SpectrochinriEa Acla, Vol. 39& No. a, pp. 151-m 1984. 0584-8547/84 $3.00 + .tXl Printed in Gmt BIitaio. Q 1984. Pergamm Pma Ltd.

Monte Carlo simulations of XRF intensities in non-homogeneous matrices

J. A. HELSEN and B. A. R. VREBOS

KULeuven, Department of Met~lku~e, G. de Croylaan 2, B-3030 Leuven, Belgium

(Received 28 July 1083; in revised form I October 1983)

Abstract-In this paper a quantitative explanation is formulated for many discrepancies in the analysis of non- homogeneous materials. Metal alloys are the examples by excellence of such materials. Fe-Q and Zn-Al alloys with a dispersed phase containing one of the elements only or both elements in a concentration different from the bulk, are simulated. Monte Carlo methods are used for generating the dispersions as well as for simulating the XRF processes.

1. INTRODUCTION

IN A PREVIOUS communication [l] we reported the simulation of the intensity of fluorescent X-rays produced in heterogeneous systems. These consisted of piles of parallel lamellae, orthogonal to the macroscopic irradiated plane. The aim was to demonstrate the influence of non-homogeneity on the intensity in order to prove that the correction algorithms commonly used in XRF analyses cannot be applied to such samples. They are all conceived for homogeneous solutions either solid or liquid. The magnitude of the deviation with respect to the homogeneous case, as calculated by the program NRLXRF [2], was shown as a function of concentration or lamellae thickness. A pile of parallel lamellae has been chosen because this geometry simplified the feasibility test of the program. Such a geometry is of course not a common feature of real samples. Although not entirely deprived of practical impor~n~, in this paper the simulation of a sample of a straightforward practical appearance is reported, in which hard spheres form the second phase randomly dispersed in the matrix. The simulations are applied to a series of samples for which the experimental verification of the Monte Carlo simulation is underway.

2. MONTE CARLA SIMULATION OF DISPERSIONS

In all simulations of this type of geometry, the common problem is the production of a dispersion with randomly distributed particles. After scanning the literature and with the common practice of quantitative metallography in mind, it was decided to produce the dispersions by a very simple procedure. The result certainly will deviate slightly from a “true” three-dimensional random distribution but, as will be shown by the magnitude of the searched effects and some ex~rimental evidence, the final correction with respect to more elaborated procedures is expected to be of minor importance.

Spheres with radius R are generated at random within acylinder of radius 2R, in such a way that the axis of the cylinder intersects every sphere. Overlap of spheres is not permitted. If overlap occurs, the last generated sphere is discarded and the next one is generated.

All spheres have the same radius R of unit length. The axis of the cylinder, in which they are generated, is forced to coincide with the path of the photons. That means that the average path-length of the photons through the spheres is a/2 * R [3]. In order to have a reasonable distribution, about 80-90 spheres are generated. The total pathlength through spheres then equals about 133 (= n/2* 85 *R).

The volume fraction V, of the dispersed phase B is given by the ratio of the length of path

B. VIEBOS and J. A. HELSEN, Spectrochim. Acta 38B, 835 (1983). T. W. CRISS, NRLXRF: a Fortran program for X-ray tluorescence analysis, COSMIC program No. DOD65

‘niversity of Athens, Georgia (1977). “. DEHOFF and F. N. RHINES, Quantitative Microscopy. McGraw-Hill, New York (1968).

751

752 J. A. HELSEN and B. A. R. VREBOS

segments through the phase considered and the total length (see Refs [1,3]):

and the total length of the cylinder will be:

The generated distribution is stored for use throughout a run but is regenerated when the composition is changed. The overall composition of the sample is determined by the voice fraction of the dispersed phase and the composition of the phases. The degree of dispersion at constant total concentration is changed by multiplying the unit length by the radius of the spheres expressed in pm. If a radius of 0.25 pm is accepted, the total length of the cylinder will be 1331 V, * 0.25. Even with high volume fractions ( VB > 0.30) the total length equals about 1OOpm which is greater than the critical length for most photons causing secondary fluorescence.

This approach is quite different from the one proposed by DOSTER and GARDNER [4] but we believe that any random distribution will give reliable results whatever the particular mechanism of generation might be.

