Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol ISSN

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1 Copyright(c)JCPDS-nternational Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol Abstract ACCURACY OF XRPD MEASUREWENT VA DFFUCTON NSTRUMENTAL MONTORNG Giovanni Berti Dipartimento di Scienze della Terra, Via S. Maria 53,56126 Pisa, taly. The.whole measurement process includes data collection, data processing and analysis and other procedures accounting for criteria useful to the evaluation of the experiemental observation and results. Reproducibility, repeatability and uncertainty are useful for evaluating the experimental observation. Precision and accuracy play a role in the results achievement from XRPD data collection. n order to gather accuracy and precision a set of operations shall be setled which are able to award physical meaning to the experimental observations. Accuracy is considered with more details because it takes into account the degree to which any given observation diverges from the true value of the quantity being observed. An example is given for the typical determination of lattice parameter. 1. ntroduction The interaction of an x-ray beam with a crystalline material substance, gives rise to a difiaction beam whose physical characteristics differ from those of the incident ray. The difyi-action is observed as sequences of intensity maximums in appropriate geometric positions depending on lattice features. The intensity distribution of each peak depends, in turn, on the shape factor of each reciprocal lattice node, which is controlled by crystallo-chemical and physical properties of the material and the lattice symmetries. The instrumentation also contributes to peak profile. n order to recognise the various effects in the diffraction signal, a set of operations shall be settled that impart physical significance to the empirical observations. n this sense the diffraction measurements requires procedure which involve sample, specimens instrument, calibration, calibration monitoring, data collection strategy, data processing methods and measurement ratings. Figure 1 reports a schematic representation of the relationships of the various aspects of a diffraction measurement for various types of analysis and field of applications [ 111. Giving physical significance to these observed entities implies defining the uncertainty and the reproducibility of the observation, the accuracy and precision of the results. n reality, reproducibility should not be expected from comparison of diffraction patterns of the raw data from distinct collections. Among the causes affecting reproducibility, the following may be enlisted: variation among laboratory and equipments, procedures for instrument calibration and calibration monitoring, environmental conditions, When the variability of these factors is maintained as small as possible (or constant, i.e. repeatability conditions are well performed) [ 121, th e variability of the observations determine the minimum uncertainty (i.e. uncertainty related to the solely random causes). This extreme condition is never reached during an interlaboratory test (Round Robin); this implies the precision of the result in an interlaboratory test is an attribute of the measurement process, not of the simple observation. Figure 2 shows the schematic concept relating uncertainty and precision when dealing with observation and measurements results.

2 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the nternational Centre for Diffraction Data (CDD). This document is provided by CDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by CDD. Usage is restricted for the purposes of education and scientific research. DXC Website CDD Website -

3 Copyright(c)JCPDS-nternational Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol The consequence is that the knowledge of the equipment functioning, specimens, calibration etc. are the proper matter for the data reduction processes. This data reduction process aims at eliminating most of the undesired effects and P ZOCEDURES RELATED TO R EFERENCE MATERALS AND Sl?ECM.EN NSPECTON SPECMEN PREPARATON 2. Working STD and Measurement Rating 3. Working Specimen PROCEDURES RELATED TO NSTRUMENT CALBRATON AND MONTORNG CALlBRATON & MONTORNG OPTCAL CALBRATON & MANTANANCE PROCEDURES OF POWDER PATTERN NTEWRF,TATON L - DATA COLLECTON STRATEGY NDCATORS OT7 OTrAT,TTy DATAPROCESSNG METEODS & EVALUATON CRTERA EVALUATON METHODS: CRlTERL4 R Bragg Peak search and heigh RV +) Pattern decomposition e x2 Profile analysis Mn and Fn Figures Rietveld of merit Data Reduction ---_ _-_- APPLCATON 0 level of interpretation nterpretation at several levels of f sophistication depending on the field of application Figure 1: Scheme of relationships relating the various aspects of the x-ray dib%action measurements. translate the experimental observation uncertainty into the result s precision of measurements. The terms of accuracy acquires special significance in relation to the measurement results if the following precepts are accepted [7].: a) the true value of a physical quantity is unknown; b) the accuracy indicates the correctness of the results (i.e. the separation of the result Corn the true value ); c) the precision indicates the interval where the probability to find the true value is high.

