Low Temp Limit: Paramagnet v. Einstein solid

[Geronimo23]

Hey everyone! So I have that the low temperature limit of a paramagnet is Ω=(Ne/N_down)^N_down while the low temperature limit of an einstein solid is Ω=(Ne/q)^q. How could I explain that these two equations are essentially the same considering their respective limits (N_down<<N and q<<N) and that oscillators in an einstein solid have an infinite number of energy levels while paramagnets have only two? Thank you so much!

 

[Vagn]

Hey everyone! So I have that the low temperature limit of a paramagnet is Ω=(Ne/N_down)^N_down while the low temperature limit of an einstein solid is Ω=(Ne/q)^q. How could I explain that these two equations are essentially the same considering their respective limits (N_down<<N and q<<N) and that oscillators in an einstein solid have an infinite number of energy levels while paramagnets have only two? Thank you so much!

Both the paramagentic material and the Einstein solid follow the binomial distribution, as any particle is either in the state of interest or is not, and so deal with factorials. In the limit of large numbers of particles, Stirling's approximation $(N!=N\ln(N)-N)$ can be applied and rearranged for that form.