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A correction method of matrix, density and thickness effects in thin samples analysed by X-ray fluorescence spectrometry
0031-6987/80/0701-0385$02.00/0
ctrochimica Acta. Vol. 3.5B pp. 3115 to 400 %kr@n-nm Ress Ltd., 1980. i’rhted in Great Britain
A correction method of matrix, density and thickness effects in thiu samples analysed by X-ray lhorescence spectrometry* P. FRIGIERI, F. ROSSI and R. TRUCCO CISE, P.O. BOX 3986, Segrate, 20100 Milano, Italy (Received 21 December 1979)
Abstract-This paper deals with X-ray fluorescence spectrometry of powdered materials deposited on collection filters. The general applicability of this method is limited because the spectral response is affected by sample density and thickness and interelement effects may occur due to the specimen composition. The present work is aimed at contributing to the solution of the above mentioned problems, and reports a mathematical correction method which changes fluorescence intensity measurement data into fluorescence intensity values free from density and thickness effects. The correction of the thickness effect was obtained by overlapping the powder sample with a standard pellet containing some reference elements. The method allows the evaluation of the thickness of the unknown samples by measuring the intensity decrease of the X-ray fluorescence radiation coming from the reference pellet and passing through the unknown sample. These data are used to correct the intensity fluorescence values of the unknown samples. A computer program to elaborate spectral data by means of different and selected mathematical algorithms was set up.
1.
INTRODU~ON
X-RAY
FLUORESCENCE spectrometry can be advantageously employed as an analytical technique for determining elementary components in powdered materials deposited on collection filters. Typical cases of this application are the analyses of the particulate suspended in air or natural waters and collected by drawing the fluid through filters of known porosity. The analysis of these samples is a fundamental step for the protection of the environment from air and natural water pollutions from industrial activities. The direct analysis of the powdered materials collected on the filter disk is sometimes necessary when the sample amount is too small to be reduced into a pellet. In this case, the use of X-ray fluorescence spectrometry without any preliminary treatments of the analytical sample allows rapid analyses and maintenance of the sample unaltered for further analyses. To get such conditions, it is necessary however to overcome some difficulties arising from the physical and chemical nature of the analytical sample. In fact the characteristic spectral response of an element is affected by several physical parameters, such as sample thickness and density, and by interelementary effects due to the sample composition. These problems can be solved by means of mathematical correction methods capable of changing the measured spectral response into intensity values independent of thickness, density, and interelementary effects.
2.
STUDY OF A
CORRECL-ION METHOD
In X-ray fluorescence analysis of a sample p having finite thickness, the expression relating the intensity I of the spectral lines of an element j to the elemental concentration is
X(1-exp[ * Paper presented s.A.(B) 35/7-A
- Ppxp ( Ci CipcL F cosec
3/* + Ci GpPi
lAi 1 cosec
$2
9
at the 21st Coll. Spectr. Int. and 8th Int. Conf. Atomic. Spectr., Cambridge (1979). 385
(1)
P. FRIGIERI, F. ROW and R. TRUCCO
386
where I& = intensity of the fluorescence radiation measured for element j in sample p; (vi - l)/vi = ionization efficiency of element j; wi = fluorescence yield of element j; gj = statistical weight relevant to element j; dQ/4n = fraction of the fluorescence radiation reaching the counter through the collimator; J(h)=spectrum emitted by the X-ray tube; edge wavelength; Amin= minimum absorption Adisc= absorption wavelength; 115= absorption coefficient of element j for the primary excitation spectrum; @T= absorption coefficient of the interfering element i for the primary excitation spectrum; pi(A,) = absorption coefficient of element i for the fluorescence radiation hj emitted by element j; Cj, = concentration of element j in sample p; C,, = concentration of element i in sample p; I& = angle between the mean direction of the primary X-ray beam and the sample surface; I,!I*= take-off angle of the spectrometer; pp = density of sample p; X, = thickness of sample p. By fixing the operating conditions of the spectrometric analysis it is possible to collect many variables of equation (1) into a variable Kj depending on the analyte j:
In addition, if we put sin &/sin I,!J~ = A, equation (1) can be rewritten in the form
c, ” = ’ CiCip/_Lt+ ACCippi(Aj)
~(1 -exp C-PpXp(ciGpPT cosec$1+CiGpPi(Aj)cosecUll.
(2)
Hence, if concentration is constant, there is an exponential relationship between the intensity of the X-ray fluorescence spectral line and the thickness of the collected material. On the other hand, equation (2) applied to element z contained in sample 4 of infinite thickness (X, = m) to the incident X-ray beam penetration, gives xi Q.L: cosec 41/r+ C CQpi(A,) cosec $2
= 0,
so that relation (2) reduces to
Let us now consider a sample of infinite thickness 4 overlapping a sample of finite thickness p; Fig. 1 shows the schematic drawing of the two samples with respect to the incident X-ray beam penetration. The intensity I,, of the fluorescence radiation A, in point x is a function of the intensity 1: of the exciting radiation:
with a proportionality
factor
the ionization
&(A,) = f(G), depending on the analyte;
the matrix
efficiency, (v, - 1)/v, ; the fluorescence
Fig. 1. Schematic drawing of two overlapping samples submitted penetration.
composition,
yield W,; and the
to incident X-ray beam
387
A correction method of matrix, density and thickness effects in thin samples
statistical weight g, ; so that the relation may be written as follows:
Since dc = I?& C i C&/-L:cosec $1 dx, we have v -1 dL,(A,) = GCzq~& -z-- w,g, cosec 1,6rdx. VZ Applying the Lambert-Beer
(4)
law to the X-ray path in the sample (Fig. 1) we can write
dl,(A,)=dMA,)
$2 1, I iCiqPi(b)PqXcosec
exp
1:=1:
exp
(
I",=IEexp
d&,, =dl,(A,)
-c
- C i C,,pyp,a cosec q$ , 1
-CiCiP~;fJqXCOSeC$,
exp
- 2
i Cippi(A,)ppacosec $2 .
(5)
From formulas (4) and (5) one obtains v -1
z VZ
d&, = W~,P& X
dR Wgz z cosec &
(6)
1 i Cippfcosec I/&+C i Cippi(Az)cosec $2
exp
X
2 i CippFcosec I+$+C i CiqcLi (A,) cosec $2
exp
II x
dx,
da being the beam fraction passing through the collimator. By integrating over x from 0 to m and collecting all constant terms into K, one obtains
X
exp
{
-PpX,
[
xi
cipl*F
cosec
J/I + xi
GpPi
(&I
co932
IL*
7
(7)
where A is still equal to the ratio sin &/sin &. Substituting equation (3) in (7), the resulting equation relates the measured fluorescence intensity Iz,P with respect to the intensity IL,4 coming from the sample q and depending on thickness and density of the interposed sample p: C i CipcLi cosec
GI+C
i CipCLi(&)
cosec
$2
.
(8)
Summarizing the above statements we see that three fundamental equations must be taken into account: (i) equation (2) referring to thick samples, (ii) equation (3) referring to thin samples, (iii) equation (8) referring to thick samples overlapping thin samples. These three equations can provide a correction method of the thickness-density effect in thin samples.
388
P. FRIGIERI, F. Row and R. TRIJCCO
The proposed method utilizes the intensity measurements obtained from four sets of samples. (a) Samples q of known composition and infinite thickness to the incident X-ray beam penetration containing element z as external reference. (b) Samples m of known composition and large thickness, containing analyte j. (c) Samples s of known composition and small thickness, containing analyte j. (d) Samples p of unknown composition and small thickness, where element j is to be determined. It is possible to apply a system of equations of type (3) to samples of type (a), where the following terms are known: I_ (measured fluorescence intensity), Ci,q (concentration values), A (geometric constant of the spectrometer), pi(&) (absorption coefficient of element i at the fluorescence wavelength A,). By this system of equations it is possible to calculate the K, and ‘& C,,,p; values, considering term Ci Cl,,& to be constant in all the samples of the same type. If C
i C,qP*.rBz,
equation (3), when applied to the samples of type (a), becomes
If the matrix of the set of the standard samples is sufficiently constant and comparable with that of the real samples (such a condition is always necessary in the analyses by the standard comparison method) it is possible to set Bj, = Bj, = Bj, = Bj.
