EVALUATION BY COMPUTER SIMULATION OF CERTAIN ERRORS IN THE THREE DIMENSIONAL TEXTURE ANALYSIS

: The influence of certain experimental errors in pole-figure determination on the accuracy of calculated coefficients of the orientation distribution func- tion has been analyzed

In his works, Bunge 3' estimates the errors of the coefu and C u without dissociating the influence of ficients F their origins. To obtain the most exact O.D.F., it is of course necessary to minimize each of the experimental errors.

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The present investigation was undertaken to examine by computer-simulating the influence of some experimental errors on the calculation of the coefficients of the pole figure. We studied more particularly the influence of: the basis of the calculation performed in this simulation. It is important not to choose these coefficients arbitrarily. In order to give them a significant role, we identify these coefficients, purposely limited to rank Z equal 22, with the corresponding coefficients of a real pole figure.
Several partitions of the pole sphere, which correspond to solid angles such as A taking the values 2 , 3 and 5 , and A8 being kept constant at 5 , were examined.
The following results may be observed regardless of the nature and texture of the specimens.
For 9 0 and for a given A the ratio Fg/F rapidly increases with . The increase of A causes a significant increase in the slope of these different curves as shown in These results show that the partition of the pole sphere 9 coefficients. Indeed, the has a sensitive influence on the F relative errors increase rapidly with A especially when and 9 are high, so that from certain values on up, the determinaof the F no longer produces meaningful results. If we want to develop the pole figure to a given rank we have to choose a partition of the pole sphere so that the errors do not rise beyond a certain level.
In practice, as shown in our figures, if we want to obcoefficients up to the tain a reasonable precision of the F rank Z equal to 22--the limit generally used in the case of a texture which does not present very sharp orientations-the partition will have to be chosen so that A be noticeably lower than 5

STATISTICAL FLUCTUATIONS
When one measures a pole figure the recorded counting rate undergoes statistical errors. The influence of these statistical errors on F were calculated by computer simulation. The total recorded number of pulses N is proportional to the density of the planes {hk} in Bragg reflexion and the counting error is . We used the following simulation scheme: The recording counting N(p,k) remained lower than 1400 impulses. The computer simulation was performed for several pole figures corresponding to samples of different textures. Figure 3 shows the absolute errors AF 9 corresponding to one of the samples. Whatever the pole figure we examined, we obtained results of this type. It may be noticed, indeed, SAMPLE Figure 3 Coefficients For each rank , the mean value of the absolute errors occurring in the case of statistical fluctuations is almost constant as shown in Figure 3. So the relative precision of the low-value coefficients F u (for instance in high rank ) is not as good as in the case of high-value coefficients F. So statistical fluctuations give rise to errors, which can be made negligible in comparison with other errors by increasing counting time, or by performing measurements with sufficiently high constant impulse number.

OVERLAPPING OF THE EXPLORED ZONE
The setting of the texture goniometer s'6 as well as of the horizontal slits of the counter determines the width A of pole sphere zones as shown by the following relation: *This normalization convention deviates from that one used in reference 3.

H A 2'R sin
where H is the aperture of the horizontal slits; R is the distance between sample and counter; and 8 is the Bragg angle. If the aperture H is too wide the zones overlap, and the measurement of the pole density becomes erroneous. This type of error was computer-simulated in the case of a 5% overlap which should be of the amount of the usual experimental error. The simulating scheme used was as follows: The analysis, which was again applied to several pole figures corresponding to samples of different nature and texture, led to the following general results: the coefficient F0 p,k /A P(p,k)dy (p,k) which is the norm of the density distribution function, grows quite logically, inasmuch as the same intensity is collected twice by overlapping.
The growth of the F coefficient disappears by normalizing the pole figure to unity. Figure 5 represents the AF errors corresponding to sample I in the case of a 5% overlapping. One may observe that, for each rank Z, the errors AF u are decreasing with 9. Furthermore, the errors are all less than a certain value (0.008 in the case of sample I). We obtain this type of results for pole figures of all kinds. Figure 6 shows the level of the mean errors (over 9 in each degree ) due to the overlapping of the explored zones in comparison with the errors occurring in the sphere partition (the highest) and in the statistical flucutations (the lowest). This result leads us to draw the same conclusion as in the previous section, and to dwell on the fact that this type of errors must be more severely corrected to improve the u coefficients by taking the greatest care precision of the F in the setting of the goniometer and the counter.
In Figure 7, the absolute values of F and of AF are plotted as a function of the rank 4, where F represents the sum of the various errors examined. It is shown that the magnitude of the error AF 9 is almost as great as the value of F calculated at high ranks of values. In order to obtain a better precision, one is tempted to develop the O.D.F. with rank Z as high as possible. Only the number of the pole figures seems to limit the serial development.
However, this work shows that experimental errors are the cause of important errors in the calculation of F coefficients. If the latter, which only represent an intermediate stage in the determination of the O.D.F. present a high uu coefficients is no longer of error the calculation of the C much significance. Practically, various precautions such as increasing the counting rate, setting the apparatus with great care and partitioning the pole sphere with A lower than five degrees are necessary in order to obtain meaningful coefficients, especially when the rank is to be higher than or equal to 22.