- Email: [email protected]
Radiation applications in art and archaeometry
Nuclear Instruments and Methods in Physics Research B 213 (2004) 683–692 www.elsevier.com/locate/nimb
Radiation applications in art and archaeometry X-ray fluorescence applications to archaeometry. Possibility of obtaining non-destructive quantitative analyses Mario Milazzo
*
Istituto di Fisica Generale Applicata, Universita degli Studi di Milano, via Celoria, 16-20133 Milan, Italy
Abstract The possibility of obtaining quantitative XRF analysis in archaeometric applications is considered in the following cases: ii(i) Examinations of metallic objects with irregular surface: coins, for instance. i(ii) Metallic objects with a natural or artificial patina on the surface. (iii) Glass or ceramic samples for which the problems for quantitative analysis rise from the non-detectability of matrix low Z elements. The fundamental parameter method for quantitative XRF analysis is based on a numerical procedure involving he relative values of XRF lines intensity. As a consequence it can be applied also to the experimental XRF spectra obtained for metallic objects if the correction for the irregular shape consists only in introducing a constant factor which does not affect the XRF intensity relative value. This is in fact possible in non-very-restrictive conditions for the experimental set up. The finenesses of coins with a superficial patina can be evaluated by resorting to the measurements of Rayleigh to Compton scattering intensity ratio at an incident energy higher than the one of characteristic X-ray. For glasses and ceramics the measurements of the Compton scattered intensity of the exciting radiation and the use of a proper scaling law make possible to evaluate the matrix absorption coefficients for all characteristic X-ray line energies. Ó 2003 Elsevier B.V. All rights reserved. PACS: 07.85 Keywords: X-ray fluorescence; Quantitative analysis; Portable instruments; Archaeometric applications
1. Introduction
*
Tel.: +39-2-5031-7438; fax: +39-2-5031-7422. E-mail address: [email protected] (M. Milazzo).
A review book has been recently published concerning radiation in art and archeometry [1] where a chapter is devoted to the ‘‘X-ray fluorescence (XRF) application to the study and
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-583X(03)01686-0
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M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
Conservation of Cultural Heritage’’ (Ferretti) and a second one to the analysis of coins and other metalwork using XRF, PIXE and activation analysis (Guerra). In fact, although analyses by characteristic X-rays do not represent the only application of radiation to the study and conservation of cultural heritage, these analyses are no doubt the most extensively used including the versions where protons (PIXE) or electrons (SEM) are employed as excitation sources. In Fig. 1 the whole of non-destructive analyses based on the use of radiations, practically covering the topics of this conference, are represented with the related probed thickness when applied to the analysis of coins, for instance. XRF is the label given to the analysis through X-ray fluorescence excited by X-rays. It adds to the general characteristics of the method – nondestructive and based on regular atomic emission
Fig. 1. Non-destructive analyses employing radiations with the correspondent experimental thicknesses for a coin. (IBA: ion beam analysis, SEM: scanning electron microscope, PIXE: proton induced X emission, PGME: proton induced gamma emission, NRA: nuclear reaction analysis, RBS: Rutherford back scattering, PAA: particle activation analysis, NAA: neutron activation analysis).
rules – the advantages of allowing comparatively simple algorithms for quantitative analysis and the possibility of setting up portable instruments. In the last 20 years, several valuable books dealing with principles of quantitative XRF analysis have been published, so that the relevant procedures can now be regarded as well established. However, the wave dispersive method for X-ray spectroscopic analysis and pellet – shaped flat samples are mainly taken into account in applications, for instance, to mineralogy or metallurgy. As regards the possibility of getting quantitative analysis in archaeometric applications, when using the energy dispersive detector for X-ray spectroscopy, the problems rising from the limited detector sensitivity in detecting low Z elements, the irregular shape or the non-homogeneous composition of the sample have generated a widespread opinion that only semi-quantitative analyses are possible in XRF applications to archaeometry. In fact, this is always true for non-homogeneous samples as, typically, painting layers. On the contrary, the problems deriving from limited sensitivity in detecting matrix light elements as well as from irregular surface under analysis can be solved in most cases. We will consider in Section 1 the problems concerning the possibility of getting quantitative XRF analysis in case of irregular shape. We will restrict ourselves to metal alloy objects and energy dispersive X-ray spectroscopy (XES). We assume that they have a microscopically homogeneous composition and, in particular, that natural or artificial processes leading to patina formation on the surface has not occurred. This is usually the case of gold coins, whereas for silver or copper alloys a patina with a thickness ranging from a few to several ten micrometers can often be found with a composition that can substantially be different from that of the bulk. The occurrence of a patina prevents us even from obtaining by XRF the correct fineness values of coins since the depth of the analysis for metals is usually less than the patina thickness. We will show in Section 2 that, as already discussed in previous works [2,3], the Rayleigh to Compton scattering intensity ratio (R/C) of monochromatic radiation of higher energy than that of characteristic X-rays can replace
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
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XRF analysis to obtain the coin fineness when a surface patina is present. As to glass and ceramic samples, the problem of quantitative XRF analysis does not rise normally from the irregular shape, but from the impossibility of evaluating the auto absorption correction due to matrix low Z elements which in most cases cannot be detected by standard solid-state portable detectors. In Section 3 we will show how to obtain quantitative XRF analysis of glasses by adding the measurement of intensity of Compton scattering, undergone by a properly chosen exciting X-ray line, to the ones of characteristic X-ray lines [4]. This method is more direct than measuring the R/C value, proposed by other authors [5], to evaluate the so called ‘‘dark matrix’’ equivalent-Z by an iterative procedure. 2. Quantitative analysis of coins Coins can be assumed as the typical case of metallic objects with irregular surface and variable size. Let us assume that the surface patina is not present so that what follows applies to gold coins (see Fig. 2) and gold jewels or metallic objects provided they are in a good state of conservation. We start by considering an experimental set up, as the one artistically shown in Fig. 3, with even angles, W1 , W2 , (for instance, 45°) for incoming and outgoing radiations with reference to the surface of a flat sample. In these symmetric conditions, the characteristic X-ray intensity, Ii , of the i element does not depend on the angle value as is evident from the well known fundamental formula [6] Ii ¼
liph I0 wi GeI Pi M ; sin W1 l ðE0 Þ lM ðEi Þ þ sin W1 sin W2
Fig. 3. Not in scale artistic representation of XRF experimental set up. (1) X-ray tube, (2) collimators and (3) X-ray solid-state detector.
