NEUTRON TEXTURE ANALYSIS OF MELT-TEXTURED YBCO BULK SAMPLES

Neutron texture measurements on YBCO bulk samples show a very 
sharp texture of the superconducting phase YBa 2 Cu 3 O 7-x with 
half-widths of less than 5°. Even with a rather coarse measurement 
grid of only 722 points per complete pole figure, satisfactory 
results for the recalculated (002) pole figures could be obtained. 
However, for a reliable calculation of a complete ODF, finer grids 
will have to be used. Due to the importance of a good alignment of 
the c-axes in the material, a quantitative analysis of the (002) 
pole figures, including an error estimation due to measurement 
grid and counting statistics, was made. An outline for the 
determination of a reliable background estimate is given.


INTRODUCTION
Bulk samples of the superconducting ceramic phase YBa2Cu307_x (orthorhombic, a= 3.821, b= 3.888, c= 11.693 for x=0.2 (JCPDS- ICDD, 1996)) with other phase additions (e.g.Y203, Y2BaCuOs, PtO2, CeO2) are potentially useful in magnetic applications.The levitation force between such an YBCO sample and a permanent magnet depends on what is referred to as the "domain structure" of YBa2Cu307_x (G6rnert, 1997).A domain in this sense is a part of the sample where superconducting currents can flow without being hindered e.g. by cracks, oxygen-deficient zones or large-angle grain boundaries.Each domain can be characterised by scanning the remnant magnetic field with a Hall probe.A domain may be smaller than a crystallographic grain.
While Hall-probe scans along a sample surface immediately measure the remnant magnetic field with good local resolution (cf.G6rnert,  1997), neutron texture analysis has the drawback of not providing any information about the spatial distribution of the potential domains in the sample.Neither can it detect intragranular cracks.It has, however, the great advantage of being a non-destructive means to analyse the whole (or at least a large part) of the bulk.With the total absorption cross sections given in the literature (Sears, 1992), the penetration depth of thermal neutrons in YBaECU307_x is found to be greater than 3 cm at A-1.344,, which is more than three orders of magnitude larger than for X-rays at the Cui wavelength.The Hall-probe method (Frangi   et al., 1994) has a penetration depth of the order of a few mm.
Although neutron texture analysis cannot reveal the actual domain structure, it allows to make hypotheses about the grain structure.Melt- textured YBCO samples frequently have grains with diameters in the cm scale (cf.G6rnert, 1997; Marinel et al., 1997).In terms of tex- ture analysis, the orientations of these grains are visible as strong texture components.Since grain boundaries with a misorientation angle greater than 10 always act as domain boundaries (Dimos et al.,  1988), an upper limit of the sizes as well as a lower limit of the number of domains in the sample can be determined by quantitative neutron texture analysis.

EXPERIMENTAL
All experiments were made on the TEX2-beamline of the GKSS Forschungszentrum GmbH with a neutron wavelength of A 1.344 using a primary Cu-(111) monochromator.Data collection time was controlled by monitor counts.Diffracted neutrons were registrated by a He 3 single detector with a 20 x 20mm 2 aperture.The initial primary neutron beam divergence of 52' was additionally reduced by a Cd mask that limited the primary beam diameter to 22 mm.The sample-detector distance was 120 cm for most of the measurements.
Two of the samples, named "fb 12" and "m209", were cylinders of 31 mm, height 15mm.A third sample, "hzj", was a cylinder of approximately the same dimensions that had been cut in two halves along its axis.
For most ofthe results presented here, a data collection grid consisting of 679 points per pole figure was used.The number of grid points per complete pole figure would be 722 since only half of the/3 circle was measured at a 90.Hence, this grid will be referred to as the 722 points grid.The solid angle attributed to each grid point was constant over the pole figure hemisphere, and the mean distance between neighbour- ing points was 5.62.For all samples, at least two pole figures were measured.The first one was the (002) pole figure, which shows no superposition of other Bragg reflections of the first order of the mono- chromator.The multiplicity of the (002) poles being only two, the number ofdistinct (002) pole density maxima is in direct correspondence with the minimum number of differently oriented domains in the sam- ple.Thus, a high concentration of (002) pole density is necessary for "good" YBCO bulk samples.The second one was the superposed (005) + ( 014) + (104) pole figure: It reflects once more the orientations of the c-axes, and the (014) and (104) pole density distributions can indir- ectly reveal the orientation of the a-and the b-axes.The reflections are relatively strong.A significant difference in the squared structure factors of the (014) and the (104) reflections exists, the former being stronger by a factor of about 2.5.Moreover, further pole figures were measured, such as the superposed (012) and ( 102) or (013)+(103)+(110) pole figures, or the (113) pole figure.There is also a squared structure factor difference of a factor of roughly two between (013) and (103).

