research papers
Coherent Xray diffraction investigation of twinned microcrystals
^{a}Departamento de Química Inorgánica, Cristalografía y Mineralogía, Universidad de Málaga, 29071 Málaga, Spain, ^{b}London Centre for Nanotechnology, University College, 17–19 Gordon Street, London WC1H 0AH, UK, and ^{c}Diamond Light Source Ltd, Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, UK
^{*}Correspondence email: g_aranda@uma.es, i.robinson@ucl.ac.uk
Coherent Xray diffraction has been used to study pseudomerohedrally twinned manganite microcrystals. The analyzed compositions were Pr_{5/8}Ca_{3/8}MnO_{3} and La_{0.275}Pr_{0.35}Ca_{3/8}MnO_{3}. The prepared loose powder was thermally attached to glass (and quartz) capillary walls by gentle heating to ensure positional stability during data collection. Many diffraction data sets were recorded and some of them were split as expected from the main observed 180° rotation around [101]. The peak splitting was measured with very high precision owing to the highresolution nature of the diffraction data, with a resolution (Δd/d) better than 2.0 × 10^{−4}. Furthermore, when these microcrystals are illuminated coherently, the different crystallographic phases of the structure factors induce interference in the form of a speckle pattern. The threedimensional speckled Bragg peak intensity distribution has been measured providing information about the twin domains within the microcrystals. Research is ongoing to invert the measured patterns. Successful phase retrieval will allow mapping out the twin domains and twin boundaries which play a key role in the physical properties.
Keywords: coherent diffraction; CXDI; pseudomerohedral twinning; perovskites; manganites.
1. Introduction
i.e. related by the socalled twin operator(s) or twin law(s) (Cahn, 1954; Santoro, 1974; Yeates, 1997). Multiplecrystal growth disorders are common, but refers to special cases where the extra symmetry element(s) must be of the kind encountered in crystal morphology (a centre of symmetry, a mirror plane or a rotation axis). If the extra is a mirror plane, called a twin plane, then this plane must be parallel to a lattice plane of the same dspacing in both domains. If the extra symmetry is a rotation axis, called a twin axis, then this axis must be parallel to a lattice row common to both domains (Cahn, 1954; Koch, 1992). The occurrence is fairly common in crystals of inorganic materials, organic compounds and macromolecular specimens, and it can be a serious complication in the determination process.
is a crystal growth anomaly in which the specimen is composed of several distinct domains whose orientations are related by one or more symmetry elements that are not a part of the spacegroup symmetry of the single crystal,Several different categories of ; Yeates, 1997). The situation in which the lattice overlap occurs in two (or one) dimensions is known as nonmerohedral or epitaxial The case in which the lattices of the domains coincide exactly in three dimensions is known as Hemihedral is a special subclass where there are just two domain orientations. The cases where the lattices overlap approximately, but not exactly, in three dimensions are referred to as pseudomerohedral This requires unusual unitcell geometries that yield a lattice with pseudosymmetry which is higher than the pointgroup symmetry (for instance the case of orthorhombic perovskites with a ≃ c, both edges deriving from the cubic parent structure).
can be defined according to the coincidence of the separated lattices (Buerger, 1960Perovskite materials display many interesting properties including ferromagnetism, piezo and ferroelectricity, multiferroic behaviour, etc. The archetypal perovskite structure is cubic but, in order to finetune the physical properties, structural distortions may take place with many materials being orthorhombic or monoclinic. Lowsymmetry perovskite crystals therefore fulfil the requisites for pseudomerohedry and they are commonly twinned. in orthorhombic perovskites has been extensively studied by a number of techniques including (TEM) (White et al., 1985; Keller & Buseck, 1994; Hervieu et al., 1996), precession electron diffraction (Ji et al., 2009), photoemission spectroscopy (Sarma et al., 2004), neutron singlecrystal diffraction (DaoudAladine et al., 2002) and Xray singlecrystal diffraction (Van Aken et al., 2002). Furthermore, Pr_{0.5}Ca_{0.5}MnO_{3} twinned crystals were studied by synchrotron (micro)diffraction to map the domains (Turner et al., 2008) but the coherence properties of the beam were not fully exploited to provide information about the in real space. To the best of our knowledge there are no other reported works dealing with the study of twinned crystals by coherent Xray diffraction.
Here we apply the method of coherent Xray diffraction imaging (CXDI), which is a rapidly advancing form of lensless microscopy (Neutze et al., 2000; Vartanyants et al., 2007; Jiang et al., 2008; Thibault et al., 2008; Nishino et al., 2009; Robinson et al., 2010, and references therein) that was opened up by the realisation that oversampled diffraction patterns may be inverted to obtain realspace images. The possibility was first pointed out by Sayre (1952) and demonstrated by Miao et al. (1999). The phase information of the diffraction pattern, which is lost in the intensity measurement, is embedded in a sufficiently sampled coherent diffraction pattern, which may allow the inversion of the diffraction data set back to an image by computational methods. In addition to obtaining threedimensional images of studied nanoparticles, CXDI may give different types of information including the quantitative strain field inside a crystal (Pfeifer et al., 2006; Robinson & Harder, 2009; Newton et al., 2010).
In this paper we focus on the use of CXDI for studying twinned microcrystals. When such a crystal is illuminated coherently, speckle patterns can form owing to the differences in amplitude and/or phase of the structure factors for different twin domains, and also from the continuous phase changes between them. Here, we show that this threedimensional speckled Bragg peak distribution can be measured providing information about the twin domains within the crystals. Research is ongoing to invert the patterns to obtain threedimensional images of the twin distribution. Threedimensional images of the twinned domains can provide information relevant to the properties of materials. An ultimate goal is that, by separating out each component, more accurate diffraction intensities may be obtained which would help in the
process of twinned crystals.2. Experimental section
Three manganite compositions were synthesized by the ceramic method as previously reported (Collado et al., 2003). The stoichiometries were Pr_{5/8}Ca_{3/8}MnO_{3}, La_{5/8}Ca_{3/8}MnO_{3} and La_{0.275}Pr_{0.35}Ca_{3/8}MnO_{3}, and they are hereafter abbreviated PCMO, LCMO and LPCMO, respectively. The three samples were characterized by highresolution laboratory Xray powder diffraction at room temperature. Powder patterns were collected on a Philips X'Pert Pro MPD diffractometer equipped with a Ge(111) primary monochromator (strictly monochromatic Cu Kα_{1} radiation) and an X'Celerator detector. characterization was carried out using a Philips CM200 microscope with the powders dispersed on copper grids.
The sample preparation step for a CXDI experiment is important as the positional stability of the particles within the beam must be ensured. In order to carry out this type of experiment for loose powders, we have developed a simple method which is valid for thermally stable samples. Dispersed crystals were attached to borosilicate glass (and also quartz) capillaries by gentle heating in a butane flame (using a cigarette lighter) up to the point where the capillary glass softens. The results are better when the heating is applied to vertically arranged capillaries (as horizontally aligned capillaries are prone to severe bending). Fig. 1 shows an optical microphotograph of a borosilicate glass capillary with the PCMO loose powder thermally attached to the wall.
Coherent Xray diffraction patterns for selected manganite microcrystals were collected at beamline I16, Diamond Light Source, which is equipped with a sixcircle kappa goniometer at 50 m from the source. The wavelength, λ = 1.55 Å (E = 8.00 keV), was selected using a Si(111) channelcut monochromator. The beam was horizontally focused down to ∼4 µm by using a Kirkpatrick–Baez (KB) mirror pair. The vertical and horizontal dimensions of the beam before these mirrors, ∼30 µm, were selected by the final beamline slits before the KB mirrors; these slits are not very precise, however. To ensure this beam was reasonably coherent, a 300 µm horizontal slit was placed in front of the beamline focusing mirror (which remained in place in the beamline). This slit is located 27 m from the source and the diffractometer is at 50 m. The horizontal coherence length at the sample is therefore increased to λD/d = 12 µm, assuming stochastic illumination of the intermediate 27 m slit. The sample was aligned by scanning the Xray absorption of the capillary with a point detector. After the alignment of the capillary, the appropriate Bragg peaks, for instance (121) at 33.17° 2θ for PCMO, were identified by using a Pilatus 100K detector (172 µm × 172 µm pixelsize detector) permanently mounted in the detector arm. After centring the reflection, this detector was moved away and the reflection was again centred in a Princeton Instruments CCD detector with optical coupling to a columnar CsI scintillator. The pixel size was 20 µm × 20 µm and it was placed far from the sample, at 1220 mm, to provide data with sufficient angular resolution. Twodimensional slices were recorded as the Bragg peak was rocked through the by rotating the sample within the beam, usually in 0.005° step size (η rocking angle) for about 0.6° although this number depends upon the width of the analyzed peak which was previously prescanned at lower resolution. For every discussed scan, the exposure time, number of accumulations and maximum intensity is given below. Coherent diffraction patterns were obtained by collating the slices of the highresolution scan to form complete threedimensional diffraction data sets. The pixel binning size and rocking step size were chosen to ensure at least three pixels per fringe, to ensure the data were oversampled.
The coherence of a beam of light is generally described by two components, the transverse and longitudinal coherence lengths, ξ_{T} and ξ_{L}, respectively (Born & Wolf, 1999; Leake et al., 2009). The transverse coherence is dependent on the size (S) of the source/undulator itself, and it is given by ξ_{T} = λD/S, where D is the distance from the source to the sample. For a typical thirdgeneration synchrotron beamline, the raw transverse coherence is ∼10 µm and ∼100 µm for the horizontal and vertical directions, respectively. By closing the beamline midpoint slit (see above), the horizontal transverse coherence length was raised to ∼12 µm, sufficient to give some limited degree of coherence beyond the ∼30 µm sample slits. On the other hand, the longitudinal coherence is dependent on the bandwidth of the monochromator, ξ_{L} = λ^{2}/2Δλ. For a silicon (111) monochromator, ξ_{L} ≃ 0.6 µm. This couples to the optical path length difference of rays through the sample. Therefore, when the crystal size is smaller than the coherence lengths, the sample is said to be in the coherent limit and it meets the required conditions for CXDI measurements. The fringe visibility can be lost by violation of any of the three coherence requirements.
3. Results
Laboratory powder diffraction data for the three manganite samples showed that they were microcrystalline single phase powders. The three phases were indexed in an orthorhombic distorted Pnma symmetry, with edges ≃ a_{c} × 2a_{c} × a_{c} where a_{c} stands for the basic perovskites cubic unitcell parameter, in agreement with previous results (Uehara et al., 1999; Collado et al., 2003). The unitcell values for the three phases are given in Table 1. It is clear from the unitcell edges given in Table 1 that these manganites are very prone to pseudomerohedral twinning.
of

The sizes, shapes and variability of the manganite particles were studied by TEM. Fig. 2 displays bright and dark field microphotographs for selected particles of PCMO. It must be noted that the dispersion in particle sizes is large, from small particles of about 0.2 µm to large particles longer than 1 µm. The shapes were also variable but most of the observed particles were elongated prisms (see Fig. 2).
Fig. 3 shows the diffraction signal of PCMO and LPCMO microcrystals, collected in the Pilatus detector, as an example. Only those well shaped rounded reflections isolated from any other diffracting signal were centred, for instance the (210) reflection of LPCMO in Fig. 3(b).
Three main types of coherent diffraction data sets were recorded on the CCD: (i) single centred, (ii) doubledcentred and (iii) multicentred peaks. Fig. 4 (top) shows an example of each type of data set as twodimensional slides. The angles used for the discussion of the results (see below) are also shown in Fig. 4(b). Fig. 4 (bottom) displays views of the threedimensional integrated patterns for the three types of microcrystals in order to highlight the widths of the measured diffraction peaks. Eleven good quality data sets were collected for PCMO and four data sets for LPCMO. It must be highlighted that we could not collect CXD data for any of the LCMO microcrystals we tried. The diffraction signal disappears from the CCD detector when centring the crystal. We speculate that the photoelectric effect may be playing an important role as this is the highest electrically conducting sample. Therefore, LCMO will not be further discussed in this work.
