Simultaneous Measurement of Pole Figures by X-Ray Diffraction using a 20-Position Sensitive Detector

Pole figure measurement is usually carried out with texture goniometers equipped with a single detector. Thereby defocalization due to the sample tilt is taken into account by using broad enough receiving slits possibly with a defocalization correction. This method does not work in line-rich diffraction spectra with profuse peak overlap, particularly at high tilt angles. In this case the whole diffraction spectrum is needed for each pole figure point {a,/}. The |ntegrated intensities are then to be obtained by a deconvolution procedure. Because of tle necessary measuring time this is virtually impossible with a single detector. Using a pottion sensitive detector complete diffraction spectra can, however, be obtained for each pole figure point in reasonable times. In the present case, a 7 linear position sensitive detector was used. The line broadening as a function of the tilt angle was measured, coordinate transformation formulae for the back-reflexion case are given and it was shown that the so obtained results exhibit smaller experimental errors than the conventional single dctector measurements.


INTRODUCTION
Texture analysis is usually carried out by pole figures measurement followed by pole figure inversion i.e. the calculation of the orientation distribution function ODF (Bunge, 1982). The accuracy of the obtained ODF is the better the higher the number of input pole figures and the smaller their experimental errors. Pole figures are maidy being measured with X-ray texture goniometers using flat samples and they are often restricted to the back-reflexion range. Pole figure inversion is then based on |ncomplete pole figures. Conventional texture goniometers are equipped with a single detector, e.g. a scintillation counter, the receiving slits of which are chosen in sucl a way that the total integrated intensity of the chosen Bragg-peak is registered (Bunge, 1986). This requires rather broad slits in order not to cut offrand 257 parts of the Bragg peak especially at Idgher sample tilt angles X. This method works quite well with materials the diffraction spectra of which contain only a low number of well-separated peaks. This is the case, for example, in the basic cubic metals. If the number of diffraction peaks is higher, however, and their angular distance is smaller, then peak overlap may occur even at low or moderate tilt angles X. In some cases it is then still possible to use this method with smaller receiving slits, taking into account that part of the broadened peak may be cut off at higher tilt angles. This error is then being corrected with the help of s defocalization correction which can be determined in a random sample. But even this method has its limitations in more complex diffraction spectra, e.g. in ceramics or geological materials. In these cases it is necessary to measure thecomplete diffraction spectrum for each pole figure point {a,/) and to apply a deconvolution procedure in order to sepaxate the increasing peak overlap with increasing tilt angle. This has to be done with sma]] receiving slits, thus increasing the measuring time considerably. At the same time, these materials often show also lower diffraction intensities and because of lower crystal symmetry the number of required pole figures is higher. The necessary measuring times would thus be prohibitively lfigh.
This situation can be improved by using a position sensitive detector/PSD/which measures the diffracted intensities I(8) in a whole 20-range simultaneously. This method has been first applied to textures analysis in the case of neutron diffraction (Btmge et al., 1982). In neutron diffraction the sample is small compared with the dimensions of the diffractometer so thai. Bragg-Brenta focalization does not play a essential role. A curved PSI) covering a 20-rme of 80 o could thus be used without defocalization problems. In X-ray diffracti, using a fiat sample in the back-reflexion technique, however, violation of the ]Iragg-Brentano condition becomes the more serious the larger the 20-range of curw.d PSD is. This was first treated by (Heizmaxm and Laruelle, 1986).
In the present investigation we used a smaller linear PSD covering A2O = 7.0 o only. In this case the violation of the Bragg-Brentano condition is rather small. On the other hand, however, the simultaneously covered 20-range is nsuMly too small to measure all required pole figures at the same time. It is then necessary to move the detector into several angular positions. Tiffs can be reMJzed either stepwise or continuously.
