Bragg Edge Determination for Accurate Lattice Parameterand Elastic StrainMeasurement

A. Steuwer (a, b), P.J. Withers (b), J.R. Santisteban (c), L. Edwards (c),G. Bruno (c), M.E. Fitzpatrick (c), M.R. Daymond (d), M.W. Johnson (d),and D. Wang (c)

(a) Department of Materials Science, University of Cambridge, Pembroke Street,Cambridge, UK

(b) Manchester Materials Science Centre, UMIST/University of Manchester,Grosvenor Street, Manchester, UK

(c) Department of Materials Engineering, The Open University, Walton Hall,Milton Keynes, UK

(d) ISIS, Rutherford Appleton Laboratory, Chilton, Didcot, UK

(Received September 7, 2000; in revised form February 6, 2001; accepted February 13, 2001)

Subject classification: 61.12.Ex; 61.12.Ld; 61.66.Dk; 62.20.Dc; S1.1; S1.2; S1.3

The transmission spectrum of thermal neutrons through a polycrystalline sample displays sudden,well-defined increases in intensity as a function of neutron wavelength. These steps, known asBragg edges, occur at the point at which the neutron wavelength exceeds the Bragg condition forcoherent scattering from the respective lattice planes, and can be easily observed using the time-of-flight method. Accurate location of these edges and determination of their magnitude and shapecan be used to extract information about the stress state, texture and phases present in the materi-al. This paper describes a method for analysing these edges singly and collectively, using a Pawley-type refinement. Furthermore, experimental trials are presented which demonstrate the utility ofthe method for the accurate measurement of lattice spacings, and thus strain. These trials includemeasuring the lattice parameter in Cu/Zn alloys as a function of Zn content, and the determinationof elastic strain of an iron rod under tensile/compressive straining. In the former case the resultsare compared with Bragg diffraction peak measurements made on HRPD and in the latter casewith conventional strain gauge measurements.

1. Introduction

Over the last two decades, neutron diffraction has played an increasingly important rolein the determination of elastic strains in materials and engineering components [1]. Incontrast to X-rays, the weak interaction of thermal neutrons with solids opens up thepossibility of monitoring strains non-destructively far inside the bulk material, even un-der working/loading conditions. Conventionally, accurate measurement of lattice spac-ing is undertaken by measuring small shifts in the angular position of diffraction peaks,using Gaussian, or in the case of the pulsed neutron technique, more complex peak-profile fitting routines. Since in the time-of-flight (TOF) method large sections of thespectrum are measured the accuracy achieved in single peak fitting can be significantlyincreased by using whole-pattern-structure-refinement, such as Rietveld refinement [2],or the less constrained Pawley-type [3, 4] refinement procedures, where many diffrac-tion peaks are fitted simultaneously, refining crystal parameters globally. In this paperwe present an extension to conventional TOF neutron diffraction in which information isextracted from the Bragg edges in the recorded transmission spectrum [5, 6], see Fig. 1.

phys. stat. sol. (a) 185, No. 2, 221–230 (2001)

In particular we present a method for fitting these edge profiles, both singly and multi-ply. Very good fits are achieved and the accuracy with which the associated edge posi-tions can be located is assessed. To examine the utility of the technique a number oftrial experiments undertaken on the PEARL beam-line at the ISIS pulsed source in thetransmission geometry are analysed and the results compared with those from comple-mentary techniques. Firstly, the measurement of lattice parameter is studied for a rangeof Cu/Zn alloys and compared with smilar measurements made on the High ResolutionPowder Diffractometer (HRPD) also at the ISIS. Secondly, the extent to which thetechnique can be used to measure strain is demonstrated through a series of measure-ments on an iron bar loaded under tension/compression. In this case the results werecompared with total strain measurements made using a strain gauge.

2. The Neutron Transmission Geometry and Time-of-Flight

At pulsed neutron sources [7], moderated neutrons of all wavelengths emerge over avery short time-pulse from the source. The short wavelength neutrons travel fasteralong the flight path from source to detector than the long wavelength neutrons. Thismeans that the wavelength of a detected neutron can be deduced from its time of arri-val, hence the name time-of-flight.

For all incident wavelengths, a subset of suitably aligned grains in a polycrystallinesample can be found that diffract neutrons at a certain angle q. However, for a givenhkl-reflection at a particular wavelength the diffraction angle reaches 2q ¼ 180�, and afraction of the incident neutrons is completely back-scattered towards the source. Inthis case the diffracting lattice planes are aligned perpendicular to the incident beam.

