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\begin{document}
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\title{ 
Quantitative determination of the effect of the harmonic component
in monochromatised synchrotron X-ray beam experiments
}
\author{
C.Q. Tran,
M. de Jonge,
Z. Barnea,
B. B. Dhal,
C. T. Chantler        \\
{\em School of Physics, University of Melbourne, Vic 3010, Australia} \\
}
\maketitle
\baselineskip=14.5pt
\begin{abstract}
Harmonic contamination has limited many synchrotron  experiments, 
often without the users realising the magnitude  of the problem. 
We  demonstrate a multiple-foil method for the quantitative 
determination of the fraction of the (333) third-order harmonic in a 
synchrotron x-ray beam monochromatised by a  monolithic silicon (111) 
channel-cut monochromator. 
The method is able to produce quantification of the effect of the harmonic
component below the 0.01\% level for the first time.


\end{abstract}
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\section{Introduction}
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Monochromators select from a given
spectrum a series of harmonics whose wavelengths satisfy the Bragg
equation for the monochromator diffracting planes.
Some harmonics can be minimized using diffracting planes with 
`forbidden'
second-order reflections as in the case of
the (111) planes of silicon and germanium monochromators.
However, third-order and higher harmonics may be present, especially when
their intensity in the source spectrum is significant.  Accurate
studies in which the monochromatic nature of the x-rays is important
require, therefore, a method for the quantitative determination of the
harmonic component in the x-ray beam.
The International Union of Crystallography
(IUCr) project to resolve discrepancies between experimentally
determined attenuation coefficients reported
that in earlier experiments `one quarter had an incident beam which
may have had second-harmonic contamination', and so were rejected.\cite{1} 
Such harmonic components occur in synchrotron and
laboratory x-ray beams and are often assumed to be insignificant
without quantification of their effect on
experimental results.

Most experimental configurations
minimise the harmonic component in the beam (e.g. by detuning), 
but the reduction of
the harmonic fraction by two orders of magnitude may still be
unacceptable for many studies.
The primary problem with many studies relating to attenuation
or scattering is alluded to by Creagh and Hubbell~\cite{1} when they
observe that `if a plot of $\ln(I/I_{0})$ against thickness\ldots does
not yield a straight line then no unique x-ray attenuation coefficient
exists and an investigation must take place to establish what is the
cause of this nonlinearity.'
Determination of the harmonic components by
absorption of the monochromatised beam by multiple foils is well known
in the literature.\cite{10} We have developed this
technique for use with synchrotron beams and ion-chamber detectors,
and extended it to provide accurate knowledge of the beam spectral
composition.  

 This approach is an essential ingredient in recent
form factor measurements.\cite{12}  The
technique relies on the log-linearity of x-ray absorption by
atomic materials.  Scattering cross-sections are often non-linear on
such a log plot, but this non-linearity makes only a small correction
in the X-ray range of energies.  
%
\section{The effect of harmonics in attenuation measurement}
%
Figure~\ref{detuning1foilu} shows the measured attenuation of eleven
sets of aluminium foils with thicknesses between 15 $\mu$m and 1
mm, in the path of an x-ray beam monochromated by a detuned,
double reflection silicon (111) channel-cut monochromator set to select 5
keV X-rays.  The slope at low thicknesses 
gives the attenuation coefficient for dominant first 
order harmonic radiation (i.e. 5 keV X-rays) while the slope for large
thicknesses gives 
the attenuation coefficient for third 
order harmonic radiation (i.e. the more penetrating 15 keV X-rays) 
The second order reflection is `forbidden'. 

% inserted figure
\begin{figure}[t]
 \vspace{5.0cm}
\special{psfile=detuning1foil2u.eps voffset=-60 hoffset=-10
         hscale=50 vscale=50 angle=0}
 \caption{\it
      The attenuation, $\ln(I/I_{0})$, as a function of the
thickness of aluminium absorber in the x-ray beam with a silicon
monochromator set to 5 keV. $\circ $ - experimental results; solid
line - curve of best fit corresponding to an admixture of $(1.09 \pm
0.02)\% $ third-order harmonic (15 keV).
    \label{detuning1foilu} }
\end{figure}
% figure inserted

