«Abstract We present a technique for determining the texture of a polycrystalline material based on the measurement of the orientation of a number of ...»
Determination of Texture From Individual Grain Orientation
John E. Blendell, Mark D. Vaudin and Edwin R. Fuller, Jr.
Materials Science and Engineering Laboratory
National Institute of Standards and Technology
Gaithersburg, MD 20899 USA
We present a technique for determining the texture of a polycrystalline material based on the
measurement of the orientation of a number of individual grains. We assumed that the sample
has fiber (i.e. axisymmetric) texture and that the texture can be characterized by a function (the March-Dollase function) with a single parameter. We simulated a large number, N, of orientation data sets, using the March-Dollase function for a total of five different texture parameters, rinit.
Using the maximum likelihood method we solved for the texture parameter, r′, that best fits each simulated data set in order to determine the distribution of r′ and evaluate the precision and accuracy with which r′ can be determined. The 90% confidence limits of the ratio r′/rinit varied as N-½ but were independent of rinit. Using the texture of slightly textured alumina as determined by x-ray diffraction we calculated the 90% confidence limits for measurements of 131 grains. The orientations of 131 grains in textured alumina were measured by electron backscatter diffraction and the texture determined from those measurements lay within these 90% confidence limits.
Introduction With the development of techniques for the rapid determination of the orientation of single grains in the surface of a polycrystalline sample1 there is the potential for determining the preferred crystallographic orientation, or texture, of a sample. Typically the crystallographic texture of a polycrystalline sample is determined by diffraction techniques. Pole figure 2 and rocking curve analysis3 are the most commonly used techniques, but for samples that exhibit 1 fiber texture, where the preferred orientation of the crystallites is axisymmetric about a sample axis (texture axis), Rietveld analysis of conventional theta - two theta scans can also be used to measure the texture4,5. All these techniques give the average texture over the area illuminated by the x-ray beam. There are cases where it may be required to measure texture over a smaller scale, either to characterize small specimens or to investigate local texture variations in a larger specimen, and in these cases, a smaller probe, such as an electron beam in a scanning electron microscope, can be used. The questions that arise are: how to compare the texture results from different techniques; and how many individual grains must be measured in order to achieve the desired accuracy and precision. In order to compare the measurements on individual grains to the results of Rietveld analysis of theta - two theta x-ray diffraction (XRD) scans, only samples with fiber texture will be considered.
Texture arises when the crystallites that make up a polycrystalline sample do not have a random arrangement of their crystallographic orientations. To describe axisymmetric texture, we define a crystallographic direction (the preferred orientation direction) that is preferentially aligned with the texture axis. The preferred orientation is typically specified as the normal, n, to a specific crystallographic plane, (hkl); the texture axis is usually a sample direction or processing axis.
The conventional measure of the degree of texture is the ratio of the volume fraction of crystallites in a textured sample with n at a specific orientation to the texture axis to the same volume fraction for a random (or untextured) sample. This ratio is called the multiple of a random distribution, MRD. If we assume that the diffracted intensity for a specific Bragg reflection in a diffraction pattern is proportional to (among other parameters) the volume fraction of crystallites correctly oriented to diffract into the Bragg peak, then one measure of the MRD profile of a polycrystalline sample is to compare the intensity of the diffraction peaks to the intensity of the same diffraction peaks for a random or untextured sample. The functional relationship of MRD to the angle between the texture axis and the orientation of the crystallites can be modeled for the case of axisymmetric rod or disk shaped crystallites.
