Paschen's law is an equation that gives the
breakdown voltage
The breakdown voltage of an insulator is the minimum voltage that causes a portion of an insulator to experience electrical breakdown and become electrically conductive.
For diodes, the breakdown voltage is the minimum reverse voltage that mak ...
, that is, the
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
necessary to start a discharge or
electric arc
An electric arc, or arc discharge, is an electrical breakdown of a gas that produces a prolonged electrical discharge. The current through a normally nonconductive medium such as air produces a plasma; the plasma may produce visible light. ...
, between two electrodes in a gas as a function of pressure and gap length.
It is named after
Friedrich Paschen
Louis Carl Heinrich Friedrich Paschen (22 January 1865 - 25 February 1947), 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 t ...
who discovered it empirically in 1889.
Paschen studied the breakdown
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
of various
gases between parallel metal plates as the gas
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
and gap
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
were varied:
* With a constant gap length, the voltage necessary to
arc across the gap decreased as the pressure was reduced and then increased gradually, exceeding its original value.
* With a constant pressure, the voltage needed to cause an arc reduced as the gap size was reduced but only to a point. As the gap was reduced further, the voltage required to cause an arc began to rise and again exceeded its original value.
For a given gas, the voltage is a function only of the product of the pressure and gap length.
The curve he found of voltage versus the pressure-gap length product ''(right)'' is called Paschen's curve. He found an equation that fit these curves, which is now called Paschen's law.
At higher pressures and gap lengths, the breakdown voltage is approximately ''proportional'' to the product of pressure and gap length, and the term Paschen's law is sometimes used to refer to this simpler relation.
However, this is only roughly true, over a limited range of the curve.
Paschen curve
Early
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
experimenters found a rather surprising behavior. An arc would sometimes take place in a long irregular path rather than at the minimal distance between the electrodes. For example, in air, at a pressure of one
atmosphere
An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A ...
, the distance for minimal
breakdown voltage
The breakdown voltage of an insulator is the minimum voltage that causes a portion of an insulator to experience electrical breakdown and become electrically conductive.
For diodes, the breakdown voltage is the minimum reverse voltage that mak ...
is about 7.5 μm. The voltage required to arc this distance is 327 V, which is insufficient to ignite the arcs for gaps that are either wider or narrower. For a 3.5 μm 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 μm distance, but would arc at a 7.5 μm distance.
Paschen found that breakdown voltage was described by the equation
:
where
is the breakdown voltage in
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defin ...
s,
is the pressure in
pascals
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defin ...
,
is the gap distance in
meters
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pr ...
,
is the
secondary-electron-emission coefficient (the number of secondary electrons produced per incident positive ion),
is the saturation ionization in the gas at a particular
(
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
/pressure), and
is related to the excitation and ionization energies.
The
constants
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ...
and
are determined experimentally and found to be roughly constant over a restricted range of
for any given gas. For example,
air
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
with an
in the range of 450 to 7500 V/(kPa·cm),
= 112.50 (kPa·cm)
−1 and
= 2737.50 V/(kPa·cm).
The graph of this equation is the Paschen curve. By differentiating it with respect to
and setting the derivative to zero, the minimal voltage can be found. This yields
:
and predicts the occurrence of a minimal breakdown voltage for
= 7.5×10
−6 m·atm. This is 327 V in
air
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
at standard atmospheric pressure at a distance of 7.5 μm.
The composition of the gas determines both the minimal arc voltage and the distance at which it occurs. For
argon
Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice a ...
, the minimal arc voltage is 137 V at a larger 12 μm. For
sulfur dioxide
Sulfur dioxide (IUPAC-recommended spelling) or sulphur dioxide (traditional Commonwealth English) is the chemical compound with the formula . It is a toxic gas responsible for the odor of burnt matches. It is released naturally by volcanic a ...
, the minimal arc voltage is 457 V at only 4.4 μm.
Long gaps
For air at
standard conditions for temperature and pressure
Standard temperature and pressure (STP) are standard sets of conditions for experimental measurements to be established to allow comparisons to be made between different sets of data. The most used standards are those of the International Union ...
(STP), the voltage needed to arc a 1-metre gap is about 3.4 MV. The intensity of the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
for this gap is therefore 3.4 MV/m.
The electric field needed to arc across the minimal-voltage gap is much greater than what is necessary to arc a gap of one metre. At large gaps (or large pd) Paschen's Law is known to fail. The
Meek Criteria for breakdown is usually used for large gaps.