The linearity of the slope is compromised even
for $-2<\ln[I/I_{0}]<0,$ where $I_{0}$ is the incident flux and $I$ the
attenuated flux. To account for the slope,
it is necessary to fit the proper equation for the harmonic fraction $x$, 
and the fundamental and higher harmonic attenuation coefficients
${\mu_{f}}$ and
${\mu_{h}}$ using $I=I_{O}[(1-x)e^{-\mu_{f}t}+xe^{-\mu_{h}t}]$ for the 
given series of thicknesses $t$.
The third order harmonic is seen by the deviation from linearity at 
low attenuation and by the inflection in the plot as the lower energy 
flux becomes heavily attenuated. However, 
the harmonic fraction is only obvious as the slope of the second 
linear part of the graph above 
log ratios of 5 or more. In this example, this corresponds
to thicknesses of 0.6 mm and to a first order attenuation log ratio of 
$\ln[I/I_{0}]\simeq -25$. 
We have probed the attenuation over a range
far exceeding the `recommended'~\cite{1} Nordfors~\cite{11} range of
$(-4<\ln[I/I_{0}]<-2)$;
corresponding to $(-30~<~\ln[I/I_{0}]~<~0)$ for the fundamental
energy. 
Sampling the linearity of the attenuation
over a much wider range than in previous investigations enables a more
extensive diagnosis of systematic effects including detector
linearity, harmonic content, dark current offsets and saturation.

Experimental results based on a 
small part of this curve will be in significant potential error, 
especially where the absorption coefficients are not known 
(and they are rarely known to better than 1\%). In 
this example, the thicknesses of the aluminium foils were not well known.
The requirement of precise knowledge of foil thickness was avoided by
using multiples of a single foil for each of these thicknesses.  In
this way the ratio of thicknesses of each of the absorbers was well
defined. 
The determined harmonic percentage is 1.09\% with an 
uncertainty of 0.02\% as the one standard deviation
uncertainty to a function fitting for both orders of radiation 
(without any assumption as to the attenuation coefficients for 
either). 

The measurements were carried out by using a metal `daisy wheel' on
whose perimeter were mounted eleven different thicknesses of foils.
These foils were placed in the beam by suitable rotation of
the daisy wheel.  This technique is accurate, reproducible and rapid.
The monitor and detector used throughout this work were matched
nitrogen gas-flow ion-chambers.  The work was performed at the bending
magnet beamline 20B of the Photon Factory synchrotron at Tsukuba.

Use of relatively calibrated sample thicknesses prevents us from
determining the attenuations ${\mu_{f}}$ and ${\mu_{h}}$ from this
measurement, but the harmonic component remains very well defined.
Any good attenuation measurement should probe
the harmonic component of the beam if it aims to be accurate at the
sub-percent level. 
Surprisingly, the harmonic contamination in numerous experiments may
exceed 10\% without the major problem being obvious to the 
experimenters. 


\section{A three-foil experiment showing the effect of the harmonic 
fraction and its energy
dependence}

To perform an actual attenuation measurement, it is
efficient to use a smaller number of carefully calibrated
thicknesses.  
A minimum of three samples of accurately known thickness
is required to simultaneously determine the harmonic percentage, 
and the fundamental and higher harmonic attenuation coefficients.  

% inserted figure
\begin{figure}[t]
 \vspace{5.0cm}
\special{psfile=3foilsearchgr3u.eps voffset=-60 hoffset=-10
         hscale=50 vscale=50 angle=0}
 \caption{\it
Effect of third-order harmonic contamination in attenuation
measurements using Si(111) monochromator and three silicon sample 
thicknesses at 5 keV.
All three measurements are consistent with a unique percentage of third-order
harmonic contamination of about 0.06\%.
\label{fig:harattn} }
\end{figure}
% figure inserted

If ${\mu_{h}}$ is provided by a separate experiment (or
theory) then use of three samples overdetermines the problem and allows for
error analysis or the possible observation of an
additional harmonic or other non-linearity.  
Figure~\ref{fig:harattn} shows the variation of the attenuation of
these three thicknesses at 5 keV when an admixture of 15 keV x-rays
(due to diffraction by the (333) plane of the silicon monochromator) is
present in the incident x-ray beam. The three points are
experimental curves, where the measurements only become consistent for 
the correct admixture of 15 keV X-rays. 
This (least-squares) fit 
can be performed using theoretical values for the
absorption of silicon for 15-keV x-rays.\cite{13,14}  
The uncertainty of 1\% in the tabulated value has
negligible effect on the uncertainty.  The three lines 
converge for a fraction 0.06\% of 15-keV x-rays.
The fitted one standard deviation
uncertainty is 0.007\%.
For this analysis, a quite thick sample is crucial.

