RFGymnasticsToolbox

Bunch manipulations with the radio-frequency system of a particle accelerator

Important: this package requires the elliptic package from https://github.com/moiseevigor/elliptic located in a subdirectory below the present one!

Check out the following papers for the theory behind the code

• V. Ziemann, Longitudinal phase-space matching between radio-frequency systems with different harmonic numbers and accelerating voltages, FREIA Report 2021-06, December 2021; see also arXiv:2112.14085;
• V. Ziemann, RF gymnastics with transfer matrices, FREIA Report 2022-03, February 2022; see also arXiv:2202.03964.

The following image shows the result of a simulation using the full non-linear equations in red, using the linearized equations in green, and using the beam matrix propagation, described in the second report, in blue.

Examples

The following examples each define a sequence of commands, such as creating a distribution, propagating a distribution, and displaying it, which is called a schedule. Check out the examples, most of the commands are reasonably self-explanatory. All commands are explained below on this page.

• ex_one_step_transfer.m, quarter-wave transfer from h1,v1 to h2,v2 via one intermediate state;
• ex_bunch_munch.m, transfer by quarter rotation at lower voltage and subsequent quarter rotation with the original settings;
• ex_phase_jump.m, transfer by moving the beam into the unstable fixed point and a subsequent 135 degree rotation;
• ex_four_step_cascade_with_munch.m, transfer via four quarter-wave steps to h4 = 2 h3 = 4 h2 = 8 h1 and a final bunch munch;
• ex_four_step_cascade_with_phase_jump.m, with a final phase jump to shorten the bunch;
• ex_four_step_cascade_with_phase_jump_fast.m, the same as the previous, but without intermediate steps;
• ex_filamentation.m, example of bunch filamentation;
• ex_rebunching_simulation.m, debunch on first harmonic and rebunch on a higher harmonis.
• ex_spider.m, make short bunches in adjacent buckets.

Support files

• create_support_functions.m, must be run once before any other function. It creates a number of anonymous functions that make writing a schedule more readable. It creates a file longitudinal_dynamics_support_functions.mat that must be present in the current subdirectory.
• run_schedule.m(schedule), this function receives the schedule and performs all calculations. It's the heart of this package.
• pendulumtracker.m, integrates the pendulum equation in terms of Jacobi elliptic functions. Requires the elliptic package from https://github.com/moiseevigor/elliptic located in a subdirectory below the present one (for the fast evaluation of elliptic functions).
• track_on_harmonic.m, shifts and scales the phase appropriately to integrate pendulum equation on a harmonic larger than the first.
• show_distribution.m, displays the distribution of sample particles.
• show_distribution_linear.m, given the longitudinal sigma matrix, this function displays the appropriate phase ellipse.
• show_projection_inset.m, creates a small inset with the longitudinal projection of the particle distribution.
• show_projections.m, creates a separate figure with the projections onto the phase and energy axis.
• show_separatrix.m, displays the separatrix suitable for the harmonic and voltage
• fwhm.m, returns the full-width at half maximum of a distribution.
• Makefile, contains commands to convert the movie file tmp.avi to tmp.mp4 and to display it (requires ffmpeg installed).

Commands understood by run_schedule()

The schedule is an array with six 6 columns and one command for each row, which starts with a code, followed by a repeat and four numbers whose interpretation depend on the code in the first column. The repeat column is just a convenient way to break up long step into a sequence of smaller steps, which makes movies very easy to create.

The first set of commands sets parameters and enables features of the simulation.

• set_parameters(Npart,Omegas) is used to define the number of particles Npart used in the simulation and the reference synchrotron frequency Omegas. It returns [100,1,Npart,Omegas,0,0]; Note that Omegas=0.5 at h=1,v=1 causes the separatrix to have a height of unity, thus setting a natural scale for the bucket half-height.

• set_vscale(vscale) set the vertical scale of the plots. It returns [200,1,vscale,0,0,0];

• set_adiabatic(Nturn) sets the number of turns Nturn to propagate the particles when simulating an adiabatic change of the voltage. It returns [300,1,Nturn,0,0,0];

• set_pause(time) sets the waiting time between plots displayed. It returns [400,1,time,0,0,0];

• show_distribution(r,g,b) displays the particle distribution. It returns [900,1,r,g,b,0];

• show_projections displays the projections of the particle distribution onto the horizontal and vertical axis. It returns [901,1,0,0,0,0];

• new_figure starts a new figure. It returns [902,1,0,0,0,0];

• save_distribution(n) saves the current particle distribution into file dist_x1.n. It returns [903,1,n,0,0,0];

• load_distribution(n) loads the previously-saved distribution from file dist_x1.n. It returns [904,1,n,0,0,0];

• start_movie(fps) starts recording a movie with frame rate fps. The file is called tmp.avi. It returns [905,1,fps,0,0,0];

• stop_movie stops recoring the move. It returns [906,1,0,0,0,0];

• save_image saves the current image to a file. It returns [907,1,0,0,0,0];

• save_images causes all images to be automatically saved. It returns [908,1,1,0,0,0];

• save_fwhm saves the full-width at half-maximum in a file for later display. It returns [909,1,1,0,0,0];

• show_inset(xpos,ypos) displays the longitudinal projection of the distribution in a small inset on the bottom right of the current plot. It returns a [910,1,xpos,ypos,0,0];

• linear_transport enables the transport of the particles with the transfer matrices discussed in the papers mentioned above, normally shown as the green dots. It returns [911,1,0,0,0,0];

• sigma_transport enables the transport of the beam (sigma) matrix with the transfer matrices mentioned in the papers above, normally shown as the blue ellipse. It returns [912,1,0,0,0,0];

The second set of parameters actually manipulates the particle distribution

• init_dist(h,v,sigphi,trunc) returns the following row [1,1,h,v,sigphi,trunc], which initializes the particle distribution with a bunch, matched to harmonic h and voltage v, having a length of sigphi degrees and the particles are sampled from a Gaussian that is truncated a trunc sigmas. If trunc is positive, all buckets are filled, if it is negative ony the one at phase zero.
• prop_dist(n,h,v,phi,dt) moves the distribution forward in n steps, each of duration dt at harmonic h and voltage v with phase shifted by phi. It returns [2, n, h, v, phi, dt]; Note that dt refers to the fraction of the revolution time at the harmonic and voltage given. Also phi refers to phase at harmonic h.
• move_dist(dphi,denergy) moves all particles by dph and denergy. It returns [4,1,dphi,denergy,0,0];
• prop_adiabatic(h1,v1,dv,v2) simulates an adiabatic (very slow) change of the RF voltage. It propagates all particles with a RF system operating at harmonic h1 and increases the voltage in steps of dv from v1 to v2. At each step Nturn (set with set_adiabatic()) synchrotron oscillations are performed before incrementing the voltage. It returns [10, 1, h1, v1, dv, v2];