Strain heterogeneity and magnetoelastic behaviour of nanocrystalline half-doped La, Ca manganite, La_0.5Ca_0.5MnO₃

Starting materials for preparation of pellets suitable for elasticity measurements were nano- and micro-crystalline powders of La_0.5Ca_0.5MnO₃ prepared originally for the studies of Sarkar et al [2] and Lahiri et al [6]. Their stoichiometry had been checked by ICPAES and found to have a La : Ca : Mn ratio of 0.507 : 0.495 : 1. Portions of four of the six powders (samples A, B, E and F in table 1 of Sarkar et al [2]) were pressed into pellets using high pressure field assisted rapid sintering (HP-FARS) apparatus following the procedure described by Anselmi-Tamburini et al [43, 44]. This procedure provides cylindrical pellets 5 mm in diameter and 1 mm thick, with masses of ∼0.12 g. Table 1 summarizes the sintering temperatures and pressures used for the pellets, as well as grain sizes of the starting powders specified by Sarkar et al [2].

It was of concern that the crystallite sizes might have been modified by the hot pressing. After completion of the elasticity measurements, therefore, the dense pellets were reground to powder and x-rayed for comparison with the starting powders. A θ–2θ Bragg–Brentano parafocusing PANalytical X'Pert PRO diffractometer, equipped with a multi-channel X'Celerator detector was used (Cu Kα, with λ = 0.15418 nm). Data were collected in the range 5–120° 2θ with a step size of 0.0084° 2θ and a counting time of about 30 s/step. The divergence slit was fixed at 0.5°. Diffraction patterns obtained in this way are illustrated in figure 1 and contain only peaks expected from orthorhombic perovskite. Table 1 gives the crystallite size of the sintered pellets determined by a total pattern refinement using MAUD software [45], which can separate the crystallite size from the RMS-microstrain effect in broadened powder diffraction peaks. The simple isotropic model was used for the evalution of crystallite size. Similar crystallite sizes to those reported by Sarkar et al [2] were obtained for samples A and B but much smaller sizes were obtained for samples E and F. The good agreement for the fine grained samples is due to the fact that the same method was used (i.e. powder diffraction line broadening). However, Sarkar et al [2] used transmission electron microscopy to measure grain size of the coarser samples, and grains observed in this way were presumably composed of aggregates of smaller crystallites or affected by grain mosaicity. The most significant point here is that crystallite sizes remained in the nano range after the hot pressing.

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Differences in lattice parameters between bulk and nano samples of La_0.5Ca_0.5MnO₃ reflect differences in the spontaneous strains associated with octahedral tilting, charge ordering and magnetic transitions. Two characteristic shear strains may be defined in terms of the lattice parameters of the Pnma structure, a_Pnma, b_Pnma and c_Pnma, with respect to the lattice parameter of the cubic reference structure, a_o, (following [46–48]):

$$\begin{eqnarray} \fl e_{{\rm tx}}& =\frac{1}{\sqrt 3}\left( {2e_{1} +e_{2} +e_{3}} \right)\nonumber\\ \fl &=\frac{1}{\sqrt 3}\bigg( 2\left( {\frac{\frac{b_{{\rm Pnma}}}{2}-a_{{\rm o}}}{a_{{\rm o}}}} \right)\nonumber\\ \fl &\quad-\left( {\frac{\frac{a_{{\rm Pnma}}}{\sqrt 2}-a_{{\rm o}}}{a_{{\rm o}}}} \right)-\left( {\frac{\frac{c_{{\rm Pnma}}}{\sqrt 2}-a_{{\rm o}}}{a_{{\rm o}}}} \right) \bigg) \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&\fl e_{{\rm 4}} =\left( {\frac{\frac{a_{{\rm Pnma}}}{\sqrt 2}-a_{{\rm o}}}{a_{{\rm o}}}} \right)-\left( {\frac{\frac{c_{{\rm Pnma}}}{\sqrt 2}-a_{{\rm o}}}{a_{{\rm o}}}} \right) \end{eqnarray} \tag{ 2 }$$

