Revised Magnetic Structure and Tricritical Behavior of the CMR Compound NaCr$_2$O$_4$ Investigated with High Resolution Neutron Diffraction and $\mu^+$SR

The mixed valence Cr compound NaCr$_2$O$_4$, synthesized using a high-pressure technique, offers a unique playground for investigating unconventional physical properties in condensed matter. In the present study, muon spin rotation/relaxation ($\mu^+$SR) and high-resolution neutron powder diffraction (NPD) measurements were carried out to clarify the true magnetic ground state of this interesting compound. Our detailed study brings new insight, allowing us to confirm the existence of a commensurate antiferromagnetic order (C-AFM) and to extract its ordered Cr moment $\mu^{\rm C}_{\rm Cr}=(4.30\pm0.01)\mu_B$. Such a value of the ordered moment is in fact compatible with the existence of high-spin Cr sites. Further, the value of the canting angle of the Cr spin axial vector is refined as $\theta_{\rm c}=(8.8\pm0.5)^{\circ}$. Employing high-quality samples in combination with time-of-flight NPD, a novel magnetic supercell was also revealed. Such supercell display an incommensurate (IC)-AFM propagation vector (0~0~${\textstyle \frac{1}{2}-}\delta$), having an ordered moment $\mu^{\rm IC}_{\rm Cr}=(2.20\pm0.03)\mu_B$. It is suggested that the C-AFM and IC-AFM modulations are due to itinerant and localized contributions to the magnetic moment, respectively. Finally, the direct measurement of the magnetic order parameter provided a value of the critical exponent $\beta = 0.245 \approx \frac{1}{4}$, suggesting a non conventional critical behavior for the magnetic phase transition in NaCr$_2$O$_4$.