![Fustion triple-product diagram Horvath](https://upload.wikimedia.org/wikipedia/commons/c/cb/Fustion_triple-product_diagram_Horvath.webp)
The Lawson criterion is a
figure of merit
A figure of merit is a quantity used to characterize the performance of a device, system or method, relative to its alternatives. Examples
*Clock rate of a CPU
* Calories per serving
*Contrast ratio of an LCD
*Frequency response of a speaker
* ...
used in
nuclear fusion
Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is manifest ...
research. It compares the rate of energy being generated by fusion reactions within the fusion fuel to the rate of energy losses to the environment. When the rate of production is higher than the rate of loss, the system will produce net energy. If enough of that energy is captured by the fuel, the system will become self-sustaining and is said to be
ignited.
The concept was first developed by
John D. Lawson in a classified 1955 paper that was declassified and published in 1957.
As originally formulated, the Lawson criterion gives a minimum required value for the product of the plasma (electron) density ''n''
e and the "energy confinement time"
that leads to net energy output.
Later analysis suggested that a more useful figure of merit is the triple product of density, confinement time, and plasma temperature ''T''. The triple product also has a minimum required value, and the name "Lawson criterion" may refer to this value.
On August 8, 2021, researchers at Lawrence Livermore National Laboratory's
National Ignition Facility
The National Ignition Facility (NIF) is a laser-based inertial confinement fusion (ICF) research device, located at Lawrence Livermore National Laboratory in Livermore, California, United States. NIF's mission is to achieve fusion ignition wit ...
in California confirmed to have produced the first-ever successful ignition of a nuclear fusion reaction surpassing the Lawson's criteria in the experiment.
Energy balance
The central concept of the Lawson criterion is an examination of the energy balance for any fusion power plant using a hot plasma. This is shown below:
Net power = Efficiency × (Fusion − Radiation loss − Conduction loss)
#Net power is the excess power beyond that needed internally for the process to proceed in any fusion power plant.
#Efficiency is how much energy is needed to drive the device and how well it collects energy from the reactions.
#Fusion is rate of energy generated by the fusion reactions.
#Radiation loss is the energy lost as light (including
X-rays
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
) leaving the plasma.
#Conduction loss is the energy lost as particles leave the plasma, carrying away energy.
Lawson calculated the fusion rate by assuming that the fusion reactor contains a hot plasma cloud which has a
Gaussian curve
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
of individual particle energies, a
Maxwell–Boltzmann distribution
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and used ...
characterized by the plasma's temperature. Based on that assumption, he estimated the first term, the fusion energy being produced, using the volumetric fusion equation.
[Lyman J Spitzer, "The Physics of Fully Ionized Gases" 1963]
Fusion = Number density of fuel A × Number density of fuel B × Cross section(Temperature) × Energy per reaction
#Fusion is the rate of fusion energy produced by the plasma
#
Number density
The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
is the density in particles per unit volume of the respective fuels (or just one fuel, in some cases)
#Cross section is a measure of the probability of a fusion event, which is based on the plasma temperature
#Energy per reaction is the energy released in each fusion reaction
This equation is typically averaged over a population of ions which has a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. The result is the amount of energy being created by the plasma at any instant in time.
Lawson then estimated
the radiation losses using the following equation:
where ''N'' is the number density of the cloud and ''T'' is the temperature. For his analysis, Lawson ignores conduction losses. In reality this is nearly impossible; practically all systems lose energy through mass leaving the plasma and carrying away its energy.
By equating radiation losses and the volumetric fusion rates, Lawson estimated the minimum temperature for the fusion for the
deuterium
Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
–
tritium
Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleus o ...
(D-T) reaction
:
to be 30 million degrees (2.6 keV), and for the deuterium–deuterium (D-D) reaction
:
to be 150 million degrees (12.9 keV).
Extensions into ''nτ''''E''
The confinement time
measures the rate at which a system loses energy to its environment. The faster the rate of loss of energy,
, the shorter the energy confinement time. It is the energy density
(energy content per unit volume) divided by the power loss density
(rate of energy loss per unit volume):
:
For a fusion reactor to operate in steady state, the fusion plasma must be maintained at a constant temperature. Thermal energy must therefore be added at the same rate the plasma loses energy in order to maintain the fusion conditions. This energy can be supplied by the fusion reactions themselves, depending on the reaction type, or by supplying additional heating through a variety of methods.
