Advantages of Neutron Diffraction in Texture Analysis

Neutron diffraction texture analysis is based on pole figure measurement followed by pole figure inversion. Neutron diffraction pole figure measurement is quite similar to that by X-rays. There are, however, several advantages in detail which are mainly due to the lower absorption coefficient. Besides these quantitative differences, there is one principle difference between the two methods. Neutron diffraction allows the magnetic texture to be measured which is not possible by X-rays. The paper gives a survey on the advantages of neutron diffraction texture analysis which are only counteracted by the limited availability and higher costs of neutron diffraction.


INTRODUCTION
The texture of a polycrystaline material is defined as the orientation distribution function of its crystallites dV/V g= ,t,, /(g) (1) sin tp dtpl, dq, dq)2 dg dg= r Thereby dV/V is the volume fraction of crystallites having a crystallographic orientation g within the limits dg.The orientation g of the crystallographic axes with respect to a sample-fixed coordin- ate system may be described by the Euler angles tpl, , tp2, or by many other orientation parameters, see e.g. Figure 1.Texture measurements can be carried out with the help of different kinds of radiation using either "imaging" or diffraction methods.These radiations are mainly visible light, X-rays, neu- trons, and electrons.
Imaging methods may be based on light and electrons whereas diffraction methods may be applied with X-rays, neutrons and electrons as is shown in Table 1.
The methods of texture analysis can also be classified according to whether they are individual crystallite methods or polycrystal methods.1) If lateral and vertical resolving power is higher than grain size, then individual orientation measurements can be used.Thereby the crystallographic orientation of a crystallite may be obtained from a) crystallographically oriented features of the crystallite seen in its image in "direct space" (imaging method) and b) the diffraction pattern ("image in reciprocal space") (diffraction method).
2) If the lateral resolving power of the method is much smaller than grain size then "collective diffraction patterns" (pole figures) can be used to obtain information about the orientation distribution.
In the first group of methods, the (discontinuous) individual orientations have to be converted somehow into a statistically relevant continuous orientation distribution function, Figure 2.
In the second method, statistically relevant continuous distribu- tion functions are directly obtained, however, not the complete distribution function depending on all three orientation parameters.The obtained pole figures are only two-dimensional "projections" taken along certain projection paths in the orientation space from which the three-dimensional distribution function has to be con- structed by mathematical methods which are comparable to the well-known "computer tomography."This is shown schematically in Figure 3 (see e.g.Bunge, 1969, 1982, 1987).
In the first group of methods statistical relevance can only be improved by increasing the number of orientation measurements i.e. by increasing the experimental effort (considerably).
In the pole figure method statistical relevance can be increased (nearly unlimited) by increasing the size of the sample, compared to grain size, provided the whole sample volume can be made to  contribute to the measurement at the same time.Hence, with the pole figure method statistically relevant results can be obtained in reasonable time.The virtual disadvantage of this method, the necessity of a mathematical inversion processs is no longer an essential obstacle since highly performant computer programs for pole figure inversion are now available.H.J. BUNGE Furthermore, pole figure measurement can be nearly completely automatized by using computer-operated texture goniometers (e.g. Puch, Klein, Bunge, 1984).The individual orientation measure- ments, on the other hand, have only been "semiautomatized" up to now.Hence, the great majority of texture determinations have been carried out by pole figure measurement followed by pole figure inversion.
As was already mentioned, pole figure measurement can, in principle, be carried out by three kinds of radiation i.e. by X-rays, neutrons, and electrons.It is the purpose of the present paper to provide an overview over the advantages and disadvantages of neutron-diffraction as a means of texture analysis.

