Neutron scattering study of the orientational disorder and phase transitions in barium carbonate

Orientational disorder of the molecular CO32− anions in BaCO3, which occurs naturally as the mineral witherite, has been studied using a combination of neutron total scattering analysed by the reverse Monte Carlo method and molecular dynamics simulations. The primary focus is on the phase transition to the cubic phase, which assumes a rocksalt structure (Strukturbericht type B1) with highly disordered orientations consistent with the mismatch between the site (m3¯m) and molecular (3/m) symmetries. Both experiment and simulation show a high degree of disorder, with the C–O bond orientation distribution never exceeding 25% variation from that of a completely uniform distribution, although there are differences between the two methods regarding the nature of these variations. Molecular dynamics simulations are also reported for the analogous phase transitions in the very important mineral calcite, CaCO3. The combination of the simulations and comparison with BaCO3 shows that the properties of calcite at all temperatures within its stability field are affected mostly by the onset of orientational disorder associated with the high-temperature cubic phase, even though this lies outside the stability field of calcite. This is a new understanding of calcite, which previously had been interpreted purely in terms of the phase transition to an intermediate partially-disordered phase. Finally, we also found that witherite itself appears to support the development of orientational disorder on heating, with the simulations showing a sequence of phase transitions that explain the much larger thermal expansion of one axis.


Introduction
Calcium carbonate, CaCO 3 , is the most significant of the nonsilicate minerals found in the Earth's crust, forming as chalk, limestone and marble, and playing an important role in both the water and carbon dioxide budgets of the planet. Calcite is well-known in the form of Iceland spar, where the beautiful single crystals with well-defined habit show the dramatic effect of double refraction. The characteristic shape of crystals of Iceland spar comes from the set of (1014) planes 6 that form the crystal faces, as shown in figure 1, because these are the electrically neutral planes in the crystal 7 . Calcite actually has a rather unusual thermal expansion, with a relatively large (for a mineral) positive linear expansivity along the trigonal [001] direction and small but negative linear expansivity along the perpendicular directions [1,2].
The large positive expansivity along the trigonal [001] direction has been associated with a probable orientational order-disorder phase transition at high temperature [3,4]. The Bragg reflections of the room-temperature phase where the Miller indices (hki ) have odd values of fall to zero on heating towards a temperature of around 1260 K [3,4]. This is consistent with a phase transition from the room temperature structure of space group R3c to that of space group R3m with a halving of the trigonal c lattice parameter. Such a transition could be achieved if the carbonate groups become orientationally disordered about axes parallel to the trigonal [001] direction. This aspect was studied by neutron powder diffraction [4], showing two interesting unexpected effects. The first was that on heating the carbonate molecular anions show a large increase in librational amplitude and no growth in the number of anions with a 60 • rotation, and secondly a large increase in the amplitude of libration about axes perpendicular to the molecular three-fold axis. This increase in the perpendicular libration was suggested to explain the rather large thermal expansion of the c axis that appeared to be associated with the phase transition.
Experimentally this was as far as was possible to get to, because the carbonate molecular anions become chemically unstable at the point of the R3c-R3m phase transition, with the rapid decomposition reaction CaCO 3 → CaO + CO 2 . Attempts to prevent this reaction, or at least raise it to a higher temperature, through the use of a pressurised CO 2 environment were unsuccessful [4]. This means that the true nature of the R3m phase and hence of the phase transition cannot be confirmed, and neither can we establish a baseline thermal expansion for the R3m phase with which to confirm that the origin of the thermal expansion of the R3c phase is associated solely with this phase transition.
Subsequent molecular dynamics simulations, which we report here, suggest that the R3c-R3m phase transition is part of a larger sequence of phase transitions that ultimately reach an orientationally disordered phase with the simple cubic rocksalt structure. Indeed, it is almost a textbook observation   [5][6][7][8]. The CO 3 groups (carbon atoms as gray spheres, oxygen atoms as red spheres) lie in planes parallel to the orthorhombic (001) plane, separated by layers of barium atoms (gold spheres). that the C and Ca ions in calcite have a distorted rocksalt arrangement.
