research papers
ENGINX: a thirdgeneration neutron strain scanner
^{a}ISIS Facility, Rutherford Appleton Laboratory, CCLRC, Chilton, Didcot, OX11 0QX, UK, and ^{b}Department of Materials Engineering, The Open University, Milton Keynes, MK7 6AA, UK
^{*}Correspondence email: jrsantisteban@cab.cnea.gov.ar
ENGINX, a new timeofflight (TOF) neutron diffractometer optimized to measure elastic strains at precise locations in bulky specimens recently commissioned at the ISIS Facility in the Rutherford Laboratory, UK, is described. Fast counting times, together with a flexible and accurate definition of the instrumental gauge volume are the main requirements of neutron strain scanning and have been addressed on ENGINX through the design of a novel TOF diffractometer with a tuneable resolution and interchangeable radial collimators. Further, the routine operation of the instrument has been optimized by creating a virtual instrument, i.e. a threedimensional computer representation of the diffractometer and samples, which assists in the planning and execution of experiments. On comparing ENGINX with its predecessor ENGIN, a 25× gain in performance is found, which has allowed the determination of stresses up to 60 mm deep in steel specimens. For comparison with constantwavelength diffractometers, special attention has been paid to the absolute number of counts recorded during the experiments. A simple expression is presented for the estimation of counting times in TOF neutron strain scanning experiments.
1. Introduction
Neutron strain scanning (NSS) is a nondestructive technique that provides insights into strain and stress fields deep within engineering components and structures (Allen et al., 1985). The popularity and maturity of NSS is reflected in a series of recent publications. An ISO standard dealing with residual stress measurement by neutron diffraction has been proposed in ISO/TTA 3:2001 (2001). A comprehensive introduction to the technique is found in the book by Hutchings et al. (2005), and a description of the stateoftheart of stress measurements using neutron and synchrotron radiation is given in the book by Fitzpatrick & Lodini (2003).
The technique was first developed in the late 1970s on conventional `allpurpose' diffractometers. The technique developed significantly over the following two decades with an increase in the user community, and a number of neutron diffractometers were converted to work almost entirely for NSS. ISIS, the world's brightest pulsed neutron source, was home to one of the first dedicated neutron stress diffractometers, ENGIN (Johnson et al., 1997). Following increased demand from the engineering community, ENGIN has recently been replaced by ENGINX (Daymond & Edwards, 2004), which was designed to provide at least an order of magnitude improvement in performance (Johnson & Daymond, 2002).
Secondgeneration neutron strain scanners were built on diffractometers originally designed to balance the competing requirements of various experiment types. This limited their utility for engineering measurements, particularly in large components. In contrast, ENGINX has been designed with the sole aim of making engineering strain measurements: essentially the accurate measurement of polycrystalline material lattice parameters, at a precisely determined location in an object. This approach has allowed considerable performance improvements to be made compared with previous instrumentation.
Here we describe ENGINX, providing a precise quantification of the gain in performance achieved through optimized neutron optics. In addition, we introduce a virtual laboratory: a computing representation of ENGINX designed to assist in the planning and execution of experiments.
2. An optimized timeofflight neutron strain scanner
2.1. Basic concepts
The aim of a timeofflight (TOF) neutron strain scanner is to define the macroscopic elastic strain tensor at definite locations in the bulk of a specimen; a schematic is shown in Fig. 1. A pulsed beam of neutrons with a wide energy range travels to the sample, where a small fraction of the beam is scattered into a detector at an angle 2θ_{B}. Assuming an the wavelength of the detected neutrons is defined from its TOF t,
with h being Planck's constant, m the neutron mass and L_{1} and L_{2} the primary and secondary flight paths, respectively. A typical spectrum diffracted by a polycrystalline material is shown in Fig. 2. Each peak corresponds to an (hkl) family of lattice planes as given by Bragg's law, = , so the dspacing is obtained from the position t_{hkl} of the peak in the TOF spectrum,
Peak positions can be precisely determined by leastsquares δ∊ = = ≃ 50 µ∊ (1 µ∊ = 10^{−6}). The elastic strain is determined from the change in the atomic interplanar distances d_{hkl}, as compared with the value d_{hkl}^{ 0} measured in the unstressed (or reference) condition, ∊_{hkl} = ( d_{hkl}  d_{hkl}^{ 0} )/d_{hkl}^{ 0}. Each diffraction peak provides information from only a small fraction of the crystallites within the sampled volume, i.e. those oriented to fulfil the Bragg condition. A good approximation to the macroscopic or `engineering' elastic strain is usually given by a weighted average of several singlepeak strains ∊_{hkl} (Kamminga et al., 2000). In TOF instruments, the engineering strain can also be approximated from the change in the average lattice parameters from Rietveld or Pawley refinements of the complete diffraction spectrum (Daymond et al., 1997; Daymond, 2004).
of the peaks, with a typical sensitivity ofThe volume of material contributing to the diffraction pattern corresponds to the intersection of the incident and diffracted beams, typically defined by slits and collimators, respectively. As neutrons penetrate quite deeply into most materials, strains within the bulk of a specimen can be nondestructively measured. The centroid of this gauge volume (typically of the order of cubic millimetres) defines the location of the measurement. In an NSS instrument, the gauge volume is fixed at a position in the laboratory, so the strain variation across the sample is explored by moving the sample using a translation stage.
The measured strain gives the component of the strain tensor along the direction of the neutron wavevector change q, which bisects the incident and diffracted beams (Fig. 1). A complete definition of the strain tensor for an unknown specimen requires measuring at least six noncoplanar directions. However, the principal strain axes can usually be inferred from the geometry of the sample, and measurements are performed only along three mutually perpendicular directions. In order to ensure that the same volume of the specimen is measured for each sample orientation, the detector is placed at approximately 90° from the incident beam, defining a near cubic gauge volume. Two perpendicular strain directions can be measured simultaneously during TOF NSS by adding a second detector opposite to the first one, aligned such that the same gauge volume is seen from both detectors.
The NSS technique is conceptually very simple but its practical application can be timeconsuming. Very long counting times are required when small gauge volumes and large penetration depths are involved, which effectively dictates the range of problems that can be studied with this technique. Withers (2004) has defined a `maximum acceptable acquisition time' for neutron and synchrotron strain scanning when measuring strain deep within materials. This is an important concept because NSS experiments are performed at large facilities, where the experimenter is granted a limited amount of beam time, under competitive review. Hence, a minimization of the experimental time required for an accurate determination of peak positions (and hence strain) is the first goal of an optimized strain scanner.