4. Realspace phase model of twinning
PCMO has orthorhombic Pnma, with lattice constants given in Table 1, which is a of the simple cubic perovskite. These perovskites are known to show three main laws for pseudomerohedral a 180° rotation around [101], 90° around [101] and a 180° rotation around [121] (Wang & Liebermann, 1993; Keller & Buseck, 1994). The `a–c' arises from the accommodation of the small difference between the a and c lattice constants, possibly forming upon cooling the crystals from a higher symmetry phase at elevated temperature. Fig. 5 shows an exaggerated view of the matching of the {101} planes at the twin boundary, running up the figure. It is clear that the a–c lattice difference corresponds to a small rotation of the twin domains, giving two sets of Bragg peaks for all reflections except along the hkh symmetry axis.
If the domains are small, their Bragg peaks will be enlarged owing to the classical `finite size' effect. If the domains are sufficiently small, the two members of the split Bragg peaks will overlap and, assuming the beam is coherent over the size of the domains, will lead to interference effects in the overlap region of the diffraction pattern. Each of the two peaks will be speckled owing to the domain arrangement within the illuminated piece of crystal and the speckles will combine together in the interference pattern. It is a general rule, for domain structures at least, that the number of speckles in the peak is approximately equal to the number of domains in the beam. The rule assumes the domains are independent, with large phase shifts between them, and applies in one, two or three dimensions. The rule can be derived by noting that the size of a speckle is the reciprocal of the size of the illuminated area, while the width of the speckle distribution is the reciprocal of the size of a typical domain.
The words `speckle' and `fringe' are used rather interchangeably in the literature. Fringes appear in optical interferometers as regular linear arrays of constructive and destructive interference, and are used as a way to measure coherence, among other things. Fringes also arise in the shapediffraction patterns of simple objects, parallel from pairs of edges or circular from round apertures or scatterers. Speckles, on the other hand, are the result of random interference between a large number of waves with arbitrary, yet well defined, amplitudes and phases. In principle their degree of modulation can be used to measure coherence, as reported by Alaimo et al. (2009) using the dynamical nearfield speckles formed by scattering from colloidal particles. The inherently random nature of domain structures owing to are therefore expected to give rise to speckles rather than simple fringes. Here, we use the term `speckle' to describe the fluctuation in the Bragg peak intensity distribution owing to interference between the (smaller) twin domains, and the term `fringe' to mean the structure owing to the (larger) crystallites. If the domain structure exists across the entire crystal, the spacing between the speckles will be determined by the crystal size. However, the overall peaks will have a breadth dependent on the domain size with the number of speckles in the peak related to the number of domains in the crystal. The formation of both speckles and fringes is a necessary consequence of having some degree of coherence in the beam.
There is considerable merit, where possible, in modelling the diffracting object as the product of a periodic structure (the ideal crystal) and a macroscopic modulation of the phase that characterizes the structures of interest (i.e. twin domains). In such a multiplication becomes a convolution, and the motif of the macroscopic object is repeated around each point in In the present case, such a treatment is complicated by the presence of two lattices. We would like to interpret the combined double Bragg peak, along with these speckles, in terms of the Fourier transform of a single object in real space. We therefore make use of the shifting property of the Fourier transform: when the object in real space is multiplied by a linear phase ramp, the diffraction pattern is shifted by an amount proportional to the ramp. In this way the two Bragg peaks of the split pair, shifted in opposite directions, can be attributed to separate regions of the crystal overlaid with oppositedirected phase ramps. This is illustrated in Fig. 5(a), where the independent lattices of the two separate domains can be described as a transformation from a single reference lattice. If the origin is chosen at the corner of the unit cells meeting at the twin plane, each is shifted by an amount proportional to its distance from the twin plane, or a linear function of its distance. This shift gives rise to a phase in the diffraction that increases linearly with distance from the twin plane, in other words a phase ramp. Therefore, through the Fourier shift theorem, the two domains give rise to opposite peak shifts in the combined diffraction pattern, seen as the splitting.
