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1. Library User Guide DAN T ABELL IT DOESN T MATTER HOW BEAUTIFUL YOUR THEORY 15 IT DOESN T MATTER HOW SMART YOU ARE Ir IT DOESN T AGREE WITH EXPERIMENT IT S WRONG RICHARD FEYNMAN THE PURPOSE OF SCIENCE IS NOT TO LEAD US TO EVERLASTING WISDOM BUT TO PLACE A LIMIT ON EVERLASTING ERROR BERTOLT BRECHT IGNORANCE IS NO EXCUSE IT S THE REAL THING IRENE PETER Polymorphic Tracking Code LIBRARY USER GUIDE DAN T ABELL TECH X CORPORATION BOULDER 2011 The writing of this manual was supported in part by the U S Department of Energy Office of Science Office of Nuclear Physics under SBIR Grant No DE FGo2 06ER84508 Copyright 2011 Tech X Corporation All rights reserved The Polymorphic Tracking Code PTC is copyright 2008 Etienne Forest and CERN All rights reserved The fibre and the integration node with their resulting linked list types the layout and the node layout are based on concepts first elaborated with J Bengtsson The node layout is similar to the Lagrangian class that Bengtsson and Forest contemplated around 1990 for the C collaboration later known as CLASSIC LEGO is a registered trademark of the LEGO Group Windodw is a registered trademark of Microsoft Corporation in the United States and other countries All other trademarks are the property of their respective owners Tech X Corporation 5621 Arapahoe Avenue Sui
2. PART 2 DO I 0 2 THIN 0 01D0 I 0 03 PART 2A CALL THIN LENS RESPLIT R1 THIN LIM LIMITS REDUCING NUMBER OF CUTS FITTING TO PREVIOUS TUNES PART 2B CALL READ COMMAND77 FIT TO BETAO RESULTS TXT COMPUTING DBETA BETA AROUND THE RING PART 2C CALL READ COMMAND77 FILL BETA1 TXT ENDDO Example 3 uses the limits from Example 2 lim 1 of 8 and lim 2 of 24 with 0 00100 KF THIN is now about 279 and KD THIN is about 197 Both are much greater than lim 2 PTC will use integration method 6 which has seven kicks per integration step Table 9 3 and the list below show the results for Part 1A of Example 3 Method 2 40 40 4 6 30 6230 Table 9 3 Results of Exam ple 3 for Part 1A Number of slices 6270 Total NST 930 Total NST due to Bend Closed Orbit o e Biggest ds 0 115885454545455 The quadrupoles and the dipoles are split using integration method 6 with a grand total of 6240 slices In Part 1B compute the tunes and betas around the ring The fractional tunes are 0 142192063715077 0 854078356314425 90 SYMPLECTIC INTEGRATION AND SPLITTING And the betas are stored for future use In Part 2A the magnet is resplit less and less finely the tune is refitted and finally the delta beta around the ring is estimated as a measure of the splitting adequacy The results for THIN 0 01 are lt DBETA BETA gt 4 025629639896395E 006 MAXI
3. mis 1 2 d0 call misalign_siamese p2 mis add true 495 elseif example 11 then Example 11 i mis 0 d0 mis 1 2 d0 mis 5 pi 8 40 call misalign girder p2 mis 500 mis 0 d0 mis 1 2 00 m call misalign_siamese p2 mis add false amp apo preserve_girder true Figure 4 8 Examples 6 end if top through 11 bottom Taylor Polymorphism and Knobs PTC supports the full usage of Taylor maps derived from the integrator for the computation of lattice functions PTC uses FPP a package of polymorphic types and tools to extract and analyze the Taylor maps Information explaining is at http mad web cern ch mad PTC_proper 5 1 POLYMORPHS A polymorph is the FORTRAN9O0 type real 8 type real 8 type taylor T active if Taylor real dp r active if real or knob integer kind 1 real 2 Taylor 3 Taylor knob 0 special integer i used for knobs real dp s scaling for knobs logical lp alloc end type real_8 is Taylor is allocated in DA package Polymorphs allow the computation of parameter dependent maps States of a Polymorph A polymorph Y can be real 1 Taylor Y amp kind 2 or a special Taylor called a knob Y amp kind 3 Computing a Taylor Map Suppose we have a closed orbit at position 1 given by six real numbers fix 1 6 We can construct the following array of six polymorphs
4. 255 allocate PSR1 DNA 2 PSR1 DNA 1 L gt L1 do i 2 2 260 5 1 1 gt PSRI DNA i 1 L next L2 end do allocate PSR2 DNA 2 PSR2 DNA 1 L gt L1 265 do i 2 2 PSR2 DNA i L gt PSR2 DNA i 1 L next L2 end do allocate Fig8 DNA 2 270 Fig8 DNA 1 L gt do i 2 2 Fig8 DNA i L gt Fig8 DNA i 1 L next L4 end do 27 allocate Col1 DNA 2 11 1 gt L5 do i 2 2 Col1 DNA i L gt Col1 DNA i 1 L next 16 end do 280 allocate Col2 DNA 2 Col2 DNA 1 L gt L5 do i 2 2 285 290 295 300 305 310 315 320 325 330 PTC GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 Col2 DNA i sL gt Col2 DNA i 1 L next L6 end do maps with polymorphs simple fit qf 1 d beans gf 1 1 qf 2 dn abe gf 3 qd 1 RM qd 2 qd 2 qd 2 name qd 4 esse gf 1 11 4 1 1 qd 1 CoLlxdna 2 L qf 2 CoLl dna 2 L qd 2 101 continue state defaultO only_4d0 1 1 1 2 call call call call call call call call call id 1 yl y2 call call nl n2 0 40 0 40 init state 2 c_ np_pol c_ np_pol is automatically computed find_orbit CoL1 fix1 1 state 1 d 6 find_orbit Col2 fix2 1 state 1 d 6 alloc yl alloc y2 alloc id alloc n1l alloc n2 alloc eq identity damap id this is permitted in ptc only not
5. The example code in this chapter is from the PTC geometry tutorial source file ptc geometry f90 which is given in appendix C The line numbers of the code in the examples refer to the line numbers of the code in the appendix This tutorial example shows how to create a map for the collider with polymorphs and knobs The first six lines of code initialize the polymorphic block for the focusing quadrupoles qf in Coll and 12 give the quadrupoles the family name QF and set their normal strength as the first parameter in the Taylor series The following six lines do the same for the defocusing quadrupoles QD qf 1 0 qf 1 name 315 Qf 1 ibn 2 qf 2 0 qf 2 name qf 2 ibn 2 qd 1 0 320 408 1 qd 1 ibn 2 qd 2 0 qd 2 name qd 2 ibn 2 The next four lines of code declare qf and qd as independent in DNA layouts L5 and L6 They perform the same function as calls to the subroutine scan_for_polymorphs If a polymorphic magnet on any fibre of the DNA layouts 15 and 16 is named QF or QD then ibn 2 becomes a knob L L L L 325 Coll dna 1 11 1 1 2 CoL 1 2 Once the knobs are set on the lattice using the scan_for_polymorphs routine or equivalent we can invoke tracking routines after the DA package has been initialized The following lines of code define the closed orbit if not o 330 101 continue state defaultO only
6. Y N TYPE INTEGRATION_NODE Pointers to POINTER 1 Body of Element parent fibre Integration Step i 0 22 7 ELEMENT Integration Step i 1 MAG P NST POINTER MAG TYPE INTEGRATION_NODE Y POINTER T2 Ld 7 Pd d 4 Pd Pointer to last child Pd Figure B 5 Type integration node and type fibre The probe tracks position X 6 and spin S The logical U is true if the particle is unstable Here is the definition of the data type PROBE 8 TYPE PROBE_8 TYPE REAL 8 X 6 TYPE SPINOR 8 S REAL DP E IJ NDIM2 NDIM2 LOGICAL U TYPE INTEGRATION NODE POINTER LOST NODE END TYPE PROBE 8 112 DATA TYPES Temporal Probe The data type temporal probe holds a probe and information relevant to time based tracking TYPE TEMPORAL_PROBE TYPE PROBE XS TYPE INTEGRATION_NODE POINTER NODE REAL DP 5 05 6 TYPE INTERNAL STATE STATE END TYPE TEMPORAL PROBE The probe XS contains the coordinates The NODE points to the integration node that the particle is in The distance DS is the approximate distance that the particle traveled inside the node assuming a drift POS 6 is the position of the particle in an absolute 3D frame STATE is the tracking state of that beam machine Temporal Beam The data type temporal beam is a collection of particles probes TYPE TEMPORAL TYPE TEMPORAL PROBE POINTER TP A 3 E
7. cell d1 qd d2 b ring PSR call survey PSR end subroutine build PSR minus subroutine build Quad for Bend PSR use madx ptc module use pointer lattice implicit none type layout target PSR real dp ang ang2 brho bl Larc Lq type fibre b call make states false exact model true default default nocavity exactmis call update states madlength false ang twopi 36 d0 360 d0 Larc 2 54948d0 brho 1 2d0 Larc ang call set_mad brho brho method 6 step 10 madkind2 drift_kick_drift ang2 ang two bl ang Larc Lq Larc sin ang2 ang2 b quadrupole B_QUAD Lq 0 40 call add b 1 0 b1 128 PTC GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 645 650 5smag spermfringe true smagp spermfringe true 0 fringe true _ true PSR 10 b PSR ring PSR call survey PSR end subroutine build_Quad_for_Bend D PTC Splitting Tutorial Source File ptc_splitting 90 This appendix contains the complete FORTRAN9O listing for the PTC example ptc_splitting This example is discussed in Symplectic Integration and Splitting chap ter 9 ptc splitting f90 program ptc splitting use run madx use pointer lattice implicit none character 48 command gino type layout pointer r1 r2 r3 r4 r5 r6 type layout pointer 1 psr2 fig8 coll
8. 9 These last two variables are kept in this order for a technical reason This ordering allows us to re tain the natural meanings positive 6 implies higher en ergy and positive implies longer path length Because of the Hamiltonian struc ture reversing their order would require us to reverse the sign on one of them Figure 2 6 Geometry and local co rdinates x y s for a generic block in PTC fact that the scale set for one element may differ from that in the preceding element e g in the case of acceleration In such cases the patching mentioned above must include not only geometric transformations but also energy transformations TRACKING STATE PTC defines a variable type called internal state Objects of this type may be used to modify certain assumptions PTC makes when it performs tracking One may for example ask PTC to track only the 4D transverse phase space variables or one may turn on or off radiation A variable of type internal state is essentially a list of flags that one may pass to PTC tracking routines to modify the algorithms used For more information see Internal States appendix A PHASE SPACE COORDINATES With the default tracking state PTC uses the six phase space variables X Pur 2 22 Here x and y denote the local transverse co rdinates and and p denote the corresponding canonical momenta divided by a scale momentum The fifth variable denot
9. are identical as a consequence the phase advance between such locations is well defined Be tween unmatched locations however the definition of phase advance can depend on the conventions used in extracting the As 18 A W Chao SLIM An early work revisited In Proceedings of the 11th European Particle Acelerator Conference Genoa Italy 23 27 June 2008 pages 2963 2967 Geneva 2008 European Physical Society 19 Overlapping fringe fields also violate the principle of magnet independence and in that case too one must exercise care 20 OVERVIEW OF PTC 20 The choice of which you are free to modify obviously affects the ac curacy of your simula tion Chapter 9 discusses how to split elements into integration nodes Figure 2 16 N 4 integra tion nodes cover an element to the parent fibre that contains it Integration nodes allow us to examine data inside a beamline element without violating the element as a fundamental unit a self contained LEGO block They allow us to resist the temptation to slice an element by hand they are the slices The data type integration_node also contains an integer that determines the type of integration node of which there are five See figure 2 16 On traversing the integration nodes that represent an element one encounters in order an entrance patch and misalignment if any which connects the geometry described by the fibre to that of the pre
10. drift D1 2 28646d0 d2 drift D2 0 4540 qf quadrupole QF 0 5d0 kf 520 Qd quadrupole QD 0 5d0 kd b rbend B Larc ang cell d1 qd 02 b d2 qf d1 PSR 525 PSR 10 cell ring PSR call survey PSR end subroutine build PSR The lines 502 506 set important internal state variables for tracking In the call to make states the boolean argument false means we are modeling a proton lattice Use true for electrons To use the full square root Hamiltonian we set the global parameter exact model to true The global variable default of type internal state is here modified from its default value by adding the following two flags nocavity which tells PTC to ignore RF cavities exactmis which tells PTC to treat misalignments exactly For more on internal state variables see appendix A Finally we set the flag madlength to false This means that PTC will use the arc length rather than the chord length to define the geometry of a rectangular bending magnet Use true to make PTC use the chord length The call in line 511 to set mad defines via brho the scale momentum p for the fibres in the layout returned by this subroutine It also specifies the type method 2 and number STEP 10 of integration steps in the body of each element We then define the five fibres needed for our basic cell Fibres d1 and d2 denote the long and short drifts Fibres qf and
11. gt PSR1 end d 1 f chart f o call translate f d 85 call compute_entrance_angle f chart f mid pl chart f mid a call rotate f a basis f chart f mid p2 gt p2 next else call append_point PSR1 p1 go end if pl gt pl next end do elements in PSR1 now in correct locations In the above block of code we point p1 and p2 respectively to the starts of DNA sequences L1 and L2 The do loop over the L1 n fibres L1 appends to PSR1 the desired fibres from L1 or L2 bends from L2 everything else from L1 If p1 points to a bend then in line 81 we append the current fibre of L2 to PSR1 This of course will always be a straight bend a B QUAD Later in line 87 we advance p2 to the next fibre in L2 If p1 does not point to a bend then in line 89 we append the current fibre of L1 to PSR1 n either case we advance in line 91 p1 to the next fibre in L1 After we append a straight bend to PSR1 we must do some extra work to place it in the correct location Since we want a given straight bend to have the same location and orientation as the corresponding rectangular bend we simply compute and perform the required translations and rotations This happens in lines 82 through 86 1 Point f to the fibre most recently appended to PSRI1 2 Compute in line 85 the vector d from the center of the newly appended straight bend f chart f o to the center of the corresponding re
12. qf d1 565 PSR PSR b d2 qf d1 8 cell d1 qd d2 b ring PSR call survey PSR 570 end subroutine build PSR minus build Quad for Bend The subroutine build Quad for Bend defines the layout shown on the right in figure 3 5 It has ten straight bends which have the same length as the rectangular bends in the PSR lattice Because this lattice contains no drifts it will require patching In addition because it contains no focusing elements the elements of this layout will actually be used only as entries in other layouts subroutine build Quad for Bend PSR 575 USe run madx use pointer lattice implicit none type layout target PSR 580 real dp ang ang2 brho bl Larc Lq type fibre b 585 Call make states false exact model true 3 3 SUBROUTINES 29 30 MODELING AN ACCELERATOR WITH PTC default default nocavity exactmis call update_states madlength false 590 ang twopi 36 d0 360 d0 Larc 2 54948d0 brho 1 2d0 Larc ang call set_mad brho brho method 6 step 10 595 madkind2 drift kick drift ang2 ang two bl ang Larc Lq Larc sin ang2 ang2 600 b quadrupole B QUAD Lq 0 40 call add b 1 0 b1 5smag spermfringe true smagps spermfringe true 605 0 fringe true fringe true PSR PSR 10 b ring PSR 610 call survey PSR end subroutine
13. type real_8 Y 6 type damap id call init state NO 0 call alloc Y 1 The CERN folder is called proper to distinguish it from the information on Most of the infor mation in the PTC proper folder is about FPP 50 TAYLOR POLYMORPHISM AND KNOBS call alloc id id 1 Y fix id The variable state describes internal states of PTC The variable id is a differential algebra map damap and id 1 constructs the identity map Then Y fix id creates the polymorphic component Y i given by Y i fix i x where x denotes the i th component of the identity map id To compute a one turn map for example we track Y around the machine here R as if it were six real numbers call track R Y 1 state At the end Y contains the Taylor map to order NO about the closed orbit fix The internal state variable state determines the exact nature of the map 1 state default0 specifies a 6 D map with cavities 2 state defaultO nocavityO specifies 6 D map with cavities skipped 3 state default only_4d0 specifies a 4 D phase space X Px Y Py 4 state default delta0 specifies a 4 D phase space X Px Y Py plus energy as the fifth variable cavities are skipped For more information about states see appendix A 5 2 KNOBS A knob is a polymorph that turns itself into a simple Taylor series when used A knob cannot be on the left side of an equal sign Knobs l
14. D B A CALL CHANGE BASIS D GLOBAL FRAME D W END SUBROUTINE FIND PATCH B The above code is a simple example of the usage of the geometric routines This geometric routine is very close to the dynamical set up of PTC Patches are written as an x rotation a rotation a 7 rotation followed by a translation transverse drift Note that the translation D B A between is expressed in the final frame W This is normal if dynamical rotations precede the translations FIND INVERSE PATCH This routine is the precise inverse of the FIND PATCH B routine The affine frame B W is the output INVERSE FIND PATCH A 3 V 3 3 D 3 ANG 3 B 3 W 3 3 SUBROUTINE INVERSE FIND PATCH A V D ANG B W USED IN MISALIGNMENTS OF SIAMESE IMPLICIT NONE REAL DP INTENT INOUT V 3 3 W 3 3 REAL DP INTENT INOUT A 3 B 3 D 3 ANG 3 REAL DP DD 3 W V CALL GEO ROT ANG 1 BASIS V CALL CHANGE BASIS D W DD GLOBAL FRAME B A DD END SUBROUTINE INVERSE_FIND_PATCH 8 2 AFFINE ROUTINES ON COMPUTER OBJECTS Affine routines on PTC s computer objects act on the affine bases of magnets siamese girders integration nodes and fibres These objects contain a myriad of affine bases to help us locate the trajectories in 3 D Within this category are two subcategories 71 72 GEOMETRIC ROUTINES 1 The quotation marks in dicate that there is really nothing standard about the so called standard implici
15. backward propagation example 12 32 BEAM data type 113 beam line expanded 113 BEAM LOCATION data type 113 beamline defined as layout 13 expanded 20 beamline element see element bend defined 7 Bengtsson Johan pioneering work 3 beta function computing 88 block see LEGO block or polymorphic block Brookhaven National Laboratory example based on 12 build PSR basic ring 23 example source code 26 138 build PSR minus build PSR minus example source code 28 build Quad for Bend example source code 29 CEBAF see Continuous Electron Beam Accelerator Facility CHANGE BASIS routine 70 CHART data type 106 chart contains misalignment 15 defined 13 15 106 CHECK NEED PATCH routine 72 check need patch routine 37 39 chromaticity computing 88 closed orbit computing 88 finding 64 global properties 57 local properties collider example 12 modeling 23 38 COMPUTE_ENTRANCE_ANGLE routine 70 compute_entrance_angle routine 33 Continuous Electron Beam Accelera tor Facility example based on 11 coordinate system see global coordi nate system or local coordi nate system cutting see splitting data structure modeling accelerators 11 data type integration_node 20 probe 21 BEAM_LOCATION 113 BEAM 113 CHART 15 106 ELEMENT 13 FIBRE_APPEARANCE 73 FIBRE 13 105 element INTEGRATION_NODE 109 LAYOUT 13 103 MAD_UNIVERSE 73 MAGNET_FRAME 106 NODE_LAYOUT 2
16. fibre defined 12 13 105 tracking routines 61 FIBRE_APPEARANCE data type 73 figure eight accelerator modeling 23 35 FIND_ORBIT routine 54 find_orbit geometric transformation 139 routine 62 FIND_ORBIT_X routine 64 FIND_PATCH routine 72 75 find patch routine 33 37 39 FIND PATCH B routine 71 flag delta 102 exactmis 101 fringe 101 madlength 27 nocavity 101 only 4d 102 para in 102 radiation 101 spin dim 102 spin only 102 spin 102 time 101 totalpath 101 114 flags internal state 101 forward propagation example 11 12 32 FPP defined 3 49 documentation 49 frame of reference girder 76 magnet 6 siamese 75 fringe flag 101 fringe field integration node 20 internal state 101 magnet 8 Fully Polymorphic Package see FPP fuzzy_split global parameter 86 GEO_ROT routine 69 GEO_TRA routine 69 geometric routines described 67 geometric transformation 140 geometry defined 8 overview 5 patching 15 geometry tutorial source file 24 44 115 girder affine frame 76 creating 44 76 defined 43 frame of reference 43 76 misaligning 43 44 77 rotating 77 translating 77 girder_frame 45 global coordinate system global frame 8 global frame described 8 geometric routines 67 positioning first element in track able layout 35 used to compute local reference frame 33 global information see also global prop erty analysis 17 define
17. leads directly to PTC s use of map based methods which emphasize a global view of the accelerator Map based methods are fundamental to the analysis of real accelerators Think of a one turn map as a piece of sophisticated diagnostic hardware that you install at some point in the ring It enables you to observe the beam turn after turn It follows that any sensible question you might ask about the beam at that point in the ring must be contained within that one turn map Indeed all of standard perturbation theory may be expressed entirely in terms of the one turn map If you desire information at some other point in the ring then you must of course install another sophisticated beam monitor at that new location Using map based methods this will be equivalent to a proper combination of the one turn map at your original location and the transfer map that connects the two locations of interest As a consequence a one turn map plus partial turn transfer maps contain the answers to all sensible questions you might ask of an accelerator model The analogy of a one turn map as a sophisticated beam monitor also implies quite correctly that no connection exists between the construction of a one turn map and its analysis The construction is done using a strictly local approach which is our only hope of getting the physics right in this complicated and messy world Once we ve obtained the one turn map we may forget where it came from and what v
18. magnetic field strength etc The resolution of this conflict lies in the fact that once we have nailed down the geometry we are then free to modify the lattice with misalignments changes to the magnetic field and the like build PSR minus The subroutine build PSR minus defines the partial ring shown in the center of figure 3 3 It begins with a bend a short drift a quadrupole and a long drift it continues with eight PSR cells and it concludes with a long drift a quadrupole a short drift and a bend Relative to the PSR lattice this partial ring is missing a short drift a quadrupole two long drifts a quadrupole and a short drift The subroutine is a straightforward modification of that for build PSR subroutine build PSR minus PSR use run use pointer lattice 535 implicit none type layout target PSR real dp ang brho kd kf Larc 540 type fibre b 41 d2 qd qf type layout cell call make states false 545 exact model true default default nocavity exactmis call update states madlength false 550 ang twopi 36 d0 360 d0 Larc 2 54948d0 brho 1 2d0 Larc ang call set mad brho brho method 6 step 10 madkind2 drift kick drift 555 kf 2 72d0 brho kd 1 92d0 brho dl drift D1 2 28646d0 560 d2 drift D2 0 4540 qf quadrupole QF 0 5d0 kf qd quadrupole QD 0 540 kd b rbend B Larc ang cell d1 qd 02 b d2
19. method 5 For more information about integration methods see Section 9 4 9 2 SPLITTING TUTORIAL SOURCE FILE The example code in this chapter is from the tutorial source file ptc splitting f90 PTC Splitting Tutorial Source File ptc splitting f90 appendix D The line numbers of the code in the examples refer to the line numbers of the code in the appendix 9 3 SPLITTING THE LATTICE This section describes the routines global parameters and arguments involved in splitting elements in the lattice Global Parameters The default values for the global parameters are in red italic type Four global parameters are involved in splitting the lattice e resplit cutting o 1 or 2 For more information see Section 9 3 sixtrack compatible true or false This global parameter enforces a second order integrator for all magnets radiation bend split true or false This global parameter splits bends with integration method 2 to improve radiation or spin results For more information about PTC s methods of integration see Section 9 1 e fuzzy split default to If this global parameter is greater than 1 0 lets some magnets with slightly too long integration steps go through i e if ds lmax0 fuzzy split Splitting Routines Mandatory arguments for the routines are in regular black type Optional argu ments are in red italic type THIN LENS RESTART R1 USEKNOB This routine resets all magnets
20. mis 5 pi 8 40 call misalign_girder p2 elseif example then mis 0 00 mis add true PTC GEOMETRY TUTORIAL SOURCE FILE ptc_geometry f90 125 mis 1 2 d0 mis 5 pi 8 40 490 call misalign_girder p2 mis mis 0 00 mis 1 2 d0 call misalign siamese p2 mis add false elseif example 10 then 495 mis 0 d0 mis 1 2 40 mis 5 pi 8 40 call misalign_girder p2 mis mis 0 00 500 mis 1 2 d0 call misalign_siamese p2 mis add true elseif example 11 then mis 0 40 mis 1 2 d0 505 mis 5 pi 8 40 call misalign_girder p2 mis mis 0 00 mis 1 2 d0 call misalign_siamese p2 mis add false amp 510 preserve_girder true end if 999 command_gino opengino call context command_gino context makes them capital 515 Call call_gino command_gino 111 command_gino mini call context command_gino context makes them capital call call gino command gino 520 vaguely necessary baloney command gino closegino call call gino command gino call ptc end end program ptc geometry 525 subroutine build PSR PSR use madx ptc module 530 use pointer lattice implicit none type layout target PSR 535 real dp ang brho kd kf Larc type fibre b 41 42 qd qf type layout cell 126 GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 540 Call make states false exact model true default default nocavity ex
21. next true call find patch f f next next false end if 100 f gt f next end do PSR1 now patched Within the do loop over all elements in PSR1 we apply the patches required between each B_QUAD or straight bend and its preceding and trailing elements No other elements require patching Finally in the following three lines of code we give layout PSR1 a formal name and we ensure that it forms a closed topological ring so that particles can circulate Note that the second two lines must both be executed to make the layout PSR1 form a closed ring The call to ring_L in line 105 sets some pointers inside the layout PSR1 that connect the end of PSR1 to the start and vice versa PSR1l name PSR 1 PSR1 closed true 105 Call ring_L PSR1 true make it a ring topologically Query At this point where in the global frame are the magnets the straight bends of layout 12212 To construct layout PSR2 for the backward propagating beam we follow essen tially the same steps as for PSR1 See the next block of code There are three significant differences i In this case all the elements are already in their proper physical locations so here we may omit the geometric computations and oper ations performed during the construction of PSR1 ii Because of the backwards propagation we initialize the pointers p1 and p2 respectively to the ends of layouts L1 and L2 and we advance those pointers not to the next
22. now bends it along the blue trajectory towards magnet 3 the first magnet in a new beamline Magnet 1 of figure 2 14 must of course appear in two fibres the first points to magnet 2 as the subsequent element while the second points to magnet 3 What about the patches Assuming the exit frame of magnet 1 and the entrance frame of 16 OVERVIEW OF PTC Figure 2 14 Patch ing elements in mul tiple beamlines Figure 2 15 Mis aligning an element magnet 2 have the same unit vectors but different origins then the patch between the fibres pointing to magnets 1 and 2 must correspond to a simple drift of the appropriate length and that is exactly what PTC will compute The patch between the fibres pointing to magnets 1 and 3 corresponds to three actions a drift of length d a co rdinate frame rotation by angle a and a translation of length h Misalignments The accelerator we design is never the one we build This fact makes it essential to investigate the margin for error in any design To simulate misalignment errors PTC uses the approach illustrated in figure 2 15 There we show the original element 2 with a light gray outline and the misaligned element 2 with a red outline To push a particle from the exit of element 1 to the entrance of element 5 PTC first applies the patch from element 1 to the original element 2 Then PTC traverses element 2 in three steps an entrance misalignment the standard element 2 and an ex
23. 0 01D0 I 0 03 9 3 SPLITTING THE LATTICE 91 CALL THIN LENS RESPLIT R1 THIN LIM LIMITS REDUCING NUMBER OF CUTS FITTING TO PREVIOUS TUNES CALL READ COMMAND77 FIT TO BETAO RESULTS 2 TXT COMPUTING DBETA BETA AROUND THE RING CALL READ COMMAND77 FILL BETA1 2 TXT ENDDO The results of the last iteration for the closed orbit are 3 129618316516444E 002 3 357101188909470E 003 0 000000000000000E 000 0 000000000000000E 000 0 000000000000000 000 0 000000000000000 000 The results of the last iteration for the stability is The results of the last iteration for the tunes are 0 142192063715077 0 854078356314425 In addition DBETA BETA 1 184251326377263E 002 MAXIMUM OF DBETA BETA 2 032348481520253E 002 We notice that the closed orbit is wild and that the fluctuation of the beta functions is twice as big as before This problem can be alleviated by enforcing a finer split of the bend This is done by setting the XBEND argument to some small value that corresponds approximately to the norm of the residual orbit Example 5 In example 5 the splitting line is replaced by this one CALL THIN_LENS_RESPLIT R1 THIN LIM LIMITS XBEND 1 D 4 The results of the final iteration steps for the closed orbit are 3 377915412576128E 003 3 670253104757456E 004 0 000000000000000E 000 0 000000000000000E 000 0 000000000000000 000 0 000000000000000E 000 The results of the final iterati
