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【正文】 points we need if we want 95% confidence limits of 177。 5%. First, we can assume for large n that t(95, n1) is approximately equal to 2, and so we want the estimated standard error, S(PP) = %. Using Eqn. (6) gives a number of points for VV = of P = 7289, considerably more than we would be able to obtain from the micrograph in Figure 1, without violating the one measurement per feature rule.Expected Error in Lineal AnalysisThe expected relative standard error can be estimated using the relationship derived by Gladman T Gladman J. Iron Steel Inst. 201 (1963) 906: (8)where n is the number of second phase particles measured. Eqn. (8) has been used to generate the data in Table 3, which shows the estimated number of second phase particles required for certain 95% confidence limits, for samples of different volume fraction second phase.Relative Error95% CLNo. of particles, nVV = VV = VV = 177。 2%19602162005000177。 5%31362592800177。 10%784648200177。 20%19616250Table 3 – Number of second phase particles needing to be measured in order to achieve various estimated relative errors of volume fraction in the lineal analysis method, determined from Eqn. (8).It can be seen from Table 3 that the values of n are less than the corresponding values of P for point counting, but n should really be pared to the number of points in the minor phase, Pminor = VVP, and on this basis point counting is statistically more efficient. In addition, when performing measurements by hand, it is quicker and more reliable to grade points as 1, or 0, rather than having to measure the intersection of the gridlines with each particle.Expected Error in Areal AnalysisHillard and Cahn Hilliard and Cahn Trans. Met. Soc. AIME 221(2) (1961) 344352 proposed a method to estimate the relative standard error in areal analysis: (9)where n is the number of minor phase areas measured, and and sA are the mean and the standard deviation of the measured areas. They further show that for a uniform structure of equal size spheres: (10)However, for more realistic structures: (11)Substitution of Eqn. (11) into Eqn. (9) gives: (12)The relative standard error in this case is therefore independent of the volume fraction second phase in the sample. Eqn. (12) has been used to generate the data in Table 4, which shows the estimated number of areas required for certain 95% confidence limits..Relative Error95% CLNo. of areas, n177。 2%15626177。 5%2500177。 10%625177。 20%156Table 4 – Number of second phase areas needing to be measured in order to achieve various estimated relative errors of volume fraction in the areal analysis method, determined from Eqn. (12).As can be seen from Table 4, the values of n are of the same order as for point counting, but once again these should really be pared to the number of points in the minor phase, Pminor = VVP, indicating that point counting is statistically more efficient.R Goodall, October 2010
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