# Why are ions and electrons at different temperatures in a plasma?

In plasmas, the collision rate among ions or electrons is much larger than the collision rate between ions and electrons. Why is that so?

• Are you talking about low collision plasmas like the solar wind or high collision plasmas like the solar photosphere? – honeste_vivere Oct 29 '14 at 19:04

There are lots of different types of plasmas.

In a thermal plasma the electrons and ions will have the same temperature.

In a non-thermal plasma the discharge is driven by some external power supply e.g. capacitatively coupled RF, inductively coupled, pulsed DC E field etc.

In a non-thermal plasma the electrons generally have a higher temperature than the ions because the energy from the RF or E field couples with the electrons more efficiently. The electrons transfer energy to the gas to sustain the plasma.

Strictly speaking non-thermal plasmas are not at equilibrium and we cannot necessarily define a temperature, but temperature is generally a useful concept to use.

Collision rates are generally lower for electrons than ions, but not always (e.g. very low energy electrons and SF6 has huge collision rate).

The reason for the temperature difference is due partly to the driving energy being mostly coupled to the electrons and the partly because the energy is not rapidly transfered by collisions to the neutral gas and ions.

# Definitions

Let us first start with some definitions of parameters, in no particular order. I will be describing elastic collisions, assuming a quasi-neutral (i.e., $n_{e} = \sum_{s} \ n_{s} \ Z_{s}$) plasma. Thus, the collisions involve long-range forces and are called Coulomb collisions.

• Constants
• Particle/Plasma Paramters
• $B_{o}$ = the magnitude of the quasi-static magnetic field
• $s$ = the letter used to identify particle species (e.g., $e$ for electrons, $i$ for ions)
• $n_{s}$ = the particle number density of species $s$
• $q_{s} = e \ Z_{s}$, or the charge of species $s$
• $m_{s}$ = the mass of species $s$
• $T_{s}$ = the average temperature of species $s$ (in a non-equilibrium ionized gas like a plasma, I am referring to expression shown at https://physics.stackexchange.com/a/218643/59023)
• $\lambda_{De} = \sqrt{ \tfrac{ \varepsilon_{o} \ k_{B} \ T_{e} }{ n_{e} \ q_{e}^{2} } }$, or the electron Debye length
• $V_{Ts} = \sqrt{ \tfrac{ 2 \ k_{B} \ T_{s} }{ m_{s} } }$, or the thermal speed of species $s$ (Note: it's specifically the most probable speed here)
• $\omega_{ps} = \sqrt{ \tfrac{ n_{s} \ q_{s}^{2} }{ \varepsilon_{o} \ m_{s} } }$, or the plasma frequency of species $s$
• $\Omega_{ps} = \tfrac{ q_{s} \ B_{o} }{ \gamma \ m_{s} }$, or the gyrofrequency (or cyclotron frequency) of species $s$, and $\gamma$ is the relativistic Lorentz factor
• Collision-Specific Parameters
• $\mu_{ss'} = \tfrac{ m_{s} \ m_{s'} }{ m_{s} + m_{s'} }$, or the reduced mass of species $s$ and $s'$
• $\lambda_{D} = \lambda_{De} \ \sqrt{ \sum_{s} \left( \tfrac{ n_{s} \ Z_{s}^{2} }{ n_{e} } \right) \ \left( \tfrac{ T_{e} }{ T_{s} } \right) }$, or a generalized Debye length
• $V_{Tss'}^{2} = V_{Ts}^{2} + V_{Ts'}^{2}$, or the average thermal speed of the species $s$ and $s'$
• $b_{min} \simeq \tfrac{ q_{s} \ q_{s'} }{ \left( 4 \ \pi \ \varepsilon_{o} \right) \ \mu_{ss'} \ V_{Tss'}^{2} }$, or the minimum impact parameter, where we have assumed the classical limit
• $\Lambda \simeq \tfrac{ 12 \ \pi \ n_{e} \ \lambda_{De}^{3} }{ Z_{i} }$, which is the plasma parameter (or Coulomb logarithm)
• $\Lambda_{ss'}$ = the generalized impact parameter integral between test particle species $s$ and moving background species $s'$

