From Wikipedia, the free encyclopedia
In
nuclear physics, nuclear fusion is a
nuclear reaction in which two or more
atomic nuclei
collide at a very high speed and join to form a new type of atomic
nucleus. During this process, matter is not conserved because some of
the matter of the fusing nuclei is converted to
photons (
energy). Fusion is the process that powers active or "
main sequence"
stars.
The fusion of two nuclei with lower masses than
iron (which, along with
nickel, has the largest
binding energy per
nucleon) generally releases energy, while the fusion of nuclei heavier than iron
absorbs energy. The opposite is true for the reverse process,
nuclear fission.
This means that fusion generally occurs for lighter elements only, and
likewise, that fission normally occurs only for heavier elements. There
are extreme
astrophysical events that can lead to short periods of fusion with heavier nuclei. This is the process that gives rise to
nucleosynthesis, the creation of the heavy elements during events such as
supernova.
Following the discovery of
quantum tunneling by
Friedrich Hund, in 1929
Robert Atkinson and
Fritz Houtermans
used the measured masses of light elements to predict that large
amounts of energy could be released by fusing small nuclei. Building
upon the
nuclear transmutation experiments by
Ernest Rutherford, carried out several years earlier, the laboratory fusion of
hydrogen isotopes was first accomplished by
Mark Oliphant in 1932. During the remainder of that decade the steps of the main cycle of nuclear fusion in stars were worked out by
Hans Bethe. Research into fusion for military purposes began in the early 1940s as part of the
Manhattan Project. Fusion was accomplished in 1951 with the
Greenhouse Item nuclear test. Nuclear fusion on a large scale in an explosion was first carried out on November 1, 1952, in the
Ivy Mike hydrogen bomb test.
Research into developing controlled
thermonuclear fusion
for civil purposes also began in earnest in the 1950s, and it continues
to this day. The present article is about the theory of fusion. For
details of the quest for controlled fusion and its history, see the
article
Fusion power.
Overview
Fusion of
deuterium with
tritium creating
helium-4, freeing a
neutron, and releasing 17.59
MeV of energy, as an appropriate amount of mass changing forms to appear as the kinetic energy of the products, in agreement with
kinetic E = Δ
mc2, where
Δm is the change in rest mass of particles.
[1]
The origin of the energy released in fusion of light elements is due to an interplay of two opposing forces, the
nuclear force which combines together protons and neutrons, and the
Coulomb force
which causes protons to repel each other. The protons are positively
charged and repel each other but they nonetheless stick together,
demonstrating the existence of another force referred to as nuclear
attraction. This force, called the strong nuclear force, overcomes
electric repulsion in a very close range. The effect of this force is
not observed outside the nucleus, hence the force has a strong
dependence on distance, making it a short-range force. The same force
also pulls the neutrons together, or neutrons and protons together.
[2] Because the nuclear force is stronger than the Coulomb force for
atomic nuclei smaller than iron and nickel, building up these nuclei from lighter nuclei by
fusion releases the extra energy from the net attraction of these particles.
For larger nuclei,
however, no energy is released, since the nuclear force is short-range
and cannot continue to act across still larger atomic nuclei. Thus,
energy is no longer released when such nuclei are made by fusion;
instead, energy is absorbed in such processes.
Fusion reactions of light elements power the
stars and produce virtually all elements in a process called
nucleosynthesis.
The fusion of lighter elements in stars releases energy (and the mass
that always accompanies it). For example, in the fusion of two hydrogen
nuclei to form helium, 0.7% of the mass is carried away from the system
in the form of kinetic energy or other forms of energy (such as
electromagnetic radiation).
[3]
Research into controlled fusion, with the aim of producing fusion
power for the production of electricity, has been conducted for over 60
years. It has been accompanied by extreme scientific and technological
difficulties, but has resulted in progress. At present, controlled
fusion reactions have been unable to produce break-even
(self-sustaining) controlled fusion reactions.
[4]
Workable designs for a reactor that theoretically will deliver ten
times more fusion energy than the amount needed to heat up plasma to
required temperatures are in development (see
ITER).