This picture is easily generalized to the full mosaic domain structure of an arbitrarily twinned crystal, which is thereby mapped onto a complex image of unit amplitude and spatially varying phase. All the leftrotated domains have phase ramps with negative slope and all the rightrotated domains have phase ramps with positive slope; where they join, the phase should be continuous (although their derivatives are not), assuming they meet at a complete unitcell boundary, but the slope reverses. The picture can be readily generalized to three dimensions: the macroscopic realspace phase is simply the threedimensional displacement of the atoms from those of the ideal reference lattice, projected onto the Qvector of the reflection under consideration. The choice of `reference' lattice is arbitrary; it simply defines the with respect to which the peak shifts are observed.
The program nearBragg (https://bl831.als.lbl.gov/~jamesh/nearBragg/ ) from James Holton was used for some initial simulations. The following description is extracted from the nearBragg manual: “…nearBragg calculates the distance from one or more source points to each `atom' and then from the `atom' to the centre of a detector pixel. The sin and cos of 2π times the number of wavelengths involved in this total distance is then added up and the amplitude and phase of the resultant wave from the whole sample (and the whole source) are obtained for each pixel. The intensity at each pixel is the modulussquare of the amplitude. The atoms are considered point scatterers with an intrinsic of `1' in all directions…”. Results from one of the simulations are shown in Fig. 5(d). This type of simulation is easy to relate to the schematic in Fig. 5(a) and it is shown for this reason. However, the phase ramp description gives the same results [as will be shown below in Fig. 6(a)] and it is computationally much faster for threedimensional objects. The phase ramp description is therefore used for the remaining simulations.
To illustrate the effect of the opposite phase ramps on the diffraction patterns, in Fig. 6 we present two simulated threedimensional objects with a double phase ramp. The calculations are carried out using a fast Fourier transform on a 256 × 256 × 128 array and an object of 48 × 24 × 24 grid points, such that each simulated twin domain has a cubic shape. The amplitudes are 1 within the object and 0 outside. The introduced phase ramps within the object are shown on a colour wheel with green representing φ = 0, blue φ = −π and red φ = +π. The corresponding amplitudes of the threedimensional diffraction pattern are depicted at the bottom of the figure showing the splitting of the peak owing to the phase structure and the intermodulation of the diffraction in between. The magnitude of the ramps, 1.5π and 2π, in Figs. 6(a) and 6(b), respectively, corresponds to an increasing angular separation of the peaks (see Fig. 6, bottom). Since the fringe spacing is fixed by the crystal size, the result is an increasing number of fringes connecting the two centres of the Bragg peaks as the slope of the ramps increases (Fig. 6b).
So far we have omitted the
from this discussion, but it is straightforward to introduce that too. In the perovskite case we have been considering that the two member peaks of the pair will have roughly the same even if they have different indices, because they are attributed to the same reflection of the cubic parent structure. In the general case where different unrelated structure factors mix together, this must be built into the complex object whose Fourier transform gives the full diffraction (speckle) pattern. Both the amplitude and phase of the are attributed to the regions of space occupied by the domain, multiplied by the phase ramp function corresponding to the domain rotation relative to the reference lattice. This picture applies equally to the interpretation of data: following phase retrieval of the diffraction pattern, the inverted threedimensional image of the crystal can be interpreted directly in terms of local structure factors and domain rotations (ramps).It is known that phase retrieval is difficult in this general case and methods still need to be developed. The best methods to date, based on Fienup's (1982) `hybrid input–output' method, still require a `realspace constraint'. The description above could conceivably be coded into a suitable algorithm, for example exploiting the slowvarying continuous property of the realspace phase image. Such a method has been used successfully by Minkevich et al. (2007) for analysing strains in semiconductor heterostructures (without twinning).
However, if phase retrieval and imaging are not possible, limited information can be extracted from the diffraction pattern itself. The speckle contrast, usually defined as the `visibility' (I_{max} − I_{min})/(I_{max} + I_{min}), is a rough estimate of the range of realspace phases present in a diffracting object. If the object is a pure, but weak, phase object, illuminated by a softedged beam, then it can be shown, by a vector summation diagram representing the Fourier transform, that the speckle contrast is proportional to the range of phases present in the object, envisaged as a realspace phase map in three dimensions. The speckle contrast can be estimated experimentally as the variance of the intensity over parts of the diffraction pattern. There is a potentially important application to the case of discussed above. When domains of different but identical lattice are present in a twinned crystal, the speckle visibility will be proportional to the crystallographic (complex) amplitude difference between the two structure factors. This might have some potential in crystallographic phase measurement. In other cases, the speckle contrast will be due to internal grain boundaries (density discontinuities) within the sample. For pseudomerohedral each component of the split Bragg peak receives its main contribution from domains with only one of the two orientations. These domains might be separated from each other by `unseen' domains with the alternative orientation, an extreme example of a density discontinuity. This is the probable reason why a relatively high speckle contrast is observed for the manganites studied here.
The important result of this section is that we can represent the complicated array of twin domains within a crystal, in which the lattice is separately defined for each domain, in terms of a phase field over the entire object. The relative positions of the domains are mapped onto phase shifts, while the local rotations are mapped onto phase ramps. This method is generally applicable, but might have little useful meaning in the limit of very small domains or the case of partially crystalline or amorphous materials.
5. Discussion
Firstly, the data shown in Fig. 3, which are representative of many recorded images, deserve a brief discussion. PCMO, as shown in Table 1, has a much larger orthorhombic ac distortion than LPCMO. This may be quantified by the lattice strain in the ac plane, S_{ac} (%),
and the corresponding values are given in Table 1. The larger distortion for PCMO is also evident in Fig. 3 as the (200) and (002) Bragg reflections are partially resolved in the powder diffraction pattern. For LPCMO, those reflections are not resolved in the powder pattern, owing to the very low orthorhombic distortion, but some studied microcrystals showed very large The data shown in Fig. 3 for LPCMO are indicative of a very large unitcell parameter distribution for some microcrystals. This may be due to inhomogeneities in the cation distribution of this but it may also be due to large microstrain owing to the rich twin structure of these compositions. This observation may be much related to its very complex lowtemperature behaviour displaying mesoscopic (Uehara et al., 1999) and persistent magnetoresistive memory effect (Levy et al., 2002).
Now we discuss the results obtained in the coherent highresolution diffraction patterns. We will focus on split Bragg reflections and we will not consider in this work the unsplit peaks as they are not likely to arise from twinned regions of the manganite microparticles. Most of this study was dedicated to recording data around the 2θ ≃ 33.1° region as it contains the most intense diffraction peaks of these manganites: (200), (121) and (020) (see Fig. 3). It must be noted that these reflections cannot be distinguished by their measured 2θ values as they have almost the same dspacings, with their angular positions slightly dependent upon the centring of the microcrystal in the beam. However, their splitting behaviour owing to the ac is quite different (see Fig. 5).
Firstly, let us consider a (121) reflection. The 2θ for the pair arising from the two twin domains must have the same value but they will be rotated by an angle, Δα_{ac}, defined as
The theoretical values for this splitting angle are also given in Table 1. For PCMO, we have collected three data sets with split peaks with an average Δα_{ac} value of 0.23 (1)°, which were measured from the diffraction data as detailed in equation (3) (see Fig. 4b),
This value is in excellent agreement with the expected angle between the axes of the two twin individuals (see Table 1). It must be noted that the measured 2θ angles for two centres were the same for this type of reflection. An example of this type of doublecentred coherent diffraction data set is shown in Fig. 4(b). This misfit angle has been previously reported (see, for instance, Wang & Liebermann, 1993) but the value was larger (∼1°) because the lattice strain in the ac plane for the studied natural perovskites, CaTiO_{3}, is larger.
