1 | <head> |
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2 | <title>Local Reference System</title> |
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3 | <!-- Changed by: Chris ISELIN, 17-Jul-1997 --> |
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4 | <!-- Changed by: Hans Grote, 25-Sep-2002 --> |
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5 | </head> |
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6 | |
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7 | <body bgcolor="#ffffff"> |
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8 | |
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9 | <center> |
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10 | EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH |
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11 | <IMG SRC="http://cern.ch/madx/icons/mx7_25.gif" align=right> |
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12 | <h2>Local Reference Systems</h2> |
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13 | </center> |
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14 | |
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15 | <h2><a name=straight>Reference System for Straight Beam Elements</a></h2> |
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16 | In straight elements the local reference system is simply translated |
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17 | by the length of the element along the local <i>s</i>-axis. |
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18 | This is true for |
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19 | <ul> |
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20 | <li><a href="drift.html">Drift space</a>, |
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21 | <li><a href="quadrupole.html">Quadrupole</a>, |
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22 | <li><a href="sextupole.html">Sextupole</a>, |
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23 | <li><a href="octupole.html">Octupole</a>, |
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24 | <li><a href="solenoid.html">Solenoid</a>, |
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25 | <li><a href="crabcavity.html">CRABCAVITY</a>, |
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26 | <li><a href="cavity.html">RF cavity</a>, |
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27 | <li><a href="separator.html">Electrostatic separator</a>, |
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28 | <li><a href="kickers.html">Closed orbit corrector</a>, |
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29 | <li><a href="monitors.html">Beam position monitor</a>. |
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30 | </ul> |
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31 | The corresponding <i>R</i>, <i>S</i> are |
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32 | <p> |
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33 | <img align=bottom src="../equations/RSstraight.gif"> |
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34 | <p> |
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35 | A rotation of the element about the <i>S</i>-axis has no effect |
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36 | on <i>R</i> and <i>S</i>, since the rotations of the reference system |
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37 | before and after the element cancel. |
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38 | <center> |
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39 | <img align=bottom src="../figures/ref_straight.gif"> |
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40 | <p> |
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41 | <b>Figure 1:</b> Reference System for Straight Beam Elements |
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42 | </center> |
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43 | |
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44 | <h2><a name=rbend>Reference System for Bending Magnets</a></h2> |
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45 | <a href="bend.html">Bending magnets</a> have a curved reference orbit. |
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46 | For both rectangular and sector bending magnets |
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47 | <p> |
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48 | <img align=bottom src="../equations/RSbend.gif"> |
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49 | <p> |
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50 | where alphais the bend angle. |
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51 | A positive bend angle represents a bend to the right, |
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52 | i.e. towards negative <i>x</i> values. |
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53 | For sector bending magnets, |
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54 | the bend radius is given by rho, |
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55 | and for rectangular bending magnets it has the value |
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56 | <p> |
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57 | rho = <i>L</i> / 2 sin(alpha/2). |
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58 | <p> |
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59 | If the magnet is rotated about the <i>s</i>-axis by an angle psi, |
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60 | <i>R</i> and <i>S</i> are transformed by |
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61 | <p> |
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62 | <i>R</i>* = <i>T R</i>, <i>S</i>* = <i>T S T<sup>-1</sup></i>. |
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63 | <p> |
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64 | where <i>T</i> is the orthogonal rotation matrix |
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65 | <p> |
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66 | <img align=bottom src="../equations/Trot.gif"> |
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67 | <p> |
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68 | The special value psi = pi/2 represents a bend down. |
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69 | |
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70 | <center> |
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71 | <a name=rbend><img align=bottom src="../figures/ref_rbend.gif"></a> |
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72 | <p> |
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73 | <b>Figure 2:</b> Reference System for Rectangular Bends; |
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74 | The signs of the pole-face rotations are positive as shown. |
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75 | </center> |
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76 | |
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77 | <center> |
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78 | <a name=sbend><img align=bottom src="../figures/ref_sbend.gif"></a> |
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79 | <p> |
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80 | <b>Figure 3:</b> Reference System for Sector Bends; |
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81 | The signs of the pole-face rotations are positive as shown. |
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82 | </center> |
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83 | |
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84 | <h2>Elements which do not Change the Local Reference</h2> |
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85 | <a href="marker.html">MARKER</a> elements do not |
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86 | affect the reference orbit. |
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87 | They are ignored for geometry calculations. |
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88 | |
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89 | <p> |
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90 | <address> |
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91 | <a href="http://www.cern.ch/Hans.Grote/hansg_sign.html">hansg</a>, |
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92 | January 24, 1997 |
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93 | </address> |
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94 | |
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95 | </body> |
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