Application of Tukey’s Test for Statistical Measurement of Percentage Indexes Derived from the Difference Between Predicted Crystallographic Data in Five Nitihf Alloys Obtained by Two Processes

— The present paper uses the Tukey test to measure the minimal significative difference (MSD) between crystallographic data predicted by non-linear mathematical models generated from graphics published by Zarinejad (2008) and Potapov (1997) on 5 nominal compositions of Ni 50 Ti 50-X Hf X .at% substitutional alloys fused by arc melting and melt spinning processes. Results show that the calculated MSD do not represent a statistically significant difference. Therefore, it is concluded that the independent variable (Hafnium content) does not change the analysed dependent variables (lattice parameters, crystalline structure volume and monoclinic angle β) for any fusing processes.


INTRODUCTION
The production of shape memory effect (SME) alloys depends sensibly on particular processes that, based on the variation of parameters involved in the equipment used (maximum melting temperature, vacuum magnitude, purge intensity, purity of the metallic charges, torch exposition time, etc), can or cannot generate different crystallographic features for samples which theoretically have the same chemical compositions. In this sense, it is important to compare the obtained results through predictive modelling functions to observe whether there are any significant statistical difference capable of indicating a great variability in crystallographic characteristics, such as: lattice parameters (a b c [Å]), cell volume (VOL [Å 3 ]) and characteristic angle β (beta [ o ]) of the monoclinic structure (space group P121/m1). In this work, they are studied as variables dependent on hafnium content, which is considered an independent variable in the nominal composition. The focus of the study is on room temperature B19' martensitic phase B19', which has a monoclinic structure (Pearson symbol mP8) and is considered by the literature as an unstable phase. In addition, the five substitutional compositions Ni50Ti50-XHfX .at% considered in this instance are X = 8, 11, 14, 17 and 20 .at%, which were obtained by two different processes: Arc Melting (Zarinejad, 2008) [1] and Melt Spinning (Potapov, 1997) [2].
All of the R 2 values were equal to 1. This makes unnecessary the calculation of Adjusted-R 2 , normally used to measure the real degree of modeling reliability. According to Table 3, since there was no standard error in the coefficients, the general error of the models is 0. For this reason, F value is not presented in table 3. Thus, it can be stated that the mathematical models adopted here are statistically significant as well as predictive for all five dependent variables observed.
Taking the four Zarinejad alloys (2008) as a reference, the residues found for each regression, in each variable of the monoclinic structure B19', are presented in table 4 and figure 5.    Considering the similarity of the composition used by Potapov (1997) regarding the composition Ni50Ti50-XHfX .at%, adopted in this work (difference of 0,2 .at% in the contents of Ni e Ti), the values of table 5 were used to proportionally calculate the same variables having as reference an alloy rich in Ni of nominal composition Ni50Ti50-XHfX .at%, quickly solidified by melt-spinning. The results are shown in Table 6:  For a graphic visualization, figure 6 presents the values calculated for the six Potapov ribbons (1997) plotted in the left column (figure 6(a), figure 6(c) and figure 6(e)) in a way that they are compared with the values predicted for the five ribbons studied here, plotted in the right column (figure 6(b), figure 6(d) and figure 6(f)).
According to the obtained numbers , the values of β angle, volume and "a" and "c" are directly proportional to the atomic percentage of hafnium from the alloy, except in the lattice parameter "b", which is inversely proportional to the content of Hf (.at%).

IV. COMPARISON OF CRYSTALLOGRAPHIC CHARACTERISTICS OF THE MARTENSITIC PHASE IN ZARINEJAD'S ALLOYS AND POTAPOV'S RIBBONS
Due to the fact that these variables are different, i.e., one-dimensional (lattice parameters) two-dimensional (β angles) and three-dimensional (monoclinic structures volumes) entities. To enable a dimensionless comparison between the predicted data from Zarinejad (2008) and Potapov (1997), it was necessary to consider the percentage indexess of the differences between the estimated values for each parameter (a, b, c, Vol e β) of the five compositions Ni50Ti50-XHfX .at%. This makes them arbitrary data. Therefore, it was necessary to organize the sample series from table 11 with the dimensionless format of table 9. Percentage indices have arbitrary units.
It is noticeable, in figure 7, that there is a concentration of minor differences close to bigger densities. This certifies that the values from both authors are close.  Table 11 -Comparison between Ni50Ti50-XHfX .at% alloys and ribbons based on the models by Zarinejad (2008) and Potapov (1997) However, there is a clear comprehension that the crystallographic characteristics depend not only on the hafnium content, but they are also sensitive to the peculiarities of the two production processes as well as the typical variables of each one of them: electric arc melting (Arc Melting) and quick solidification Melt Spinning. Depending on how the peculiarities involving each one of the processes are treated, one can directly influence the main properties of the samples such as: homogeneity, amount of residual stress and minimization of oxidation, among others. These physical and mechanic properties, among others are derived from atomic arrangementknown as crystallographic structures (or crystalline structures).
It can be mentioned as variables of these processes: the number of times that the bulk was melted, exposure time to the torch, type of material of the mold , efficiency of the applied vacuum, rotating speed of the copper flywheel in quickly solidified ribbons etc. Hence, all the possibilities of variation and instrumental errors generate conditions of solidification which interfere in the micro structure of the alloys and obtained ribbons.

V. CONCLUSIONS
The analysis of variance (ANOVA) is an unilateral test based on the F-Snedecor table, which after calculating the F-value, it measures whether it is inside or outside of the acceptance area, according to the following hypothesis test: { H 0 : μ Potapov = μ Zarinejad H 1 : there is at least one difference between the means As it is an OneWay ANOVA, the only factor (independent variable) is the atomic percentage of Hf (.at%). As a rule, in terms of null hypothesis (H0), the calculated Fvalue is inside the area of acceptance, i.e., minor than critical F. Otherwise, if H0 is rejected, the alternative hypothesis (H1) will be accepted having necessarily, the F-value calculated outside the area of acceptance, i.e., bigger than the critical F.
In this case, we considered that F(5%); DF1 (DFW); DF2 (DFR) = 3,12 (tabulated), according to the standard table for Tukey's Test (α=0.05) [5]. According to Table 12, as the calculated F = 3.13 is bigger than the critical tabulated value, we concluded that there is a difference between the averages of both treatment groups: Potapov and Zarinejad. This conclusion is also confirmed by the P-value, which is smaller than 0.05. The lower the P-value, lower is the possibility for H0 be true. It is important to note that DF1 [horizontal = numerator] is the degree of freedom between groups (DFW) and that DF2 [vertical = denominator] is the degree of freedom of the residues (DFR). Therefore, the null hypothesis is rejected (H0). However, to ascertain whether this difference is statistically significant, it is necessary to perform a parametric test to certify this significance. This decision was taken based on the normality test of Anderson-Darling [6], which predicts as "normal" the set of 25 data organized in table 9, as shown in figure 8: