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<title>Continuum/background Models</title>
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<div class=Section1>
<p class=MsoNormal style='text-align:justify'><b style='mso-bidi-font-weight:
normal'><span lang=EN-US>PyMCA � X-Ray Spectrum Analysis in Python<o:p></o:p></span></b></p>
<p class=MsoNormal style='text-align:justify'><span lang=FR style='mso-ansi-language:
FR'>V. A. Sol� and E. Papillon.<o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>European
Syncrothon Radiation Facility (ESRF), 38043 Grenoble Cedex.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><b style='mso-bidi-font-weight:
normal'><span lang=EN-US>1. Introduction<o:p></o:p></span></b></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>Up to date most, if not all, spectrum fitting for X-ray fluorescence
measurements at the ESRF (ID13, ID18F, ID21, ID22) has been performed using
externally supplied software (generally based on the AXIL software developed at
the University of Antwerp, Belgium). Whilst this software is fairly robust and
reliable, we have very little influence over its development and consequently
its direct integration into our control system and subsequent data analysis
routines is not straightforward. A further limitation is that we can not
distribute that software to our user community.</span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>A versatile non-linear least-squares fitting application had been
already developed as part of the tools of the BLISS group at the ESRF and had
been used among others by the NewPlot visualization package. That fitting
application, based on the Levenberg-Marquardt algorithm, is implemented
entirely in Python, thus ensuring a high level of platform compatibility and
straightforward integration with the ESRF control system. The logical step to
follow was to write a dedicated function to describe the x-ray fluorescence
spectra and feed that function to the fitting module.</span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The need to have an easy way to setup the configuration parameters
of the fit, led to the development of a complete visualization and data
analysis tool, PyMCA, that relies on Qt and Qwt to build its graphical
interface and plotting routines. Nevertheless, the fitting code can run in
prompt/batch mode fully independent of any graphical package, and its output
file, can be used by other python module (also GUI independent) to
automatically generate a fully detailed HTML report that can be visualized by
any browser.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><b style='mso-bidi-font-weight:
normal'><span lang=EN-US>2. Algorithms</span></b></p>
<p class=MsoNormal style='text-align:justify'><i style='mso-bidi-font-style:
normal'><span lang=EN-US>2.1 Continuum/background Models<o:p></o:p></span></i></p>
<p class=MsoNormal style='text-align:justify'><u><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></u></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The continuum is modeled in two possible ways: estimation or
fitting. In the former the estimated background is subtracted from the
experimental data prior to the least-squares fitting of the fluorescence peaks.
In the fitting mode the continuum is described by an analytical function which
enters into the least-squares fitting algorithm. </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>For the time being PyMCA implements one of each of the models.
Background can be estimated thru an iterative smoothing procedure in which the
content of each channel is compared against the average of the content of its</span><span
style='mso-ansi-language:EN-GB'> neighbours at a distance of <i
style='mso-bidi-font-style:normal'>i</i> channels</span><span lang=EN-US>. If
the content is above the average, it is replaced by the average. In order to
speed up the procedure, <i style='mso-bidi-font-style:normal'>i</i> can be
taken as a fraction of the peaks full-width-half-maximum (FWHM) at the
beginning of the iterative process, being one at the end of it. The only
analytical function currently supported to describe the background is a linear
polynomial that can also be used in conjunction with the stripping procedure.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><i style='mso-bidi-font-style:
normal'><span lang=EN-US>2.2 Peak Shape Model<o:p></o:p></span></i></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The following is drawn primarily from a description of Least-Squares
fitting of XRF spectra in <b style='mso-bidi-font-weight:normal'>[1]</b></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The response function of most solid-state detectors is predominantly
Gaussian. In certain instances it may be necessary to resort to more
complicated models such as Voigt or Hypermet <b style='mso-bidi-font-weight:
normal'>[2]</b> functions.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>A Gaussian peak is characterized by three parameters: the position,
width, and height or area. It is desirable to describe the peak in terms of its
area rather than its height because the area is directly related to the number
of x-ray photons detected, while the height depends on the spectrometer
resolution. The first approximation to the profile of a single peak is then
given by:</span></p>
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<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
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</xml><![endif]--><span style='mso-tab-count:1'>������ </span>(1)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>where A is the
peak area (counts ), </span><span lang=EN-US style='font-family:Symbol'>s</span><span
lang=EN-US> the width of the Gaussian expressed in channels and x<sub>0</sub>
the location of the peak maximum. The FWHM is related to </span><span
lang=EN-US style='font-family:Symbol'>s</span><span lang=EN-US> by FWHM = 2.355
</span><span lang=EN-US style='font-family:Symbol'>s</span><span lang=EN-US>.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>In Equation (1) the peak area is a linear parameter; the width and
position are non-linear parameters. This implies that a nonlinear least-squares
procedure is required to find optimum values for the latter two parameters.
