Pole distribution

The function Fh(y) defined by Equation (11) is called the reduced pole distribution (pole figure). Hereafter we will call it, simply, the pole distribution (or pole density), because />i,(y) will be used very rarely. The pole distribution is centrosymmetric and for crystal and sample symmetry higher than triclinic it... 

The traditional method to measure the pole distributions becomes unsatisfactory if the peaks are overlapped. This happens for low symmetry compounds or when the sample contains many phases. In addition, by using a position sensitive detector or neutron time-of-flight diffraction, a large segment or the whole pattern can be recorded simultaneously, and using only a small number of peaks, a large volume of information is lost. To eliminate these drawbacks, Wenk, Matthies and Lutterotti proposed a combination of the WIMV procedure (or other inversion method) with the Rietveld method - more exactly with the Le Bail (Chapter 8) routine for peak intensity extraction. In this combined method it is presumed that the structural parameters, or... 

Those interested mostly in structure determination from powder diffraction see the texture problem differently. The presence of the preferred orientation makes a good pattern fitting difficult or even impossible and, consequently, a procedure is needed to correct for the texture effect in the Rietveld codes. For that it is not necessary to find the ODF, but to have a reliable model of the pole distribution whose parameters are refined together with the structure and other parameters. 

Later Von Dreele implemented the general description of texture by spherical harmonics in GSAS. Von Dreele proved that, by using this description, beside the robustness of the texture correction in the Rietveld method it is also possible to perform a reliable quantitative texture analysis. He measured by neutron time-of-flight diffraction a standard calcite sample previously used for a texture round robin. The patterns from different detector banks and sample orientations were processed by GSAS, refining the harmonic coefficients simultaneously with the structural and other parameters. Six pole distributions calculated from the refined harmonic coefficients and used as input in the... 

In eonneetion with the implementation in the Rietveld codes, the Dollase March model and the spherical harmonics approach, for pole distributions determination, is developed in the next two parts. The problem of pole figure inversion is outside the scope of this chapter. 

According to equation (14.160) of Bunge, the pole distribution function defined by Equation (5) becomes ... 

The reduced pole distribution Ph(y) accessible to the diffraction measurement is obtained from Equation (30) according to Equation (11). When h is changed in —h, O pass into re — O and pass into tt + By using the property Equation (34) one obtains for p h(y) Equation (30) with a supplementary factor (—1) inside the sum over 1. Consequently, in Ph(y) only the terms with I even remain from Equation (30). It is convenient to rearrange Equation (30) to contain only real functions and positive indices m, n. By using Equations (29) and (33), P fy) becomes ... 

Selection Rules for all Laue Classes. The selection rules for the harmonic coefficients are derived from the invariance of the pole distribution to the operations of the crystal and sample Laue groups. The invariance conditions are applied to every function t/ (h, y) from Equation (36), as the terms of different / in this equation are independent. If we compare Equations (38) and (39) with (37) we observe that they have an identical structure. On the other side the sample and the crystal coordinate systems were similarly defined. As a consequence the selection rules for the coefficients of", flf", and respectively y) ", resulting from the sample symmetry must be identical with the selection rules for the coefficients A P and resulting from the crystal symmetry, if the sample and the crystal Laue groups are the same. The exception is the case of cylindrical sample symmetry that has no correspondence with the crystal symmetry. In this case, only the coefficients af and y l are different from zero, if they are not forbidden by the crystal symmetry. 

After application of the selection rules the pole distribution Equation (36) becomes ... 

Once the series convergence is reached the pole distributions can be calculated from the refined harmonic coefficients. The standard error of the calculated pole distribution is given by ... 

For the determination of the orientation distribution function it is necessary to record diffraction patterns successively by rotating the sample on a goniometer, as was shown in Section 12.1.4.2. The patterns must be measured in a large number of points (, y) scattered more or less uniformly on a hemisphere. It is difficult to evaluate beforehand how many such points are necessary for a reliable determination of the ODE. For a calcite sample previously used in a texture round robin Von Dreele recorded neutron time of flight diffraction patterns in about 50 points (T, y). All patterns were processed by GSAS simultaneously and six pole distributions calculated from the refined harmonic coefficients were further used as input in the WIMV inversion routine. An ODF similar to those obtained in the texture round robin resulted, but its dependence on the number of points in the space (T, y) was not examined. 

