[430] | 1 | <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> |
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| 2 | <html> |
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| 3 | <head> |
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| 4 | <title>Variables in MAD</title> |
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| 5 | <!-- Changed by: Chris ISELIN, 17-Jul-1997 --><!-- Changed by: Hans Grote, 10-Jun-2002 --> |
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| 6 | </head> |
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| 7 | <body bgcolor="#ffffff"> |
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| 8 | <center> |
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| 9 | EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH |
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| 10 | <img SRC="http://cern.ch/madx/icons/mx7_25.gif" align="right"> |
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| 11 | <h2>Variables</h2> |
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| 12 | </center> |
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| 13 | <h5> |
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| 14 | For each variable the physical units are listed in square brackets. |
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| 15 | </h5> |
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| 16 | <h3><a name="canon">Canonical Variables Describing Orbits</a></h3> |
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| 17 | MAD uses the following canonical variables |
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| 18 | to describe the motion of particles: |
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| 19 | <ul> |
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| 20 | <li>X: |
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| 21 | Horizontal position <i>x</i> of the (closed) orbit, |
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| 22 | referred to the ideal orbit [m]. |
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| 23 | </li> |
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| 24 | <li>PX: |
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| 25 | Horizontal canonical momentum <i>p<sub>x</sub></i> |
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| 26 | of the (closed) orbit referred to the ideal orbit, |
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| 27 | divided by the reference momentum: |
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| 28 | PX = <i>p<sub>x</sub> / p<sub>0</sub></i>, [1]. |
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| 29 | </li> |
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| 30 | <li>Y: |
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| 31 | Vertical position <i>y</i> of the (closed) orbit, |
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| 32 | referred to the ideal orbit [m]. |
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| 33 | </li> |
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| 34 | <li>PY: |
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| 35 | Vertical canonical momentum <i>p<sub>y</sub></i> |
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| 36 | of the (closed) orbit referred to the ideal orbit, |
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| 37 | divided by the reference momentum: |
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| 38 | PY = <i>p<sub>x</sub> / p<sub>0</sub></i>, [1]. |
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| 39 | </li> |
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| 40 | <li>T: |
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| 41 | Velocity of light times the negative time difference |
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| 42 | with respect to the reference particle: |
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| 43 | T = <i> - c t</i>, [m]. |
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| 44 | A positive T means that the particle arrives ahead |
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| 45 | of the reference particle. |
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| 46 | </li> |
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| 47 | <li>PT: |
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| 48 | Energy error, divided by the reference momentum times the velocity |
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| 49 | of light: |
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| 50 | PT = delta(<i>E</i>) / <i>p<sub>s</sub> c</i>, [1]. |
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| 51 | This value is only non-zero when synchrotron motion is present. |
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| 52 | It describes the deviation of the particle from the orbit of a |
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| 53 | particle with the momentum error DELTAP. |
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| 54 | </li> |
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| 55 | <li>DELTAP: |
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| 56 | Difference of the reference momentum and the design momentum, |
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| 57 | divided by the reference momentum: |
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| 58 | DELTAP = delta(<i>p</i>) / <i>p<sub>0</sub></i>, [1]. |
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| 59 | This quantity is used to <a href="defects.html">normalize</a> |
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| 60 | all element strengths. |
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| 61 | </li> |
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| 62 | </ul> |
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| 63 | The independent variable is: |
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| 64 | <ul> |
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| 65 | <li><a name="s">S</a>: |
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| 66 | Arc length <i>s</i> along the reference orbit, [m]. |
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| 67 | </li> |
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| 68 | </ul> |
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| 69 | In the limit of fully relativistic particles |
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| 70 | (gamma >> 1, <i>v = c</i>, <i>p c = E</i>), |