ð1Þ
where I0 wi G eI Pi lph
Fig. 2. Archaic gold Greek coins (Museo Archeologico di Milano): VI sec. BC.
lM ðEÞ E0 , E i
exiting radiation intensity i-element relative content geometrical factor intrinsic detector efficiency product of atomic factors for i-element massive photoelectric absorption coefficient of i-element
massive total absorption coefficient of sample matrix at energy E exciting radiation and characteristic XRF energies
In case of an irregular sample, we must substitute an unknown geometric factor value G0 for G which results from the calibration with a flat standard sample. We stress that the so called method of fundamental parameters for XRF quantitative analysis
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is based on the relative values of characteristic X-ray line intensities obtained by dividing the single intensities by the sum of intensities of all the spectrum lines which, obviously, do not depend on any geometric factor. However, as a further consequence of the irregular shape, we have two uneven effective values for the angles. This very fact prevents us from eliminating the angle dependence in the formula (1). Resorting to the expedient of rotating the coin during the measurement, however, we can assume that the effective angles, even if unknown, turn to be equal. So, at the only condition of employing a symmetric experimental set up, we recover in a very simple way the condition of independence from the angle values also in case of irregular objects. We note that keeping on rotating the sample during the XRF measurement is not necessary. Summing up the results of two subsequent measurements with up and down positions of the coin is in fact equivalent to a single measurement with even effective angles for incoming and outgoing radiations. In this way the only consequence of the irregular shape we have been left with is the unknown value of G0 factor which does not affect the relative values of X-ray line intensities. Evidently, rotating the portable instrument instead of the sample might be easier in some cases. We recall that the equation correlating the experimental XRF intensity, Ii , with the related concentration for the generic i element in the most general case of polychromatic exciting radiation includes secondary excitation effects. It is not a linear equation since it contains the integration on the exciting spectrum energy of a function which includes the logarithmic terms coming out from the volume integration of secondary excitation effects. We assume for the moment that the terms giving secondary effect contributions to XRF intensity do not depend on the angles. The easiest way to get to a solution of the equation set for the n elements in the sample turns out to be a numerical iterative procedure [6] starting from fixing the initial relative concentration set ðw1 Þ1 . . . ðwn Þ1 on the basis of the correspondence rule, where we denote by IðiÞ the X-ray fluorescent intensity of a pure i element sample produced in the same experimental conditions:
Ii IðiÞ
ðwi Þ1 ¼ P
In n IðnÞ
¼ R0i :
ð2Þ
This set is entered into the equations describing the theoretical dependence for experimental intensities on intensity and energy of exciting radiation and auto absorption and secondary excitation effects. We obtain a new set of ‘‘theoretical’’ values, ðRi Þ1 , for relative intensities which are different from the experimental ones. The second concentration set is obtained by putting ðwi Þ2 ¼
R0i ðwi Þ1 : ðRi Þ1
ð3Þ
And again wi values are scaled to unity and used to calculate the next intensity set. The process is iterated until the set of wi values differ by less than some arbitrary amount, say 0.1%, between two successive calculations. The last wi set is accepted as the correct composition. We stress the fact that the application of the method of numerical solution does not depend on the eventual introduction of a geometrical correcting factor, but only on the difficulty of finding out the analytical solution. In other words, it is not typical of analyses of irregular objects since it applies to pellet analysis for instance in geological or mineralogical applications. We can consider it a lucky circumstance from which we can affirm that irregular shape of metallic objects, provided they are microscopically homogeneous, is not per se a cause of impossibility to get XRF quantitative analysis. Let us examine now in detail how the angle values enter into the equation term describing secondary excitation. It is well known [7] that secondary excitation effects can be calculated by a (triple) integration extended to the examined volume of the sample. The final formula containing both the direct and secondary contributions for a selected excitation energy E0 , is [7] liph 1 ei Pi Gwi M sin W1 l ðE0 Þ lM ðEi Þ þ sin W1 sin W2 " # n X 1þ wj dij ;
Ii ¼ I0
j¼1
ð4Þ
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
where lj ðE0 Þ Dij Pj li ðEj Þ Aij ðE0 Þ; dij ¼ li ðE0 Þ 2 8 > > < 1 lM ðE0 Þ Aij ¼ ln 1 þ M > l ðEj Þ sin W1 lM ðE0 Þ > : sin W1
9 > > =
þ
1 lM ðEi Þ ; ln 1 þ > lM ðEj Þ sin W2 > lM ðEi Þ ; sin W2
Dij ¼ 0;
if E0 6 Eiabs ;
Dij ¼ 1;
if E0 > Ejabs ;
ð5Þ
Ej > Eiabs :
By the way, we note that this integration can be obtained analytically only considering infinite integration volume [8] and this has never been put into evidence in the literature. However, for metals this means an integration over so short distances that the assumption of constant angles over the integration volume is correct. The weak dependence of the contribution to XRF intensity due to secondary excitation on the angle values is given by the lij terms. We note that for the exciting energy we always have E0 > Ei , therefore lM ðE0 Þ lM ðEi Þ and from the trivial approximation lnð1 þ xÞ ffi x, if x 1, in practice, we obtain that the first term does not depend on angle W1 . So we can assume that we have only the dependence on W2 in the form 1 lM ðEi Þ ln 1 þ ; lM ðEj Þ sin W2 lM ðEi Þ sin W2 as we have already said, it is a very weak dependence. Practically, with the condition lðE0 Þ lðEi Þ, the only dependence on geometry is given in Eq. (4) by the factor G0 ðsin W2 = sin W1 Þ. Assuming that the angle effective values, W1 and W2 , turn out to be the same in the measurements, we come back to the situation where the irregular shape is only accounted for by factor G0 . For completenessÕ sake, we consider also the rather unusual case that the portable set up has not been designed with even angles or is not pos-
687
sible to rotate its position relative to the object. Assuming once again E0 Ei then the only consequence will be that the corrective factor is G0 ðsin W2 = sin W1 Þ, being W1 6¼ W2 this time. Even if the above inequality for energies cancelling the dependence from W1 , cannot be assumed for the lower energies of the exciting spectrum, nevertheless we have that the additional dependence of intensity on angles due to the secondary effects is very weak. Experimentally it was proved [9] that, for instance, for an Au–Cu alloy disk with 30% Cu contents with 35 KV X-ray tube voltage we have a variation of about 6% in the ratio ICu =IAu for W2 going from 45° to 75°. It has been now several years since the programme QXAS-AXIL (Quantitative X-Ray Analysis System) based on the fundamental parameter method was published. The preliminary operation of this programme is a best fit procedure for background subtraction and overlapping peak deconvolution. It can directly be used for the analysis of XRF spectra of irregular metallic objects and quantitative analysis can be obtained in archaeometric applications if the non-veryrestrictive experimental conditions we have examined are fulfilled. We must take into account, however, that a very high precision is normally not required. Results from quantitative XRF analyses of gold archaic Greek coins are shown in Table 1.
3. Measurement of coin fineness In case a patina of several ten l thickness is present on metal surface, we can employ an exciting radiation conveniently more energetic than the characteristic X-ray fluorescence and consider the measure of Rayleigh to Compton scattering intensities ratio, R/C, to probe the inner material [10]. The measurement of the R/C value offers the advantage of being a method of analysis independent on the sample geometry. For quantitative calculations we might make use of theoretical scattering cross-sections of Rayleigh and Compton effects. But in the X-ray energy range coherent
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M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
Table 1 XRF quantitative analyses for archaic gold Greek coins (VI–VII B.C.) Sample 1 Au% Ag% Cu% qXRF qSG R
Sample 2
Sample 3
Sample 4
Sample 5
R
O
R
O
R
O
R
O
R
O
50.1 46.6 3.3
51.3 46.9 1.8
69.9 28.7 1.4
67.2 31.2 1.6
63 35.8 1.2
64.5 33.9 1.6
66.8 32.4 0.8
66.3 32.9 0.8
56.2 43 0.8
56.9 42.1 1
13.5 13.1 1.03
15.2 14.1 1.03
14.8 13.6 1.08
15.2 14.8 1.03
14.1 13.3 1.06
R ¼ reverse, O ¼ obverse, qXRF ¼ density (g/cm3 ) as resulting from XRF analysis, qSG ¼ density as resulting from weighing in water, R ¼ qXRF =qSG .
scattering cross-sections have been neither calculated nor systematically measured with sufficient precision. Considering, for instance, the differential Rayleigh cross-sections for 60 keV incident energy of Cu and Ag we have found a disagreement of about 10 and 6 factors respectively, respectively between the experiment [11] and the published data [12]. It is therefore convenient to resort directly to the comparison with the experimental measurements of the intensities of X-rays scattered from pure element standards. We have already shown [2] that we can assume with a good approximation a linear dependence of R/C on the relative contents. Silver contents in Cu–Ag alloy of several modern coins obtained with this method are shown in Table 2. The c line emitted by an Am241 source (10 mC) was used (Ec ¼ 59:54 keV). The Ag relative content values in the last column have been obtained from gamma absorption measurements. Note that the XRF results are not correct in some cases. Table 2 Silver content values obtained by R/C (relative error ±5%, h ¼ 131° ± 1°), XRF and c absorption for modern coins. Ec ¼ 59:54 (Am241 source) Sample
Nominal
R/C
XRF
Gamma Abs.