Although we cannot access this state in calcite, it appears that a similar orientationally-disordered rocksalt phase exists in BaCO 3 [9]. Actually at ambient conditions BaCO 3 is found as the mineral witherite [5][6][7][8], which has the aragonite structure adopted by CaCO 3 , as shown in figure 2 (orthorhombic symmetry, space group Pmcn; details of the crystal structure will be reported below) at a slight high pressure. At a temperature of 1084 K [6] it transforms to a rhombohedral phase of symmetry either R3m or R3m; the former is ordered with all carbonate anions the same orientation, and the latter is necessarily disordered with respect to the orientations of the carbonate anions. Subsequently at a temperature of 1255 K [6]  it transforms to the disordered rocksalt cubic phase [9], until eventually undergoing the decomposition reaction BaCO 3 → BaO + CO 2 .
In this paper we use the method of neutron total scattering coupled with the reverse Monte Carlo method to investigate the atomic structure of the orientationally disordered cubic phase of BaCO 3 . We also report supporting molecular dynamics simulations to provide the wider context and to give a point of comparison. It is to be hoped that as a result of this work we will have a much better understanding of the properties experienced at ambient conditions by the important life-giving mineral calcite.

Molecular dynamics simulations
Molecular dynamics (MD) simulations were performed using the DL_POLY code [10]. The potential energy function of Archer et al [11] was used with rigid CO 3 molecules and rigid ions. This was chosen in preference to other alternatives in the literature [12] because it was been specifically tuned to account for the vibrational dynamics of calcite, and in particular to reproduce the soft phonon branch associated with the high-pressure phase transition, while retaining full stability, and it was designed to allow for a wide range of divalent cations.
Simulation boxes with orthogonal axes of approximate length 50 Å were prepared using the data2config/ RMCreate program [13]. One axis is along the trigonal [001] axis, one along the trigonal [100] axis, and the third along the orthogonal [210] axis. MD simulations were also performed using configurations from the witherite crystal structure.
In all cases we used time steps of 0.005 ps. Simulations were equilibrated for 10 000 steps (50 ps) and run for 50 000 steps (250 ps) to generate configurations for analysis. We used constant-stress [14,15] and constant temperature [16,17] thermodynamic ensembles.

Neutron powder diffraction and total scattering measurements
For the experimental work reported here BaCO 3 was obtained commercially. A sample was loaded into a thin-walled vanadium can, which was itself placed into a vacuum furnace. Neutron scattering measurements were performed on the GEM instrument at the ISIS spallation neutron facility [18]. Long measurements were performed for total scattering analysis, with shorter measurements for Rietveld analysis. For data correction, measurements were also made of the empty instrument, empty furnace, and empty sample can within the furnace, together with a calibration measurement of a vanadium rod.
The data were prepared in a form for Rietveld refinement using the MANTID software [19]. The total scattering data were corrected using the GUDRUN code [20]. The end result was the scattering function i(Q), where Q is the modulus of the scattering vector, and the Fourier transform of i(Q), the pair distribution function (PDF), has the In these equations the partial function g mn (r) is defined such that the number of atoms of type n lying within a spherical shell of thickness dr at a distance r from an atom of type m is given as 4πr 2 dr × c n ρ × g mn (r). The function g mn (r) is effectively a multiplicative factor that determines how much the distribution of atoms of type n differs from random at specific distances from atoms of type m; it has a value equal to 1 when the distribution at a distance of r gives the same density within the spherical shell as that for a random distribution. ρ is the atomic number density. The other quantities are c m to represent the fractional concentration of atoms of type m, and b m as the coherent scattering length of atom of type m. Further discussion of these functions, and comparisons with other functions in common use, has been given by Keen [21].

