1.IntroductionThe study of mixed-valent manganites such as La1-xCaxMnO3 has aroused much interest because these compounds display an unusual coexistence of charge-ordered antiferromagnetic and metallic ferromagnetic phases [1]. Charge ordering has been described as a localization of Mn 3+ and Mn 4+ ions which occurs as the specimen is cooled below the charge ordering transition temperature indicated by a sudden upturn in the resistivity as the sample is cooled [2]. This transition is accompanied by the appearance of superlattice reflections in X-ray, neutron and electron diffraction experiments indicating that the crystal structure has become modulated [2, 3]. The resistivity anomaly occurs at the same temperature at which the magnetic susceptibility shows a sudden downturn on cooling and therefore this phase has been described as antiferromagnetic and charge ordered [2].In his review of modulated structures, Bak [4] distinguishes three types of modulation. The first is a ‘commensurate’ structure where the periodicity of the modulation is related to the periodicity of the undistorted lattice by a rational fraction. The second is an ‘incommensurate’ structure where the periodicity of the modulation is unrelated to that of the undistorted lattice. Finally there is a ‘chaotic’ structure where the modulation has no long range periodicity but is composed of a random mixture of small subunits of varying lengths, the length of each subunit being an integer multiple of the size of the undistorted unit cell. The appearance of a diffraction pattern from such a structure is treated mathematically by Shao [5]. Notice that these definitions are different to those normally used when referring to the ‘charge ordering’ modulation observed in the manganites where ‘commensurate’ describes a modulation whose periodicity is an integer (rather than a rational) multiple of the undistorted unit cell and ‘incommensurate’ describes any modulation which is not commensurate [3, 6].It is well known that when x?>?0.5, the periodicity of the modulation is primarily determined by the calcium doping concentration and the modulation wavevector q, expressed as a fraction of the reciprocal lattice vector a*, broadly follows the relationship q?=?1-x where q is approximately collinear with the a* direction (labelling the unmodulated structure as orthorhombic Pnma) [3, 7]. Thus, the modulation in La1/2Ca1/2MnO3 shows a doubling of the undistorted cell, La1/3Ca2/3MnO3 shows a tripling and La1/4Ca3/4MnO3 shows a quadrupling [6]. This immediately begs the question of what happens in-between these compositions. Few authors investigate compositions that do not give integer period modulations but it is generally accepted that the modulation is ‘chaotic’ in the sense defined by Bak [4]. Mori et al.[6] state that ‘?...?such cases show that charge ordering appears as a proper mixture of the two adjacent distinct commensurate configurations according to the lever rule. For example, a sample with x?=?5/8, between 1/2 and 2/3?...?is found to exhibit a fine mixture of 25% 2a and 75% 3a. The minority 2a configuration?...?appears as incoherent stacking-fault defects in the otherwise perfect 3a configuration’. In this paper we show that far from forming a ‘chaotic’ structure, the modulation observed in La1-xCaxMnO3 has a uniform periodicity even on the smallest scales.2.Mathematical treatment of a structural modulationWe briefly describe the appearance of a diffraction pattern from a sinusoidally modulated crystal structure using the derivation given by Giacovazzo et al.[8]. Although the modulation observed in the manganites may not be sinusoidal, it has a uniform periodicity on an atomic scale and this treatment represents the principal Fourier component of the modulation. As such, we use it as a guide to interpreting diffraction patterns taken from modulated manganite crystals.In a perfect, unmodulated crystal, the position of a unit cell is defined by the lattice vector ru?=?ua + vb + wc and the position of the jth atom within the unit cell is given by Rj. Thus, a given atom is located by the position of the unit cell and the position of the atom within the unit cell thus: We consider deforming the lattice with a sinusoidal modulation of amplitude, Aj, phase, ?j, and wavevector, q. The positions of the atoms are now given by A calculation of the structure factors that arise from such a modulated structure shows that diffraction peaks occur at H'?=?ha* + kb* + lc* + mq (where h, k, l, m are integers) and have structure factors given by where Jm is the mth order spherical Bessel function and fj is the scattering factor of atom j.This analysis shows that diffraction peaks occur in the same positions as in the unmodulated structure, g = ha* + kb* + lc*. However, each of these so-called ‘parent’ reflections is decorated with satellite reflections which occur at integer multiples of the modulation wavevector on either side of each parent reflection, that is at g± mq. The intensity of the satellite reflections falls rapidly as the order, m, of the satellite reflection increases as described by equation (3).If the modulation is transverse as predominantly occurs in the manganites (see section 4.4), q is perpendicular to Aj and if q is parallel to a*, as is broadly the case for the manganites, H ' is parallel to q if k and l are 0. So for this case, no superlattice reflections should occur on the k?