Effects of sample transparency in powder diffraction

Transcription

1 Effects of sample transparency in powder diffraction Derk Reefman Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, the Netherlands (Received 26 June 1995; accepted 15 December 1995) The effects of sample transparency in a powder diffraction experiment are discussed, and formulas including effects from horizontal divergence are derived. It is shown that the classical formula by Alexander [L. E. Alexander J. Appl. Phys. 21, 126 (1950)] is only a limiting case for zero horizontal divergence. It is also shown that in some cases extinction effects can play a role. The impact of these effects on both the line profile as well as the integrated intensity and experiments illustrating these phenomena, are discussed International Centre for Diffraction Data. Key words: Line profile, profile analysis, absorption, transparency, extinction I. INTRODUCTION The importance of the analysis of line profiles is becoming increasingly more recognized, as the demands which are set in materials science become more severe. These include not only the classical size-strain analysis, which can often adequately be performed by a Warren-Averbach analysis (Warren, 1950, 1952), but details about the origin of the (micro)strain as well, such as defect densities, etc. Nevertheless, even when only interested in a Warren-Averbach-like analysis, the accuracy of the numbers obtained by this method depends strongly on the quality of the data, which, in turn, is, for a large part, determined by the "instrumental profile" g(26) (Klug and Alexander, 1974). This term may be a bit misleading, however, as from its definition, it comprises all effects which change the original, undistorted sample signal f(20) to the measured response h{26). Indeed, much work has been spent on devising experimental methods to obtain an instrument function (Scardi et al., 1994), where the instrument function is set equal to the observed line profile for an "ideal" (i.e., almost free of size and strain effects) sample. Although, indeed, the imperfectness of the instrument gives a large contribution to g{28), the specimen itself may contribute to the aberrations as well. Among these aberrations, texture is probably the one most intensively studied (Bunge, 1982). It certainly has the advantage of being detected relatively easily. Effects that are less easy to detect are surface roughness and sample inhomogeneity. These effects have been studied less systematically (Hermann and Ermrich, 1992; Reefman, 1995), but can be ignored in cases of low microabsorption. However, in the latter case, the large sample transparency causes serious deviations from the real sample profile f(26) as well: The profile gets asymmetrically broadened toward low angles, which causes the center of gravity to shift to lower 28 values. For that reason, the most recent attempts for the determination of g(26) have been performed on a thin layer of Si powder on a wafer of single-crystalline Si (Van Berkum, et al. 1995). Up until now, the phenomenon of sample transparency has been discussed only on the basis of heuristic arguments (Alexander, 1950). In the present article, a full derivation of the profile function g(20) due to sample transparency will be given. Moreover, the reduction of the sample transparency due to extinction in the particles making up the powder is elaborated on as well, and the implications of these effects are discussed. The resulting profile function g(20) will be compared with data of a thin A1 2 O 3 powder layer on a Si wafer, and data on bulk A1 2 O 3. II. THE MODEL In the following, a derivation will be given for the detected intensity / as a function of 26, where the incident intensity is set equal to unity. The explicit dependence of the intensity / on 29 will be dropped to avoid cumbersome notation. Then the function / is equal to g(26), but still contains information on the absolute scales because it is not normalized. A typical situation in a transparent powder is depicted in Figure 1, where both source and detector are assumed to be line-shaped (i.e., have zero thickness), and axial divergence is negligible. First, suppose that the diffractometer is set to a nominal incidence angle o>,-, and detection angle w 0 such that o>,- + a>o=20, where 26 is the actual Bragg angle of the material. It is easily verified, that in that case only a single point is in diffraction (namely, the point on the goniometer axis). All other points of the focussing circle lie above the sample surface. Now suppose that the diffractometer is set to a nominal incidence angle &>,, and detection angle w 0, such that Wj + o) 0 <26. It then is still possible to find a point of reflection P, but now at a depth q, where the incident ray is reflected in the detector. In fact, as can be verified from Figure 1, it is even possible to find a whole curve of points which are in reflection position because of the finite horizontal divergence. This is in contrast to the case where tt, + cu o =20, where only a single reflection point can be found. However, the intensity is decreased by an exponential factor e~^sl+s2 \ where /u, is the inverse of the decay length of the X rays, and s^,s 2 are defined in Figure l(b). It can be shown (see Appendix A) that the total path length s x + s 2 can be written as sin 20 [sin( G> 0 +/3) + sin(to,-+ a) ]; /3=26-(co i +a)-co 0, (1) where a and /3 are defined in Figure 1. To first order in a and ys (a,/3 < a>i,w 0 ) with R equal to the radius of the goniometer, we find for l x and l Powder Diffraction 11 (2), June /96/11 (2)/107/7/$ JCPDS-ICDD 107

2 FC (b) Figure 1. (a) Geometrical model for sample transparency used in the calculations. F is the focus, C is the goniometer axis, and S is the receiving slit. All points in the sample which are on the circle FC through F, P, and S, and are irridiated (given a certain horizontal divergence) can contribute to the intensity detected in S. (b) Enlargement of the path of the rays in the sample with the notation used in the text. Ra sin a>j+ a cos «, ' Rfi (2) sin (i) 0 + /3 cos w 0 ' Because the total detected intensity / tot is due to the contributions of all possible divergence angles a, we integrate Eq. (1) over a: I-rJ a. where the integration extends over all allowed values of a (which not necessarily equals the horizontal divergence see below). To simplify this integration, we expand Eq. (1) to first order in a, obtaining da (3) R r a sin28\s i +ac i (r-a)c 0 ), (4) where S,' = sin(2# «,-), 5,- = sin &>,, and correspondingly for the cosines and w 0 ; r=20 o>,- a> 0. Now suppose that the only scans that we are interested in are coupled 8-28 scans, centered around an incidence angle (o. We thus can write for w, and w 0 : = (26 )+S. (5) We now have that 28 is the difference between the diffractometer setting a>, + <w 0 and the real Bragg angle 20. As discussed in the context of Figure 1, this means that the detected intensity is nonzero only for negative (or zero) values of S. Furthermore, we assume that the profile originating from the optics used and the sample has a small width. Therefore, the region of S we are interested in is small and we can expand in 8, which leaves Eq. (4) as a polynomial in 8. We can now formally perform the integration in Eq. (3), where we have to distinguish the case w= 8, and arbitrary w. The calculations for arbitrary a> are more cumbersome and will be done in Appendix C. Here, we will set w= 8 and only retain terms up to second order in 8: 4fiS 2 \ (6) where S = sin8, C=cos 6>, and fi'=fj,r/sm 28. Note, that here we have dropped any indices from C and S, as now incidence and exit angle are equal. We can now expand again in 8, obtaining (to second order) j /, _ \ ^ii'(4s~ss CIS) ("j\ 1 V "2 \>" v'/ We now have to bother about the integration limits, a x and a 2. These limits are determined by the fact that the length l t + l 2 should be larger than zero in order to be meaningful since otherwise the diffraction point on the circle FC in Figure 1 lies above the sample surface. Setting /j + / 2 = 0 an d solving for a, we obtain a=-8±sl8 2-8(C 2 S+CS-CS8 2-8S). (8) Assuming that second and higher powers of 8 can be discarded (which introduces an error of approximately 5% for linewidths of 1 and quadratically less for smaller widths) we can write (9) 108 Powder Diffr., Vol. 11, No. 2, June 1996 Derk Reefman 108

3 1.00' s Figure 2. Comparison of Eq. (6) and the result by Alexander (Alexander, 1950) for a coupled 6-26 diffractometer with a nominal Bragg angle of 20=25. Long dashes: Eq. (6) with Oo=l ; /JL=\OQ cm" 1 ; medium dashes: Oo=l/4 ; /x=100 cm" 1 ; short dashes: 00=1/32 ; /u.