【正文】 to these directions. The effect of this is seen in the deep drawing operation that constitutes the first stage of the process to produce aluminium cans. As shown in Figure 2, a strong rolling texture results in the aluminium deforming more along the certain directions, resulting in “ears” forming at the top of the drawn cup. This can be a source of significant waste in can production if sheet texture is not controlled, as this excess material is unusable and is cut away. Figure 2 – Deep drawn aluminium cups, showing the effects of, left, weak rolling texture and, right, strong rolling texture on the deformation observed. How is Texture Measured? In modern materials science texture is normally measured by a diffraction of xrays, electrons or neutrons from the crystallographic planes. XRays The most mon method of measuring texture uses xray diffraction and is known as the “Schultz reflection method”. The apparatus used is known as a fourangle diffractometer or a Eulerian cradle, see figure 3. The source of xrays and the detector are oriented so that a particular value of 2θ (the angle between the source and the detector) is specified. This means that only diffraction from a single set of planes, with a particular spacing will be measured. The sample is tilted and rotated systematically, so that all angular orientations are investigated. When the lattice plane specified by the 2θ value is in the right orientation, it will diffract and the detector will record the reflection. For a polycrystalline material, the intensity of detected xrays will increase when there are more grains in that specific orientation, and the intensity for any angle is proportional to the volume fraction of crystallites with that orientation. Areas of high and low intensity suggest a preferred orientation, while constant intensity at all angles would occur in a random polycrystalline aggregate. Quantification of Microstructure and TextureTexture R Goodall, October 2022 13 Figure 3 – A schematic diagram of a four angle diffractometer for texture determination. In some materials the bulk and surface textures may be different。 the equatorial radius, a, and the radius along the axis of rotation, c. See Analysis of Size Distribution in Planar Sections If we have measured the sizes of a number of individual features, for example by the method of equivalent circles, we will be able to plot a histogram of their sizes and thereby obtain some information about the distribution of feature size. However, from what we have learned above, we cannot expect this distribution to be identical to the true distribution of feature sizes in 3D. As indicated by the schematic diagram in Figure 2, random planar sections through a microstructure containing spherical particles are more likely to intersect with the larger ones (from now on in this lecture, it will be assumed that we are working with second phases that approximate in shape to spheres, as this is simpler, although the same reasoning and similar analysis apply to particles of other shapes). Figure 2 – A schematic diagram of the intersection of various planar sections through a 3D distribution of spheres of different diameter. Quantification of Microstructure and TextureTrue Size and Size Distributions of Second Phases R Goodall, October 2022 3 Scheil2 developed a method of converting the observed distribution of circles in a planar section into the volume distribution of spheres. The central theory was that the observed circular section diameters will range in size from 0 to D, the diameter of the largest sphere. Circles of diameter D could only be observed when the section cut through the centre of the largest sphere. The probability that this would occur can be calculated for different distributions of different sized spheres, and the residual probability can be assigned to the appearance of the smaller size groups of circles. The process can then be repeated for the next largest circles and so on until a theoretical distribution of spheres is found that matches the observed distribution of circles. This method is effective, but leads to large uncertainties for the smallest size groups of spheres. Improved versions of the method have been developed by Schwartz3, Saltykov4 and Woodhead (reported in 5). These methods develop a matrix of coefficients ?(i, j) for the number of circles in size group i arising from spheres in size group j from probability distributions of randomly sectioning spheres. By matrix inversion coefficients ?(j, i) are formed, which can be used to derive the number of spheres in size group j from the numbers of circles in the various size groups i. Values of these coefficients are given in Tables and in Higginson and Sellars’ book. There is a slight difference in the coefficients given by Scwartz’s and Saltykov’s method and that of Woodhead, as in the former case the spheres are assumed to be of discrete sizes, whereas in the latter their sizes are spread over the range. These coefficients are applied to determine the number of spheres per unit volume in the size range j, NV (j). ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?kNkjjNjjjNjjjNAAAV ,...11,1 ??? ??????? (3) where ? is the size interval, NA (i) is the number of circles per unit area in the size range i and k is the number of groups. The effect of applying these methods to experimental data is shown in Figure 3. Figure 3 – The effect of the SchwartzSaltykov (SS) and Woodhead (W) analyses applied to experimentally measured distributions of particle diameters in a planar section. From the histogram in Figure 3 it is clear that the experimental measurements underestimate the true size of the features (as was seen at the start of this lecture) and also give the incorrect distribution. The Schw