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**Paschen's law** is an equation that gives the breakdown voltage, that is, the voltage necessary to start a discharge or electric arc, between two electrodes in a gas as a function of pressure and gap length.^{ [2] }^{ [3] } It is named after Friedrich Paschen who discovered it empirically in 1889.^{ [4] }

The **breakdown voltage** of an insulator is the minimum voltage that causes a portion of an insulator to become electrically conductive.

**Voltage**, **electric potential difference**, **electric pressure **or **electric tension** is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named *volt*. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for *volt* uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by ∆*V*, but more often simply as *V*, for instance in the context of Ohm's or Kirchhoff's circuit laws.

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. An arc discharge is characterized by a lower voltage than a glow discharge and relies on thermionic emission of electrons from the electrodes supporting the arc. An archaic term is **voltaic arc**, as used in the phrase "voltaic arc lamp".

- Paschen curve
- Long gaps
- Physical mechanism
- Derivation
- Basics
- Impact ionization
- Breakdown voltage
- Plasma ignition
- Conclusions, validity
- Effects with different gases
- See also
- References
- External links

Paschen studied the breakdown voltage of various gases between parallel metal plates as the gas pressure and gap distance were varied:

**Gas** is one of the four fundamental states of matter. A pure gas may be made up of individual atoms, elemental molecules made from one type of atom, or compound molecules made from a variety of atoms. A gas mixture, such as air, contains a variety of pure gases. What distinguishes a gas from liquids and solids is the vast separation of the individual gas particles. This separation usually makes a colorless gas invisible to the human observer. The interaction of gas particles in the presence of electric and gravitational fields are considered negligible, as indicated by the constant velocity vectors in the image.

**Pressure** is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

**Distance** is a numerical 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. In most cases, "distance from A to B" is interchangeable with "distance from B to A". In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and is a way of describing what it means for elements of some space to be "close to" or "far away from" each other.

- 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.^{ [2] }^{ [3] } 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.^{ [3] }

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.^{ [5] } However, this is only roughly true, over a limited range of the curve.

Early vacuum 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, the distance for minimal breakdown voltage 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.

**Vacuum** is space devoid of matter. The word stems 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 discuss ideal test results that would occur in a *perfect* vacuum, which they sometimes simply call "vacuum" or **free space**, and use the term **partial vacuum** to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is lower than atmospheric pressure. The Latin term * in vacuo* is used to describe an object that is surrounded by a vacuum.

The **standard atmosphere** is a unit of pressure defined as 101325 Pa. It is sometimes used as a *reference* or *standard pressure*. It is approximately equal to the atmospheric pressure at sea level.

Paschen found that breakdown voltage was described by the equation^{ [1] }

where is the breakdown voltage in volts, is the pressure in pascals, is the gap distance in meters, 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/pressure), and is related to the excitation and ionization energies.

The constants and are determined experimentally and found to be roughly constant over a restricted range of for any given gas. For example, air with an in the range of 450 to 7500 V/(kPa·cm), = 112.50 (kPa·cm)^{−1} and = 2737.50 V/(kPa·cm).^{ [6] }

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 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, the minimal arc voltage is 137 V at a larger 12 μm. For sulfur dioxide, the minimal arc voltage is 457 V at only 4.4 μm.

For air at standard conditions for temperature and pressure (STP), the voltage needed to arc a 1-metre gap is about 3.4 MV.^{ [7] } The intensity of the electric 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. 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-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^{ [8] } 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.

The mean free path 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 at 1 atm 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 needed to dislodge an electron from nitrogen 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 then leads to avalanche breakdown, and an arc takes place from the cascade of released electrons.^{ [9] }

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.

To calculate the breakthrough voltage, a homogeneous electrical field is assumed. This is the case in a parallel-plate capacitor 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^{ [10] }. 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:

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 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 molecules which electrons can hit. It can be calculated using the equation of state of the ideal gas

- (: 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 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 :

For the parallel-plate capacitor we have , 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^{ [11] } and whose formula was first derived by Townsend in,^{ [12] } section 227:

- with

Plasma ignition 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 their way. So according to the definition of this relation must be fulfilled:

If is used instead of (5) one gets for the breakdown voltage

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 background. - 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. 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 etc.). The experimental, reproducible determination of the factor is therefore nearly impossible.
- The electrical field is homogeneous.

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 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.

In probability theory and statistics, the **exponential distribution** is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

**Compton scattering**, discovered by Arthur Holly Compton, is the scattering of a photon by a charged particle, usually an electron. It results in a decrease in energy of the photon, called the **Compton effect**. Part of the energy of the photon is transferred to the recoiling electron. **Inverse Compton scattering** occurs when a charged particle transfers part of its energy to a photon.

**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 is called an ion. Ionization can result from the loss of an electron after collisions with subatomic particles, collisions with other atoms, molecules and ions, or through the interaction with electromagnetic radiation. Heterolytic bond cleavage and heterolytic substitution reactions can result in the formation of ion pairs. Ionization can occur through radioactive decay by the internal conversion process, in which an excited nucleus transfers its energy to one of the inner-shell electrons causing it to be ejected.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

In probability theory and statistics, the **Weibull distribution** is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution.

In economics and econometrics, the **Cobb–Douglas production function** is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs and the amount of output that can be produced by those inputs. The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947.

