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1. end time shift_cO value fsnij time number The particle tracking is activated by the tracker section There are three possibilities to define the time end fena of the calculation either directly by the command end time c0 fena or incremental by delta time c0 dtena tend tpart or by reference to a field boundary with marker see position section end time marker marker name end time shift cO tshifr The last option defines the time end relative to the time tm when the reference particle travels through the field boundary which is specified by marker name lend tp l shift If the time grid is defined by an external file see section track step command time grid file the tracking is executed until the last grid value before or equal to the end time is reached Otherwise the recursive time step algorithm generates time steps that end exactly at tena The tracker section can be called more then once Before a consecutive call of the tracker section it is for example possible to modify track_step parameters or to call the monitor section 28 ref particle l shift tart tp lend time of particle distribution end time specified before tracking in tracker Example tracker end time marker d4b end time shift _c0 0 10 29 4 6 Section forces identifier argument none projected csr p to p csr g to p csr p tom csr g tom unit type shape sphere ellipsoid2. sigma long value o1 relative file length number gt 0 relative long value Gitrel number gt 0 sigma rad value 0 file length number gt 0 sigma vert value oy relative file length number gt 0 relative vert value Ovrel number gt 0 sigma file file name character string shield shield max use old mesh parl par2 integer gt 0 par4 number par5 number wake file file name 30 character string 4 6 1 Introduction to the Different CSR Models CSRtrack provides several models for the calculation of self forces The type of the model is specified by the command type with the possible choices none projected csr p to p csr g to p csr p tom csr g tom The parameter none causes particle tracking without self forces If type is set to projected CSRtrack uses a simple and very efficient model that neglects transverse dimensions of the source distribution transverse forces the transverse dependency of longitudinal forces as well as space charge effects This model is based on a gaussian sub bunch approach all source particles are replaced by sub bunches with corresponding strengths and longitudinal offsets Therefore the three dimensional distribution of point particles is approximated by a smooth line charge density The calculation of longitudinal fields neglects deformations of t
3. out are specified as targets for in and output Now the lattice is defined see section lattice lattice dipole ist dipole position rho 0 0 psi 0 0 marker dla properties r 8 4 position rho 0 5 psi 0 0 marker d1b dipole 2nd dipole position rho 1 0 psi 0 0 marker d2a properties r 8 4 position rho 1 5 psi 0 0 marker d2b dipole 3rd dipole position rho 2 5 psi 0 0 marker d3a properties r 8 4 position rho 3 0 psi 0 0 marker d3b dipole 4th dipole position rho 3 5 psi 0 0 marker d4a properties r 8 4 position rho 4 0 psi 0 0 marker d4b And describes a four dipole magnet chicane Ea ea BL EZ with the following parameters chicane bend magnet length projected drift length proj B1 gt B2 and B3 gt B4 drift length B2 gt B3 bend radius momentum compaction The particles distribution to be tracked is specified next particles reference momentum reference particle reference point x 0 0 reference point y 0 0 reference point phi 0 0 format fmt1 array file name in particles fmt1 After some options to change the reference system the distribution is referenced to the file we have already seen in the in folder The format is fmtl in this case see section particles The example consists of a Gaussian particle distribution 1000 particles Beam parameters are energy 511 MeV charge 0 833 nC bunch length
4. WinZip i and open it You should see the following M rae window Se Ve Geass Open Favorites Add Extract View CheckOut Wizard Modified Size Ratio Pack Path 19 10 2 496 6 53 234 E A 1920 2 3 111 80 635 in_particles fmt1 19 10 2 176 6 90 16 9 in Extracting all files into a new folder called CSRtrack Test here should lead to the following if you explore the folder Total 3 files 661KB csRtrack_Test File Edit View Favorites Tools Help 5 P search E Folders flv Address C Documents and Settings imberg Desktop CSRtrack_Test J The folder in contains the particle File and Folder Tasks distribution file in_particles fmtl see Make a new folder E aar Dee section particles the out folder is EB Shre his ole csrtrack 1 100 cerka 7 empty The CSRtrack executable expects to find a csrtrk in file in the E same folder it is located That file can E Wy documents be a complete description of the run or it may refer to other input files see Command File Other Places Double clicking the executable runs CSRtrack and the following i Fie Edit View Favorites Tools Help files should appear in the out s seach gt roves ile folder s Address C Documents and Settings imberg Desktop CSRtrack_Test out E end fmt3 x_0010 fmt3 E x_0025 fmt3 File and Folder Tasks gt latou
5. red behind blue N p_to_p g to_p proj hor vs long red behind blue 100 Pr 100 g 3 3 0 100 200 100 0 100 200 wum p_to_p g _to_p proj f trans vs long 100 4 fred behind blue 0 A a z A cehweitoeaieg Noa gt gt ae Wee 4 100 ae gt anana one ra Pe a Manenecses 200 300 Caption 43 5 3 Example for Position Averaging Pulse Smoothing Technique longitudinal phase space o 00 N 100000 a 0 02 0 03 100 5000 wol Without position averaging lt q z rT 6a 1 4m 5 Il 3 oom 0 0 14 wm 1000 C100 50 0 50 100 44 5000 with position averaging 4000 m M 5000 ai o 1 4 um o 9 14 um 1000 C100 5 4 Fixed Bugs In Version 1 0 The sub bunch length for the self force of type projected was slightly wrong the longitudinal self field Ea is calculated as convolution of the longitudinal 1d current J with the CSR field Ea ie of a Gaussian sub bunch The Id current J is computed on an equidistant mesh by substitution of all point particles by sub bunches of the length o V2 By mistake the sub bunch length for the calculation of E z WaS Oj instead of o I V2 Therefore the effective sub bunch length was J 0 1 220 instead of o This has been fixed 6 Bibliography 1 E Saldin E Schneidmiller M Yurkov Radiative Interaction of Electrons in a Bunch Moving in an Undulator NIM A417 1998 158 168 2 M Dohlus Two Methods
6. out e aaa end fmt3 x_0002 fmt3 x_0010 fmt3 x_0018 fr latout dat x_0003 fmt3 x_0011 fmt3 x_0019 frr E log txt x_0004 fmt3 x_0012 fmt3 x_0020 frr p1 fmt3 x_0005 fmt3 x_0013 fmt3 x_0021 frr 4 ps_viewer_data mat x_0006 fmt3 x_0014 fmt3 x_0022 frr steps dat x_0007 fmt3 x_0015 fmt3 x_0023 frr sub_bunch dat x_0008 fmt3 x_0016 fmt3 x_0024 frr x_0001 fmt3 x_0009 fmt3 x_0017 fmt3 x_0025 frr lt F Files oftype All Files v Cancel The ps_ viewer reads phase space distributions saved in the CSRtrack formats fmt1 like the input file or fmt3 Open the x_0001 fmt3 file and the longitudinal phase space at this position will appear after a message box There are 52 files from x_0001 fmt3 to x_0052 fmt3 has informed you that there are 52 files along the beam line A Movie button pops up when there are more than 3 files of such type and you can push it to see a movie of how the longitudinal phase space evolves along the chicane Other phase space projections can be plotted by using the pull down menus at the axis of the plot The following shows the horizontal phase space at the beginning right and end left of the chicane the slider or the editing field can be used to go back and forth between the different x_nnnn files ps_viewer File Plots Options CSRtrack Phase Space Viewer
7. started after the closing bracket 3 3 Global Commands There is only one global command in version 1 0 global_command file name filename filename lt name of nested input file gt The file command opens a nested input file The commands in this file are processed in the same way as that in the command file csrtrk in Maximal 10 nested files can be opened at once Each nested file has to end with lt CR gt After the processing of a nested file CSRtrack continues processing of commands in files with higher level or of commands in csrtrk in CSRtrack searches the input file filename either in the root directory with csrtrk in or if specified in the input directory that is defined by the section iopath 13 4 CSRtrack Sections section name io path file io lattice lattice definition particles definition of particle distribution track step Time grid iterative tracking tracker tracking forces model for self forces monitor monitor online monitor 4 1 Section io path online monitor The io path section is used to specify the input and output directories If these directories are unspecified input and output files are read from or written to the root directory with esrtrk in identifier argument unit type input directory character string output directory character string file name character string
8. 0 marker d3a properties r 1 66275 position rho 2 963 psi 0 0 marker d3b dipole 4 dipole position rho 3 463 psi 0 0 marker d4a properties r 1 66275 position rho 3 963 psi 0 0 marker d4b 15 field boundaries reference trajectory 4 2 1 Definition of Field Boundaries Subsection position identifier argument unit type rho value length number psi value angle number deltas value length number delta psi value angle number marker marker name character string duty yes no The position section is used to define field boundaries They can be defined absolute or relative a Absolute definition of field boundaries The field boundary is defined in polar coordinates by the parameters rho and psi The definition of the first field boundary has to be absolute The reference trajectory before the first field boundary is identical to the x axis The first reference point is the intersection of the first reference plane and the x axis The rest of the reference trajectory and all later reference points are recursively defined by dipoles and their curvature radii Ve field boundary 16 b Relative definition of field boundaries If the position and direction of the reference trajectory are defined for one field boundary the position and direction of the intersection with the next field boundary is uniquely defined by the path length difference delta_s and the curvature radiu
9. 2 t2 t2 t2 t2 t2 t2 t i h Via Fa Sm En Ain ae h m f v m TE L t3 t3 t3 t3 t3 t3 t3 t3 t3 t h m h m Va Ea Sm Em Am tA m h m v m L m Lin The first column t t2 t is the time of the monitor event unit time The next eleven rows describe the position momentum charge and force in the same way as for the corresponding input format fmt3 40 4 7 2 online monitor type subbunch Writes a file containing the sub bunch dimensions along the beam line 4 7 3 online monitor type steps Writes a file containing the longitudinal position at the end of each calculation step the tracking error at this position and the number of tracking iterations 41 5 Appendix 5 1 CSRtrack Units dimension normalization length lm time 1m co momentum leV co charge 1C angle 5 2 Results for Example Case Comparison of calculations with force types p_to_p g to p and projected The data points for force type p_to_p are plotted in red for g_to_p in blue and for projected in 1 deg green Usually the red points are hidden by the blue ones bunch current T T T csr_p_to_p csr_g to p projected M red behind blue gt curent kA 60 p_to_p g to_p proj de vs long red behind blue 0 02 0 02 s um 0 005 42 p_to_p ato p pro red behind blue delta_de vs long p_to_p g to_p proj h vshor
10. CSRtrack Version 1 2 User s Manual M Dohlus and T Limberg Contact Martin Dohlus desy de Torsten Limberg desy de Table of Contents 1 Introduction and Code Improvements sssi paise i Ee Ee e EEEE EEE E E E E E E a ai 3 Isl Changes in Version Ilienea etei NEEE EE TEE EA S EEE EE N E e 3 1 2 Changes ini Version V2 ionennrceiiope enin e e E tease egies ty e n R n R ee meiheete ce 3 2 First Steps to run CSRtrack on a WINDOWS XP Computer ececccecccesecesecsceeseeeseeeneeecesecnseenseeesecnaecaecnaecaeeaeeenes 4 2 1 Download Installation and a First Rum ccccccccccsccccssscececssececeessececesssececseeeceesseeeceessececsseeeceeseseceeaeeeenseeeeees 4 22 The Example Input Pil ci seernes E e E n e E e EE a re E EEA ESEA Ee E FETES NARE 5 23 SPlOMINGARESUIES 2s2 cosets tsheons Ss hac E T A E AE E tind t2ePin nll oo 9 2 4 Calculation of the example with other models for the CSR fields forces ccceseeseesseeeeeeseeeeeeseeeeeenseeneees 11 PE O With FOrce Lype Pp to 2piscseccstes ats E A E A E A E Gat laateetet ss 11 242 With FOrce hype T E TE E eis cteet A E E ha eteatin Olay 11 2 5 Typical CPU Time and File Structure for the Example ccccecscessseescessceeeeeseceaeceneceecaeeeseeeseeeeeeeeeseresreneees 12 3 CSRtrack Command Stuc ture 2s 74 5002s a a a ten ai a a ten atts Seocge tok obntaantopeteate 13 S 1eCommmand Piles 5321 ots scste AAA E E E E EEA 13 ES ATO EE E EEA E A fas
11. Choose Files Vimberg Desktop C Rtrack_example_1 out x_0001 fmt o Desktop CSRtrack_example_1 tout x_0052 fmt3 File File ooo PE as 4 r 52 View Slices Beam Dispiay z x10 pho RMS 2 42e 005 rad i RMS 4 46e 005 rad v7 E A n ao 1 2i 1 5 A 0 5 0 0 5 1 5 RMS 1 02e 004 m M RMS 5 52e 005 m o BEN r Beam Parameters Emittance 1 00e 006 m rad Emittance 2 35e 006 m rad beta 10 4 alpha 2 26 beta 1 3 alpha 0 316 Message Window Plot phase space data When normal transverse phase space like x x here hor for horizontal and p_hor for normalized horizontal momentum is plotted the emittance and the optics is calculated else only the RMS values 10 2 4 Calculation of the example with other models for the CSR fields forces 2 4 1 With Force Type p to p The command file is the same as for force type projected with exception of the force definition forces type csr p to p shape ellipsoid sigma long 5 3e 6 sigma rad 33 0e 6 sigma vert 50 0e 6 2 4 2 With Force Type g to p The command file is the same as for force type projected with exception of the force definition forces type csr g to p shape ellipsoid sigma long 5 3e 6 sigma rad 33 0e 6 sigma vert 50 0e 6 The dimensions of the sub bunches are summarized in the following table Sub Bunch Dimensions sub bunch length sub bunch width hori
