Symmetry and strain analysis of structural phase transitions in ${\text{Pr}}_{0.48}{\text{Ca}}_{0.52}{\text{MnO}}_3$

Structural evolution as a function of temperature through the $Pnma\leftrightarrow \text{incommensurate}$ (IC) phase transition in ${\text{Pr}}_{0.48}{\text{Ca}}_{0.52}{\text{MnO}}_3$ perovskite has been analyzed from the perspectives of symmetry and strain. The structure and stability of both phases are shown to depend on combinations of order parameters which have symmetries associated with irreducible representations ${\text{M}}_3^{+}$, ${\text{R}}_4^{+}$, ${\text{M}}_2^{+}$, ${\Gamma }_3^{+}$ and ${\Sigma }_2$ of space group $Pm\overline{3}m$. The physical origin of these can be understood in terms of octahedral tilting, cooperative Jahn-Teller distortions and charge order/Zener polaron ordering. The ${\text{M}}_2^{+}$ order parameter describes the Jahn-Teller ordering scheme which develops in ${\text{LaMnO}}_3$ while the ${\Gamma }_3^{+}$ order parameter relates to an ordering scheme in which the unique axes of the distorted octahedra are all aligned in the same direction. Irrep ${\Sigma }_2$ contains two components with gradient coupling and provides the symmetry-breaking mechanism by which the IC transition can occur. Each order parameter couples with macroscopic spontaneous strains in a manner that depends strictly on symmetry and this leads to specific interactions between the order parameters through their coupling with common strains. In order to establish the extent and importance of this coupling, symmetry-adapted strains have been extracted from a new set of lattice parameters obtained by high-resolution powder neutron diffraction in the temperature interval 10–1373 K. It is found that the predominant strain of the incommensurate structure (up to $\sim 2.5\mathrm{\%}$) is a tetragonal shear strain which arises by bilinear coupling with the ${\Gamma }_3^{+}$ order parameter. This combination is probably responsible for most of the energy reduction accompanying the $Pnma\leftrightarrow \text{IC}$ transition and also gives it some characteristics typical of a pseudoproper ferroelastic transition. Strain coupling promotes mean-field behavior and the evolution of the symmetry-breaking order parameter can be described by a standard Landau tricritical solution, $q^4\propto \left(T_{\text{c}}-T\right)$ with $T_{\text{c}}=237\pm 2\text{K}$. Octahedral tilting at high temperatures is closely similar to tilting in the $Pnma$ structure of other perovskites, such as ${\text{SrZrO}}_3$. This is accompanied by a degree of Jahn-Teller ordering on the basis of the ${\text{M}}_2^{+}$ scheme below $\sim 775\text{K}$ but is replaced by the ${\Gamma }_3^{+}$ scheme below $T_{\text{c}}$. In contrast with the tilting and Jahn-Teller effects, magnetic ordering at the Néel temperature $\left(\sim 180\text{K}\right)$ is accompanied by only the slightest volume strain and is not likely to influence the evolution of the other order parameters to any significant extent, therefore. An additional change in the volume strain below $\sim 85\text{K}$ is perhaps related to changes in magnetic structure at lower temperatures. Line broadening in powder diffraction patterns collected in the temperature interval $\sim 150–260\text{K}$ appears to be related to the presence of ferroelastic twins arising from octahedral tilting and draws attention to the fact that the $Pnma\leftrightarrow \text{IC}$ transition takes place in a material which already contains heterogeneities. Finally, correlation of the repeat distance of the IC structure with ${\Gamma }_3^{+}$ distortions of ${\text{MnO}}_6$ octahedra shows that the nature of the IC structure itself is also determined essentially by geometrical factors and strain.

Figure 1

Relationship between unit-cell axes for $Pnma$ and $Pnm{2}_1$ structures and reference axes X, Y, and Z.

Figure 2

Lattice parameter and volume variations in ${\text{Pr}}_{0.48}{\text{Ca}}_{0.52}{\text{MnO}}_3$ as a function of temperature, refined under $Pnma$ or $R\overline{3}c$ symmetry. The highest temperature values of $a$ and $\alpha$ are for the pseudocubic rhombohedral cell. The broken line in (b) is a fit of the baseline function given in the text to data for the $Pnma$ structure between 300 and 490 K with the saturation temperature, ${\Theta }_{\text{so}}$, fixed at 150 K.

