American Mineralogist, Volume 58, pages 1069-1072, 1973
Behaviorof X-ray Mass Absorption
Coefficientsnear AbsorptionEdges:
Reynolds'Method Revisited
Dnvrp Werxrn
Dep'qrtment of Geological Sciences, Hqrvsrd Uniuersity,
Cambridge, Massachusetts 02 138
Abstract
Reynolds' (1967) method for relating the mass absorption coefficient on the long wavelength
side of a major element absorption edge to the intensity of the emission line of the element
causing the absorption discontinuity is shown to be a special case of a more general relation.
The systematic positive errors noted by Reynolds in using his method are consistent in sign
and of similar magnitude to errors generated by neglecting the general relation. If the ratio
of the mass absorption coefficient on the short wavelength side to that on the long wavelength
side of a major element's absorption edge is plotted against the peak intensity of this element'
the plots allow use of the general relation and permit simultaneous comparisons to be made
to more than one standard. The technique is used to best advantage (in common geologic applications) in the analysis of elements of atomic numbers from calcium to iron.
present,Reynolds' (1.967) suggestionfor calculating
mass absorption coefficientsin a wavelengthregion
Quantitative X-ray fluorescenceanalysisrequires
major elementabsorpthat matrix absorption effectsbe either uniform or on a long wavelengthside of a
coefficient
on the short wavethe
edge
when
tion
known if nonuniform when comparing X-ray emisA more genwelcome.
known
is
most
is
length
side
sion line intensitiesof standardsand unknowns.The
presented
problem
now
is
this
treatment
of
eral
use of standardsvery similar to the unknowns or
treatment,
Reynolds'
comparison
with
a
and
includes
the use of high dilution and/or heavy absorbers
for a better solution to
are techniqueswhich make use of uniform absorp- followed by a suggestion
problem.
the
tion qualities of standards and unknowns. The
absorption characteristicsneed not be specifically
Discussion of Theory and Comparison Methods
known. The use of internal standardsor the use of
In generalthe rnagnitudeof the disruption of the
iterative calculations based on the content of all
major elementspresent or the use of mass absorp- smooth variation with wavelength of mass absorption coefficientsestimated from either transmitted tion coefficient of a sample in wavelength regions
or scatteredradiation are methodswhich correct for near the absorptionedgeof a major elementpresent
non-uniform matrix absorption effects.
in that sample will be a function of how much of
Reynolds (1963) describeda very useful tech- that element is present in that sample. If p(,fe) is
nique for determining the X-rav mass absorption the mass absorptioncoefficientof the samplein the
coefficient of a rock (for wavelengthregions near region of high absorptionrelative to the edge (short
the primary X-ray beam not disrupted by major wavelengthside) and lr(rn) is the mass absorption
element absorption edges) by measuringCompton- coefficientof the samplein the region of the emitted
scatteredradiation generatedby the primary X-ray line associatedwith the absorptionedge (long wavebeam interacting with the sample.This method has length side), then since the intensity In of the
been checked by Powell, Skinner, and Walker emissionline of the elementcausingthe absorption
(1969) and found to give results of high accuracy discontinuity increaseswith increasing amounts of
and precision. Inasmuch as there are elementsof that elementpresent,we should expect that lp(x^) /
interest with emission lines of longer wavelength p(rE)l should increasewith Ip.
Liebhafsky,Pfeiffer,Winslow andZemany( 1960)
than the absorption edge of the heaviest element
Introduction
1069
1070
MINERALOGICAL
and Norrish and Chappell (1967) among others
havejustified the following expressionfor the intensity of an element'semissionline In as a function of
the weight fraction of the element present XB and
the absorptioncoefficientsof the sampleat the wavelengths of the absorbedand emitted radiation.
X-I"K
p ( t r e ) . s e cd * p ( t r n ) . s e c 0
Is:
NOTES
tuting for XB involved an implicit assumptionwhich
is not strictly correct for a situation involving a
major element absorption edge in the wavelength
range of interest.The implicit but incorrect assumption of Reynolds' equation (7) could have been
statedthis wav.