(10)
Applying equation (3) to samples of type (b) one has cim Bi +A Ci GmFi(Ai) ’
&,,=Ki
(11)
In this case, since terms 4,,,, Cl,,, A and pi(&) are known, it is possible to calculate Bi and Ki. As to samples of type (c), with a treatment similar to that just mentioned, one obtains
Gs 4S=K,Bj+AxiCiSpi(Aj)-
Bj cosec $I+ xi Cispi(hi) cosec $2
II). (12)
Since the terms &, Kj, CL,,,Bj, A, pi(&), cosec a+l+and cosec & are known, one can calculate the term psx,. Hence, from the analysis of samples of type (a) of known composition, for the overlapping of thin standard samples of type (c), equation (6) becomes B, cosec $I+ xi Ci,pi(A,) cosec $2
I)
.
The values of I=,+,,ItP4,K,, PG,, % G,p, cosec +I, cosec & and pi(AZ) being known, one can now calculate the term K,. After obtaining the values of the terms B,, K,, Kj and Kz from the analysis of standard samples, it is possible to calculate both 4% and c,P for an unknown sample by two successive measurements: the intensity of X-ray fluorescence directly coming from the unknown thin sample, and the intensity of X-ray
A correction method of matrix, density and thickness effects in thin samples
389
fluorescence coming from the thick standard sample overlapping the unknown thin sample. The values thus obtained are introduced into equation (2) and (8), respectively. If we take equation (3) into account, it turns out that ci Ciq& = BZ, i.e. a constant for a given analyte, and
a quantity depending on both the element and the analytical sample, which can be calculated from the wavelength, concentration and absorption coefficient values of the analytes. The term A, defined as the sin $&in & ratio, and equal to unity in the spectrometer used;is therefore not taken into account and included into the term o,,,. Finally, the term AT is introduced into the formulation to get w=,~ and B, values independent of the measurements units specific of absorption coefficients and concentrations. On the basis of the above assumptions, equation (3) becomes: (14) which shows the behaviour of the measured fluorescence intensity values with respect to the elemental concentration data. From this relation, and taking into account the net values with background corrections, it follows that the intensity value is zero when the corresponding concentration value is zero; from an experimental viewpoint it is, indeed, more convenient to disregard the background corrections and to introduce into equation (14) a correction coefficient D,, depending on the analyte, that, together with K,, B, and A,, can be obtained mathematically from the experimental values relevant to a set of samples of known composition and thickness infinite to X-ray penetration. Equation (14) then becomes
Writing this equation in the equivalent
form
Lq (B, + A%,, ) = Kc,,
+ 0,
and recalling the theoretical linear dependence between the intensity and the concentration values, one can develop an iterative calculation procedure as follows. (1) Setting B, = 1 and AT = b one determines the terms K, and D, by a least square method. (2) Using K, and D, just determined one considers K,C,,, + D, as a known term and determines B, and AT by a least square method. (3) Considering the term B, + A:o+ one calculates by means of the above determined B, and AZ, and recalculates the K, and D, values; (4) Then it is necessary to repeat from (2) until the pairs (K,, 0,) and (B,, A:) stabilize. Applying this calculation method to a set of thick samples of known composition, we observed that the previously described process stabilizes within a few interations. In the same way as for equation (3), it is possible to write equation (12) in the form J = Kic;S-t-Di { 1-exp IS B, + Aywis
[ -p,x,(B,
cosec+,,+AFwj,
cosec &)1).
(16)
Moreover, it is apparent that, since I,$ is equal to & and term pS is always related to term x,, by setting X, = x,p, cosec $,
P. FRIGIERI,F. ROSSI and R. TRUCCO
390
one obtains -X,(Bj+Ayo,)]}
Iis = K,qs:Di(l-exp[ Bj + Aj tijs
(17)
Also equation (13) can be rewritten, in agreement with the general formulations generate equations (15) and (17); from equation (15) one then obtains
J~,(B,+A%J,,)-Q
c
zq
so that equation (13), where I,,jK, I
zs
that
3
KZ is an approximation
=KT[l=,(B~+A:w=,)-D~]. exp[_x KZ
of Cr,q, becomes +A*o
(B
s
z
)] L
rq
08)
’
Before describing the calculation procedure, it is better to define some assumptions in detail. It can be seen that in equation (17) term X, relevant to the thickness-density characteristics of sample s, is independent of the analyte in practice since the determination of X, values is performed by measuring the fluorescence radiation of a given element, this numerical value, proportional to the thickness-density parameter, also depends on the analyte. Therefore this value will be denoted in the following by Xis. As to equation (18), the K, value seems to depend on the reference element only, but actually the value depends on the choice of the analyte and the reference elements. Since one can use the same reference element for the correction of the various analytes the term K with the symbol of the analyte and written as Ki will be added hereafter. Then, equation (18) can be rewritten as
Equations (15)-(19) can be used in a computation program for the corrections thickness, density and matrix effects in samples of small thickness. 3.
of
CALCULATION METHOD
The mathematical correction method proposed to transform the spectral response data into values independent of the thickness effect in the analysis of powdered samples with finite thickness to the incident X-ray beam penetration can be divided into six fundamental calculation steps. (i) In this step, the parameters AT, B,, D,, K,, qq are determined from both the concentration (C,,,) and the resulting intensity (I,,,) values of the reference pellets overlapping a blank filter. The values of the absorption coefficients (CL)are derived from the literature [l], the equation used is equation (15). (ii) Parameters AT, Bi, Dj, Ki, q,, are determined from both the concentration (q,,) and the intensity (&,,) values coming from the standard pellets as directly read. Reference is made again to an equation of the type of equation (15), obviously related to the analytes. (iii) In this step, parameters X,,,, o+~ are determined by means of the values of AT, Bj, Kj, Dj and C,,,, Ii,,, the last two referring to the concentrations and the intensities of the standard filters respectively. The following formula, derived from equation (17), is used
Xjs = -A log l- IjsBi+AY@js](Bj+ AToj,). D,
+
I
K.c.
J
(20)
JS
[l] H. A. LIEBHAFSKY, H. G. PFEIFFER, E. H. WINSLOW and P. D. ZEIMANY, X-Ray, Analytical Chemisrryp. 525. Wiley Interscience,New York (1972).
Electrons, and
A correction method of matrix, density and thickness effects in thin samples
391
(iv) The I$ values are determined by means of the AT, B,, D,, K,,I,,,, T$,,to_ and I_ values, where I,,, denotes the intensities of the reference pellets read behind a standard filter. The expression used is the following:
(21) which is directly obtained from equation (18), and can be rewritten, on the basis of the equation (15), as follows:
(22) (v) The Xj,, values are calculated by means of the AT, B,,D,,K,,Iz,p, &, o,,~, Kj values where & denotes the intensities of the reference pellets read behind the unknown filters. The formula used is
Up
” =-Alogq[I,,(B, +A&J-
Q] I
(B, + A%,,)-‘,
(23)
obtained from equation (18) relevant to the set of samples p. Taking equation (15) into account, we can write equation (23) in the form Xi, = -A log &
[
1
(Bz+A~w,,)-'.
I
=I
(iv) In this last step of the calculation process the values of the concentrations of unknown samples are determined. The following parameters are used: AT, Bj,Di,Kj, &,,Xiqp, where the A,, denote the intensities of the unknown filters. The expression used is the following:
l&=
Di+K,Cip Bj + AywjP
11- exp [-qP (Bi + ATwjp)lI,
(25)
obtained from equation (17) for the set of unknown samples.
4. CHECK OF THE METHOD To test the applicability of the correction method to samples of small thickness, three sets of filters having various amounts (15, 25 and 50 mg) of deposited powders with different composition were prepared. To this aim weighted amounts of powder were suspended in 20 cm3 of distilled water; the suspended solid was then accurately subdivided into small size particles by an ultrasonic device, and collected on inert supports by filtration. The filter disks having 50 mm diameter and 0.8 pm porosity were placed on a glass plate and fastened with a metal ring to avoid changes in their planarity during the drying phase. In addition to the filter disks, 7 mixtures containing vanadium, molybdenum and arsenic at different concentration values were also prepared and used as external reference samples. The powder, added with boric acid as binder, was converted into self-supporting pellets. All these standard samples were directly analysed by X-ray fluorescence spectrometry in order to check the possible systematic errors during the preparation process. By using an automatic computation program for an IBM 370/125 computer, the plots of the intensity values vs the elemental concentration values of the materials deposited on the filters were obtained. The results are reported in Figs; 2-6. It is apparent that the intensity values are functions of both the element concentration and the deposited material weight, and that the obtained plots could be fitted by three
P. FFUGIEFU, F. ROW
392
and
R. TRUWO
0.40 -
0.36 -
0.32 -
0.28 T v) 0.24 0 ‘0 x lo 0.20 7
x x x
x
x
Fe
concentration,
%
Fig. 2
0.60 -
054-
0.48 -
0.42 x 5
0.36 -
x
i 9 x (D
Y
x
x
x
0.30-
+ x
5
.g 0.24 *
x
x
+
x
0.18 -
+
i
+
+
Ii
0.12 -
x
.
