Effects of Element Misalignments on Accelerator Performance Andrea - PowerPoint PPT Presentation

Effects of Element Misalignments on Accelerator Performance Andrea Latina (CERN) andrea.latina@cern.ch 2 nd PACMAN Workshop - Debrecen, Hungary - June 13-15, 2016

Table of Contents ◮ Recap of beam dynamics ◮ Why is the emittance important? ◮ Effects of static misalignments ◮ Mitigation techniques ◮ Integrated performance ◮ Conclusions 2/19 A.Latina - 2nd PACMAN Workshop

Recap of beam dynamics (I) : Forces and kicks Phase-space coordinates of a single particle: � x , � T x ′ , y ′ , δ = P − P 0 ∆ t , y , P 0 [m] [rad] [m] [rad] [m/c] [#] with the transverse angles defined as: x ′ = P x [MeV/c] y ′ = P y [MeV/c] P z [MeV/c] ; P z [MeV/c] Recall the Lorentz force, F = q ( E + v × B ) . A transverse force over a length ∆ s imparts a transverse momentum ∆ P ⊥ = F ⊥ ∆ t : 1 ∆ P ⊥ [MeV/c] = F ⊥ [MeV/m] V z [c] ∆ s [m] which translates into a transverse kick, e.g. ∆ x ′ along the x axis: ∆ x ′ [rad]= ∆ P x [MeV/c] x ′ ( s +∆ s ) = P x +∆ P x = x ′ ( s ) + ∆ x ′ with P z P z [MeV/c] 3/19 A.Latina - 2nd PACMAN Workshop

Recap of beam dynamics (II) : Twiss parameters The particle motion in a periodic lattice is described via the solution of the Hill’s equation: x ( s ) = √ ε � β ( s ) cos ( µ ( s ) + µ 0 ) ◮ β ( s ) , beta function, and µ ( s ) , phase advance (see Twiss parameters), are lattice properties : � s ds ′ µ ( s ) = β ( s ′ ) 0 ◮ µ 0 , the initial phase, and ε , the Courant-Snyder invariant (or “action”), are a particle properties Particle transport:      �  0 β j � ∆ x ′ ( cos µ + α i sin µ ) β j β i sin µ           β i 0   � �     X j = · X i +  �  ∆ y ′       ( α i − α j ) cos µ − ( 1 + α i α j ) sin µ β i       ( cos µ − α j sin µ )   0   � β j β i β j 0 i � �� � Mi → j (actually a 6 × 6 matrix) IMPORTANT: the β -function amplifies unwanted transverse kicks. 4/19 A.Latina - 2nd PACMAN Workshop

Recap of beam dynamics (III) : Beam emittance The Twiss parameters define the beam ellipse: ε = γ x 2 + 2 α x x ′ + β x ′ 2 ◮ The ellipse amplitude, ε , , is a particle property (called Courant-Snyder invariant or “action”) ◮ For an ensemble of particles we define the (geometric) beam emittance , ǫ , a quantity proportional to the area of the ellipse The geometric emittance, ǫ , is defined as: � � � � x , x ′ �� � x 2 � � x ′ 2 � − � x x ′ � 2 ǫ geom = det cov = From which we can compute beam size and divergence: � σ x = βǫ geom � ǫ geom σ x ′ = β (recall the normalised emittance, ǫ norm = β rel γ rel ǫ geom ) 5/19 A.Latina - 2nd PACMAN Workshop

Effects of kicks on the emittance Nominal emittance: � � � x 2 � � x ′ 2 � − � x x ′ � 2 = (if x and x ′ are uncorellated) � x 2 � � x ′ 2 � ǫ 0 = In presence of transverse kicks, ∆ x ′ , the emittance transforms ǫ 0 → ǫ : � � ( x ′ + ∆ x ′ ) 2 � � � x 2 � � x ′ 2 � � + � x 2 � � ∆ x ′ � 2 � x 2 � ǫ = ≈ � � � �� � ǫ 2 0 � � x 2 � � ∆ x ′ � 2 1 + ✟✟ = ǫ 0 � x 2 � ✟✟ � x ′ 2 � � � � ∆ x ′ � 2 1 + 1 ≈ ǫ 0 2 � x ′ 2 � From which derives the emittance growth: σ 2 ∆ ǫ = ǫ − ǫ 0 = 1 ∆ x ′ ∆ ǫ ∝ σ 2 ⇒ ∆ x ′ σ 2 ǫ 0 ǫ 0 2 x ′ For example a quadrupole: ∆ x ′ = K 1 L q ∆ x misalign ≡ ∆ x misalign ∆ ǫ ∝ σ 2 ⇒ misalign f length 6/19 A.Latina - 2nd PACMAN Workshop

Why is the emittance important? ◮ In a particle collider the number of collision is given by the luminosity: N 2 n b f r L = H D 4 πσ ⋆ x σ ⋆ y with � σ ⋆ ⊥ = β ⋆ ⊥ ǫ ⊥ geom ⊥ = x or y • HD , disruption parameter (enhances the luminosity) ; N , number of particles per bunch ; nb , number of bunches per train ; fr , repetition frequency ; σ⋆ x , y transverse beam sizes at the interaction point (IP) ◮ CLIC emittance growth budgets, in the main linac, due to static misalignments: ∆ ǫ x =60 nm and ∆ ǫ y =10 nm in the horizontal and vertical axes, respectively. 7/19 A.Latina - 2nd PACMAN Workshop

