What is meant by oscillator in Einstein solids? When we are studying the Einstein solid, it is said that each solid has $N$ number of oscillators, but my doubt is what does it means. Are oscillators atoms? Or degree of freedom of each atoms? For example, an atom has four oscillators, what does it means?
 A: 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 $V_{n}(\vec{r}_{n})=\frac{1}{2}M\omega_{0}^{2}(\vec{r}_{n}-\vec{a}_{n})^{2}$, 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 $T\rightarrow0$.  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.)
