Paschen's law

The Paschen's law , named after Friedrich Paschen , who established experimentally this context, in 1889, states that the breakdown voltage is a function of the product of gas pressure and electrode spacing ( striking distance ) when the conditions for the Townsend mechanism are met ( homogeneous field , negligible space charge). The equation that John Sealy Townsend first derived is

${\displaystyle V={\frac {B}{\ln(A\cdot p\cdot d)-\ln \left[\ln(1+\gamma ^{-1})\right]}}\cdot p\cdot d}$

in which

• ${\displaystyle p}$ the gas pressure
• ${\displaystyle d}$ the electrode gap
• ${\displaystyle \gamma }$the 3rd  Townsend coefficient
• ${\displaystyle A}$and constants derived below ${\displaystyle B}$

represent.

Course of the ignition voltage V versus 
pressure times the distance  p · d
for different gases
in a double logarithmic representation

The Paschen curve is the graphic representation of Paschen's law. It has a minimum for small values, which is approx. 7.3 bar · µm for air at 340 V and approx. 3.5 bar · µm for SF ₆ at 507 V. Above the minimum one speaks of long penetration. There the curve behaves linearly . In this area, either the field strength caused by the voltage drops or the mean free path of the particles is reduced by the pressure. Below this, in the so-called near breakdown, the breakdown voltage rises steeply again. This is because the distance is too small or the pressure for impact ionization is too low. With impact ionization is no longer possible. ${\displaystyle pd}$ ${\displaystyle Bpd}$ ${\displaystyle \lambda }$ ${\displaystyle d\leq \lambda }$

However, there are indications that the Paschen curve is not valid below 3 µm and that the breakdown voltage continues to drop.

Physical background

In addition to the perfect vacuum, there are always atoms and a few free electrons and ions between two electrodes . The charged particles are accelerated by the electric field between the electrodes. The ions are much heavier and larger than the electrons, so they are only accelerated slowly and quickly collide with other atoms or ions again. However, the electrons can be accelerated to a speed that gives them enough energy to ionize them when they strike an atom (impact ionization). The resulting free electrons are in turn accelerated and generate even more free electrons, so that an avalanche effect sets in.

An electrical breakdown occurs at the earliest when the free electrons are accelerated to an energy sufficient to have ionized at least one atom on the way to the anode. The applied voltage must therefore reach a certain value, which is called the breakdown voltage. This obviously depends on the ionization energy of the gas atoms. The achievable energy of an electron depends on its mean free path, the distance it covers before it hits an atom. The longer this path, the higher the energy from the acceleration. The free path depends on the size of the atoms and their density, i.e. also on temperature and pressure.

Values ​​of constants

Typical values ​​for the constants A and B of some gases:

gas		A in ${\textstyle \mathrm {\frac {1}{Pa\,m}} }$	B in ${\textstyle \mathrm {\frac {V}{Pa\,m}} }$	validity ${\textstyle {\frac {E}{p}}/\mathrm {\frac {V}{Pa\,m}} }$	source
air		10.95	273.8	75-600
nitrogen	N 2	9.00	256.5	75-450
hydrogen	H 2	3.83	104.1	15-450
helium	Hey	2.25	25.5	15-100
argon	Ar	10.20	176.3	75-450
carbon dioxide	CO 2	15.00	349.5	375-750

Derivation

Basics

In order to calculate the breakdown voltage, one assumes a plate capacitor with the plate spacing . The cathode is at the point . So one can assume a homogeneous electric field between the plates. ${\displaystyle d}$ ${\displaystyle x=0}$

For impact ionization it is a prerequisite that the electron energy is greater than the ionization energy of the gas atoms that are between the plates. The number of ionizations will occur per path length . is known as the first Townsend coefficient because it was introduced by Townsend in section 17. The change in the current of the electrons can be described as follows for the plate capacitor structure: ${\displaystyle E_{e}}$ ${\displaystyle E_{I}}$ ${\displaystyle x}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \Gamma _{e}}$

${\displaystyle \Gamma _{e}(x=d)=\Gamma _{e}(x=0)\,\mathrm {e} ^{\alpha d}\qquad \qquad (1)}$

(The number of free electrons on the anode is the number of free electrons on the cathode that has increased due to impact ionization. The larger and / or is, the more free electrons are generated.) ${\displaystyle d}$ ${\displaystyle \alpha }$

The number of free electrons generated during discharge is

${\displaystyle \Gamma _{e}(d)-\Gamma _{e}(0)=\Gamma _{e}(0)\left(\mathrm {e} ^{\alpha d}-1\right)\qquad \qquad (2)}$

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

${\displaystyle \Gamma _{i}(0)-\Gamma _{i}(d)=\Gamma _{e}(0)\left(\mathrm {e} ^{\alpha d}-1\right)\qquad \qquad (3)}$

