[430] | 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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