where the value of a_o can be taken as (a_Pnmab_Pnmac_Pnma/4)^1/3. Variations of these shear strains as a function of temperature are shown in figure 2 for samples A and F, using original lattice parameter data of Sarkar et al [2]. For comparison, shear strains for bulk samples have been added using the lattice parameters given by Radaelli et al [7]. These reproduce the substantial differences in structural evolution between the nanocrystalline sample A and the more coarsely crystalline sample F already reported by Sarkar et al [2], but also show differences between sample F and the bulk sample of Radaelli et al [7]. In particular, the onset of changes in e_tx occurs at a lower temperature during heating of sample F than in the bulk sample, and there is an additional anomaly in e₄ at ∼130 K. Unpublished high resolution neutron powder diffraction data of Chatterji and co-workers from sample F do not show the anomaly in e₄ and show the intensity of antiferromagnetic ordering reflections decreasing to zero at ∼155 K. For present purposes, it is assumed that sample F has properties that are intermediate between the nano and bulk materials, consistent with the new determination of crystallite size as ∼136 nm (table 1).

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The lattice parameter data of Sarkar et al [2] are not of sufficient resolution to determine the evolution of volume strains because they are substantially smaller than the shear strains. Some idea of their likely magnitude and sign is given by data for other sources, however. For Pr_0.48Ca_0.52MnO₃ there is a small but significant positive volume strain of up to ∼0.002 accompanying charge ordering [50]. On the other hand, ferromagnetic ordering in La_0.75Ca_0.25MnO₃ is accompanied by a negative volume strain which, from the data in figure 1 of Radaelli et al [37] is estimated to reach ∼−0.002. The antipathetic relationship between ferromagnetism and charge order/orbital order must at least be enhanced by the opposite sign of these volume strains.

Magnetic properties of the four sintered pellets used for RUS measurements were characterized using a MPMS-XL 7 T Superconducting Quantum Interference Device (SQuID) magnetometer. Many details of this measuring system are given by McElfrish et al [51]. Each sample was held in an arbitrary orientation within the sample chamber and two types of measurements made. Magnetic moments were determined in nominally zero field and in a 5 T field at 0.01 K intervals between 5 and 300 K, during cooling and then heating. The magnetic moment was then measured in hysteresis loops with 0.03 kOe intervals between +3 and −3 kOe at temperatures of 300, 250, 200, 150, 75 and 5 K in a cooling sequence.

Figure 3 shows data for magnetic moment during cooling and then heating in zero field and at 5 T. The magnetic properties of samples A and B are closely similar, consistent with the determination of crystallite size (34 nm and 42 nm, respectively). Both show patterns of evolution expected for a paramagnetic–ferromagnetic transition, with slight hysteresis, and only a further additional small anomaly near 42 K. If the small hysteresis is not due to thermal lag in the instrument, values of T_c for cooling and heating at zero field would be ∼268 and ∼284 K for A and ∼270 and ∼278 K for B, as estimated from the expanded view shown in figure 3(c). Both at zero field and under the influence of a 5 T field, there are irregularities in the magnetic moment at temperatures between 300 and 150 K during cooling but these are not present during heating (figures 3(a) and (b)). Sample F (136 nm) does not show the characteristic pattern of ferromagnetic ordering followed by a ferromagnetic–antiferromagnetic transition reported elsewhere for bulk samples in either zero field or in an applied field (e.g. [1, 12, 37, 52–55]). Instead it appears to have a paramagnetic–ferromagnetic transition near 260 K in zero field (T_c ∼ 257 K during cooling and ∼262 K during heating, as estimated from figure 3(c)), followed by a very obvious hysteresis pattern with limits of ∼45 and ∼155 K which is perhaps analogous to the hysteresis between limits of 125 and 180 K that is attributed to the ferromagnetic–antiferromagnetic transition in bulk samples (e.g. [1, 55]). The characteristic pattern for bulk samples involves lower moments due to the antiferromagnetic structure, however, while sample F has consistently increasing moment with falling temperature. There is a further increase in moment near 42 K, as opposed to the decrease seen in zero field data for samples A, B and E. Sample E (75 nm) has variations different from those of A, B, F, but still with a trend consistent with a paramagnetic–ferromagnetic transition near 270 K in zero field (T_c ∼ 267 K during cooling and ∼275 K during heating, estimated from figure 3). Details of the anomaly near 42 K are considered in section 3.