24. 1 0 call alloc y call alloc normal call alloc id id 1 y x id call track lattice y 1 istate normal y write 6 a 3 2x f9 6 tunes normal stune Chromaticity Anharmonicity 6 2 S DEPENDENT GLOBAL QUANTITIES Betatron Amplitude X zero x 5 delta 58 COMPUTING ACCELERATOR PROPERTIES call find_orbit lattice x 1 istate 1 d 7 call init istate 1 0 call alloc y call alloc normal call alloc id id 1 y x id call track lattice y 1 istate normal y y X normal a_t gt latticesstart track this normalizing transformation beta_x y 1 sub 10 2 y 1 sub 01 2 beta y y 3 sub 0010 2 y 3 sub 0001 2 write 6 a 2 2x f9 6 beta x beta y beta x beta y do j 1 latticesn call track lattice y j 1 istate beta x y 1 sub 10 2 y 1 sub 01 2 beta y y 3 sub 0010 2 y 3 sub 0001 2 write 6 a 2 2x f9 6 beta x beta y beta x beta y p gt end do Dispersion zero x 5 delta call find_orbit lattice x 1 istate 1 d 7 call alloc id call alloc disp call alloc xt call alloc eta call alloc y call alloc normal id 1 y x id type type damap type damap type real 8 dimension 6 type real 8 dimension 6 type normalform call track lattice y 1 default normal y y
25. 472 QF 2 1 483 483 487 QD 2 1 1412 1412 1416 Table 9 9 Results of Exam B 2 1 1422 1422 1426 ple 9 when useknob true If we perform the same call with useknob false then the magnets with knob true are masked and the other magnets are split CALL THIN_LENS_RESPLIT R1 THIN EVEN TRUE LMAX0 0 005D0 XBEND 1 D 4 USEKNOB FALSE Table 9 10 shows the results Name Method Steps 1 2 D1 2 1 1 1 5 D1 2 1 924 1 928 D2 2 92 112 160 207 QF 6 102 818 871 923 QD 6 102 934 987 1039 Table 9 10 Results of Exam B 4 510 1136 1393 1649 ple 9 when useknob false 9 4 OTHER SPLITTING ROUTINES 95 9 4 OTHER SPLITTING ROUTINES This section discusses PTC integration methods 1 3 and 5 Splitting a Single Fibre This routine splits one fibre RECUT KIND7 ONE FIBRE LMAXO DRIFT In the presence of space charge or for other reasons it may be more important to observe the beam at many locations approximately than to use a high order integration method PTC provides integration method 1 for certain magnets Drift Kick Drift Strict Talman Interpretation For this type of algorithm the exact model true and the exact model false can be switched into method 1 if and only if the original method was method 2 i e second order integrator Basically negative propagators drifts are not acceptable this is the strict Talman interpretation to which PTC does not adhere except in the
26. 6 b mag name smagsspssmethod 5 1 pause 2 Talman algorithm example 3 write 6 write 6 Talman algorithm example 3 write 6 thin 0 001d0 cut like crazy call thin lens resplit rl1 thin l im limits ptc command file could be a mad x command or whatever call read ptc command77 fill beta0 txt computing tune and beta around the rir write 6 write 6 now reducing the number of steps and refitting write 6 x PTC SPLITTING TUTORIAL SOURCE FILE ptc_splitting f90 131 do i 0 2 thin 0 0190 i 0 03 call thin_lens_resplit r1 thin lim limits reducing number of cuts call read_ptc_command77 fit_to_betaO_results txt fitting to previous tunes 85 call read ptc command77 fill betal txt computing dbeta beta around the ring end do pause 3 go write 6 write 6 call append empty layout m u number 2 95 Call set up m usend rl gt m_u end exact true method drift kick drift Call build PSR r1 exact method write 6 write 6 now reducing the number of steps and refitting write 6 105 do 0 2 thin 0 0140 i 0 03 call thin_lens_resplit r1 thin reducing number of cuts call read_ptc_command77 fit_to_betaO_results_2 txt fitting to previous tunes call read ptc command77 fill betal 2 txt computing dbeta beta around the ring 10 end do pause 4 write
27. 6cas 1 the entrance fringe field integration 1 one of N steps in the body of the element integration node 6cas o the exit fringe field integration node 6cas 2 the exit patch misalignment tilt of its parent fibre integration node7ocas 2 See figure B 5 NEXT and PREVIOUS are the pointers necessary to create the NODE LAYOUT which is patterned closely on the LAYOUT PARENT FIBRE points to the parent fibre which engendered the INTEGRATION NODE POS is the position in the NODE LAYOUT POS IN FIBRE is the position in the PARENT FIBRE TEAPOT LIKE indicates whether the internal frame of reference is curved or straight This is useful to move approximately inside an integration node step For S S 1 contains the ideal arc length position LD e S 2 contains the local integration position 0 lt 5 2 L e S 3 contains the total L up to that integration node e S 4 contains the length of the integration step DL FIBRE MAG L FIBRE MAG Probe The data type probe holds information about the location of a particle TYPE PROBE REAL DP X 6 TYPE SPINOR S LOGICAL U TYPE INTEGRATION NODE POINTER END TYPE PROBE LOST NODE 5 B 2 TIME BASED TRACKING 111 Type INTEGRATION_NODE Entrance Patches cas 1 N N amp Type FIBRE Pointer to first child Entrance Fringes cas 1 N N Parent Fibre x s m Xy 4 INTEGER POINTER POS
28. Call alloc af b smag amp ssiamese frame call find patch b smagsspssf sa b mag p f ent amp a pl mag girder_framexent amp 8smags ssiamese b smag ssiamese frame sangle 430 43 Write 6 Example from the manual 1 11 read 5 example 124 PTC GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 440 445 450 455 460 465 470 475 480 485 call move_to Col2 p2 7 if example 1 then mis 0 00 mis 5 pi 8 d0 call misalign_girder p2 elseif example 2 then mis 0 40 mis 1 2 0d0 call misalign_girder p2 elseif example 3 then mis 0 00 mis 5 pi 8 d0 call misalign_girder p2 mis 0 d0 mis 1 2 d0 call misalign_girder p2 elseif example 4 then mis 0 d0 mis 5 pi 8 d0 call misalign_girder p2 mis 0 00 mis 1 2 d0 call misalign_girder p2 elseif example 5 then mis 0 40 mis 1 2 40 mis 5 pi 8 d0 call misalign_girder p2 elseif example 6 then mis 0 d0 mis 1 2 d0 mis mis mis add mis false mis mis add true mis call misalign_siamese p2 mis elseif example then mis 0 d0 mis 1 2 d0 call misalign_siamese p2 mis mis 0 00 mis 1 2 d0 mis 5 pi 8 d0 call misalign_girder p2 elseif example 8 then mis 0 d0 mis 1 2 d0 add mis false call misalign_siamese p2 mis mis 0 40 mis 1 2 d0
29. DURING CIRCULAR SCANNING TYPE INTEGRATION_NODE POINTER END_GROUND STORE THE GROUNDED VALUE OF END DURING CIRCULAR SCANNING TYPE LAYOUT POINTER PARENT_LAYOUT TYPE ORBIT_LATTICE POINTER ORBIT_LATTICE END TYPE NODE_LAYOUT Beam The purpose of the node layout is to track a beam We now define the data type BEAM TYPE BEAM TYPE REAL_8 POINTER Y REAL DP POINTER X REAL DP POINTER SIGMA DX BBPAR ORBIT LOGICAL LP POINTER U LOCATION POINTER POS INTEGER POINTER N LOST INTEGER POINTER CHARGE LOGICAL LP POINTER TIME INSTEAD OF S LOGICAL LP POINTER BEAM BEAM BBORBIT END TYPE BEAM We also define the data type BEAM LOCATION TYPE BEAM LOCATION TYPE INTEGRATION NODE POINTER NODE END TYPE BEAM_LOCATION LOST is the number of lost particles 114 DATA TYPES In a normal mode the BEAM is pushed integration node to integration node It can also be pushed from a location S1 to S2 Additionally time tracking is possible with TOTALPATH true In that case the position of the particle is given by specifying the actual integration node immediately proceeding the particle and the distance from that integration node measured in the integration variable L The node is located using the pointer POS Node and the distance is located in X Drifting back and forth to that position is done approximately with a drift either in C
30. PSR PSR 10 cell ring PSR call survey PSR call clean up 133 134 PTC SPLITTING TUTORIAL SOURCE FILE ptc_splitting f90 end subroutine build_PSR 235 Bibliography D P Barber Heinemann and Ripken A canonical 8 dimensional formalism for classical spin orbit motion in storage rings I A new pair of canonical spin variables Z Phys C 64 1 117 142 Mar 1994 Dor 10 1007 1557243 A W Chao SLIM An early work revisited In Proceedings of the 11th European Particle Acelerator Conference Genoa Italy 23 27 June 2008 pages 2963 2967 Geneva 2008 European Physical Society E Forest A Hamiltonian free description of single particle dynamics for hope lessly complex periodic systems J Math Phys 31 5 1133 1144 May 1990 DOI 10 1063 1 528795 Forest Locally accurate dynamical Euclidean group Phys Rev E 55 4 4665 4674 Apr 1997 DOI 10 1103 PhysRevE 55 4665 Forest Beam Dynamics A New Attitude and Framework volume 8 of The Physics and Technology of Particle and Photon Beams Harwood Academic Publishers Ams terdam 1998 Forest Geometric integration for particle accelerators J Phys A Math Gen 39 19 5321 5377 May 2006 10 1088 0305 4470 39 19 503 Forest and K Hirata A contemporary guide to beam dynamics Technical Report KEK 92 12 KEK Tsukuba Japan Aug 1992 Forest F Schmidt and E McIntosh Introduction to the Po
31. TRA 69 INVERSE FIND PATCH 71 KILL PARA 52 LOCATE TEMPORAL BEAM 112 MAKE STATES 101 MISALIGN FIBRE 81 MISALIGN GIRDER 77 MISALIGN SIAMESE 79 ORIGINAL P TO PTC 112 POSITION TEMPORAL BEAM 112 RECUT KIND7 ONE 95 97 ROTATE FIBRE 74 ROTATE FRAME 69 ROTATE GIRDER 77 ROTATE LAYOUT 74 ROTATE MAGNET 74 ROTATE_SIAMESE 75 SET ELEMENT 52 SET TPSAFIT 52 THIN LENS RESPLIT 86 LENS RESTART 86 TRACK BEAM 63 TRACK REF 64 TRACK NODE PROBE 62 TRACK NODE V 63 TRACK NODE X 63 TRACK PROBE2 62 TRACK PROBE X 63 TRACK PROBE 62 TRACK TEMPORAL BEAM 64 TRACK TIME 64 TRACK 50 54 TRANSLATE FIBRE 73 TRANSLATE FRAME 69 TRANSLATE GIRDER 77 TRANSLATE LAYOUT 73 TRANSLATE MAGNET 74 TRANSLATE SIAMESE 76 UPDATE STATES 101 allocate af 45 allocate 40 append empty layout 31 32 36 38 append point 33 36 38 check need patch 37 39 compute entrance angle 33 routine 143 144 routines find_orbit 62 find_patch 33 37 39 make_states 27 misalign_girder 47 misalign_siamese 47 move_to 44 rotate 33 35 scan_for_polymorphs 51 53 set_mad 27 survey 28 track 61 translate 33 FIND_PATCH_B 71 routines affine 67 71 72 dynamical 82 geometric 67 object oriented 62 splitting elements into integration nodes 86 95 standard tracking 61 time based tracking 64 tracking 61 trackin
32. Table 9 7 Results B 4 52 297 325 352 of Example 7 for resplit_cutting 2 Now all the magnets are split FIB Optional Argument Continuing with the previous example we make the following call Example 8 WRITE 6 KEYWORD EXAMPLE 8 CALL THIN LENS RESTART R1 CALL THIN LENS RESPLIT R1 THIN EVEN TRUE LMAX0 0 005D0 XBENDz1 D 4 FIB D1 NEXT Table 9 8 shows the results Name Method Steps 1 2 D1 2 46 1 26 50 D1 2 458 167 26 628 D2 2 10 67 74 80 QF 6 12 151 159 166 QD 6 12 679 687 694 B 4 52 709 737 764 Table 9 8 Results of Exam ple 8 94 SYMPLECTIC INTEGRATION AND SPLITTING Only the fibre D1_NEXT is affected All the others are left intact This allows the splitting of a single fibre USEKNOB Optional Argument We can use the type pol_block to produce a splitting of the lattice Example 9 FAM 0 FAMSSNAME D1 R1 FAM CALL THIN LENS RESTART R1 PUTS BACK METHOD 2 AND NST 1 EVERYWHERE CALL THIN_LENS_RESPLIT R1 THIN EVEN TRUE LMAX0 0 005D0 XBEND 1 D 4 USEKNOB TRUE If useknob is true then the magnets with the knob flag on are split assuming no other flags prevent it In this particular case resplit cutting 2 therefore the drifts will be split on the basis of LMAX0 0 00500 Table 9 9 shows the results Name Method Steps 1 2 D1 2 458 1 232 462 D1 2 455 458 232 949 D2 2 1 468 468
33. This routine finds the closed orbit Find_orbit_x 7 5 CLOSED ORBIT ROUTINE 65 FIND_ORBIT_X R FIX STATE eps TURNS fibrel nodel Here fibrel and nodel are integer positions The routine finds the fixed point for TURNS turns one turn if not specified The argument eps real dp number is used to do numerical differentiation typically 1 d 6 works For a no cavity fixed point fix 5 must contain the energy variable EIGHT Geometric Routines PTC s geometric routines are divided into three categories affine routines on pure geometry affine routines on computer objects dynamical routines Mandatory arguments for the geometric routines are in regular black type Optional arguments are in red italic type 8 1 AFFINE ROUTINES ON PURE GEOMETRY Affine routines on pure geometry act on a pure geometrical affine basis A 3 and V 3 3 where A represents the coordinates of a point in the global frame and V 3 3 the coordinates of a triad of vectors Theory Consider a point a and vector basis v1 75 73 attached to a solid a magnet for example See figure 8 1 V a t 3 s R TU 3 The vector can be expressed as follows i Figure 8 1 Rotating point a and vector basis 01 02 03 by R 68 GEOMETRIC ROUTINES The basis vectors v can be expressed in terms of a global basis that is PTC s global frame 0 Ven Suppose we rotat
34. about the move Fibre The data type fibre contains a pointer to a beamline element as well as pointers to the fibres that precede and follow it along a particle path When strung together in a linked list a sequence of fibres defines a beamline The fact that different fibres can point to the same element means that separate beamlines can share common elements and this gives PTC the capability to model complex accelerator topologies A fibre contains among other items a pointer to an element pointers to the previous and next fibres along a beamline a pointer to a chart that locates the element within the global reference frame a pointer to a patch that connects the elements of successive fibres geometrically energetically and temporally the direction of propagation through the element Layout The data type ayout represents a beamline as a doubly linked list of fibres It follows the particle path by specifying the order of the fibres which each point to an actual The use of a doubly linked list enables both forward and reverse propagation through a given sequence of beamline elements 14 OVERVIEW OF PTC 13 Note that a layout need not represent an entire ma chine it may represent just a piece of an machine The full accelerator would then be built from several layouts Figure 2 10 A layout with a linked list of fi bres pointing to magnets As before we say mag net but it may mean any type
35. allows the girder to have an existence independent of the fibres themselves One can move fibres within a girder during its creation while keeping this frame fixed ROTATE GIRDER and TRANSLATE GIRDER The routines operate exactly as the siamese routines do and therefore do not require any special description The routines describe design displacements of the girder and therefore the affine frames A ENT and B EXI are moved together ROTATE_GIRDER S2 ANG OMEGA ORDER BASIS PATCH PREC TRANSLATE_GIRDER S2 D ORDER BASIS PATCH PREC Misalignment Routines PTC has three fundamental misalignment routines related to a single magnet a siamese and a girder The single magnet and the siamese are defined with respect to their fibre po sition and are thus inherently similar The girder has a special frame of reference independent of the fibre If one is not careful a single magnet or a siamese misalignment may break the girder To avoid this problem we start with the girder misalignment MISALIGN_GIRDER MISALIGN_GIRDER S2 S1 OMEGA BASIS ADD To understand how the misalignments affect single magnets siamese and girders we examine the following example o girder example 1 girder after rotation and additive misalignment example 2 misalign siamese followed by misalign girder example 3 misalign girder followed by misalign siamese example 4 misalign siamese with PRESERVE_GIRDER true In these exam
36. any of the maintenance routines This allows the routine MOVE TO which locates an actual fibre to find its target faster In our case speed is not of great importance We tend to traverse a LAYOUT in the order of tracking 1 S BASED TRACKING Actual nodes of type fibre representing magnet number i Pointer to next node Pointer to previous node lt 2 Special nodes of type fibre Null Pointer at the start and the end of the list lt Linked cut in tracking a ring 5 or one sphere topology gt Linked replacing lt in the case of S Fibre The type FIBRE is recursively defined This allows the creation of a linked list One can think of a linked list as a chain data potentially hangs on each link In our case the fundamental datum is the object CHART of data type CHART This contains three actual charts affine frames of reference one at each end of the fibre and one in the middle TYPE FIBRE BELOW ARE THE DATA CARRIED BY THE NODE INTEGER POINTER DIR TYPE PATCH POINTER PATCH Figure B 1 A layout is a linked list of fibres 105 106 DATA TYPES TYPE CHART POINTER CHART TYPE ELEMENT POINTER MAG ELEMENTP POINTER END OF DATA POINTER TO THE MAGNETS ON EACH SIDE OF THIS NODE TYPE FIBRE POINTER PREVIOUS TYPE FIBRE POINTER NEXT POINTING TO PARENT LAYOUT AND PARENT FIBRE DATA TYPE LAYOUT POINTER PARE
37. append empty layout m u call set up m u send L6 gt m us send 65 Call build PSR L6 DNA sequence 6 We now have six DNA sequences or non trackable layouts in our DNA database Moreover each beamline element appears just once in that database The PTC type element contains an object doko from the Japanese word for where that enables PTC to keep track of which fibres point to a given beamline element This object doko allows PTC to use a given element multiple times in the database of trackable layouts We shall see several examples of this in the next section 3 5 MODELING COMPLEX ACCELERATOR TOPOLOGIES In the previous section we created the layouts in the left hand column of table 3 1 We now set about creating the layouts right hand column of table 3 1 for the three accelerator models shown in figure 3 2 the ring with forward and reverse propagation the figure eight lattice and the collider Complex accelerator topolo gies typically have a one to many correspondence between some of the beamline elements stored in the DNA sequences of the DNA database and the fibres in the trackable layouts By the end of this section you should understand how to set up such correspondences Ring with Forward and Reverse Propagation Here we model an accelerator that carries a pair of counter propagating beams fig ure 3 5 also middle lattice in figure 3 2 The two beams will propagate through dif ferent layouts PRS1 f
38. build Quad for Bend Except for some changes in the type declarations the code through line 595 in this subroutine is identical to that in the previous two subroutines The straight bend is created in lines 597 through 606 We first compute the length Lq as the chord length of the PSR s rectangular bend To achieve the desired straight reference frames we define the straight bend as a zero strength quadrupole Then to make this a dipole we set in line 602 the normalized dipole strength of b to the computed value b1 Finally we set some flags to give this magnet the fringe fields appropriate to a rectangular bend Query After the call to survey in line 611 where in the global frame are the 7 They lie all in a straight magnets that this subroutine defines 7 line one after the other starting at the global ori gin and extending toa POPULATING THE DNA DATABASE point 10Lq out along the z axis Making this lay out look like the right Using the subroutines described in the previous section we shall populate our hand layout of figure 3 3 database with six DNA sequences non trackable layouts L1 L2 L3 L4 L5 will require more work ang 6 Note that we shall want each element in our three accelerator models to appear once and only once in the DNA sequence portion of the DNA database This uniqueness will reflect the uniqueness of the individual beamline elements in our accelerators Layouts L1 and
39. case of switching to integration method 1 Figure 9 1 shows a pictorial example Method 2 cyan bars Method 2 blue bars Method 2 green lines between separate 4 steps indicate 4 kicks blue bars indicate 5 drifts Fi 1 Drift kick drift Method 1 red bars Method 1 blue bars Method 1 green lines between 2 separate 16 steps indicate 4 kicks red bars indicate 16 drifts and 2 The blue bars and green lines represent the original drift kick drift of integration method 2 The figure shows four kicks and five drifts There are four steps separated by the cyan bars The new integration method 1 split is simply a splitting of the drifts There are now 16 drifts but still four kicks Obviously the results should be the same to machine precision Example with the 36 degree bend of the PSR Los Alamos BE SBEND B 2 54948D0 TWOPI 36 D0 360 D0 X 0 001D0 CALL TRACK BE X DEFAULT WRITE 6 BE SMAG P METHOD 5 WRITE 6 X CALL RECUT KIND7 ONE BE 2 54948D0 16 FALSE X 0 001D0 CALL TRACK BE X DEFAULT 96 SYMPLECTIC INTEGRATION AND SPLITTING Figure 9 2 Matrix kick matrix for integration methods 2 4 and 6 WRITE 6 BE MAG P METHOD 6 5 6 X X 0 001D0 CALL ALLOC Y Y X CALL TRACK BE Y DEFAULT X Y WRITE 6 BE MAG P METHOD 6 5 WRITE 6 X PAUSE 888 Integration results are 2 4 962104445927857E 003
40. check need patch pl pl next 1 d 10 pos if pos 0 call find patch p1l pl next next false pl gt 1 end do Coll lower collider ring layout 10 Col2 upper collider ring layout 11 d zero d 3 40 d0 call translate L6 d a zero a 2 pi call rotate L5 L5 start chartxf a a call move_to L6 pl B pos d 1 L5 end chart f sb call translate L5 d call append empty layout m u Coll gt m_u end pl gt L start do i 1 L6 n call append_point Coll pl pl gt pl next end do write 6 Collider 1 has Coll n fibres 11 Collider 1 Collxclosed true call ring l Coll true make it a ring topologically call append empty layout m_u 12 gt pl gt L start next next do i 1 6 write 6 1 call append_point Col2 1 Col2 end dir 1 pl gt 1 5 119 120 PTC GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 end do pl gt L5 start 235 do i 1 L3 n call append point Col2 p1 pl gt pl next end do 240 write 6 Collider 2 has Col2 n fibres Col2 name Collider 2 Col2 closed true call ring_l Col2 true make it a ring topologically 245 pl gt Col2 start do i 1 Col2 n call check_need_patch pl pl next 1 d 10 pos if pos 0 call find_patch p1 pl next next false pl gt pl next 250 end do
41. col2 type fibre pointer p b f qf qd dl d2 1o integer i pos j mf example logical lp doneit real dp a 3 d 3 x 6 b0 mis 6 thin type real 8 y1 6 y2 6 real dp fix1 6 fix2 6 15 type normalform n1 n2 type damap id type internal state state type taylor eq 4 type gmap g 20 integer method limits 2 logical exact Lmax 100 d0 2 Use info true I user stuff one layout necessary before starting GUI call ptc ini append 130 PTC SPLITTING TUTORIAL SOURCE FILE ptc splitting f90 30 35 40 45 50 55 60 65 70 75 80 use_info true call append_empty_layout m_u number 1 call set_up m_u end rl gt m_u end exact false method drift_kick_drift call build PSR r1 exact method call move to rl qf gf pos call move 1 qd qd pos call move to rl b b pos first respliting example 1 thin 0 0140 limits 1 2 100000 call thin_lens_resplit rl thin lim limits write 6 qf smag amp name qf mag p method qf mag p nst write 6 qd smag amp name qd mag p method qd mag sp snst write 6 b mag name 0 5 pause 1 second respliting 11111 example 2 thin 0 01d0 limits 1 8 limits 2 24 call thin lens resplit rl1 thin lim limits write 6 qf smag amp name qf mag p method qf mag p nst write 6 qd smag amp name qd mag p method qd mag sp snst write
42. elements of PTC amenable to explicit symplectic integration 9 1 PHILOSOPHY PTC s philosophy for symplectic integration which is based on the work of Richard Talman involves six steps 1 Split the elements in the lattice into integration nodes using one of PTC s integration methods For a list of PTC s integration methods see Section 9 1 2 Fit all the stuff you would normally fit using your matching routines 3 Examine the resulting lattice functions and perhaps some short term dynamic aperture 4 If your results are satisfactory reduce the number of integration nodes the sophistication of the integration method or both Then go back to step 1 5 If your results are not satisfactory increase the number of integration nodes the sophistication of the integration method or both Then go back to step 1 6 After oscillating between steps 4 and 5 make up your mind and call that the lattice For your particular lattice store all the information at step 6 so that you do not have to repeat the process In this chapter we describe step 1 in detail The other steps depend on the results you want for your simulation Example 3 in Section gives an example the Talman algorithm of PTC code that performs these steps Integration Methods PTC has six integration methods In the drift kick drift D K D case PTC has three methods of integration method 2 the naive second order method which has one kick per integratio
43. fibre which rotates all the frames of the magnet If patch true then all the appearances of this magnet stored in DOKO are patched provided the norm of the patches is greater than PREC It is advisable to set PREC to a not too small number for example 10710 to avoid useless patches If there is no DNA that is if PTC runs in pure compatibility mode with standard codes then patches are on the parent fibre of the magnet TRANSLATE MAGNET This routine operates like ROTATE MAGNET described above TRANSLATE MAGNET R D ORDER BASIS PATCH PREC 8 2 AFFINE ROUTINES ON COMPUTER OBJECTS 75 Operations on Siamese For a discussion of siamese see 4 1 Siamese are tied together by the pointer SIAMESE which sits on fibre mag siamese To create a siamese structure we make a circular linked list of magnets For example suppose three DNA fibres f1 f2 and f3 must be tied together This can be done as follows 1 gt 2 f2 mag siamese gt f 3 mag f3 mag siamese gt f 1 mag Siamese Frame of Reference Because a siamese does not have its own frame of reference it is advisable to set up a so called affine_frame on the siamese In the above example one picks up any magnet of the siamese for example f1 mag and calls the routine CALL ALLOC_AF F1 MAG SIAMESE_FRAME CALL FIND_PATCH F1 CHART F A FI CHART FENT A ENT amp F1 MAG STAMESE_FRAME D F1 MAG STAMESE_FRAME ANGLE The siamese fra
44. fpp id fix2 closed orbit added to map track Coll yl 1 state unary activates knobs track Col2 y2 1 state yl normal forms abused of language permitted by ptc y2 normally one should do gt damap y normal form damap write 6 tunes 1 write 6 nl stune 1 2 write 6 tunes 2 write 6 n2 tune 1 2 eq 1 eq 2 nl dhdj v 1 0 254d0 1 2 0 255d0 121 122 GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 335 340 345 350 355 360 365 370 375 380 385 eq 3 n2 dhdj v 1 0 130d0 eq 4 n2 dhdj v 2 0 360d0 do 1 4 lt call kanalnummer mf eq txt do i 1 4 call daprint eq i mf end do close mf call 111 1 call kill y2 call kill id call kill n1 call kill n2 call kill eq call init 1 4 call alloc g 4 call kanalnummer mf eq txt do 1 4 call read g amp sv i mf end do close mf 9 g oo 1 tpsafit 1 4 g set_tpsafit true set_element true Collxdna 1 L qf 1 Coll dna 1 L qd 1 11 2 qf 2 11 2 qd 2 set_tpsafit set_element call kill g write 6 more read 5 i if i 1 goto 101 call 111 011 4 1 1 call kill 011 4 2 1 o c c n set up Siamese and Girder call move_to Coll pl
45. full square root Hamiltonian The next example shows how to set the default internal state environment with the NOCAVITY and EXACTMIS flags CALL MAKE STATES FALSE EXACT MODEL TRUE 102 INTERNAL STATES DEFAULT DEFAULT NOCAVITY EXACTMIS CALL UPDATE_STATES The following six flags are strictly related to TPSA calculations 1 If PARA_IN is specified TPSA knobs are included in the calculation It is activated by a unary on a state for example TRACK PSR Y 1 DEFAULT 2 If ONLY_4D is specified then neither path length nor time is a TPSA variable This means that the phase space dimension in the normal form will be 4 X Px Y Py Also X 5 will not be TPSA unless DELTA is also specified see next the item This flag has no effect on PTC tracking PTC always tracks six phase space variables 3 If DELTA is specified then ONLY_4D is also true However in this case X 5 is the fifth TPSA variable X Y Py plus Energy as the fifth variable The phase space dimension in the normal form will also be 4 momentum compaction cannot be computed 4 SPIN 5 SPIN ONLY 6 SPIN DIM This example shows how to set the default internal state environment with a four dimensional phase space in the normal form DEFAULT DEFAULT ONLY_4D The next example shows how to set the default internal state environment with a four dimensional phase space in the normal form and with energy as the fifth dimension TPSA var
46. if one uses the variables 2 3b Here x 1 p denotes the curvature defined by the geometry of the element and A denotes the longitudinal component of the vector potential Figure 2 6 indicates the variables including the longitudinal co rdinate s For straight elements of course 0 If one uses the variables 2 2 then there is effectively an additional term that subtracts the default path length or flight time across that element One may if desired have PTC track using an approximation of one of the Hamiltonians 2 4 i e using truncated expansions for the square root and the vector potential One might for example do this to speed the computation but of course the validity of the expansion for your particular problem should be verified with careful testing When integrating across an element described by one of the Hamiltonians 2 4 one typically splits the Hamiltonian into two integrable pieces If those pieces are for example the two terms of 2 42 then this is referred to as drift kick split Since the motion of a particle in a uniform dipole is integrable an alternative split involves writing A x y as a sum of two pieces the part that produces a uniform dipole field and everything else Then the dipole field part is added to the drift term of the Hamiltonian and we obtain a bend kick split One may ask PTC to use one or the other of these splitting methods STRUCTURES FOR PARTICLES AND BEAMs Bec
47. mag p method qf mag p nst qf tl pos qf tm pos qf t2 write 6 qd mag name qd mag p method qd mag p nst qd tl pos qd tm pos qd t2 write 6 b mag name mag p method mag p nst tl pos tm pos 1 2 150 write 6 T keyword example 7 resplit cutting call puts back method 2 and nst 1 everywhere thin 0 0140 call thin_lens_resplit rl thin even true lmax0 0 05d0 xbend 1 d 4 r 155 Call make node layout r1 write 6 a8 2x 5 i14 8x dl mag sname 1 8 1 41 0 5 11 1 write 6 a8 2x 5 i4 8x d2 mag sname 1 8 d2 mag p method d2 mag p nst d2 t write 6 a8 2x 5 i14 8x qf mag sname 1 8 qf mag p method qf mag p nst qf tl write 6 a8 2x 5 i4 8x qd mag name 1 8 qd mag p method qd mag p nst qd t 160 write 6 a8 2x 5 14 8x b mag name 1 8 mag p method mag p nst t 1 resplit_cutting call thin lens restart r1 puts back method 2 and nst 1 everywhere thin 0 01d0 call thin lens resplit rl1 thin even true lmax0 0 05d0 xbend 1 d 4 r 165 call make node layout r1 write 6 a8 2x 5 i14 8x dl mag sname 1 8 dl smagssp smethod 41 0 5 d1 t write 6 a8 2x 5 i4 8x d2 mag sname 1 8 d2 mag p method d2 mag p nst d2 t write 6 a8 2x 5 i14 8x qf mag sname 1 8 qf mag p method qf mag p nst qf tl write 6 a8 2x 5 i4 8x qd mag name 1 8 22 qd mag p n
48. n2 dhdj v 2 0 360d0 do 1 4 360 eq i lt c_ npara end do call kanalnummer mf eq txt do i 1 4 365 call daprint eq i mf end do close mf 370 375 380 385 390 call call call call call call call call call do i call read g v i mf kill y1 kill y2 kill id kill n1 kill n2 kill eq init 1 4 alloc g 4 kanalnummer mf eq txt 1 4 end do close mf g g oo 1 tpsafit 1 4 set tpsafit set element 11 dna 1 L Coll Coll 11 dna 1 L dna 2 L dna 2 L set_tpsafit set_element call kill g 9 true true qf 1 qd 1 qf 2 qd 2 false false 5 3 TUTORIAL EXAMPLE 55 We need to kill the knobs after we compute them the two calls to kill para kill the knobs in DNA layouts L5 and L6 395 read 5 i if i 1 goto 101 call kill_para Coll dna 1 l call kill_para Coll dna 2 l write 6 more SIX Computing Accelerator Properties This chapter explains how to compute global and local accelerator properties and provides examples of the code required for the computations 6 1 GLOBAL SCALARS Global scalars apply to the accelerator as a whole They can be derived only after the complete accelerator has been modeled and they are the same at any point on the closed orbit Tunes X zero x 5 delta call find orbit lattice x 1 istate 1 d 7 call init istate
49. or insert a new element A virtue of the linked list is that one may easily add or rearrange elements in the lattice However such a simple approach does not suffice when an accelerator topology reuses the magnets in different sequences different directions or both To see the difficulties associated with a recirculating beam consider Figure 2 8 which shows a machine similar to the Continuous Electron Beam Accelerator Facility CEBAF at Thomas Jefferson National Laboratory It illustrates the concept of particles circulating through a varying sequence of magnets The particles start out traveling through an injector from magnet M1 through magnet M4 During the 19 D P Barber K A Heine mann and G Ripken A canonical 8 dimensional formalism for classical spin orbit motion in stor age rings I A new pair of canonical spin vari ables Z Phys C 64 1 117 142 Mar 1994 DOI 10 1007 BFo1557243 1 R I McLachlan and G R W Quispel Splitting methods Acta Numer 11 341 434 Jan 2002 DOT 10 1017 50962492902000053 12 Forest Beam Dynam ics A New Attitude and Framework volume 8 of The Physics and Technology of Particle and Photon Beams Harwood Academic Pub lishers Amsterdam 1998 and Forest Geometric integration for particle accelerators J Phys A Math Gen 39 19 5321 5377 May 2006 DOI 10 1088 0305 4470 39 19 S03 M1 M8 M2 M7 M3 M6 M4 M5 Figure 2 7 Fo
50. p2 respectively to the 67 and fibres of layouts Coll and 12 Those two fibres contain the pair of elements we wish to group together as a siamese Each of those elements p1 mag and p2 mag contains an element pointer called siamese In lines 388 and 389 we now set each element s siamese pointing to the other element thus creating a circular linked list that ties these two elements together In a similar fashion the next four lines link together as a siamese the two bends at the right hand end of the common straight Those elements belong to the 4 and 7048 fibres respectively of layouts Coll and Col2 call move 011 pl 67 call move to Col2 p2 7 1 gt p2 mag 2 gt 1 390 Call move to Coll pl 4 call move to Col2 p2 70 1 gt p2 mag 2 gt 1 Our next goal is to group together onto a girder the above two siamese all the intervening elements shared by layouts Coll and 12 In addition we will add one more element to our girder the bend in the 14 fibre of layout Col2 see upper left of figure 4 4 We do this not because this example seems likely from an engineer s perspective but because we want to illustrate the flexibility of PTC s approach In 4 2 BUILDING SIAMESE GIRDERS AND THEIR REFERENCE FRAMES 45 particular we wish to emphasize the fact that elements tied to a common gir
51. probe temporal probe and temporal beam node layout including beam B 2 TIME BASED TRACKING 100 FIBRE SMAG P F O 3 FIBRE MAG P F MID 3 3 FIBRE SMAG P F EXI 3 3 FIBRE26MAG GP76F 6A 3 hos 75 3 2 Ly FIERE CHART P EXS 3 y y 3 3 FEATS A AEREACHARTAFYENT3 3 3 3 di zt p E Fi Plane attached to magnet Figure B 4 Misaligned d igure B 4 Misaligne LM ais IIS ii layout planar fibre in three dimen sions Integration Node Integration nodes allow us to look at any point in the layout One of the most important applications is to permit collective forces Moreover we can extend the coordinates of a particle to permit first order time based tracking within the higher order s framework To do so we add to two coordinates to the particle Z x px y P Y t p t 68 p_n The coordinate ds is the distance from the beginning of the integration node measured in the coordinate s used by the integrator The variable p 7 is a pointer to the integration node in which the particle finds itself at time 1 During normal s based tracking s is always zero In a time tracking mode a drift is assumed between nodes to estimate the time of a particle the result s is computed Since inverse drift
52. reference frame of one element and the entrance reference frame of the subsequent element As a consequence a patch is a property not of an element but of a fibre and it must be computed after placing both elements in their final locations As a concrete example consider the set of elements illustrated in figure 2 13 There we see three elements separated by a pair of drifts We know the locations of these elements in particular the reference frames attached to these elements because of the information stored in the corresponding charts The second drift is explicitly defined as the third beamline element The first drift however is not explicitly defined The exit frame of element 1 and the entrance frame of element 2 do not coincide and this indicates the need for a patch The remaining elements all have exit entrance frame pairs that coincide and hence no additional patches are required element 1 particle ele path ment 2 drift Patch required Patch tequireg Figure 2 13 Patching ele ments in a single beamline As another example of patching figure 2 14 shows three magnets in a recirculator A high energy particle passing through magnet follows the red trajectory to magnet 2 After exiting magnet 2 the particle continues around the machine and returns with a lower energy after deceleration by appropriately phased rf fields to magnet 1 Because the particle has less energy magnet 1 the separator
53. represents the full machine illustrated in figure 2 8 or 2 10 or 2 12 Evidently a trackable layout may be constructed from one or more DNA sequences 4 In general once we have populated the DNA database either from within PTC or by importing external lattice descriptions we can use PTC to string together these basic layouts to create the additional layouts we need for tracking These latter T1 2 2 MODELING ACCELERATOR TOPOLOGIES 15 through TN say contain fibres that point to elements in the DNA database For the recirculator shown in figure 2 8 we would create one beamline layout T1 with 26 fibres each pointing to an element in the DNA database For the collider shown in figure 2 9 we would create two beamline layouts T1 and T2 one layout for each ring Chart and Patch PTC uses the new data types chart and patch to locate connect and move the elements of a beamline The generic capability which we term patching connects the exit frame of one element to the entrance frame of the subsequent element The data type chart contains a pointer to a frame of reference actually the collection of three frames illustrated in figure 2 1 that locates a fibre really the element the fibre points to within PTC s global three dimensional reference frame The chart also contains information that describes if present any misalignment of the fibre The data type patch contains information about the about the relation between the exit
54. string of siamese and this permits automatic patching TRANSLATE_SIAMESE This routine functions like the above rotate_siamese with the same priorities concerning the optional variable BASIS TRANSLATE SIAMESE 52 0 ORDER BASIS PATCH PREC Operations on Girders For a discussion of girders see 4 1 To create a girder structure we make a circular linked list of magnets Let us create a girder structure with the three siamese f1 f2 and f3 above In addition let us put a single magnet f0 on the girder fO smags amp sgirder gt fl mag flxmag girder gt f2 mag f2 mag girder gt f3 mag f3 smags amp girder gt f0 mag Four magnets are on the girder three are in a siamese as well as on the girder Girder Frame of Reference Unlike a siamese a girder has a frame of reference For example we may elect to put the frame of reference on 70 CALL ALLOC_AF F0 MAG GIRDER_FRAME GIRDER TRUE Then we set the following two affine frames of the girder to the same value 8 2 AFFINE ROUTINES ON COMPUTER OBJECTS FO MAG GIRDER_FRAME ENT ENT FOQ MAG GIRDER_FRAME A FO MAG GIRDER_FRAME EXI ENT FO MAG GIRDER_FRAME B The affine frame A ENT can be any convenient frame chosen by the people who align the girder on the floor of the machine If a girder is misaligned the affine frame B EXI contains the new position of the girder If the misalignments are removed then B EXI coincides with A ENT This
55. the desired origin a and basis same as for the girder And the last two arguments are the computed translation and angle which we store in our siamese frame _ and b mag siamese_frame angle In a similar fashion the remaining lines in this block of code define the reference frame for the other siamese 415 a pl smag amp girder frame a a 3 a 3 5 40 call move to Coll b 67 call alloc_af b mag siamese_ frame call find_patch b magxp sf a b mag p f ent amp 420 a pl smag amp girder amp smag ssiamese 0 0 5 frame sangle pl smag sgirder a 3 a 3 5 d0 call move to Coll b 4 425 Call alloc af b smagssiamese frame call find_patch b magxp sf a b mag p f ent amp a pl smag amp girder amp smag ssiamese 0 0 5 frame sangle 4 3 EXAMPLES OF MISALIGNMENTS 47 4 3 EXAMPLES OF MISALIGNMENTS The remaining blocks of code in this chapter show some examples of misalignment operations on the girder and siamese with variations in their order and options In the margin are corresponding figures that illustrate the effect of the different misalignments For comparison figure 4 6 shows the same portion of the collider with no misalignments To make the effects clear we have made very exaggerated misalignments 71 8 22 5 for rotations and 2m for
56. to be compared with an actual layout It identities families or single elements using the name part of the name or the vorname first name of an element to make certain variables knobs The last name is the family for example QF type pol block character nlp name integer n name character vp vorname types for setting magnet using global array TPSAfit real dp dimension pointer TPSAfit logical lp pointer set TPSAfit logical lp pointer set element types for parameter dependence integer npara should not be used anymore knob index integer ian nmax ibn nmax real dp san nmax sbn nmax integer ivolt ifreq iphas integer ib sol scales for knobs 52 TAYLOR POLYMORPHISM AND KNOBS real dp svolt sfreq sphas real dp sb sol user defined functions type pol_block_sagan SAGAN end type pol_block Consider the following pol_block qf af 0 initialize the pol block qf name QF specify a family name qf ibn 2 1 set normal quad strength as first parameter If we call the routine scan for polymorphs R qf or qf then the DNA layout R is scanned If a polymorphic magnet on any fibre of the layout R is named QF then bn 2 becomes a knob In our example bn 2 bn 2 r qf sbn 2 X fpp npara qf ibn 2 bn 2 bn 2 r X fpp_npara 1 The index fpp_npara depends on the state as explained above One may also specify an in
57. top of one another The cell with the straight bend however requires patching regardless of direction or charge because the adjacent drifts have frames that are rotated with repect to those of the straight bend When you see the particu lar numbers in a few pages you may recognize this cell as that of the Los Alamos Proton Storage Ring 24 MODELING AN ACCELERATOR WITH PTC Figure 3 1 Basic cells for the three accelerator models Figure 3 2 Accelerator models Arrows indi cate the direction of mo tion of particle beams in the constituent layouts cell for figure eight and collider cell for ring with forward and reverse propagation The basic ring see the build_PSR example in figure 3 3 on page 26 has ten cells It has 7o fibres pointing to 70 elements The ring with forward and reverse propagation has 140 fibres pointing to 140 elements The figure eight and the collider each have 140 fibres pointing to 134 elements In each of those models the upper and lower rings share the elements at the start of one cell long drift quadrupole short drift and the elements at the end of another cell short drift quadrupole long drift figure eight Fig8 forward and reverse rings PSR1 and PSR2 collider Col1 and 12 Our three tutorial examples are not real accelerators They do not have particle injectors or dumps and two of the examples are not physically po
58. type universe PTC 25 UPDATE STATES routine 101 variable phase space 10 z ptc geometry f90 geometry tutorial source file 44 53 146 z ptc splitting f90 z ptc splitting f90 z ptc splitting f90 splitting tutorial source file 86
59. which one may define zin and Zout nothing more A beamline element is defined by a block with three reference frames together with a model or integrator which describes how to propagate a particle from the face with the entrance reference frame to the face with the exit reference frame element block model The model transforms the phase space variables from entrance to exit of the block according to 2 1 In other words it represents the physics of the device Note that the model also incorporates any approximations we make As an example a block with a bend geometry might actually be a simple drift keeping the transverse momenta invariant and changing only the positions Or it might be a composition of blocks describing the body of a magnet and its fringe field regions An element might also have non Hamiltonian effects such as radiation The list of possible models is endless In general a model is defined by one or more blocks internal to the model a model for each internal block the equation of motion in each internal block together with its integration method and associated number of integration steps IN SUMMARY we define a beamline element locally the definition whether the element is on a work bench or in an accelerator is determined solely by the characteristics of that element And once we have defined a model for our element we stick with it the LEGO block is inviolate Like a physical magnet it e
60. x normal A t gt latticesstart do j 1 lattice n call track lattice y j 1 default id 0 xt y disp xt id x xt 1 disp id disp xt disp xt eta x xt 6 2 S DEPENDENT GLOBAL QUANTITIES 59 dispx eta 1 sub 00001 dispy eta 3 sub 00001 write 6 a 2 2x f9 6 disp x disp y dispx dispy end do Phase Advance R Afi2 2 See caveat at end of 8 2 3 X zero x 5 delta call find orbit lattice x 1 istate 1 d 7 call init istate 1 0 call alloc y call alloc normal call alloc id id 1 y x id call track lattice y 1 istate normal y y X normal a_t theta_prev zero phi zero write 6 a 2 2x f9 6 CS phase adv phi 1 2 gt latticesstart do j 1 latticesn call track lattice y j 1 istate theta 1 atan2 y 1 sub 01 y 1 sub 10 twopi theta 2 atan2 y 3 sub 0001 y 3 sub 0010 twopi do k 1 2 if theta k lt zero and abs theta k gt tiny then theta k theta k one end if dphi k theta k theta_prev k M IN Ma Mn Figure 6 1 One turn and partial turn transfer maps Figure 6 2 This graphic illustrates the essential relationships between the one turn map and the normal form at two different locations in a ring lattice 60 COMPUTING ACCELERATOR PROPERT
61. 0 113 PATCH 15 107 PROBE_8 111 PROBE 110 TEMPORAL_BEAM 112 TEMPORAL_PROBE 21 112 THREE_D_INFO 64 pol_block 51 data types described 103 database see DNA database default global variable 101 delta flag 102 DNA see DNA array DNA database or DNA sequence DNA array described 40 DNA database m_u global variable 25 defined 14 example source code 26 populating 30 73 storing trackable layouts 40 DNA sequence defined 14 example source code 26 populating DNA database 30 73 DOKO pointer 73 doko described 32 36 37 73 drift defined 7 drift kick drift integration methods 85 dynamical group discussed 82 dynamical routines described 82 ELEMENT data type 13 element bend 7 element frame composed of integration nodes 20 defined 8 13 doko for multiple use 32 drift 7 misaligning 77 splitting into integration nodes 85 86 element frame see element reference frame element reference frame described 6 energy flag 102 patch 72 specifying 27 entrance frame see entrance reference frame entrance fringe field integration node 20 entrance patch integration node 20 entrance reference frame described 6 Euclidean group discussed 82 pseudo 83 exact_model global parameter 27 90 95 exactmis flag 101 exit frame see exit reference frame exit fringe field integration node 20 exit patch integration node 20 exit reference frame described 6 FIBRE data type 105
62. 0 pseudo Euclidean group discussed 83 defined 3 features 3 integrator 6 overview 5 source file 24 44 53 86 115 129 universe 25 ptc geometry f90 geometry tutorial source file 24 115 ptc splitting f90 splitting tutorial source file 129 radiation internal state 101 tracking routines on fibres 61 tracking routines on integration nodes 62 radiation flag 101 radiation bend split global parameter 86 recirculator recirculator example 11 patching beamlines 15 RECUT_KIND7_ONE routine 95 97 recutting see splitting reference frame described 6 patching 15 reference frames connecting 8 reference orbit see reference trajectory reference trajectory not used by PTC 6 Relativistic Heavy Ion Collider example based on 12 resplit_cutting global parameter 86 resplitting see splitting RF cavity internal state 101 RHIC see Relativistic Heavy Ion Col lider ring with forward and reverse propa gation modeling 23 32 rotate routine 33 35 ROTATE_FIBRE routine 74 ROTATE_ FRAME routine 69 ROTATE_GIRDER routine 77 ROTATE_LAYOUT routine 74 ROTATE_MAGNET routine 74 ROTATE SIAMESE routine 75 rotation order of 70 78 rotation routines described 69 74 routine ALLOC_AF 75 76 APPEND_EMPTY 73 APPEND_FIBRE 73 APPEND_POINT 73 CHANGE BASIS 70 CHECK NEED PATCH 72 COMPUTE ENTRANCE ANGLE 70 FIND ORBIT X 64 FIND ORBIT 54 FIND PATCH 72 75 GEO ROT 69 GEO
63. 000000000000000E 003 1 16 962104445927856E 003 000000000000000E 003 1 16 962104445927856E 003 000000000000000E 003 ORIGINAL METHOD AND 253543855340728E 003 000000000000000E 003 253543855340728E 003 000000000000000E 003 253543855340728E 003 000000000000000E 003 NUMBER OF STEPS 3 546933066933068E 003 2 523695791724136 003 3 546933066933066 003 2 523695791724136 003 3 546933066933066 003 2 523695791724136 003 Note that method 2 has four integration steps and method 1 has 16 integration steps A larger number of integration steps for method 1 does not change the accuracy Matrix Kick Matrix Integration methods 2 4 and 6 switching to methods 1 3 and 5 PTC has a matrix kick matrix method where the matrix is energy independent The delta dependence is buried in the kick Our comments here apply to exact_model false Integration methods 2 4 and 6 shown in figure 9 2 always split the integra tion step using matrices of equal length These methods are the so called biased integration methods Although they have existed in accelerators since the days of SSC recently they have been discussed at length by McLachlan and also by Laskar and Robutel Kick Kick Kick Kick Matrix Matrix Matrix Matrix Matrix B Matrix _ Matrix lll Matrix 1 6 1 6 7 90 12 90 7 90 2 3 32 90 32 90 Euler like Simpson like Bode like 9 4 OTHER SPLITTING ROUTINES 97 In accelerators integrati
64. 1 2 d0 call misalign_girder p2 mis add true elseif example 5 then Example 5 455 mis 0 d0 mis 1 2 d0 mis 5 pi 8 40 call misalign_girder p2 mis Figure 4 6 No misalign ment A 3 4 D Figure 4 7 Examples 1 top through 5 bottom 48 LINKING MAGNETS TOGETHER AND MOVING THEM AS A GROUP The examples above all misalign only the girder In the following examples we apply a misalignment also to one of the siamese Note that the difference between examples 7 and 8 is solely to the optional argument add in lines 470 and 478 elseif example 6 then Example 6 460 mis 0 40 mis 1 2 d0 call misalign_siamese p2 mis elseif example 7 then Example 7 mis 0 40 465 mis 1 2 d0 call misalign_siamese p2 mis mis 0 00 6 mis 1 2 d0 mis 5 pi 8 40 470 call misalign girder p2 mis add false elseif example 8 then Example 8 mis 0 d0 mis 1 2 40 call misalign_siamese p2 mis 7 475 mis 0 40 mis 1 2 d0 mis 5 pi 8 d0 call misalign_girder p2 mis add true elseif example 9 then Example 9 m S 480 mis 0 d0 mis 1 2 40 p n mis 5 8 00 call misalign_girder p2 mis mis 0 d0 485 mis 1 2 40 5 call misalign_siamese p2 mis add false elseif example 10 then Example 10 mis 0 40 ur mis 1 2 d0 490 mis 5 pi 8 d0 call misalign girder p2 mis mis 0 d0
65. 1 FIBRE2 NODE1 NODE2 TRACK NODE PROBE XS K FIBRE1 FIBRE2 NODE1 NODE2 These are all calls to the same routine The fibres and the nodes are actual pointers to the objects One turn can be done as follows 7 3 TRACKING ROUTINES ON 3 D INFORMATION THROUGH AN INTEGRATION NODE TRACK PROBE XS K NODE1 T NODE2 T PREVIOUS In the TRACK_PROBE_X routine the fibres and the nodes are pointers to the objects The routine wraps the TRACK_PROBE routine shown above TRACK PROBE X R X K U T FIBRE1 FIBRE2 NODE1 NODE2 Routines for Tracking either Real or Real 8 these routines wrap the above routines and therefore support radiation beam beam and s dependent apertures TRACK NODE X TRACK NODE X T X K TRACK PROBE X TRACK PROBE X R X K T FIBRE1 FIBRE2 NODE1 NODE2 U LOGICAL where TRUE indicates UNSTABLE optional T is a pointer to the fibre where the particle is lost optional TRACK BEAM This routine tracks a beam of particles TRACK BEAM R B K T FIBRE1 FIBRE2 NODE1 NODE2 For the type definition of BEAM see Section B 2 7 3 TRACKING ROUTINES ON 3 D INFORMATION THROUGH AN INTEGRATION NODE These routines track three dimensional information through an integration node Track node v The TRACK NODE V routine tracks a trajectory and records its three dimensional position for plotting at the beginning and the end of a node TRACK NODE V T V K REF 63 64 TRACKING ROUTINES T i
66. 16 2 3 Analyzing an Accelerator to Understand its Properties 16 Local versus Global Information 17 Polymorphs and Normal Form 18 2 4 Modeling Particle Interactions 19 Integration Node 19 Node Layout 20 Probe and Temporal Probe 20 Time based Tracking 21 3 Modeling an Accelerator with PTC 23 3 1 Accelerator Models 23 3 2 Geometry Tutorial Source File 24 Initial Code 25 3 3 Subroutines 26 build PSR 26 build PSR minus 28 build Quad for Bend 29 X CONTENTS 3 4 Populating the DNA Database 30 3 5 Modeling Complex Accelerator Topologies 32 Ring with Forward and Reverse Propagation 32 Figure Eight 35 Collider 38 3 6 DNA Arrays 40 Linking Magnets Together and Moving Them as a Group 43 4 1 Siamese and Girders 43 4 2 Building Siamese Girders and their Reference Frames 44 4 3 Examples of Misalignments 47 Taylor Polymorphism and Knobs 49 5 1 Polymorphs 49 States of a Polymorph 49 Computing a Taylor Map 49 5 2 Knobs 50 Using Knobs 50 Creating Knobs 51 Polymorphic Blocks 51 5 3 Tutorial Example 53 Computing Accelerator Properties 57 6 1 Global Scalars 57 Tunes 57 Chromaticity 57 Anharmonicity 57 6 2 s Dependent Global Quantities 57 Betatron Amplitude 57 Dispersion 58 Phase Advance 59 Beam Envelope 60 6 3 Local Quantities 60 Tracking Routines 61 7 1 Standard Tracking Routines on Fibres 61 Track 61 Find_orbit 62 7 2 Tracking Routines on Integration Nodes 62 Routines for Tracking either Probe or Probe_
67. 19 1 19 B 2 15 1 15 Table 9 1 Results of Exam ple 1 KF THIN is about 27 9 which is close to the 27 integration steps that result from Example 1 and KD THIN is about 19 7 which is close to the 19 integration steps that result from Example 1 Example 1 behaves as expected it splits according to quadrupole strength The method stayed 2 i e second order integrator This is not efficient if the number of steps must be large Therefore we must teach the splitting algorithm to switch to the fourth order or sixth order integrator We discuss how to do this in Section 9 3 In the bend we also look at the amount of natural horizontal focusing vertical focusing or both In this example a small ring with bends of 36 degrees the bends are unusually strong The global parameter radiation bend split and the optional argument xbend can be used to affect the bends 88 SYMPLECTIC INTEGRATION AND SPLITTING Table 9 2 Results of Example 2 LIM Optional Argument The optional array 1 2 determines when PTC should choose the second order fourth order or sixth order integration method lim 1 gt KL THIN second order method e lim 2 gt KL THIN gt lim 1 fourth order method e KL THIN gt lim 2 sixth order method Example 2 Let us rerun the previous case with a different limit array THIN 0 01D0 LIMITS 1 8 LIMITS 2 24 CALL THIN LENS RESPLIT R1 THIN LIM LIMITS WRITE 6
68. 35 Matching L3 to L4 36 Collider 38 A pair of elements linked together as a siamese 43 Incorrect and correct rotations of a siamese 43 43 44 45 4 6 47 6 1 6 2 A trio of elements linked together as a girder that has its own reference frame 44 Collider interaction region The numbers show the indices of a few of the fibres within the corresponding layout 44 Collider interaction region The small cross hairs indicate the frame location for each siamese 46 No misalignment 47 Examples 1 top through 5 bottom 47 Examples 6 top through 11 bottom 48 One turn and partial turn transfer maps 59 This graphic illustrates the essential relationships between the one turn map and the normal form at two different locations in a ring lattice 59 Rotating point a and vector basis v1 02 03 by R 67 Rotating and translating the frames of a magnet 70 Example o girder 78 Example 1 girder after 22 5 degree rotation 78 Example 1 girder after an additive misalignment 79 Example 2 misalign siamese followed by misalign girder 80 Example 3 misalign girder followed by misalign siamese 80 Example 4 misalign siamese with preserve_girder true 81 Pseudo Euclidean maps 84 Drift kick drift for integration methods 1 and 2 95 Matrix kick matrix for integration methods 2 4 and 6 96 A layout is a linked list of fibres 105 Three charts attached to an element 107 Misalignments for a element 108 Misalign
69. 5 0 40 in the siamese always returns the siamese to its girder position not to fibre position MISALIGN_FIBRE This routine behaves like the siamese routine but acts on a single fibre MISALIGN FIBRE S2 S1 OMEGA BASIS ADD PRESERVE_GIRDER Note If the MISALIGN_FIBRE routine is used on a siamese it mildly breaks the siamese One can imagine very tiny errors internal to a siamese and bigger errors on the siamese and yet bigger errors on a girder containing siamese and individual elements Therefore the following sequence of commands is acceptable if B is an element part that is of a siamese and part of a girder CALL MISALIGN_FIBRE B MIS1 CALL MISALIGN_SIAMESE B MIS2 ADD TRUE CALL MISALIGN_GIRDER B MIS3 ADD TRUE Here we would imagine MIS1 lt MIS2 lt MIS3 Notice that ADD TRUE on a siamese does not break a girder 82 GEOMETRIC ROUTINES CALL MISALIGN_GIRDER B MIS3 CALL MISALIGN_SIAMESE B MIS2 ADD FALSE PRESERVE_GIRDER TRUE CALL MISALIGN_FIBRE B MIS1 ADD TRUE 8 3 DYNAMICAL ROUTINES Dynamical routines perform geometric rotations and translations which act on the tracked object itself PTC has an exact dynamical group and an approximate dynamical group It is remarkable that the approximate dynamical rotations and translations also form a group but it is not isomorphic to the affine Euclidean group The ultimate goal of PTC is to propagate particles and maps thanks to Taylor po
70. 6 write 6 sbend orbit small problem example 5 15 Write 6 x write 6 write 6 now reducing the number of steps and refitting with xbend 1 d 4 write 6 x do i 0 2 120 thin 0 01d0 ix 0 03 call thin_lens_resplit r1 thin xbend 1 d 4 reducing number of cuts call read_ptc_command77 fit_to_betaO_results_2 txt fitting to previous tunes call read ptc command77 fill betal 2 txt computing dbeta beta around the ring end do 125 pause 5 write 6 even example 6 call move 1 d1 dl pos call move 1 42 d2 pos 130 Call move 1 qf qf pos call move to rl qd pos 132 SPLITTING TUTORIAL SOURCE FILE ptc splitting f90 call move to rl b b pos call thin lens restart r1 puts back method 2 and nst 1 everywhere 135 thin 0 01d0 call thin lens resplit rl thin even true xbend 1 d 4 reducing number call make node layout r1 write 6 qf smag amp name qf mag p method qf mag p nst qf tl pos qf tm pos qf t2 write 6 qd mag name qd mag p method qd mag p nst qd tl pos qd tm pos qd t2 140 write 6 b mag name mag p method mag p nst tl pos tm pos 1 2 call thin lens restart r1 puts back method 2 and nst 1 everywhere thin 0 0140 call thin_lens_resplit r1 thin lim limits xbend 1 d 4 reducing number of 145 Call make node layout r1 write 6 qf smag amp name qf