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| 71 | the variables T, PT used here agree with the longitudinal variables |
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| 72 | used in <a href="bibliography.html#transport">[TRANSPORT]</a>. |
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| 73 | This means that T becomes the negative path length difference, |
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| 74 | while PT becomes the fractional momentum error. |
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| 75 | The reference momentum <i>p<sub>s</sub></i> |
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| 76 | must be constant in order to keep the system canonical. |
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| 77 | <h3><a name="normal">Normalised Variables and other Derived Quantities</a></h3> |
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| 78 | <ul> |
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| 79 | <li>XN: |
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| 80 | The normalised horizontal displacement |
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| 81 | <p>XN = <i>x<sub>n</sub></i> = Re(<i>E<sub>1</sub><sup>T</sup> S Z</i>), |
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| 82 | [sqrt(m)]. |
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| 83 | </p> |
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| 84 | <p></p> |
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| 85 | </li> |
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| 86 | <li>PXN: |
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| 87 | The normalised horizontal transverse momentum |
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| 88 | <p>PXN = <i>x<sub>n</sub></i> = Im(<i>E<sub>1</sub><sup>T</sup> S Z</i>), |
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| 89 | [sqrt(m)]. |
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| 90 | </p> |
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| 91 | <p></p> |
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| 92 | </li> |
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| 93 | <li>WX: |
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| 94 | The horizontal Courant-Snyder invariant |
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| 95 | <p>WX = sqrt(<i>x<sub>n</sub><sup>2</sup> + p<sub>xn</sub><sup>2</sup></i>), |
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| 96 | [m]. |
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| 97 | </p> |
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| 98 | <p></p> |
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| 99 | </li> |
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| 100 | <li>PHIX: |
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| 101 | The horizontal phase |
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| 102 | <p>PHIX = - atan(<i>p<sub>xn</sub> / x<sub>n</sub></i>) / 2 pi [1]. |
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| 103 | </p> |
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| 104 | <p></p> |
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| 105 | </li> |
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| 106 | <li>YN: |
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| 107 | The normalised vertical displacement |
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| 108 | <p>YN = <i>x<sub>n</sub></i> = Re(<i>E<sub>2</sub><sup>T</sup> S Z</i>), |
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| 109 | [sqrt(m)]. |
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| 110 | </p> |
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| 111 | <p></p> |
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| 112 | </li> |
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| 113 | <li>PYN: |
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| 114 | The normalised vertical transverse momentum |
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| 115 | <p>PYN = <i>x<sub>n</sub></i> = Im(<i>E<sub>2</sub><sup>T</sup> S Z</i>), |
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| 116 | [sqrt(m)]. |
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| 117 | </p> |
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| 118 | <p></p> |
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| 119 | </li> |
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| 120 | <li>WY: |
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| 121 | The vertical Courant-Snyder invariant |
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| 122 | <p>WY = sqrt(<i>y<sub>n</sub><sup>2</sup> + p<sub>yn</sub><sup>2</sup></i>), |
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| 123 | [m]. |
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| 124 | </p> |
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| 125 | <p></p> |
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| 126 | </li> |
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| 127 | <li>PHIY: |
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| 128 | The vertical phase |
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| 129 | <p>PHIY = - atan(<i>p<sub>yn</sub> / y<sub>n</sub></i>) / 2 pi [1]. |
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| 130 | </p> |
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| 131 | <p></p> |
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| 132 | </li> |
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| 133 | <li>TN: |
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| 134 | The normalised longitudinal displacement |
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| 135 | <p>TN = <i>x<sub>n</sub></i> = Re(<i>E<sub>3</sub><sup>T</sup> S Z</i>), |
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| 136 | [sqrt(m)]. |
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| 137 | </p> |
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| 138 | <p></p> |
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| 139 | </li> |
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| 140 | <li>PTN: |
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| 141 | The normalised longitudinal transverse momentum |
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| 142 | <p>PTN = <i>x<sub>n</sub></i> = Im(<i>E<sub>3</sub><sup>T</sup> S Z</i>), |
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| 143 | [sqrt(m)]. |
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| 144 | </p> |