Further, a precise knowledge of the location of the gauge volume within the sample is paramount to NSS experiments. An international round robin for the standardization of NSS (Webster, 2000; Daymond et al., 2001) unexpectedly revealed that one of the main sources of differences between the participating laboratories resulted from positioning errors, rather than from uncertainties in the determination of interplanar distances. Aligning a specimen is relatively easy for simple geometries such as plates or cylinders, but it is nontrivial with real components presenting more complex shapes. As a result, the time spent on alignments often represents an important fraction of allocated beam time. Thus, a second aim for an optimized neutron strain scanner is a precise yet flexible definition of the instrumental gauge volume, ideally in conjunction with fast and accurate alignment procedures.
2.2. Counting time for a TOF neutron strain scanner
The uncertainty in the measured strain is proportional to the uncertainty in the determination of the position of the Bragg peaks. Withers et al. (2001) have shown that the time T required to measure the position of an arbitrary shaped peak on zerobackground with a strain precision δ∊ is
where I_{hkl} is the integrated peak intensity per unit time, t_{hkl} is the peak position and δt_{hkl} the peak width. The inverse of T can be used as a figure of merit for the purpose of optimizing a neutron strain scanner (Johnson & Daymond, 2002). On the righthand side of equation (3), T is expressed in terms of the relative peak width δ_{∊}t_{hkl} = (): a dimensionless parameter that allows direct comparison between peaks appearing at different TOF, as well as with the required strain precision δ∊. Throughout this paper the notation δ_{∊}X will be used to represent the dimensionless relative width of the probability distribution of the variable X.
The relative peak width, or resolution, of a TOF diffractometer is (Windsor, 1981)
where (δ_{∊}t)_{mod} is the moderator contribution to the resolution, and are the divergence in the diffraction plane for the incident and detected beams, respectively, and (δ_{∊}d_{hkl}) is the sampleinduced broadening appearing from microstresses or gradients in composition (Snyder et al., 1999). The divergence perpendicular to the diffraction plane does not affect the peak width.
The moderator contribution to the peak width is a crucial characteristic of a TOF diffractometer, as it usually dictates the overall resolution of the instrument. It represents the time distribution of the neutron pulse due to the neutron thermalization process. Its shape and dependence on neutron wavelength has been extensively discussed by Ikeda & Carpenter (1985). Its contribution to the resolution is inversely proportional to the total neutron flight path L = L_{1} + L_{2},
where Δt_{0} is the intrinsic width of the neutron pulse: a measure of the time spent by the neutrons in the moderator.
The integrated intensity of a TOF diffraction peak (Buras, 1963) depends on the scattering power and absorption of the sample as well as on instrumental parameters, such as the incident neutron flux Φ_{0} (expressed in neutrons s^{−1} cm^{−2} Å^{−1} steradian^{−1}), the total divergence in the diffraction plane, the incidentbeam divergence perpendicular to the diffraction plane , the size of the illuminated gauge volume δV, and the detector characteristics:
where h_{det} and %_{det} are the detector height and efficiency, respectively. The beam attenuation is given by the absorption coefficient μ_{hkl}, and l, the total flight path inside the sample indicated by the dotted line in Fig. 1. The factor P_{hkl} depends only on the material,
with v_{0} the volume per atom, F_{hkl}^{ 2} the (including the Debye–Waller factor) and m_{hkl} the multiplicity of the reflection. A precise definition of the illuminated gauge volume δV is given in the next section.
Thus the counting time required to achieve a δ∊ uncertainty in strain, expressed in terms of the experimental parameters, is
This expression gives some guidelines for the design of a TOF neutron strain scanner. Fig. 3(a) shows the dependence of T on the horizontal incidentbeam divergence , for different divergences of the diffracted beam. The calculations are for the case of negligible sample broadening (δ_{∊}d_{hkl} = 0) and δ_{∊}t_{mod} = 0.001. For small divergences of the diffracted beam ( = 0.0005), a minimum exists at ≃ 0.002, i.e. twice the moderator broadening. As the detector collimation is relaxed ( = 0.001, 0.002), the minimum shifts to lower incidentbeam divergences, and the counting time increases. The condition for the minimum is
Thus, in an optimized instrument, the angular contribution to the resolution must match the combined contributions from moderator and sample: a principle usually adopted in instrument design (Windsor, 1981). This means that the resolution of an optimized instrument is ultimately determined by the moderator contribution (δ_{∊}t)_{mod}, i.e. by the choice of moderator and total flight path. However, the sample broadening should also be taken into account at the design stage. This can be seen in Fig. 3(b), showing the increase in counting times due to sampleinduced broadening. As indicated by the vertical dotted line, an instrument having an incident divergence optimized for a negligible sample broadening would become rather inefficient for samples presenting broad peaks, because counting times increase quite sharply for highly collimated incident beams. Therefore, this calls for an instrument with a variable divergence, which can be adjusted to match the peak broadening introduced by the sample, since typically the incident flux can be increased if we are willing to increase the divergence. In practical terms, the moderator contribution to the resolution is minimized by the choice of moderator and a long incident flight path. Increasing the secondary flight path L_{2} is not effective because there is an associated decay in intensity of the form (1/L_{2})^{2}. By contrast, the loss of neutrons caused by increasing L_{1} can be kept relatively low by bringing the neutrons along a neutron guide. The actual choice of L_{1} will depend on a variety of parameters, including building costs and the materials likely to be studied with the diffractometer.
2.3. The gauge volume in a TOF neutron strain scanner
The measured strain corresponds to an average over the volume δV and solid angle δΩ sampled. The instrument gauge volume (IGV) is the volume of space defined by the neutron beam paths through the defining apertures, taking into account the beam intensity profile and divergence. Fig. 4(a) schematically shows the main experimental parameters affecting the shape and size of the IGV. The cross section of the neutron beam is defined by slits of height S_{y} and width S_{x} located at a distance S_{z} from the geometric centre of the diffractometer. The incident beam has a divergence in the diffraction plane and perpendicular to it. Due to this divergence, the beam cross section is larger at the IGV centre than at the slits and hence the edges of the IGV are blurred. The edge profile is described by an error function, with a width defined by the divergence and slit distance. Withers et al. (2000) have shown that the full width at halfmaximum (FWHM) of the incidentbeam profile at the IGV centre is
Hence, the effective dimensions of the beam, as defined by the FWHM, increase slowly with the slit distance S_{z}, even when the actual cross section of the beam increases markedly.
The size of the IGV along the beam direction Δz is usually defined by a collimator. According to equation (9), an instrument with a moderator resolution of 0.001 requires a diffracted beam with a divergence ≃ 0.002 radians in the diffraction plane. For detectors located at a secondary flight path of 1.5 m, this gives a detector width δz ≃ 3 mm. The height of the detector δy can be increased considerably (∼200 mm) as the divergence perpendicular to the diffraction plane does not affect the resolution, and the height is restricted primarily by the requirement not to cut the curving Debye rings at 2θ_{B} ≠ 90°. In TOF NSS, count rates are usually increased by using large arrays of detectors, covering horizontal and vertical angular ranges of about 30°. The TOF spectra recorded by each individual detector are transformed into a common dspacing scale and added together in a single diffraction spectrum, in a process commonly known as electronic time focusing (Jorgensen et al., 1989). Larger angular arrays are not feasible for NSS due to space and access requirements, and due to the increased uncertainty in the measured strain (Daymond, 2001).
Radial collimators made of thin absorbing blades (Withers et al., 2000) are used to collimate the diffracted beam and ensure that all detectors look at exactly the same volume of the sample (Fig. 4b). The shape of the IGV and its dependence on the radialcollimator parameters can be calculated by simulations of the neutron trajectories (Wang et al., 2000). The predicted intensity profile along the beam direction is triangular for the line passing through the focal point, and gradually becomes a Gaussian profile at increasing distances from this point. The FWHM of these distributions can be changed without altering the collimation , by assembling blades of equal length at different distances from the focal point (l_{c}^{ 1} and l_{c}^{ 2} in Fig. 4b). Hence, a variable gauge volume can be achieved with two pairs of adjustable slits to define the incidentbeam cross section, and a number of radial collimators of fixed collimation assembled at different distances from the focal point.
The previous calculations represent the ideal performance of slits and collimators. In practice, these may be affected by manufacturing details and by intensity variations across the beam, so the actual resolution should be determined experimentally.
From the present discussion, it follows that a precise definition of the IGV is a rather complex matter. For operational purposes, we define the IGV as a cuboid of sides Δx, Δy and Δz, so the illuminated gauge volume becomes δV = ΔxΔyΔz. The dimensions of Δx and Δy are the FWHM values given by equation (10), and Δz is the width of the intensity profile seen from the collimators. For measurements performed near surfaces, the IGV may be only partially immersed within the specimen, and the effective centroid of the IGV differs from its geometrical centre. The sampled gauge volume (SGV) is the threedimensional absorption weighted intersection between the IGV and the sample.
Finally, the uncertainty in the solid angle ascribed to the measured strain direction is given by the solid angle explored by the q vectors of all individual detector elements. For a detector bank spread over relatively small angular ranges, Δ(2θ_{B}) in the horizontal plane and Δϕ out of this plane, the uncertainty in the q direction is half of the detector's angular spread in the horizontal plane, and nearly equal to the angular coverage out of this plane. This large angular coverage also means that even for a single diffraction peak, the actual number of crystallites sampled by a TOF diffractometer is much larger than those sampled on a reactor. This is because the angular 2θ_{B} acceptance of a constantwavelength neutron strain scanner is typically of the order of 0.1°, in comparison with the 20–30° range covered by a TOF detection bank.
3. The ENGINX diffractometer
Following the requirements discussed above, ENGINX has been designed with a tuneable resolution and a variable gauge volume. This demanded novel solutions in terms of neutron optics and detector design, as described in this section.
The ISIS methane moderator was chosen for ENGINX due to its combination of narrow pulse width and high flux over the 1–3 Å wavelength range relevant to most engineering materials. After a comprehensive optimization, Johnson & Daymond (2002) showed that for this moderator the counting times are minimized with a primary flight path of ∼50 m, and a diffracted beam with a horizontal divergence of 0.002 radians.
The neutrons are brought from the moderator to the sample position by means of neutron guides. Firstly, supermirror (m = 2) coated metal in the ISIS S8 primary beam shutter improves the uptake of neutrons from the moderator. Then a glass supermirrorcoated neutron guide (m = 3, 60 mm high × 25 mm wide) transports the neutrons to the sample position. From 4 m to 37.5 m from the moderator, the guide is curved in the horizontal plane with a radius of 5 km, away from the proton beam. This curvature improves the signal to noise ratio of the instrument by removing the highenergy neutrons and gamma rays produced during the spallation process. However, this also limits the minimum wavelength at the sample position to around 0.5 Å. A straight neutron guide from 37 m to 48.5 m, i.e. ending at 1.5 m before the sample position, removes any asymmetry in the neutron profile induced by the curved part of the guide.
The horizontal divergence of the beam, , at the end of the neutron guide is ∼0.005 radians Å^{−1}, or ∼5000 µ∊ Å^{−1}, clearly larger than the angular resolution specified in the design. Thus, both the horizontal and the vertical divergence of the incident beam can be adjusted with two sets of slits inserted in the straight part of the neutron guide, at 4 m and 1.5 m from the sample position, labelled s_{4} and s_{1.5}, respectively. Figs. 5(a) and 5(b) show the effect of varying the slits' opening on the symmetric width and intensity of the 111 and 311 diffraction peaks measured on an AISI 306H stainlesssteel sample. Both slits are opened by the same horizontal width, whilst the height is kept fixed at 45 mm. In both cases there is a sharp increase at 12.5 mm, i.e. when the opening corresponds to half of the guide width. The effect on counting time estimated with equation (3) is shown in Fig. 5(c). The instrument is optimized at somewhere between 12 and 15 mm. These experimental results were used to validate Monte Carlo simulations of the incidentbeam optics performed with the software package McStas (Nielsen & Lefmann, 2000). Results of the simulations are shown by the solid and dotted lines in Figs. 5(a) and 5(b). The calculated horizontal divergence of the incident beam at the sample position is shown by the solid and dotted curves in Fig. 5(d). A sharp increase in divergence is found at 12.5 mm. Inspection of the simulation results reveals that this increase is linked to the appearance of satellite peaks in the divergence distribution, complicating the tuning of the divergence. These satellite peaks disappear when the opening of s_{1.5} is reduced, whilst still allowing a direct line of sight of the s_{4} slit from the sample position. The dashed–dotted line in Fig. 5(d) shows the results of the simulations for an arrangement with s_{1.5} (mm) = s_{4}(1.5/4) + 2 (mm) (now with s_{4} in the abscissa). For this configuration, the divergence behaves linearly up to ∼20 mm, where an increase in the slope is found, again, due to small satellite peaks.
The actual size of the IGV is defined by a set of motorized slits capable of producing a beam having a horizontal dimension of 0.3–10 mm, and a vertical dimension of 0.3–30 mm. These slits can be moved along a rail aligned to the beam direction, allowing change of the distance S_{z} to the sample between 0 and 120 mm. For small gauge volumes, these slits can also have a noticeable effect on the incident divergence (Santisteban, 2005).
