ASTM E1361-02(2021)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E1361 − 02 (Reapproved 2021)
Standard Guide for
Correction of Interelement Effects in X-Ray Spectrometric
Analysis
This standard is issued under the fixed designation E1361; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 3. Terminology
3.1 For definitions of terms used in this guide, refer to
1.1 This guide is an introduction to mathematical proce-
Terminology E135.
dures for correction of interelement (matrix) effects in quanti-
tative X-ray spectrometric analysis.
3.2 Definitions of Terms Specific to This Standard:
1.1.1 Theproceduresdescribedcorrectonlyfortheinterele-
3.2.1 absorption edge—the maximum wavelength (mini-
ment effect(s) arising from a homogeneous chemical compo-
mum X-ray photon energy) that can expel an electron from a
sition of the specimen. Effects related to either particle size, or
given level in an atom of a given element.
mineralogical or metallurgical phases in a specimen are not
3.2.2 analyte—an element in the specimen to be determined
treated.
by measurement.
1.1.2 These procedures apply to both wavelength and
3.2.3 characteristic radiation—X radiation produced by an
energy-dispersive X-ray spectrometry where the specimen is
element in the specimen as a result of electron transitions
considered to be infinitely thick, flat, and homogeneous with
between different atomic shells.
respect to the depth of penetration of the exciting X-rays (1).
3.2.4 coherent (Rayleigh) scatter—the emission of energy
1.2 This document is not intended to be a comprehensive
from a loosely bound electron that has undergone collision
treatment of the many different techniques employed to com-
with an incident X-ray photon and has been caused to vibrate.
pensateforinterelementeffects.ConsultRefs (2-5)fordescrip-
The vibration is at the same frequency as the incident photon
tions of other commonly used techniques such as standard
and the photon loses no energy. (See 3.2.7.)
addition, internal standardization, etc.
3.2.5 dead-time—time interval during which the X-ray de-
1.3 This international standard was developed in accor-
tection system, after having responded to an incident photon,
dance with internationally recognized principles on standard-
cannot respond properly to a successive incident photon.
ization established in the Decision on Principles for the
3.2.6 fluorescence yield—a ratio of the number of photons
Development of International Standards, Guides and Recom-
of all X-ray lines in a particular series divided by the number
mendations issued by the World Trade Organization Technical
of shell vacancies originally produced.
Barriers to Trade (TBT) Committee.
3.2.7 incoherent (Compton) scatter—theemissionofenergy
from a loosely bound electron that has undergone collision
2. Referenced Documents
withanincidentphotonandtheelectronhasrecoiledunderthe
2.1 ASTM Standards:
impact, carrying away some of the energy of the photon.
E135Terminology Relating to Analytical Chemistry for
3.2.8 influence coeffıcient—designated by α (β, γ, δ and
Metals, Ores, and Related Materials
other Greek letters are also used in certain mathematical
models), a correction factor for converting apparent mass
fractions to actual mass fractions in a specimen. Other terms
This guide is under the jurisdiction of ASTM Committee E01 on Analytical
commonly used are alpha coefficient and interelement effect
ChemistryforMetals,Ores,andRelatedMaterialsandisthedirectresponsibilityof
coefficient.
Subcommittee E01.20 on Fundamental Practices.
Current edition approved March 15, 2021. Published April 2021. Originally
3.2.9 mass absorption coeffıcient—designated by µ, an
ε1
approved in 1990. Last previous edition approved in 2014 as E1361–02(2014) .
atomic property of each element which expresses the X-ray
DOI: 10.1520/E1361-02R21.
absorption per unit mass per unit area, cm /g.
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
this standard.
3.2.10 primary absorption—absorption of incident X-rays
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
by the specimen.The extent of primary absorption depends on
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
the composition of the specimen and the X-ray source primary
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. spectral distribution.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1361 − 02 (2021)
3.2.11 primary spectral distribution—the output X-ray
spectral distribution usually from an X-ray tube. The X-ray
continuum is usually expressed in units of absolute intensity
per unit wavelength per electron per unit solid angle.
3.2.12 relative intensity—the ratio of an analyte X-ray line
intensity measured from the specimen to that of the pure
analyte element. It is sometimes expressed relative to the
analyte element in a multi-component reference material.
3.2.13 secondary absorption—the absorption of the charac-
teristicXradiationproducedinthespecimenbyallelementsin
the specimen.
3.2.14 secondary fluorescence (enhancement)—the genera-
tionofX-raysfromtheanalytecausedbycharacteristicX-rays
from other elements in the sample whose energies are greater
than the absorption edge of the analyte.
Curve A—Linear calibration curve.
Curve B—Absorption of analyte by matrix. For example, R versus C in
3.2.15 X-ray source—an excitation source which produces
Ni Ni
Ni-Fe binary alloys where nickel is the analyte element and iron is the matrix
X-rayssuchasanX-raytube,radioactiveisotope,orsecondary
element.
target emitter.
Curve C—Negative absorption of analyte by matrix. For example, R versus
Ni
C in Ni-Al alloys where nickel is the analyte element and aluminum is the
Ni
matrix element.
4. Significance and Use
Curve D—Enhancement of analyte by matrix. For example, R versus C in
Fe Fe
Fe-Ni alloys where iron is the analyte element and nickel is the matrix ele-
4.1 Accuracy in quantitative X-ray spectrometric analysis
ment.
depends upon adequate accounting for interelement effects
either through sample preparation or through mathematical
FIG. 1 Interelement Effects in X-Ray Fluorescence Analysis
correction procedures, or both. This guide is intended to serve
as an introduction to users of X-ray fluorescence correction
methods. For this reason, only selected mathematical models
relatively constant. In general, Curve B is obtained when the
for correcting interelement effects are presented. The reader is
absorptionbythematrixelementsinthespecimenofeitherthe
referredtoseveraltextsforamorecomprehensivetreatmentof
primary X-rays or analyte characteristic X-rays, or both, is
the subject (2-7).
greater than the absorption by the analyte alone. This second-
ary absorption effect is often referred to simply as absorption.
5. Description of Interelement Effects
The magnitude of the displacement of Curve B from CurveA
5.1 Matrix effects in X-ray spectrometry are caused by
in Fig. 1, for example, is typical of the strong absorption of
absorption and enhancement of X-rays in the specimen. Pri-
nickel K-L (K ) X-rays in Fe-Ni alloys. Curve C represents
2,3 α
mary absorption occurs as the specimen absorbs the X -rays
the general case where the matrix elements in the specimen
from the source. The extent of primary absorption depends on
absorb the primary X-rays or characteristic X-rays, or both, to
thecompositionofthespecimen,theoutputenergydistribution
a lesser degree than the analyte alone. This type of secondary
oftheexcitingsource,suchasanX-raytube,andthegeometry
absorption is often referred to as negative absorption. The
of the spectrometer. Secondary absorption occurs as the char-
magnitude of the displacement of Curve C from Curve A in
acteristic X radiation produced in the specimen is absorbed by
Fig. 1, for example, is typical of alloys in which the atomic
the elements in the specimen. When matrix elements emit
number of the matrix element (for example, aluminum) is
characteristicX-raylinesthatlieontheshort-wavelength(high
much lower than the analyte (for example, nickel). Curve D in
energy) side of the analyte absorption edge, the analyte can be
Fig. 1 illustrates an enhancement effect as defined previously,
excited to emit characteristic radiation in addition to that
and represents in this case the enhancement of iron K-L (K )
2,3 α
excited directly by the X-ray source. This is called secondary
X-rays by nickel K-L (K ) X-rays in Fe-Ni binaries.
2,3 α
fluorescence or enhancement.
NOTE 1—The relative intensity rather than absolute intensity of the
5.2 These effects can be represented as shown in Fig. 1
analytewillbeusedinthisdocumentforpurposesofconvenience.Itisnot
usingbinaryalloysasexamples.Whenmatrixeffectsareeither
meant to imply that measurement of the pure element is required, unless
under special circumstances as described in 9.1.
negligible or constant, Curve A in Fig. 1 would be obtained.
That is, a plot of analyte relative intensity (corrected for
6. General Comments Concerning Interelement
background, dead-time, etc.) versus analyte mass fraction
Correction Procedures
wouldyieldastraightlineoverawidemassfractionrangeand
would be independent of the other elements present in the 6.1 Historically, the development of mathematical methods
specimen (Note 1). Linear relationships often exist in thin for correction of interelement effects has evolved into two
specimens, or in cases where the matrix composition is approaches, which are currently employed in quantitative
constant. Low alloy steels, for example, exhibit constant X-ray analysis.When the field of X-ray spectrometric analysis
interelement effects in that the mass fractions of the minor was new, researchers proposed mathematical expressions,
constituents vary, but the major constituent, iron, remains which required prior knowledge of corrective factors called
E1361 − 02 (2021)
LT LT
influence coefficients or alphas prior to analysis of the speci- C 5 R 11α C 1α C (5)
~ !
k k ki i kj j
mens. These factors were usually determined experimentally
Therefore, six alpha coefficients are required to solve for the
by regression analysis using reference materials, and for this
mass fractions C, C, and C (see Appendix X1). Once the
i j k
reason are typically referred to as empirical or semi-empirical
influence coefficients are determined, Eq 3-5 can be solved for
procedures (see 7.1.3, 7.2, and 7.8). During the late 1960s,
the unknown mass fractions with a computer using iterative
another approach was introduced which involved the calcula-
techniques (see Appendix X2).
tion of interelement corrections directly from first principles
7.1.3 Determination of Influence (Alpha) Coeffıcients from
expressions such as those given in Section 8. First principles
Regression Analysis—Alpha coefficients can be obtained ex-
expressions are derived from basic physical principles and
perimentallyusingregressionanalysisofreferencematerialsin
contain physical constants and parameters, for example, which
which the elements to be measured are known and cover a
include absorption coefficients, fluorescence yields, primary
broad mass fraction range.An example of this method is given
spectral distributions, and spectrometer geometry. Fundamen-
in X1.1.1 of Appendix X1. Eq 1 can be rewritten for a binary
tal parameters method is a term commonly used to describe
specimen in the form:
interelement correction procedures based on first principle
R
equations (see Section 8).
~C /R ! 2 1 5 α C (6)
i i ij j
R
6.2 In recent years, several researchers have proposed
where: α =influence coefficient obtained by regression
ij
fundamental parameters methods to correct measured X-ray
analysis. A plot of (C/R)−1 versus C gives a straight line
i i j
R
intensities directly for interelement effects or, alternatively,
with slope α (see Fig. X1.1 of Appendix X1). Note that the
ij
proposed mathematical expressions in which influence coeffi-
superscript LT is replaced by R because alphas obtained by
cients are calculated from first principles (see Sections 7 and
regression analysis of multi-component reference materials do
LT
8). Such influence coefficient expressions are referred to as
notgenerallyhavethesamevaluesas α (asdeterminedfrom
ij
fundamental influence coefficient methods.
first principles calculations). This does not present a problem
generally in the results of analysis if the reference materials
7. Influence Coefficient Correction Procedures
bracket each of the analyte elements over the mass fraction
ranges that exist in the specimen(s). Best results are obtained
7.1 The Lachance-Traill Equation:
only when the specimens and reference materials are of the
7.1.1 Forthepurposesofthisguide,itisinstructivetobegin
same type. The weakness of the multiple-regression technique
with one of the simplest, yet fundamental, correction models
asappliedinX-rayanalysisisthattheaccuracyoftheinfluence
within certain limits. Referring to Fig. 1, either Curve B or C
coefficientsobtainedisnotknownunlessverified,forexample,
(thatis,absorptiononly)canberepresentedmathematicallyby
from first principles calculations. As the number of compo-
a hyperbolic expression such as the Lachance-Traill equation
nentsinaspecimenincreases,thisbecomesmoreofaproblem.
(LT) (8).Forabinaryspecimencontainingelements iand j,the
Results of analysis should be checked for accuracy by incor-
LT equation is:
poratingreferencematerialsintheanalysisschemeandtreating
LT
C 5 R ~11α C ! (1)
i i ij j
themasunknownspecimens.Comparisonoftheknownvalues
with those found by analysis should give acceptable
where:
agreement, if the influence coefficients are sufficiently accu-
C = mass fraction of analyte i,
i
rate. This test is valid only when reference materials analyzed
C = mass fraction of matrix element j,
j
as unknowns are not included in the set of reference materials
R = the analyte intensity in the specimen expressed as a
i
from which the influence coefficients were obtained.
ratio to the pure analyte element, and
LT
α = the influence coefficient, a constant. 7.1.4 Determination of Influence Coeffıcients from First
ij
Principles—Influence coefficients can be calculated from fun-
The subscript i denotes the analyte and the subscript j
LT
damentalparametersexpressions(seeX1.1.3ofAppendixX1).
denotes the matrix element. The subscript in α denotes the
ij
This is usually done by arbitrarily considering the composition
influence of matrix element j on the analyte i in the binary
of a complex specimen to be made up of the analyte and one
specimen. The LT superscript denotes that the influence coef-
matrix element at a time (for example, a series of binary
ficient is that coefficient in the LT equation. The magnitude of
elements, or compounds such as oxides). In this way, a series
the displacement of Curves B and C from Curve A is
LT
of influence coefficients are calculated assuming hypothetical
represented by α which takes on positive values for B type
ij
compositions for the binary series of elements or compounds
curves and negative values for C type curves.
that comprise the specimen(s). The hypothetical compositions
7.1.2 ThegeneralformoftheLTequationwhenextendedto
can be selected at certain well-defined limits. Details of this
multicomponent specimens is:
procedure are given in 9.3.
LT
C 5 R ~11 α C ! (2)
i i ( ij j
7.1.5 Use of Relative Intensities in Correction Methods—As
stated in Note 1, relative intensities are used for purposes of
For a ternary system, for example, containing elements i, j
convenience in most correction methods. This does not mean
and k, three equations can be written wherein each of the
that the pure element is required in the analysis unless it is the
elements are considered analytes in turn:
only reference material available. In that case, only fundamen-
LT LT
C 5 R 11α C 1α C (3)
~ !
i i ij j ik k
tal parameters methods would apply. If influence coefficients
LT LT
C 5 R ~11α C 1α C ! (4)
j j ji i jk k are obtained by regression methods from reference materials,
E1361 − 02 (2021)
then R can be expressed relative to a multi-component where:
i
reference material. Eq 6 can be rewritten in the form for
C = the analyte mass fraction in the fused specimen,
i
regression analysis as follows:
C = the mass fraction of the flux (for example, Li B O ),
f 2 4 7
R' α = influence coefficient which describes the absorption
C /R' 2 1 5 α C (7) if
~ !
i i ij j
effect of the flux on the analyte i, and
where:
R' = the relative intensity of the analyte in the fused
i
R' = analyte intensity in the specimen expressed as a ratio specimen to the intensity of the analyte in a fused
i
to a reference material in which the mass fraction of reference material.
i is less than 1.0, and
R' Various equations have been used in which the alpha
α = influence coefficient obtained by regression analysis.
ij
correctiondefinedaboveismodifiedbyincorporatingtheeffect
of a constant term. For example, the alphas in fused systems
R'
The terms R' and α can be related to the corresponding
i ij
can be modified by including the mass fraction of flux which
terms in Eq 6 by means of the following:
remainsessentiallyconstant.Thatis,theterm α /(1+ α C)in
ij if f
M
R' k 5 R (8) Eq 10 can be referred to as a modified alpha, α . The loss or
i i i ij
R
gain in mass on fusion can also be included in the alpha terms
α
ij
R'
α 5 (9)
ij
(Note 2). Modified alphas have also been used for non-fused
k
i
specimens in briquette form, such as minerals, to express the
where:
correction in terms of the metal oxides rather than the metals
k = a constant.
i themselves.
7.1.6 Limitations of the Lachance-Traill Equation:
NOTE 2—Under the action of heat and flux during fusion, the specimen
7.1.6.1 For the purposes of this guide, it is convenient to
willeitherloseorgainmassdependingontherelativeamountsofvolatile
classify the types of specimens most often analyzed by using
matterandreducedspeciesitcontains.Therefore,thetermslossonfusion
(LOF) and gain on fusion (GOF) are used to describe this behavior. It is
X-ray spectrometric methods into three categories: (1) metals,
commontoseethetermlossonignition(LOI)usedincorrectlytodescribe
(2) pressed minerals or powders, and (3) diluted samples such
this behavior.