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Figure 4 shows hysteresis curves for samples A, E and F. Data for sample B are not shown as they are closely similar to those of sample A. Both sets are consistent with paramagnetism at 300 K and ferromagnetism at T ⩽ 250 K. Sample F appears to be paramagnetic at 300 and 250 K, but the pattern at 5 K is more nearly typical of ferromagnetism, though saturation magnetization is not reached at 3 kOe. At intermediate temperatures (200, 150, 75 K) the loops also remain open, which is suggestive of metamagnetic transitions and/or the response of a mixture of phases with different magnetic properties. Typical paramagnetic behaviour of sample E is limited to 300 K. At 250, 200 and 5 K the forms of the hysteresis curves are consistent with ferromagnetism, while the loops remain open to high fields at 150 and 75 K.

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The physics of RUS has been described by Migliori and Sarrao [56], and the Cambridge helium flow cryostat used to collect spectra between ∼5 and 300 K is described in McKnight et al [57]. The hot pressed discs of samples A, B and E (∼0.12 g) were used directly. Pellet F was broken during depressurization and the piece used for RUS measurements was about half of the disc (0.055 g). Each disc was placed with its faces directly in contact with the transducers in an atmosphere of a few mbar helium to allow heat transfer to the sample. Spectra were collected in a sequence of 30 K intervals during cooling from 280 to 10 K, with a 20 min dwell time at each temperature to allow thermal equilibration, followed by heating up to 305 K in 5 K steps with the same dwell time. Pellet F was investigated in more detail in a second run through the ranges 265–190 K (5 K steps during cooling and heating) and 20–60 K (2 K steps during heating). A representative section of the spectra collected during heating is shown in figure 5. Individual spectra contained 65 000 data points in the frequency range 0.05–1.2 MHz and were transferred to the software package IGOR PRO (Wavemetrics) for analysis. Frequency, f, and the width at half maximum height, Δf, of selected peaks were determined by fitting them with an asymmetric Lorentzian function. Acoustic resonances of a small polycrystalline oxide sample are dominated by shearing rather than breathing motions and f² for the peaks scales closely with the shear modulus in most cases (e.g. [58]). Anelastic loss is characterized in terms of the inverse mechanical quality factor, Q⁻¹, as given by Q⁻¹ = Δf/f.

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In order to compare the elastic behaviour of separated nanocrystalline grains with that of the hot pressed pellets, the same four starting powders that had been used for the hot pressed pellets were also mixed with CsI powder that had been dried at ∼110 °C. Mixtures of ∼65 wt% CsI + 35 wt% sample powder for B, E and F and ∼84 wt% CsI + 16 wt% powder for sample A were ground together by hand in an agate mortar and pestle and then mechanically pressed in an infrared pellet press to ∼31 MPa. This produced discs 13 mm in diameter, ∼2 mm thick and with mass ∼1.17 g for B, E, F, and ∼1.31 g for A. The discs were found to give measurable resonance spectra when placed with their large flat faces in contact with the transducers. RUS spectra containing 130 000 points in the range 0.05–1.2 MHz were collected in 5 K steps during cooling and heating between 290 and 140 K, with a settle time of 20 min for thermal equilibration at each temperature. As illustrated by a representative portion of spectra from the CsI pellet containing powder of sample F in figure 6, a characteristic feature of all spectra collected in this way is hysteresis, such that the pellet is elastically slightly softer during heating than during cooling. This is believed to be due to some relaxation of grain boundaries between the two phases in the pellets, but does not detract from the internal consistency in the sense that the influence of the sample powder can be clearly seen in both the cooling and heating sequences. Figure 6 shows resonance peaks which have the same break in trend near 225 K that is seen in spectra from the hot-pressed pellets.