**Electrical breakdown** or **dielectric breakdown** is when current flows through an electrical insulator when the voltage applied across it exceeds the breakdown voltage. This results in the insulator becoming electrically conductive. Electrical breakdown may be a momentary event, or may lead to a continuous arc if protective devices fail to interrupt the current in a power circuit.

The **Klein–Nishina formula** gives the differential cross section of photons scattered from a single free electron in lowest order of quantum electrodynamics. At low frequencies this yields Thomson scattering; at higher frequencies this yields Compton scattering.

In mathematical logic and type theory, the **λ-cube** is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to:

The **proportional counter** is a type of gaseous ionization detector device used to measure particles of ionizing radiation. The key feature is its ability to measure the energy of incident radiation, by producing a detector output pulse that is *proportional* to the radiation energy absorbed by the detector due to an ionizing event; hence the detector's name. It is widely used where energy levels of incident radiation must be known, such as in the discrimination between alpha and beta particles, or accurate measurement of X-ray radiation dose.

The **Pearson distribution** is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.

In statistics and information theory, a **maximum entropy probability distribution** has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.

In general relativity, a **geodesic** generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

In continuum mechanics, the **finite strain theory**—also called **large strain theory**, or **large deformation theory**—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.

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 and free additional electrons. The result is an avalanche multiplication that permits electrical conduction through the gas. The discharge requires a source of free electrons and a significant electric field; without both, the phenomenon does not occur.

The **gaseous detection device-GDD** is a method and apparatus for the detection of signals in the gaseous environment of an environmental scanning electron microscope (ESEM) and all scanned beam type of instruments that allow a minimum gas pressure for the detector to operate.

**Pitzer equations** are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water. They were first described by physical chemist Kenneth Pitzer. The parameters of the Pitzer equations are linear combinations of parameters, of a virial expansion of the excess Gibbs free energy, which characterise interactions amongst ions and solvent. The derivation is thermodynamically rigorous at a given level of expansion. The parameters may be derived from various experimental data such as the osmotic coefficient, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion activity coefficients and water activities in solutions of high ionic strength for which the Debye–Hückel theory is no longer adequate. They are more rigorous than the equations of specific ion interaction theory, but Pitzer parameters are more difficult to determine experimentally than SIT parameters.

In mathematics, **Ricci calculus** constitutes the rules of index notation and manipulation for tensors and tensor fields. It is also the modern name for what used to be called the **absolute differential calculus**, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.

**Heat transfer physics** describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is also transformed (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

- 1 2 Lieberman, Michael A.; Lichtenberg, Allan J. (2005).
*Principles of plasma discharges and materials processing*(2nd ed.). Hoboken, N.J.: Wiley-Interscience. 546. ISBN 978-0471005773. OCLC 59760348. - 1 2 "Paschen's Law".
*Merriam-Webster Online Dictionary*. Merriam-Webster, Inc. 2013. Retrieved June 9, 2017. - 1 2 3 Wadhwa, C.L. (2007).
*High Voltage Engineering*(2nd ed.). New Age International. pp. 10–12. ISBN 978-8122418590. - ↑ Friedrich Paschen (1889). "
*Ueber die zum Funkenübergang in Luft, Wasserstoff und Kohlensäure bei verschiedenen Drucken erforderliche Potentialdifferenz*(On the potential difference required for spark initiation in air, hydrogen, and carbon dioxide at different pressures)".*Annalen der Physik*.**273**(5): 69–75. Bibcode:1889AnP...273...69P. doi:10.1002/andp.18892730505. hdl:2027/uc1.$b624756. - ↑ Graf, Rudolf F. (1999).
*Modern Dictionary of Electronics*(7th ed.). Newnes. p. 542. ISBN 978-0750698665. - ↑ Husain, E.; Nema, R. (August 1982). "Analysis of Paschen Curves for air, N2 and SF6 Using the Townsend Breakdown Equation".
*IEEE Transactions on Electrical Insulation*. EI-17 (4): 350–353. doi:10.1109/TEI.1982.298506. - ↑ Tipler, Paul (1987).
*College physics*. New York, NY: Worth Publishers. p. 467. ISBN 978-0879012687. - ↑ Emmanouel Hourdakis; Brian J. Simonds & Neil M. Zimmerman (2006). "Submicron gap capacitor for measurement of breakdown voltage in air".
*Rev. Sci. Instrum*.**77**(3): 034702–034702–4. Bibcode:2006RScI...77c4702H. doi:10.1063/1.2185149. - ↑ Electrical Discharges-How the spark, glow and arc work.
- ↑ J. Townsend, [The Theory of Ionization of Gases by Collision http://www.worldcat.org/wcpa/oclc/8460026]. Constable, 1910. Section 17.
- ↑ Paschen, F. (1889). "Ueber die zum Funkenübergang in Luft, Wasserstoff und Kohlensäure bei verschiedenen Drucken erforderliche Potentialdifferenz".
*Annalen der Physik*.**273**(5): 69–96. Bibcode:1889AnP...273...69P. doi:10.1002/andp.18892730505. hdl:2027/uc1.$b624756. - ↑ J. Townsend, Electricity in Gases. Clarendon Press, 1915. Online: http://www.worldcat.org/wcpa/oclc/4294747

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