If pole figures are being measured with a PSD the diffraction vectors of simultaneously measured reflexions are parMle] to different sample directions. Hence, in a fixed sample position, different pole figure points {a,} are thus measured in the different pole figures. This is not a serious problem which can easily be treated by an appropriate coordinate transformation from the angles {),' } of the Fulerian cradle to the pole figure angles {a,/3}. Nevertheless, the usual equal angular scan AX, A leads then to an equal angular scan Aa, Aft only in on....e of the pole figures. Hence, new scanning modes have to be applied in PSD texture measurement. With the small linear PSD used in the present work also this transformation is much smaller than in the case of a large curved PSD.
Hence, the small linear PSD seems to be a good compromise between reduction of measuring time on the one hand and increasing defocalization effects on the other.

THE X-RAY DIFFRACTION ASSEMBLY
The hardware system for the X-ray diffraction measurements used in this project, shown schematicly in Fig.l, consists in principle of two main functional units. A measuring unit contains the SIEMENS X-Ray Texture Difractometer DS00/TX equipped with the DACO-MP/TX microprocessor and a controlling-analysing unit based on the DIGITAL MicroVAX II multitask, multifunctional host computer with necessary peripheral equipment. The X-ray dif['ractometer contains a radiation block wherein a copper tube anode at the voltage of 40 kV with a nickel filter emits a characteristic A'= wavelength of A=1.54056 -A, a texture goniometer with a Eulerian cradle carrying a sample to be investigated and a detection block including a position sensitive detector together with the required electronics and a gas controlling unit. The texture goniometer used for these measurements allows a precise driving and supervising of all four goniometer circles. The horizontal circles of the diffractometer namely the 20-drive and the w-drive are equipped with step motors allowing a driving accuracy of 0.001 in the single step. The position sensitive detector is mounted on the 20-arm together the optimal operating conditions, i.e. gas press.ure 7.0 bar, gas consumption 0.2 1/h and detector voltage 3.4 kV, the following detection parameters are reached. The linear location resolution is better than 100 #m what corresponds to 0.01 on the 20-circle as the maximal angular resolving power. There are, however, possibilities to work at a weaker resolution, for example 0.1 as it is usually chosen in the case of pole figure measurements. The detection quantum yield for the Cu K-line is about 50%, this means that the exposition time has to be increased comparing to scintitation detector. The detection homogeneity fault of the PSD detector is less than 0.5% over the entire electrode length and the detection ]Jnearity assumed as a channel-to-peak allocation is i channel onto 0.01 . The PSD-detector is followed by a multichannel analysing system/MCA/ with energy discrimination stage, location discrimination stage, multichannel analyser and an automatic step counter relating the stepmotors run to the angular detector displacement and counts channelizing in the MCA system. The used MCA version is characterized by 8192 channels, counts capacity per channel is 2-109 and maximal access frequency is 400 kHz. Both the texture goniometer and the mttltichannel analyser are equipped with their own microprocessors, so the simultaneous goniometer driving and multichannel data processing with on-line data presentation are ordy then possible when a multifunctional fast computer is employed.

SIMULTANEOUS SCANNING METHOD IN BACK-REFLEXlON TECHNIQUE
It was already mentioned above that the simultaneous llcasurement of pole figures with a position sensitive detector requires a modified scanning technique compared to that used in the sequential measuring method. Conventionally, pole figures axe measured one by one with the diffraction condition i:2i being always fulfilled.
The detector 2O-location corresponds to the peak profile maximum and the diffraction vector (the bisectrix between the iJLcident and reflected beam) remains each time parallel to the sample normal vector. The mathematical algorithm describing the coordinates transformation from goniometer angles (X, ) to pole figure angles {a,/) was first given by  for neutron diffraction using spherical samples and transmission technique, in X-ray diffraction the back-reflexion technique is being used which requires an adaptation of this method. The mounting of a fiat sample on the saple holder of the Eulerian cradle for the back-reflexion technique is depicted in Fig.2. Fig.3 and. 4 illustrate the geometrical conditions of the simultaneous pole figures analysis. The angle w of the sample is chosen in such a way that the diffraction vector of the middle peak 2i is perpendicular to the sample surface. For all other diffraction angles this is not the case. The sample is then scanned through the angles X and as is shown in Fig.3. The pole figures are represented in terms of the polar coordinates (a,/} of the diffraction vector with respect to the sample coordinate system. These angles (a/ correspond directly to the scanning angles {X, } only for the middle angle 2. For all other 0-angles different from Oi=w it appears an offset of the diffraction vector Iw > 0. Then each new sample positioning {X, } has to be appropriately transformed into pole figure points {a, fl}(h) as follows; We consider two cases/9 < 0i and 0 > 0; where 0i = w.
