Paschen's law
Encyclopedia
Paschen's Law, named after Friedrich Paschen
, was first stated in 1889. He studied the breakdown voltage
of gas
between parallel plates as a function of pressure
and gap distance
. The voltage necessary to arc
across the gap decreased up to a point as the pressure was reduced. It then increased, gradually exceeding its original value. He also found that for normal pressure the voltage needed to cause an arc reduced with the gap size, but only up to a point. As the gap was reduced further, the required voltage began to rise and again, exceeded its original value.
experimenters found a rather surprising behavior. An arc would sometimes take place in a long irregular path rather than at the minimum distance between the electrodes. For example, at a pressure of 10-3 atmospheres
, the distance for minimum breakdown voltage is about 7.5 mm. The voltage required to arc that distance is 327 V and is greater for gaps that are either wider or narrower. For a 3.75 mm gap, the required voltage is 533 V, nearly twice as much. If 500 V were applied, it would not be sufficient to arc at the 2.85 mm distance, but would arc at a 7.5 mm distance.
Paschen found that breakdown voltage was described by the equation
Where
is the breakdown voltage, 
is the pressure, and 
is the gap distance. The constants
and 
depend upon the composition of the gas. For air at standard atmospheric pressure
of 101 kPa,
= 43.6×106 V/(atm·m) and 
= 12.8.
The graph of this equation is the Paschen curve. By differentiating it with respect to
and setting the derivative to zero, the minimum voltage can be found. This yields
and predicts the occurrence of a minimum breakdown voltage for
= 7.5×10-6 m·atm. This is 327 V in air at standard atmospheric pressure at a distance of 7.5 µm. The composition of the gas determines both the minimum arc voltage and the distance at which it occurs. For argon
, the minimum arc voltage is 137 V at a larger 12 µm. For sulfur dioxide
, the minimum arc voltage is 457 V at only 4.4 µm.
For air at STP
, the voltage needed to arc a 1 meter gap is about 3.4 MV. The intensity of the electric field
for this gap is therefore 3.4 MV/m. The electric field needed to arc across the minimum voltage gap is much greater than that necessary to arc a gap of one meter. For a 7.5 µm gap the arc voltage is 327 V which is 43 MV/m. This is about 13 times greater than the field strength for the 1 meter gap. The phenomenon is well verified experimentally and is referred to as the Paschen minimum. The equation loses accuracy for gaps under about 10 µm in air at one atmosphere
and incorrectly predicts an infinite arc voltage at a gap of about 2.7 micrometers. Breakdown voltage can also differ from the Paschen curve prediction for very small electrode gaps when field emission from the cathode surface becomes important.
of a molecule in a gas is the average distance between its collision with other molecules. This is inversely proportional to the pressure of the gas. In air the mean free path of molecules is about 96 nm. Since electrons are much smaller, their average distance between colliding with molecules is about 5.64 times longer or about 0.5 µm. This is a substantial fraction of the 7.5 µm spacing between the electrodes for minimum arc voltage. If the electron is in an electric field of 43 MV/m, it will be accelerated and acquire 21.5 electron volt
s of energy in 0.5 µm of travel in the direction of the field. The first ionization energy
needed to dislodge an electron from nitrogen
is about 15 eV
. The accelerated electron will acquire more than enough energy to ionize a nitrogen atom. This liberated electron will in turn be accelerated which will lead to another collision. A chain reaction then leads to avalanche breakdown
and an arc takes place from the cascade of released electrons.
More collisions will take place in the electron path between the electrodes in a higher pressure gas. When the pressure-gap product
is high, an electron will collide with many different gas molecules as it travels from the cathode to the anode. Each of the collisions randomizes the electron direction, so the electron is not always being accelerated by the electric field
-- sometimes it travels back towards the cathode and is decelerated by the field.
Collisions reduce the electron's energy and make it more difficult for it to ionize a molecule. Energy losses from a greater number of collisions require larger voltages for the electrons to accumulate sufficient energy to ionize many gas molecules, which is required to produce an avalanche breakdown
.
On the left side of the Paschen minimum, the
product is small. The electron mean free path can become long compared to the gap between the electrodes. In this case, the electrons might gain lots of energy, but have fewer ionizing collisions. A greater voltage is therefore required to assure ionization of enough gas molecules to start an avalanche.
setup. The electrodes may have the distance
. The cathode is located at the point 
.
To get impact ionization
the electron energy
must become greater than the ionization energy 
of the gas atoms between the plates. Per length of path 
a number of 
ionizations will occur. 
is known as the first Townsend coefficient as it was introduced by Townsend in , section 17. The increase of the electron current 
can be described for the assumed setup
as
(So the number of free electrons at the anode is equal to the number of free electrons at the cathode that were multiplied by impact ionization. The larger
and/or 
the more free electrons are created.)
