Landau model for the elastic properties of the quasi-one-dimensional antiferromagnetic compound<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="normal">Cs</mml:mi><mml:mi mathvariant="normal">Ni</mml:mi><mml:msub><mml:mi mathvariant="normal">Cl</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math>

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 $\ensuremath{\beta}$ associated with the order parameter of the elliptical state below $T=4.10\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ is $\ensuremath{\beta}=0.33$. 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 $7.0\phantom{\rule{0.3em}{0ex}}\mathrm{kbar}$, 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 $d{T}_{{N}_{1}}∕dP=0.17\phantom{\rule{0.3em}{0ex}}\mathrm{K}∕\mathrm{kbar}$ and $d{T}_{{N}_{2}}∕dP=0.065\phantom{\rule{0.3em}{0ex}}\mathrm{K}∕\mathrm{kbar}$, 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, $d{J}_{\ensuremath{\Vert}}∕dP=0.45\phantom{\rule{0.3em}{0ex}}\mathrm{kbar}$. 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.

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