P(tra)
p(tru)
(l)
where d is the anglebetweenthe primary beam and
the normal to the sample, f is the angle between
the secondarybeam and the normal to the sample,
16 is the intensity of the primary radiation, and K
is the yield factor which is a measureof excitation,
detection, and collimation efficiency.
Equation (1) can be rearrangedto give
P(tre)n
P(trn)n
(8)
If for the sample only the element in question
has an absorption edge in the wavelength region
of interest,the expression[p(Ie)n/p(In)nJ can be
considereda constant,and the absorptionexpression
in equation (2) can be recast into the following
form:
p ( t r a ) ' s edc * p ( X B ) . s e c f : p ( t r 6 )
,"
:
f_
ft-
[p(\a).sec 0 * p(trn).sec6]
The mass absorption coefficient of a sample at
any wavelength is a linear function of the weight
fraction of the components of the sample times
the mass absorption coefficient of the component
of the sample.
p(Xa) :
Xo.p(X-o)' + (1 -
X,).p(tr.o)^
(3)
where p,,(,la)s and p(,1,a)* refer to the mass absorption coefficients of the element causing the edge in
question and the remainder of the sample, respectively. Rewriting:
p(tr")n.(l -
Xu) :
p(\a) -
p(tra)o.Xn
(4)
Similarly for the wavelength of emission:
p(tr')".(l -
X') :
p ( t r o )-
p(tr,)".Xu
(5)
: p(r").45*
p(xo)
'
- ' l r ( A e ) n * x"{p(r^)"
-l'
""
(6)
At this point we can substitute (2) for XB and
obtain after some rearrangement:
: u!l^J-+ l!- (-^^^ r p(\e)
4p
- ' s e c^^^u
"1/
p(tr") p0")" - t r \secI t
p(ro)o\
L , , - p(tro)r
'\r(Ie)n
ffil
l
d/
(e)
Equation (2) could be then simplified:
x":
Io
- ,l
p(tro)*
,
+ffi
ift.p( Ie) \secd
^\
sec0/
( 10)
This is only strictly valid when there is no major
element absorption edge between ),e and ,\a, hence
use of this expressionwill introduce errors when
dealing with major elements.Substitutionof expression (10) for Xt of equation (6) gives:
p(tro): p(tro)n-r Ie ( ^^^ ^ - p(tro)n^^^ \
seco)
p(x')
p(trr)e ' /oK \secd + ffi
.n r)
|
- p(r')''ffi}
'{r(r^)"
Dividing (a) by (5) and rearanging:
P(Io)*\
- p(tre)e
ffi/
L
, p(xo)o
d * ffi'sec
\sec
Q)
tttl
Equation ( 11) above is analogousto equation
(8) of Reynolds (L967). It can be seen that
(11) linearly relates 1n to [p.(Ie)/p(rr)], since
[,"(r")n/r,(In)n] can be regarded as a constant.
Equation (11) differs from equation (7) only in
the expressionfor the slopeof the line. Equation (7)
is not linear since the slope dependson [p(Ie)/
/.,(IE)1. The expressionsfor the slope differ by a
factor {:
(r2)
Q)
Reynolds' (1967) equation (6) is analogousto
It can be seen that. for small amounts of an
equation (6) above.Reynolds'approachto substi- elementpresent,[p ( r.") /rr( rr ) ] approaches
[p(re ) n/
MINERALOGICAL
p(rn)nl and { approaches1. Evidently for small
amounts of the major element, equation ( 11) is a
viable approximation.The errors introduced by this
approximationtend to cause [p(fe)/rr(rr)] to be
evaluated at less than its true value. Consequently
p(ro) is overestimatedand so is Xr. Note that
Equation (2) must be used to find Xs rather than
Equation (10) when Xr refers to a major element.