.
+
.
.
.
+
.
0.06 -
0
*,
, 2.00
,
, 4.00
,
, 6.00
,
,
,
8.00 Cr
,
,
,
,
12.00
10.00
concentration,
, 14.00
)
, 16.00
,
, 18.00
,
(
20.00
%
Fig. 3 Figs. 2-6. Behaviour of the fluorescence intensity function of the % concentration in sets of powdered deposit; + 25 mg deposit;
of the elements Fe, Cr, Mn, Cu, Zr as a materials deposited on filter disks. X 50 mg A 15 mg deposit.
393
A correction method of matrix, density and thickness effects in thin samples
2.40
-
2.24
-
2.08
-
x x 1.92 x
Tu)
x 1.76 -
x
Y Q * N
1.60 x Y x
3 .G m
+
1.44 + x
l
+
+
x 1.28 +
x
+ .
+
x
. .
+ 1.12 -
x x II x
0.96 -
0.80~
,
z
t
,
,
2.00
0
$
+
+
I
b
, 4.00
,
.
+
.
+ .:.a .
:
A
, 6.00
,
I 8.00 Mn
I
I 10.00
I
I 12.00
concentration,
I
I 14.00
I
I 16.00
I
I 18.00
I
1 20.00
%
Fig. 4
100.00 -
x 90.00
-
80.00
-
x
x
x
x 70.00
-
60.00
-
x x
7,
+
+
+
‘: Q x
x 50.00
+
-
r z .F v)
+
x + 40.00
-
+
I x
*
.
. .
+
x 30.00
-
+
::
20.00
x
L + .
J4 0
,
, 2.00
.
l
.
+
+
,
f
l
I
I 8.00
II
* +
x 10.00 -
. .
;
x
.
*
I
I 6.00
t
I 4.00
Cu
11 10.00
concentration,
Fig. 5
’ 12.00 %
I”“” 14.00
16.00
18.00
20.00
394
P. FRIGIERI,F. Ross
and R. TRU~~O
x
5
I
x
Y o_
(;: 50.00
x ::
+
+
x
30.00-
+
x
x
+
. .
+
:: 20.00-
i+ 1
1 0
+: ++
++ +
60.00
+ _.
. .
. .
:-
I I I I I I I I I I I',,, 6.00 6.00 10.00 12.00 14.00 2.00 4.00 Zr concentration,%
I1 16.00
I ,I 16.00 2000
Fig.6
curves with different slopes for 15, 25 and 50 mg of deposited powder, respectively. These results are in agreement with the data previously obtained [2-51. Some preliminary tests showed the applicability of the method and its agreement with the theoretical assumptions. Measurements of the fluorescence intensity coming from the reference elements present in the pellets overlapping the set of samples with different amount of deposited materials, show a decrease of the intensity with an increase in the thickness, with an exponential behaviour between the considered parameters, according to equation (19). Figures 7-9 report the values of the spectral intensity measured for the reference elements present in a set of pellets with the same concentration vs the weight of the powder deposited on the filters. To confirm the possibilities of correction of the proposed method, one can analyse the residuals obtained by using equation (15), instead of the straight-line equation, for the best fit of the experimental data relevant to the sets of samples of infinite thickness (reference and standard pellets). In the set of samples used in these tests, for instance, the mean error in the case of iron is 15% along the straight-line and decreases to 2% along the curve; similarly for chromium, the error decreases from 10 to 3%. The validity of the method is further proved by calculations, according to equation (24), of the Xj,P of a set of samples with known amounts of deposited materials. In the case of iron, the X,, assumes different values depending on the amount of deposited material: it is 0.25, 0.45 and 0.85 for deposited amounts of 15, 25 and 50 mg respectively. Similarly, the X,-, values corresponding to these amounts of deposited material are 0.20, 0.31 and 0.66 respectively. In a preliminary way, also the validity of the theoretical assumptions on the Kj value [2] P. FRIGIERI,R. TRUCCO, R. ANZANI and E. CARE~A, Chim. bad. 54, 12 (1972). [3] A. ROLLA, P. FFUGIERI,A. GIRELLI, and R. TRUCCO, Chim. Ind. 55, 8 (1973). [4] P. FRIGIERI and R. TRUCCO, X-Ray Spectrom. 3, 40 (1974). [S] P. FRIGIERI,R. TRUCCO and E. CARETTA,X-Ray Specfrom. 4, 28 (1975).
A correction method of matrix, density and thickness effects in thin samples
98.00
94.00
90.00
66.00
T 82.00 ': Q " 78.00
.
70.00-
I :I
66.00-
. ; i . .
62.0058.00
I
I
6.00
0
I
I,
12.00
I1 18.00
I1 24.00
I I I I I I I I a I I 30.00 36.00 42.00 48.00 54.00 60.00
Deposited amount of MO,
mg
Fig. 7
690.00-
630.00-
I
7 45O.OON m 5
4
3 390.005
t .
z
.
330.00-
t . .
270.00-
t . : &
i t
150.00 -
f: . . .
90.00
2 . .
.
210.00-
; .
I 0
I I I I I I I I a I I I I I I I I I1 30.00 36.00 42.00 48.00 54.00 60.00 6.00 12.00 18.00 24.00 Deposited amount of V, mg
Fig. 8 Figs. 7-9. Relationship between the K, intensities of molybdenum, vanadium and arsenic and the deposited amounts.
395
396
P. FRIGIERI,F. Row
200.00
190.00
and R. TRUCCO
-
-
r v) 160.00 7 B x Ic 150.00 5 .g 140.00cn
1 f
. :
.
f t
. .
b .
130.00 -
t
120.00 -
I10.00 -
l00.00,
, 0
,
6.00
,
‘
12.00
,
,
,
,
,
18.00 24.00
,
,
30.00
Deposited amount
,
1
,
I
I
36.00 42.00 48.00
of As,
I
I
54.00
I
I
60.00
mg
Fig.9
was tested. In fact, from equation (21) it is possible to obtain the Kj,S values for each analysed standard sample, whereas the method exploits a generic Kj value valid for the whole set of samples and depending only on the pair of elements, reference and analyte, involved. Calculating the single values of Ki*,and their average value, the KFe for iron was 27.6 with a standard deviation of 3.2 and, similarly, the resulting value of K,, for chromium was 28.2 with a standard deviation of 2.9. All the preliminary tests giving satisfactory resuhs, the complete computation code was set up. The application of the calculation procedure to a set of samples consisting of filters having different deposits of oxide mixture confirms the validity of the proposed method. Table 1 reports the output data when the analysis was made on 7 samples of known composition analysed by comparison with 20 standard samples; the ‘apparent concentrations’ are the concentration values calculated using a linear calibration curve. 5. CONCLUSIONS
The analysis of the results obtained showed that the validity of the proposed method is strongly affected by the numerical values given to the absorption coefficients and used for the o calculation. These values being always rather uncertain, the analyst himself will recalculate the absorption coefficient values from the experimental data after the whole method has been optimized. These conditions are reached with the proper choice of the operating conditions, such as standard and reference samples and pairs of analyte and reference elements, and using a set of thin samples of known chemical composition. The proposed method was applied to the analysis of airborn particulate and river sediments collected on filter disks.