3. MONTE CARLO SIMULATION OF THE FLUORESCENCE PRWESSES

The Monte Carlo program, used for the simulations, adopted several variance reduction techniques [5]. As a result the calculation time was drastically decreased, but the program length increased by a factor 10 (flow sheet in Appendix).

An incident photon might be absorbed in P’. P’ is selected according to the following general equation:

x = (-l/~)ln~Rnd) (I)

where x is the pathlength PP’ (Fig. l), p the linear absorption coefficient and Rnd a random number uniformly distributed between 0 and 1. As a consequence of variance reduction, all fluoresced lines of all elements present, are generated: the probability (P,,,) that a certain characteristic X-ray photon is generated, is calculated from the composition and the fundamental parameters (mass absorption coelkients, fluorescence yields, absorption jump

Fig. 1. Generation of the dispersion: solid line: cylinder axis, dashed line: cylinder axis after rotation to the direction of the nascent photon; P: point of incidence; P’, P” and P”’ : interaction points.

[4] J. D. DOSTER and R. P. GARDNER, X-ray Spectrom. 11, 173 (1982). [S] J. M. HAMMERSLEY and D. C. HANDSCOMB, Monte Carto Methods. Methuen, London (1967).

Monte Carlo simulations of XRF intensities 753

ratios and radiative tmnsition probabi~ties of the considered elements) [6]. Pgtn is the weight associated with the photon. Next, far each photon the probability that it leaves the sample (P,) with the given exit angle, is calculated from Lambert’s law (from P’ or P“ in Fig. 1). Muitipli~tion of Psa by P,, yields the probab~ity of detection (an ideal detector is assumed).

For the secondary fluorescence all characteristic photons are generated and taken into consideration: they follow a random direction, and may interact at some selected place. In case of a Fe-Cr-Ni alloy, the secondary photon e.g. Ni K/3, excited at P” of Fig. 1, may subsequently excite Fe or Cr (or better Fe and Cr because in the variance reduction all processes are always genemt~ with their appropriate weights). This photon excites Fe and Cr, and for each of these photons the probability of generation is deter~ned. The weight of these photons is then given by PBen (Ni I$)* P,,,(I), since Ps_, (Ni KP) is the weight of the incident photon and P,,,(I) is the probability that Fe I&, K, or Cr K, or K, are generated. Again, the Pw values are calculated for all fluoresced lines and stored. The tertiary fluorescence is treated in the same way: e.g. a Fe K, (weight: Psm (Ni K,& Ppen (Fe KB) may generate Cr radiation. In order to reduce the variance of the results, every iuc~dent photon is forced to generate the whole series of events (Primary, secondary and tertiary fluorescence).

Every incident photon starts at a randomly selected point P (Fig. 1) along the cylinder axis, mentioned above. The axis is positioned on the route followed by that photon and, subsequently a point of interaction is selected according to Eqn (1) (P’ in Fig. 1). For the calculation of Pm, which is dependent on the linear absorption coefficients of the different phases, the cylinder axis is rotated around the point of interaction as shown in Fig. 1 so that it coincides with the photon path according to the given exit angle. P, is calculated as a function of the linear absorption coefficients of the different phases along the path.

After each inter~tion, the direction of the nascent photon is chosen at random in order to simulate higher order fluorescence. The new photon continues to travel through the axis of the same originally generated cylinder but rotated around the point of interaction to the direction followed by the new photon.

Subsequently, the mean of all probab~ities is calculated for every characteristic hue and expressed relative to the similar probability calculated for the pure element under the same excitation conditions together with an estimate of the standard deviation.

