Appendix 1 TEXTURE A1.1 REPRESENTATION OF TEXTURE

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1 Appendix 1 TEXTURE The crystallographic orientation or texture is an important parameter describing the microstructure of a crystalline material. Traditionally, textures have been determined by x-ray diffraction and represented by pole figures, but in recent years, new methods of texture representation and determination have also become widely used. The purpose of this appendix is to provide the non-specialist with sufficient information to understand the discussions of texture in the book. In this appendix we examine how textures are obtained experimentally and displayed. Some of the experimental methods used, such as EBSD, are also important methods of quantitative microstructural characterisation, and these aspects are considered in appendix 2. A1.1 REPRESENTATION OF TEXTURE A simple treatment of texture representation is given in this section, and further details may be found in Hatherly and Hutchinson (1979), Cahn (1991b) and Randle and Engler (2000). Detailed accounts of texture representation with particular reference to orientation distribution functions (ODF) may be found in the works of Bunge (e.g. Bunge 1982), and Randle (2003) discusses texture representation with particular reference to microtextures. 527

2 528 Recrystallization A1.1.1 Pole figures A pole figure is a stereographic projection which shows the distribution of a particular crystallographic direction in the assembly of grains that constitutes the specimen. If it is to have meaning it must also contain some reference directions that relate to the material itself. Traditionally these directions refer to the forming process, e.g. the drawing direction in wires or the rolling direction etc. in rolled sheets. Idealized pole figures for a drawn wire and a rolled sheet are given in figure A1.1. In figure A1.1a, the drawing direction of a wire specimen is shown at the top and the distribution of <100> directions indicates that the grains have these directions at 45 and 90 to the wire axis. This axis, is therefore, parallel to a <110> direction and the texture is described as a <110> fibre texture. Figure A1.1b refers to a rolled sheet. The orthogonal specimen axes, rolling direction (RD), transverse direction (TD) and sheet normal direction (ND) are plotted with ND at the centre and RD at the top. Once again the distribution of the <100> directions is shown. There is a concentration of these directions at RD and in the plane defined by ND and TD at 45 to ND. Such a texture is described by the notation {110}<001> which states that planes of the form {110} are parallel to the surface of the sheet and directions of the form <001> are parallel to the rolling direction. The distribution of intensity in real pole figures is much more diffuse than shown above. The intensity distribution is usually represented by contour lines with values 1 to n times R where R is the value associated with a specimen of completely random orientation. The texture of heavily rolled copper (fig. 3.1) shows the typical spread observed. A1.1.2 Inverse pole figures The inverse pole figure is particularly useful for deformation processes such as wire drawing or extrusion, which require the specification of only a single axis. The Fig. A1.1. Idealized 100 pole figures for: (a) drawn wire, showing <110> fibre texture, (b) rolled sheet, showing {110}<001> rolling texture.

3 Texture 529 frequency with which a particular crystallographic direction coincides with the specimen axis is plotted in a single triangle of a stereographic projection (fig. 3.14). Rolling textures can also be described by inverse pole figures but in this case two or sometimes three separate plots are presented, one being used for each of the principal strain axes ND, RD (and TD if required). This method is used much more frequently for bcc steels than for fcc materials. A1.1.3 Orientation distribution functions and Euler space The description of texture by pole figures is incomplete. The information provided refers only to the statistical distribution of a single direction and there is no way of using this to obtain the complete orientation of individual grains or volume elements. A better description is given by the ODF which describes the orientation of all the discrete volumes in the aggregate. ODF analysis was developed originally for materials with cubic crystallography and orthorhombic sample symmetry, i.e. for sheet products. There have been a few studies of hexagonal metals but most of the literature and most of what follows refer to rolled materials with fcc or bcc structures. Some explanation of the formalism used to describe the ODF is necessary but no description of the mathematics involved in generating the ODF will be given. The interested reader is referred instead to the definitive work of Bunge (1982). Consider the case of a rolled sheet in which a particular volume element has the orientation (hkl)[uvw]. The orientation of this element can be described in terms of three Euler angles. Several different notations have been used to define these angles, but that of Bunge is most common and will be used here. The crystallographic axes are represented in the normal way in a standard projection (fig. A1.2) and the specimen orientation is specified by the reference directions ND and RD. The angles and 2 completely specify the direction ND. RD lies in the plane normal to ND and the angle 1 completely specifies the direction RD. Because three variables have been used to define (hkl)[uvw], the ODF can only be displayed as a three dimensional plot with the Fig. A1.2. Definition of Euler angles used for rolled sheet.