Rietveld analysis
The Rietveld refinements of the different structures of BaCO 3 were performed using the GSAS/EXPGUI [22,23] and GSAS II [24] programs. A representative fit for the ordered witherite phase is shown in figure 3. Results for lattice parameters are given in table 1, for atomic fractional coordinates in  table 2, and for atomic displacement parameters in tables 3 and 4. The table reports refinements in two different space groups for the intermediate temperature phase; we found that both gave equivalent quality fits for data from all neutron banks except the highest-resolution bank of angle 154.40 • . As shown in figure 4, the R3m model gives a better fit and correct relative peak heights. Table 4 shows the model refined with R3m space group has a large Ba atomic displacement parameter, which is comparable with the C atomic displacement parameter. However, in the witherite phase and the cubic phase, the heavy atom Ba always has a much smaller atomic displacement parameter than the light atom C. Meanwhile, the R3m models have atomic displacement parameters more consistent with other phases. Therefore, we believe R3m is more likely to be the space group of the intermediate-temperature phase, in spite of the discussion given by Antao and Hassan [6].

Reverse Monte Carlo method
The reverse Monte Carlo method was performed using the RMCprofile code [25]. For most of the work we used v6.7, but for the intermediate temperature phase we used a completely new version in prototype form, provisionally assigned version number 7, which is being designed for simulation of more than a single phase 8 . Because of decomposition at high temperature, our first experimental measurements were performed at high temperature, but we lost the first sample due to a malfunction within the furnace. In fact this sample had survived intact on heating into the high-temperature cubic phase with no decomposition detected. The data for the intermediatetemperature phase were obtained on a second sample. Unfortunately, in this case we did see the onset of decomposition, with clear Bragg peaks associated with the presence of BaO. The strategy for the RMC analysis of the intermediatetemperature phase was to use two configurations, alternatively moving one atom at random in each configuration. The relative weights of each phase to the total scattering, PDF [26] and Bragg scattering were determined by the phase fraction determined by the prior Rietveld analysis. As for the MD simulations, starting configurations of the BaCO 3 phases, with orthogonal axes of approximate length 50 Å, were prepared using the data2config/RMCreate program [13], using the structures generated by the Rietveld refinement analysis at each temperature and lattice parameters constrained to these values. Minimum distances and interatomic potentials for the C-O bond and O-C-O angles were applied as constraints in the RMC simulations. Taking the RMC simulation of the cubic phase as the example, the minimum allowed distances for the C-C, C-O, C-Ba, O-O, O-Ba and Ba-Ba contacts were 3.8, 1.0, 2.6, 1.93, 2.25 and Table 2. Atomic fractional coordinates for the various phases of BaCO 3 as indicated in the table. In the space group Pmcm by symmetry x = 1/4 for Ba, C and O1. In the R3m phase x = y = 0 for Ba and C, and y = −x for O. In the R3m phase Ba has x = y = 0 and z = 1 4 , C has x = y = z = 0 and O has y = z = 0. In the cubic Fm3m phase Ba has coordinates (0, 0, 1 2 ), C has coordinates (0, 0, 0), and O has x = 0 and y = −z. These results were obtained by Rietveld refinement of the diffraction data from the GEM diffractometer at ISIS.
0.1257 (6) 0.6012 (6) 0.136(4) 0.6012 (6)  900 0.0840 (9) 0.5664 (7) 0.140 (5) 0.5664 (7)  950 0.0795 (8) 0.5681(9) 0.1124 (3) 0.5637 (9)  R3m T (4) 3.8 Å respectively. The C-O bond potential was a Morse function with 1.29 Å as the equilibrium distance and 3.5 eV as the difference in energy between the lowest energy and the energy at infinite distance. The O-C-O angle potential was set with 120 • as the equilibrium angle and 5.0 eV as the corresponding force constant. Each RMC simulation was run until the average number of successful moves for every atom in the BaCO 3 phase was nearly 200, with a maximum single move distance of 0.05 Å. The BaO configurations used with the multiphase analysis of the intermediate-temperature phase were prepared in the same way and were of similar sizes. BaO has the standard rocksalt structure, space group Fm3m.
