291 19
contribution of the count statistics in the standard spectrum
to the overall measurement precision. The ideal standard to
maximize concentration is a pure element, but for those ele-
ments whose physical and chemical properties prohibit them
from being used in the pure state, for example, gaseous ele-
ments such as Cl, F, I, low melting elements such as Ga or In,
or pure elements which deteriorate under electron bombard-
ment, for example, P, S, a binary compound can be used, for
example, GaP, FeS 2 , CuS, KCl, etc.
19.2 Uncertainties in k-ratios
All k-ratio measurements must have associated uncertainty
estimates. The primary source of uncertainty in a k-ratio
measurement is typically count statistics although instru-
mental instability also can contribute. X-ray emission is a
classic example of a Poisson process or a random process
described by a negative exponential distribution.
Negative exponential distributions are interesting because
they are “memoryless.” For a sequence of events described by
a negative exponential distribution, the likelihood of an
event’s occurring in an interval τ is equally as likely regard-
less of when the previous event occurred. Just because an
event hasn’t occurred for a long time doesn’t make an event
any more likely in the subsequent time interval. In fact, the
most probable time for the next event is immediately follow-
ing the previous.
If X-rays are measured at an average rate R, the average
number of X-rays that will be measured over a time t is
N = R · t. Since the X-ray events occur randomly dispersed
in time, the actual number measured in a time t will rarely
ever be exactly N = R · t. Instead, 68.2 % of the time the
actual number measured will fall within the interval (N –
ΔN, N + ΔN) where ΔN ~ N1/2 when N is large (usually true
for X-ray counts). This interval is often called the “one
sigma” interval. The one-sigma fractional uncertainty is
thus N/N1/2 = 1/N1/2, which for constant R decreases as t
increased. This is to say that generally, it is possible to make
more precise measurements by spending more time making
the measurement. All else remaining constant, for example,
instrument stability and specimen stability under electron
bombardment, a measurement taken for a duration of 4 t
will have twice the precision of a measurement take for t.
Poisson statistics apply to both the WDS and EDS mea-
surement processes. For WDS, the on-peak and background
measurements all have associated Poissonian statistical
uncertainties. For EDS, each channel in the spectrum has an
associated Poissonian statistical uncertainty. In both cases,
the statistical uncertainties must be taken into account care-
fully so that an estimate of the measurement precision can be
associated with the k-ratio.
The best practices for calculating and reporting measure-
ment uncertainties are described in the ISO Guide to
Uncertainty in Measurement (ISO 2008 ). Marinenko and
Leigh ( 2010 ) applied the ISO GUM to the problem of k-ratios
and quantitative corrections in X-ray microanalytical mea-
surements. In the case of WDS measurements, the applica-
tion of ISO GUM is relatively straightforward and the details
are in Marinenko and Leigh ( 2010 ). For EDS measurements,
the process is more complicated. If uncertainties are associ-
ated with each channel in the standard and unknown spec-
tra, the k-ratio uncertainties are obtained as part of the
process of weighted linear squares fitting.19.3 Sets of k-ratios
Typically, a single compositional measurement consists of
the measurement of a number of k-ratios—typically one or
more per element in the unknown. The k-ratios in the set are
usually all collected under the same measurement conditions
but need not be. It is possible to collect individual k-ratios at
different beam energies, probe doses or even on different
detectors (e.g., multiple wavelength dispersive spectrometers
or multiple EDS with different isolation windows).
There may more than one k-ratio per element. Particularly
when the data is collected on an energy dispersive spectrom-
eter, more than one distinct characteristic peak per element
may be present. For period 4 transition metals, the K and L
line families are usually both present. In higher Z elements,
both the L and M families may be present. This redundancy
provides a question – Which k-ratio should be used in the
composition calculation?
While it is in theory possible to use all the redundant
information simultaneously to determine the composition,
standard practice is to select the k-ratio which is likely to pro-
duce the most accurate measurement. The selection is non-
trivial as it involves difficult to characterize aspects of the
measurement and correction procedures. Historically, select-
ing the optimal X-ray peak has been something of an art.
There are rules-of-thumb, but they involve subtle compro-
mises and deep intuition.
This subject is discussed in more detail in Appendix 19.A.
For the moment, we will assume that one k-ratio has been
selected for each measured element.kk={}Z:eZ∈lements
(19.3)Our task then becomes converting this set of k-ratios into an
estimate of the unknown material’s composition.CC=∈{}Z:eZ lements
(19.4)To get the most accurate trace and minor constituent
measurements, it is best to average together many
k-ratios from distinct measurements before applying the
matrix correction. Don’t truncate negative k-ratios before
you average or you’ll bias your results in the positive
direction.TIP19.3 · Sets of k-ratios