The ITER facility is expected to finish its construction phase in 2019.
It will start commissioning the reactor that same year and initiate
plasma experiments in 2020, but is not expected to begin full
deuterium-tritium fusion until 2027.
[5]
It takes considerable energy to force nuclei to fuse, even those of the lightest element,
hydrogen.
This is because all nuclei have a positive charge due to their protons,
and as like charges repel, nuclei strongly resist being put close
together. Accelerated to high speeds, they can overcome this
electrostatic repulsion and be forced close enough for the attractive
nuclear force to be sufficiently strong to achieve fusion. The fusion of lighter nuclei, which creates a heavier nucleus and often a
free neutron or proton, generally releases more energy than it takes to force the nuclei together; this is an
exothermic process that can produce self-sustaining reactions. The US
National Ignition Facility, which uses laser-driven
inertial confinement fusion, is thought to be capable of break-even fusion.
The first large-scale laser target experiments were performed in June 2009 and ignition experiments began in early 2011.
[6][7]
Energy released in most
nuclear reactions is much larger than in
chemical reactions, because the
binding energy that holds a nucleus together is far greater than the energy that holds
electrons to a nucleus. For example, the
ionization energy gained by adding an electron to a hydrogen nucleus is
13.6 eV—less than one-millionth of the
17.6 MeV released in the
deuterium–
tritium (D–T) reaction shown in the diagram to the right (one
gram of matter would release
339 GJ of energy). Fusion reactions have an
energy density many times greater than
nuclear fission; the reactions produce far greater energy per unit of mass even though
individual fission reactions are generally much more energetic than
individual fusion ones, which are themselves millions of times more energetic than chemical reactions. Only
direct conversion of
mass into energy, such as that caused by the
annihilatory collision of
matter and
antimatter, is more energetic per unit of mass than nuclear fusion.
Requirements
Details and supporting references on the material in this section can
be found in textbooks on nuclear physics or nuclear fusion.
[8]
A substantial energy barrier of electrostatic forces must be overcome
before fusion can occur. At large distances, two naked nuclei repel one
another because of the repulsive
electrostatic force between their
positively charged
protons. If two nuclei can be brought close enough together, however,
the electrostatic repulsion can be overcome by the attractive
nuclear force, which is stronger at close distances.
When a
nucleon such as a
proton or
neutron
is added to a nucleus, the nuclear force attracts it to other nucleons,
but primarily to its immediate neighbours due to the short range of the
force. The nucleons in the interior of a nucleus have more neighboring
nucleons than those on the surface. Since smaller nuclei have a larger
surface area-to-volume ratio, the binding energy per nucleon due to the
nuclear force
generally increases with the size of the nucleus but approaches a
limiting value corresponding to that of a nucleus with a diameter of
about four nucleons. It is important to keep in mind that the above
picture is a
toy model because nucleons are
quantum objects,
and so, for example, since two neutrons in a nucleus are identical to
each other, distinguishing one from the other, such as which one is in
the interior and which is on the surface, is in fact meaningless, and
the inclusion of quantum mechanics is necessary for proper calculations.
The electrostatic force, on the other hand, is an
inverse-square force, so a proton added to a nucleus will feel an electrostatic repulsion from
all
the other protons in the nucleus. The electrostatic energy per nucleon
due to the electrostatic force thus increases without limit as nuclei
get larger.
The
electrostatic force between the positively charged nuclei is repulsive, but when the separation is small enough, the attractive
nuclear force
is stronger. Therefore, the prerequisite for fusion is that the nuclei
have enough kinetic energy that they can approach each other despite the
electrostatic repulsion.
The net result of these opposing forces is that the binding energy
per nucleon generally increases with increasing size, up to the elements
iron and
nickel, and then decreases for heavier nuclei. Eventually, the
binding energy
becomes negative and very heavy nuclei (all with more than 208
nucleons, corresponding to a diameter of about 6 nucleons) are not
stable. The four most tightly bound nuclei, in decreasing order of
binding energy per nucleon, are
62Ni,
58Fe,
56Fe, and
60Ni.