% inserted figure
\begin{figure}[t]
 \vspace{5.0cm}
\special{psfile=simodelexptgr3u.eps voffset=-50 hoffset=-30
         hscale=50 vscale=50 angle=0}
 \caption{\it
      A harmonic component measurement with three well calibrated
thicknesses provides a constant and reliable indicator of accuracy in
attenuation measurements.  Error bars are given by the thickness of
the line.
    \label{fig:simodelexpt} }
\end{figure}
% figure inserted

Figure~\ref{fig:simodelexpt} shows the fitted $\ln(I/I_{0})$ curves of
the three-foil measurements as functions of thickness, with decreasing 
harmonic contamination as the energy is raised.\cite{12}
The final uncertainty depends on the accuracy 
of the
sample thicknesses and the counting statistics. 
Larger attenuation ranges also yield higher accuracy.

% inserted figure
\begin{figure}[t]
 \vspace{5.0cm}
\special{psfile=harpercentgr3u.eps voffset=-60 hoffset=-10
         hscale=50 vscale=50 angle=0}
 \caption{\it
Energy variation of the fraction of harmonic contamination in the 
example chosen (at maximum detuning).
    \label{fig:harpercent} }
\end{figure}
% figure inserted

Figure~\ref{fig:harpercent} shows the percentage of third-order
harmonic as a function of the energy of the fundamental x-rays.  The
energy dependence of the percentage of harmonic contamination detected
in our experiment is due to the fundamental output spectrum of the
synchrotron, the intensity of the monochromator diffraction of x-rays
of the various energies (with the given detuning current and its
effect on the suppression of the harmonic), and the relative detection
efficiency of the detector for the fundamental and harmonic x-rays.
Using this technique it is possible to correct for
very small fraction of harmonic contamination eg. $0.01\%$ at 5.6
keV. 

In other work,\cite{15} we show that the likelihood of residual harmonic 
contamination significantly affecting synchrotron experiments is much 
higher than often assumed, especially on an ID line where the 
third harmonic can often have high intrinsic flux and be transmitted 
 with minimal loss of intensity
 by any monochromator crystals and through any windows or other 
attenuation path.

The multiple-foil method can be used as a sensitive
diagnostic method for the quantitative determination of the fraction
of harmonic radiation in a monochromatised x-ray beam.  The method can
simultaneously also provide quantitative information about nonlinear
detector response such as may occur at high counting rates. The
determination of the harmonic fraction is the result of a number of
measurements and thus is insensitive to errors in the thickness
determination of the individual foils.
The simplicity of the method and the ease with which it
can be automated renders it suitable as a diagnostic 
test in a large variety of
experiments. 

\section{Acknowledgements}

We acknowledge encouragement from D. C. Creagh.
The work was performed at the Australian National Beamline Facility 
with
support from the Australian Synchrotron Research Program, which is 
funded
by the Commonwealth of Australia under the Major National Research
Facilities program.

%
\begin{thebibliography}{99}
  
\bibitem{1} DC Creagh, JH Hubbell, Acta Cryst.{\bf7} 102-112 (1987).

\bibitem{10} Z. Barnea, J. Mohyla, J. Appl.  Cryst.
{\bf7} 298-299 (1974).

\bibitem{11} B. Nordfors B., Arkiv f$\ddot{u}$r Fysik {\bf
18} 37-47 (1960).

\bibitem{12} CT Chantler, CQ Tran, D Paterson, DJ Cookson \& Z. Barnea
Phys.  Lett.  A {\bf 286} 338-346 (2001).

\bibitem{13} C.T. Chantler, J. Phys.  Chem.  Ref.  Data {\bf
24} 71-643 (1995).

\bibitem{14} C.T. Chantler, J. Phys.  Chem.  Ref.  Data {\bf
29} 597-1048 (2000).

\bibitem{15} C.Q. Tran, Z. Barnea, M. de Jonge, B.B. Dhal, 
D. Paterson, D.J. Cookson, C.T. Chantler, 
submitted X-ray Spectoscopy (2002).

\end{thebibliography}

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