1. PTC ½ day – Experience in PS2 and SPS H. Bartosik, Y. Papaphilippou

2. 1 PTC ½ day – Experience in PS2 and SPS PTC – Introduction (I) PTC (Polymorphic Tracking Code) is a tracking code – Written by E. Forest (“Introduction to the Polymorphic Tracking Code”, 2002) – Based on symplectic integration using maps – Order of symplectic integrator is defined by user  allows to choose number of integration steps – PTC is a very powerful tool not only for tracking itself PTC normal form analysis allows to compute lattice parameters up to any order – Hamiltonian driving terms – Chromaticities – Anharmonicities (detuning with amplitude coefficients) – … – This cannot be done with MADX alone … PTC provides a generic way to take fringe fields into account – Hard-edge kicks in either side of the magnets (see E. Forest book) – It would be nice to be able to include map from measured field profiles 1

3. 2 PTC ½ day – Experience in PS2 and SPS PTC – Introduction (II) “Exact” (no expansion in momentum error) treatment of Hamiltonian – Can be of great important for small machines – Option “Exact” usually used for PS2 studies – Example drift space: user can choose between – The map of the exact Hamiltonian is non-linear in the momenta!!! Magnetic errors can be assigned to “thick elements” – As opposed to MADX, multipole errors can be assigned to thick elements instead of inserting thin lens kicks as used in MADX – Allows for better treatment of multipole errors PTC is very well suited for studying nonlinear dynamics in machines for which a detailed magnetic model exists (for example the LHC) However, PTC is not (yet) optimized for speed – Tracking with PTC is quite slow (many calls of “IF” routines) – Tracking with PTC a few times slower than with MADX 2 Exact HamiltonianExpanded Hamiltonian

4. 3 PTC ½ day – Experience in PS2 and SPS PTC as library in MADX PTC is used as a library in MADX, i.e. MADX can be used as front end for PTC – Definition of lattice in MADX (lattice can also be defined in PTC directly, however it is more complicated) Usual procedure for using PTC: – Creating “PTC-Layout” from existing accelerator structure within MADX – Normal form analysis or tracking can be done with PTC – Read the output back into MADX and provide to user No “matching” routine available in PTC – Matching is usually done in MADX – Matching of PTC parameters (like higher order chromaticity) can be done in MADX by calling PTC within a macro Some features are not sufficiently documented and thus are not easy to use or not known to exist – Users would benefit a lot from a better documentation of PTC and MADX as front end for PTC 3

5. 4 PTC ½ day – Experience in PS2 and SPS Tracking with PTC Using PTC for PS2 tracking studies – Define lattice in MADX – Match working point, chromaticity, …, in MADX (with macro using PTC in some cases) – Define multipole errors, misalign magnets in MADX and assign in PTC Simplest example: Tracking studies for finding dynamic aperture 4 Include errors

6. 5 PTC ½ day – Experience in PS2 and SPS Frequency Maps – PS2 lattices Plot “tune diffusion coefficient” d for generating Frequency map – Diffusion map shows d as function of initial condition for tracking in configuration space – Frequency map shows d as function of tune – Useful for identifying resonances – Distinguish regular motion (small tune diffusion) from chaotic motion (large tune diffusion) 5 tunes in first (second) half of total turns

7. 6 PTC ½ day – Experience in PS2 and SPS Amplitude detuning – PS2 lattices Normal form analysis in PTC can be used to calculate higher order anharmonicities (coefficients for detuning with amplitude) – Sufficiently far away from strong resonances and for sufficiently small amplitudes (non-chaotic regime), anharmonicities can be used to reconstruct detuning with amplitude as found by tracking Example of PS2 lattice including misalignments and fictitious error table for particles up to 3σ 6 Analytic representation using anharmonicities up to higher order Nominal tune Off momentum tunes Nominal tune TrackingFrom anharmonicities