71. 6 Collider 1 has Coll n fibres 11 Collider 1 Coll closed true call ring L Coll true make it a ring topologically In the next block of code we construct layout 12 We call append empty layout again on m_u set the layout pointer 12 to point to this new layout and then locate in line 227 the first short drift in layout L6 To populate 12 we now in lines 228 233 march backwards appending the six elements in the straight section at the top of layout L6 short drift defocusing quadrupole two long drifts focusing quadrupole and short drift While doing this we inform PTC in line 231 that this layout traverses those elements in the opposite direction Finally in lines 234 238 we append in order all the fibres of L5 to Col2 22 Call append empty layout m u Col2 gt m usend pl gt L start next next do i 1 6 write 6 1 230 call append point Col2 pl Col2 end dir 1 pl gt pl previous end do pl gt L5 start 235 do i 1 L5 n call append_point Col2 1 pl gt pl next end do In this last block of code for Col2 we give our new layout a formal name Collider 2 ensure that it is topologically closed and apply any necessary patches This layout requires patches only where fibre dir switches sign As in our earlier layout Fig8 this happens at the two ends of the common straight section where they join the bends of the upper ring 240
72. 67 call move_to Col2 p2 7 1 gt p2 mag PTC GEOMETRY TUTORIAL SOURCE FILE ptc_geometry f90 123 2 gt 1 call move_to Coll pl 4 call move_to Col2 p2 70 1 gt p2 mag 390 2 gt pl smag call move to Coll pl 68 f gt pl remember start of girder linked list do i 2 7 395 p2 gt pl next pl mag girders gt p2 mag pl gt pl next end do call move to Col2 p2 7 pl mag girders gt 1 plxmag siamese girders gt p2 mag p2 mag girders gt 2 call move_to Coll pl 67 call move_to Col2 p2 14 405 pl mag girders gt p2 mag p2 mag girders gt f mag call move_to Coll pl 1 call alloc_af pl mag girder_frame girder true 410 1 _ pl smag sparent_fibreschart f ent pl mag girder_frame a pl mag parent_fibreschart f a plxmag girder_framesexi pl smags sparent fibresscharts sf sent plxmag girder_framesb pl magsparent_fibre schartxf a 415 a pl magsgirder_frame xa a 3 a 3 5 d0 call move_to Coll b 67 call alloc_af b mag siamese_frame call find patch b smagsspssf sa b mag p f ent amp 420 a pl smag amp girder amp _ b smag ssiamese frame sangle pl smag amp sgirder a 3 a 3 5 d0 call move to Coll b 4 425
73. 8 62 Routines for Tracking either Real or Real_8 63 7 3 Tracking Routines on 3 D Information through an Integration Node 63 Track node v 63 7 4 lime based Tracking Routines 64 Track time 64 Track temporal beam 64 7 5 Closed Orbit Routine 64 Find orbit x 64 Geometric Routines 67 8 1 Affine Routines on Pure Geometry 67 Theory 67 Descriptions of the Routines 69 8 2 Affine Routines on Computer Objects 71 Affine Routines on Fibrous Structures 72 Misalignment Routines 77 8 3 Dynamical Routines 82 Exact Patching and Exact Misalignments Dynamical Group 82 Inexact Patching and Exact Misalignments 83 Symplectic Integration and Splitting 85 9 1 Philosophy 85 Integration Methods 85 9 2 Splitting Tutorial Source File 86 CONTENTS Xi 9 3 Splitting the Lattice 86 Global Parameters 86 Splitting Routines 86 Arguments 87 9 4 Other Splitting Routines 95 Splitting a Single Fibre 95 Splitting an Entire Lattice 97 Appendices 99 A Internal States 101 B Data Types 103 B 1 S based Tracking 103 Layout 103 Fibre 105 Chart 106 Patch 107 B 2 Time based Tracking 108 Integration Node 109 Node Layout 113 PTC Geometry Tutorial Source File ptc geometry f90 115 D PTC Splitting Tutorial Source File ptc splitting f90 129 Bibliography 135 Index 137 List of Figures 2 9 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 21 2 12 2 19 2 14 2 15 2 10 2 17 3d 8 2 3 3 34 3 5 3 6 3 7 4 1 4 2 LEGO bloc
74. AGNET FRAME TYPE MAGNET FRAME POINTER DIMENSION POINTER DIMENSION ENT 1 S BASED TRACKING REAL DP POINTER DIMENSION 0 REAL DP POINTER DIMENSION MID REAL DP POINTER DIMENSION B REAL DP POINTER DIMENSION EXI END TYPE MAGNET_ FRAME Figure B 2 shows the three charts affine frames of reference attached to an element Middle Chart Entrance Chart ENT 3 MID 1 3 Exit Chart ENT 1 3 EXI 1 3 A 3 M lt 3 EXI 3 3 LengthL 7 MID 3 3 O 3 Element Particle Particle motion motion The variables A ENT 0 MID and B EXI define the three charts The vari ables L and ALPHA characterize the gray plane of the magnet At the center of this plane lies the chart at This is where a magnet is compressed for misalignment purposes The TRACK routine sends a ray from the cyan entrance chart A ENT through the middle chart 0 MIS to the green exit chart B EXI Misalignment The CHART contains the misalignment information for a fibre which of course points to an element The variables D IN ANG IN and D OUT ANG OUT are the displacements and rota tions at the entrance and exit of the fibre In figure B 3 the purple arrows represent D IN and D OUT The dotted purple arcs represent the angles ANG IN and ANG OUT Figure B 4 provides a three dimensional view of a misaligned planar fibre The p
75. C when it hands a particle from one element to the next allow PTC to 2 1 TRACKING PARTICLES THROUGH AN ACCELERATOR 7 handle arbitrary changes in the placement of beamline elements On entering an element a particle has a certain phase space location with respect to the entrance frame PTC tracks the particle as it traverses the element computing the particle s phase space coordinates with respect to the element s exit frame Knowing the relation between the exit frame of the current element and the entrance frame of the subsequent element PTC uses the transformations of the dynamical Euclidean group to hand the particle to the next element Particle Motion Reference Frame Local Coordinate System for Element Particle d Motion Entrance Reference Frame The beamline element represented in figure 2 1 is a drift a block with parallel entrance and exit faces to which local Cartesian reference frames are attached The entrance and exit reference frames have the same unit vectors The line that links the two frames has a length L and it is perpendicular to both those frames Note that for the purposes of this discussion any beamline element quadrupole sextupole solenoid etc that does not bend the design orbit is a drift A LEGO block element might also be a bend a block with two faces that have parallel y axes vertical and x axes horizontal that meet at an angle 0 The two x axes and the line joining the two ori
76. EOMETRY TUTORIAL SOURCE FILE ptc geometry f90 read bends from L2 call append point PSR1 p2 f gt PSR1 end d 1 f chart f o call translate f d call compute entrance angle f schartssf mid pl chart f mid a call rotate f a basis f chart f mid p2 gt 2 1 lt call append_point PSR1 p1 end if pl gt pl next end do elements in PSR1 now in correct locations f gt PSR1 start do i 1 PSR1 n if f mag name B QUAD then call find patch fssprevious f next true call find patch f f next next false end if f gt f next end do PSR1 now patched 1 PSR 1 PSR1 closed true call ring_L PSR1 true make it a ring topologically PSR2 backward ring layout 8 call append empty layout m u PSR2 gt m pl gt Ll xend p2 gt L2 end do i 1 11 if pl smag amp name B then call append_point PSR2 p2 p2 gt p2 previous else call append point PSR2 1 end if f gt PSR2 end dir 1 charge 1 pl gt pl previous end do f gt PSR2 start do i 1 PSR2 n if f mag name B QUAD then call find patch fssprevious f next true 117 118 PTC GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 135 140 145 150 155 160 165 170 175 180 call find_patch f f next next false end if f gt f next end do PSR2 na
77. FIND PATCH EL1 EL2 NEXT NEXT ENERGY PATCH PRECi EL1 and EL2 NEXT PREC is a small real dp number If the norm of the geometric patch is smaller than PREC then the patch is ignored If energy patch is true then it compares the design momenta of the magnets in both fibres If the magnitude of difference is greater than PREC then an energy patch is put on CHECK NEED PATCH This routine checks whether a patch is needed without actually applying it to the layout The routine returns the integer PATCH NEEDED If zero it is not needed CHECK NEED PATCH EL1 EL2 NEXT PREC PATCH NEEDED Fibre Content Before going any further we remind the reader of the frames contained within a fibre e the frames of the fibre itself located in type chart that is fibre chart f the frames of the magnet fibre mag p f and its polymorphic twin e the frames of the integration nodes associated with this fibre magnet if they are present the frame of a girder that might be tagged on this magnet 8 2 AFFINE ROUTINES ON COMPUTER OBJECTS Subroutines Invoking the Magnet and Relying on the DNA When complex structures are constructed in PTC it is preferable to follow a strict discipline First various layouts with no cloning of magnets are created in a mad universe of data type MAD_UNIVERSE The code MAD X calls this universe M_U These no clone layouts contain the actual magnet database of the accelerator complex Therefore we
78. IES if dphi k lt zero and abs dphi k gt tiny then dphi k dphi k one end if phi k phi k dphi k end do theta_prev theta write 6 a 2 2x f9 6 CS phase adv phi 1 2 p gt Beam Envelope Text and example code here 6 3 LOCAL QUANTITIES Local properties do not apply to the accelerator as a whole They are derived from individual magnets and they differ at different points on the closed orbit The trajectory of a particle through a magnet is local it is derivable from the individual magnet irrespective of the magnet s position in the accelerator SEVEN Tracking Routines PTC s tracking routines are divided into four categories standard tracking routines on fibres tracking routines on integration nodes tracking routines on 3 D information through an integration node time based tracking routines A fifth section documents the closed orbit routine Mandatory arguments for the tracking routines are in regular black type Op tional arguments in red italic type Positions are normally specified by I1 I2 fibrel fibre2 nodel or node2 Generally if only x1 is present x I fibre or node this produces a one turn map from position x1 back to position x1 if the layout is closed If the layout is opened then it goes to the end of the line If x1 and x2 are present the routine tracks through 1 all the way to the front of x2 x2 not i
79. L2 will provide the elements for the concentric rings accelerator trackable layouts PSR1 and PSR2 See figure 3 2 Layouts L3 and L4 will provide the elements for the figure eight accelerator trackable layout Fig8 And layouts L5 and L6 will provide the elements for the collider trackable layouts COL1 and 3 4 POPULATING THE DNA DATABASE 31 Non trackable layouts Trackable layouts L1 70 elements build_PSR PSR1 created from L1 and L2 L2 10 elements build Quad for Bend PSR2 created from L1 and L2 L3 64 elements build PSR minus Fig8 created from L3 and L4 L4 70 elements build PSR Coll created from L6 L5 64 elements build PSR minus Col2 created from L5 and L6 L6 70 elements build PSR COL2 Table 3 1 summarizes the layouts we plan to create six DNA sequences or non trackable layouts and five trackable layouts Query As initially constructed by build PSR layouts L1 L4 and L6 are identical So also are layouts L3 and L5 Why don t we avoid this duplication The following four lines create the first DNA sequence call append empty layout m u DNA sequence 1 call set up m u send L1 gt m_u end 4o Call build_PSR L1 The call to append_empty_layout appends an empty layout to the global linked list of layouts m_u The layout pointer m_u end points to this empty layout The second line initializes this new layout and the third line makes L1 point to it Finally the call to build_PSR populates la
80. LES THROUGH AN ACCELERATOR 9 Local Coordinate System for Element 2 Particle Motion Exit Reference Frame Element Sig Reference bay Frame k UN 8 Entrance Reference J Frame D Reference Frame Element p Reference Frame Local Coordinate Particle System for Element 1 Motion R4 Coordinate System Entrance Reference Frame the blocks faces must be parallel and the frames on the two faces must line up Connecting a horizontal bend with a horizontal drift as for elements By and D in figure 2 5 is easy because the x and y axes of the two elements match Connecting a horizontal drift with a vertical bend as for elements D and By in figure 2 5 is not as straight forward their local x and y axes do not match We need in essence a new type of LEGO block one with an x y rotation of angle In figure 2 5 of course 90 To build an arbitrary accelerator we shall require a full complement of such additional LEGO blocks This is the subject of patching which uses geometric transformations to connect the exit frame of one beamline element to the entrance frame of a subsequent element Most significantly patching enables us to position beamline elements wherever we want them We discuss this in more detail later in Chart and Patch page 15 Particle Tracking Now that we have the basic building blocks and the geometric transformations to fit them together we can
81. MAXo Optional Argument The Lmax0 optional argument must be used in conjunction with the resplit cutting global parameter set to 1 or 2 The resplit cutting global parameter is normally defaulted to zero If resplit_cutting 1 the ordinary magnets are left as is However the drifts are split so that the maximum length of an integration node cannot exceed 1 Example 7 call move to rl1 d1 D1 pos call move to rl d1l next DI1 pos call move to r1 d2 1 Put the pointer for search back at position 1 call move to r1 d2 D2 pos call move to rl qf QF pos call move to rl1 qd QD pos call move to rl b B pos resplit_cutting 1 call THIN LENS restart r1 thin 0 01d0 CALL THIN_LENS_resplit R1 THIN even true lmax0 0 05d0 xbend 1 d 4 Table 9 6 shows the results of the above run 9 3 SPLITTING THE LATTICE 93 Name Method Steps 1 2 D1 2 46 1 26 102 D1 2 46 103 26 152 D2 2 10 57 64 70 QF 6 4 95 99 102 QD 6 2 203 206 208 Table 9 6 Results B 4 6 223 228 232 of Example 7 for resplit_cutting 1 Notice that only the drifts D1 and D2 were split If resplit_cutting 2 then all the magnets and the drifts are split to achieve a maximum length of 1 This is useful in space charge calculations See table 9 7 Name Method Steps 1 2 D1 2 46 1 26 166 D1 2 46 167 26 216 D2 2 10 67 74 80 QF 6 12 151 159 166 QD 6 12 267 275 282
82. MUM OF DBETA BETA 6 034509389788590E 006 The results for THIN 0 04 are lt DBETA BETA gt 2 065365900468319E 003 MAXIMUM OF DBETA BETA 4 014532098810251E 003 The results for THIN 0 07 are lt DBETA BETA gt 5 779446473894797 003 MAXIMUM OF DBETA BETA 1 068563516341058 002 One can also look at the chromaticities For example the chromaticities at the beginning were 2 87009384223620 2 03916389491785 At the end the chromaticities are 2 86244921970493 2 03671376872069 At this point we can freeze the lattice for good Of course refitting the tune was only an example in other lattices we may want to rematch a more complete set of properties dispersion alphas phase advances etc XBEND Optional Argument This section discusses usage of the XBEND optional argument when exact true The lattice uses sector bends Because this is an ideal lattice the tune should be the same regardless of whether with exact model true or false Therefore we will fit to the same tune as before passing it through the same algorithm Example 4 WRITE 6 WRITE 6 x SBEND ORBIT SMALL PROBLEM EXAMPLE 4 WRITE 6 CALL APPEND_EMPTY_LAYOUT M_U NUMBER 2 CALL SET UP M U SsEND R1 gt M_U END EXACT TRUE METHOD DRIFT KICK DRIFT CALL RUN_PSR R1 EXACT METHOD WRITE 6 6 NOW REDUCING THE NUMBER OF STEPS AND REFITTING WRITE 6 DO I 0 2 THIN
83. NT 3 3 POC TOTAL TIME INTEGER N TYPE INTEGRATION NODE POINTER C POINTER CLOSE TO A 3 TYPE INTERNAL STATE STATE END TYPE TEMPORAL A temporal beam must be allocated CALL ALLOC TB N POC POC is the reference momentum of the beam We must set the initial conditions of the layout CALL POSITION_TEMPORAL_BEAM LAYOUT TB STATE The subroutine POSITION TEMPORAL BEAM locates the beam in three dimensions and stores TBSTP i 9 P0S 1 3 2 X Y Z which is PTC s global frame calls the LOCATE TEMPORAL BEAM routine which in turn calls the ORIGINAL P TO PTC routine which adjusts the momenta The beam s values of TB TP i XS X 1 6 are now in PTC s local coordinates 2 TIME BASED TRACKING 113 Node Layout The data type NODE_LAYOUT is a linked list of nodes which represents an expanded beam line The nodes do not contain new data their fibres contain their data The type NODE_LAYOUT is defined as follows TYPE NODE_LAYOUT CHARACTER 120 POINTER NAME IDENTIFICATION INTEGER POINTER INDEX IDENTIFICATION LOGICAL LP POINTER CLOSED INTEGER POINTER N TOTAL ELEMENT IN THE CHAIN POINTERS OF LINK LAYOUT INTEGER POINTER LASTPOS POSITION OF LAST VISITED TYPE INTEGRATION NODE POINTER LAST LAST VISITED INTEGRATION_NODE POINTER END TYPE INTEGRATION_NODE POINTER START TYPE INTEGRATION_NODE POINTER START_GROUND STORE THE GROUNDED VALUE OF START
84. NT LAYOUT TYPE INFO POINTER I TYPE INTEGRATION NODE POINTER T1 T2 FIRST AND LAST INTEGRATION NODE CHILDREN CORRESPOUNDING TO PATCHES TYPE INTEGRATION NODE POINTER TM MIDDLE INTEGRATION NODE INTEGER POINTER 05 POSITION IN LAYOUT NEW STUFF POINTER BETAO GAMMAOI GAMBET MASS POC INTEGER POINTER CHARGE REAL dDP POINTER AG TO TIE LAYOUTS TYPE FIBRE POINTER P TYPE FIBRE POINTER N INTEGER POINTER LOC END TYPE FIBRE The type definition includes the beam element MAG and its polymorphic version MAGP MAG is the generic magnet to which we attach a single particle propagator MAGP is almost a carbon copy of MAG The integer DIR defines the direction of propagation through the fibre We can enter the magnet either from the front or from the back POC and BETAO define a preferential frame of reference for the energy of this fibre It is usually the same as the energy of the element MAG T1 and T2 are pointers to the first and last integration node children The integer POS locates the fibre in the node layout See figure B 5 Chart Type CHART contains the information that locates an element in three dimensional space at the position where we want it to be TYPE CHART FRAME POINTER F FIBRE MISALIGNMENTS REAL DP DIMENSION POINTER D_IN ANG_IN REAL DP DIMENSION POINTER D OUT ANG_OUT END TYPE CHART CHART refers to the data type M
85. PTC and the motivation behind many of the decisions that went into its design Many of the examples in this manual are based on code he provided Al Kemp of Impact Technical Publications helped with the overall organisation and the initial draft In addition several of my colleagues at Tech X commented on various parts of this document George Bell Andrey Sobol and David Bruhwiler Very special thanks go to Desmond Barber for his support encouragement and detailed reviews of the manuscript His keen ear for the English language broad knowledge of accelerator physics and consistent willingness to help have improved this document in more ways than I can possibly note I am deeply in debt to him Financial support for the writing of this document was provided in part by the U S Department of Energy Office of Science Office of Nuclear Physics under SBIR Grant No DE FGo2 06ER84508 Introduction 11 PTC FPP The Polymorphic Tracking Code PTC is a library of ForTRAN90 data structures and subroutines for integrating the equations of orbital and spin motion for particles in modern accelerators and storage rings The data structures hold the material that we will mould into physically correct and consistent three dimensional com puter models of the complex topologies found in machines such as colliders and recirculating linacs including the effects of errors in both location and strength of th
86. PTC provides and we illustrate their use by applying them to the collider layouts 11 and 12 of the previous chapter 4 1 SIAMESE AND GIRDERS A siamese consists of elements that are linked together so that one can move them as a group Those elements may be in the same or different layouts The elements in a siamese are often parallel as in a collider but they may be in different arcs of a recirculator arcs for example that share common cryogenics To create a siamese we build a circular linked list containing the several elements we wish to tie together See figure 4 1 When moving a siamese PTC traverses the linked list moving all the siamese elements in concert Note that doing this properly i e preserving the geometric relations between the siamese elements requires the use of a common reference frame Figure 4 2 for example illustrates incorrect and correct rotations of a pair of elements linked together as a siamese In the left hand graphic of that figure the same rotation applied to the separate elements breaks the geometry of the siamese In the right hand graphic the use of a common reference frame when rotating the elements preserves the geometry When we link the elements together as a siamese and then ask PTC to rotate the siamese as opposed to the individual elements P TC takes care of the details and preserves the internal geometric constraints We discuss geometric operations applied to siamese later in
87. QF MAG P METHOD QF MAG P NST WRITE 6 QD MAG NAME QD MAG P METHOD QD MAG P NST Note that KF THIN which is about 27 9 is greater than lim 2 which is 24 KD THIN which is about 19 7 is less than 1 2 Both are greater than lim 1 Table 9 2 shows the results Element Method Steps Kicks Step Total Kicks QF 6 3 7 21 QD 4 6 3 18 B 4 5 3 15 PTC uses integration method 6 for element QF which results in three integration steps with seven kicks per step a total of 21 kicks PTC uses integration method 4 for element QD which results in six integration steps with three kicks per step a total of 18 kicks As we can see the number of kicks has remained more or less constant while the integration method has increased in order Example 3 This PTC code is an example of the six step PTC philosophy for splitting see 9 1 which we are calling the Talman algorithm The example reuses Example 2 s limit array TALMAN ALGORITHM EXAMPLE WRITE 6 WRITE 6 TALMAN ALGORITHM EXAMPLE 3 WRITE 6 PART 1A 9 3 SPLITTING THE LATTICE 89 THIN 0 001D0 CUT LIKE CRAZY CALL THIN_LENS_RESPLIT R1 THIN LIM LIMITS PTC COMMAND FILE COULD BE A MAD X COMMAND OR WHATEVER COMPUTING TUNE AND BETA AROUND THE RING PART 1B CALL READ COMMAND77 FILL_BETAO TXT WRITE 6 WRITE 6 NOW REDUCING THE NUMBER OF STEPS AND REFITTING WRITE 6
88. SOCIATED CHILD THIN LENS LAYOUT MAD_UNIVERSE POINTER PARENT UNIVERSE TYPE LAYOUT ARRAY POINTER DNA TYPE SIAMESE ARRAY POINTER GIRDER TYPE JUNCTION ARRAY POINTER CON2 END TYPE LAYOUT The first lines of the TYPE definition contain data specific to the layout itself data concerning steps of integration statistics Besides the fibres the two most important quantities in this layout are N and CLOSED The variable N is the number of fibres in the layout N is updated each time magnets are inserted or deleted The boolean CLOSED refers to the topology of the so called base space or in accelerator parlance the s variable Is the variable s periodic a ring or is the variable s defining an interval a straight beam line In the case of a ring CLOSED is set to true otherwise it is set to false This does not happen automatically the user must set it to the desired value The pointer T points to the layout s its associated NODE LAYOUT This pointer is normally grounded because PTC does not systematically create node layouts or thin layouts When not grounded the pointer contains the expanded representation displayed on the left side of figure B 5 Figure B 1 illustrates the roles of the pointer variables The figure shows a layout with four elements Obviously the number of elements in an actual beam line could be enormous First of all a linked list must have at least one pointer this allows the cr
89. SR2 Fig8 Coll Col2 15 type fibre pointer pl p2 b f type internal state state type pol block qf 2 qd 2 type normalform n1 n2 20 type damap id type taylor eq 4 type gmap g 25 Lmax 100 40 use_info true user stuff one layout necessary before starting GUI 116 PTC GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 30 35 40 45 50 55 60 65 70 75 call ptc_ini_no_append thin 1 call thin_lens_resplit m_u end thin xbend 1 d 10 call append_empty_layout m_u DNA sequence 1 call set_up m_u end L1 gt m_u end call build_PSR L1 call append_empty_layout m_u DNA sequence 2 call set_up m_u end L2 gt m_u end call build_Quad_for_Bend L2 call append_empty_layout m_u DNA sequence 3 call set_up m_u end L3 gt m_u end call build_PSR_minus L3 call append_empty_layout m_u DNA sequence 4 call set_up m_u end L4 gt m_u end call build_PSR L4 call append_empty_layout m_u DNA sequence 5 call set_up m_u end L5 gt m_u end call build_PSR_minus L5 call append_empty_layout m_u DNA sequence 6 call set_up m_u end L6 gt m_u end call build_PSR L6 PSR1 forward ring layout 7 call append empty layout m_u PSR1 gt m_u end pl gt 1155 p2 gt L2 start do i 1 11 if pl mag name B then 80 85 90 95 100 105 110 115 120 125 130 PTC G
90. UIDE Where to Start Reading The Overview of PTC chapter 2 presents concepts important for understanding PTC In depth discussions of all these concepts can be found in several publications by Etienne Forest Etienne Forest A Hamiltonian free description of single particle dynamics for hopelessly complex periodic systems J Math Phys 31 5 1133 1144 May 1990 DOT 10 1063 1 528795 Etienne Forest and Kohji Hirata A contemporary guide to beam dynamics Technical Report KEK 92 12 KEK Tsukuba Japan August 1992 Etienne Forest Locally accurate dynamical Euclidean group Phys Rev E 55 4 4665 4674 April 1997 DOT 10 1103 PhysRevE 55 4665 Etienne Forest Beam Dynamics A New Attitude and Framework volume 8 of The Physics and Technology of Particle and Photon Beams Harwood Academic Publishers Amsterdam 1998 Etienne Forest Geometric integration for particle accelerators J Phys A Math Gen 39 19 5321 5377 May 2006 10 1088 0305 4470 39 19 503 If you want to get started right away begin with chapter 3 Modeling an Accelerator with PTC and refer back to the overview as needed while working with the tutorial examples Overview of PTC This overview describes the PTC approach to simulating and analyzing particle accelerators At the heart of PTC s accelerator simulation code is an integrator capable of tracking particles through various kinds of beamli
91. Write 6 x Collider 2 has Col2 n fibres Col2 name Collider 2 Col2 closed true call ring_L Col2 true make it a ring topologically 245 Pl gt Col2 start do i 1 Col2 n 40 MODELING AN ACCELERATOR WITH PTC call check need patch pl pl next 1 d 10 pos if pos 0 call find patch pl pl next next false pl gt pl next 250 end do The trackable layouts Coll and Col2 are now complete They have 140 fibres pointing to 134 elements in DNA sequences L6 and L5 Twelve of the fibres in Col1 and Col2 point to the six elements which are common to the upper and lower rings 3 6 DNA ARRAYS The following code constitutes a bit of housekeeping We have created a set of six layouts L1 L6 that form the core of our DNA database What we do here is tell each of the trackable layouts we have created PSR1 PSR2 Fig8 Coll and Col2 which DNA sequences they use This information is stored in the array DNA that is part of the data held in each PTC ayout In line 258 we record in PSR1 the fact that DNA sequence L1 is used by PSR1 Then line 260 records the fact that PSR1 also uses DNA sequence L2 Note that this latter line makes use of the linked list character of our DNA database Recall see page 25 that m u is a linked list of layouts Since PSR1 DNA 1 L already points to L1 see line 258 and since L2 is the next layout in the DNA database then PSR1 DNA 1 L next points to L2 In this very simple ca
92. _4d0 11 1 0 d0 54 TAYLOR POLYMORPHISM AND KNOBS 1 2 0 d0 335 Call init state 2 c_ np_pol snp pol is automatically computed The 2 is automatically computed above counting the number of DNA variables 1 4 This means the Taylor series now has eight variables call find orbit CoL1 fix1 1 state 1 d 6 call find orbit Col2 fix2 1 state 1 d 6 call alloc yl call alloc y2 340 Call alloc id call 11 1 call alloc n2 call alloc eq id 1 identity damap 345 yl id this is permitted in ptc only not fpp y2 id fix2 closed orbit added to map The plus sign in the next two lines of code activates the knobs If we remove the plus sign PTC will ignore the knobs call track Coll yl 1 state unary activates knobs call track Col2 y2 1 state After accounting for knobs the code computes the tunes with and without knobs Equations 1 and 2 compute the tunes for col1 equations 3 and 4 compute the tunes for col2 The first number is the difference between the goal and what we have obtained which should be as close to o as possible nl yl normal forms abused of language permitted by ptc 350 n2 y2 normally one should do gt damap y normalform damap write 6 tunes 1 write 6 1 1 2 write 6 tunes 2 write 6 n2 tune 1 2 355 eq 1 nl dhdj v 1 0 254d0 eq 2 nl dhdj v 2 0 255d0 eq 3 n2 dhdj v 1 0 130d0 eq 4
93. above Given a software library that implements a truncated power series algebra TPSA one can use polymorphism and operator overloading to turn any particle tracking code into a map generation code Rather than propagating a particle through the accelerator one may instead propagate an array of truncated power series TPS that represents the particle and its phase space neighborhood The result of such tracking is a one turn map for that region of phase space The polymorphism of PTC makes it possible to do this in a transparent manner THE CONCEPT OF A NORMAL FORM is absolutely central to the modern view of accelerator analysis In the very simplest case a one turn map reduces to a matrix M and we factor it in the form of a similarity transformation Here N denotes a normal form and A denotes the matrix that transforms between N and M Continuing our simplest example N generates rotations in each of the separate phase planes and A converts the circles of normal form co rdinates to generic phase space ellipses If we go to another location in the accelerator the one turn map will differ but it will have the same normal form A N A71 In other words N is an invariant of the ring Global scalars by which we mean global quantities that do not depend on location around the accelerator may be derived from N The s dependent global quantities SDGQs may be derived from the transformations A A Thus fo
