Texture of Crystalline Solids1976, Vol. 2, pp. 95-111

O1976 Gordon & Breach Science Publishers, Ltd.Printed in Scotland

QUANTITATIVE DETERMINATION OF PREFERRED ORIENTATIONBY ENERGY-DISPERSIVE X-RAY DIFFRACTION

L. GERWARD, S. LEHN, and G. CHRISTIANSENLaboratory of Applied Physic8 III, Building 30?,Technical .University of Denmark, DK-2800 Lyngby,Denmar

(Received February I, 1976)

Abstract" The use of energy-dispersive X-ray dif-fraction for quantitative determination of preferredorientations in polycrystalline specimens is ana-lysed. The method is applied to determinations ofrolling texture and fibre texture. The-adaptabilityof the method to in 8itu studies is demonstrated byobservations of texture changes simultaneous withthe deformation of a specimen in a tension test.

1 INTRODUCTION

The purpose of the present work is to develop energy-dispersive X-ray diffractometry into a useful tool forquantitative analysis of preferred orientation. Energy-dispersive diffraction, introduced in 1968, 1,2 has someunique advantages compared with standard angle-dispersivediffraction but requires some changes in the analyticalprocedures generally used.

In energy-dispersive diffraction a beam of poly-chromatic X-rays is incident on the crystalline specimen.The photon-energy spectrum of the scattered X-rays ismeasured by a semiconductor detector connected to a mul-tichannel pulse-height analyser. In this way severalBragg reflections are recorded simultaneously. The in-formation obtained with a fixed scattering angle and afixed specimen orientation is sufficient for the deter-mination of an inverse pole figure. Alternatively, sev-eral normal pole figures can be obtained with a fixedscattering angle by varying the specimen orientation.

95

96 L. GERWARD, S. LEHN, AND G. CHRISTIANSEN

Laine and Lhteenmki used an energy-dispersivemethod to study preferred orientation in splat-cooledcadmium. They used the intensity ratio of the 002 dif-fraction line and the Cd Ks fluorescence line as a meas-ure of the preferred orientation. Szpunar, Ojanen andLaine 4 showed that the spectrum of diffraction lines canbe used for the construction of inverse pole figures. Asimilar method using neutron diffraction and time-of-flight analysis was proposed by Szpunar, O14s, Buras,Sosnowski and Pietras.

The quantitative determination of rolling textureand fibre texture using energy-dispersive diffraction isdeveloped in the present work. The theoretical formulafor the integrated intensity in terms of the photon en-ergy is derived starting with an ideal non-absorbing spec-imen without texture and then applying various correc-tions. In particular, the absorption in a real specimenaffects the observed intensities to a large extent. Thismakes it difficult to compare the X-ray patterns for dif-ferent positions of the specimen. In addition, the ab-sorption is strongly wavelength dependent. This is acomplication in energy-dispersive diffraction where acontinuous X-ray spectrum is used. The X-ray absorptioncoefficient for a given specimen may vary by almost twopowers of ten within the useful wavelength range.

The determination of inverse pole figures is ham-pered by problems associated with normalizing the data.In this work it is shown how the normalizing can be im-proved by optimizing the scattering angle and by sep-arating the intensity contributions of the overlappingdiffraction peaks.

The energy-dispersive method is well suited for n8.tu studies on specimens subjected to various controlledenvironments. This has been demonstrated in a dynamicexperiment- changes in the texture of a recrystallizedaluminium wire have been observed by energy-dispersivediffraction analysis simultaneous with the deformationof the specimen in i tension test machine.

2. ENERGY-DISPERSIVE DIFFRACTOMETRY

Figure 1 shows the principle of energy-dispersiveX-ray diffractometry. Using this method the Bragg equa-tion can be written in the following form

(Ed)hk sin 8o hc/2, (i)

where Ehk Z X-ray photon energy, dhk i interplanar

spacing, @0 fixed Bragg angle, h Planck’s constant

and c velocity of light. For practical purposes it

DETER/INATION OF PREFERRED ORIENTATION 97

is convenient to notice that the constant on the right-hand side of Equation (I) equals hc/2 6.199 keV.A ifE is expressed in keV and d in .

X- ray

S

Sampletable

Multi-channelpulse heightanaiyser

Sample//

s ,@

"’ Semi-conductordetector

FIGURE I. Principle of the energy-dispersive X-raymethod. Sl, $2, $3 and S slits defining the beampaths, 28o fixed scattering angle.