1 Dollar USA 1 Peso Mexico 20 Rappen 500 Lit. 5 cent. USA 10 Rappen 5 Lit. 5 Rappen
90 50 15 83.5 35 10 83.5 5
92.5 50.1 15.3 84.7 35.9 11.6 84 6.6
91 71.9 15.5 89 49 – 92 –
– 49.8 ± 1.3 16.8 ± 0.96 83.00 34.4 ± 1.8 – – –
4. Quantitative analysis of glass and ceramics As is well known, the possibility of obtaining quantitative analysis is limited for glasses and ceramics by the low efficiency in detecting low Z elements typical of energy dispersive X ray detectors which depends on the comparatively thick protective Be window. We assume a value of 24 lm (1 ml) for this thickness which can realistically be considered strong enough to offer a reasonable safety degree in the portable instruments for which the possibility of accidental breaks are more probable than for laboratory set up. The lowest limit for the Z value of the detectable element corresponding to 1 mil window results to be Z ¼ 14 (Si; ka ¼ 1:4 keV). X-ray fluorescence secondary excitation by minority heavy elements does not occur in glasses and ceramics since the matrix composition is dominantly formed by light elements. So, the simpler formula (1) applies where the correction to X-ray intensity due to matrix absorption is to be taken into account. Being the matrix composition unknown, this is the so called ‘‘dark’’ matrix problem [5]. As already noted, the Rayleigh to Compton scattering ratio – R/C – is strongly dependent on the Z value and the idea of evaluating the matrix absorption effects by measuring this value was proposed by Van Espen and He [5]. We have also already noted that this ratio is independent on any geometrical correction effect. More important: this ratio does not depend on the sample thickness and it might result of some use in case of examination of enamel layers, for instance.
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
The method of resorting to R/C measurement in order to obtain quantitative XRF analysis by energy dispersive spectroscopy is clearly explained in the above quoted paper. Basically, for a given incident X-ray energy E0 , the experimental R/C value includes the contributions coming from the (dark matrix) low Z elements and from the (detected) n high Z elements. The mathematical expression of R/C depends on the relative contents of the matrix, w0 , and of the n elements, w1 . . . wn . The same relative contents figure in the expression giving the XRF experimental intensity for each measured X-ray line. Accordingly, lM ðEÞ must be substituted in formula (1) by the sum of the corresponding contributions. Dividing formula (1) by the one corresponding to the XRF intensity IðiÞ for a pure element i we obtain Ii wi ðli ðE0 Þ þ li ðEi ÞÞ Pn : ¼ IðiÞ w0 ðl0 ðE0 Þ þ l0 ðEi ÞÞ þ i¼1 wi ðli ðE0 Þ þ li ðEi ÞÞ ð6Þ The absorption coefficient value attributed to the matrix corresponds to the matrix effective Z value. It must be the same as the one which corresponds theoretically to the contribution of the matrix to R/C value which best fits the experimental result. This is obtained by a numerical double iteration scheme from which the effective – or equivalent – Z value for the matrix of the examined sample is also obtained. We note that a subdirectory based on this method is included in the already quoted AXIL programme for glass and ceramic quantitative XRF analysis. However, we estimate simpler the method, already proposed [13], which it is based on considering that in the range of our exciting X-ray energies the massive Compton scattering coefficient can be assumed constant for all low Z elements for which Z=A ¼ 1=2 (Table 3). As a consequence, limiting ourselves to the measurement of C intensity, we have that the contribution to it deriving from the matrix light elements depends on the universal value for the massive Compton scattering coefficient lC0 ðE0 Þ, while the contributions from the other elements depend on the relevant Compton scattering coef-
689
Table 3 Massive Compton scattering coefficients of low Z elements normalized to the value for silicon Element (Z)
E0 ¼ 9:885 keV (Ge Ka )
E0 ¼ 17:476 keV (Mo Ka )
E0 ¼ 26:355 keV (Sb Ka )
C (6) O (8) Na (11) Mg (12) Al (13) S (16) K (19) Ca (20)
1.262 1.206 1.019 1.028 0.981 0.972 0.928 0.953
1.155 1.111 1.007 1.022 0.985 0.978 0.926 0.941
1.108 1.081 0.986 1.007 0.973 0.980 0.932 0.953
ficients lCi ðE0 Þ found in the literature [12] which can be considered fundamental parameters as well as the others used in the method. As regards the matrix total absorption cofficient, l0 ðE0 Þ, let us start from the Compton scattered intensity of incident radiation from the sample P lC0 ðE0 Þw0 þ ni¼1 lCi ðE0 Þwi C¼K : ð7Þ lðE0 Þ lðEC Þ þ sin w1 sin w2 Using a scaling law, for the Compton scattered photon energy EC we can write (8) n lðEC Þ EC ¼ ¼ b; ð8Þ lðE0 Þ E0 and we obtain
P lC0 ðE0 Þw0 þ ni¼1 lCi ðE0 Þwi 1 b lðE0 Þ þ sin w1 sin w2 P n lC ðE0 Þw0 þ i¼1 lCi ðE0 Þwi : ¼ K0 0 lðE0 Þ
C¼K
ð9Þ
K 0 is a constant value that can experimentally be obtained by measuring C from a silicon sample, for instance. We will comment on how to choose the correct value for n in the scaling low. Its choice is not critical in the above case since C ffi E0 . PE n Since lðE0 Þ ¼ w0 l0 ðE0 Þ þ i¼1 li ðE0 Þwi Eq. (9) becomes Pn C C l ðE Þw 0 l0 ðE0 Þw0 þ Pi¼1 i 0 i : C¼K ð10Þ l0 ðE0 Þw0 þ ni¼1 li ðE0 Þwi
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M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
To find out l0 ðEi Þ from l0 ðE0 Þ, we write nSiO 2 E0 l0 ðE0 Þ ¼ ai ðE0 Þl0 ðE0 Þ; l0 ðEi Þ ¼ Ei ð11Þ where the nSiO2 exponent is given by the angular factor of the straight line on log scale obtained for lðEÞ from Ei to E0 considering a SiO2 matrix. Therefore, Eq. (6) can be rewritten in the following form: , Ii ¼ wi IðiÞ ðli ðE0 Þ þ li ðEi ÞÞ þ
n X
w0 ð1 þ ai ðE0 ÞÞl0 ðE0 Þ !