[9] Even though the
nickel isotope,
62Ni, is more stable, the
iron isotope 56Fe is an
order of magnitude more common. This is due to the fact that there is no easy way for stars to create
62Ni through the alpha process.
An exception to this general trend is the
helium-4 nucleus, whose binding energy is higher than that of
lithium, the next heaviest element. This is because protons and neutrons are
fermions, which according to the
Pauli exclusion principle
cannot exist in the same nucleus in exactly the same state. Each proton
or neutron energy state in a nucleus can accommodate both a spin up
particle and a spin down particle. Helium-4 has an anomalously large
binding energy because its nucleus consists of two protons and two
neutrons, so all four of its nucleons can be in the ground state. Any
additional nucleons would have to go into higher energy states. Indeed,
the helium-4 nucleus is so tightly bound that it is commonly treated as a
single particle in nuclear physics, namely, the
alpha particle.
The situation is similar if two nuclei are brought together. As they
approach each other, all the protons in one nucleus repel all the
protons in the other. Not until the two nuclei actually come in contact
can the strong
nuclear force
take over. Consequently, even when the final energy state is lower,
there is a large energy barrier that must first be overcome. It is
called the
Coulomb barrier.
The Coulomb barrier is smallest for isotopes of hydrogen, as their nuclei contain only a single positive charge. A
diproton
is not stable, so neutrons must also be involved, ideally in such a way
that a helium nucleus, with its extremely tight binding, is one of the
products.
Using
deuterium-tritium fuel, the resulting energy barrier is about 0.1 MeV. In comparison, the energy needed to remove an
electron from
hydrogen is 13.6 eV, about 7500 times less energy. The (intermediate) result of the fusion is an unstable
5He nucleus, which immediately ejects a neutron with 14.1 MeV. The recoil energy of the remaining
4He
nucleus is 3.5 MeV, so the total energy liberated is 17.6 MeV. This is
many times more than what was needed to overcome the energy barrier.
The fusion reaction rate increases rapidly with temperature until it
maximizes and then gradually drops off. The DT rate peaks at a lower
temperature (about 70 keV, or 800 million kelvin) and at a higher value
than other reactions commonly considered for fusion energy.
The reaction
cross section
σ is a measure of the probability of a fusion reaction as a function of
the relative velocity of the two reactant nuclei. If the reactants have
a distribution of velocities, e.g. a thermal distribution, then it is
useful to perform an average over the distributions of the product of
cross section and velocity. This average is called the 'reactivity',
denoted <σv>. The reaction rate (fusions per volume per time) is
<σv> times the product of the reactant number densities:
If a species of nuclei is reacting with itself, such as the DD reaction, then the product
must be replaced by
.
increases from virtually zero at room temperatures up to meaningful magnitudes at temperatures of
10–
100 keV. At these temperatures, well above typical
ionization energies (13.6 eV in the hydrogen case), the fusion reactants exist in a
plasma state.
The significance of
as a function of temperature in a device with a particular energy
confinement time is found by considering the
Lawson criterion.
This is an extremely challenging barrier to overcome on Earth, which
explains why fusion research has taken many years to reach the current
high state of technical prowess.
[10]
Methods for achieving fusion
If the matter is sufficiently heated (hence being
plasma),
the fusion reaction may occur due to collisions with extreme thermal
kinetic energies of the particles. In the form of thermonuclear weapons,
thermonuclear fusion is the only fusion technique so far to yield
undeniably large amounts of useful
fusion energy. Usable amounts of thermonuclear fusion energy released in a controlled manner have yet to be achieved.
Inertial confinement fusion
Inertial confinement fusion (
ICF) is a type of
fusion energy
research that attempts to initiate nuclear fusion reactions by heating
and compressing a fuel target, typically in the form of a pellet that
most often contains a mixture of
deuterium and
tritium.
Beam-beam or beam-target fusion
If the energy to initiate the reaction comes from
accelerating one of the nuclei, the process is called
beam-target fusion; if both nuclei are accelerated, it is
beam-beam fusion.
