DEVELOPMENT OF A METHODOLOGY FOR ANALYSIS OF THE CRYSTALLINE QUALITY OF SINGLE CRYSTALS

Transcription

1 DEVELOPMENT OF A METHODOLOGY FOR ANALYSIS OF THE CRYSTALLINE QUALITY OF SINGLE CRYSTALS S. METAIRON, C.B.R. PARENTE, V.L. MAZZOCCHI Instituto de Pesquisas Energéticas e Nucleares (IPEN-CNEN/SP), São Paulo, SP T.J. LEMAIRE Universidade Estadual de Feira de Santana (UEFS), Feira de Santana, BA Brazil Abstract This work aims to establish a methodology for the analysis of the crystalline quality of single crystals. It shows that from neutron diffraction tridimensional rocking curves it is possible to determine the intrinsic full widths at half maximum (FWHM) of the crystalline domains of a crystal, as well as the relative intensities of such domains and the angular distances between them. For the development of the method, a tridimensional I x ω x χ rocking curve has been obtained with neutrons from a mosaic aluminum crystal. The intensity I was measured as rocking curves by turning the crystal around the ω-axis of a goniometer, one curve for each angular position χ obtained by step-scanning this angle in a convenient interval. The set of individual bidimensional I x ω rocking curves formed the tridimensional I x ω x χ rocking curve for the aluminum crystal. The tridimensional rocking curve was fitted by Gaussians and deconvolved from the instrumental broadening in both directions ω and χ. The instrumental broadenings were obtained with a perfect lithium fluoride (LiF) crystal from rocking curves measured around the ω and χ-axes. Owing to an enhanced Lorentz factor in direction χ, which excessively enlarged the rocking curves in that direction, the χ scale had to be shrunk by a correction factor. The shrinkage turned the FWHM of domains in such direction equivalent to those found in direction ω. The construction of a contour map with individualized domains, on the basis of the tridimensional rocking curve of the aluminum crystal, made it easier to determine the characteristics of each domain. This contour map showed five domains. They were characterized in relation to the FWHM, relative intensity and angular distance between them. 1. Introduction Recently, a study of the crystalline quality of Czochralski grown barium lithium fluoride (BaLiF 3 ) single crystals has been published [1]. In the study, certain characteristics of intensity curves (rocking curves), obtained by neutron diffraction, are correlated to several parameters involved in the growth method. As pointed out by authors, neutrons are better than x-rays for this purpose since they give information about the bulk of the crystal. A neutron beam has, in general, a large cross-sectional area and a sample can be completely immersed in it. Since neutrons are high-penetrating particles for most materials, it is possible to observe simultaneously all crystalline domains of a sample. An x-ray beam, on the other hand, is infinitely less penetrating than a neutron beam and has, in general, a small cross-sectional area. In this case, if the crystal is large, the x-ray diffraction gives information about a very small portion of the crystal [1]. Unless the beam impinges upon a domain boundary, a single domain is observed in an x-ray rocking curve. Although the methodology employed in the above study led to useful results, it is not suitable for a complete characterization of the crystalline domains of a sample. This present work establishes a methodology for the analysis of the crystalline quality of single crystals. It uses tridimensional rocking curves formed by several bidimensional ones, measured by neutron diffraction. A bidimensional I x ω rocking curve is obtained by turning the crystal around the ω-axis of an Eulerian cradle [2] while the intensity I of an hkl reflection is measured. A tridimensional I x ω x χ rocking curve is formed by a set of I x ω rocking curves measured at different values of χ, such values obtained by stepping this angle in a convenient interval. Figure 1 is a schematic representation of the rotations used in the obtainment of a tridimensional rocking curve. 1

2 Feixe Incidente Incident beam Diffracted beam c q w q Crystal FIG. 1. Schematic representation of the rotations ω and χ used in the obtainment of a tridimensional rocking curve. A bidimensional rocking curve shows, in general, a few crystalline domains. Such domains appear as overlapping peaks compounding the curve. Except for those cases where the domains are closely aligned in the direction of the measurement, only a tridimensional rocking curve can show all domains of a crystal. They appear located at different positions in the ω.χ grid. In principle, it is possible to determine the number of domains, their relative intensities, mean breadth of each one and angular dispersion between them. From a macroscopic point of view, determination of these characteristics corresponds to an evaluation of the crystalline quality of the crystal under study. In this work, the evaluation of the crystalline quality is made by determining the intrinsic full width at half maximum (FWHM) of each crystalline domain, the angular distances and the relative intensities between them. 2. Experimental In order to obtain the tridimensional rocking curve of the aluminum crystal, an experimental arrangement was assembled in the IPEN neutron diffractometer installed at the IEA-R1 2 MW research reactor. In this arrangement, the monochromatic-beam Soller collimator was substituted by a special collimator which limits both vertical and horizontal divergences to a few minutes of arc. It is the same collimator used to perform neutron multiple diffraction measurements [3, 4]. For this work, vertical and horizontal plates were removed. Removal of plates was done in an alternate way. The number of plates was reduced from 15 to 7 increasing, consequently, vertical and horizontal divergences from circa 10 to 20. The arrangement also included a five-circle goniometer normally used in the diffractometer. This goniometer resembles an Eulerian cradle except that it has an extra axis (Σ) appropriate to be used in neutron multiple diffraction measurements [3]. The measurements were carried out with a mosaic aluminum (single) crystal, shaped as a square plate 8 x 8 x 1.5 cm 3 and oriented with (111) planes parallel to the 8 x 8 cm 2 face. The crystal was mounted on a goniometer-head attached to the ϕ-axis of the five-circle goniometer. A careful orientation of the crystal in the neutron beam, in order to obtain the maximum intensity of reflection 111 from the 8 x 8 cm 2 face, preceded the measurements of the bidimensional I x ω rocking curves. Such measurements were done scanning ω in steps of 0.05, 2 min of counting time each step. To obtain the tridimensional rocking curve, the angle χ was varied in steps of 0.5. The scanning widths, in both directions ω and χ, were made large enough to extend the background, on both sides of the curves, for a few degrees. Counting time, steps and scanning widths, used in the measurements, were chosen to assure that the curves had the quality needed for the application of fitting and deconvolution processes. To determine the instrumental broadenings, several crystals were analyzed. A typical analysis consisted of the measurement of single rocking curves, in both directions ω and χ, followed by a fitting of Gaussians in order to characterize the crystalline domains. Such analyses showed that most of the crystals were mosaic with several crystalline domains. Only a few had characteristics of a perfect crystal. Amongst them a lithium fluoride (LiF) single crystal was chosen, albeit not without some difficulties. This LiF crystal was grown by the Czochralski method [5]. It is a crystal boule ca. 5.1 cm long with a square basis 2.8 x 2.8 cm 2 and oriented with the 100 direction along the axis of the 2

