INVESTIGATION OF INHOMOGENEOUS SURFACE TEXTURES WITH CONSTANT INFORMATION DEPTH : PART 1 : FUNDAMENTALS

Amethod is described which allows to measure pole figures under the condition ofconstant information depth. This is achieved by using wand x-tilt simultaneously. In order to obtain satisfactory integrated intensities the use of a position sensitive detector is nearly indispensable. The functional relationship betweenwand X for constantinformationdepth and the transformation of Euler angles {o, , o2} to pole figure angles {/3} are given as well as the necessary intensity correction. The use ofo-tilt leads to blind areas in the centre of pole figures which can, however, be compensated by increasing the number of pole figures and using appropriate ODF programs. The method was tested with composite samples obtained by stacking aluminium foils withknown textures but with exchanging rolling and transverse direction ofthe uppermost foil. The results will be shown in the second part of the paper.


INTRODUCTION
Texture analysis is mainly based on the diffraction of X-rays which have a penetration depth in most investigated materials in the order of O-O0 tm. Hence, the method actually "sees" the texture ofthe material Corresponding author. 22 J. T. BONARSKI et al. in a layer of this order of magnitude. In many cases one is interested in the bulk texture ofthe material. Hence, a great number ofmethods have been developed on how the actually measured texture can be made representative for the bulk texture e.g. by placing the investigated layer in such a way that it cuts through different zones of the material.
On the other hand, the surface sensitivity ofX-ray diffraction can also be used to study texture inhomogeneities. One way of doing that is to remove surface layers step-by-step and thus to measure texture profiles. This method is, on the one hand, cumbersome and time consuming and, on the other hand, it must still presume that the texture is homogeneous in each investigated layer (Tomov et al., 1992;Tomov, 1994).
With the development of surface techniques, texture gradients have attracted great interest. Hence, methods are needed by which texture gradients can be measured non-destructively and, if possible, with a resolving power better than the penetration depth (Tarasiuk et al., 1994;Tizliouine et al., 1994;Chateigner et al., 1994). Considerations on this line are not new. Particularly, the penetration depth can be changed by changing the wavelength of the radiation used. If synchrotron radiation is available, this allows even a continuous variation of the penetration depth. It can then be tried to deduce the texture profile from a set of measurements with (continuously) varying penetration depth.
In the conventional method of pole figure measurement in reflection technique the sample is tilted through the angle X of the Eulerian cradle in order to obtain the pole figure values in the required range 0 < a < amax-Each pole figure angle c is thus measured with another sample tilt X-Penetration of X-rays must then be considered in the diffraction plane which forms the angle X with the sample normal direction. This leads to a different information depth from which the reflected intensity comes for each sample tilt angle. If the texture gradient to be measured is in the same order ofmagnitude then each angle a of the pole figure thus corresponds to another "actual" texture. Hence, the pole figure values are not consistent with each other. The same also holds for the pole figures ofdifferent (h k 1). IfODFs are calculated from such inconsistent pole figures the result can only be some (not very well defined) average of the texture over these varying information depths.
In order to avoid these errors, it is necessary to measure pole figures with constant information depth. This can be achieved by varying simultaneously the tilt angles X and which is easily possible with SURFACE TEXTURES 23 modern texture goniometers (Bonarski et al., 1994;Szpunar and Blandford, 1994). On the other hand, w-tilt is not as convenient as pure x-tilt in as far as it requires more sophisticated techniques for measuring integrated intensities from distorted diffraction peaks and transformation of the measured intensity into pole densities required for the subsequent ODF analysis. It has, however, been shown that pole densities can be obtained even from diffraction peaks with strongly distorted profile by using a position sensitive detector. It is the purpose of the present work to develop a method of pole figure measurement and subsequefit ODF analysis under the conditions of constant information depth and, eventually, to deduce the texture surface profile from measurements with different information depth.

Texture Inhomogeneity
The texture of a polycrystalline material is defined by the volume fraction of crystallites having a particular orientation g within the orientation element dg: =f(g); g {qOl bqo2}.
(1) Strictly speaking, this definition assumes that all crystallites in the total volume V of the material are included (Bunge, 1982b).
If we cut several smaller samples out of the material we may find a different texture in each of them. One reason for that may be an insufficient grain statistics. This will, however, not be considered here. (In this case the texture will vary "erratically" with the location r at which the sample was taken.) We consider a systematic dependence of the texture on the location r: dV/Vr f(g, r); r (x x2 x3}. (2) Thereby r is the location of the volume V, in which the texture is being considered (Fig. 1). Thenf(g, r) is called the local texture compared with the global texture of the material (Bunge, 1982a): We consider particularly the case of flat samples, the texture of which is homogeneous in the sample plane {Xl, x2}. Inhomogeneity ofthe texture is only assumed in the direction x3 i.e. the sample normal direction. Particularly, we admit that the texture may vary strongly in the vicinity ofthe sample surface. We further assume that we measure the texture by X-ray diffraction using a wavelength which has a penetration depth in the sample comparable with the range of strong variation of the texture.
In this case the variation ofpenetration depth with the Bragg angle t9 and the sample orientation angles {vX qo} is superposed on the inhomogeneity ofthe texture. This requires that both effects be treated together. We define pole figures as a function of the depth x3 below the sample surface ( Fig. 2): (4) P(hkl) (tX, X3) kl)_l_{a/} Thereby {a } are the polar angles of the diffraction vector S with respect to the sample coordinate system KA.
The beam reflected in the layer at x3 has passed the total length l(x3) in the sample. This length is proportional to x3 with a geometrical factor p SURFACE TEXTURES 25 f(g,x) X 2 FIGURE 2 X-ray diffraction in a sample with inhomogeneous texture near the surface. Sample coordinate system KA. The diffraction vector S defined by the incident and the reflected beam has the polar coordinates {a 3} with respect to KA. The texture in the layer parallel to the surface at the depth x3 is f(g, x3). where N(hkl is a normalization factor, F is the irradiated area on the sample surface. The total measured intensity is then given by the integral In the conventional pole figure measuring technique e(hkl)(OQ) is assumed to be independent of x3. The integral over x3 is then to be carried out only over the exponential term. Together with Fthis gives the absorption factor which is independent of the texture. If, however, e(hkl)(Ot) depends on x3 then Eq. (7) can no longer be split into two independent factors.
The pole figure angles {cz fl} depend on the sample orientation {w X qo} and if we assume a measuring technique using a position sensitive detector, then they may also depend on . Having chosen (h k 1), z9 is fixed; having also chosen {a fl}, there is still one degree of freedom in choosing the angles {w X qo}. Let us assume that this can be used to keep the geometrical factor p independent of(h k 1) and {cz/3}, then both sides of Eq. (4) can be integrated over x3 similar to the pole figure in Eq. (7). In order to keep the normalization conditions of pole figures and the texture function we define an averaged pole figure P(hkl) (00) #" p" P(hkl) (0/, X3)-e -px3 dx3, and analogously the averaged texture ](g) lz. p. f(g, x3) e -upx3 dx3.
The factor #p is the inverse of the integral over the exponential factor, and one obtains in analogy to Eq. (4) q g) db.
Equation (10) can be solved for f(g) in the same way as in the case of homogeneous textures with given pole figures P(hkt)(cB). Particularly SURFACE TEXTURES 27 the approaches of solving Eq. (10) with incomplete pole figures are also applicable. Hence, we can expect to obtain an integrated texture according to the definition, Eq. (9) which is, however, different from the global texture defined in Eq.
(3). Figure 3 illustrates schematically the influence ofthe exponential factor in Eq. (9) on the integralf(g). We may approximate the integral by extending it only up to the "information depth" X at which the exponential factor has decreased to e as is also illustrated in Fig. 3.
The information depth is defined (Bonarski et al., 1994) by where the value of e defines the assumed limit for the weakest beam which is still included in the integration. The integral Eq. (9) can then be approximated by Particularly, we shall use, later on, e 0.01. Then the neglected part of the integral is in the order of 1% which is better than the usually reached accuracy of pole figure measurement. The local texture f(g, x3) as a function of x3 can be deduced (under certain conditions) from the averaged texturef(g) by varying the factor #p i.e. the information depth X from which the major part of the average comes. This can be done in two ways i.e. by varying p with the help of an appropriate choice of the angles {w X o} or by varying # via different wavelengths, or by both methods together.

Texture Measurement with Constant Information Depth
In order to measure pole figures the Bragg angle Ohkl must be chosen and the diffraction vector Shkl must be brought into all required orientations {c/3} in the sample specified in the sample coordinate system KA (Fig. 2).
For the sake ofsimplicity, we consider only one Bragg angle Ohklat a time as is shown in Fig. 4

(a) The o-x Relationship for Constant Information Depth
With these definitions of the goniometer angles {w X qo} the geometrical factor p(ghktCOXO) is given by the expression sin (0 + to) + sin (0 w) P sin(O+ )-sin(O to) cos X" (13) It is independent of the rotation qo of the sample about its normal direction. The condition pconst., on the one hand, defines an information depth X according to Eq. (11) and, on the other hand, it fixes a functional relationship between the two sample tilt angles w and X (with 0 given by (h k 1)) according to Eq. (13) which may be written in the form O3 O.)(X)p:const" The influence of sample tilt X in the conventional symmetrical Bragg-Brentano condition (w=0) on the information depth X0.01 defined according to Eq. (11) is shown as an example in Fig. 5 for the (1 1) reflection ofA1 with CoKa radiation. It is essentially the cosine function. In this condition it is X c. Hence, each circle c const, ofa pole figure, measured this way, would correspond to another information depth X and hence to another texture average according to Eq. (12). Ifthe texture is inhomogeneous then such a "pole figure" would not correspond to any texture at all. In fact, it would not be consistent in itself.
Combining both X-and w-tilt according to Eq. (13) and inserting the p-values in Eq. (11) gives information depths according to Fig. 6. It is immediately seen from the figure that it is now possible to move along an "equilevel" path of this profile i.e. a path ofconstant information depth. The curve of Fig. 5 is a one-dimensional section along w 0 of this profile. Figure 7 shows some paths w w(X)p according to Eq. (14) for some p-values.
arcsin(cosw (sin X/sin a)) #0 180 arcsin(cos w (sin X/sin a)) ( > 0), where -7r/2 < w < +7r/2 and 0 < X < 7r/2. Thereby is related to X depth is considered. It is seen that the transformation relationship {X qo}={a fl} is independent of 9. This is due to the definition of w with respect to the symmetric Bragg-Brentano orientation of the sample. If a position sensitive detector is used as is indicated in Fig. 4, measuring several reflections at the same time, then this definition of o would require another o for each reflection. If we apply the transformation relationship (Eqs. (15a) and (15b)) to the conventional equal angular scan with respect to the goniometer angles X and qo with w=w(X) according to Eq. (14) then the scanning lattice AX, Aqo according to Fig. 9 (top) is "wound up" to spiral paths as is indicated in Fig. 9 (bottom). It is also seen that there is a blind area in the centre of the pole figure which cannot be measured under the condition p const, i.e. using the relationship a; a;(X). If this blind area is not too large, however, the ODF can still be obtained from incomplete pole figures of the type Fig. 9 (bottom) if only an appropriate ODFcalculation program is available (Bunge, 1982b;Pawlik, 1986;Dahlem-Klein et al., 1993).
This relationship is shown in Fig. 10.
where b(tg) is the peak shape function (Wcislak et al., 1993(Wcislak et al., , 1996. In the symmetrical Bragg-Brentano condition i.e. v=x=O the focusing condition can be quite well fulfilled. In the conventional texture measurement only x-tilt is being used. Thereby only one line of the SURFACE TEXTURES 37 sample remains on the focusing circle, upper and lower part of the sample move inwards and outwards respectively so that symmetrical peak broadening is observed. It has been shown that in this case the peak shape function could be quite well approximated by a Gaussian distribution function bhkl(9) bmax" e-((a-a)/'), where 00 tghkl -at AI9 is the centre of the shifted peak, tr is a broadening parameter depending on the sample tilt angle X as well as on the diffraction angle tghk given by the Bragg equation A 2dhkl" sin hkl" Using w-tilt additionally to x-tilt, broadening becomes asymmetric so that the symmetric Gauss function is no longer a good approximation. Nevertheless, in a first approximation the Gauss function can still be used as is shown in Fig. 11. The case of asymmetric conditions w # 0 is very sensitive to a correct sample adjustment in the goniometer as well as to the intensity distribution in the primary beam.

CONCLUSIONS
The conventional method of pole figure measurement using sample x-tilt leads to an "information depth" X which is different for different tilt angles X c. The information depths also depend on the Bragg angle OhkI.
If the texture is inhomogeneous in a depth comparable with the information depths the different pole figures used for ODF calculation are then no longer compatible with each other and the pole figures are even inconsistent in themselves because of the a-dependence of the information depth. Using a combination of X-and w-tilt, however, it is possible to obtain pole figure measurements with constant information depth. Pole figures obtained in this way correspond to an averaged texture f(g) (depending on the actually used information depth). If measurements with different information depths are combined, it is possible (under certain assumptions) to determine the depth profile ofthe texture. blind area in the centre of the pole figures. This can, however, be compensated by using appropriate ODF programs for incomplete pole figures. Furthermore, a geometrical intensity correction is needed depending on the tilt angle v. Finally, w-tilt leads to much more "distorted" peak profiles than x-tilt. It is, however, possible to "handle" such peak profiles with appropriate peak profile functions. In a first approximation, especially for small w and X angles, Gauss functions lead already to acceptable results. Pole figure measurement with peak profile analysis can conveniently be done with a position sensitive detector.