An Investigation into the Accuracy of the Vector Method by Comparison with ECP Measurements

In an attempt to investigate the accuracy of the vector method for crystal texture analysis, a comparison has been made between the inversion result of the pole figure made by X-ray studies using the VM and the inversion result of the pole figure made by ECP. A comparison has been made between the inversion by the pole figure generated by direct measurement of orientations by ECP and the actual orientation distribution (measured by ECP) displayed in the same mode. The materials studied were recrystallized Fe-3% Si and Fe-50% Ni. The main findings were:


INTRODUCTION
In order to compare the vector method (VM) and the harmonic method (HM) for texture analysis, experimental pole figures made by X-ray measurement or model textures are commonly utilized (Kitagawa et al., 1982;Pospiech et al., 1984;Matsuo et al., 1985, Esling et al., 1987).
When experimental pole figures are used, the accuracy and completeness of the X-ray measurement is of extreme importance (Schaben, 1984).The method of making complete pole figures by combining the reflection and transmission pole figures is one of the most important factors in achieving high accuracy in the inversion of pole figures by the VM (Shimizu et al., 1984).When model pole figures are used, the method of generating a suitable intensity distribution for the model orientation is also an important factor in the accuracy of the inversion of pole figures (Matsuo et al., 1985).A Gaussian distribution is usually used.When experimental pole figures are utilized for the inversion either by VM or HM, it is rather difficult to say which method results in the greater accuracy as the "true texture" contained in the pole figure is not known.
If the "true texture" of the specimen can be determined by means other than X-ray measurements, then the real accuracy of the VM can be obtained by comparing the results of the VM analysis with this "true texture."Recently it has become possible to analyze orientations of grains larger than 5/m in diameter very accurately by the SEM-ECC-ECP technique.Computer processing allows rapid analysis (Shimizu et al., 1984).It is now possible to determine the accurate texture distribution in a specimen by analyzing orientations and sizes of several hundreds of grains by ECP without utilizing X-ray measurements.The texture analyzed by ECP doesn't include "ghost."In connection with this, the present authors have tried to evaluate the VM using texture obtained by ECP as the "true texture."There is, however, some limitation to the accuracy of the "true texture" compared with the texture obtained by the X-ray diffraction method, because it is impossible to measure all the orientations of grains by ECP over the same area measured by the X-ray diffraction method.The definitions of the terms used in the present paper is summarized in         The angles to were calculated from (HKL)(UVW) of each grain measured by ECP.
Orientation distribution is then shown in the same mode as the vector method taking account of the area of the each grain measured by ECP.
The angles (ai, fl) (i 1, 2, 3) of (100) PFX respectively.The inversions of pole figures of PFE and PFX by VM are called ODVE and ODVX, respectively.The "true texture" measured by ECP and displayed in the same format as VM is called ODE which is represented as a function of W, to and , which are directly obtained from orientations of individual grains and their areas by ECP.It is now possible to evaluate the accuracy of VM by comparing these three-dimensional orientational distributions, ODE, ODVE and ODVX, respectively.The terms p, to and are the same as those defined in the VM.
Comparison of ODE and ODVX was made by utilizing a Fe-3% Si specimen.As for the comparison of the ODE and ODVE, Fe-3% Si and Fe-50% Ni specimens were used.In order to detemine the accuracy of the inversion of the minor textural component by the VM, the results inverted by VM and those obtained by ECP were compared by considering the inte_nsity distribution of the Goss orientation ({110}(001)) rotated around the ND axis in a primary recrystallized Fe-3% Si.
2 Comparison of ODE and ODVX in Fe-3% Si alloy The material used was cold rolled and recrystallized Fe-3% Si sheet with a grain size distribution of between 30 and 300/,m as shown in Figure 4(c).The measurement area of the pole figure determined by the X-ray diffraction method (PFX) is 15 mm as shown in Figure 1.The area selected for ECP measurement was made up of nine blocks of the size about 5 mm x 5 mm to evenly cover the area measured by X-ray.From each block, the orientations of the 100 or so grains present were measured by ECP.The total number of grain orientations measured was 967, which corresponds to 7.1% of the area measured by the X-ray diffraction method.The pole figure obtained by X-ray measurement (PFX) of the area shown in Figure 1 is given in Figure 2. The inversion result (ODVX) of this pole figure by the VM is shown in Figure 3.
The (HKL)[UVW] of individual grains are obtained by ECP.These are shown in the form of the (100) pole figure and the inverse pole figures in Figure 4(a) and (b), respectively.The areas of the grains are not considered in these figures.When normalized, (HKL)[UVW] are expressed as (hokolo)[UoVoWo], p, to and are obtained as follows; p tan-l(-ho/ko) to tan-(-ko/lo) sin-l(-vo + k0w0(1 +/0)-1).The Box number of the individual grain, defined by the VM, is determined from p and to.Based on the Box number, and the area of the individual grains, ODE expressed in Ar(;; m(,asured Are;.me,;s,rcd by I).y ion TD Figure 1 Schematic illustration of the area measured by ECP and X-ray diffraction method.Numbers in the figure denote the areas in which grain orientations were measured by ECP.exactly the same format as the VM is obtained.ODE is regarded as nearly representing "true texture" in the present paper.ODE based on the measurements of 967 grains are shown in Figure 5 in the symmetrical mode.The pole figure (PFE) made from the (HKL)[UVW] of individual grains and their areas is shown in Figure 6.PFE (Figure 6) and PFX (Figure 2) are in good qualitative  agreement.This suggests that the texture measured by ECP is a good representation of the "true texture" in the area measured by the X-ray diffraction method.A comparison of the mean intensities of the individual Box numbers obtained by ODE and ODVX is shown in Figure 7. Fairly good agreement is obtained.This result indicates that the accuracy of the VM in generating the inverse pole figure is good.The intensity distribution along angle in the Box number 1, 29 and 36 equivalent to the minor, medium and major texture components in this specimen, are shown in Figure 8.With the exception of Box number 36, agreement is not good.The numbers of grains measured by ECP were 116, 43 and 17 for Box 36, 29 and 1, respectively.The low number of grains analyzed in Box numbers   1 and 29 in ECP measurements is a possible cause of the larger discrepancy at these positions.This suggests that it is not appropri- ate to evaluate the accuracy of the minor texture component by VM as the numbers of grains measured by ECP is not sufficient for the statistical analysis.ODVE inverted by VM from the (100) pole figure in Fe-3% Si (PFE shown in Figure 6 made by ECP) is shown in Figure 9. Figure 10 shows a comparison of the mean intensities of the individual Box numbers obtained by ODE and ODVE in Fe-3% Si.As seen previously in Figure 7, these data are also in very good agreement.