3 boule. Its basis has rounded vertices. A brief description of the analysis done for the LiF crystal is presented below. With the crystal positioned in the maximal intensity of reflection 200, single rocking curves were measured in directions ω and χ. After the rocking curve in direction ω was measured the crystal was rotated 90, by means of the ϕ-axis of the goniometer, before the measurement of the rocking curve in direction χ. This was done to assure that approximately the same region of the crystal was used to obtain the instrumental broadenings. A careful fitting of Gaussians showed that the crystal had several domains. Despite the number of domains in direction χ was greater than that of direction ω, indicating that the regions were actually different, the crystal was used for the determination of the instrumental broadenings. This decision was based on the fact that both rocking curves showed domains with small FWHM values characterizing a perfect crystal. Although the LiF crystal was assumed to be a perfect crystal, presence of some slight mosaicity was not entirely discarded. Table I lists the results found in the fits. They correspond to the parameters associated to each one of the fitted Gaussians, namely its position, maximal intensity and FWHM (β in the Table). The values for the standard deviation η are also listed. They are related to β by the well-known formula η = β/2(2ln2) 1/2. The smallest value of β was chosen as the instrumental broadening, in each direction. They appear in italics in Table I. TABLE I. RESULTS FOUND IN THE FITS OF THE ROCKING CURVES MEASURED WITH THE LIF CRYSTAL DIRECTIO N POSITION MAX. INTENSITY β η ω 1 = h 1 = 218 β 1 = η 1 = ω w 2 = h 2 = 2941 b 2 = h 2 = ω 3 = h 3 = 646 β 3 = η 3 = χ 1 = h 1 = 1537 β 1 = η 1 = χ 2 = h 2 = 543 β 2 = η 2 = χ c 3 = h 3 = 1054 b 3 = h 3 = χ 4 = h 4 = 3359 β 4 = η 4 = χ 5 = h 5 = 1563 β 5 = η 5 = χ 6 = h 6 = 1423 β 6 = η 6 = DATA TREATMENT The purpose of the data treatment was to obtain the intrinsic crystalline domains of the aluminum crystal. The procedure adopted to attain this purpose was first to fit Gaussians to the experimental bidimensional rocking curves. To obtain better results in the fits, an interpolation [6] of extra points between the experimental points was performed for both directions ω and χ. After the fits, the instrumental broadening was deconvolved from each fitted Gaussian. Both processes were first applied to the rocking curves obtained for direction ω. They were then applied to the rocking curves for direction χ, these latter ones extracted from the set of former ones Fitting of Gaussians The fits were performed by means of a computer program which adjusts Gaussians to an experimental curve, using a least-squares method in the fitting process. It is not worthwhile to give here a more detailed description of this program. It belongs to a well-known class of computer programs which makes the fitting of adequate mathematical functions to curves given by points. The program gives as output all three parameters of each fitted Gaussian, namely position, maximal 3

4 intensity and FWHM. It also gives the area under each Gaussian and the resulting fitted curve by its points. Parameters and areas are accompanied by their corresponding standard deviations. An overall standard deviation for the fitting itself is also given. A serious drawback of these sort of programs is that, in certain cases, the number of Gaussians to be specified for the fit becomes undefined. This happens when the curve to be fitted has several overlapping peaks with a poor separation between them. It is the case of this work. During the fits of the numerous rocking curves it was verified that, after a certain number of Gaussians, the agreement between experimental and fitted curves experienced no substantial improvement. When the number of Gaussians becomes too large, the fit becomes erratic although without spoiling the agreement. A plot of the overall standard deviation σ versus number n of Gaussians, obtained by fitting 1,2,3,...,n Gaussians to a same rocking curve, shows how the agreement behaves with the increasing of n. Figures 2 a., b., and c. were obtained by fitting, successively, up to 5 Gaussians to the rocking curves I x ω measured at χ = 89.5, 90, 90.5, respectively. It is remarkable how σ changes by ca. two orders of magnitude when n passes from 1 to 3. From 3 onwards there is no change in the order of magnitude. In this case, a decision about the number of Gaussians to be adopted, 3, 4 or even 5, still depends on a qualitative evaluation of the fit. In this evaluation it is taken into account how a Gaussian is altered when passing from a rocking curve to a neighbor one. It should be noted that a tridimensional crystalline domain appears as Gaussians in those bidimensional rocking curves which pass it through. For a nondeformed domain the parameters of such Gaussians, in particular the maximal intensity and FWHM, change monotonically along by both sides of its maximum. To be ascertain that the results of the fitting process are reliable, the successive fits must be checked continuously in order to verify if the domains follow the above behavior. It should be mentioned that the procedure for the determination of n must be applied again if domains appear or disappear in the sequence of rocking curves. Only as an illustration, in the case of Fig. 2 three Gaussians were chosen to initiate the fits a. Standard deviation (σ) b c Number of Gaussians (n) FIG. 2. Overall standard deviation (σ) vs. number of Gaussians (n) obtained by fitting, successively, up to 5 Gaussians to the rocking curves I x ω measured at χ = 89.5 (a.), 90 (b.) and 90.5 (c.). 4

