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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!--Converted with LaTeX2HTML 2008 (1.71)
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
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<HEAD>
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<TITLE>Observables</TITLE>
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<META NAME="description" CONTENT="Observables">
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<META NAME="keywords" CONTENT="slsDetectors-FAQ">
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<BR>
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HREF="node50.html">Basic binning</A>
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HREF="node47.html">Introduction</A>
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HREF="node48.html">Notation</A>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<H3><A NAME="SECTION00621200000000000000">
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Observables</A>
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</H3>
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The physical observable of interest in any scattering experiment is [1-3] the differential cross section
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<P><!-- MATH
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\begin{displaymath}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\bf {\sigma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="31" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
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SRC="img65.png"
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ALT="$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ens...
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...ma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}}
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$">
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</DIV><P>
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</P>
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as a function of direction <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img66.png"
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ALT="$ \Omega$">
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.
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To measure that directly we should operate with zero-width point detectors, with instant measurement and unit incident intensity.
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Practically
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the quantity we can actually measure - putting a detector in a position covering a certain
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solid angle for a certain time with a certain incident intensity - is
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<P><!-- MATH
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\begin{displaymath}
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{I_0}\Delta t \Delta\Omega{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\bf {\sigma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="89" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
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SRC="img67.png"
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ALT="$\displaystyle {I_0}\Delta t \Delta\Omega{\ensuremath{\displaystyle{\frac{{\ensu...
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...ma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}}
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$">
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</DIV><P>
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</P>
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If <IMG
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WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img68.png"
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ALT="$ \Delta t$">
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, <!-- MATH
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$\Delta\Omega$
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-->
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<IMG
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WIDTH="29" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img69.png"
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ALT="$ \Delta\Omega$">
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are small and known and <IMG
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WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img70.png"
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ALT="$ I_0$">
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is separately monitored,
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we can (have to) normalize the observations by simply dividing them out.
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<P>
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Specifically for the powder diffraction field, historically, this is not usually done because
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- as it is normally true with anode sources and point detectors and usual procedures -
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the counting times <IMG
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WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img68.png"
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ALT="$ \Delta t$">
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, the solid angle width <!-- MATH
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$\Delta\Omega\propto \Delta {\ensuremath{{2\theta}}}$
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-->
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<IMG
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WIDTH="80" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img71.png"
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ALT="$ \Delta\Omega\propto \Delta {\ensuremath{{2\theta}}}$">
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and the incident intensity <IMG
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WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img70.png"
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ALT="$ I_0$">
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are considered
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constant and therefore go into some 'global scaling' constant that is usually considered arbitrary.
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<P>
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However, as we have more sophisticated acquisition methods,
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we may need revert to the original approach and consider the
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counts divided by time and angular width as the real observable.
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<P>
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<HR>
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<BR>
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<B> Next:</B> <A NAME="tex2html795"
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HREF="node50.html">Basic binning</A>
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<B> Up:</B> <A NAME="tex2html791"
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HREF="node47.html">Introduction</A>
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<B> Previous:</B> <A NAME="tex2html787"
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HREF="node48.html">Notation</A>
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HREF="node1.html">Contents</A></B>
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<!--End of Navigation Panel-->
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<ADDRESS>
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Thattil Dhanya
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2018-03-12
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</ADDRESS>
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</HTML>
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