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As set out in detail in Carpenter et al [50, 60], with respect to Pnma as the parent space group the order parameter for the commensurate structure of half doped manganites transforms as the irreducible representation (irrep) X₁. This has two components, such that (a,0) gives the P2₁/m structure and (a, a) gives Pnm2₁. The transition would be improper ferroelastic or co-elastic, respectively. However, treatment in terms of a single order parameter conceals the separate contributions which might develop from charge ordering, cooperative Jahn–Teller distortions and octahedral tilting. With respect to a parent cubic structure with space group Pm ${\rm \bar{{3}}}m$ these transform as Σ₂ (charge order), ${\rm M}_{2}^{+}$ and $\Gamma_{3}^{+}$ (cooperative Jahn–Teller), ${\rm M}_{3}^{+}$ and ${\rm R}_{4}^{+}$ (octahedral tilting). Shear strains e_tx and e₄ couple with each order parameter component, q, as set out in equations (10) and (11) for Pnm2₁ in Carpenter et al [50]. Coupling of e_tx with the $\Gamma_{3}^{+}$ order parameter is bilinear, i.e. with the form λe_txq where λ defines the coupling strength, and the other couplings are linear quadratic, i.e. λ eq². With respect to anomalies of shear elastic constants, the linear-quadratic contributions will give a stepwise softening below the transition point (improper ferroelastic or co-elastic), while the bilinear term will give non-linear softening as the transition point is approached from both above and below (pseudoproper ferroelastic) [61–64]. From figure 2 it is clear that the dominant strain in coarse grained samples is e_tx and from figure 7 it is clear that the dominant softening mechanism has the form expected for a pseudoproper ferroelastic transition. The dominant spontaneous strain must therefore be due to the bilinear coupling between e_tx and the $\Gamma_{3}^{+}$ order parameter which gives elastic softening of the form

$$\begin{equation} \left( {C_{11} -C_{12}} \right)=\left( {C_{11}^{{\rm o}} -C_{12}^{{\rm o}}} \right)\left( {\frac{T-T_{{\rm c}\Gamma}^{\ast}}{T-T_{{\rm c}\Gamma}}} \right). \end{equation} \tag{ 3 }$$

$T_{{\rm c}\Gamma}^{\ast}$ is the transition temperature for a transition driven by the $\Gamma_{3}^{+}$ order parameter and is related to the unrenormalized critical temperature T_cΓ according to (from [60])

$$\begin{equation} T_{c\Gamma}^{\ast} =T_{c\Gamma} +\frac{\lambda^{2}}{a\textstyle{1 \over 2}\left( {C_{11}^{{\rm o}} -C_{12}^{{\rm o}}} \right)}. \end{equation} \tag{ 4 }$$

In equations (3) and (4), λ is the coefficient for the bilinear coupling term, a is the coefficient for the second order term in a Landau expansion and $\left( {C_{11}^{{\rm o}} -C_{12}^{{\rm o}}} \right)$ is a shear elastic constant for the cubic parent structure without influence from the phase transition. The shear modulus depends also on other shear elastic constants, specifically C₄₄ in the case of a cubic phase, but the bulk modulus is expected to display a step wise softening due to linear quadratic coupling of volume strain with any of the order parameters, as shown schematically in figure 1 of Carpenter et al [60]. Equation (4) has the same form as derived from consideration of Jahn–Teller theory, and softening of the shear modulus above T_co has been used to estimate a value of $T_{c\Gamma}^{\ast} -T_{c\Gamma} ~=2\,{\rm K}$ for La_0.5Ca_0.5MnO₃ [27].

The strain evolution and elastic properties obtained by RUS for the incommensurate structure of La_0.5Ca_0.5MnO₃ are similar in form to those previously obtained for Pr_0.48Ca_0.52MnO₃, which has a nearly identical incommensurate structure stable below ∼235 K [65–67]. Comparison of figure 7 of Carpenter et al [50] with figure 2 of the present work shows that values of |e_tx| for bulk samples increase by ∼0.025 through T_co, and | e₄| changes by ∼0.003 in both cases. This implies that the strength of coupling between shear strains and the Σ₂, ${\rm M}_{2}^{+}$, $\Gamma_{3}^{+}$, ${\rm M}_{3}^{+}$ and ${\rm R}_{4}^{+}$ order parameter components is broadly similar, if not the same, for bulk samples of both materials. Data for the shear modulus and Q⁻¹ are reproduced in figure 10 to show that, while the form of the elastic anomaly in Pr_0.48Ca_0.52MnO₃ is the same as of f² for sample F, the magnitude of softening as the transition is approached from above is less, as is the recovery below T_co. This is consistent with a larger value of $T_{c\Gamma}^{\ast} -T_{c\Gamma}$ (57 ± 24 K) estimated by applying equation (4) to the data for Pr_0.48Ca_0.52MnO₃ [60], as opposed to 2 K referred to above for La_0.5Ca_0.5MnO₃. If the coupling strength (λ) is about the same, it follows that the effective values of a and/or $\left( {C_{11}^{{\rm o}} -C_{12}^{{\rm o}}} \right)$ in equation (4) would be larger. In other words, replacing Pr by La may cause the entropy associated with the Jahn–Teller component of the transition and/or the bare elastic constants to increase. This remains somewhat speculative, however, because although values of the shear modulus reported by Zheng et al [26]. are comparable with those obtained here, the ultrasonic data of Zheng et al [30]. show a much steeper recovery below T_co and are rather similar to those of Pr_0.48Ca_0.52MnO₃. It is possible that data for the elastic properties of sample F are not fully representative of those of a bulk sample of La_0.5Ca_0.5MnO₃.

