1. Astra A Space Charge Tracking Algorithm
Klaus Flöttmann
DESY
Zeuthen

2. General
Program development started in ’96 based on a precursor by Ch. Stolzenburg
written in Fortran 90, runs on:
Windows PC
LINUX PC
SUN UNIX
MAC OS
~ users

3. Astra design philosophy
the program is user oriented, i.e. it is designed
to be easy to use
to be robust against user errors
to avoid ambiguous parameters
Astra supports parameters scans.
Astra is complemented by graphics and analysis tools.

4. Astra design philosophy
in order to avoid the development of diverging versions only executables are distributed.
to give the user insight into what is going on in the code a number of tools are provided. Examples will be shown in the following.

5. Space Charge Calculation
a grid consisting of rings (transversely) and slices (longitudinally) is set up for the space charge calculation.
the grid size matches the dimensions of the bunch, with some additional grid cells added automatically outside of the bunch.

6. Space Charge Calculation
the space charge field is calculated in the average rest frame of the bunch assuming a constant charge density inside the ring elements.
remaining non-relativistic velocity components in the average rest frame can be taken into account.
in the center of the bunch statistical problems may arise due to the small cell volume despite the higher charge density.

7. uniform charge density, 1000 macro-particles,
Example: radial electric space charge field of a bunch as calculated by Astra, plotted with fieldplot
uniform charge density,
1000 macro-particles,
10x10 grid cells
grid lines
bunch extension

8. Example: radial electric space charge field of a bunch as calculated by Astra, plotted with fieldplot
uniform charge density,
1000 macro-particles,
10x10 grid cells
increased cell size in the center

9. Example: radial electric space charge field of a bunch as calculated by Astra, plotted with fieldplot
uniform charge density,
5000 macro-particles,
10x10 grid cells
adjusted cell size

10. Space Charge Calculation
The grid and the space charge field are scaled as the bunch propagates through the beam line according to varying bunch dimensions, charge and energy.
When the scaling factor exceed a user defined limit a new space charge calculation is automatically initiated.

11. Example: development of electric space charge field as seen by probe particles. (Calculation with Astra, plotted with lineplot.)
L-band rf gun
5000 macro-particles,
10x10 grid cells

12. Example: development of electric space charge field as seen by probe particles. (Calculation with Astra, plotted with lineplot.)
L-band rf gun
5000 macro-particles,
10x10 grid cells

13. Example: development of space charge scaling factors in Astra.
L-band rf gun
5000 macro-particles,
10x10 grid cells
Cavity shape

14. Example: development of time steps.
L-band rf gun
5000 macro-particles,
10x10 grid cells

15. Emission of particles from the cathode

16. Emission of particles from the cathode
Random distribution with clusters and voids.

17. Emission of particles from the cathode
Regular distribution with mis-matched grid.

18. Emission of particles from the cathode
Quasi random distribution based on a Hammersley sequence.

19. Emission of particles from the cathode
Particles are sorted w. r. t. their emission time. After the emission of a single particle
the space charge field is scaled with:
When a complete time step is fulfilled a new space charge field calculation is initiated.
Some resolution below a time step.

20. Emission of particles from the cathode
The grid is set up successively as the particles come out of the cathode.
Equivalently a mirror charge bunch is generated

21. Example: Space Charge field at the cathode during emission.
1 nC,
3.0 mm dia.
Plate capacitor:

22. Example: Space Charge field at the cathode during emission.
space charge limited emission:
5 nC launched
2.3 nC emitted
3.0 mm dia.

23. Emission of particles from the cathode
In Astra it is not necessary to assign the same charge to all macro particles.
This allows to simulate the emission self-consistently, e.g. when the quantum efficiency of the cathode depends on the applied accelerating field, so-called Schottky effect.

24. Example: Charge vs. phase of an rf gun.
1 nC, 40 MV/m

25. Example: Charge vs. phase of an rf gun.
1 nC, 40 MV/m
with strong
Schottky effect

26. Example: Charge vs. phase of an rf gun.
1 nC, 40 MV/m
without space charge
Ekin = 0.0 eV
Ekin = 0.25 eV
Ekin = 0.5 eV
Ekin = 1.0 eV

27. Astra graphics tools
for the data visualization and analyses three graphics programs are provided
fieldplot
lineplot
postpro
the plot programs are menu controlled

28. Some menu pages of graphic programs

29. Slice emittance plot

30. Core emittance plot

31. Acknowledgement
Thanks to all colleagues who contributed to the program development, especially: Christoph Stolzenburg, Phillipe Piot, Bagrat Grigorian and Sven Reiche.