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1<head>
2<title>Sign Conventions</title>
3<!-- Changed by: Chris ISELIN, 17-Jul-1997 -->
4<!-- Changed by: Hans Grote, 10-Jun-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>Sign Conventions for Magnetic Fields</h2>
13</center>
14
15The MAD program uses the following Taylor expansion
16for the field on the mid-plane <i>y</i>=0, described in
17<a href="bibliography.html#slac75">SLAC-75</a>:
18<p>
19<img align=bottom src="../equations/Taylor_field.gif">
20<p>
21Note the factorial in the denominator.
22The field coefficients have the following meaning:
23<ul>
24<li><i>B</i><sub>0</sub>: Dipole field, with a positive value in the
25positive <i>y</i> direction;
26a positive field bends a positively charged particle to the right.
27<li><i>B</i><sub>1</sub>: Quadrupole coefficient
28<p>
29<i>B</i><sub>1</sub> = (del <i>B<sub>y</sub></i> / del <i>x</i>);
30<p>
31a positive value corresponds to horizontal focussing of a positively
32charged particle.
33<li><i>B</i><sub>2</sub>: Sextupole coefficient
34<p>
35<i>B</i><sub>2</sub> =
36(del<sup>2</sup> <i>B<sub>y</sub></i> / del <i>x</i><sup>2</sup>).
37<p>
38<li><i>B</i><sub>3</sub>: Octupole coefficient
39<p>
40<i>B</i><sub>3</sub> =
41(del<sup>3</sup> <i>B<sub>y</sub></i> / del <i>x</i><sup>3</sup>).
42<p>
43</ul>
44Using this expansion and the curvature <i>h</i> of the reference orbit,
45the longitudinal component of the vector potential to order 4 is:
46<p>
47<img align=bottom src="../equations/Taylor_A_s.gif">
48<p>
49Taking curl <i>A</i> in curvilinear coordinates,
50the field components can be computed as
51<p>
52<img align=bottom src="../equations/Taylor_B.gif">
53<p>
54It can be easily verified that both curl <i>B</i> and div <i>B</i>
55are zero to the order of the <i>B</i><sub>3</sub> term.
56Introducing the magnetic rigidity <i>B</i>rho,
57the multipole coefficients are computed as
58<p>
59<i>K<sub>n</sub></i> = <i>e B<sub>n</sub> / p<sub>s</sub></i> =
60<i>B<sub>n</sub> / B</i> rho.
61<p>
62
63<p>
64<address>
65<a href="http://www.cern.ch/Hans.Grote/hansg_sign.html">hansg</a>,
66June 17, 2002
67</address>
68
69</body>
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