# In a plasma, can the ion temperature become larger than the electron temperature?

Let's assume a hydrogen gas, initially at low temperature

The density is low enough so that heating of this gas by inverse bremsstrahlung can be neglected.

Now if a high-intensity radiation source interacts with this gas the electrons are heated due to photo-ionization and keep the energy difference $kh\nu-\Delta E$ where "k" represents the number of photons and $\Delta E$ is the ionization energy. At the same time the ions are heated as well. Let me put it this way:

The electron heating/cooling rate due to (multi)-photo ionization and the corresponding reverse process (superscript "b") is (assume instantly thermalized electrons):

$$\Theta_e=(\sum_{Z,i,j}k_{MPI}(k)[H_i^{Z+}]-k_{MPI}^b(k)[H_j^{(Z+1)+}]n_e)(kh\nu-\Delta E)\frac{2}{3k_Bn_e}$$ and the ionic heating/cooling is $$\Theta_i=(\sum_{Z,i,j}k_{MPI}(k)[H_i^{Z+}]-k_{MPI}^b(k)[H_j^{(Z+1)+}]n_e)\Delta E\frac{2}{3k_B\sum_{Z,i}[H_i^{Z+}]}$$

Where $[H_i^{Z+}]$ represents the concentration of Hydrogen ions in state $i$ and degree of ionization $Z$. $\Delta E=E_{(Z+1)+,j}-E_{Z+,i}$ (Ionization potential from $H_i^{Z+}$ to state $H_j^{(Z+1)+}$)and $k_{MPI}$ is the ionization rate.

This implys that for rather low-frequency radiation (VIS) the heating rate of the ions might become larger thatn the electrons' heating rate. Further, the bigger the free-electron density, the lower this heating rate becomes. That is, the ionic temperature might become larger than the electronic.

Now I'm wondering if this is allowed at all? Can the ioni temperature of a plasma become larger than the electron temperature?

• Yes, there is nothing wrong with $T_{e}$ > $T_{i}$ (e.g., see physics.stackexchange.com/q/143738/59023 ). – honeste_vivere Nov 30 '17 at 2:20
• I agree..but what about Ti>Te – OD IUM Nov 30 '17 at 9:43
• Also perfectly fine. Since plasmas are often collisionless, there is really no reason to think that $T_{e}$ must equal $T_{i}$. – honeste_vivere Nov 30 '17 at 14:11