2 The model MRD function chosen for this work is the March-Dollase function5 that has been incorporated in software packages (such as GSAS6 - a Rietveld technique) used to analyze powder diffraction data. In these techniques, a number of functions which model the experimental and sample conditions are used to calculate a model diffraction pattern. The parameters in each model are adjusted to minimize the error between the experimental data and the pattern calculated from the model functions. The function, P(r, α), models the MRD for
crystallites with orientation n at an angle α to the texture axis using a single parameter r:
P(r, α ) = [ r 2 cos 2α + r -1 sin 2 α ] −3 / 2 (1)
We set M = P(r, 0) = r-3; M is the MRD at α = 0, and it is frequently used as a parameter for describing crystallographic texture. For a random sample the probability of a given crystallite orientation is uniform over orientation space. The probability of a volume element with n lying at an angle α to the texture axis is proportional to sin α. Thus, for the textured specimen, the probability of a given crystallite orientation is P(r,α) sin α.
Recently developed techniques allow the measurement of the crystallographic orientation of individual grains in the surface of a polycrystalline sample1. Such data can be used to determine the texture of the sample by fitting the data to Eq. ( 1 ). One method of doing this requires the data to first be put into bins to form a histogram. The March-Dollase equation is then fitted to the histogram data by allowing r to vary. This technique may lead to poor results if the number of grains measured is low; in addition, the results are dependent on the bin width.
An alternative technique for fitting the experimental data is the Maximum-Likelihood Method 7.
If we assume that the sample MRD distribution is given by the March-Dollase function, Eq. ( 1 ), then the probability of making any single grain measurement resulting in a orientation of αi, is P(r,αi)sin(αi), which we will call the March-Dollase distribution. The measurements of the orientations, αi, of the individual grains (the angle of the crystallographic axis to the texture axis) give a set of N orientations that are assumed to be randomly taken from the entire sample orientation distribution with parameter value r. We assume that the texture of the sample is homogeneous and that the grains on the polished surface examined in the SEM are representative of the bulk grains. While this may not be true for a fired surface, it will be the case for an internal 3 section. Given an experimentally measured (or simulated) set of orientations, taken from a population with an unknown degree of texture, r, the problem is to determine an estimate of the degree of texture of the entire sample orientation distribution. The estimate, r′, is the one that maximizes the likelihood that the sample set came from a population with degree of texture, r′.
The relationship of r′ to r and the confidence limits on r′ will be investigated, and related to the number of orientation measurements in the set.
In the method of maximum likelihood, the value of the estimator, M′, that has the highest probability is assumed to be the best value for M, the parameter for the whole population. In order to find the maximum of the likelihood function, the derivative of Eq. ( 4 ) with respect to M′ is set equal to zero and then solved for M′.
Simulation In order to determine the conditions where the maximum likelihood method yields a better estimate of the texture than fitting the data to a histogram, and to determine if there is any bias in the technique, we have simulated experimental results using a Monte Carlo method. First a random set of angles, αi, is chosen from a March-Dollase distribution with a given texture parameter, rinit or Minit. Then the maximum likelihood technique is used to find an estimator of the texture parameter, r′, for that set of angles. This process is repeated a large number of times and the average and distribution of the texture estimator is compared to the initial texture parameter, rinit.
results in a set of angles that fit the March-Dollase distribution.
Eq. ( 6 ) can be solved and inverted to give cos(αi) for an initial texture parameter, Minit.
The variable in Eq. 8 is the ratio M′/Minit, therefore, the results are expected to fall on a master curve independent of N or Minit. We also fitted the simulated data by putting the data in bins and fitting the resultant histogram. Bin widths were varied from 15° to 3.75° and the March-Dollase distribution was fitted to the data using a least squares analysis.
Results Sets of αi were generated for M=8, 27, 64 and 125 (corresponding to r = 1 2, 1 3, 1 4, 1 5 ). The sets contained 10, 20, 50, 100, 200, 500, 1000 or 10000 grain orientations. For each set, M′/Minit was found using Newton’s method to solve Eq. ( 8 ) and this was repeated 10000 times. The mean of M′/Minit and the 5% and 95% limits were determined and are shown in Figure 1. It is seen that the data normalized by the initial Minit of the simulation lie on a single curve. Also, there is a bias in the calculated values of M′ / M init for small numbers of grain orientations measured. For 10 orientations measured, the maximum likelihood method gave an estimate of M′ / M init that is 4% too small but the error decreases to 2% for 20 orientations measured and decreases to less than 1% when more than 40 orientations are measured.
