# Issue with Einstein solid

1. Nov 17, 2009

### aihaike

Hi guys,

Einstein potential is define as

$$U_{e}=\sum_{i}\alpha_{i}(r_{i}-r0_{i})$$.

The partition function of the Hamiltonian

$$H=U_{e}+\frac{p^{2}}{2m}$$

is given by

$$Q=\left(\frac{2m_{i}}{\alpha_{i}}\left(\frac{\pi}{\beta h}\right)^{2}\right)^{\frac{3}{2}}$$

Which gives rises to the free energy

$$A=-\frac{3}{2\beta}\sum_{i=1}^{N}\ln\left[\frac{2m_{i}}{\alpha_{i}}\left(\frac{\pi}{\beta h}\right)^{2}\right]$$

Ok, now suppose we have two species in the system.
We can reformulate the partition function introducing $$\omega_{i}$$ and $$T_{\mathrm{E}_{i}}$$ for each species difine as

$$\omega_{i}=\sqrt{\frac{2\alpha_{i}}{m_{i}}}\quad\mathrm{and}\quad T_{\mathrm{E}_{i}}=\frac{h\omega_{i}}{2\pi k}$$

And it comes

$$A=3N_{1}kT\ln\left(\frac{T_{\mathrm{E}_{1}}}{T}\right)+3N_{2}kT\ln\left(\frac{T_{\mathrm{E}_{2}}}{T}\right)$$

where $$N_{1}$$ and $$N_{2}$$ are the number of atom of each species.

We also can take the quantum version of the partition function defines as

$$Q={\left(\frac{\exp\left(-\frac{T_{\mathrm{E}_{1}}}{2T}\right)}{1-\exp\left(-\frac{T_{\mathrm{E_{1}}}}{T}\right)}\right)^{3N_{1}}} {\left(\frac{\exp\left(-\frac{T_{\mathrm{E}_{2}}}{2T}\right)}{1-\exp\left(-\frac{T_{\mathrm{E_{2}}}}{T}\right)}\right)^{3N_{2}}}$$

Now comes my question.
On one hand it seems to me that the Einstein temperature $$T_{\mathrm{E}$$ is a characteristic is the system but it seems also depends on the system temperature.

I'm working on silica (SiO2) and I use to compute for each species $$\alpha$$ from the mean square displacement (u) calculated from a NVT monte carlo simulation (with the initial structure from the average coordinates of NPT simulation) using a standard 2-body potential with the formula

$$\alpha=\frac{3KT}{2u}$$

The I calculate the corresponding frequency and Einstein temperature

$$\omega=\sqrt{\frac{2\alpha}{m}}\quad\mathrm{and}\quad T_{\mathrm{E}}=\frac{h\omega}{2\pi k}$$

And the $$T_{\mathrm{E}$$ I get does not correspond to my apply temperature.
May be I should call it vibrational temperature instead of Einstein temperature.

I'm very confused, if we want to calculate the free energy of a crystal based on this model, what value of $$T_{\mathrm{E}$$ shoud we take? Is there table somewhere?
I also wonder if $$T_{\mathrm{E}$$ and $$\alpha$$ are really related actually.