POLE FIGURE MEASUREMENT
A pole figure is defined as the orientation distribution function of an individual crystal direction h perpendicular to a reflecting lattice plane (hkl) dy dy sin ct do dfl Thereby dV/V is the volume fraction of crystals the direction h of which is parallel to the sample direction y.In the conventional method of pole figure measurement one incident and one reflected beam are being used, the bisectrix of which defines the diffraction vector s.Pole figure measurement then requires to bring all sample directions y---one after the other--into the diffraction direction s and each time to measure the diffracted intensity.This method requires a diffractometer in order to fix the Bragg-angle 0 and a sample rotation device which is usually a Eulerian cradle shown schematically in Figure 4b.The Eulerian cradle allows to rotate the sample through three angles which are usually denoted by toXtp as is also shown in Figure 4b.For pole figure measurement, the sample has to be rotated through two angles (fl).Hence, different scanning procedures may be applied e.g. by using only the angles ) and or the angles to and ) respectively.(This corresponds to the reflection and transmission method in X-ray diffraction although the principle distinction between these two methods does not apply to H.J. BUNGE neutron diffraction.)In the conventional method of pole figure measurement the whole angular range 0 -< te -< 90 , 0 -< fl -< 360 is to be scanned in steps of Ate, Aft (which may be constant in the whole range or not).In each sample position the reflected intensity is to be measured and the result is to be represented in the form of a continuous pole density function as defined in Eq. ( 2).The whole procedure is then to be repeated with the next pole figure, i.e. with another Bragg-angle.One thus obtains a set of pole figures which provide the input data for the pole figure inversion process shown schematically in Figure 3.The whole measuring procedure can be carried out automatically when a computer operated texture gonio- meter is available.This method is very similar in X-ray diffraction and neutron diffraction.
It should be mentioned here, that in electron diffraction a different method can be used in which the sample is only rotated through one angle.A second angle is obtained by deflecting incident and/or reflected beam magnetically which is not possible with X-rays and neutrons (see e.g.Schwarzer and Weiland, 1986).
Pole figure measurement with all three types of radiation is normally carried out in the angular dispersive mode.This means that monochromatic radiation is being used and reflection at the various lattice planes (with different lattice plane spacings d) is distinguished by different Bragg angles according to Bragg's law n 2d(hkl sin O(hkl   (3)  In the case of X-rays the spectrum consists of two parts, a continuous (white) spectrum (the Bremsspectrum) and the charac- teristic line spectrum Figure 5.The peak intensity of the latter one is usually much higher than that of the white spectrum.In most cases of X-ray texture analysis, it is thus sufficient either to work with the spectrum as it is or to use only a fl-filter.A crystal monochromator is then not necessary.The diffraction of the white radiation contributes a small part to the background scattering which is being taken into account by background measurements taken at both sides of the correct Bragg angle.The background is often taken as a function of the pole figure angle te but not as a function of/3.This is certainly not strictly correct but it is often a sufficient compromise.
The spectrum of thermal neutrons does not contain a characteris- 5 Figure 5 Neutrons Spectral characteristics of X-rays and neutrons.
tic line spectrum as is seen in Figure 5. Hence, it is then necessary to use a monochromator.Typical monochromator materials are copper, aluminium, lead, silicon, germanium and graphite which select a wave length according to Bragg's law Eq.(3).In order to change the wave length it is thus necessary to change the monochr- omator or to change the Bragg angle 0 or both.In this case the reflected beam (i.e. the incident beam of the texture goniometer) has to pass through the monochromator shielding at different angles and the position of the goniometer itself has to be changed.This requires a higher technical effort than to work with just one "take off angle."Hence, the available wavelengths are often restricted to only a few (notwithstanding the continuous nature of the neutron spectrum).
According to Bragg's law a monochromator may allow a shorter wavelength to pass through (an integer fraction )/n).For the case of texture analysis this is, however, not a serious problem.It leads only to the superposition of pole figures (hkl) and (nh, nk, nl)   which are identical.
The angular divergence of incident and reflected beam of the monochromator determines the spectral width A3. of the monochro- matized beam and this, in turn, gives rise to a certain line broadening in the measured diffraction spectrum of the sample.On the other hand, the higher the divergencies and the higher A, the higher is the measured intensity and hence, the shorter is the required measuring time.For texture analysis in most cases a rather strong instrumental line broadening can be tolerated (at least if material, such as metals, with not too linerich spectra are being studied).Hence, an optimization of the divergencies is used in texture analysis which may be different from the requirements in other diffraction methods (e.g.high-resolution structure analysis).
This must be taken into consideration when multi-purpose diffractometers are being used for texture analysis.These diffractometers are mostly optimized for crystal structure analysis.

TEXTURE ANALYSIS BY NEUTRON DIFFRACTION
The three kinds of radiation, X-rays, neutrons, and electrons have greatly different penetration depths in matter due to different absorption coefficients.At the same time, also the lateral resolving power in the three methods is different as is shown schematically in Table 2. Accordingly, the irradiated sample volume may be different by nearly twenty orders of magnitude for electron diffraction and neutron diffraction.
Even if one takes into account that in X-ray diffraction texture measurement an additional sample integration may be applied which increases the irradiated sample size without decreasing the  angular resolving power (see e.g.Bunge and Puch, 1984) then still a difference of about three orders of magnitude in sample size remains between neutron and X-ray diffraction.This bigger sample size constitutes one of the main advantages of neutron diffraction compared with X-ray diffraction texture analysis.
After neutron diffraction became available with the advent of nuclear reactors after 1945, it was applied to texture measurement for the first time by Brockhouse in 1953.At that time, one of the major advantages of neutron diffraction over X-ray diffraction for texture analysisthe much higher accuracy of the results--could, however, not yet be exploited.In fact, it could not even be proven, since quantitative accuracy figures for texture measurements were not yet available.Hence, the major disadvantage of neutron diffraction, its much higher experimental effort dominated, such that virtually no more texture measurements were carried out by neutron diffraction in the following fifteen years.
A reliability criterion for pole figures based on the series expansion method (Bunge, 1966) showed lateron that neutron diffraction pole figures were much more accurate than those obtained by X-rays (Schlifer, 1968).Furthermore, this higher accuracy was actually needed for the calculation of ODF from pole figures.Hence, Bunge and Tobisch (!968) and Bunge, Tobisch and   Sonntag (1971) determined the ODF of cold rolled copper from neutron diffraction pole figures.Textures of a-Brass were deter- mined by Bunge and Tobisch (1972) and Bunge, Tobisch and  Miicklich (1974) and the rolling and recrystallization textures of low carbon steels were investigated, this way, by Schlifer and Bunge  1974 and Bunge, Schleusener and Schlifer (1974).The methodical details as well as the advantages of neutron diffraction .fortexture analysis were reviewed by Szpunar (1976) and by Kleinstiick et al.  (1976).From this time on, neutron diffraction texture analysis was continuously carried out in several neutron diffraction laboratories including magnetic scattering (Szpunar et al., 1968; Stott and   Hutchinson, 1973; Hennig et al., 1981).
The high-speed PSD method was applied to dynamic recrystalliza- tion investigations by Jensen et al. (1981) and a detailed review was given by Welch (1986).The time-of-flight method was especially developed by Szpunar et al. (1968), Nosik et al. (1979), Feldmann et  al. (1981), Betzl et al. 1984 and a review of this method is contained in the paper by Feldmann in this volume.

LARGE GRAIN SAMPLES
The statistical relevance of pole figure measurement is limited by the number of grains involved in each pole figure point.This number depends on the angular resolving power and on the total number of grains in the irradiated sample volume.Comparing X-ray and neutron measurements with the same angular resolving power, it is thus possible to obtain the same statistical relevance in neutron diffraction samples having bigger grain size by one order of magnitude in diameter (i.e. three orders in grain volume).

TEXTURE INHOMOGENEITY AND GLOBAL TEXTURES
The texture of a material is often inhomogeneous, i.e.It is different in samples taken from different places in the material, as is shown schematically in Figure 6a.It is then necessary to distinguish between local textures and the global texture of the whole material.Inhomogeneities may be divided roughly into macro-and microin- homogeneities, examples of which are shown in Figure 6b and c respectively.An important macroinhomogenity is, for instance, the variation of the texture of a sheet from surface to the interior, Figure 6b.A prominent microinhomogeneity occurs in the shear bands, Figure 6c.In Table 3 it is shown how local and global textures in the macro-and microscale can be distinguished by the three kinds of radiation.The global texture on a macroscale is needed for instance when macroscopic properties of the material are being considered.If E(g) is any physical property of a crystallite which depends on its orientation g (e.g.resistance to plastic deformation) then the corresponding macroscopic property is   given by (4) Thereby f(g) is the orientation distribution of all crystals of the macroscopic sample i.e. the global texture of the material.
In the case of plastic aniostropy as a fundamental property for deep drawing sheet, thicknesses in the range of several milimeters are often to be considered.The plastic properties of such materials are correlated with the global texture obtained by the x-ray composite sample method or directly by neutron diffraction.The surface texture measured by the x-ray back-reflection method gives 1.0 0.8 10 composite _2.
[1.2 0* 10' 20' 30* 40* 50* 60' 70* B0* 90* Angle to roiling direction Figure 7 Variation of the r-value of plastic anisotropy in the sheet plane of a steel sample measured mechanically (circles) and calculated from texture measure- ments in the whole volume (composite sample) and the surface (back-reflection).quite different results.This is illustrated in Figure 7. Hence, neutron diffraction is especially advantageous for technological samples.

ACCURACY OF THE MEASUREMENT
Pole figure measurement is based on intensity measurements of the reflected beam.The intensity depends on the volume fraction of crystallites in reflection position (the pole density value which is to be measured) and the absorption of incident and reflected beam in the sample.This has to be taken into account by an absorption correction factor A e -''x (5 where # is the linear absorption coefficient, and x is the total path of incident and reflected beam in the sample.The absorption factor A may be calculated and used as a correction factor if the shape of the sample (and hence the path x) is correctly known.If the shape of the sample is only incorrectly known, then the absorption factor and hence the resulting pole density values are falsified.Differentiating 0 Neutrons Figure 8 The accuracies dx/x required for a neutron and an X-ray sample in order to obtain a resulting accuracy of 5% in the correction factor and hence in the pole density values.H.J. BUNGE Eq. ( 5) with respect to x gives dA dx Ax (6)   x Hence, the relative error dA/A of the correction factor depends on Figure 9 The spherical sample method in neutron diffraction texture analysis.
a) The absorption coefficient A is independent of the pole figure angles a/3.b) a "spherical sample" may have an approximate form.
NEUTRON DIFFRACTION IN TEXTURE ANALYSIS 281 the relative error dx/x of the sample shape but it also increases with / x.In the case of neutron samples usually much lower/ x values are being used than in the case of X-rays and also the sample size x is much higher for neutrons than for X-rays.Hence the possible tolerance dx of the sample size is much higher for neutrons than for X-rays.Two typical examples are compared in Figure 8, where dA/A 5% is assumed.Hence, the accuracy of neutron texture measurements can easily be made better by an order of magnitude compared to X-ray measurements.
Figure 10 A "spherical" sample for neutron texture analysis.
H.J. BUNGE The highest accuracy can be obtained when a spherical sample is being used (Tobisch, Bunge 1972).Then the absorption factor A is independent of the pole figure angles (, fl) as is shown in Figure 9a.Because of the low/, x-value in Eq. ( 6) a "sphere" may have the shape of Figure 9b or it may even be a cylinder as in Figure 10.The resulting accuracy is shown in Figure 11 where error coefficients AF are drawn as a function of the order A. It is seen that neutron diffraction with the spherical sample method gives much better results than X-ray diffraction.Especially, the strong increase of these coefficients at low A-values does not occur.It has been shown that the increase is due to systematic errors, Figure 12

MULTIPHASE MATERIALS
A particularly strong influence of the absorption factor on X-ray texture measurements occurs in the case of multiphase materials, if the shape and arrangement of the phases is anisotropic.In Figure 13 the absorption factor of a directionally solidified Pb-Sn eutectic is shown as a function of the angle , of the diffraction plane relative to the lamellae plane (Bunge, Liu, Hanneforth, 1987).It is seen that in this case the influence of absorption is so strong that texture measurements by X-ray diffraction become virtually impossible.
O ROTATION kJI6LE Y Figure 13 Anisotropic absorption of X-rays in lamellar structures, a) Metall- ographic section of a directionally solidified Pb-Sn eutectic b) Absorption factor as a function of the angle ), between lamellar plane and diffraction plane.
Figure 14 Metallographic section of a 90% extruded AI-Pb composite.
The situation is similar in highly deformed two-phase materials e.g.90% extruded A1-Pb composites as shown in Figure 14 (Brokmeier, B6cker and Bunge, 1988).Also in this case the X-ray absorption factor varies by as much as 25%.Hence, X-ray texture measure- ments are falsified by this amount.In neutron diffraction this influence is in the range of 0.1%.Figure 15 shows inverse pole figures of AI and Pb of a highly extruded AI-Pb composite obtained by neutron diffraction.These measurements are free of errors due to anisotropic absorption.

SMALL VOLUME FRACTIONS OF SECOND PHASES
It is interesting to study the texture of a phase which is present in a multiphase material only in small volume fractions e.g.below 1%. 13.9

3.3
Hgare 15 Inverse pole figures of the AI and Pb-phase of pure metals and composites extruded 90% measured by neutron diffractiotL penetration depth increased correct reduced volume fraction Figure 16 Possible influences of sample preparation on small particles of a second phase in the sample surface.

RLPHR
Pole figure of the copper phase of an extruded AI-Cu composite with 1% Cu.
In this case, X-ray texture analysis must assume that the area fraction of the phase in the investigated surface is the same as the volume fraction.With a small volume fraction this may not be the case simply because of poor statistics.Furthermore because of different properties of the phases, the surface of the particles may not be flat as is shown schematically in Figure 16.Hence, X-ray texture measurements in phases of less than about 2 or 3% become virtually impossible.In neutron diffraction, on the other hand, the whole sample volume contributes to the measurement and artefacts at the surface do not play any important role.Hence, neutron texture measurements can be carried out with reasonable accuracy well below 1 Vol.%. Figure 17 shows for instance the pole figure of the copper phase of an A1-Cu composite with 1% Cu.Recently measurement with 0.1% of copper have also been carried out.

SAMPLE PROTECTION
Another advantage of the high penetration depth of neutrons compared with X-rays is that it is easily possible to provide a 288 H.J. BUNGE neutron sample with a protective coating.This may be interesting when studying textures in oxygen-sensitive or hygroscopic materials.
Similarly, different kinds of environments may be easily applied to a texture sample, e.g.high temperatures.In this case, the Eulerian cradle can be easily shielded from the heating device with shields the thicknesses of which are in the millimeter range.

INFLUENCE OF THE ATOMIC NUMBER
It was already mentioned, that the absorption coefficient of neutr- ons is smaller than that of X-rays by two or three orders of magnitude.This allows much bigger samples to be used.On the other hand, also the scattering factors are much smaller.Hence, it is not only possible but, in most cases, also necessary to use bigger samples.
Absorption coefficient and scattering factor are shown in Figure 18 and 19 respectively (see e.g.Bacon, 1975; Squires, 1978).They are not only smaller than those for X-rays.Also the variation of these quantities with atomic numbers is quite different in the two cases.Both quantities show a systematic, strong increase with atomic number for X-rays which is not the case for neutrons.This has some consequences for neutron texture analysis.In neutron diffraction, neighbouring elements may have quite different scatter- ing factors.Hence it is easily possible to study ordering effects in solid solutions of such elements.As far as texture measurements are concerned this may be an important advantage when directional ordering in polycrystalline materials is to be studied, for instance in iron-nickel alloys.Directional ordering is strongly related to the texture of the material.The situation is also quite different for the light elements.Hence, texture studies of light element phases (especially in the presence of other phases containing heavy elements) will be much easier with neutrons than with X-rays.
As a consequence of different scattering factors of the atoms also the structure factors of a crystalline phase may be strongly different in X-ray and neutron diffraction.In texture analysis this may be important, for instance, if a particular structure factor is zero in one of the methods.The corresponding pole figure cannot be measured in this case.An example is the important (0001) pole figure of Figure 18 The absorption factor for X-rays and neutrons as a function of atomic number.
A1203 ceramics.Its X-ray intensity is lower than 1% of (11.3) which has the highest intensity.Hence, this pole figure cannot be measured by X-rays.The corresponding neutron diffraction inten- sity is, however, in the order of magnitude of about 10% of the maximum value.Hence it can well be measured by neutron diffraction.This is of great interest since A1203 substrates some- times have a strong (0001) fibre texture which is well expressed in the (0001) pole figure.

INFLUENCE OF THE ISOTOPE
The neutron scattering factors for various isotopes of the same element are usually quite different (till to opposite sign of the scattering factor).A particularly striking example of this effect is observed in hydrogen.Hydrogen itself has a low coherent scattering factor but shows strong incoherent scattering whereas the opposite is true for deuterium.As a consequence of this, neutron diffraction texture analysis in natural salt samples may be strongly complicated by the often observed presence of moisture in these materials.On the other hand, texture studies in ice, for instance, are facilitated if it is possible to use deuterated ice.The scattering factor for X-rays and neutrons as a function of sin ANGULAR DEPENDENCE OF THE SCATTERING FACTOR X-ray scattering takes place in the whole volume of an atom whereas the most important part of neutron scattering comes from the nucleus the dimension of which is neglegible compared with the wavelength of the neutrons.Hence, the X-ray scattering depends on sin 0/ whereas neutron scattering is independent of this para- meter, Figure 20.This difference is important for texture analysis especially when pole figures are to be converted into ODF.In this case the resolving power is the better the more pole figures can be used.Neutron diffraction then allows to measure also high index pole figures with a satisfactory accuracy.This is especially important for materials with low crystal symmetries.

POSITION SENSITIVE DETECTORS
In the conventional method of pole figure measurement the detector is placed at the Bragg angle 0 with respect to the incident 292 H. J, BUNGE beam and an apperture is being used which allows the total (integrated) intensity of the diffraction peak to be measured.This way, one pole figure is obtained at a time.Then the position of the detector is changed to the next Bragg-position.Hence, the neces- sary pole figures are being measured sequentially.With the help of a position sensitive detector the whole O-spectrum can be measured at the same time.Figure 21 shows the instrument D1B at Grenoble, equipped with a position sensitive detector (Allemand et al., 1975).
It is thus possible to measure all required pole figures simultaneously at the same time.This method provides two advantages: 1) the measuring time is greatly reduced 2) Overlapping pole figures can be separated This latter point is especially important if materials with line-rich diffraction spectra are to be investiaged.Such spectra may be due Figure 21 The instrument D1B at Grenoble used as a texture goniometer with position sensitive detector.b) high lattice constants c) multiphase materials Hence, this method is particularly suited to the study of: a) intermetallic phases b) ceramics c) minerals d) rocks e) composites Figure 22a shows for instance a felspare spectrum measured this way (Bunge, Wenk, Pannetier, 1982).An enlarged part of the spectrum is shown in Figure 22b.It is seen that line separation is possible by line profile analysis.It is also seen that a complete profile analysis improves the determination of background scatter- ing and hence improves the accuracy of the measured integrated a) intensity (Jansen, Schiifer and Will, 1986).Figure 22c finally shows another part of the spectrum.Here the overlapping of different diffraction peaks is total.These peaks cannot be separated "experimentally" by line profile analysis.They can however be separated "theoretically" in the process of ODF analysis using the special methods of "overlapping pole figures."This is possible for "intraphase" overlapping (Dahms and Bunge, 1987) as well as for "interphase" overlapping (Dahms, Brokmeier, Seute and Bunge,  1988).

5-104
When a position sensitive detector is being used the plane of the Eulerian cradle (plane of the ;t-rotation) can only be in "symmetry position" for one specific Bragg angle e.g. the angle 0,, in Figure 23.Only for this angle, the x-circle passes through north and south pole of the pole sphere.For all other O-angles ;t and t# rotation gives a pole figure with a "blind area" in the vincinity of north-and south pole as is shown in Figure 24.Hence, in order to obtain complete pole figures an additional scan is necessary in order fill in the blind area and a coordinate transformation is necessary by which the angles to;t of the Eulerian cradle are transformed into the pole figure angles crfl (Bunge, Wenk, Pannetier, 1982).Some of  the felspare pole figures measured this way are shown in Figure 25 (Wenk, Bunge, Jansen and Pannetier, 1986).
In Figure 21 the position sensitive detector was used in "Bragg angle orientation ."In this orientation it measures one pole figure point of several Bragg angles at the same time.It can, however, also be used in "pole figure orientation" as is shown in Figure 26a (Juul-Jensen and Kjems, 1983).In this case the wavelength must be  H.J. BUNGE chosen such that 2 90.The detector then measures a whole line of pole figure points but only in one pole figure as is shown in Figure 26b.In order to scan a whole pole figure completely a greatly reduced number of steps is then sufficient, as is shown in Figure 26d.This method has especially been applied for rapid pole figure measurment, e.g.real-time recystallization and phase trans- formation stuides (e.g.Juul-Jensen, 1986).

WAVE LENGTH DISPERSIVE METHOD
Beside the normal angular dispersive measurement also a wave length dispersive method can be used.In this case, white radiation is being used.The reflection at different lattice planes is then obtained with different wavelengths but at the same Bragg angle n i(hkl 2d(hkl sin # (7) The wavelength of neutrons is correlated with their velocity h Hence, wavelength measurement can be obtained by time-of-flight measurement and this, in turn, requires to fix the starting time of a neutron.Time of flight measurement requires narrow pulses of neutrons following each other in rather long time intervals.A pulsed beam can be obtained mainly in two ways: a) a continuous beam can be "chopped" into short pulses using a chopper b) The neutrons can be directly produced in pulses e.g. in a pulse reactor or by a pulsed spallation source.
It is thus necessary to measure the time of each incoming neutron registered in the detector.The "time spectrum" obtained with one neutron pulse thus corresponds with the wavelength spectrum and hence with the d-value spectrum.Such spectra are then to be added up in a multi-channel analyzer for a sufficient number of neutron pulses.Finally a time-of-flight spectrum is obtained as shown for instance in Figure 27 (Feldmann, 1986).measured for all necessary sample orientations in order to obtain all pole figures at the same time.Thus far, this method is similar to the method using a position sensitive detector.In contrast to the latter one, however, all pole figures are measured with the same Bragg-angle, i.e. in "symmetry position" as in the sequential angular dispersive method.Hence, a blind area does not occur in this method and no transformation from to;t to cfl is necessary.

COMBINATION OF PSD AND TOF METHOD
The three methods mentioned above may be combined into one method i.e. two-dimensional area of a large number of pole figures.Hence, this method is the most economic one.

MAGNETIC TEXTURE ANALYSIS
A second part of neutron scattering is due to magnetic interaction of the neutron's magnetic dipole moment with the distribution of the magnetic moment of the electrons in the atom.Hence, this part of neutron scattering depends on sin t/ in a similar way as X-ray scattering, Figure 29 (see e.g.Bacon, 1975).In the case of non-polarized neutrons the magnetic part of the reflected intensity is proportional to q2 sin 2 a' (9) where a is the angle between magnetization direction and the diffraction vector s i.e. the normal direction to the reflecting lattice plane as is seen in Figure 30.Magnetic neutron scattering is related to the magnetic crystal structure of the material which may be different from the "chemical" crystal structure.The most important 0'0 0'I 0.2 0": 0.4 (sinO.)l,(I0' cm-*) X-rays Neutrons 0"5 Figure 29 The magnetic scattering factor of neutrons depends on sin /.
cases of magnetic structures are the ferromagnetic and antifer- romagnetic case.In the first case all magnetic moments of the atoms are parallel to a certain crystal direction m, in the second case they are mutually antiparallel.This means that in the first case the unit cell is the same as the chemical unit cell but crystal symmetry is different and in the second case unit cell and symmetry both are changed.From the view point of texture analysis the ferromagnetic case is most important.With an applied magnetic diffraction vector rnognefizofion Figure 341 The magnetic scattering of unpolarized neutrons depends on the angle tr between magnetization direction and normal direction to the reflecting lattice plane.
H.J. BUNGE field H the magnetization direction rn may be parallel to any crystal direction h.Two limiting cases may be particularly considered.In the case of zero magnetic field the magnetization direction is parallel to one of several crystallographically equivalent directions hg.In the case of saturation it is parallel to the field direction rn II hg H 0 (10) rn II/-/ /-/oo In the second case the magnetization direction is uniquely fixed in each crystal.It is parallel to the same sample direction but to different crystal directions which are however uniquely determined by crystal orientation g.In the first case the distribution of magnetization directions over the crystal directions hg will generally depend on the magnetic history of the material.Different directions hg may occur in one and the same crystallite which is then divided into magnetic domains.In this case the "magnetic texture" can be easily defined as the orientation distribution function of the domains.We specify a crystal coordinate sytem K, the xa-axis of which is parallel to the magnetization direction hg of the considered domaine.The orientation of this coordinate system with respect to the sample coordinate system K, is described by the rotation gM.
The magnetic texture is then the volume fraction of domains having an orientation gM within the angular range dg 1 fM(g) 11) This definition of the magnetic texture is completely analogous to Eq. (1), only the symmetry of the magnetic texture function is different from that of the crystallographic texture.As an example, in cubic ferromagnets the magnetization direction is often parallel to the cubic axes h (100) (12) The magnetic crystal symmetry is then tetragonal instead of cubic.Also the sample symmetry may be different from the conventional sample symmetry due to the distribution of magnetization directions over the various h-directions.Also the pole figures of this magnetic texture are defined in the same way as in the crystallographic texture (see e.g.Miicklich, Hennig, Bouillot and Matthies, 1984)   dV/V P(y) (13) dy with the only difference that now h is the normal direction to the reflecting lattice plane (hkl) indexed in the magnetic coordinate system K i.e. with the lower "magnetic" crystal symmetry (e.g.tetragonal).The pole figure P(y) can, however, not be measured directly by neutron diffraction.Rather all those pole figures are systematically superposed which belong to crystallographically equiv- alent directions hi.
Ph}(Y) q2 p, (y) (14) i=1 The superposition factors q2 are known according to Eq. ( 9).Pole figure inversion, i.e. the calculation of the ODF fVt(g) from pole figures P(y) can be carried out with superposed pole figures Ph}(Y) instead of the no-superposed ones (see e.g.Dahms and   Bunge 1987).Hence, information about the magnetic texture can be obtained from the magnetic part of neutron diffraction pole figures (see also Bunge (in print), 1989).