wi ðli ðE0 Þ þ li ðEi ÞÞ :
ð12Þ
i¼1
The only unknown parameter, l0 ðE0 Þ, can be deduced from Eq. (10) obtaining Pn wi ½K 0 lCi ðE0 Þ Cli ðE0 Þ l0 ðE0 Þ ¼ i¼1 w0 C Pn K 0 lC0 ðE0 Þ w i ci ¼ i¼1 þ þ C; ð13Þ C w0 where both the quantities K 0 lCi ðE0 Þ Cli ðE0 Þ K 0 lC0 ðE0 Þ and C ¼ C C are evaluated using K 0 , and Compton experimental intensity, C. P Taking into account the condition ni¼0 wi ¼ 1 with algebraic elaborations that we omit and
cj ¼
calling l i ¼ li ðE0 Þ þ li ðEi Þ, we finally obtain for each element i the linear equation , ! n X Ii ¼ wi IðiÞ li 1 wi Cð1 þ ai ðE0 ÞÞ i¼1
þ
n X
!
wi ½ci ð1 þ ai ðE0 ÞÞ þ
l i
ð14Þ
:
i¼1
We note that the use of a scaling law to evaluate the massive absorption coefficient of the matrix for each characteristic X-ray energy is the only price to pay to have a linear equation set which can directly be solved to obtain the relative contents of minority elements. The scaling law can properly be chosen on the basis of preliminary information on the matrix composition (glass, ceramic, earth, . . .). In Table 4 we show for example the scaling factors ai ðE0 Þ for selected E0 and Ei values for SiO2 and two standard glasses. The exponent n values which appear in the scaling laws are practically the same as SiO2 in all cases. We can consider a valid approximation to resort to the scaling law of SiO2 , whatever be the glass. At last, we consider the portable instruments based on X-ray tube for which this method must be extended to the case when a continuous spectrum is used for excitation. The only expedient in this case is that a characteristic X-ray line from the anode be evident in the X-ray tube spectrum at an energy high enough to be used for the Compton scattering measurement. In comparison with the
Table 4 Scaling factors ai ðE0 Þ for selected E0 and Ei values for SiO2 and two standard glasses Glass matrix composition SiO2 SG55 Standard SiO2 0.687, Al2 O3 0.011, Na2 O 0.102, K2 O 0.070, SO3 0.003, B2 O3 0.0022, F 0.009, CaO 0.0005, MgO 0.0001 CER1 Standard SiO2 0.645, Al2 O3 0.216, Na2 O 0.008, K2 O 0.002, CaO 0.0009, MgO 0.0004
E0 ¼ 9:885 keV (Ge Ka ) Ei ¼ 3:691 keV (Ca Ka ) n ai ðE0 Þ ¼ EE0i ¼ 16:49 ai ðE0 Þ ¼
ai ðE0 Þ ¼
n E0 Ei
n E0 Ei
E0 ¼ 14:163 keV (Sr Ka ) Ei ¼ 6:403 keV (Fe Ka ) n ai ðE0 Þ ¼ EE0i ¼ 10:09
¼ 16:76
ai ðE0 Þ ¼
¼ 16:63
ai ðE0 Þ ¼
n E0 Ei
n E0 Ei
E0 ¼ 26:355 keV (Sb Ka ) Ei ¼ 13:38 keV (Rb Ka ) n ai ðE0 Þ ¼ EE0i ¼ 6:15
¼ 10:14
ai ðE0 Þ ¼
¼ 10:14
ai ðE0 Þ ¼
n E0 Ei
n E0 Ei
¼ 6:10
¼ 6:13
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
691
Table 5 Results of XRF analyses of some glass standards supplied by Stazione Sperimentale del Vetro di Murano Element
Experimental
Certification
Ag Sr Pb Zn Cu Ni Co Fe Mn Ti Ca
239 ± 8 ppm 488 ± 15 ppm 412 ± 13 ppm 430 ± 13 ppm 429 ± 13 ppm 449 ± 14 ppm 380 ± 11 ppm 449 ± 14 ppm 479 ± 14 ppm 427 ± 13 ppm 8.32 ± 0.25%
249 ppm 508 ppm 428 ppm 438 ppm 438 ppm 458 ppm 388 ppm 458 ppm 488 ppm 438 ppm 8.54 %
Zn Fe Ba Ca K Cd Se
55,417 ± 1663 ppm 246 ± 7 ppm 23,618 ± 710 ppm 377 ± 12 ppm 56,783 ± 1704 ppm 4099 ± 123 ppm 67 ± 2 ppm
55,907 ppm 249 ppm 23951 ppm 384 ppm 57,767 ppm 4175 ppm 70 ppm
Zn Ni Co Fe Mn Ti Ba Ca K Zr Sr
14,169 ± 425 ppm 2080 ± 63 ppm 1539 ± 47 ppm 1837 ± 55 ppm 4727 ± 142 ppm 11,186 ± 340 ppm 23,730 ± 712 ppm 687 ± 21 ppm 1777 ± 54 ppm 8325 ± 250 ppm 242 ± 8 ppm
13,931 ppm 2058 ppm 1520 ppm 1868 ppm 4802 ppm 13,614 ppm 23,158 ppm 679 ppm 1752 ppm 8443 ppm 253 ppm
Zn Cu Ni Co Fe Mn Ba Ca K Sr Pb As
69 ± 11 ppm 93 ± 15 ppm 4001 ± 599 ppm 442 ± 71 ppm 239 ± 40 ppm 1850 ± 308 ppm 25,800 ± 3700 ppm 2.50 ± 0.46% 9.01 ± 1.43% 437 ± 26 ppm 6.51 ± 0.17% 1285 ± 114 ppm
– 70 ppm 4100 ppm 520 ppm 340 ppm 2050 ppm 25,000 ppm 2.85% 8.38% – 6.88% 1200 ppm
Standard NB10
Standard SG55
Standard CER1
Standard P66
monochromatic excitation case, a minor complication derives from applying the scaling law to evaluate the matrix absorption coefficients for
several exciting energies corresponding to the subdivision of the X-ray spectrum into finite interval values [7].
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Some examples of quantitative XRF analyses of standard glassed obtained with this method are shown in Table 5.
References [1] D.C. Creagh, D.A. Bradley (Eds.), Radiation in Art and Archeometry, Elsevier, 2000. [2] L. Confalonieri, R. Crippa, M. Milazzo, Appl. Radiat. Isotopes 38 (1987) 139. [3] C. Bui, M. Milazzo, M. Monichino, Nucl. Instr. and Meth. B 28 (1987) 88. [4] L. Bonizzoni, A. Galli, M. Milazzo, X-Ray Spectrom. 29 (2000) 443.
[5] F. He, P. Van Espen, Nucl. Instr. and Meth. A 299 (1990) 580. [6] J.W. Criss, L.S. Birkc, Anal. Chem. 40 (1968) 1080. [7] R.M. Rousseau, X-Ray Spectrom. 13 (1984) 115. [8] C. Bui, M. Milazzo, C. Sironi, X-Ray Spectrom. 22 (1993) 17. [9] M. Milazzo, C. Cicardi, X-Ray Spectrom. 26 (1997) 211. [10] G.E. Gigante, S. Sciuti, Int. J. Appl. Radiat. Isotopes 35 (1984) 481. [11] C. Bui, M. Milazzo Il, Nuovo Cimento 2 (1989) 655. [12] M.J. Berger, J.H. Hubble, XCOM: Photon Cross Section Database, NIST Standard Reference Database 8 (XGAM) NBSIR 87-3597. [13] L. Bonizzoni, A. Galli, M. Milazzo, X-Ray Spectrom. 31 (2002) 35.
