Specific heat of Na$_{1-x}$V$_2$O$_5$ single crystals

Temperature dependences of the specific heat $C$ and the magnetic susceptibility $\chi$ of Na$_{1-x}$V$_2$O$_5$ single crystals ($x=0, 0.01, 0.02, 0.03$, and 0.04) are studied. In NaV$_2$O$_5$, the transition to the spin-gap state ($T_c=34\,$ K) is accompanied by a sharp decrease in $\chi$, while $C$ exhibits a $\lambda$-shaped anomaly. At low temperatures, the specific heat of NaV$_2$O$_5$ is approximated by the sum of phonon $\sim T^3$ and magnon $\sim {\rm exp}(-\Delta/T)$ contributions, which makes it possible to estimate the Debye temperature $\Theta_D=336\,$ K and the gap in the magnetic excitation spectrum $\Delta=112\,$ K. With the departure from stoichiometry, the anomalies observed in the behavior of $\chi$ and $C$ are spread and shifted to lower temperatures. The low-temperature specific heat of nonstoichiometric samples is determined by the sum of phonon and magnon components and the contribution due to the presence of defects. The values of magnetic entropy characterizing the phase transitions in Na<sub class="a-plus-plus">1-$x$ </sub>V<sub class="a-plus-plus">2</sub>O<sub class="a-plus-plus">5</sub> are calculated.