For illustration, the Lawson criterion for the D-T reaction will be derived here, but the same principle can be applied to other fusion fuels. It will also be assumed that all species have the same temperature, that there are no ions present other than fuel ions (no impurities and no helium ash), and that D and T are present in the optimal 50-50 mixture. Ion density then equals electron density and the energy density of both electrons and ions together is given by
:
where
is the temperature in electronvolt (eV) and
is the particle density.
The volume rate
(reactions per volume per time) of fusion reactions is
:
where
is the fusion
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
,
is the
relative velocity
The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.
Classical mechanics
In one dimension (non-relativistic)
We begin with relative motion in the classi ...
, and
denotes an average over the
Maxwellian velocity distribution at the temperature
.
The volume rate of heating by fusion is
times
, the energy of the charged fusion products (the neutrons cannot help to heat the plasma). In the case of the D-T reaction,
.
The Lawson criterion requires that fusion heating exceeds the losses:
:
Substituting in known quantities yields:
:
Rearranging the equation produces:
The quantity
is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product
. This is the Lawson criterion.
For the
deuterium
Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
–
tritium
Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleus o ...
reaction, the physical value is at least
:
The minimum of the product occurs near
.
Extension into the "triple product"
A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, ''nTτ''
''E''. For most confinement concepts, whether
inertial
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
,
mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
, or toroidal confinement, the density and temperature can be varied over a fairly wide range, but the maximum attainable pressure ''p'' is a constant. When such is the case, the fusion power density is proportional to ''p''
2<σ''v''>/''T''
2. The maximum fusion power available from a given machine is therefore reached at the temperature ''T'' where <σ''v''>/''T''
2 is a maximum. By continuation of the above derivation, the following inequality is readily obtained:
:
The quantity
is also a function of temperature with an absolute minimum at a slightly lower temperature than
.
For the D-T reaction, the minimum occurs at ''T'' = 14 keV. The average <σ''v''> in this temperature region can be approximated as
:
so the minimum value of the triple product value at ''T'' = 14 keV is about
:
This number has not yet been achieved in any reactor, although the latest generations of machines have come close.
JT-60
JT-60 (short for Japan Torus-60) is a large research tokamak, the flagship of Japan's magnetic fusion program, previously run by the Japan Atomic Energy Research Institute (JAERI) and currently run by the Japan Atomic Energy Agency's (JAEA) Naka ...
reported 1.53x10
21 keV.s.m
−3. For instance, the
TFTR
The Tokamak Fusion Test Reactor (TFTR) was an experimental tokamak built at Princeton Plasma Physics Laboratory (PPPL) circa 1980 and entering service in 1982. TFTR was designed with the explicit goal of reaching scientific breakeven, the point wh ...
has achieved the densities and energy lifetimes needed to achieve Lawson at the temperatures it can create, but it cannot create those temperatures at the same time.
ITER
ITER (initially the International Thermonuclear Experimental Reactor, ''iter'' meaning "the way" or "the path" in Latin) is an international nuclear fusion research and engineering megaproject aimed at creating energy by replicating, on Earth ...
aims to do both.
As for
tokamak
A tokamak (; russian: токамáк; otk, 𐱃𐰸𐰢𐰴, Toḳamaḳ) is a device which uses a powerful magnetic field to confine plasma in the shape of a torus. The tokamak is one of several types of magnetic confinement devices being d ...
s, there is a special motivation for using the triple product. Empirically, the energy confinement time τ
''E'' is found to be nearly proportional to ''n''
1/3/''P''
2/3. In an ignited plasma near the optimum temperature, the heating power ''P'' equals fusion power and therefore is proportional to ''n''
2''T''
2. The triple product scales as
:
The triple product is only weakly dependent on temperature as ''T''
-1/3. This makes the triple product an adequate measure of the efficiency of the confinement scheme.
Inertial confinement
The Lawson criterion applies to
inertial confinement fusion
Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with thermonuclear fuel. In modern machines, the targets are small spherical pellets about the size of ...