4 530 Recrystallization three Euler angles as axes as shown in fig. A1.3a. For rolled fcc materials the data are normally shown as a series of slices taken through the three dimensional ODF space at 2 ¼ 0, 5, ; as shown in fig. A1.3b. Equations A1 A9 define the relationship between Euler angles and Miller indices for cubic materials. In order to obtain consistent results, equations A1.1 to A1.6 should be used to obtain Miller indices from Euler angles, and equations A1.7 to A1.9 to obtain Euler angles from Miller indices. h ¼ sin sin 2 k ¼ sin cos 2 l ¼ cos u ¼ cos 1 cos 2 sin 1 sin 2 cos v ¼ cos 1 sin 2 sin 1 cos 2 cos w ¼ sin 1 sin tan cos 2 ¼ k l tan 2 ¼ h k cos tan 1 ¼ lw ku hv ða1:1þ ða1:2þ ða1:3þ ða1:4þ ða1:5þ ða1:6þ ða1:7þ ða1:8þ ða1:9þ Table A1.1 gives the Euler angles for {110}<112> and {110}<001>, two of the orientations that are commonly used to describe the textures of fcc metals, and fig. A1.3b indicates where these occur in ODF space. It should be noted that the general Fig. A1.3. (a) Location of Euler angles in ODF space, (b) ODF sections showing location of {110}<112> (filled circles), and {110}<001> (open circles) orientations.

5 Texture 531 Table A1.1 Euler angles for some texture components. Component 1 2 {110}<112> {110}<001> orientation {hkl}<uvw> appears more than once in the customary cube of Euler space. The use of ODFs allows a more quantitative description of textures than is possible with pole figures. Although the interpretation of a full ODF (e.g. fig. 3.3) is not immediately apparent to the non-specialist, there are, in any material only a relatively few important orientations (e.g. tables 3.1 and 3.3) and these are readily identified in the ODF sections (e.g. fig. A1.3b). More importantly, the ODF allows the identification of texture fibres as shown in figure 3.4, and quantitative plots of intensity along these fibres (e.g. fig. 3.5) provide very detailed information. The volume fractions of any texture components (often defined as orientations within 10 or 15 of the ideal) may also be readily calculated from the ODF data. Such simple yet quantitative representations of the data (e.g. table 3.2, figures ) may be compared directly with theoretical predictions or may form part of the specification of an industrial product. Despite the benefits derived from the use of ODFs and their general acceptance, there are a number of disadvantages associated with the use of Euler space (e.g. Randle 2003). Each orientation appears three times in the conventional cube (table A1.1 and fig. A1.3b). The population of Euler space by a random array of orientations is very distorted. If ¼ 0 the orientation is determined by ( 1 þ 2 ) and all points in the plane ¼ 0 and having the same value of ( 1 þ 2 ) represent the same orientation. This is particularly confusing with respect to orientations of the form {001}<hk0>. Significant fibres in the texture often lie on curves in Euler space (fig. 3.4) and may be difficult to recognise. A1.1.4 Rodrigues-Frank space Some of the problems discussed above can be overcome by the use of other threedimensional representations and of these the most suitable appears to be that advocated by Frank (1988), and based on the analysis of Rodrigues (1840). A brief description

6 532 Recrystallization follows, but for more detailed accounts the reader is referred to the text by Randle (2003) or to Frank s paper. The concept of an angle/axis of rotation is widely used to describe the misorientation relationship between neighbouring grains ( 4.2). In order to express the absolute orientation of a crystal, the reference crystal is taken to be the standard cube crystal orientation. If the axis is defined by a vector L and the rotation angle by the required relationship is given by the so-called Rodrigues vector R ¼ L tanð=2þ ða1:10þ The three orthogonal axes R 1,R 2,R 3 define a Rodrigues-Frank (R-F) space in which all possible angle/axis combinations are found. There are 24 possible R vectors and in practice that with the smallest rotation angle is used. By definition this has the smallest R vector and it follows that all such R vectors lie close to the origin of R-F space. A full set of equivalent orientations lies in a fundamental zone of R-F space but in the case of high symmetry crystals only a small part of this zone is needed. Figure A1.4 shows the form of the fundamental zone for cubic crystals and details of the reduced volumes that suffice for high symmetry. The major advantages of this method of representation have been summarized by Randle (2003): Each orientation appears only once in the fundamental zone. Rotations about a common axis fall on a straight line. This means that the identification of fibre components is simple. The volume element of R-F space is much more homogeneous than is the case for Euler space. The axes of R-F space coincide with those of the specimen. It is important to remember that R-F space is three dimensional and although there has been some use of sections taken through the reduced zones, these are much more Fig. A1.4. The fundamental zone of Rodrigues-Frank space for holosymmetric cubic crystals. Internal volumes define 1/8th and 1/48th of the fundamental zone, (after Randle 2003).