Radiation applications in art and archaeometry X-ray fluorescence applications to archaeometry. Possibility of obtaining non-destructive quantitative analyses Mario Milazzo
*
Istituto di Fisica Generale Applicata, Universita degli Studi di Milano, via Celoria, 16-20133 Milan, Italy
Abstract The possibility of obtaining quantitative XRF analysis in archaeometric applications is considered in the following cases: ii(i) Examinations of metallic objects with irregular surface: coins, for instance. i(ii) Metallic objects with a natural or artificial patina on the surface. (iii) Glass or ceramic samples for which the problems for quantitative analysis rise from the non-detectability of matrix low Z elements. The fundamental parameter method for quantitative XRF analysis is based on a numerical procedure involving he relative values of XRF lines intensity. As a consequence it can be applied also to the experimental XRF spectra obtained for metallic objects if the correction for the irregular shape consists only in introducing a constant factor which does not affect the XRF intensity relative value. This is in fact possible in non-very-restrictive conditions for the experimental set up. The finenesses of coins with a superficial patina can be evaluated by resorting to the measurements of Rayleigh to Compton scattering intensity ratio at an incident energy higher than the one of characteristic X-ray. For glasses and ceramics the measurements of the Compton scattered intensity of the exciting radiation and the use of a proper scaling law make possible to evaluate the matrix absorption coefficients for all characteristic X-ray line energies. Ó 2003 Elsevier B.V. All rights reserved. PACS: 07.85 Keywords: X-ray fluorescence; Quantitative analysis; Portable instruments; Archaeometric applications
1. Introduction
*
Tel.: +39-2-5031-7438; fax: +39-2-5031-7422. E-mail address: [email protected] (M. Milazzo).
A review book has been recently published concerning radiation in art and archeometry [1] where a chapter is devoted to the ‘‘X-ray fluorescence (XRF) application to the study and
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-583X(03)01686-0
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M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
Conservation of Cultural Heritage’’ (Ferretti) and a second one to the analysis of coins and other metalwork using XRF, PIXE and activation analysis (Guerra). In fact, although analyses by characteristic X-rays do not represent the only application of radiation to the study and conservation of cultural heritage, these analyses are no doubt the most extensively used including the versions where protons (PIXE) or electrons (SEM) are employed as excitation sources. In Fig. 1 the whole of non-destructive analyses based on the use of radiations, practically covering the topics of this conference, are represented with the related probed thickness when applied to the analysis of coins, for instance. XRF is the label given to the analysis through X-ray fluorescence excited by X-rays. It adds to the general characteristics of the method – nondestructive and based on regular atomic emission
Fig. 1. Non-destructive analyses employing radiations with the correspondent experimental thicknesses for a coin. (IBA: ion beam analysis, SEM: scanning electron microscope, PIXE: proton induced X emission, PGME: proton induced gamma emission, NRA: nuclear reaction analysis, RBS: Rutherford back scattering, PAA: particle activation analysis, NAA: neutron activation analysis).
rules – the advantages of allowing comparatively simple algorithms for quantitative analysis and the possibility of setting up portable instruments. In the last 20 years, several valuable books dealing with principles of quantitative XRF analysis have been published, so that the relevant procedures can now be regarded as well established. However, the wave dispersive method for X-ray spectroscopic analysis and pellet – shaped flat samples are mainly taken into account in applications, for instance, to mineralogy or metallurgy. As regards the possibility of getting quantitative analysis in archaeometric applications, when using the energy dispersive detector for X-ray spectroscopy, the problems rising from the limited detector sensitivity in detecting low Z elements, the irregular shape or the non-homogeneous composition of the sample have generated a widespread opinion that only semi-quantitative analyses are possible in XRF applications to archaeometry. In fact, this is always true for non-homogeneous samples as, typically, painting layers. On the contrary, the problems deriving from limited sensitivity in detecting matrix light elements as well as from irregular surface under analysis can be solved in most cases. We will consider in Section 1 the problems concerning the possibility of getting quantitative XRF analysis in case of irregular shape. We will restrict ourselves to metal alloy objects and energy dispersive X-ray spectroscopy (XES). We assume that they have a microscopically homogeneous composition and, in particular, that natural or artificial processes leading to patina formation on the surface has not occurred. This is usually the case of gold coins, whereas for silver or copper alloys a patina with a thickness ranging from a few to several ten micrometers can often be found with a composition that can substantially be different from that of the bulk. The occurrence of a patina prevents us even from obtaining by XRF the correct fineness values of coins since the depth of the analysis for metals is usually less than the patina thickness. We will show in Section 2 that, as already discussed in previous works [2,3], the Rayleigh to Compton scattering intensity ratio (R/C) of monochromatic radiation of higher energy than that of characteristic X-rays can replace
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
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XRF analysis to obtain the coin fineness when a surface patina is present. As to glass and ceramic samples, the problem of quantitative XRF analysis does not rise normally from the irregular shape, but from the impossibility of evaluating the auto absorption correction due to matrix low Z elements which in most cases cannot be detected by standard solid-state portable detectors. In Section 3 we will show how to obtain quantitative XRF analysis of glasses by adding the measurement of intensity of Compton scattering, undergone by a properly chosen exciting X-ray line, to the ones of characteristic X-ray lines [4]. This method is more direct than measuring the R/C value, proposed by other authors [5], to evaluate the so called ‘‘dark matrix’’ equivalent-Z by an iterative procedure. 2. Quantitative analysis of coins Coins can be assumed as the typical case of metallic objects with irregular surface and variable size. Let us assume that the surface patina is not present so that what follows applies to gold coins (see Fig. 2) and gold jewels or metallic objects provided they are in a good state of conservation. We start by considering an experimental set up, as the one artistically shown in Fig. 3, with even angles, W1 , W2 , (for instance, 45°) for incoming and outgoing radiations with reference to the surface of a flat sample. In these symmetric conditions, the characteristic X-ray intensity, Ii , of the i element does not depend on the angle value as is evident from the well known fundamental formula [6] Ii ¼
liph I0 wi GeI Pi M ; sin W1 l ðE0 Þ lM ðEi Þ þ sin W1 sin W2
Fig. 3. Not in scale artistic representation of XRF experimental set up. (1) X-ray tube, (2) collimators and (3) X-ray solid-state detector.