= W0 cm" 1. The drawn line is the exponential dependence according to Eq. (11). All curves have been scaled to a maximum value of unity. Figure 3. Results of Eq. (13), for a 26 value of 25. Drawn line: a o =l ; /U=100 cm" 1 ; long dashes: OQ=1/2 \ /ii=100 cm" 1 ; medium dashes: a o =l ; /U=50 cm" 1 ; short dashes: oio=l/2 0 ; /x=50 cm" 1. The classical result is a straight line that coincides with the results of Eq. (13) for 6=0. This, in turn, leads to the following expressions for the integration limits: a x = Max( -a o,-s-\l- CSS), a 2 = Min(a 0, - 8+ yj-css), (10) where 2a 0 equals the horizontal divergence. We thus have obtained the formal result for the line profile for transparent samples. This result clearly deviates from the expression given by Alexander (1950): I*e 4fl ' s, (11) which is independent of both the horizontal divergence and 26. In Figure 2 the expression by Alexander is compared to Eq. (6). For R a value of 17 cm is used here and furtheron. Clearly, two main differences can be observed. First, for the same absorption coefficient, the width of the curves is rather different: Eq. (6) produces a result which is narrower than the the Alexander expression for small divergences, and broader than the Alexander expression for larger divergences. Second, the profile displays that at a position exactly equal to 26, the intensity is zero. This is physically correct, as for a certain ray there is only one position in the sample which reflects a ray in the detector see also Figure 1. For lower angles, the diffraction volume increases steeply while the diffraction depth is negligible. For still lower angles, the diffraction volume keeps increasing. The average diffraction depth, however, becomes larger as well, thus finally causing an almost purely exponential decrease in intensity. This is also the physical reason for the difference between the result by Alexander and the presently derived one; in the derivation of Eq. (11) it is implicitly assumed that the total reflecting area for a certain depth remains constant. In fact, this limit is reached when the limit of zero horizontal divergence is approached. Indeed, when in Eq. (10) a 0 >0, a x,a 2 >a 0. Thus, apart from a constant factor 2a 0, Eq. (11) represents the first order in S to Eq. (6). This limit is demonstrated in Figure 2, where the shape of the profile for a horizontal divergence of 1/32 hardly differs from an ordinary exponential. Note that this does not necessarily imply that the area under both curves is equal! When the material is highly transparent (e.g., graphite) the influence of the second-order term in Eq. (6) for large A26 will become still more pronounced, causing the curve to drop faster than simple exponential. A useful quantity is the total integrated intensity 7 int : Ant = (12) If we assume that the decay length arises solely from absorption effects, it can be easily inferred from Eq. (11) that the classical result is I im^l/fi, which is 26independent under the appropriate conditions (Klug and Alexander, 1974). Using Eq. (6), however, in calculating the integrated intensity (see Appendix B), a 26 dependence is present: int 4/x' cs + ^T372{r(3/2,4a O 7r')- - J^l {T(2Aa 0t i')-r[2a(a 0 +CS) f i']}. (13) In Figure 3 we compare the classical result with Eq. (13). For a o =l ; /*=100 cm" 1 and a o =l/2 ;,11=100 cm" 1 the curves behave almost classically, i.e., a divergence slit twice as large produces a 6 independent intensity which is twice as large. However, though the curve for a o =l ; H= 100 cm" 1 is horizontal on the scale of the plot, the curve for the smaller divergence slit is clearly not. This effect is even more pronounced for the set of curves for /i=50 cm" 1, 109 Powder Diffr., Vol. 11, No. 2, June 1996 Sample transparency in powder diffraction 109

4 where the 6 dependence has become drastic. Fortunately, such highly transparent samples do not occur frequently, but one should be aware of these anomalies. III. EXTINCTION In the previous section, the presence of the inverse decay length fi was not explicitly attributed to any particular physical effect though ordinarily this factor is interpreted as the microabsorption coefficient. For powder samples with crystallites of not too bad a quality, with reflectivities in the percent range, extinction effects will start to play a role as well, and will reduce the effective decay length in the sample. A simple estimation of this effect follows below. Suppose, that a single X-ray beam traverses along the path s [see Figure l(b)]. Furthermore, if the probability that a single crystallite is in reflecting position (which not necessarily means that the diffracted beam is recorded by the detector as well!) is denoted by p and the ray is completely reflected, the probability P that the beam reaches the Nth particle is given by We now have to change from the number of particles to length scales which can be done when introducing statistical averages (Reefman, 1995) for the distribution of particles in the powder. If we assume that the powder can be modeled as a collection of spheres of diameter d, and the powder is not far from close packing, we can use a result which can easily be inferred from a close packing; -1/3 _ ^ i ago _ (15) d d' Now that we have introduced statistics, we can map Eq. (14) simply to a relation for the intensity left after an X-ray beam has traversed a length s: /-(l-p) 1-39 *'"". (16) Finally, we can change the base of the exponent to e, and introduce a reflection coefficient R c as well, which measures the fraction of the intensity which is reflected by a single crystallite. By doing so, we replace the probability p that a single ray is reflected by R c p: I- pr c = ^p (17) where we also used that the probability pr c usually is much smaller than unity. It should be mentioned, however, that this statistical approach breaks down whenever the reflection coefficient R c becomes large, because the number of particles involved in that case is too small to justify some mean field procedure. Some numerical estimates indicate that R c should be smaller than 0.2 in order to have a reasonably accurate description. Now we can see that the functional form of this expression is the same as the one used for microabsorption. To get the flavour of the importance of this effect, the inverse decay length can be calculated according to Eq. (17) to obtain /i e ~30 cm" 1 for realistic values of p=0.15, R c =0.05, and d=3 /urn. This is a value which becomes non-negligible compared to ordinary absorption coefficients //., and shows that one has to be aware of this effect. It should be noted, however, that it is an entirely different phenomenon. Although its effect on the line profile is exactly the same as the effect of microabsorption, the effect on the total integrated intensity is completely different which, in fact, is the border between the kinematical approximation and the more sophisticated dynamical approach. IV. EXPERIMENT In this section, results of measurements on powdered A1 2 O 3 (corundum, JCPDS ) will be presented. The powder has been annealed at 1300 C for 12 h, to obtain high quality crystals. Longer annealing times or annealing at a higher temperature did not result in a smaller linewidth, whereas a sample annealed for only 5 h displayed a broader line. Fractions of different size distributions (average size 4.5 and 18 /am) were obtained by sieving, after which the size distribution was determined with flotation/sedimentation. The widths of the size distributions were approximately 15% of the average. This material exhibits a representative, though fairly large, transparency; the absorption coefficient (which will be denoted as fi a from now for reasons of clarity) fi a = 129 cm" 1 (compare silicon with a fi a of 140 cm" 1 ). To distinguish transparency broadening from other effects, two kinds of samples have been prepared: a bulk specimen, of virtually infinite thickness, and a thin layer of powder on a silicon single-crystalline wafer. The latter specimens have been prepared according to the recipe of Van Berkum (1995). The thickness of the layer should be such that no residual transparency broadening can be observed for these samples. The bulk specimens had a packing density n of 74 (±2)%. Because of the high transparency of A1 2 O 3, this gives an absorption coefficient fx, a of the sample of 0.74 X129 cm" 1 =96 cm" 1 (Reefman, 1995). To subdivide the transparency effects from extinction effects, samples have been prepared for a powder of 18 /an diameter and particles of 4.5 fxxa diameter, for which we expect the reflectivity to be lower. Measurements have been performed on the (012) reflection of corundum (corresponding to a 20 value of for Cu Ka^ radiation). The measurements have been carried out on a Philips X'pert diffractometer. In Figure 4 we have depicted the profiles for the bulk sample with a grain size of 18 /xm diameter, measured with a receiving slit of 0.1 mm and divergence slits of 1 and 1/2, deconvoluted with the thin layer sample measured under the same conditions. A secondary monochromator was not used since the exact influence on the opening angle of detection is unclear. Because of this deconvolution technique, we have separated the microabsorption and extinction effects from all other linebroadening effects like size broadening and instrumental artifacts. Also, strain effects can be ignored, because the hardness of corundum prevents the presence of any strain due to the different mechanical conditions for the bulk and the thin layer sample. Most clearly is the extra linebroadening introduced due to the sample transparency, and it shows that this is a nonnegligible effect for this particular sample. Also, the profiles display a different width; the profile measured with a slit of 1 has a full width at half maximum (FWHM) of 0.06, 110 Powder Diffr., Vol. 11, No. 2, June 1996 Derk Reefman 110