# Coulomb Collisions

If we assume only $90^{\circ}$ scattering (i.e., no small-angle collisions), then one can define a reference collision frequency between test particle species $s$ and moving background species $s'$, given by: $$\nu_{ o,ss' }\left( v \right) \simeq \frac{ 4 \ \pi \ n_{s} \ q_{s}^{2} \ q_{s'}^{2} }{ \left( 4 \ \pi \ \varepsilon_{o} \right)^{2} \ m_{s}^{2} \ v^{3} } \ \ln \Lambda_{ss'} \tag{1}$$ where $v$ is the speed of moving background species $s'$ relative to test particle species $s$ and $\Lambda_{ss'}$ is given by: \begin{align} \ln \Lambda_{ss'} & = \int_{b_{min}}^{b_{max}} \ \frac{ db }{ b } = \ln \left( \frac{ \lambda_{D} }{ b_{min} } \right) \tag{2a} \\ \Lambda_{ss'} & \simeq \frac{ \left( 4 \pi \varepsilon_{o} \right) \mu_{ss'} \ V_{Tss'}^{2} }{ \sqrt{2} \ Z_{s} \ Z_{s'} \ e^{2} } \left[ \left( \frac{ \omega_{ps} }{ V_{Ts} } \right)^{2} + \left( \frac{ \omega_{ps'} }{ V_{Ts'} } \right)^{2} \right]^{-1/2} \tag{2b} \end{align} where Equation 2b is the classical limit of the last expression in Equation 2a.

The collision rate in Equation 1 is more often expressed as Maxwellian-averages in the Lorentz collision model for electron-proton, proton-proton, and electron-electron collisions [e.g., see Wilson et al., 2018 and references therein], given by: \begin{align} \nu_{ep} & = \frac{ 2 \ \sqrt{ 4 \pi } \ n_{p} \ e^{4} }{ 3 \left( 4 \pi \varepsilon_{o} \right)^{2} \ \mu_{ep}^{2} \ V_{Tep}^{3} } \ \ln \Lambda_{ep} \tag{3a} \\ \nu_{pp} & = \frac{ 4 \ \sqrt{ \pi } \ n_{p} \ e^{4} }{ 3 \left( 4 \pi \varepsilon_{o} \right)^{2} \ m_{p}^{2} \ V_{Tp}^{3} } \ \ln \Lambda_{pp} \tag{3b} \\ \nu_{ee} & = \frac{ 4 \ \sqrt{ \pi } \ n_{e} \ e^{4} }{ 3 \left( 4 \pi \varepsilon_{o} \right)^{2} \ m_{e}^{2} \ V_{Te}^{3} } \ \ln \Lambda_{ee} \tag{3c} \end{align}

In plasma, collision rate among ions or electrons is much larger than the collision rate between ions and electrons...

No, I think you have it backwards. If the only collisions are $90^{\circ}$ Coulomb collisions, then the collision rate between electrons and ions is higher than both the single species collision rates, as can be seen from Equations 3a--3c above.

...why is that so?

This can be understood by looking at the expression for $b_{min}$. For electron-proton collisions with $T_{e} \sim T_{i}$, then we have $\mu_{ei} \sim m_{e} \ll m_{i}$ and $V_{Tei} \sim V_{Te} \gg V_{Ti}$, which results in $b_{min,ei} \rightarrow \tfrac{ e^{2} }{ \left( 4 \ \pi \ \varepsilon_{o} \right) \ m_{e} \ V_{Te}^{2} } \sim b_{min,ee}$. For proton-proton collisions, $\mu_{ei} \sim m_{i}$ and $V_{Tei} \sim V_{Ti}$, which results in $b_{min,ii} \rightarrow \tfrac{ e^{2} }{ \left( 4 \ \pi \ \varepsilon_{o} \right) \ m_{i} \ V_{Ti}^{2} } \sim \left( \tfrac{ T_{e} }{ T_{i} } \right) b_{min,ei}$.

The larger(smaller) the minimum impact parameter, the smaller(larger) the $\Lambda_{ss'}$ contribution to the collision rate. The ions are also much slower on average than the electrons. This means that the probability of interacting with another ion in a given unit time is smaller as well, since it takes them longer to move the same distance as the much lighter electrons. Thus, the collision rates are effectively controlled by the higher mobility of the electrons.

why ions and electrons are at different temperatures in plasma?