A correction method of matrix, density and thickness effects in thin samples
397
Table 1. Results obtained when the analysis was made on seven samples of known composition analysed by comparison with twenty standard samples
Analyte
Apparent concentration Sample (“A)
Calculated concentration W)
Expected concentration (%)
Difference of calcu- Difference of aplated and expected parent and expected concentration concentration
Fe Fe Fe Fe Fe Fe Fe
1 2 3 4 5 6 7
3.79 7.65 9.11 8.35 4.23 2.45 1.61
3.90 6.84 8.10 8.27 4.52 2.69 1.79
4.00 6.00 7.00 8.00 5.00 3.00 2.00
-0.10 +0.84 +1.10 +0.27 -0.48 -0.31 -0.21
-0.21 +1.65 +2.11 +0.3s -0.77 -0.55 -0.39
Cr Cr Cr Cr Cr Cr Cr
1 2 3 4 5 6 7
5.20 8.10 9.30 14.12 1.95 1.69 12.83
5.61 7.55 8.70 11.28 2.40 1.85 11.34
6.00 7.00 8.00 9.00 3.00 2.00 10.00
-0.39 +0.55 +0.70 +2.28 -0.60 -0.15 +1.34
-0.80 +1.10 +1.30 +5.12 -1.05 -0.31 + 2.83
Mn Mn Mn Mn Mn Mn Mn
1 2 3 4 5 6 7
5.46 7.03 9.3s 1.20 1.78 12.2s 12.17
5.61 7.06 9.37 1.41 1.77 11.54 11.81
6.00 7.00 9.00 4.00 2.00 10.00 11.00
-0.39 +0.06 +0.37 -2.59 -0.23 +1.54 +0.81
-0.54 +0.03 +0.35 -2.80 -0.22 +2.2s +1.17
cu cu cu cu cu cu cu
1 2 3 4 5 6 7
8.17 11.99 5.25 4.98 10.29 13.21 IO.55
8.55 11.62 5.23 5.43 10.90 13.20 9.87
9.00 11.00 5.00 6.00 12.00 13.00 9.00
-0.45 +0.62 +0.23 -0.57 -1.10 +0.20 +0.87
-0.83 +0.99 +0.25 -1.02 -1.71 +0.21 +1.55
Zr Zr Zr Zr Zr Zr Zr
1 2 3 4 5 6 7
13.28 5.50 11.62 9.21 8.94 7.22 2.12
11.54 4.72 8.42 8.59 10.71 7.12 2.51
10.00 4.00 6.00 8.00 13.00 7.00 3.00
+1.54 +0.72 -t-2.42 +0.59 -2.29 +0.12 -0.49
+3.28 +1.50 +S.62 +1.21 -4.06 +0.22 -0.88
APPENDIX It has been shown that parameters A *, B, D, K for the reference elements and analytes can be obtained from expression (15), the values of I, C and o being known. First one of the unknown parameters must be normalized, since, if AT, B,, D,, K, is a solution of the problem, CIA?, aB,, mD1, aK, (where a is arbitrary and different from zero) is also a solution of the same problem. In this calculation one applies the least square technique and determine the minimum of the following function
where M is the number of samples under analysis. Equation (Al) is a non-linear function in the unknown parameters and can be solved by an iterative procedure, i.e. by determining a sequence of values (Xx) such that, in case of convergence, one obtains lim {Xx} = X*, x-++C= where X* is exactly the wanted solution.
P. FRIGIERI, F. ROSSI and R. TRUCCO
398
Each term X, is usually denoted as a function of the previous term of a sequence Xx_,; hence X, = gk(XK_J. The initial value X0 is then to be chosen independently of the iterative technique employed, because, though necessary for starting the iterative process, it does not depend on it. The initial value X,, is generally denoted as “guess” and in its choice lies one of the most critical problems of the calculation, since the known theorems guarantee convergence only if X0 is close “enough” to the searched solution X*. In the processing program set up for the correction of matrix effects in thin samples analysed by X-ray fluorescence, the procedure adopted allows the automatic determination of the “guess” for problem (Al) to be obtained. A linear behaviour is assumed between intensity and concentration, and D and K values are determined by the least square technique using by the relationship I,=D+KC,
(A2)
By substituting these values in equations: (B + Ao,)I,
= D + KC4
(A3)
A and B are then calculated, and, from them, D and K values are calculated from equation (A3) again. The cycle is repeated until stable values of A *, B, D, K are obtained (as a matter of fact the unknown parameters are only three, since in an underlying assumption D = 1). It is evident from the experimental checks that this procedure yield a “guess” fit for the solution of the problem under consideration when the value of K is thus obtained for each sample of series (C), as shown in the paper, account is given to its mean value obtained after discarding, by a suitable “Student test”, the data not included in a convenient range of values. As regards the determination of the unknown concentration, the expression relating the intensity values to the concentration values can be rewritten as follows (Bi + Ajwi,p)li,p - (KjC,,, + Dj){l - exp [-Xj,,(Bj + Ai~j,p)]} = 0.
(A4)
Since
the unknown terms in equation (A4) are (&; C2,p;. . . ; CN,,. Rewriting the expression for j, which varies from 1 to N, one obtains a non-linear system of N equations in N unknown terms, viz. F,,,(C,., 1.
. . CN.~)= 0 (A5)
bN,,(C,,,. .CN.p)=0. Working out such a system for each analysed sample, it is possible to identify all values of the unknown concentrations. In this case also it is necessary to determine the convenient initial “guess”; two methods are at present being developed for the automatic determination of the “guess” Method 1 Interrupting obtain
the sequence development
of the exponential
12{I - I+
lj,, = :,? I
,.P
of equation (A4) at the first order term we
Xj,p(Bj + +j,p)J’
I
from which immediately follows
If index j is varied from 1 to N and the index p from 1 to Mp, the complete set of initial estimates of all unknown concentrations is obtained. Method 2 Disregarding the exponential term in equation (A4) and substituting a convenient expression for w~,~,we obtain N (Ah)
A correction method of matrix, density and thickness effects in thin samples
399
Setting
we find
Keeping index p fixed and varying j from 1 to N we obtain the following system from the equation (A7) A(P) . X(P)= B(P),
hnf
(A@
(n,l) (n,U
where A@#,
L) = .$!!;
X(P)(L) = CL,
;
B(p)(L) = Q,LP’
Solving the system (A8) for all values of p, one obtains the complete set of initial estimates of the unknown concentrations. Other methods
suggested
Other methods for the determination of the unknown concentration from equation (A4) can be suggested. One of them consists of solving the mentioned equation as a fixed-point problem. If 4i.p= (Bi + Aj i /+Cx,p) K=l and @i,p = (Di + KjCj,p),
equation (A4) can be rewritten in the form
= @j,pIl-exp [-X,p~~,pI>-B~~j,p 4j.p f ~+,rG,p l-=1
W)
and from this C;.,p=
Ajj,pIl - exp[-Xjptj,pl).-Tj-(QJj,p - ~,pPj,j)(Fj,,)-l~
(AlO)
where *i,p
= @ji,d@kP~i.,)
and rj = B,I(AjFi,j), Equation (AlO) contains the unknown terms Ci,,; . . ; CNp, and if it is rewritten varying the index j from 1 to N leads to the following system x, = g,(x,, x2,
x3.
. . . , x,)
(All)
i.e. a system of type X = F(X)
(Al3
where XER”
and
F:R”+R”.
Solving system (All) is similar to determining a vector X* such that X* = F(X*). The solution method suggested by the literature [6] is the so-called fixed-point technique, which consists in the following iterative processes.
[6] J. M. ORTEGA, Numerical Analysis.
A Second Course, p. 140. Academic Press, New York (1972).
P. FRIGIERI, F. Ross1 and R. TRUCCO
400
A Y
__._-_-_-
Fig. Al. Graphic interpretation
Y=FtXI
of the fixed-point procedure. Y = X is the straight-line.
Y = F(X) is the function and
(a) An initial estimate X,, of the solution is supplied; (b) The sequence X,,, = F(xk) is constructed; (c) Under conditions essentially involving the variation field of the partial derivatives of function F, the convergence of sequence {&} into the value X*, i.e. the solution of the system, is guaranteed. An interesting graphic interpretation of the problem (A12) and of the fixed-point procedure occurs when F:R + W (that is, when n = 1); system (A12) is solved by intersecting the curve Y = F(x) and the straight line Y = X (Fig. Al). Using a fixed-point calculation procedure, the iterative technique can be expressed as follows Ci$“’ = hj?{l-exp
[-Xj,,(Bj + +~,“d)I}-Tj
- (&‘+ CWlr. .)(cL..)-’ ,,P 1.1 I.1 . I.P
(A13)
When index j increases, equation (A13) can be ‘up-dated’ by substituting the oi,r values determined by the unknown concentrations as they are being calculated. It is possible to set up a further method to develop the problem with the ‘fixed-point’ technique. From the equation (A9) it follows
Qj,p= A$j,pwj.p+ @j,pexp (-Xj,p . 6j.p)+ Bj1j.p and recalling that ajj,r = Dj + KjCj.r, it is one obtains Cj,p = [4,&j,, + exp (-Xj,&,p) -
D/l($)-‘.