8. 7 PTC ½ day – Experience in PS2 and SPS Nonlinear chromaticity - SPS Normal form analysis in PTC provides directly nonlinear chromaticity up to any order Example SPS: Establish “effective” machine model by matching multipole errors of main magnets to measured chromaticity – Was done in the past using MAD (G. Arduini et al., EPAC02) or using response matrix approach using the calculations of PTC (R. Tomas et al., PAC07) – Can be done very conveniently by matching “directly” higher order chromaticities using PTC macro in MADX 7 SPS Q20 optics

9. 8 PTC ½ day – Experience in PS2 and SPS Space charge simulations – PTC-ORBIT? ORBIT is a (macro particle) space charge simulation code developed for SNS – Written in a modular way  can be extended or combined with other codes – Is compatible with parallel computing on a cluster – Is used at CERN for simulations of the PSB in combination with LINAC4 Lattice imperfections and nonlinearities might play a key role for space charge effects – Defining magnet errors and misalignments in ORBIT is not easy – Properties of the lattice is not easily comparable with MADX simulations Idea of combining the power of MADX-PTC with the space charge calculations of ORBIT  PTC-ORBIT (A. Molodozhentsev et al.) Basic idea: start with MADX-PTC (or PTC) – Generate lattice, match optics – Study impact of misalignments and magnet errors on single particle motion with MADX-PTC – In case of existing machine: Develop “effective” machine model which is capable of reproducing the measured imperfections and non-linear effects (closed orbit, higher order chromaticity, amplitude detuning, resonances, …) – Dump lattice to PTC flat file 8

10. 9 PTC ½ day – Experience in PS2 and SPS PTC-ORBIT Use PTC-ORBIT for space charge simulations – The exact same lattice as used in MADX-PTC can be used for the tracking in space charge simulations  full control of the lattice (instead of creating the lattice in ORBIT itself with all the complications) – Many of the ORBIT standard routines are available in PTC-ORBIT First attempts with PTC-ORBIT were made with the SPS, PSB and PS lattices in 2010 – Still in progress of optimizing PTC parameters (obtaining good machine models) and implementing new features necessary for the CERN injector (time varying fields, double harmonic RF, travelling wave cavities, …) – No robust and trustable simulation for CERN machines results up to now Down sides of PTC-ORBIT – Documentation: manual of ORBIT is from 1999 – PTC-ORBIT is still in development  some features don’t work – Up to now, a lengthy procedure of executing various small codes has to be followed in order to generate input files in special formats needed for running PTC-ORBIT – Tracking with PTC is slower than using the built in tracking of ORBIT 9

11. 1010 PTC ½ day – Experience in PS2 and SPS PTC-ORBIT – Simulation procedure Use MADX PTC for generating lattice and do all the matching – Taking advantage of all the usual features of MADX PTC Create a PTC-flat file with all the lattice definitions – Define how many places should be foreseen for the insertion of space charge calculation nodes – The modifications of the lattice (including error assignments for example) are finished here – Exception: the feature of time varying fields in PTC-ORBIT allow to modify magnet strengths even at a later stage in the execution of PTC-ORBIT Prepare “acceleration table” – This table contains the evolution of the magnetic field (energy) and the parameters for the RF-cavities (phases, voltages) Use PTC-ORBIT for space charge simulations – Running PTC-ORBIT is very similar to running ORBIT itself, but basically all routines related to the lattice are (or have to be) replaced by special routines for invoking PTC – Many of the standard ORBIT features can be used also in PTC-ORBIT 10

12. 1 PTC ½ day – Experience in PS2 and SPS Summary PTC is a tracking code which provides powerful tools as for example normal form analysis – Very interesting for nonlinear dynamics analysis and lattice optimization – Very useful for developing effective nonlinear machine model PTC is usually used as library in MADX – Slower than other tracking codes – Big overhead of generating PTC environment, possibly several times per job execution One of the big problems is documentation PTC-ORBIT is a very interesting development for future space charge studies – Still needs some further improvement and development – One of the main issues is again documentation 11