94. actmis call update states madlength false 545 ang twopi 36 d0 360 d0 Larc 2 54948d0 brho 1 2040 Larc ang call set_mad brho brho method 2 step 10 550 madkind2 drift_kick_drift kf 2 72d0 brho kd 1 92d0 brho 555 01 drift D1 2 28646d0 d2 drift D2 0 45d0 qf quadrupole QF 0 5d0 kf qd quadrupole QD 0 540 kd b rbend B Larc ang cell d1 qd d2 b d2 qf d1 PSR PSR 10 cell ring PSR 565 Call survey PSR end subroutine build PSR 570 Subroutine build PSR minus PSR use madx ptc module use pointer lattice implicit none 575 type layout target PSR real dp ang brho kd kf Larc type fibre b dl 42 qd qf type layout cell call make_states false exact_model true default default nocavity exactmis 585 call update states madlength false ang twopi 36 d0 360 d0 Larc 2 54948d0 590 595 605 610 615 620 625 630 635 640 PTC GEOMETRY TUTORIAL SOURCE FILE ptc_geometry f90 127 brho 1 2d0 Larc ang call set_mad brho brho method 6 step 10 madkind2 drift_kick_drift kf 2 7200 brho kd 1 92d0 brho dl drift D1 2 28646d0 d2 drift D2 0 4540 af quadrupole QF 0 540 kf qd quadrupole QD 0 540 kd b rbend B Larc ang cell d1 qd d2 b d2 qf d1 PSR PSR b d2 qf d1 8
95. al frame However the tracking is in local variables We need routines that are able to connect geometrically both points of view 8 1 AFFINE ROUTINES ON PURE GEOMETRY 69 Descriptions of the Routines This section describes the affine routines on pure geometry GEO_ROT GEO ROT V 3 3 W 3 3 A 3 B 3 ANG 3 BASIS 3 3 This routine exactly reproduces the theory in the previous section BASIS is an optional variable which is set equal to the described in Section 8 1 The real array ANG 3 is used to define R Rz ang 3 Ry ang Q Rx ang 1 GEO ROT V 3 3 W 3 3 ANG 3 BASIS 3 3 This routine is the same as the previous routine except that A and B are not needed GEO ROT V 3 3 A 3 ANG 3 I BASIS 3 3 Here the final W and B are copied back into V and A The integer I can be 1 to produce the inverse rotation R GEO ROT V 3 3 ANG 3 I BASIS 3 3 This routine is the same as the previous routine without the A vector GEO TRA GEO TRA A 3 V 3 3 3 D I A is translated by D expressed in the basis V I 1 result diVijej 1 The result is put back in A that is A Ax VID Rotating and Translating the Frames of a Magnet FRAME F OMEGA ANG ORDER BASIS 3 3 F is of type magnet frame and contains three affine frames tied to the magnet F A 3 F ENT 3 F O 3 F7oMID 3 3 FHEXI 3 The entire cont
96. al probe defined 21 TEMPORAL_BEAM data type 112 TEMPORAL_ PROBE data type 112 THIN_LENS_RESPLIT routine 86 THIN_LENS_RESTART routine 86 Thomas Jefferson National Laboratory example based on 11 three dimensional information data type 64 THREE_D_INFO data type 64 time flag 101 time based tracking data types 108 described 21 integration node 20 21 topology see accelerator topology totalpath flag 101 114 TRACK routine 50 54 track routine 61 TRACK_BEAM routine 63 TRACK_FILL_REF routine 64 TRACK_NODE_ PROBE routine 62 TRACK_NODE_V routine 63 TRACK_NODE_X routine 63 TRACK_PROBE routine 62 TRACK_PROBE2 routine 62 TRACK_PROBE_X routine 63 TRACK_TEMPORAL_BEAM routine 64 TRACK_TIME z_ptc_geometry f90 145 routine 64 tracking see also s based tracking or time based tracking automatic 15 tracking routines 3 D information through integra tion node 63 beam of particles 63 fibres 61 integration nodes 62 63 list of 61 object oriented 62 radiation 61 62 spin 61 62 standard 61 time based 64 trajectory reference 6 transformation see geometric transfor mation translate routine 33 TRANSLATE FIBRE routine 73 TRANSLATE FRAME routine 69 TRANSLATE GIRDER routine 77 TRANSLATE LAYOUT routine 73 TRANSLATE MAGNET routine 74 TRANSLATE SIAMESE routine 76 translation routines described 69 73 tune computing 54 88 type see data
97. analysis tool 3 phase space dimensions 102 one turn map 17 analysis independent of construc tion 17 142 only_4d computing 50 creating 61 tracking 62 only_4d flag 102 ORIGINAL_P_TO_PTC routine 112 overview PTC 5 para_in flag 102 particle interaction 19 21 space charge kick 21 tracking 6 particle dynamics local 8 PATCH data type 107 patch checking whether needed 37 39 72 defined 13 15 107 inserting 33 37 39 71 72 patching CHECK_NEED_PATCH routine 72 FIND_PATCH routine 72 INVERSE_FIND_PATCH routine 71 check_need_patch routine 37 39 find_patch routine 37 39 defined 9 energy 72 exact 82 find_patch routine 33 FIND_PATCH_B routine 71 inexact 83 reference frames 15 routines 72 patching routines described 72 perturbation theory derived via one turn map 17 phase space flag 102 variables 10 Poincar map FPP 3 pointer DOKO 73 SIAMESE 75 radiation bend split pol block data type 51 polymorph defined 49 knob 50 states 49 tutorial source file 53 polymorphic block described 51 removing from layout 52 setting values 52 Polymorphic Tracking Code see PTC POSITION TEMPORAL routine 112 PROBE data type 110 probe see also temporal probe defined 21 110 tracking routines 62 PROBE 8 data type 111 propagation see backward propaga tion or forward propagation property accelerator 16 57 global 17 57 local 17 6
98. and spin the use of maps derived from the integrator via operator overloading and polymorphism for computing the full range of accelerator properties includ ing the parameter dependence of these properties History of Latter half of 1980s at SSC CDG Berz introduces automatic differentiation and develops DA package in 77 Forest builds L1EL1B on top of DA thus bringing the power of map methods to the analysis of maps produced by integrators 1 When we say PTC we shall almost always mean PTC and FPP combined but you keep in mind the distinction 4 INTRODUCTION Early 1990s J Bengtsson pioneers the use of run time polymorphism to greatly simplify and improve the process of making run time changes to for example computing parameter dependence Forest and Bengtsson together develop the ideas from which derive the fundamental building blocks of PTC Forest develops the dynamical Euclidean group Middle to late 1990s Forest develops LrELrs into FPP using LBNL versions of DA LiELr1s FPP replaces the real variable type real 8 with a new type called real_8 to produce Taylor series for analysis Forest also develops PTC proper the integrator Together these tools compose PTC Middle to late 2000s Forest adds spin dynamics to PTC L Yang develops a C replacement for Berz s DA package Where to Obtain PTC http www takafumi org etienne ptc 12 PTC LIBRARY USER G
99. arious frames of reference were used to compute it We now concentrate on asking sensible questions of our beam monitor Local versus Global Information PTC provides you with two kinds of information local and global Information about an element or a particle in an element is called local if it can be obtained independently of any knowledge of the element s position in an accelerator It could be obtained even if the element were a prototype sitting on a test bench in your lab The field strength at a particular location in the element is an obvious example of local information Another example is a particle s trajectory through a magnet When for example an electron appears at the entrance of a quadrupole we can predict its local trajectory without any knowledge of where the electron came from or where it is going Yet another example of local information is the amount of synchrotron radiation emitted by a particle as it traverses a magnet Information is called global if it can be obtained only after constructing an accelerator The dynamic aperture is an example of global information because it makes sense only in the context of circulating particles around an accelerator ring Other examples include tune closed orbit normal mode decomposition tune shift resonance damping partition number chromaticity linear lattice function nonlinear distortion function anharmonicity equilibrium emittance short mid and long term stability a
100. artesian or Polar coordinates depending on the variable TEAPOT LIKE of the type INTEGRATION NODE TYPE BEAM REAL DP POINTER X N 1 7 LOGICAL LP POINTER U N LOCATION POINTER POS N INTEGER POINTER N LOST INTEGER POINTER CHARGE LOGICAL LP POINTER TIME INSTEAD OF S END TYPE BEAM The array X N 1 7 contains N macro particles whose PTC co rdinates are located in X N 1 6 The array U N contains a stability flag true is unstable LOST is the number of lost particles In the additionally paragraph the third sentence originally read as follows The node is located using the pointer POS N Node and the distance is located in X N 7 PTC Geometry Tutorial Source File ptc_geometry f90 This appendix contains the complete FORTRAN90 listing for the PTC example ptc geometry Most of this example is discussed in Modeling an Accelerator with PTC chapter 3 More advanced aspects of this example are discussed in Linking Magnets Together and Moving Them as a Group chapter 4 ptc geometry f90 program ptc geometry use madx ptc module use pointer lattice implicit none character 48 command gino logical lp doit integer i j mf pos example real dp 10 real dp dimension 3 a d real dp dimension 6 fix2 mis x type real 8 dimension 6 yl y2 type layout pointer L1 L2 L3 L4 L5 L6 type layout pointer PSR1 P
101. ate flags allow you to control certain aspects of PTC s behavior based on the needs of your simulation For example you could run one simulation with the RADIATION flag off and another simulation with the flag on By comparing the results you would answer the question How important is radiation to my simulation Here are six internal state flags and the PTC behavior each flag controls 1 TOTALPATH ensures a computation of the total path length or total time of flight 2 TIME selects time of flight rather than path length cT to be precise RADIATION turns on classical radiation 4 NOCAVITY forces the code to ignore RF cavities It has also implications on the normal form if performed in three degrees of freedom 5 FRINGE turns on quadrupole fringe fields based on the b and az components in the element 6 EXACTMIS forces the misalignments to be treated exactly 55 PTC has special internal state called DEFAULT that selects path length rather than time of flight does not ensure a computation of the total path length turns off classical radiation includes RF cavities turns off quadrupole fringe fields and does not treat misalignments exactly This example shows how to set the default internal state environment described above CALL MAKE_STATES FALSE EXACT_MODEL TRUE DEFAULT DEFAULT CALL UPDATE STATES MAKE STATES is TRUE for electrons and FALSE for protons When EXACT MODEL is TRUE PTC uses the
102. ause PTC tracks particles with respect to local frames of reference a set of values for as an example the six phase space variables 2 2a will not mean too much unless one knows the context in particular where in the lattice the particle is located When you need PTC to keep track of such information for you it provides some additional structures The type beam holds an N x 6 array for the phase space variables as well as a pointer that tells you where in the lattice that beam is located There is also a type probe that holds not only phase space data but also data about a particle s spin state Data Structures for Modeling Accelerator Topologies PTC data structures fully account for the three dimensional structure of a lattice and potential topological complexities such as those found in colliders and recirculators In this section we discuss a simple lattice and then two complex lattices A very simple lattice is shown in Figure 2 7 which illustrates basic forward propagation of particles through a sequence of magnets that never varies Each magnet appears once in the sequence and particles go magnet to magnet from M1 through M8 At that point they re nter the magnet M1 A simple linked list of magnets suffices to model this basic topology ring linked list M 1 M2 M3 M4 M5 M6 M7 M8 One might instead use an array of magnets but that approach would require us to recreate and reallocate the array whenever we wish to add
103. begin to think about how to track particles Here we present some essential information about particle tracking in PTC In keeping with its LEGO block philosophy PTC tracks particles with respect to local frames of reference The local phase space co rdinates together with PTC s knowledge of the local frames allow the interested user to reconstruct full 3D particle trajectories with respect to the global frame In fact doing exactly that is useful for checking that one has the constructed the correct lattice topology in PTC Unirs PTC measures all lengths in meters and all angles in radians It provides the constant twopi to simplify the conversion from degrees 7 Particle momenta are scaled a value This scale momentum is usually set when defining the lattice see for example the discussion of set mad on page 27 A subtle point involves the Figure 2 4 Two LEGO blocks elements on a base global frame By Bu D so fo Figure 2 5 Connecting two horizontal LEGO blocks and a vertical LEGO block 6 Forest Locally accurate dynamical Euclidean group Phys Rev E 55 4 4665 4674 Apr 1997 10 1103 PhysRevE 55 4665 7 Many other constants are defined in the module precision constants 10 OVERVIEW OF PTC 8 PTC can turn all six of the variables into Taylor series or it can omit two of them Setting an in ternal state flag to only four dimensions imple ments the latter behavior
104. bits through complex topologies Inside of PTC is a set of modules referred to collectively as the Full Polymorphic Package FPP These routines extend the capabilities of PTC making it possible not only to track particles but also to propagate maps PTC uses FPP to compute the maps one may analyze for accelerator properties of interest to us lattice functions and the like Analyzing an Accelerator to Understand its Properties 2 3 discusses this topic Finally when we track particles in accelerators we must sometimes account for particle interactions Modeling Particle Interactions 2 4 discusses this topic For in depth discussions of the topics introduced in this overview see the Bibliog raphy page 135 Here correctly means in a manner that respects the physical reality Forest Y Nogiwa and F Schmidt The FPP and libraries In Int Conf Accel Phys 2006 pages 17 21 and E Forest Y Nogiwa and F Schmidt The FPP documentation In Int Conf Accel Phys 2006 pages 191 193 6 OVERVIEW OF PTC 3 PTC also has the capability to perform first order time based tracking See Modeling Particle Interactions 2 4 PTC and Reference Trajectories A reference trajectory or reference orbit describes the ideal path of a particle through an ideal accelerator Accelerator modeling codes that use a reference trajectory simulate particle orbits as deviations from that ideal Howeve
105. but to the previous fibres in their respective layouts See lines 117 and 124 iii We must add the information that the beam described by this layout traverses its elements in the reverse direction line 122 and has particles with negative charge line 123 13 PSR2 backward ring layout 8 call append empty layout m 110 PSR2 gt m_u end pl gt 11 p2 gt L2 end do i 1 119 115 if pl mag name B then call append_point PSR2 p2 p2 gt p2 previous else call append point PSR2 pl 120 end if f gt PSR2 end dir 1 charge 1 1 gt pl previous 125 end do 3 5 MODELING COMPLEX ACCELERATOR TOPOLOGIES 35 f gt PSR2 start do i 1 PSR2 n if f mag name B QUAD then 130 call find_patch f previous f next true call find_patch f f next next false end if f gt f next end do 135 PSR2 name PSR 2 PSR2 closed true call ring_L PSR2 true make it a ring topologically Trackable layouts PSR1 and PSR2 are now complete Both contain 70 fibres pointing to the same set of 70 elements in DNA sequences L1 and L2 Figure Eight Here we model a figure eight lattice the left hand lattice in figure 3 2 It carries a single beam clockwise forward around the lower ring then counter clockwise backward around the upper ring The two rings share several elements in one straight section See figure 3 6 The single trackable layout Fig8 we shall construct fro
106. ceding fibre in the linked list an entrance fringe field which contains approximate fringe effects if any N body integration nodes representing the body of element an exit fringe field which contains approximate fringe effects if any an exit patch which connects the geometry described by the fibre to that of the following fibre in the linked list any number N of body entrance fringe integration nodes here 6 integration node entrance patch integration node e d 24 Using s based tracking we can obtain particle data at the entrance and exit of each integration node in the body of a magnet that is at the seven black lines in figure 2 16 In addition we can obtain first order information about the location within a node of a particle at some fixed time see Time based Tracking The integration nodes contain no duplicate data about the elements they have no existence apart from the beamline elements they represent Because the data in the integration nodes reside on the fibres which point to the elements PTC is able to carry over changes affecting elements e g a change in the magnetic field and changes affecting the fibres e g misalignments to the integration nodes exit fringe integration node exit patch integration node Node Layout The data type node_layout represents a beamline as a linked list of integration_nodes It includes pointers to the first and last integration nodes in
107. come this difficulty PTC provides what it refers to as a DNA database This database is first populated with a set of layouts whose fibres point to unique elements In other words no element appears more than once via the fibres in the set of DNA layouts This much may if desired be done by importing from an external lattice description These simple layouts we shall refer to them as DNA sequences may then be used to construct complex layouts of the form shown in figure 2 10 As a concrete example consider how we might do this for the accelerator shown in figure 2 8 or 2 10 We first create a set of layouts in which no magnet appears twice and which have no magnets in common The logical course in this case is to create the eight layouts L1 through L8 shown in figure 2 11 Note that each of the eighteen magnets appears just once in this database Here we show each of the DNA sequences in a different color and these colors match those used in figure 2 12 which shows the accelerator with the different DNA sequences indicated by the corresponding colors F5 F13 21 5 F6 14 22 6 1 2 gt 2 F3 M3 gt 4 7 7 F8 M8 F10 F18 26 10 F9 F17 25 9 Having created the eight DNA sequences in our DNA database we can now use PTC to string these basic layouts together to create the trackable layout that
108. ctangular Read f chart f o as bend in L1 1 7 0 f chart frame origin 3 Translate the newly appended fibre line 84 4 Compute in line 85 the rotation a from the frame attached to the middle of the newly appended straight bend f chart f mid to the frame attached to the middle of the corresponding rectangular bend in L1 pl chart f mid 11 Because rotations do not 5 Rotate the newly appended fibre about its center f chart f o by angles a commute their order is which are given with respect to the frame attached to the middle of that element important E TC performs rotations in the rt mid order x axis y axis Z axis When the above do loop terminates the beamline elements of PSR1 are all in their correct locations However recall the discussion concerning figure 3 1 the straight bends of PSR1 all require patching This happens in the following block of code f gt PSR1 start 95 do i 1 PSR1 n if f mag name B QUAD then 34 MODELING AN ACCELERATOR WITH PTC 12 The geometry operations performed on the bend fibres of PSR1 applied ul timately to the elements in L2 Hence those ele ments are now oriented as in the right hand lay out of figure 3 3 with the global origin centered be tween the adjacent ends of the top two magnets 13 Both direction and charge are properties of fibres and both default to 1 call find_patch f previous f
109. d 17 global parameter exact_model 27 90 95 fuzzy split 86 radiation bend split 86 resplit cutting 86 sixtrack compatible 86 global property see also global infor mation described 57 global variable default 101 lmax 26 _ 25 31 integration philosophy 85 process 85 88 steps 85 88 integration methods described 85 95 96 integration node defined 20 109 setting maximum length 26 splitting 85 86 linked list tracking routines 62 63 integration nodes specifying number of 27 INTEGRATION NODE data type 109 integrator described 6 overview 5 splitting elements into integration nodes 85 86 Taylor map 3 49 internal states described 101 example source code 27 setting 50 INVERSE FIND PATCH routine 71 JLab see Thomas Jefferson National Laboratory kick integration method 85 96 space charge 21 KILL PARA routine 52 knob creating 51 defined 50 flag 102 tutorial source file 53 using 50 94 Large Hadron Collider bore magnets 43 LAYOUT data type 103 layout see also node layout defined 15 103 global frame 8 non trackable 14 30 trackable 14 30 32 40 LEGO block analogy 6 8 LHC see Large Hadron Collider Lie algebra discussed 82 Lie operators discussed 83 linked list m u global variable 25 1 appearances of magnet 73 circular 43 example for fibres 12 13 example for magnets 11 integration nodes 20 layout 103 node la
110. der 15 Yes this sounds much like the description for layout Fig8 Be sure to note the differences 1 Call ring L Fig8 true make it topologically closed pl gt Fig8 start do i 1 Fig8 n call check need patch pl pl next 1 d 10 pos 195 if pos 0 call find patch pl pl next next false pl gt pl next end do Trackable layout Fig8 is now complete It has 140 fibres pointing to 134 elements in DNA sequences L4 and L3 Twelve of the fibres in Fig8 point to the six elements four drifts and two quadrupoles at the top of L4 which are common to the upper and lower rings See figure 3 6 Collider Here we model a collider figure 3 8 also the right hand lattice in figure 3 2 which has clockwise propagating beams in each of its two rings This model comprises two layouts Col1 and Col2 which we shall construct from DNA sequences L5 and L6 see table 3 1 For the lower ring we use the full ten cell PSR lattice in layout L6 For the upper ring we use all of layout L5 PRS_minus and we re use six of the fibres in layout 16 15 In addition so that this layout does not overlap our previous layouts PSR1 PSR2 and Fig8 we shall place it 40 m distant in the positive z direction The construction of layouts Col1 and 12 takes place in the following four blocks of code We describe each in detail In this first block of code we place all the beamline elements of layouts Coli and 12 in their correct locatio
111. der need not be adjacent Nevertheless tied to a common substrate they will move as a unit The next block of code links together the elements we want on our girder As for the case of siamese every PTC element contains an element pointer called girders and we use this to construct the linked list that ties our girder elements together Pointing p1 to the short drift at the left hand end of the common straight section in fibre 68 of layout 011 we deal first with the girder elements common to layouts Coll and 12 In lines 396 400 we march along the straight section pointing p2 to the next fibre linking the elements in p1 and p2 line 398 and then advancing pl After that loop terminates both p1 and p2 point to fibre 4 in layout Col1 and we have linked together the elements of the common straight section plus that last bend call move to Coll pl 68 395 f gt pl remember start of girder linked list do i 2 7 p2 gt pl next pl mag girders gt p2 mag pl gt pl next end do call move_to Col2 p2 7 pl smag amp girders gt 1 pl smag amp siamese sgirders gt p2 mag p2 smag amp sgirders gt 2 405 Call move to Coll pl 67 call move_to Col2 p2 14 plxmag girders gt p2 mag p2 mag girders gt f mag In the rest of this block of code we link in the remaining four bends we want on our girder 7 14 and 7o from 12 and 67 from 11 Because these
112. der the following sequence of calls where the fibre pointer B is pointing to a member of the left siamese MIS 0 d0 MIS 1 2 d0 CALL MISALIGN_SIAMESE B MIS ADD FALSE MIS 0 d0 MIS 1 2 d0 MIS 5 PI 8 d0 CALL MISALIGN_GIRDER B MIS ADD TRUE Figure 8 6 shows the result Example 3 Now let us switch the order 79 80 GEOMETRIC ROUTINES Figure 8 6 Example 2 misalign siamese fol lowed by misalign girder MIS 0 d0 MIS 1 2 d0 MIS 5 PI 8 d0 CALL MISALIGN_GIRDER B MIS ADD FALSE MIS 0 d0 MIS 1 2 d0 CALL MISALIGN SIAMESE B MIS ADD FALSE Here we purposely made ADD FALSE on the siamese call since naively one expects the position to be relative to the girder on which it is attached The results should be the same but they are not as we see in figure 8 7 Figure 8 7 Example 3 misalign girder followed by misalign siamese the magnets store their effective misalignments in one array Therefore the siamese has no knowledge of being on a girder ADD FALSE sends the siamese back to its original fibre 8 2 AFFINE ROUTINES ON COMPUTER OBJECTS 81 Example 4 This command avoids sending the siamese back to its original fibre prior to the misalignment MIS 0 d0 MIS 1 2 d0 CALL MISALIGN_SIAMESE B MIS ADD FALSE PRESERVE_GIRDER TRUE Figure 8 8 shows the result Figure 8 8 Example 4 misalign siamese with preserve_girder true The same command with 1
113. displacements The various misalignments are specified by the six component vector mis the first three components describe translation while the last three describe rotation in PTC order This code is probably only somewhat self explanatory Nevertheless we present it here for you to look at and mull over with only a little in the way of comments A detailed explanation is given in chapter 8 Geometric Routines Concerning line 431 note that bend 7 i e the bend contained in fibre 7 of layout 12 is present in both the girder and the left hand siamese which are the parts we misalign in the examples given here That bend is not directly associated with either the girder frame or the siamese frame But it is indirectly associated via the linked lists that define the girder and the siamese PTC can always find the appropriate girder or siamese frame write 6 Example from the manual 1 11 read 5 example call move_to Col2 p2 7 if example 1 then Example 1 mis 0 40 mis 5 pi 8 40 435 call misalign_girder p2 mis elseif example 2 then Example 2 mis 0 40 mis 1 2 040 call misalign_girder p2 mis 440 elseif example 3 then Example 3 mis 0 40 mis 5 8 40 call misalign_girder p2 mis mis 0 40 445 mis 1 2 d0 call misalign_girder p2 mis add false elseif example 4 then Example 4 mis 0 40 mis 5 pi 8 d0 450 call misalign_girder p2 mis mis 0 40 mis
114. e the solid by a rotation defined by its action on the basis 01 92 03 Wi Riv j Then we ask the following question what are the components of w in the global basis in terms of the component array Vi Note The components of b in the frame w1 w2 w3 which is the image of a upon rotation of the magnet are also given aj because this point is fixed in the magnet Solution Riv j jk k Thus we have W RV The components of the vector in the global frame are then by ai Ri Viner gt b RV a ij We now address a slightly harder problem which is essential in PTC The rotation R instead of being defined in the frame v1 v2 0 might be defined on a totally different frame 14 12 Therefore prior to rotating the basis 01 02 03 we must express it in the frame 11 U2 a Y aiViUg Un ik ikn ikn We can now apply the rotation R defined on the basis u1 u2 be y a VipUnkRnmum iknm iknm Thus the final result of W is W Vu Ru The point b in global coordinates is b Y aW gt b k Notice that if V LI we regain the previous result PTC factors the rotation R in the form R RyRx Using local variables results the need to go back and forth between frames The magnets must be placed within the glob
115. e various machine elements We use the subroutines to actually construct the computer model of a machine and then integrate through it including the geo metric transformations translations and rotations of machine elements connecting elements to form trackable beamlines and tracking the orbital and spin degrees of freedom In a word PTC is an integrator The Fully Polymorphic Package FPP is a package of polymorphic types and tools that provide PTC with facilities for analysis In particular FPP implements a polymorphic Taylor type hence the P in PTC that can change shape at execution time This Taylor type makes it possible for FPP to extract a Poincar map from PTC or some other integrator Moreover FPP provides the tools to analyse the resulting map The most common and most important tool is the normal form with this at hand one can compute tunes lattice functions and nonlinear extensions of these and all other standard quantities of accelerator theory Indeed the combination of PTC and FPP gives access to all of standard perturbation theory on complicated accelerator lattice designs In a nutshell then the three central features of are e the FORTRAN9O0 types and code that facilitate fully three dimensional placement of beamline elements and construction of non simple accelerator topologies colliders recirculators dog bones and combinations of these topologies the polymorphic integration routines for orbit