12. Concept The particles section is used to define the position momentum and charge of a particle distribution in a absolute pecified coordinate system xp yp Zp The first particle of this distribution is called reference particle The reference particle is treated as all other particles but some input and output parameters are defined relative to the reference particle Eg A time event may be defined by the transition of the reference particle through a field boundary or the coordinates and momenta of other particles are given as increment to that of the reference particle The reference momentum relates the lattice settings to absolute field strengths The reference momentum is not necessarily the momentum of the reference particle The properties of the particle distribution as well as the absolute time are defined in an array that is assigned to array This assignment is terminated by the closing bracket of the section Therefore array has to be the last assignment b Yp Zp coordinates The xp yp Zp coordinate system of the particle definition is related to the coordinate system of the lattice definition by 20 Yp particles g x ref J ref p Xref Jcosp y Y ref Jsin se Y y x i sin g y y Jcosp Z Z c reference momentum The reference momentum is used to relate the curvature radii and strengths settings r strength in lattice to magnetic field strengths reference momentum value p re
13. Example io path input data bc2 in output data bc2 out logfile log txt has the same effect as io path logfile log txt input data bc2 in output data bc2 out The logfile is written to data bce2_ 100 out log txt 4 2 Section lattice subsection name dipole definition of dipoles 14 quadrupole definition of quadrupoles multipole definition of multipoles a Concept CSRtrack supports magnetic dipole and multipole fields that are defined in specified x y z coordinates The range of these fields is defined by two field boundaries that are perpendicular to the xy plane The dipole field between field boundaries is constant and parallel to the z axis Together with the dipoles a reference trajectory is defined that is composed by arcs and lines and lies in the xy plane The first part of the reference trajectory coincides with the x axis The definition of each lattice element has three parts the definition of the first field boundary the definition of element properties and the definition of the second field boundary field boundaries reference trajectory b Example lattice dipole 1 dipole position rho 0 0 psi 0 0 marker dla properties r 1 66275 position rho 0 5 psi 0 0 marker d1b dipole 2 dipole position rho 1 0 psi 0 0 marker d2a properties r 1 66275 position rho 1 5 psi 0 0 marker d2b dipole 3 dipole position rho 2 463 psi 0
14. I I r X y r Z P X P 7 r P Z q Ov OPS Oph OpV dz os On OV Ops Oph Opv q os On Ov Ops Oph Opv 4q n n n The structure of format 2 is the same as for format 1 Position and momentum of the reference particle are defined as before For all other particles the position and momentum differences to the reference particle are given in s h v coordinates The direction of the s axis is defined by the direction of the reference particle The orthogonal h and v directions follow from u p reference _ particle TD omii pana u u 5 u u u x u r r s u h u vu P P e ps u ph u Opv U Astra Format format astra Astra coordinates are converted to CSRtrack coordinates by the permutation z x y astra gt Xp Yp Zp e Example particles reference momentum reference particle reference point x 0 0 0 0 reference point y 22 reference point phi 0 0 format fmt1 array file name particles bc2 fmt1 4 4 Section track step identifier precondition argument yes no unit type iterative value it integer gt error per ct value err time 1 number gt 0 error weight mo mentum value Winom number gt 0 ct step min value tnin time number gt 0 ct step max value tax time number gt 0 ct step first value thirst time number gt 0 duty steps yes no increase
15. ai Rech esse gO ae ea ase R san eee se eee Sk ses Ree 34 4 6 0 Setting Sub bunch Sizes from File eoria neee EE EEEE E nie ae AR 35 A 6 Je Exampl esn E a E R E A E E E ante oe 36 4 7 Sections monitor aNd ONLINE MONLCOLK ccccsccsesesssevensacsessesscusnsenssiestenschcesdasevstsesesteosensabecguoesteuseses 36 AT V Online monitor type phase e ern n e e E O EAE E E A a eee 38 4 2online Monitor type Subbunich seeretari E E E E E E E EA R O RR E S 41 457 3e Onlmne Monitor type Steps eene A E RAE E O E E E ERE 41 De APP On 1D GE EEEE ES E EE EEE EE ET SN E E A 42 5 1 CSRtrack Units eieo a EEEE EEA EA lees A A E eens obs O E a E ets 42 5 2 Results for Example Case scarno a A E conse E E E E R AE E 42 5 3 Example for Position Averaging Pulse Smoothing Technique c ccccesccssessseeceesceesceeeceseceseceseeneeeeeeaeeeneeees 44 DAL EDLER BAIN EA 2 sete ESEE N E AEE E E TNE A EA ET cours haloes cds 45 6 Biblio graphy AEAEE EE E E A ET A E A E E E 45 1 Introduction and Code Improvements The CSRtrack code tracks particle ensembles through beam lines with arbitrary geometry The field calculation in CSRtrack offers different algorithms to choose from from the fast projected 1 D method to the most rigorous the three dimensional integration over 3D Gaussian sub bunch distributions see section forces The 3D field calculations take Coherent Synchrotron Radiation fields into account as well as intra bunch field
16. ative to the rms length of the particle distribution by sigma long relative relative long Oire or it is set by an input file with file name file sigma file sigma long 31 The last two possibilities allow to use sub bunch dimensions that depend on time The calculation of fields with time dependent sub bunches neglects the change of the bunch length at retarded times The use of bunch dimension files is described below 4 6 2 1 Current Smoothing Gauss Filter and Position Averaging The input for the 1d current smoothing algorithm are the longitudinal position z and charge q of each particle The continuous current is calculated as I z gt 4 hlz 2 0 V2 with Meo Jeol 2 The sub bunch length o is controlled by sigma_long or if specified by a data file see previous paragraph Usually the longitudinal positions z and Z are identical For non systematic phase space distributions random initial distribution identical charge of all particles a slight manipulation of the longitudinal particle positions helps to reduce the noise This is for example possible by position averaging m 1 a Ie with m M 2 u v m supposed the particles are sorted by the longitudinal position CSRtrack uses the superposition of two averaging operations 2 1 Sy ag 2 3 3 Position averaging is activated by setting the optional parameter par1 in the forces section to 1 The value of M has to be assi
17. d time c0 fend or by reference to a field boundary with marker start time marker marker name start time shift _ c0 dtar end time marker marker name end time shift c0 had The second possibility defines the time relative to the time tm when the reference particle travels through the field boundary which is specified by marker name l start tp F distart and or lend tp T dtena The time step is defined by time_step_c0 fse all The argument a11 has the effect that all points on the time grid are monitored 37 monitor grid t start a t step t end C e EE monitor events 1 2 3 4 5 6 e Name convention for output files Phase monitors of the full particle distribution type phase particle al11 create for each monitor event an individual file The file name is derived from the argument of the name command and the event number as follows name lt string gt lt extension gt 1 monitor event gt file nme lt string gt 0001 lt extension gt 2 monitor event gt filename lt string gt 0002 lt extension gt 3 monitor event file name lt string gt 0003 lt extension gt The numbering of monitor events depends on the definition of tstart and tstep that can be defined individually for different monitors Therefore events with the same number that have been recorded by different monitors do not necessarily coincide f Example online monitor name x fmt3 type phase format fmt3 partic
18. e the tracker section has been called b monitor section name file name character string format fmt1 fmt2 fmt3 The name of the output file is defined by name file name The file formats are described below c online monitor section identifier argument unit type name file name character string type phase subbunch 36 steps format fmt1 fmt2 fmt3 particle value m integer start time c0 value tsart time number now start_time marker marker name character string start_time_shift_c0 value dtsrari time number end time c0 value tena time number end time marker marker name character string end time shift value dtena time number time step _ c0 value tstep time number all There are three different monitor types type phase subbunch steps The type phase is used to monitor properties of an individual particle or of the complete distribution d Monitor Events Online monitors are served during the tracking each time when a monitor event is fulfilled In principle monitor events are defined by the start time tstarn end time tena and time step tstep but the so defined monitor grid does usually not coincide with the time grid defined in the section track_step Therefore monitor events are related always to the next following point on the time grid tstar and tena can be specified either directly by start time c0 fstr NOW en
19. factor value finc number gt 1 arc factor value farc number gt 1 time grid file file name character string steps tolerance value tol time number gt 0 a Concept CSRtrack calculates self forces on a time grid and interpolates them linearly for particle tracking The section track _step is used to control the time grid as well as the tracking from one grid point to the next There are two possibilities to determine the time grid or the widths of time steps either an external file defines the grid directly or the step widths are set recursively For a new force calculation the phase space coordinates all particles have to be known but they depend on the force that has to be determined This implicit problem is solved recursively iterative tracking The time steps and the parameters for iterative tracking are crucial for the accuracy of the calculation The track _step settings can be redefined eg Before a consecutive call of the tracker section b time grid file The time grid can be defined directly by the uses of a file with time grid values The filename and a tolerance parameter are set by the following commands time grid file file name steps tolerance value tol CSRtrack expects an ascii input file with the specified name in the input or root directory see io path It reads a list of time grid values one value per input line with the time unit 1 m Co 23 time g
20. ference particle average Either the reference momentum is pecified directly by value or it is set to the momentum of the reference particle or to the average momentum of all particles d format and array Three different input formats are available to define particle distributions Format 1 format fmtl1 array r ly r I r Vs X Ya Key Px PY PZ d OX OV amp OPX Y OP q amp px Y Z q OX OY Oy OPX OPY Pn An The arguments of the array command are processed in the same way as the rest of the command file Therefore the input can be directed to an other input file by the global command ile name filename and comments as well as all CSRtrack separators are valid The end of the argument list is defined by the closing bracket of the particles section In format 1 CSRtrack expects 7 N 1 numerical arguments The first number defines the time of the distribution The next six values 11 F r3 r4 rs re have no meaning and do not affect the result of the calculation The 21 triplets Xj Y Zr PX PV Pzr define the position and momentum of the reference particle in Xp Yp Zp coordinates Q is the charge of the reference particle Each further particle is defined by an additional set of numbers x dy 6Z PXp OPYis OPZi Gi With the position xr dx yr y zr zi the momentum px x PYr DY PZr Opzi and the charge qi Format 2 format fmt2 array r ly I
21. for the Calculation of CSR Fields TESLA FEL 2003 05 3 M Dohlus A Kabel T Limberg Efficient Field Calculation of 3D Bunches on General Trajectories NIM A445 2000 338 342 45
22. gned to a second optional parameter par2 Example forces type projected sigma long 0 000008 parl 1 par2 10000 The effect of the position averaging on the beam distribution is shown in the appendix 32 4 6 2 2 Projected Force and Wake per Length In combination with the self force of type projected it is possible to define and use a position independent longitudinal wake This can be used to estimate the effect of the resistive wall wake The wake per length is defined by an ascii file with a table in the following format each line has to end with lt CR gt Each line that includes the comment character is ignored The numerical input one number per line is read from all other lines The first number of the table is the rms sub bunch length o for which the wake potential has been calculated The second number is the step width 6 table of the table and the next two integers i and i define the longitudinal position of the first and last point in the table 6 4 1 5 Oriel The next 1 i i numbers are the values of wake table unit V Cm In the calculation loop that is processed in the tracker section CSRtrack determines for each step the sub bunch length o as specified and calculates the longitudinal current z as described above If the sub bunch length o is smaller than o J2 the wake Woy is calculated by a convolution If O is larger than required CSRtrack uses the tabulated wake table wi
23. he phase space position of the reference particle along the beam line are dumped in single files while the x_n files contain the phase space coordinates for all particles at positions 10 cm apart from start to end time markert end time shift _c0 m The method to calculate CSR fields and the parameters of the sub bunches are chosen in the forces command force definition forces type projected sigma long 5 3e 6 In this example the 1 D projected field calculation method is used with a longitudinal size of 5 3um See chapter forces for details and other field solvers Finally the range for the tracking calculation and the numerical parameters for the self consistent iterative particle tracking are specified track step precondition yes iterative 2 error per ct 0 001 error weight momentum 0 0 ct step min 0 02 ct step max 0 10 ct step first 0 10 increase factor 1 5 arc factor 0 3 duty steps yes tracker end time marker d4b end time shift c0 1 00 The track step command specifies that iterative tracking will be performed until two iterations are done or the error criterion is reached The parameter ct_ step min and the ones below control the step width algorithm For details see section track_step The tracker command tells the code to track to the marker d4b which is associated with the end of the last bend see the lattice definition above plus an additional time interval which correspo
