Paschen's law

The Paschen's law, named after Friedrich Paschen, who established this relationship experimentally in 1889, states that in a homogeneous field, the breakdown voltage is a function of the product of gas pressure and electrode spacing. The equation that John Sealy Townsend herleitete first is,

Wherein the gas pressure, the electrode spacing and the second Townsend coefficient representing. A and B are constants which are derived in the following.

The Paschen curve is a plot of the Paschen law. It has a minimum of small values ​​, which is air at 340 V at about 7.3 bar · microns and SF6 at 507 V at about 3.5 bar · micron. Above the minimum one speaks of the wide breakdown. There, the curve is linear with. In this area is reduced, either caused by the voltage field strength or the mean free path of the particles is reduced by the pressure. Below, in the so-called Nahdurchschlag, the breakdown voltage increases again steeply. This is because the distance is too small or the pressure to the impact ionization is too low. When impact ionization is no longer possible.

However, there are indications that the Paschen curve is not valid below 3 microns and the breakdown voltage drops again.

Physical background

Between two electrodes are always in perfect vacuum except atoms and even getting a few free electrons and ions. By the electric field between the electrodes, the charged particles are accelerated. The ions are much heavier and larger than the electrons are thus accelerated slowly and quickly collide with other atoms or ions. However, the electrons can be accelerated to a speed which gives them sufficient energy to ionize an atom that upon striking ( impact ionization ). The resulting free electrons are again accelerated and still produce more free electrons, so that an avalanche effect starts.

An electrical breakdown thus occurs earlier on when the free electrons are accelerated to an energy sufficient that they have ionized on the way to the anode at least one atom. Thus, the applied voltage must reach a certain value, the breakdown voltage is known. This is obviously dependent on the ionization of the gas atoms. The achievable energy of an electron depends on its mean free path, the path that it travels until it encounters an atom. The longer this distance, the higher the energy due to the acceleration. The free path length is dependent on the size of the atoms and the density of which, including the temperature and pressure.

Derivation

Basics

To calculate the breakdown voltage, one starts from a plate capacitor with the plate spacing. The cathode is located at the point. It can thus be of a uniform electric field between the plates.

For impact ionization, it is provided that the beam energy is greater than the ionization energy of the gas atoms that are located between the plates. The number of ionizations will occur per path length. is known as the first Townsend coefficient, since it was introduced by Townsend, section 17. The change of the current of electrons can thus be described for the plate capacitor structure like this:

( The number of free electrons on the anode is thus the number of free electrons on the cathode, which has increased by impact ionization. The larger and / or is, the more free electrons are generated. )

The number of generated free electrons in the discharge is

Neglecting that can be ionized atoms several times, the number of ions generated is equal to the number of generated free electrons:

Is the stream of ions. Thus, the discharge does not go out immediately, free electrons must be generated on the cathode surface. This is possible because the ions to the cathode knock out secondary electrons upon impact. ( For very high applied voltages can also field emission occur. ) Without field emission, one can write

Wherein the number of the electrons is, the knocks an incident ion in section. This is referred to as a second Townsend coefficient. Adopted to obtain a relationship between the Townsend coefficient by substituting (4) into (3) and transforms:

Impact ionization

The question now is how big. The number of ionizations depends on how likely it is that an electron hits an ion. This probability is the ratio of the area of the cross section of an impact between the electron and ion relative to the total standing at disposal area through which the electron can fly:

As the second part of the equation illustrates, one can express the probability as a ratio of the distance traveled by the electron path length to the mean free path ( before re- ionization occurs).

Is the number of electrons, because each may encounter. The number can be combined with the equation of state of the ideal gas

Express. As adjacent sketches show is. Since the radius of the electron relative to the radius of an ion can be neglected, it is easier to. If you use this relationship sets (7 ) into ( 6 ) and formed by order, one obtains

Where the factor was introduced for convenience only.

The change in the current of not yet collided electrons at each waypoint can be seen as

Express. This differential equation can be easily solved:

The probability that is, so that at the point, no shock has occurred, is

According to this definition is the number of ionizations per path length and thus the ratio of the probability at which the mean free path of the ions is no collision has taken place, to the mean free path of the electrons:

It was considered that the energy that can accommodate a charged particle between a shock depends on the electric field strength and the charge:

Breakdown voltage

Applies to the capacitor plate, wherein the applied voltage. Since it was assumed that a single ionization is the elementary charge. It is now (13) and (8 ) into (12) and receives

Substituting this in (5 ) and formed according to, one obtains the Paschen's law for the breakdown voltage, which were first investigated by Paschen in and its equation was first derived by Townsend in, section 227:

Plasma ignition

Plasma ignition in the definition of Townsend ( Townsend discharge ) means that the plasma reaches a point where there is a fire by itself, independent of an external source of free electrons. This means that the electrons reach the anode to the cathode at a distance, while at least one atom on the path must then ionized. So this relationship to be fulfilled according to the definition of need:

If one uses instead of ( 5), one obtains for the breakdown voltage

Conclusion / validity

So the Paschen's law assumes that

• It, free electrons on the cathode is already before the ignition (), which can be accelerated to cause impact ionization. Such so-called Seedelektronen can be generated by ionization by cosmic background radiation.
• The generation of additional free electrons occurs only by impact ionization. So the Paschen's law does not apply when external electron sources are available. This can, for example, Be light that generates secondary electrons by the photoelectric effect. This must be considered in experiments.
• An ionized atom leads to only ever a free electron. Multiple ionizations occur, however, in practice forever.
• Free electrons are generated on the cathode surface by the impinging ions. However, the number of electrons thus generated is heavily dependent on the cathode material, the surface texture (roughness, impurities) and the ambient conditions (temperature, humidity, etc.) dependent. The experimental determination of the factor is hardly reproducible possible.
• The electric field is homogeneous.