Beta (plasma physics)
Encyclopedia
The beta of a plasma
, symbolized by β, is the ratio of the plasma pressure
(p = n
kB T
) to the magnetic pressure
(pmag = B
²/2μ0
). The term is commonly used in studies of the Sun and Earth's magnetic field
, and in the field of fusion power
designs.
In the fusion power field, plasma is often confined using large superconducting magnet
s that are very expensive. Since the temperature of the fuel scales with pressure, reactors attempt to reach the highest pressures possible. The costs of large magnets roughly scales like β½. Therefore beta can be thought of as a ratio of money in to money out for a reactor, and beta can be thought of (very approximately) as an economic indicator of reactor efficiency. To make an economically useful reactor, betas better than 5% are needed.
The same term is also used when discussing the interactions of the solar wind
with various magnetic fields. For example, the beta in the corona of the Sun is about 1%
occurs when the nuclei of two atoms approach closely enough for the nuclear force
to pull them together into a single larger nucleus. The strong force is opposed by the electrostatic force created by the positive charge of the nuclei's proton
s, pushing the nuclei apart. The amount of energy that is needed to overcome this repulsion is known as the Coulomb barrier
. The amount of energy released by the fusion reaction when it occurs may be greater or less than the Coulomb barrier. Generally, lighter nuclei with a smaller number of protons and greater number of neutron
s will have the greatest ratio of energy released to energy required, and the majority of fusion power
research focusses on the use of deuterium
and tritium
, two isotope
s of hydrogen
.
Even using these isotopes, the Coulomb barrier is large enough that the nuclei must be given great amounts of energy before they will fuse. Although there are a number of ways to do this, the simplest is to simply heat the gas mixture, which, according to the Maxwell–Boltzmann distribution, will result in a small number of particles with the required energy even when the gas as a whole is relatively "cool" compared to the Coulomb barrier energy. In the case of the D-T mixture, rapid fusion will occur when the gas is heated to about 100 million degrees.
into a plasma, a mixture of electron
s and nuclei form a globally neutral gas. As the particles within the gas are charged, this allows them to be manipulated by electric or magnetic fields. This gives rise to the majority of controlled fusion concepts.
Even if this temperature is reached, the gas will be constantly losing energy to its surroundings (cooling off). This gives rise to the concept of the "confinement time", the amount of time the plasma is maintained at the required temperature. However, the fusion reactions might deposit their energy back into the plasma, heating it back up, which is a function of the density of the plasma. These considerations are combined in the Lawson criterion
, or its modern form, the fusion triple product. In order to be efficient, the rate of fusion energy being deposited into the reactor would ideally be greater than the rate of loss to the surroundings, a conduction known as "ignition".
(MCF) reactor designs, the plasma is confined within a vacuum chamber using a series of magnetic fields. These fields are normally created using a combination of electromagnet
s and electrical currents running through the plasma itself. Systems using only magnets are generally built using the stellarator
approach, while those using current only are the pinch
machines. The most studied approach since the 1970s is the tokamak
, where the fields generated by the external magnets and internal current are roughly equal in magnitude.
In all of these machines, the density of the particles in the plasma is very low, often described as a "poor vacuum". This limits its approach to the triple product along the temperature and time axis. This requires magnetic fields on the order of tens of Teslas
, currents in the megaampere, and confinement times on the order of tens of seconds. Generating currents of this magnitude is relatively simple, and a number of devices from large banks of capacitor
s to homopolar generator
s have been used. However, generating the required magnetic fields is another issue, generally requiring expensive superconducting magnet
s. For any given reactor design, the cost is generally dominated by the cost of the magnets.
is normally measured in terms of the total magnetic field. However, in any real-world design, the strength of the field varies over the volume of the plasma, so to be specific, the average beta is sometimes referred to as the "beta toroidal". In the tokamak design the total field is a combination of the external toroidal field and the current-induced poloidal one, so the "beta poloidal" is sometimes used to compare the relative strengths of these fields. And as the external magnetic field is the driver of reactor cost, "beta external" is used to consider just this contribution.
is always smaller than 1 (otherwise it would collapse). Ideally, a MCF device would want to approach this limit as closely as possible, as this would imply the minimum amount of magnetic force needed for confinement. In practice, it is difficult to come even close to this, and production machines generally operate at betas around 0.1, or 10%. The record was set by the START
device at 0.4, or 40%.
These low achievable betas are due to instabilities in the plasma generated through the interaction of the fields and the motion of the particles due to the induced current. As the amount of current is increased in relation to the external field, these instabilities become uncontrollable. In early pinch experiments the current dominated the field components and the kink and sausage instabilities were common, today collectively referred to as "low-n instabilities". As the relative strength of the external magnetic field is increased, these simple instabilities are damped out, but at a critical field other "high-n instabilities" will invariably appear, notably the ballooning mode. For any given reactor design, there is a limit to the beta it can sustain. As beta is a measure of economic merit, a practical reactor must be able to sustain a beta above some critical value, which is calculated to be around 5%.
