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Plasma is often described as a separate phase of matter. The defining property of plasmas are they they are gases of freely moving charged particles, allowing large scale electric and magnetic phenomena to dominate their behavior. One of the ways that we pick out phase transitions is to look at how the temperature varies with input heat (heat capacity), with transitions occurring when the energy is going into driving a phase transition instead of increasing the kinetic motion in the medium (e.g. driving water molecules out of the liquid phase when boiling water).

With that in mind, if I take gas made of atoms of a single element (for concreteness, say carbon or oxygen) and proceed to heat it, does it exhibit a phase transition of the sort described above between ionization states? If so, are those ionization states considered separate phases, or some kind of "sub-phases" like the different phases in solid ice?

If no phase transition between ionization states occurs, then does that mean there is also no phase transition between neutral gas and plasma? If there is, then what separates that first ionization transition from the remaining ones?

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    $\begingroup$ When you heat a gas to the point of ionization, it doesn't just go to one ionization state (unless it's hydrogen, of course); it populates all ionization states with number densities described by the Saha equation for each particular state. $\endgroup$ – probably_someone Dec 11 '17 at 20:04
  • $\begingroup$ @probably_someone is that really enough to say there's no phase transition? If I place pure water in an enclosed container then it will exhibit a phase equilibrium between water and gas that depends on temperature and pressure, and as I heat it the balance shifts to gas from liquid, but that doesn't mean there isn't a phase transition. $\endgroup$ – Sean E. Lake Dec 11 '17 at 20:43
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    $\begingroup$ The difference here is that in water, you have some order parameter (in this case, the density) that becomes discontinuous at a critical temperature. This parameter doesn't exist for plasma; in every measurable variable, the system evolves continuously as temperature is increased. This process is usually what's referred to as an analytic crossover. Small source: web.mit.edu/16.unified/www/FALL/thermodynamics/notes/… $\endgroup$ – probably_someone Dec 11 '17 at 21:46
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As I have just explained in a different answer, there are commonly three criteria that must be fulfilled to call a system a plasma:

  1. $\lambda_D\ll L$, with $\lambda_D$ the Debye length and $L$ the spatial dimension of the plasma. As you have described it in the question, a plasma consists of charged particles (and neutrals) and the interaction between the particles is given by the Coulomb interaction. This is the major difference to a gas, where you simply have direct collisions. The result are long-range interactions in a plasma which do not occur in a gas. Coming back to the Debye length, it is the distance after which the electric field of a test charge is screened by plasma particles.
  2. $N_D\gg 1$, with $N_D$ the number of particles in the Debye sphere. The Debye sphere has a radius of $\lambda_D$ and contains all the plasma particles screening the field of the test charge we put into the plasma.
  3. $\omega_{pe}\tau_{0e}>1$, with $\omega_{pe}$ the electron plasma frequency and $\tau_{0e}$ the collision time between neutrals and electrons. The plasma frequency is often described as the most fundamental time-scale in plasma physics. It is the frequency with which the particles collectively oscillate when you apply an external electric field (think of a harmonic oscillation). The highest frequency and thus the shortest time scale is given by the electrons as they can react much faster to external fields (due to their smaller mass compared to the ions). Another way to write this condition would be $\tau_{pe}\ll \tau_{0e}$, i.e. the collision time must be much larger than the oscillation time.

As you have guessed it, the answer to your question is that there is no clear phase transition to the plasma state. Calling a plasma the fourth state of matter is therefore somewhat misleading. Also, if you increase the level of ionization, you simply increase the number density of electrons and the energy of the plasma particles. That might change the Debye length and the plasma frequency (the former depends on electron temperature and density, the latter only on the density) but there will not be an abrupt transition.

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