7 Texture 533 difficult to comprehend than the sections of Euler space shown in fig. A1.3b, and this has limited the use of R-F space for the representation of bulk textures. In addition, unlike the methods discussed above the data cannot be directly extracted from bulk x-ray data (Becker and Panchanadeeswaran 1989). However, it does have some potential advantages for the representation of misorientation data (Randle 2003). A1.1.5 Misorientations The increasing availability and sophistication of equipment for measuring single orientations ( A1.3) has led to a considerable interest in the misorientations that exist across grain boundaries and the association of these misorientations with textures. The difference between a normal texture orientation and a misorientation is simply that in the former case the external axes of the specimen provide the frame of reference, whilst in the latter the axes of one of the grains serves this purpose. The misorientation parameters can be expressed in a number of ways. Misorientations may be expressed as Euler angles and displayed in Euler space as misorientation distribution functions (MODF). The axes of misorientation may be represented on inverse pole figures. In this case the angle of misorientation () is conveniently plotted on an axis orthogonal to the inverse pole figure, and the data presented as sections of constant. The misorientations may be represented in R-F space. A1.2 MEASUREMENT OF MACROTEXTURE A1.2.1 X-ray diffraction The most commonly used X-ray techniques are those developed by Schulz (1949). Most measurements involve materials that have been rolled or annealed and originally two separate methods of examination, involving back reflection and transmission techniques, were required to obtain a complete pole figure. Nowadays the transmission method is rarely used and useful pole figures (covering an area of up to 85 from the centre) are obtained by the back reflection technique. If complete pole figures are required these can be recalculated from an ODF which has been obtained from a number of partial pole figures. A typical specimen is some 25 mm square with a flat surface, and the specimen must be thick enough (>0.2 mm) to prevent penetration of the incident X-ray beam. The specimen is mounted in a two-circle goniometer (fig. A1.5) which permits simultaneously, a rotation through an angle,, about its normal and a rotation,, about an orthogonal axis that lies in the plane defined by that normal and the incident and diffracted beams. These beams which are restricted by a series of slits are set at the appropriate Bragg angles for diffraction from the required plane. The intensity of the diffracted beam is measured by normal counting methods and normalised to that obtained from a randomly oriented standard specimen. Because of the absorption effects that develop as approaches 90 this technique provides a partial pole figure that

8 534 Recrystallization Fig. A1.5. The reflection method for pole figure determination. extends only some from the centre. For a full account of a modern, computer controlled goniometer see Hirsch et al. (1984). Although most texture determinations are made with X-ray equipment and the Schulz back reflection method, there are some severe limitations involved. The most important of these is the small volume of material actually examined. The depth to which the incident beam penetrates (and from which the diffracted beams emerge) is governed principally by the wavelength of the X-rays used and the absorption coefficient of the specimen material and is rarely greater than 0.1 mm. It has been pointed out many times in this book that the most prominent feature of a deformed metal is the heterogeneity of the microstructure and it will be clear that there must always be some doubt as to whether or not an X-ray based texture result is truly representative of a rolled specimen. In many rolled products the texture varies through the thickness of the sheet, and in most cases texture studies are made on mid-plane sections. Because recrystallization does not necessarily occur homogeneously in such a material similar doubts must also apply to annealed specimens. The time to acquire the data depends on the material, the resolution required, and the strength of the texture, but typically a single partial pole figure is collected in 1 hr. The very high intensities of X-rays emitted by synchrotrons enable rapid data collection (Szpunar and Davies 1984), and it is possible to investigate changes taking place during the annealing of deformed materials. If an ODF is required, this is calculated by deconvoluting the data of 3 4 separate pole figures from a sample. These need not be complete and modern practice uses only the Schulz back reflection method. The separate pole figures are determined sequentially and in some cases the goniometer head is capable of holding several specimens so that continuous overnight data collection is possible. The raw data are used to derive an orientation distribution function, f, for a particular orientation, usually by a series expansion method, and the combination of all possible f values gives the ODF. There are a number of problems arising in the calculation of ODFs from pole figures. One of these is the so-called ghost problem which affected all of the early ODFs and manifests itself by the generation of components that are known not to be present in the