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| 145 | <p></p> |
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| 146 | </li> |
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| 147 | <li>WT: |
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| 148 | The longitudinal invariant |
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| 149 | <p>WT = sqrt(<i>t<sub>n</sub><sup>2</sup> + p<sub>tn</sub><sup>2</sup></i>), |
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| 150 | [m]. |
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| 151 | </p> |
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| 152 | <p></p> |
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| 153 | </li> |
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| 154 | <li>PHIT: |
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| 155 | The longitudinal phase |
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| 156 | <p>PHIT = + atan(<i>p<sub>tn</sub> / t<sub>n</sub></i>) / 2 pi [1]. |
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| 157 | </p> |
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| 158 | <p></p> |
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| 159 | </li> |
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| 160 | </ul> |
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| 161 | in the above formulas <i>Z</i> is the phase space vector |
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| 162 | <p><i>Z = ( x, p<sub>x</sub>, y, p<sub>y</sub>, t, p<sub>t</sub>)<sup>T</sup></i>. |
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| 163 | </p> |
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| 164 | <p>the matrix <i>S</i> is the ``symplectic unit matrix'' |
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| 165 | </p> |
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| 166 | <p><img src="../equations/S_matrix.gif" align="bottom"> |
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| 167 | </p> |
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| 168 | <p>and the vectors <i>E<sub>i</sub></i> are the three complex |
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| 169 | eigenvectors. |
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| 170 | </p> |
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| 171 | <h3><a name="linear">Linear Lattice Functions (Optical Functions)</a></h3> |
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| 172 | Several MAD commands refer to linear lattice functions. |
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| 173 | Since MAD uses the canonical momenta (<i>p<sub>x</sub></i>, <i>p<sub>y</sub></i>) |
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| 174 | instead of the slopes |
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| 175 | (<i>x</i>', <i>y</i>'), |
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| 176 | their definitions differ slightly from those |
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| 177 | in <a href="bibliography.html#courant">[Courant and Snyder]</a>. |
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| 178 | Notice that in MAD-X PT substitutes DELTAP as longitudinal variable. |
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| 179 | Dispersive and chromatic functions are hence derivatives with respects |
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| 180 | to PT. Being PT=BETA*DELTAP, where BETA is the relativistic Lorentz |
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| 181 | factor, those functions must be multiplied by BETA a number of time |
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| 182 | equal to the order of the derivative. The linear lattice functions are |
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| 183 | known to MAD under the following names: |
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| 184 | <ul> |
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| 185 | <li>BETX: |
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| 186 | Amplitude function beta<sub><i>x</i></sub>, [m]. |
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| 187 | </li> |
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| 188 | <li>ALFX: |
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| 189 | Correlation function alpha<sub><i>x</i></sub>, [1]: |
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| 190 | <p>ALFX = alpha<sub><i>x</i></sub> = - 1/2 * (del beta<sub><i>x</i></sub> |
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| 191 | / del <i>s</i>). |
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| 192 | </p> |
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| 193 | <p></p> |
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| 194 | </li> |
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| 195 | <li>MUX: |
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| 196 | Phase function mu<sub><i>x</i></sub>, [2pi]: |
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| 197 | <p>MUX = mu<sub><i>x</i></sub> = integral (d<i>s</i> / beta<sub><i>x</i></sub>). |
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| 198 | </p> |
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| 199 | <p></p> |
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| 200 | </li> |
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| 201 | <li>DX: |
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| 202 | Dispersion <i>D<sub>x</sub></i> of <i>x</i>, [m]: |
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| 203 | <p>DX = <i>D<sub>x</sub></i> = (del <i>x</i> / del PT). |
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| 204 | </p> |
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| 205 | <p></p> |
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| 206 | </li> |
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| 207 | <li>DPX: |
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| 208 | Dispersion <i>D<sub>px</sub></i> of <i>p<sub>x</sub></i>, [1]: |
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| 209 | <p>DPX = <i>D<sub>px</sub></i> = (del <i>p<sub>x</sub></i> / del |
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| 210 | PT) / <i>p<sub>s</sub></i>. |
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| 211 | </p> |
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| 212 | <p></p> |
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| 213 | </li> |
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| 214 | <li>BETY: |
