## 2 Methods
In this case, the maximum height of the surface protrusions varies, respectively, from Rz = 40-80 to 6-10 \(\upmu\)m. Since the height of the micro protrusion is much less than the interelectrode gap (of 0.01 m or more), for calculation of electric field of such protrusion the electrostatics task has being solved about the conductive ellipsoid in an external field parallel to one of the main axes of the ellipsoid [23]. This is an equivalent to a semi-ellipsoidal protrusion on one of the flat electrodes, or to the gap between two electrodes parallel to each other (Figure 5) which exceeds protrusion height. If the \(x\) is the axis of the protrusion in the form of a semi-ellipsoid of rotation perpendicular to the electrode, where \(x\) = 0, then the field strength on the extension of this axis when \(x\) is greater than the height of the protrusion \(h\) is [22]:  
\[E(x)=E_{av}\left[(1-\frac{arth\frac{c}{x}-\frac{c}{x}}{arth\frac{c}{x}-\frac{c} {x}})+\frac{1}{(arth\frac{c}{h}-\frac{c}{h})(\frac{x^{2}}{c^{2}}-1)\frac{x}{c }}\right], \tag{6}\]  
where \(E_{av}\) is the average electric field strength created in the cathode region of a glow discharge by a cathodic potential drop; \(c\)--half the distance between the ellipsoid focuses located on the x axis (Figure 5), determined, according to [24], as \(c=\sqrt{a^{2}-b^{2}}\). The term in square brackets in this expression describes the field strength increasing due to the presence of a protrusion. Denote it by \(\beta_{E}\), then the expression (6) can be written as \(E(x)=E_{av}\ \beta_{E}\). The term in parentheses of expression (6) is always less than one and on the surface of the protrusion at \(x=h\) vanishes. Therefore, the main is the second term, the value of which is maximum at minimum x, i.e., at \(x=h\) and at the \(x=d_{c}\) simultaneously. Figure 4 shows the dependence of the field enhancement \(\beta_{E}\) on the ratio \(h/d_{c}\). Expression (6) describes an increase in the field strength on a flat smooth cathode from a protrusion having a smooth peak with a considerable radius. For a hemispherical protrusion having the same semi axes a = b, the field enhancement is \(\beta_{E}\approx 3\)[25]. The results of calculating \(\beta_{E}\) from expression (6) for surface protrusions in the form of semi-ellipsoidal extended along the x-axis are shown in Figure 3:0 The graph also shows that, depending on the degree of protrusions elongation, the electric field enhancement at their apex can reach up to 10-30. Figure 3:1 Scheme of a semi-elliptical protrusion on the surface of a flat cathode: a, b are the semi axes of the ellipsoid; \(h\)—the height of the protrusion; \(d_{c}\)—the length of the cathodic potential drop region. In practice, after machining protrusions are formed on the surface, having an acute-angled shape, with a radius at the apex within the unit fractions of micrometers (Figure 3). The acute-angled protrusions contribute dramatically to a more intense electric field distortions and an additional increase in its intensity nearby the protrusion. Such an increase in electric field strength can be estimated by the coefficient [21]:  
Figure 3:2  
where \(h\)--the height of the protrusion; \(r\)--the curvature radius on the peak of the protrusion. For protrusions with a height of 20-40 \(\upmu\)m or more with a radius of vertex curvature from fractions to units of micrometers (Figure 3), the value of \(\upmu\) can reach up to \(\mu\geq 10\). Then the field strength near the vertices of such protrusions will be  
Figure 3:3  
Under conditions of increased gas pressure, the boundary of the region of cathodic potential drop declines sharply and approaches the vertices of micro protrusions roughness \(x=d_{c}\approx h\). As a result, the local field strength near the vertices of such protrusions can reach E(x) \(\geq 10^{9}\) V/m.