Wavelengths of characteristic lines and mass absorption coefficients are obtained from MCMASTER et al. [‘7]. The fluorescent yields are calculated with the ~lynomial given by BAMBYNEK et al. [S], so that all spectral data are identical to those used by the NRLXRF program. As a test, the relative intensities for homogeneous samples are calculated by NRLXRF and compared with the results of Monte Carlo simulations (with both phases having the same composition). The results are given in Table 2 for 3 compositions of Fe-Cr-Ni ternary alloys. These include ternary fluorescence effects. The agreement may be quoted as excellent and is true over the whole concentration range as is shown in Fig. 2 by the dotted line (NRLXRF) and the black squares ~simulation~. The pure element intensity, which is used in the calculation of the relative intensities, can be calculated by the SHERMAN

Table 1. Number of photons generated for one incident photon

primary secondary tertiary

Ni K, and K, 1 0 0

Fe K, and K, 1 2 0

Cr K, and K, 1 4 4

[a] R. P. GARDNER and A. R, HAWTHORNE, X-ray Spectrom. 4,138 (1975).

[7] W. MCMASTER, M. DELGRANDE, J. MALLET and J. HUE~EL, Compilation of X-ray cross sections. Univ. California J_awrence Radiation Laboratory Report UCPL-50174 (1969).

181 W. BAMBYNEK, B. CRASEMANN, R. W. FINK, H. U. FREUND, H. M&K, C. LX Swm, R. E. PRICE and P. VWCGOPALA RAO, Rev. h&xi. Phys. 44,716 (1972).

754 J. A. HEL~EN and B. A. R. VREBOS

Table 2. Comparison of intensities calculate-d by Monte Carlo simulation with computations by NRLXRF

Composition

(wt. %) Iron Chromium Nickel Fe Cr Ni A B A B A B

33 33 33* 0.2430 0.2420 0.3628 0.3528 0.2661 0.2142 t 0.2599 0.2586 0.3679 0.3569 0.2362 0.2344

20 40 40 0.1379 0.1399 0.4185 0.4103 0.2732 0.2128 0.1510 0.1508 0.4242 0.4147 0.2976 0.2969

40 20 40 0.3403 0.3408 0.2298 0.2214 0.2604 0.2605 0.3570 0.3573 0.2356 0.2249 0.2822 0.2832

A and B stands for Monte Carlo and NRLXRF respectively. * and + mean intensity of K, and K,.

o 20.0 pm A iO.Opm x 5.0pm + 2.5pm 0 l.Opm l Homogeneous

Fe

I I I I I I I I

0.1 0.2 03 04 0.5 0.6 0.7 0.8 0.9

Concentration

Fig. 2. Relative intensity of Fe K, of a Fe-Q alloy vs weight fraction with particle radius as parameter,

eq~tion [9] as the elemental distribution is by definition homogeneous. Obviously, it can also be simulated as described for the heterogeneous samples and these values converge with the Sherman ones.

[9] J. SHERMAN, Spectrochim. Acta 17,283 (1955).

Monte Carlo simulations of XRF intensities

4. RESULTS A&D Drscussron

755

The simulations were performed on binary alloys of iron and chromium, so that these results for dispersions of spherical particles can directly be compared with those reported earlier for lamellar segregation [ 1). The photon energy is in all eases 10 keV with incident and exit angles equal to 45”. Simulations with white spectra will be done in the future. White radiation, however, is not expected to affect the magnitude of the effect. The experimental evidence, from which this work originated, was obtained using white radiation but full quantitative verification is in progress on different alloys and dispersions. A very interesting example is the system aluminium-zinc. These alloys are subject to a full characterization of their properties in this department and they exhibit a very peculiar microscopic structure. Scattered radiation is also neglected since it is of minor importance compared to the fluorescence process.

The intensities relative to the pure element for iron or chromic in chromic or iron are plotted in Figs 2 and 3 as a function of concentration with grain size as greeter. The relative intensities for homogeneous samples are calculated by NRLXRF (dotted line in Fig. 2) as well as simulate by the Monte Carlo program but with both phases having the same composition (black squares in Fig. 2). The coincidence between both approaches is remarkable. On the left hand from the centre in Figs 2 and 3 dispersions are represented with one element as matrix, on the right hand the matrix is the other element. The central part is empty for obvious reasons. For iron the influence of grain size is rather important on the whole concentration range, while for chromium the strongest effect appears below 30x,

s ~amogs~eous 6 25.0pm

Cr

I t I I I i I I I

0.1 0.2 0.3 0.4 0.5 0.6 O.? 0 6 0.9

Concentration

Fig. 3. Relative intensity of Cr I&, of a Fe-Cr alloy vs weight fraction with particle radius as parameter.

