CoF2: a model system for magnetoelastic coupling and elastic softening mechanisms associated with paramagnetic ↔ antiferromagnetic phase transitions

Resonant ultrasound spectroscopy has been used to monitor variations in the elastic and anelastic behaviour of polycrystalline CoF2 through the temperature interval 10–290 K and in the frequency range ∼0.4–2 MHz. Marked softening, particularly of the shear modulus, and a peak in attenuation occur as the Néel point (TN = 39 K) is approached from both high and low temperatures. Although the effective thermodynamic behaviour can be represented semiquantitatively with a Bragg–Williams model for a system with spin 1/2, the magnetoelastic coupling follows a pattern which is closely analogous to that of a Landau tricritical transition which is co-elastic in character. Analysis of lattice parameter data from the literature confirms that linear spontaneous strains scale with the square of the magnetic order parameter and combine to give effective shear and volume strains on the order of 1‰. Softening of the shear modulus at T > TN is attributed to coupling of acoustic modes with dynamical local ordering of spins and can be represented by a Vogel–Fulcher expression. At T < TN the coupling of strains with the antiferromagnetic order parameter leads to softening of the shear modulus by up to ∼2%, but this is accompanied by a small and frequency-dependent acoustic loss. The loss mechanism is attributed to spin–lattice relaxations under the influence of externally applied dynamic shear stress. CoF2 provides a reference or end-member behaviour against which the likely antiferromagnetic component of magnetoelastic behaviour in more complex multiferroic materials, with additional displacive instabilities, Jahn–Teller effects and ferroelastic microstructures, can be compared.


Introduction
Magnetic ordering transitions are typically accompanied by small lattice distortions which give rise to anomalies in elastic properties that have been characterized in a wide variety of Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. materials including metals, intermetallics, oxides, sulphides and halides [1][2][3]. In descriptions of structural phase transitions, the distortions would be referred to as spontaneous strain, but in magnetism the effects are usually described in terms of magnetostriction, exchange striction or magnetoelastic coupling. Recent intense interest in multiferroic materials has brought back into focus the importance of magnetoelastic effects at least in part because they provide a mechanism by which separate instabilities can become coupled in a single, homogeneous phase or in two-phase mixtures. For example, the direct magnetoelectric effect, in which magnetic order induces a change in ferroelectric order, and vice versa, is generally acknowledged to be weak but would be significantly enhanced if strains due to the magnetic order overlapped with strains due to ferroelectric displacements. As in the extreme example of colossal magneto-resistance, where manganite perovskites can display ferromagnetism, antiferromagnetism, octahedral tilting, cooperative Jahn-Teller distortions, charge ordering and, perhaps also, ferroelectricity (e.g. [4,5]), many properties of potential interest for device materials depend on multiple instabilities. Whatever their origin might be in detail at an atomic scale, it is inevitable that strain accompanying magnetic, electronic and structural phase transitions will provide a permeating influence over wide ranges of parameter space, in terms of promoting relatively long range correlation lengths, providing coupling between different order parameters, producing strong grain size dependences in the nano region, and enhancing local heterogeneity related to transformation microstructures such as twin walls.
Against this broader picture, the present study was designed to investigate strain relaxation behaviour in one of the simplest possible circumstances, namely, antiferromagnetic ordering without coupling to symmetry-breaking shear strain and, hence, also without ferroelastic twinning. CoF 2 has been described as a model antiferromagnet elsewhere [6], and its particular value here is that the influence of factors other than coupling between a magnetic order parameter and co-elastic strain can be excluded. This forms part of a more extensive effort to characterize coupling behaviour, from the perspectives of strain and elasticity, in systems with different combinations of magnetic/structural transitions (e.g. KMnF 3 [7], Fe x O [8], metal-organic frameworks [9], (Pr, Ca)MnO 3 [10,11], BiFeO 3 [12]). The key point is that elastic properties are susceptibilities with respect to strain, analogous to magnetic and dielectric susceptibilities with respect to magnetic and electric dipoles. As such they vary by orders of magnitude more than the strains themselves and are highly sensitive to the symmetry changes involved and the thermodynamic character of the transitions. In addition, anelastic losses can provide insights into the dynamics of both intrinsic (order parameter relaxation, fluctuations) and extrinsic (transformation microstructure) contributions. Resonant Ultrasound Spectroscopy (RUS) provides a convenient and effective method for measuring elastic and anelastic properties of small samples simultaneously over wide temperature intervals, while Landau theory provides perhaps the most effective phenomenological basis for then relating observed changes in elastic properties to specific patterns of strain coupling and order parameter evolution.