as aqueous solutions, fusions with borate salts, and oils. When
a sample is fused in a fixed sample-to-flux ratio to produce a
7.1.6.2 If the influence coefficient in the Lachance-Traill
glass disk, or when a powdered sample is mixed in a fixed
equation is calculated from first principles as a function of
sample-to-binder ratio and pressed to produce a briquette,
mass fraction assuming absorption only, it can be shown that
LT
physical and chemical differences among materials are corre-
α is not a constant but varies with matrix mass fraction
ij
spondingly decreased and the magnitudes of the interelement
depending on the atomic number of each matrix element. This
effects are reduced and stabilized. Since enhancement effects
is illustrated in Table 1, for example, for a selected series of
are usually negligible in these systems, the LT equation is
binary specimens in which iron is the analyte. Note that in
sufficiently accurate in many applications for making interele-
some cases (for example, α ), the influence coefficient is
FeMg
ment corrections. It has also been shown that the LT equation
nearly constant whereas, for others (for example, α ), the
FeCo
is in agreement with first principles calculations when applied
influence coefficient exhibits a wide variation and even
tofusedspecimens(thatis,atleast1partsample+6partsflux
LT
changessign.Inpractice,thisvariationin α doesnotpresent
ij
dilutions or greater). For fused specimens, an equation can be
problems when the specimen composition varies over a rela-
written according to Lachance (9) as follows:
tively small range, and enhancement effects are absent. This
α
ij source of error is also minimized to some degree when type
C 5 R' ~11α C ! 11 C 1… (10)
i i if f F F G j G
11α C
if f reference materials are used which reasonably bracket the
A
TABLE 1 Alpha Coefficients for Analyte Iron in Binary Systems Computed Using Fundamental Parameters Equations
α
Fej
C O(8) Mg(12) Al(13) Si(14) Ca(20) Ti(22) Cr(24) Mn(25) Co(27) Ni(28) Cu(29) Zn(30) As(33) Nb(41) Mo(42) Sn(50)
Fe
0.01 −0.841 −0.52 −0.39 −0.25 0.93 1.46 2.08 −0.10 −0.18 −0.44 −0.42 −0.36 −0.13 0.74 0.86 2.10
0.02 − 0.840 − 0.52 − 0.39 − 0.25 0.93 1.46 2.08 − 0.10 − 0.17 − 0.44 − 0.41 − 0.35 − 0.13 0.74 0.86 2.10
0.05 − 0.839 − 0.51 − 0.39 − 0.25 0.93 1.46 2.09 − 0.10 − 0.15 − 0.42 − 0.41 − 0.35 − 0.12 0.74 0.86 2.10
0.10 − 0.838 − 0.51 − 0.39 − 0.25 0.93 1.46 2.09 − 0.10 − 0.14 − 0.40 − 0.39 − 0.34 − 0.12 0.75 0.86 2.10
0.20 − 0.835 − 0.51 − 0.38 − 0.24 0.94 1.47 2.10 − 0.10 − 0.11 − 0.36 − 0.37 − 0.32 − 0.11 0.76 0.87 2.11
0.50 −0.832 −0.50 −0.37 −0.22 0.96 1.50 2.13 −0.10 −0.04 −0.27 −0.31 −0.28 −0.08 0.78 0.90 2.14
0.80 − 0.831 − 0.49 − 0.36 − 0.21 1.01 1.55 2.19 − 0.10 0.00 − 0.20 − 0.25 − 0.24 − 0.05 0.83 0.94 2.20
0.90 − 0.830 − 0.48 − 0.35 − 0.20 1.03 1.58 2.23 −0.10 0.01 − 0.18 − 0.23 − 0.23 − 0.04 0.85 0.96 2.25
0.95 − 0.830 − 0.48 − 0.35 − 0.20 1.05 1.60 2.26 − 0.10 0.02 −0.17 −0.23 −0.22 −0.03 0.86 0.98 2.28
0.98 − 0.830 − 0.48 − 0.35 − 0.20 1.06 1.62 2.29 − 0.10 0.02 − 0.17 − 0.22 − 0.22 − 0.03 0.87 0.98 2.30
0.99 −0.830 −0.48 −0.35 −0.20 1.06 1.62 2.29 − 0.10 0.02 − 0.16 − 0.22 − 0.21 − 0.02 0.87 0.99 2.31
A
Data used by permission from G. R. Lachance, Geological Survey of Canada. The values represent the effect of the element listed at the top of each column on the
analyte Fe for each mass fraction of Fe listed in the first column.