(A14)
The unknown terms in (A14) are the concentrations Cl+; f&,;. . ; C”,r; so by simply varying index j from 1 to N, a system completely analogous to system (All) is set up. Use can also be made of the same iterative fixed-point technique as that previously described. Acknowledgements-This work has been performed in the frame of a CISE/ENEL collaboration programme (Contract FLUOR 39/300). The computation codes relevant to the correction programmes are therefore the property of ENEL.
ctrochimica Acta. Vol. 3.5B pp. 3115 to 400 %kr@n-nm Ress Ltd., 1980. i’rhted in Great Britain
A correction method of matrix, density and thickness effects in thiu samples analysed by X-ray lhorescence spectrometry* P. FRIGIERI, F. ROSSI and R. TRUCCO CISE, P.O. BOX 3986, Segrate, 20100 Milano, Italy (Received 21 December 1979)
Abstract-This paper deals with X-ray fluorescence spectrometry of powdered materials deposited on collection filters. The general applicability of this method is limited because the spectral response is affected by sample density and thickness and interelement effects may occur due to the specimen composition. The present work is aimed at contributing to the solution of the above mentioned problems, and reports a mathematical correction method which changes fluorescence intensity measurement data into fluorescence intensity values free from density and thickness effects. The correction of the thickness effect was obtained by overlapping the powder sample with a standard pellet containing some reference elements. The method allows the evaluation of the thickness of the unknown samples by measuring the intensity decrease of the X-ray fluorescence radiation coming from the reference pellet and passing through the unknown sample. These data are used to correct the intensity fluorescence values of the unknown samples. A computer program to elaborate spectral data by means of different and selected mathematical algorithms was set up.
1.
INTRODU~ON
X-RAY
FLUORESCENCE spectrometry can be advantageously employed as an analytical technique for determining elementary components in powdered materials deposited on collection filters. Typical cases of this application are the analyses of the particulate suspended in air or natural waters and collected by drawing the fluid through filters of known porosity. The analysis of these samples is a fundamental step for the protection of the environment from air and natural water pollutions from industrial activities. The direct analysis of the powdered materials collected on the filter disk is sometimes necessary when the sample amount is too small to be reduced into a pellet. In this case, the use of X-ray fluorescence spectrometry without any preliminary treatments of the analytical sample allows rapid analyses and maintenance of the sample unaltered for further analyses. To get such conditions, it is necessary however to overcome some difficulties arising from the physical and chemical nature of the analytical sample. In fact the characteristic spectral response of an element is affected by several physical parameters, such as sample thickness and density, and by interelementary effects due to the sample composition. These problems can be solved by means of mathematical correction methods capable of changing the measured spectral response into intensity values independent of thickness, density, and interelementary effects.
2.
STUDY OF A
CORRECL-ION METHOD
In X-ray fluorescence analysis of a sample p having finite thickness, the expression relating the intensity I of the spectral lines of an element j to the elemental concentration is
X(1-exp[ * Paper presented s.A.(B) 35/7-A
- Ppxp ( Ci CipcL F cosec
3/* + Ci GpPi
lAi 1 cosec
$2
9
at the 21st Coll. Spectr. Int. and 8th Int. Conf. Atomic. Spectr., Cambridge (1979). 385
(1)
P. FRIGIERI, F. ROW and R. TRUCCO
386
where I& = intensity of the fluorescence radiation measured for element j in sample p; (vi - l)/vi = ionization efficiency of element j; wi = fluorescence yield of element j; gj = statistical weight relevant to element j; dQ/4n = fraction of the fluorescence radiation reaching the counter through the collimator; J(h)=spectrum emitted by the X-ray tube; edge wavelength; Amin= minimum absorption Adisc= absorption wavelength; 115= absorption coefficient of element j for the primary excitation spectrum; @T= absorption coefficient of the interfering element i for the primary excitation spectrum; pi(A,) = absorption coefficient of element i for the fluorescence radiation hj emitted by element j; Cj, = concentration of element j in sample p; C,, = concentration of element i in sample p; I& = angle between the mean direction of the primary X-ray beam and the sample surface; I,!I*= take-off angle of the spectrometer; pp = density of sample p; X, = thickness of sample p. By fixing the operating conditions of the spectrometric analysis it is possible to collect many variables of equation (1) into a variable Kj depending on the analyte j:
In addition, if we put sin &/sin I,!J~ = A, equation (1) can be rewritten in the form
c, ” = ’ CiCip/_Lt+ ACCippi(Aj)
~(1 -exp C-PpXp(ciGpPT cosec$1+CiGpPi(Aj)cosecUll.
(2)
Hence, if concentration is constant, there is an exponential relationship between the intensity of the X-ray fluorescence spectral line and the thickness of the collected material. On the other hand, equation (2) applied to element z contained in sample 4 of infinite thickness (X, = m) to the incident X-ray beam penetration, gives xi Q.L: cosec 41/r+ C CQpi(A,) cosec $2
= 0,
so that relation (2) reduces to
Let us now consider a sample of infinite thickness 4 overlapping a sample of finite thickness p; Fig. 1 shows the schematic drawing of the two samples with respect to the incident X-ray beam penetration. The intensity I,, of the fluorescence radiation A, in point x is a function of the intensity 1: of the exciting radiation:
with a proportionality
factor
the ionization
&(A,) = f(G), depending on the analyte;
the matrix
efficiency, (v, - 1)/v, ; the fluorescence
Fig. 1. Schematic drawing of two overlapping samples submitted penetration.
composition,
yield W,; and the
to incident X-ray beam
387
A correction method of matrix, density and thickness effects in thin samples
statistical weight g, ; so that the relation may be written as follows:
Since dc = I?& C i C&/-L:cosec $1 dx, we have v -1 dL,(A,) = GCzq~& -z-- w,g, cosec 1,6rdx. VZ Applying the Lambert-Beer
(4)
law to the X-ray path in the sample (Fig. 1) we can write
dl,(A,)=dMA,)
$2 1, I iCiqPi(b)PqXcosec
exp
1:=1:
exp
(
I",=IEexp
d&,, =dl,(A,)
-c
- C i C,,pyp,a cosec q$ , 1
-CiCiP~;fJqXCOSeC$,
exp
- 2
i Cippi(A,)ppacosec $2 .
(5)
From formulas (4) and (5) one obtains v -1
z VZ
d&, = W~,P& X
dR Wgz z cosec &
(6)
1 i Cippfcosec I/&+C i Cippi(Az)cosec $2
exp
X
2 i CippFcosec I+$+C i CiqcLi (A,) cosec $2
exp
II x
dx,
da being the beam fraction passing through the collimator. By integrating over x from 0 to m and collecting all constant terms into K, one obtains
X
exp
{
-PpX,
[
xi
cipl*F
cosec
J/I + xi
GpPi
(&I
co932
IL*
7
(7)
where A is still equal to the ratio sin &/sin &. Substituting equation (3) in (7), the resulting equation relates the measured fluorescence intensity Iz,P with respect to the intensity IL,4 coming from the sample q and depending on thickness and density of the interposed sample p: C i CipcLi cosec
GI+C
i CipCLi(&)
cosec
$2
.
(8)
Summarizing the above statements we see that three fundamental equations must be taken into account: (i) equation (2) referring to thick samples, (ii) equation (3) referring to thin samples, (iii) equation (8) referring to thick samples overlapping thin samples. These three equations can provide a correction method of the thickness-density effect in thin samples.
388
P. FRIGIERI, F. Row and R. TRIJCCO
The proposed method utilizes the intensity measurements obtained from four sets of samples. (a) Samples q of known composition and infinite thickness to the incident X-ray beam penetration containing element z as external reference. (b) Samples m of known composition and large thickness, containing analyte j. (c) Samples s of known composition and small thickness, containing analyte j. (d) Samples p of unknown composition and small thickness, where element j is to be determined. It is possible to apply a system of equations of type (3) to samples of type (a), where the following terms are known: I_ (measured fluorescence intensity), Ci,q (concentration values), A (geometric constant of the spectrometer), pi(&) (absorption coefficient of element i at the fluorescence wavelength A,). By this system of equations it is possible to calculate the K, and ‘& C,,,p; values, considering term Ci Cl,,& to be constant in all the samples of the same type. If C
i C,qP*.rBz,
equation (3), when applied to the samples of type (a), becomes
If the matrix of the set of the standard samples is sufficiently constant and comparable with that of the real samples (such a condition is always necessary in the analyses by the standard comparison method) it is possible to set Bj, = Bj, = Bj, = Bj.
(10)
Applying equation (3) to samples of type (b) one has cim Bi +A Ci GmFi(Ai) ’
&,,=Ki
(11)
In this case, since terms 4,,,, Cl,,, A and pi(&) are known, it is possible to calculate Bi and Ki. As to samples of type (c), with a treatment similar to that just mentioned, one obtains
Gs 4S=K,Bj+AxiCiSpi(Aj)-
Bj cosec $I+ xi Cispi(hi) cosec $2
II). (12)
Since the terms &, Kj, CL,,,Bj, A, pi(&), cosec a+l+and cosec & are known, one can calculate the term psx,. Hence, from the analysis of samples of type (a) of known composition, for the overlapping of thin standard samples of type (c), equation (6) becomes B, cosec $I+ xi Ci,pi(A,) cosec $2
I)
.
The values of I=,+,,ItP4,K,, PG,, % G,p, cosec +I, cosec & and pi(AZ) being known, one can now calculate the term K,. After obtaining the values of the terms B,, K,, Kj and Kz from the analysis of standard samples, it is possible to calculate both 4% and c,P for an unknown sample by two successive measurements: the intensity of X-ray fluorescence directly coming from the unknown thin sample, and the intensity of X-ray
A correction method of matrix, density and thickness effects in thin samples
389
fluorescence coming from the thick standard sample overlapping the unknown thin sample. The values thus obtained are introduced into equation (2) and (8), respectively. If we take equation (3) into account, it turns out that ci Ciq& = BZ, i.e. a constant for a given analyte, and
a quantity depending on both the element and the analytical sample, which can be calculated from the wavelength, concentration and absorption coefficient values of the analytes. The term A, defined as the sin $&in & ratio, and equal to unity in the spectrometer used;is therefore not taken into account and included into the term o,,,. Finally, the term AT is introduced into the formulation to get w=,~ and B, values independent of the measurements units specific of absorption coefficients and concentrations. On the basis of the above assumptions, equation (3) becomes: (14) which shows the behaviour of the measured fluorescence intensity values with respect to the elemental concentration data. From this relation, and taking into account the net values with background corrections, it follows that the intensity value is zero when the corresponding concentration value is zero; from an experimental viewpoint it is, indeed, more convenient to disregard the background corrections and to introduce into equation (14) a correction coefficient D,, depending on the analyte, that, together with K,, B, and A,, can be obtained mathematically from the experimental values relevant to a set of samples of known composition and thickness infinite to X-ray penetration. Equation (14) then becomes
Writing this equation in the equivalent
form
Lq (B, + A%,, ) = Kc,,
+ 0,
and recalling the theoretical linear dependence between the intensity and the concentration values, one can develop an iterative calculation procedure as follows. (1) Setting B, = 1 and AT = b one determines the terms K, and D, by a least square method. (2) Using K, and D, just determined one considers K,C,,, + D, as a known term and determines B, and AT by a least square method. (3) Considering the term B, + A:o+ one calculates by means of the above determined B, and AZ, and recalculates the K, and D, values; (4) Then it is necessary to repeat from (2) until the pairs (K,, 0,) and (B,, A:) stabilize. Applying this calculation method to a set of thick samples of known composition, we observed that the previously described process stabilizes within a few interations. In the same way as for equation (3), it is possible to write equation (12) in the form J = Kic;S-t-Di { 1-exp IS B, + Aywis
[ -p,x,(B,
cosec+,,+AFwj,
cosec &)1).
(16)
Moreover, it is apparent that, since I,$ is equal to & and term pS is always related to term x,, by setting X, = x,p, cosec $,
P. FRIGIERI,F. ROSSI and R. TRUCCO
390
one obtains -X,(Bj+Ayo,)]}
Iis = K,qs:Di(l-exp[ Bj + Aj tijs
(17)
Also equation (13) can be rewritten, in agreement with the general formulations generate equations (15) and (17); from equation (15) one then obtains
J~,(B,+A%J,,)-Q
c
zq
so that equation (13), where I,,jK, I
zs
that
3
KZ is an approximation
=KT[l=,(B~+A:w=,)-D~]. exp[_x KZ
of Cr,q, becomes +A*o
(B
s
z
)] L
rq
08)
’
Before describing the calculation procedure, it is better to define some assumptions in detail. It can be seen that in equation (17) term X, relevant to the thickness-density characteristics of sample s, is independent of the analyte in practice since the determination of X, values is performed by measuring the fluorescence radiation of a given element, this numerical value, proportional to the thickness-density parameter, also depends on the analyte. Therefore this value will be denoted in the following by Xis. As to equation (18), the K, value seems to depend on the reference element only, but actually the value depends on the choice of the analyte and the reference elements. Since one can use the same reference element for the correction of the various analytes the term K with the symbol of the analyte and written as Ki will be added hereafter. Then, equation (18) can be rewritten as
Equations (15)-(19) can be used in a computation program for the corrections thickness, density and matrix effects in samples of small thickness. 3.
of
CALCULATION METHOD
The mathematical correction method proposed to transform the spectral response data into values independent of the thickness effect in the analysis of powdered samples with finite thickness to the incident X-ray beam penetration can be divided into six fundamental calculation steps. (i) In this step, the parameters AT, B,, D,, K,, qq are determined from both the concentration (C,,,) and the resulting intensity (I,,,) values of the reference pellets overlapping a blank filter. The values of the absorption coefficients (CL)are derived from the literature [l], the equation used is equation (15). (ii) Parameters AT, Bi, Dj, Ki, q,, are determined from both the concentration (q,,) and the intensity (&,,) values coming from the standard pellets as directly read. Reference is made again to an equation of the type of equation (15), obviously related to the analytes. (iii) In this step, parameters X,,,, o+~ are determined by means of the values of AT, Bj, Kj, Dj and C,,,, Ii,,, the last two referring to the concentrations and the intensities of the standard filters respectively. The following formula, derived from equation (17), is used
Xjs = -A log l- IjsBi+AY@js](Bj+ AToj,). D,
+
I
K.c.
J
(20)
JS
[l] H. A. LIEBHAFSKY, H. G. PFEIFFER, E. H. WINSLOW and P. D. ZEIMANY, X-Ray, Analytical Chemisrryp. 525. Wiley Interscience,New York (1972).
Electrons, and
A correction method of matrix, density and thickness effects in thin samples
391
(iv) The I$ values are determined by means of the AT, B,, D,, K,,I,,,, T$,,to_ and I_ values, where I,,, denotes the intensities of the reference pellets read behind a standard filter. The expression used is the following:
(21) which is directly obtained from equation (18), and can be rewritten, on the basis of the equation (15), as follows:
(22) (v) The Xj,, values are calculated by means of the AT, B,,D,,K,,Iz,p, &, o,,~, Kj values where & denotes the intensities of the reference pellets read behind the unknown filters. The formula used is
Up
” =-Alogq[I,,(B, +A&J-
Q] I
(B, + A%,,)-‘,
(23)
obtained from equation (18) relevant to the set of samples p. Taking equation (15) into account, we can write equation (23) in the form Xi, = -A log &
[
1
(Bz+A~w,,)-'.
I
=I
(iv) In this last step of the calculation process the values of the concentrations of unknown samples are determined. The following parameters are used: AT, Bj,Di,Kj, &,,Xiqp, where the A,, denote the intensities of the unknown filters. The expression used is the following:
l&=
Di+K,Cip Bj + AywjP
11- exp [-qP (Bi + ATwjp)lI,
(25)
obtained from equation (17) for the set of unknown samples.
4. CHECK OF THE METHOD To test the applicability of the correction method to samples of small thickness, three sets of filters having various amounts (15, 25 and 50 mg) of deposited powders with different composition were prepared. To this aim weighted amounts of powder were suspended in 20 cm3 of distilled water; the suspended solid was then accurately subdivided into small size particles by an ultrasonic device, and collected on inert supports by filtration. The filter disks having 50 mm diameter and 0.8 pm porosity were placed on a glass plate and fastened with a metal ring to avoid changes in their planarity during the drying phase. In addition to the filter disks, 7 mixtures containing vanadium, molybdenum and arsenic at different concentration values were also prepared and used as external reference samples. The powder, added with boric acid as binder, was converted into self-supporting pellets. All these standard samples were directly analysed by X-ray fluorescence spectrometry in order to check the possible systematic errors during the preparation process. By using an automatic computation program for an IBM 370/125 computer, the plots of the intensity values vs the elemental concentration values of the materials deposited on the filters were obtained. The results are reported in Figs; 2-6. It is apparent that the intensity values are functions of both the element concentration and the deposited material weight, and that the obtained plots could be fitted by three
P. FFUGIEFU, F. ROW
392
and
R. TRUWO
0.40 -
0.36 -
0.32 -
0.28 T v) 0.24 0 ‘0 x lo 0.20 7
x x x
x
x
Fe
concentration,
%
Fig. 2
0.60 -
054-
0.48 -
0.42 x 5
0.36 -
x
i 9 x (D
Y
x
x
x
0.30-
+ x
5
.g 0.24 *
x
x
+
x
0.18 -
+
i
+
+
Ii
0.12 -
x
.
