1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Issue with Einstein solid

  1. Nov 17, 2009 #1
    Hi guys,

    Einstein potential is define as


    The partition function of the Hamiltonian


    is given by

    [tex]Q=\left(\frac{2m_{i}}{\alpha_{i}}\left(\frac{\pi}{\beta h}\right)^{2}\right)^{\frac{3}{2}}[/tex]

    Which gives rises to the free energy

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

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

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

    And it comes


    where [tex]N_{1}[/tex] and [tex]N_{2}[/tex] are the number of atom of each species.

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

    [tex]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 [tex]T_{\mathrm{E}[/tex] 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 [tex]\alpha[/tex] 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


    The I calculate the corresponding frequency and Einstein temperature

    [tex]\omega=\sqrt{\frac{2\alpha}{m}}\quad\mathrm{and}\quad T_{\mathrm{E}}=\frac{h\omega}{2\pi k}[/tex]

    And the [tex]T_{\mathrm{E}[/tex] 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 [tex]T_{\mathrm{E}[/tex] shoud we take? Is there table somewhere?
    I also wonder if [tex]T_{\mathrm{E}[/tex] and [tex]\alpha[/tex] are really related actually.
    Any comments are welcome.
    Thanks in advance,

  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted