X-Ray Fluorescence Analytical Techniques - CNSTN : Centre ...

The relative error resulting from applying equation (II.3) instead of the exact equation
does not exceed 5% when the total mass per unit area is lower than:
0.1
( ) ( )
µ E cos ecφ+µ E cos ecψ
o i
; (II.6)
where µ(Eo) and µ(Ei) are the total mass attenuation coefficients for the whole specimen at the
energy of primary radiation (Eo) and the energy of characteristic X-rays of the ith element
(Ei), respectively; φ is the effective angle of incidence of the primary exciting beam; and ψ is
the effective take-off angle of characteristic X-rays. The total mass attenuation coefficient
µ(E) for the whole specimen at the energy E is given by the mixture rule:
n
µ ( E) = ∑ Wj µ j ( E)
, (II.7)
j=
1
where Wj and µj(E) are the weight fraction and the mass attenuation coefficient of the jth
element present in the sample, respectively, and n is the total number of the elements in the
sample. A major feature of the thin sample technique is that the intensity of characteristic Xrays,
Ithin, depends linearly on the concentration of the ith element; it is equivalent to the fact
that the so-called matrix effects can safely be neglected.
The values of the constant Si (called the sensitivity factors), which are necessary to
convert the measured intensity of the characteristic X-rays into mass concentrations, can be
determined either experimentally as the slope of the straight calibration line for the ith
element obtained on the basis of thin homogeneous standard samples or semi-empirically
based on both the experimentally determined (G/sin φ)IoEo value and the relevant fundamental
parameters (τi(Eo), ωi, ρi and ji). Also the detector efficiency ε(Ei) can b determined either
experimentally or theoretically based on the parameters of a given detector. In multi-element
XRF analysis, the calibration process can be greatly simplified because the elemental
sensitivities Si vary as a smooth function with atomic number.
Various homogeneous standard samples are now commercially available from several
manufactures. In many cases, one can also produce synthetic laboratory standard according to
the actual needs and possibilities; for example, by precipitating known quantities of elements
in solution, and filtering off as a thin layer or on a filter membrane.
V.2 Intermediate Thickness Samples
Intermediate thickness samples are defined as those samples whose masses per unit area
fulfil following inequality:
mthin < m< mthick
, (II.8)
where mthick is the mass of the so-called infinitely thick or saturated sample, above which
practically no further increase in the intensity of the characteristic radiation will be observed
as the sample thickness is increased, given by:
mthick
4.61
= . (II.9)
µ E cos ecφ+µ E cos ecψ
( ) ( )
o i
Intermediate thickness samples can be preferable to thick specimens because less
material is required, remaining uncertainties in the knowledge of the mass attenuation
coefficients have a smaller effect on the analysis results, the sensitivity is more favourable for
low-Z elements, and secondary enhancement effects are less important. In practice, samples
of intermediate thickness are used when the investigated material is scarce and does not allow