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I am considering the spin state's effect on the charge carrier's lifetime, I was told that high spin and low spin state could coexist when the crystal field splitting energy and the pairing energy are close, but I am not sure how to estimate the ratio between high spin state and low spin state, is there any theory I could adopt to analyze this kind of situation?
Here is some background I find:

whether a complex is high spin or low spin depends on two main factors: the crystal field splitting energy and the pairing energy. The electrons will take the path of least resistance--the path that requires the least amount of energy. If the paring energy is greater than Δ , then electrons will move to a higher energy orbital because it takes less energy. If the pairing energy is less than Δ , then the electrons will pair up rather than moving singly to a higher energy orbital.

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    $\begingroup$ en.wikipedia.org/wiki/Boltzmann_distribution is your friend. By the way, you should use > for quoting instead of `...`. $\endgroup$ Commented Jul 23, 2021 at 15:59
  • $\begingroup$ So, the spin state distribution fits the Boltzmann distribution? could you give more hints? $\endgroup$
    – Jack
    Commented Jul 23, 2021 at 16:29
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    $\begingroup$ If you've not seen the Boltzmann distribution before, I'd suggest looking it up in a textbook on physical chemistry, or something similar. You won't get an answer on the Internet that's better than that. (Once you understand it, it's a fairly straightforward application of the formula $n_i / n_j = \exp[-(E_i - E_j)/kT]$. After all, the ratio you're after is exactly given by $n_i / n_j$, right?) $\endgroup$ Commented Jul 23, 2021 at 16:31

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