ð1Þ
where I0 wi G eI Pi lph
Fig. 2. Archaic gold Greek coins (Museo Archeologico di Milano): VI sec. BC.
lM ðEÞ E0 , E i
exiting radiation intensity i-element relative content geometrical factor intrinsic detector efficiency product of atomic factors for i-element massive photoelectric absorption coefficient of i-element
massive total absorption coefficient of sample matrix at energy E exciting radiation and characteristic XRF energies
In case of an irregular sample, we must substitute an unknown geometric factor value G0 for G which results from the calibration with a flat standard sample. We stress that the so called method of fundamental parameters for XRF quantitative analysis
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is based on the relative values of characteristic X-ray line intensities obtained by dividing the single intensities by the sum of intensities of all the spectrum lines which, obviously, do not depend on any geometric factor. However, as a further consequence of the irregular shape, we have two uneven effective values for the angles. This very fact prevents us from eliminating the angle dependence in the formula (1). Resorting to the expedient of rotating the coin during the measurement, however, we can assume that the effective angles, even if unknown, turn to be equal. So, at the only condition of employing a symmetric experimental set up, we recover in a very simple way the condition of independence from the angle values also in case of irregular objects. We note that keeping on rotating the sample during the XRF measurement is not necessary. Summing up the results of two subsequent measurements with up and down positions of the coin is in fact equivalent to a single measurement with even effective angles for incoming and outgoing radiations. In this way the only consequence of the irregular shape we have been left with is the unknown value of G0 factor which does not affect the relative values of X-ray line intensities. Evidently, rotating the portable instrument instead of the sample might be easier in some cases. We recall that the equation correlating the experimental XRF intensity, Ii , with the related concentration for the generic i element in the most general case of polychromatic exciting radiation includes secondary excitation effects. It is not a linear equation since it contains the integration on the exciting spectrum energy of a function which includes the logarithmic terms coming out from the volume integration of secondary excitation effects. We assume for the moment that the terms giving secondary effect contributions to XRF intensity do not depend on the angles. The easiest way to get to a solution of the equation set for the n elements in the sample turns out to be a numerical iterative procedure [6] starting from fixing the initial relative concentration set ðw1 Þ1 . . . ðwn Þ1 on the basis of the correspondence rule, where we denote by IðiÞ the X-ray fluorescent intensity of a pure i element sample produced in the same experimental conditions:
Ii IðiÞ
ðwi Þ1 ¼ P
In n IðnÞ
¼ R0i :
ð2Þ
This set is entered into the equations describing the theoretical dependence for experimental intensities on intensity and energy of exciting radiation and auto absorption and secondary excitation effects. We obtain a new set of ‘‘theoretical’’ values, ðRi Þ1 , for relative intensities which are different from the experimental ones. The second concentration set is obtained by putting ðwi Þ2 ¼
R0i ðwi Þ1 : ðRi Þ1
ð3Þ
And again wi values are scaled to unity and used to calculate the next intensity set. The process is iterated until the set of wi values differ by less than some arbitrary amount, say 0.1%, between two successive calculations. The last wi set is accepted as the correct composition. We stress the fact that the application of the method of numerical solution does not depend on the eventual introduction of a geometrical correcting factor, but only on the difficulty of finding out the analytical solution. In other words, it is not typical of analyses of irregular objects since it applies to pellet analysis for instance in geological or mineralogical applications. We can consider it a lucky circumstance from which we can affirm that irregular shape of metallic objects, provided they are microscopically homogeneous, is not per se a cause of impossibility to get XRF quantitative analysis. Let us examine now in detail how the angle values enter into the equation term describing secondary excitation. It is well known [7] that secondary excitation effects can be calculated by a (triple) integration extended to the examined volume of the sample. The final formula containing both the direct and secondary contributions for a selected excitation energy E0 , is [7] liph 1 ei Pi Gwi M sin W1 l ðE0 Þ lM ðEi Þ þ sin W1 sin W2 " # n X 1þ wj dij ;
Ii ¼ I0
j¼1
ð4Þ
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
where lj ðE0 Þ Dij Pj li ðEj Þ Aij ðE0 Þ; dij ¼ li ðE0 Þ 2 8 > > < 1 lM ðE0 Þ Aij ¼ ln 1 þ M > l ðEj Þ sin W1 lM ðE0 Þ > : sin W1
9 > > =
þ
1 lM ðEi Þ ; ln 1 þ > lM ðEj Þ sin W2 > lM ðEi Þ ; sin W2
Dij ¼ 0;
if E0 6 Eiabs ;
Dij ¼ 1;
if E0 > Ejabs ;
ð5Þ
Ej > Eiabs :
By the way, we note that this integration can be obtained analytically only considering infinite integration volume [8] and this has never been put into evidence in the literature. However, for metals this means an integration over so short distances that the assumption of constant angles over the integration volume is correct. The weak dependence of the contribution to XRF intensity due to secondary excitation on the angle values is given by the lij terms. We note that for the exciting energy we always have E0 > Ei , therefore lM ðE0 Þ lM ðEi Þ and from the trivial approximation lnð1 þ xÞ ffi x, if x 1, in practice, we obtain that the first term does not depend on angle W1 . So we can assume that we have only the dependence on W2 in the form 1 lM ðEi Þ ln 1 þ ; lM ðEj Þ sin W2 lM ðEi Þ sin W2 as we have already said, it is a very weak dependence. Practically, with the condition lðE0 Þ lðEi Þ, the only dependence on geometry is given in Eq. (4) by the factor G0 ðsin W2 = sin W1 Þ. Assuming that the angle effective values, W1 and W2 , turn out to be the same in the measurements, we come back to the situation where the irregular shape is only accounted for by factor G0 . For completenessÕ sake, we consider also the rather unusual case that the portable set up has not been designed with even angles or is not pos-
687
sible to rotate its position relative to the object. Assuming once again E0 Ei then the only consequence will be that the corrective factor is G0 ðsin W2 = sin W1 Þ, being W1 6¼ W2 this time. Even if the above inequality for energies cancelling the dependence from W1 , cannot be assumed for the lower energies of the exciting spectrum, nevertheless we have that the additional dependence of intensity on angles due to the secondary effects is very weak. Experimentally it was proved [9] that, for instance, for an Au–Cu alloy disk with 30% Cu contents with 35 KV X-ray tube voltage we have a variation of about 6% in the ratio ICu =IAu for W2 going from 45° to 75°. It has been now several years since the programme QXAS-AXIL (Quantitative X-Ray Analysis System) based on the fundamental parameter method was published. The preliminary operation of this programme is a best fit procedure for background subtraction and overlapping peak deconvolution. It can directly be used for the analysis of XRF spectra of irregular metallic objects and quantitative analysis can be obtained in archaeometric applications if the non-veryrestrictive experimental conditions we have examined are fulfilled. We must take into account, however, that a very high precision is normally not required. Results from quantitative XRF analyses of gold archaic Greek coins are shown in Table 1.
3. Measurement of coin fineness In case a patina of several ten l thickness is present on metal surface, we can employ an exciting radiation conveniently more energetic than the characteristic X-ray fluorescence and consider the measure of Rayleigh to Compton scattering intensities ratio, R/C, to probe the inner material [10]. The measurement of the R/C value offers the advantage of being a method of analysis independent on the sample geometry. For quantitative calculations we might make use of theoretical scattering cross-sections of Rayleigh and Compton effects. But in the X-ray energy range coherent
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M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
Table 1 XRF quantitative analyses for archaic gold Greek coins (VI–VII B.C.) Sample 1 Au% Ag% Cu% qXRF qSG R
Sample 2
Sample 3
Sample 4
Sample 5
R
O
R
O
R
O
R
O
R
O
50.1 46.6 3.3
51.3 46.9 1.8
69.9 28.7 1.4
67.2 31.2 1.6
63 35.8 1.2
64.5 33.9 1.6
66.8 32.4 0.8
66.3 32.9 0.8
56.2 43 0.8
56.9 42.1 1
13.5 13.1 1.03
15.2 14.1 1.03
14.8 13.6 1.08
15.2 14.8 1.03
14.1 13.3 1.06
R ¼ reverse, O ¼ obverse, qXRF ¼ density (g/cm3 ) as resulting from XRF analysis, qSG ¼ density as resulting from weighing in water, R ¼ qXRF =qSG .
scattering cross-sections have been neither calculated nor systematically measured with sufficient precision. Considering, for instance, the differential Rayleigh cross-sections for 60 keV incident energy of Cu and Ag we have found a disagreement of about 10 and 6 factors respectively, respectively between the experiment [11] and the published data [12]. It is therefore convenient to resort directly to the comparison with the experimental measurements of the intensities of X-rays scattered from pure element standards. We have already shown [2] that we can assume with a good approximation a linear dependence of R/C on the relative contents. Silver contents in Cu–Ag alloy of several modern coins obtained with this method are shown in Table 2. The c line emitted by an Am241 source (10 mC) was used (Ec ¼ 59:54 keV). The Ag relative content values in the last column have been obtained from gamma absorption measurements. Note that the XRF results are not correct in some cases. Table 2 Silver content values obtained by R/C (relative error ±5%, h ¼ 131° ± 1°), XRF and c absorption for modern coins. Ec ¼ 59:54 (Am241 source) Sample
Nominal
R/C
XRF
Gamma Abs.