The quality of the RMC fits to data are shown for the example of the witherite phase at room temperature (at this temperature the sample is single phase) in figure 5. We consider these to be satisfactory. Similar quality was obtained for all temperatures, including for the analyses where we performed two-phase RMC.

Simulations starting in the calcite structure
The temperature-dependence of the lattice parameters of both CaCO 3 and BaCO 3 , obtained by MD simulation and starting in the ordered calcite form, are shown in figure 6. The simulation box gives two orthogonal evaluations of the trigonal a lattice parameter. To highlight the transition to the cubic phase, we actually scale the lattice parameters as a = √ 2a and c = c/2 √ 3 to give the values corresponding to those of the cubic phase a lattice parameter. In both cases the transitions to the cubic form is very clear-at temperatures of 1665 ± 10 K and 2535 ± 10 K for BaCO 3 and CaCO 3 respectively-as also are the transitions between the low-temperature ordered phases to the disordered R3m phases at temperatures of 905 ± 10 K and 1985 ± 10 K for BaCO 3 and CaCO 3 respectively. We note that the simulations over-estimate the R3c-R3m transition temperature in calcite by about 60%, and the transition to the cubic phase of BaCO 3 is over-estimated by around 30%. However, transition temperatures are very hard to predict from models derived from equilibrium properties only. What is encouraging is that the transition temperatures in BaCO 3 are lower than in CaCO 3 , as expected. What is interesting, however, is that the background thermal expansion around the R3c-R3m transition in both cases is mostly associated with the approach to the transition to the cubic form, and not, as previously assumed [3,4], due to the ordering associated with the R3c-R3m transition. Figure 6 also shows the strains calculated by comparing the lattice parameters with the extrapolation from the cubic phase, it Table 3. Atomic displacement parameters for the witherite phase of BaCO 3 . These results were obtained by Rietveld refinement of the diffraction data from the GEM diffractometer at ISIS.      can be seen that these are extraordinarily large for a phase transition, reaching over 20%.
Configurations from the MD simulation of each phase are shown in figure 7. The barium and calcium atoms are not shown in these figures in order to make the orientational behaviour of carbonate groups prominent. At low temperature, the carbonate anions in both BaCO 3 and CaCO 3 are orientationally ordered, although it is clear that the carbonate groups in BaCO 3 have spontaneously reoriented to give an ordered structure that is different from both calcite and witherite. As the temperature increases on heating towards   A similar trend exists in BaCO 3 , albeit with different orientations of the carbonate molecular ion as noted above, and for which the C-O bond orientation distribution is shown in figure 9. At 200 K, there are 2 preferred orientations i.e. two concentrated bright dots at the equator lie viewing from the [100] direction. As the temperature increases to 1000 K, shown in figures 9(a)-(d), the C-O orientation becomes more widely spread while the two preferred orientations are still distinguishable (brighter than other solid angles). At the intermediate-temperature phase, as shown in figures 9(e) and (f), the C-O orientation become more evenly spread around the equator and more progressively spread towards the poles.
The C-O bond orientation distribution plots from molecular dynamics simulations suggest that in both CaCO 3 and BaCO 3 the carbonate ions are orientationally ordered at the low temperature phases. Their rotational and librational vibration amplitudes increase with increasing temperatures resulting in a larger spread of C-O bond orientation distribution. In the intermediate-temperature phase, full disorder with respect to rotation about the threefold axis is achieved and libration out of the molecular plane increases, before the orientations of the carbonate ions become fully disordered at the cubic phase.

Simulations starting in the witherite structure
We also performed simulations of BaCO 3 starting in the witherite phase. Over the time scale of the simulations the structure will retain the basic layout of the sites for the Ba and C atoms, but will allow reorientations of the CO 3 groups as seen in the simulations of BaCO 3 starting in the calcite structure.
