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1<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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3<!--Converted with LaTeX2HTML 2002-2-1 (1.70)
4original version by:  Nikos Drakos, CBLU, University of Leeds
5* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
6* with significant contributions from:
7  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
8<HTML>
9<HEAD>
10<TITLE>ibs</TITLE>
11<META NAME="description" CONTENT="ibs">
12<META NAME="keywords" CONTENT="ibs">
13<META NAME="resource-type" CONTENT="document">
14<META NAME="distribution" CONTENT="global">
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16<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
17<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
18
19<LINK REL="STYLESHEET" HREF="ibs.css">
20
21</HEAD>
22
23<BODY >
24
25<P>
26
27<P>
28<H1><A NAME="SECTION00000000000000000000">
29The Intra-Beam Scattering Module (IBS)</A>
30</H1><FONT SIZE="+1">
31<P>
32
33<P><BR>
34<FONT SIZE="+1"><HR></FONT>
35<P><P><BR>
36<FONT SIZE="+1"></FONT>
37<P>
38<FONT SIZE="+1">As emphasized by its name, the Intra-Beam Scattering module (IBS) computes the
39contribution to emittance growth rates due to Coulomb scattering of particles
40within relativistic beams.
41The formalism used in this module is that derived by J.D. Bjorken and
42S.K.Mtingwa [<a href="../Introduction/bibliography.html#bm1">Bjorken and Mtingwa</a>] in 1982,
43using the expansion of M. Conte and M. Martini [<a href="../Introduction/bibliography.html#conte">Conte and Martini</a>] in 1985,
44generalized to the case of nonzero vertical dispersion.
45The IBS module uses the beta and dispersion provided by the TWISS command. 
46The IBS module does not include a consistent treatment of linear betatron coupling.
47The present implementation of the IBS module in MAD-X is described in a forthcoming note
48[<a>Antoniou and Zimmermann</a>] (2012).
49
50<P>
51Contrary to other IBS-routines, the Bjorken-Mtingwa formalism takes into
52account the variation of the lattice parameters around the machine, rather than using average values. Consequently, the knowledge of the optical functions of the machine is required. In MAD-X, this is achieved with the ``<B>twiss</B>'' command.
53</FONT>
54<P>
55<FONT SIZE="+1">It is well known that the intra-beam scattering growth times behave like:
56</FONT><BR><P></P>
57<DIV ALIGN="CENTER">
58<!-- MATH
59 \begin{displaymath}
60\frac{1}{\tau_i} \quad = \quad C_i \times \frac{N}{\gamma \epsilon_x \epsilon_y \epsilon_s} \qquad (i = x, y, s)
61\end{displaymath}
62 -->
63
64<IMG
65 WIDTH="289" HEIGHT="47" BORDER="0"
66 SRC="img1.png"
67 ALT="\begin{displaymath}
68\frac{1}{\tau_i} \quad = \quad C_i \times \frac{N}{\gamma \epsilon_x \epsilon_y \epsilon_s} \qquad (i = x, y, s)
69\end{displaymath}">
70</DIV><BR CLEAR="ALL">
71<P></P><FONT SIZE="+1">
72where C<IMG
73 WIDTH="10" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
74 SRC="img2.png"
75 ALT="$_i$"> accounts for some constants and the integrals for the scattering functions, N is the number of particles in the bunch, <IMG
76 WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
77 SRC="img3.png"
78 ALT="$\gamma$"> is the relativistic factor and <IMG
79 WIDTH="17" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
80 SRC="img4.png"
81 ALT="$\epsilon_i$"> are the normalized emittances in the horizontal, vertical and longitudinal plane respectively.
82It thus follows that the second required input is a description of the beam parameters, which is achieved via the ``<B>beam</B>'' command (see below).
83</FONT>
84<P>
85<FONT SIZE="+1">Once the optical functions and the beam parameters have been defined, the evaluation of the scattering growth times follows via the ``<B>ibs</B>'' command.
86The ``<B>ibs</B>'' command should be immediately preceded by a call of ``<B>twiss</B>''.
87In particular, the ``<B>emit</B>'' command should be followed by another call of ``<B>twiss</B>'' before ``<B>ibs</B>'' is used.   
88
89<P>
90If ``<B>twiss</B>'' calculates the optical functions at the end of each element ``<B>ibs</B>'' performs a linear extrapolation to determine their values
91at the center of the elements. If ``<B>twiss</B>'' already computes the optical functions at the center of each element ``<B>ibs</B>'' uses these
92values directly without making any interpolation.
93<P>
94
95The logical follow-up of the MAD-X commands is illustrated in the two examples provided with the IBS-module.
96</FONT>
97<P>
98
99<H1><A NAME="SECTION00010000000000000000">
100Input of the beam parameters</A>
101</H1><FONT SIZE="+1">
102This section briefly describes the parameters which have to be present in the
103``<B>beam</B>'' command in order to run the IBS-module:
104</FONT>
105<P>
106
107<H2><A NAME="SECTION00011000000000000000">
108Type of particle</A>
109</H2><FONT SIZE="+1">
110The parameter ``<B>particle=</B>'' is mandatory. It can take one of the following <B>three</B> values: <B>proton, electron or ion</B>.