116. eation of a simple linked list rather than a doubly linked list In our case this role is played by the pointer PREVIOUS In a simple linked list each node has the red PREVIOUS pointer The pointer END points to the end of the list in our case fibre 4 Then the list is traversed backwards using the pointer PREVIOUS A more complex structure is required to handle two way traversing To do this we use the pointer START At each node we add the green pointer NEXT The green pointer points to the next element until it finally ends on the START pointer The START pointer as well as the END pointer are nullified or grounded Thus one can perform an ASSOCIATED check on either NODE NEXT or NODE PREVIOUS to determine if the extremities of the list have been reached We want to have a circular linked list for tracking a ring When circular the magenta link pointing to the grounded START is cut The last fibre s green pointer now points to fibre 1 blue link Thus the list is made circular in the forward direction The purpose of the START GROUND pointer is to remember the location of the grounded START pointer to re establish the ordinary two way terminated list if necessary The same type of operation is performed on the magenta link pointing to the END pointer The list is the fully circular PTC routines permit a programmer to toggle back and fourth between terminated and circular lists The LAST pointer remembers the last fibre which was accessed by
117. ed planar fibre in three dimensions 109 Type integration_node and type fibre 111 LIST OF FIGURES Xxiii List of Tables 8 1 9 1 9 2 9 3 9 4 9 5 9 7 9 8 9 9 9 10 DNA database for the PTC geometry tutorial 31 Results of Example 1 87 Results of Example 2 88 Results of Example 3 for Part 1A 89 Results of Example 6 when even is not specified 92 Results of Example 6 when even true 92 Results of Example 7 for resplit_cutting 1 93 Results of Example 7 for resplit_cutting 2 93 Results of Example 8 93 Results of Example 9 when useknob true 94 Results of Example 9 when useknob false 94 Note to the Reader This manual describes Etienne Forest s PTC a software library of data structures and tools for both integrating and analysing the orbital and spin motion of particles in modern accelerators and storage rings I believe that PTC is an important contribution that should be more widely known understood and used by the accelerator physics community hence my efforts here I will therefore greatly appreciate your reporting any difficulties you have with this document errors inconsistencies infelicities of language etc to dabell txcorp com Acknowledgements First and foremost I must thank Etienne Forest Not only is he the author of PTC hence sine qua non he has also answered innumerable questions over the years and helped me to understand both the structure of
118. ements in DNA sequence L4 knows that it is pointed to by two different fibres in trackable layout Fig8 Note that we need not initialize the pointer p1 to L4 start the last execution in line 157 of p1 gt 1 returns p1 to the beginning of the layout 160 write 6 1 call append point Fig8 pl pl gt pl next write 6 1 3 5 MODELING COMPLEX ACCELERATOR TOPOLOGIES 37 call append point Fig8 1 165 pl gt 1 write 6 1 call append point Fig8 1 In the next block of code we append the fibres of layout L3 to Fig8 Since our beam traverses L3 in the reverse direction we initialize the fibre pointer p1 to L3 end line 169 and we advance that pointer to the previous fibre line 174 During this process we take care of two other tasks We tell PTC that these elements are traversed in the reverse direction line 172 and if the element is a bend we reverse the sign of the magnetic field line 173 The particle charge remains the same default value 1 pl gt L3 end 170 do i 1 L3 n call append_point Fig8 pl Fig8 end dir 1 if pl mag name B pl mag bn 1 pl mag bn 1 pl gt 1 175 end do To complete the upper ring and hence our trackable layout Fig8 three fibres re main to be appended the last three fibres of the lower ring L4 This is accomplished in the next block of code First in line 177
119. ent of F is rotated as shown in figure 8 2 70 GEOMETRIC ROUTINES Figure 8 2 Rotating and translating the frames of a magnet Origin TRANSLATE FRAME F D ORDER BASIS 3 3 CALL CHANGE BASIS D BASIS DD GLOBAL FRAME ORDER DD ORDER DD 0 ORDER DD 79 99 ge ow gt nou ou The frame is simply translated by D The translation D is expressed in BASIS if present CHANGE BASIS CHANGE BASIS A 3 V 3 3 B 3 W 3 3 The component vector A expressed in the basis V is re expressed as B using basis W B is the output of this subroutine aiVinex b Wirek b WV a ik ik COMPUTE_ENTRANCE_ANGLE COMPUTE ENTRANCE ANGLE V 3 3 W 3 3 ANG 3 This is a crucial routine Given two frames V and W which most likely represent a magnet or a beam line the routine computes a rotation R in the standard PTC order that is Rz ang 3 Ry ang 2 Rx ang 1 such that V is transformed into W It is the reverse routine from GEO_ROT 8 2 AFFINE ROUTINES ON COMPUTER OBJECTS FIND_PATCH_B This routine connects the affine frame A V to the affine frame B W FIND PATCH B A 3 V 3 3 B 3 W 3 3 D 3 ANG 3 SUBROUTINE FIND PATCH B A V B W D ANG FINDS PATCH BETWEEN V AND W INTERFACED LATER FOR FIBRES IMPLICIT NONE REAL DP INTENT INOUT V 3 3 W 3 3 REAL DP INTENT INOUT A 3 B 3 D 3 ANG 3 CALL COMPUTE ANGLE V W ANG
120. entrance frame coincides with the exit frame of the previous element For the PSR lattice the reference frames for the bends and quadrupoles line up exactly with those of their adjacent drifts As a consequence this lattice requires no patching We have therefore finished building the PSR lattice in PTC FEW READERS WILL COMPLAIN that the set of lines 517 through 522 violates the philosophy of PTC and they have a point This may seem subtle but we urge all other readers to invest the time required to comprehend this point Look for example at the definition of the bend given in line 521 After careful consideration of this definition and what happens on the succeeding lines the reader should recognize that this definition actually serves a dual purpose On the one hand the setting of arc length and bend angle define the geometry of the element On the other hand this line also serves to define the physics of this element To verify this claim query PTC for the magnetic field of this element write 6 a f7 4 PSR B field b mag bn 1 brho will do the trick You will learn that PTC already knows the magnetic field has a value of 1 2 T This information of course requires a knowledge of not only the geometry but also the particle mass and energy Here is the apparent violation of the PTC philospophy which aims to separate the geometric description of beamline elements location reference frames etc from their physical content
121. es the relative momentum deviation Po and the sixth variable denotes the path length deviation One may modify the tracking state so as to change the phase space variables If desired one may set a flag that causes PTC to compute flight time rather than path length In this case the phase space co rdinates are X Px amp ct 2 2b Here the fifth variable denotes the scaled energy deviation where E is the energy associated with po In other words if po mc Boo then mc y And the sixth variable ct is the time of flight deviation multiplied by the speed of light One may set a separate flag that causes PTC to compute the total path length or flight time rather than the deviation In this case the phase space variables will be either x L 2 3a or x Px 2 30 where L and T denote respectively the total path length and total flight time HAMILTONIANS In accord with its LEGO block philosophy PTC tracks a particle across an element using a Hamiltonian that is local to that element The Hamiltonian used by PTC in the body of an element no fringe field then has the simple form Lxx 04 5 p2 pd 1 As x y aci 0 if one uses the variables 2 3a or 2 1 Fc 24 2 14 Kor As x y 2 4b 0 2 1 TRACKING PARTICLES THROUGH AN ACCELERATOR 11
122. et users set up parameters that can be changed without having to recom pile Using Knobs Example The bn 2 quadrupole component of a magnet is made into the first knob 99 bn 2 kind 3 bn 2 i 2 5 99 1 1 At execution time during an operation involving bn 2 the knob becomes the following Taylor map bn 2 bn 2 r 2 5 X_j In the map above j npara_fpp bn 2 i The integer npara_fpp depends on the state It equals the minimal number of variables compatible with the state selected In the four state examples above 5 2 KNOBS 51 1 npara_fpp 6 7 2 6 7 3 _ 4 5 4 npara_fpp 5 gt 6 Creating Knobs While FPP has a routine to help users make a knob PTC has tailored routines to put knobs into a layout and remove knobs from a layout The types subroutines and routines are type pol block subroutine scan for polymorphs R B orR B unary as in state to activate knobs in a track routine TPSAfit 1 lnv array set TPSAfit and set element logicals subroutine kill para L R For more information about the unary used to activate knobs in a track routine see Internal States appendix A Polymorphic Blocks This section discusses type pol block setting values for polymorphic blocks and removing polymorphic blocks from layouts Type pol block This data type creates an object
123. follow different trajectories The particle interactions communicate this information to the tail and now particles in the tail also follow different trajectories their behavior differs because of a change in a magnet they have yet to see This scenario tells us that if our beam is spatially extended and self interacting we cannot define an isolated propagator for each element In other words particle interactions violate the LEGO block approach to modeling accelerators To fit within the philosophy of PTC beamline elements must be independent of one another This idealization upon which most tracking codes rely should be borne in mind when using PTC or any of those other codes to simulate beams with self interacting particles In the rest of this section we discuss some of the data types useful for including particle interactions integration_node node_layout probe and temporal_probe We also describe the basic idea behind the time based tracking capability of PTC Integration Node The data type integration_node represents a step of integration in PTC In particular this data type includes entrance and exit reference frames pointers to the previous and next integration nodes along a particle path see Node Layout and a pointer 16 This should already be part of your routine 17 One example is the so called phase advance At matched locations in a ring the normalizing transforma tions chosen with common conventions
124. four indices are distinct we simplify the following description instead of saying for example the bend in fibre 4 of 11 we shall say simply bend 4 First we move p2 to bend 7 Line 402 then links bend 4 to bend 7o because 1 already points to the latter Line 403 links bend 70 to bend 7 and line 404 links bend 7 to bend 67 because p2 mag siamese already points to the latter It now remains for us to link in bend 14 and then close our linked list of girder elements To do this we first point 1 to bend and p2 to bend 14 Then line 407 adds the girder link from one to the other of those two elements Finally line 408 closes the linked list We now have a pair of siamese and a girder defined in our collider Our next task is to define appropriate siamese and girder frames Though we may locate those frames wherever we wish a sensible choice for the girder might be to make it coincide with the entrance frame of the first fibre in layout 11 at the center of our collider s common straight section We accomplish this in the next block of code After moving p1 to the first fibre in layout 11 we then in line 410 allocate mem ory for the special dual reference frame needed by our girder pl mag girder_frame This has four pieces we need to define a and b for the origins of the original and current or misaligned frames and ent and exi for the bases of those two frames In the remaining four
125. g fibres 61 tracking integration nodes 62 63 s based tracking data types 103 described 6 integration node 20 scan_for_polymorphs routine 51 53 SDGQ s dependent global quantity 18 sequence see DNA sequence SET_ELEMENT routine 52 set_mad routine 27 SET_TPSAFIT routine 52 SIAMESE pointer 75 siamese affine frame 75 creating 44 75 defined 43 frame of reference 43 75 misaligning 43 44 77 rotating 75 translating 76 Taylor type siamese_frame 46 sixtrack_compatible global parameter 86 source file geometry tutorial 24 44 115 polymorphs and knobs tutorial 33 splitting tutorial 86 129 space charge kick applying 21 used with time based tracking 21 spin flag 102 tracking routines on fibres 61 tracking routines on integration nodes 62 spin flag 102 spin dim flag 102 spin only flag 102 splitting drifts 97 elements into integration nodes 85 86 lattices 97 routines 86 95 tutorial source file 86 129 stability computing 88 state internal 27 50 survey routine 28 symplectic integration see integration symplectic integrator see integrator Talman Richard algorithm 85 88 integration process 85 88 philosophy for symplectic inte gration 85 strict interpretation of drift kick drift 95 Taylor map computing 49 derived from integrator 3 49 Taylor polymorphism described 49 Taylor type temporal beam polymorphic 3 temporal beam defined 112 tempor
126. gins form a plane perpendicular to the two faces An arc of circle of length L passes perpendicularly through the origin of both faces The purpose of this element is to bend an incoming particle trajectory by approximately angle 0 The internal details of the element whether it is a sector bend a bend with an irregular field etc is for this discussion immaterial Figure 2 2 illustrates these two fundamental types of LEGO blocks with the entrance and exit reference frames for their local co rdinate systems The solid arrows show the direction of the local y axes and the dashed arrows show the direction of the local x axes In summary the bend is characterized by a bending radius p a length L and two faces each with a reference frame The drift is characterized by a length L and also two faces each with a reference frame The bend is the fundamental block used in the composition of complex blocks the drift of course is just a special case of the bend WE ARE INTERESTED in the deterministic motion through each LEGO block repre sented by the transformation 2 1 where f denotes a vector of generally nonlinear functions connecting the local phase space co rdinate Zin defined in the entrance frame to the local phase space co rdinate Zout in the exit frame The construction of such a transformation depends on internal considerations that are not needed for an understanding of the LEGO block approach The LEGO block for a single ele
127. he appearances of a magnet and rotate the frames on all integration nodes if present TRANSLATE_FIBRE R D 3 ORDER BASIS 3 3 DOGIRDER Here R is a fibre to be translated by D Dogirder true forces the translation of the girder frame if present TRANSLATE_LAYOUT This routine 73 74 GEOMETRIC ROUTINES scans the layout from position I1 to I2 inclusive if present I1 and I2 are defaulted to 1 and R N respectively calls TRANSLATE FIBRE with dogirder true on each fibre uses the DOKO construct if associated to locate all the appearances of a magnet and rotate the frames on all integration nodes if present TRANSLATE LAYOUT R D 11 I2 ORDER BASIS Rotation Routines with No Automatic Patching This section describes rotation routines that do not automatically patch fibres together ROTATE FIBRE The comments that apply to TRANSLATE FIBRE also apply to ROTATE FIBRE ROTATE FIBRE OMEGA ANG ORDER BASIS DOGIRDER ROTATE LAYOUT The comments that apply to TRANSLATE LAYOUT also apply to ROTATE LAYOUT ROTATE_LAYOUT R OMEGA ANG 11 I2 ORDER BASIS DNA Designed Rotation Routines with Automatic Patching This section describes rotation routines that automatically patch fibres together The routines have been designed to rotate magnets stored in the DNA database ROTATE MAGNET ROTATE MAGNET ANG OMEGA ORDER BASIS PATCH PREC R is a magnet The routine rotates parent
128. iable DEFAULT DEFAULT DELTA Data Types This appendix contains descriptions and examples of the FORTRAN9O data that PTC uses for s based tracking and time based tracking B 1 S BASED TRACKING This section discusses the following PTC data types for s based tracking and gives the FoRTRAN90O definitions for several important types layout fibre chart including magnet frame e patch Layout The LAYOUT type is a doubly linked list whose nodes of data type FIBRE contain the magnet the local charts and the patches TYPE LAYOUT CHARACTER 120 POINTER NAME IDENTIFICATION INTEGER POINTER INDEX HARMONIC_NUMBER IDENTIFICATION CHARGE SIGN LOGICAL LP POINTER CLOSED INTEGER POINTER N TOTAL ELEMENT IN THE CHAIN INTEGER POINTER NTHIN NUMBER IF THIN LENSES IN COLLECTION FOR SPEED ESTIMATES REAL DP POINTER THIN PARAMETER USED FOR AUTOMATIC CUTTING INTO THIN LENS POINTERS OF LINK LAYOUT INTEGER POINTER LASTPOS POSITION OF LAST VISITED TYPE FIBRE POINTER LAST LAST VISITED TYPE FIBRE POINTER END TYPE FIBRE POINTER START TYPE FIBRE POINTER START_GROUND STORE THE GROUNDED VALUE OF START DURING CIRCULAR SCANNING TYPE FIBRE POINTER END_GROUND STORE THE GROUNDED VALUE OF END DURING CIRCULAR SCANNING 104 DATA TYPES TYPE LAYOUT POINTER NEXT TYPE LAYOUT POINTER PREVIOUS TYPE NODE LAYOUT POINTER T AS
129. icle is in at a given time PTC can then compute a first order accurate location inside that node for the particle at that time after SC kick no SC kick Figure 2 17 Applying a space charge kick at time PTC s time based tracking capability allows us to obtain a snapshot of the beam at a fixed time Figure 2 17 illustrates this approach to applying a space charge kick at time T 1 PTC checks node by node the entrance and exit times of a particle to determine the integration node for which fentrance texit After locating the appropriate node PTC returns the particle to the node entrance At this point in figure 2 17 because falls between times f and the particle is on the blue curve at the point labeled with time fp 2 PTC determines the time difference fentrance Then using the par ticle s position and momentum at the node entrance as an initial condition and assuming the current integration node to be a drift PTC computes a position and momentum for the particle at time It records the shift s in the temporal probe If the element is a drift the position of the particle at time T is exact If the element is not a drift the position of the particle at time is a close approximation At this point in figure 2 17 the particle is at the point labeled with time 3 After completing the above two steps for all particles in the beam you can apply space charge kicks to your part
130. icles At this point in figure 2 17 the particle remains at the point labeled with time but it has a new momentum as indicated by the angle between the light blue arrow and the red arrow 22 OVERVIEW OF PTC 4 PTC now drifts the particle with its new momentum back to the entrance of the integration node At this point in figure 2 17 the particle is back in the 52 plane but now on the red curve at the point labeled with time 4 5 PTC now continues tracking the particle with its new momentum seeking the integration node for which fentrance and texit bracket the time for the next space charge kick THREE Modeling an Accelerator with PTC This chapter explains how to use PTC to model the geometry for a range of accelerators To that end we introduce three accelerator topologies and show how to define their geometries using PTC After going through these models in detail you should have a good understanding of how to use PTC to model your own accelerator designs If some of your elements move together as units because they re tied to a girder for example then you will also want to absorb the information presented in chapter 4 Linking Magnets Together and Moving Them as a Group In 3 1 we describe briefly the accelerator topologies we shall model In 3 2 we introduce the PTC source code for our examples The three accelerator topologies use some common structures and we describe the subroutines for the
131. in line 148 the fibre pointer p1 to the first bend B in L4 compute in line 149 the vector d from the exit of the last bend in L3 L3 end chart f b to the entrance of the first bend in L4 pl chart f a and then translate in line 150 layout L3 by vector d Fig8 figure eight lattice layout 9 d zero d 3 40 d0 call translate L4 d 145 d Zero a 2 pi call rotate L3 L3 start chart f a a call move to L4 pl B pos d 1 L3 end chart f sb 15 0 Call translate L3 d At this point the beamline elements for the figure eight lattice are all correctly located and oriented In the next block of code we first call append empty layout on m u and set the layout pointer Fig8 to point to this new layout Then in lines 154 158 we append in order all the fibres of L4 to Fig8 call append empty layout m u Fig8 gt m_u end pl gt L4 start 155 do i 1 L4 n call append point Fig8 p1 pl gt pl next end do We now have the complete lower ring In the next block of code we start the upper ring by appending to Fig8 the first three fibres of the lower ring These three new fibres in Fig8 will of course point to the same physical elements as do the first three fibres in Fig8 a long drift a defocusing quadrupole and a short drift During the calls to append_point PTC will add this information to the dokos of the corresponding elements in L4 As a consequence each of those three el
132. it misalignment PTC then applies the patch from element 2 to element 3 If we remove the misalignment which translates and rotates element 2 the element returns to its design position in the fibre element 1 7 particle Meng 3 path lement 5 23 ANALYZING AN ACCELERATOR TO UNDERSTAND ITS PROPERTIES In this overview chapter the first two sections Tracking Particles through an Accelerator and Modeling Accelerator Topologies discuss the essential concepts and data structures used by PTC to construct a computer model of an accelerator and simulate particle trajectories through that model Your next activity is likely to be that of asking questions of your simulated accelerator Where is the particle in that magnet What are the Twiss parameters at this location in the ring What are the horizontal and vertical tunes of this machine How much synchrotron radiation does my particle emit while traversing this magnet Such questions are the domain of analysis 2 3 ANALYZING AN ACCELERATOR TO UNDERSTAND ITS PROPERTIES 17 At the heart of the software design decisions made for PTC lies the desire to analyze real accelerators The fact that real accelerators are complicated beasts with many individual magnets performing many individual functions leads directly to PTC s simulating accelerators using the LEGO block approach which emphasizes a local view of beamline elements and particle tracking Similarly the desire to analyze
133. k element with reference frames for the entrance element body and exit 7 Two LEGO blocks drift D and bend B 7 Particle trajectories through drift and bend LEGO blocks 8 Two LEGO blocks elements on a base global frame 9 Connecting two horizontal LEGO blocks and a vertical LEGO block 9 Geometry and local co rdinates x y s for a generic block in PTC 10 Forward propagation of particles in a circular accelerator 11 A recirculator illustrates particles traveling through a varying sequence of magnets 12 Particles circulating in different directions through an accelerator 12 A layout with a linked list of fibres pointing to magnets 14 DNA database with eight DNA sequences L1 through L8 The arrows represent the links in the doubly linked list of fibres that constitutes a layout Each fibre points to contains the indicated magnet 1 M18 14 Eight layouts or DNA sequences 14 Patching elements in a single beamline 15 Patching elements in multiple beamlines 16 Misaligning an element 16 N 4 integration nodes cover an element 20 Applying a space charge kick at time T 21 Basic cells for the three accelerator models 24 Accelerator models Arrows indicate the direction of motion of particle beams in the constituent layouts 24 Subroutines for creating DNA sequences 26 Geometry of the rectangular bend 27 Ring with forward and reverse propagation 32 Fig8 fibres pointing to elements in L3 and L4
134. lected in the two linked lists one linked list for each of the two particle beams 2 2 MODELING ACCELERATOR TOPOLOGIES To set up a linked list of containers with pointers to the elements in a beamline PTC employs three different data types element fibre and layout PTC also provides what it refers to as a DNA database which one may populate with ayouts that one may use to construct complex accelerator topologies Finally for the proper positioning of elements in the accelerator PTC uses two additional data types chart and patch We discuss all these concepts in this section Element The data type element represents a beamline element in a machine It contains information about the element type dipole quadrupole rf cavity etc its physical properties length strength etc and its location and orientation in space In PTC it is the fundamental object PTC preserves each element s physical and mathematical integrity to ensure that it can be positioned repositioned or misaligned as a whole At the same time PTC provides the capability for one to look inside the element to see for example details of a particle trajectory One should therefore never feel compelled to split an element in half In addition to information about the element itself the data type element includes information about which fibres point to it This information is used by PTC to ensure that if an element is moved then all beamlines containing it know
135. lines of this block of code we set these components equal to the corresponding parts origin or basis of the entrance reference frame of the ele 1 Read alloc af as allocate affine frame Here the origin of a ref erence frame refers to a three dimensional vector that stores the co rdinates of the local frame s origin with respect to the global frame The basis refers to a 3 x 3 matrix whose rows contain the orthogonal unit vectors of the local frame written with respect to the global frame See chapter 8 for more details 46 LINKING MAGNETS TOGETHER AND MOVING THEM AS A GROUP 4 YY Coll Figure 4 5 Collider interac tion region The small cross hairs indicate the frame location for each siamese 3 This means the siamese frame will be attached to bend 67 but one could equally well attach it to bend 7 the other magnet in this siamese ment in fibre p1 This information is held in pl mag parent_fibre chart f which you might read as the magnet frame attached to the chart associated with this element s parent fibre call move to Coll pl 1 Call alloc_af pl mag girder_frame girder true pl mags amp sgirder frame a pl smag amp sparent fibreschart sf sa pl magssgirder frames sent pl smags amp sparent fibresschartssf sent pl magssgirder frame sb plxmag parent_fibreschart f a pl magssgirder pl magsparent_fibre schartsf ent As set here the original and c
136. lymorphic Tracking Code Technical Report KEK 2002 3 KEK Tsukuba Japan 2002 Forest Y Nogiwa and F Schmidt The FPP and PTC libraries In Int Conf Accel Phys 2006 pages 17 21 Forest Y Nogiwa and F Schmidt The FPP documentation In Int Conf Accel Phys 2006 pages 191 193 Int Conf Accel Phys 2006 Proceedings of the 9th International Computational Acceler ator Physics Conference Chamonix France 2 6 October 2006 2006 McLachlan and G R W Quispel Splitting methods Acta Numer 11 341 434 Jan 2002 DOI 10 1017 50962492902000053 Index s dependent global quantity defined 18 3 D information see three dimensional information accelerator analysis 16 modeling 11 13 modeling tutorial 23 properties 16 57 accelerator topology collider 12 complex 11 32 model of collider 23 38 model of figure eight 23 35 model of ring with forward and reverse propagation 23 32 recirculating 11 simple 11 affine basis geometric routines 67 71 affine frame attached to element 107 dynamical group 82 girder 76 siamese 75 affine routines on computer objects 71 on fibrous structures 72 on pure geometry 67 theory 67 affine_frame 43 ALLOC_AF routine 75 76 allocate routine 40 allocate_af routine 45 APPEND_EMPTY routine 73 append empty layout routine 31 32 36 38 APPEND FIBRE routine 73 APPEND POINT routine 73 append point routine 33 36 38
137. lymorphism Placing geometric objects in three dimensions is worthless unless our rotations and translations can be translated into dynamical equivalents acting on rays and on spin It is useful first to look at the Lie algebra acting on the t based dynamics because it contains within it as a subgroup the affine part discussed above Exact Patching and Exact Misalignments Dynamical Group PTC first computes patching and misalignments geometrically The connection between two affine frames is always expressed as follows through pure geometric computations Connection T dx dy dz o Rz o Ry o Rx This product of operators is in the usual matrix ordering Therefore the rotation in the x axis acts first followed by the y axis etc The rotations are computed using COMPUTE ENTRANCE ANGLE We factor the rotation in that manner because the operators Ry and are drifts in polar coordinates and are therefore nonlinear For example the rotation R is a pole face rotation dubbed prot by Dragt in the code MARYLIE The formula is given by x 2 x Where p Peat s pi Py cosa amp tana Po Py pxcosa pzsin Xpy tana Py Py TN x d pr tana gt 1 P tana P Here t p form a canonical pair In PTC p t form a canonical pair 8 3 DYNAMICAL ROUTINES 83 We get the rotation Ry from by interchanging x and y Both rotations rotate the magnet towards the directi