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| 215 | Amplitude function beta<sub><i>y</i></sub>, [m]. |
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| 216 | </li> |
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| 217 | <li>ALFY: |
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| 218 | Correlation function alpha<sub><i>y</i></sub>, [1]. |
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| 219 | <p>ALFY = alpha<sub><i>y</i></sub> = - 1/2 * (del beta<sub><i>y</i></sub> |
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| 220 | / del <i>s</i>). |
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| 221 | </p> |
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| 222 | <p></p> |
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| 223 | </li> |
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| 224 | <li>MUY: |
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| 225 | Phase function mu<sub><i>y</i></sub>, [2pi]. |
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| 226 | <p>MUY = mu<sub><i>y</i></sub> = integral (d<i>s</i> / beta<sub><i>y</i></sub>). |
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| 227 | </p> |
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| 228 | <p></p> |
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| 229 | </li> |
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| 230 | <li>DY: |
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| 231 | Dispersion <i>D<sub>y</sub></i> of <i>y</i>, [m]: |
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| 232 | <p>DY = <i>D<sub>y</sub></i> = (del <i>y</i> / del PT). |
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| 233 | </p> |
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| 234 | <p></p> |
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| 235 | </li> |
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| 236 | <li>DPY: |
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| 237 | Dispersion <i>D<sub>px</sub></i> of <i>p<sub>x</sub></i>, [1]: |
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| 238 | <p>DPY = <i>D<sub>py</sub></i> = (del <i>p<sub>y</sub></i> / del |
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| 239 | PT) / <i>p<sub>s</sub></i>. |
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| 240 | </p> |
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| 241 | <p></p> |
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| 242 | </li> |
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| 243 | <li>R11, R12, R21, R22: |
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| 244 | Coupling Matrix |
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| 245 | <p></p> |
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| 246 | </li> |
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| 247 | <li>ENERGY: The total energy per particle in GeV. If given, it must |
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| 248 | be |
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| 249 | greater then the particle mass. |
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| 250 | <p></p> |
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| 251 | </li> |
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| 252 | </ul> |
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| 253 | <!-- The TWISS table also defines the following expressions which --> |
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| 254 | <!-- can be used in plots:--><!--ul--> |
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| 255 | <!--li--><!-- GAMX = (1 + ALFX*ALFX) / BETX, --><!--li--><!-- GAMY = (1 + ALFY*ALFY) / BETY, --> |
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| 256 | <!--li--><!-- SIGX = SQRT(BETX * EX), the vertical r.m.s. half-width of the beam, --> |
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| 257 | <!--li--><!-- SIGY = SQRT(BETY * EY), the vertical r.m.s. half-height of the beam. --><!--/ul--> |
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| 258 | <h3><a name="chrom">Chromatic Functions</a></h3> |
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| 259 | Several MAD commands refer to the chromatic functions. |
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| 260 | (<i>p<sub>x</sub></i>, <i>p<sub>y</sub></i>) instead of the slopes |
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| 261 | (<i>x</i>', <i>y</i>'), |
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| 262 | their definitions differ slightly from those |
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| 263 | in <a href="bibliography.html#montague">[Montague]</a>. |
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| 264 | Notice that in MAD-X PT substitutes DELTAP as longitudinal variable. |
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| 265 | Dispersive and chromatic functions are hence derivatives with respects |
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| 266 | to PT. Being PT=BETA*DELTAP, where BETA is the relativistic Lorentz |
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| 267 | factor, those functions must be multiplied by BETA a number of time |
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| 268 | equal to the order of the derivative. The chromatic functions are known |
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| 269 | to MAD under the following names: |
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| 270 | <p><font color="#ff0000"><i>Please note that this option is needed |
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| 271 | for a proper calculation of the chromaticities in the |
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| 272 | presence of coupling!</i></font></p> |
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| 273 | <ul> |
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| 274 | <li>WX: |
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| 275 | Chromatic amplitude function <i>W<sub>x</sub></i>, [1]: |
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| 276 | <p>WX = <i>W<sub>x</sub></i> = sqrt(<i>a<sub>x</sub><sup>2</sup> + |
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| 277 | b<sub>x</sub><sup>2</sup></i>), |
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| 278 | </p> |
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| 279 | <p><i>a<sub>x</sub></i> = (del beta<sub><i>x</i></sub> / del PT) / |
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| 280 | beta<sub><i>x</i></sub>, |
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| 281 | </p> |
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| 282 | <p><i>b<sub>x</sub></i> = |
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| 283 | (del alpha<sub><i>x</i></sub> / del PT) - (alpha<sub><i>x</i></sub> / |
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| 284 | beta<sub><i>x</i></sub>) * |
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| 285 | (del beta<sub><i>x</i></sub> / del PT). |
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| 286 | </p> |
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| 287 | <p></p> |
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| 288 | </li> |
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| 289 | <li>PHIX: |
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| 290 | Chromatic phase function Phi<sub><i>x</i></sub>, [2pi]: |
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| 291 | <p>PHIX = Phi<sub><i>x</i></sub> = atan(<i>a<sub>x</sub> / b<sub>x</sub></i>). |
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| 292 | </p> |
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| 293 | <p></p> |
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| 294 | </li> |
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| 295 | <li>DMUX: |
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| 296 | Chromatic derivative of phase function mu<sub><i>x</i></sub>, [2pi]: |
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| 297 | <p>DMUX = (del mu<sub><i>x</i></sub> / del PT). |
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| 298 | </p> |
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| 299 | <p></p> |
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| 300 | </li> |
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| 301 | <li>DDX: |
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| 302 | Chromatic derivative of dispersion <i>D<sub>x</sub></i>, [m]: |
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| 303 | <p>DDX = 1/2 * (del<sup>2</sup><i>x</i> / del PT<sup>2</sup>). |
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| 304 | </p> |
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| 305 | <p></p> |
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| 306 | </li> |
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| 307 | <li>DDPX: |
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| 308 | Chromatic derivative of dispersion <i>D<sub>px</sub></i>, [1]: |
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| 309 | <p>DDPX = 1/2 * (del<sup>2</sup><i>p<sub>x</sub></i> / del PT<sup>2</sup>) |
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| 310 | / <i>p<sub>s</sub></i>. |
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| 311 | </p> |
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| 312 | <p></p> |
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| 313 | </li> |
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| 314 | <li>WY: |
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| 315 | Chromatic amplitude function <i>W<sub>y</sub></i>, [1]: |
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| 316 | <p>WY = <i>W<sub>y</sub></i> = sqrt(<i>a<sub>y</sub><sup>2</sup> + |
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| 317 | b<sub>y</sub><sup>2</sup></i>), |
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| 318 | </p> |
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| 319 | <p><i>a<sub>y</sub></i> = (del beta<sub><i>y</i></sub> / del PT) / |
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| 320 | beta<sub><i>y</i></sub>, |
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| 321 | </p> |
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| 322 | <p><i>b<sub>y</sub></i> = |
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| 323 | (del alpha<sub><i>y</i></sub> / del PT) - (alpha<sub><i>y</i></sub> / |
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| 324 | beta<sub><i>y</i></sub>) * |
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| 325 | (del beta<sub><i>y</i></sub> / del PT). |
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| 326 | </p> |
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| 327 | <p></p> |
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| 328 | </li> |
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| 329 | <li>PHIY: |
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| 330 | Chromatic phase function Phi<sub><i>y</i></sub>, [2pi]: |
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| 331 | <p>PHIY = Phi<sub><i>y</i></sub> = atan(<i>a<sub>y</sub> / b<sub>y</sub></i>). |
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| 332 | </p> |
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| 333 | <p></p> |
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| 334 | </li> |
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| 335 | <li>DMUY: |
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| 336 | Chromatic derivative of phase function mu<sub><i>y</i></sub>, [2pi]: |
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| 337 | <p>DMUY = (del mu<sub><i>y</i></sub> / del PT). |
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| 338 | </p> |
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| 339 | <p></p> |
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| 340 | </li> |
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| 341 | <li>DDY: |
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| 342 | Chromatic derivative of dispersion <i>D<sub>y</sub></i>, [m]: |
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| 343 | <p>DDY = 1/2 * (del<sup>2</sup><i>y</i> / del PT<sup>2</sup>). |
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| 344 | </p> |
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| 345 | <p></p> |
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| 346 | </li> |
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| 347 | <li>DDPY: |
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| 348 | Chromatic derivative of dispersion <i>D<sub>py</sub></i>, [1]: |
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| 349 | <p>DDPY = 1/2 * (del<sup>2</sup><i>p<sub>y</sub></i> / del PT<sup>2</sup>) |
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| 350 | / <i>p<sub>s</sub></i>. |
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| 351 | </p> |
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| 352 | <p></p> |