=?l?=?0 row of a diffraction pattern. Note that this assumes that diffraction is kinematic (i.e. the incident radiation only undergoes single scattering events so that the intensity of the diffracted beams are proportional to the squared modulus of their structure factors). Electron diffraction involves multiple scattering (dynamical diffraction) so that it is necessary to tilt the specimen to diffraction conditions where multiple scattering is minimised in order to approximate the kinematic case as we have done in section 4.4 [9].3.Experimental details3.1.Sample synthesis and preparationThe samples of La1-xCaxMnO3 used in this investigation were synthesised by A. J. Williams at the Chemical Laboratory, Cambridge University. They were prepared by repeated grinding, pressing and sintering of La2O3, CaCO3 and MnO2 in stoichiometric proportions. The La2O3 was heated overnight prior to use in order to dehydrate it. Each sample was prepared by initially heating at 950°C for 12 hours to decarboxylate the CaCO3, and then 1350°?C for 12 hours. Each sample was then reground, repelleted and heated at 1350°?C for 4 days; and then reground, repelleted and reheated at 1300°C for 2 days. X-ray powder diffraction confirmed the presence of a single phase. Large area (10?×?10?µm) energy dispersive X-ray analysis using a scanning electron microscope showed that the La/Ca ratio was constant to within the error of the measurements (dx?=?±?2 %). The samples were prepared for transmission electron microscopy by conventional mechanical polishing and argon ion thinning at liquid nitrogen temperatures using a Gatan Duo Mill ion thinner. Samples with compositions x?=?0.50, 0.52, 0.58, 0.67 and 0.71 were used in this investigation.3.2.Transmission electron microscopyTransmission electron microscopy was carried out using Philips CM30 and CM300FEG electron microscopes. A Gatan liquid nitrogen-cooled specimen stage was used to cool the samples to 90?K. Images and diffraction patterns were recorded on CCD cameras and photographic plates.4.Results4.1.Uniform periodicity on all length scalesThe concept that the modulation is made up of small, integer period sub-units was derived from lattice images taken using high resolution electron microscopy [7]. Such lattice images are not directly interpretable and one cannot easily assign different atomic species, let alone different ionic valencies, to the features observed in these images as is sometimes claimed. Only the periodicity of the structure can be directly interpreted and this is readily determined from fast Fourier transforms of high resolution images. When we carried out this procedure, we found that for compositions x???1/2, 2/3, 3/4 the periodicity of the modulation remained at a non-integer multiple of a irrespective of the area of the image that was investigated.The most rigorous test performed in this work involved using convergent beam electron diffraction [9] which allowed the periodicity of the lattice to be determined on very small length scales. This experiment was conducted at 90?K, the lowest temperature accessible with the liquid nitrogen cooled specimen stage and more than 120?K below the ‘charge ordering temperature’ at which anomalies occur in resistivity and magnetisation data [1]. It has been stated explicitly [6] that the charge ordering modulation should be composed of integer periods at this temperature. The composition x?=?0.52 was chosen for this experiment as it is close to the composition x?=?0.5 and so the modulation should be composed mostly of sub-units of size 2a which outnumber those of size 3a by a factor of 7.3 (assuming that the modulation is composed only of subunits of size 2a and 3a). Thus, subunits of 3a should be 7.8?nm apart on average, separated by continuous runs of subunits of 2a. If other sub-units such as 4a and 5a are present, the average length of a continuous run of subunits of 2a will be even larger. Thus, diffraction patterns taken from regions of less than 7.8?nm should mostly show the periodicity 2a (or a wavevector q?=?1/2).Convergent beam electron diffraction patterns with a probe size of 3.6?nm (full width half maximum) were then taken from La0.48Ca0.52MnO3, an example of which is shown in figure 1a. None showed a wavevector of q?=?1/2 or 1/3 and instead a uniform modulation of wavevector q?=?0.473?±?0.005 was observed. This agrees with the wavevector q?=?0.468?±?0.003 obtained from a selected area diffraction pattern taken from the same grain using an aperture of diameter 500?nm (as seen in the object plane) shown in figure 1b.Figure 1. Diffraction patterns on different length scales. (a) A convergent beam electron diffraction (CBED) pattern from a La0.48Ca0.52MnO3 grain at 90?K. Parent reflections are delineated by solid lines. The random mixture model predicts that x?=?1/2 sub-units should outnumber x?=?2/3 sub-units by a factor of 7.3 such that the latter are on average 7.8?nm apart. The CBED probe FWHM is 3.6?nm (6.7 room temperature unit cells). In all CBED patterns acquired, q?