9 Texture 535 texture (Matthies 1979, Lu cke et al. 1981). In addition the peak intensities may be reduced by as much as 10 30%. This difficulty has its origin in the use of the series expansion technique to calculate the ODF from the experimental pole figures. Such expansions have both odd and even terms but assumptions made about symmetry led initially to the use of only the even coefficients and the appearance of the non-existent ghost peaks in the ODF. Ghosts in bcc textures are less significant than in fcc and the nature of the texture components is such that the ghosts appear only in high intensity regions. Because of this they cannot usually be recognised. Exceptions occur only for the {112}<110> and {001}<110> components. These problems do not exist if the ODFs are constructed from individual orientation measurements, such as from EBSD, as discussed in A A1.2.2 Neutron diffraction The availability of thermal neutrons with a wavelength of 1A provides an opportunity for the use of neutron diffraction in texture studies. Because the absorption of neutrons in most metals is low it is then possible to use large specimens. A steel specimen of thickness 10 mm will absorb about 20% of a typical neutron beam whereas a 0.1 mm specimen will absorb >90% of a similar X-ray beam. Neutron beam techniques are therefore useful for the examination of coarse grained materials and for gathering information from the full thickness of an inhomogeneous material. In some cases it is possible to examine directly the texture changes occurring during annealing (see, for example, Juul Jensen et al. 1984). A1.3 MEASUREMENT OF MICROTEXTURE There are many occasions when it is desirable to obtain data about the local array of orientations present in particular parts of the specimen, and the textures within small specified volumes are generally referred to as microtextures. By linking the orientation and spatial parameters this approach provides a more complete description of the specimen, and, as outlined in appendix 2 can also provide detailed quantitative microstructural data. There is a range of methods available, each with its own particular application, and further details may be found in the reviews of Humphreys (1988b), Randle (2003), Schwarzer (1993) and Randle and Engler (2000). A1.3.1 Optical methods The optical techniques used by geologists and mineralogists and applied to transparent non-cubic minerals are described in standard mineralogical texts. Using plane polarised light in a transmission optical microscope equipped with a universal stage (goniometer), the specimen is manipulated until extinction of a grain or subgrain is achieved, and hence the crystallographic direction parallel to the optic axis of the microscope is determined.

10 536 Recrystallization Surface films such as anodic films, whose thickness or surface topography are dependent on the orientation of the underlying crystalline material may sometimes be used to obtain information about orientations. In the case of cubic metals, such as aluminium, the high symmetry of the material makes it difficult to obtain unambiguous data from an anodised specimen (Saetre et al. 1986b). However for hexagonal metals such as magnesium or titanium, the orientation of the basal plane may be obtained with the use of a polarising reflection microscope (Couling and Pearsall 1957). A1.3.2 Deep etching This etching technique (Ko hlhoff et al. 1988b) has been used with considerable success by Duggan, Ko hlhoff and their collaborators (e.g. Duggan et al. 1993) to study copper and copper alloys. The heavily etched surface is examined at a magnification in the range X in an optical or scanning electron microscope, and a typical example is seen in figure The basis of the technique is that the {111} planes are attacked more slowly than others so that a relief structure of tilted {111} planes is developed. The lines of the internal structure within the grains define the intersections of the {111} planes and are therefore <110> directions. The various etched patterns are characteristic of the crystallographic orientation and may be readily identified to an accuracy of 5 and with a spatial resolution of 10 mm. A1.3.3 Transmission electron microscopy (TEM) Techniques for the determination of the orientation of small regions of a specimen in the transmission electron microscope from spot or Kikuchi line diffraction patterns have been well established for many years and are fully discussed in textbooks on electron microscopy. However, their use for the determination of local textures, which requires the acquisition and solution of many diffraction patterns has increased with the availability of computer based on-line techniques for the rapid solution of the patterns (e.g. Schwarzer and Weiland 1984). A Single orientations Transmission electron diffraction is particularly suited to applications requiring high spatial resolution, as electron microscopes can produce beams of less than 10 nm diameter. Thus the technique is suitable for the examination of very small cell or subgrain sizes in deformed materials. The crystallographic orientation may be determined from the diffraction spots, but because of relaxation of the Bragg diffracting conditions in thin specimens, the angular resolution is usually in the range 2 5 (Duggan and Jones 1977). The orientation may be determined to a much higher accuracy (<0.1 ) using the Kikuchi line patterns in a convergent beam diffraction pattern. The Kikuchi patterns are normally analysed according to the method of Heimendahl et al. (1964), a similar technique to that used for EBSD patterns (A1.3.4). However, because fully automatic indexing of TEM Kikuchi patterns is unreliable because of the large nonsystematic intensity variations in the patterns (see e.g. Randle and Engler 2000), the analysis is normally carried out semi-automatically on-line, and thus the number of measurements obtainable may be small.

