What is meant by oscillator in Einstein solids?

The Einstein model of a solid treats each atom as being the mass element in a three-dimensional harmonic oscillator potential that keeps the atom confined to the vicinity of its equilibrium position. Each individual atoms experiences a potential; the $n$th atom feels $V_{n}(vec{r}_{n})=frac{1}{2}Momega_{0}^{2}(vec{r}_{n}-vec{a}_{n})^{2}$, where $vec{a}_{n}$ is the equilibrium position of $n$th atom in the crystal lattice. For a lattice of $N$ atoms, this means $N$ completely independent three-dimensional oscillator potentials (equivalent to $3N$ independent one-dimensional oscillators), all with the same frequency.

The Einstein model is satisfactory as a crude model for the high-temperature heat capacity of solids, and it also satisfies the Third Law of Thermodynamics, that the heat capacity approaches zero as $Trightarrow0$. However, the approach of $C_{V}rightarrow0$ is qualitatively wrong, since the individual atoms are assumed to oscillate completely independently, whereas in a real solid the behavior at low temperatures is dominated by the behavior of low-frequency collective modes—long-wavelength phonons in which many atoms move together. A much more realistic model of a solid is the Debye model, which models the solid as having $3N$ normal modes, with frequencies given by the $3N$ lowest models of vibration of an elastic continuum. (The Debye model is not quantitatively accurate at intermediate temperatures, but it handles the low- and high-$T$ regimes quite accurately.)