Through the 1980s the understanding of the high-n instabilities grew considerably. Shafranov and Yurchenko first published on the issue in 1971 in a general discussion of tokamak design, but it was the work by Wesson and Sykes in 1983 and Francis Troyon in 1984 that developed these concepts fully. Troyon's considerations, or the "Troyon limit", closely matched the real-world performance of existing machines. It has since become so widely used that it is often known simply as the beta limit.
The Troyon limit is given as:
Where I is the induced current,
is the external magnetic field, and a is the minor radius of the tokamak (see torus
for an explanation of the directions).
was determined numerically, and is normally given as 0.028 if I is measured in megaamperes. However, it is also common to use 2.8 if 
is expressed as a percentage.
Given that the Troyon limit suggested a beta around 2.5 to 4%, and a practical reactor had to have a beta around 5%, the Troyon limit was a serious concern when it was introduced. However, it was found that
changed dramatically with the shape of the plasma, and non-circular systems would have much better performance. Experiments on the DIII-D machine (the second D referring to the cross-sectional shape of the plasma) demonstrated higher performance, and the spherical tokamak
design outperformed the Troyon limit by about 10 times.
with the magnetic fields of the Sun
or Earth
. In this case, the betas of these natural phenomena are generally much smaller than those seen in reactor designs; the Sun's corona
has a beta around 1%. Active regions have much higher beta, over 1 in some cases, which makes the area unstable.
Plasma (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...
, symbolized by β, is the ratio of the plasma 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 :...
(p = n
Number density
In physics, astronomy, and chemistry, number density is an intensive quantity used to describe the degree of concentration of countable objects in the three-dimensional physical space...
kB T
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
) to the magnetic pressure
Magnetic pressure
Magnetic pressure is an energy density associated with the magnetic field. It is identical to any other physical pressure except that it is carried by the magnetic field rather than kinetic energy of the gas molecules. Interplay between magnetic pressure and ordinary gas pressure is important to...
(pmag = B
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
²/2μ0
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
). The term is commonly used in studies of the Sun and Earth's magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
, and in the field of fusion power
Fusion power
Fusion power is the power generated by nuclear fusion processes. In fusion reactions two light atomic nuclei fuse together to form a heavier nucleus . In doing so they release a comparatively large amount of energy arising from the binding energy due to the strong nuclear force which is manifested...
designs.
In the fusion power field, plasma is often confined using large superconducting magnet
Superconducting magnet
A superconducting magnet is an electromagnet made from coils of superconducting wire. They must be cooled to cryogenic temperatures during operation. In its superconducting state the wire can conduct much larger electric currents than ordinary wire, creating intense magnetic fields...
s that are very expensive. Since the temperature of the fuel scales with pressure, reactors attempt to reach the highest pressures possible. The costs of large magnets roughly scales like β½. Therefore beta can be thought of as a ratio of money in to money out for a reactor, and beta can be thought of (very approximately) as an economic indicator of reactor efficiency. To make an economically useful reactor, betas better than 5% are needed.
The same term is also used when discussing the interactions of the solar wind
Solar wind
The solar wind is a stream of charged particles ejected from the upper atmosphere of the Sun. It mostly consists of electrons and protons with energies usually between 1.5 and 10 keV. The stream of particles varies in temperature and speed over time...
with various magnetic fields. For example, the beta in the corona of the Sun is about 1%
Fusion basics
Nuclear fusionNuclear fusion
Nuclear fusion is the process by which two or more atomic nuclei join together, or "fuse", to form a single heavier nucleus. This is usually accompanied by the release or absorption of large quantities of energy...
occurs when the nuclei of two atoms approach closely enough for the nuclear force
Nuclear force
The nuclear force is the force between two or more nucleons. It is responsible for binding of protons and neutrons into atomic nuclei. The energy released causes the masses of nuclei to be less than the total mass of the protons and neutrons which form them...
to pull them together into a single larger nucleus. The strong force is opposed by the electrostatic force created by the positive charge of the nuclei's proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s, pushing the nuclei apart. The amount of energy that is needed to overcome this repulsion is known as the Coulomb barrier
Coulomb barrier
The Coulomb barrier, named after Coulomb's law, which is named after physicist Charles-Augustin de Coulomb , is the energy barrier due to electrostatic interaction that two nuclei need to overcome so they can get close enough to undergo a nuclear reaction...