The simulated lattice parameters are shown in figure 10 and compared with the experimental values derived from table 1. In figure 10 we make use of the fact that the atomic structure of the witherite (aragonite) phase has the Ba atoms in an approximate hexagonal close-packed arrangement, with the C atoms in the corresponding octahedral sites to give a structure that is derived from the ideal type structure of NiAs (Strukturbericht type B8 1 ). On this basis, the a lattice parameter of the orthorhombic phase will correspond to that of the hexagonal phase, and the b lattice parameter, representing an orthogonal direction will correspond to a distance equal to √ 3 times the hexagonal a lattice parameter. The c lattice parameter reflects the spacing between the close-packed layers in the hexagonal close-packed arrangement, and will have ideal value when c = 8/3a. Thus in figure 10 we plot a, b/ √ 3 and c/ 8/3. The hexagonal symmetry is broken by the ordering of the orientations of the carbonate groups, but on heating in the simulation we see a clear phase transition to a hexagonal structure as the carbonate groups develop disorder first with orientations about the molecular three-fold axis and then as full three-dimensional disorder.  . The scaling is explained in the text, and is chosen so that the scaled a and b lattice parameters will come together in value with a phase transition from the orthorhombic to disordered hexagonal structure, and the scaled c lattice parameter will become closer in value as the disordered hexagonal phase approaches the ideal close-packed structure.
The experimental lattice parameters for the witherite phase, shown in figure 10, follow the simulated values up to the point of the transition to the rhombohedral phase. Given the structural similarities between the Strukturbericht type B 1 and B8 1 structures, the energies of the two phases will be very similar. In general phase transitions or metastable co-existence of these types of structures is common. It is therefore not surprising that as the experimental witherite phase begins to disorder, it switches structure to a distorted B 1 type structure rather than following the behaviour seen in the simulation. What is interesting from figure 10 is the close correspondence between the simulated and experimental lattice parameters. The experimental coefficients of the thermal expansion at higher temperatures in the witherite phase are α a = 5 × 10 −6 K −1 , α b = 13 × 10 −6 K −1 , and α c = 82 × 10 −6 K −1 . The relatively large experimental value of α c can therefore be understood in terms of the onset of three-dimensional disorder, and the smaller temperature variation of the a and b lattice parameters can be understood in terms of the convergence of these quantities in the simulated transition to a hexagonal phase. Figure 11 show representative layers of the configurations from the RMC analysis, which in the cases of the witherite phase and the intermediate rhombohedral phase are view down the directions perpendicular to the alignment of the molecular three-fold axes in the crystal structures. In comparison with figure 7 it will noted there are bond-bending distortions of the carbonate molecular ions here that are not seen in the MD configurations. This is because the carbonate ions are held rigidly fixed in shape in the MD simulations but are clearly allowed to distort in reality and as seen in the RMC simulations.   Figure 12(a) shows the distribution of C-O bond orientations within the RMC configurations for the witherite phase, and the corresponding distribution from the MD simulations is shown in figure 13(b). The distributions for the RMC models for the two rhombohedral-phase starting points are shown in figures 12(c) and (d). As compared with the MD (figure 9), the distribution shows more motion of the bonds corresponding to rotations about axes perpendicular to the carbonate three-fold rotation axis. This is consistent with the extraordinary large values of the atomic displacement parameter U 33 values given in table 3, and is fully consistent with the phase behaviour discussed earlier as revealed by the MD simulations.