ENGINX has two detector banks, centred at 2θ_{B} = ±90° to the incident beam and ∼1.53 m from the IGV. The detector banks cover ±16° in the horizontal plane and ±21° in the vertical plane. Each detector bank is made up of five units stacked vertically, each unit consisting of 240 scintillators. Each scintillator is 196 mm high by 3 mm wide, providing a horizontal angular resolution of ∼0.002 radians. A new detector design was required in order to achieve this spatial resolution efficiently (Schooneveld & Rhodes, 2003). The detectors are made of ZnS/^{6}Li scintillator material, coded via fibre optics to an array of photomultipliers tubes. Each bank covers a total detector area of 1.4 m^{2}, which represents about 5% of the total 4π solid angle.
The dimension of the IGV along the beam is defined using radial collimators: a technology first designed and implemented on ENGIN (Johnson et al., 1997). Whilst ENGIN used a single gauge dimension, ENGINX uses five sets of removable radial collimators providing 0.5, 1, 2, 3 and 4 mm gaugewidth options. These sizes represent typical dimensions for spatial strain scanning experiments, as macroscopic strain distributions usually scale with the object size (Edwards & Santisteban, 2002). Additionally, the 4 mm gauge is ideal for realtime in situ studies, maximizing the IGV to reduce count times, whilst keeping the background low by eliminating neutrons scattered from sampleenvironment equipment. The collimator mounting system enables removal of a collimator to provide space for very bulky samples, with the sample still on the instrument.
Each radial collimator is composed of 160 vanes spanning 32° on the horizontal plane, assembled at different distances from the focal point. The vanes are 350 mm long and 50 µm thick, made of a 12 µm polyethylene foil covered by a paint containing gadolinium oxide particles. The collimators have l_{c} distances of 100, 160, 310, 400 and 490 mm, to provide the gauge sizes of 0.5, 1, 2, 3 and 4 mm, respectively. A horizontal cross section of the IGV for the 4 mm collimator is shown in Fig. 6(a), measured by horizontally scanning a 0.25 mm nylon thread across the beam. Line profiles for the 2 and 4 mm collimators along the beam direction, i.e. the z axis of Fig. 4(a), are shown in Fig. 6(b). Both profiles are properly fitted by Gaussian distributions, with widths of (2.05 ± 0.05) mm and (3.95 ± 0.05) mm, respectively. The profiles normal to the beam direction (i.e. along the x axis) are shown in Fig. 6(c). The edges of these distributions are well described by error functions with a broadening of 0.73 mm. The widths of these distributions, as given by the distance between the edges, were (2.3 ± 0.1) mm and (3.95 ± 0.05) mm, respectively.
The combined effects of these design features, along with the significant (2×) increase in the detector solid angle, has resulted in a new instrument that greatly exceeds the performance of its predecessor, ENGIN. A demonstration of this is given by Fig. 7(a), which compares a cerium oxide (200) diffraction peak measured on ENGIN with its equivalent on ENGINX, normalized by the volume of the IGV. The gains of ENGIN over ENGINX can be quantified by the intensity and width of the peaks: the two parameters involved in the figure of merit. Fig. 7(b) shows the experimentally defined resolution of both instruments for the wavelength range of interest. As seen in the graph, on ENGINX the moderator and geometrical contributions are matched, whilst on ENGIN the moderator dictates the overall resolution. For ENGINX, a simple squares sum of moderator and geometrical contributions results in a nearly constant resolution of 1300 µ∊.
The gain in intensity is shown in Fig. 7(c). The curves correspond to the normalized count rate for a vanadium sample. The ENGINX incidentspectrum intensity is zero at short wavelengths due to the curvature of the neutron guide; the useable range is ∼0.5–6 Å. Overall, over an order of magnitude increase in intensity has been achieved on ENGINX. However, we note that for any single measurement, the wavelength range accessible to ENGINX is smaller than for ENGIN, even when both instruments look at the same moderator. This is because for ENGIN, located 16 m away from the moderator, all the wavelength range of interest is available with the 50 Hz pulse rate of the ISIS source; but this is no longer possible on ENGINX. Due to its 50 m flight path, at 50 Hz the usable wavelength window is reduced to just over 1.5 Å. Thus, a slower pulse rate is required if wider wavelength ranges are to be exploited. In effect, the full range of the ENGINX spectrum displayed in Fig. 7(c) is only available with the instrument running at 12.5 Hz. This is accomplished on ENGINX by means of two sets of disc choppers located at 6 m and 9 m from the moderator. The frequency (50, 25, 16.7 and 12.5 Hz) and phase shift of the choppers relative to the neutron source pulse rate can be independently changed, providing great versatility for shaping the incident spectrum. In order to reduce the opening and closing time of the choppers, a novel design consisting of two counterrotating discs was developed for this project (Galsworthy et al., 2003).
Hence, the comparison between ENGIN and ENGINX can be performed in either of two ways, based on the capability of the instruments to measure: (i) an individual peak or, (ii) all peaks within a predefined wavelength range. Fig. 7(c) compares the intensities of ENGIN and ENGINX for the latter case, with ENGINX running at 25 Hz, which is the optimal choice for macroscopic strain scanning in most engineering materials. We emphasize that only a userselected 2.5 Å interval of the displayed incident spectrum would be readily available at ENGINX at this pulse rate. Table 1 provides a more practical comparison between the performances of ENGINX (at 25 Hz) and ENGIN, in terms of the counting time taken by typical macroscopic strain scanning experiments in aluminium and steel samples. More than one order of magnitude decrease in counting times has been achieved. The experimental values given for the spatial scanning results are rough estimates based on a 50 µ∊ uncertainty returned from a of diffraction data. For each category, measurements have been made on several different samples with various alloy contents and textures, with path lengths close to the values given; count times have been extrapolated to the conditions given. The ENGINX experimental times are compared with estimates described in the following section.

Finally, the ENGINX positioning table is capable of holding a sample weighing up to 1.5 tonnes. It can move the sample in the x, y, z and ω axes, with ranges of ±250 mm in x and y, 700 mm in z, and 370° in ω, with a nominal accuracy of 10 µm/100 mm for a 0.5 tonne sample. In order to measure materials under applied loads, an in situ 100 kN hydraulic stress rig is also available on ENGINX, of identical design to that described by Daymond & Priesmeyer (2002). The sample temperatures can range from room temperature to 1273 K within atmosphere or using a radiant furnace (Daymond & Withers, 1996). Table 1 also lists typical gains in performance achieved by ENGINX for in situ loading experiments. For this case, the data collection times correspond to the five most intense firstorder peaks in a randomly textured material, measured to 50 µ∊ uncertainty.
4. Counting times on a TOF strain scanner
A reliable estimation of experimental counting times for other materials, geometries and spatial resolutions than those given in Table 1 would be very useful in assessing the feasibility of specific experiments on a TOF strain scanner. Besides this, it would enable an efficient use of the limited experimental time by allowing the optimization of the count times at each measurement position, as recently discussed for constantwavelength diffractometers by Withers (2004).