138. m DNA sequences L3 and L4 see table 3 1 For the lower ring we use the full ten cell PSR lattice in layout L4 For the upper ring we use all of layout L3 PRS_minus and we re use six of the fibres in layout L4 In addition so that this layout does not overlap our previous layouts PSR1 and PSR2 we shall place it 4om distant in the negative z direction 64 fibres in layout fig8 point to 64 fibres in DNA layout L3 12 fibres in layout fig8 point to 6 fibres in DNA layout L4 64 fibres in layout fig8 point to 64 fibres in DNA layout L4 The construction of layout Fig8 takes place in the following six blocks of code We describe each in detail Figure 3 6 Fig8 fibres pointing to elements in L3 and L4 36 MODELING AN ACCELERATOR WITH PTC Figure 3 7 Match ing L3 to L4 14 Each of those elements already knows it belongs to a fibre in layout L4 In this first block of code we translate layout L4 a distance 40m in the negative 2 direction lines 142 144 We then lines 145 147 rotate layout L3 180 about the entrance of its first element L3 start chart f a This rotates the gap in L3 down to the bottom Now note that the bend to the right of this gap after the rotation belongs to the last fibre in L3 To finish placing L3 in the correct location we want the end of its last bend to coincide with the entrance of the first bend in the lower ring L4 See figure 5 7 We therefore point
139. me PSR 2 PSR2 closed true call ring l PSR2 true make it a ring topologically Fig8 figure eight lattice layout 9 d zero d 3 40 d0 call translate L4 d a zero a 2 pi call rotate L3 L3 start chart f a a call move_to L4 pl B pos d plxchart f a L3 end chart f sb call translate L3 d call append empty layout m_u Fig8 gt m_u end pl gt L4 start do i 1 L4 n call append point Fig8 1 pl gt pl next end do write 6 1 call append point Fig8 pl pl gt pl next write 6 1 call append_point Fig8 1 pl gt pl next write 6 1 call append_point Fig8 1 pl gt L3 end do i 1 L3 n call append_point Fig8 p1 Fig8 end dir 1 if pl mag name pl mag bn 1 pl mag bn 1 pl gt 1 end do pl gt L4 end previous previous write 6 1 call append_point Fig8 pl pl gt pl next write 6 1 185 190 195 200 205 210 215 220 225 230 PTC GEOMETRY TUTORIAL SOURCE FILE ptc geometry f90 call append point Fig8 p1 pl gt 1 write 6 1 call append point Fig8 p1 write 6 Fig8 has Fig8 n fibres Fig8 name Figure Eight Fig8 closed true call ring l Fig8 true make it topologically closed pl gt Fig8 start do i 1 Fig8 n call
140. me is located in relative coordinates from the entrance of the fibre that contains the magnet on which siamese_frame is attached green objects above Generally we may know the desired frame of reference in absolute coordinates given by the red A 3 and ENT 3 3 above The relative translation and rotation can be computed and the result is stored in the blue variables ROTATE_SIAMESE This routine rotates the siamese by a set of angles ang 1 3 in the usual PTC order ROTATE_SIAMESE S2 ANG OMEGA ORDER BASIS PATCH PREC S2 is any fibre that contains an element of the siamese string The intricate usage of the optional variables OMEGA ORDER BASIS is best explained by displaying the actual code CALL FIND_AFFINE_SIAMESE 52 FOUND IF FOUND CALL FIND FRAME SIAMESE CN B EXI ADDZMY FALSE IF PRESENT BASIS THEN BASIST BASIS ELSE IF FOUND THEN BASIST EXI ELSE BASIST GLOBAL FRAME ENDIF 76 GEOMETRIC ROUTINES ENDIF IF PRESENT OMEGA THEN ELSE IF FOUND THEN OMEGAT B ELSE OMEGAT GLOBAL_ORIGIN ENDIF ENDIF If OMEGA is present then it is used If BASIS is present it is also used Normally one would expect both to be present or both to be absent PTC does not check for this If they are not present PTC looks for a siamese frame which is then used if it exists Otherwise the global frame is used This routine calls the equivalent magnet routines over the entire
141. ment might in fact comprise a Zin Zout fina Forest Locally accurate dynamical Euclidean group Phys Rev E 55 4 4665 4674 Apr 1997 DOI 10 1103 PhysRevE 55 4665 Figure 2 1 LEGO block element with reference frames for the entrance element body and exit Note that calling the x and y axes horizontal and vertical reflects the bias of accelerator physicists towards flat rings with vertical dipole fields What we locally i e within the element call the x axis need not be horizontal See for example the discussion of the vertical bend indicated in figure 2 5 zT gt Figure 2 2 Two LEGO blocks drift D and bend B 8 OVERVIEW OF PTC Figure 2 3 Particle trajecto ries through drift and bend LEGO blocks 5 Appendix A discusses PTC s internal state vari ables which describe the characteristic dynam ics you can choose for your accelerator model sequence of simpler blocks This structure dictated by the internal geometry of the element may be hidden In figure 2 3 we show a pair of particle trajectories in drift and bend LEGO blocks passing from the entrance face through to the exit face As suggested by the trajectories illustrated the internal structure may be very complicated think of a wiggler but seen from the outside our LEGO blocks are quite simply blocks with two faces that define local co rdinates with respect to
142. n step 11 the early days of kick codes physicists believed that the drift kick drift D K D kick model was massively inadequate because it requires a large number of steps to achieve decent convergence and thus the correct tunes Such codes were restricted to special applications such as radiation and spin calculations Chao s code SLIM The reason for this state of affairs was two fold First a technical reason We did not have high order symplectic integrators Second a philosophical reason People did not understand their own approaches which contradicted the inadequacy of kick codes The technical issue was resolved by Ruth and later by a cabal of authors including Neri Forest and finally Yoshida We can use high order symplectic integrators on our usual MADS Hamiltonian for the body of the magnet Almost simultaneously the code TEAPOT emerged from the suffocating entrails of the SSC CDG TEAPOT integrated exactly using a D K D method or more accurately a PROT KICK PROT method the combined function S bend TEAPOT was a second 86 SYMPLECTIC INTEGRATION AND SPLITTING e method 4 the Ruth Neri Yoshida fourth order method which has three kicks per integration step method 6 the Yoshida sixth order method which has seven kicks per integra tion step In the matrix kick matrix M K M case PTC also has three methods of integra tion method 1 e method 3
143. ncluded The only exception to all of this is the time based tracking routine 7 1 STANDARD TRACKING ROUTINES ON FIBRES These routines do not support spin and radiation Track track R X I1 I2 K X is an array of six real dp or REAL 8 I1 and 12 are the position of the fibres Result TRACK FLAG 11 12 Result Logical true indicates stable false indicates unstable track C X K CHARGE This routine tracks through the fibre C 62 TRACKING ROUTINES Find_orbit find_orbit R FIX LOC STATE eps TURNS The find_orbit routine works in the same way as the subroutine find_orbit_x but on the fibre structure without radiation For more information about the sub routine find_orbit_x see Find_orbit_x LOC is the integer location in the layout R 7 2 TRACKING ROUTINES ON INTEGRATION NODES These routines support spin and radiation Routines for Tracking either Probe or Probe_8 For the type definitions of probe and probe_8 see Probe TRACK PROBE2 TRACK PROBE2 R XS K I1 I2 11 12 NODE POSITION I1only implies a one turn map K INTERNAL STATE TRACK PROBE TRACK PROBE R XS K FIBRE1 FIBRE2 NODE1 NODE2 FIBRE1 FIBRE2 NODE1 NODE2 are all integer positions of either the fibre or the integration node TRACK NODE PROBE TRACK NODE PROBE T XS K T is an integration node Object Oriented Routines TRACK PROBE2 XS K FIBREI FIBRE2 NODE1 NODE2 TRACK PROBE 5
144. nd much more Note that all local quantities are governed solely by the particulars of an element and the underlying equations of motion Particle information local field strengths and the Lorentz equation are all you need to know for the computation of local 15 Forest A Hamiltonian free description of single particle dynamics for hopelessly complex periodic systems J Math Phys 31 5 1133 1144 May 1990 10 1063 1 528795 18 OVERVIEW OF PTC quantities Global quantities on the other hand are governed by the fact that an accelerator ring is designed to circulate particles in stable orbits for many turns They result from our efforts to interpret the one turn map Polymorphs and Normal Form The simplest simulation of particle trajectories in an accelerator represents a particle as an array z of real numbers which constitutes the phase space co rdinates and any other quantities spin for example that need to be tracked One then tracks the particle through the accelerator by integrating the appropriate equations of motion with initial conditions given by 27 If we wish to know the behavior of nearby particles then we could of course launch other particles with initial conditions zi and study how the results vary with However the polymorphic aspect of PTC enables us to do this in a way that gives us access to all of standard perturbation theory including the kinds of global information mentioned
145. ne elements A particle going through a beamline element sees only the local magnetic field and PTC uses a local reference frame for describing that magnetic field the frame most appropriate for the geometry of that type of element To track through a series of beamline elements PTC locates the elements with respect to each other using geometric transformations that connect the reference frame of one element to the reference frame of neighboring elements Those two pieces the integrator and the geometric transformations give PTC the capability to model an accelerator with arbitrarily placed beamline elements In tracking orbits through particle accelerators we want the capability of mod eling not only simple linear and ring accelerators but also complex topologies for example the Continuous Electron Beam Accelerator Facility CEBAF at the Jefferson National Laboratory JLab To model a complex topology correctlyt PTC uses a data structure that captures the location of the physical elements as well as the topology of the beam trajectory It is that data structure that makes it possible for us to modify a single PTC beamline element in our simulation code even when more than one distinct beam trajectory traverses that element See the discussion of Modeling Accelerator Topologies 2 2 PTC itself handles the integrator the geometric transformations and the associ ated data structures which provide the capability to track particle or
146. ns We translate layout L6 a distance in the positive z direction We then rotate layout L5 180 about the entrance of its first element L5 start chart f a This rotates the gap in L5 down to the bottom To finish placing L5 in the correct location we want the end of its last bend to coincide with the entrance of the first bend in the lower ring L6 We therefore point layout pointer p1 to the first bend B in L6 compute the vector d from the exit of the last bend in L5 L5 end chart f b to the entrance of the first bend in L6 pl chart f a and then translate layout L5 by that vector d 200 Coll lower collider ring layout 10 Col2 upper collider ring layout 11 d zero d 3 40 d0 call translate L6 d 205 d Zero a 2 pi call rotate L5 L5 start chart f a call move to L6 pl B pos d plxchart sf a L5 end chart f sb 210 Call translate L5 d In the next block of code we construct layout Col1 We call append_empty_ layout on m_u and set the layout pointer Coll to point to this new layout Then we append in order all the fibres of L6 to 11 Finally we give our new layout a formal name Collider 1 and ensure that it is topologically closed This layout does not require patching 3 5 MODELING COMPLEX ACCELERATOR TOPOLOGIES 39 call append empty layout m u Coll gt m usend pl gt L start 215 do i 1 L6 n call append point Coll pl pl gt pl next end do 220 write
147. of beamline element L1 MI M2 4 L2 M5 M6 L3 M7 M8 L4 M9 M10 L5 11 12 L6 MI3 M14 L7 M15 M16 L8 M17 M18 Figure 2 11 DNA database with eight DNA sequences L1 through L8 The ar rows represent the links in the doubly linked list of fibres that constitutes a layout Each fibre points to contains the indi cated magnet M1 M 18 Figure 2 12 Eight lay outs or DNA sequences 14 By contrast we may sometimes refer to the DNA sequences high level building blocks of our accelerator model as non trackable layouts element in the beamline In addition a layout defines the direction of propagation through the sequence of fibres and whether the particles recirculate F5 F13 F21 gt M5 1 gt 1 2 gt 2 F3 M3 F4 MA Fo F14 F22 2 M6 oO Mz Mas 7 7 8 gt 8 F10 F18 26 gt 10 F9 F17 F25 M9 As an example consider the accelerator illustrated in figure 2 8 We could con struct it as indicated in figure 2 10 which shows PTC layout with a linked list of 26 fibres F pointing to the 18 magnets 4 MOST ACCELERATOR MODELING CODES do not as PTC does enforce a dichotomy between fibres and elements As a consequence it can be difficult to import lattice descriptions from other codes into PTC in a manner consistent with PTC s approach to modeling accelerators To over
148. on for the stability is The results of the final iteration for the tunes are 0 142192063715077 0 854078356314425 In addition lt DBETA BETA gt 3 542348342695508 003 MAXIMUM OF DBETA BETA 5 551255781973659E 003 EVEN Optional Argument There are three possibilities for the EVEN optional argument EVEN true enforces an even split EVEN false enforces an odd split e EVEN is not in the call statement an even or odd split is acceptable 92 SYMPLECTIC INTEGRATION AND SPLITTING Table 9 4 Results of Example 6 when even is not specified Table 9 5 Results of Ex ample 6 when even true Example 6 EVEN true is important if the motion must be observed at the center of a magnet This is useful during matching procedures in particular The data in table 9 4 show the indifferent splitting T1 is a pointer to the integration node at the beginning of the fibre TM points to the center if it exists T2 points to the end of the fibre TM and T1 are identical when the number of steps is odd there is no center of the magnet and by default TM points to 1 Name Method Steps 1 2 QF 6 3 35 35 41 QD 4 6 52 57 61 B 4 5 67 67 75 The data in table 9 5 shows that when we use EVEN true TM is always different from 1 TM points directly at the center of the magnet Name Method Steps 1 2 QF 6 4 39 43 46 QD 4 6 59 64 68 B 4 6 75 80 84 L
149. on methods 2 4 and 6 have the advantage of producing the exact tune for a typical ideal machine The effect of the matrix on the kicks is of order 2 4 or 6 although all three methods are truly second order integrators strictly speaking the word biased refers to this uneven way of ordering the perturbation due to the kicks Hamiltonian that produced the matrices Let us retry the above example with the Bode like integrator 6 a ORIGINAL METHOD AND NUMBER OF STEPS 3 966179536791447E 003 1 252157150179758E 003 3 546933066933068 003 1 000000000000000E 003 1 000000000000000 003 2 529326133576445E 003 5 16 3 966179536791447E 003 1 252157150179758E 003 3 546933066933068 003 1 000000000000000E 003 1 000000000000000 003 2 529326133576445E 003 5 16 3 966179536791447E 003 1 252157150179758E 003 3 546933066933068 003 1 000000000000000E 003 1 000000000000000 003 2 529326133576445E 003 Please note that exact model true works only with straight elements if the matrix kick matrix method kind7 is selected Bends should use the drift kick drift method integration method 2 if exact model true Drifts Drifts can be split using RECUT KIND7 ONE D1 0 45D0 16 D0 TRUE Splitting an Entire Lattice This routine applies RECUT KIND7 ONE over the entire layout RECUT KIND7 LAYOUT LMAXO DRIFT Appendices Internal States PTC contains a series of flags held in the global variable DEFAULT These internal st
150. on of propagation that is towards the z direction The rotation is the usual affine rotation along the z axis It is linear and transforms x y and px identically leaving t pt untouched It rotates the x axis of the magnet towards its y axis The Lie operators for the three translations and rotations are given by p Ty i pxt py c a 8 pe pes dui yy 8 p y xy qup Lz It is remarkable that these Lie operators have the same Lie algebra as the ordinary affine or time Lie operators Lx Ly Lz Lx Py Pz Lx Pz Py Ly Le Le Pe Ly px pz Ly Ly Lz gt Lz Py The Lie groups are therefore locally isomorphic Of course one notices that prot Rx and Ry has a divergence at a 90 It is not possible in the lens or s paradigm to rotate a magnet map by 90 and get meaningful propagators Therefore the dynamical group is locally isomorphic Inexact Patching and Exact Misalignments PTC provides for an emasculated pseudo Euclidean group The Lie operators are obtained by expanding the dynamical maps keeping the energy dependence exact This is in tune with the exact_model false option Ty iPx Ty 2 2 Tz 71 09 c N Be DL y 1 8 Ly x 1 Lz This group has
151. or the forward propagating beam and PRS2 for the backward propagating beam but the two trackable layouts must of course share the same ring with the same beamline elements They will just have different directions of propagation and oppositely charged particles There is a further complication this ring uses straight bends instead of rect angular bends see figure 3 1 In order to place these elements in their proper locations we will use the lattice of layout L1 as a guide taking the straight bends from layout L2 9 The final result will be a pair of layouts PSR1 and PSR2 each of which refer to fibres in two different DNA sequences L1 and L2 We begin by calling append empty layout on m_u and then setting the layout pointer PSR1 to point to this new layout 72 PSR1 forward ring layout 7 call append empty layout m u PSR1 gt m_u end We shall populate this layout with a linked list of fibres taken appropriately from layouts L1 and L2 Two fibre pointers p1 and p2 will step through these two layouts and a third fibre pointer f will keep track of our location in PSR1 Elements taken from L1 have the correct geometric relations but the straight bends taken from L2 will have to be moved to their correct locations 3 5 MODELING COMPLEX ACCELERATOR TOPOLOGIES 33 pl gt Ll start p2 gt L2 start do i 1 11 if pl mag name B then 80 read bends from L2 call append_point PSR1 p2 f
152. osition FIBRE CHART F A 3 and the vector triad basis FIBRE CHART F ENT 3 3 specify the fibre s desired location Patch The data type PATCH contains variables used in the main tracking loop to perform certain adjustments if the exit chart of one element does not connect smoothly with the entrance chart of the element that follows it for example when transferring from one beam line to another TYPE PATCH INTEGER 2 POINTER PATCH Figure B 2 Three charts attached to an element 107 108 DATA TYPES ANG_OUT Length L Element s Desired Location Particle Particle motion motion Figure B 3 Misalign ments for a element IF TRUE SPACIAL PATCHES NEEDED INTEGER POINTER A_X1 A_X2 FOR ROTATION OF PI AT ENTRANCE 1 DEFAULT 1 INTEGER POINTER B_X1 B_X2 FOR ROTATION OF PI AT EXIT 1 DEFAULT 1 REAL DP DIMENSION POINTER A_D B_D ENTRANCE AND EXIT TRANSLATIONS A_D 3 REAL DP DIMENSION POINTER A_ANG B_ANG ENTRANCE AND EXIT ROTATIONS A_ANG 3 INTEGER 2 POINTER ENERGY IF TRUE ENERGY PATCHES NEEDED INTEGER 2 POINTER TIME IF TRUE TIME PATCHES NEEDED POINTER A_T B_T TIME SHIFT NEEDED SOMETIMES WHEN RELATIVE TIME IS USED END TYPE PATCH B 2 TIME BASED TRACKING This section discusses the following PTC data types for time based tracking and gives the FORTRANgo code integration node including
153. ples we assume that a girder frame of reference has been defined otherwise the girder becomes simply a giant siamese 77 78 GEOMETRIC ROUTINES Figure 8 3 Ex ample o girder Example o On the image in figure 8 3 the red and the cyan magnets are part of a single girder The origin of the girder frame of reference is located in the middle of the red drift and displayed with an orange cross The cyan magnets form two siamese one on the left and one on the right The array MIS 1 6 contains the actual misalignments the translations in MIS 1 3 and the rotations in MIS 4 6 They are applied in the standard PTC order rotation around the x then the y and finally the z axis followed by the translation Example 1 Now we rotate the girder by 22 5 degrees as shown in figure 8 4 Figure 8 4 Exam ple 1 girder after 22 5 degree rotation This was done with the command MIS 0 d0 MIS 5 PI 8 d0 CALL MISALIGN_GIRDER B MIS ADD FALSE If we follow this command by 8 2 AFFINE ROUTINES ON COMPUTER OBJECTS MIS 0 d0 MIS 1 2 d0 CALL MISALIGN_GIRDER B MIS ADD TRUE The ADD TRUE indicates that the second misalignment of the girder is additive The misalignment is in the direction of the rotated girder as shown in figure 8 5 Figure 8 5 Example 1 girder after an additive misalignment MISALIGN_SIAMESE MISALIGN SIAMESE S2 S1 OMEGA BASIS ADD PRESERVE_GIRDER Example 2 Let us consi
154. qd denote the focusing and defocusing quadrupoles And fibre b denotes the rectangular bending magnet Note that it is defined by its arc length and bend angle here Larc and ang respectively See figure 3 4 The symbols in quotes e g QF are the names given to these elements We may now define the variable cell of type layout as the appropriate sequence of fibres 3 3 SUBROUTINES 27 For other particles use the ratio of particle mass to electron mass as the argument of make states Figure 3 4 Geometry of the rectangular bend 5 Because a layout may have any number of these ele ments the names actually define classes of elements 28 MODELING AN ACCELERATOR WITH PTC 6 How to ask this of PTC will become clear later in this chapter See for exam ple page 33 especially the discussion of moving and rotating elements Finally the layout PSR returned by this subroutine is defined as 10 cells and line 525 modifies PSR to make it a ring i e closed At this point line 525 the elements of the PSR lattice are present and in the correct sequence of fibres in layout PSR But should we ask for the location of those elements we would discover that all of them have their entrance frames at the global origin In other words they are all stacked on top of one another at the global origin The call to survey causes PTC to loop through the fibres in PSR moving each element so that its
155. r a particle traveling through a real accelerator sees only local magnetic and electric fields and PTC respects this physical reality When pushing a particle through say a quadrupole PTC makes no distinction between that magnet sitting on a lab bench and that magnet in a beamline it simply integrates the particle motion in a frame local to that quadrupole As we Shall see an essential component perhaps the essential component of PTC is the collection of geometry routines that handle transformations between different frames of reference This approach has the added benefit that even large displacements e g a quadrupole shifted to act as a combined function bend can be handled in the same consistent and physically correct manner 2 1 TRACKING PARTICLES THROUGH AN ACCELERATOR This section discusses four important concepts that enable PTC to perform accurate simulations of particles through an arbitrarily complex accelerator blocks e geometric transformations particle tracking e data structures for modeling accelerator topologies Blocks PTC uses the longitudinal co rdinate generically denoted s as the independent variable for integrating particle trajectories The particulars of PTC s s based or element based tracking serve to maintain the physical and mathematical integrity of each element in a beamline and moreover make it possible for PTC to provide information about particles inside an element again without
156. r example one extracts tunes from N and linear lattice functions from the As Two very important benefits derive from a normal form factorization of the one turn map The first benefit is that this factorization generalizes in a straightforward manner to nonlinear maps We may thus write the map M z as the composition M z AoNoA z 2 5 where now M A and N all denote nonlinear functions on phase space As in the matrix case the normal form N z yields the global scalars and the transformation A z yields the SDGQs N for example contains not only the tunes but also the nonlinear information of how the tunes vary with amplitude The second benefit is 2 4 MODELING PARTICLE INTERACTIONS 19 that we may use the same code that produced the one turn map M z to propagate the normalizing transformation A z Doing exactly this using the polymorphic capabilities of PTC of course we can compute SDGQs at many locations around the ring As with all things wonderful some caveats attach to the normal form decompo sition 2 5 The first caveat is associated with the approximate nature of the work we do Unless our system is extremely special we can know M z only up through some finite order Moreover because of the complicated nature of particle orbits in accelerators the expansion is generally asymptotic As a consequence we should examine all analyses in the light of common sense and actual particle tracking The second cavea
157. rder If applying a different misalignment to the girder PTC starts with the original frame If adding to an existing misalignment PTC uses the misaligned frame as its starting point If we remove the misalignment PTC easily returns the girder to its original position in the lattice We discuss geometric operations applied to girders later in this chapter See also Operations on Girders page 76 Do note that linking together a group of elements on a girder does not mean those elements may no longer move with respect to one another It is only the geometric operations that apply specifically to girders that will preserve the geometric relations between the girder elements A similar comment applies for the siamese In this chapter we show how to create a pair of siamese and a girder in the collider trackable layouts 11 and Col2 from chapter 3 as well as how to misalign them The code in this chapter is from the PTC geometry tutorial source file ptc geometry f90 which is given in appendix C The line numbers of the code shown here refer to the line numbers of the code in that appendix 4 2 BUILDING SIAMESE GIRDERS AND THEIR REFERENCE FRAMES At each end of the straight section shared by layouts 11 and 12 see figure 4 4 is a pair of overlapping bend magnets In the first block of code below we group each pair of bends as a siamese For the pair at the left hand end of the straight we set in the first two lines pointers p1 and
158. refer to these layouts as the DNA of the complex Then trackable structures are appended after the DNA Since the magnets of these structures must be in the DNA they are created with the APPEND_POINT routine rather than the standard APPEND_EMPTY or APPEND_FIBRE used for the DNA production When the APPEND_POINT routine is used a magnet will automatically retain mem ory of its various fibre appearances though a data type called FIBRE_APPEARANCE TYPE FIBRE_APPEARANCE TYPE FIBRE POINTER PARENT_FIBRE TYPE FIBRE_APPEARANCE POINTER NEXT END TYPE FIBRE APPEARANCE Each magnet of the DNA contains a pointer called TYPE FIBRE APPEARANCE POINTER DOKO which constitutes a linked list storing all the appearances of this magnet in the pointer parent fibre The linked list is terminated or grounded at the last fibre appearance If more trackable structures are created this linked list is extended for each magnet There are all sorts of reasons why we may have multiple appearances of a DNA magnet recirculation common section of colliders or even multiple trackable structures of the same physical object It is through the DOKO construct that the following routines know where all the magnets are located Translation Routines with No Automatic Patching This section describes translation routines that do not automatically patch fibres together TRANSLATE_FIBRE This routine uses the DOKO construct if associated to locate all t
159. rward prop agation of particles in a circular accelerator In this discussion we use the word magnet whether or not it is actually a magnet a drift or any other beamline element 12 OVERVIEW OF PTC Figure 2 8 A recircula tor illustrates particles traveling through a vary ing sequence of magnets Figure 2 9 Particles circu lating in different direc tions through an accelerator first trip circulating around the accelerator the particles go through the sequence of magnets M5 6 M7 M8 M9 M10 M11 and M12 During their second trip around the accelerator the particles go through the sequence of magnets M5 M6 M13 M14 M9 M10 M15 and M16 And during their last trip the particles go through the sequence of magnets M5 M6 M17 M18 M9 and M10 The particles are then dumped Note that magnets M5 M6 M9 and M10 appear in all three circuits M1 M2 M3 M4 M5 M16 M12 M15 M11 M10 A linked list of magnets cannot properly represent the physics of this situation In a linked list magnet M6 for example must point to only one magnet It cannot point to M7 M13 and M17 unless we create two clones of M6 Creating clones however is a bad idea If we wish to adjust the position or strength of magnet M6 such adjustments would have be made three times once in the magnet itself and once in each of its two clones A be
160. s an integration node V is of type THREE D INFO The type definition is given below REF TRUE or FALSE If REF TRUE then the results of the TRACK FILL REF routine are used The ray is magnified by V SCALE see type THREE D INFO below with respect to a trajectory computed and stored by the TRACK FILL REF routine which tracks the ray FIX from fibre I1 back to fibre I1 TRACK FILL REF R FIX Il1 K Here is the data type definition for three dimensional information TYPE THREE D INFO REAL DP A 3 B 3 REAL DP ENT 3 3 EXI 3 3 REAL DP WX WY REAL DP 0 3 MID 3 3 CENTRE OF ENTRANCE AND EXIT FACES REAL DP REFERENCE RAY 6 ENTRANCE AND EXIT FRAMES FOR DRAWING MAGNET FACES WIDTH OF BOX FOR PLOTTING PURPOSES FRAMES AT THE POINT OF TRACKING REAL DP X 6 RAY TRACKED WITH REFERENCE_RAY USING A REAL DP R0 3 R 3 RAY POSITION GLOBAL RETURNED REAL DP SCALE MAGNIFICATION USING REFERENCE_RAY LOGICAL LP U 2 UNSTABLE FLAG FOR BOTH RAY AND REFERENCE_RAY END TYPE THREE_D_INFO 7 4 TIME BASED TRACKING ROUTINES These routines provide time based tracking of temporal probes and temporal beams Track time TRACK TIME XT DT K XT is a temporal probe For the type definition of TEMPORAL PROBE see Section B 2 Track temporal beam TRACK TEMPORAL BEAM B DT STATE B is a temporal beam For the type definition of TEMPORAL see Section B 2 7 5 CLOSED ORBIT ROUTINE