${\displaystyle \Gamma _{i}}$is the stream of ions. So that the discharge does not go out again immediately, free electrons have to be generated on the cathode surface. This is possible because the ions knock out secondary electrons when they hit the cathode . (For very high applied voltages, field emission can also occur.) One can write without field emission

${\displaystyle \Gamma _{e}(0)=\gamma \Gamma _{i}(0)\qquad \qquad (4)}$

where is the number of electrons that an impacting ion knocks out in the section. This is known as the third Townsend coefficient. Suppose that a relationship between the Townsend coefficients is obtained by plugging (4) into (3) and transforming: ${\displaystyle \gamma }$ ${\displaystyle \Gamma _{i}(d)=0}$

${\displaystyle \alpha d=\ln \left(1+{\frac {1}{\gamma }}\right)\qquad \qquad (5)}$

Impact ionization

The question now is how big is. The number of ionizations depends on how likely it is that an electron will hit an ion. This probability is the ratio of the area of ​​the cross-section of a collision between electron and ion in relation to the total available area through which the electron can fly: ${\displaystyle \alpha }$ ${\displaystyle P}$ ${\displaystyle \sigma }$ ${\displaystyle A}$

${\displaystyle P={\frac {N\sigma }{A}}={\frac {x}{\lambda }}\qquad \qquad (6)}$

As the second part of the equation makes clear, the probability can also be expressed as the ratio of the distance covered by the electron to the mean free path (before ionization occurs again). ${\displaystyle x}$ ${\displaystyle \lambda }$

Illustration of the cross section : If the center of particle b enters the blue circle, it collides with particle a . The area of ​​the circle is therefore the cross section and its radius is the sum of the radii of the particles. ${\displaystyle \sigma }$ ${\displaystyle r}$

${\displaystyle N}$is the number of electrons, because everyone can collide. The number can be calculated with the equation of state of the ideal gas

${\displaystyle pV=Nk_{\mathrm {B} }T\qquad \qquad (7)}$

( : Pressure : volume : Boltzmann constant , : Temperature) ${\displaystyle p}$ ${\displaystyle V}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle T}$

express. As the adjacent sketch shows, is . Since the radius of an electron can be neglected compared to the radius of an ion , it simplifies to . If you use this relationship, insert (7) into (6) and transform it , you get ${\displaystyle \sigma =\pi (r_{a}+r_{b})^{2}}$ ${\displaystyle r_{I}}$ ${\displaystyle \sigma =\pi r_{I}^{2}}$ ${\displaystyle \lambda }$

${\displaystyle \lambda ={\frac {k_{\mathrm {B} }T}{\pi r_{I}^{2}p}}={\frac {1}{Lp}}\qquad \qquad (8)}$

the factor was only introduced for the sake of clarity. ${\displaystyle L}$

The change in the current of electrons that have not yet collided at each waypoint can be described as ${\displaystyle x}$

${\displaystyle \mathrm {d} \Gamma _{e}(x)=-\Gamma _{e}(x)\,{\frac {\mathrm {d} x}{\lambda _{e}}}\qquad \qquad (9)}$

express. This differential equation can easily be solved:

${\displaystyle \Gamma _{e}(x)=\Gamma _{e}(0)\,\exp {\left(-{\frac {x}{\lambda _{e}}}\right)}\qquad \qquad (10)}$

The probability that it is , that is, that there has not yet been a shock at this point is ${\displaystyle \lambda >x}$ ${\displaystyle x}$

${\displaystyle P(\lambda >x)={\frac {\Gamma _{e}(x)}{\Gamma _{e}(0)}}=\exp {\left(-{\frac {x}{\lambda _{e}}}\right)}\qquad \qquad (11)}$

According to its definition, the number of ionizations per path length and thus the ratio of the probability at which no collision has yet occurred in the mean free path of the ions to the mean free path of the electrons is: ${\displaystyle \alpha }$

${\displaystyle \alpha ={\frac {P(\lambda >\lambda _{I})}{\lambda _{e}}}={\frac {1}{\lambda _{e}}}\exp \left(-{\frac {\lambda _{I}}{\lambda _{e}}}\right)={\frac {1}{\lambda _{e}}}\exp \left(-{\frac {E_{I}}{E_{e}}}\right)\qquad \qquad (12)}$

It was considered that the energy that a charged particle can absorb between two collisions depends on the electric field strength and the charge : ${\displaystyle E}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle Q}$

${\displaystyle E=\lambda Q{\mathcal {E}}\qquad \qquad (13)}$

Breakdown voltage

The following applies to the plate capacitor , where is the applied voltage. Since a simple ionization was assumed, is the elementary charge . One can now insert (13) and (8) in (12) and get ${\displaystyle E={\frac {U}{d}}}$ ${\displaystyle U}$ ${\displaystyle Q}$ ${\displaystyle e}$