Linear least-squares fitting method can be used assuming the position and width
of the peak are know with high accuracy from calibration.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>To describe part of a measured spectrum, the fitting function must
contain a number of such functions, one for each peak. For 10 elements and 2
peaks (K</span><span lang=EN-US style='font-family:Symbol'>a</span><span
lang=EN-US> and K</span><span lang=EN-US style='font-family:Symbol'>b</span><span
lang=EN-US>) per element we would need to optimize 60 parameters. It is highly
unlikely that such a nonlinear least-squares fit will terminate successfully at
the global minimum. To overcome this problem the fitting function can be
written in a different way as shown in the next paragraph.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><i style='mso-bidi-font-style:
normal'><span lang=EN-US>2.3 FWHM and Position of peaks<o:p></o:p></span></i></p>
<p class=MsoNormal style='text-align:justify'><u><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></u></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The first step is to abandon the idea of optimizing peak and
position of all peaks independently. The energies of the X-ray fluorescence
lines are typically known with an accuracy of l eV or better. The pattern of
peaks observed in the spectrum is directly related to the elements present in
the sample.</span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>Based on these elements we can predict all of the X-ray lines that
constitute the spectrum and their energies. The peak fitting function is therefore
written in terms of energy rather than channel number.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>Defining ZERO as the energy of channel 0 and expressing the spectrum
GAIN in eV/channel, the energy of channel <i style='mso-bidi-font-style:normal'>i</i>
is given by:</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:140.25pt;height:20.25pt'
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<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>and the
normalized Gaussian peak can be written as:</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
id="_x0000_i1026" type="#_x0000_t75" style='width:185.25pt;height:50.25pt'
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<v:imagedata src="./PyMCA_files/image007.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=247 height=67
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</xml><![endif]--><span style='mso-tab-count:1'>��������� </span>(3)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>with <i
style='mso-bidi-font-style:normal'>E<sub>j</sub></i> the energy (in eV) of the
x-ray line and<i style='mso-bidi-font-style:normal'> s</i> the peak width given
by:</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
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<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>In this equation
NOISE is the electronic contribution to the peak width (typical 80-100 eV FWHM)
with the factor 2.3548 to convert to </span><span lang=EN-US style='font-family:
Symbol'>s</span><span lang=EN-US> units, FANO is the Fano factor (~0.114) and
3.85 the energy required to produce an electron-hole pair in silicon. </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The least squares fit optimizes ZERO, GAIN, NOISE and FANO for the
entire spectrum (fitting region), thus for all peaks simultaneously. One can,
after the fit, calculate the position of the peak (using eq. 2) and the width
of the peak using eq. 4. (s in sigma of the peak in eV!!!), convert to channels
via the factor GAIN and to FWHM via the factor 2.3548. </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><i style='mso-bidi-font-style:
normal'><span lang=EN-US>2.4 Element Line Groups<o:p></o:p></span></i></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>A further simplification that can be applied to reduce the number of
fitting parameters is to model entire elements rather than single peaks. Some
lines can be considered as being grouped together such as the K</span><span
lang=EN-US style='font-family:Symbol'>a</span><sub><span lang=EN-US>1</span></sub><span
lang=EN-US>, K</span><span lang=EN-US style='font-family:Symbol'>a</span><sub><span
lang=EN-US>2</span></sub><span lang=EN-US> doublets or even all K lines of an
element. A single area parameter A representing the total number of counts in