161. s are exactly known this reduces to the normal s based tracking when collective effects are absent One immediate application is the tracking of several macro particles in a recircu lator with important wake field effects between the macro particles time ordering of the bunches is crucial and painlessly done in our framework The data type INTEGRATION NODE is defined as follows TYPE INTEGRATION NODE 110 DATA TYPES 1 This is similar to the sur vey L of MADS in the sense that for a real Cartesian bend the distance between the two parallel faces is used In other words it is the sum around the ring of the integration variables whatever they may be For a MAD type RBEND it is still LD because these bends are really sectors with wedges 1 Succ INTEGER POINTER pos_in_fibre CAS INTEGER POINTER pos real dp POINTER S real dp POINTER ref real dp pointer ent a real dp pointer exi b real dp POINTER delta_rad_in real dp POINTER delta_rad_out INTEGER POINTER TEAPOT_LIKE TYPE INTEGRATION_NODE POINTER NEXT TYPE INTEGRATION_NODE POINTER PREVIOUS TYPE NODE_LAYOUT POINTER PARENT_NODE_LAYOUT TYPE FIBRE POINTER PARENT_FIBRE TYPE EXTRA WORK POINTER WORK TYPE BEAM BEAM NODE POINTER BB END TYPE INTEGRATION NODE Each integration node is either the entrance patch misalignment tilt of its parent fibre integration node
162. se in 3 3 Then in 3 4 and 3 5 we construct our DNA sequences and show in detail how to construct our three accelerator designs We conclude in 3 6 with some housekeeping for our DNA database 3 1 ACCELERATOR MODELS Figure 3 2 illustrates the three accelerator topologies we shall model figure eight ring with both forward and reverse propagation e collider All three machines are based on a ten cell ring with each cell composed of seven elements a drift a focusing quadrupole a short drift a dipole a short drift a defocusing quadrupole and another drift In the figure eight and collider examples the dipole is a rectangular bend In the ring with forward and reverse propagation however the dipole is what we shall call for want of a better term a straight bend with the quotation marks By this we shall mean a rectangular bending magnet whose entrance and exit reference frames are parallel to one another and orthogonal to the main axis of the magnet This implies see Chart and Patch page 15 that any cell containing a straight bend will require patching Figure 3 1 illustrates the two basic cells We outline the quadrupoles in blue the rectangular bend in red and the straight bend in green The black lines indicate the drifts The cell with the rectangular bend does not require patching unless the direction or charge change because the reference frames of adjacent elements lie on
163. se you could of course replace lines 259 261 with the single statement PSR1 DNA 2 L gt L2 The remaining lines in this block of code populate the DNA arrays for the other trackable layouts allocate 1 2 PSR1 DNA 1 L gt 11 do i 2 2 260 5 1 1 gt PSRI DNA i 1 L next L2 allocate PSR2 DNA 2 PSR2 DNA 1 L gt L1 265 do i 2 2 PSR2 DNA i L gt PSR2 DNA i 1 L next L2 end do allocate Fig8 DNA 2 270 Fig8 DNA 1 L gt do i 2 2 Fig8 DNA i L gt Fig8 DNA i 1 L next L4 end do 27 allocate Col1 DNA 2 Col1 DNA 1 L gt L5 do i 2 2 Col1 DNA i L gt Col1 DNA i 1 L next L6 end do 280 allocate Col2 DNA 2 3 6 DNA ARRAYS 41 Col2 DNA 1 L gt L5 do i 2 2 Col2 DNA i L gt Col2 DNA i 1 L next L6 285 end do FOUR Linking Magnets Together and Moving Them as a Group In many accelerator designs there are groups of elements that move together as a unit The Large Hadron Collider for example has dual bore magnets that carry the two counter propagating beams through the cryostats A more common example is a group of elements assembled onto a single substrate When misalignments are applied to such units our accelerator modeling code should respect the internal geometric constraints PTC allows us to link elements together to model groups of elements that move as units In this chapter we describe the tools
164. seven generators for convenience The Lie algebra differs from the original algebra of the Euclidean group as follows Ls Ly 0 Lx py Tz 0 1 emasculated pz Ly Px 0 1 84 GEOMETRIC ROUTINES Why do we care about the approximate Euclidean group The reason is speed if a fast post processor to PTC is written Let us assume that we have represented a large machine with exact model false and the drift kick drift option Figure 8 9 schematically shows two magnets separated by a drift Figure 8 9 Pseudo Euclidean maps The drifts 1 3 6 are in red Some of the drifts 1 6 are part of the integration scheme The misalignments are made of our six pseudo Euclidean operators they are in cyan 2 4 Finally a patch 4 is needed it is in magenta The blue and cyan operators contain our six pseudo Euclidean maps Therefore to go from 1 to 6 in the figure may involve over 20 maps By using the group properties this can be reduced to six maps NINE Symplectic Integration and Splitting PTC generally attempts to integrate the maps of each magnet using a symplectic integrator The current release of PTC supports two exceptions to explicit symplectic integration the traveling wave cavity used in linear accelerators and the pancake type for fitted B fields Support for other exceptions could be added to be PTC exact wigglers and exact helical dipoles exact solenoids etc This chapter discusses the
165. ssible because some of their elements overlap The goal of these examples is to introduce the principal PTC concepts involved in modeling the geometry of an accelerator After learning these concepts we can use PTC to model complex real world accelerators 3 2 GEOMETRY TUTORIAL SOURCE FILE The example code in this chapter is from the PTC geometry tutorial source file ptc geometry f90 which is given in appendix C The line numbers of the code in the examples refer to the line numbers of the code in that appendix Initial Code The initial code in the ptc_geometry f90 source file includes type declarations and performs some initialization 3 2 GEOMETRY TUTORIAL SOURCE FILE 25 ptc_geometry f90 This program describes the geometry of several PTC lattices program ptc_geometry use run_madx use pointer_lattice implicit none character 48 command gino logical lp doit integer i j mf pos example real dp bO 10 real dp dimension 3 a d real dp dimension 6 fix2 mis x type real 8 dimension 6 yl y2 type layout pointer type layout pointer 15 type fibre pointer pl p2 b f type internal state state type pol block qf 2 qd 2 type normalform n1 n2 20 type damap id type taylor eq 4 type gmap g 25 Lmax 100 d0 use info true L1 L2 L3 L4 L5 L6 PSR1 PSR2 Fig8 Coll Col2 user stuff one layout necessary before star
166. st qd t1 170 Write 6 a8 2x 5 i14 8x b mag name 1 8 1 999 command_gino opengino call context command_gino context makes them capital call call_gino command_gino 175 111 command_gino mini call context command_gino context makes them capital call call_gino command_gino 180 command gino closegino call call gino command gino 185 190 195 200 205 210 215 220 225 230 PTC SPLITTING TUTORIAL SOURCE FILE ptc splitting f90 call ptc end end program ptc splitting subroutine build PSR PSR exactTF imethod use run madx use pointer lattice implicit none type layout target PSR logical lp exactTF integer imethod real dp ang brho kd kf larc lq type fibre b 41 42 qd qf type layout cell call make states false default default nocavity exactmis call update states madlength false exact model exactTF ang twopi 36 d0 360 d0 larc 2 54948d0 brho 1 2d0 larc ang call set mad brho brho method 6 step 100 madkind2 imethod kf 2 72d0 brho kd 1 92d0 brho lq 0 540 write 6 a kf lq kd lq write 6 kf lq kd lq dl drift D1 2 28646d0 d2 drift D2 0 4540 qf quadrupole QF lq kf qd quadrupole QD lq kd rbend B larc ang b sbend B larc ang cell d1 d2 b d2 qf d1
167. t geometry of MADS It is worth pointing out that the code MAD of CERN and the code SAD of KEK have different implicit geometry once vertical magnets are in voked nothing is standard Affine Routines on Fibrous Structures Displacements that correspond to the design positioning and thus requiring patching Misalignment Routines Misalignments representing errors that do not require patching The misalignments displace the magnet away from a fibre that con tains it The misalignments also displace girders away from their original position Affine Routines on Fibrous Structures We discussed in the previous section PTC s geometric tools on the affine frame This is useful in giving a pictorial representation of a ring in 3 D One can imagine linking PTC with a CAD program equipped with a magnet widget containing at least one affine frame say the cord frame 0 3 MID 3 3 of a PTC magnet Of course PTC is dynamically a more complex structure than just magnets fibres integration nodes layouts etc All these objects have affine frames attached to them and we must be able to move them We describe first the routines that displace PTC structures away from a standard MADS configuration These are used in the generation of non standard systems Patching Routines The central power of PTC is its ability to place magnets in arbitrary positions To do so the concept of a fibre with a patch is necessary FIND PATCH
168. t is associated with the fact that the transformation A z is not unique If B denotes any transformation that commutes with N then replacing A z in 2 5 with A o B z yields the same result In other words the particular A we choose to work with depends on certain conventions For physical quantities associated with A z this freedom in the definition of A necessarily has no effect But some quantities that people discuss do in fact depend on the particular choice for the normalizing transformation A Such quantities needless to say cannot be physical and should be used with great care or not at all 2 4 MODELING PARTICLE INTERACTIONS This overview chapter has until now discussed concepts related to single particle dynamics In this section we introduce some concepts related to the inclusion of particle interactions in PTC simulations Some of these concepts however apply also to single particle dynamics and hence we urge even readers not interested in space charge to at least skim this section We begin with a caveat The inclusion of particle interactions violates the philosophy of PTC To see this consider a beam of self interacting particles at a moment when the head has entered a magnet and the tail has not Particles in the head and tail communicate via the self interactions and hence the dynamics of particles in the head and tail affect one another If we place the magnet in a different location particles in the head will
169. te A Boulder CO 80303 http www txcorp com info txcorp com Typeset 15 05 on 29 July 2011 using the memoir class in 2e Short contents Short contents vii Contents ix List of Figures xii List of Tables xiv Note to the Reader xv Acknowledgements xvii 1 Introduction 3 2 Overview of PTC 5 3 Modeling an Accelerator with PTC 23 4 Linking Magnets Together and Moving Them as a Group 43 5 Taylor Polymorphism and Knobs 49 6 Computing Accelerator Properties 57 7 Tracking Routines 61 8 Geometric Routines 67 9 Symplectic Integration and Splitting 85 Appendices gt 99 A Internal States 101 B Data Types 103 PTC Geometry Tutorial Source File ptc geometry f90 115 D Splitting Tutorial Source File ptc splitting f90 129 SHORT CONTENTS Bibliography gt 135 Index 137 Contents Short contents vii Contents ix List of Figures xii List of Tables xiv Note to the Reader xv Acknowledgements xvii 1 Introduction 3 11 PTC and FPP 3 History of PTC 3 Where to Obtain PTC 4 1 2 PTC Library User Guide 4 Where to Start Reading 4 2 Overview of PTC 5 PTC and Reference Trajectories 6 2 1 Tracking Particles through an Accelerator 6 Blocks 6 Geometric Transformations 8 Particle Tracking 9 Data Structures for Modeling Accelerator Topologies 11 2 2 Modeling Accelerator Topologies 13 Element 13 Fibre 13 Layout 13 Chart and Patch 15 Misalignments
170. teger qf n_name If for example qf n_name 2 then a polymorph is set if the magnet name matches QFxxxxxxxxxxxxxxx where x denotes any character Once the knobs are set on the lattice using the routine scan for polymorphs tracking routines can be invoked after the DA package has been initialized Setting Values using Polymorphic Blocks Polymorphs allow for the computation of parameter dependent maps These maps can be analyzed by various methods including normal forms From these maps one may attempt to fit certain computed quantities by modifying the parameters of the polymorphs on the ring This is done as follows with scan for polymorphs or the sign assignment The global parameter set TPSAfit turns the scan for polymorphs routine into a routine that inputs the array TPSAfit 1 C pol into variables that the pol blocks make into knobs Note that the knobs exist only in the polymorphic version of the magnet located at fibres amp magp The polymorphic version is copied into the real magnet fibre mag if set element is true set TPSAfit true set element true 11 1 qf 1 11 1 qd 1 CoL1 DNA 2 L qf 2 CoL1 DNA 2 L qd 2 set_element false set TPSAfit false Removing Polymorphic Blocks from Layouts To remove a polymorphic block from a layout use the subroutines 5 3 TUTORIAL EXAMPLE 53 call kill_para Col1 DNA 1 L call kill_para Col1 DNA 2 L 5 3 TUTORIAL EXAMPLE
171. the beamline as well as a pointer to the parent ayout To access the integration nodes of a particular fibre one may step through them making use the fact that the data type fibre includes pointers to the first last and middle integration nodes for the corresponding element A node layout is not created automatically when you create a layout You must ask PTC to create it and populate the associated reference frames Probe and Temporal Probe The data type probe represents a particle In particular it contains for a given particle both the orbital phase space data and a spinor for the spin data Should the 2 4 MODELING PARTICLE INTERACTIONS 21 particle become lost the probe contains a logical in which to record this fact as well a pointer to the integration_node in which the particle loss occured The data type temporal_probe contains a probe together with information rele vant to the time based tracking capability of PTC This includes a pointer to the integration_node that currently contains the particle Time based Tracking Any high order s based integrator may be converted to a first order time based integrator provided information is available about the three dimensional environment When we ask PTC to perform time based tracking PTC augments its s based information with temporal information PTC s tracking remains fundamentally s based but with the additional information it can determine which integration node a part
172. this chapter See also Operations on Siamese page 75 A siamese does not have an independent reference frame instead a siamese frame is defined in terms of translations and rotations with respect to the frame of one of its constituent elements Misalignments of a siamese are then specified in a similar way with respect to the siamese frame If we zero the misalignments the siamese returns to its original location I Figure 4 1 A pair of ele ments linked together as a siamese Figure 4 2 Incorrect and correct rotations of a siamese 44 LINKING MAGNETS TOGETHER AND MOVING THEM AS A GROUP 56 5 5 Figure 4 3 A trio of ele ments linked together as a girder that has its own reference frame Figure 4 4 Collider inter action region The num bers show the indices of a few of the fibres within the corresponding layout A girder is a collection of siamese and regular elements tied to a substrate so that one can move them as a unit Like a siamese we construct a girder as a circular linked list containing the elements belonging to the common substrate See figure 4 3 Unlike a siamese a girder typically has its own reference frame independent of any element on the girder This is actually a pair of reference frames stored in the PTC data type affine_frame The two frames specify location and orientation for the original and misaligned girder This structure simplifies the process of misaligning a gi
173. ting GUI call ptc ini no append The module run madx tells to use the madx ptc module which defines the global variable m u This variable which we shall use frequently denotes a linked list of layouts that will constitute our DNA database The gino mentioned on line 6 refers to a Windows based graphical user interface developed for PTC by Etienne Forest Its use is not required 2 Think Map Universe 3 We use the linked list aspect of m only briefly in 8 5 6 DNA Arrays PTC uses the following type declarations for modeling accelerator topologies real dp for double precision real numbers type layout for layouts type fibre for fibres type internal state for setting the characteristic dynamics desired of your accelerator see Internal States appendix A These data types will be discussed in this chapter PTC uses the following type declarations for analyzing accelerator properties type real 8 for polymorphs type pol block for polymorphic blocks 26 MODELING AN ACCELERATOR WITH PTC type normalform for normal forms type damap for differential algebra maps type taylor for Taylor maps type gmap for a vector of Taylor maps These latter data types will be described in chapter 5 Taylor Polymorphism and Knobs The global variable Lmax defines the maximum length here 100 meters of an integration node For more information about splitting elemen
174. to second order integration with one step THIN LENS RESPLIT R THIN EVEN LIM 1 2 LMAXO XBEND USEKNOB This routine splits all the magnets in the lattice into integration nodes or thin lenses To restrict the action of THIN LENS RESTART and THIN LENS RESPLIT to specific fibres or groups of magnets use the optional arguments FIB and USEKNOB For more information see Section 9 3 9 3 SPLITTING THE LATTICE 87 Arguments This section describes the arguments for the THIN_LENS_RESTART and THIN_LENS_RESPLIT routines or Argument The name of the layout to be split Optional Argument THIN The optional argument THIN describes an approximate integrated quadrupole strength for which a single integration node or thin lens in the body of an el ement should be used For example the quadrupoles QF and QD in Example 1 below have an integrated strength of KF L 0 279309637578521 KD L 0 197159744173073 Example 1 For our build psr layout see build_PSR page 26 we make following calls CALL MOVE_TO R1 QF QF POS CALL MOVE TO R1 QD QD PO0S 0 0100 LIMITS 1 2 100000 CALL THIN LENS RESPLIT R1 THIN LIM LIMITS WRITE 6 0 QF MAG P NST WRITE 6 QD MAG sP METHOD 00 6 5 NST is the number of integration steps Table 9 1 shows the results Element Method Steps Kicks Step Total Kicks QF 2 27 1 27 QD 2
175. ts into integration nodes see Symplectic Integration and Splitting chapter 9 This initial code also on line 29 initializes a layout this must be done before starting PTC s Gino based graphical user interface for Windows 3 3 SUBROUTINES The end of the ptc geometry 90 source file contains three subroutines we use to create the DNA sequences non trackable layouts for our DNA database Figure 3 3 shows the layouts that the three subroutines create In subsequent sections of this chapter we describe how to use the layouts from this database or pieces of these layouts to form the accelerator models of figure 3 2 e amp build PSR build PSR minus build Quad for Bend Figure 3 3 Subroutines for creating DNA sequences build PSR The subroutine build PSR creates the basic ten cell ring lattice shown on the left in figure 3 3 490 Subroutine build PSR PSR use run madx use pointer lattice implicit none 495 type layout target PSR real dp ang brho kd kf Larc type fibre b 41 42 qd qf type layout cell call make states false exact model true default default nocavity exactmis 505 Call update states madlength false ang twopi 36 d0 360 d0 Larc 2 54948d0 510 brho 1 2d0 Larc ang call set mad brho brho method 2 step 10 madkind2 drift kick drift kf 2 7240 brho 515 kd 1 92d0 brho d1
176. tter solution separates the magnets from the linked list that tracks the path of particles through the magnets PTC does this with a linked list of containers called fibres each with a pointer to the appropriate magnet Adjustments to the position or strength of a magnet are made once and are automatically taken into account each time a container in the linked list points to that magnet We discuss the linked list of containers with pointers to magnets in Modeling Accelerator Topologies 2 2 M1 M2 M3 M4 What about a collider Figure 2 9 based on the intersecting rings of the Relativistic Heavy Ion Collider RHIC at Brookhaven National Laboratory BNL shows particles circulating in both directions forward and backward propagation through some of the same magnets For example the particles following the blue path go though one set of magnets in the sequence M1 M2 M3 M4 Meanwhile the 2 2 MODELING ACCELERATOR TOPOLOGIES 13 particles following the yellow path traverse the same set of magnets in the reverse order M4 M3 M2 M1 Using an array or a linked list with clones of magnets creates the same problems as before and PTC alleviates the difficulties by using doubly linked lists of containers that follow the particle paths containers which have inside them pointers to the appropriate magnets Changes to the strength or position of say magnet M3 are made once in the object for magnet M3 and are immediately ref
177. urrent frames are identical so this girder is in its design location Later if we ask PTC to misalign this girder it will modify girder_frame b and girder_frame exi leaving girder and girder_frame ent to record the design location and orientation of our girder In the following block of code we also define reference frames for the two siamese In this case we choose to locate the siamese frames along the axis of the common straight section five meters from the center of that straight and oriented parallel to the girder s frame of reference See the cross hairs in figure 4 5 Consider first the left hand siamese We compute in lines 415 416 the desired origin of our siamese frame Then in the following two lines we move fibre pointer b to bend 67 and allocate the necessary memory Here is where a siamese and girder differ When PTC allocates a girder frame by setting girder true as in line 410 above it allocates memory for the data a ent b and exi For the siamese frame the default as in line 418 below is girder false PTC allocates memory for the data d and angle which specify not the frame itself but rather how to get there translation and angle from the entrance frame of the local element Our next step is therefore to compute d and angle This we do in line 419 with a call to find_patch The first two arguments are respectively the origin and basis of bend 67 s entrance frame The second two arguments are
178. violating the element s physical and mathematical integrity To explain how PTC does this we use an analogy with LEGO blocks Each different type of beamline element in an accelerator is analogous to a different type of LEGO block in the sense that LEGO blocks are self contained objects A magnet for example is a self contained object that produces a local field that affects local particle trajectories Local geometric considerations e g the shape and symmetry of the magnetic field determine the co rdinate system and frame of reference for a local Hamiltonian that describes the magnet We choose the co rdinate system and frame of reference so as to push particles through the element as simply as possible If individual magnets have conflicting geometries we patch them together In other words we use local co rdinate transformations to connect particle trajectories between the differing geometries The rest of the accelerator does not and should not affect our choice of reference frame for any given beamline element LOCAL CO RDINATE SYSTEM attached to each beamline element permits us to propagate physical quantities across the elements That internal structure is the province of the integrator In addition each element includes three reference frames a reference frame at the entrance a reference frame at the midpoint the element reference frame and a reference frame at the exit See figure 2 1 These extra frames used by PT
179. we point p1 to the fibre containing the short drift near the end of layout L4 We then append to Fig8 that fibre and the next two During the calls to append_point PTC will add the appropriate information to the dokos of the corresponding elements pl gt L4 end previous previous write 6 1 call append_point Fig8 1 180 pl gt pl next write 6 pl mag name call append_point Fig8 p1 pl gt pl next write 6 1 185 call append point Fig8 p1 Finally in the last block of code for Fig8 we give our new layout a formal name Figure Eight ensure that it is topologically closed and apply any necessary patches The call to check need patch line 194 returns the integer pos equal to zero if no patch is needed A non zero value indicates the type of patch required By using check need patch we can apply patches only where necessary Our figure eight lattice requires patching because some of the constituent elements are traversed in the reverse direction In particular there are adjacent fibres f which have opposite values of f dir This happens between the short drift at end of the common straight section and the first bend of the upper ring and also between the last bend of the upper ring and the subsequent short drift write 6 Fig8 has Fig8 n fibres Fig8 name Figure Eight Fig8 closed true 38 MODELING AN ACCELERATOR WITH PTC Figure 3 8 Colli
180. xhibits identical properties under identical conditions A significant virtue of our computer based element however is that it is free to report about what is happening inside Geometric Transformations Now that we have defined our two basic types of beamline elements drift and bend the next step is to fit them together We need geometric transformations that connect the exit reference frame of one beamline element to the entrance reference frame of a subsequent element Continuing the LEGO block analogy we put the individual LEGO blocks on a base which represents the accelerator model s global frame Once we know the location of each LEGO block with respect to the global frame we know their locations with respect to to one another Figure 2 4 shows two such LEGO blocks and their reference frames on a base which represents the layout of beamline elements in the global frame of an accelerator With the LEGO blocks now on the global frame we require transformations that translate the phase space co rdinates in the exit reference frame of one LEGO block to the phase space co rdinates in the entrance reference frame of the next LEGO block To construct for example a recirculating accelerator out of LEGO blocks we connect bends and drifts one after another to model the desired machine On reaching the last LEGO block we require that the last block s exit face coincide with the first entrance face This means that 2 1 TRACKING PARTIC
181. yout 113 imax global variable 26 local coordinate system defined 6 local information see also local prop erty analysis 17 defined 17 local property see also local informa tion described 60 LOCATE_TEMPORAL_BEAM routine 112 m_u global variable 25 31 MAD universe see PTC universe MAD8 discussed 72 MAD_UNIVERSE data type 73 madlength flag 27 madx_ptc_module module 25 magnet in girder 43 siamese 43 magnet frame defined 106 magnet based tracking see s based track ing MAGNET_FRAME data type 106 magnets linking together 43 moving as group 43 MAKE_STATES routine 101 make_states routine 27 map see one turn map Poincar map or Taylor map map based methods 17 one turn map 141 global 17 matrix kick matrix integration methods 86 96 MISALIGN FIBRE routine 81 MISALIGN GIRDER routine 77 misalign girder routine 47 MISALIGN SIAMESE routine 79 misalign siamese routine 47 misalignment contained in chart 107 described 16 element 77 exact 82 girder 43 44 77 inexact 83 internal state 101 routines 77 siamese 43 44 77 misalignment routines described 77 modeling accelerator topologies 11 12 23 32 particle interactions 19 module madx_ptc_module 25 momenta adjusting 112 move_to routine 44 nocavity flag 101 node see integration node node layout defined 20 113 NODE_LAYOUT data type 113 normal form computing 54 FPP
182. yout L1 with the 70 elements shown for the PSR in figure 3 3 We repeat this process five times for layouts L2 L3 L4 L5 and L6 For L2 we call build_Quad_for_Bend which populates that layout with 10 elements the straight bends For L3 we call build_PSR_minus which populates that layout with 64 elements And for L4 L5 and L6 we repeat the calls to build_PSR and build_PSR_minus call append_empty_layout m_u DNA sequence 2 call set_up m_u end L2 gt m_u end 45 Call build Quad for Bend L2 call append empty layout m u DNA sequence 3 call set up m L3 gt m_u end 50 Call build PSR minus L3 call append empty layout m u DNA sequence 4 call set up m L4 gt m_u end 55 Call build_PSR L4 call append_empty_layout m_u DNA sequence 5 call set_up m_u end L5 gt m_u end Table 3 1 DNA database for the PTC geometry tutorial 8 Because we want to be able to modify move mis power etc the individual elements We do not want changes made to COL1 for example to appear in Fig8 32 MODELING AN ACCELERATOR WITH PTC Figure 3 5 Ring with forward and re verse propagation 9 The point of this compli cation is to illustrate some of PTC s geometry oper ations and the process of patching It should also we hope emphasize the LEGO block concepts of PTC cf figure 2 3 and the associated discussion Call build PSR minus L5 call

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