Reynolds (1967) noted that systematic positive
errors relative to acceptedvalues were encountered
in his analyticalwork. We have just seenthat neglect
of the general relationship will introduce systematic
positive error. This error can be calculated for
various materials,wavelengths,and spectrometerdesigns.For the example of iron in Reynolds' Xnn-5
spectrometerthe error introduced in G-l (1.33
percent Fe) is negligible, and indeed his data for
G-1 show no particular bias, but show relative errors
of up to 8 percent. On the other hand Reynolds'
iron-aluminum oxide mixtures (6-12 percent Fe)
show a positive bias with "overall errors in the
vicinity of. *4 percent" and a spread of relative
errors of about 8 .percent.The relative error introduced by neglectof the generalexpressionis calculated to be about 2 percent. Consideringthe 8 percent spread,this effect may be a substantialpart of
the 4 percent positive bias.
When ir(,1,e) has been determined by Compton
scattering (Reynolds, 1963), the value of p(.1,n)is
easily obtained. To determinethe amount of major
element in the unknown sample, an equation of
the form
x,"
- :
{'l lpq^)l sec d-t p0s)-'secd]. ,,"
g]
1"" [p(trr)".sec
0 f p(trs)".sec
(13)
is used where superscriptsu and s refer to unknown
and standard. If the wavelengthregion around ),r
contains trace elementswhich have no major element absorption discontinuity intervening between
the emissionand absorptionwavelengths,then equation (13) reduces to the simpler form used by
@
o7
;
I
. ---tica-t
on"
-)
.-/
G-2
/-./
DTS-I
Graphical Method
The preceding discussionhas suggestedthat the
relation betweenr[rr0e)/p(rr)l and 1B is nearly
linear. For selectedranges of [r,(Ir)/p(ru)] the
curves are indistinguishablefrom linear, and for low
16the slopeof the curve follows from equation ( 11).
A useful way to avoid systematicbias in determining [r,.(r.e)/p(IE)] and to allow comparisonof the
sample to several standards simultaneouslyis to
plot Lr(Ia),/,p(rr)l versus18. The resultingcurve,
in practice indistinguishablefrom a straight line
although the slope may not be given by Equation
(11), may be used to find [r,(r.o)/p(rr)] for a
sample.Figure 1 contains some examplesof curves
of [p(rr)/p(trn)] versus In. For conveniencein
using such curves, calculated values of p(.1,a),
p(trr), and [p(Ia)//,(IE)] for a variety of absorp
tion edges are tabulated for the uscs standardsin
Table 1. Henrich's (1966) mass absorption coefficients and Fleischer's (1969) and Flanagan's
(1969) tabulations of recommendedanalysesfor
the uscs standardsare used to compute the values
in Table 1.
to71
NOTES
12345679
lcrKq
l o o sC P S
e
A
f,,
u,
,.<4w_r
AGV-I
a
GSP-I
I FeKc( KCPS
FIc. 1. Graphs demonstratinglinear relation between 1o
and [p(rn)/p(re)] for limited ranges of composition.Data
are for pressedpowder samples having 30 percent by weight
cellulose as a binder. (A) Shows relations at the calcium
absorption edge. The three points for W-l correspond to
different percentagesof cellulose binder. (B) Shows relations at the iron absorption edge. The dashed line is drawn
through SiO, and the value of [p(0.93)/p(1.94)] which
was calculated for W-l by Reynolds' method. This line
helps estimate the amount of systematic error introduced by
not using the solid line.
to12
MINERALOGICAL
NOTES
Tesls 1. Mass Absorption Coefficient Data for U.S.G.S. Standards
w-l
v ( .927)
y (I.937)
v (2.7 48)
u (3.359)
t J( 3 . 7 4 2 )
p (7.126)
u (8.336)
10.82
7 6. 2 0
I99 .7
336.7
388.5
990.9
1182.
v( .927)/v(1.937)
u(1.e37)/v(2.748)
v (2.7 4el/v (3.359)
u ( 3 . 3 5 9) , / u 1 3 . 7 4 2 )
v ( 3 . 7 4 2 ' /) r 1 7. I 2 5 1
v(7.126)/v (8.336)
L 6. 2 5
I 3. 32
2]-3.7
278.6
368.2
1233.
L527.
