|
The particle tracking is not done via
the Turtle-native matrix formalism, but by solving the differential equation of motion in
each short solenoid segment with the 4th order Runge-Kutta method (see F. Foroughi: Equation of motion of a
charged particle in a magnetic and electrical field. PSI internal report, may 1990). The
type code element 28 (Solenoid) is defined as following:
where dz is the length of the solenoid segment, and Bzi and Bri are the solenoid field expansion coefficients for the solenoid magnetic field in z- and r-direction at the beginning of each solenoid segment:
where AR is the aperture radius. If AR is negative, then the aperture constraint is not observed for this solenoid segment. The above formulas for the longitudinal and radial solenoid field expansions are empirical, but probably approximate values extracted from solenoid field maps of solenoids without iron shielding computed with TOSCA to an accuracy of better than a percent up to the aperture radius at all z-coordinates. Between the different type code 28. instructions there is no limitation on intervening instructions of any type code (e.g. histograms or a whole degrader. There are no checks; the user himself has to decide if these intervening instructions inside the solenoid are making sense.) In order to scale a given solenoid field-map computed for an arbitrary solenoid field strength (given by the values Bzi and Bri) to a desired strength the following type code 16 has been added:
All used solenoid field strengths are multiplied with this factor. The coefficients Bzi and Bri may be computed with MINUIT-fits by using axial and radial distributions read out from a computed or measured (x,y,z)-map file.
As an example an ironless solenoid of 1 m length and 20 cm diameter is used for focusing
protons of 0.1 GeV/c at a distance of 2.41 m (symmetric layout). The DOS-formatted solenoid field map
sol10.mfo (has to be converted to binary form {sol10.map} with mfotomap.exe
[Windows], mfotomap [Linux] or mfotomap.app [Mac OS X - X11]) was computed with
the program TOSCA by V. Vrankovic (PSI), who also furnished the sample code mapread (for Linux),
mapread.exe (for Windows) and mapread.app (for Mac OS X - X11) to read out the (Bx,By,Bz)-field
vector at any (x,y,z)-point (view and download).
The code mapread was modified (mapread2 for Linux, mapread2.exe for Windows or mapread2.app
for Mac OS X - X11) to give either r- (= x-direction at y=0) or z-field distributions for different z-coordinates
suitable as input files for MINUIT in order to fit the needed coefficients either in z-
(view and download)
or x- (view and download)
direction. The tailored MINUIT routines fitx.exe / fitz.exe (for Windows), fitx /
fitz (for Linux) or fitx.app / fitz.app (for Mac OS X -X11) was furnished by
R. Scheuermann (PSI). For the different fitx.dat / fitz.dat (= xnnn.lis / znnn.lis) data files
the input files fitx.in / fitz.in may have to be edited in order to adjust the 3 start values
and step sizes for different z-coordinates.
(With ASCII files under Linux and Mac OS X - X11 pay attention to have only LF at the end of each line.
The programs crlftolf.exe [Windows] crlftolf [Linux] and crlftolf.app
[Mac OS X - X11] transform DOS/Windows ASCII-formatted files into UNIX-formatted ASCII files {without CR},
whereas the opposite is done with the program lftocrlf.exe under Windows.)
Sample run of the program mapread2 (mapread2.exe or mapread2.app):
$ crlftolf sol10.mfo (only needed for Linux and Mac OS X - X11)
sol10.mfo: initial length = 789002, new length = 772214
16788 <CR>s found in file
$ mfotomap sol10.mfo sol10.map
conversion done.
$ mapread2 sol10.map
Rotation angle -> 0.0 (rot.all)
Reflection indices -> 1 1 -1
Reference point -> 0.000 0.000 0.000
Starting point -> -140.000 -140.000 -118.000
End point -> 140.000 140.000 118.000
Step size -> 1.000 1.000 1.000
Input direction x or z (def=x):
Input z (def= 0.0): 55.
Input x0, xmax, xinc (def= 0. 10. 1.)
0.0 0.00 0.01
1.0 12.91 0.13
2.0 25.85 0.26
3.0 38.81 0.39
4.0 51.78 0.52
5.0 64.67 0.65
6.0 77.26 0.77
7.0 89.18 0.89
8.0 99.82 1.00
9.0 108.30 1.08
10.0 113.62 1.14
Input direction x or z (def=x):
.
.
.