756 J. A. HELSEN and B. A. R. VREBO~

0 5 IO 15 20 25

90.0 % 80.0% 70.0 % 60.0 % 40 .o % 30.0 % 20 .o % IO ‘0 %

Fe

-Radius f&m)

Fig. 4. Relative,intensity of Fe& in Fe-0 alloy YS radius of the iron particles.

Although less pronounced than for iron, the effect remains sizable down to 10 pm. For iron in the concentration below 25 % the intensity is about twice the value for the homogeneous sample. Fu~he~ore, even for a very fine dispersion (R = lpm) the intensity of the Fe K, is about 10 y0 higher compared to the homogeneous sample. The ma~itude of the effect of the particle radius is clearly demonstrated in Fig. 4 with concentration as parameter. The partner, causing secondary fluorescence, suffers the strongest interaction with granulometry as explained in [ 11.

The dashed line in Fig. 2 gives the intensity as function of con~ntration for homogeneous samples. The correction algorithm proposed by LACHANCE and TRAILL [lo] was applied for the absorption correction of iron:

C,/R, = 1 +a*c, where C,, and Cc, are the con~ntrations of Fe and Cr respectively expressed as weight fraction and R, is the relative intensity of Fe K,.

The values of the slopes of C&./R,, vs C,, for each radius, i.e. the influence coefficient with radius as parameter, are plotted against radius in Fig. 5. The change is quite spectacular. As is clear from Fig. 3 the same effects do exist for chromic but are much smaber.

10 15 Radius (pm)

Fig. 5. Influence coefficient for Fe K, in Fe-Cr alloy vs radius of the dispersed particles.

[IO] G. R. LACHANCE and R. J. TRAILL, Can. Spectrosc. t&43 (1966).

Monte Carlo simulations of XRF intensities 757

This shows one way to deal with such heterogeneous samples: applying the classic correction algorithms but using an influence coefficient which is segregation dependent (or as described here, radius dependent).

As explained above, the dispersions are not generated 3”dimensionally. The closest neighbour sphere is easily found in our generation algorithm but far more compli~t~ 3-dimensionally. Moreover, several 3dimensional generation algorithms exist between which we were unable to make a funded choice. In order to have an ides of the inffuence of the distribution of particle size, as an extreme case a series of Fe-Cr dispersions have been compared to dispersions with all spheres on the cylinder axis at equal interdistances. Table 3 represents the results. For the extremes for iron the difference amounts up to a few percent. In the final result, however, when for example used in the Lachance-Trail1 algorithm, it will be of minor importance relative to the effect of the particle diameter.

Table 3. Comparisod of intensities for random and ordered distributions

K,: intensities relative to pure element Chromium Iron

Radius (pm) Rnd Fix Rnd Fix

1 0.7675 0.1776 0.1317 0.1264 2.5 0.7711 0.7830 0.1516 0.1486 5 0.76% 0.7728 0.1753 0.1739

Std dev. (%) 0.1 0.5

The dispersions have been made up with spheres of equal radius. We anticipate that a distribution of radii within a given range will attenuate the effects a little.

Iron~hromium alloys are of str~~htforw~d importance. Since our first co~uni- cation’[ 13, DOSTER and GARDNER [4] published an excellent paper with analogous observa- tions and ~lc~ations. Our approach is a bit different because we consider also the fluorescence contribution in and by the matrix. The discrepancy between the results for the homogeneous case and the lower limit of the non-homogeneous case we observed when using parallel lamellae[l], is apparently also present in DOSTER and GARDNER’S results. In the present results, however, a very good agreement between the calculated homogeneous data (NRLXRF) and the non-homogeneous data with both phases having the same composition is obtained (Table 2). As a matter of fact, a smooth transition from dispersion to solid

a.9 aan a

t 13 0

0 78.0 % x 22.0%

0 5 IO 15 20 2

Radius @ml

Fig. 6. Relative intensity of Al ( x ) and Zn( KI ) K, m .a. n-Al alloy vs particie radius.

758 J. A. HELSW and B. A. R. VREIKX

solution is observed when R -+ 0 (Fig. 4). This gives additional confidence in the present approach, although refinements are undoubtedly necessary and will be considered in the future on the transit to the development of a new additional term in a correction algorithm.

Although Cr-Fe alloys are very interesting, quanti~tively the influence of segregation is still more extreme in the case of zinc-aluminium alloys as exhibited by Fig. 6. The impossibility of obtaining any reliable quantitative results by XRF, forced us to look more carefully to these alloys. The reasons for the spectroscopical discrepancies are quite obvious now. It is remarkable that the influence of segregation is visible down to the very small radii.

To conclude we may simply state that the difficulties encountered in the XRF analytical practice for some alloys are well understood and quantitatively explained in terms of segregation. These effects seem to have been neglected in the last decade of spectral practice. The development of a suitable correction pr;ocedure will be the next and final aim of this study.

Acknowledgment-I.W.O.N.L., the institute for stimulation of scientific research in industry and agronomy, is gratefully acknowl~g~ by B. VREBOS for granting him a research fellowship.

APPENDIX

Flow sheet of the computer pro~ru~ The algorithm we used in making tracks for photons through a dispersion (matrix

-I- randomly distributed particles) without generating a “true” dispersion has the advantage to reduce considerably the computation time and the number of coordinates necessary to locate and characterize a site of interaction.

The mechanism of selection is already described in Section 2. Physically, two arrays are stored in memory during the computations. One contains the coordinate of the projection of the centre of the spheres on the axis of the cylinder with radius 2R, while the other is used to store the chord length through each of the spheres.

Two coordinates are necessary: one is the absolute depth in the sample (necessary for the computation of P,,), the other is a coordinate on the cylinder axis. For ease of computation as third “coordinate” a serial number for each sphere is used. The distance from any point to the next sphere can easily be calculated from the coordinates on the cylinder axis. It is not necessary to consider all spheres in order to find the closest neighbour.

The direction is determined, only by the direction cosine with respect to the z-axis, perpendicular to the macro~opic surface of the sample, since there is no need to consider the x and r location.

The depth of a new point can be calculated from the former interaction point, the distance travelled and the direction cosine.

The flow sheet of the Monte Carlo program is given in Fig. Al. The INPUT module handles the composition and the volume fraction of the phases, the

energy(-spectrum) of the incident photons, incidence and exit angles and the two arrays just referred to.

Next, the fundamental parameters for each element and the photon energy (incident and fluorescent) are read from external data-files on disks and stored in core memory.

The simulation of the history for each incident photon starts with the selection of a random coordinate on the cylinder axis (P in Fig. 1). Then, the distance over which the photon travels without interaction is determined, according to Eqn (1). If a phase boundary is encountered, a new travel distance is determined in the same way as before but using the p-value appropriate to that particular phase.

At the interaction point (P’ in Fig. I), for all elements present al1 lines are generated with their respective probabilities (P en) For each of these photons, two effects are considered: firstly, the probability that they 6. ridge over the distance to the sample surface without being absorbed (P&J is calculated. Secondly, a random direction is chosen and a new point of interaction is determined (P” in Fig. l), for the calculation of the secondary fluorescence,

Tertiary fluorescence is treated in the same way as indicated in Fig. Al. The function of the OUTPUT module is quite clear: it merely computes the arithmetic

mean of the probabilities and estimates the standard deviation.

Monte Carlo situations of XRF intensities 759

c START ‘1

INPUT geometry and composition fundamental parameters

OUTPUT calculate means of probabilities and standard deviation

STOP

I Select position of interaction

t I

ALL ELEMENTS

calculate P,, and Pest

NEXT ELEMENT >

NEXT PHOTON

Fig. 1A.

The standard deviation is calculated, using the well known equation

where the sums cPi and CPF are calculated, using double precision, to avoid truncation errors when the difference is calculated. The n in the denominator is the number of photons (with the energy considered) which escape the sample. This is sound statistics, since the final result of Monte Carlo simulation is the calculation of an average probability. Furthermore, for simulations with a large number of photons, the relative standard deviation, calculated from the above mentioned formulae by dividing s by the average probability, is in good accordance with the (in XRF practice) more familiar expression, using the square root of the intensity.