24. he retarded density function The method csr _p_to_p replaces all source particles by three dimensional gaussian sub bunches with individual strength and trajectory but with the same shape It neglects vertical offsets and vertical particle motion As all point to point or more precise sub bunch to point interactions have to be calculated the numerical effort increases quadratically with the number of particles The method esr_g_to_p is based on the same sub bunch approach and the same point to point interactions but it uses a pseudo green s function for the field of a sub bunch Before each computation of point to point interactions the electromagnetic field is calculated on a mesh in the horizontal plane for a typical sub bunch The trajectory of other sub bunches can be fitted to the trajectory of the typical sub bunch by a coordinate transformation The same transformation is used to calculate the electromagnetic fields of other sub bunches from the meshed field New in version 1 2 are csr_p to mand csr _g to m They have the same effect as csr p to pandcsr_g to p but the forces are calculated from meshed electromagnetic field values 4 6 2 Parameters for the projected Force This model uses one dimensional Gaussian sub bunches There are three possibilities to define their longitudinal rms length Either the longitudinal size oj is set directly by the command sigma long Oj or it is defined rel
25. ify shielding by perfect conducting horizontal planes shield defines the distance h between the conducting planes and shield max defines the maximal vertical distance to which mirror charges are taken into account 34 mirror charges path Work in progress the mesh of the Green s function is not optimized for wave fronts that are created by mirror charges therefore the global mesh density is increased This solution is not optimal concerning mesh resolution and computational efficiency Attention Shielding causes a dispersive propagation of electromagnetic fields a time grid see track step that works for calculations without shielding is not necessarily sufficient with shielding and otherwise it is recommended to observe the particle coordinates and forces at few test particles for all time steps see online monitor type phase particle m and to adjust the grid if required 4 6 6 Setting Sub bunch Sizes from File Sub bunch sizes can be defined directly by file The file name is set with the command sigma file filename CSRtrack expects an ascii input file with the specified name in the input or root directory see io path It reads a list of values for time and dimensions units time length length length ft On On y to Oj712 O72 Ov2 tz O713 073 0v3 t4 O14 O46 Ov4 th Oj n Orn Ovn Each input line has to start with four numbers the rest is ignored The sub bunch dimensions are calculated b
26. in 80 um bunch length out 20 um peak current out SkA horizontal twiss parameters in normalized emittance 1 mm mrad alpha 2 2 beta particle distribution 10m number of particles 997 number of slices 83 particles per slice 12 reference particle without charge sub bunch length 5 3 um sub bunch width horizontal 33 um sub bunch width vertical 50 um The output of CSRtrack is prompted by so called monitors Online monitors write data to file during the tracking offline monitors save data at the beginning or the end see monitor online monitor name sub bunch dat type subbunch start time c0 now end time marker d4b end time shift c0 2 0 time step c0O all online monitor name steps dat type steps start time c0 now end time marker d4b end time shift c0 2 0 time step c0O all online monitor name p1 fmt3 type phase format fmt3 particle 1 start time c0 now end time marker d4b end time shift c0 1 0 time step c0O all online monitor name x fmt3 type phase format fmt3 particle all start time c0 now end time marker d4b end time shift c0 1 0 time step c0 0 10 where name specifies the file name s to write the data to accordingly the files named sub_bunch dat etc appear in our out folder Details can be found in the section monitor in the example above data describing sub bunch length time step width and t
27. ion between f f t and f f t As f depends on unknown phase space coordinates Xph 2 Xpn t2 they are estimated by ae and are improved to C by iterative tracking The first estimation x25 is calculated with force f 0 X ph 2 first step 25 one step fy t x force_type n l e n Ly 5X 5n2 X ph 2 f force_type error tfi Xpn 1 CSRtrack repeats the iterative tracking until the error criterion set by error per ct is fulfilled or the maximal number of iterations set by iterative is reached iterative criterion first step l 2 1 error force_type ph 2 y ph 2 xn _ g n l f b one step x tf X ph 1 X x error gt err t t force_type ph 2 ph 2 projected n and n lt it El yes f f X jt PI As the field computation for force type projected is very efficient CSRtrack calculates X oho ee f f t x projected for this type and uses x N f sia oa otherwise The force _type is set by the command type in the forces section e Iterative tracking with precondition The forces section provides several models for the calculation of self forces e g type projected csr_p_to p csr_g to p As the projected model needs much less numerical effort then the other models it could be helpful to use phase space coordinates x that have been computed by iterative tracking with this 26 method to improve the
28. le all start time c0 now end time marker d4b end time shift c0 1 0 time step c0 0 02 4 7 1 online monitor type phase Writes files containing phase space information of all or selected particles a format fmtl particle all This format is compatible to the corresponding input format of the particles section The output file can be used as input file for a later restarted calculation Each line of the output file is written with the fortran format 7 1x e22 15 The last six numbers in the first line r1 r2 3 F4 rs r6 are set to zero b format fmt2 particle all This format is compatible to the corresponding input format of the particles section The output file can be used as input file for a later 38 restarted calculation Each line of the output file is written with the fortran format 7 1x e22 15 The last six numbers in the first line r1 72 73 F4 F5 re are set to zero c format fmt3 particle all Each line of the output file is written with the fortran format 10 1x e22 15 2 1x 11 The output file has the following structure t y 0 0 0 0 0 ese c oo 0 0 A k vy 0 amp a Ja Jm Ja ba La h bE t G a h Ja Jsa Jas L L h h v vy s amp a fa fas fis Les La h h v v S oe d Jia Tes Sos L L n n n n n t is the time of the distribution y is the Lorentz factor y that corresponds to the reference momentum see particles section Os o and o are the longitudinal radial and vertical r
