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Homogeneity Testing at the Micrometer Scale Dennis Harries

Department of Analytical Mineralogy , Institute of Geosciences, Friedrich Schiller University , Jena , Germany

Abstract: The testing of homogeneity and the quantification of chemical heterogeneity in reference materials for microanalysis is an important step in obtaining reliable standards and correct analytical results. This article describes a statistical method for evaluating homogeneity at the micrometer scale using electron probe x-ray analysis. Operation of the method is demonstrated by analysis of a natural Ca phosphate mineral standard. The results and discussion highlight several important factors and concepts, in particular the importance of a suffi ciently high number of spot analyses, the role of an uncertainty budget, and the critical level at which heterogeneity can be detected.


Homogeneous reference materials (standards) are a key to accurate and reproducible elemental microanalysis by energy- and wavelength-dispersive x-ray spectrometry in the scanning electron microscope (SEM) and the electron probe microanalyzer (EPMA). Standards are needed for quantifi- cation (converting x-ray counts to mass fractions) and for quality control of analysis results (testing on known composi- tions). Because the spatial resolution of the electron beam allows analysis of volumes down to a few µm 3 , any reference material with a known composition used as a microanalysis standard ideally must have the same composition regardless of where on the standard the measurements are made: it must be homogeneous.

In principle, perfect homogeneity in materials does not exist. Modern aberration-corrected scanning transmission electron microscopes equipped with energy-dispersive x-ray detectors and electron energy loss spectrometers are capable of detecting elemental differences at atomic levels. For the more common use of x-ray microanalysis of bulk specimens, the scale and level of heterogeneity that can be accepted for a given purpose must be defined. Tests for homogeneity are then employed to answer the question of whether a material is “fit for the purpose.” This question can be answered positively when signifi cant heterogeneity cannot be detected with the measurement method under its chosen conditions of operation.

While standards are oſt en pure elements, homogeneous multi-element standards are relatively easy to produce if stoichiometric compounds with a limited number of components are used (for example, pure oxides). However, as soon as components substitute for each other, as in glasses or solid solutions, chemical heterogeneity becomes a concern. T is is the case for natural minerals that may have formed under changing physicochemical conditions, but it also concerns synthetic materials where homogeneous doping or alloying may be difficult. For these reasons, testing for homogeneity is an important step in qualifying a material as an analysis standard.

Homogeneity testing of a potential standard requires the investigator to discriminate between the elemental variations


due to true compositional differences in the sample and those variations that are related to the instrument and the measurement process itself. The latter variations are the “precision” of the analysis, which represents the statistically random scatter in the results. Whether or not heterogeneity among different locations on the sample can be detected depends on the magnitude of the chemical variations relative to this omnipresent scatter; hence, the instrumental precision has to be known.

The easiest approach to determining the instrumental precision is to repeatedly analyze the same spot on the sample. Then the comparison of different locations on the sample by “analysis of variance” (ANOVA) allows the statis- tical separation of instrumental scatter from compositional variations with a given degree of confi dence [ 1 , 2 ]. However, in practical microanalysis this approach is oſt en problematic because repeated analyses of a single spot may result in progressive degradation (or even loss) of the analyzed volume. In electron probe microanalysis this degradation may result from the build-up of surface contamination, induced diff usion, or structural damage and decomposition. In consequence, the apparent instrumental variances from repeated measurements on single spots are often not suitable for a sound statistical analysis. T is was recognized early in the history of electron probe microanalysis, and counting statistics traditionally have been used to obtain the instrumental precision [ 3 ]. T e method outlined here follows this approach: T e instrumental precision of a single measurement is the standard deviation solely derived from measured x-ray counts; the variance of the measurement is the standard deviation squared. Calculations are usually more conveniently done with variances, while the standard deviations are usually stated because they preserve the units of the measured quantity. Both are expressions of the uncertainty of the measurement. Beginning with [ 3 ], geological and technological reference materials have been characterized by an index derived as the ratio of the observed standard deviation to the standard deviation (instrumental precision) predicted by counting statistics [ 4, 5, 6 ]. However, the interpretation of this “sigma ratio,” or “homoge- neity index,” has been based on subjective experience: usually materials with a homogeneity index of H < 3 have been accepted as suitable standards. However, this is statistically not sound, and a revised interpretation of the homogeneity index has been given only recently [ 7 ]. Based on this new evaluation, this article describes an improved statistical method for determining the degree of homogeneity in a bulk specimen at the micrometer level of spatial resolution, given the instrumental precision of the analysis based on counting statistics.

Materials and Methods

As a candidate reference material for use as a microprobe standard, natural crystals of the mineral fluorapatite from

doi: 10.1017/S1551929516001115 • 2017 January

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