4 Copyright(c)JCPDS-nternational Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol A. STAGE OF DATA COLLECTON senaration of the observed from the true value [ uncertaintv] true qalue observation \ \, \ 1 1 \ \\\ B. STAGE OF DATA REDUC&N 1 a. \ \ / \ Reduced separation \ \\ [ 1, &qcision] [ uncertaiq,.] i-7 -jr, F, best estimated observation value b. Reduced separation : precision 1 [ uncertainty ] 1 [ 11 best estimated observation value Figure 2: Schematic representation between the observation uncertainty, accuracy and precision in a physical experiment. n the second case (b), the precision on the best estimed value is so low that the separation between observed and best estimed is compatible with zero. The significance of the last two statements, which refers to the absolute lack of knowledge set out in the first, makes ambiguous the measurement concept. However, this ambiguity can be avoided by simulating the physical phenomenon as well as the experiment and the operations necessary for the observation (equipment, environment condition, etc.). This operation implies to substitute the true value by its best estimated value. Systematic effects produced during observation on the quantities in question can be reduced. As a corollary, it is to be accepted that results, corrected from the systematic effects, are closer to the true value of the quantity, and therefore more accurate than the raw observation. This last statement has clear evidence in Table Diffraction nstrumental Monitoring (DM) in relation to measurement accuracy DM implements a constrained optimisation of three models: l the instrument contribution to the peak position l the instrument contribution to the peak broadening, l the d-spacing of the crystal lattice.

5 Copyright(c)JCPDS-nternational Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol The parameters, which are in common to the three models, are constrained to minimise the same x2 function and use its linearity property. Other minor models and constraints are used in orther to keep the number of parameters low. Restraints on the parameters complete the system, thus performing a constrained and restrained modelling optimisation. Both constraints on models and restraints on the variability interval of parameters guarantee the achievement of the best estimated values for each of the involved parameters. The approach is based on the Mathematical Theory of Powder Diffraction developed by A.J.C. Wilson (1963) [lo]. Details on the modelling, the optimisation system, the usability and other features are outlined elsewhere by Berti (1993) [2], Berti et al. (1995) [6], Berti (1999a) [4]. Among the various features, DM determines, through an a-posteriori process, the effective (not nominal) values of individual parameters of the instrumentation under run conditions. Moreover, well performing corrections proposed as the result of the modelling process are visualised as correction curves. These corrections are required for minimizing the effect of instrument contribution to the peak position. When the corrected positions (Wilson angles) of peaks straddle around the expected ones (Bragg angles), the effect of systematic contributions to the peak position are minimised and the good performance of the correction is reached. Zero calibration, specimen surface displacement, the compactness of the specimen in the sampleholder, axial and equatorial divergence focal spot, diffractometer radius are the most common systematic contribution (error). The goal of DM is the evaluation of these undesired effects r 0.1 T B D 0.05 Fe a _ - L ,tf :, 3 1 u 1, Figure 3: Curves of the correction model for minimising the effect of instrument contribution to the peak positioning. The symbol 20 represents the expected Bragg angle (X-axis) and A28 (Y-axis) is the separation of data Corn the expected. The dash line (A20 = 0) is the zero line of the separation; the gray line is the correction model; The filled squares are the separation from the expected Bragg angles of the observed ones; the open squares are the separation Corn the zero line of the angles computed after the correction model (Wilson angle). Berti et al. (1995) [6] and Berti, Enea (1998) [3]. and their removal from the collected data. This evaluation implyes impart accuracy to the difiaction measurement results. Figure 3 shows two characteristic curves of the results of two different diffractomenters in a given experiment. These two difiactometers are here conventionally labelled with B and D and are a part of a more general work, which involves several other diffractometers [ 51. The curves of Figure 3a show the required corrections to the experiemental observation (black squares ) bringing the Wilson angles (open squares) to straddling around the expected Bragg angles. The curve of Fig.3a has a predominant effect similar to a cotangent with the

6 Copyright(c)JCPDS-nternational Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol addition of other smaller adjustments; this cotangential behaviour suggests the axial divergency, related to the Soller slits, might contribute to the peak positioning, with other minor systematic effects. On the contrary, Figure 3b shows an apparent perfect correspondence between the Wilson and Bragg angles. The curve in this case does not show any characteristic behavior and a very small separation from the zero line. 3. Accuracy of lattice parameter determination and refinement Among the various types of analysis involving X-ray Powder Diffraction (XRPD) data, the reduction from systematic errors is mostly cogent for the determination of lattice parameters. The determination of lattice parameters of a sample of KC1 from data collected on two distinct diffractometers (B and D) is here shortly outlined. Table 1 reports the results and summarises the details of the refinement process obtained by following the Appleman method [l]. The Appleman refinement method uses the geometrical and symmetry features of the Laue class related to the lattice under investigation and does not alow correction from systematic effects. t is here adopted as the complement of DM for the purpose of the lattice parameter refinement. Table 1: determination of the lattice parameter obtained from the Appleman procedure with (Q ) and without (a~ ) the correction model of DM. a0 a$ SD SD R R V-C V-C N N B ~10-~ ~10-~ 7x10- l 7 3 D ~10-~ x1o-7 8x The values ao for the diffractometer B and D were firstly calculated without using the correction model of DM and then a were computed after considering the correction model. The 15 most intense reflections of KC1 were included in the calculation. The 28 angles were deduced from the profile analysis using the pseudo-voigt as best fitting function for each of the two peaks of the doublet ~2. The adopted processing system is DSVAR93 [5]. When the starting data set of the Appleman procedure contains the angular values including the correction model (Wilson angles), the resulting refinement process is improved. Such a comparison is evident in Table 1: 1. no standard deviation (SD) is related to the refined parameter, 1. the variance-covariance matrix (V-C) values are smaller than those calculated without considering the correction model 2. no rejection of peaks (R) is slagged from running the Appleman procedure 3. the number of required cycles for reaching the converegnce is smaller. More details on the refinement Appleman procedure are in Appleman and Evans (1973). [ Concluding Remarks The adoption of a suitable data reduction system improups accuracy in the transition from observation to measurement results of a physical quantity. The knowledge of the instrumental contribution and the exhaustiveness of the mathematical model both play important roles in defining the achievable accuracy. When the best estimate value is a known quantity, an estimation of the achievable accuracy can be obtained directly from the observation, by

7 Copyright(c)JCPDS-nternational Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol properly taring the instrument and adopting the best suited diffractometer alignement for the specific application purpose. This accuracy is an observation attribute and is useful sometime for routine analysis. Sometimes the use of an internal standard is helpful. A second use of the term accuracy exists, which is an attribute of the best estimate value (the measurement result). t depends on the model completeness adopted to describe the unknown observable quantity. A significant distinction exists between observation and the true value of the observable quantity ; this separation passes through instrument calibration, experiment set-up and modeling. From a practical point of view the Dif?action nstrumental Monitoring is a method enabling the accuracy controls of XRPD measurement. Such a control is carried out through successive steps and the adoption of curves similar to the calibration curves proposed by Jenkins (1989a) [S] and [9]. These curves have been obtained from the correction model and allow the user to define the effective characteristics of the instrument [5]. The significance of X-ray dieaction measurements implies specifications shall be given for the various procedural implementation steps. Protocols are expected for gathering to certifiable measurement results. These protocols shall concern: the instrument alignments, calibration maintenance and monitoring, sample and specimens treatment, data collection strategy and data processing as well. The generality of this concept adapts to the most field of application and types of analysis of modem x-ray powder difiactometry References [l]appleman D.E., Evans H.T.Jr. (1973); job 9214: ndexing and Least square Refinement of Powder Difiaction Data. U.S. Geol. Comp. Centr.20. Ditfraction data: Results from a Round Robin Project. Powder Diffraction, 11, , [2] Berti G. (1993); Variance and Centroid optimisation in X-ray powder difiaction analysis, Powder DiBaction, 8, [3] Berti G., Enea A. (1998); DSVAR96: a New Software tool for the Diffraction nstrumental Monitoring. Proceed. Eurp. Stand: Past and Present need at the first Centenary of X-Ray discovery, [4] Berti G. (1999a); A method for routine comparison of XRPD Measurements through Removal of some systhematic Effects, Powder Diffraction - Submitted paper. [5] Berti G. (1999b); Application of Diffraction h&rumental Monitoring to the Analysis of Diffraction Pattern from a Round Robin project on KCl. Powder Dieaction, Submitted paper. [6] Berti G., Giubbilini S.,Tognoni E., (1995) Disvar93: A software Package for determining systematic effects in X-ray powder diffiractometry. Powder DiEaction 10, 104,111. [7] Bevington P.H. Data Reduction and Error Analysis for the Physical Science; McGraw Hill (1974) [8] Jenkins R., Snyder R.L.. ntroduction to X-ray Powder Diffractometry J. Wiley & Sons, nc [9] Jenkins R. (1989); in Modern Powder Difiaction, Reviews in Mineralogy 20, [lo]wilson A.J.C.. Mathematical Theory of X-Ray powder Diffraction Philips Tech. Lib [ 1 l] Document UNYPND - GL diffiattometria - Pisa Nov [12] S and BS5497 Part