ASTM E1361-02(2014)e1

Standard Guide for Correction of Interelement Effects in X-Ray Spectrometric Analysis

SIGNIFICANCE AND USE
4.1 Accuracy in quantitative X-ray spectrometric analysis depends upon adequate accounting for interelement effects either through sample preparation or through mathematical 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 for correcting interelement effects are presented. The reader is referred to several texts for a more comprehensive treatment of the subject  (2-7).
SCOPE
1.1 This guide is an introduction to mathematical procedures for correction of interelement (matrix) effects in quantitative X-ray spectrometric analysis.  
1.1.1 The procedures described correct only for the interelement effect(s) arising from a homogeneous chemical composition of the specimen. Effects related to either particle size, or mineralogical or metallurgical phases in a specimen are not treated.  
1.1.2 These procedures apply to both wavelength and energy-dispersive X-ray spectrometry where the specimen is considered to be infinitely thick, flat, and homogeneous with respect to the depth of penetration of the exciting X-rays (1).2  
1.2 This document is not intended to be a comprehensive treatment of the many different techniques employed to compensate for interelement effects. Consult Refs (2-5) for descriptions of other commonly used techniques such as standard addition, internal standardization, etc.

General Information

• Status: Historical
• Publication Date: 14-Nov-2014
• ICS: 71.040.50 - Physicochemical methods of analysis
• Technical Committee: E01 - Analytical Chemistry for Metals, Ores, and Related Materials
• Drafting Committee: E01.20 - Fundamental Practices

Standards Content (Sample)

NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
´1
Designation: E1361 − 02 (Reapproved 2014)
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.
ε NOTE—Editorial corrections were made throughout in April 2015.
1. Scope 3.2.1 absorption edge—the maximum wavelength (mini-
mum X-ray photon energy) that can expel an electron from a
1.1 This guide is an introduction to mathematical proce-
given level in an atom of a given element.
dures for correction of interelement (matrix) effects in quanti-
tative X-ray spectrometric analysis. 3.2.2 analyte—an element in the specimen to be determined
by measurement.
1.1.1 Theproceduresdescribedcorrectonlyfortheinterele-
ment effect(s) arising from a homogeneous chemical compo-
3.2.3 characteristic radiation—X radiation produced by an
sition of the specimen. Effects related to either particle size, or
element in the specimen as a result of electron transitions
mineralogical or metallurgical phases in a specimen are not
between different atomic shells.
treated.
3.2.4 coherent (Rayleigh) scatter—the emission of energy
1.1.2 These procedures apply to both wavelength and
from a loosely bound electron that has undergone collision
energy-dispersive X-ray spectrometry where the specimen is
with an incident X-ray photon and has been caused to vibrate.
considered to be infinitely thick, flat, and homogeneous with
2 The vibration is at the same frequency as the incident photon
respect to the depth of penetration of the exciting X-rays (1).
and the photon loses no energy. (See 3.2.7.)
1.2 This document is not intended to be a comprehensive
3.2.5 dead-time—time interval during which the X-ray de-
treatment of the many different techniques employed to com-
tection system, after having responded to an incident photon,
pensateforinterelementeffects.ConsultRefs (2-5)fordescrip-
cannot respond properly to a successive incident photon.
tions of other commonly used techniques such as standard
3.2.6 fluorescence yield—a ratio of the number of photons
addition, internal standardization, etc.
of all X-ray lines in a particular series divided by the number
2. Referenced Documents of shell vacancies originally produced.
2.1 ASTM Standards: 3.2.7 incoherent (Compton) scatter—theemissionofenergy
from a loosely bound electron that has undergone collision
E135Terminology Relating to Analytical Chemistry for
Metals, Ores, and Related Materials withanincidentphotonandtheelectronhasrecoiledunderthe
impact, carrying away some of the energy of the photon.