CoF 2 is part of a family of metal difluorides, including MnF 2 , FeF 2 and NiF 2 , which crystallize in the rutile structure (space group P4 2 /mnm). It orders as an antiferromagnet below T N ≈ 39 K, with spins aligning parallel to the c-axis, and there is no associated change in crystallographic space group [13][14][15][16][17][18][19][20]. The Co 2+ ions have a high spin configuration t 5 2g e 2 g (spin 3/2), with an orbitally triply degenerate ground state and, therefore, unquenched orbital angular momentum. This leads, in detail, to relatively complex magnetic properties (e.g. [14][15][16][21][22][23]), but the antiferromagnetic ordering transition itself can be represented simply in terms of a ground state which is only doubly degenerate, with effective spin 1/2 [14,16,21]. This is supported by heat capacity measurements which show that the total entropy change is near to R ln 2, i.e. for spin 1/2, rather than R ln 4, which would expected for spin 3/2 [24,25]. High spin Co 2+ ions would also normally be expected to be Jahn-Teller active but octahedra in the rutile structure are already distorted, with orthorhombic geometry. Apical Co-F bonds of CoF 6 octahedra in CoF 2 at room temperature are ∼2% shorter than the equatorial Co-F bonds, and the F-Co-F bond angles for equatorial F are 79 • rather than 90 • [26]. Moreira et al [27] have estimated that ∼5-10% of the energy reduction associated with this deformation from regular octahedral geometry can be attributed to the Jahn-Teller effect. This does not appear to have any direct bearing on the antiferromagnetic ordering transition since the difference in bond lengths between apical and equatorial M-F bonds is almost identical at 295 and 15 K [17].
A recent high resolution neutron powder diffraction study of CoF 2 has shown that the antiferromagnetic ordering transition is accompanied by small lattice distortions [19], and these form the starting point for understanding anomalies in the elastic properties obtained by RUS. Evolution of the macroscopic order parameter follows (1 − T /T N ) β with β ≈ 0.31 [18,19]. The only other instability which is known to be not far away in pressure-temperature space is related to a soft optic mode with B 1g symmetry. In CaCl 2 and CaBr 2 this gives rise to a tetragonal (P4 2 /mnm) ↔ orthorhombic (Pnnm) transition as a function of temperature [28][29][30][31][32], while the same transition occurs in MgF 2 as a function of pressure [33]. The transition in CaCl 2 has changes in elastic properties which match those expected for a pseudoproper ferroelastic transition [34], such as have been investigated more extensively for the equivalent transition in stishovite, SiO 2 [35][36][37][38][39][40][41]. As shown below, however, there does not seem to be any overt evidence for the influence of this transition on the evolution of elastic properties through T N in CoF 2 . Finally, the lattice parameter data of Chatterji et al [19] show a tail in strain above the Néel temperature, indicative of some degree of dynamic and/or static short range order ahead of the magnetic transition.

Sample description and experimental methods
The RUS method has been described in detail elsewhere [42]. A small sample, usually in the shape of a parallelepiped with dimensions in the range ∼1-5 mm, is held lightly between two piezoelectric transducers. The first transducer is excited at constant amplitude across a range of frequencies in the vicinity of 1 MHz, which in turn causes the sample to resonate at particular frequencies. The second transducer detects these resonances, or normal modes of vibration of the sample. The square of a given resonance frequency is directly proportional to the elastic constants associated with the normal mode involved [42]. In the low temperature head of the Cambridge instrument [43], the sample is placed across a pair of faces between the two transducers in a mount which is lowered vertically into a helium flow cryostat. A few mbar of helium are added to the sample chamber to allow heat exchange between the sample and the cryostat. Absolute values of temperature are believed to be accurate to within ±1 K, as checked against the transition temperature of SrTiO 3 , and temperature stability during data collection is ±0.1 K or better.
The powder sample of CoF 2 used as a starting material was purchased from Sigma Aldrich. A polycrystalline pellet was prepared by first pressing some of the powder into a thin disc and then firing it in air at 600 • C. Two parallelepipeds, with dimensions 1.745 × 4.189 × 3.717 mm 3 (0.122 g) and 1.689 × 3.741 × 3.741 mm 3 (0.1062 g), were cut from the fired pellet using a fine annular diamond saw. These are referred to below as samples 1 and 2, respectively. RUS data were collected in cooling and heating sequences. Sample 1 was cooled from 290 to 110 K in 30 K steps with a 20 min equilibration time at each temperature before data collection, followed by 110 → 10 K in 10 K steps. It was then heated from 10 to 30 K in 2 K steps, 30 to 50 K in 1 K steps, 50 to 110 K in 2 K steps and 110 to 290 K in 5 K steps, all with a 15 min settle time for thermal equilibration. Each spectrum contained 65 000 data points in the frequency range 100-1200 kHz. Data collection for sample 2 followed exactly the same temperature steps, except that 130 000 data points were collected in the frequency range 100-2000 kHz.