RECALCULATION METHOD
The iterative series expansion method proposed by Dahms and Bunge (1989) was used to obtain recalculated pole figures as well as the C- coefficients ofthe orientation distribution function (ODF).This method allows to use a comparatively high series expansion degree even with rather few experimental pole figures.In the results presented here, the maximum expansion degree was lmax=34, using three or four experimental pole figures.Orthorhombic crystal and triclinic sample symmetries were assumed.The method also allows to include over- lapped pole figures when information about the relative powder intensities of the overlapping reflections is supplied.A maximum number of three reflections per pole figure could be treated here.

RESULTS
The agreement between the measured and the recalculated pole figures was quite satisfactory, with a range of RP values between below four and 14.A strong preferred orientation was found in all of the samples, the texture indices being 311,145 and 71 for the samples "hzj", "m209" and "fb 12" (for a definition of the texture index see e.g.Bunge, 1969).
In the plots, the angle denoted by "Phi" is the angle commonly referred to as "" (see Bunge, 1986).For the angle "a", the goniometer angle X will be used synonymously.The solid circles in the pole figure plots are at X 30 and X 60. (005) + ( 014) + (104) (right) of the sample "hzj".In Fig. (b), the latter has been decomposed into its three constituents by the iterative series expansion method.The strong pole density maximum near the centre is attributed mainly to the (005) reflection, but also to the (014) reflection.Correspondingly, in the recalculated (002) i.e. (005) pole figure, there is a slightly increased pole density at the positions of the ( 014) and ( 104) maxima which has no correspondence in Fig. l(a).Thus, the nearly central maximum in the recalculated (014) pole figure is probably an artefact of the recalculation which may have its origin in slightly wrong superposition factors for ( 014) and ( 005). (8)
Another feature of the texture is the overlap of the recalculated (014) poles with the (104) poles.The overlap of these two poles shows that there are two texture components which have a common c-axis orien- tation but differ from each other by a 90orotation around (001).This will be a result ofthe formation oftwin lamellae along planes ofthe (110) type, which is frequently observed in the material (Diko et al., 1996).However, Fig.(b) also shows that preferentially the (104) poles seem to form the maxima at qo 0 and qo 180 , and the (014) poles those at qo=90 and o= 270.Apparently, there is no complete symmetry concerning the distribution of the a-and the b-axes in the texture.
The most conspicuous o-dependence of the intensities of individual maxima is visible in the superposed (013) + ( 103) + (110) pole figure (see Fig. 2).However, as will be discussed below, this is more likely to be an effect of a too coarse measurement grid.
The samples "hzj" and "m209" have been reexamined in a later experimental period, using more accurate data collection grids, i.e. 2.5 x 2.5 for sample "hzj" and 2 2 for sample "m209".For reasons of limited allocated time, the measured areas were restricted to the maxima of the ( 013)+ (103) intensity distribution (these poles can be distinguished from the (110) poles with the help of the (002) pole figure ) as well as to those of (014) + ( 104) and ( 005).The aim was to determine the half-widths and intensities of these maxima with greater accuracy.
For the sample "hzj", Fig. 2 shows new results for the (013) and ( 103) maxima (left).On the fight side of Fig. 2, the (013) + ( 013) + (110) pole figure measured with the 722 points grid is shown.In the latter, there is an obvious difference in the intensities between the maxima at o 0/ o 180 and those at o 90/o 270.In the former, no such difference is apparent.Also, the integrated intensities of the four individual max- ima have been calculated.For the measurement with the 722 points grid, they were found to be about 25% lower at o--90 and o 270 than at qo 0 and qo 180.For the finer grid, they still vary about 10%, but do not show the same systematic dependence on o.
