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Dhanya Thattil c477970cf0 manual updated
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<H1><A NAME="SECTION00610000000000000000"></A><A NAME="sec:angcal"></A>
<BR>
How is the channel number coverted into angle?
</H1>
<P>
Mythen II modules are composed by 1280 pixels, each having width p=0.05&nbsp;mm, and numbered with j=0,..,1279.
Angles are counted counterclockwise from the beam direction. For the m-th module, the angle <!-- MATH
$\alpha_{jm}$
-->
<IMG
WIDTH="33" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img41.png"
ALT="$ \alpha_{jm}$">
of its j-th pixel center
can be determined using the three geometric parameters <IMG
WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$ R_m$">
&nbsp;[mm], <IMG
WIDTH="28" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$ \Phi_m$">
&nbsp;[deg], <IMG
WIDTH="30" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$ D_m$">
&nbsp;[mm], as in Fig.&nbsp;<A HREF="#acon"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
The detector group uses instead the 3 parameters center <IMG
WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
ALT="$ c_m$">
&nbsp;[ ], offset <IMG
WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$ o_m$">
&nbsp;[deg], conversion <IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.png"
ALT="$ k_m$">
&nbsp;[ ].
The law with the 3 geometric parameter is
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\alpha_{jm}=\Phi_m-{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}\arctan{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m-pj}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}}\right)}}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="285" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
ALT="$\displaystyle \alpha_{jm}=\Phi_m-{\ensuremath{\left({{\ensuremath{\displaystyle...
...remath{\displaystyle{D_m-pj}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}}\right)}}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(5.1)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
The corresponding law using DG's parameters is
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\alpha_{jm}=o_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}c_mk_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}\arctan{\ensuremath{\left[{{\ensuremath{\left({j-c_m}\right)}}k_m}\right]}}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="391" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
SRC="img49.png"
ALT="$\displaystyle \alpha_{jm}=o_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\f...
...t)}}\arctan{\ensuremath{\left[{{\ensuremath{\left({j-c_m}\right)}}k_m}\right]}}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(5.2)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
One can convert the two forms by equating separately the term out of the arctan and the argument of arctan for two different values of j.
It results
<BR>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{eqnarray}
c_m&=&{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{p}}}}}}};\\
k_m&=&{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{R_m}}}}}}};\\
o_m&=&\Phi_m-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}.
\end{eqnarray}
-->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$\displaystyle c_m$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="38" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{p}}}}}}};$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5.3)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img53.png"
ALT="$\displaystyle k_m$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="37" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img54.png"
ALT="$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{R_m}}}}}}};$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5.4)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$\displaystyle o_m$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="109" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
SRC="img56.png"
ALT="$\displaystyle \Phi_m-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl...
...\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}.$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5.5)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL">
Conversely,
<BR>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{eqnarray}
\Phi_m&=&o_m+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}c_mk_m;\\
R_m&=&{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{k_m}}}}}}};\\
D_m&=&c_m p.
\end{eqnarray}
-->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="28" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$\displaystyle \Phi_m$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="115" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$\displaystyle o_m+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}c_mk_m;$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5.6)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img59.png"
ALT="$\displaystyle R_m$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="33" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{k_m}}}}}}};$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5.7)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="30" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img61.png"
ALT="$\displaystyle D_m$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="36" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img62.png"
ALT="$\displaystyle c_m p.$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5.8)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL">
<P>
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<ADDRESS>
Thattil Dhanya
2018-09-28
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