Generally the Coulomb collision rates are much slower than many other rates except for regions like the solar chromosphere or in some lab plasmas. For instance, if we use the following typical solar wind parameters $n_{e}$ ~ 6 $cm^{-3}$, $T_{e}$ ~ 15 eV, and $Z_{i} = 1$, then $\nu_{ei} \sim 8 \times 10^{-6} \# \ s^{-1}$ or an effective collision time scale, $\tau_{ei} \sim 1.25 \times 10^{5} s$. The corresponding time scale for effective collisions with a magnetically turbulent spectrum (e.g., Kolmogorov magnetic power spectrum is $\propto k^{5/3}$) is ~1-10 seconds or ~4-5 orders of magnitude faster. The effective collision rates between particles and electromagnetic waves is even faster, with up to 7 orders of magnitude more collisions per unit time than equivalent electron-ion Coulomb collisions (e.g., http://adsabs.harvard.edu/abs/2007PhRvL..99d1101W).

For the ions and electrons to equilibriate (i.e., $T_{e} \sim T_{i}$), the Coulomb collision rates would need to dominate to allow for an equipartition of energy among the various species. Most plasmas are considered weakly collisional or collisionless, which means that the Coulomb collision rates are negligible compared to other processes that would affect the velocity distributions.

Thus, it is generally the case that $T_{e} \neq T_{i}$ in most plasmas unless there are significant collisions.

[I removed some extraneous equations from the background section and added references.]

why ions and electrons are at different temperatures in plasma?

A recent statistical study of 10 years of measurements [e.g., Wilson et al., 2018] found that the electron-to-proton temperature ratio is ~1.64 on average (median value is ~1.27) in the solar wind near Earth. They also noted that the effective collision rates introduced by electromagnetic and electrostatic fluctuations can be upwards of seven orders of magnitude higher than any particle-particle Coulomb collision rate. Thus, while Coulomb collisions may slowly act to try and relax a plasma to thermal equilibrium, other, faster processes can act to drive it away.

# References

• R. Hernandez and E. Marsch, J. Geophys. Res. 90, pp. 11062, doi:10.1029/JA090iA11p11062, 1985.
• R.W. Schunk, Planet. Space Sci. 23, pp. 437--485, doi:10.1016/0032-0633(75)90118-X, 1975.
• R.W. Schunk, Rev. Geophys. Space Phys. 15, pp. 429--445, doi:10.1029/RG015i004p00429, 1977.
• L.B. Wilson III et al., Astrophys. J. Suppl. 236(2), pp. 41, doi:10.3847/1538-4365/aab71c, 2018.

Consider a plasma that just's been formed and then left alone. Being far from equilibrium, the plasma will evolve towards an equilibrium state. At this stage, it's not very useful to characterize the plasma with a temperature because the velocity distribution would bear little resemblance to a Boltzmann distribution, or really any kind of distribution function with meaningful moments.

Let's assume that the plasma is being driven towards equilibrium by collision events. The particles will lose energy during each inelastic collision event, with some being more inelastic than others. The degree of inelasticity will depend significantly on the mass ratio: ion-ion collisions and electron-electron collisions will be inelastic compared to ion-electron collisions.

Since the $i-i$ and $e-e$ collisions are relatively inelastic, it often occurs that the ions and electrons quickly develop their own "temperatures". The ion velocity distribution rapidly approaches a Boltzmann distribution with temperature, and the electrons also rapidly approach a Boltzmann distribution with some different temperature. If you were to superpose the two, you'd get a funny-looking distribution with a tall top and wide tails, i.e. not global equilibrium.

The "centeredness" or "tailedness" (basically kurtosis) of the total distribution tells you how far from equilibrium the plasma is. The further the system is from equilibrium, the faster it will try to equilibrate via collisions. So if the temperatures are very different, the $i-e$ collision frequency will be high. If the temperatures are not too different, the $i-e$ will be lower, but still higher than the $e-e$ and $i-i$ systems that have already approximately equilibrated with themselves.

Note: In laboratory experiments, it's common to produce plasmas whose ions and electrons do not settle to a Boltzmann distribution. In that case, the kurtosis argument flies out the window. Another major complication arises if the electrons and ions have different flow speeds, where so-called "two-stream" or "bump-on-tail" instabilities become important in the equilibration physics.