Homogeneity Testing

Figure 2 : Schematic probability density functions of the homogeneity index for a heterogeneous material (maximum probability density at H > 1). Most of the tests fail the homogeneity criterion, but some tests are below the critical homoge- neity index H crit . This erroneous non-detection of heterogeneity is a type II error. Increasing the number of measurements N decreases the width of the curve (its center stays fi xed) and decreases H crit . This decreases the probability of a type II error.

example of a diff erent material is found in the electronic annex of [ 7 ].

Figure 1 : Schematic probability density functions of the homogeneity index for a perfectly homogeneous material (maximum probability density at H = 1). A signifi cance criterion (critical homogeneity index H crit ) is chosen by accepting that a certain number (usually 5%) of all tests fail despite the null hypothesis is true (type I error). (a) A curve assuming N = 30 measurements. The width of the curve indicates the inherent uncertainty of the homogeneity index. (b) A curve assuming N = 300 measurements. The uncertainty of the homogeneity index is strongly reduced and H crit is reduced as well.

Imilchil in Morocco were investigated. Fluorapatite is a halogen- bearing calcium phosphate, ideally Ca 5 (PO 4 ) 3 F, and can substitute a wide range of elements into its structure. It is also one of the more beam-sensitive minerals and known for loss of fl uorine during electron beam analysis [ 8 ]. It was analyzed using wavelength- dispersive electron probe x-ray spectrometry on a JEOL JXA-8900 (15 kV, 30 nA beam current, 14 µm spot diameter). A worked

30

Instrumental precision . In the case that only one measurement per analysis spot is practically feasible in order to avoid specimen degradation, signifi cance criteria for detectable heterogeneity can be derived in a statistically sound fashion if the counting statistics of x-ray photons is the only signif- icant contribution to the instrumental precision. Hence, the instrumental precision then would be the standard deviation derived from the number of counting events for a particular element x-ray peak. T is limitation of instrumental precision to counting statistics is only valid if other sources of instrumental random noise are insignificant. Such a source may be, for example, the alignment of the analyzer crystal in a mechanical wavelength-dispersive spectrometer or variations in the beam current measurement—basically any parameter that is set and reset before and aſt er the acquisition of each data point. Because technological development has achieved highly reproducible mechanics and electronics, these sources of scatter in the results are usually negligible. Drift . A much bigger problem is instrumental drift. In this case instrumental variations occur not randomly but in a time-constrained manner, for example in relation to changing temperature. Instrumental driſt must be avoided or properly corrected, and the time series of analyses acquired must be screened meticulously for the presence of variations that correlate with time. Statistical methods for assessing this have been suggested [ 9 ]. The technique presented here assumes that driſt is absent and any instrumental contribution

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Figure 2 : Schematic probability density functions of the homogeneity index for a heterogeneous material (maximum probability density at H > 1). Most of the tests fail the homogeneity criterion, but some tests are below the critical homoge- neity index H crit . This erroneous non-detection of heterogeneity is a type II error. Increasing the number of measurements N decreases the width of the curve (its center stays fi xed) and decreases H crit . This decreases the probability of a type II error.

example of a diff erent material is found in the electronic annex of [ 7 ].

Figure 1 : Schematic probability density functions of the homogeneity index for a perfectly homogeneous material (maximum probability density at H = 1). A signifi cance criterion (critical homogeneity index H crit ) is chosen by accepting that a certain number (usually 5%) of all tests fail despite the null hypothesis is true (type I error). (a) A curve assuming N = 30 measurements. The width of the curve indicates the inherent uncertainty of the homogeneity index. (b) A curve assuming N = 300 measurements. The uncertainty of the homogeneity index is strongly reduced and H crit is reduced as well.

Imilchil in Morocco were investigated. Fluorapatite is a halogen- bearing calcium phosphate, ideally Ca 5 (PO 4 ) 3 F, and can substitute a wide range of elements into its structure. It is also one of the more beam-sensitive minerals and known for loss of fl uorine during electron beam analysis [ 8 ]. It was analyzed using wavelength- dispersive electron probe x-ray spectrometry on a JEOL JXA-8900 (15 kV, 30 nA beam current, 14 µm spot diameter). A worked

30

Instrumental precision . In the case that only one measurement per analysis spot is practically feasible in order to avoid specimen degradation, signifi cance criteria for detectable heterogeneity can be derived in a statistically sound fashion if the counting statistics of x-ray photons is the only signif- icant contribution to the instrumental precision. Hence, the instrumental precision then would be the standard deviation derived from the number of counting events for a particular element x-ray peak. T is limitation of instrumental precision to counting statistics is only valid if other sources of instrumental random noise are insignificant. Such a source may be, for example, the alignment of the analyzer crystal in a mechanical wavelength-dispersive spectrometer or variations in the beam current measurement—basically any parameter that is set and reset before and aſt er the acquisition of each data point. Because technological development has achieved highly reproducible mechanics and electronics, these sources of scatter in the results are usually negligible. Drift . A much bigger problem is instrumental drift. In this case instrumental variations occur not randomly but in a time-constrained manner, for example in relation to changing temperature. Instrumental driſt must be avoided or properly corrected, and the time series of analyses acquired must be screened meticulously for the presence of variations that correlate with time. Statistical methods for assessing this have been suggested [ 9 ]. The technique presented here assumes that driſt is absent and any instrumental contribution

www.microscopy-today.com • 2017 January

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