The 90% confidence limits are seen to decrease as the number of orientations measured increases. The accuracy (standard deviation) of M′/Minit was found to vary approximately as N-½ which suggests that the main source of error is random error and not a systematic error due to the maximum likelihood technique. The slope of the standard deviation as a function of N was
-0.518. When only the values for more than 100 orientations are used, the slope decreased to
6 For fitting the March-Dollase distribution to a histogram of the data, the results were found to be dependent on the bin width chosen. The value of M′/Minit was always smaller than the corresponding value from the maximum likelihood method except for the case of a 15° bin size and an Minit value of 125 (rinit =0.2). Thus, the results were dependent on the initial texture value chosen for the simulation and did not fall on a single curve when normalized by Minit. When both the bin size and data set are small, it is likely that some bins will have no orientations which makes the fitting inaccurate. When analyzing highly textured samples, the orientation density may change significantly across the bin so that the center of the bin will not accurately represent the average value of the data in the bin. While there are methods for adjusting the bin size and location based upon the data, these methods may introduce artefacts which can be avoided by using the maximum likelihood method.
The maximum likelihood method was used to calculate the degree of texture for a set of 131 grain orientations measured using backscattered Kukuchi patterns (EBSP) generated in a SEM8.
The sample was a polycrystalline Al2O3 substrate (SRM 19769) and the fired surface was examined. The grain size was from 1 µm to 10 µm and the orientation measurements were taken every 100 µm, so that the sampling position was chosen at random, no grain was sampled twice and there was no knowledge of the size of each sampled grain. The EBSP data was analyzed using both the maximum likelihood method, and by putting the data into bins and fitting the March-Dollase distribution to the resulting histogram. From the histogram data, M′=6.81 (r=0.5277); and from the maximum likelihood method, M′=6.20 (r=0.5443).
For comparison, the sample texture was also measured using an x-ray technique. Measurement of weak  texture in Al2O3 cannot simply be performed by measuring the intensity diffracted by the basal planes using the 0006 or 000.12 peaks since both those peaks have low structure factors and are extremely weak; therefore they can only be used in rocking curve or single pole figure measurements on alumina samples with considerable texture. Therefore, the texture was measured by performing a Rietveld refinement of standard theta-two theta x-ray diffraction data using GSAS. From the Rietveld refinement, the texture parameter M was found to be 4.28 (r=0.616) indicating that the sample has some small degree of  texture for a texture axis normal to the surface of the substrate.
7 In order to estimate the confidence limits, the simulation was run for the case of 131 orientations with a Minit of 6.20. The average of 10,000 iterations was M = 6.14 (r=0.5460) and the normalized 90% confidence limits for 131 orientations are M′(5%)/Minit= 0.723 and M′(95%)/Minit= 1.373. This implies that based on the measured M′, the true M of the sample, with a 90% confidence limit, would lie between 4.46 and 8.00. It is seen that the M value determined from the measurement of the orientation of individual grains is above the measured x-ray M value. For an Minit of 4.28 (r = 0.616) the simulation yielded an M of 4.07 (r = 0.626) with 90% confidence limits of M = +2.23, -1.29 (r = +0.1504, -0.1278, respectively). Thus for measurements of 131 grains from a sample with a texture parameter of 4.28, 90% of the time, the measurements would lie between M=2.78 and M=6.30.
It is seen that the single measurements we have made is within the range of expected values of the texture parameter for the sample, but it is at the high end of the range. This reflects the errors associated with the small number of grains and also may be due to the differences between the surface grains measured by EBSP and the grains measured by x-ray diffraction. Thus the discrepancy between the x-ray and EBSP measurements of the texture parameter is within the expected variation of the techniques and does not indicate a difference in texture.