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