Pole density

In Refs. [292, 293], the BEN treatment followed by the (100) growth was undertaken to identify the most influential parameter for oriented nucleation in the two-step process. The Si(lOO) substrate was placed on a graphite holder. The film characterization was done using the (220) pole density maxima in the (220)XPF diagram, and the FWHM was selected as an orientational parameter. 

Fig. 9-8 (111) pole figure for an imperfect [100] fiber texture. F.A. = fiber axis. Cross-hatched areas are areas of high (111) pole density. 

Because of its rotational symmetry a pole figure of a fiber texture displays redundant information, in the sense that the pole density along any longitude line (meridian) is the same as along any other. Thus a plot of pole density vs. angle between 0 and 90° is a simpler description of the texture for the texture shown in Fig. 9-8, such a plot would show a single maximum at 54.7°. 

A plot of R vs. a is given in Fig. 9-15 for typical values involved in the 111 reflection from aluminum with Cu Aa radiation, namely,/It = 1.0 and 0 = 19.25°. (Values of 100/i are tabulated in [G.l 1, vol. 2, p. 307] and values of 1 /R by Taylor [G.19].) Figure 9-15 shows that the integrated intensity of the reflection decreases as a increases in the clockwise direction from zero, even for a specimen containing randomly oriented grains. In the measurement of preferred orientation, it is therefore necessary to divide each measured intensity by the appropriate value of the correction factor R in order to arrive at a figure proportional to the pole density. From the way in which the correction factor R was derived, it follows that we must measure the integrated intensity of the diffracted beam. To do this with a fixed... 

Diffracted intensities must be divided by S, which is independent of n, to give values proportional to pole density. The correction is less severe (5 closer to 1), the larger the value of 9 it is therefore advantageous to measure a higher order of the hkl reflection measured in transmission. The specimen holder can be identical with that used in the transmission method. 

A transmission method yields pole densities covering the outer part of the pole figure, from a = 0 to about 50°. A reflection method covers the inner part, from a = about 40° to 90°. The pole densities are in arbitrary units, either directly measured diffracted intensities or corrected intensities, depending on the method... 

Once one set of data is normalized to match the other set, numbers proportional to pole density can be written on the pole figure at each point at which a measurement was made. Contour lines are then drawn at selected levels to connect points of the same pole density, and the result is a pole figure such as Fig. 9-19. Many, but not all, textures are symmetrical with respect to reflection planes normal to the rolling and transverse directions, and many published pole figures have been determined from measurements made only in one quadrant, with the other quadrants found by assuming symmetry, without supporting data. 

Fig. 9-19 (111) pole figure of alpha brass sheet (70 Cu-30 Zn), cold rolled to a reduction in thickness of 95 percent. Pole densities in arbitrary units. The outer parts of all four quadrants were determined experimentally the inner parts of the upper right and lower left quadrants were measured, and the other two constructed by reflection. The solid triangles show the (110) [fl2] orientation. Hu, Sperry, and Beck [9.17]. 

Diffracted intensities, proportional to pole densities, may be put on a times-random basis by comparing them with intensities diffracted by a random specimen [9.20]. The random specimen should be of the same material as the textured specimen and, for a transmission method, it should have the same value of fit if not, a correction has to be made that will depend on the transmission method involved. The random specimen itself is usually made by compressing and sintering a powder [9.11, 9.12]. The randomness of grain orientation in this specimen must be checked by determining its diffraction pattern with a diffractometer in the usual way the measured integrated intensities of all lines should agree with those calculated by Eq. (4-21). 

Fig. 9-20 (110) pole figure of recrystallized commercial low-carbon (0.04 percent) sheet steel, aluminum Mled, 0.9 mm thick. Pole densities in times random units. Determined by a reflection method from composite specimens see text under General. Bunge and Roberts [9.18]. 

As mentioned in Sec. 9-6, if a wire or rod has a true fiber texture, its pole figure will have rotational symmetry about the fiber axis and will resemble Fig. 9-8. We therefore have to measure pole density only along a single radius. The angle between the pole N and the fiber axis F.A. is usually called (j), rather than a, when dealing with fiber textures. 

The Field and Merchant method may be used to measure pole density, and two specimens are required to cover the entire 90° range of 0 ... 

Diffracted intensities are to be divided by W to obtain numbers proportional to pole density. 

When the diffracted intensities given by each method have been divided by fFand normalized in the region of overlap, we have a set of numbers / proportional to pole density. Figure 9-22 shows an /, (j) curve obtained in this way for the inside texture of cold-drawn aluminum wire. The peaks at = 0 and 70° are due to the strong [ill] component of the texture and the peak at 55° to a weak [100] component. 

By analysis of an I, (j> pole density curve we can (a) put pole densities on a times-random basis and (b) determine the relative amounts of the components in a double fiber texture [9.36-9.38]. 