The SODF is not accessible directly in diffraction measurements but the strain pole distribution given by Equation (80). The strain pole distribution is for the SODF the equivalent of the pole distribution for texture with an important difference in place of one distribution, six separate SODFs in a well-defined linear combination [Equation (67)] are projected on the space ( P, y). The strain pole distribution given by Equation (80) contains as a normalizing factor the texture pole distribution Pbiy) that is not accessible to the diffraction measurements. This can be replaced by the reduced pole distribution because the peak positions for —h and h are not distinguishable. Therefore, the strain pole distribution becomes ... 

For textured samples the relation between the peak shifts and sin F becomes nonlinear and analytical expressions can be found only by approximating the texture pole distribution by d functions on some prominent sample directions. This could be a rough approximation, especially if the grain elastic interactions are not of the Reuss type, and numerical calculations of the diffraction stress factors are preferable. 

Arguments for recent developments of the spherical harmonics approach for the analysis of the macroscopic strain and stress by diffraction were presented in Section 12.2.3. Resuming, the classical models describing the intergranular strains and stresses are too rough and in many cases cannot explain the strongly nonlinear dependence of the diffraction peak shift on sin even if the texture is accounted for. A possible solution to this problem is to renounce to any physical model to describe the crystallite interactions and to find the strain/ stress orientation distribution functions SODF by inverting the measured strain pole distributions ( h(y)). The SODF fully describe the strain and stress state of the sample. 

The absorption correction (correcting for the effects arising from variations in irradiated volume as well as in beam path length within the sample) is important, and one way to ensure its accuracy is to obtain a reference sample having the same constitution and shape as the test sample but known to be isotropic and to make sure that the intensity measured with this reference sample is truly constant after the absorption correction is applied. From the intensity /( ) or /( , ) obtained after the absorption correction, the pole distribution r( ) or t( , d>), for a sample of uniaxial or biaxial orientation, respectively, is evaluated according to... 

For biaxial orientation the pole distribution t(, 4>) may be visualized as a density distribution defined on the surface of a sphere. The method of stereographic projection is then used to transcribe the density distribution from the spherical surface onto a sheet of paper. The contour map thus obtained is called a pole figure. 

The pole distribution t ) or t ( , O) and the orientation parameter/ are defined and measured for individual poles. The pole orientation parameter by itself does not specify the state of orientation of crystallites in the sample. The orientation of an individual crystallite in space is uniquely, defined when the orientations of at least two nonparallel directions associated with the crystallite are given. Thus, in an effort to specify the average orientation of crystallite in a sample, one may evaluate the orientation parameters fa and / of two nonparallel poles, a and b, and represent... 

The distribution of crystallite orientation can therefore be expressed as a function of Euler angles w(a, ft, y) defined for 0 < a < 2n, 0 < j3 < jt, and 0 < y < 2n. The question raised above can now be rephrased as follows (1) When the pole distributions /, , < ) are known for a finite number of poles j = 1, 2, 3,.. ., v, is it possible to derive w(a, /, y)l A related question is (2) When w(a, /J, y) is known, is it possible to calculate t(, < ) for any crystallographic plane (hkl) If the answers to these two questions are both affirmative, it follows that when the pole distributions for a finite number of poles are experimentally determined, the pole distribution for any other pole can be calculated. This last possibility is a useful one, since this implies that if, for example, the intensity of (001) reflection is too weak to allow direct experimental determination of the orientation distribution of polymer chain backbones, measurements of (100), (010), (110), etc., might allow the (001) orientation distribution to be derived indirectly. 

The observed pole distribution tj (0, 0) may be expanded in a series of spherical harmonics... 

Ackermanns formula improves the traditional pole assignment standard algorithm of SISO system. The open loop eigenpolynomial of system is not requisite. Ackermanns formula is used for designing the control law u, and the desired closed-loop pole is obtained from designed ideal pole distribution with differential transformation method. 

All case studies are devoted to capacitive poles distributed in space. 

Each loop contains the relations of mutual influence between the two poles distributed spatially that constitute the two masses. The loop in dotted lines features the influence of mass 2 on mass 1 and the loop in solid line features the other influence. The horizontal upper paths express the influences between state variables (global level) and the slanted lower paths express the reciprocal influences between spatially reduced variables that are the gravitational field g and the gravitational displacement G (for which the name gravitization is more appropriate see Remarks [1] and [2]). (For building a loop, these influences have been reversed This is a gravitational displacement... 

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