.
+
.
.
.
+
.
0.06 -
0
*,
, 2.00
,
, 4.00
,
, 6.00
,
,
,
8.00 Cr
,
,
,
,
12.00
10.00
concentration,
, 14.00
)
, 16.00
,
, 18.00
,
(
20.00
%
Fig. 3 Figs. 2-6. Behaviour of the fluorescence intensity function of the % concentration in sets of powdered deposit; + 25 mg deposit;
of the elements Fe, Cr, Mn, Cu, Zr as a materials deposited on filter disks. X 50 mg A 15 mg deposit.
393
A correction method of matrix, density and thickness effects in thin samples
2.40
-
2.24
-
2.08
-
x x 1.92 x
Tu)
x 1.76 -
x
Y Q * N
1.60 x Y x
3 .G m
+
1.44 + x
l
+
+
x 1.28 +
x
+ .
+
x
. .
+ 1.12 -
x x II x
0.96 -
0.80~
,
z
t
,
,
2.00
0
$
+
+
I
b
, 4.00
,
.
+
.
+ .:.a .
:
A
, 6.00
,
I 8.00 Mn
I
I 10.00
I
I 12.00
concentration,
I
I 14.00
I
I 16.00
I
I 18.00
I
1 20.00
%
Fig. 4
100.00 -
x 90.00
-
80.00
-
x
x
x
x 70.00
-
60.00
-
x x
7,
+
+
+
‘: Q x
x 50.00
+
-
r z .F v)
+
x + 40.00
-
+
I x
*
.
. .
+
x 30.00
-
+
::
20.00
x
L + .
J4 0
,
, 2.00
.
l
.
+
+
,
f
l
I
I 8.00
II
* +
x 10.00 -
. .
;
x
.
*
I
I 6.00
t
I 4.00
Cu
11 10.00
concentration,
Fig. 5
’ 12.00 %
I”“” 14.00
16.00
18.00
20.00
394
P. FRIGIERI,F. Ross
and R. TRU~~O
x
5
I
x
Y o_
(;: 50.00
x ::
+
+
x
30.00-
+
x
x
+
. .
+
:: 20.00-
i+ 1
1 0
+: ++
++ +
60.00
+ _.
. .
. .
:-
I I I I I I I I I I I',,, 6.00 6.00 10.00 12.00 14.00 2.00 4.00 Zr concentration,%
I1 16.00
I ,I 16.00 2000
Fig.6
curves with different slopes for 15, 25 and 50 mg of deposited powder, respectively. These results are in agreement with the data previously obtained [2-51. Some preliminary tests showed the applicability of the method and its agreement with the theoretical assumptions. Measurements of the fluorescence intensity coming from the reference elements present in the pellets overlapping the set of samples with different amount of deposited materials, show a decrease of the intensity with an increase in the thickness, with an exponential behaviour between the considered parameters, according to equation (19). Figures 7-9 report the values of the spectral intensity measured for the reference elements present in a set of pellets with the same concentration vs the weight of the powder deposited on the filters. To confirm the possibilities of correction of the proposed method, one can analyse the residuals obtained by using equation (15), instead of the straight-line equation, for the best fit of the experimental data relevant to the sets of samples of infinite thickness (reference and standard pellets). In the set of samples used in these tests, for instance, the mean error in the case of iron is 15% along the straight-line and decreases to 2% along the curve; similarly for chromium, the error decreases from 10 to 3%. The validity of the method is further proved by calculations, according to equation (24), of the Xj,P of a set of samples with known amounts of deposited materials. In the case of iron, the X,, assumes different values depending on the amount of deposited material: it is 0.25, 0.45 and 0.85 for deposited amounts of 15, 25 and 50 mg respectively. Similarly, the X,-, values corresponding to these amounts of deposited material are 0.20, 0.31 and 0.66 respectively. In a preliminary way, also the validity of the theoretical assumptions on the Kj value [2] P. FRIGIERI,R. TRUCCO, R. ANZANI and E. CARE~A, Chim. bad. 54, 12 (1972). [3] A. ROLLA, P. FFUGIERI,A. GIRELLI, and R. TRUCCO, Chim. Ind. 55, 8 (1973). [4] P. FRIGIERI and R. TRUCCO, X-Ray Spectrom. 3, 40 (1974). [S] P. FRIGIERI,R. TRUCCO and E. CARETTA,X-Ray Specfrom. 4, 28 (1975).
A correction method of matrix, density and thickness effects in thin samples
98.00
94.00
90.00
66.00
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.
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I
6.00
0
I
I,
12.00
I1 18.00
I1 24.00
I I I I I I I I a I I 30.00 36.00 42.00 48.00 54.00 60.00
Deposited amount of MO,
mg
Fig. 7
690.00-
630.00-
I
7 45O.OON m 5
4
3 390.005
t .
z
.
330.00-
t . .
270.00-
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.
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I 0
I I I I I I I I a I I I I I I I I I1 30.00 36.00 42.00 48.00 54.00 60.00 6.00 12.00 18.00 24.00 Deposited amount of V, mg
Fig. 8 Figs. 7-9. Relationship between the K, intensities of molybdenum, vanadium and arsenic and the deposited amounts.
395
396
P. FRIGIERI,F. Row
200.00
190.00
and R. TRUCCO
-
-
r v) 160.00 7 B x Ic 150.00 5 .g 140.00cn
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130.00 -
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Deposited amount
,
1
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36.00 42.00 48.00
of As,
I
I
54.00
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60.00
mg
Fig.9
was tested. In fact, from equation (21) it is possible to obtain the Kj,S values for each analysed standard sample, whereas the method exploits a generic Kj value valid for the whole set of samples and depending only on the pair of elements, reference and analyte, involved. Calculating the single values of Ki*,and their average value, the KFe for iron was 27.6 with a standard deviation of 3.2 and, similarly, the resulting value of K,, for chromium was 28.2 with a standard deviation of 2.9. All the preliminary tests giving satisfactory resuhs, the complete computation code was set up. The application of the calculation procedure to a set of samples consisting of filters having different deposits of oxide mixture confirms the validity of the proposed method. Table 1 reports the output data when the analysis was made on 7 samples of known composition analysed by comparison with 20 standard samples; the ‘apparent concentrations’ are the concentration values calculated using a linear calibration curve. 5. CONCLUSIONS
The analysis of the results obtained showed that the validity of the proposed method is strongly affected by the numerical values given to the absorption coefficients and used for the o calculation. These values being always rather uncertain, the analyst himself will recalculate the absorption coefficient values from the experimental data after the whole method has been optimized. These conditions are reached with the proper choice of the operating conditions, such as standard and reference samples and pairs of analyte and reference elements, and using a set of thin samples of known chemical composition. The proposed method was applied to the analysis of airborn particulate and river sediments collected on filter disks.
