the gap and the resulting modification to the original field E0. For simplicity
the space charge at the head of the avalanche is assumed concentrated within
a spherical volume, with the negative charge ahead because of the higher
electron mobility. The field is enhanced in front of the head of the avalanche
with field lines from the anode terminating at the head. Further back in the
avalanche, the field between the electrons and the ions left behind reduced
the applied field E0. Still further back the field between the cathode and the
positive ions is enhanced again. The field distortion becomes noticeable with a
carrier number n > 106. For instance, in nitrogen with d D 2 cm, p D 760 torr,
˛ ³ 7 and E0/p ³ 40 V/torr cm, the field distortion is about 1 per cent, leading
to 5 per cent change in ˛. If the distortion of ¾D1 per cent prevailed in the entire
gap it would lead to a doubling of the avalanche size, but as the distortion is
only significant in the immediate vicinity of the avalanche head it has still an
insignificant effect. However, if the carrier number in the avalanche reaches
n ³ 108 the space charge field becomes of the same magnitude as the applied
field and may lead to the initiation of a streamer. The space charge fields play
an important role in the mechanism of corona and spark discharges in nonuniform
field gaps. For analytical treatment of space charge field distortion
the reader is referred to reference 12.
In the Townsend spark mechanism discussed in the previous section the
gap current grows as a result of ionization by electron impact in the gas
and electron emission at the cathode by positive ion impact. According to
this theory, formative time lag of the spark should be at best equal to the
electron transit time ti. In air at pressures around atmospheric and above
pd > 103 torr cm the experimentally determined time lags have been found
to be much shorter than ti. Furthermore, cloud chamber photographs of
avalanche development have shown22 that under certain conditions the space
charge developed in an avalanche is capable of transforming the avalanche into
channels of ionization known as streamers that lead to rapid development of
breakdown. From measurements of the prebreakdown current growth23 and
the minimum breakdown strength it has been found that the transformation
from avalanche to streamer generally occurs when the charge within the
avalanche head (Fig. 5.15) reaches a critical value of n0 exp[˛xc] ³ 108 or
˛xc ³ 18–20, where xc is the length of the avalanche path in field direction
when it reaches the critical size. If xc is larger than the gap length xc > d
then the initiation of streamers is unlikely. Typical cloud chamber photographs
of electron avalanche and streamer development are shown in Figs 5.16(a) to
(d). In (a) the discharge has been arrested before reaching the critical size
¾108, giving the avalanche the classical ‘carrot’ shape. In (b) the avalanche
has grown beyond the critical size, its head has opened up indicating ionization
around the original avalanche head and a cathode directed streamer starts. This
continues (c, d) till a plasma channel connects cathode and anode. The early
cloud chamber results have led Raether22 to postulate the development of two