5 « e Figure 4. Deconvoluted profiles of the A1 2 O 3 (012) reflection (see text). Receiving slit of 0.1 mm. Drawn line: 1/2 divergence slit. Dashes: the same, but 1 divergence slit Figure 5. Logarithmic plot of the deconvoluted lineprofile (see text). Drawn line: sample A; dashes: sample B. The straight lines through the low-angle part are linear regressions. whereas the curve measured with a 1/2 divergence slit has a FWHM of This is in accordance with the ideas developed in Sec. II (see Figure 2), where for an absorption coefficient of 100 cm" 1 approximately the same values are obtained for the FWHMs. The maxima of the profiles do not coincide, but the measurement of the line position is not accurate enough to attribute this to any physical effect. In fact, from Figure 2 it can be inferred that a shift of maximum should occur, but in the other direction, as displayed in Figure 4. Moreover, also the shape of the measured profile does not map exactly on the calculated profile. This may well be due to correlations between the optics of the instrument and the transparency of the sample. In the current analysis, such correlations have been ignored. In Figure 5 we have depicted on a logarithmic scale the deconvolution of the profile of the measured bulk sample with the measured thin layer sample, both for the small particles (4.5 fim) and the larger particles (18 /im). The measurements were performed using a 0.1-mm receiving slit, a 1/2 divergence slit, and a secondary monochromator to reduce the background. As the divergence is constant now, the exact influence of the monochromator on the opening angle of detection has become unimportant. On the logarithmic scale the expected approximate exponential behavior of the line profile is clearly visible (i.e., the low angle part of the line profile is a straight line on the logarithmic plot in Figure 5). From the slope of the linear part, the effective decay length can be extracted. Note that the effective absorption coefficient /x is the sum of microabsorption and extinction effects: /J,= /j, a + /M e. For the sample consisting of the large particles (sample A, drawn line), the effective absorption coefficient obtained by linear regression of the profile from to is 120 (±7) cm" 1. For the sample consisting of the smaller particles (sample B, dashed line), however, the effective absorption coefficient is only 99 (±6) cm* 1, which thus confirms the expectation that extinction plays a role in the sample with the larger particles. Moreover, as is evident from Figure 5, the line width of sample B is increased with respect to sample A. This is not due to the finite size effect (which has been removed because of the deconvolution), but solely arises from the increased second moment of the line profile for samples with smaller /A'S. From the measured decay lengths, we can estimate the reflectivities for the two powders A and B, using Eq. (17). When using a multiplicity of 12 for the (012) reflection of corundum and a divergence slit of 1/2, the probability p is estimated to be roughly 6% (de Wolff, 1959), ignoring any effects due to texture. Using this value for p, a reflectivity of approximately 0.3 is obtained for sample A which is a fairly high value, but not unreasonable in view of the good crystal quality of the crystallites. However, one may wonder whether the statistical approach adopted in Sec. Ill still holds for these large reflectivities. For sample B, the reflectivity is reduced drastically, and amounts to approximately This is an order of magnitude smaller than for the large particles, though the diameter is only a factor of 4 smaller. This reflects the better average crystal quality of the larger crystals. It is expected, that for even smaller particles the extinction effects become unobservable. V. CONCLUSIONS A calculation of the transparency effects in powder diffraction has been given. Though the agreement with experimentally determined profiles is not perfect, the correct trends are predicted, like a divergence-dependent width of the profile. This implies that the conventional functional form (Al- 111 Powder Diffr., Vol. 11, No. 2, June 1996 Sample transparency in powder diffraction 111

