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In textbooks, the collision frequency $Z_{ab}$ between two species "a" and "b" is always given as something like:

$$Z_{ab} = n_an_b\sigma_{ab} \Bigg(\frac{8 k T}{\pi \mu}\Bigg)^{1/2}= n_an_b\sigma_{ab} \bar{C}_{rel}$$

where $n_a$ and $n_b$ are the number density of species a and b respectively, $\sigma$ is the collision cross-section, and the reduced mass $\mu = m_am_b/(m_a+m_b)$. Given that the mean speed of a species "i" is given by $\bar{C}_i = \Big(\frac{8 k T}{\pi m_i}\Big)^{1/2}$, then, algebraically, the first equation is equivalent to:

$$Z_{ab} = n_an_b\sigma_{ab} \sqrt{\bar{C}_a^2 + \bar{C}_b^2}$$

Is there a reason that this is not a valid interpretation of the collision frequency? I have not found a single source which presents either the collision rate $Z_{ab}$, or the mean relative speed $\bar{C}_{rel}$ in this way.

beans
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  • Although the following is for a plasma, the derivation starts from the same place: https://physics.stackexchange.com/a/268594/59023 – honeste_vivere Oct 29 '21 at 13:53
  • Thanks, Thats the first time I've seen someone else refer to the mean speed of both species in this way! My application is also plasma based, but far lower temperature than them. – beans Oct 29 '21 at 14:02
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    Yes, the relevant speed is mean average thermal speed between the two species, for a statistical representation assuming Maxwellians. I imagine this changes if the velocity distributions are not Maxwellians since the relevant speed changes. – honeste_vivere Oct 29 '21 at 14:46
  • @honeste_vivere , I just noticed you are the only person who commented on my other post as well. Thank you. – beans Oct 29 '21 at 16:15

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