3. Terminology
3.2.8 influence coeffıcient—designated by α (β, γ, δ and
3.1 For definitions of terms used in this guide, refer to other Greek letters are also used in certain mathematical
Terminology E135. models), a correction factor for converting apparent mass
fractions to actual mass fractions in a specimen. Other terms
3.2 Definitions of Terms Specific to This Standard:
commonly used are alpha coefficient and interelement effect
coefficient.
This guide is under the jurisdiction of ASTM Committee E01 on Analytical 3.2.9 mass absorption coeffıcient—designated by µ, an
ChemistryforMetals,Ores,andRelatedMaterialsandisthedirectresponsibilityof
atomic property of each element which expresses the X-ray
Subcommittee E01.20 on Fundamental Practices.
absorption per unit mass per unit area, cm /g.
Current edition approved Nov. 15, 2014. Published April 2015. Originally
approved in 1990. Last previous edition approved in 2007 as E1361–02 (2007).
3.2.10 primary absorption—absorption of incident X-rays
DOI: 10.1520/E1361-02R14E01.
by the specimen.The extent of primary absorption depends on
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
the composition of the specimen and the X-ray source primary
this standard.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or spectral distribution.
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
3.2.11 primary spectral distribution—the output X-ray
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. spectral distribution usually from an X-ray tube. The X-ray
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
´1
E1361 − 02 (2014)
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.
3.2.15 X-ray source—an excitation source which produces
X-rayssuchasanX-raytube,radioactiveisotope,orsecondary
Curve A—Linear calibration curve.
Curve B—Absorption of analyte by matrix. For example, R versus C in
target emitter.
Ni Ni
Ni-Fe binary alloys where nickel is the analyte element and iron is the matrix
element.
4. Significance and Use
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
4.1 Accuracy in quantitative X-ray spectrometric analysis
matrix element.
depends upon adequate accounting for interelement effects
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-
either through sample preparation or through mathematical
ment.
correction procedures, or both. This guide is intended to serve
as an introduction to users of X-ray fluorescence correction
FIG. 1 Interelement Effects in X-Ray Fluorescence Analysis
methods. For this reason, only selected mathematical models
for correcting interelement effects are presented. The reader is
primary X-rays or analyte characteristic X-rays, or both, is
referredtoseveraltextsforamorecomprehensivetreatmentof
greater than the absorption by the analyte alone. This second-
the subject (2-7).
ary absorption effect is often referred to simply as absorption.
The magnitude of the displacement of Curve B from CurveA
5. Description of Interelement Effects
in Fig. 1, for example, is typical of the strong absorption of
5.1 Matrix effects in X-ray spectrometry are caused by
nickel K-L (K ) X-rays in Fe-Ni alloys. Curve C represents
2,3 α
absorption and enhancement of X-rays in the specimen. Pri-
the general case where the matrix elements in the specimen
mary absorption occurs as the specimen absorbs the X -rays
absorb the primary X-rays or characteristic X-rays, or both, to
from the source. The extent of primary absorption depends on
a lesser degree than the analyte alone. This type of secondary
thecompositionofthespecimen,theoutputenergydistribution
absorption is often referred to as negative absorption. The
oftheexcitingsource,suchasanX-raytube,andthegeometry
magnitude of the displacement of Curve C from Curve A in
of the spectrometer. Secondary absorption occurs as the char-
Fig. 1, for example, is typical of alloys in which the atomic
acteristic X radiation produced in the specimen is absorbed by
number of the matrix element (for example, aluminum) is
the elements in the specimen. When matrix elements emit
much lower than the analyte (for example, nickel). Curve D in
characteristicX-raylinesthatlieontheshort-wavelength(high
Fig. 1 illustrates an enhancement effect as defined previously,
energy) side of the analyte absorption edge, the analyte can be
and represents in this case the enhancement of iron K-L (K )
2,3 α
excited to emit characteristic radiation in addition to that
X-rays by nickel K-L (K ) X-rays in Fe-Ni binaries.