E1361 − 02 (2021)
composition of the specimen(s). However, it should be recog- 7.3.1 The Claisse-Quintin equation (CQ) can be described
nizedthatforsometypesofsamples,whichhaveabroadrange as an extension of the Lachance-Traill equation to include
LT
of concentration, assumption of a constant α could lead to enhancement effects and can be written for a binary according
ij
inaccurate results. For example, in the cement industry, low to Refs 13, 14 as follows:
dilutions (for example, typically 1 part sample+2 parts flux)
C 5 R 11 α 1α C C (12)
~ !
i i ij ijj j j
(
@ #
have been employed to analyze cement and geological mate- n21
rials. Low dilutions are used to maximize the analyte intensity LT
where α + α C = α . The term α + α C allows for
ij ijj j ij ij ijj j
LT
for trace constituents.At such low dilutions, it has been shown
linearvariationof α withcomposition.AccordingtoClaisse
ij
by Moore (10) that a modified form of Eq 1 gives more
and Quintin (13) and Tertian (14), the interelement effect
accurate results. This modified or exponential form of Eq 1 is
correctionforternaryandmorecomplexsamplesisnotstrictly
also described in ASTM suggested methods (see E-2 SM
equal to a weighted sum of binary corrections. This phenom-
10-20, E-2 SM 10-26, and E-2 SM 10-34). In 7.2 – 7.7,
enon is referred to as a third element or cross-effect. For a
severalequationswillbedescribedwhichtakeintoaccountthe
ternary, the total correction for the interelement effects of j and
LT
variability in α with mass fraction, and are fundamentally
ij
k on the analyte i is given by Claisse and Quintin (13) as:
more accurate than Eq 1 because they also include correction
11~α 1α C !C 1~α 1α C ! C 1α C C (13)
ij ijj j j ik ikk k k ijk j k
for enhancement effects.
The binary correction terms for the effect of j on i and k on
7.2 The Rasberry-Heinrich Equation— Rasberry and Hein-
i are (α + α C) C and (α + α C ) C , respectively. The
ij ijj j j ik ikk k k
rich (RH) (11) proposed an empirical method to correct for
higher order term α C C is introduced to correct for the
ijk j k
both strong absorption and strong enhancement effects present
simultaneous presence of both j and k. The term α is called
ijk
in alloys such as Fe-Ni-Cr. The general expression can be
a cross-product coefficient. Tertian (15) has discussed in detail
written as follows:
the cross-effect and has introduced a term, ε, calculated from
n n
B
ik
first principles to correct for it. The contribution of the
C 5 R 11 A C 1 ·C (11)
F G
i i ij j k
( (
11C
~ !
j k i
cross-effect or cross-product term to the total correction is
relatively small, however, compared to the binary coefficient
where:
terms, but it can be significant.
A = a constant used when the significant effect of element
ij
7.3.2 ThegeneralformoftheClaisse-Quintinequationfora
jon iisabsorption;insuchcasesthecorresponding B
ik
multicomponent specimen can be written according to Ref 13
values are zero (and Eq 11 reduces to the Lachance-
as:
Traill equation), and
B = a constant used when the predominant effect of ele-
ik
C 5 R 11 ~α 1α C ! C 1 α C C (14)
i i ( ij ijj M j ( ( ijk j k
@ #
jfi1 j k
ment kon iisenhancement;thenthecorresponding A
ij
values are zero.
where C =sum of all elements in the specimen except i.
M
The binary coefficients, α and α , can be calculated from first
Eq 11 has given good results for analyses of Fe-Ni-Cr ij ijj
principles, usually at hypothetical compositions of C =0.20
ternary alloys. These authors obtained the coefficients by i
and 0.80, and C =0.80 and 0.20, respectively. The cross-
j
regression analysis of data from a series of Fe-Ni, and Fe-Cr,
product coefficient, α , is calculated at C =0.30, C =0.35,
and Ni-Cr binaries, and a series of Fe-Ni-Cr ternary reference ijk i j
and C =0.35.
k
materials, which covered a broad range of mass fractions from
essentially zero to 0.99. For Fe-Ni binaries, the enhancement
7.4 The Algorithm of Lachance (COLA):
B
ik 7.4.1 The comprehensive Lachance algorithm (COLA) pro-
term thatis, ·C gives values for the effect of Ni(k)on
S D
k
11C
~ !
i posed by Lachance (16) corrects for both absorption and
Fe(i) that are in reasonably good agreement with those pre-
enhancement effects over a broad range of mass fraction. The
dicted from first principles calculations over a broad range of
general form of the COLA expression is given as follows:
mass fraction. Further examination by several researchers of
C 5 R 11 α' C 1 α C C (15)
the accuracy of the RH equation for interelement effect
i i ij j ijk j k
( ( (
~ !
j j k
correction in other ferrous as well as non-ferrous binary alloys
The coefficient α' can be computed from the equation:
ij
reveal wide discrepancies when these coefficients are com-
pared to those obtained from first principles calculations. Even α C
2 M
α' 5 α 1 (16)
ij 1
modification of the enhancement term cannot overcome some
11α 1 2 C
~ !