Bulk (K ) and shear (µ) moduli were determined by matching observed peak frequencies with calculated frequencies using the DRS software [42] and assuming an isotropic medium. The result of fitting to the frequencies of 26 resonance peaks in a spectrum collected at 284 K from sample 2 was µ = 38.51 ± ∼0.03 GPa and K = 103 ± ∼1 GPa, with an rms error on the fitting of 0.25%. This compares with 39, 83.6 GPa (Hashin-Shtrikman bounds from single crystal data of [44] in [45]) and 37.7, 110.4 GPa (Voigt-Reuss-Hill values from single crystal data listed in [46]). The density from the dimensions and mass of the sample was 4.493 g cm −3 , implying a porosity (which includes some contribution of slightly damaged edges of the parallelepiped) of 2.2% when compared with a theoretical density of 4.592 g cm −3 calculated from lattice parameters. Correction for this using the equations of Ledbetter et al [47] gave µ = 40.1, K = 110.0 GPa as the best estimate of absolute values for a fully dense polycrystalline sample. Determination of absolute values of the elastic properties was not a prime consideration in the present study, so the porosity correction was not made to all the data. Similarly, no correction was made for thermal expansion of the sample because the change would be very small and the effects of magnetic ordering can be seen in the uncorrected data.
All spectra were transferred to the software package Igor Pro (WaveMetrics) for analysis. Peak positions and widths at half height were determined for a selection of peaks by fitting with an asymmetric Lorentzian function. The mechanical quality factor, Q, was calculated using the relationship Q = f / f , where f is the peak frequency and f is the width of the peak at half its maximum height. The inverse of the mechanical quality factor, Q −1 , is a measure of acoustic dissipation in the sample. Segments of RUS spectra obtained during heating, stacked in proportion to the temperature at which they were collected ((a) from sample 1, (b) from sample 2). The left axis is really amplitude, but is labelled as temperature. Peaks with the lowest frequencies occur in the spectra collected at ∼40 K which are shown in black.

Results
Figure 1(a) shows segments of the raw RUS spectra from sample 1 stacked in proportion to the temperature at which they were collected (∼80 K at the top of the stack and ∼12 K at the bottom). Figure 1(b) shows similar data from sample 2. Antiferromagnetic ordering is clearly marked by elastic softening as T → T N from both sides. A slight change in the trend near 50 K in spectra from sample 1 is not reproduced in the spectra from sample 2 and is therefore considered to be an artefact, due either to a defect such as a small crack, or in some way to the data collection sequence. The antiferromagnetic ordering transition is clearly accompanied also by changes in resonance peak widths. Figure 2 shows the evolution with temperature of f 2 and Q −1 for several representative peaks at different frequencies in spectra collected from sample 2. Their f 2 values have been scaled to be the same at room temperature, for easy comparison, and since the resonances mainly involve shearing motions, they are indicative primarily of the evolution of  the shear modulus. Softening evident in the primary spectra (figure 1) is asymmetric, occurring more steeply as T → T N from below than from above. There is a break between ∼39 and ∼41 K; this is taken to mark the Néel point which is given below as 39.5 K. The onset of softening with falling temperature in the stability field of the high temperature phase occurs between ∼100 and ∼150 K. At first view, differences between f 2 data for the separate resonance peaks do not appear to depend systematically on frequency and are probably due in part to different contributions from breathing modes, related to the bulk modulus. Variations in Q −1 are limited to a slight premonitory effect within a few degrees above T N , a steep increase at the transition point and then a decay back to the same low values as for the high temperature structure by ∼20 K. The magnitude of the peak in Q −1 at ∼39 K varies systematically with the frequency of the resonance peak from which the values were determined, with the largest maximum value at the highest frequency and the lowest maximum value at the lowest frequency. Figure 3 shows the variation of the bulk and shear moduli as a function of temperature for both samples. Rms errors for the fits using 15-20 peaks were <∼0.5% and estimated uncertainties for K and µ were ±1.5% and ±0.15%, respectively. As expected, the shear modulus shows the same pattern of softening associated with the antiferromagnetic ordering transition as is seen for individual resonance peaks in figure 2. The maximum amount of softening is ∼2.5% but this reduces to ∼0.5% at ∼10 K. The evolution of µ below T N is essentially the same in both samples. Data for K are more scattered, with the observed anomalies below T N close to the level of the estimated experimental uncertainties and apparently not reproducible. In particular, they provide no obvious evidence for softening of the bulk modulus on either side of the transition point.

Strain analysis, order parameter evolution and elastic softening below T N
Although there are insufficient data to produce a quantitative description of the elastic anomalies, their form can at least be predicted on the basis of a conventional Landau expansion in the magnetic order parameter, Q m , and coupling with spontaneous strains. In order for the structure to become antiferromagnetic with spins parallel to the c-axis and no breaking of crystallographic symmetry, the transition must proceed according to irrep m + 2 of parent magnetic space group P4 2 /mnm1 , leading to magnetic sub group P4 2 /mnm . The group theory program ISOTROPY [48] has been used to determine the permitted terms in a conventional Landau expansion for the excess free energy, G, due to this symmetry change as where a, b and c are normal Landau coefficients, T c is the critical temperature, e 1 − e 6 are spontaneous strains, λ 1 − λ 6 are strain/order parameter coupling coefficients, and C o ik are the bare elastic constants, i.e. those excluding the effect of the phase transition. s is a saturation temperature for the order parameter, with the coth function giving the correct form of variation as T → 0 K [49].