11 Texture 537 A Pole figures A microtexture method which does not require the measurement of individual diffraction patterns has been developed by Humphreys (1983) and Weiland and Schwarzer (1984). Using this method, which is implemented on a TEM, a pole figure is obtained from a selected area, typically 5 10 mm in diameter, of a thin specimen. A eucentrically-mounted specimen is tilted through an angle of 50 in steps of 2, and at each tilt the intensity around a low index Debye ring is measured by scanning the beam over a transmission electron detector. The technique is fully automated and has a similar geometry to the transmission x-ray method. Using this method, partial pole figures are obtained from the selected areas in a few minutes. The technique is ideally suited to the study of highly deformed materials in which the number of cells or subgrains is too large for the acquisition of individual TEM patterns to be feasible. It has been used to study the orientations near second-phase particles and the texture of shear bands. A1.3.4 Electron Backscatter Diffraction (EBSD) As discussed above, the disadvantage of obtaining complete texture data (e.g. ODFs) from pole figures is that errors are introduced when the pole figures are deconvoluted to obtain the orientation distribution functions. However, if individual orientations are measured, then the ODFs can be obtained directly from a sufficient number of measurements. Methods based on TEM can only provide data from small volumes of material, whereas individual orientation measurements from EBSD in the SEM can be obtained rapidly from either small areas of interest or over the whole surface of a sample. With the improved spatial resolution for EBSD now possible in Field Emission Gun SEMs (FEGSEM), EBSD is likely to become the standard method of texture measurement in the near future. A The EBSD technique Electron backscattered diffraction (EBSD) is based on the acquisition of diffraction patterns from bulk samples in the scanning electron microscope, and although such patterns were first obtained over 40 years ago, it was the work of Dingley (e.g. Dingley and Randle 1992), who pioneered the use of low light TV cameras for pattern acquisition and on-line pattern solution, which stimulated widespread interest in the technique, leading to the development of commercially available systems; recent reviews include those of Randle and Engler (2000) and Humphreys (2001). A more recent innovation has been the use of EBSD in conjunction with Field Emission Gun Scanning Electron Microscopes (FEGSEM) (Humphreys and Brough 1999), and the consequent increase in spatial resolution has further extended the range of applications of EBSD. The EBSD acquisition hardware generally comprises a sensitive CCD camera, and an image processing system for pattern averaging and background subtraction. Figure A1.6 is a schematic diagram showing the main components of an EBSD system. The EBSD acquisition software will control the data acquisition, solve the diffraction patterns and store the data. Further software is required to analyse, manipulate and display the data.

12 538 Recrystallization Fig. A1.6. Schematic diagram of a typical EBSD installation in an SEM. EBSD is carried out on a specimen which is tilted between 60 and 70 from the horizontal, and a series of data points are obtained by rastering the beam across the sample. These data points may be plotted as pixels to form an orientation map, and several examples of this are presented in this book. Because the beam moves off the optic axis during such a scan, errors in the absolute orientation are introduced at very low magnifications and beam defocussing due to the highly tilted sample may lead to a loss of spatial resolution (Humphreys 1999b). For these reasons, beam scanning is generally limited to areas of less than mm. In order to avoid these errors when measuring larger areas, a stationary electron beam is used and the specimen is moved relative to the beam with stage stepping motors controlled by the EBSD software. The disadvantage of stage scanning is that it is much slower than beam scanning, and the time for stage movement is typically 1 second. In addition, the positional accuracy of stage scanning using a normal SEM stage is not high, and stage scanning is most suitable for scan steps larger than 1 mm. There are a number of factors which must be taken into account when deciding if EBSD can be successfully used for a particular investigation. The specimen The backscattered electron signal increases with the atomic number (z)of the material. The quality of the diffraction pattern increases with z and the spatial resolution may also improve with increasing z. The specimen surface must be carefully prepared (e.g. electropolished) so as to eliminate any artefacts introduced by sample preparation. The speed of data acquisition The time to acquire a data point during a scan depends on the slowest of three operations: The time required to obtain an analysable diffraction pattern. This depends primarily on the material and microscope operating conditions. The time required to analyse the pattern. This depends on the processing speed of the computer, the speed of the pattern-solving algorithm and the number of lines in