2,3 α
excited directly by the X-ray source. This is called secondary
NOTE 1—The relative intensity rather than absolute intensity of the
fluorescence or enhancement.
analytewillbeusedinthisdocumentforpurposesofconvenience.Itisnot
5.2 These effects can be represented as shown in Fig. 1
meant to imply that measurement of the pure element is required, unless
usingbinaryalloysasexamples.Whenmatrixeffectsareeither under special circumstances as described in 9.1.
negligible or constant, Curve A in Fig. 1 would be obtained.
6. General Comments Concerning Interelement
That is, a plot of analyte relative intensity (corrected for
Correction Procedures
background, dead-time, etc.) versus analyte mass fraction
wouldyieldastraightlineoverawidemassfractionrangeand 6.1 Historically, the development of mathematical methods
would be independent of the other elements present in the for correction of interelement effects has evolved into two
specimen (Note 1). Linear relationships often exist in thin approaches, which are currently employed in quantitative
specimens, or in cases where the matrix composition is X-ray analysis.When the field of X-ray spectrometric analysis
constant. Low alloy steels, for example, exhibit constant was new, researchers proposed mathematical expressions,
interelement effects in that the mass fractions of the minor which required prior knowledge of corrective factors called
constituents vary, but the major constituent, iron, remains influence coefficients or alphas prior to analysis of the speci-
relatively constant. In general, Curve B is obtained when the mens. These factors were usually determined experimentally
absorptionbythematrixelementsinthespecimenofeitherthe by regression analysis using reference materials, and for this
´1
E1361 − 02 (2014)
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
~C /R ! 2 1 5 α C (6)
i i ij j
equations (see Section 8).
R
where: α =influence coefficient obtained by regression
6.2 In recent years, several researchers have proposed
ij
analysis. A plot of (C/R)−1 versus C gives a straight line
fundamental parameters methods to correct measured X-ray
i i j
R
with slope α (see Fig. X1.1 of Appendix X1). Note that the
intensities directly for interelement effects or, alternatively,
ij
proposed mathematical expressions in which influence coeffi- superscript LT is replaced by R because alphas obtained by
regression analysis of multi-component reference materials do
cients are calculated from first principles (see Sections 7 and
LT
8). Such influence coefficient expressions are referred to as notgenerallyhavethesamevaluesas α (asdeterminedfrom
ij
first principles calculations). This does not present a problem
fundamental influence coefficient methods.
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
7.1 The Lachance-Traill Equation:
ranges that exist in the specimen(s). Best results are obtained
7.1.1 Forthepurposesofthisguide,itisinstructivetobegin
only when the specimens and reference materials are of the
with one of the simplest, yet fundamental, correction models
same type. The weakness of the multiple-regression technique
within certain limits. Referring to Fig. 1, either Curve B or C
asappliedinX-rayanalysisisthattheaccuracyoftheinfluence
(thatis,absorptiononly)canberepresentedmathematicallyby
coefficientsobtainedisnotknownunlessverified,forexample,
a hyperbolic expression such as the Lachance-Traill equation
from first principles calculations. As the number of compo-
(LT) (8).Forabinaryspecimencontainingelements iand j,the
nentsinaspecimenincreases,thisbecomesmoreofaproblem.
LT equation is:
Results of analysis should be checked for accuracy by incor-
LT
poratingreferencematerialsintheanalysisschemeandtreating
C 5 R 11α C (1)
~ !
i i ij j
themasunknownspecimens.Comparisonoftheknownvalues
where:
with those found by analysis should give acceptable
C = mass fraction of analyte i,
agreement, if the influence coefficients are sufficiently accu-
i
C = mass fraction of matrix element j,
j rate. This test is valid only when reference materials analyzed
R = the analyte intensity in the specimen expressed as a
i
as unknowns are not included in the set of reference materials
ratio to the pure analyte element, and
from which the influence coefficients were obtained.