. The amount of energy released by the fusion reaction when it occurs may be greater or less than the Coulomb barrier. Generally, lighter nuclei with a smaller number of protons and greater number of neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s will have the greatest ratio of energy released to energy required, and the majority of fusion power
Fusion power
Fusion power is the power generated by nuclear fusion processes. In fusion reactions two light atomic nuclei fuse together to form a heavier nucleus . In doing so they release a comparatively large amount of energy arising from the binding energy due to the strong nuclear force which is manifested...
research focusses on the use of deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...
and tritium
Tritium
Tritium is a radioactive isotope of hydrogen. The nucleus of tritium contains one proton and two neutrons, whereas the nucleus of protium contains one proton and no neutrons...
, two isotope
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...
s of hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
.
Even using these isotopes, the Coulomb barrier is large enough that the nuclei must be given great amounts of energy before they will fuse. Although there are a number of ways to do this, the simplest is to simply heat the gas mixture, which, according to the Maxwell–Boltzmann distribution, will result in a small number of particles with the required energy even when the gas as a whole is relatively "cool" compared to the Coulomb barrier energy. In the case of the D-T mixture, rapid fusion will occur when the gas is heated to about 100 million degrees.
Confinement
This temperature is well beyond the physical limits of any material container that might contain the gasses, which has led to a number of different approaches to solving this problem. The main approach relies on the nature of the fuel at high temperatures. When the fusion fuel gasses are heated to the temperatures required for rapid fusion, they will be completely ionizeIonization
Ionization is the process of converting an atom or molecule into an ion by adding or removing charged particles such as electrons or other ions. This is often confused with dissociation. A substance may dissociate without necessarily producing ions. As an example, the molecules of table sugar...
into a plasma, a mixture of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s and nuclei form a globally neutral gas. As the particles within the gas are charged, this allows them to be manipulated by electric or magnetic fields. This gives rise to the majority of controlled fusion concepts.
Even if this temperature is reached, the gas will be constantly losing energy to its surroundings (cooling off). This gives rise to the concept of the "confinement time", the amount of time the plasma is maintained at the required temperature. However, the fusion reactions might deposit their energy back into the plasma, heating it back up, which is a function of the density of the plasma. These considerations are combined in the Lawson criterion
Lawson criterion
In nuclear fusion research, the Lawson criterion, first derived on fusion reactors by John D. Lawson in 1955 and published in 1957, is an important general measure of a system that defines the conditions needed for a fusion reactor to reach ignition, that is, that the heating of the plasma by the...
, or its modern form, the fusion triple product. In order to be efficient, the rate of fusion energy being deposited into the reactor would ideally be greater than the rate of loss to the surroundings, a conduction known as "ignition".
MCF approach
In magnetic confinement fusionMagnetic confinement fusion
Magnetic confinement fusion is an approach to generating fusion power that uses magnetic fields to confine the hot fusion fuel in the form of a plasma. Magnetic confinement is one of two major branches of fusion energy research, the other being inertial confinement fusion. The magnetic approach is...
(MCF) reactor designs, the plasma is confined within a vacuum chamber using a series of magnetic fields. These fields are normally created using a combination of electromagnet
Electromagnet
An electromagnet is a type of magnet in which the magnetic field is produced by the flow of electric current. The magnetic field disappears when the current is turned off...
s and electrical currents running through the plasma itself. Systems using only magnets are generally built using the stellarator
Stellarator
A stellarator is a device used to confine a hot plasma with magnetic fields in order to sustain a controlled nuclear fusion reaction. It is one of the earliest controlled fusion devices, first invented by Lyman Spitzer in 1950 and built the next year at what later became the Princeton Plasma...
approach, while those using current only are the pinch
Pinch (plasma physics)
A pinch is the compression of an electrically conducting filament by magnetic forces. The conductor is usually a plasma, but could also be a solid or liquid metal...
machines. The most studied approach since the 1970s is the tokamak
Tokamak
A tokamak is a device using a magnetic field to confine a plasma in the shape of a torus . Achieving a stable plasma equilibrium requires magnetic field lines that move around the torus in a helical shape...
, where the fields generated by the external magnets and internal current are roughly equal in magnitude.
In all of these machines, the density of the particles in the plasma is very low, often described as a "poor vacuum". This limits its approach to the triple product along the temperature and time axis. This requires magnetic fields on the order of tens of Teslas
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...
, currents in the megaampere, and confinement times on the order of tens of seconds. Generating currents of this magnitude is relatively simple, and a number of devices from large banks of capacitor
Capacitor
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...
s to homopolar generator
Homopolar generator
A homopolar generator is a DC electrical generator comprising an electrically conductive disc rotating in a plane perpendicular to a uniform static magnetic field. A potential difference is created between the center of the disc and the rim, the electrical polarity depending on the direction of...
s have been used. However, generating the required magnetic fields is another issue, generally requiring expensive superconducting magnet
Superconducting magnet
A superconducting magnet is an electromagnet made from coils of superconducting wire. They must be cooled to cryogenic temperatures during operation. In its superconducting state the wire can conduct much larger electric currents than ordinary wire, creating intense magnetic fields...
s. For any given reactor design, the cost is generally dominated by the cost of the magnets.
Beta
Given that the magnets are a dominant factor in reactor design, and that density and temperature combine to produce pressure, the ratio of the magnetic energy to the pressure of the plasma naturally becomes a useful figure of merit when comparing MCF designs. In effect, the ratio illustrates how effectively a design confines its plasma. This ratio, beta, is widely used in the fusion field: is normally measured in terms of the total magnetic field. However, in any real-world design, the strength of the field varies over the volume of the plasma, so to be specific, the average beta is sometimes referred to as the "beta toroidal". In the tokamak design the total field is a combination of the external toroidal field and the current-induced poloidal one, so the "beta poloidal" is sometimes used to compare the relative strengths of these fields. And as the external magnetic field is the driver of reactor cost, "beta external" is used to consider just this contribution.Troyon beta limit
For a stable plasma, is always smaller than 1 (otherwise it would collapse). Ideally, a MCF device would want to approach this limit as closely as possible, as this would imply the minimum amount of magnetic force needed for confinement. In practice, it is difficult to come even close to this, and production machines generally operate at betas around 0.1, or 10%. The record was set by the STARTSmall Tight Aspect Ratio Tokamak
The Small Tight Aspect Ratio Tokamak, or START was a nuclear fusion experiment that used magnetic confinement to hold plasma. The experiment began at the Culham Science Centre in the United Kingdom in 1991 and was retired in 1998. It was built as a low cost design, largely using parts already...
device at 0.4, or 40%.
These low achievable betas are due to instabilities in the plasma generated through the interaction of the fields and the motion of the particles due to the induced current. As the amount of current is increased in relation to the external field, these instabilities become uncontrollable. In early pinch experiments the current dominated the field components and the kink and sausage instabilities were common, today collectively referred to as "low-n instabilities". As the relative strength of the external magnetic field is increased, these simple instabilities are damped out, but at a critical field other "high-n instabilities" will invariably appear, notably the ballooning mode. For any given reactor design, there is a limit to the beta it can sustain. As beta is a measure of economic merit, a practical reactor must be able to sustain a beta above some critical value, which is calculated to be around 5%.
Through the 1980s the understanding of the high-n instabilities grew considerably. Shafranov and Yurchenko first published on the issue in 1971 in a general discussion of tokamak design, but it was the work by Wesson and Sykes in 1983 and Francis Troyon in 1984 that developed these concepts fully. Troyon's considerations, or the "Troyon limit", closely matched the real-world performance of existing machines. It has since become so widely used that it is often known simply as the beta limit.
The Troyon limit is given as:
Where I is the induced current,
is the external magnetic field, and a is the minor radius of the tokamak (see torusTorus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
for an explanation of the directions).
was determined numerically, and is normally given as 0.028 if I is measured in megaamperes. However, it is also common to use 2.8 if is expressed as a percentage.Given that the Troyon limit suggested a beta around 2.5 to 4%, and a practical reactor had to have a beta around 5%, the Troyon limit was a serious concern when it was introduced. However, it was found that
changed dramatically with the shape of the plasma, and non-circular systems would have much better performance. Experiments on the DIII-D machine (the second D referring to the cross-sectional shape of the plasma) demonstrated higher performance, and the spherical tokamakSpherical tokamak
A spherical tokamak is a type of fusion power device based on the tokamak principle. It is notable for its very narrow profile, or "aspect ratio". A traditional tokamak has a toroidal confinement area that gives it an overall shape similar to a donut, complete with a large hole in the middle...
design outperformed the Troyon limit by about 10 times.
Astrophysics
Beta is also sometimes used when discussing the interaction of plasma in space with different magnetic fields. A common example is the interaction of the solar windSolar wind
The solar wind is a stream of charged particles ejected from the upper atmosphere of the Sun. It mostly consists of electrons and protons with energies usually between 1.5 and 10 keV. The stream of particles varies in temperature and speed over time...
with the magnetic fields of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
or Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
. In this case, the betas of these natural phenomena are generally much smaller than those seen in reactor designs; the Sun's corona
Corona
A corona is a type of plasma "atmosphere" of the Sun or other celestial body, extending millions of kilometers into space, most easily seen during a total solar eclipse, but also observable in a coronagraph...
has a beta around 1%. Active regions have much higher beta, over 1 in some cases, which makes the area unstable.