The number of created electrons is
Neglecting possible multiple ionizations of the same atom, the number of created ions is the same as the number of created electrons:
is the ion current. To keep the discharge going on, free electrons must be created at the cathode surface. This is possible because the ions hitting the cathode release secondary electrons
at the impact. (For very large applied voltages also field electron emission can occur.) Without field emission, we can write
where
is the mean number of generated secondary electrons per ion. This is also known as the second Townsend coefficient. Assuming that 
one gets the relation between the Townsend coefficients by putting (4) into (3) and transforming:
? The number of ionization depends upon the probability that an electron hits an ion. This probability 
is the relation of the cross-sectional
area of a collision between electron and ion
in relation to the overall area 
that is available for the electron to fly through:
As expressed by the second part of the equation, it is also possible to express the probability as relation of the path traveled by the electron
to the mean free path
(distance at which another collision occurs).
is the number of electrons because every electron can hit. It can be calculated using the equation of state of the ideal gas
The adjoining sketch illustrates that
. As the radius of an electron can be neglected compared to the radius of an ion 
it simplifies to 
. Using this relation, putting (7) into (6) and transforming to 
one gets
where the factor
was only introduced for a better overview.
The alteration of the current of not yet collided electrons at every point in the path
can be expressed as
This differential equation can easily be solved:
The probability that
(that there was not yet a collision at the point 
) is
According to its definition
is the number of ionizations per length of path and thus the relation of the probability that there was no collision in the mean free path of the ions, and the mean free path of the electrons:
It was hereby considered that the energy
that a charged particle can get between a collision depends on the electric field
strength
and the charge 
:
, where 
is the applied voltage. As a single ionization was assumed 
is the elementary charge
. We can now put (13) and (8) into (12) and get
Putting this into (5) and transforming to
we get the Paschen law for the breakdown voltage 
that was first investigated by Paschen in and whose formula was first derived by Townsend in , section 227:
in definition of Townsend (Townsend discharge
) is a self-sustaining discharge, independent of an external source of free electrons. This means that electrons from the cathode can reach the anode in the distance
and ionize at least one atom on its way. So according to the definition of 
this relation must be fulfilled:
If
is used instead of (5) one gets for the breakdown voltage
and tend to have smaller diameters. This gives them a greater mean free path length.
Ionization potentials differ between molecules as well as the speed that they recapture electrons after they have been knocked out of orbit. All three effects change the number of collisions needed to cause an exponential growth in free electrons. These free electrons are necessary to cause an arc.
Friedrich Paschen
Louis Karl Heinrich Friedrich Paschen , was a German physicist, known for his work on electrical discharges. He is also known for the Paschen series, a series of hydrogen spectral lines in the infrared region that he first observed in 1908...
, was first stated in 1889. He studied the breakdown voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
of gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
between parallel plates as a function of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
and gap distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
. The voltage necessary to arc
Electric arc
An electric arc is an electrical breakdown of a gas which produces an ongoing plasma discharge, resulting from a current flowing through normally nonconductive media such as air. A synonym is arc discharge. An arc discharge is characterized by a lower voltage than a glow discharge, and relies on...
across the gap decreased up to a point as the pressure was reduced. It then increased, gradually exceeding its original value. He also found that for normal pressure the voltage needed to cause an arc reduced with the gap size, but only up to a point. As the gap was reduced further, the required voltage began to rise and again, exceeded its original value.
Paschen curve
Early vacuumVacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
experimenters found a rather surprising behavior. An arc would sometimes take place in a long irregular path rather than at the minimum distance between the electrodes. For example, at a pressure of 10-3 atmospheres
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...
, the distance for minimum breakdown voltage is about 7.5 mm. The voltage required to arc that distance is 327 V and is greater for gaps that are either wider or narrower. For a 3.75 mm gap, the required voltage is 533 V, nearly twice as much. If 500 V were applied, it would not be sufficient to arc at the 2.85 mm distance, but would arc at a 7.5 mm distance.
Paschen found that breakdown voltage was described by the equation
Where
is the breakdown voltage, is the pressure, and is the gap distance. The constantsCoefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
and depend upon the composition of the gas. For air at standard atmospheric pressureAtmospheric pressure
Atmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...
of 101 kPa,
= 43.6×106 V/(atm·m) and = 12.8.The graph of this equation is the Paschen curve. By differentiating it with respect to
and setting the derivative to zero, the minimum voltage can be found. This yields
and predicts the occurrence of a minimum breakdown voltage for
= 7.5×10-6 m·atm. This is 327 V in air at standard atmospheric pressure at a distance of 7.5 µm. The composition of the gas determines both the minimum arc voltage and the distance at which it occurs. For argonArgon
Argon is a chemical element represented by the symbol Ar. Argon has atomic number 18 and is the third element in group 18 of the periodic table . Argon is the third most common gas in the Earth's atmosphere, at 0.93%, making it more common than carbon dioxide...
, the minimum arc voltage is 137 V at a larger 12 µm. For sulfur dioxide
Sulfur dioxide
Sulfur dioxide is the chemical compound with the formula . It is released by volcanoes and in various industrial processes. Since coal and petroleum often contain sulfur compounds, their combustion generates sulfur dioxide unless the sulfur compounds are removed before burning the fuel...
, the minimum arc voltage is 457 V at only 4.4 µm.
For air at STP
Standard conditions for temperature and pressure
Standard condition for temperature and pressure are standard sets of conditions for experimental measurements established to allow comparisons to be made between different sets of data...
, the voltage needed to arc a 1 meter gap is about 3.4 MV. The intensity of the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
for this gap is therefore 3.4 MV/m. The electric field needed to arc across the minimum voltage gap is much greater than that necessary to arc a gap of one meter. For a 7.5 µm gap the arc voltage is 327 V which is 43 MV/m. This is about 13 times greater than the field strength for the 1 meter gap. The phenomenon is well verified experimentally and is referred to as the Paschen minimum. The equation loses accuracy for gaps under about 10 µm in air at one atmosphere
and incorrectly predicts an infinite arc voltage at a gap of about 2.7 micrometers. Breakdown voltage can also differ from the Paschen curve prediction for very small electrode gaps when field emission from the cathode surface becomes important.
Physical mechanism
The mean free pathMean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
of a molecule in a gas is the average distance between its collision with other molecules. This is inversely proportional to the pressure of the gas. In air the mean free path of molecules is about 96 nm. Since electrons are much smaller, their average distance between colliding with molecules is about 5.64 times longer or about 0.5 µm. This is a substantial fraction of the 7.5 µm spacing between the electrodes for minimum arc voltage. If the electron is in an electric field of 43 MV/m, it will be accelerated and acquire 21.5 electron volt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...
s of energy in 0.5 µm of travel in the direction of the field. The first ionization energy
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...
needed to dislodge an electron from nitrogen
Nitrogen
Nitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...
is about 15 eV
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...
. The accelerated electron will acquire more than enough energy to ionize a nitrogen atom. This liberated electron will in turn be accelerated which will lead to another collision. A chain reaction then leads to avalanche breakdown
Townsend discharge
The Townsend discharge is a gas ionization process where an initially very small amount of free electrons, accelerated by a sufficiently strong electric field, give rise to electrical conduction through a gas by avalanche multiplication: when the number of free charges drops or the electric field...
and an arc takes place from the cascade of released electrons.
More collisions will take place in the electron path between the electrodes in a higher pressure gas. When the pressure-gap product
is high, an electron will collide with many different gas molecules as it travels from the cathode to the anode. Each of the collisions randomizes the electron direction, so the electron is not always being accelerated by the electric fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
-- sometimes it travels back towards the cathode and is decelerated by the field.
Collisions reduce the electron's energy and make it more difficult for it to ionize a molecule. Energy losses from a greater number of collisions require larger voltages for the electrons to accumulate sufficient energy to ionize many gas molecules, which is required to produce an avalanche breakdown
Electron avalanche
An electron avalanche is a process in which a number of free electrons in a medium are subjected to strong acceleration by an electric field, ionizing the medium's atoms by collision , thereby forming "new" electrons to undergo the same process in successive cycles...
.
On the left side of the Paschen minimum, the
product is small. The electron mean free path can become long compared to the gap between the electrodes. In this case, the electrons might gain lots of energy, but have fewer ionizing collisions. A greater voltage is therefore required to assure ionization of enough gas molecules to start an avalanche.Basics
To calculate the breakthrough voltage a homogeneous electrical field is assumed. This is the case in a parallel plate capacitorCapacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
setup. The electrodes may have the distance
. The cathode is located at the point .To get impact ionization
Impact ionization
Impact ionization is the process in a material by which one energetic charge carrier can lose energy by the creation of other charge carriers...
the electron energy
must become greater than the ionization energy of the gas atoms between the plates. Per length of path a number of ionizations will occur. is known as the first Townsend coefficient as it was introduced by Townsend in , section 17. The increase of the electron current can be described for the assumed setupas
(So the number of free electrons at the anode is equal to the number of free electrons at the cathode that were multiplied by impact ionization. The larger
and/or the more free electrons are created.)The number of created electrons is
Neglecting possible multiple ionizations of the same atom, the number of created ions is the same as the number of created electrons:
is the ion current. To keep the discharge going on, free electrons must be created at the cathode surface. This is possible because the ions hitting the cathode release secondary electronsSecondary electrons
Secondary electrons are electrons generated as ionization products. They are called 'secondary' because they are generated by other radiation . This radiation can be in the form of ions, electrons, or photons with sufficiently high energy, i.e. exceeding the ionization potential...