An estimation of the counting time required for a desired strain precision was given in equation (3). For many engineering specimens, the sampleinduced broadening is small and the peak width is dominated by the instrument resolution, hence the count times are effectively dictated by the peak count rate, i.e. equation (6). Although useful for instrument design purposes, for experimental design it is convenient to group all the instrumentdependent parameters in equation (6) into a single factor, Φ_{instr},
giving the peak intensity in terms of only four contributions. Φ_{instr} is a wavelengthdependent factor containing the incident neutron flux, the efficiency and the solid angle of the detection bank. This term is strongly instrumentdependent, so interpolation of an experimental lookup table is the easiest way to determine it. The Φ_{instr} functions for ENGINX and ENGIN are presented in Fig. 7(c). The wavelength dependence of Φ_{instr} was defined by measuring a vanadium sample (assuming it is a perfectly elastic incoherent scatterer), whilst the scaling factor was obtained from reference Fe and Al specimens. The factor P_{hkl} depends on the material. The experimental counting rates for a series of technologically relevant materials are shown in Fig. 8(a), together with the values calculated using equation (11). The reported count rate corresponds to a 1 mm^{3} gauge volume, located 1 mm under the surface of the specimen, measured on powder specimens. The agreement between experimental and calculated values is very good, varying over almost three orders of magnitude. The count rate in actual engineering specimens will be somewhat different, due to the higher density of solid specimens and the likely presence of texture.
The third factor affecting the count rate is the attenuation of the neutron beam within the specimen. Fig. 8(b) shows such a decrease in peak intensities, in this case for the line CC′ of the stainlesssteel specimen described in the next section. The attenuation in all three peaks is well described by leastsquares fits with an law, but with slightly different attenuation coefficients: μ_{111} = (0.119 ± 0.03) cm^{−1}, μ_{200} = (0.106 ± 0.03) cm^{−1}, μ_{220} = (0.111 ± 0.02) cm^{−1}. These differences are due to the dependence of the total material cross section on neutron wavelength. A diffraction peak measured on a TOF diffraction bank contains contributions from neutrons from a range of wavelengths, from λ_{min} = to λ_{max} = , where θ_{min} and θ_{max} are the minimum and maximum Bragg angles in the detector bank. The material is given by μ = Nσ_{tot}, with N the number of atoms per unit volume and σ_{tot} the material's microscopic total cross section. The total cross section of polycrystalline materials has a highly complex dependence on neutron wavelength; however, the μ_{hkl} for a particular reflection can be calculated by averaging the total cross section over the associated wavelength range (Wang et al., 2001):
Equation (12) for stainless steel gives μ_{111} = 0.1164 cm^{−1}, μ_{200} = 0.1069 cm^{−1} and μ_{220} = 0.1113 cm^{−1}, agreeing well with the measured values. This variation in the should be accounted for when estimating counting times deep in specimens. For instance, due to this variation, the count rate of the 111 peak is 75% larger than that of 200 for a neutron path of 12 mm, but both peaks have essentially the same count rate for a 52 mm path length. It is worth noting that the μ_{hkl} depends on texture through the total cross section.
The last factor that may affect the peak intensities is the partial filling of the IGV, particularly for scans near the surface of a specimen. Based on the description of the IGV in §2.3, the count rate near a surface can be calculated by replacing δV in equation (11) by the SGV δV′, corresponding to the intersection between the IGV and the sample. Fig. 8(c) shows an intensity profile for a line scan across a 25.4 mm cube filled with iron powder, together with calculated values. The good agreement between the experiment and the simulated values supports the use of a cuboid to describe the IGV in intensity calculations, instead of the more complex distributions presented in Fig. 6.
Thus, we can estimate the counting time required to measure the position of a single diffraction peak on ENGINX with a resolution δ∊:
In practice the utility of equation (13) will be limited by the presence of any texture. Moreover, in TOF neutron strain scanning experiments, the value of the macroscopic strain is obtained from the analysis of all the peaks appearing within the accessible wavelength range. That is, the average strain across all the crystallites sampled by a TOF experiment is derived from the variation of the crystallographic lattice parameters, obtained from a Rietveld or Pawley leastsquares of the full diffraction pattern (Daymond et al., 1997; Daymond, 2004). The effect of measuring more than one peak is effectively to lower the uncertainty in the experimental strain, hence reducing counting times (Johnson & Daymond, 2002),
where the sum is over all the hkl peaks accessible to the experiment, and we have assumed that the resolution of an optimized TOF diffractometer is nearly constant (δ_{∊}t_{hkl} = δ_{∊}t).
Predicted counting times using equations (13) and (14) are compared with the corresponding experiments in Table 1, using strain uncertainties of 50 µ∊ and 70 µ∊ for spatial scanning; and requiring the five most intense peaks to be defined within 50 µ∊ for in situ loading. For both types of experiment, the estimated counting times agree well with the predictions from equations (13) and (14). All counting times were calculated assuming a constant instrument resolution δ_{∊}t_{hkl} = 1300 µ∊ and a sample broadening δ_{∊}d_{hkl} = 100 µ∊. Predictions for other materials and depths are presented in Table 2. The predicted counting times are for 50 µ∊ strain accuracy on ENGINX running at 25 Hz, using gauge volumes of 2 × 2 × 2 mm^{3} and 4 × 4 × 4 mm^{3}. In general, the counting times for the fullpattern analysis are better estimates of the actual experimental times, as the total count rate is less sensitive to the presence of texture.

We note that equation (13) is only correct for an isolated peak with no background. In the presence of a nonnegligible background level, the time T required to achieve the same strain accuracy increases by the penalty factor [1 + 2(2^{1/2})b/h_{hkl}], where h_{hkl} is the height of a Gaussian peak and b is the background level (Withers et al., 2001; Withers, 2004). The factor of 2(2^{1/2}) in this expression is replaced by slightly different factors for nonGaussian peak shapes (Withers et al., 2001). This correction will be important for materials with a very large cross section, or for measurements performed deep within a specimen. For the materials listed in the table, the effect of is only significant for titanium, where b/h_{hkl} for the peak is 0.05, and is even larger for the other reflections. On the other hand, the increasing importance of background at larger depths is due to the contribution of multiple scattered neutrons mainly within the sample. We have experimentally found that the background level is about a third of the peak height for 4 × 4 × 4 mm^{3} of stainless steel at a depth of 50 mm. Due to the complex nature of the TOF peak shape, and the fact that the background varies as a function of wavelength, a more quantitative description of the influence of background in TOF diffractometers on count times is beyond the scope of this work.