COMPARISON OF ODE WITH ODVE IN FE-50% Ni ALLOY
Three specimens (1A, 5A and 1D) of primary and secondary recrystallized Fe-50% Ni alloy were utilized in this investigation.
The chemical compositions and the experimental procedure for the preparation of the specimens have been described previously (Harase et al., 1987).The number of grains analyzed by ECP were 206, 515 and 275 for specimen 1D, 5A, and 1A respectively.The orientations of these grains are shown in Figures 11, 12 and 13 in the form of (100) pole figure and in standard stereographic triangles.In these figures each point represents one orientation and the grain area has not been considered.In contrast with Fe-3% Si alloy used in the previous section, each specimen has several strong preferred orientations.
Pole figures which take account of the grain area of these three  (a) (100) pole figure, (b) Standard stereographic triangle projections (inverse pole figures).The areas of grains are not taken into consideration in these figures.specimens (PFE) are shown in Figure 14.Specimen 5A shows very good agreement and specimen 1A shows a fairly good agreement, however, in specimen 1D the discrepancies at Box numbers 15, 20 and 21 are considerable.This is considered to be due to the sharpness of the texture in 1D for the following reasons.
In the process of making PFE from the original orientation measurements by ECP, the intensity of an individual point in  the pole figure is derived by averaging the intensities of several surrounding points.
The intensity distribution of the original orientation measured by ECP is given in a Gaussian distribution in PFE.While the intensity of a point in ODE represents the original intensity measured by ECP, the intensity of the same point in ODVE is affected by the method used to generate the pole figure.This is either done by inversion using the vector method or by  considering the intensity of a point in ODVE to newly correspond to the average of the intensities of neighbouring points.Therefore the discrepancy between ODE and ODVE might increase when the texture is sharper.The total intensities of the three Box numbers (15, 20 and 21) agree well having values of 14.8 and 12.9 for ODE and ODVE respectively.
4 INVESTIGATIONS INTO THE ACCURACY OF THE DERIVED MINOR TEXTURE COMPONENTS BY VM It has been pointed out elsewhere (Matsuo et al., 1985) that the VM is not so reliable in the analysis of minor texture components, however, these conclusions were derived from an investigation utilizing model orientations.It has already been shown that one of the weak points of the VM is the large discrepancy appearing between the inverted texture by the VM and the given texture when model textures are utilized.In this connection, {ll0}(uvw) orientations in a primary recrystallized Fe-3% Si was used as the minor texture component.This is considered to be very important for the development of secondary recrystallization texture in this material.The average grain size of the specimen investigated was 7.5 #m.PFX and ODVX are shown in Figures 18 & 19, respec- tively.A surface area of 10 mm x 10 mm was scanned by ECP mode perpendicular to the rolling direction (RD) at each 100 #m interval.
When the patterns indicating the {ll0}(uvw) were detected during scanning, an ECP of the grain was taken and its grain size was measured by ECC mode.As the mean grain size of the specimen is 7.5 #m the total area scanned by ECP is estimated about 8% of the area of 10 mm x 10 mm.For the analysis of ECP, the rotation angle around ND, RD and TD axes is measured from the ECP directly instead of obtaining (HKL)[UVW] from the pattern because the rotation angle of {110}( 001) is important in this analysis.Counting the areas and numbers of grains belonging to the same ranges of rotation angle, the relative intensities were calculated and compared with ODVX inverted by the VM.The intensity distribution of {ll0}(uvw) orientation around ND axis corresponding to No. 29 Box of 180 to 90 and 0 to 90 in Figure 19 is shown in Figure 20.In the VM, each Box has an approximately 6 spread in the p and the to angles.Therefore, the orientation rotated more than 6 around the RD or the TD axis is classified either in Box No. 22 or Box No. 30, respectively.Among the analyzed results by ECP, orientations within 6 from RD, TD axis were selected.The relative intensities around the ND axis are shown in Figure 21.The broken line in this figure is taken from Figure 20 for comparison.The shape of the relative intensities along angle for both lines are in good agreement.