In the case O < 0i (designated by 0/ in Figure 3)  It is to be mentioned that with the used PSD version the transformed values {a, fl} were always very near to {X, }-With an angular detector range of 7 in 2/9circle, the maximum deviation of 2/9 from 20 is 5=3.5 and hence, the maximum deviation of the ditfraction vector from its middle position is only 5=1.75 . The carried out computer sinulation showed that la-XI < 0.1 which can be neglected compared with the scanning step Aa = 5. The deviation in/9-coordinate was Somewhat larger. It was found that in an extreme situation i.e. at maximal sample tilt X = 75 and maximal diffraction vector offset I/9-0i = 1.75 , the rotation deviation Ifl-1 < 3.20 which is comparable with the used scarmlng step A = 3.6.
The transformation formulae were checked experimentally using silver single crystal which was cut parallel to the (001) plane. The (111) pole figure was measured in the middle position of diffraction vector so that 2/?(1) = 20i. The pole figure is shown in Fig.5. The same pole figure was measured again with 2-offset of the 20 reflex from 20i position. Plotting directly the Eu]erian cradle angles X, without coordinates correction gives the pole figure Fig.6. One sees that the (111) poles are broadened, shifted and lost their symmetries. Then the transformation {X, } ---' {a,} was carried out. The so obtained pole figure was practically indistinguishable from that of Fig.5 and its maxima were repositioned identically as for the pattern pole figure from the middle position.

POLE FIGURES SCANNING MODES
If the required 20-range is not larger than the viewing angle of the PSD i.e. 7 in our case, then the whole data collection can be carried out in the stationary detector mode. The middle of the detector is fixed in the required 2-position. The Eulerian cradle angles {X, } are then step-scanned in equiangular intervals e.g. AX 5,A 3.6. The pole figures angles {a:fl}(hkl) are then obtained with the transformation formulae 2)-( 5 ).
If the required 2-range is larger than 70 then the whole 2-range has to be composed of several 7-intervals. Since the transformation formulae (2)-(5) contain the detector middle position w, each measurement has to be performed in the stationary detector mode. Thereby the sequence of (X, } scans and 2-scans can be changed. It is thus possible to fix 2 and scan {X, } through all required positions, then to sltli 2 to the next position and again scan {X, } and so on until the whole required 2 age is covered. It is, however, equally possible to fix at first {X, } and steps't 2 in 7-intervals then to go to the next {X, } position and again step-scan 2Larca. After the whole measurement is finished the two results are identical. Both t|ese variants can be executed with the goniometer controlling program. Pole figure measurement is, however, also possible in he "moving detector mode". Thereby the detector moves continuously through the required 28-range and the smnple together with the Eulerian cradle moves in the coupled regime on the w-circle, while the {X, } position remains fixed. In this case the" PSD controlling software automatically considers the actual middle position of the PSD assigning a location within the detector to a particular channel of the MCA. This procedure is the same as is used in fast PSD powder diffraction. After one run the MCA contains the whole 2spectrum (larger than 7). If this method is applied to pole figure measurement, however one has to take into account the variable middle position w of the detector in equations (2)-(5). Hence, with the same {X, } values variables .{a,} are obtained when a particular diffraction peak (hkl) moves from 28/ through 28 to 28/, see Fig.3 and 4. Hence, the stored intensity in a particular MCA channel corresponds to an integral over a small line in the pole figure centered at {a, }={X, }. As long as the length of this line is comparable with the scanning steps AX, A or Aa, A then this integration does not reduce the accuracy of the obtained pole figure. Quite the contrary, the integration may even be advantageous since it improves the grain statistics during the measurement. In the "moving detector mode" the subroutine executing the transformations (2)-(5) is thus not activated.