5. The ENGINX virtual laboratory
In parallel with the optimization of the neutron optics, improvements in sample positioning have been achieved by using detailed threedimensional models of the samples, and a pair of theodolites for precise alignment on the instrument. However, computing help is essential for full exploitation of these advances in instrumentation, as well as to monitor the experimental progress and results. Otherwise measurements would be made in a conservative manner, and the final information gathered would be less than could be optimally achieved. With this in mind, the processes of planning, alignment and data analysis have been simplified by writing SSCANSS (James et al., 2002; James et al., 2004), a computer program that: (i) provides computer aids in setting up the initial measuring strategy, (ii) automates sample alignment and all routine aspects of the measurement process, (iii) provides data analysis in near realtime, allowing decisions on changes to the measurement strategy.
These requirements have been achieved by the creation of a `virtual laboratory', i.e. a threedimensional representation of the laboratory and the sample that can be easily manipulated through a graphical user interface, as shown in Fig. 9. The threedimensional model of the laboratory includes the collimators, the slits and the positioning table, with the IGV at the centre of the laboratory system. The dimensions of the IGV can be changed by changing the slit apertures and collimators. The threedimensional model of the sample can be produced within the program from basic primitive objects (cuboids, cylinders, pipes, etc.), exemplified by the simple model used in Figs. 9(a) and 9(c) to describe the 190 kg pipe shown in place in the instrument in Fig. 9(b). For more complex shapes or regions requiring improved spatial resolution, a precise threedimensional model can be created using a coordinatemeasuring machine (CMM), also available on the ENGINX facility. Fig. 9(d) shows such a model for the region of interest of the pipe, i.e. in the vicinity of a weld repair. The ENGINX CMM is an LK HC90 model provided with a Metris laser head for fast scanning of large samples. The measurement zone of the CMM limits a single measurement scan to around 0.5 × 0.5 × 1 m; however, multiple overlapping scans can be built up to deal with larger objects. The resolution of the laser scanner in determining the position of a single point in a surface is around 5 µm, but a smoothly varying surface can be defined to a substantially better accuracy than this. Alternatively, the sample model can be imported from external computeraided design (CAD) software packages. The SSCANSS package is written in IDL (Research Systems, 2004); the implementation also makes use of the Open Genie software suite (MoretonSmith et al., 1996).
5.1. Experiment planning
The points to be measured are defined interactively in a graphical visualization of the sample. To do so, the program presents cross sections of the specimen where the points of interest can be selected using a mouse. Alternatively, the measurement points can be defined by importing their (x, y, z) coordinates in the sample system. The crosses displayed in Fig. 9(d) represent a complete scan to be performed during the experiment.
Different sample orientations provide different components of the strain tensor. The program allows the exploration of alternative experimental arrangements, and provides warnings in case of collisions. The effects of beam attenuation for each configuration are easily explored by rotating the sample in the virtual laboratory, as shown in Figs. 9(a) and 9(c) for the axial and radial strain components, respectively. The program can also provide the estimated counting time expected for each orientation using the expressions introduced in the previous section.
5.2. Sample alignment
In ENGINX, the positioner movements required to bring samples of arbitrary complexity to the correct position for measurement may be generated automatically. This facility requires the determination of the transformation that relates the measurement positions in the sample and laboratory coordinate systems. This transformation is expressed in terms of a 4 × 4 matrix, S, in homogeneous coordinates. The use of homogeneous coordinates (Foley et al., 1997) is standard in many areas of computer modelling and is convenient as it allows translation, which must normally by represented by an addition, to be represented by a matrix multiplication. In order to determine S, the positions of a number of fiducial points on the surface of the sample are measured using a pair of theodolites. A leastsquares procedure is then used to find the transformation matrix that most closely maps the fiducial points on the sample to their measured positions in the laboratory. Since the sample is a rigid body, a minimum of three noncollinear fiducial points are sufficient for this purpose, though using a larger number may improve accuracy. The transformation matrix so determined provides the position of the sample corresponding to the initial position of the ENGINX positioning table, which is described by a further transformation matrix P. It follows that if the positioner were now moved to (x = y = z = ω = 0), the sample position would be given by the transformation S_{0} = P^{−1}S. The transformation matrix, S_{i}, for any subsequent table position may now be calculated as
where P_{i} is the transformation matrix describing the new position of the table.
Since the angular orientation of the positioning table is generally prescribed by the requirement that a particular strain component be measured, the matrix P_{i} is unique and is determined by the translation needed to bring the required measurement point to the gauge volume.
The alignment procedure described above is purely optical insofar as no reference is made to the position of the neutron beam. As a check of the position of the sample with respect to the beam, and in order to take account of slight beam misalignments, the diffracted beam intensity is measured as entrance/exit scans are performed. The experimental intensity profile can be used to estimate the location of the surface of the sample, or, alternatively it may be compared directly with a SSCANSS predicted profile, such as the solid lines shown in Fig. 8(c). Any discrepancy due to beam misalignment is then incorporated into SSCANSS as an offset, enabling the scan to proceed as previously defined.
The theodolites are placed at beam height. One of theodolites makes an angle of 135° with the incident beam, i.e. is aligned to q_{2} in Fig. 1, and the other is more arbitrarily positioned, making an angle of approximately 30°. The models of the sample, the laboratory and the transformation matrices are attached to the experimental data files for a precise reconstruction of the instrumental arrangement for inspection and analysis at a later stage.
5.3. Experiment execution and data analysis
Provided with the transformation between sample and laboratory coordinate systems, the program can drive the positioning table in order to visit all of the points within the scan. At each measurement position, the spectra recorded by the individual detectors are timefocused, as described above. The program provides automatic singlepeak and fullpattern GSAS (Von Dreele et al., 1982). As most samples are textured, the is performed leaving the peak intensities unconstrained, as described by Pawley (1981). To ensure the convergence of the initial guesses of the lattice parameters are obtained from singlepeak of the most intense peak.
of the diffraction spectra specially devised for strain determination, through a library of common engineering materials. This enables researchers who are not experts in crystallography to keep pace with the experiment, and to be able to modify the experimental plan in the light of experience gained during earlier parts of the measuring process. Singlepeak fits use a peak profile consisting of the convolution of a truncated exponential with a Voigt function. Fullpattern refinements use the computer codeThe program can calculate the centroid of the SGV (as given by the geometric approximation of §2.3) and display the IGV in a threedimensional representation of the sample, as well as provide the actual direction of the measured strain in the sample coordinate system.
