Planar DPF – would it work?

[Ferret]

I thought a bit about this. Instead of the classical cylindrical structure of the DPF, one could use a planar structure. The same electrodes, the same placement, same cross-section, only the electrodes are flat this time. The plasma sheaths (one above, one below) run to the end of electrodes, then jump over the electrodes, then the two sheaths attract by way of electromagnetic force between currents and make contact, initiating a planar pinch. At the same time currents attract each other inside the pinch, such that the planar pinch separates into several axial pinches, quite similar to the classical DPF pinch, only there are more of them.

What do you think, would it work to induce nuclear fusion? How could we derive the distance at which pinches form? Could the distance be preset by a saw-shaped edge of the electrodes?

I expect the current in each pinch to be comparable to the one in a classical DPF, only being more pinches means a higher total current spread over the flat electrodes. At the same time, there being no external axial structure to induce axial symmetry to the pinches, these may not point in precisely the same direction. This effect may be mitigated with an external magnetic field.

Overall, the structure is simpler and probably cheaper than a classical DPF, although the differences in price may not be too big. On the other side, if one wants to obtain several pinches instead of one, as it may be necessary in a space propulsion system, a planar DPF may offer advantages.

[asymmetric_implosion]

The most common example of this approach is so-called planar wire arrays. Rather than arrange a group of wires in concentric cylindrical shells, they are arranged in two straight planes or other planar geometries. The JxB force drives the wires to some central axis. The problem appears to be that the wires arrive at different times. Rather than getting all the mass arriving on the axis at the same time which leads to a dense pinch, you have waves of mass arriving. You can’t compress the plasma as tightly leading to a cold pinch which is bad for the application of wire arrays (soft x-ray production). Planar wire arrays are one of the fads of the Z-pinch community right now so the literature is filled with tomes on the subject.

One might argue that a planar sheet of gas is different. My concern is that when the sheet breaks up into a few pinches, you are splitting the current between each pinch. Say you get an equal division of current in two pinches. The radiation yield (neutrons, alphas, x-ray) scales with the current to some power (Yield=a*Current^n). In the equation, n tends to vary from 3-6 for fusion and a is all over the place (1E-4 to 0.1). The typical value of n is around 4. You divide the current into two equal pinches sharing half the current. Thus, each pinch is emitting ~6% of the radiation a single pinch with all the current. You have two pinches so you get a total of ~12% of the radiation yield from a single pinch in a concentric geometry if a is the same. The exact value of a is not easy to get. My feeling, but not proven, is a will drop for the parallel pinch case because you are dividing the pinch inductance into a parallel paths reducing what is commonly referred to as the current bite. Small current bite tends to correlate with small a. The problem is far more complicated but you need to read the published literature for a more complete description.

If you argue you can drive more current per shot, you might have a hope of increasing the yield by increasing the number of pinches. However, you can drive more current in a cylinder than you can a set of parallel plates for the same gap between anode and cathode due to inductance.

[Ferret]

Well, that’s exactly what happens when more pinches are formed. The total current has to provide a high enough intensity to each pinch. This can be done if the current is high enough and it is quite constant during the formation of pinches. The pinches would then be hot even if the plasma sheath were not moving at the same speed, as long as the plasma feeding each pinch arrives at the same time. Since the pinches separate due to plasma instabilities, how could the distance at which they form be estimated? On the other side, the number of pinches may also depend on the current intensity, which is a factor in plasma instabilities.

[asymmetric_implosion]

Ferret wrote: Well, that’s exactly what happens when more pinches are formed. The total current has to provide a high enough intensity to each pinch. This can be done if the current is high enough and it is quite constant during the formation of pinches. The pinches would then be hot even if the plasma sheath were not moving at the same speed, as long as the plasma feeding each pinch arrives at the same time. Since the pinches separate due to plasma instabilities, how could the distance at which they form be estimated? On the other side, the number of pinches may also depend on the current intensity, which is a factor in plasma instabilities.

Let me put it this way, if you build a pulse power circuit you will always get more radiation yield (fusion, x-ray, etc) from a single pinch rather than groups of pinches in parallel. This is a well studied and well understood problem. People have suggested this in various forms for years and every experiment has proven this is a bad approach when maximizing yield is king. When you want to increase the current on a plasma focus or other pulse power device, you generally need to increase the charge voltage. One can argue that you can streamline the pulse power system but there is a limit. Say you operate near the minimum bank impedance such as FoFu-1, you can only increase your current to support multiple pinches at the previous single pinch current by increasing the bank voltage. In a capacitor bank, the energy stored increases with the square of the voltage (E_stor=0.5*C*V^2 where C is the capacitance, V is the voltage and E_stor is the energy stored). It turns out that the peak current is linearly related to the charge voltage in an RLC circuit (and yes, the trend holds with the dynamic properties of the plasma as long as you don’t screw around with the electrodes too much). Take the two relationships together and your current increases with square root of energy. Take the case of two pinches running in parallel. You want to push 3 MA through each pinch. Compared to a 3 MA single pinch circuit, you need to store 4 times the energy to drive the total current of 6 MA. By splitting the current in two paths, you get a yield twice the 3 MA value because you have two pinches. Overall, you multiply Q by 0.5 because you have increased the energy stored by 4 and increased the yield only by 2. (Q=Yield/Store).

Now, take the case of increasing the pulse power to 6 MA with a single pinch. Using conventional scaling laws with Y~I^4, the yield for a 6 MA machine with a single pinch is 16X larger than the 3 MA machine with a single pinch. You still increased the stored energy by a factor of 4. The net result is multiplying Q by 4 on the single pinch at 6 MA.

Regardless of your geometry, planar, multiple cylinders arranged in a pattern, etc, you always lose in Q. You also make the electrode heating problem worse by adding more energy. You’ve increased the dissipation area by ~2X and increased the energy by 4X.

The other point you state about the plasma will break up into pinches due to instabilities is not correct. Consider the modes of plasma formation. You start with a glow discharge, like fluorescent light bulbs. It is largely homogenous. As you increase the current, the gas becomes susceptible to other forces like JxB, the force that drives implosion and pinch formation. The plasma will constrict into an arc. Now, you may say that is what I’m talking about but step back. The plasma carried during the axial phase is already an arc. How can an arc break up into many arcs? The opposite is usually observed. If you start with multiple arcs, they tend to coalesce into a single arc. Why would a planar geometry drive the plasma in the opposite direction. One observes regions in a pinch that are different from other regions but the structure is still a single pinch. If you want, the pinch can be thought of as a group of micro pinches in series. An individual arc can carry huge currents well into the mega amp range without any problems so it doesn’t need to break up. It will be susceptible to instabilities like Magneto Raleigh Taylor (m=0) and kinking (m=1) but it tends to stay together as a single body. I can explode or re-strike but it is still a single current carrying structure. If you want to start with multiple arcs and implode each of them into pinches, it is a large breakdown voltage with a fast rise time that matters. This approach is used in low inductance, high current switches to create multiple arc channels. The spec for the switches I use is 4 kV/ns with breakdown at 20 kV. You would need something similar to create multiple arcs in a plasma focus.

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