at the impact. (For very large applied voltages also field electron emission can occur.) Without field emission, we can write
where
is the mean number of generated secondary electrons per ion. This is also known as the second Townsend coefficient. Assuming that one gets the relation between the Townsend coefficients by putting (4) into (3) and transforming:Impact ionization
What is the amount of? The number of ionization depends upon the probability that an electron hits an ion. This probability is the relation of the cross-sectionalCross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....
area of a collision between electron and ion
in relation to the overall area that is available for the electron to fly through:As expressed by the second part of the equation, it is also possible to express the probability as relation of the path traveled by the electron
to the mean free pathMean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
(distance at which another collision occurs). is the number of electrons because every electron can hit. It can be calculated using the equation of state of the ideal gasIdeal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
- (
: pressure, 
: volume, 
: Boltzmann constant, 
: temperature)
The adjoining sketch illustrates that
. As the radius of an electron can be neglected compared to the radius of an ion it simplifies to . Using this relation, putting (7) into (6) and transforming to one getswhere the factor
was only introduced for a better overview.The alteration of the current of not yet collided electrons at every point in the path
can be expressed asThis differential equation can easily be solved:
The probability that
(that there was not yet a collision at the point ) isAccording to its definition
is the number of ionizations per length of path and thus the relation of the probability that there was no collision in the mean free path of the ions, and the mean free path of the electrons:It was hereby considered that the energy
that a charged particle can get between a collision depends on the electric field Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
strength
and the charge :Breakdown voltage
For the parallel-plate capacitor we have, where is the applied voltage. As a single ionization was assumed is the elementary chargeElementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
. We can now put (13) and (8) into (12) and getPutting this into (5) and transforming to
we get the Paschen law for the breakdown voltage that was first investigated by Paschen in and whose formula was first derived by Townsend in , section 227:
- with
Plasma ignition
Plasma ignitionPlasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
in definition of Townsend (Townsend discharge
Townsend discharge
The Townsend discharge is a gas ionization process where an initially very small amount of free electrons, accelerated by a sufficiently strong electric field, give rise to electrical conduction through a gas by avalanche multiplication: when the number of free charges drops or the electric field...
) is a self-sustaining discharge, independent of an external source of free electrons. This means that electrons from the cathode can reach the anode in the distance
and ionize at least one atom on its way. So according to the definition of this relation must be fulfilled:If
is used instead of (5) one gets for the breakdown voltageConclusions / Validity
Paschen's law requires that- there are already free electrons at the cathode (
) which can be accelerated to trigger impact ionization. Such so-called seed electrons can be created by ionization by cosmic x-ray backgroundX-ray backgroundThe observed X-ray background is thought to result from, at the "soft" end , Galactic X-ray emission , and, at the "hard" end , from a combination of many unresolved X-ray sources outside of the Milky Way .The galactic X-ray background is produced largely by emission from hot gas in the Local...
. - the creation of further free electrons is only achieved by impact ionization. Thus Paschen's law is not valid if there are external electron sources. This can for example be a light source creating secondary electrons via the photoelectric effectPhotoelectric effectIn the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...
. This has to be considered in experiments. - each ionized atom leads to only one free electron. But multiple ionizations occur always in practice.
- free electrons at the cathode surface are created by the impacting ions. The problem is that the number of thereby created electrons strongly depends on the material of the cathode, its surface (roughness, impurities) and the environmental conditions (temperature, humidityHumidityHumidity is a term for the amount of water vapor in the air, and can refer to any one of several measurements of humidity. Formally, humid air is not "moist air" but a mixture of water vapor and other constituents of air, and humidity is defined in terms of the water content of this mixture,...
etc.). The experimental, reproducible determination of the factor
is therefore nearly impossible. - the electrical field is homogeneous.
Effects with different gases
Different gases will have different mean free paths for molecules and electrons. This is because different molecules have different diameters. Noble gases like helium and argon are monoatomicMonatomic gas
In physics and chemistry, monatomic is a combination of the words "mono" and "atomic," and means "single atom." It is usually applied to gases: a monatomic gas is one in which atoms are not bound to each other....
and tend to have smaller diameters. This gives them a greater mean free path length.
Ionization potentials differ between molecules as well as the speed that they recapture electrons after they have been knocked out of orbit. All three effects change the number of collisions needed to cause an exponential growth in free electrons. These free electrons are necessary to cause an arc.