Turtle input file for the solenoid represented by the map sol10.map. Note that the z-field is symmetric to the midplane whereas the r-field is anti-symmetric to this plane (This means that the type-code 28 sequence starts with negative radial solenoid field values because the axial solenoid field values are positive).
/Test solenoid ray tracing tc28/ 1000000 1. .1 150. .1 150. 0. 0.0 0.1 /BEAM/ ; 13. 10. ; 16. 190. 0. 100. ; 16. 50. 0.01445 /SCAL/ ; 28. 0.08 4.04 -0.0015 0.0 -0.0637 0.0 0.210 -10.0 /H1/ ; 28. 0.10 5.30 -0.0023 0.0 -0.0924 0.0 0.210 -10.0 /H2/ ; 28. 0.10 7.81 -0.0048 0.000002 -0.158 0.0 0.210 -10.0 /H3/ ; 28. 0.10 12.35 -0.0109 0.000006 -0.323 0.0416 0.224 -10.0 /H4/ ; 28. 0.10 21.73 -0.0306 0.000026 -0.730 0.168 0.210 -10.0 /H5/ ; 28. 0.05 45.11 -0.1093 0.000139 -2.096 0.832 0.202 -10.0 /H6/ ; 28. 0.05 71.26 -0.2295 0.000326 -3.959 1.868 0.203 -10.0 /H7/ ; 28. 0.05 121.50 -0.4844 0.000537 -7.514 2.007 0.252 -10.0 /H8/ ; 28. 0.03 172.70 -0.6834 -0.000026 -10.56 0.602 0.372 -10.0 /H9/ ; 28. 0.02 219.05 -0.7169 -0.00185 -12.98 0.0128 0.715 -10.0 /H10/ ; 28. 0.03 305.56 -0.0771 -0.009892 -16.37 -0.5038 0.483 10.0 /H11/ ; 28. 0.02 374.73 0.0 0.0 -17.01 -0.427 0.585 10.0 /H12/ ; 28. 0.03 474.34 0.4393 0.006371 -15.40 -0.5266 0.390 10.0 /H13/ ; 28. 0.02 530.47 0.7066 0.00197 -12.97 0.0129 0.715 10.0 /H14/ ; 28. 0.03 596.07 0.6185 -0.000278 -9.467 1.131 0.314 10.0 /H15/ ; 28. 0.02 627.83 0.4838 -0.000523 -7.49 1.954 0.254 10.0 /H16/ ; 28. 0.05 677.88 0.2303 -0.000325 -3.889 1.600 0.213 10.0 /H17/ ; 28. 0.05 703.77 0.1104 -0.000140 -2.051 0.740 0.210 10.0 /H18/ ; 28. 0.05 718.03 0.0569 -0.000059 -1.150 0.343 0.210 10.0 /H19/ ; 28. 0.05 726.33 0.0321 -0.000027 -0.674 0.163 0.210 10.0 /H20/ ; 28. 0.05 731.32 0.0197 -0.000013 -0.403 0.0846 0.210 10.0 /H21/ ; 28. 0.05 734.28 0.0134 -0.000005 -0.230 0.0723 0.210 10.0 /H22/ ; 28. 0.05 735.85 0.0107 -0.000005 -0.1 0.0010 0.210 10.0 /H23/ ; 28. 0.05 736.34 0.0098 -0.000004 0.0 0.0 0.210 10.0 /H24/ ; 28. 0.05 735.85 0.0107 -0.000005 0.1 -0.0010 0.210 10.0 /H23/ ; 28. 0.05 734.28 0.0134 -0.000005 0.230 -0.0723 0.210 10.0 /H22/ ; 28. 0.05 731.32 0.0197 -0.000013 0.403 -0.0846 0.210 10.0 /H21/ ; 28. 0.05 726.33 0.0321 -0.000027 0.674 -0.163 0.210 10.0 /H20/ ; 28. 0.05 718.03 0.0569 -0.000059 1.150 -0.343 0.210 10.0 /H19/ ; 28. 0.05 703.77 0.1104 -0.000140 2.051 -0.740 0.210 10.0 /H18/ ; 28. 0.05 677.88 0.2303 -0.000325 3.889 -1.600 0.213 10.0 /H17/ ; 28. 0.02 627.83 0.4838 -0.000523 7.49 -1.954 0.254 10.0 /H16/ ; 28. 0.03 596.07 0.6185 -0.000278 9.467 -1.131 0.314 10.0 /H15/ ; 28. 0.02 530.47 0.7066 0.00197 12.97 -0.0129 0.715 10.0 /H14/ ; 28. 0.03 474.34 0.4393 0.006371 15.40 +0.5266 0.390 10.0 /H13/ ; 28. 0.02 374.73 0.0 0.0 17.01 +0.427 0.585 10.0 /H12/ ; 28. 