=?1/2 (indicated by arrows and dashed lines) or q?=?1/3 modulations are never seen and instead q?=?0.473?±?0.005. (b)?Diffraction pattern from the same grain taken with a selected area aperture of diameter 500?nm (as measured in the object plane). The wavevector q?=?0.468?±?0.003. This matches the CBED value within experimental error.This shows that the ‘charge ordering’ modulation observed in La1-xCaxMnO3 does not form a ‘chaotic’ structure (in the sense defined by Bak [4]) but has a uniform periodicity on all lengthscales and this periodicity need not be an integer multiple of the undistorted unit cell. This result throws doubt on the concept of charge order as localised Mn 3+ and Mn 4+ ions as discussed in section 5.4.2.Variation of wavevector with dopingIt is well known that the structural modulation observed when charge ordering occurs is found to have a wavevector related to the composition by q?=?1-x at low temperatures [7]. Figures 2a and 2b show diffraction patterns for compositions x?=?0.52 and 0.71 taken at 90?K. First order (m?=?1) superlattice reflections can be clearly seen at g?±?q and weaker, second order (m?=?2) reflections can be seen at g?±?2q. Figure 2c is a plot of the wavevector observed at 90?K versus composition, constructed from these and other diffraction patterns. It can be seen that the wavevector appears to follow the general trend q?=?1-x but there are significant departures from this relationship. It should be noted that all but one of the points show wavevectors which are smaller than the wavevector predicted by the formula q?=?1-x possibly suggesting that the wavevector tends towards the value given by 1-x as the temperature is lowered but the transition may be inhibited and the wavevector may remain frozen in a non-equilibrium state as discussed in section 4.8.Figure 2. Wavevector at 90?K observed at different doping concentrations by electron diffraction. (a) A diffraction pattern from the [010] zone axis of La0.48Ca0.52MnO3. Superlattice reflection can be observed at g?±??q and g ±??2q. (b) A similar diffraction pattern for La0.29Ca0.71MnO3. (c) A graph of wavevector against Ca concentration. Circles with error bars are from this work, squares from Chen et al.[10] and triangles from Chen et al.[7]. All but one of the points lie below the expected value indicated by the straight line. The errors in the measurement of the wavevector were determined by measuring the wavevector at several different places in the same diffraction pattern and finding the standard error of these measurements. The error in the Ca concentration is that estimated for the macroscopic concentration. The variation in concentration from one grain to another may be greater than this.4.3.TwinningSharp and strong reflections with wavevector q were observed at 90?K at all compositions investigated between 0.5?=?x?=?0.71. In this work and that of other authors [11], q is found to be nearly collinear with the a* axis of the crystal. The diffraction pattern in figure 3a appears to show the modulation wavevector parallel to both the a* and c* directions but as shown by the dark field images in (b) and (c), this is caused by perpendicular twins contributing to the diffraction pattern.Figure 3. Diffraction pattern and dark field images taken from La0.48Ca0.52MnO3 at 90?K. (a) appears to show the modulation wavevector in both the a* and c* directions. However, taking dark field images from each type of reflection (at positions 002?+?q) shown in (b) and (c) demonstrates that the pattern seen in (a) is in fact the superposition of two crystallographic twins related by a 90° rotation.Several features of the twins are immediately apparent. The first is the speckled contrast observed in the dark field image. This appears to indicate that some regions of the crystal are charge ordered (and appear bright) whilst others are not charge ordered (and appear dark). In fact it is shown in section 4.7 that at this temperature the whole area is charge ordered and the speckle probably arises from small out of plane variations of the wavevector. It can also be seen from these images that the straight sections of the twin boundaries are approximately parallel to the [101] and [01] directions as has also been observed by Chen and Cheong [12]. The twin boundaries also have large non-crystallographic curved sections indicating that the energy of the boundary is low.It has been shown [11] that these twins are not caused by charge ordering and are present at room temperature. The twins are related by a 90° rotation of the a and c axes but are very difficult to distinguish parallel to the [010] zone axis at room temperature because a and c are very similar in magnitude. However, Wang et al.