29. longitudinal and radial step width of the equidistant mesh The step widths are As sbox oj Ar rbox o with oj the longitudinal and radial rms dimension of the sub bunches The default values are sbox 0 5 rbox 0 5 Mesh field values are used only for cells with at least four particles observers per cell otherwise the field values are calculated directly The additional output per calculation step looks like this ctstep 0 03000 gt ct 10 77925 step error step time 4 98611017E 05 boxes 1961 e points 5027 p points 7886 step _error step time 0 00198496122 boxes 1960 e_points 5026 p points 7887 step error step time 6 39205063E 07 ctstep 0 02581 gt ct 10 80506 step error step time 5 22485114E 05 boxes 1992 poirte 5070 p points 7712 step _error step time 0 00203230079 where boxes is the number of mesh cells with at least four particles that are calculated e points is the number of edge points of these cells and p points is the number of particles observers that are calculated directly Sometimes it is possible to reuse the Green s function for forces csr_g to _p and csr _g to m for successive tracking iterations e g if sufficient memory space is available to store the function on the complete mesh The command use_old mesh yes no can be used to enable disable this possibility 4 6 5 Shielding the commands shield and shield _ maxe are used to spec
30. me of the monitor event unit time The next seven rows describe the position momentum and charge in the same way as for the corresponding input format fmt1 x Yp Zp coordinates This means the coordinates of the reference particle m 1 are absolute the coordinates of all other particles m gt 1 are relative to the reference particle e format fmt2 particle m For each monitor event a new line with the actual phase space information of particle m is added to the output file The file is of the following structure 21 t1 t1 t1 1 t1 t s h v Ops oph pv Am m m m t2 t2 t2 t2 t2 t2 t Os on v Ops Oph OPV Ain m m m m t3 t3 t3 t3 t3 t3 t OS on V Op S Op h Op V Ain m m The first column t f2 t3 is the time of the monitor event unit time The next seven rows describe the position momentum and charge in the same way as for the corresponding input format fmt2 This means the coordinates of the reference particle m 1 are absolute the coordinates of all other particles m gt 1 are relative to the reference particle U U U base f format fmt3 particle m For each monitor event a new line with the actual phase space coordinates and forces of particle m is added to the output file The file is of the following structure t1 tl t1 tl t1 t1 tl t1 tl t h ha S e Ain Je 7 L L m m m m m sn hm m s m t m t2 t2 t
31. ms sub bunch dimensions that were used for the force calculation see forces section The particle coordinates h vi si and slopes h v are defined relative to the reference trajectory and the reference particle h i h i are horizontal parameters v v are the offset and slope in vertical or z direction and s is the path length difference with respect to the reference particle de is the relative energy deviation normalized to the energy that corresponds to the reference momentum fsi fhi fvi are the force components with U p D u u XU u U XU f Fei fini Uy In the present version of CSRtrack there is no input format available that allows to set or reset the source and test flags Lsi Lsi Therefore they are always set 1 The source flag determines if a certain particle contributes to the generation of self fields or not A particle is tracked with self and external forces if the test flag is set otherwise only external fields affect the motion of this particle A atan h ref particle 39 d format fmt1 particle m For each monitor event a new line with the actual phase space information of particle m is added to the output file The file is of the following structure t1 t1 t1 1 t1 t1 fi Ox OZ Opx OPY m OPZn Am m m ty n On n Oln Yn Pn Mm m m n AO DO AO D Spy Spe qp m m The first column ti t2 t3 is the ti
32. nds to the reference particle traveling the length of 1 m with speed c0 end_time_ shift _c0 1 00 For details see section tracker Finally another monitor command writes the phase space at the end of the tracking to the file end fmt3 monitor format fmt3 name end fmt3 The calculation is started by running CSRtrack in the directory with the input file The particle distribution is read from in particles in fmt1 and all output files are written to the directory out This should create the same files as in the folder out_ solved 2 3 Plotting Results A MATLAB GUI called CSRtrack_ps_ viewer is available to plot CSRtrack results Download the file ps_viewer zip and extract the files in a folder ps_viewer Open MATLAB choose that folder as working directory and enter ps_viewer on the MATLAB command line You should see the following GUI ps_viewer File Plots Options CSRtrack Phase Space Viewer Choose Files Browse Browse View Slices Beam Display 1 0 8 06 0 4 0 2 m Beam Parameters Message Window Done Initializing Basically there are two axes to plot phase space projections for comparison Use the Browse button to open one of the result files you have in the out folder in your CSRtrack test folder The browse window should show the following files Choose a Phasespace Data File Lookin
33. rid from file tolerance first step last step part lend time of particle distribution end time specified before tracking in tracker 4 CSRtrack processes time grid values between the actual time associated to the particle distribution tpart to the end time tena that is defined in the section tracker Time grid values are ignored if they need steps smaller than the tolerance parameter The tolerance parameter tol is used to avoid a first step of zero length or to avoid extremely small steps For the preparation of the time grid the user has to take care about the position of the particles distribution in the lattice The steps monitor see online monitor type steps can be used to support this task E g a new time grid can be derived from an old one that was generated by a calculation with recursive step widths control c Recursive calculation of time steps The automatic time stepping is active if no time grid file is specified It is controlled by the parameters ct step max mar ct_step min fmin ct step first first duty steps yes no increase factor finc arc factor fare If duty_steps is set to no or if no field boundary is passed during the track step CSRtrack uses time steps longer or equal tmin The maximal time step is limited by min tin tarce With tore fare 3 24R 0 With Re the actual curvature radius of the trajectory and Oms the actual rms length of the particle distribu
