The integration must be taken along the line of the highest field strength.

**BREAKDOWN IN NON-UNIFORM FIELDS**

where d is the gap length. The integration must be taken along the line of the highest field strength.

The
expression is valid also for higher pressures if the field is only slightly
non-uniform.

In
strongly divergent fields there will be at first a region of high values of E/p
over which α/p > 0. When the field falls below a given strength E_{c} the integral

ceases to exist.

Townsend
mechanism then loses its validity when the criterion relies solely on the γ
effect, especially when the field strength at the cathode is low.

In
reality breakdown (or inception of discharge) is still possible if one takes
into account photo ionization processes. The criterion condition for breakdown
(or inception of discharge) for the general case may be represented to take
into account the non-uniform distribution of

where N_{cr}
is the critical electron concentration in an avalanche giving rise to
initiation of a streamer (it was shown to be approx. 10^{8}), x_{c}
is the path of avalanche to reach this size and d the gap length.

Figure
2.9 illustrates the case of a strongly divergent field in a positive point
plane gap. Equation (2.15) is applicable to the calculation of breakdown or
discharge inception voltage, depending on whether direct breakdown occurs or
only corona.

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