CONCLUSIONS
The orientation distributions (ODE) in recrystallized Fe-3% Si and Fe-50% Ni have been analyzed using both ECP and inversion results of pole figures by the VM.The results of this comparison are: ODE made from orientations and area measurements of 967 grains in Fe-3% Si and ODVX inverted by VM from pole Figure2(100) pole figure (PFX) made by X-ray measurement of recrystallized Fe-3% Si.

Figure 3
Figure 3 Three dimensional texture representation (ODVX) inverted by VM from the pole figure (Figure 2).
Figure 4 Orientation distribution and grain size distribution of 967 grains analyzed by ECP and ECC.(a) (100) pole figure, (b) Standard stereographic triangle projections (inverse pole figures), (c) Grain size distribution.The area of the grains are not taken into account in the (100) pole figure and the inverse pole figure.
Figure $ Three dimensional texture (ODE) of Fe-3% Si represented in the mode utilized in the VM based on the orientation measurement by ECP.The areas of the grains and grain orientations are taken into consideration in this representation.

Figure 6 (
Figure 6 (100) pole figure (PFE) of Fe-3% Si made by orientation measurements by ECP.The areas of the grains are taken into consideration in this figure.

Figure 7 Figure 8
Figure 7 Comparison of the mean intensities in each Box between ODE and ODVX.
of the mean intensities in each Box between ODE and ODVE in Fe-3% Si.

Figure 11
Figure 11 Orientation distribution of 206 grains analyzed by ECP in the specimen 1D.(a) (100) pole figure, (b) Standard stcrcographic triangle projections (inverse pole figures).The areas of grains are not taken into consideration in these figures.

Figure 1
Figure 1 Orientation distribution of 275 grains measured by ECP in specimen 1A.(a) (100) pole figure, (b) Standard stereographic triangle projections (inverse pole figures).The areas of grains are not taken into consideration in these figures.

Figure 15
Figure 15 Comparison of mean intensities in each Box between ODE and ODVE (specimen 1D).

Figure 17
Figure 17 Comparison of mean intensities in each Box between ODE and ODVE (specimen 5A).

Figure 20
Figure20 angle about ND axis Intensity distribution of {ll0}(uvw) along angle analyzed by VM.
The pole figures made from ECP measurement and X-ray measurement are called

Table
Definition of symbols utilized in this paper The mean intensities of the individual Box obtained by ODE and ODVE are shown in Figures 15, 16 and 17 for specimens 1D, 1A and 5A respectively.