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Comparison of Q⁻¹ in figure 10 shows that a common feature for all the samples is the increase in loss below ∼100 K. The peak in Q⁻¹ at ∼75 K for Pr_0.48Ca_0.52MnO₃ was tentatively ascribed to a freezing process with an activation energy of ∼5–10 kJ mol⁻¹ which might be related the motion of polarons as part of the response of the incommensurate structure to the application of external stress [60]. Essentially the same loss peak appears to occur at ∼85 K in data from sample F and, even though Q⁻¹ variations are not constrained, there is an increase in loss through the same temperature range for the other samples of La_0.5Ca_0.5MnO₃. However, two notable differences remain. Firstly, the 40 K anomaly is seen only in the data for La_0.5Ca_0.5MnO₃. As Pr_0.48Ca_0.52MnO₃ is purely antiferromagnetic at low temperatures, it follows that this anomaly is due to the ferromagnetic phase of La_0.5Ca_0.5MnO₃. Secondly, Q⁻¹ at T > T_co drops to low values for Pr_0.48Ca_0.52MnO₃ but remains high for La_0.5Ca_0.5MnO₃, signifying that some degree of relaxational disorder with strain coupling remains at high temperatures in hot pressed pellets of the latter.

Suppression of the changes in shear strain at crystallite sizes of ∼30–40 nm is accompanied by the expected reduction in the magnitude of the recovery of the shear modulus below T_co but not by complete elimination of the elastic anomaly. Softening as T → T_co from above appears to remain almost the same but recovery occurs in the form of a broad, rounded minimum rather than the concave-down form expected for pseudoproper ferroelastic behaviour. The macroscopic shear strain e_tx arising from coupling with the $\Gamma_{3}^{+}$ order parameter is suppressed by reducing grain size, but some local ordering apparently still remains and a convenient analogy is perhaps with the difference between a ferroelectric transition and freezing of dynamical polar nano regions in a relaxor ferroelectric. The signature of the latter with respect to strain and elastic properties is a frequency-dependent minimum in the elastic constants above the freezing point and Vogel-Fulcher dynamics through it. This is seen in RUS data for Pb(Mg_1/3Nb_2/3)O₃, for example [68]. There are hints in the present data for samples A and B of dispersion with respect to resonance frequencies, such that the same may apply also to La_0.5Ca_0.5MnO₃, but the quality of the data is not quite sufficient to determine this definitively. Evidence of local Jahn–Teller distortions of MnO₆ octahedra in nanocrystalline grains is provided also from XAFS investigation of sample A by Lahiri et al [6].

Variations in elastic properties of the hot pressed pellets appear to be the same as for the powders pressed into pellets with CsI, suggesting that the measured elastic properties are intrinsic to the individual grains. There is, however, some contrast in magnetic properties between the pellets produced by hot pressing for the present work relative to those reported by Sarkar et al [2] for pellets produced by sintering at one atmosphere. The magnetic moment shown in figure 8 of Sarkar et al [2] for F increases below ∼250 K, due to ferromagnetic ordering, followed by a decrease with hysteresis below ∼200 K characteristic of the ferromagnetic–antiferromagnetic transition. Data for the pellet of F shown in figure 3 show a steady increase in magnetic moment below ∼260 K but no further reduction and a hysteresis interval at lower temperatures. The difference must be due to near surface effects when grains are in close proximity, since hot pressing will produce a denser ceramic with a higher proportion of grain boundaries in close contact.

There are no elastic anomalies in data collected above ∼40 K from the hot pressed pellets which correlate with changes in magnetic moment shown in figure 3 or from the CsI pellets which correlate with changes in magnetic moment shown in figure 8 of Sarkar et al [2]. This confirms the conclusion from the incommensurate structure of Pr_0.48Ca_0.52MnO₃ that coupling between shear strain and the magnetic order parameter is weak.

