Advantages of Neutron Diffraction in Texturedownloads.hindawi.com/archive/1989/756108.pdfconstitutes one of the main advantages of neutron diffraction comparedwithX-raydiffraction

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Textures and Microstructures, 1989, Vol. 10, pp. 265-307Reprints available directly from the publisherPhotocopying permitted by license only(C) 1989 Gordon and Breach Science Publishers Inc.Printed in the United Kingdom

Advantages of NeutronDiffraction in Texture AnalysisH. J. BUNGEDepartment of Physical Metallurgy, Technical University of Clausthal, FRG

(Received 3 February, 1989)

Neutron diffraction texture analysis is based on pole figure measurement followed bypole figure inversion. Neutron diffraction pole figure measurement is quite similar tothat by X-rays. There are, however, several advantages in detail which are mainlydue to the lower absorption coefficient. Besides these quantitative differences, thereis one principle difference between the two methods. Neutron diffraction allows themagnetic texture to be measured which is not possible by X-rays. The paper gives asurvey on the advantages of neutron diffraction texture analysis which are onlycounteracted by the limited availability and higher costs of neutron diffraction.

KEY WORDS: Pole figure measurement, low absorption coefficient, positionsensitive detector, time-of-flight-method, magnetic texture analysis.

INTRODUCTION

The texture of a polycrystaline material is defined as the orientationdistribution function of its crystallites

dV/V g= ,t,,/(g) (1)

sin tpdtpl, dq, dq)2

dgdg= r

Thereby dV/V is the volume fraction of crystallites having acrystallographic orientation g within the limits dg. The orientation gof the crystallographic axes with respect to a sample-fixed coordin-ate system may be described by the Euler angles tpl, , tp2, or bymany other orientation parameters, see e.g. Figure 1.

265

266 H.J. BUNGE

normoldireclion

transversedirection

rolting

//direction

Angle cz

.90%50%

20%

Angle (x

Figure 1 The texture is the orientation distribution function of the crystallites. Itdepends on three orientation parameters.

Texture measurements can be carried out with the help ofdifferent kinds of radiation using either "imaging" or diffractionmethods. These radiations are mainly visible light, X-rays, neu-trons, and electrons.

Imaging methods may be based on light and electrons whereasdiffraction methods may be applied with X-rays, neutrons andelectrons as is shown in Table 1.The methods of texture analysis can also be classified according to

whether they are individual crystallite methods or polycrystalmethods.

NEUTRON DIFFRACTION IN TEXTURE ANALYSIS 267

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

Radiation Texture measurement byimaging methods diffraction methods

Light xX-rays xNeutrons xElectrons x x

1) If lateral and vertical resolving power is higher than grain size,then individual orientation measurements can be used. Thereby thecrystallographic orientation of a crystallite may be obtained from a)crystallographically oriented features of the crystallite seen in itsimage in "direct space" (imaging method) and b) the diffractionpattern ("image in reciprocal space") (diffraction method).

2) If the lateral resolving power of the method is much smallerthan grain size then "collective diffraction patterns" (pole figures)can be used to obtain information about the orientationdistribution.

In the first group of methods, the (discontinuous) individualorientations have to be converted somehow into a statisticallyrelevant continuous orientation distribution function, Figure 2.

In the second method, statistically relevant continuous distribu-tion functions are directly obtained, however, not the completedistribution 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 fromwhich the three-dimensional distribution function has to be con-structed by mathematical methods which are comparable to thewell-known "computer tomography." This is shown schematically inFigure 3 (see e.g. Bunge, 1969, 1982, 1987).

In the first group of methods statistical relevance can only beimproved by increasing the number of orientation measurementsi.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 tograin size, provided the whole sample volume can be made to

268 H.J. BUNGE

a) o

/

c) o

ligare 2 Texture determination from individual orientation measurements, a) Theorientation of an individual grain represented in Euler space, b) A "cloud" oforientation points, c) The continuous orientation density distribution.


Figare 3 Texture determination from pole figures.

contribute to the measurement at the same time. Hence, with thepole figure method statistically relevant results can be obtained inreasonable time. The virtual disadvantage of this method, thenecessity of a mathematical inversion processs is no longer anessential obstacle since highly performant computer programs forpole figure inversion are now available.

270 H.J. BUNGE

Furthermore, pole figure measurement can be nearly completelyautomatized 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 tonow. Hence, the great majority of texture determinations have beencarried out by pole figure measurement followed by pole figureinversion.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 toprovide an overview over the advantages and disadvantages ofneutron-diffraction as a means of texture analysis.

POLE FIGURE MEASUREMENT

A pole figure is defined as the orientation distribution function of anindividual crystal direction h perpendicular to a reflecting latticeplane (hkl)

dV/V y { a, fl}Ph(Y) (2)dy dy sin ct do dflThereby dV/V is the volume fraction of crystals the direction h ofwhich is parallel to the sample direction y. In the conventionalmethod of pole figure measurement one incident and one reflectedbeam are being used, the bisectrix of which defines the diffractionvector s. Pole figure measurement then requires to bring all sampledirections y---one after the other--into the diffraction direction sand each time to measure the diffracted intensity. This methodrequires a diffractometer in order to fix the Bragg-angle 0 and asample rotation device which is usually a Eulerian cradle shownschematically in Figure 4b. The Eulerian cradle allows to rotate thesample through three angles which are usually denoted by toXtp as isalso shown in Figure 4b. For pole figure measurement, the samplehas to be rotated through two angles (fl). Hence, differentscanning procedures may be applied e.g. by using only the angles )and or the angles to and ) respectively. (This corresponds to thereflection and transmission method in X-ray diffraction although theprinciple distinction between these two methods does not apply to


a)

b)

X

Euleriancradle

diffractometer-drive

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

272 H.J. BUNGE

neutron diffraction.) In the conventional method of pole figuremeasurement the whole angular range 0 -< te -< 90, 0 -< fl -< 360 isto be scanned in steps of Ate, Aft (which may be constant in thewhole range or not). In each sample position the reflected intensityis to be measured and the result is to be represented in the form ofa continuous pole density function as defined in Eq. (2). The wholeprocedure is then to be repeated with the next pole figure, i.e. withanother Bragg-angle. One thus obtains a set of pole figures whichprovide the input data for the pole figure inversion process shownschematically in Figure 3. The whole measuring procedure can becarried out automatically when a computer operated texture gonio-meter is available. This method is very similar in X-ray diffractionand neutron diffraction.

It should be mentioned here, that in electron diffraction adifferent method can be used in which the sample is only rotatedthrough one angle. A second angle is obtained by deflecting incidentand/or reflected beam magnetically which is not possible withX-rays and neutrons (see e.g. Schwarzer and Weiland, 1986).

Pole figure measurement with all three types of radiation isnormally carried out in the angular dispersive mode. This meansthat monochromatic radiation is being used and reflection at thevarious lattice planes (with different lattice plane spacings d) isdistinguished 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, acontinuous (white) spectrum (the Bremsspectrum) and the charac-teristic line spectrum Figure 5. The peak intensity of the latter oneis usually much higher than that of the white spectrum. In mostcases of X-ray texture analysis, it is thus sufficient either to workwith the spectrum as it is or to use only a fl-filter. A crystalmonochromator is then not necessary. The diffraction of the whiteradiation contributes a small part to the background scatteringwhich is being taken into account by background measurementstaken at both sides of the correct Bragg angle. The background isoften taken as a function of the pole figure angle te but not as afunction of/3. This is certainly not strictly correct but it is often asufficient 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 necessaryto use a monochromator. Typical monochromator materials arecopper, aluminium, lead, silicon, germanium and graphite whichselect a wave length according to Bragg’s law Eq. (3). In order tochange the wave length it is thus necessary to change the monochr-omator or to change the Bragg angle 0 or both. In this case thereflected beam (i.e. the incident beam of the texture goniometer)has to pass through the monochromator shielding at different anglesand the position of the goniometer itself has to be changed. Thisrequires a higher technical effort than to work with just one "takeoff angle." Hence, the available wavelengths are often restricted toonly a few (notwithstanding the continuous nature of the neutronspectrum).According to Bragg’s law a monochromator may allow a shorter

wavelength to pass through (an integer fraction )/n). For the caseof texture analysis this is, however, not a serious problem. It leadsonly 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-

274 H.J. BUNGE

matized beam and this, in turn, gives rise to a certain linebroadening in the measured diffraction spectrum of the sample. Onthe other hand, the higher the divergencies and the higher A, thehigher is the measured intensity and hence, the shorter is therequired measuring time. For texture analysis in most cases a ratherstrong instrumental line broadening can be tolerated (at least ifmaterial, such as metals, with not too linerich spectra are beingstudied). Hence, an optimization of the divergencies is used intexture analysis which may be different from the requirements inother diffraction methods (e.g. high-resolution structure analysis).This must be taken into consideration when multi-purposediffractometers are being used for texture analysis. Thesediffractometers are mostly optimized for crystal structure analysis.

TEXTURE ANALYSIS BY NEUTRON DIFFRACTION

The three kinds of radiation, X-rays, neutrons, and electrons havegreatly different penetration depths in matter due to differentabsorption coefficients. At the same time, also the lateral resolvingpower in the three methods is different as is shown schematically inTable 2. Accordingly, the irradiated sample volume may bedifferent by nearly twenty orders of magnitude for electrondiffraction and neutron diffraction.Even if one takes into account that in X-ray diffraction texture

measurement an additional sample integration may be appliedwhich increases the irradiated sample size without decreasing the

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

Penetrationdepth [mm]Lateralresolvingpower [mm]Samplevolume[mm3]

Neutrons

10

10

10

X-rays

10-1_10-2

1_10-2

10-1_10-6

Electrons

10-4

10-3_10-6

10-1o_10-16


angular resolving power (see e.g. Bunge and Puch, 1984) then still adifference of about three orders of magnitude in sample size remainsbetween neutron and X-ray diffraction. This bigger sample sizeconstitutes one of the main advantages of neutron diffractioncompared with X-ray diffraction texture analysis.

After neutron diffraction became available with the advent ofnuclear reactors after 1945, it was applied to texture measurementfor the first time by Brockhouse in 1953. At that time, one of themajor advantages of neutron diffraction over X-ray diffraction fortexture 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 werenot yet available. Hence, the major disadvantage of neutrondiffraction, its much higher experimental effort dominated, suchthat virtually no more texture measurements were carried out byneutron diffraction in the following fifteen years.A reliability criterion for pole figures based on the series

expansion method (Bunge, 1966) showed lateron that neutrondiffraction pole figures were much more accurate than thoseobtained by X-rays (Schlifer, 1968). Furthermore, this higheraccuracy was actually needed for the calculation of ODF from polefigures. Hence, Bunge and Tobisch (!968) and Bunge, Tobisch andSonntag (1971) determined the ODF of cold rolled copper fromneutron diffraction pole figures. Textures of a-Brass were deter-mined by Bunge and Tobisch (1972) and Bunge, Tobisch andMiicklich (1974) and the rolling and recrystallization textures of lowcarbon steels were investigated, this way, by Schlifer and Bunge1974 and Bunge, Schleusener and Schlifer (1974). The methodicaldetails as well as the advantages of neutron diffraction .for textureanalysis were reviewed by Szpunar (1976) and by Kleinstiick et al.(1976). From this time on, neutron diffraction texture analysis wascontinuously carried out in several neutron diffraction laboratoriesincluding magnetic scattering (Szpunar et al., 1968; Stott andHutchinson, 1973; Hennig et al., 1981).

Besides metallic materials, also non-metallics were studied (Wal-ter et al. 1981, Brokmeier 1982) and the method proved to beespecially valuable for geological studies (Bunge, Wenk, Pannetier1982, Bankwitz et al. 1984, Htifler, Schiller, Will 1986, H6fler, Will1986, Drechsler et al. 1984). Texture development in steels was

276 H.J. BUNGE

studied by Schreiber et al. (1978) and by Klimanek et al. (1981).The high-speed PSD method was applied to dynamic recrystalliza-tion investigations by Jensen et al. (1981) and a detailed review wasgiven by Welch (1986). The time-of-flight method was especiallydeveloped by Szpunar et al. (1968), Nosik et al. (1979), Feldmann etal. (1981), Betzl et al. 1984 and a review of this method is containedin the paper by Feldmann in this volume.

LARGE GRAIN SAMPLES

The statistical relevance of pole figure measurement is limited bythe number of grains involved in each pole figure point. Thisnumber depends on the angular resolving power and on the totalnumber of grains in the irradiated sample volume. Comparing X-rayand neutron measurements with the same angular resolving power,it is thus possible to obtain the same statistical relevance in neutrondiffraction samples having bigger grain size by one order ofmagnitude 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 differentin samples taken from different places in the material, as is shownschematically in Figure 6a. It is then necessary to distinguishbetween 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 crespectively. An important macroinhomogenity is, for instance, thevariation of the texture of a sheet from surface to the interior,Figure 6b. A prominent microinhomogeneity occurs in the shearbands, Figure 6c. In Table 3 it is shown how local and globaltextures in the macro- and microscale can be distinguished by thethree kinds of radiation. The global texture on a macroscale isneeded for instance when macroscopic properties of the materialare being considered. If E(g) is any physical property of acrystallite which depends on its orientation g (e.g. resistance toplastic deformation) then the corresponding macroscopic property is


a) Iota[

texture

\ ,v/

gtoba[texture

b)

c)

N eu on \\\\\\\’\\

sarnp e

X-ray sample

Figure 6 Inhomogenity of textures, a) Definition of local and global texture.

b) Macroinhomogeneity. c) Microinhomogeneity (shear band).

278 H.J. BUNGE

Table 3 Detection of macro- and microinhomogeneitiesby the three kinds of radiation

Inhomogeneities Neutrons X-rays Electrons

macro global local localmicro global global local

given by

(4)

Thereby f(g) is the orientation distribution of all crystals of themacroscopic sample i.e. the global texture of the material.

In the case of plastic aniostropy as a fundamental property fordeep drawing sheet, thicknesses in the range of several milimetersare often to be considered. The plastic properties of such materialsare correlated with the global texture obtained by the x-raycomposite sample method or directly by neutron diffraction. Thesurface 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 asteel 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 technologicalsamples.

ACCURACY OF THE MEASUREMENT

Pole figure measurement is based on intensity measurements of thereflected beam. The intensity depends on the volume fraction ofcrystallites in reflection position (the pole density value which is tobe measured) and the absorption of incident and reflected beam inthe sample. This has to be taken into account by an absorptioncorrection factor

A e-’’x (5)where # is the linear absorption coefficient, and x is the total path ofincident and reflected beam in the sample. The absorption factor Amay be calculated and used as a correction factor if the shape of thesample (and hence the path x) is correctly known. If the shape ofthe sample is only incorrectly known, then the absorption factor andhence 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 orderto obtain a resulting accuracy of 5% in the correction factor and hence in the poledensity values.

280 H.J. BUNGE

Eq. (5) with respect to x gives

dA dxA - x (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.


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 valuesare being used than in the case of X-rays and also the sample sizex is much higher for neutrons than for X-rays. Hence the possibletolerance dx of the sample size is much higher for neutrons than forX-rays. Two typical examples are compared in Figure 8, wheredA/A 5% is assumed. Hence, the accuracy of neutron texturemeasurements can easily be made better by an order of magnitudecompared to X-ray measurements.

Figure 10 A "spherical" sample for neutron texture analysis.

282 H.J. BUNGE

The highest accuracy can be obtained when a spherical sample isbeing used (Tobisch, Bunge 1972). Then the absorption factor A isindependent of the pole figure angles (, fl) as is shown in Figure9a. Because of the low/, x-value in Eq. (6) a "sphere" may havethe 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 seenthat neutron diffraction with the spherical sample method givesmuch better results than X-ray diffraction. Especially, the strongincrease of these coefficients at low A-values does not occur. It hasbeen shown that the increase is due to systematic errors, Figure 12

0.12

0.10

008

006

00/,

. 0.02

\. Bockreflecti0n

\/Iransnission\ Spherical

/Sample

X X X

SphericalJ o o

Sampteo

Neutrons

, 8 12 16 ZO

Order

ligare 11 Error coefficients AF for pole figure measurements: X-ray transmission-

reflection method; X-ray spherical sample method; neutron spherical samplemethod.


0"10

0"08

0"06

0’04

0"02

error

statist=cal error

0 4 8 12 16

x

Order hFigure 12 Statistical and systematical errors of pole figures expressed in the errorcoefficients IAFI

which are particularly detrimental when ODF are to be calculatedfrom pole figure measurements.

MULTIPHASE MATERIALS

A particularly strong influence of the absorption factor on X-raytexture measurements occurs in the case of multiphase materials, ifthe shape and arrangement of the phases is anisotropic. In Figure 13the absorption factor of a directionally solidified Pb-Sn eutectic isshown as a function of the angle , of the diffraction plane relative tothe lamellae plane (Bunge, Liu, Hanneforth, 1987). It is seen thatin this case the influence of absorption is so strong that texturemeasurements by X-ray diffraction become virtually impossible.

284 H.J. BUNGE

18 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 afunction 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 absorptionfactor varies by as much as 25%. Hence, X-ray texture measure-ments are falsified by this amount. In neutron diffraction thisinfluence is in the range of 0.1%. Figure 15 shows inverse polefigures of AI and Pb of a highly extruded AI-Pb composite obtainedby neutron diffraction. These measurements are free of errors dueto anisotropic absorption.

SMALL VOLUME FRACTIONS OF SECOND PHASES

It is interesting to study the texture of a phase which is present in amultiphase material only in small volume fractions e.g. below 1%.

286 H.J. BUNGE

13.9

3.3

Hgare 15 Inverse pole figures of the AI and Pb-phase of pure metals andcomposites 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 secondphase in the sample surface.


Figure 17

CU {llll

0 15 30 5 60 75 90

RLPHRPole 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 areafraction of the phase in the investigated surface is the same as thevolume fraction. With a small volume fraction this may not be thecase simply because of poor statistics. Furthermore because ofdifferent properties of the phases, the surface of the particles maynot be flat as is shown schematically in Figure 16. Hence, X-raytexture measurements in phases of less than about 2 or 3% becomevirtually impossible. In neutron diffraction, on the other hand, thewhole sample volume contributes to the measurement and artefactsat the surface do not play any important role. Hence, neutrontexture measurements can be carried out with reasonable accuracywell below 1 Vol. %. Figure 17 shows for instance the pole figure ofthe copper phase of an A1-Cu composite with 1% Cu. Recentlymeasurement with 0.1% of copper have also been carried out.

SAMPLE PROTECTION

Another advantage of the high penetration depth of neutronscompared 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 interestingwhen studying textures in oxygen-sensitive or hygroscopic materials.Similarly, different kinds of environments may be easily applied to atexture sample, e.g. high temperatures. In this case, the Euleriancradle can be easily shielded from the heating device with shieldsthe 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 ofmagnitude. This allows much bigger samples to be used. On theother hand, also the scattering factors are much smaller. Hence, it isnot only possible but, in most cases, also necessary to use biggersamples.Absorption coefficient and scattering factor are shown in Figure

18 and 19 respectively (see e.g. Bacon, 1975; Squires, 1978). Theyare not only smaller than those for X-rays. Also the variation ofthese quantities with atomic numbers is quite different in the twocases. Both quantities show a systematic, strong increase withatomic number for X-rays which is not the case for neutrons. Thishas some consequences for neutron texture analysis. In neutrondiffraction, neighbouring elements may have quite different scatter-ing factors. Hence it is easily possible to study ordering effects insolid solutions of such elements. As far as texture measurements areconcerned this may be an important advantage when directionalordering in polycrystalline materials is to be studied, for instance iniron-nickel alloys. Directional ordering is strongly related to thetexture of the material. The situation is also quite different for thelight elements. Hence, texture studies of light element phases(especially in the presence of other phases containing heavyelements) 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 differentin X-ray and neutron diffraction. In texture analysis this may beimportant, for instance, if a particular structure factor is zero in oneof the methods. The corresponding pole figure cannot be measuredin this case. An example is the important (0001) pole figure of


100

0()

00

0 0 00 0 0

000 0 o

00 C 0 0

00

0

0)0( l 00

0

Ooo Neutrons o

Ne C Zn Z Sn N:I Yb Hg0 10 20 30 40 50 60 70 80

Atomic number Z

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 bemeasured by X-rays. The corresponding neutron diffraction inten-sity is, however, in the order of magnitude of about 10% of themaximum value. Hence it can well be measured by neutrondiffraction. This is of great interest since A1203 substrates some-times have a strong (0001) fibre texture which is well expressed inthe (0001) pole figure.