[11] point out that they can be distinguished by taking diffraction patterns parallel to the [100] and [001] zone axes. They find that the charge ordering wavevector always lies close to a* and is never seen parallel to the c* direction.4.4.Transverse modulationWhen tilting the specimen away from the zone axis in order to perform dark field imaging, it was found that the superlattice reflections became very weak if the specimen was tilted about the a-axis to give a strong systematic row along a* as illustrated in figure 4. Equation (3) shows that if the amplitude of the modulation, A, is perpendicular to a*, then superlattice reflections along the a* direction are kinematically forbidden. Since q is parallel to a*, figure 4 shows that the amplitude of the modulation is approximately perpendicular to the wavevector and thus the wave has a predominantly transverse nature. This observation has also been made by other authors using neutron and X-ray diffraction [3].Figure 4. Transverse modulation in La1-xCaxMnO3. (a) A diffraction pattern in which the crystal has been tilted to give a strong systematic row parallel to a* to minimise the effects of double diffraction. Superlattice reflections can be seen apparently along both a* and c* directions but this is in fact due to twinning (see figure 4). It can be seen that although both sets of reflections off the row have comparable intensities (see for example the reflections around 20 2), reflections perpendicular to the a* row are far stronger than those along the row as shown by the white arrows. (b) Summary of this information. Filled circles show kinematically allowed reflections and empty circles show kinematically forbidden reflections.4.5.Rotation of the wavevectorAlthough the wavevector q usually lies very close to the a* direction, significant departures have been observed. The largest departure observed in this investigation was in La0.29Ca0.71MnO3 as shown by the diffraction pattern in the inset of figure 5. It shows that q is rotated about the [010] axis by an angle of 1.9° from a* towards the c* direction. This can also be seen in the high resolution image from the same area if viewed at a glancing angle parallel to the superlattice rows. The image also demonstrates that the rotation is intrinsic to the modulation and not caused by faulting. This also throws doubt on the standard models for charge ordering which predict that the wavevector is strictly parallel to a*[6] as discussed in section 5.Figure 5. High-resolution image taken from La0.29Ca0.71MnO3 at 90?K parallel to the [010] zone axis. The inset shows a diffraction pattern from the same region. The wavevector q?=?0.27 and is rotated 1.9° from the a* direction. Note that 2nd order superlattice reflections can be seen at 2q.4.6.Variation of the wavevector with temperatureFigure 6a shows the variation with temperature of the wavevector observed in La0.48Ca0.52MnO3. Other authors have remarked on the large reduction in wavevector that occurs in the manganites when compared with other charge density wave systems [10]. Surprisingly, the modulation is still present at room temperature (293?K) as shown in figure 6b, well above the charge ordering transition temperature of 230?K obtained from resistivity and magnetisation measurements performed on this particular sample [13], although the amplitude of the modulation is greatly reduced.Figure 6. Variation of wavevector with temperature in La0.48Ca0.52MnO3. Measurements were taken first on warming from 90?K to 240?K and then on cooling back to 90?K. The point at 293?K is from a different region of sample. The errors in the measurement of the wavevector were determined by measuring the wavevector at several different places in the same diffraction pattern and finding the standard error of these measurements. (b) A diffraction pattern from La0.48Ca0.52MnO3 taken at room temperature. (c) The same diffraction pattern after high pass filtering to enhance the superlattice reflections (two of which are marked by arrows). The region of specimen from which this pattern was recorded was twinned as in figure 3 and so superlattice reflections can be seen in two perpendicular directions.It can also be seen that the superlattice reflections are broader at room temperature than at 90?K. Estimates of the correlation length of the charge ordered regions can be found by finding the full width half maximum of the charge order reflections and finding the reciprocal of this quantity. This analysis showed that the correlation length at 90?K is 34?±?7?nm and at 297?K it is 3.6?±?0.2?nm.Surprisingly, dark field images discussed in section 4.7 showed that the correlation length does not reflect the sizes of charge ordered clusters within an unmodulated background. Instead, the entire grain appears to be charge ordered and the correlation length indicates the distance over which the wavevector remains constant.In charge density wave systems and other systems that show modulated structures, the wavevector q is frequently seen to change in discrete jumps as a function of temperature (called a ‘staircase’). The conventional explanation for this is based on the results of the Frenkel–Kontorova model which predicts that the wavevector locks into plateaus of stability which occur when q is a rational fraction of the reciprocal lattice vector [4]. As far as the system studied in this investigation is concerned, this does not appear to be the explanation for the structure of the q(T) diagram shown in figure 6a.In the warming curve, it appears that the wavevector has locked-in at low temperatures and has the value q?=?0.438?±?0.001. The nearest rational fraction value to this is 7/16?=?0.4375.When cooled again, the modulation no longer stabilises at q?