the line group can then be fitted. </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The spectrum of an element can then be represented by:</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
id="_x0000_i1033" type="#_x0000_t75" style='width:114pt;height:36.75pt' o:ole="">
<v:imagedata src="./PyMCA_files/image011.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=152 height=49
src="./PyMCA_files/image012.gif" v:shapes="_x0000_i1033"><![endif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1033"
DrawAspect="Content" ObjectID="_1160990290">
</o:OLEObject>
</xml><![endif]--><span style='mso-tab-count:2'>��������������������� </span>(5)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>where G are the
Gaussians for the various lines with energy <i style='mso-bidi-font-style:normal'>E<sub>j</sub></i>
and <i style='mso-bidi-font-style:normal'>R<sub>j</sub></i> the relative
intensities of the lines. The summation runs over all lines in the group (<i
style='mso-bidi-font-style:normal'>N<sub>p</sub></i> ) with </span><span
lang=EN-US style='font-family:Symbol'>S</span><i style='mso-bidi-font-style:
normal'><span lang=EN-US>R<sub>j</sub></span></i><span lang=EN-US>=1. The
transition probabilities of all lines originating from a vacancy in the same
(sub-)shell (K, L<i style='mso-bidi-font-style:normal'><sub>I</sub></i> , L<i
style='mso-bidi-font-style:normal'><sub>II</sub></i> ...) are constants,
independent of the excitation. However, the relative intensities depend on the
absorption in the sample and in the detector windows. To take this into
account, the x-ray attenuation must be included in Equation (5). The relative
intensity ratios are obtained by multiplying the transition probabilities with
an </span><span style='mso-ansi-language:EN-GB'>absorption correction term:</span><span
lang=EN-US> </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
id="_x0000_i1034" type="#_x0000_t75" style='width:95.25pt;height:48.75pt'
o:ole="">
<v:imagedata src="./PyMCA_files/image013.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=127 height=65
src="./PyMCA_files/image014.gif" v:shapes="_x0000_i1034"><![endif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1034"
DrawAspect="Content" ObjectID="_1160990291">
</o:OLEObject>
</xml><![endif]--><span style='mso-tab-count:3'>��������������������������� </span>(6)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The absorption correction term<i style='mso-bidi-font-style:normal'>
T<sub>a</sub></i> (E), used in the equation (6) includes the x-ray attenuation
in all layers and windows between the sample surface and the active area of the
detector. </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><i style='mso-bidi-font-style:
normal'><span lang=EN-US>2.5 Sum and Escape Peaks<o:p></o:p></span></i></p>
<p class=MsoNormal style='text-align:justify'><i style='mso-bidi-font-style:
normal'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></i></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>The escape fraction <i style='mso-bidi-font-style:normal'>f</i> is
defined as the number of counts in the escape peak <i style='mso-bidi-font-style:
normal'>N<sub>e</sub></i> divided by the number of detected counts (escape +
parent). Assuming normal incidence to the detector and escape only from the
front surface, the following formula can be derived for the escape fraction
(Reed S.J.B., Ware N.G., .J. Phys. E 5 (1972) 582)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
id="_x0000_i1035" type="#_x0000_t75" style='width:234pt;height:39.75pt' o:ole="">
<v:imagedata src="./PyMCA_files/image015.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=312 height=53
src="./PyMCA_files/image016.gif" v:shapes="_x0000_i1035"><![endif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1035"
DrawAspect="Content" ObjectID="_1160990292">
</o:OLEObject>
</xml><![endif]--><span style='mso-tab-count:1'>����� </span>(7)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>where </span><span
lang=EN-US style='font-family:Symbol'>m</span><sub><span lang=EN-US>I</span></sub><span
lang=EN-US> and </span><span lang=EN-US style='font-family:Symbol'>m</span><sub><span
lang=EN-US>K</span></sub><span lang=EN-US> are the mass attenuation
coefficients of silicon for the impinging and the Si K x-ray fluorescence
radiation respectively.; </span><span lang=EN-US style='font-family:Symbol'>w</span><sub><span
