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

Later attempts were made to find an empirical equation that more accurately accounted
for the real relationship between measured X-ray intensity and specimen concentration.
Claisse and Quintin took the original Sherman equation (Equation VI.5) and modeled for
polychromatic incident radiation by taking the superposition of mass absorption coefficients
at multiple energies.
Though there is no direct theoretical support of this, it was generally found that the
LaChance and Traill equation accounted for minor enhancement of X-ray intensities with
negative alpha coefficients. Rasberry and Heinrich observed that strongly enhancing elements
in binary mixtures yielded a concentration/intensity plot that did not follow the hyperbolic
dependence of the LaChance and Traill equation. This led them to propose a modified form of
the equation where a new term was to be used in place of the LaChance and Traill alpha
coefficient for analytes causing significant secondary enhancement:
⎛ ∑ βij
C j ⎞
i≠j Ci = R
⎜
i 1 ijC ⎟
∑
⎜
+ α j +
i≠j 1 C ⎟
⎜ + i ⎟
⎝ ⎠
. (VI.12)
By the middle of the seventies, other forms of the alpha correction models had been
proposed. Most notable are the equations of Tertian who, observing that alpha coefficients are
more properly not constant with specimen composition, proposed forms of the LaChance and
Traill, and Rasberry and Heinrich equations utilizing alpha coefficients that were linear
functions of element concentration Ci. Later, Tertian also showed that for a binary system, his
modified form of the Rasberry and Heinrich equation reduced to the Claisse and Quintin
equation.
III.3 Fundamental Parameters Method
Sherman’s equation (Equation VI.7) expresses the intensity of a characteristic X-ray
fluoresced from an element contained in a specimen of known composition. By determining
the concentrations of elements required to produce the measured set of intensities the
composition of a specimen can be determined. The direct use of Sherman’s equation is termed
‘the fundamental parameters method’. Instrument and measurement geometry effects are
removed by measuring characteristic line intensities emanating from standards of known
composition. Since this equation accounts for all absorption and enhancement, in theory only
one standard is required for each element. It should be noted that the standard should also
account for reflection from the surface of the specimen. As such, the surface texture of the
standard should be similar to that of the unknown.
Equation (VI.7) requires a knowledge of all elements contained in the specimen, the
values of the total mass absorption and mass photoabsorption coefficients of each of these
elements, and the step ratios of the mass photoabsorption coefficients at the absorption edges
of the measured characteristic lines. A knowledge of the incident X-ray tube intensity
distribution is also required. To account for secondary enhancement in the specimen, a
knowledge of shell fluorescent yields and line transition probabilities are required.
Criss and Birks were among the first to utilize the full fundamental parameters method.
They were able to obtain uncertainties in concentrations for nickel and iron-base alloys
between 0.1% and 1.7%. Aside from the requirement for significant computing power to
evaluate the above integrals, the method is limited by the accuracy of the fundamental
parameters themselves, and how well the tube spectrum is known. Determining a tube spectral
distribution is no trivial matter. Due to the intensity of the primary radiation, direct
measurement is not feasible. A common approach was to measure the reflected distribution