=?0.438?±?0.001 but goes straight through this value, reaching q?=?0.446?±?0.001 (for which the nearest rational fraction is 17/38?=?0.447). We do not expect a plateau which was formerly stable to become unstable during thermal cycling. We conclude therefore that the graph of q(T) does not follow a staircase structure whereby the wavevector locks into plateaus of stability as the temperature is changed. Instead, we believe that the modulation wavevector tends towards q?=?1-x via a nucleation and growth process as the temperature falls but becomes pinned at non-equilibrium wavevectors giving rise to apparent plateaus. Pinning of the wavevector at non-equilibrium values also explains why different regions of the same sample show different wavevectors as shown in figure 2c. This explanation is consistent with the fact that changes in wavevector occur by a nucleation and growth process as discussed in section 4.8.4.7.Spatial variations of the wavevectorA very accurate way to measure spatial variations in the wavevector is to take dark field images which include not only the wavevector q but also the conjugate wavevector g – q in the objective aperture (see figure 7a). Interference fringes are then seen with a wavevector e?=?g-2q. It should be noted that this procedure is often performed accidentally if the two superlattice reflections lie close to one another and the resulting interference fringes have been mistaken for soliton arrays [14]. This method is very sensitive to small changes in the angle that the wavevector makes with the parent reflections and any changes in the magnitude of q. It can be seen from the figure that the closer q approaches q?=?1/2, the higher this sensitivity becomes.Figure 7. Spatial variations of the wavevector q in La0.48Ca0.52MnO3 at 90?K. (a) The definition of the angles in the diffraction pattern and the placement of the objective aperture. (b) A region from the dark field image with fringes spaced by 1/(g-2q). Notice the changes in magnitude and direction of the wavevector that occur on a scale of tens of nanometres. (c) A diffraction pattern from the same area.Experimentally one observes the wavelength of the interference fringes ??=1/? and the angle ? as defined in figure 7. These are related to q and ? via:Figure 7b shows a region from a grain of size about?0.5?µm in La0.48Ca0.52MnO3. Statistics were gathered from the whole grain using images like figure 7b, by taking fast Fourier transforms from 80 regions of size 50?nm. This gave the result that the average wavevector was ?=?0.450 and the standard deviation of q was ?q?=?0.004. For this sample the average wavevector was parallel to the a* axis and varied about this mean orientation with an angular standard deviation of ???=?0.56°.The most noticeable feature of the dark field images is that they have dark and bright patches of size 10–50?nm. It may be thought at first that the bright patches refer to the modulated structure and the dark patches are areas which are not modulated but this proved not to be the case. Convergent beam electron diffraction patterns with a probe size of 4?nm were taken from 20 regions and although the black patches can be as large as?50?nm in size, the diffraction patterns always showed the modulation. Taking dark field images and tilting the specimen by about 1° showed that dark patches could become bright and bright patches dark as shown for La0.5Ca0.5MnO3 in figure 8a and 8b. Given that the angle made by the modulation can vary in the ac plane by about 0.5°, it seems reasonable to assume that it can also vary by a similar amount in the b direction. If this is the case, it explains the mottled appearance of the dark field image. Different patches have different components of the wavevector in the b direction and thus have slightly different orientations for exciting strong Bragg diffraction. Small tilts of the crystal excite different patches as illustrated schematically in figure 8c.Figure 8. Dark field images from La0.5Ca0.5MnO3 at 90?K. (a) A dark field image taken from the 002?+?q superlattice reflection. Bright regions are outlined. (b) A dark field image from the same region using the same reflection but tilted slightly (~?1°) with respect to (a). The same regions are outlined but it can be seen that they are no longer bright. An explanation for this effect is illustrated schematically in (c) and (d). The left hand panel of (c) shows a cross section of a diffraction pattern with two domains contributing to it. One domain has a wavevector q1 oriented along a*, the other has a wavevector q2 with a small b* component. The specimen is oriented so that the Ewald sphere intersects q1 and so the domain containing q1 is bright as illustrated in the right hand panel. In (d), the specimen is tilted slightly so that the domain containing q2 is bright.4.8.Changes in wavevector by nucleation and growthIn section 4.6, we postulated that the changes in wavevector take place by a nucleation and growth process. Figure 9 confirms this hypothesis by showing two large areas (>0.1?µm) with different wavevectors coexisting in the same crystallite of La0.5Ca0.5MnO3 at 90?K. The interface between these two areas is marked by a dashed line and we postulate that if the sample were further cooled, this interface would move in the direction shown by the arrow so that the area with wavevector q?