Secondly, let us consider a (200) [or (002)] type reflection. It is clear that, for a single domain crystal, if one reflection is observed on the detector the other cannot be measured as it is perpendicular. However, when the (200) reflection of a pseudomerohedrally twinned microcrystal is being measured, the (002) diffraction from the other twin domain will also be present close to that diffraction angle. The 2θ difference between the two centres of the peaks will be given by the difference in the dspacing values of those planes. So, for the (200)/(002) peaks, we can define Δ2θ_{(200)} which will be the difference of the diffraction angles of the (200) and (002) twinrelated diffraction peaks. The values of this angle, for the studied manganite compositions, are also given in Table 1. The angle has been measured for two PCMO microcrystals just by converting the Δy pixel distance to 2θ angle, see Fig. 4(b) (using the sampletodetector distance). Fig. 7(a) shows one example of this type of reflection, and the measured Δ2θ_{(200)} angle, 0.15°, is in very good agreement with the expected one, 0.14° (see Table 1). Furthermore, these centres are also displaced in η and γ angles (see Fig. 4b). We can determine the Δα_{(200)} angle, 0.106°, which was also in very good agreement with the expected one, (1 − a/c)/2 = 0.116°.
Furthermore, the (200)/(002) splitting described just above can be generalized to any reflections having h ≠ l indexes. We were able to collect coherent diffraction data for a (weak) highorder reflection (220) diffracting at 40.86° (2θ). Fig. 7(b) shows one slide of the diffraction data set revealing the (220)/(022) double peak owing to pseudomerohedral The measured Δ2θ_{(220)} angle, 0.15°, is relatively close to the expected one, 0.12°. Furthermore, the measured Δα angle between the two centres, 0.11°, is also fully consistent with the twin description. Finally, it must be noted that curved fringes connecting the two peak centres, see Fig. 7(b), were also observed in other coherent Xray diffraction data sets.
The consequences of S_{ac} lattice difference (see Table 1) makes the peak splitting very small for both type of reflections. For LPCMO microcrystals, five data sets were measured and two of them have clear split peaks. Fig. 7(c) shows an image of a split peak. The measured Δ2θ_{(200)} angle, 0.022°, is larger than the expected value, 0.010° (see Table 1). The measured Δα angle between the two centres was 0.057°. Although the split angles are somewhat larger that the expected ones, for the average composition, these observations are totally consistent with this twin description. Furthermore, this is to be expected because the (average) composition of the studied microcrystal may be not exactly the same as that of the powder.
in the diffraction data sets described above are also applicable to LPCMO microcrystals. However, to distinguish between (121) and (200) type reflections is not straightforward because the very lowSince the beam was made coherent during the experiment by use of a coherencedefining aperture and then focused to obtain sufficient intensity from the micrometresized crystals, the speckles and fringes of the diffraction patterns of split Bragg peaks were found to interfere with each other and give a single merged coherent diffraction pattern. From the fringes spacing, an estimation of the crystal sizes can be made,
where D is the sampletodetector distance, n is the average number of the pixel for the repeating fringes, and p_{s} is the pixel size of the CCD detector. We have applied equation (4) to many fringe sets such as those shown in Figs. 4 and 7. The average sizes ranged between 450 and 900 nm for different microcrystals. Sizes close to 700 nm were obtained very frequently. These sizes are consistent with those observed by (Fig. 2). It must also be noted that nonregular fringe spacing has been observed for several crystals. This could be due to very strained crystals as reported previously (Cha et al., 2010).