Rhombohedral phase
Although in our previous discussion on the Rietveld analysis we remarked that we believe that the rhombohedral phase is more likely to have the disordered R3m structure than that with Figure 13. Partial pair distribution functions g mn (r) for cubic BaCO 3 from RMC (blue lines) and MD (black curves) for each of the atomic pairs. Note that we do not include the low-r part for the C-O and O-O functions because the peaks in the MD simulations are δ-functions since we used rigid carbonate molecular anions. Due to the simulation transition temperature being higher than experiment, the PDFs for experiment are at 1273 K and for the simulation at 2600 K. A small rescaling of the values of r for the MD data was applied to bring the peaks into best registry, accounting for imperfections in the interatomic potentials and the effects of thermal expansion. space group R3m, we explored using both the R3m and R3m structures as the starting configurations for the RMC analysis of this phase. In fact we found that both give similar quality fits to the three sets of data, and the structures remained the same throughout the RMC simulation. Thus RMC is not giving us a preference for one structure over the other. Figure 12(c) shows the C-O bond orientation distribution for the R3m phase, and that for the R3m phase is shown in figure 12(d). Similar to the RMC and MD results shown in figures 12(a) and (b) for the room-temperature witherite phase, the vertical libration is more prominent than the horizontal rotation movement. This is consistent with the atomic displacements parameters obtained from Rietveld refinement as show in tables 3 and 4; although the values of U 11 and U 22 increase dramatically in the rhombohedral phase, they remain smaller than the value of U 33 .

Cubic phase
The various partial pair distribution functions g(r) from both RMC and MD configurations for cubic BaCO 3 are shown in Table 5. Values of the coefficients c in the expansion of the C-O bond orientational distribution function P(Ω) as defined by equation (3). These values were obtained by averaging over several atomic configurations as described in the text. Note that c 0 = 1.  The more interesting functions are those involving the oxygen atoms, because these will reflect the degree of disorder, and here the agreement between MD and RMC is considered to be good. Small differences will be a reflection of some small differences in bond orientational distribution function described next.
In the cubic phase of BaCO 3 , and theoretically of CaCO 3 also, the carbonate groups (molecular point group symmetry 3/m) occupy sites of the full octahedral symmetry (point group m3m). The lower symmetry of the molecular anion implies that there must be orientational disorder. Such cases of orientationally disordered crystals are common; examples are KCN [27,28], and CBr 4 and SF 6 [29]. It is common to characterise the atomic structure in terms of the orientational distribution of the component bonds of the molecules, which in this case means the angular distribution of the C-O bonds. It is then common to represent this as a distribution function in terms of polar angles Ω = (θ, φ), defined mathematically as an expansion in terms of the Kubic harmonics, which are formed as symmetry-adapted combinations of spherical harmonics, and are defined in the appendix A for convenience [29,30]. The bond orientational distribution function is thus defined as In diffraction experiments the coefficients c can be fit in a structure refinement [29]. On the other hand, in a simulation that gives a configuration of atoms we can obtain the values of the coefficients as c = K , a result that follows directly from the orthornormality of the functions K (Ω). The average is performed over all bonds in the atomic configuration, and for higher accuracy (as done here) can also be averaged over several independent configurations. Values of c for the distribution of C-O bonds in the cubic phase of BaCO 3 as obtained from both RMC and MD are given in table 5, together with corresponding values for calcite from MD. Figure 14 shows the C-O bond orientational distribution function for cubic BaCO 3 calculated from the RMC configurations. We display both the bond distribution function calculated point-by-point from the histogram of bond angles, and as calculated from equation (3). The value P(Ω) = 1/4π corresponds to a uniform distribution function. It is seen that is some preference for alignment away from the 100 directions, but only of the order of just over 10% variation away from the uniform distribution. Other orientationally-disordered crystals can show much more structure in the bond orientational distribution function [29]. Figure 15 shows the C-O bond orientational distribution functions for cubic BaCO 3 and CaCO 3 calculated from the MD configurations. These distributions are more uniform than in the RMC configurations, but the phase transitions are at higher temperatures and so these distributions represent correspondingly higher temperatures. In fact the distributions become more uniform on heating, since over the temperature range 1500-3000 K the value of the coefficient c 4 for cubic BaCO 3 falls by about one half. The distributions in both BaCO 3 and CaCO 3 differ from the RMC in that they show slight preferences for orientations along the 100 directions rather than away from these directions, but the effects are actually very slight.