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| 353 | </li> |
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| 354 | </ul> |
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| 355 | <h3><a name="summ">Variables in the SUMM Table</a></h3> |
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| 356 | After a successful TWISS command a summary table |
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| 357 | is created which contains the following variables: |
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| 358 | <ul> |
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| 359 | <li>LENGTH: |
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| 360 | The length of the machine, [m]. |
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| 361 | <p></p> |
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| 362 | </li> |
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| 363 | <li>ORBIT5: |
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| 364 | The T (= <i>c t</i>, [m]) component of the closed orbit. |
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| 365 | <p></p> |
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| 366 | </li> |
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| 367 | <li>ALFA: |
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| 368 | The momentum compaction alpha<sub>p</sub>, [1]. |
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| 369 | <p></p> |
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| 370 | </li> |
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| 371 | <li>GAMMATR: |
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| 372 | The transition energy gamma<sub>transition</sub>, [1]. |
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| 373 | <p></p> |
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| 374 | </li> |
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| 375 | <li>Q1: |
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| 376 | The horizontal tune <i>Q<sub>1</sub></i> [1]. |
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| 377 | <p></p> |
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| 378 | </li> |
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| 379 | <li>DQ1: |
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| 380 | The horizontal chromaticity dq<sub><i>1</i></sub>, [1]: |
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| 381 | <p>DQ1 = dq<sub><i>1</i></sub> = (del <i>Q<sub>1</sub></i> / del |
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| 382 | PT). |
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| 383 | </p> |
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| 384 | <p></p> |
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| 385 | </li> |
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| 386 | <li>BETXMAX: |
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| 387 | The largest horizontal beta<sub><i>x</i></sub>, [m]. |
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| 388 | <p></p> |
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| 389 | </li> |
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| 390 | <li>DXMAX: |
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| 391 | The largest horizontal dispersion [m]. |
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| 392 | <p></p> |
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| 393 | </li> |
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| 394 | <li>DXRMS: |
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| 395 | The r.m.s. of the horizontal dispersion [m]. |
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| 396 | <p></p> |
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| 397 | </li> |
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| 398 | <li>XCOMAX: |
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| 399 | The maximum of the horizontal closed orbit deviation [m]. |
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| 400 | <p></p> |
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| 401 | </li> |
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| 402 | <li>XRMS: |
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| 403 | The r.m.s. of the horizontal closed orbit deviation [m]. |
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| 404 | <p></p> |
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| 405 | </li> |
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| 406 | <li>Q2: |
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| 407 | The vertical tune <i>Q<sub>2</sub></i> [1]. |
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| 408 | <p></p> |
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| 409 | </li> |
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| 410 | <li>DQ2: |
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| 411 | The vertical chromaticity dq<sub><i>2</i></sub>, [1]: |
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| 412 | <p>DQ2 = dq<sub><i>2</i></sub> = (del <i>Q<sub>2</sub></i> / del |
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| 413 | PT). |
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| 414 | </p> |
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| 415 | <p></p> |
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| 416 | </li> |
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| 417 | <li>BETYMAX: |
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| 418 | The largest vertical beta<sub><i>y</i></sub>, [m]. |
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| 419 | <p></p> |
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| 420 | </li> |
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| 421 | <li>DYMAX: |
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| 422 | The largest vertical dispersion [m]. |
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| 423 | <p></p> |
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| 424 | </li> |
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| 425 | <li>DYRMS: |
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| 426 | The r.m.s. of the vertical dispersion [m]. |
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| 427 | <p></p> |
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| 428 | </li> |
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| 429 | <li>YCOMAX: |
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| 430 | The maximum of the vertical closed orbit deviation [m]. |