3 M
of these limitations, as discussed by Tertian (12). For these
where α , α , and α are constants. The concept of cross-
1 2 3
reasons, the RH equation is not considered to be generally
product coefficients as given by Claisse and Quintin (see Eq
applicable, but it is satisfactory for making corrections in
14) is retained and included in Eq 15. The three constants (α ,
Fe-Ni-Cr alloys assuming availability of proper reference
α , and α)in Eq 16 are calculated from first principles using
2 3
materials.
hypothetical binary samples. For example, in alloy systems, α
7.3 The Claisse-Quintin Equation:
is the value of the coefficient at the C =1.0 limit (in practice
i
computed at C =0.999; and C =0.001). The value for α is
i j 2
the range within which α' will vary when the concentration of
4 ij
Suggested Methods for Analysis of Metals, Ores, and Related Materials, 9th
the analyte decreases to the C =0.0 limit (in practice, com-
ed.,ASTMInternationalHeadquarters,100BarrHarborDrive,POBoxC700,West i
Conshohocken, PA 19428-2959, 1992, pp. 507-573. puted from two binaries where C =0.001 and 0.999; and
i
E1361 − 02 (2021)
C =0.999 and 0.001, respectively). The α term expresses the where:
j 3
rate with which α' is made to vary hyperbolically within the
ij a = intercept,
o
two limits stated. In practice, it is generally computed from
a = slope, and
i
I = net intensity measured in counts per unit time.
threebinarieswhereC =0.001,0.5,and0.999;andC =0.999,
i j i
0.5, and 0.001, respectively. Since α can take on positive,
3 The terms a , a, and I are instrument-dependent parameters
o i i
zero, or negative values, α' can be computed for the entire
ij
and considered separate from the physical parameters mani-
dj
composition range from C =1.0 down to 0.0. The cross-
i fested in α .
ij
product coefficients α are calculated at the same levels as in
ijk 7.6.2 For a series of specimens containing n elements in
Eq 14.
whichtheconcentrationsofeachanalytevaryoverarange,De
7.4.2 For multi-element assay of alloys, all coefficients in Jongh’s method requires that the influence coefficients be
calculated at an average composition for each element (for
Eq15arecalculated.Foroxidespecimenssuchascementsand
¯ ¯ ¯
example,C ,C ,.C wherej =1,2,3,.n)inthespecimens.
powdered rocks, α is very small and in practice is usually
1 2 n
Both absorption and enhancement effects are treated by this
equated to zero. Eq 15 then reduces to the Claisse-Quintin Eq
method. An interesting feature of the method is that one
14. For fused specimens, another simplification can be made
element can be arbitrarily eliminated from the correction
because the mass fraction of the fluxing agent is the major
procedure so there is no need to measure it. For example, in
constituent and can be held relatively constant. In this case α ,
ferrousalloys,ironisoftenthemajorconstituentandisusually
α , and α are very small and in practice are also equated to
3 ijk
LT
determined by difference, and therefore, can be eliminated
zero, so that α reduces to α . Hypothetical binary standards
ij ij
LT
from the correction procedure. For details on the mathematical
are used to calculate α where C is taken at the mid-range
ij i
procedure used to eliminate a component from the analysis,
of the analyte concentration (for example, C = 0.5 and
i
refer to the original publication.
C =0.5) in the specimen.
j
7.7 MethodofBroll&Tertian—TheexpressionofBrolland
7.4.3 Asignificant improvement was obtained using COLA
LT
Tertian (22, 23) allows for variation of α in the Lachance-
rather than the CQ equation for the analysis of iron in a series ij
Traillequationtoaccountforbothabsorptionandenhancement
of Fe-Ni alloys (17). This is believed to be due to the term α
LT
effects. The term α in the LT equation is replaced by
ij
(1− C)in α' inEq16whichallowsfornonlinearvariationin
j ij
effective influence coefficients as follows:
α' withcompositionratherthanalinearvariationdescribedby
ij
the CQ relation. For this reason, the COLA equation is more C
i
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