We know that the charges in between the electrodes separated by a distance d increase by a factor e-power(αd) when field between electrodes is uniform.

**STREAMER OR KANAL MECHANISM OF
SPARK**

We know
that the charges in between the electrodes separated by a distance d increase
by a factor e^{αd} when field between electrodes is uniform. This is
valid only if we assume that the field E_{0} = V/d is not affected by
the space charges of electrons and positive ions. Raether has observed that if
the charge concentration is higher than 10^{6} but lower than 10^{8}
the growth of an avalanche is weakened i.e., dn/dx < e^{αd}.

Whenever
the concentration exceeds 10^{8}, the avalanche current is followed by
steep rise in current and breakdown of the gap takes place. The weakening of
the avalanche at lower concentration and rapid growth of avalanche at higher
concentration have been attributed to the modification of the electric field E_{0}
due to the space charge field. Fig. 2.6 shows the electric field around an
avalanche as it progresses along the gap and the resultant field i.e., the
superposition of the space charge field and the original field E_{0}.
Since the electrons have higher mobility, the space charge at the head of the
avalanche is considered to be negative and is assumed to be concentrated within
a spherical volume. It can be seen from Fig. 2.6 that the filed at the head of
the avalanche is strengthened.

The field
between the two assumed charge centres i.e., the electrons and positive ions is
decreased as the field due to the charge centres opposes the main field E_{0}
and again the field between the positive space charge centre and the cathode is
strengthened as the space charge field aids the main field E_{0} in
this region. It has been observed that if the charge carrier number exceeds 10^{6},
the field distortion becomes noticeable. If the distortion of field is of 1%,
it would lead to a doubling of the avalanche but as the field distortion is
only near the head of the avalanche, it does not have significance on the
discharge phenomenon. However, if the charge carrier exceeds 10^{8},
the space charge field becomes almost of the same magnitude as the main field E_{0}
and hence it may lead to initiation of a streamer. The space charge field,
therefore, plays a very important role in the mechanism of electric discharge
in a non-uniform gap.

Townsend
suggested that the electric spark discharge is due to the ionization of gas
molecule by the electron impact and release of electrons from cathode due to
positive ion bombardment at the cathode. According to this theory, the
formative time lag of the spark should be at best equal to the electron transit
time t_{r}. At pressures around atmospheric and above p.d. > 10^{3}
Torr-cm, the experimentally determined time lags have been found to be much
shorter than t_{r}. Study of the photographs of the avalanche
development has also shown 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. It has also been observed through measurement that the
transformation from avalanche to streamer generally takes place when the charge
within the avalanche head reaches a critical value of

n_{0}e^{αx}
≈ 10^{8} or αX_{c} ≈ 18 to 20

where Xc
is the length of the avalanche parth in field direction when it reaches the
critical size. If the gap length d < Xc, the initiation of streamer is
unlikely.

The
short-time lags associated with the discharge development led Raether and
independently Meek and Meek and Loeb to the advancement of the theory of
streamer of Kanal mechanism for spark formation, in which the secondary
mechanism results from photo ionization of gas molecules and is independent of
the electrodes.

Raether
and Meek have proposed that when the avalanche in the gap reaches a certain
critical size the combined space charge field and externally applied field E_{0}
lead to intense ionization and excitation of the gas particles in front of the
avalanche head. There is recombination of electrons and positive ion resulting
in generation of photons and these photons in turn generate secondary electrons
by the photo ionization process. These electrons under the influence of the
electric field develop into secondary avalanches as shown in Fig. 2.9. Since
photons travel with velocity of light, the process leads to a rapid development
of conduction channel across the gap.

Raether
after thorough experimental investigation developed an empirical relation for
the streamer spark criterion of the form

where E_{r}
is the radial field due to space charge and E_{0} is the externally
applied field. Now for transformation of avalanche into a streamer E_{r}
≈ E

Therefore, αx_{c} = 17.7 + ln x_{c}

For a
uniform field gap, breakdown voltage through streamer mechanism is obtained on
the assumption that the transition from avalanche to streamer occurs when the
avalanche has just crossed the gap. The equation above, therefore, becomes

αd = 17.7
+ ln d

When the
critical length X_{c} ≥ d minimum breakdown by streamer mechanism is
brought about.

The
condition X_{c} = d gives the smallest value of α to produce streamer
breakdown.

Meek
suggested that the transition from avalanche to streamer takes place when the
radial field about the positive space charge in an electron avalanche attains a
value of the order of the externally applied field. He showed that the value of
the radial field can be obtained by using the expression.

where x
is the distance in cm which the avalanche has progressed, p the gas pressure in
Torr and α the Townsend coefficient of ionization by
electrons corresponding to the applied field E. The minimum breakdown voltage
is assumed to correspond to the condition when the avalanche has crossed the
gap of length d and the space charge field E_{r} approaches the
externally applied field i.e., at x = d, E_{r} = E. Substituting these
values in the above equation, we have

The
experimentally determined values of α/p and the corresponding E/p are used to
solve the above equation using trial and error method. Values of α/p
corresponding to E/p at a given pressure are chosen until the equation is
satisfied.

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