Neither is there a o-dependent periodicity ofthe integrated intensifies of the non-(005) poles in the superposed (014)+( 104)+(005) pole 8aal;)le: hz:J Nax.65.99 iq:)le: (013)+( 103  figure, neither for the more accurate data nor for those measured with the 722 points grid.It must then be concluded that the result of the recalculation based on the latter was strongly influenced by the apparent periodicity in the intensity distribution of the superposed (013)+ ( 013) + (110) pole figure.
But, as the results obtained with the finer grid show, this is rather an effect ofa too coarse data collection grid that must have missed the peaks of some of the ( 013) + ( 103) maxima and not a real feature ofthe texture of this sample.Further below, the effect of a too coarse measurement grid will be quantitatively discussed.
For the (013) + ( 103) maxima of the sample "m209", the results obtained with the finer grid are shown in Fig. 3.For the individual maxima, half-widths in A X and in sin(x)A99 have been determined.with a 2.5x 2.5 (sample "hzj") resp.2x 2 (sample "m209") measurement grid.The values are the average of AX and sin(x) Ao.For (014)+( 104) and ( 013 graphically from more detailed plots.Describing the maxima in terms of half-widths presupposes that they are single peaks which can be approximated by Gaussian bell curves.For some of the more scattered maxima, e.g.those for (013)+ (103) in the sample "m209", this is not strictly the case (of.Fig. 3).The discrepancy between the half-widths for (013) + ( 013) and ( 014) + (104) in the sample "m209" can also have its origin in the 20-dependence of the instrumental resolution for the angle X-The (013)+ ( 103) reflection was measured at 20= 28.4 and ( 014) + ( 104) at 20 33.4The instrumental resolution for X is found to improve with 20, which can explain why the half-width AX is larger for (013) + ( 103) than for (014) + ( 104), and thus leads to the higher average value.
A Quantitative Evaluation of Texture Components with a Common c-axis Orientation Given the importance ofa good alignment ofthe crystallographic c-axes (Jin et al., 1988), a quantitative analysis of the (002) orientation distribution has been made.In the experimental (002) pole figures of the samples "hzj", "m209" and "fb 12", as shown in Fig. 4, solid angle areas comprising the main maxima have been marked by dashed lines.The volume fractions of texture components having a c-axis orien- tation within one of these marked areas, denoted by "v/', can be cal- culated as follows (cf.Bunge, 1986): (1) Here, h (002).P(y-') is then the (002) pole density normalised over the measured hemisphere (thus, the normalisation factor in Eq. ( 1) is (1/270 instead of (1/470 as in (Bunge, 1986)).fi is the marked solid angle area in the (002) pole figure, f is a direction in the sample, given by the two pole figure angles (c, fl), and df is a solid angle element around .
(2)   angle areas where volume fractions arc to be calculated have been marked in the experimental pole figures.
The sum is over all grid points lying within the area and ')i and d is the element of solid angle attributed to the grid point "f'.The pole density PK(.) is then given by P7(g) 27r(I0)-BK) E(I(fik)-BK) d)Tk, (3) k where I0is the total intensity measured at grid point "f'.BK is the background intensity, which is assumed to be constant all over the pole figure .A geometrical consideration revealed that due to the low absorption ofthermal neutrons in our material, there was no necessity to correct the measured intensities for effects like absorption or secondary beam broadening at the sample and instrument geometries used here.The volume fractions of the solid angle areas marked in Fig. 4 are shown in Table II.The same area limits were used for the recalculated pole figures.All values are given in percent ofirradiated sample volume.
From Eqs. ( 2) and ( 3), an error analysis can be attempted.Its results were denoted by "or" in Table II.Two effects were taken into account: (a) the coarseness of the 722 points measurement grid leading to wrong values of the integrated intensities, and (b) the error due to counting statistics.The two aspects are considered as uncorrelated, and the resulting total cr is then given by (dd + O'c2ount) 1/2" The individual con- tributions of (a) and (b) are shown in Table III.
For the quantitative estimation of (a), a simulation method was used.It was assumed that the intensity data shown in Fig. 2 (left) and in Fig. 3 for (013)+ (013) were measured with the best possible grid resolution.Measurements using the 722 points grid were then simulated by TABLE II Volume fractions of the solid angle areas assigned in Fig. 4 for both experimental and recalculated (002) pole figures.Values are given in percent of irra- diated sample volume.Sigma is an estimated standard deviation, as explained in the  interpolating these data at the positions ofthe 722 grid points.A total of 4000 simulation cycles per pole figure was run.For each cycle, a relative orientation of sample and grid was determined by three Eulerian angles (qo,b, qo2) chosen randomly within the intervals [00,360] for [00,2.5]for b and [0 , 360] for qo2.This was meant to represent the arbitrariness of the /3=0-position as well as a possible slight mis- alignment of the cylinder axis for each of the simulated measurements.
As a result, it was found that the expected relative error (square root of variance divided by mean value) of the integrated intensities of the individual maxima was of the order of 10%.The same procedure was applied to the (014)+ (104) and the (005) maxima of both samples, leading to a similar value.
Due to the decrease in instrumental resolution at lower 20 values, the relative error for a (002) maximum, measured at 20= 13.2 , can be expected to be somewhat smaller than, e.g., for a (005) maximum appearing at 20 33.4.On new data, half-widths in A X were found to increase by a factor of 1.3 for (002) with respect to (005).Nevertheless, for the estimation of trgd, a relative error of 10% of the integrated (002) intensities will be assumed.
In order to apply this result to the calculated volume fractions, it was assumed that only in a few distinct areas of the (002) pole figure, like those marked in Fig. 4, the pole density is greater than zero.The sum of the volume fractions attributed to these areas will then be one.Each of these areas should contain exactly one maximum.All maxima should have equal half-widths, so that the effect ofa coarse measurement grid is the same for all of them.In practice, these conditions will not always be strictly fulfilled (cf.Fig. 4).Thus, it has to be stressed here that the O'grid values shown in Table III should be taken as estimates, and not as exact figures.
When, due to a coarse measurement grid, the relative error of the integrated intensity of each of the solid angle areas in a (002) pole figure is 10%, the inaccuracy thus introduced on the volume fraction "v[' attributed to area No. "' is (cf.Appendix): In order to estimate the effect of (b), i.e. the counting statistics, as a source of error for the determination of volume fractions, all the mea- sured intensities were taken as uncorrelated variables.Also, the back- ground intensity at each grid point was considered as an independent variable.The error-propagation rule was then applied to Eq. ( 2).
The Influence of an Inaccurately Determined Mean Background Value The crucial importance of an accurate knowledge of the mean back- ground intensity (assumed to be constant all over the pole figure) can be seen by inserting Eq. (3) into Eq.(2) and then deriving Eq. ( 2) with respect to BK, i.e. the mean background intensity (of.Appendix).The following quantitative relationship is also illustrated in Fig. 5: v ---.B----" \BK-1 1- (5) Equation ( 5) is approximately valid only for small values of ABK/BK, where the change of BK in the denominator can be neglected.It shows the effect of a shift in the background estimate, denoted by ABK, on the volume fraction attributed to the area No. "t", vi.As in Eq. ( 2), fli is the solid angle over which the intensity was integrated., i.e. l)i/2r is the fraction of the pole figure hemisphere occupied by the maximum in question.(I) is the mean value of intensity measured on the pole figure, including the background.As Fig. 5 illustrates, the change of v is always to the negative when ABK/BK is negative, i.e. when the background is estimated too low.In this case, the excess of intensity that remains in a (002) pole figure after the background correction will be interpreted as a texture component with randomly distributed c-axis.Sharp maxima FIGURE 5 Effect of a shift in the background estimate (ABK) on the volume fraction vi.Sharp maxima (low values of fi/27rvi) are more sensitive to shifts of the assumed background than diffuse maxima.Weak Bragg reflections measured with a high background will have a (I)/BK ratio only slightly greater than one and thus require a very accurate determination of the background.
