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Homogeneity Testing


56% of the total uncertainty (corresponding to s h <0.013 wt%). Table 2 shows the detailed analysis of another crystal (crystal s7.5 in Table 1 ) that showed an average Ce content of 0.378 wt% with a total uncertainty of 0.038 wt%. In this case heterogeneity was signifi cant (in Figure 4 to the right of the critical value). T e contribution of heterogeneity to the total uncertainty was determined to be 84% (corresponding to s h = 0.032 wt%). T e dashed curves in Figure 4 display the standard deviation of s h , which can be approximated as:


(Equation 5)


These curves illustrate that below the critical value (left of the red line) the uncertainty strongly increases. Hence, the critical homogeneity index and the parameters derived from it, as explained above, serve as a detection limit for heterogeneity at a chosen level of signifi cance given by α .


Discussion T e example given in Figure 4 shows that the quantifi cation of heterogeneity needs to consider a detection limit. Obtaining the uncertainty due to heterogeneity s h from the uncertainty budget off ers a quantitative parameter, but the signifi cance of s h and its uncertainty strongly depend on the number of analysis spots and the analytical precision. For example, a short counting time leads to a poor precision and to a large contribution of counting statistics to the uncertainty budget. In this case the relative contribution of heterogeneity to the uncertainty budget is low, and heterogeneity may be below the detection limit discussed above. Because compositional heterogeneity is a fi xed characteristic of the sample, an increase in counting time or beam currents will decrease the counting statistical uncertainty and increase the relative contribution of heterogeneity to a value above the detection limit. Hence, perception of homoge- neity or heterogeneity depends on the analytical method of investigation and its conditions of operation. An EPMA routine tailored for a more precise determination of Ce in fl uorapatite might detect heterogeneity even in those crystals that passed the homogeneity test in the example above. The statistical evaluation discussed is potentially applicable to a wide range of analytical techniques. A standard suitably homogeneous for electron probe x-ray analysis would be adequate for methods with poorer spatial resolution and similar precision but may require re-testing for use with a method with better spatial resolution (for example, thin specimen analysis in STEM) or better precision at low concentrations (for example, SIMS). From a practical perspective the correct determination


of s Pois can be a challenge. In the case of electron probe microanalysis it is not sufficient to just consider the counts above the background because the counts of the background measurements also contribute to s Pois [ 7 ]. At high precision, matrix effects may produce apparent heterogeneity of a homogeneously distributed element A if a heterogeneously distributed element B aff ects the measured x-ray intensity of A. Matrix correction procedures should account for this.


2017 January • www.microscopy-today.com 35 Conclusion


In electron probe x-ray microanalysis, homogeneity testing using the homogeneity index based on counting statistics off ers the advantage that only one analysis per sample spot is required. Methods requiring multiple measurements on the same spot are prone to errors from degradation of beam-sensitive samples such as many geological materials and glasses. T e sound statis- tical evaluation shown here is applicable to all kinds of reference materials (alloys, glasses, natural and synthetic crystals, etc.) and may be applied to other microanalytical techniques in which photon or ion counting takes place. T e level at which heterogeneity may be detected and quantifi ed depends strongly on the number of measurements N .


References [1] TPJ Linsinger et al ., Accredit Qual Assur 6 ( 2001 ) 20 – 25 . [2] RB Marinenko et al ., National Bureau of Standards Special Publication 260 –6 5 ( 1979 ).


[3] FR Boyd et al ., Carnegie Institution of Washington Year Book 67 ( 1967 ) 210 – 15 .


[4] E Jarosewich , JA Nelen, and JA Norberg , Geostandard Newslett 4 ( 1980 ) 43 – 47 .


[5] E Jarosewich and IG MacIntyre , J Sediment Petrol 53 ( 1983 ) 677 – 78 .


[6] P Carpenter et al ., J Res Natl Inst Stan 107 ( 2002 ) 703 – 18 . [7] D Harries , Chemie der Erde 74 ( 2014 ) 375 – 84 . [8] JC Stormer et al ., Am Mineral 78 ( 1993 ) 641 – 48 . [9] V Liebich et al ., Fresen Z Anal Chem 335 ( 1989 ) 945 – 53 .


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