Dynamics of microparticles in vacuum breakdown: Cranberg’s scenario updated by numerical modeling

Microparticles (MP) and thermofield emission in vacuum are mainly caused by the roughness present at the surface of electrodes holding a high voltage. They can act as a trigger for breakdown, especially under high vacuum. This theoretical study discusses the interactions between one MP and the thermofield emission electron current as well as the consequences on the MP’s transit. Starting from Cranberg’s assumptions, new phenomena have been taken into account such as MP charge variation due to the secondary electron emission induced by energetic electron bombardment. Hence, the present model can be solved only numerically. Four scenarios have been identified based on the results, depending on the electron emission current from the cathode roughness (tip) and the size of the MP released at the anode, namely (i) one way; (ii) back and forth; (iii) oscillation; and (iv) vaporization. A crash study of the MP on the cathode shows that the electron emission can decrease if the MP covers the thermoemissive tip, i.e., if the MP is larger than the tip size—a phenomenon often called “conditioning”—and helping to increase the voltage holding in vacuum without breakdown.

Figure 1

Sketch showing the phenomena occurring in the interelectrode gap. Both the tip (on the cathode) and the microparticle (MP) are at micrometer ($\mu \mathrm{m}$) scale, while the anode-cathode gap is in the mm range (the figure is not to scale). The dashed lines denote the trajectories of thermofield emitted electrons from the tip due to the high anode-cathode voltage.

Figure 2

Representation of the three characteristic zones of a specific electric field, as used in this model. Zone I: field enhancement close to the emissive tip extending on twice the tip height (on the figure is indicated the larger value of this zone for a tip of $10\mu \mathrm{m}$ height and $0.97\mu \mathrm{m}$ radius at its base); zone II: zone of uniform field (gap lying from 0.5 to 5 mm); zone III: spherical field produced by the charge of the MP surrounding it. The figure is not drawn to scale.

Figure 3

Magnification of the vicinity of the tip taken with an elliptic shape. The tip is represented with its thermal reservoir (thermostat) which is the cathode. The inset shows the interelectrode gap (the size of the tip is not to scale).

Figure 4

2D distributions of the electric field (a), current density (b), and temperature (c) calculated for an imposed uniform macroscopic field ${\mathrm{E}}_{\text{macro}}=5\times {10}^7\mathrm{V}/\mathrm{m}$ and the big tip. Surface normal current density is represented in (b′), the inset of panel (b).

Figure 5

(a) 2D polar distribution of the normalized surface number density of electrons impinging the anode (color scale of top panel) and (b) the same distribution expressed versus radial position.

Figure 6

Primary electron trajectories passing close to the MP for two velocities. The deflection occurs only very close of the MP.

Figure 7

Secondary electron emission induced by primary energetic electrons for a spherical MP. The location where a secondary electron is created is marked by a star.

Figure 8

Secondary electron yield induced by electron bombardment for a $10\mu \mathrm{m}$ radius MP.

Figure 9

The escaping coefficient of a secondary electron versus the MP charge number for MPs of different sizes.

Figure 10

Model flowchart of the 3D MP dynamics. Two models are connected, one corresponds to thermofield electron emission from an apex (red) and the second to the MP trajectory (blue). SEE denotes the secondary electron emission.

Figure 11

Dynamics regimes of a test MP for different emission currents. The rows represent the time evolution of: (a) the flight distance (z) from anode ($500\mu \mathrm{m}$) towards the cathode ($\mathrm{z}=0$); (b) the charge ($\times {10}^6\mathrm{e}$) of the MP; and (c) the MP radius ($R_{\mathrm{MP}}$). By columns, the first corresponds to “one way” trajectory from the anode to the cathode and low tip current. The second column corresponds to a MP back-attracted to the anode. The third column corresponds to an oscillating MP before reaching the tip. The fourth column represents an MP finishing fully evaporated due to the very high primary current. For each current, ${\mathrm{V}}_{\text{app}}=25\mathrm{kV}$ and $\mathrm{d}=0.5\mathrm{mm}$ have been kept fixed as well as the initial particle radius. Animations for each regime are available on-line [58].

Figure 12

Different regimes of MPs dynamics versus their sizes. The figure corresponds to a constant voltage ${\mathrm{V}}_{\text{app}}=25\mathrm{kV}$ and two gap distances (a) $\mathrm{d}=0.5\mathrm{mm}$ and (b) $\mathrm{d}=50\mathrm{mm}$. The horizontal lines corresponds to the maximum emitted current by three tips, the largest one ($H_{\text{tip}}=10\mu \mathrm{m}$, red), the average size (${\mathrm{H}}_{\text{tip}}=2\mu \mathrm{m}$, green and $0.5\mu \mathrm{m}$, blue) and the smallest one ($H_{\text{tip}}=0.2\mu \mathrm{m}$, cyan).

Figure 13

Comparison of the velocity at the entrance of zone I for ${\mathrm{I}}_{\text{tip}}=1.0\times {10}^{-7}\mathrm{A}$ and the threshold velocity obtained for ${\mathrm{Q}}_{\text{cathode}}$ for different $R_{\mathrm{MP}}$.

Figure 14

Comparison of the charge necessary to stop and reverse the MP trajectory (${\mathrm{Q}}_{\text{return}}$) and the charge of MP in equilibrium with the cathode for different. $R_{\mathrm{MP}}$.

Figure 15

(a) Sketch of the crash of a large MP touching the tip. (b) Off-axis study. Δ is a reduced parameter, related to the size of the tip divided by the MP radius ($\mathrm{\Delta }=\frac{d_{\mathrm{int}}}{R_{MP}}$).

Figure 16

Kinetic energy of the MP during the crash with the tip versus the MP radius, for $\mathrm{I}=0.1\mu \mathrm{A}$, ${\mathrm{V}}_{\text{app}}=25\mathrm{kV}$, and $\mathrm{d}=0.5\mathrm{mm}$.