290 H.J. BUNGE

x-roys

x lO-Z3cm2O

18

16

Neutrons

x lOZ6cm22OO

oCo

Cu

x-to

MO AuC,

180Hg,,,10

140

120

100

8O

6O

,o -404]O o o o’ OoOZno o2G- -,., c o 20

"0 000n o -..- 0

I0 20 30 40 50 60 70 80Atomic Number Z

Figure 19 The scattering cross section of X-rays and neutrons as a function of theatomic number.

INFLUENCE OF THE ISOTOPE

The neutron scattering factors for various isotopes of the sameelement are usually quite different (till to opposite sign of thescattering factor). A particularly striking example of this effect isobserved in hydrogen. Hydrogen itself has a low coherent scatteringfactor but shows strong incoherent scattering whereas the oppositeis true for deuterium. As a consequence of this, neutron diffractiontexture analysis in natural salt samples may be strongly complicatedby the often observed presence of moisture in these materials. Onthe other hand, texture studies in ice, for instance, are facilitated ifit is possible to use deuterated ice.


Figure 20

1.2

1.0Neutrons

X-toys

0.2 0.4 0.6 0.8 1.0 1.2sin a.,q"

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 atomwhereas the most important part of neutron scattering comes fromthe nucleus the dimension of which is neglegible compared with thewavelength of the neutrons. Hence, the X-ray scattering depends onsin 0/ whereas neutron scattering is independent of this para-meter, Figure 20. This difference is important for texture analysisespecially when pole figures are to be converted into ODF. In thiscase the resolving power is the better the more pole figures can beused. Neutron diffraction then allows to measure also high indexpole figures with a satisfactory accuracy. This is especially importantfor materials with low crystal symmetries.

POSITION SENSITIVE DETECTORS

In the conventional method of pole figure measurement thedetector 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. Thisway, one pole figure is obtained at a time. Then the position of thedetector is changed to the next Bragg-position. Hence, the neces-sary pole figures are being measured sequentially. With the help of aposition sensitive detector the whole O-spectrum can be measuredat 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 figuressimultaneously at the same time. This method provides twoadvantages:

1) the measuring time is greatly reduced

2) Overlapping pole figures can be separated

This latter point is especially important if materials with line-richdiffraction spectra are to be investiaged. Such spectra may be dueto:

a) low symmetry

reactor

II beom shutterfilter

monochromator

monitorcollimator

detectorshietding

detector

diffractometerm-drive

sample

Eulerioncradle

Figure 21 The instrument D1B at Grenoble used as a texture goniometer withposition sensitive detector.


b) high lattice constants

c) multiphase materials

Hence, this method is particularly suited to the study of:

a) intermetallic phasesb) ceramicsc) mineralsd) rockse) composites

Figure 22a shows for instance a felspare spectrum measured thisway (Bunge, Wenk, Pannetier, 1982). An enlarged part of thespectrum is shown in Figure 22b. It is seen that line separation ispossible by line profile analysis. It is also seen that a completeprofile analysis improves the determination of background scatter-ing and hence improves the accuracy of the measured integrated

a) 5-104

I0 20" 30" t,O’ 50’Brogg ongle

Figure 22 Diffraction spectrum of a felspare sample, a) the whole spectrum.

294 H.J. BUNGE

b)

22-

(OZO)(O01)

10’ 11 12"

Brogg angle

c)

5 105 (01i) (0il)(200)

(0) (30)120i) 11i)

15 18 19’ ZO Z18ragg angle 0

Figure 22 (Continued) b) partially overlapped peaks, c) totally overlapped peaks.


intensity (Jansen, Schiifer and Will, 1986). Figure 22c finally showsanother part of the spectrum. Here the overlapping of differentdiffraction peaks is total. These peaks cannot be separated "ex-perimentally" by line profile analysis. They can however beseparated "theoretically" in the process of ODF analysis using thespecial 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).When a position sensitive detector is being used the plane of the

Eulerian cradle (plane of the ;t-rotation) can only be in "symmetryposition" for one specific Bragg angle e.g. the angle 0,, in Figure23. Only for this angle, the x-circle passes through north and southpole of the pole sphere. For all other O-angles ;t and t# rotationgives a pole figure with a "blind area" in the vincinity of north- andsouth pole as is shown in Figure 24. Hence, in order to obtaincomplete pole figures an additional scan is necessary in order fill inthe blind area and a coordinate transformation is necessary bywhich the angles to;t of the Eulerian cradle are transformed intothe pole figure angles crfl (Bunge, Wenk, Pannetier, 1982). Some of

\\m:

Eulerian ’’’craddle 0(,J.) /,.>

incident] beam

Figttre 23 Adjustment of the Eulerian cradle in "symmetry position" for one of theBragg angles 0,.

296 H.J. BUNGE

Figure 24 Pole figure scan for "off-symmetry" Bragg angles showing blind areas.

Figure 25 Four pole figures of a felspare sample.


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 "Braggangle orientation ." In this orientation it measures one pole figurepoint 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

a) Oebye-Scherrer b)Kegel 90

Reflexions-K

Primir-

Zhler Erfat.erBereich Potfigur

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.

298 H.J. BUNGE

chosen such that 2 90. The detector then measures a whole lineof pole figure points but only in one pole figure as is shown inFigure 26b. In order to scan a whole pole figure completely agreatly reduced number of steps is then sufficient, as is shown inFigure 26d. This method has especially been applied for rapid polefigure 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 wavelength dispersive method can be used. In this case, white radiationis being used. The reflection at different lattice planes is thenobtained 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(8)

m.u

Hence, wavelength measurement can be obtained by time-of-flightmeasurement and this, in turn, requires to fix the starting time of aneutron. Time of flight measurement requires narrow pulses ofneutrons following each other in rather long time intervals. Apulsed beam can be obtained mainly in two ways:

a) a continuous beam can be "chopped" into short pulses using achopper

b) The neutrons can be directly produced in pulses e.g. in a pulsereactor or by a pulsed spallation source.

It is thus necessary to measure the time of each incoming neutronregistered in the detector. The "time spectrum" obtained with oneneutron pulse thus corresponds with the wavelength spectrum andhence with the d-value spectrum. Such spectra are then to be addedup in a multi-channel analyzer for a sufficient number of neutronpulses. Finally a time-of-flight spectrum is obtained as shown forinstance in Figure 27 (Feldmann, 1986). Such spectra have to be


100 200 300 400 500 800 700 800 gO0

N

Figure 27 Time-of-Flight spectrum of copper.

measured for all necessary sample orientations in order to obtain allpole figures at the same time. Thus far, this method is similar to themethod using a position sensitive detector. In contrast to the latterone, however, all pole figures are measured with the sameBragg-angle, i.e. in "symmetry position" as in the sequentialangular dispersive method. Hence, a blind area does not occur inthis 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 onemethod i.e.

a) PSD in Bragg-angle direction

b) PSD in pole figure direction

c) TOF method

This requires a pulsed neutron source and a two-dimensionalposition sensitive detector which is at the same time also time-sensitive. This method shown in Figure 28 has been used byVergamini and Wenk (1988). It gives---at the same time--a

300 H.J. BUNGE

OETECTOR .COLLIMAI’OR

10 BEAMSTOP

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

two-dimensional area of a large number of pole figures. Hence, thismethod is the most economic one.

MAGNETIC TEXTURE ANALYSIS

A second part of neutron scattering is due to magnetic interactionof the neutron’s magnetic dipole moment with the distribution ofthe magnetic moment of the electrons in the atom. Hence, this partof neutron scattering depends on sin t/ in a similar way as X-rayscattering, Figure 29 (see e.g. Bacon, 1975). In the case ofnon-polarized neutrons the magnetic part of the reflected intensityis proportional to

q2 sin2 a’ (9)

where a is the angle between magnetization direction and thediffraction vector s i.e. the normal direction to the reflecting latticeplane as is seen in Figure 30. Magnetic neutron scattering is relatedto the magnetic crystal structure of the material which may bedifferent 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 theatoms are parallel to a certain crystal direction m, in the secondcase they are mutually antiparallel. This means that in the first casethe unit cell is the same as the chemical unit cell but crystalsymmetry is different and in the second case unit cell and symmetryboth are changed. From the view point of texture analysis theferromagnetic 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.

302 H.J. BUNGE

field H the magnetization direction rn may be parallel to any crystaldirection h. Two limiting cases may be particularly considered. Inthe case ofzero magnetic field the magnetization direction is parallel toone of several crystallographically equivalent directions hg. In thecase 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 ineach crystal. It is parallel to the same sample direction but todifferent crystal directions which are however uniquely determinedby crystal orientation g. In the first case the distribution ofmagnetization directions over the crystal directions hg will generallydepend on the magnetic history of the material. Different directionshg may occur in one and the same crystallite which is then dividedinto magnetic domains. In this case the "magnetic texture" can beeasily defined as the orientation distribution function of thedomains. We specify a crystal coordinate sytem K, the xa-axis ofwhich is parallel to the magnetization direction hg of the considereddomaine. The orientation of this coordinate system with respect tothe sample coordinate system K, is described by the rotation gM.The magnetic texture is then the volume fraction of domains havingan orientation gM within the angular range dg1

fM(g) dV(gm)/VdgM (11)

This definition of the magnetic texture is completely analogous toEq. (1), only the symmetry of the magnetic texture function isdifferent from that of the crystallographic texture. As an example,in cubic ferromagnets the magnetization direction is often parallelto 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 conventionalsample symmetry due to the distribution of magnetization directionsover the various h-directions. Also the pole figures of this magnetictexture are defined in the same way as in the crystallographic


texture (see e.g. Miicklich, Hennig, Bouillot and Matthies, 1984)dV/VP(y) (13)dy

with the only difference that now h is the normal direction to thereflecting lattice plane (hkl) indexed in the magnetic coordinatesystem K i.e. with the lower "magnetic" crystal symmetry (e.g.tetragonal). The pole figure P(y) can, however, not be measureddirectly by neutron diffraction. Rather all those pole figures aresystematically 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). Polefigure inversion, i.e. the calculation of the ODF fVt(g) from polefigures P(y) can be carried out with superposed pole figuresPh}(Y) instead of the no-superposed ones (see e.g. Dahms andBunge 1987). Hence, information about the magnetic texture can beobtained from the magnetic part of neutron diffraction pole figures(see also Bunge (in print), 1989).

CONCLUSIONS

Texture measurements can be carried ou by neutron diffractionmuch in the same way as by X-ray diffraction, i.e. by pole figuremeasurement followed by pole figure inversion. Because of thespecific properties of the two radiations there are, however, somedifferences between the two method as far as the quantitativerelations are considered. These differences are mainly in favour ofneutron diffraction, whereas the main drawback of this radiation(compared to X-rays) is its limited availability and high cost ofneutron diffraction equipment and especially of the neutron source.The advantages of neutron diffraction are, first of all, due to thelower absorption coefficient. As a consequence of this, much biggersamples can be used.

304 H.J. BUNGE

--This in turn leads to a much better grain statistics. Hence,samples with one order of magnitude bigger grain sizes can besuccesfully 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 figuresmay 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 whichare present with only very low volume fractions.

raThe high penetration depth allows the global texture oftechnologically 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 scatteringfactor independent of sin /. This allows high-index pole figures tobe easily measured.

Neutron diffractometers are often equipped with position sensitivedetectors which have not yet become standard equipment in X-raydiffraction. A PSD in O-direction allows the measurement of manypole figures at the same time. Furthermore, it allows the seperationof overlapping peaks which is particularly valuable for materialswith big unit cells and low symmetries as well as multiphasematerias. Finally, neutron scattering contains a magnetic part whichallows magnetic textures to be measured which is by no meanspossible using X-ray diffraction.

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