LT
α = the influence coefficient, a constant.
ij
7.1.4 Determination of Influence Coeffıcients from First
The subscript i denotes the analyte and the subscript j
Principles—Influence coefficients can be calculated from fun-
LT
denotes the matrix element. The subscript in α denotes the
ij
damentalparametersexpressions(seeX1.1.3ofAppendixX1).
influence of matrix element j on the analyte i in the binary
This is usually done by arbitrarily considering the composition
specimen. The LT superscript denotes that the influence coef-
of a complex specimen to be made up of the analyte and one
ficient is that coefficient in the LT equation. The magnitude of
matrix element at a time (for example, a series of binary
the displacement of Curves B and C from Curve A is
elements, or compounds such as oxides). In this way, a series
LT
represented by α which takes on positive values f
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
´1
Designation: E1361 − 02 (Reapproved 2007) E1361 − 02 (Reapproved 2014)
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. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Editorial corrections were made throughout in April 2015.
1. Scope
1.1 This guide is an introduction to mathematical procedures for correction of interelement (matrix) effects in quantitative X-ray
spectrometric analysis.
1.1.1 The procedures described correct only for the interelement effect(s) arising from a homogeneous chemical composition
of the specimen. Effects related to either particle size, or mineralogical or metallurgical phases in a specimen are not treated.
1.1.2 These procedures apply to both wavelength and energy-dispersive X-ray spectrometry where the specimen is considered
to be infinitely thick, flat, and homogeneous with respect to the depth of penetration of the exciting X rays X-rays (1).
1.2 This document is not intended to be a comprehensive treatment of the many different techniques employed to compensate
for interelement effects. Consult Refs (2-5) for descriptions of other commonly used techniques such as standard addition, internal
standardization, etc.
2. Referenced Documents
2.1 ASTM Standards:
E135 Terminology Relating to Analytical Chemistry for Metals, Ores, and Related Materials
3. Terminology
3.1 For definitions of terms used in this guide, refer to Terminology E135.
3.2 Definitions of Terms Specific to This Standard:
3.2.1 absorption edge—the maximum wavelength (minimum X-ray photon energy) that can expel an electron from a given level
in an atom of a given element.
3.2.2 analyte—an element in the specimen whose concentration is to be determined.determined by measurement.
3.2.3 characteristic radiation—X radiation produced by an element in the specimen as a result of electron transitions between
different atomic shells.
3.2.4 coherent (Rayleigh) scatter—the emission of energy from a loosely bound electron that has undergone collision with an
incident X-ray photon and has been caused to vibrate. The vibration is at the same frequency as the incident photon and the photon
loses no energy. (See 3.2.7.)
3.2.5 dead-time—time interval during which the X-ray detection system, after having responded to an incident photon, cannot
respond properly to a successive incident photon.
3.2.6 fluorescence yield—a ratio of the number of photons of all X-ray lines in a particular series divided by the number of shell
vacancies originally produced.
3.2.7 incoherent (Compton) scatter—the emission of energy from a loosely bound electron that has undergone collision with an
incident photon and the electron has recoiled under the impact, carrying away some of the energy of the photon.
This guide is under the jurisdiction of ASTM Committee E01 on Analytical Chemistry for Metals, Ores, and Related Materials and is the direct responsibility of
Subcommittee E01.20 on Fundamental Practices.
Current edition approved Jan. 15, 2007Nov. 15, 2014. Published January 2007April 2015. Originally approved in 1990. Last previous edition approved in 20022007 as
E1361 – 02.E1361 – 02 (2007). DOI: 10.1520/E1361-02R07.10.1520/E1361-02R14E01.
The boldface numbers in parentheses refer to the list of references at the end of this standard.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
´1
E1361 − 02 (2014)
3.2.8 influence coeffıcient—designated by α (β, γ, δ and 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 commonly used are
alpha coefficient and interelement effect coefficient.
3.2.9 mass absorption coeffıcient—designated by μ, an atomic property of each element which expresses the X-ray absorption
per unit mass per unit area, cm /g.