DEFOCALIZATION BROADENING
When the sample is being tilted through the angle X( a) then only one line of its surface can remain on the focussing circle. The upper part of the illuminated area is, for instance, behind it whereas the lower part is in front of it. Hence, the whole peak is no longer focussed at the detector position. The actual shape of the broadened line depends on the vertical extension of the illuminated area on the sample surface and it. may even depend on the intensity distribution of the irradiating beam within this area. Besides this. the broadening also depends on the Bragg angle/9. Fig.7 exemplifies a peak broadening for different tilt angles X(, a).
The exact form of this broadening depends on the particular choice of the prime/i'y divergence slits. This line broadening has to be taken into account when integrated peak intensities are to be determined. As long as the peaks of different reflex.ions (hld) are wide enough separated one can simply integrate the intensity over the whole peak profile (i.e. sum up the counts of the corresponding channels of the MCA). It is, however, also possible to chose an increasing integration interval still covering a whole broadened peak profile with increasing tilt angle X( a). This latt0r method was chosen in the present case. Fig.Sa shows the pole figures of the (lll)-reflex of a cold rolled copper sheet, measured with the PSD in stationary l,et'tor mode, integrating over the appropriate peak width A2/9 (the surface area 11ldt;r the peak envelope) i.e. an adapted number of channels for each tilt angle {-a). The results are compared with an evaluation of the same primary data itegrating over a constant 2-interval, namely A2/9 1.1 = const, corresponding to tle maximtun receiving slit width ava;.lable in a scintillation counter measurement, Flg.Sb. For a usually used angular resolving power of 0.1/charmel for pole figure measurenent this corresponds to 11 channels of the MCA. Finally a measurement with A2/? = 0.5 const. (corresponding to 5 MCA channels) was simulated Fig.Sc. ]n Figure 8a the whole peak profile is integrated up to the maximum tilt angle. The sinulated "scintillation counter measurement" shows already "cut off" effects at high tilt. angles and the sinulated "5 channels" measurement depicts this effect to an even higher degree. The pole figure data of Fig.8 were then used as input data for ODF analysis. Tlis allows to calculate the error coefficients AC; (averaged over # and shown in Fig.9. It is seen that the 5 channels measurement shows higher error values particularly for low -values which is due to systematic errors in the pole figures i,e. the cut off effect, at higher tilt angles. The error coefficients of the "scintillation counter" and "variable A2-interval" measurement Fig.9 are still distinguishable. This shows that the routine measurement of pole figures with scintillation counter (without defocalization correction) gives already quite good results due to the least squares averaging of the pole figure inversion method. Nevertheless are these two pole figures as raw measured data still clearly distinguishable at higher tilt angles. Fig.10 illustrates difference pole figures, i.e. differences between pole figure data in the case of "variable A2O-interval" minus "11 channels" and "variable A2-interval" minus "5 channels" respectively. All local maxima and contour levels were presented in counts/sec to avoid averaging effects which always appear in the normalization procedure. It is seen that the results are identical or similar in the center of the pole figures and that the differences increase with increasing tilt angle. Finally, Fig.ll shows the recalculated pole figures which are a least squares compromise between the errors contained in all pole figures used for the ODF calculation. It is seen that the recalculated pole figures show more details especially at the higher tilt angles. Fig.8-11 conftrm the advantage of the use of PSD measurements in textttre analysis compared with the conventional single detector method. In the present example of sufficiently separated peaks the same results could of course also be attained with the single detector metha.d. In the case of line-rich diffraction spectra with -overlap, however, the PSD method is indispensable. A deconvolution of the broadened peak shapes according to Fig.7 with possible a-overlap has then to be involved in the evaluation programs.   (111), (200) and (220)-pole figures raw" data (a) variable A20-interval "adapted channels summation", (b) constant A20-interval "11-channe!s", (c) constant A2/-interval "5-channe!s" (a) "variable 528-interval" minus "constant interval ll-channe!s", (b) "variable 20-interval" minus "constant interval 5-channels"