Homogeneity Testing

Figure 4 : Results of the homogeneity studies on 36 apatite crystals. The composition of each crystal (shown as dots) was measured 30 times along a diagonal profi le across the crystal. Shown is the contribution of heterogeneity to the total combined uncertainty of the average Ce mass fraction, either as absolute value ( s h ) or relative value ( s h,rel ). In 16 crystals signifi cant heterogeneity was detected; they plot to the right of the critical values of s h and s h,rel (red solid vertical line), which can be computed from the critical homogeneity index H crit . The ordinate shows the uncertainty of the uncertainty due to heterogeneity. It becomes very large below the critical values, which serve as a detection limit for heterogeneity. Increasing the number of measurements N decreases the expected uncertainty of s h (dashed curves).

to its components comprised of the variance due to hetero- geneity of the sample ( s2

measurement process, which, in this case, is reduced to the Poisson variance:

h ) and the variance due to the (Equation 3)

Based on this simple relationship, it is possible to defi ne the relative contribution of heterogeneity to the total uncertainty of a reference value ( s h,rel ), a parameter that can be related to the homogeneity index:

(Equation 4)

T e parameter s h provides information on the extent of the compositional variations of the sample. T e interval ±2 s h around the measured mass fraction of an element covers about 95% (2 sigma) of the sample’s compositional variations. Because s h and s Pois cannot be separated with absolute certainty, s h contains some uncertainty that depends on the number of measure- ments N (this is shown in the Results section). T is approach is limited by the non-linear relationship between H and s h,rel resulting in a bias that yields apparently large contributions of compositional heterogeneity even if materials are almost perfectly homogeneous [ 7 ]. In such a case only an upper limit of possibly present heterogeneity can be stated: a detection limit of heterogeneity. At N =10 a homogeneous sample that passes the homogeneity test ( H < H crit ) may show an apparent contribution of heterogeneity to the uncertainty budget of up to 68%. Similar to stating the detection limit for a non-detected element, this s h,rel indicates the upper limit of relative heterogeneity that may

2017 January • www.microscopy-today.com

be present. To obtain this number, H crit is calculated as a scaled quantile of the chi-squared distribution with a signifi cance level α = 0.05 and N - 1 degrees of freedom via Equation 2. If H crit replaces H in Equation 4, then the upper limit of s h,rel results, and the percentage can be calculated by multiplying it with 100%. Because the uncertainty of the homogeneity index is primarily determined by the number of measurements N , this limit can be reduced by increasing N. In order to state that the contribution of compositional heterogeneity to the total uncertainty budget is less than 30%, a homogeneity test with N = 577 measurements has to be passed ( H crit = 1.048). For 20% this number increases to more than 3,000 measurements. T e homogeneity index and the uncertainty budget not only allow decisions of whether or not heteroge-

neity has been detected—in the sense of a detection limit for elemental variations instead of the mere presence of a chemical element—they allow the quantifi cation of heterogeneity if it is detected. Because s c and s Pois are obtained from the measure- ments, s h can be calculated easily from Equation 3.

Results

Figure 3 shows silicon and cerium x-ray intensity maps of one crystal of the fl uorapatite candidate reference material. In this example the distribution of Ce was studied, which replaces Ca in a coupled substitution with Si (replacing phosphorus) or Na (replacing calcium) for charge balance. While the variation in Si is clearly visible, the signifi cance of heterogeneity in the content of Ce is diffi cult to judge by the eye. Figure 4 displays the wavelength-dispersive electron probe results of the Ce content of the apatite crystals. Each location on the polished crystal section was analyzed only once. Out of 36 crystals, each measured at N = 30 locations, 16 samples showed detectable heterogeneity of Ce under the measurement conditions used. T e critical homogeneity index and the corresponding values of s h and s h,rel defi ne the level at which heterogeneity becomes signifi cant at the α = 0.05 level of confi cence. Below H crit , the uncertainty of the heterogeneity contribution s h to the total combined uncertainty strongly increases. A minimum of uncertainty at any given N is obtained if s h equals s Pois , and the overall uncertainty can be reduced by increasing N .

One apatite crystal marked in Figure 4 showed an average Ce content of 0.215 wt% with a standard deviation ( s c ) of 0.020 wt% (crystal s6.3 in Table 1 ). In this case, heterogeneity among the 30 measurement locations could not be detected: the data point is to the leſt of the critical homogeneity index (red vertical line), which was derived from Equation 2 (N = 30 and α = 0.05) and converted to s h,rel using Equation 4. T erefore, it is possible to state that the uncertainty due to heterogeneity is less than

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Figure 4 : Results of the homogeneity studies on 36 apatite crystals. The composition of each crystal (shown as dots) was measured 30 times along a diagonal profi le across the crystal. Shown is the contribution of heterogeneity to the total combined uncertainty of the average Ce mass fraction, either as absolute value ( s h ) or relative value ( s h,rel ). In 16 crystals signifi cant heterogeneity was detected; they plot to the right of the critical values of s h and s h,rel (red solid vertical line), which can be computed from the critical homogeneity index H crit . The ordinate shows the uncertainty of the uncertainty due to heterogeneity. It becomes very large below the critical values, which serve as a detection limit for heterogeneity. Increasing the number of measurements N decreases the expected uncertainty of s h (dashed curves).

to its components comprised of the variance due to hetero- geneity of the sample ( s2

measurement process, which, in this case, is reduced to the Poisson variance:

h ) and the variance due to the (Equation 3)

Based on this simple relationship, it is possible to defi ne the relative contribution of heterogeneity to the total uncertainty of a reference value ( s h,rel ), a parameter that can be related to the homogeneity index:

(Equation 4)

T e parameter s h provides information on the extent of the compositional variations of the sample. T e interval ±2 s h around the measured mass fraction of an element covers about 95% (2 sigma) of the sample’s compositional variations. Because s h and s Pois cannot be separated with absolute certainty, s h contains some uncertainty that depends on the number of measure- ments N (this is shown in the Results section). T is approach is limited by the non-linear relationship between H and s h,rel resulting in a bias that yields apparently large contributions of compositional heterogeneity even if materials are almost perfectly homogeneous [ 7 ]. In such a case only an upper limit of possibly present heterogeneity can be stated: a detection limit of heterogeneity. At N =10 a homogeneous sample that passes the homogeneity test ( H < H crit ) may show an apparent contribution of heterogeneity to the uncertainty budget of up to 68%. Similar to stating the detection limit for a non-detected element, this s h,rel indicates the upper limit of relative heterogeneity that may

2017 January • www.microscopy-today.com

be present. To obtain this number, H crit is calculated as a scaled quantile of the chi-squared distribution with a signifi cance level α = 0.05 and N - 1 degrees of freedom via Equation 2. If H crit replaces H in Equation 4, then the upper limit of s h,rel results, and the percentage can be calculated by multiplying it with 100%. Because the uncertainty of the homogeneity index is primarily determined by the number of measurements N , this limit can be reduced by increasing N. In order to state that the contribution of compositional heterogeneity to the total uncertainty budget is less than 30%, a homogeneity test with N = 577 measurements has to be passed ( H crit = 1.048). For 20% this number increases to more than 3,000 measurements. T e homogeneity index and the uncertainty budget not only allow decisions of whether or not heteroge-

neity has been detected—in the sense of a detection limit for elemental variations instead of the mere presence of a chemical element—they allow the quantifi cation of heterogeneity if it is detected. Because s c and s Pois are obtained from the measure- ments, s h can be calculated easily from Equation 3.

Results

Figure 3 shows silicon and cerium x-ray intensity maps of one crystal of the fl uorapatite candidate reference material. In this example the distribution of Ce was studied, which replaces Ca in a coupled substitution with Si (replacing phosphorus) or Na (replacing calcium) for charge balance. While the variation in Si is clearly visible, the signifi cance of heterogeneity in the content of Ce is diffi cult to judge by the eye. Figure 4 displays the wavelength-dispersive electron probe results of the Ce content of the apatite crystals. Each location on the polished crystal section was analyzed only once. Out of 36 crystals, each measured at N = 30 locations, 16 samples showed detectable heterogeneity of Ce under the measurement conditions used. T e critical homogeneity index and the corresponding values of s h and s h,rel defi ne the level at which heterogeneity becomes signifi cant at the α = 0.05 level of confi cence. Below H crit , the uncertainty of the heterogeneity contribution s h to the total combined uncertainty strongly increases. A minimum of uncertainty at any given N is obtained if s h equals s Pois , and the overall uncertainty can be reduced by increasing N .

One apatite crystal marked in Figure 4 showed an average Ce content of 0.215 wt% with a standard deviation ( s c ) of 0.020 wt% (crystal s6.3 in Table 1 ). In this case, heterogeneity among the 30 measurement locations could not be detected: the data point is to the leſt of the critical homogeneity index (red vertical line), which was derived from Equation 2 (N = 30 and α = 0.05) and converted to s h,rel using Equation 4. T erefore, it is possible to state that the uncertainty due to heterogeneity is less than

33

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