lang=EN-US>K</span></sub><span lang=EN-US> is the K-shell fluorescence yield
and r the K-shell jump ratio of silicon. The calculated escape fraction is in
very good agreement with the experimentally determined values for impinging
photons up to 15keV. The area, relative to the area of the parent peak can be
calculated from the escape fraction: </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
id="_x0000_i1036" type="#_x0000_t75" style='width:75.75pt;height:35.25pt'
o:ole="">
<v:imagedata src="./PyMCA_files/image017.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=101 height=47
src="./PyMCA_files/image018.gif" v:shapes="_x0000_i1036"><![endif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1036"
DrawAspect="Content" ObjectID="_1160990293">
</o:OLEObject>
</xml><![endif]--><span style='mso-tab-count:5'>���������������������������������������������������������� </span>(8)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>Si escape peaks can be modeled by a Gaussian at energy 1.742 keV
below the parent peak. Including the escape peaks, the description of the
fluorescence of element becomes</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal align=center style='text-align:center'><span lang=EN-US><sub><!--[if gte vml 1]><v:shape
id="_x0000_i1037" type="#_x0000_t75" style='width:210pt;height:36.75pt' o:ole="">
<v:imagedata src="./PyMCA_files/image019.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=280 height=49
src="./PyMCA_files/image020.gif" v:shapes="_x0000_i1037"><![endif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1037"
DrawAspect="Content" ObjectID="_1160990294">
</o:OLEObject>
</xml><![endif]--><span style='mso-tab-count:2'>������������� </span>(9)</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>where G
represents the Gaussian fitting function and <i style='mso-bidi-font-style:
normal'>N<sub>esc</sub></i> the energy of the escaped photon. </span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>So they are not fitted as truly independent peak, but they are part
of the multiplet. For spectra obtained with a Ge detector one needs to account
in a similar way for both the Ge-K</span><span lang=EN-US style='font-family:
Symbol'>a</span><span lang=EN-US> and the Ge-K</span><span lang=EN-US
style='font-family:Symbol'>b</span><span lang=EN-US><span style="mso-spacerun:
yes">� </span>escape peaks for elements above arsenic.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>Summing correction is performed by using a very intuitive approach.
Since any of the peaks can be detected simultaneously with any of the other
peaks, one can calculate the summing contribution of channel <i
style='mso-bidi-font-style:normal'>i</i>, simply shifting the whole calculated
spectrum by i channels and multiplying it by the calculated content of the
channel times a fitted parameter. This is then repeated for all the points of
the spectrum. The physical meaning of that parameter, for a time acquisition of
one second, could be interpreted as the minimum time, measured in seconds,
needed by the acquisition system to distinguish two photons individually and
not consider them as simultaneous.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><b style='mso-bidi-font-weight:
normal'><span lang=EN-US>3. Conclusion<o:p></o:p></span></b></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify;text-indent:36.0pt'><span
lang=EN-US>Despite being at its early stages, PyMca and its fitting engine
already implement most of the needs of x-ray fluorescence spectroscopy. It is
fast (~1 second per complex spectrum with < 1 GHz processors), portable (it
already runs on Solaris, Linux and Windows) and can be freely distributed.
Current developments are focused on the implementation of alternative continuum
algorithms.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><b style='mso-bidi-font-weight:
normal'><span lang=EN-US>References<o:p></o:p></span></b></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>[1] R.E. Van
Grieken, A.A. Markowicz. Handbook of X-ray Spectrometry, Chapter 4: Spectrum
Evaluation,<span style="mso-spacerun: yes">� </span>Ed. Marcel Dekker, New York
1993.</span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span lang=EN-US>[2]<span
style="mso-spacerun: yes">� </span>G.W. Phillips and K.W. Marlow, NIM 137
(1976) 525 � 536.</span></p>
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