Effect of magnet misalignments ◮ "Feed-down" effect: A misaligned magnet of order N behaves like a magnet of order N − 1 : ◮ a misaligned quadrupole gives a dipolar kick, ◮ a misaligned sextupole gives a quadrupolar kick, etc. etc. ◮ Effect of a misaligned quadrupole (the most frequently used type of magnet!) ◮ deflects the beam trajectory, excites betatron oscillations, introduces unwanted dispersion ◮ Unwanted dispersion is bad ! Because it adds a position-momentum correlation, and increases the beam size: x ( s ) → x ( s ) + D ( s ) ∆ P P 0 D 2 σ 2 σ 2 x → σ 2 x + ∆ P P 0 ���� ( energy spread ) 2 8/19 A.Latina - 2nd PACMAN Workshop

Effect of accelerator structure misalignments Particles traversing a structure with an offset excite Wakefields: ◮ transverse effect: z -dependent deflection ∆ x ′ = W ⊥ ( z ) Ne 2 L S ∆ x P 0 ◮ W ⊥ ( z ) , transverse wakefield function, [V/pC/m/mm] ◮ N , number of particles per bunch ◮ L S , structure length ◮ z , particle distance from bunch head ◮ ∆ x , transverse offset beam ← → structure ◮ longitudinal effect: z -dependendent energy loss ∆ P z = W � ( z ) Ne 2 L S ◮ W � ( z ) , longitudinal wakefield function, [V/pC/m] ◮ independent from misalignment ◮ Their effect can be felt by particles at ◮ short-range (same bunch) → emittance growth ◮ long-range (bunch to bunch) → beam breakup 9/19 A.Latina - 2nd PACMAN Workshop

Effect of BPM misalignments Misaligned BPMs affect the beam, indirectly: ◮ they do not deflect the beam, but can compromise the effectiveness of optimisation techniques ◮ (and actually they can also deflect the beam, e.g. high-resolution “cavity BPMs”, which can create Wakefields) Besides, off-centred BPMs can display: ◮ loss of resolution, or scaling errors: x read = α scaling x real ⇒ BBA, Beam-Based Alignment, can cure this problem (e.g. DFS, Dispersion-Free Steering) . Other errors that affect all elements are: ◮ angle errors: offsets in ∆ x ′ and ∆ y ′ ◮ roll errors: rotations around the beam axis ∆ φ 10/19 A.Latina - 2nd PACMAN Workshop

Beam-based alignment techniques 1. Simplest: ◮ Quad-shunting: each quadrupole is moved until the magnetic centre is determined ◮ One-to-one : transverse kickers are used to steer the beam through the centre of the BPMs 2. Dispersion-Free Steering (DFS) / Wakefield-Free Steering (WFS) ◮ DFS: the presence of dispersion is detected and measured, and transverse kickers are used to counteract it ◮ WFS: the presence of wakefields is detected and measured, and transverse kickers are used to counteract it 3. RF Alignment (specific to CLIC) ◮ Relative beam offset in the accelerating structures is measured, and structures are moved to reduce it ◮ Others: ◮ Emittance tunng bumps, coupling correction ◮ MICADO: similar to one-to-one, but picks the best correctors ◮ Kick-Minimisation: useful when DFS cannot easily be applied (e.g. ILC turnaround loops, ...) 11/19 A.Latina - 2nd PACMAN Workshop

Example of emittance growth in the CLIC main linac Misalignment of all components, σ RMS = 10 µ m ◮ Initial emittance is 10 nm ◮ Emittance growth is enormous... One simple mitigation technique is used: “One-to- one” steering, but it’s far from being sufficient... “One-to-one” steering: Dispersive orbit ◮ Each corrector is used to steer the beam (PLACET Simulation, courtesy of D.Schulte) through the centre of the following BPM ◮ Orbit makes its way through the accelerator, but dispersion is still present... 12/19 A.Latina - 2nd PACMAN Workshop

Beam-based Alignment: Dispersion-Free Steering (DFS) Principle of DFS: 1. Measure the dispersion by altering the beam energy (or scaling the magnets) 2. Compute the correction which minimise the dispersion Graphically: That is, minimise: χ 2 = ( y ∆ E , i − y i ) 2 + β 2 � � y 2 i + ω 2 � θ 2 m bpms bpms corrs In practice, one needs to solve the system of equations:   θ 1     b − b 0 R .   ω ( η − η 0 ) = ω D .      .  0 β I θ m � �� � � �� � � �� � measured observables response matrices unknowns 13/19 A.Latina - 2nd PACMAN Workshop

Example of DFS in the CLIC main linac Minimise: � � 2 + β 2 � χ 2 = � y 2 + ω 2 � θ 2 y ∆ E , i − yi i m corrs bpms bpms or, solve the system of equations:   θ 1     b − b 0 R  .    ω ( η − η 0 ) = ω D .       . 0   β I θ m � �� � � �� � response matrices � �� � measured observables unknowns 10000 QUADs CAVs BPMs 1000 RES ALL ∆ε y / ε y [%] 100 10 1 0.1 (PLACET Simulation, courtesy of D.Schulte) 0.01 0.1 1 10 100 1000 10000 ω [#] 14/19 A.Latina - 2nd PACMAN Workshop

Example of RF Alignment in the CLIC main linac RF Alignment: ◮ Each structure is equipped with a wakefield monitor ◮ Up to eight structures on one movable girders Before correction: After correction: (PLACET Simulation, courtesy of D.Schulte) 15/19 A.Latina - 2nd PACMAN Workshop