1 Dollar USA 1 Peso Mexico 20 Rappen 500 Lit. 5 cent. USA 10 Rappen 5 Lit. 5 Rappen
90 50 15 83.5 35 10 83.5 5
92.5 50.1 15.3 84.7 35.9 11.6 84 6.6
91 71.9 15.5 89 49 – 92 –
– 49.8 ± 1.3 16.8 ± 0.96 83.00 34.4 ± 1.8 – – –
4. Quantitative analysis of glass and ceramics As is well known, the possibility of obtaining quantitative analysis is limited for glasses and ceramics by the low efficiency in detecting low Z elements typical of energy dispersive X ray detectors which depends on the comparatively thick protective Be window. We assume a value of 24 lm (1 ml) for this thickness which can realistically be considered strong enough to offer a reasonable safety degree in the portable instruments for which the possibility of accidental breaks are more probable than for laboratory set up. The lowest limit for the Z value of the detectable element corresponding to 1 mil window results to be Z ¼ 14 (Si; ka ¼ 1:4 keV). X-ray fluorescence secondary excitation by minority heavy elements does not occur in glasses and ceramics since the matrix composition is dominantly formed by light elements. So, the simpler formula (1) applies where the correction to X-ray intensity due to matrix absorption is to be taken into account. Being the matrix composition unknown, this is the so called ‘‘dark’’ matrix problem [5]. As already noted, the Rayleigh to Compton scattering ratio – R/C – is strongly dependent on the Z value and the idea of evaluating the matrix absorption effects by measuring this value was proposed by Van Espen and He [5]. We have also already noted that this ratio is independent on any geometrical correction effect. More important: this ratio does not depend on the sample thickness and it might result of some use in case of examination of enamel layers, for instance.
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
The method of resorting to R/C measurement in order to obtain quantitative XRF analysis by energy dispersive spectroscopy is clearly explained in the above quoted paper. Basically, for a given incident X-ray energy E0 , the experimental R/C value includes the contributions coming from the (dark matrix) low Z elements and from the (detected) n high Z elements. The mathematical expression of R/C depends on the relative contents of the matrix, w0 , and of the n elements, w1 . . . wn . The same relative contents figure in the expression giving the XRF experimental intensity for each measured X-ray line. Accordingly, lM ðEÞ must be substituted in formula (1) by the sum of the corresponding contributions. Dividing formula (1) by the one corresponding to the XRF intensity IðiÞ for a pure element i we obtain Ii wi ðli ðE0 Þ þ li ðEi ÞÞ Pn : ¼ IðiÞ w0 ðl0 ðE0 Þ þ l0 ðEi ÞÞ þ i¼1 wi ðli ðE0 Þ þ li ðEi ÞÞ ð6Þ The absorption coefficient value attributed to the matrix corresponds to the matrix effective Z value. It must be the same as the one which corresponds theoretically to the contribution of the matrix to R/C value which best fits the experimental result. This is obtained by a numerical double iteration scheme from which the effective – or equivalent – Z value for the matrix of the examined sample is also obtained. We note that a subdirectory based on this method is included in the already quoted AXIL programme for glass and ceramic quantitative XRF analysis. However, we estimate simpler the method, already proposed [13], which it is based on considering that in the range of our exciting X-ray energies the massive Compton scattering coefficient can be assumed constant for all low Z elements for which Z=A ¼ 1=2 (Table 3). As a consequence, limiting ourselves to the measurement of C intensity, we have that the contribution to it deriving from the matrix light elements depends on the universal value for the massive Compton scattering coefficient lC0 ðE0 Þ, while the contributions from the other elements depend on the relevant Compton scattering coef-
689
Table 3 Massive Compton scattering coefficients of low Z elements normalized to the value for silicon Element (Z)
E0 ¼ 9:885 keV (Ge Ka )
E0 ¼ 17:476 keV (Mo Ka )
E0 ¼ 26:355 keV (Sb Ka )
C (6) O (8) Na (11) Mg (12) Al (13) S (16) K (19) Ca (20)
1.262 1.206 1.019 1.028 0.981 0.972 0.928 0.953
1.155 1.111 1.007 1.022 0.985 0.978 0.926 0.941
1.108 1.081 0.986 1.007 0.973 0.980 0.932 0.953
ficients lCi ðE0 Þ found in the literature [12] which can be considered fundamental parameters as well as the others used in the method. As regards the matrix total absorption cofficient, l0 ðE0 Þ, let us start from the Compton scattered intensity of incident radiation from the sample P lC0 ðE0 Þw0 þ ni¼1 lCi ðE0 Þwi C¼K : ð7Þ lðE0 Þ lðEC Þ þ sin w1 sin w2 Using a scaling law, for the Compton scattered photon energy EC we can write (8) n lðEC Þ EC ¼ ¼ b; ð8Þ lðE0 Þ E0 and we obtain
P lC0 ðE0 Þw0 þ ni¼1 lCi ðE0 Þwi 1 b lðE0 Þ þ sin w1 sin w2 P n lC ðE0 Þw0 þ i¼1 lCi ðE0 Þwi : ¼ K0 0 lðE0 Þ
C¼K
ð9Þ
K 0 is a constant value that can experimentally be obtained by measuring C from a silicon sample, for instance. We will comment on how to choose the correct value for n in the scaling low. Its choice is not critical in the above case since C ffi E0 . PE n Since lðE0 Þ ¼ w0 l0 ðE0 Þ þ i¼1 li ðE0 Þwi Eq. (9) becomes Pn C C l ðE Þw 0 l0 ðE0 Þw0 þ Pi¼1 i 0 i : C¼K ð10Þ l0 ðE0 Þw0 þ ni¼1 li ðE0 Þwi
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To find out l0 ðEi Þ from l0 ðE0 Þ, we write nSiO 2 E0 l0 ðE0 Þ ¼ ai ðE0 Þl0 ðE0 Þ; l0 ðEi Þ ¼ Ei ð11Þ where the nSiO2 exponent is given by the angular factor of the straight line on log scale obtained for lðEÞ from Ei to E0 considering a SiO2 matrix. Therefore, Eq. (6) can be rewritten in the following form: , Ii ¼ wi IðiÞ ðli ðE0 Þ þ li ðEi ÞÞ þ
n X
w0 ð1 þ ai ðE0 ÞÞl0 ðE0 Þ !