CONCLUSIONS
Texture measurements can be carried ou by neutron diffraction much in the same way as by X-ray diffraction, i.e. by pole figure measurement followed by pole figure inversion.Because of the specific properties of the two radiations there are, however, some differences between the two method as far as the quantitative relations are considered.These differences are mainly in favour of neutron diffraction, whereas the main drawback of this radiation (compared to X-rays) is its limited availability and high cost of neutron diffraction equipment and especially of the neutron source.The advantages of neutron diffraction are, first of all, due to the lower absorption coefficient.As a consequence of this, much bigger samples can be used.
H.J. BUNGE --This in turn leads to a much better grain statistics.Hence, samples with one order of magnitude bigger grain sizes can be succesfully studied by neutrons.
raThe remaining absorption correction is usually much smaller with neutrons and can be calculated with a much higer precission.
Hence, the experimental errors of neutron diffraction pole figures may be an order of magnitude lower than those of X-ray diffraction.This is particularly valuable for high precision ODF calculations.
--The lower error allows texture analyses of second phases which are present with only very low volume fractions.
raThe high penetration depth allows the global texture of technologically interesting work pieces to be measured directly.
--There is virtually no anisotropic absorption effect in anisotropic polyphase materials as is the case in X-ray diffraction.
--A further advantage of neutron diffraction is due to a scattering factor independent of sin /.This allows high-index pole figures to be easily measured.
Neutron diffractometers are often equipped with position sensitive detectors which have not yet become standard equipment in X-ray diffraction.A PSD in O-direction allows the measurement of many pole figures at the same time.Furthermore, it allows the seperation of overlapping peaks which is particularly valuable for materials with big unit cells and low symmetries as well as multiphase materias.Finally, neutron scattering contains a magnetic part which allows magnetic textures to be measured which is by no means possible using X-ray diffraction.

Figure 1
Figure1The texture is the orientation distribution function of the crystallites.It depends on three orientation parameters.
determination from individual orientation measurements, a) The orientation of an individual grain represented in Euler space, b) A "cloud" of orientation points, c) The continuous orientation density distribution.

Figare 4
Figare 4 Principles of pole figure measurement using a Eulerian cradle.a) Diffraction of an incident beam in a polycrystalline sample, b) The Eulerian cradle.

Figure 12
Figure 12 Statistical and systematical errors of pole figures expressed in the error coefficients IAFI

Figure 19
Figure19The scattering cross section of X-rays and neutrons as a function of the atomic number. Figure20

Figure 22 (
Figure 22 Diffraction spectrum of a felspare sample, a) the whole spectrum.
Figttre 23 Adjustment of the Eulerian cradle in "symmetry position" for one of the Bragg angles 0,.

Figure 25
Figure 25 Four pole figures of a felspare sample.

Figure 26 A
Figure 26 A position sensitive detector used in "pole figure orientation".a) Positioning of the detector at 20 =90.b) Scanned line in the pole figure, c) Complete pole figure scan.

Figure 28
Figure28Texture measurement with a two-dimensional TOF-method.

Table 1
Texture measurement with different radiations, using imaging and diffraction methods

Table 2
Penetration depth, lateral resolving power, and sample volume for the three kinds of radiation

Table 3
Detection of macro-and microinhomogeneities by the three kinds of radiation