with high pole densities covering a small solid angle are more sensitive to a background misestimation than diffuse maxima.Also, weak Bragg reflections measured with a high background will have a (I)/BK ratio only slightly greater than one and thus require a very accurate deter- mination of the background.Although our (002) pole figures are good examples for the latter case, the effect of a possible misestimation of the mean background could not be included in the quantitative error con- siderations in Table II because ofthe difficulty to guess the inaccuracy of our mean background estimate.
The determination ofthe background intensity is not a trivial task in a material such as YBCO, where many reflections of the main phase YBa2Cu307_x ("YBCO-123") overlap and where a second phase, Y2BaCuO5 ("YBCO-211"), is also present.In melt-textured material, the weight percentage ofYBCO-211 is usually ofthe order ofabout 30%.Ifa multiphase texture analysis is not intended ornot possible, an overlap ofa reflection ofthe 211-phase with a reflection ofthe.123-phase will increase the background in the pole figure ofthe latter, provided there is no strong texture of the 211-phase (recent publications (Chateigner et al., 1997;  Endo and Shiohara, 1997) hint to a preferred orientation of the 211- phase, qualified as "weak" by Chateigner et al. (1997); so this point will have to be investigated).A background value taken at a two-theta angle somewhat off the Bragg angle will then be too small.A possible way out of this problem is to take the background value directly from the measured pole figure data.This can be done with the help of a Poisson distribution fit, which works well when there are only a few and well-defined maxima in the pole figure, as it is the case for the (002) pole figure of the sample "hzj" (see Fig. 6).
If there are more maxima present, if different Bragg reflections overlap in one pole figure, or if the texture is smoother, a method has to be devised to carry out the statistics only over the points ofthe pole figure where there is no Bragg contribution ofthe 123-phase.These data points have then to be sorted out, which can be done by assuming that the maxima are coherent in terms of solid angle and can thus be "cut out" from the data set.
However, it should be noticed that trying to find the best approx- imation for the background by this method cannot distinguish between actual background and a texture component of random orientation, i.e. a so-called "fon", of the 123-phase.This is why separate background measurements should be done in any case.To decide whether or not there might be a "fon" component in the 123-texture, the measured background values and the result of the poisson fit for every pole figure can be compared.Ifthe latter is always greater than the former and if the difference is always in correlation with the structure factors of the respective Bragg reflections, then that can be a hint to the existence of a "fon" component in the 123-texture.then lead to Eq. ( 4).When there is more than one maximum in the integration area fj, then 2 with y Intpartial Int;, Crgrid (Intj)= 0.1 (yIntpartial) 1/2 where the sum is over the number of partial maxima in f.Then, agd(Int) < 0.1 Int because of Intpartial > 0. ffgrid(Vj) will then also be smaller.

Figure
Figure l(a) shows the experimental pole figures (002) (left) and

FIGURE 6
FIGURE 6Poisson distribution fit over counting rates in the experimental (002) pole figure of sample "hzj", where the maxima have been excluded.The vertical axis is scaled in terms of probability.Empiric values are denoted by rhombae, fitted values by triangles.
The average values are shown in TableI.These values were determined

TABLE
Full widths at half maximum values obtained

TABLE III
Contributions of counting statistics and measurement grid to the total expected error denoted by "sigma" in TableII.All values are given in percent of irradiated sample volume