wi ðli ðE0 Þ þ li ðEi ÞÞ :
ð12Þ
i¼1
The only unknown parameter, l0 ðE0 Þ, can be deduced from Eq. (10) obtaining Pn wi ½K 0 lCi ðE0 Þ Cli ðE0 Þ l0 ðE0 Þ ¼ i¼1 w0 C Pn K 0 lC0 ðE0 Þ w i ci ¼ i¼1 þ þ C; ð13Þ C w0 where both the quantities K 0 lCi ðE0 Þ Cli ðE0 Þ K 0 lC0 ðE0 Þ and C ¼ C C are evaluated using K 0 , and Compton experimental intensity, C. P Taking into account the condition ni¼0 wi ¼ 1 with algebraic elaborations that we omit and
cj ¼
calling l i ¼ li ðE0 Þ þ li ðEi Þ, we finally obtain for each element i the linear equation , ! n X Ii ¼ wi IðiÞ li 1 wi Cð1 þ ai ðE0 ÞÞ i¼1
þ
n X
!
wi ½ci ð1 þ ai ðE0 ÞÞ þ
l i
ð14Þ
:
i¼1
We note that the use of a scaling law to evaluate the massive absorption coefficient of the matrix for each characteristic X-ray energy is the only price to pay to have a linear equation set which can directly be solved to obtain the relative contents of minority elements. The scaling law can properly be chosen on the basis of preliminary information on the matrix composition (glass, ceramic, earth, . . .). In Table 4 we show for example the scaling factors ai ðE0 Þ for selected E0 and Ei values for SiO2 and two standard glasses. The exponent n values which appear in the scaling laws are practically the same as SiO2 in all cases. We can consider a valid approximation to resort to the scaling law of SiO2 , whatever be the glass. At last, we consider the portable instruments based on X-ray tube for which this method must be extended to the case when a continuous spectrum is used for excitation. The only expedient in this case is that a characteristic X-ray line from the anode be evident in the X-ray tube spectrum at an energy high enough to be used for the Compton scattering measurement. In comparison with the
Table 4 Scaling factors ai ðE0 Þ for selected E0 and Ei values for SiO2 and two standard glasses Glass matrix composition SiO2 SG55 Standard SiO2 0.687, Al2 O3 0.011, Na2 O 0.102, K2 O 0.070, SO3 0.003, B2 O3 0.0022, F 0.009, CaO 0.0005, MgO 0.0001 CER1 Standard SiO2 0.645, Al2 O3 0.216, Na2 O 0.008, K2 O 0.002, CaO 0.0009, MgO 0.0004
E0 ¼ 9:885 keV (Ge Ka ) Ei ¼ 3:691 keV (Ca Ka ) n ai ðE0 Þ ¼ EE0i ¼ 16:49 ai ðE0 Þ ¼
ai ðE0 Þ ¼
n E0 Ei
n E0 Ei
E0 ¼ 14:163 keV (Sr Ka ) Ei ¼ 6:403 keV (Fe Ka ) n ai ðE0 Þ ¼ EE0i ¼ 10:09
¼ 16:76
ai ðE0 Þ ¼
¼ 16:63
ai ðE0 Þ ¼
n E0 Ei
n E0 Ei
E0 ¼ 26:355 keV (Sb Ka ) Ei ¼ 13:38 keV (Rb Ka ) n ai ðE0 Þ ¼ EE0i ¼ 6:15
¼ 10:14
ai ðE0 Þ ¼
¼ 10:14
ai ðE0 Þ ¼
n E0 Ei
n E0 Ei
¼ 6:10
¼ 6:13
M. Milazzo / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 683–692
691
Table 5 Results of XRF analyses of some glass standards supplied by Stazione Sperimentale del Vetro di Murano Element
Experimental
Certification
Ag Sr Pb Zn Cu Ni Co Fe Mn Ti Ca
239 ± 8 ppm 488 ± 15 ppm 412 ± 13 ppm 430 ± 13 ppm 429 ± 13 ppm 449 ± 14 ppm 380 ± 11 ppm 449 ± 14 ppm 479 ± 14 ppm 427 ± 13 ppm 8.32 ± 0.25%
249 ppm 508 ppm 428 ppm 438 ppm 438 ppm 458 ppm 388 ppm 458 ppm 488 ppm 438 ppm 8.54 %
Zn Fe Ba Ca K Cd Se
55,417 ± 1663 ppm 246 ± 7 ppm 23,618 ± 710 ppm 377 ± 12 ppm 56,783 ± 1704 ppm 4099 ± 123 ppm 67 ± 2 ppm
55,907 ppm 249 ppm 23951 ppm 384 ppm 57,767 ppm 4175 ppm 70 ppm
Zn Ni Co Fe Mn Ti Ba Ca K Zr Sr
14,169 ± 425 ppm 2080 ± 63 ppm 1539 ± 47 ppm 1837 ± 55 ppm 4727 ± 142 ppm 11,186 ± 340 ppm 23,730 ± 712 ppm 687 ± 21 ppm 1777 ± 54 ppm 8325 ± 250 ppm 242 ± 8 ppm
13,931 ppm 2058 ppm 1520 ppm 1868 ppm 4802 ppm 13,614 ppm 23,158 ppm 679 ppm 1752 ppm 8443 ppm 253 ppm
Zn Cu Ni Co Fe Mn Ba Ca K Sr Pb As
69 ± 11 ppm 93 ± 15 ppm 4001 ± 599 ppm 442 ± 71 ppm 239 ± 40 ppm 1850 ± 308 ppm 25,800 ± 3700 ppm 2.50 ± 0.46% 9.01 ± 1.43% 437 ± 26 ppm 6.51 ± 0.17% 1285 ± 114 ppm
– 70 ppm 4100 ppm 520 ppm 340 ppm 2050 ppm 25,000 ppm 2.85% 8.38% – 6.88% 1200 ppm
Standard NB10
Standard SG55
Standard CER1
Standard P66
monochromatic excitation case, a minor complication derives from applying the scaling law to evaluate the matrix absorption coefficients for
several exciting energies corresponding to the subdivision of the X-ray spectrum into finite interval values [7].
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Some examples of quantitative XRF analyses of standard glassed obtained with this method are shown in Table 5.
References [1] D.C. Creagh, D.A. Bradley (Eds.), Radiation in Art and Archeometry, Elsevier, 2000. [2] L. Confalonieri, R. Crippa, M. Milazzo, Appl. Radiat. Isotopes 38 (1987) 139. [3] C. Bui, M. Milazzo, M. Monichino, Nucl. Instr. and Meth. B 28 (1987) 88. [4] L. Bonizzoni, A. Galli, M. Milazzo, X-Ray Spectrom. 29 (2000) 443.
[5] F. He, P. Van Espen, Nucl. Instr. and Meth. A 299 (1990) 580. [6] J.W. Criss, L.S. Birkc, Anal. Chem. 40 (1968) 1080. [7] R.M. Rousseau, X-Ray Spectrom. 13 (1984) 115. [8] C. Bui, M. Milazzo, C. Sironi, X-Ray Spectrom. 22 (1993) 17. [9] M. Milazzo, C. Cicardi, X-Ray Spectrom. 26 (1997) 211. [10] G.E. Gigante, S. Sciuti, Int. J. Appl. Radiat. Isotopes 35 (1984) 481. [11] C. Bui, M. Milazzo Il, Nuovo Cimento 2 (1989) 655. [12] M.J. Berger, J.H. Hubble, XCOM: Photon Cross Section Database, NIST Standard Reference Database 8 (XGAM) NBSIR 87-3597. [13] L. Bonizzoni, A. Galli, M. Milazzo, X-Ray Spectrom. 31 (2002) 35.