OPTIMIZATION OF TEXTURE MEASUREMENTS . IV . THE INFLUENCE OF THE GRAIN-SIZE DISTRIBUTION ON THE QUALITY OF TEXTURE MEASUREMENTS

It is the focus ofattentionthat texture experiments deal with real samples which consist ofa finite number of crystallites of different size. Because of this the main sample-induced statistics (grain statistics and grain-size statistics) have essential influence on the quality of experimental data. This article quantitatively analyzes the dependence ofthe integral error on the parameters of the mentioned statistics and the approximation goodness.


INTRODUCTION
In texture measurements there exist several sources of statistical errors induced by the investigated sample.This article concentrates on two sources as the most interesting ones from the viewpoint of the author and Mucklich and Klimanek (1994): Grain statistics arising when a sample with finite number of grains is taken for texture measurements from the sample multitude (e.g., a geological sample taken from a rock).
Grain-size statistics related to the distribution of grains over size or volume.
V. LUZIN They influence the experimental results during texture measurements increasing the total statistical errors of pole figures (PFs) or the orien- tation distribution function (ODF).
The aim of this article is to show the impact of these two statistics on the quality of the measured texture data and reveal their relationship with an earlier introduced concept ofan "optimal" texture measurement (Luzin, 1997a), which provides the best feasible quality of experimental data.

FUNDAMENTALS
To analyze the quality of experimental PF, one needs to compare quantitatively the true PF pt fs pt hi (Y) hi(Y) d (y)   (1) related to the sample multitude and the experimental PF h, (.., N) fn ps (37, N)K(jT,.Fj., {P)) dw() hi (2) determined on the measurement grid F--{y} with the number of grid points J-#F.Experimental PF is expressed in the form of a sampling PF: N P,(fi, N) Z VnS(.7-fin), n--1 N 6(37 fi,)dw07 1, V Z V,.
(3) 4r n=l Pi (f, N) represents a sample with N grains (individual orientations).It is assumed that each grain is represented by the 6-function.The integral kernel K(fi,., {p}), system of local supports {fj} together with the measurement grid F {yj.} reflect the experimental conditions of the given texture experiment.For the sake of simplicity, {.} can be referred to the system of elementary patches on PF produced by the detector window and K(y, , {p)) to the detector sensitivity.
(4) It is the sum of expectation of local weighted errors obtained by con- ventional summing (sensitivity level e 0.1).It can be reduced to P,(') h, 07)K07, )Tn, {P}) dw07) } 47r p 07) dw07) IJll pj' (5) where fP Prob{)7 Pgh, (37) dwO 7) (6)   and the random variable is the volume fraction of grains with the poles )Tn E fj.The first term in (5) does not depend on the sample statistics and represents the approx- imation errors.The second term appears due to sample statistics.In the framework of an equal grain model (Vn V0, n 1,..., N, V NVo) for a sample consisting of N grains we get (Luzin, 1997a) under the assumption that 1/llfll, )7 v, K(37, ., {p}) 0, jT# fj. ( 9) Further, we assume that the grain volumes do not have constant value and are distributed with some distribution density p(w).fp(w)aw= 1. (10) Next, the purpose is to show how this assumption changes formula (8).

CALCULATION OF THE R-VALUE IN THE CASE OF THE GRAIN-VOLUME DISTRIBUTION
Ensuing analysis is connected with the distribution of the random variable where Y]n 1 -Jc" 2 n t" + n, ]N-n Y]N Y]n ?']n+l "+" f/n+2 -I--- (12)   Here n is the integer number (n 1,..., N) and l, z,... ,ru are the independent random quantities identically distributed with the density p(x).For definiteness sake we take as this -distribution the gamma distribution cVxv-le-x p,,(x)-(u) u>0, 0_<x<o, (13) (I'(u) is the gamma function) with the following properties.The expec- tation and dispersion of this distribution are E{r/} a, D{r/} -ff (14) and the parameter w--D{r/---( 15) can be interpreted, in the sense of the grain-volume distribution, as the squared ratio of the average grain volume V0 to the spread parameter A V ofgrain volumes around the average value.The gamma distribution is chosen because it is good with respect to convolution Pa,v * Pa,2 Pa,v.+2.
(16) So, the quantities 2n and -]N--n have similar gamma distributions, onvxnve-ax, After intermediate calculations it turns out that the distribution o(x) of the quantity n is xnV ," p(x) B(nu, (N-n)u) (1 X) (N-n)u-0 X 1, (18) i.e. it is the beta distribution B(x,s, t) with the parameters s nu and

N+
This equation can be reduced to (8) simply putting AV =0.When N >> 6, Eq. ( 24) gives the sample statistical error 1-p2l PJ N(1 + 6). (25 It means that the sample statistical error in ( 24) is greater than a similar one in (8) by the factor (1 + 6).Thus, the influence of grain-volume distribution on the PF statistical errors can be interpreted as a decrease in the grain number (1 + 6) times.The main point of ( 24) is that the result does not depend on the grain-volume distribution features: it depends only on the first two moments of the distribution (expectation and dispersion).It should be emphasized that this is valid for the gamma distribution considered as the grain-volume distribution.It is conventional to work mainly with the grain-size distribution.It can be recalculated from the grain-volume distribution with the help of the relation v as3, where v is the grain volume, s is the grain size, a is the constant which depends on the grain shape.Thus recalculated grain-size distribution is also shown in Fig. for a 0.5 (a is equal to 0.52 for a spherical grain).It looks like a typical distribution of the grain size.
To illustrate the statistical properties of n, in Fig. 2 its distribution density (beta distribution) is plotted for several sets ofparameters.It can be clearly seen how the dispersion of n changes as the number of grains in the sample N increases.
For the described conditions the dependence of the R-value (for Pc00, cubic crystal symmetry) on the grid parameter A is presented in Fig. 3 for different numbers of grains.The texture model is quite simple (Luzin, 1997a), it is the one Gauss component {0 , 0 , 0} with HWHM 19.7.Two types of R-value curves are plotted: with and without taking into consideration the grain-volume distribution.

CONCLUDING REMARKS
1.The gamma distribution is very convenient for solving the posed problem and it provides the possibility of an analytical description of v. LUZIN the influence of the grain-volume distribution on the PF statistical errors.The other candidates (Lognormal, Maxwell, Weibull dis- tributions, etc.) do not provide the simple analytical description.
2. Dispersion ofgrain volumes leads to an increase in the statistical error of experimental PF (ODF) and is described by rather simple for- mulas ( 24) and (26).So, the quality of experimental PF for a sample with the finite number of grains N decreases because of the spreading of grain volumes. 3. Introducing the grain-volume distribution slifts the value of the optimal grid parameter to greater values and the minimal achieved R-value becomes higher.It is safe to assume that the optimal smooth- ing procedure (Luzin, 1997b) guarantees attainment of this minimal R-value (for the left branch of the R-value curves in Fig. 3).4. When the grain-volume (-size) distribution is experimentally found, in ( 24) and (26) the mean (value) and the standard deviation can be used instead of the expectation and the dispersion, respectively.

FIGURE 3
FIGURE3 The dependences of R-value on the grid parameter for different number of grains.(gsd,grain-size distribution).