Table Of ContentAADDVVAANNCCEESS IINN
XX--RRAAYY AANNAALLYYSSIISS
VVoolluummee 3355BB
A Continuation Order Plan is available for this series. A continuation order will bring delivery of
each new volume immediately upon publication. Volumes are billed only upon actual shipment.
For further information please contact the publisher.
ADVANCES IN
X-RA Y ANAL YSIS
Volume 35B
Edited by
Charles S. Barrett John V. GiIfrich
University of Denver SachslFreeman Associates
Denver, Colorado Washington, D. C.
Ting C. Huang Ron Jenkins
IBM Almaden Research Center ICPDS-ICDD
San Iose, Califomia Swarthmore, Pennsylvania
Gregory J. McCarthy Paul K. Predecki
North Dakota State University University of Denver
Fargo, North Dakota Denver, Colorado
Richard Ryon Deane K. Smith
Lawrence Livermore National Laboratory Pennsylvania State University
Livermore, California University Park, Pennsylvania
Sponsored by
University of Denver Department of Engineering
and
JCPDS - International Centre for Diffraction Data
Springer Science+Business Media, LLC
The Library of Congress cataloged the first volume of this titIe as folIows:
Conference on Application of X-ray Analysis.
Proceedings 6th- 1957- [Denver]
v. iIIus. 24-28 Cßl. annual.
No proceedings published for the first 5 conferences.
Vals. for 1958- caIIed also: Advances in X-ray analysis, v. 2-
Proceedings for 1957 issued by the conference under an earlier name: Conference
on Industrial Applications of X-ray Analysis. Other slight variations in name of
conference.
Vol. for 1957 published by the University of Denver, Denver Research Institute,
Metallurgy Division.
VoIs. for 1958- distributed by Plenum Press, New York.
Conferences sponsored by University of Denver , Denver Research Institute.
1. X-rays-Industrial applications-Congresses. I. Denver University. Denver
Research Institute H. TitIe: Advances in X-ray analysis.
TA406.5.C6 58-35928
ISBN 978-1-4613-6532-7 ISBN 978-1-4615-3460-0 (eBook)
DOI 10.1007/978-1-4615-3460-0
Proceedings of combined First Pacific-International Congress
on X-Ray Analytical Methods (pICXAM) and Fortieth Annual Conference
on Applications of X-Ray Analysis, held August 7-16, 1991,
in Hilo and Honolulu, Hawaii
© Springer Science+Business Media New York 1992
Originally published by Plenum Press in 1992
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, e1ectronic, mechanical, photocopying, microfIlming,
recording, or otherwise, without written permission from the Publisher
MATHEMATICAL CORRECTION PROCEDURES IN XRF -
THE LONG AND THE SHORT
Gerald R. Lachance
Nepean, Ontario, Canada
The processes of absorption and enhancement in x-ray fluorescence
spectrometry preclude the direct conversion of measured intensities to
concentrations unless the specimens being analyzed are practically identical to the
reference materials that were used as standards for calibration. While well known
techniques of spectrometry such as the use of internal standards or techniques
specific to x-ray fluorescence such as Compton scatter may be used in some
instances, the analysis of alloys for example is not amenable to these techniques.
The goal of mathematical correction procedures is to compensate adequately for
matrix effects and therefore to extend the concentration range within which
analysis may be accomplished. The evolution of the concepts and expressions that
have been proposed for this purpose spans some 50 years, is somewhat fascinating
and examined in outline after a brief nostalgic look at where it all began.
Having set out to "examine spectra of a few elements in greater detail,"
Moseley! describes how he ". .. used as targets a number of substances mounted on
a truck inside an exhausted tube ... " with provision for ". .. each target to be
brought in turn into the line of fire." Of the elements chosen as forming a
continuous series (Z 20-30) ". .. calcium alone gave any trouble. In this case ... the
layer of lime which covered the surface of the metal gave off such a quantity of gas
that the x-ray could only be excited for a second or two at-a-time." Prophetically,
Moseley concluded that "the prevalence of lines due to impurities suggest that this
may prove a powerful method of chemical analysis."
In a second paper2, the instrumentation is described: ". .. the aluminium
trolley that carries the targets can be drawn to and fro by means of silk fishing
lines wound on brass bobbins. The slit (which defines the x-ray beam) should be
fixed exactly opposite the focus-spot of the cathode stream, though a slight error
can be remedied by deflecting the cathode rays with a magnet." Consider some of
Moseley's comments: ". .. the examination of keltium would be of exceptional
interest, as no place ... , ... lines due to impurities were frequently present but
caused little trouble except in the rare-earth group ... The x-ray spectrum of
praesodymia showed that it consisted roughly 50% La, 35% Ce and 15% Pr ... "
Advances in X-Ray Analysis. Vol. 35
Edited by C.S. Barrett et al. • Plenum Press. New York. 1992 693
694 X. MATHEMATICAL METHODS IN XRS
Some twenty years later, von Hevest determined lead in a number of alloys.
The procedure may be summarized as follows:
- one sample of zinc with 0.087% Pb used as standard
-equal exposures made with other samples
-Pb concentration calculated by direct ratio
with the comment, "Results for the alloys of unknown lead content ranged from
0.064 to 0.008 percent and, although no analysis by another method was carried
out, these values were in good agreement with other evidence as to the
composition of the samples." Thus was the stage set for x-ray fluorescence to
emerge as a technique for quantitative elemental determination and the derivation
of mathematical expressions for converting intensities to concentration. For easier
comparison, a common set of symbols (see Appendix) is used rather than those in
the original expressions.
One of the first theoretically derived expressions relating emitted intensity
to specimen composition is due to von Hamos.4 For the restricted analytical
context: binary specimen irradiated by a monochromatic incident source and
secondary fluorescence emission (enhancement) is not present, the emitted
intensity is given by
(1)
with the variable K; defined as a function of Cj, J1.j' J1.j and instrument geometry
only. Current practice for influence coefficient models tends towards the
calculation of the RECIPROCAL of K; as a function of all the elements EXCEPT Cj.
Some nine years later, Sherman5 derived the following expression for a
monochromatic incident source wavelength A
IllI' C. g.
l. = I I (2)
I • •
Cj Ili + Cj Ilj + ...
where
gj is a proportionality constant
IliI = Ili (A) esc V'
Il *i = Ili (A) esc V' + Ili (Ai) esc V"
*
Il j = Ilj (A) esc V' + Ilj (Ai) esc V"
thereby extending Eq (1) to multi-element systems. The terms for absorption of
the incident and emergent x-radiations were combined for each element
individually, the asterisk indicating that the path lengths of each having been
taken into consideration. Sherman6 and Shiraiwa and Fujin07 further developed
the theory of emitted intensity as a function of specimen composition and
fundamental parameters for a polychromatic incident beam but their treatment is
beyond the scope of this presentation. It incorporates an expression for
enhancement previously reported, without proof, by Gillam and Heal.8 However it
is interesting to note that in a discussion of the problem of extending
G. R. LACHANCE 695
monochromatic to polychromatic formulations, Sherman concluded "Instead of
measuring intensity. .. as counts per second, it is advantageous to represent the
intensity by the reciprocal of the counting rate, i.e., time required for the counter
to reach a predetermined fIxed count." This led to the expression
* *
{k)
+ (Cj) IJ. j + IJ. k + (3)
C. * C. *
IJ. i IJ. i
I I
It is noted that tj / t(i) = I(i) / Ij , the reciprocal of the generally current practice of
expressing relative intensity as
(4)
Beattie and Brissey9 retained Sherman's defInition of relative intensity and
reformulated Eq (3) yielding
*
IJ..
- I) C. + (-1.*. ) C. + ... = 0 (5)
I J
lJ.i
Faced with the prospect of working with a polychromatic incident source and
lacking the practical means of dealing with integrals over the wavelength range of
interest, Beattie and Brissey opted to calculate the values of the influence
coefficients from experimental intensity data using the expression
Kij = CC.j (1I(iJ. - I) (6)
J I
which are then introduced in Eq (5)
- (I(IiiJ -I)C. + Kij Cj +... = 0 (7)
I
For the determination of tantalum in niobium oxides, Campbell and CarllO
proposed the following expression to correct for the matrix effect due to the
presence of impurities
(8)
where Cj.expr is the value determined experimentally from a calibration of tantalum
oxide in niobium oxide and C is given by
j
l. (jor specimen)
CJ. = JI j(for 1 % C) , (or Cj determined chemically).
It was observed that the experimental data agreed qualitatively with those
expected from calculations of the linear absorption coefficients for the various
matrices. Mitchellll working with the same oxide system also generated influence
coefficient values experimentally (termed absorption-enhancement indices) and
examined their pattern for a given analyte by observing plots of indices versus
matrix element atomic number.
696 X. MATHEMATICAL METHODS IN XRS
A quite different concept was proposed by Lucas-Tooth and Price12 for
compensating mathematically for matrix effects. Here the individual corrections
are a function of intensities of the matrix elements rather than their
concentrations. In a subsequent work, Lucas-Tooth and Pyne13 proposed a model
in which apparent concentrations are substituted for the intensities of the matrix
elements. In either case the influence coefficients are obtained by regression of
experimental data. The coefficients are empirical, i.e., have little or no physical
meaning, and it is generally acknowledged that this approach is only applicable
over very limited concentration ranges.
In the mid 60's Lachance and Traill14 proposed a correction model that
circumvents the over-definition problem of the Beattie-Brissey model Eq (5),
namely
(9)
with the stipulation that the values of the influence coefficients may be calculated
provided the assumption can be made that a single wavelength may represent (i.e.,
be equivalent to) a polychromatic incident beam. This was an oversimplification
that was subsequently abandoned in favor of the concept that coefficients may be
Q
related to 'fundamental influence coefficients,' thus retaining a close link to the
basic principles of x-ray physics.
From 1968 on, the emergence of the fundamental parameters approach
proposed by Criss and Birks15 provided a method for the correction of matrix
effects based on physical principles of x-ray fluorescence emission and led to the re
evaluation of older influence coefficient models. Criss and Birks compared two
mathematical approaches for corrected matrix effects: the empirical regression
method which had been in use for many years, and a new method that accounts
for matrix effects by means of measured spectral distributions of x-ray tube
generated incident sources, mass attenuation coefficients, fluorescence yields, etc.
The model used for the regression method was that of Beattie and Brissey [Eq (5)]
but simplified, updated and optimized for computer evaluation. The coefficients
are determined using experimental data by solving sets of simultaneous equations
of the type
(10)
For a three component system, nine influence coefficients are required and
unknowns are analyzed by rewriting the above equation in the following form and
q, ...
solving for C
j,
etc., with the stipulation that Cj + Cj + ~ = 1.0
It was observed that influence coefficients selected in this way do not represent
the effect of one element on another but are the best set of numbers to describe
the intensities measured.
G. R. LACHANCE 697
The fundamental parameters approach for the correction of matrix effects
proposed by Criss and Birks is based on the concept of using either measured or
calculated spectral distributions of x-ray tubes and replacing integrals by
summations over a discrete number of effective wavelength intervals. This led to
the expression for primary fluorescence emission.
~ Ilj(A) leA) 11 A
P. = g C L.." (11)
, I I A 1l .. (A) esc lJr' + 1l .. (A) esc lJr"
where
I (A) is the incident intensity for the interval 11 A
:E
II. . = i C Il,; :E C, = 1.0
j
and to the expression (abridged) that includes both primary and secondary
fluorescence emission, i.e., enhancement present.
1
Pi + Si = Pi (1 + eij Cj) (12)
An iteration procedure is used to determine the weight fractions in unknowns.
The measured relative intensities are scaled to 1.0 and used as the first estimate of
composition. A set of theoretical relative intensities are calculated and compared
to the measured values. The differences are used to make a better estimate of
composition. The process is repeated until for some set of assumed concentrations,
the calculated relative intensities agree with the measured values.
Subsequently it was shown, Lachance,16 that the expressions for Pi + Si and
P(i) could be combined leading to the formalism
-1 1
Pi + Si = p(i! Ci Xj,a Cj + Xj,. Cj (13)
where ~ .. and ~.e are the sums of the prorated absorption and enhancement
monochromatic terms, respectively. Thus, the primary and secondary fluorescence
emitted intensity is equal to product P(I) C minus the sum of each individual
1
absorption effect plus the sum of each individual enhancement effect. Solving the
P(i) C leads to
i 1 -7
p(i! Cj = Pi + Si + Xj,a Cj Xj,. Cj (14)
which can be interpreted as 'the intensity corrected for matrix effect is equal to
the emitted intensity plus a correction term that compensates for the decrease in
intensity due to absorption minus a correction term that compensates for the
increased intensity due to secondary enhancement.' Defining ~.n = ~ .• -~.e and
substituting in Eq (14) gives
E
PC' C. = P. + S. + x. c. (15)
',' , , j loll 1
which can also be expressed as
P. + S, ~ X.
C. = -'- - [1 + L.." __1, _"_ C.] (16)
I Per) j Pi + Sj 1
698 X. MATHEMATICAL METHODS IN XRS
or
-7
Cj = Rj [1 + njj C) (17)
i.e., Criss and Birks' fundamental parameters equation expressed in influence
coefficient formalism. Two concepts that have been proposed in the domain of
fundamental influence coefficients are examined next, namely the option to choose
an element other than the analyte for elimination in the correction for matrix
effect term, and the option to calculate influence coefficients specific to absorption
and enhancement.
Given that the experimental determination of influence coefficients requires
a large number of standards and that these are not always available, de Jongh17,18
proposed a model wherein an element other than the analyte is arbitrarily chosen
for elimination in contrast to Eq (17) which is based on the elimination of the
analyte. The process is described as follows:
q / R, is calculated using a fundamental parameters expression for an
average composition ...
q / R, is calculated with respect to the average composition due to 0.1%
change of element j, leading to the computation of delta-coefficients.
The elimination of a selected components yields beta-coefficient values,
Betas are converted to 'ALPHAS' so that they may be used in expressions
similar to Eq (17), i.e., .
-7
Cj = Rj [1 + njjn C) (18)
where n is retained as the symbol for the influence coefficients to indicate that the
coefficients are in the domain of fundamental influence coefficients. The subscript
ijn refers to the influence of an exchange of j and n on the value of C R, of the
j /
analyte i. It is noted that j n in the summation term Eq (18) whereas j i in the
¢ ¢
summation term Eq (17).
The fact that Eq (14) may be broken down to
p(.I ) C.I = P.I + Lj x.lo a CJ. (19a)
and
(19b)
and transformed to
Cj = R/ (1 +Lj Qjj,p C) (20a)
and
+L
Rj = R/ (1 j ejj,p C) (20b)
where
P. X. X.
R.I ' =P-('/). , a jj,P =~P..' e jj,P =-hP!.., .
I