It was observed that the split peaks, discussed above, have different intensities. This can be explained by smaller or fewer domains for one orientation. In order to study the possible effects of the twin domain size, several simulations have been carried out. Fig. 8 displays the threedimensional coherent diffraction patterns of three twinned objects modelled as explained above. Fig. 8(a) displays the pattern for a twin pair with a twin being half the size of its sister. Fig. 8(b) displays a similar study but for a twin being four times smaller. Finally, Fig. 8(c) shows the simulation for a twin being eight times smaller than its sister. The phase ramp for these simulations is the same as that employed in Fig. 6(a). The centre of symmetry evident in Fig. 6 is lost, as expected, because the twins have different sizes but the intermodulation signal between the two peaks is retained. Furthermore, the diffraction peaks arising from the twins with the smaller sizes are wider. As the domain size decreases, with respect to its twin sister, its fringe spacing becomes broader (see Fig. 8c) as, with only one domain of each type, the fringe spacing is determined by the size of the domains. This might be an alternative explanation for nonregular fringe spacing.
In addition to the different size of the twin domains, complex strain patterns may take place within the twinned microcrystals. A TEM study of this work (see Fig. 2) and many other reports about these types of manganites (see, for instance, Uehara et al., 1999; Fäth et al., 1999; Kim et al., 2000) showed very complex microstructures and behaviours. The coherent diffraction patterns showed a curved set of fringes [see, for instance, Fig. 7(b)] and fringe branches without a constant spacing. In order to justify these observations, we have carried out simulation studies imposing a more complex phase change within the particles. Fig. 9 shows three examples of this type of simulation. In this case the phase is not constant with respect to the faces of the microcrystal but they have a curvature modelled by a parabolic function. The phase ramp is similar to that of Fig. 6(a) and the magnitude of the curvature has been varied. As can be seen, Fig. 9(a) has a small curvature and its coherent diffraction pattern is quite similar to that reported in Fig. 6(a). However, as curvature of the phase (strain) within the crystal increases (see Fig. 9b) the intermodulation signal increases in intensity. Furthermore, when strain is more relevant (see Fig. 9c) curvature of the fringes develops. The interplay between nonflat faces, large strain fields and different twin domain sizes is believed to generate the complex fringe/speckle patterns observed in this study.
Finally, simulations were also carried out in order to study the role of the number of domains in the threedimensional coherent diffraction patterns. Fig. 10 displays the coherent diffraction signal corresponding to a twinned crystal with four alternative domains (ABAB). In addition to the split peak, owing to the two types of domains (A and B), three speckles are evident within each diffraction maxima (see bottom of Fig. 10) for this twin arrangement. The simulations can be interpreted as follows: (i) the number of diffraction peaks is the number of opposite phase ramps (two in this case); (ii) the number of speckles within a peak (three in this case) is the overall number of domains minus one, the speckles within each peak arising from the ABA and BAB configurations, respectively. This can be readily generalized to n domains within a crystal but the diffraction from a given specimen may be complex owing to the possible existence of domains related by different twin laws, not only 180° rotation around [101], and the complex strain pattern within the microcrystal. A multitwinned microcrystal would give complex threedimensional patterns such as that shown in Fig. 4(c).
In summary, using coherent Xray diffraction, Bragg peaks of pseudomerohedral twinned manganite microcrystals have been measured and they were broken up into fringes and speckles. This is consistent with a new theory of the connection between the
and the phase of the domain that contributes to a given Bragg peak. It is straightforward to show that the number of speckles in the Bragg peak is indicative of the number of domains in the beam. Our new understanding of the origin of this speckle shows that the visibility is proportional to the phase shift between the domains. This method represents an entirely new way of measuring the domains mismatch and it may allow sensitive testing of strain in crystals. In the future, successful phase retrieval will allow mapping out the twin domains and twin boundaries which play a key role in many physical properties like magnetoresistance.Acknowledgements
Diamond Light Source is thanked for the provision of Xray synchrotron diffraction beam time, and Gareth Nisbet for his assistance during the experiment. We also thank James Holton for providing a copy of the nearBragg program. The work at Malaga was financed by MAT200907016 research grant and at UCL under EPSRC grant EP/F020767/1 and an ERC Advanced grant.
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