5 3.2. Deconvolution of Gaussians In general, deconvolution of an experimental curve is a process with several drawbacks. Its application renders good results only in special cases and after a careful data treatment [6]. In this work the deconvolution involves only Gaussians, i.e. a Gaussian is deconvolved from another Gaussian, both represented by their parameters. It should be noted that the resulting deconvolved curve is also a Gaussian. For this case the deconvolution becomes a straightforward process, without any additional difficulty. In a general case, where the n Gaussians fitted to a rocking curve are consecutively deconvolved, the intensity of the deconvolved rocking curve I d (θ) is given by [7]: I n 2 d ( θ) = Ad,i exp[ αd,i( θ θd,i) ] i= 1 (1) The unknown parameters are given by the following expressions: d,i = 1 α exp,i 1 1 α α K A θ d,i d,i = A = θ exp,i exp,i 2 2 ( α ) 1/ ( ) 1/ d,i + αk / π A K In the above expressions α exp,i and α K are calculated by α = 1/2η 2 where η assumes the value of the standard deviation of the ith fitted Gaussian (η exp,i ) and of the instrumental broadening in the direction of deconvolution (η K ), respectively; A exp,i is the maximal intensity of such a Gaussian and θ exp,i its angular position; A K is the maximal intensity of the instrumental broadening. This last parameter is calculated by A K = 1/η K.(2π) 1/2 and it is such that the area under the Gaussian representing the instrumental broadening becomes equal to the unity, i.e. + A K 2 exp( α θ )dθ = 1 K A computer program has been written using above formulas. As implicit in formula (1), all fitted Gaussians of a bidimensional rocking curve can be deconvolved in a single run of the program. Nevertheless, in the run, the deconvolution process is applied individually to each Gaussian according to the sequence they were listed in the input data. The program gives as output the data of the deconvolved curve as well as the parameters of each deconvolved Gaussian. It was first applied to the bidimensional I x ω rocking curves obtained for the aluminum crystal, after the fits of Gaussians. The instrumental broadening in direction ω was used in this case. Bidimensional I x χ rocking curves were then extracted from the set formed by the deconvolved I x ω rocking curves, one for each value of ω in its interval of variation. In a similar manner, by using the instrumental broadening in direction χ, a set of deconvolved I x χ rocking curves was obtained. It should be noted that each interpolated point generated a new rocking curve in the transversal direction. This fact, which is valid for both directions ω and χ, caused a substantial increase in the number of rocking curves to be fitted and deconvolved. Both sets of deconvolved bidimensional rocking curves together formed the deconvolved tridimensional I x ω x χ rocking curve for the aluminum crystal. Figure 3a. shows such a rocking curve constructed with the raw experimental data. In Fig. 3b., the rocking curve was reconstructed after application of the data treatment described above. Interpolation of points is clearly noticed by comparing both figures. 5

6 3.3 Contour maps A tridimensional plot, like that of Fig. 3b., does not permit a through visualization of the rocking curve. Unless the plot is rotated around a vertical axis and several plots are shown, some details of the curve are always missing. For a better visualization of a tridimensional rocking curve, a contour map can be constructed with the same data used in a tridimensional plot a b Intensity Intensity χ (degrees) ω (degrees) χ (degrees) ω (degrees) FIG. 3. Tridimensional I x ω x χ rocking curves for the aluminum crystal constructed with the raw experimental data (a.) and after interpolation and deconvolution (b.) Shrinking the c scale During the determination of the instrumental broadenings it was observed that the breadths of the rocking curves, measured in directions ω and χ, were very different. In direction χ the breadth was circa 16 times that of direction ω. Such a large difference could not be attributed to intrinsic characteristics of the crystal. In fact, as mentioned before, the crystal was turned 90 o for the measurement in direction χ. This assured that essentially a same crystalline region was observed in the measurements. As a matter of fact, the enlargement of the rocking curve in direction χ is due to an enhancement of the Lorentz factor [2,7] in such direction. This factor depends on the inclination of the trajectory, described by the reciprocal-lattice point during a rocking of the crystal, with respect to the surface of the Ewald sphere. The more inclined the trajectory, the more the rocking curve is enlarged. In a rocking ω, the trajectory is almost perpendicular to the sphere. In this case, a small enlargement occurs. In a rocking χ, on the other hand, the reciprocal lattice point approximates to the surface of the sphere in a grazing incidence. In this case, a very enlarged rocking curve results [7,8]. In order to reduce the enlargement in direction χ, a correction factor was determined. This factor provokes a shrinkage of the χ scale such that the breadths of the bidimensional domains in one direction become of the same order of magnitude as those in the other direction. The correction factor was calculated by dividing the FWHM of a rocking curve measured in direction ω by the FWHM of a rocking curve measured in direction χ. These rocking curves were measured in a same crystalline direction in the aluminum crystal, according to the procedure adopted in the determination of the instrumental broadenings. The value for this correction factor resulted f corr = The shrinkage of the χ scale is done by calculating the correct χ using the expression χ corr = (χ - χ max ).f corr. Figure 4 is the contour map constructed for the aluminum crystal. It was plotted in a shrunken χ scale. Clearly, the ω scale continues to be the same as those of Figs 3a. and 3b. 6