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| 431 | <p></p> |
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| 432 | </li> |
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| 433 | <li>YCORMS: |
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| 434 | The r.m.s. of the vertical closed orbit deviation [m]. |
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| 435 | <p></p> |
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| 436 | </li> |
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| 437 | <li>DELTAP: |
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| 438 | Energy difference, |
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| 439 | divided by the reference momentum times the velocity of light, [1]: |
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| 440 | <p>DELTAP = delta(<i>E</i>) / <i>p<sub>s</sub> c</i>.</p> |
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| 441 | </li> |
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| 442 | </ul> |
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| 443 | Notice that in MAD-X PT substitutes DELTAP as longitudinal variable. |
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| 444 | Dispersive and chromatic functions are hence derivatives with respects |
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| 445 | to PT. Being PT=BETA*DELTAP, where BETA is the relativistic Lorentz |
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| 446 | factor, those functions must be multiplied by BETA a number of time |
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| 447 | equal to the order of the derivative. |
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| 448 | <h3><a name="track">Variables in the TRACK Table</a></h3> |
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| 449 | The command RUN writes tables with the following variables: |
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| 450 | <ul> |
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| 451 | <li>X: |
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| 452 | Horizontal position <i>x</i> of the orbit, |
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| 453 | referred to the ideal orbit [m]. |
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| 454 | </li> |
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| 455 | <li>PX: |
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| 456 | Horizontal canonical momentum <i>p<sub>x</sub></i> |
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| 457 | of the orbit referred to the ideal orbit, divided by the reference |
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| 458 | momentum. |
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| 459 | </li> |
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| 460 | <li>Y: |
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| 461 | Vertical position <i>y</i> of the orbit, referred to the ideal orbit |
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| 462 | [m]. |
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| 463 | </li> |
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| 464 | <li>PY: |
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| 465 | Vertical canonical momentum <i>p<sub>x</sub></i> |
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| 466 | of the orbit referred to the ideal orbit, divided by the reference |
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| 467 | momentum. |
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| 468 | </li> |
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| 469 | <li>T: |
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| 470 | Velocity of light times the negative time difference |
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| 471 | with respect to the reference particle, [m]. |
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| 472 | A positive T means that the particle arrives ahead of the reference |
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| 473 | particle. |
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| 474 | </li> |
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| 475 | <li>PT: |
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| 476 | Energy difference, |
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| 477 | divided by the reference momentum times the velocity of light, [1]. |
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| 478 | </li> |
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| 479 | </ul> |
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| 480 | When tracking Lyapunov companions (not yet implemented), |
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| 481 | the TRACK table defines the following dependent expressions: |
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| 482 | <ul> |
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| 483 | <li>DISTANCE: |
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| 484 | the relative Lyapunov distance between the two particles. |
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| 485 | </li> |
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| 486 | <li>LYAPUNOV: |
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| 487 | the estimated Lyapunov Exponent. |
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| 488 | </li> |
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| 489 | <li>LOGDIST: |
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| 490 | the natural logarithm of the relative distance. |
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| 491 | </li> |
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| 492 | <li>LOGTURNS: |
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| 493 | the natural logarithm of the turn number. |
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| 494 | </li> |
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| 495 | </ul> |
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| 496 | <address> |
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| 497 | <a href="http://www.cern.ch/Hans.Grote/hansg_sign.html">hansg</a>, |
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| 498 | January 24, 1997. Revised in February 2007.<br> |
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| 499 | </address> |
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| 500 | </body> |
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| 501 | </html> |
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