(ICF) as well as to
magnetic confinement fusion
Magnetic confinement fusion is an approach to generate thermonuclear fusion power that uses magnetic fields to confine fusion fuel in the form of a plasma. Magnetic confinement is one of two major branches of fusion energy research, along with i ...
(MCF) but in the inertial case it is more usefully expressed in a different form. A good approximation for the inertial confinement time
is the time that it takes an ion to travel over a distance ''R'' at its
thermal speed
:
where ''m''
''i'' denotes mean ionic mass. The inertial confinement time
can thus be approximated as
:
By substitution of the above expression into relationship (), we obtain
:
This product must be greater than a value related to the minimum of ''T''
3/2/<σv>. The same requirement is traditionally expressed in terms of mass density ''ρ'' = <''nm''
i>:
:
Satisfaction of this criterion at the density of solid D-T (0.2 g/cm
3) would require a laser pulse of implausibly large energy. Assuming the energy required scales with the mass of the fusion plasma (''E''
laser ~ ''ρR''
3 ~ ''ρ''
−2), compressing the fuel to 10
3 or 10
4 times solid density would reduce the energy required by a factor of 10
6 or 10
8, bringing it into a realistic range. With a compression by 10
3, the compressed density will be 200 g/cm
3, and the compressed radius can be as small as 0.05 mm. The radius of the fuel before compression would be 0.5 mm. The initial pellet will be perhaps twice as large since most of the mass will be
ablated during the compression.
The fusion power times density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burn-up of the fuel is probably more useful. The burn-up should be proportional to the specific reaction rate (''n''
2<''σv''>) times the confinement time (which scales as ''T''
-1/2) divided by the particle density ''n'':
:
Thus the optimum temperature for inertial confinement fusion maximises <σv>/''T''
3/2, which is slightly higher than the optimum temperature for magnetic confinement.
Non-thermal systems
Lawson's analysis is based on the rate of fusion and loss of energy in a thermalized plasma. There is a class of fusion machines that do not use thermalized plasmas but instead directly accelerate individual ions to the required energies. The best-known examples are the
migma Migma, sometimes migmatron or migmacell, was a proposed colliding beam fusion reactor designed by Bogdan Maglich in 1969. Migma uses self-intersecting beams of ions from small particle accelerators to force the ions to fuse. Similar systems using l ...
,
fusor
A fusor is a device that uses an electric field to heat ions to nuclear fusion conditions. The machine induces a voltage between two metal cages, inside a vacuum. Positive ions fall down this voltage drop, building up speed. If they collide in ...
and
polywell
The polywell is a proposed design for a fusion reactor using an electric field to heat ions to fusion conditions.
The design is related to the fusor, the high beta fusion reactor, the magnetic mirror, and the biconic cusp. A set of electromagn ...
.
When applied to the fusor, Lawson's analysis is used as an argument that conduction and radiation losses are the key impediments to reaching net power. Fusors use a voltage drop to accelerate and collide ions, resulting in fusion. The voltage drop is generated by wire cages, and these cages conduct away particles.
Polywell
The polywell is a proposed design for a fusion reactor using an electric field to heat ions to fusion conditions.
The design is related to the fusor, the high beta fusion reactor, the magnetic mirror, and the biconic cusp. A set of electromagn ...
s are improvements on this design, designed to reduce conduction losses by removing the wire cages which cause them. Regardless, it is argued that radiation is still a major impediment.
[Todd H. Rider, "Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium" Physics of Plasmas, April 1997, Volume 4, Issue 4, pp. 1039–1046.]
See also
*
Fusion energy gain factor
A fusion energy gain factor, usually expressed with the symbol ''Q'', is the ratio of fusion power produced in a nuclear fusion reactor to the power required to maintain the plasma in steady state. The condition of ''Q'' = 1, when the power bein ...
(''Q'')
*
List of plasma (physics) articles
This is a list of plasma physics topics.
A
* Ablation
* Abradable coating
* Abraham–Lorentz force
* Absorption band
* Accretion disk
* Active galactic nucleus
* Adiabatic invariant
* ADITYA (tokamak)
* Aeronomy
* Afterglow plasma
* Airg ...
Notes
External links
Mathematical derivation archived 2019 fro
{{fusion power
Fusion power