The spontaneous strains are expected to evolve with Q m according to and these relationships can be tested with measurements of lattice parameters as a function of temperature. Data of Chatterji et al [19] have been reanalysed in this context, as set out in the appendix, to show that the observed strains have values up to 0.001, that they scale with the intensities of  (1).
superlattice reflections in powder neutron diffraction patterns (∝ Q 2 m ), and that the order parameter follows a pattern which can be described effectively as being close to tricritical (figures A.1-A.3). In addition, there is a clear tail in e 1 which extends to ∼70 K above T N , indicative of a degree of short range ordering ahead of the phase transition (figure A.2).
Softening of the shear modulus appears to be typical of a phase transition with strain/order parameter coupling, and can be tested against the normal expectations predicted using the equation of Slonczewski and Thomas [50]: This gives the expressions for individual elastic constants listed in table 1 (following [39,51]). Coupling terms of the form e 2 Q 2 , such as λ 6 e 2 6 Q 2 m , give the variation of the elastic constants more simply by applying C ik = ∂ 2 G/∂e i ∂e k , hence giving C 66 = C o 66 + 2λ 6 Q 2 m . Expressions for the Voigt limit, µ V and K V , illustrate how the separate coupling parameters contribute to the softening: and Terms originating from coupling of the form λe 2 Q 2 would be expected to contribute to softening or stiffening in proportion to Q 2 m , but this is usually a small effect in comparison to terms originating from coupling of the form λeQ 2 which include the inverse susceptibility, . For a second order transition, Q 2 χ is constant and the softening is therefore expected to be discontinuous by a fixed amount. For a tricritical transition softening at the transition point is also expected to be discontinuous, but with a non-linear recovery as temperature reduces in the stability field of the low symmetry phase. The amount of softening in the present case clearly depends on the coefficients, λ 1 and λ 3 , and values of these can be estimated from equations (2) and (3) figure 3 are of sufficient precision to show that the pattern of softening predicted on this overall basis for the shear modulus is consistent with a strain/order parameter relaxation mechanism at a phase transition which is approximately tricritical in character.
Discrimination between models for the evolution of the order parameter, as shown in figure A.3, depends in part on the form of the excess entropy, S. Equation (1) is based on the assumption of displacive character, for which The order/disorder limit would have the excess entropy as purely configurational, for which higher order terms in Q m are required if it is to be expressed as a series expansion. As set out in the appendix, heat capacity data from Catalano and Stout [24] have been reanalysed and used to confirm that the thermodynamic behaviour is indeed well represented by order/disorder character, the order parameter evolution is similar in form to that of Landau tricritical, and there is a degree of precursor ordering ahead of the Néel point.

Precursor effects at T > T N
Tails in the data for e 1 ( figure 5(b)) and C p (figure A.4) at T > T N signify premonitory effects, indicative of short range ordering that has stronger correlations within the a-b plane than in the c-direction [19]. This premonitory ordering is presumed also to account for the elastic softening evident in the data shown in figures 2 and 3 over the same temperature interval. In order to characterize the softening more quantitatively, excess values of the shear modulus have been obtained by first fitting a baseline, f 2 o , with the form of equation (A.1) to f 2 data in the temperature interval 128-286 K for the resonance mode with frequency ∼445 kHz at room temperature (figure 2). Differences between this fit, extrapolated down to 10 K, and the observed values are shown as | f 2 / f 2 o | in figure 4. Values of the saturation temperature, θ s , from the fitting were 157 K for the 445 kHz peak and between 241 and 253 K for the other peaks. In this context equation (A.1) merely provides a convenient description of reducing slope as T → 0 K, which is comparable in form to a description by the Varshni equation (e.g. see [52]). Data for Q −1 , the excess of Q −1 with respect to a straight line fit to values above ∼50 K, have been added to figure 4, excluding high values that were due to saturation of the amplifier.
The complete data in figure 4 show that the evolution of changes in f 2 , i.e. changes predominantly in the shear modulus, is independent of frequency at T > T N and that the softening is not accompanied by any detectable increase in acoustic loss. For improper ferroelastic or co-elastic transitions driven by a soft optic phonon, such softening is generally understood in terms of local fluctuations and can be described by a power law of the form C ik = A ik |T − T c | K [53][54][55][56][57].
