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Explore the theory and effects of wakefields on collimators, including resistive and geometric properties, simulations, and application to accelerators. Learn about losses, kicks, and modes in various structures.

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## Collimator wake-fields

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**Collimator wake-fields**Wake fields in collimators General information Types of wake potentials Geometric Steep Tapered Dielectric layers Resistive Uniform Transient Surface roughness Merlin simulations Geometric Resistive Collimator wakefields - G.Kurevlev Manchester**Generalinformation about Wakefields applicable to**collimators and waveguides General theory When charged particle is moving through linac or any other structure in accelerator it generates fields in the structure. Reasons of the fields can be different: limited resistivity of the pipes, dielectric layers on the walls, macro obstacles or micro obstacles due to surface roughness. The generated fields will affect any trailing particle following the leading one. In general the fields depend on both particles positions so the momentum change is (Stupakov): And we can define wakes We can integrate over the structure fields generated by the bunch to get total effect of fields on particles from the same or the next bunch. Total effects are usually expressed in loss and kick factors. Loss factor is the integrated energy (momentum) loss by the bunch over passing the structure. Kick factor is usually average transverse angle gained by the bunch passing the structure. Fields of the leading particle are calculated assuming there are no transverse displacements of the particle during passing the structure due to the Wake fields. Collimator wakefields - G.Kurevlev Manchester**Generalinformation about Wakefields applicable to**collimators and waveguides Superposition Superposition principle allows us to calculate one particle fields and then integrate over all the other particles in the bunch so: We can split bunch of relativistic particles into transverse slices. Si – slice position in the bunch. Wake fields of the slice can be received by integration over transverse distribution of the particles in the slice. Wake fields of the bunch can be received by convolution over longitudinal distribution of the particles in the bunch. Slices cannot mix so we can express all the field components in a frequency domain in the following way: Useful but not obligatory simplification - ultrarelativictic limit: Additional simplification without specific geometry assumed – relation between longitudinal and transverse wakes stated as Panofsky-Wenzel theorem Collimator wakefields - G.Kurevlev Manchester**Generalinformation about Wakefields applicable to**collimators and waveguides Uniform structures For uniform structures we don't need to integrate over longitudinal direction of the structure as the fields don't depend on this so we need only get the field itself. Wake fields are estimated over the unit length of the structure. Different units of measurements for Wakes used here (1/L). Consequently to get usual loss or kick factors we need to multiply by L. Collimator wakefields - G.Kurevlev Manchester**Generalinformation about Wakefields applicable to**collimators and waveguides Modes in axially symmetric structures All the filed components depend on the angle by exp(lθ). Collimator wakefields - G.Kurevlev Manchester**Geometric wakefields**Linked with macro obstacles in the structures such collimators as a whole. In steep collimators diffraction theory applicable so we have Circular case (Chao) With higher modes and correction confirmed by MAFIA and Merlin simulations Rectangular case (Stupakov) No higher modes yet and for m=1 Tapered cases Circular (Yokoya) Rectangular (Stupakov) Collimator wakefields - G.Kurevlev Manchester**Dielectric wakefields**Investigated as new collective acceleration methods (Park, Hirshfield (2000)) Usual approach is similar to resistive wakes - field matching technique on boundaries The difference is that we don’t need to count energy loss in the walls but have waves with different speed in different media. Cn-normalization constant that can be received by orthonormality relation and from the source currents. Collimator wakefields - G.Kurevlev Manchester**Resistive wakefields**Linked with finite conductivity of the metal walls of the pipe. Analysis usually limited by uniform structures Circular case well investigated Chao Bane, Sands – with corrections for small s and for a.c. conductivity Flat and elliptic cases (Piwinski; Yokoya) Rectangular case and elliptic case (Gluckstern) – only monopole and dipole Form factors in comparison with round case Transition effects were studied recently for short bunches and for the same circular case by Ivanyan, Tsakanov; Glukstern; Stupakov It was demonstrated how the potential evolves to uniform one received in Bane, Sands paper when there is a transition from infinite to finite conductivity pipe at some z. Collimator wakefields - G.Kurevlev Manchester**Resistive wakefields**Rectangular case again Known approach for rectangular case – perturbation theory (Gluckstern; Palumbo). Following Chao’s analysis but in Cartesian coordinates we can get similar expression for all the fields. Ignoring space charge in ultrarelativistic limit we will get constant longitudinal electric field in the pipe. Integrating this over frequency we will get the same expression for 0 order longitudinal wake as in circular case. 1st order solutions based on Poisson equation and Leontovich boundary condition. Form factors received for rectangular case are quite similar to elliptical In fact we can utilize standard eigenfunctions approach developed for dielectric layered waveguides for rectangular case. Different set of eigenfunction - so we need to use different orthonormality relations to get normalization constant. This is under investigation now. Collimator wakefields - G.Kurevlev Manchester**Merlin simulation**Merlin - optical code with phenomenological wake fields to get the effect on beam transport through the linac and BDS Merlin had a process for handling wake fields. We inserted Wake functions. Bunch slices for wakes integration prepared and integration already implemented Momentum change on slice i Collimator wakefields - G.Kurevlev Manchester**Merlin simulation**Geometric wake for steep collimator, kick factor SLAC experiment Merlin simulation with up to 50 modes Beam parameters used in experiment and simulation: Beam (bunch) charge Q_total = 2e+10 e - number of electrons in the bunch - Ne = 2e+10Beam energy p0 = 1.19 GeVLattice horizontal betta function betta_x = 3 mHorizontal emittance emit_x = 0.36 mm, emit_y = 0.16 mm ,Lattice vertical betta function betta_y = 10 m (we used these ones to have minimal emittance grow - there are no exact values from experiment)Longitudinal bunch size sigma_z = 0.65 mmSquared aperture of the spoiler with the side width 38 mm (half width to use in wake formulas b1 = 1.9 mm) Collimator wakefields - G.Kurevlev Manchester**Merlin simulation**Resistive SLAC experiment Merlin simulations for 5 modes (1-Cu, 2- Ti) Collimator wakefields - G.Kurevlev Manchester**Merlin simulation**Dielectric wakes, in progress Collimator wakefields - G.Kurevlev Manchester

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