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1<head>
2<title>Known Defects of MAD8 and MAD-X</title>
3<!-- Changed by: Chris ISELIN, 24-Jan-1997 -->
4<!-- Changed by: Hans Grote, 25-Sep-2002 -->
5</head>
6
7<body bgcolor="#ffffff">
8
9<center>
10EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
11<IMG SRC="http://cern.ch/madx/icons/mx7_25.gif" align=right>
12<h2>Known Differences to Other Programs</h2>
13</center>
14
15<h2>Definitions</h2>
16MAD uses full 6-by6-matrices to allow coupling effects to
17be treated, and the canonical variable set
18(<i>x</i>, <i>p<sub>x</sub> / p<sub>0</sub></i>),
19(<i>y</i>, <i>p<sub>y</sub> / p<sub>0</sub></i>),
20(<i>-ct</i>, delta(<i>E</i>) / <i>p<sub>0</sub> c</i>),
21as opposed to other programs most of which use the set
22(<i>x</i>, <i>x</i>'), (<i>y</i>, <i>y</i>'),
23(-delta(<i>s</i>), delta(<i>p</i>)/<i>p<sub>0</sub></i>).
24Like <a href="bibliography.html#dragt">[Dragt]</a>,
25MAD uses the relative energy error <i>p<sub>y</sub> / p<sub>0</sub></i>,
26which is equal the relative momentum error
27delta = delta(<i>p</i>)/<i>p<sub>0</sub></i> multiplied by beta = v/c.
28<p>
29As from Version 8.13, MAD8 uses an additional <b>constant</b> momentum
30error delta<sub><i>s</i></sub> in all optical calculations.
31The transfer maps contain the <b>exact</b> dependence upon this value;
32therefore the tunes for large deviations can be computed with high
33accuracy as opposed to previous versions.
34<p>
35The choice of canonical variables in MAD still
36leads to slightly different definitions of the lattice functions.
37In MAD the Courant-Snyder invariants in
38<a href="bibliography.html#courant">[Courant and Snyder]</a> take the form
39<p>
40W<sub>x</sub> = gamma<sub><i>x</i></sub> <i>x</i><sup>2</sup> -
412 alpha<sub><i>x</i></sub> <i>x p<sub>x</sub></i> +
42beta<sub><i>x</i></sub> <i>p<sub>x</sub></i><sup>2</sup>
43<p>
44Comparison to the original form
45<p>
46W<sub>x</sub> = gamma<sub><i>x</i></sub> <i>x</i><sup>2</sup> -
472 alpha<sub><i>x</i></sub> <i>x x</i>' +
48beta<sub><i>x</i></sub> <i>x</i>'<sup>2</sup>
49<p>
50shows that the orbit functions cannot be the same.
51A more detailed analysis, using
52<p>
53<i>x</i>' = <i>p<sub>x</sub></i> / (1 + delta)
54<p>
55shows that all formulas can be made consistent
56by defining the MAD orbit functions as
57<p>
58beta<sub><i>x</i>M</sub> = beta<sub><i>x</i>C</sub> * (1 + delta),
59alpha<sub><i>x</i>M</sub> = alpha<sub><i>x</i>C</sub>,
60gammaa<sub><i>x</i>M</sub> = gamma<sub><i>x</i>C</sub> / (1 + delta),
61<p>
62For constant delta<sub><i>s</i></sub> along the beam line and
63delta = 0, the lattice functions are the same.
64In a machine where delta varies along the circumference,
65e.g. in a linear accelerator or in an electron-positron storage ring,
66the definition of the Courant-Snyder invariants must be generalised.
67The MAD invariants have the advantage that they
68remain invariants along the beam line even for variable delta.
69<p>
70With the new method this problem occurs in
71<a href="../twiss/twiss.html">Twiss module</a>
72only for non-constant delta.
73
74<h2>Treatment of Energy Error in TWISS</h2>
75It has been noted in
76<a href="bibliography.html#ruggiero">[Milutinovic and Ruggiero]</a>
77that MAD returned tunes which are too low for non-zero delta.
78The difference was found to be quadratic in delta with a negative coefficient.
79This problem has been eliminated thanks to the new treatment
80of momentum errors from MAD8 Version 8.13 onwards.
81
82<p>
83<address>
84<a href="http://www.cern.ch/Hans.Grote/hansg_sign.html">hansg</a>,
85January 24, 1997
86</address>
87
88</body>
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