7 90,4 χ (degrees) 90,2 90,0 89,8 89,6 13,2 13,4 13,6 13,8 14,0 14,2 14,4 14,6 ω (degrees) FIG. 4. Contour map constructed for the aluminum crystal after deconvolution in both directions ω and χ. The χ scale was shrunk by using χ max = 90 and a correction factor fcorr = Individualization of domains The contour map of Fig. 4 contains all essential information to allow characterization of the aluminum crystal, concerning its crystalline quality. Nevertheless, it is not a useful source of information when the individual characterization of the crystalline domains and the relationships between them are sought. To reach this objective, a contour map was constructed with tridimensional data of each domain. These data were collected from the outputs of the computer program used for the deconvolution of the bidimensional rocking curves. As mentioned before, the output includes the parameters of each deconvolved Gaussian. Following a same domain through a sequence of bidimensional rocking curves and using its parameters, given for each curve, it became possible to construct a contour map for the domain. It is worth mentioning that the direction of the rocking curves used in the procedure is irrelevant, provided a same direction is used for the entire contour map. The procedure was extended to all rocking curves of the crystal, such that a contour map including all domains could be constructed. Such a contour map is shown in Fig. 5. It gathers the five tridimensional domains found in the analysis of the aluminum crystal. FIG. 5. Contour map for the aluminum crystal constructed with individualized domains. 7

8 4. Results Using the individualized domains of Fig. 5, angular distances between them, their maximal intensities and FWHM could be determined. Below are presented the procedures used and results found in the determinations FWHM of domains In this work, it was assumed as representatives of the FWHM of a tridimensional crystalline domain those values determined for directions ω and χ. These values are simply determined by first tracing orthogonal straight lines intersecting at the point of maximal intensity of the domain. The lines are traced parallel to axes ω and χ. When the lines encounter the contour line corresponding to half of the maximal intensity, the FWHM in both directions ω and χ are determined. Figure 6 shows domain no. 3 of Fig. 5. At right side of Fig. 6, the values in the scale for the contour lines corresponding to the maximal intensity (35787) and half the maximal intensity (17883) were used in the determination. With these values and following the procedure above described the FWHM for both directions were determined. They are indicated in the Figure by β ω and β χ and their values are listed in Table II, together with the values for the other domains. The corresponding values for η ω and η χ are also listed. 90,3 90,2 χ (degrees) 90,1 90,0 89,9 89,8 b c ,7 b w 13,6 13,7 13,8 13,9 ω (degrees) FIG. 6. Determination of the FWHM of domain no. 3 in Fig. 5, in both directions ω and χ (βω and βχ, respectively). TABLE II. FWHM AND STANDARD DEVIATION FOR BOTH DIRECTIONS ω AND χ, DETERMINED FOR THE FIVE CRYSTALLINE DOMAINS OF THE ALUMINUM CRYSTAL Domain β ω β χ η ω η χ o o o 0,053 o o o o o o o o o o o o o o o o o 8

9 4.2. Angular distances between domains In a spherical triangle, whose sides are the arcs u, v, w and opposite angles U, V, W, respectively, arc w can be related to the others by the cosine law [7]: cosw = cosu.cosv + sinu.sinv.cosw (2) Using expression (2) to calculate the angular distance between two domains of Fig. 5, it becomes: cos = cos ω.cos χ + sin ω.sin χ.cosw (3) where ω and χ are the ω- and χ-components of. It should be understood that, to find these angular distances in the contour map, the points of maximal intensity of the domains must be localized in the ω.χ grid. Since ω- and χ-axes form an orthogonal system, cosw = 0 and expression (3) is reduced to: cos = cos ω.cos χ (4) Using expression (4), the different distances between the five domains of the aluminum crystal were calculated. The distances for the 10 possible combinations, as well as the corresponding ω and χ, are listed in Table III. TABLE III. ANGULAR DISTANCES ( ) BETWEEN THE FIVE DOMAINS OF THE ALUMINUM CRYSTAL Combination ω χ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 4.3. Relative intensities The intensity of a domain relative to another one was calculated by using the areas of the bidimensional deconvolved Gaussians which pass through the points of maximal intensity of both domains. Since the tridimensional domains exhibit a certain asymmetry, the two Gaussians oriented in directions ω and χ were considered for each domain. The areas were calculated according to the formula A = I max.β, where the maximal intensity (I max ) and the FWHM (β) were given in the output of the program used in the deconvolutions. Finally, the arithmetic mean A m of the two areas was calculated and used in the determination of the relative intensity I r between two domains. Table IV lists, for each domain, the areas A ω and A χ obtained for directions ω and χ, respectively, the mean area A m and the relative intensities I r between domains. 9

10 TABLE IV. RELATIVE INTENSITIES I r BETWEEN THE DOMAINS OF THE ALUMINUM CRYSTAL Domain A ω A χ A m I r (%) ACKNOWLEDGEMENTS The authors are indebted to Dr. Sonia L. Baldochi and MSc. Izilda M. Ranieri, from the Crystal Growth Laboratory at IPEN-CNEN/SP, for providing most of the crystals used in this study. They also thank the financial support given by the International Atomic Energy Agency (IAEA), under project no. 6974/R1/RB, and the Conselho Nacional de Desenvolvimento Cientíco e Tecnológico (CNPq), under contract no /93-5-FA. One of the authors (SM) acknowledges the grant of a fellowship by CNPq, process no /94-1. This fellowship permitted her to obtain a Master of Science degree in the area Applications of Nuclear Technology. REFERENCES [1] BALDOCHI, S.L.; MAZZOCCHI, V.L., PARENTE, C.B.R.; MORATO, S.P., Study of the Crystalline Quality of Czochralski Grown Barium Lithium Fluoride Single Crystals, Mat. Res. Bull. 29, 12 (1994) [2] ARNDT, U.W.; WILLIS, B.T.M., Single Crystal Diffractometry Cambridge at the University Press (1966). [3] MAZZOCCHI, V.L.; PARENTE, C.B.R., Study of β-quartz by Neutron Multiple Diffraction, J. Appl. Cryst. 27 (1994) [4] MAZZOCCHI, V.L.; PARENTE, C.B.R., Refinement of the Ferri- and Paramagnetic Phases of Magnetite from Neutron Multiple Diffraction Data, J. Appl. Cryst. 31 (1998) [5] HAGENMULLER, P. (Ed.), Preparative Methods in Solid State Chemistry New York and London, Academic Press (1972). [6] PRESS, W.H.; TEUKOLSKY, S.A.; VETTERLING, W.T.; FLANNERY, B.P., Numerical Recipes in Fortran: the art of scientific computing. 2nd ed. Cambridge: Cambridge University Press (1994). [7] METAIRON, S., Desenvolvimento de uma metodologia de análise da qualidade cristalina de monocristais. MSc Thesis, Universidade de São Paulo, São Paulo (1999). [8] ROSSMANITH, E., Comparison of an Umweganregung Pattern, Measured on an Automatic Single-Crystal Diffractometer, With Calculated Patterns, Acta Cryst. A42 (1986)