s) = γ 0 7/17 D. Pellegrini - Practical Lattice Design

8 Let s find the location of the waist: α = 0 the location of the point of smallest beam size, β Beam waist: α (s) = α 0 γ 0s = 0 s = α0 γ 0 Beam size at that point } γ (l) = γ 0 α (l) = 0 γ (l) = 1 + α2 (l) β (l) This beta, at l = l waist, is also called beta star : = 1 β (l) = l waist β min = 1 γ 0 β = β min It s at l = l waist that the interaction point (IP) is located. 8/17 D. Pellegrini - Practical Lattice Design

9 A drift space with L = l waist : the Low β-insertion We can assume we have a symmetry point at a distance l waist : β (s) = β 0 2α 0s + γ 0s 2, at α (s) = 0 β = 1 γ 0 On each side of the symmetry point we have β grows quadratically with s. β (s) = β + s2 β A drift space at the interaction point, with length L = l waist, is called low-β insertion : 9/17 D. Pellegrini - Practical Lattice Design

10 Phase advance in a low-β insertion We have: β (s) = β + s2 β The phase advance across the straight section is: µ = Lwaist L waist ds β + s2 β which is close to µ = π for L waist β. = 2 arctan L waist β In other words: in the interaction region the tune increases by half an integer! 10/17 D. Pellegrini - Practical Lattice Design

11 Crossing angle At the LHC the bunch spacing can go as small as 25 ns or 7.5 m. Therefore the beams need to be separated to avoid unwanted collision. However introducing a crossing angle also reduces the bunch overlap, with impact on the luminosity: L Xing = LS; S = 1 ( σ 1 + z σ x tan θ 2 The minimum separation between the two beams is determined by the mutual perturbation due to the long range encounters. 11/17 D. Pellegrini - Practical Lattice Design ) 2

12 Combining low-β and beam separation: a big headache! Both dipoles and quadrupoles need to be close to the IP, not always integrable into the detector. 12/17 D. Pellegrini - Practical Lattice Design

13 Reference orbit and closed orbit The beams, that normally are in two separated pipes, are brought on the same reference orbit by the separating dipoles, D1 and D2. The triplet is centered on the reference orbit (same for both beams). Orbit correctors are used to create orbit bumps (horizontal or vertical) to separate the beams. The closed orbit differs from the reference orbit. Bumps are not only for the crossing angle, but also for IP shifts. These are used when the relative beam positions need to be scanned (e.g. to find collisions), or to compensate for ground motion, keeping the collision spot at the center of the detector. 13/17 D. Pellegrini - Practical Lattice Design

14 Some Guidelines 1. Calculate the periodic solution in the arc 2. Start from the IP, introduce the drift space needed for the insertion device (detector...) 3. Install a quadrupole triplet (or doublet?) fix the aperture requirements and the achievable field gradient 4. Set the desired beta*, drive the triplet at high field, so that the beam is focused back 5. Introduce additional quadrupoles to match the beam parameters to the values at the beginning of the arc Parameters to be optimized & matched to the periodic solution: β x α x D x µ x β y α y D y µ y (D is normally accepted at the IP) 8 (at least) individually powered quad magnets are needed to match the insertion 14/17 D. Pellegrini - Practical Lattice Design

15 2 in 1 Synchrotron and Collider While it may look natural to have the same machine performing both the acceleration and the collisions for physics, this is not without additional complications! The beam size shrinks significantly during the acceleration from 450 GeV to 7000 GeV (remember ε n = γ rel ε is constant), this means that at injection energy it will not be possible to accommodate low β. We need a dedicated optics for injection, which will be modified after reaching top energy and/or during the energy ramp. To achieve the β squeeze, many match points for a fine granularity of β values are computed, making sure that the magnetic strengths vary in a smooth way between them. 15/17 D. Pellegrini - Practical Lattice Design

16 Pushing even further: telescopic squeeze One should avoid flipping the polarity of the magnets, always maintaining some strengths in them in order to avoid uncertainties from the hysteresis cycle. At the LHC although the triplet aperture would allow for smaller β, one runs out of steam due to the limited length of the matching sections and the available strength. The solution comes from freezing the matching section and let the β mismatch propagate along the arc (remember: at top energy we have extra aperture gained during the acceleration), fixing the optics on the other side of the arc. With a proper phase advance per cell (eg. 90 deg or 120 deg) the arc periodicity is anyway restored after few cells. Exercise: show that the transfer matrix of 4 FODO cells tuned at 90 deg is the identity matrix. 16/17 D. Pellegrini - Practical Lattice Design

17 Designing the IR: not just optics! The IR has to accomodate a number of devices: Quadrupoles and dipoles for the beam squeeze and separation Higher order magnets and correctors to compensate for field errors Instrumentation (BPM) and kickers for orbit control and correction Absorbers for the debris coming from the IP, important to limit the irradiation of components and/or the background to the experiments Collimators for the protection of the triplet in case of failure of some machine component external to the IR For more info: LHC Design Report, CERN /17 D. Pellegrini - Practical Lattice Design