222 A. Steuwer et al.: Bragg Edge Determination for Accurate Lattice Parameter

Fig. 1. Typical TOF transmission spectrum at a spallation source, in this case for iron (Fe) powderof 12 mm thickness, compared with the incident spectrum. The powder spectrum clearly displaysthe Bragg edges and the underlying bcc crystal symmetry. The incident spectrum shows the charac-teristic Maxwellian distribution from the moderation process and has been scaled down to fit thegraph. Edges visible in the incident spectrum arise due to transmission through the evacuatedbeam guide windows made of aluminium

For longer wavelengths, diffraction from that set of lattice plane spacings cannot occur.This is accompanied by a sudden step, the Bragg edge, in the transmitted intensity. As aresult, each Bragg edge indicates a particular lattice spacing of the crystal structure ofthe sample [5], as shown in Fig. 1. Analogous to diffraction the distribution of edgesover the spectrum characterises the crystal structure, and a change in the position of aBragg edge corresponds to a change in the lattice spacing for planes normal to thebeam and hence enables the determination of strains in this direction.

2.1 Time-of-flight Bragg diffraction

The fundamental equation of diffraction is the Bragg equation

l ¼ 2d sin q : ð1Þ

The spectrum of diffraction peaks is usually obtained during a q=2q-scan using a singleknown incident wavelength, in which strains are determined by a shift in the angularpeak position, e ¼ �cot q Dq. In the time-of-flight method, however, the scattering an-gle is kept constant and instead, the spectrum is recorded as a function of neutronwavelength l ¼ h=p, where h is Planck’s constant, and p the neutron momentum. Sincethe flight path length L on a TOF neutron beam-line is constant, and neutrons of allwavelengths emerge from the spallation process at the same time, fast neutrons arriveat the detector earlier than slow ones and the neutron wave-length can be expressed asa function of time. Explicitly, the relationship between neutron velocity v, wavelength land detection time t (time-of-flight) is given by v ¼ L=t ¼ p=m ¼ h=lm, where m de-notes the neutron mass. The scattering angle in transmission equals 2q ¼ p and insert-ing the above expressions in the Bragg equation one obtains a linear equation describ-ing the Bragg condition in pulsed neutron transmission:

d ¼ h

2mLt ; ð2Þ

where d is the lattice spacing. Since eq. (2) is a simple linear relationship betweenlattice spacing and the time-of-flight, the definition of strains in this framework simplyextends to the strain

ehkl ¼dhkl � d0

hkl

d0hkl

� Dt

t

� �hkl

; ð3Þ

where d0hkl is the unstrained lattice spacing of reflection hkl. Strains in the direction

parallel to the beam can therefore be determined by measuring the relative shift inposition of individual Bragg edges in the time-of-flight transmission spectrum. It isworth pointing out that all the Bragg edges (from different sets of grains) in the time-of-flight spectrum measure strain in this particular direction.

2.2 Neutron transmission

Formally, neutron transmission records the response of the material as a function of theincident wavelength, and the transmitted spectrum TðlÞ reflects the macroscopic [8]total scattering cross section SðlÞ of the sample TðlÞ ¼ ItrðlÞ=IiðlÞ ¼ exp ð�SðlÞ xÞwhere Ii; Itr denote the incident and transmitted intensity, and x the sample thickness.The expression for the scattering cross-section for a particular set of lattice planes is

phys. stat. sol. (a) 185, No. 2 (2001) 223

taken from [5] and is, neglecting numerical factors for simplicity, proportional to the squareof the neutron wavelength l2, the square of the structure factor F2

hkl, the number of unit

cells N and the d-spacing of a particular reflection dhkl SðlÞ � Nl2

2

Phkld�l

2

F2hkldhkl , where this

expression has to be summed over all sets of crystal planes and orders of reflection whichare consistent with the Bragg condition, Eq. (2). Figure 2 shows the comparison betweenthe theoretical and an experimental cross-section TðlÞ for a standard Fe-powder. The theo-retical cross-section has been calculated using the Fortran routines called CRIPO [9].

2.3 The Bragg edge profile

For most engineering materials coherent elastic scattering predominates in the thermalregime, and the transmission profile can be understood in terms of the elastic scatteringevents which remove neutrons from the transmitted beam. As a result the shape of theedge, hðt; t0Þ, can be described by subtracting the integral of all the (elastic) diffractionevents which remove intensity from the just before given hkl-reflection (in other words,of scattering events that occur at wavelengths slightly smaller than twice the spacing ofthe hkl-lattice planes, since l ¼ 2d sin 90� ¼ 2d):

hðt; t0Þ �Ðt

�1gðt0; t0Þ dt0 ; ð4Þ

where gðt; t0Þ is the normalised diffraction peak-profile of the particular neutron source, e.g.a normalised Gaussian, and t0 denotes the time-of-flight position of a particular hkl-reflec-tion. Diffraction peak shapes are generally a convolution of several instrumental and mate-rial contributions. Whereas peaks at sources which are in thermodynamical equilibrium,such as nuclear reactors, usually consist of symmetric Gaussian and Lorentzian parts, peak

224 A. Steuwer et al.: Bragg Edge Determination for Accurate Lattice Parameter

Fig. 2. The measured total cross-section S compared with the theoretical total scattering cross-sec-tion for iron powder

shapes at pulsed sources display a characteristic asymmetry which reflects the brief mod-eration process of the burst of fast neutrons emitted in the spallation event. This asymme-try, see Fig. 3, is purely instrumental in character and can be observed and calibratedequally well in diffraction and transmission, for both peaks and edges respectively.

For the PEARL peak profile, a convolution of a Voigt function and decaying expo-nentials [4], the full integral Eq. (4) of the diffraction peak proves difficult to solveanalytically and a simplified approach, based on an edge profile first suggested byKropff et al. [10], has been adopted. In this approach Bragg edges in the spectrum arefitted to a function based on the integral of the convolution (�) of a normalised Gauss-ian Gðt; sÞ of width r with a normalised, decaying exponential truncated for negativetimes E ¼ HðtÞ exp ð�t=aÞ a�1, where HðtÞ is the Heavyside step-function. The trun-cated exponential introduces the profile asymmetry and is itself wavelength dependent.The edge profile function reads

hðt; t0; s;aÞ ¼Ðt

�1GðsÞ � EðaÞð Þ ðt0; t0Þ dt0 ð5Þ

and can be solved in closed form using error-functions. Figure 3 shows a typical TOFBragg edge fitted with the function of Eq. (5). Since in many cases one is interested inmeasuring relative changes in the edge position, the absolute (true) value of the edgeposition in terms of d-spacing can often be neglected.

2.4 Single and multiple edge refinement

For engineering purposes, it is important to know strains to an accuracy of the order ofDd=d ¼ 0:01% ¼ 100 me. In order to achieve this accuracy, the position of the individualedge has to be determined very precisely. This task can be best performed by fitting thedata with the help of a least-square algorithm on a computer, and special software for

phys. stat. sol. (a) 185, No. 2 (2001) 225

Fig. 3. The refinement of the Fe-211 edge using Eq. (6). This graph clearly shows the high degreeof asymmetry observed. The model function, based on the Kropff profile, fits the data very well.The dashed line indicates the edge position given by the refinement. The plot at the bottom indi-cates the difference function, enlarged by a factor of two. Also shown is the peak obtained bydifferentiation of the edge (inset)

this purpose has been developed. In the refinement process for a single Bragg edge, theprofile function hðt; t0Þ acts as a smooth switch-function to interpolate in the immediatevicinity of an edge between two linear functions of a point of equal origin. The modelrefinement function is given by

ycðtÞ ¼ a1 þ a2ðt � a6Þ þ ða5 � a2Þ hðt � a6; t0; s;aÞ ðt � a6Þ ð6Þand the edge is refined only in a small interval around the edge as shown in Fig. 4.

The model function requires seven parameters to fit an individual edge, four ofwhich bear direct physical meaning (the amplitude can be calculated). Thus, for a suffi-ciently small interval around the edge the model function can be regarded as a localapproximation to the total scattering cross section.

The software package developed for this purpose is also capable of fitting the Braggedges collectively in a Pawley-type multiple-edge refinement. The Pawley method isclassified as a pattern decomposition method. Unlike the Rietveld method which aimsat describing all physical features of the spectrum, in the Pawley method the observedspectrum is separated into each individual Bragg components without reference to astructural model [11]. Only the peak positions in the Pawley method are constrained byadjustable unit-cell parameters. This has the advantage that only a minimal knowledgeof structural parameters is required and physical factors such as site-occupancy or ther-mal coupling are neglected. During the refinement process this is accomplished by al-lowing the parameters describing certain features to vary unconstrained, e.g. height,width. In our case, when refining a set of Bragg edges collectively, Eq. (6) is used tomodel each edge, but the individual edge position is constrained by the (globally re-fined) unit-cell parameters. (Optionally, the software routine uses a set of calibratedvalues for the instrumental asymmetry parameter a rather than allowing it to vary un-constrained.) Also, only the regions around the individual edges are being used in therefinement, as shown in Fig. 4.

226 A. Steuwer et al.: Bragg Edge Determination for Accurate Lattice Parameter

Fig. 4. An example of a Pawley-type multiple edge refinement of the transmission spectrum of ironpowder. The refinement includes the thirteen lowest reflections. The lattice parameter (expressedin TOF) was refined with a relative error of Dt=t ¼ 7:4 10�6. The difference curve (enlarged by afactor two) is plotted at the bottom

In principle elastic anisotropy should be considered for each edge during a multipleedge refinement. However, it has been shown [12] that the elastic macrostrain calcu-lated from lattice parameter changes in a Rietveld refinement without accounting forhkl dependent anisotropy are almost indentical to the bulk elastic response. To thisend, multiple edge refinement represents a very good approximation to the bulk elasticbehaviour of the material, and elastic anisotropy can be neglected. Furthermore, multi-ple edge refinement is very robust to poor counting statistics, thus enabling a reductionof counting times.

Tabulated values for the asymmetry parameter a measured from a calibration experi-ment on a stress-free Fe-powder using both diffraction (ENGIN) and transmission havebeen employed in the refinement. The uncertainty in the refined lattice parameter isestimated by the fitting routine, and was often better than the desired accuracy of100 me.

3. The Trial Experiments

A series of basic experiments have been undertaken in order to establish the validityand accuracy of neutron transmission Bragg edge fitting procedures, comparing the re-sults with those obtained by different techniques and on other instruments. The generalexperimental setup is depicted in Fig. 5. On the PEARL beam-line at ISIS (RutherfordAppleton Laboratories, UK), the thermal spectrum of wavelengths ranges from 0.5 to4 �A and is received over a time-scale of 2 ms, equal to the pulse rate of 50 Hz. Thelength of the flight path on ENGIN from source to detector is approximatelyL ¼ 15:4 m, and the conversion factor between time-of-flight and d-spacing of Eq. (2)is therefore ðh=2mLÞ ¼ 1:18 10�8 ms. The effective area of the transmission detectoris 10 mm2 and typical measurement times were a few hours at 180 mAh beam current.

3.1 Vegard’s law

The first experiment was designed to validate the method for determining the latticeparameter on a relatively coarse scale. The aim was to measure the change in latticeparameter of a binary alloy as a function of the volume fraction of the constituentelements. Vegard’s law predicts a linear interpolation between the lattice parameters ofthe two elements. The material used, a-brass, was based on copper with varying de-grees of zinc additions, ranging from 0% to 20%. This mixture is known to be a solid

phys. stat. sol. (a) 185, No. 2 (2001) 227

Fig. 5. The typical geometry of TOFtransmission experiment showingthe transmission detector and theENGIN diffraction detectors whichhave been used for calibration pur-poses

solution with fcc crystal symmetry and obeying Vegard’s law over this range. The samesamples had previously been measured on the high resolution powder diffractometer(HRPD) at ISIS [13] providing accurate results for a direct comparison. The resultsshown in Fig. 6 were obtained by using nine edges in the refinement process, but it wasobserved that single edge refinement produced similar results.

3.2 The strain gauge

In the previous example, the lattice parameter was seen to vary by 2% over the alloyrange. Because accurate elastic strain measurements rely on detecting very smallchanges in lattice parameter (e.g. 0.01% strain), strain measurements represent a verygood way of evaluating further the capability to make accurate lattice spacing measure-ments. A simple experiment was designed to assess the transmission method againstaccurate strain measurements made by strain gauges. A 5 5 mm2 cross-section ironrod was mounted in a stress rig and subjected to a number of different loads, bothtensile and compressive. The applied load was controlled by a strain gauge attached tothe sample, and the different values of applied strain were ð0;�1000;�1850Þ me – alllying within the elastic regime of the material. The sample was then placed in the beamsuch that beam and applied strain vector were perpendicular (see Fig. 5). The appliedload resulted in a uniaxial stress sx, which in turn produces a contraction (or tension)proportional to Poisson’s ratio n in the other principal directions according to

ez ¼1E

½sz � nðsx þ syÞ�: ð7Þ

Hence, for iron the Poisson strain measured in transmission should be around�n ¼ �0:28 of the induced strain. The analysis was carried out by refining individualedges, as well as by using multiple edge refinement, where the lattice parameter refine-ment was over several edges, as described in the previous section. The results are

228 A. Steuwer et al.: Bragg Edge Determination for Accurate Lattice Parameter

Fig. 6. The change of the alloy lattice parameter with varying Zn content as measured with thetransmission detector using nine edges in the refinement process compared with the results fromHRPD. The results lie on a straight line (slope 1:01 � 0:01) indicating that the transmission methodhas successfully reproduced the HRPD results

shown in Fig. 7. The slope of the graph gives the experimental value for the Poissonratio, and whilst individual edge refinement showed the expected anisotropy, the multi-ple edge refinement reproduced the expected bulk value of n ¼ 0:28, Fig. 7.

4. Discussion

In this article we have concentrated on locating the position of the edges as precisely aspossible by fitting a semi-phenomenological edge profile. There are many factors, such astexture, absorption, inelastic scattering, which determine the exact profile of a transmis-sion spectrum, and a fit which attempts to fit all aspects may sacrifice fitting accuracy atthe edges in favour of a better fit in other regions of the profile. For this reason wehave fitted only over the edge regions themselves, approximating the total cross-sectionlocally. To some extent this is analogous to a Pawley fit for conventional Rietveld meth-ods used to fit diffraction data. At any rate the fits show very low residuals, despite thesmall number of parameters used in the fitting, indicating that the peak shape chosen isadequate for the statistical quality of the data. The demonstration experiments havehighlighted a number of points: Firstly, because the line of the beam through the sam-ple to the detector is unique for a transmission measurement it is possible to build apixellated detector array. This provides the possibility of simultaneous, spatial imagingof strain or lattice parameter in plane samples (such as plates, weld sections) as well asimaging the extent of phase transformations in inhomogeneous objects (such as acrosswelds). Furthermore, while the time needed to collect data of sufficient quality to accu-rately determine peak position for strain determination is lengthy (1 h or more for afew mm2 detector), phase identification could be carried out much more quickly. This

phys. stat. sol. (a) 185, No. 2 (2001) 229

Fig. 7. The measured Poisson strain vs. the applied axial strain for the strained iron sample. Thelinear line denotes the expected values calculated with a Poisson’s ratio of n ¼ 0:28, compared withthe measured points, where thirteen edges have been included in the refinement. Some resultsfrom single edge fitting of low reflection edges, clearly displaying elastic anisotropy, are shown(inset) for completeness

might open up the opportunity for mapping or tracking phase changes in two dimen-sions.

The most obvious characteristic of the technique is that the lattice parameter orstrain is an average over the transmission path of the beam. This combined with prob-lems of penetration for very thick or non-trivially shaped objects mean that the techni-que is best suited to plates and other essentially two dimensional stress fields. For thelatter, however, this implies that the measured values, e.g. of strain, are closer to themacroscopic properties of the sample. In addition because the path length is the samewherever the sample is placed along the beam path, unlike other diffraction methods,the inferred lattice spacing is not subject to accurate knowledge of sample locationalong the beam. Further experiments evaluating the applicability of X-ray methods de-veloped for characterisation of biaxial stress fields, e.g. the sin2y-method, are currentlyundertaken.

5. Conclusions

The illustrative examples in this paper have shown that the location of Bragg edges canbe determined to sufficient accuracy for basic crystal structure identification, latticeparameter determination and strain measurement. Increased measurement accuracyover that feasible on single transmission edges is achieved using a routine which carriesout a Pawley type multi-edge refinement. The Bragg edge transmission method used inconjunction with these analysis routines opens up the possibility of strain imaging ex-periments in the future. To this end an imaging 2D detector is currently being built.

Acknowledgements The authors acknowledge the financial support from the EPSRC.We would also like to thank Prof. Priesmeyer and S. Vogel for useful discussions. Theexperimental results were obtained at the ENGIN spectrometer of the ISIS facility atthe Rutherford-Appleton Laboratory, UK. Also, we wish to thank M. Dutta whohelped undertake many of the experiments, and N. Rhodes and E. N. Schooneveld(ISIS) for the technical development of the transmission detector.

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230 A. Steuwer et al.: Bragg Edge Determination for Accurate Lattice Parameter