Hello.
I'm studying ICP (Inductively Coupled Plasma) and reading famous paper, "Plasma Sources Sci. Techno.l. 1 (1992) 179-186, "A simple analysis of an inductive RF discharge", R B Piejak, V A Godyak and B M Alexandrovich".
In this paper, It is shown that higher frequency operation results in lower power transfer efficiency to the plasma as shown in Fig. 3.
The paper says that it is due to that higher w/v (ratio of solenoid current frequency to electron collisional frequency) requires stronger electric field maintaining plasma so higher current is necessary.
At first, I agreed this as I thought higher frequency results in shorter time of electron acceleration time before collision stronger field is required.
However, when I considered "Faraday's law of induction" I found that this may be not good explanation. For the same current amplitude flowing through the solenoid, induced electric field amplitude is proportional to the current frequency (Faraday's law of induction) so high frequency operation naturally results in stronger electric field for the same current amplitude. Thus...shorter time of energy gain of electrons in acceleration phase is automatically compensated by the stronger field such that the total energy gain of the electron in the acceleration phase in high frequency will be the same to that of in lower frequency.
Did I miss something? The Fig. 3 in the paper so clearly shows frequency dependency of the efficiency so there may be something I don't see now.
Hello,
Yes your consideration of Faraday's law is ok. But you may think in the following way.
It can be realized that at high frequency, the displacement current a dE/dt ~ wE (E is the electric field), through the sheaths would increase.
Because of the enhancement of the current through the sheath, electron heating can be changed. Even if you are using ICP, it is no longer an ICP loop rather power may be coupled electrostatically (or you may say capacitively).
Consider the power coupled to plasmas by the low frequency (LF) and high frequency (HF) be PLF and PHF, respectively. Additionally, the discharge current varies proportionally to the square of the excitation frequencies. This dependency causes the plasma impedance ZHF
When you consider an oscillating field applied to the plasma, something like E=E_0cos(wt), the electron current present an inertia, due also to collision. If you consider the Boltzmann equation for free electrons, the field to be considered in the equation is an effective one, given by
E=E0(cos(wt)+w/ve sin(wt)/(1+w^2/ve^2)
Which makes the field to be reduced as w/ve increases.
But this is only one aspect of the problem.
Another aspect is that related to the concept of efficiency.
In general the efficiency is the ratio between the energy of going in a given channel to the energy furnished to the system or to the power supply. In a plasma in general, the efficiency is considered the ionization efficiency.
The reduction of ionization rate is not linear with the electric field (E/N), but, below a given limit, it varies almost exponentially (see Capriati er al. Plasma Chem. Plasma Proc. 1991).
Moreover, part of the energy goes in internal degrees of freedom, part of it can be in ionization or dissociation, part is lost by radiation and another part is thermalized.
Also temperature increase is another channel. Therefore one must have clean what are the channels to be included in the efficiency calculation.
Moreover, I think that also the efficiency of power supply should be included in the calculation, which can depend on the generated frequency.
Efficiency calculation in a plasma is a complex concept!
To be short:
I think, you haven't considered electron density variation.
At first (relatively low power) more electron/neutral collisions leads to more electron density which leads to even higher power transfer. At this stage, your power is transferred to the old electrons and more and more of the new ones as the power gets higher. And then power transfer efficiency gets higher.
But it doesn’t last forever because:
When you go up to a certain power you have freed almost all the easy electrons and therefore the efficiency of electron/neutral collisions gets lower. At this stage, your power is transferred to the old electrons plus fewer and fewer of the new ones as the power gets higher. And then power transfer efficiency gets lower.
Your thinking is ok when you have the same electron density.
Hello. I'm sorry to lately reply your kind comments!
The answers givens to me looks much more complicated than I had originally expected. In order to get good understanding of what they means, I think It may be better to ask specific questions one by one.
For Bibhuti B Sahu (is there a way to mark someone like Facebook here?), you said sheath and I'm confusing why you said this terminology. As far as I know the sheath is ion cloud surrounding the electrode due to very difference between electrons and ions in terms of their speed. So I suspected that you're thinking our ICP as coil is immersed in the plasma. Actually our ICP has coil surrounding a quartz cube in which plasma is generated. So I don't know your comments are still valid for our case.
For Gianpiero Colonna, I really want to know how to obtain the equation of effective field in the plasma in your comments. It would be very helpful if you provide me the source of the equation or a way of its derivation as well as the specific location of the paper you mentioned. I couldn't find the paper.
For Rami Ljazouli, you emphasized importance of number of major energy carriers (charged particles) in the efficiency calculation. I agree this but you're actually away from the intent of my question that was actually asking about frequency dependency of power efficiency.
Yes sorry,
The remark was that you haven't considered that in plasma, through collisions, electrons create other electrons. That defines big part of power transfer.
You should consider the mean free path in plasma. Higher frequency leads shorter electron oscillation which leads to fewer collisions (even though they have the same energy but the probability of collisions is lower).
But very low frequency is not good either, because the electrons don't gain enough energy before colliding.
I hope I answered your question this time
Rami
Thanks Rami Ljazouli.
Yes, I haven't considered ratio of net electron moving distance in acceleration phase of electric field induced by the current through the coil to the mean free path determined by species densities (and also species temperatures) in the plasma. As I said in my question, total energy gain of the electron in the acceleration phase will be the same for both low and high frequencies due to Faraday's law of induction (this is single particle picture so the effective field mentioned by Gianpiero Colonna is not considered yet). Since energy gain of the particle is nothing but the integral of force applied to the particle over the particle's moving distance (It is work done to the particle by the field), stronger field amplitude in high frequency must result in shorter moving distance in the acceleration phase. And shorter moving distance means lower chance of collisions to other particles.
Your idea makes me feel that I have clearer view:) Thanks you again!
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