A correction method of matrix, density and thickness effects in thin samples
397
Table 1. Results obtained when the analysis was made on seven samples of known composition analysed by comparison with twenty standard samples
Analyte
Apparent concentration Sample (“A)
Calculated concentration W)
Expected concentration (%)
Difference of calcu- Difference of aplated and expected parent and expected concentration concentration
Fe Fe Fe Fe Fe Fe Fe
1 2 3 4 5 6 7
3.79 7.65 9.11 8.35 4.23 2.45 1.61
3.90 6.84 8.10 8.27 4.52 2.69 1.79
4.00 6.00 7.00 8.00 5.00 3.00 2.00
-0.10 +0.84 +1.10 +0.27 -0.48 -0.31 -0.21
-0.21 +1.65 +2.11 +0.3s -0.77 -0.55 -0.39
Cr Cr Cr Cr Cr Cr Cr
1 2 3 4 5 6 7
5.20 8.10 9.30 14.12 1.95 1.69 12.83
5.61 7.55 8.70 11.28 2.40 1.85 11.34
6.00 7.00 8.00 9.00 3.00 2.00 10.00
-0.39 +0.55 +0.70 +2.28 -0.60 -0.15 +1.34
-0.80 +1.10 +1.30 +5.12 -1.05 -0.31 + 2.83
Mn Mn Mn Mn Mn Mn Mn
1 2 3 4 5 6 7
5.46 7.03 9.3s 1.20 1.78 12.2s 12.17
5.61 7.06 9.37 1.41 1.77 11.54 11.81
6.00 7.00 9.00 4.00 2.00 10.00 11.00
-0.39 +0.06 +0.37 -2.59 -0.23 +1.54 +0.81
-0.54 +0.03 +0.35 -2.80 -0.22 +2.2s +1.17
cu cu cu cu cu cu cu
1 2 3 4 5 6 7
8.17 11.99 5.25 4.98 10.29 13.21 IO.55
8.55 11.62 5.23 5.43 10.90 13.20 9.87
9.00 11.00 5.00 6.00 12.00 13.00 9.00
-0.45 +0.62 +0.23 -0.57 -1.10 +0.20 +0.87
-0.83 +0.99 +0.25 -1.02 -1.71 +0.21 +1.55
Zr Zr Zr Zr Zr Zr Zr
1 2 3 4 5 6 7
13.28 5.50 11.62 9.21 8.94 7.22 2.12
11.54 4.72 8.42 8.59 10.71 7.12 2.51
10.00 4.00 6.00 8.00 13.00 7.00 3.00
+1.54 +0.72 -t-2.42 +0.59 -2.29 +0.12 -0.49
+3.28 +1.50 +S.62 +1.21 -4.06 +0.22 -0.88
APPENDIX It has been shown that parameters A *, B, D, K for the reference elements and analytes can be obtained from expression (15), the values of I, C and o being known. First one of the unknown parameters must be normalized, since, if AT, B,, D,, K, is a solution of the problem, CIA?, aB,, mD1, aK, (where a is arbitrary and different from zero) is also a solution of the same problem. In this calculation one applies the least square technique and determine the minimum of the following function
where M is the number of samples under analysis. Equation (Al) is a non-linear function in the unknown parameters and can be solved by an iterative procedure, i.e. by determining a sequence of values (Xx) such that, in case of convergence, one obtains lim {Xx} = X*, x-++C= where X* is exactly the wanted solution.
P. FRIGIERI, F. ROSSI and R. TRUCCO
398
Each term X, is usually denoted as a function of the previous term of a sequence Xx_,; hence X, = gk(XK_J. The initial value X0 is then to be chosen independently of the iterative technique employed, because, though necessary for starting the iterative process, it does not depend on it. The initial value X,, is generally denoted as “guess” and in its choice lies one of the most critical problems of the calculation, since the known theorems guarantee convergence only if X0 is close “enough” to the searched solution X*. In the processing program set up for the correction of matrix effects in thin samples analysed by X-ray fluorescence, the procedure adopted allows the automatic determination of the “guess” for problem (Al) to be obtained. A linear behaviour is assumed between intensity and concentration, and D and K values are determined by the least square technique using by the relationship I,=D+KC,
(A2)
By substituting these values in equations: (B + Ao,)I,
= D + KC4
(A3)
A and B are then calculated, and, from them, D and K values are calculated from equation (A3) again. The cycle is repeated until stable values of A *, B, D, K are obtained (as a matter of fact the unknown parameters are only three, since in an underlying assumption D = 1). It is evident from the experimental checks that this procedure yield a “guess” fit for the solution of the problem under consideration when the value of K is thus obtained for each sample of series (C), as shown in the paper, account is given to its mean value obtained after discarding, by a suitable “Student test”, the data not included in a convenient range of values. As regards the determination of the unknown concentration, the expression relating the intensity values to the concentration values can be rewritten as follows (Bi + Ajwi,p)li,p - (KjC,,, + Dj){l - exp [-Xj,,(Bj + Ai~j,p)]} = 0.
(A4)
Since
the unknown terms in equation (A4) are (&; C2,p;. . . ; CN,,. Rewriting the expression for j, which varies from 1 to N, one obtains a non-linear system of N equations in N unknown terms, viz. F,,,(C,., 1.
. . CN.~)= 0 (A5)
bN,,(C,,,. .CN.p)=0. Working out such a system for each analysed sample, it is possible to identify all values of the unknown concentrations. In this case also it is necessary to determine the convenient initial “guess”; two methods are at present being developed for the automatic determination of the “guess” Method 1 Interrupting obtain
the sequence development
of the exponential
12{I - I+
lj,, = :,? I
,.P
of equation (A4) at the first order term we
Xj,p(Bj + +j,p)J’
I
from which immediately follows
If index j is varied from 1 to N and the index p from 1 to Mp, the complete set of initial estimates of all unknown concentrations is obtained. Method 2 Disregarding the exponential term in equation (A4) and substituting a convenient expression for w~,~,we obtain N (Ah)
A correction method of matrix, density and thickness effects in thin samples
399
Setting
we find
Keeping index p fixed and varying j from 1 to N we obtain the following system from the equation (A7) A(P) . X(P)= B(P),
hnf
(A@
(n,l) (n,U
where A@#,
L) = .$!!;
X(P)(L) = CL,
;
B(p)(L) = Q,LP’
Solving the system (A8) for all values of p, one obtains the complete set of initial estimates of the unknown concentrations. Other methods
suggested
Other methods for the determination of the unknown concentration from equation (A4) can be suggested. One of them consists of solving the mentioned equation as a fixed-point problem. If 4i.p= (Bi + Aj i /+Cx,p) K=l and @i,p = (Di + KjCj,p),
equation (A4) can be rewritten in the form
= @j,pIl-exp [-X,p~~,pI>-B~~j,p 4j.p f ~+,rG,p l-=1
W)
and from this C;.,p=
Ajj,pIl - exp[-Xjptj,pl).-Tj-(QJj,p - ~,pPj,j)(Fj,,)-l~
(AlO)
where *i,p
= @ji,d@kP~i.,)
and rj = B,I(AjFi,j), Equation (AlO) contains the unknown terms Ci,,; . . ; CNp, and if it is rewritten varying the index j from 1 to N leads to the following system x, = g,(x,, x2,
x3.
. . . , x,)
(All)
i.e. a system of type X = F(X)
(Al3
where XER”
and
F:R”+R”.
Solving system (All) is similar to determining a vector X* such that X* = F(X*). The solution method suggested by the literature [6] is the so-called fixed-point technique, which consists in the following iterative processes.
[6] J. M. ORTEGA, Numerical Analysis.
A Second Course, p. 140. Academic Press, New York (1972).
P. FRIGIERI, F. Ross1 and R. TRUCCO
400
A Y
__._-_-_-
Fig. Al. Graphic interpretation
Y=FtXI
of the fixed-point procedure. Y = X is the straight-line.
Y = F(X) is the function and
(a) An initial estimate X,, of the solution is supplied; (b) The sequence X,,, = F(xk) is constructed; (c) Under conditions essentially involving the variation field of the partial derivatives of function F, the convergence of sequence {&} into the value X*, i.e. the solution of the system, is guaranteed. An interesting graphic interpretation of the problem (A12) and of the fixed-point procedure occurs when F:R + W (that is, when n = 1); system (A12) is solved by intersecting the curve Y = F(x) and the straight line Y = X (Fig. Al). Using a fixed-point calculation procedure, the iterative technique can be expressed as follows Ci$“’ = hj?{l-exp
[-Xj,,(Bj + +~,“d)I}-Tj
- (&‘+ CWlr. .)(cL..)-’ ,,P 1.1 I.1 . I.P
(A13)
When index j increases, equation (A13) can be ‘up-dated’ by substituting the oi,r values determined by the unknown concentrations as they are being calculated. It is possible to set up a further method to develop the problem with the ‘fixed-point’ technique. From the equation (A9) it follows
Qj,p= A$j,pwj.p+ @j,pexp (-Xj,p . 6j.p)+ Bj1j.p and recalling that ajj,r = Dj + KjCj.r, it is one obtains Cj,p = [4,&j,, + exp (-Xj,&,p) -
D/l($)-‘.
(A14)
The unknown terms in (A14) are the concentrations Cl+; f&,;. . ; C”,r; so by simply varying index j from 1 to N, a system completely analogous to system (All) is set up. Use can also be made of the same iterative fixed-point technique as that previously described. Acknowledgements-This work has been performed in the frame of a CISE/ENEL collaboration programme (Contract FLUOR 39/300). The computation codes relevant to the correction programmes are therefore the property of ENEL.