5.4. Example
We briefly describe a case study, namely a map of the elastic strain near a repair weld, in order to illustrate the new capabilities that ENGINX and SSCANSS have opened within this field. Repair welds are introduced into structures either to remedy initial fabrication defects found by routine inspection, or to rectify inservice degradation of components. We have worked in collaboration with British Energy to determine the residual stresses in repair welds in large components (Bouchard et al., 2005). A typical example is the repairwelded component shown in Fig. 9(b), and schematically depicted in Figs. 10(a) and 10(b). The pipe was fabricated by manual metal arc welding of two exservice 316H stainlesssteel powerstation steam headers. In order to study the effect of a typical repair process, a section of the original weld was removed, and subsequently filled with new material using the same process. Scans of the full strain tensor were performed along lines BB′ and CC′ at the middle and end of the heataffected zone (HAZ) of the repair; and along lines DD′ and EE′ on equivalent locations on the original weld. In addition, we measured maps of the axial strain on the planes containing these lines. Critical to this work has been the use of laser CMM scanned models within SSCANSS, allowing manipulation of the virtual model for a detailed preplanning of the experiment. A visual representation of the measurement positions is shown in Fig. 9(d). Before the development of the SSCANSS software, setting up large complex specimens could take days of beam time and the calculation of beam exposure times was a hit or miss affair based on experience.
A map of the measured axial strain in the HAZ of the repair is shown in Fig. 10(d). The map reveals two main concentrations of tensile strain beneath the outer surface of the pipe at the stopend and about midway between BB′ and CC′. These shortrange stress concentrations revealed the importance of stopend effects for a correct description of the repair process. Each point was measured for 7 min, so the full map was completed in about 7 h. Considering only exposure times, a similar map on ENGIN would have taken nearly a week. The detailed information provided by the axial strain maps would not have been accessible in a reasonable time scale for such a large industrially relevant component before ENGINX.
6. Conclusions
In this paper we have described ENGINX, an instrument that represents the current stateoftheart for NSS, which has been specifically designed to perform measurements of interplanar distances at precise locations within bulky specimens as fast as possible. Two very different aspects of the instrument have been separately optimized. Firstly, the neutron optics was designed using a figure of merit that describes the effect of all instrumental parameters on counting times, and a flexible definition of gauge volumes and divergence was provided. Secondly, the routine operation of the instrument was optimized by including ancillary hardware (theodolites, CMM), in combination with a threedimensional computer representation of the instrument.
Examples of the reduction in counting times between ENGINX and its predecessor ENGIN, as a result of the optimized neutron optics, have demonstrated improvements in performance between 15× and 40×. The variation in performance results from qualitative differences between experiments, as in NSS we are mainly interested in the average response of the material, whilst for in situ loading we are concerned with shifts of the individual peaks. The improvements in neutron optics have been presented in Fig. 7, which compares the count rates and of ENGINX and ENGIN. The theoretical improvements in counting time of ENGINX over ENGIN can be calculated with equation (3). For an experiment where we were only interested in the position of a single diffraction peak, ENGINX could run at a 50 Hz pulse rate and the reduction in counting would be up to 100×. However, in practice ENGINX must run at a lower pulse rate (typically 25 Hz) in order to measure the position of several peaks. Due to the opening and closing times of the choppers, the actual TOF range accessible with ENGINX operating at lower pulse rates is smaller than the nominal TOF period, so the 40× improvement observed for in situ loading is reasonable. For strain scanning, such counting time estimates are not straightforward, as the uncertainty comes from a of the diffraction data. Nevertheless, the counting times reported in Table 1, estimated by including the contribution from all peaks within the available TOF range, agree quite well with the experimental times. For this case, the typical reductions in counting times for ENGINX running at 25 Hz are ∼20×.
Strain measurements performed on TOF neutron strain scanners are sometimes compared or combined with experiments performed on constantwavelength strain scanners (Stelmukh et al., 2002). Therefore, in this paper we have paid special attention to presenting absolute values for the experimental count rates, properly normalized by the instrumental gauge volume to allow comparison between the techniques. It must be noted, however, that a proper evaluation of the instrument performance cannot be based on count rates or resolution only, but on the ability to tackle specific problems in strain analysis. Consider for example the case of NSS in nickel, as a direct comparison can be performed with Ni powder data measured at a constantwavelength strain scanner at Chalk River, Canada (Browne, 2001). The reported count rate for the 311 reflection (0.4 counts s^{−1} mm^{−3}) is about twice the count rate for the same reflection measured on ENGINX running at 25 Hz, but only a fourth of the added count rate including all available reflections (1.6 counts s^{−1} mm^{−3}). Considering that the resolution of the instrument used in that work (∼2500 µ∊) is about twice that of ENGINX (∼1300 µ∊), and that two strain directions can be measured simultaneously in ENGINX, we estimate a ∼12× reduction in counting times. Even larger improvements are achieved for materials with longer d spacings. For instance, measuring the (0002) planes for zirconium alloys requires an incident wavelength of 3.6 Å (at 2θ_{B} ≃ 90°), readily available on ENGINX, but not available at constantwavelength strain scanners. As a result, the (0004) planes are studied instead, which results in an order of magnitude decrease in the diffracting power of the material.
In addition to the gains in count rates, ENGINX allows a very flexible definition of the IGV and the incidentbeam divergence. In particular, we have discussed how the divergence can be tuned for samples presenting very broad peaks, and its effect on the shape and size of the IGV. We have confirmed that, for operational purposes, the gauge volume can be effectively represented as a cuboid having sides defined by the height, width and position of the three pairs of slits.
The improvements achieved by the development of the ENGINX virtual laboratory software are more difficult to quantify. However, the impact has been evident in two areas: (i) it has allowed novice users to perform strain scanning experiments successfully with little training time, and (ii) it has opened the possibility of more complex scans and improved spatial mapping experiments. For instance, the strain map reported here would not have been practically possible on ENGIN, not only because of the long experimental times involved, but also due to the complexity of positioning the gauge volume at precise locations below a highly irregular surface. Another contribution of the SSCANSS virtual laboratory has thus been in the calculation of the centroid of the SGV for measurements performed near surfaces, allowing a precise definition of the measurement position.
The maximum accessible depth is an important feature of a neutron strain scanner. On ENGINX, the maximum path lengths measured so far are 60 mm in stainless steel (Hossain et al., 2006), 60 mm in nickel superalloys (Rist et al., 2006) and 220 mm in aluminium. An approximate estimation of the maximum depth achievable for other materials for a given set of strain uncertainty, gauge volume and maximum counting time can be inferred from Table 2. For more precise estimations, the sample texture, the the peak width, and the background rate should all be properly defined. At present, the easiest way to do this is still to perform a test measurement prior to the actual experiment.
As a result of all these improvements in performance, maps of the elastic strain in irregular bulky specimens (1–1000 kg) can be easily measured with ENGINX within several hours.
Acknowledgements
The authors would like to thank Ed Oliver and Jude Dann for their help during much of the experimental work, Mike Johnson for guidance and support, Dave Maxwell for technical assistance, and John Bouchard for kindly allowing the publication of the results on the repaired pipe. This work has been funded by an EPSRC grant and CCLRC grant No. GR/M51963.