It takes into account non-uniformity in the electric field and formation of streamers due to the build up of charge within the gap that can occur over long distances. For a 7.5 μm gap the arc voltage is 327 V, which is 43 MV/m. This is about 14 times greater than the field strength for the 1.5-metre 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 micrometres. 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 path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
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, given constant temperature. In air at STP the mean free path of molecules is about 96 nm. Since electrons are much smaller, their average distance between colliding with molecules is about 5.6 times longer, or about 0.5 μm. This is a substantial fraction of the 7.5 μm spacing between the electrodes for minimal arc voltage. If the electron is in an electric field of 43 MV/m, it will be accelerated and acquire 21.5
eV of energy in 0.5 μm of travel in the direction of the field. The first
ionization energy
Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule ...
needed to dislodge an electron from
nitrogen
Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...
molecule is about 15.6 eV. The accelerated electron will acquire more than enough energy to ionize a nitrogen molecule. This liberated electron will in turn be accelerated, which will lead to another collision. A
chain reaction
A chain reaction is a sequence of reactions where a reactive product or by-product causes additional reactions to take place. In a chain reaction, positive feedback leads to a self-amplifying chain of events.
Chain reactions are one way that sys ...
then leads to
avalanche breakdown
Avalanche breakdown (or avalanche effect) is a phenomenon that can occur in both insulating and semiconducting materials. It is a form of electric current multiplication that can allow very large currents within materials which are otherwise good ...
, 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
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical 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
Avalanche breakdown (or avalanche effect) is a phenomenon that can occur in both insulating and semiconducting materials. It is a form of electric current multiplication that can allow very large currents within materials which are otherwise good ...
.
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 large amounts of energy, but have fewer ionizing collisions. A greater voltage is therefore required to assure ionization of enough gas molecules to start an avalanche.
Derivation
Basics
To calculate the breakthrough voltage, a homogeneous electrical field is assumed. This is the case in a parallel-plate
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
setup. The electrodes may have the distance
. The cathode is located at the point
.
To get
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. For example, in semiconductors, an electron (or hole) with enough kinetic energy can knock a bound ...
, 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.
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
Secondary electrons are electrons generated as ionization products. They are called 'secondary' because they are generated by other radiation (the ''primary'' radiation
In physics, radiation is the emission or transmission of energy in th ...
at the impact. (For very large applied voltages also
field electron emission
Field electron emission, also known as field emission (FE) and electron field emission, is emission of electrons induced by an electrostatic field. The most common context is field emission from a solid surface into a vacuum. However, field emissio ...
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 () into () and transforming:
Impact ionization
What is the amount of
? The number of ionization depends upon the probability that an electron hits a gas molecule. This probability
is the relation of the
cross-sectional
Cross-sectional data, or a cross section of a study population, in statistics and econometrics, is a type of data collected by observing many subjects (such as individuals, firms, countries, or regions) at the one point or period of time. The analy ...
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
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
(distance at which another collision occurs).
is the number of molecules which electrons can hit. It can be calculated using the equation of state of the
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
:
(: pressure, : volume, : Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, : 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 () into () 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
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
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 charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
. We can now put () and () into () 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
[
]
:
with
Plasma ignition
Plasma ignition in the definition of Townsend (
Townsend discharge
The Townsend discharge or Townsend avalanche is a gas ionisation process where free electrons are accelerated by an electric field, collide with gas molecules, and consequently free additional electrons. Those electrons are in turn accelerated an ...
) 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 their way. So according to the definition of
this relation must be fulfilled:
If
is used instead of () one gets for the breakdown voltage
Conclusions, 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 natural radioactivity or cosmic rays.
* 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 by the
photoelectric effect
The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...
. This has to be considered in experiments.
* Each ionized atom leads to only one free electron. However, 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,
humidity
Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation, dew, or fog to be present.
Humidity dep ...
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
monatomic
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 a gas in which atoms are not bound to each other. Examples at standard conditions ...
and tend to have smaller diameters. This gives them greater mean free paths.
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.
See also
*
Atmospheric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, ...
*
Breakdown voltage
The breakdown voltage of an insulator is the minimum voltage that causes a portion of an insulator to experience electrical breakdown and become electrically conductive.
For diodes, the breakdown voltage is the minimum reverse voltage that mak ...
*
Dielectric strength
In physics, the term dielectric strength has the following meanings:
*for a pure electrically insulating material, the maximum electric field that the material can withstand under ideal conditions without undergoing electrical breakdown and beco ...
*
Townsend discharge
The Townsend discharge or Townsend avalanche is a gas ionisation process where free electrons are accelerated by an electric field, collide with gas molecules, and consequently free additional electrons. Those electrons are in turn accelerated an ...
References
External links
Electrical breakdown limits for MEMS{{Webarchive, url=https://web.archive.org/web/20111016014127/http://home.earthlink.net/~jimlux/hv/paschen.htm , date=2011-10-16
Paschen's law calculatorBreakdown Voltage vs. PressureElectrical Breakdown of Low Pressure GasesPressure Dependence of Plasma Structure in Microwave Gas Breakdown at 110GHz
Electrical discharge in gases
Electrochemistry
Electrostatics
Electrical breakdown
Plasma physics