Electrode Plasma Injection

We know that when we used a very clean magnet, we rarely saw breakdowns after initial pumpdown. However if we painted or superglued the surface of the magnet, arcing continued for days if not weeks. This suggests that some outgassing may be needed to trigger the arcs, which is consistent with the negative slope of the "Paschen" breakdown voltage vs (pressure x distance) plot. That is, under clean conditions, other mechanisms operate to discharge the plasma space charge so that a steady state glow is obtained. When outgassing occurs, it necessarly produces a higher density near the magnet, thereby temporarily making the magnetic flux tube the lowest resistance path to ground leading to an arc discharge.

Despite this weak dependence on space charge, we can analyse the discharge as if it were a steady state phenomenon in a constant pressure atmosphere. If the discharges are due to breakdown caused by avalanche breakdown by electrons, then the maximum electric field is approximately the voltage needed to accelerate an electron to at least the first ionization potential of the background gas within a distance less than the mean free path, or otherwise collisions would thermalize the electron energy below that needed for ionization.

$$E_b = V_i /\lambda \\$$ (1)

where $E_b$ is the breakdown electric field strength, and $V_i$ is the first ionization potential for the dominant neutral gas species, and $\lambda$ is the collisional mean free path for this species at this pressure and temperature. Putting in numbers for N$_2$ obtained from the CRC, we get $V_i=15.5V$, and $\lambda=0.7mm$ at 100mTorr and 20$^\circ$. Thus $E_b=22kV/m$, which given the length of a typical arc of around 5 cm, is roughly 1 kV.

We can get a more accurate estimate by using Paschen's law for gas breakdown which can be written:

$$E_b = p B/( C + ln(p d))$$ (2)

where $p$ is the pressure in Torr, $d$ is the arcing distance in centimeters, and the two constants are $B=365$V/(cm Torr), and $C=1.18$. Again, plugging in the values gives $E_b=7.5 kV/m$ or 375V over 5cm, roughly consistent with our earlier crude estimate. Note that plotting Paschen's law as $E_b$ vs. $p d$ has minimum around $p d = 0.7$ Torr-cm, so that we are operating to the left of the minimum. This means that outgassing will decrease the breakdown voltage and lead to a discharge, in agreement with our observations.

Now this derived breakdown voltage is comparable to the $\sim600 V$ applied to the electrode, or the $\sim 400 V$ applied to the magnet directly. However, the arc follows the field line to the magnetic equator, it does not connect to the ground plane, which in the biassed dipole configuration is at least 13 cm below the magnet, so that the vacuum DC applied electric field cannot account for the observed field-aligned discharges that terminate in the annular plasma. A naive estimate might suppose that the magnetic equator is roughly a quarter of the distance to the ground plane and should therefore be at approximately 300 V potential, or a mere 100 V difference between the magnet and annulus, which is insufficient to create a breakdown. That is, it appears that positive charge stored in the annular plasma increases the potential difference between the magnet and the equator by 100's of Volts, leading to arc discharges. It is this charge that we estimate to be 300 nC giving us a 3 nF estimate for the capacitance of the magnetic field at this pressure.

At first glance this system seems to violate Joule's Law, because we obtain a higher voltage out of the system than we apply to form the glow discharge. But on further thought, any reactive circuit element can behave this way, as in a Cockroft-Walton accelerator. In the first configuration, we were unable to maintain the electrode current injection for more than a few seconds before we exceeded the current limit of our power supply, which was still three orders of magnitude longer than a typical arc, giving us confidence that we are observing a steady state, not transient response. This was confirmed by limiting the discharge current from the magnet with a large resistor, and observing a steady state DC glow discharge plasma with superposed arcing. Thus we differ from usual voltage multipliers that involve reactive circuit elements in that the dipole magnet is in a DC glow discharge without any explicit time variation in the input. That is, we have constructed a DC-DC converter without any modulation frequency.