${\displaystyle \alpha =L\cdot p\,\exp \left(-{\frac {L\cdot p\cdot d\cdot E_{I}}{eU}}\right)\qquad \qquad (14)}$

If you put this in (5) and transform it , you get Paschen's law for the breakdown voltage , which was first investigated by Paschen in and whose equation was first derived from Townsend in, section 227: ${\displaystyle U}$ ${\displaystyle U_{\mathrm {Durchschlag} }}$

${\displaystyle U_{\mathrm {Durchschlag} }={\frac {L\cdot p\cdot d\cdot E_{I}}{e\left(\ln(L\cdot p\cdot d)-\ln \left(\ln \left(1+\gamma ^{-1}\right)\right)\right)}}\qquad \qquad (15)}$

With ${\displaystyle \textstyle L={\frac {\pi r_{I}^{2}}{k_{\mathrm {B} }T}}}$

The constants explained at the beginning and are thus: ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \textstyle A=L={\frac {\pi r_{I}^{2}}{k_{\mathrm {B} }T}}}$

${\displaystyle \textstyle B={\frac {L\cdot E_{I}}{e}}={\frac {\pi r_{I}^{2}E_{I}}{Tk_{\mathrm {B} }e}}}$

Plasma ignition

Plasma ignition as defined by Townsend (Townsend discharge) means that the plasma reaches a point where it burns by itself, regardless of an external source of free electrons. This means that the electrons of the cathode reach the anode at a distance and must have ionized at least one atom on the way there. According to the definition of , this relationship must be fulfilled: ${\displaystyle d}$ ${\displaystyle \alpha }$

${\displaystyle \alpha d\geq 1\qquad \qquad (16)}$

If one uses instead of (5), one gets for the breakdown voltage ${\displaystyle \alpha d=1}$

${\displaystyle U_{\text{Durchschlag Townsend}}={\frac {L\cdot p\cdot d\cdot E_{I}}{e\cdot \ln(L\cdot p\cdot d)}}={\frac {d\cdot E_{I}}{e\cdot \lambda _{e}\,\ln \left({\frac {d}{\lambda _{e}}}\right)}}\qquad \qquad (17)}$

Conclusion / validity

Paschen's law therefore assumes that

• there are already free electrons on the cathode before ignition ( ), which can be accelerated to trigger the impact ionization. Such so-called seed electrons can be generated by ionization by cosmic background radiation . ${\displaystyle \Gamma _{e}(x=0)\neq 0}$
• the generation of further free electrons only happens through impact ionization. Paschen's law does not apply when external electron sources are present. This can e.g. B. be light that generates secondary electrons through the photoelectric effect . This must be taken into account in experiments.
• an ionized atom only leads to one free electron. However, multiple ionizations always occur in practice.
• free electrons are generated on the cathode surface by the impacting ions. The number of electrons generated in the process is, however, strongly dependent on the cathode material, its surface properties ( roughness , impurities) and the ambient conditions (temperature, humidity, etc.). The experimental determination of the factor is therefore hardly possible in a reproducible manner. ${\displaystyle \gamma }$
• the electric field is homogeneous.

Individual evidence

1. ↑ Andreas Küchler: High voltage technology . 2nd Edition. Springer-Verlag, Berlin / Heidelberg 2005, ISBN 978-3-540-78412-8 , pp. 159 .
2. ↑ Paperback of electrical energy technology: with 102 tables . ISBN 3-446-40475-9 , pp. 289 ( limited preview in Google Book search).
3. ↑ Emmanouel Hourdakis, Brian J. Simonds, and Neil M. Zimmerman: Submicron gap capacitor for measurement of breakdown voltage in air . In: Rev. Sci. Instrum. . 77, No. 3, 2006, p. 034702. doi : 10.1063 / 1.2185149 .
4. ↑ ^{a b c d e f} Jane Lehr, Pralhad Ron: Electrical Breakdown in Gases . In: Foundations of Pulsed Power Technology . John Wiley & Sons, Inc., 2017, ISBN 978-1-118-88650-2 , pp. 369-438 , doi : 10.1002 / 9781118886502.ch8 ( wiley.com [accessed September 14, 2017]).
5. ^ J. Townsend: The Theory of Ionization of Gases by Collision . Constable, 1910.
6. ↑ F. Paschen, “About the potential difference required for spark transfer in air, hydrogen and carbonic acid at different pressures,” Annalen der Physik, vol. 273, no. 5, pp. 69 - 96, 1889. doi: 10.1002 / andp.18892730505
7. ^ J. Townsend: Electricity in Gases. Clarendon Press, 1915.