6 exander, 1950) which is often used for transparency effects, and which does not exhibit these features, cannot be used for high precision measurements, which are needed, e.g., in detailed size-strain analyses. Although in the present investigations the actual need for such a profile was absent, because the transparency effects could be separated from other effects by use of a thin layer sample, this may not be the case in general as it is often impossible to prepare such a thin layer sample. In those cases the newly developed functional form can be used. It also appears that under some circumstances, the sample transparency may be effected by extinction effects. These effects are shown to be undistinguishable from absorption effects as far as the line profile is regarded, but have a different influence on the reflected intensity. Experiments show that these effects may play a role only in powder samples of high quality, and that the relative importance of this effect can be estimated with the use of the model derived in this article. Finally, the profile function g(2ff) described here incorporating different artifacts arising from the sample itself, and not from the optics, may eventually be used in conjunction with sample independent line profile analysis programs. ACKNOWLEDGMENTS The author acknowledges H. in't Veld and W. Keur for kindly providing the samples, and J.G.M. van Berkum for a critical reading of the manuscript. r 2 = I a ds[s+yl-scs+a o ]e^' s, J -a-cs -C I. ds[2a o ]e V s (22) J -a If we now assume that CS>a 0 (which is quite a natural assumption) and e~ M CS <1, we can ignore /, and 7 2. After some further cumbersome manipulation, we arrive finally at the desired result: CS {r(2,4a 0/ a)-r[2,4(a 0 +C5) M ]}. (23) Here, F(a,z) is the incomplete gamma function. For large values of /x', we can conveniently expand the incomplete gamma function as (Abramowitz and Stegun, 1972): r(a,z)~z a ~ l e~ z, (24) which makes evaluation of the expression for / int fairly straightforward. APPENDIX A: TOTAL PATH LENGTH From Figure l(b), we can write tan(«,+ a)= t 1 Solving for q we find APPENDIX C: ARBITRARY to For arbitrary a>, the expression for the total intensity becomes more cumbersome, and we have = -. (18) 1= 1,/i'(b 2 l4a-c) erf "* tan(w,+ a) + tan(«0 + /3) ' v (10) Now, making use of the relation a>j + a+w o +/3=26 and standard trigonometric relations, we find Eq. (1): sin P=20-(toi+a)-(o 0. ' (20) where -erf- c(c-cy (C-C')-S S' S C" C CS (25) APPENDIX B: TOTAL INTEGRATED INTENSITY To calculate the total integrated intensity, we have to evaluate the following integral: J -oo (S)dS. (21) According to Eq. (10) we have to split the integral into four separate parts /j,/ 2,/ 3,/ 4 : f-a -a-cs = J -=o 28 (26) where S' = sin(26-(o); 5 = sin w and correspondingly for the cosines. Abramowitz, M., and Stegun, I. A. (1972). Handbook of Mathematical Functions (Dover, New York). Alexander, L. E. (1950). "Geometrical Factors Affecting the Contours of X-Ray Spectrometer Maxima, n. Factors Causing Broadening," J. Appl. Phys. 21, 126. Bunge, H. J. (1982). Texture Analysis in Materials Science (Butterworths, London). 112 Powder Diffr., Vol. 11, No. 2, June 1996 Derk Reefman 112

7 P. M. de Wolff, Taylor, J. M., and Parrish, W. (1959). "Experimental Study of Effect of Crystallite Size Statistics on X-Ray Diffractometer Intensities," J. Appl. Phys. 30, 63. Herman, H., and Ermrich, M. (1992). "Microabsorption Corrections," Proc. Int. Conf., "Accuracy in Powder Diffraction II." Klug, H. P., and Alexander, L. E. (1974). X-Ray Diffraction Procedures (Wiley, New York), 2nd ed. Reefman, D. (1995). "Calculation of the effective absorption coefficient in multiphase powders," Textures Microstruct. (to be published). Scardi, P., Lutterotti, L., and Maistrelli, P. (1994). "Experimental determination of the instrumental broadening in the Bragg-Brentano geometry," Powder Diffr. 9, 180. J. G. M. van Berkum, Sprang, G. J. M., Keijser, Th. H. de, Delhez, R., and Sonneveld, E. J. (1995). "The optimum standard specimen for X-Ray diffraction line-profile analysis," Powder Diffr. 10, 129. Warren, B. E., and Averbach, B. L. (1950). "The Effect of Cold-Work Distortion on X-Ray Patterns," J. Appl. Phys. 21, 595. Warren, B. E., and Averbach, B. L. (1952). "The Separation of Cold-Work Distortion and Particle Size Broadening in X-Ray Patterns," J. Appl. Phys. 23, Powder Diffr., Vol. 11, No. 2, June 1996 Sample transparency in powder diffraction 113