A (Plasma Physics) Spatial uniformity of particle species

[IonReactor]

Early in Bellan he asks us to consider a finite-temperature plasma and assume that the ion and electron densities are initially equal and spatially uniform. He approaches the problem of calculating the Debye length by considering each species of particle, σ
, as being a fluid so that the equation of motion for each species (after some approximations and assumptions) is

$$0≈−n_{\sigma}q_{\sigma}∇ϕ−∇P_{\sigma}$$

where $P_{\sigma}=κT_{\sigma}n_{\sigma}$ which because we assumed that the temperature is spatially uniform gives us

$$0≈−n_{\sigma}q_{\sigma}∇ϕ−κT_{\sigma}∇n_{\sigma}$$

But didn't we also assume that $n_{\sigma}$ was also spatially uniform? Why then are we taking a gradient of it?

It seems to me like what we have done is write an operator $−q_{\sigma}∇ϕ−κT_{\sigma}∇$ and used it to look for homogeneous solutions of the species density but I am still confused as to why we are taking a gradient of a quantity that we assumed to be spatially uniform and I hope that one of you good people will have some insight

 

[mselectromagnetic]

I'm wondering why there is an initial assumption about the temperature being spatially uniform? Are you looking for an idealized quantification for this environment? I ask, because though this is not my area of focus, I have a practical background with plasma deposition & cleaning. In using an oxygen plasma within an argon sheath for cleaning stainless steel, even in a forced gas flow, we noticed a species separation that seemed to produce results along a gradient (when we analyzed "cleaned" samples with a spectrometer seconds after passing under the plasma.) On the deposition side, when sputtering gold for deposition on flexible circuits, we were often plagued with non-uniform results that seemed to come from several factors like variations in vapor capacitance due to very tiny temperature variations due to turbulence currents within the chamber. Please don't punish me for relating this to the real world...just offering some experience and possible reasons for needing to calculate along a gradient.

 

[IonReactor]

Yes I think the reason for the assumption of finite temperature is for an idealized quantification of the environment. The example you offer is intriguing and since this expression comes from a physics textbook I believe it would make sense to use the real world as a reason to break from mathematical rigor. However, I'm not sure if it's the reason that Dr. Bellan had in mind.

I'm curious, in the example that you gave of non-uniform gold sputtering if the fluctuations in vapor capacitance were due to tiny temperature variations that were due to turbulence currents, what were the turbulence currents due to?

 

[mselectromagnetic]

We never felt totally sure of the reasons for varying capacitance in the vapor, but after numerous experiments with thermal probe placements, as well as capacitance measurements, we theorized that the actual sputtering process caused particles to enter the chamber at fairly high velocity, where they temporarily "bunched up" with other particles that were an "older" part of the vapor (lower velocity). I'd love to see the results if the chamber were located in a no-gravity environment. Due to plasmon surface oscillations, we felt that some part of the entering flow might be in a sort of harmonic resonance with older particles, but the electrical attraction of the target eventually overcame those attractions. We were looking for a single atom layer of deposition, but that was only achieved randomly. The problem was reduced to acceptable deposition performance by locating the targets further away, and below the sputtering target.

 

[alantheastronomer]

You're right - the number density gradient is zero, which means the pressure gradient is zero, which means the gradient of the potential should also be zero, and we get a trivial solution where the fluid is motionless. But to determine the Debye length, we move from looking at the plasma as a whole to looking at the environment of just one ion, where the number density of electrons which surround it due to their attraction to the ion, falls off to the average value with distance. This is what I think Bellan failed to mention. What are Bellan's subsequent equations and conclusions? What does he arrive at for the Debye length?