References
Allen, W., Andreani, C., Hutchings M. T. & Windsor, C. G. (1985). Adv. Phys. 34, 445–473.
Bouchard, P. J., George, D., Santisteban, J. R., Bruno, G., Dutta, M., Edwards, L., Kingston, E. & Smith, D. J. (2005). Int. J. Pressure Vessels Piping, 82, 299–310. Web of Science CrossRef CAS
Browne, P. A. (2001). PhD thesis, University of Salford, UK.
Buras, B. (1963). Nukleonika, 8, 259–260. CAS
Daymond, M. R. (2001). Physica B, 301, 221–226. Web of Science CrossRef CAS
Daymond, M. R. (2004). J. Appl. Phys. 96, 4263–4272. Web of Science CrossRef CAS
Daymond, M. R., Bourke, M. A. M., Von Dreele, R. B., Clausen, B. & Lorentzen, T. (1997). J. Appl. Phys. 82, 1554–1562. CrossRef CAS Web of Science
Daymond, M. R. & Edwards, L. (2004). Neutron News, 15, 24–28. CrossRef
Daymond, M. R., Johnson, M. W. & Sivia, D. S. (2001). J. Strain Anal. 37, 73–85. Web of Science CrossRef
Daymond, M. R. & Priesmeyer, H. G. (2002). Acta Mater. 50, 1613–1626. Web of Science CrossRef CAS
Daymond, M. R. & Withers, P. J. (1996). Scr. Met. 35(6), 717–720. CrossRef
Edwards, L & Santisteban, J. R. (2002). Appl. Phy. A Mater. Sci. Process. 74S, 1424–1426. CrossRef
Fitzpatrick, M. E. & Lodini, A. (2003). Analysis of Residual Stress by Diffraction using Neutron and Synchrotron Radiation. London: Taylor and Francis.
Foley, J. D., van Dam, A., Feiner, S. K. & Hughes, J. F. (1997). Computer Graphics: Principles and Practice, in C. New York: Addison Wesley.
Galsworthy, P., Abbley, D., Bowden, Z. A., Brind, M. S., Lay, S. & Wakefield, S. (2003). Proc. ICANS XVI, Vol. 1. ISSN 1433559X.
Hossain, S., Truman, C. E., Smith, D. J. & Daymond, M. R. (2006). Int. J. Mech. Sci. 48, 235–243. Web of Science CrossRef
Hutchings, M. T., Withers, P. J., Holden, T. M. & Lorentzen, T. (2005). Introduction to the Characterization of Residual Stress by Neutron Diffraction. Boca Raton: CRC Press, Taylor and Francis.
Ikeda, S. & Carpenter, J. M. (1985). Nucl. Instrum. Methods A, 239, 536–544. CrossRef Web of Science
ISO/TTA 3:2001 (2001). Polycrystalline Materials – Determination of Residual Stresses by Neutron Diffraction. International Organization for Standardization.
James, J. A., Santisteban, J. R., Daymond, M.R. & Edwards, L. (2002). Proc. NOBUGS2002, NIST, Gaithersburg, USA, http://arXiv.org/abs/condmat/0210432 .
James, J. A., Santisteban, J. R., Daymond, M. R. & Edwards, L. (2004). Physica B, 350, E743–E746. Web of Science CrossRef CAS
Johnson, M. W. & Daymond, M. R. (2002). J. Appl. Cryst. 35, 49–57. Web of Science CrossRef CAS IUCr Journals
Johnson, M. W., Edwards, L. & Withers, P. J. (1997). Physica B, 234, 1141–1143. CrossRef Web of Science
Jorgensen, J. D., Faber, J. Jr, Carpenter, J. M., Crawford, R. K., Haumann, J. R., Hitterman, R. L., Kleb, R., Ostrowski, G. E., Rotella, F. J. & Worlton, T. G. (1989). J. Appl. Cryst. 22, 321–333. CrossRef CAS Web of Science IUCr Journals
Kamminga, J. D., De Keijser, T. H., Mittemeijer, E. J. & Delhez, R. (2000). J. Appl. Cryst. 33, 1059–1066. Web of Science CrossRef CAS IUCr Journals
MoretonSmith, C. M., Johnston, S. & Akeroyd, F. A. (1996). J. Neutron Res. 4, 41–47.
Nielsen, K. & Lefmann, K. (2000). Physica B, 283, 426–432. Web of Science CrossRef CAS
Pawley, G. S. (1981). J. Appl. Cryst. 14, 357–361. CrossRef CAS Web of Science IUCr Journals
Research Systems Inc. (2004). Interactive Data Language Version 6.1. (http://www.ittvision.com/idl .)
Rist, M. A., Tin, S., Roder, B. A., James, J. A. & Daymond, M. R. (2006). Met. Mater. Trans. A, 37A, 459–467. CrossRef CAS
Santisteban, J. R. (2005). J. Appl. Cryst. 38, 934–944. Web of Science CrossRef CAS IUCr Journals
Schooneveld, E. M. & Rhodes, N. J. (2003). Proc. ICANS XVI, Vol. 1, pp. 251–258. ISSN 1433559X.
Snyder, R. L., Fiala, J. & Bunge, H. J. (1999). Defects and Microstructure Analysis by Diffraction. Oxford University Press.
Stelmukh, V., Edwards, L., Santisteban, J. R., Ganguly. S. & Fitzpatrick, M. E. (2002). Mater. Sci. Forum, 404–407, 599–604. CrossRef CAS
Von Dreele, R. B., Jorgensen, J. D. & Windsor, C. G. (1982). J. Appl. Cryst. 15, 581–589. CrossRef CAS Web of Science IUCr Journals
Wang, D. Q., Santisteban, J. R. & Edwards, L. (2001). Nucl. Instrum. Methods A, 460, 381–390. Web of Science CrossRef CAS
Wang, D. Q., Wang, X. L., Robertson, J. L. & Hubbard, C. R. (2000). J. Appl. Cryst. 33, 334–337. Web of Science CrossRef CAS IUCr Journals
Webster, G. W. (2000). Editor. VAMAS Report 38. ISSN 10162186.
Windsor, C. G. (1981). Pulsed Neutron Scattering. London: Taylor and Francis.
Withers, P. J. (2004). J. Appl. Cryst. 37, 596–606. Web of Science CrossRef CAS IUCr Journals
Withers, P. J., Daymond, M. R. & Johnson, M. W. (2001). J. Appl. Cryst. 34, 737–743. Web of Science CrossRef CAS IUCr Journals
Withers, P. J., Johnson, M. W. & Wright, J. S. (2000). Physica B, 292, 273–285. Web of Science CrossRef CAS
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.