34. s similar to the well known space charge fields on straight trajectories but on curved paths not cleanly separable from radiative fields any longer Tracking is done in absolute coordinates through a magnet lattice defined by magnet field boundaries see section lattice using a self consistent algorithm CSRtrack handles dipole quadrupole and multipole magnets RF sections are not implemented yet Tracking through long straight RF sections is better left to codes like elegant or ASTRA depending on the importance of space charge force 1 1 Changes in Version 1 1 New types of CSR field calculation methods csr p tom csr g tom see section forces The forces are calculated using meshed electromagnetic field values which is useful when tracking big numbers of particles since the cpu time scales linear and not quadratically with the number of particles 1 2 Changes in Version 1 2 Current smoothing for projected force with a Gauss filter and position averaging Added the possibility for the force type projected to introduce a user defined wake field to model for instance the resistive wake of a magnet chicane vacuum chamber Bug fix for self force of type projected see appendix 2 First Steps to run CSRtrack on a WINDOWS XP Computer 2 1 Download Installation and a First Run Download the file CSRtrack_example1 zip from http www desy de xfel beam csrtrack index html
35. s r if a dipole is bounded The orientation of the field boundary is either specified by psi or delta_psi or it is perpendicular to the reference trajectory Psi defines the absolute orientation in the same way as for the absolute definition and delta_psi defines the angle between field boundary and the plane perpendicular to the trajectory at the intersection point reference trajectory field boundaries hon ba poo i 1 1 1 1 i 1 A a t 1 1 1 1 1 1 1 delta_s path length difference c marker The identifier marker is used to assign a name to a field boundary The marker names can be used to specify time events eg The instantaneous time when the reference particle see particle definition passes a field boundary This identifier is optional its argument is a character string d duty The identifier duty can be used to affect the step width control of the tracking algorithm see track step This identifier is optional its argument is yes or no If duty is not specified it is set to yes for field boundaries of dipoles and to no for the rest e Example position delta_s 0 2 marker quad_in duty no 4 2 2 Branch Section Dipole position definition of field boundaries properties dipole properties a Subsection position See Definition of Field Boundaries b Subsection properties r sets the curvature radius of the reference trajectory In combination with the reference momen
36. start estimation for other methods This option is activated or deactivated by the command precondition yes no f Error criterion The accuracy parameter is set by the command error per ct err error weight momentum Wmom The relative error citerion is error X 5 ia lt t t err ma Ca pra oJ w i n ntl _ error X pn 9 9X pho C se pe nor e pe pe with s h and p the longitudinal horizontal and momentum offset of all particles with respect to the reference particle The operator x averages phase space coordinates without weighting by particle charges g Example track step ct step min 0 02 ct step max 0 20 ct step first 0 20 increase factor 2 0 arc factor 0 3 duty steps yes iterative 2 error per ct 0 001 error weight momentum 0 1 precondition yes The sum of the relative errors of all track steps from tpar start time to tena is below err tena tpart For a chicane with a path length of about 5m the simulated time interval will be similar 5 m c Therefore the sum of relative errors is below 0 005 for err 0 001 Note that the parameter err controls errors due to iterative tracking and not the error related to the quality of the time grid 27 4 5 Section tracker identifier argument unit type end_time_c0 value tend time number delta_time_c0 value dtena time number end_time_marker marker name character string
37. system v h s with its origin in the intersection point of the reference trajectory and the field boundary field boundaries reference trajectory s longitudinal h horizontal v vertical z The magnetic field between the field boundaries is independent on the longitudinal coordinate In complex notation the horizontal and vertical components B and B of the field are Bee P _ h n J v v Me q n 1 with q the particle charge p the reference momentum n the azimuthal order a the multipole strength ho vo the horizontal and vertical offset and the skew angle These parameters are related to the properties identifiers by poles 2n 4 for quadrupoles strength 4 horizontal_offset ho vertical_offset Vo alpha The parameters horizontal offset vertical offset and alpha need not to be specified Their default is zero 19 c Example multipole position delta s 0 2 marker quad in duty no properties poles 4 strength 0 100 alpha 0 horizontal offset 0 vertical offset 0 position delta s 0 5 marker quad out duty no 4 3 Assignment Section particles identifier argument unit type format fmt1 fmt2 astra reference momentum value p momentum number reference partice average reference point x value xref length number reference point y value y ef length number reference point phi value angle number array a
38. t dat x_0011 fmt3 x_0026 fmt3 log txt x_0012 fmt3 x_0027 fmt3 Bco x_0040 fmt3 x_0041 fmt3 x_0042 fmt3 SEE is iB Make a new folder pi fmt3 x_0013 fmt3 x_0028 fmt3_ _ x_0043 fmt3 Publish this folder to the steps dat x_0014 fmt3 x_0029 fmt3_ x_0044 fmt3 Web sub_bunch dat x_0015 fmt3 m aaa m x_0045 fmt3 E Share this folder x_0001 fmt3 5 x_0016 fmt3 3 x_0031 fmt3 x_0046 fmt3 Now you ran CSRtrack successfully x 0002 5 x_0017 fmt3 lx onsets x_0047 fmt3 s x_0003 fmt3 x_0018 fmt3 x_0033 fmt3 x_0048 fmt3 for the first time For a better f omerpices S xcoooafmes E Cooisfm3 Ex oo34fmt3 x 0049 mt3 x_0005 fmt3 x_0020 fmt3 x_0035 fmt3_ _ x_0050 fmt3 under standing what you calculated cSRtrack_Test x_0006 fmt3 x_0021 fmt3 x_0036 fmt3 x_0051 fmt3 j My Documents x_0007 fmt3 x_0022 fmt3 x_0037 fmt3 x_0052 fmt3 and how the output 1S generated My Computer x_o008 fmt3 x_0023 fmt3 x_0038 fmt3 x_0009 fmt3 x_0024 fmt3 x_0039 fmt3 my Network Places let s look into the csrtrk in file 2 2 The Example Input File It starts with the specification of in and output paths see section io path starting the path in the folder where the executable is located io path input in output out logfile log txt So our folders in and
39. ta Staats de say E E E EE EEA 13 3 32 Global Command Si na e heh a a a e tetas tg Sai e a a a aa 13 BAERI RST TE S AO EEEE AAE AE E E A AEE 14 A A RS Te ntaa 10 Pa A a EAEE E A E A A E 14 4 2 Section Labelc esis civil iin E R ERE EN O E E nile 14 4 2 1 Definition of Field Boundaries Subsection position ssssssessseseessreeessesersreseesessteresseserseeseesessesnessesees 16 4 22 Branch Section Dipole cccccss cccesslecceecsegetees cate iE E E E E E R A E a i 17 4 2 3 Branch Section quadrupole isis recnncnninni n ii die E EEE E E E E 18 4 2 4 Branch Section multipole siisii en iE aE E E A AEE a i a 18 4 3 Assignment Section part icl Sisecnnennrein ie e e E E E e EE E E E RE i 20 44 Section track Stepiirasisicnsinaiiaiiiveeela ine E meee eae aie ees 23 AS Sections CLACKESL ienr n heii aed E EER ote eee centage pied lea ales est aseeeo 28 A 6 Section fOrCeS ina ied K ivi ented mete os ened ee eee eee eae 30 4 6 1 Introduction to the Different CSR Models ccceecssssssseeceseseeeseceeeseceeeeceaeceeseecaeeeccnevseceaeeeeeaecaeeeeeneeeres 31 4 6 2 Parameters for the projected FOrce csecsen iiie seriis c ii a SEE ii e EE a NE Eia siias 31 4 6 3 Parameters for the csr_p_to_p and csr_g_to_p Forces sesssesessessesesssreressesetsreseesessteresseseeseesersessrenessesees 33 4 6 4 Additional Parameters for the csr_p_to_m and csr_g to_M Forces 0 0 0 ceeeeseceseesceeeeeeeeeeeceeeeeeeeeseeeeensees 34 4 6 5 ICAI ei
