The thermal expansivity of a single crystal of EuO was determined in the temperature range 25 to 250°K by a differential-strain-gauge method. The temperature of the peak in the $\lambda$ curve of expansivity is 69.2°K, in agreement with the specific-heat measurements. After correcting for the normal lattice expansivity using the Grüneisen theory, we observe that the resulting magnetoelastic component of expansivity ${\alpha }_{\mathrm{me}}$ obeys a magnetic Grüneisen law, being proportional to the magnetic specific heat $C_m$ over wide ranges of temperature both above and below the $\lambda$ transition. Europium oxide can therefore be characterized by a temperature-independent "magnetic" Grüneisen constant, $\frac{\partial \ln U_m}{\partial \ln V}=-5.3\mathrm{}$, given by $-3\mathrm{}{\alpha }_{\mathrm{me}}{C_m}^{-1\mathrm{}}{B}_T$, where the isothermal bulk modulus $B_T=1.07\mathrm{}\times {10}^{+12\mathrm{}}$ dyn/${\mathrm{cm}}^2$. The lattice Grüneisen constant, $\frac{\partial \ln U_l}{\partial \ln V}=1.9\mathrm{}$, was similarly derived from the data at temperatures well above the $\lambda$ anomaly. For $U_m$, the internal magnetic energy, we also derive the variation with temperature $\frac{U_m\mathrm{}\left(T\right)\mathrm{}}{U_m\mathrm{}\left(0\right)\mathrm{}}$, the variation with pressure $\frac{\partial \ln U_m}{\partial P}=4.9\mathrm{}\times {10}^{-12\mathrm{}}$ ${\mathrm{dyn}}^{-1\mathrm{}}$ ${\mathrm{cm}}^2$, and the value $U_m\mathrm{}\left(0\right)=-4.9\mathrm{}\times {10}^8$ erg/${\mathrm{cm}}^3$ at 0°K. Comparison with results of other experiments and with theories based on the Heisenberg Hamiltonian is also presented. The model of D. C. Mattis and T. D. Schultz and of E. Pytte is consistent with the observed proportionality between $C_m$ and ${\alpha }_{\mathrm{me}}$. A more general model proposed by E. R. Callen and H. B. Callen includes magnetoelastic coupling of unequal strengths to first- and second-nearest neighbors. When the second-neighbor interaction is weaker than the first, this model is also consistent with a single effective magnetic Grüneisen constant not only because the model then differs only slightly from a special case of that of Mattis and Schultz and of Pytte, but also because the spin correlation functions ${〈\mathrm{S}·{\mathrm{S}}^{\prime }〉}_{1\mathrm{s}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ and ${〈\mathrm{S}·{\mathrm{S}}^{\prime }〉}_{2\mathrm{n}\mathrm{d}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ appear to be nearly proportional to each other over a wide temperature range.

• Received 31 March 1967

DOI:https://doi.org/10.1103/PhysRev.160.413