In this work we have used the continuous X-ray spec-trum from a tungsten tube and a Si(Li) detector. A moredetailed description of the equipment is given elsewhere.The experimental diffraction peaks have been correctedfor the background and fitted to a Gaussian shape in or-der to calculate the integrated intensities, half widthsand peak positions. Using Curve fitting it has also beenpossible to separate diffraction peaks which are partlyoverlapping.

3. INTEGRATED INTENSITIES

For a powder specimen the integrated intensity, i.e.,the total diffracted power, in a complete Debye-Scherrerring using the energy-dispersive method is given by

(l+cos 2 2e 0)2 N2V[i0 (I)X j JFJ 2]hki 8 sin 80re cot e0 AS0, (2)

where re 2.82 x l0-I cm the classical electron rad-ius, N number of crystal unit cells per unit volume,V volume of irradiated material, i0 (I) intensity perunit wavelength range of the incident beam, j multi-plicity factor, F structure factor and AS0 diver-gence of the X-ray beam.

Equation (2) has been derived with the assumptionsthat the orientations of the crystallites are randomlydistributed and that absorption and extinction are

98 L. GERWARD, S. LEHN AND G. CHRISTIANSEN

negligible.* The subscript hkl indicates that the quan-tities I, j and F depend on the reflection hkl.

Preferred orientations are described by the orien-tation distribution Phki(’ This is proportional to

the volume fraction of material having normals to a givenfamily of lattice planes (hki) oriented in a small solidangle d in the angular range to + d, 8 to + d,relative to some fixed directions in the specimen. Theorientation distribution is unity for a random specimen.

Absorption is taken into account by multiplyingEquation (2) with the absorption factor A(, 80, ,8)In the energy-dispersive method, where the wavelengthis variable, the absorption factor depends on the wave-length of the diffracted X-rays as well as on the dif-fraction geometry.

It is generally very difficult to apply extinctioncorrections. In the energy-dispersive method it is evenmore complicated due to the wavelength dependence. Inthis work we assume that the grain size of the polycrys-talline specimen is sufficiently small (of the order ofa few m) so that the extinction can be neglected.

The diffracted intensity is recorded as a functionof the photon energy by the semiconductor detector sys-tem. It is therefore convenient to express the inte-grated intensities on the photon energy scale using

i0 (E) i0 (I) (12/hc)i0 (I) (hc/E 2)i0 (I) (3)

where i0 (E) intensity per unit energy range of the in-cident beam.

Finally, we take into account that the detector re-cords the fraction D/2 r sin 28o (D height of detectorwindow and r distance between sample and detector) ofthe total intensity in the Debye-Scherrer ring.

Inserting the corrections discussed above in Equa-tion (2) one obtains the following expression for theintegrated intensity, Ihk, to be used in the energy-

dispersive analysis**:

*Furthermore the incident beam is supposed to be un-polarized. Polarization effects may have to be consideredfor the shortest wavelengths of the incident spectrum.

**Another obvious correction is that due to the spec-tral response of the detector. However, if the same de-tector is used throughout the experiments one can regardthe intensity i0 as the effective intensity as seen bythe detector.

DETERMINATION OF PREFERRED ORIENTATION 99

Ihkz=(hc) 3r 2 D N2V[i0 (E)Ee r

l+cos 228 o AO2j IF 2A(E,@0 ,e,8)P(e,8) ]hkZ-32 sinO0

(4)

Equation (4) may be written more concisely as

IhkiC[i0 (E)E-2 jlFI 2A(E,O0,,8)p(,8)]hkZ (5)

where C is a constant depending on the diffractiongeometry.

4. DETERMINATION OF INVERSE POLE FIGURES

In the energy-dispersive method a number of reflec-tions hkZ are recorded simultaneously in a given diffrac-tion geometry. For example, a wire specimen can be ar-ranged so that the wire axis is parallel to the scatter-ing vector.* In this way lattice planes having normalsalong the wire axis contribute to the diffracted inten-sities. The observed diffraction pattern can be directlyused for quantitative analysis by the construction of theinverse pole figure according to Harris’ method 9 withsome modifications due to the use of a continuous X-rayspectrum.

The integrated intensities of the specimen are com-pared with those of a random specimen. The ratio of thecorresponding intensities, corrected for absorption, willbe called Xhk Z. It follows from Equation (5) that

(I r C___Xhk Ir hkZ Cr Phk (6)

where the subscript r denotes the random specimen.The orientation distribution for a given family of

lattice planes (hkZ) is normalized in real space by

2- Phki(’8) d I, h,k,Z const., (7)

*The scattering vector is parallel to u- u0,

where u unit vector in the direction of the scattered

beam and u0 unit vector in the direction of the inci-

dent beam.

I00 L. GERWARD, S. LEHN AND G. CHRISTIANSEN

where the integration is performed over all orientationsrelative to the specimen. In reciprocal space one hasto resort to an approximate normalization by assumingthat for a large number of reflections in a given dif-fraction geometry one has

1 7. Phki(e’8) i, ,8 const.nhki

(8)

where n number of p-values considered. It should benoted that only one p-value is counted when several or-ders of reflection from a particular family of latticeplanes are observed.

The p-values associated with the direction parallelto the scattering vector can now be obtained using Equa-tion (6) and the normalizing condition, Equation (8)

PhkZ Xhk n (9)

’ Xh’k’ Z’h’k’i’

In structures of high symmetry it often occurs thatdifferent families of lattice planes have the same inter-planar spacing. The corresponding diffraction lines ap-pear at the same photon energy according to Equation (i).In many cases the intensity contributions from each ofthe reflections can be separated because the p-value forone family of lattice planes is known from another orderof reflection.

Consider two reflections, hlklll and h2k22, whichcoincide in the diffraction pattern. According to Equa-tion (5) the measured integrated intensity is given by*

Ii,2 C[io (E)E-21FI 2 A] ,2 x (jPl + j2P2)- (i0)

For the random specimen one has a similar expression butwith p p2 i. The ratio of the intensities from thespecimen and the random specimen, corrected for the ab-sorption is then

*Here we assume that the two reflections have thesame structure factor. In non-centrosymmetric crystalsthe structure factors may differ due to dispersion ef-fects and Equation (i0) should be modified.

DETERMINATION OF PREFERRED ORIENTATION 101

(I Ar) C jPl + j2Pz jlxl + j2x2X ,2 I r A ,2 Cr 91 + 92 jl + 92 (Ii)

where x and x2 are the ratios, which would have beenobserved if the reflections had not coincided, If xl isknown from another order of reflection then x2 can becalculated from Equation (ii). Examples are given inthe following sections.

5. ROLLING TEXTURE

Determination of rolling texture has been exempli-fied using a cold-rolled thin foil of e-iron (puritybetter than 99.9%). The thickness of the foil has beendetermined as 15.6 m. Foils of this kind are suppliedas X-ray filters in sheets of 20 mm x 170 mm in size byJohnson & Mat they Chemicals Ltd., London. Photographi-cally recorded Debye-Scherrer rings were used to controlthe grain size.

Figure 2 shows the diffraction pattern for an ironpowder used as the random specimen. Figures 2b and 2cshow the corresponding patterns for the iron sheet ori-ented with the rolling direction parallel to the scat-ering vector (Figure 2b) and with the surface normal par-allel to the scattering vector (Figure 2c). The quali-tative features of the texture can be found immediatelyby visual inspection of the spectra. Notice for examplethat the 022 reflection is very strong in Figure 2b butalmost absent in Figure 2c.

The 011 and 002 diffraction lines have been omittedbecause they overlap the tungsten L-lines of the X-rayspectrum. However, P0 i can be determined from the 022and 044 reflections, which are present in the spectrum.In the same way P00 can be determined from the 004 re-flection. The contributions of the 033/114 and the006/244 peaks can be separated according to the methoddescribed in Section 4. The reflections up to and in-cluding 136 have been used. In this way the p-valuescan be normalized using 13 points in the inverse polefigure.

It has been found that a scattering angle about28o 50 deg is the best choice in the present case. Atthis angle the largest number of useful diffraction peaksis situated within the observable energy range.

Absorption factors for a specimen in the shape of asheet large enough to intercept the entire incident beamare given in the International Tables for X-ray Crystal-lography I and in many textbooks on X-ray crystallography.

102 L. GERWARD, S. LEHN AND G. CHRISTIANSEN

OO

’T’"

I.-- 4 -,’-’ 0 c 0". ..,. uol’.ll eo

2

10 20 30 40

E keV --terns o iron. 20 50 . (a) Random spec+/-men(powder). (b) Rolle seet, roll+/-ng irect+/-on par-allel to scattering vector. (c) Rolled seet, sur-face normal parallel to scattering vector. Noticethe presence of a strong texture.

The photon energy dependence of the X-ray absorptioncoefficient has been taken into account using the relation

(E)/p k E-m (12)

where (E) linear absorption coefficient, p densityof mass and k and m are constants as long as no absorp-tion edge is included in the energy range.

DETERMINATION OF PREFERRED ORIENTATION

Linear regression analysis of the absorption datagiven in the International Tableslof X-Ray Crystallo-graphy (Vo.l. IV) t0 has given k .321 x i04, m 2.820(p/p in units of cm2/g) and the coefficient of determina-tion r 2 0.99998 for E < 7.11 key (iron K-absorptionedge) and k 9 019 x 104 0.99991m 2.728 and rfor E > 7.11 keV.

Figures 3a and 3b show the inverse pole figures inthe rolling direction and the normal direction, respec-tively. The highest probabilities are observed for{iii} planes parallel to the sheet surface and <ii0>directions parallel to the rolling direction but othertexture systems can also be observed in the pole figures.

N

0.10

// R.D.

0",111&O.

L!o

0.0

0.34O.O3 2 5

0.0

0.15 0.21 7.2

001,>101

FIGURE 3a. Inverse pole figure of iron. Rollingdirecion.

The reproducibility of the p-values has been esti-mated in two ways: (i) The p-values determined from dif-ferent orders of reflections available in the same dif-fraction pattern have been found to agree within 10%;(2) Using the diffraction patterns recorded at threedifferent scattering angles and normalizing them with

104 L. GERWARD, S. LEHN AND G. CHRISTIANSEN

N

/Z3

1

./0.0

0.050.70

0.(?&O o

2.5 0.04 0.0 0.0

001101

FIGURE 3b. Inverse pole figure of iron. Normaldirection.

the same number of reflections we find that the corre-sponding p-values agree within 15%.

6. FIBRE TEXTURE

The determination of inverse pole figures are ofparticular value in the analysis of axisymmetric speci-mens. Figure 4 shows a diffraction pattern from an alum-inium wire. The wire axis is placed parallel to thescattering vector. It is obvious that the wire has astrong [iii] texture. The very characteristic featuresare that the 002 reflection has almost disappeared fromthe wire sample, and that the relative intensities ofthe 113 and 222 peaks are reversed compared with therandom specimen.

For the random specimen a capillary filled withaluminium powder was chosen. Absorption factors for therandom specimen and the wire oriented with the wire axisperpendicular to the scattering vector are given in the

DETERMINATION OF PREFERRED ORIENTATION

133A024

004 IV 115f....., / \ 224333

15 20 25 30 35

E(keV) =

FIGURE 4a. Energy-dispersive X-ray diffraction pat-tern of aluminum. 280 32 Random specimen(powder)

International Tables for X-Ray Crystallography. 10

The absorption factor for a wire specimen orientedwith its axis parallel to the scattering vector has beencalculated in this work using the definition of A: TheX-rays reflected by the specimen are reduced by a factbr

A V exp[-(p + q)]dr (13)

where V volume of irradiated material, p and q lengthsof the paths of the incident and reflected beams, re-spectively, in the crystal.

Figure 5 shows the diffraction geometry. It followsfrom the symmetry that the sum of the beam paths are thechord of an ellipse. Equation (13) can then be written

106 L. GERWARD, S. LEHN AND G. CHRISTIANSEN

222

F_

022

]5

115133 22433

004 1024 l\ __a

20 25 30 35

E(keV) :

FIGURE 4b. Energy-dispersive X-ray diffraction pat-tern of aluminium. 28o 32 Wire, wire axisparallel to scattering vector.

R2].,R /i- y2/R2) dy, (14)A 2R i ’2/R 2 exp( cos 80

-R

where R radius of the wire. Equation (14) has beensolved by numerical integration.

The linear absorption coefficient for the photonenergy of each of the diffraction lines was calculatedaccording to Equation (12) using k 2 266 x 104 m2. 937 (/p in cm2/g) and r 2 0. 99996.

The inverse pole figure for the direction parallelto the wire axis is shown in Figure 6. The 333/115 con-tributions have been separated according to the methoddescribed in Section 4. The p-values have been normalize(using 9 points.

DETERMINATION OF PREFERRED ORIENTATIONS lo7

)0

FIGURE 5. Diffraction geometry for calculating theabsorption factor of a wire specimen. Wire axisparallel to scattering vector. (Notation, see text.)

7. A DYNAMIC EXPERIMENT

The advantages of a fixed scattering angle and arapid data acquisition make the energy-dispersive methodsuitable for texture studies in dynamic experiments.This has been demonstrated in a tension test wherechanges in the texture are observed simultaneously wfththe deformation of the specimen.

Figure 7 shows the experimental arrangement. Thespecimen is an aluminium wire of the same kind as theone used in Section 5. The X-ray tube and the detector

108 L. GERWARD, S. LEHN AND G. CHRISTIANSEN

0. 7 1.." 0.13.

0,53. 0.21.

0.09 0.03

101FIGURE 6. Inverse pole figure of aluminium wire.Direction parallel to wire axis.

were inclined about 15 with respect to the vertical andplaced close to an Instron Model 1026 tension test machine.The wire specimen was mounted vertically and loaded insteps of about 0.5 kg until break occurred. After eachstep a diffraction pattern was recorded in a countingtime of i00 s.

The texture of the wire as it was received from themanufacturer appeared to be insensitive to the deforma-tion. No changes in the diffraction patterns were ob-served until break occurred. Therefore a new series ofexperiments was performed with samples, which were heattreated at 550C before the tension test.

The heat treatment resulted in a texture which wassensitive to the deformation. Figures 8a to 8c showsome diffraction patterns recorded during a tension test.At about one half of the maximum load one observes agradual decrease of the 222 peak and a corresponding in-crease of the 133/024 peak (the two lines are not re-solved in the spectra).

The experiments have shown that the energy-disper-sive method can rather easily be adapted to dynamicstudies. In this way, the method could be used for linecontrol of the texture in production processes, such as

DETERMINATION OF PREFERRED ORIENTATION lO9

F

FIGURE 7. Experimental arrangement (schematic) forX-ray diffraction analysis simultaneous with tensiontest of wire specimen. F applied force, MCAmultichannel pulse-height analyser.

the forming of a wire or a sheet. With the X-ray tubeand detector mounted close to a heat chamber the devel-opment of recrystallization texture could be followedin a similar way as a function of temperature and time.

ACKNOWLEDGMENTS

The work reported here was made possible by a grantfrom Statens teknisk-videnskabelige Fond. This financial

ii0 L. GERWARD, S. LEHN AND G. CHRISTIANSEN

111 022

133222 024

115

o 260 460 e6o 800

Channel number

FIGURE 8. Energy-dispersive X-ray diffraction pat-terns recorded at different stages of the deforma-tion of an aluminium wire. 28o = 30 (a) Load4.0 kg. (b) Load 5.1 kg. (c) Load 6.5 kg. Noticethe changes in texture. Break occurred at the load10.5 kg.

support is gratefully acknowledged. The authors wish tothank Professors L. Lindegaard-Andersen .and B. Buras forreading the manuscript and for valuable comments. Theyare also indebted to B. Ribe for technical assistance.

REFERENCES

l B. C. Giessen and G. E. Gordon, Se.nee, 159, 973(1968).B. Buras, J. Chwaszczewska, S. Szarras and Z. Szmid,Institute of Nuclear Research (Poland), Report No.894/II/PS (1968)E. Laine and I. Lhteenmki, J. Mate. Sc., 6, 1418(1971).J. Szpunar, M. Ojanen and E. Laine, Z. MetaZZkc.,65, 221 (1974).. Szpunar, A. O14s, B. Buras, I. Sosnowska and A.Pietras, Nuceon.k, 13, iiii (1968)B. Buras, J. Staun Olsen, L. Gerward, B. Selsmarkand A. Lindegaard Andersen, Aot Cyst., A31_, 327(1975).B. Buras and L. Gerward, bd., A31, 372 (1975).O. Alstrup, L. Gerward, B. Selsmark, B. Buras andJ. Staun Olsen, bd., A31, S 234 (1975).

DETERMINATION OF PREFERRED ORIENTATION iii

G. B. Harris, Phil. Mag., 4_3, 113 (1952); see alsoC. S. Barrett and T. B. Massalski, Structure ofMetals, 3rd ed., McGraw-Hill, New York (1966)International Tables for X-Ray Crystallography,Vol. IiI (1962) and Vol. IV (1974) Kynoch Press,Birmingham.