顯微結(jié)構(gòu)的量化處理及組織分析的教程lecturen(2)-資料下載頁
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【正文】 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 Analysis The expected relative standard error can be estimated using the relationship derived by Gladman2: ? ? ? ?22 12LLnVVSVV ?????????? (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 Error 95% CL No. of particles, n VV = VV = VV = 177。 2% 19602 16200 5000 177。 5% 3136 2592 800 177。 10% 784 648 200 177。 20% 196 162 50 Table 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 Analysis Hillard and Cahn3 proposed a method to estimate the relative standard error in areal analysis: ? ????????? ???????????????? 22 11 AnVVS AVV ? (9) 2 T Gladman J. Iron Steel Inst. 201 (1963) 906 3 Hilliard and Cahn Trans. Met. Soc. AIME 221(2) (1961) 344352 Quantification of Microstructure and Texture Volume Fraction from Planar Sections R Goodall, October 2022 7 where n is the number of minor phase areas measured, and A and ?A are the mean and the standard deviation of the measured areas. They further show that for a uniform structure of equal size spheres: ??????? AA? (10) However, for more realistic structures: ??????? AA? (11) Substitution of Eqn. (11) into Eqn. (9) gives: ? ?nVVS VV ? (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 Error 95% CL No. of areas, n 177。 2% 15626 177。 5% 2500 177。 10% 625 177。 20% 156 Table 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.
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