Specific heat of sodium-vanadium bronzesNaxV2O5between 0.5 and 50 K

The specific heat of ${\mathrm{Na}}_x{\mathrm{V}}_2{\mathrm{O}}_5$, $x=0.2, 0.25, 0.3, 0.36, \mathrm{and} 0.41$, and of pure ${\mathrm{V}}_2$ ${\mathrm{O}}_5$ single crystals, was measured between 0.5 and 50 K in zero magnetic field as well as in fields of different intensities up to 2.8 T. The specific heat of pure ${\mathrm{V}}_2$ ${\mathrm{O}}_5$ shows a well-defined maximum in $\frac{C}{T^3}$; this indicates the presence of low-lying modes which can be described by Einstein oscillators. The lowest Einstein frequency ${\nu }_E$ obtained from the specific-heat data is 76 ${\mathrm{cm}}^{-1\mathrm{}}$ while the Debye temperature ${\Theta }_D=164\mathrm{}$ K. The addition of sodium to ${\mathrm{V}}_2$ ${\mathrm{O}}_5$ enhances the specific heat below 7 K considerably. The excess specific heat is linear in temperature and is magnetic field dependent and can be described by a simple model of many three-level systems with spin $S=1\mathrm{}$. However, this model is at the moment not understood theoretically. This is the same difficulty encountered in spin-glasses which also contain an excess specific heat rather linear in temperature over a considerable temperature range.