Landau model for the elastic properties of the quasi-one-dimensional antiferromagnetic compound $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$

We present a Landau model that accounts for the unusual elastic properties of the quasi-one-dimensional antiferromagnetic compound $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$. The model’s predictions are tested using published strain measurements and recent high resolution sound velocity measurements realized as a function of temperature and pressure. In particular, we derive an analytical expression which indicates how the low temperature dependence of the elastic constants is related to the critical order parameters. A close inspection of the temperature dependence of $C_{33}$ indicates that the value of the critical exponent $\beta$ associated with the order parameter of the elliptical state below $T=4.10\mathrm{K}$ is $\beta =0.33$. The obtained value agrees with the expected critical behavior of $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$ in the absence of an external magnetic field (conventional three-dimensional $XY$ criticality). Using sound velocity measurements under hydrostatic pressures up to $7.0\mathrm{kbar}$, we also derive the pressure-temperature phase diagram of $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$. With increasing pressure, we observe that the critical temperatures $T_{N_1}$ and $T_{N_2}$ increase at a rate of $d{T}_{N_1}∕dP=0.17\mathrm{K}∕\mathrm{kbar}$ and $d{T}_{N_2}∕dP=0.065\mathrm{K}∕\mathrm{kbar}$, respectively. Finally, taking advantage of the magnetoelastic coupling in the paramagnetic state, we could estimate the pressure dependence of the interchain exchange coupling along the $c$ axis, $d{J}_{\Vert }∕dP=0.45\mathrm{kbar}$. All these results seem to indicate that the Ising-type anisotropy and the quasi-one-dimensional character of $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$ are both enhanced with pressure.

Figure 1

Schematic magnetic phase diagram of $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$ for $H\Vert \widehat{c}$. As shown by Plumer et al. (Ref. 15), the theoretical predictions agree very well with the experimental observations.

Figure 2

Relative variation of the elastic constants as a function of temperature, (a) and (b) show $C_{33}$ and $C_{11}$, respectively. Symbols represent the data points while the continuous line corresponds to the theoretical prediction obtained using Eqs. (10, 11). The dashed line is obtained by taking into account higher coupling terms, Eqs. (12, 13). The inset in (a) shows $C_{33}$ as a function of the reduced temperature $\tau =1-T∕T_{N_2}$ on a log-log scale. The value of the critical exponent obtained from the fit is $\beta =0.33\pm 0.02$.

Figure 3

Comparison between experimental and theoretical predictions for the low temperature thermal expansion of $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$ [along the $c$ axis (a) and in the basal plane (b)]. The experimental results (symbols) correspond to data obtained by Rayne et al. (Ref. 16) while the lines correspond to the mean field predictions obtained by taking the temperature derivative of Eqs. (7, 8).

Figure 4

Comparison of the relative variation of the sound velocity (a) measured as a function temperature at constant pressures, using longitudinal waves propagating along the hexagonal axis of $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$. (b) shows the pressure-temperature phase diagram derived from sound velocity measurements.

Figure 5

Relative variation of the sound velocity as a function of temperature for longitudinal waves propagating along the hexagonal axis of $\mathrm{Cs}\mathrm{Ni}{\mathrm{Cl}}_3$. The different curves represent results obtained at different pressures. The inset shows the contribution associated with the magnetoelastic coupling.

Figure 6

Relative variation of $C_{11}$ as a function of temperature. The inset shows the contribution possibly associated with the magnetoelastic coupling in the basal plane.