Magnetic ordering in samples A, B and E is predominantly ferromagnetic. In detail the transition from para- to ferromagnetism in the hot-pressed pellets occurs near 270 K, with first order character implied by the hysteresis between heating and cooling (figure 3(c)). This mirrors the first order character of the paramagnetic–ferromagnetic transition seen in bulk samples of La_0.7Ca_0.3MnO₃ (e.g. [69–71]), but there are contrasting features as well. Most notably, a bulk sample of La_0.7Ca_0.3MnO₃ does not show the elastic softening as T → T_c from above and there is a steep increase in stiffness at the transition point [69]. The same pattern of stiffening is seen for both longitudinal and transverse acoustic waves in La_0.67Ca_0.33MnO₃ [38, 39] and resembles the pattern expected from biquadratic coupling of strain with the magnetic order parameter (λe²m²), i.e. stiffening (or softening, depending on the sign of the coupling coefficient, λ) which scales with m² as seen in the case of YMnO₃ [72]. This difference reinforces the view that the elastic properties of nanocrystalline La_0.5Ca_0.5MnO₃ as a function of temperature are not determined by the magnetic transition but are indicative of local ordering of Jahn–Teller distorted octahedra. There are small irregularities in the temperature dependence of magnetic moments measured during cooling in a 5 T field—suggesting that the ferromagnetic ordering is not necessarily homogeneous over the entire temperature interval between ∼40 and ∼270 K (figure 9 shows the data up to 70 K).

There are a number of reports of changes in magnetic structure of half doped manganites at low temperatures. For example, a peak in ac magnetic susceptibility at 41 K in Pr_0.5Ca_0.5MnO₃ has been explained in terms of a reentrant spin glass transition that leads to a low temperature state with ferromagnetic and spin glass clusters in an antiferromagnetic matrix [73]. The reported frequency dependence is rather slight, such that the peak shifts by only 0.2 K between 0.01 and 3 kHz. Doping of La_1−x Ca_x MnO₃ with Y also leads to a frequency dependent anomaly in the magnetic susceptibility near 40 K. In this case the shift in maximum susceptibility is several K for a frequency change of 0.001 to 1 kHz [74, 75]. Glass-like arrested states have also been proposed for bulk samples of La_0.5Ca_0.5MnO₃ at low temperatures but the distinct anomaly in magnetic behaviour near 35 K was ascribed to a change from metastable ferromagnetic order to equilibrium antiferromagnetic order [49, 76, 77]. In nanowires of La_0.5Sr_0.5MnO₃ a magnetic anomaly at 42 K was attributed to freezing behaviour of surface spins [78]. By way of contrast, the changes in magnetic moment near 40 K shown in figure 9 perhaps more closely resemble the pattern expected for a first order transition between structures with different magnetic ordering states. In the case of F, a peak in acoustic loss measured at ∼150 kHz occurs at the same temperature as the anomaly in dc magnetic moment, ruling out the possibility of any substantial dispersion with respect to frequency that would be expected for a freezing process. In this case, the data for f² just show a break in slope rather than an increase with falling temperature that would signify a classical Debye loss mechanism of the type well illustrated by the data for f² and Q⁻¹ near 85 K. There must be some strain contrast across the transition and the loss mechanism perhaps involves motion under stress of interfaces between coexisting phases which also have slightly different elastic constants.

From the hysteresis curves in figure 4 it is clear that sample A (and B) has predominantly ferromagnetic structures above and below 41 K, so the transition is between one ferromagnetic structure and another. Changes in magnetic moment across the transition presumably reflect some or all of differences in moment, changes in the proportion of any relict antiferromagnetic moment and changes in domain structure. These are less obvious in the 5 T data (figure 9), which show only a break in slope of the moment at ∼40 K, presumably reflecting saturation magnetization.