0.03 305.56 -0.0771 -0.009892 16.37 +0.5038 0.483 10.0 /H11/ ; 28. 0.02 219.05 -0.7169 -0.00185 12.98 -0.0128 0.715 -10.0 /H10/ ; 28. 0.03 172.70 -0.6834 -0.000026 10.56 -0.602 0.372 -10.0 /H9/ ; 28. 0.05 121.50 -0.4844 0.000537 7.514 -2.007 0.252 -10.0 /H8/ ; 28. 0.05 71.26 -0.2295 0.000326 3.959 -1.868 0.203 -10.0 /H7/ ; 28. 0.05 45.11 -0.1093 0.000139 2.096 -0.832 0.202 -10.0 /H6/ ; 28. 0.10 21.73 -0.0306 0.000026 0.730 -0.168 0.210 -10.0 /H5/ ; 28. 0.10 12.35 -0.0109 0.000006 0.323 -0.0416 0.224 -10.0 /H4/ ; 28. 0.10 7.81 -0.0048 0.000002 0.158 0.0 0.210 -10.0 /H3/ ; 28. 0.10 5.30 -0.0023 0.0 0.0924 0.0 0.210 -10.0 /H2/ ; 28. 0.08 4.04 -0.0015 0.0 0.0637 0.0 0.210 -10.0 /H1/ ; 28. 0.0 4.04 -0.0015 0.0 0.0637 0.0 0.210 -10.0 /H0/ ; 51. 1. -10. 10. 1. ; 52. 3. -10. 10. 1. ; 51. 1. -.5 .5 .025 ; 52. 3. -.5 .5 .025 ; 51. 1. -.5 .5 .025 ; 52. 2. -150. 150. 7.5 ; 51. 3. -.5 .5 .025 ; 52. 4. -150. 150. 7.5 ; SENTINEL SENTINEL
The following picture shows the 4 requested contour plots. Well recognizable are
the spiraling shapes of the contour lines in the (x,x')- and (y,y')-phase space,
which are due to higher order aberrations. The spot size diameter is almost 2 times
larger than with first order computations only (see below).
In order to compare the above results with computations done for the same solenoid represented via the traditional type code 19 (1st order only), the following input code may be used:
/Test solenoid ray tracing tc19/ 100000 1. .1 150. .1 150. .0 .0 0.1 /BEAM/ ; 16. 190. 0. 100. ; 13. 10. ; 3. .25 ; 3. .25 ; 3. .18 ; 6. 1. 10. 3. 10. ; 19. 0. 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 6. 1. 10. 3. 10. ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. .1 10.211 /H/ ; 19. 0. 10.211 /H/ ; 6. 1. 10. 3. 10. ; 3. .25 ; 3. .25 ; 3. .18 /END/ ; 51. 1. -10. 10. 1. ; 52. 3. -10. 10. 1. ; 51. 1. -.5 .5 .025 ; 52. 3. -.5 .5 .025 ; 51. 1. -.5 .5 .025 ; 52. 2. -150. 150. 7.5 ; 51. 3. -.5 .5 .025 ; 52. 4. -150. 150. 7.5 ; SENTINEL SENTINEL
The following picture shows the 4 requested contour plots:
Addendum:
If paraxial rays represented with the beam- and shift-instructions:
1.0 .001 .001 .001 .001 .0 .0 .1 /BEAM/ ; 7.0 x0. 0. y0. 0. 0. 0. ;are tracked through the solenoid, then the location of the picture points at the end of the solenoid field are rotated by about 90 degrees around the z-axis. This corresponds to the formula [see A. J. Dragt: Numerical third-order transfer map for solenoid in Nuclear Instruments and Methods in Physics Research A298 (1990) 441-459, formulas (15), (37) and (50)]:
Theta = Integral(Bz * dz) / (2 * Bero) = (735.85 * 0.01445 kG * 1 m) / (2. * 33.33 * 0.1 kGm) = 1.6 (=90 deg)
Back to:Recent Turtle modification history
Last updated by
Urs Rohrer on 8-Feb-2006