111For proton and electron, the parameter ``particle'' is the only one to be defined.
112In case <B>ion</B> is used, two additional parameters have to be
113defined, namely ``<B>mass=</B>'', which is typically the number of
114nucleons for the corresponding ion multiplied by <B>nmass</B> the unified atomic
115mass unit [0.931494013 GeV/(c**2)] , and ``<B>charge=</B>'' for the number of charges.
116
117<H2><A NAME="SECTION00012000000000000000">
118The energy</A>
119</H2><FONT SIZE="+1">
120The definition of the energy (total, kinetic, total energy of the ions or energy per nucleon) is a difficult one.
121In the present approach, the energy is the <B>total</B> energy of the
122particle. For ions, the expected input is the <B>proton equivalent</B>
123energy, i.e. the total energy a proton would have when circulating in
124the defined machine. As an illustration, in the LHC, protons will be
125injected with an energy of 450&nbsp;GeV. Consequently, to evaluate the
126growth times for Lead ions at injection in the LHC, one has to input
127<B>energy=450*charge</B>. Therefore the above example of Lead at the
128LHC injection energy may look as follows in the MAD-X input language:
129</FONT><DIV ALIGN="CENTER"><FONT SIZE="+1">
130<B>
131nucleon=208;
132charge=82;
133beam,particle=ion,charge=charge,energy=450*charge,mass=nucleon*nmass;</B></FONT></DIV>
134
135An important check for the correctness of the input is the printed value of the relativistic factor <IMG
136 WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
137 SRC="img3.png"
138 ALT="$\gamma$">. The latter should correspond to:
139</FONT><BR><P></P>
140<DIV ALIGN="CENTER">
141<!-- MATH
142 \begin{displaymath}
143\gamma_{ion} \quad = \quad \gamma_{proton} \times \frac{charge}{nucleon}
144\end{displaymath}
145 -->
146
147<IMG
148 WIDTH="220" HEIGHT="42" BORDER="0"
149 SRC="img5.png"
150 ALT="\begin{displaymath}
151\gamma_{ion} \quad = \quad \gamma_{proton} \times \frac{charge}{nucleon}
152\end{displaymath}">
153</DIV><BR CLEAR="ALL">
154<P></P>
155<H2><A NAME="SECTION00013000000000000000">
156The number of particles</A>
157</H2><FONT SIZE="+1">
158The number of particles (or number of ions) is defined with the parameter
159``<B>npart=</B>''.
160</FONT>
161<P>
162
163<H2><A NAME="SECTION00014000000000000000">
164Beam sizes - Emittances</A>
165</H2><FONT SIZE="+1">
166This part of the input is used to define the normalized emittances (horizontal, vertical and longitudinal). The required parameters are the <B>physical</B> transverse emittances (<B>ex=</B> and <B>ey=</B> [<IMG
167 WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
168 SRC="img6.png" ALT="$\pi$">m]), the longitudinal emittance (<B>ET=</B>) which is defined   
169as the product of the bunch length (<B>sigt=</B> [m]) and the relative energy spread (<B>sige=</B>). If only
170the longitudinal emittance is defined (and not the <B>sigt</B> and <B>sige</B> as well), an RF cavity is also necessary. Otherwise, the bunch length (<B>sigt</B>) and the relative energy spread (<B>sige</B>) should also be defined.
171
172</FONT>
173<P>
174
175<H1><A NAME="SECTION00020000000000000000">
176File Attribute</A>
177</H1><FONT SIZE="+1">
178If FILE="file_name" appears MAD-X produces a table and writes on a
179file for each element of the machine: ELEMENT NAME, Position S [m],
180DELS [m] (Length Difference of consecutive Elements in the Table),
181TLI (Longitudinal growth time), TXI (Horizontal growth time), TYI (Vertical growth time),
182BETX [m], ALFX [1], DX [m], DPX [1], BETY [m], ALFY [1], DY [m], DPY [1].
183
184<H1><A NAME="SECTION00020000000000000000">
185Features</A>
186</H1><FONT SIZE="+1">
187The average growth rates in [sec] are defined as variables called ibs.tx, ibs.ty, ibs.tl for the horizontal, vertical and longitudinal growth times respectively. One can access them simply by calling them after the ibs commant is called. </p> 
188Example: </p> 
189 ibs; </p> 
190 Tx=ibs.tx;</p> 
191 This defines a variable Tx which is the average horizontal growth rate in seconds.
192</FONT>
193<P>
194 </FONT>
195
196<H1><A NAME="SECTION00030000000000000000">
197Examples</A>
198</H1><FONT SIZE="+1">
199The two examples provided for the module Intra-Beam Scattering illustrate the commands required to run the module. The two examples have been selected such as to highlight the differences between a computation for protons and that for ions. Both examples compute the IBS growth times at injection into the LHC. The examples are located <a href="http://cern.ch/madx/madx.old">here</a>.
200</FONT>
201<P>
202 </FONT>
203
204<P>
205</DL>
206<P>
207<BR><HR>
208<ADDRESS>
209Frank Schmidt
2102003-05-23
211</ADDRESS>
212</BODY>
213</HTML>
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