Discussion and conclusion
The key objective of this paper was to identify the orientational disorder in the cubic phase of BaCO 3 , which we have shown by both RMC and MD to be almost completely uniform. Given the shape of the CO 3 molecular anion, it is not surprising that this might be so. We see from lower-temperature MD data and earlier crystallographic data on CaCO 3 that although the C-O bond has a preference to point along symmetry directions in the hexagonal basal plane, it has a large growth in out-of-plane orientations on heating.
The existence of the cubic phase of BaCO 3 that is reflected also in the MD simulations of both systems leads to a new interpretation of the observed phase transition in CaCO 3 . Previously it was assumed that the main phase transition was from space groups R3c to R3m on heating, with the onset of orientational disorder about the three-fold axis leading to halving of the crystallographic repeat distance parallel to the three-fold axis. The onset of decomposition at the phase transition meant that it was impossible to confirm the existence of the higher-symmetry phase, but loss of certain Bragg peaks on heating to the transition are only consistent with this explanation. What was previously unexplained was the fact that on heating towards the phase transition the largest orientational fluctuations were associated with tilting of the three-fold axis away from the plane normal rather than about this axis. This is seen also in the MD simulations of the R3c in calcite, figure 8. Furthermore, without data on the lattice parameters for temperatures above the phase transition it was hard to analyse the strains associated with ordering. What emerges from the MD results presented here is that the variation of the lattice parameters with temperature is affected primarily by the spontaneous strains associated with the transition to the cubic phase and not between the two rhombohedral phases. This is a completely new viewpoint on the phase behaviour of the important mineral calcite.

Postscript: some personal comments by the corresponding author
The corresponding author (MTD) held his first post-doctoral research position in the Department of Physics at the University of Edinburgh during the time when Roger Cowley was head of department. Edinburgh Physics was an exciting environment at that time. The condensed matter group had a large grant covering the activities of all the research-active staff, mostly in the general area of phase transitions. My personal interest was in molecular crystals, working with Stuart Pawley, but from my PhD I had been inspired by the work of Roger, and his supervisor Bill Cochran (also in Edinburgh at the time). Prior to coming to Edinburgh I had studied hard (and bought the resultant book of) the three papers of Roger and Alastair Bruce on phase transitions in the journal Advances in Physics [31][32][33]. I have also been helped throughout my career by Roger's analysis of the acoustic modes that accompany ferroelastic phase transitions [34], and have used the measurement of the soft mode in SrTiO 3 performed by Roger in collaboration with Bill Buyers and Gerald Dolling [35].
Roger and I were co-authors on just one paper, concerning the ferroelastic phase transition at 755 K in Na 2 CO 3 [36], another carbonate material. This transition is a second-order transition between hexagonal and monoclinic space groups. The symmetry change is such that there is a softening of one set of acoustic modes lying for all wave vectors within a plane in reciprocal space. Previously Roger and Andreas Meyer had shown that such a case is very similar to melting in two dimensions [37], and the amazing thing was to indeed see all the Bragg peaks except those with Miller indices of the form (00 ) disappear at the phase transition, to be replaced by broad peaks of diffuse scattering [36,38,39], fully consistent with the theory [37]. It was characteristic of Roger's modesty that he declined to be co-author on the other papers in the series by Mark Harris, as his postdoctoral assistant, and I. Moreover, Roger showed a lot of interest in our work on the anomalous inelastic scattering of radiation in calcite [40,41] and the related system NaNO 3 [42] carried out by Mark Harris and I, but again never requesting to appear as co-author.
It is fair to say that Roger had become one of my heroes as a young research student, before I subsequently had the privilege to work in his department, and he has remained as such throughout my whole career. And while it is said that you should never meet your heroes, because they often disappoint you, this was definitely not the case with Roger.