40. th modification and warning Therefore the longitudinal self field is calculated by the following convolution E 2 f z T7 a E yz b W yala with a 1 par4 and b 1 par5 The name of the file with the tabulated wake as well as the parameters par4 and par5 can be set in the forces section The definition of a wake table is optional the default values of par4 and par5 are zero Example forces type projected sigma long 0 000008 parl 1 par2 10000 shield 0 008 wake file wake cu flat 2x4mm dat 4 6 3 Parameters for the csr p to p and csr g to p Forces These models use Gaussian sub bunches that are either spheres or ellipsoids shape sphere ellipsoid The radius of spherical sub bunches is defined in the same way as the longitudinal dimension of sub bunches of the projected force Ellipsoidal bunches have three rms parameters the longitudinal size oj the horizontal or radial size o and the vertical size oy oj and ox are either set directly or relative to the corresponding rms dimension of the particle distribution or by an 33 input file with sub bunch dimensions o is either defined directly of by file The calculation of fields with time dependent sub bunches neglects the change of the bunch length at retarded times The use of bunch dimension files is described below 4 6 4 Additional Parameters for the csr p_ to m and csr g tom Forces Beyond the parameters above sbox and rbox can be used to modify the
41. tion It starts with a step of the length tjrs and increases it for each new step by the factor farc until it is limited by MiN tnin tarce The time dependency of forces is usually slowly compared to the bunch length Transition processes are typically of the order of the formation time c 3 24R o This is different if the bunch shape or the curvature radius change rapidly The duty steps command is used to consider fast transient processes especially of radial forces at field boundaries CSRtrack uses extra grid points for the transition of field boundaries if duty steps is set to yes At these points the complete particle distribution is either directly before or directly after a field boundary Duty steps at a particular field boundary can be disabled by the duty command in the position section The recursive step algorithm is forward looking it uses steps that are 24 shorter than allowed if this helps to avoid a very short step before the end point defined in section tracker or a duty point fib recursive automatic time steps duty steps first step last step tart lend time of particle distribution end time specified before tracking in tracker d Iterative tracking To integrate the equation of motion from on grid point t to the next t2 CSRtrack needs the self forces f t to all particles in the complete time interval t lt t lt t2 They are approximated by a linear interpolat
42. tum see particles the strength of the dipole field is uniquely determined To take 17 into account vertical effects z direction by edge pecifie the tracking algorithm applies a vertical kick proportional to the offset from x y plane at the field boundaries field boundaries reference trajectory c Example dipole 1 dipole position rho 0 0 psi 0 0 marker dla properties r 1 66275 position rho 0 5 psi 0 0 marker d1b 4 2 3 Branch Section quadrupole position definition of field boundaries properties quadrupole properties The definition of quadrupoles is identical to that of multipoles that is described in the next subsection The only difference is the parameter poles that obsolete Example quadrupole position delta s 0 2 marker quad in duty no properties strength 0 100 alpha 0 horizontal offset 0 vertical offset 0 position delta s 0 5 marker quad out duty no 4 2 4 Branch Section multipole position definition of field boundaries 18 properties multipole properties a Subsection position See Definition of Field Boundaries b Subsection properties identifier argument unit type strength value a length n number alpha value a angle number horizontal offset value ho length number vertical offset value vo length number poles value 2n integer The magnetic multipole field is defined in a local specified coordinate
43. y linear interpolation along the time axis see track _step time grid The interpolated longitudinal dimension is used if the argument file is assigned to sigma_long Corresponding assignments can be used for the transverse dimensions The output file created by the steps monitor see online monitor type steps can be used as sub bunch dimension file e g to perform calculations with exactly the same dimensions a in an earlier run 35 4 6 7 Examples The CSR field of ellipsoidally shaped sub bunches is calculated with the Green s function method csr_g_to_p The bunch length at different positions along the beam line is read from the file my_file dat the horizontal size is fixed and the vertical scales with the vertical size of the beam forces type csr g to p shape ellipsoid sigma file my file dat sigma long file sigma rad 0 0003 sigma vert relative relative vert 1 0 If meshed forces and shielding are used the general format of the forces section will look like this forces type May sbox coon BX SEE use old mesh shield shield max 4 7 Sections monitor and online monitor a Concept The monitor section is used to write particles properties and forces before or after a tracking calculation to file while monitors defined in the online monitor section are served during the tracking As fields and forces are calculated during the tracking which is activated in the tracker section they are undefined befor
44. zontal sub bunch width vertical Results for the calculations with the different force types are plotted in the appendix 11 2 5 Typical CPU Time and File Structure for the Example Calculation time on a PC from 2004 for the projected method is less then 1minute for the g_to_p method 30 minutes and for the direct p_to_p method 8 5 hours csrtrk in in particles fmt1 log txt latout dat sub_bunch dat steps dat x_0001 fmt3 x_0052 fmt3 end fmt3 12 3 CSRtrack Command Structure 3 1 Command File CSRtrack reads all commands from the ascii input file csrtrk in It has the following structure commands exit lt CR gt with commands command commands command comment global_command section comment text lt CR gt separator lt blank gt lt CR gt The input file can be used to specify and open further input files see ile The length of command lines in input files is limited to 400 characters 3 2 Sections section name section_body A section call causes three activities 1 The section is initialized after the opening bracket 2 Global commands and specific section commands are valid in the section body Section specific commands are either assignment statements or nested sections The assignment of section parameters can be done in any succession The only exception is the particles definition see section particles 3 The section action is
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