Log-log plots of the data for | f 2 / f 2 o | and (T − 39.5), are distinctly non-linear in the present case, however. An alternative variation due to short range ordering is shown by relaxor ferroelectrics, such as Pb(Mg 1/3 Nb 2/3 )O 3 (PMN), where the softening is attributed to coupling between dynamical polar nano-regions and acoustic phonons. As a pure piece of empiricism it was found that softening at high temperatures in PMN can be described using a Vogel-Fulcher type of equation that is usually used to describe a freezing process according to τ is a relaxation time, τ o is the inverse of the attempt frequency, U is an effective activation energy, k B is the Boltzmann constant and T f is a characteristic freezing temperature. In this case, the low loss implies ωτ 1, where ω(=2π f ) is the angular frequency at which the measurement was made. For PMN the relaxation time was replaced by the change in shear modulus (figure 10 of [58]). For CoF 2 the change in f 2 of a single RUS mode is used in place of the change in shear modulus, but the same empiricism provides an equally good description of the premonitory softening in CoF 2 . The black curve in figure 4 is for | f 2 / f 2 o | = 0.0001 exp(95/(T − 15)). In the absence of any physical justification, this has not been explored further but, if real, the value of U/k B = 95 K would imply an effective activation energy barrier of ∼0.8 kJ mol −1 or ∼0.008 eV, with an effective zero-frequency freezing temperature of ∼15 K.
The form and magnitude of dynamical local ordering at T > T N which gives rise to the elastic softening also gives the tail in strain, e 1 , and the two effects are closely interdependent. As shown in figure 5, e 2 1 scales closely with | f 2 / f 2 o |, implying a relationship e 2 1 ∝ µ ∝ | f 2 / f 2 o |. The macroscopic order parameter Q m is strictly zero, but the average degree of local ordering can be represented by a short range order parameter, σ m . In the simplest case, both the strain and the elastic softening would behave as excess properties and would then be expected to scale in some simple way with σ m . There is no change in symmetry involved, so there are no constraints on the form of coupling between σ m and e 1 or e 3 , which can be λσ m e, λσ m e 2 , λσ 2 m e 2 , λσ 2 m e, etc. A term λσ m e 2 1 added to the elastic energy 1 2 C o 11 e 2 1 will cause a renormalization of the elastic constant, C 11 , as The observed relationships in figure 5 could then result if the contribution to softening of the shear modulus comes mainly from C 11 (together with C 22 and C 12 ) and if the same coupling term gives rise to a dependence σ m ∝ e 2 1 . There is no equivalent tail in e 3 , implying that the coupling coefficient for an equivalent term λσ m e 2 3 is small and, presumably, that the softening of C 33 would also be small. According to this treatment, the tail in e 2 1 and the variation of | f 2 / f 2 o | are measures of the dynamical average of the degree of short range order as T → T N . It should also be pointed out that additional contributions to the shear modulus may come from C 44 , C 66 , etc, through terms such as λσ 2 m e 2 4 and λσ 2 m e 2 6 , but in MnF 2 at least, the precursor softening of C 44 and C 66 is very small [59].

Acoustic attenuation
Attenuation of acoustic waves in the vicinity of a magnetic ordering transition might be expected to follow a power law dependence on the reduced temperature as [1,52], where α is the attenuation coefficient (∝Q −1 ). This has been used to describe the pattern of attenuation in a temperature interval of up to a few degrees above the Néel point of Data from the same temperature interval for the peak near 445 kHz, which includes a larger dependence on the bulk modulus, give η = 0.6. The attenuation mechanism, at least for temperatures within ∼1 K of T N , has been attributed to energy density fluctuations [3]. Over the relatively narrow range of frequencies observed in the present study, there is too much scatter in the data to test the expected frequency dependence, but the ω 2 dependency has been observed for MnF 2 (e.g. [1,60]). In the absence of any ferroelastic twin microstructure, the acoustic attenuation at T < T N is more likely to be intrinsic. A log-log plot for data between 12.8 and 38.8 K from the heating sequence (figure 6(a)) shows that a conventional power law does not provide a good description over the entire temperature range below T N . The dashed line in figure 6(a) has a slope of 0.6 which provides a reasonable fit for T ∼28-39 K, but this extends well away from the temperature interval expected for critical fluctuations. It compares with 0.29 ± 0.03 [60] and 0.13 [62] reported for immediately below T N from measurements at ∼10-60 MHz in MnF 2 and 0.55 ± 0.06 at ∼0.1-1 MHz in Fe 2 O 3 [67]. In spite of the scatter in the data, there appears to be a discernible dispersion with frequency, as shown by data for Q −1 in figure 6(b) at three temperatures immediately below T N (37.6, 36.6, and 35.6 K). These variations are expected to conform to normal Debye-like behaviour, where M U is the unrelaxed modulus, M R the relaxed modulus and M o =(M U M R ) 1/2 . The observations are that Q −1 , the loss component associated with the antiferromagnetic structure, increases with increasing frequency in a manner that may not be far from linear ( figure 6(b)), which is consistent with ωτ 1. The relaxation time immediately above T N in MnF 2 is ∼3 × 10 −9 , as determined by Kawasaki and Ikushima [68] and Moran and Lüthi [69] from measurements of acoustic velocity and attenuation at 10 MHz. At 1 MHz this would give ωτ ≈ 0.02.
For a small change in the observed modulus, δ M = M U − M(ωτ ), equations (12)- (14) can be combined to give which may be used to say something about the relaxation times at T < T N . For direct comparability it is necessary to divide the values of Q −1 given above by √ 3, to take account of the fact that Q −1 obtained from widths of the amplitude of resonance peaks is a factor of ∼ √ 3 larger than the true value [70][71][72][73]. Using Q −1 = 0.0015/ √ 3 as representing Q −1 at T N when measured at ∼1882 kHz would give δ M/M = 0.02 as the expected amount of anelastic softening if the relaxation time of the loss mechanism remained at 3 × 10 −9 s. However, this is an order of magnitude larger than the frequency-dependent variations of f 2 / f 2 o shown in figure 6(b), signifying that the loss mechanism below T N has a relaxation time which is at least an order of magnitude slower than just above T N . In order to obtain absolute values for τ , it is necessary to estimate values for the equivalent of the unrelaxed modulus, M U , but the data for f 2 / f 2 o are too scattered for this. (A substantial part of the experimental uncertainty arises from the choice of baseline f 2 o and the relatively narrow range of frequencies which can be obtained by RUS.) Nevertheless, the relatively steep decline in Q −1 with decreasing temperature in the interval of ∼20 K below T N is most likely due to a significant lowering of the relaxation time with decreasing temperature and this is counter to what would be expected for a thermally activated loss mechanism. As also proposed by Moran and Lüthi [69], this is more in line with the Landau-Khalatnikov relation for critical slowing down [74] where T c is the critical temperature (T N in this case) and τ o is a constant. If τ scales approximately with Q −1 , this would give a slope of −1 in figure 6(a) and a dashed line with this slope is included as a guide to the eye. The inverse susceptibility, χ −1 = ∂ 2 G/∂ Q 2 m , would go linearly to zero at T N for an order parameter evolution which follows Landau tricritical behaviour and this could be the dominant factor in determining the temperature dependence of the relaxation time over a temperature interval of at least 20 K below T N , which is well beyond the expected range of any critical fluctuations. The most straightforward model is then of spin-lattice coupling in which there is an adjustment of Q m to an applied stress through the strain/order parameter coupling terms. The restoring force would depend on χ −1 , though a small thermal barrier could still operate. In other words, the small strain induced in the RUS experiment, which may be ∼10 −6 [72], would simply cause an adjustment in the degree of magnetic order with a small phase lag.

Discussion
Apart from there being different driving mechanisms, i.e. antiferromagnetic ordering rather than softening of an optic phonon, variations in the strain, elastic and anelastic properties which accompany the phase transition in CoF 2 are remarkably similar in form to those shown by quartz at the β (hexagonal) ↔ α (trigonal) transition [51,75,76]. Both transitions are co-elastic, i.e. they do not involve a symmetry-breaking (shear) strain, both have non-symmetry-breaking strains which scale with the square of the order parameter, and the order parameter evolution with temperature in each case can be represented as being close to tricritical. Both show non-linear softening of elastic constants below the transition point and small increases in acoustic dissipation across a narrow temperature interval around the transition point. The Slonczewski-Thomas softening mechanism involves relaxation of the order parameter in response to changes of strain when an external stress is applied. To be observed, this relaxation must occur on a timescale which is shorter than that of the strain response, <∼10 −6 s in this case. All the data presented here are consistent with this pattern of behaviour. The order parameter susceptibility would not be expected to be that for a displacive system, and an order-disorder model not dissimilar from the Bragg-Williams model of a spin 1/2 system would produce a more nearly quantitative alternative.
Differences in the magnitude of softening between quartz and CoF 2 (∼2% softening of the shear modulus of CoF 2 but ∼80% and ∼8% for K and µ, respectively, in quartz [76]), and between magnetic ordering versus displacive systems more generally, can be understood in terms of the strength of the strain coupling and the magnitudes of the entropies involved. For example, the amount of softening at a second order co-elastic transition scales with λ 2 /b, where b is the Landau fourth order coefficient, excluding renormalization by coupling with strain, and is close to being equal to aT c (a is the Landau coefficient for the second order term, T c the transition temperature). For a tricritical transition, b ≈ 0 and c = aT c , where c is the coefficient for the sixth order term. Spontaneous strains are about an order of magnitude larger in quartz than they are in CoF 2 , so λ 2 will be approximately two orders of magnitude greater. In quartz, the total excess entropy is ∼ 4.9 J K −1 mol −1 [75], in comparison with a total excess entropy for CoF 2 of ∼ 4 J K −1 mol −1 so, as a first approximation, the a coefficients will be similar. T c for the quartz transition is ∼840 K, in comparison with 39 K, so the c coefficient will be comparably larger, and this will reduce the difference in expected softening by approximately one order of magnitude, which is not far from what is observed.
Displacive systems more typically have smaller excess entropies and strains of a few . For example, symmetrybreaking shear strains up to ∼−0.002 and volume strains up to ∼0.005 accompany the (soft mode) octahedral tilting transition in LaAlO 3 , and the total excess entropy is <2 J K −1 mol −1 [77]. Softening of the shear and bulk moduli below the second order transition at T c = 817 K amounts to ∼40% and ∼25% respectively [72]. Tilting transitions in SrZrO 3 are accompanied by shear and volume strains of up to ∼±0.003, ∼0.002, respectively, together with softening of the shear and bulk moduli by up to ∼40% and ∼8% [78]. The effective (non-symmetry-breaking) shear strain accompanying antiferromagnetic ordering in CoF 2 can be expressed in terms of the tetragonal strain e t = 1 √ 3 (2e 3 − e 1 − e 2 ) which reaches a maximum value of ∼0.002, and the volume strain (2e 1 + e 3 ) reaches a maximum value of ∼0.001. Thus the shear strains are comparable, but the volume strain is smaller. The associated elastic anomalies should be smaller, due to the larger excess entropy, but greater for the shear modulus than for the bulk modulus, exactly as observed.
Given that the general pattern of softening shown by the bulk elastic properties of CoF 2 conforms to the Slonczewski-Thomas mechanism, it must be expected that the variations of single crystal elastic constants at T < T N will be described correctly by the equations in table 1. While there are no available single crystal data for CoF 2 , data for effectively the same transition in MnF 2 [59] support this view. Each of C 44 , C 66 and 1/2 (C 11 − C 12 ) show continuous variations through the Néel point with increasing differences from the bare elastic constants which would scale with Q 2 , but C 11 and C 33 show the characteristic dip due to the influence of the inverse susceptibility, χ. Some influence of pseudoproper ferroelastic softening of 1/2 (C 11 − C 12 ), across a temperature interval of at least ∼300 K as T → T N from above, is also seen however. This is attributable to the P4 2 /mnm ↔ Pnnm instability somewhere not far away in pressure-temperature parameter space, since a soft optic mode has been observed by Schleck et al [79]. Contributions from Jahn-Teller cooperative transitions to the elastic softening can be ruled out in the light of the structural data which shows significant distortions of the CoF 6 octahedra that are closely similar at 15 and 300 K [17].
Acoustic dissipation associated with shear modes at the β ↔ α transition in polycrystalline quartz is also low, and, as in CoF 2 , rises to a distinct maximum at the transition temperature [76]. Softening as T → T c from above is not accompanied by any obvious tail in the strain [75], however, and the power law dependency is entirely consistent with a phonon mechanism and fluctuations. In contrast, the pattern of softening in CoF 2 , which occurs in the temperature interval where there is a tail in the strain, is represented better by a Vogel-Fulcher expression. The softening mechanism is presumed to involve dynamical order/disorder of spin orientations which couple with the acoustic modes, perhaps with a low thermal barrier for reversal of some spin orientations within relatively well ordered clusters. The drop in relaxation time below T N is consistent with a different loss mechanism operating, and the simplest mechanism which might give the observed temperature dependence would involve spinlattice coupling and restoring forces responding to an applied stress which depend substantially on the susceptibility of the magnetic order parameter.
Finally, an additional magnetoelastic contribution could be due to piezomagnetism. Previous theoretical studies [80] have predicted that CoF 2 might have a significant piezomagnetic effect based on considerations of the crystal symmetry [81]. A linear compression in the a-b plane would give rise to a magnetization along the c-axis, which is also the axis for sublattice alignment in the antiferromagnetic state. Recently it was suggested that the piezomagnetic effect might cause a linear relationship between the strain and the induced ferromagnetic moment [82]. If this holds true for the present case, the relationship between the antiferromagnetic order parameter, Q AFM , and induced ferromagnetic order parameter, Q FM , would be Q FM ∝ Q 2 AFM , since the strain is proportional to the square of the antiferromagnetic order parameter. However there is no direct evidence for a separate contribution from piezomagnetism in the data presented here.

Conclusions
In combination, elastic and anelastic properties measured by RUS for CoF 2 in the present study, together with lattice parameter and heat capacity data from the literature, reveal patterns of spin-lattice coupling and dynamics which are likely to be characteristic of aspects of magnetic transitions in insulating oxides. In particular: (1) Elastic softening at T < T N can be understood in terms of the same phenomenological strain/order parameter coupling as occurs in displacive systems. Although the order parameter relates to spin ordering, it evolves in a manner that is not far from the Landau tricritical solution for a displacive system. Differences in the magnitudes of the softening can then be understood in terms simply of differences in the strength of coupling, which tends to be weaker in magnetic systems than for displacive transitions. (2) It is proposed that elastic softening, C, at T > T N can be understood in terms of dynamical ordering of spin orientations, possibly with a small activation energy barrier for local reorientations, and coupling of these with acoustic modes. The short range ordering is sufficient to give a measurable tail in strain, e, which scales simply as e 2 ∝ C. (3) Relaxation times for spin-lattice coupling associated with the antiferromagnetic ordering transition in CoF 2 are sufficiently fast that, when measured at ∼1 MHz by RUS, there is no evidence of acoustic attenuation in the dynamical region above T N . Slowing down of the spinlattice coupling as T → T N is sufficient to result in a small but significant peak in acoustic attenuation. At T < T N , the relaxation times appear to show a weak temperature dependence which correlates, at least qualitatively, with the intrinsic order parameter susceptibility. (4) CoF 2 provides a model for the likely antiferromagnetic part of magnetoelastic behaviour in more complex multiferroic materials with additional displacive instabilities, Jahn-Teller effects and ferroelastic microstructures. A complete understanding of this behaviour remains incomplete, however, in view of the fact that in other selected systems, such as hexagonal YMnO 3 [83] and the organic radical β-p-NCC6F4CNSSN [82], antiferromagnetic ordering is accompanied by elastic stiffening, rather than softening.

Acknowledgments
RUS facilities were established in Cambridge through a grant from the Natural Environment Research Council of Great Britain to MAC, which is gratefully acknowledged (NE/B505738/1). C J Howard is thanked for discussions and comments on the manuscript.

Appendix. Strain coupling and order parameter evolution
In order to analyse the spontaneous strains associated with antiferromagnetic ordering in CoF 2 , a baseline of the form [49,84] a o = a 1 + a 2 θ s coth θ s T (A.1) was first fit to the c lattice parameter data from Chatterji et al [19]. This provided a constrained value for θ s to use in the subsequent fitting of the a parameter ( figure A.1).
Values of e 1 and e 3 were calculated in the usual way from the excess in the lattice parameters below the magnetic transition, i.e.  [19]), have been added to confirm that this treatment gives the expected dependence on Q 2 m . The correlation for e 3 is close over the entire temperature interval, while e 1 correlates closely below ∼38 K but has a tail above T N that extends up to ∼120 K. Correlations between the trends shown for e 1 and e 3 with respect to I 100 are more robust than in the original analysis of Chatterji et al [19] probably because of the choice of baseline used here for calculating the strains.
Variations of the equilibrium order parameter derived from equation (1)   where A is a constant, and n = 2 for a second order transition or n = 4 when it is tricritical. Using I 100 to represent Q 2 m , fits of equation (A.2) in figure A.3(a) show that the evolution of the order parameter can be adequately represented by a Landau tricritical solution with s = 95 K. Order parameter variations derived from the Brillouin function for systems with spin 3/2 or spin 1/2 [86,87] do not fit the data as well ( figure A.3(b)). A second order solution (n = 2) does not quite give as good a description as the tricritical solution (n = 4). Also shown are equivalent intensity data for the antiferromagnetic transition in FeF 2 from Chatterji et al [19], which can be described using the same function for tricritical character, with s = 60 K (T c = 79 K). Strempfer et al [16] showed that Ising behaviour also provides a reasonable representation of the sublattice magnetization (their figure 4).
Equation (A.2) represents the solution for a displacive system but, in this case, the behaviour is close to the order/disorder limit as can be shown from considerations of the excess entropy. Heat capacity data from Catalano and Stout [24] have been used to determine the observed excess entropy associated with the magnetic phase transition. A baseline of the form of equation (A.1) was fit to the low temperature heat capacity data as shown in figure A.4. This gave an excess heat capacity, C p , from which the excess entropy and enthalpy were calculated as 2) to neutron diffraction intensity data (I 100 ∝ Q 2 m ) for CoF 2 (red crosses) and FeF 2 (blue circles), for n = 4 (a) and n = 2 (b). For CoF 2 the fit was to data from figure A.2 between 2.5 K and T c = 39 K, giving a proportionality constant (with I 100 in place of Q 2 m ) of = 3.5794 × 10 5 and s = 95.48 K (n = 4) or 2.827 × 10 5 , 90.70 K (n = 2). Intensity data for FeF 2 were taken from Chatterji et al [19]; the fit for n = 4 gave a proportionality constant of 1.7763 × 10 [9] and s = 60.17 K. Also shown (right axis, bottom) are solutions to the Brillouin function for systems with S = 3/2 and 1/2.
Variations of the excess entropy and enthalpy obtained in this way are shown in figures A.5(a) and (b), respectively, along with I 100 (∝ Q 2 m ). The measured excess entropy does not scale linearly with the measured variation of Q 2 m and the susceptibility, χ, derived from equation (1) would not provide a quantitative description of the elastic softening, therefore. The simplest model which would reproduce, semiquantitatively, the variations of Q 2 m , H and S is one site (spin = 1/2, n = 1) Bragg-Williams ordering. The order parameter would be expected to vary according to which is the solution derived from the Brillouin function for spin 1/2 and is closer to the variation of Landau tricritical than Landau second order [88].  where n = 1 and R is the gas constant. Values of Q m estimated from the I 100 data, scaled to 1 at low temperature, have been used to calculate S BW as shown in figure A.5(a). Although the model values are somewhat higher than values derived from the excess heat capacity (figure A.5(b)), the form of variation is correct. Using T c = 39 K and Q m = 1 for complete order, equation (A.4) gives | H BW | = 162 J mol −1 . This is higher than 'observed' but the form is again correct. A degree of short range ordering ahead of the Néel point, not included because of the choice of baseline used to determine C p , would contribute to the differences between observed and calculated values for both the excess enthalpy and entropy.