I q (n
.r420
.3816
.5931
.8563
.3922
.8384
.3900
.7 669
. r)of,
.2987
.8071
GSP-I
AGV-1
PCC-1
DTS-I
I1.23
76,54
196.8
326.6
386.9
I045.
t23I.
t2.25
78.19
200.8
333.1
3 8 4. 0
1055.
125I.
13.45
78.L2
198.8
304.1
375.7
rr52.
1349.
12.04
60.70
157.5
270.8
355.5
1430.
2205.
L2.39
6r.92
159.5
2 7 7, 7
374.7
1511.
2338.
.1468
.3888
.6026
.8442
.3704
.8489
I ((r
.3894
.6029
.8575
.3640
.8434
. 1 72 2
.3930
.5536
.8095
.3260
.8546
. r983
.3855
.5814
.7 409
.2555
.6487
.2001
.388I
.5746
. 7 41 0
.2480
.6465
BCR-I
I7.09
80.8r
2 0 0. 2
289.9
37r.0
1 2 1 0.
1525.
.2I15
.4035
.6908
.7813
.3057
.7932
determining all the elements present and that it
requires a bare minimum of sample preparation.
Reynolds:
Xou
'- :
1rr".P(X")".
Xo"
(14)
1r" p(trp)"
Applications and Limitations of; the Technique
Reynolds'(1963, 1967) Comptonscatteringtechnique for determining mass absorption coefficients
coupled with the proposed method for extending
the absorption coemcientsacross absorption edges
is general in application but may be used to best
advantagein situationsthat do not require (or where
one is not able to) determineall the major elements
presentand where simple rock-powdersamplesmay
be used. This extension of Reynolds' technique is
probably used to best advantagein the analysis of
the elementscalcium through iron in rock samples.
The technique could be usefully applied in many
other specific situations, for example the analysis
of iron, copper, and zinc in sphalerite ores.
Instrumental factors conspire to limit the usefulness of the technique for atornic numbers below
calcium. For the lighter elementsa vacuum spectrograph is required which is normally capable of
determining most of the rock forming major elements.For this situation someproaedureof iterative
calculation such as suggestedby Norrish and Chap
pell (1967) is most useful. In addition, for the
lighter elements,grain size and absorption property
contrastsbetweenmineral constituentsoften require
fusion for hornogenization,which introduces the
possibility of using dilution andfor heavy absorbers
with little extra effort.
The principal advantagesof this techniqueare that
it allows determinationof specific elementswithout
Acknowledgments
This work was completed under the auspicesof the Committee on Experimental Geology and Geophysics of Harvard
University. I gratefully acknowledge the support of Professor J. F. Hays (NASA Grant NGR 22-0O7-175). The
comments of Professor C. W. Burnham have improved
the manuscript.
References
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Geochim. Cosmochim. Acta, 33, 8l-120.
M. (1969) U. S. Geological Survey standardsFLETscHER,
I. Additional data on rocks G-l and W-1, 1965-1967'
G eochim. Co'smochim.Acta, 33, 65-7 9.
HErNnrcn, K. F. J. (1966) X-ray absorption uncertainty.
In,T. D. McKinley, K. F. J. Heinrich, and D. B. Wittry,
Eds. The Electron Microprobe. John Wiley & Sons,
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LrcsHersrv, H. A., H. G. Prnrrrnn, E. H. WrNsrow, eNo
P. D. ZevreNY (1960) X-ray Abrcrption and Emissiott
in Analytical Chemistry. John Wiley & Sons, Inc., N.Y.
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N.Y., pp. 16l-214.
Powerr, I. L., W. R. SrINNrn, lNr D. Werren (1969)
Whole-rock Rb-Sr age of metasedimentary rocks below
the Stillwater Complex, Montana. GeoI. Soc. Am. Bull.
80. 1605-1612.
REyNoLDs,R. C. (1963) Matrix corrections in trace element analysis by X-ray fluorescence: Estimation of the
mass absorption coefficient by Compton scattering. Amer.
Mineral. 4t. 1133-1143.
(1967) Estimation of mass absorption coefficientby
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Manuscript receiued, February 7, 1973; accepted
for publication, April 6, 1973.