Not all samples of La_0.5Ca_0.5MnO₃ seem to show the same behaviour at low temperatures. For example, there are anomalies similar to those reported here near 40 K in the magnetic data of Levy et al [1] but not in the data of Sarkar et al [2], Rozenberg et al [3], Jirak et al [4] or Freitas et al [55], Chen et al [79] found an anomaly in the evolution of the longitudinal modulus at ∼50 K which is similar to that observed here in f² for acoustic resonances, and more marked changes at the same temperature in samples with higher Ca-contents. Several possible explanations may be offered but this issue is not resolved here. Firstly, some impurity phase(s) might be present in some samples. For example, Mn₃O₄ has prominent magnetoelastic transitions near 40 K (e.g. [80, 81]), though there does not appear to be evidence for its presence in powder diffraction patterns. Secondly, there could be differences of stoichiometry. For example, increasing the number of oxygen vacancies causes an anomaly to appear near 40 K in zero field cooled magnetic moments, in comparison with a stoichiometric sample with three oxygen atoms per formula unit [19]. However, the reduction in oxygen content is also accompanied by a reduction in the magnetic ordering temperature below 240 K and, in the present study at least, it is has been found that the 40 K anomaly occurs in samples with T_c ∼ 260 − 280 K. Sarkar et al [2] reported that their samples all had some oxygen deficiency, ∼La_0.5Ca_0.5MnO_2.98 for the finest grain sizes and slightly greater non-stoichiometry for the coarsest sample, but it is possible that this was changed in the hot-pressing process. Finally, grain size effects, particularly with respect to magnetism, depend substantially on the changing ratio of material in the surface to material within the bulk in the usual way, but these can be modified by interactions between grains. Samples with higher density, such as produced by hot pressing, have a higher proportion of welded grain boundaries across which there should be relatively strong magnetic interactions, giving rise to the possibilities of changing patterns of stability in nanocrystalline samples.

Jahn–Teller and charge ordering are suppressed both by application of a magnetic field and by reducing grain size, but the resulting ferromagnetic phases are not the same with respect to their macroscopic strain. It follows that other contributions to the structural state also cannot be the same, in particular because shear strains are coupled to octahedral tilting. Figure 2 contains values for symmetry-adapted shear strains calculated from lattice parameters in Siruguri et al [49] for a bulk sample measured at 9 K in zero field and in an applied field of 7 T. In zero field the sample is antiferromagnetic, with strains that have values which are indistinguishable from those calculated from the original data of Radaelli et al [7]. At 7 T these become more typical of an orthorhombic perovskite with distortions from cubic lattice geometry associated only with octahedral tilting. In particular, e_tx is nearly 3% for the structural state with cooperative Jahn–Teller distortions and only a few ‰ when this is suppressed. The comparison is essentially the same as between Pr_0.52Ca_0.48MnO₃ and SrZrO₃ in Carpenter et al [50]. On the other hand, values of e_tx and e₄ for the ferromagnetic nanocrystalline sample are ∼2% and show no evidence of changes through T_co or T_c (figure 2). A homogeneous ferromagnetic state with no Jahn–Teller ordering can be represented by La_0.75Ca_0.25MnO₃ for which lattice parameters of Radaelli et al [37] give e_tx and e₄ values of ∼−0.003. This is again consistent with the effects of coupling between strain and octahedral tilting alone. The primary data show that the main strain associated with ferromagnetic ordering is a volume strain which reaches a maximum value of ∼−0.002 in this case. Large shear strains in nanocrystalline sample of La_0.5Ca_0.5MnO₃ are presumably due to octahedral tilting alone, but the octahedra must on average be more distorted than at compositions where charge order/orbital order does not develop.

Comparison above of the elastic properties through the ferromagnetic transition in samples A and B with the more typical effects of magnetoelastic coupling seen in La_0.7Ca_0.3MnO₃ and La_0.67Ca_0.33MnO₃ also confirms that the structural state of the nanomaterials is somehow different with respect to strain. The minimum in shear modulus near T_co seen in the data for samples A and B has been interpreted here as indicative of local ordering of Jahn–Teller distorted octahedra, but without the development of additional macroscopic strain. This appears to be analogous to the suppression of strain accompanying tilting due to the introduction of A-site cation/vacancy disorder in the perovskite La_0.6Sr_0.1TiO₃. Local heterogeneous strain fields arising from the disorder hinder the development of a coherent macroscopic strain but do not suppress the tilting [82]. The length scale of these local heterogeneities is likely to be comparable with that seen associated with replacing one cation by another of different size, i.e. ∼ 10–20 Å [83]. If, as is proposed and raised as a possibility by Sarkar et al [2], nanocrystals of La_0.5Ca_0.5MnO₃ contain heterogeneous shear strains locally within them, they would not be expected to show identical physical properties to thin films in which a homogeneous strain is imposed by the substrate.