Figure 3

Octahedral tilt angles determined from refined atomic coordinates of the average $Pnma$ structure. Triangles are data for the high-temperature rhombohedral structure in $R\overline{3}c$. Dots are data for ${\text{SrZrO}}_3$ from McKnight et al. (Ref. 37).

Figure 4

Mn-O bond lengths determined from refined atomic coordinates of the average $Pnma$ structure. ${\text{Mn-O}}_1$ is aligned approximately parallel to [010] while ${\text{Mn-O}}_{21}$, ${\text{Mn-O}}_{22}$ lie approximately within the (010) plane.

Figure 5

Segments of neutron powder-diffraction patterns stacked according to the temperatures at which they were collected. Note the prominent asymmetric line broadening that develops just below the $Pnma\leftrightarrow \text{IC}$ transition temperature of $\sim 235\text{K}$. Some asymmetric broadening remains in the pattern collected at 10 K.

Figure 6

Peak width and $d$-spacing variations with temperature of two representative peaks in powder-diffraction patterns. (a) and (b) are for 040 and 202 peaks, indexed under $Pnma$ symmetry. (c) and (d) are for overlapping 022 and 220 reflections; changes in the apparent peak width are due to shifts in the positions of the individual peaks in this case.

Figure 7

Shear strains determined from the lattice parameter data shown in Fig. 2. The broken line represents $e_{tx}$ arising from octahedral tilting alone. The tilting behavior of ${\text{Pr}}_{0.48}{\text{Ca}}_{0.52}{\text{MnO}}_3$ is closely similar to that of ${\text{SrZrO}}_3$, as shown by comparison with data from McKnight et al. (Ref. 37).

Figure 8

Volume strain, $V_{\text{s}}$, and excess shear strain, $\Delta e_4$, due to the $Pnma\leftrightarrow \text{IC}$ transition. Strain $\Delta e_{tx}$ is the tetragonal shear strain after removal of the calculated contribution of octahedral tilting.

Figure 9

Strain/strain relationships for strains arising at the $Pnma\leftrightarrow \text{IC}$ transition and excluding the influence of octahedral tilting. Linear correlations are consistent with the expected relationships $V_{\text{s}}\propto \Delta e_4\propto \Delta e_{\text{tx}}\propto q_{\text{IC}}^2$ with some precursor $e_{\text{tx}}$ strain ahead of the transition point.

Figure 10

Variation with temperature of (strains) (Ref. 2) which are expected to scale as $q_{\text{IC}}^4$. The solid line is a fit of an equation with the from of Eq. (23), showing that the $Pnma\leftrightarrow \text{IC}$ transition can be described as Landau tricritical with $T_{\text{c}}=237\text{K}$, ${\Theta }_{\text{s}}=276\text{K}$.

Figure 11

Schematic variations in the principal order parameters through the $Pnma\leftrightarrow \text{IC}$ transition. $q_2$ and $q_4$ are tilting order parameters and $q_{\text{tx}}$, $q_{2\text{JT}}$ refer to two different Jahn-Teller ordering schemes, ${\Gamma }_3^{+}$ and ${\text{M}}_2^{+}$, respectively. The incommensurate order parameter, $q_{\text{IC}}$, scales with $q_{\text{tx}}$ as $q_{\text{IC}}^2\propto q_{\text{tx}}$. Below $T_{\text{c}}$ there is a crossover between the two Jahn-Teller ordering schemes of deformed octahedra. This is accompanied by changes in the tilt angles.

Figure 12

Details of peak shapes for selected reflections at different temperatures through the $Pnma\leftrightarrow \text{IC}$ transition. Intermediate temperatures are marked by strong asymmetric line broadening. Indexing is for the $Pnma$ lattice.

Figure 13

Comparison of the repeat distance of the IC structure, as expressed through the parameter $q/a^{\ast }$ [data of Sánchez et al. (Ref. 50)] with the ${\text{Mn-O}}_1$ bond length. The temperature dependences are indistinguishable, implying that the repeat length of the IC structure depends essentially on geometrical factors.