Fig. 9-22 (111) pole density / (full curve) and / sin (dashed) as a function of 4> for a cold-drawn aluminum wire, reduced in area 95 percent by drawing, and etched to 80 percent of the as-drawn diameter. Final specimen diameter 1.3 mm, Cr Ka radiation, 222 reflection. Freda et o/. [9.35]. 

If Ir is the pole density of a random specimen, then n = /,(2w). Therefore... 

This relation is valid whether the n poles are distributed randomly on the sphere or in some preferred manner, and it enables us to find from measurements on a textured specimen. From experimental /, data we construct a curve of / sin vs. shown dashed in Fig. 9-22, determine its average ordinate, and find /,. from Eq. (9-16). Once Ir is known (14.3 units for this wire), the I, pole density curve can be put on a times-random basis (right-hand ordinate). (Because the angular aperture of the counter slit is not small relative to when is small, the true pole density I near = 0 can only be approximated [9.37]. We therefore extrapolate the / sin curve near = 0 rather than extend it to zero, as is mathematically indicated.)... 

Note that this result was obtained without making any use of the measurements made at 4> values less than 40°. Thus a complete pole density curve is not necessary for the evaluation of a texture, provided the texture is sharp enough to produce well resolved peaks in the high- region. This is a fortunate circumstance, because high- measurements require little or no specimen preparation. 

Figure 9-24(a) is an inverse pole figure for the inside texture of an extruded aluminum rod, showing the density distribution of the rod axis on a times-random basis. It was derived by a trial-and-error method [9.36] from pole density curves, as in Fig. 9-22, for the ((X)l), (111), and (113) poles. We note concentrations of the rod axis at [001] and [111], indicating a double fiber texture the volume fractions of the [001] and [ill] components were estimated as 0.53 and 0.47, respectively. Note that an inverse pole figure shows immediately the crystallographic direction of the.scatter. In this double texture, there is a larger scatter of each component toward one another than toward [011]. 

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... 

According to Equation (10) the diffracted intensity of a textured polycrystal is the diffracted intensity of the randomly oriented polycrystal multiplied by the pole density in the direction of the scattering vector in sample. The pole density Ph(y) is the unique function connected with the preferred orientation that is accessible to a direct measurement by diffraction. 

Fig. 3. Pole figures of rubrene films prepared at low deposition rate are measured at qz = 0.86A-1 (a) and qz = 1.19A-1 (b). The calculated pole densities of individual crystal orientations with (141), (131) and (121) are denoted by , and, respectively. High deposition rate films are measured at qz = 1,45A (c) and qz = 1.72A-1 (d). The calculated pole densities due to (001), (010) of the orthorhombic phase and (010) of the triclinic phase are denoted by v, a and o, respectively. The single high intensity spots are due to single crystalline mica substrate.

Normalization of a Pole Figure — Pole Density. As already explained one measurement of I i x) is proportionnal to the volume fraction of the material whose normals to a given family (hkl) are parallel to a given y specimen direction. The power of the x ray source, the size of the beam, the distance of the specimen to both the source and the detector, the reflectivity of the diffracting planes, etc. determine this proportionality factor. If one wishes to compare several pole figures either measured under different conditions or on different goniometers it is necessary to normalize the data once they have been corrected for the previously described effects. One defines, for that purpose, the pole density. The pole density Phi ( 7 x) reads,... 

With such a definition, a specimen without preferred orientations, i.e. a random specimen will have a pole density Phi ((/ , x) = 1 whatever the ((/ , x) considered direction and whatever the hi reflection considered. One can also note that the integral over the half sphere of the pole figure can be directly determined if the complete pole figure is available. As explained previously, this is not the case for pole figures measured in reflection mode which are necessarily incomplete. 

Once again there are two ways to obtain the required information which is here the normalization factor Ni. The first is again based on measurements of a random specimen under exactly the same experimental conditions as for the pole figure measurements. The pole density is then deduced through the relation,... 

Instead of measuring a random specimen, which is often difficult to fabricate, one can use a mathematical method. Several methods have been proposed to calculate the normalization coefficients. Most of them use the correlations which must exist between several hi pole figures of a given specimen [5-7], i.e. they use the fact a texture function exists which is related to the pole densities. This last point will be considered in the next section. Explaining these methods of determination of the normalization coefficients will require to many equations and the interested reader is refered to the above mentioned literature. 

A normalized experimental pole figure, i.e. a pole density, is a function defined on a sphere. In order to present a picture of such a function one uses first the... 

Figure 6 shows the (111) and (200) pole figures of a deformed aluminum sheet. Once again there are clearly preferred orientations but no simple description is yet possible for these complicated pole densities. 

There is of course a relation between the ODF of a given material and the pole densities or pole figures. This relation, called the fundamental equation of the texture analysis, has an integral form. 

Both the ODF, /( ), and the pole densities, Ph y) can be expanded on the basis of orthogonal functions which are well adapted to the problem. The ODF... 

The texture of a given material is then characterized with the set of the coefficients they are the unknowns in the texture analysis by using the harmonic method. Similarly the pole densities can be expanded into series. 

---