=?0.48 would grow at the expense of the area with wavevector q?=?0.46 and that, if kinetics allowed, further wavevectors would nucleate and grow until the whole grain eventually had a wavevector q?=?0.5 as given by the composition.Figure 9. Coexistence of two different wavevectors in La0.5Ca0.5MnO3 at 90?K. The bulk of the grain has a wavevector of q?=?0.46 but a wavevector of q?=?0.48 is observed at one edge of the grain and the interface between these two is shown by the dashed line. If the sample were further cooled, we postulate that the interface would propagate across the grain as indicated by the arrow.We notice that the area with wavevector q?=?0.48 occurs near the edge of the grain which is normal to the wavevector. The diagrams in figure 10 illustrate why this might be. Panel (a) shows the case of a new wavevector growing within the bulk of the material. There is an energy cost associated with forming an interface between the regions with different wavevector. Panel (b) shows that there is an energy saving if the new wavevector nucleates at the edge of the grain since the size of the interface is reduced. There is, however, a large mismatch between the two lattices if the interface runs parallel to the wavevector. Panel (c) shows that the minimum energy configuration is for nucleation to occur at an edge or linear defect and for the interface to be normal to the wavevector.Figure 10. Schematic diagram of the nucleation of a new region with a different wavevector. The square represents a grain and the black and white lines show peaks and troughs of the modulation. (a) The new wavevector forming as a nucleus within the bulk of the grain. (b) The new wavevector nucleating at an edge with its wavevector parallel to the edge. (c) The new wavevector nucleating at an edge with its wavevector perpendicular to the edge.4.9.Defects in the charge density waveWhere a region with a particular wavevector meets another region with a different wavevector, there must be misfit ‘dislocations’ at the interface. Such dislocations are frequently observed in dark field images and figure 11 shows two examples together with a schematic representing this type of ‘dislocation’ in three dimensions. Such ‘dislocations’ seem to be intrinsic to the modulated structure since bright and dark field images from parent lattice reflections do not show dislocations or similar faults in the same areas.Figure 11. Dislocations in the structural modulation. (a) and (b) show faults reminiscent of edge dislocations in the structural modulation observed in La0.5Ca0.5MnO3 at 90?K and (c) shows a schematic of this type of fault in three dimensions. Dotted lines show the position of minima in the wave and solid lines the position of maxima. A dislocation occurs where two waves of different wavelength meet. Arrows indicate the ‘nodes’ of the dislocations.5.Discussion and conclusionsThe idea that the ‘charge ordering’ modulation should be composed of supercells which are an integer multiple of the undistorted unit cell is a direct result of assuming that the modulation is the result of a localisation of Mn 3+ and Mn 4+ ions. We have shown that the structural modulation need not be in registry with the parent lattice and it is therefore likely that the accompanying charge modulation is not in registry with the parent lattice as well. If this is the case, then individual manganese ions should adopt a range of valencies, not merely integer values. The possibilities go beyond this: it is not necessarily the manganese ions which have the mixed valency. Indeed, Hartree–Fock simulations which determine the charge distribution from the structure of the material suggest that it is the oxygen valency which varies [15]. Nor is it necessary that an entire electron be transferred during the charge ordering transition and valence bond sums by Daoud-Aladine et al.[16] suggest that it may in fact be as little as 1/100 of the electron charge.The facts that the modulation need not be in registry with the parent lattice and also that the modulation wavevector need not be collinear with the a* direction lead us to think that the modulation is weakly coupled to the parent lattice. We therefore consider it more likely that the structural distortion is the result of a charge density wave. The mechanism usually used to account for the occurrence of a charge density wave is Fermi surface nesting although other mechanisms have been suggested for different materials all of which are labelled as charge density wave systems [17, 18, 19]. We note that resistivity versus temperature curves for ‘charge ordered’ manganites do not show a metal–insulator transition but then neither do many curves for other charge density wave systems (see, for example, Fogle and Perlstein [20], Jiang et al.[21], Lue et al.[22], Naito and Tanaka [23], Rossnagel et al.[24]).Our results suggest that the manganites may be pseudogap [25] materials where electrons are present at the Fermi level but are small in number, in agreement with observations made by optical spectroscopy for La1-xCaxMnO3, 0.5