13 Texture 539 the pattern required for a solution. Software which recognises that subsequent similar patterns do not require analysis may lead to significant increases in speed. The time to reposition the beam or stage, which as discussed above is negligible for beam scanning but may be greater than 1s for stage scanning. Spatial resolution If the area of the sample contributing to a diffraction pattern contains more than one crystallographic orientation, e.g. a grain boundary region, a single crystal diffraction pattern is not obtained, the automated pattern solving routines may fail and the pattern will not be indexed. The spatial resolution therefore depends on the size of the electron probe and the nature of the material. Due to the tilt of the sample, the spatial resolution parallel to the axis of tilt ( A ) is typically around three times better than that perpendicular to the tilt axis ( P ). The small intense beam produced in a FEGSEM gives a significant advantage over the conventional W-filament electron gun. Angular resolution The absolute orientation of a crystallite, is typically obtained with an accuracy of 2, depending on the sample alignment and EBSD operating conditions. However, the relative orientation between adjacent data points is related to the precision with which the orientations of data points within the same crystallite can be measured. The corresponding angular resolution is 1, but this can be significantly improved by data averaging methods (Humphreys et al. 2001a). The current status of EBSD When routinely acquiring data for quantitative metallography in a standard SEM and using commercially available EBSD equipment, the most relevant parameters which affect the quantity and quality of the data are material, pattern acquisition time, effective spatial resolution and relative angular precision, and the current values of these parameters are summarised in table A1.2 for W-filament and FEG microscopes. Table A1.2 Summary of typical EBSD performance for various metals in W-filament and FEG microscopes. Sample and microscope type Spatial resolution (nm) A P Raw data Angular precision (degrees) Data averaging Time/pattern (s) Beam scan Al W FEG Brass W FEG iron W FEG Stage scan

14 540 Recrystallization A Obtaining textures by EBSD EBSD is most commonly used to obtain local crystallographic or microstructural information (see also appendix 2). However, it is being increasingly used to obtain the bulk texture from a sample. The sample, which should be representative of the bulk material, is often polished on the RD-ND plane because in rolled material this section samples the microstructure better than the rolling plane. A specimen which is 15 mm in the rolling direction, cut from 3 mm sheet in a material of grain size 50 mm would reveal grains on its surface. Diffraction patterns are obtained from a grid of points covering the entire specimen or a selected region. From these data the orientation distributions are obtained and these can be displayed as pole figures or ODFs, or alternatively the volume fraction of material approximating to selected ideal texture components may be calculated. If a texture representative of the bulk material is to be obtained then it is important that data are obtainable from all parts of the microstructure. A heavily deformed material may contain cells or subgrains which are below the resolution of a W-filament SEM, and this may result in an unacceptably low fraction of indexed diffraction patterns. In addition, if the cell size depends on grain orientation, certain orientations will be sampled more efficiently than others and measured texture will be incorrect. For such materials, it may be necessary to use a FEGSEM to achieve an acceptable level of pattern solution. If the technique described above is to be used to determine the bulk texture of a specimen then the number of data points required to produce a statistically significant orientation distribution function needs consideration. Experiments have indicated that the number of orientation determinations required is in the range (Wright and Kocks 1996, Hutchinson et al. 1999a). If these individual orientations are used directly, then the pole figures or orientation distribution functions may be noisy, and the data may be smoothed by convolution with a Gaussian of half width 1 5. Such a texture determination using 2000 points and carried out by stage scanning will take only 30 min, which should be compared with the 4 5 hrs required for the multiple polefigure x-ray analysis discussed in A We therefore conclude that for suitable materials, bulk texture determination by EBSD, in addition to being more accurate, may offer a significant time saving over conventional x-ray analysis. In many rolled materials the deformation or recrystallization textures vary through the sheet thickness. In such circumstances, suitable data grids on a single ND-RD section specimen may be analysed by EBSD to provide through-thickness texture data at a fraction of the time which would be required for conventional x-ray analysis of several specimens sectioned parallel to the rolling plane and ground to the required depths.