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Issue with Einstein solid

  1. Nov 17, 2009 #1
    Hi guys,

    Einstein potential is define as

    [tex]U_{e}=\sum_{i}\alpha_{i}(r_{i}-r0_{i})[/tex].

    The partition function of the Hamiltonian

    [tex]H=U_{e}+\frac{p^{2}}{2m}[/tex]

    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

    [tex]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)[/tex]

    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}}}
    [/tex]

    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

    [tex]\alpha=\frac{3KT}{2u}[/tex]

    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,

    Eric.
     
  2. jcsd
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