3.2.10 primary absorption—absorption of incident X rays X-rays by the specimen. The extent of primary absorption depends
on the composition of the specimen and the X-ray source primary spectral distribution.
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 characteristic X radiation produced in the specimen by all elements in the
specimen.
3.2.14 secondary fluorescence (enhancement)—the generation of X rays X-rays from the analyte caused by characteristic X rays
X-rays from other elements in the sample whose energies are greater than the absorption edge of the analyte.
3.2.15 mass fraction—a concentration unit expressed as a ratio of the mass of analyte to the total mass.
3.2.15 X-ray source—an excitation source which produces X rays X-rays such as an X-ray tube, radioactive isotope, or
secondary target emitter.
4. Significance and Use
4.1 Accuracy in quantitative X-ray spectrometric analysis depends upon adequate accounting for interelement effects either
through sample preparation or through mathematical 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 for correcting
interelement effects are presented. The reader is referred to several texts for a more comprehensive treatment of the subject (2-7).
5. Description of Interelement Effects
5.1 Matrix effects in X-ray spectrometry are caused by absorption and enhancement of X rays X-rays in the specimen. Primary
absorption occurs as the specimen absorbs the X -rays from the source. The extent of primary absorption depends on the
composition of the specimen, the output energy distribution of the exciting source, such as an X-ray tube, and the geometry of the
spectrometer. Secondary absorption occurs as the characteristic X radiation produced in the specimen is absorbed by the elements
in the specimen. When matrix elements emit characteristic X-ray lines that lie on the short-wavelength (high energy) side of the
analyte absorption edge, the analyte can be excited to emit characteristic radiation in addition to that excited directly by the X-ray
source. This is called secondary fluorescence or enhancement.
5.2 These effects can be represented as shown in Fig. 1 using binary alloys as examples. When matrix effects are either
negligible or constant, Curve A in Fig. 1 would be obtained. That is, a plot of analyte relative intensity (corrected for background,
dead-time, etc.) versus analyte mass fraction would yield a straight line over a wide mass fraction range and would be independent
of the other elements present in the specimen (Note 1). Linear relationships often exist in thin specimens, or in cases where the
matrix composition is constant. Low alloy steels, for example, exhibit constant interelement effects in that the mass fractions of
the minor constituents vary, but the major constituent, iron, remains relatively constant. In general, Curve B is obtained when the
absorption by the matrix elements in the specimen of either the primary X rays X-rays or analyte characteristic X rays, X-rays,
or both, is greater than the absorption by the analyte alone. This secondary absorption effect is often referred to simply as
absorption. The magnitude of the displacement of Curve B from Curve A in Fig. 1, for example, is typical of the strong absorption
of nickel nickel K-L (K ) X rays X-rays in Fe-Ni alloys. Curve C represents the general case where the matrix elements in the
2,3 α
specimen absorb the primary X rays X-rays or characteristic X rays, X-rays, or both, to a lesser degree than the analyte alone. This
type of secondary absorption is often referred to as negative absorption. The magnitude of the displacement of Curve C from Curve
A in Fig. 1, for example, is typical of alloys in which the atomic number of the matrix element (for example, aluminum) is much
lower than the analyte (for example, nickel). Curve D in Fig. 1 illustrates an enhancement effect as defined previously, and
represents in this case the enhancement of iron K-L (K ) X rays X-rays by nickel K-L (K ) X rays X-rays in Fe-Ni binaries.
2,3 α 2,3 α
NOTE 1—The relative intensity rather than absolute intensity of the analyte will be used in this document for purposes of convenience. It is not meant
to imply that measurement of the pure element is required, unless under special circumstances as described in 9.1.
6. General Comments Concerning Interelement Correction Procedures
6.1 Historically, the development of mathematical methods for correction of interelement effects has evolved into two
approaches, which are currently employed in quantitative X-ray analysis. When the field of X-ray spectrometric analysis was new,
researchers proposed mathematical expressions, which required prior knowledge of corrective factors called influence coefficients
or alphas prior to analysis of the specimens. These factors were usually determined experimentally by regression analysis using
´1
E1361 − 02 (2014)
Curve A—Linear calibration curve.
Curve B—Absorption of analyte by matrix. For example, R versus C in
Ni Ni
Ni-Fe binary alloys where nickel is the analyte element and iron is the matrix
element.
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.
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
element.
FIG. 1 Interelement Effects in X-Ray Fluorescence Analysis
reference materials, and for this reason are typically referred to as empirical or semi-empirical procedures (see 7.1.3, 7.2, and 7.8).
During the late 1960s, another approach was introduced which involved the calculation of interelement corrections directly from
first principles expressions such as those given in Section 8. First principles expressions are derived from basic physical principles
and contain physical constants and parameters, for example, which include absorption coefficients, fluorescence yields, primary
spectral distributions, and spectrometer geometry. Fundamental parameters method is a term commonly used to describe
interelement correction procedures based on first principle equations (see Section 8).
6.2 In recent years, several workersresearchers have proposed fundamental parameters methods to correct measured X-ray
intensities directly for interelement effects or, alternatively, proposed mathematical expressions in which influence coefficients are
calculated from first principles (see Sections 7 and 8). Such influence coefficient expressions are referred to as fundamental
influence coefficient methods.
7. Influence Coefficient Correction Procedures
7.1 The Lachance-Traill Equation:
7.1.1 For the purposes of this guide, it is instructive to begin with one of the simplest, yet fundamental, correction models within
certain limits. Referring to Fig. 1, either Curve B or C (that is, absorption only) can be represented mathematically by a hyperbolic
expression such as the Lachance-Traill equation (LT) (8). For a binary specimen containing elements i and j, the LT equation is:
LT
C 5 R 11α C (1)
~ !
i i ij j
where:
C = mass fraction of analyte i,
i
C = mass fraction of matrix element j,
j
R = the analyte intensity in the specimen expressed as a ratio to the pure analyte element, and
i
LT
α = the influence coefficient, a constant.
ij
LT
The subscript i denotes the analyte and the subscript j denotes the matrix element. The subscript in α denotes the influence
ij
of matrix element j on the analyte i in the binary specimen. The LT superscript denotes that the influence coefficient is that
LT
coefficient in the LT equation. The magnitude of the displacement of Curves B and C from Curve A is represented by α which
ij
takes on positive values for B type curves and negative values for C type curves.
7.1.2 The general form of the LT equation when extended to multicomponent specimens is:
LT
C 5 R ~11 α C ! (2)
i i ( ij j
For a ternary system, for example, containing elements i, j and k, three equations can be written wherein each of the elements
are considered analytes in turn:
´1
E1361 − 02 (2014)
LT LT
C 5 R 11α C 1α C (3)
~ !
i i ij j ik k
LT LT
C 5 R 11α C 1α C ! (4)
~
j j ji i jk k
LT LT
C 5 R ~11α C 1α C ! (5)
k k ki i kj j
Therefore, six alpha coefficients are required to solve for the mass fractions C , C , and C (see Appendix X1). Once the influence
i j k
coefficients are determined, Eq 3-5 can be solved for the unknown mass fractions with a computer using iterative techniques (see
Appendix X2).
7.1.3 Determination of Influence (Alpha) Coeffıcients from Regression Analysis—Alpha coefficients can be obtained experi-
mentally using regression analysis of reference materials in which the elements to be measured are known and cover a broad mass
fraction range. An example of this method is given in X1.1.1 of Appendix X1. Eq 1 can be rewritten for a binary specimen in the
form:
R
~C /R !2 15 α C (6)
i i ij j
R
where: α = influence coefficient obtained by regression analysis. A plot of (C /R ) − 1 versus C gives a straight line with slope
ij i i j
R
α (see Fig. X1.1 of Appendix X1). Note that the superscript LT is replaced by R because alphas obtained by regression analysis
ij
LT
of multi-component reference materials do not generally have the same values as α (as determined from
...