Crystallite orientation distribution function

The diffraction measurement determines the orientation distribution f(0) or t( , O) of a pole. In most cases, however, we need to know the orientation distribution of crystallites in the sample. The question is then whether it is possible to deduce the crystallite orientation distribution once we have experimentally determined the pole orientation distributions for some finite number of different (hkl) poles. Before we can answer this question, we have to examine and define more clearly the relationship between the crystallite orientation distribution function and the pole orientation distribution functions. 

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 

Equation (3.82) follows from the orthonormal properties of spherical harmonics that can be stated as 

81) the summation with respect to / is for all integer values 0,1,2, etc. However, as mentioned earlier, x-ray and neutron diffraction cannot distinguish the positive and negative directions of a pole, and as a result t( , 4 ) is centrosymmetric. Since Pf1 (x) is an odd function of x when / is odd, the coefficient Tim evaluated according to (3.82) 

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V Vu w(a,p,y) w(N, r) W t) wq volume of a polymer segment. 6.1.1.3 scattering volume. 1.2.2 unit cell volume. 3.3.1 crystallite orientation distribution function. 3.6.3 end-to-end distribution of a Gaussian chain. 5.2.1 [5.12] slit-length weighting function. 5.6.1 constant value of W(t) with infinite slit approximation. 5.6.3... 

Figure 2. Inelastic neutron spectra from 36Ar monolayers adsorbed on Graf oil at 5 K (9). Curves plotted immediately below data are computed spectra for two different particle-orientation distribution functions (a) in-plane configuration with momentum transfer Q parallel to the preferred orientation of the graphite basal planes (b) out-of-plane configuration with Q perpendicular to the preferred basal planes. Curves at the bottom represent the calculated contribution to the observed spectra caused by in-plane scattering from misoriented crystallites.

Pseudo-affine model, the deformation process of polymers in cold drawing is very different from that in the rubbery state. Elements of the structure, such as crystallites, may retain their identity during deformation. In this case, a rather simple deformation scheme [12] can be used to calculate the orientation distribution function. The material is assumed to consist of transversely isotropic units whose symmetry axes rotate on stretching in the same way as lines joining pairs of points in the bulk material. The model is similar to the affine model but ignores changes in length of the units that would be required. The second moment of the orientation function is simply shown to be ... 

A complete description of the texture (or preferred orientation) is formulated as a probability for finding a particular crystallite orientation within the sample this is the orientation distribution function (ODF). For an ideally random powder the ODF is the same everywhere (ODF=l) while for a textured sample the ODF will have positive values both less and greater than unity. This ODF can be used to formulate a correction to the Bragg intensities via a fourdimensional surface general axis equation) that depends on both the direction in reciprocal space and the direction in sample coordinates ... 

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. 

A series of articles by Brandolini, Dybowski and co-workers [25-29] reported the use of NMR for static samples of PTFE to obtain information on orientation distribution of crystallites following deformation by tensile stress. The spectra were obtained using the MREV8 pulse sequence to remove homonuclear dipolar coupling so as to observe chemical shift dispersion as a function both of the draw ratio and of the angle between the draw direction and applied magnetic field (Fig. 18.5). The moments of the spectra were used [25-28] to determine the order parameters of the samples up to (Pg(cos 0)). The orientation distribution functions were then calculated... 

The crystallite orientation distribution w(a, / , y) can similarly be expanded in a series. Since the distribution is a function of three variables, the series expansion requires orthonormal functions more general than the spherical harmonics. The generalized spherical harmonics to be used are now of the form45... 

Figure 11 shows the change in the uniaxial orientation distribution function of crystallites, w((, 0, ri) with the extension ratio. This function was calculated using equation (4) with a finite series expansion, with the coefficients Wiq determined from equation (11), / taking a maximum value of 18. As can be seen from Figure 11, two populous regions appear in the orientation distribution i.e, one at 9 a little larger than 30° and rj of 90°, and the other at 0 a little smaller than 30° and rj of 0°. It seems... 

Figure 11 Change in uniaxial orientation distribution function of crystallites, w(, 0, >/), with the extension ratio of the bulk...

This polymer is of special interest due to its commercial importance as an electroactive material e.g. as piezo- or pyro-electric films). The molecule is highly polar and the material may exist in five crystal forms in three of which the molecular dipoles are parallel.It is evident from a study of the early literature (see ref. 10 for a review) that the dielectric behaviour of this polymer is the most complex of all the linear polymer systems. Studies have been made for partially crystalline samples having different crystal forms, different degrees of orientation and crystallinity, and electrical and thermal histories. Unoriented materials may have degrees of crystallinity up to 50% so they are composites of crystalline and amorphous regions with the attendant complications of relating the measured permittivities to the volume fractions of the phases, the permittivities of the phases, including and for the crystals, and the orientation distribution function of the crystallites, as has... 

Preferred orientation effects are addressed by introducing the preferred orientation factor in Eq. 2.65 and/or by proper care in the preparation of the powdered specimen. The former may be quite difficult and even impossible when preferred orientation effects are severe. Therefore, every attempt should be made to physically increase randomness of particle distributions in the sample to be examined during a powder diffraction experiment. The sample preparation will be discussed in Chapter 3, and in this seetion we will discuss the modelling of the preferred orientation by various functions approximating the radial distribution of the crystallite orientations. 

It should be noted that the orientation functions do not specify the distribution of crystallite orientation. More than one distribution may produce the same orientation functions. In the case of monoclinic polypropylene, the u-axis makes an angle of 99.3° with the c-axis, and Equation 3.11 does not hold. For convenience, it is customary to define an axis, which is not a true crystallographic axis but is one perpendicular to both the b- and c-axes. Then the orientation of the a-axis may be determined from Equation 3.11,/, and f. 

The arcing observed in the latter X-ray diffraction pattern is typical of moderately oriented crystalline material. For the PTV fibers, the orientation distribution of crystallites was about 18° (full width at half maximum) with respect to drawing direction for the PDMPV fibers, the full width at half maximum about 8°. A quantitative measure of the degree of orientation may be obtained from the angular half-width at half height of the intensity distribution. For PDMPV, the 8° angular spread corresponds to an orientation function... 

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