Accelerator-based light-ion fusion is a technique using particle
accelerators to achieve particle kinetic energies sufficient to induce
light-ion fusion reactions. Accelerating light ions is relatively easy,
and can be done in an efficient manner—all it takes is a vacuum tube, a
pair of electrodes, and a high-voltage transformer; fusion can be
observed with as little as 10 kV between electrodes. The key problem
with accelerator-based fusion (and with cold targets in general) is that
fusion cross sections are many orders of magnitude lower than Coulomb
interaction cross sections. Therefore the vast majority of ions end up
expending their energy on
bremsstrahlung and ionization of atoms of the target. Devices referred to as sealed-tube
neutron generators
are particularly relevant to this discussion. These small devices are
miniature particle accelerators filled with deuterium and tritium gas in
an arrangement that allows ions of these nuclei to be accelerated
against hydride targets, also containing deuterium and tritium, where
fusion takes place. Hundreds of neutron generators are produced annually
for use in the petroleum industry where they are used in measurement
equipment for locating and mapping oil reserves.
Muon-catalyzed fusion
Muon-catalyzed fusion is a well-established and reproducible fusion process that occurs at ordinary temperatures. It was studied in detail by
Steven Jones in the early 1980s. Net energy production from this reaction cannot occur because of the high energy required to create
muons, their short 2.2 µs
half-life, and the high chance that a muon will bind to the new
alpha particle and thus stop catalyzing fusion.
[11]
Other principles
Some other confinement principles have been investigated, some of
them have been confirmed to run nuclear fusion while having lesser
expectation of eventually being able to produce net power, others have
not yet been shown to produce fusion.
Sonofusion or
bubble fusion, a controversial variation on the
sonoluminescence
theme, suggests that acoustic shock waves, creating temporary bubbles
(cavitation) that expand and collapse shortly after creation, can
produce temperatures and pressures sufficient for nuclear fusion.
[12]
The
Farnsworth–Hirsch fusor
is a tabletop device in which fusion occurs. This fusion comes from
high effective temperatures produced by electrostatic acceleration of
ions.
The
Polywell
is a non-thermodynamic equilibrium machine that uses electrostatic
confinement to accelerate ions into a center where they fuse together.
Antimatter-initialized fusion uses small amounts of
antimatter to trigger a tiny fusion explosion. This has been studied primarily in the context of making
nuclear pulse propulsion, and
pure fusion bombs feasible. This is not near becoming a practical power source, due to the cost of manufacturing antimatter alone.
Pyroelectric fusion was reported in April 2005 by a team at
UCLA. The scientists used a
pyroelectric crystal heated from −34 to 7 °C (−29 to 45 °F), combined with a
tungsten needle to produce an
electric field of about 25 gigavolts per meter to ionize and accelerate
deuterium nuclei into an
erbium deuteride target. At the estimated energy levels,
[13] the
D-D fusion reaction may occur, producing
helium-3 and a 2.45 MeV
neutron.
Although it makes a useful neutron generator, the apparatus is not
intended for power generation since it requires far more energy than it
produces.
[14][15][16][17]
Hybrid nuclear fusion-fission (hybrid nuclear power) is a proposed means of generating
power by use of a combination of nuclear fusion and
fission processes. The concept dates to the 1950s, and was briefly advocated by
Hans Bethe
during the 1970s, but largely remained unexplored until a revival of
interest in 2009, due to the delays in the realization of pure fusion.
[18] Project PACER, carried out at
Los Alamos National Laboratory (LANL) in the mid-1970s, explored the possibility of a fusion power system that would involve exploding small
hydrogen bombs
(fusion bombs) inside an underground cavity. As an energy source, the
system is the only fusion power system that could be demonstrated to
work using existing technology. However it would also require a large,
continuous supply of nuclear bombs, making the economics of such a
system rather questionable.
Important reactions
Astrophysical reaction chains
The
CNO cycle dominates in stars heavier than the Sun.
The most important fusion process in nature is the one that powers stars. The net result is the fusion of four
protons into one
alpha particle, with the release of two
positrons, two
neutrinos
(which changes two of the protons into neutrons), and energy, but
several individual reactions are involved, depending on the mass of the
star. For stars the size of the sun or smaller, the
proton-proton chain dominates. In heavier stars, the
CNO cycle is more important. Both types of processes are responsible for the creation of new elements as part of
stellar nucleosynthesis.
At the temperatures and densities in stellar cores the rates of
fusion reactions are notoriously slow. For example, at solar core
temperature (
T ≈ 15 MK) and density (160 g/cm
3), the energy release rate is only 276 μW/cm
3—about a quarter of the volumetric rate at which a resting human body generates heat.
[19]
Thus, reproduction of stellar core conditions in a lab for nuclear
fusion power production is completely impractical. Because nuclear
reaction rates strongly depend on temperature (exp(−
E/
kT)),
achieving reasonable power levels in terrestrial fusion reactors
requires 10–100 times higher temperatures (compared to stellar
interiors):
T ≈ 0.1–1.0 GK.
Criteria and candidates for terrestrial reactions
In artificial fusion, the primary fuel is not constrained to be
protons and higher temperatures can be used, so reactions with larger
cross-sections are chosen. This implies a lower
Lawson criterion,
and therefore less startup effort. Another concern is the production of
neutrons, which activate the reactor structure radiologically, but also
have the advantages of allowing volumetric extraction of the fusion
energy and
tritium breeding. Reactions that release no neutrons are referred to as
aneutronic.
To be a useful energy source, a fusion reaction must satisfy several criteria. It must:
- Be exothermic: This limits the reactants to the low Z (number of protons) side of the curve of binding energy. It also makes helium 4He the most common product because of its extraordinarily tight binding, although 3He and 3H also show up.
- Involve low Z nuclei: This is because the electrostatic repulsion must be overcome before the nuclei are close enough to fuse.
- Have two reactants: At anything less than stellar densities,
three body collisions are too improbable. In inertial confinement, both
stellar densities and temperatures are exceeded to compensate for the
shortcomings of the third parameter of the Lawson criterion, ICF's very
short confinement time.
- Have two or more products: This allows simultaneous conservation of energy and momentum without relying on the electromagnetic force.
- Conserve both protons and neutrons: The cross sections for the weak interaction are too small.
Few reactions meet these criteria. The following are those with the largest cross sections:
[citation needed]
-
For reactions with two products, the energy is divided between them
in inverse proportion to their masses, as shown. In most reactions with
three products, the distribution of energy varies. For reactions that
can result in more than one set of products, the branching ratios are
given.
Some reaction candidates can be eliminated at once.
[20] The D-
6Li reaction has no advantage compared to
p+-
11
5B because it is roughly as difficult to burn but produces substantially more neutrons through
2
1D-
2
1D side reactions. There is also a
p+-
7
3Li reaction, but the cross section is far too low, except possibly when
Ti
> 1 MeV, but at such high temperatures an endothermic, direct
neutron-producing reaction also becomes very significant. Finally there
is also a
p+-
9
4Be reaction, which is not only difficult to burn, but
9
4Be can be easily induced to split into two alpha particles and a neutron.
In addition to the fusion reactions, the following reactions with
neutrons are important in order to "breed" tritium in "dry" fusion bombs
and some proposed fusion reactors:
-
The latter of the two equations was unknown when the U.S. conducted the
Castle Bravo
fusion bomb test in 1954. Being just the second fusion bomb ever tested
(and the first to use lithium), the designers of the Castle Bravo
"Shrimp" had understood the usefulness of Lithium-6 in tritium
production, but had failed to recognize that Lithium-7 fission would
greatly increase the yield of the bomb. While Li-7 has a small neutron
cross-section for low neutron energies, it has a higher cross section
above 5 MeV.
[21]
Li-7 also undergoes a chain reaction due to its release of a neutron
after fissioning. The 15 Mt yield was 150% greater than the predicted 6
Mt and caused casualties from the fallout generated.
To evaluate the usefulness of these reactions, in addition to the
reactants, the products, and the energy released, one needs to know
something about the cross section. Any given fusion device has a maximum
plasma pressure it can sustain, and an economical device would always
operate near this maximum. Given this pressure, the largest fusion
output is obtained when the temperature is chosen so that <σv>/T
2 is a maximum. This is also the temperature at which the value of the triple product
nTτ required for
ignition is a minimum, since that required value is inversely proportional to <σv>/T
2 (see
Lawson criterion).
(A plasma is "ignited" if the fusion reactions produce enough power to
maintain the temperature without external heating.) This optimum
temperature and the value of <σv>/T
2 at that temperature is given for a few of these reactions in the following table.
fuel |
T [keV] |
<σv>/T2 [m3/s/keV2] |
2
1D-3
1T |
13.6 |
1.24×10−24 |
2
1D-2
1D |
15 |
1.28×10−26 |
2
1D-3
2He |
58 |
2.24×10−26 |
p+-6
3Li |
66 |
1.46×10−27 |
p+-11
5B |
123 |
3.01×10−27 |
Note that many of the reactions form chains. For instance, a reactor fueled with
3
1T and
3
2He creates some
2
1D, which is then possible to use in the
2
1D-
3
2He reaction if the energies are "right". An elegant idea is to combine the reactions (8) and (9). The
3
2He from reaction (8) can react with
6
3Li in reaction (9) before completely thermalizing. This
produces an energetic proton, which in turn undergoes reaction (8)
before thermalizing. Detailed analysis shows that this idea would not
work well,
[citation needed] but it is a good example of a case where the usual assumption of a
Maxwellian plasma is not appropriate.
Neutronicity, confinement requirement, and power density
Any of the reactions above can in principle be the basis of
fusion power
production. In addition to the temperature and cross section discussed
above, we must consider the total energy of the fusion products
Efus, the energy of the charged fusion products
Ech, and the atomic number
Z of the non-hydrogenic reactant.
Specification of the
2
1D-
2
1D reaction entails some difficulties, though. To begin
with, one must average over the two branches (2i) and (2ii). More
difficult is to decide how to treat the
3
1T and
3
2He products.
3
1T burns so well in a deuterium plasma that it is almost impossible to extract from the plasma. The
2
1D-
3
2He reaction is optimized at a much higher temperature, so the burnup at the optimum
2
1D-
2
1D temperature may be low, so it seems reasonable to assume the
3
1T but not the
3
2He gets burned up and adds its energy to the net reaction. Thus the total reaction would be the sum of (2i), (2ii), and (1):
- 5 2
1D → 4
2He + 2 n0 + 3
2He + p+, Efus = 4.03+17.6+3.27 = 24.9 MeV, Ech = 4.03+3.5+0.82 = 8.35 MeV.
We count the
2
1D-
2
1D fusion energy
per D-D reaction (not per pair of deuterium atoms) as
Efus = (4.03 MeV + 17.6 MeV)×50% + (3.27 MeV)×50% = 12.5 MeV and the energy in charged particles as
Ech
= (4.03 MeV + 3.5 MeV)×50% + (0.82 MeV)×50% = 4.2 MeV. (Note: if the
tritium ion reacts with a deuteron while it still has a large kinetic
energy, then the kinetic energy of the helium-4 produced may be quite
different from 3.5 MeV, so this calculation of energy in charged
particles is only approximate.)
Another unique aspect of the
2
1D-
2
1D reaction is that there is only one reactant, which must be taken into account when calculating the reaction rate.
With this choice, we tabulate parameters for four of the most important reactions
fuel |
Z |
Efus [MeV] |
Ech [MeV] |
neutronicity |
2
1D-3
1T |
1 |
17.6 |
3.5 |
0.80 |
2
1D-2
1D |
1 |
12.5 |
4.2 |
0.66 |
2
1D-3
2He |
2 |
18.3 |
18.3 |
~0.05 |
p+-11
5B |
5 |
8.7 |
8.7 |
~0.001 |
The last column is the
neutronicity
of the reaction, the fraction of the fusion energy released as
neutrons. This is an important indicator of the magnitude of the
problems associated with neutrons like radiation damage, biological
shielding, remote handling, and safety. For the first two reactions it
is calculated as (
Efus-
Ech)/
Efus.
For the last two reactions, where this calculation would give zero, the
values quoted are rough estimates based on side reactions that produce
neutrons in a plasma in thermal equilibrium.
Of course, the reactants should also be mixed in the optimal
proportions. This is the case when each reactant ion plus its associated
electrons accounts for half the pressure. Assuming that the total
pressure is fixed, this means that density of the non-hydrogenic ion is
smaller than that of the hydrogenic ion by a factor 2/(
Z+1).
Therefore the rate for these reactions is reduced by the same factor, on
top of any differences in the values of <σv>/T
2. On the other hand, because the
2
1D-
2
1D reaction has only one reactant, its rate is twice as
high as when the fuel is divided between two different hydrogenic
species, thus creating a more efficient reaction.
Thus there is a "penalty" of (2/(Z+1)) for non-hydrogenic fuels
arising from the fact that they require more electrons, which take up
pressure without participating in the fusion reaction. (It is usually a
good assumption that the electron temperature will be nearly equal to
the ion temperature. Some authors, however discuss the possibility that
the electrons could be maintained substantially colder than the ions. In
such a case, known as a "hot ion mode", the "penalty" would not apply.)
There is at the same time a "bonus" of a factor 2 for
2
1D-
2
1D because each ion can react with any of the other ions, not just a fraction of them.
We can now compare these reactions in the following table.
fuel |
<σv>/T2 |
penalty/bonus |
reactivity |
Lawson criterion |
power density (W/m3/kPa2) |
relation of power density |
2
1D-3
1T |
1.24×10−24 |
1 |
1 |
1 |
34 |
1 |
2
1D-2
1D |
1.28×10−26 |
2 |
48 |
30 |
0.5 |
68 |
2
1D-3
2He |
2.24×10−26 |
2/3 |
83 |
16 |
0.43 |
80 |
p+-6
3Li |
1.46×10−27 |
1/2 |
1700 |
|
0.005 |
6800 |
p+-11
5B |
3.01×10−27 |
1/3 |
1240 |
500 |
0.014 |
2500 |
The maximum value of <σv>/T
2 is taken from a
previous table. The "penalty/bonus" factor is that related to a
non-hydrogenic reactant or a single-species reaction. The values in the
column "reactivity" are found by dividing 1.24
×10
−24
by the product of the second and third columns. It indicates the factor
by which the other reactions occur more slowly than the
2
1D-
3
1T reaction under comparable conditions. The column "
Lawson criterion" weights these results with
Ech
and gives an indication of how much more difficult it is to achieve
ignition with these reactions, relative to the difficulty for the
2
1D-
3
1T reaction. The last column is labeled "power density" and weights the practical reactivity with
Efus. It indicates how much lower the fusion power density of the other reactions is compared to the
2
1D-
3
1T reaction and can be considered a measure of the economic potential.
Bremsstrahlung losses in quasineutral, isotropic plasmas
The ions undergoing fusion in many systems will essentially never occur alone but will be mixed with
electrons that in aggregate neutralize the ions' bulk
electrical charge and form a
plasma.
The electrons will generally have a temperature comparable to or
greater than that of the ions, so they will collide with the ions and
emit
x-ray radiation of 10–30 keV energy (
Bremsstrahlung). The Sun and stars are
opaque to x-rays, but essentially any terrestrial fusion reactor will be
optically thin
for x-rays of this energy range. X-rays are difficult to reflect but
they are effectively absorbed (and converted into heat) in less than mm
thickness of stainless steel (which is part of a reactor's shield). The
ratio of fusion power produced to x-ray radiation lost to walls is an
important figure of merit. This ratio is generally maximized at a much
higher temperature than that which maximizes the power density (see the
previous subsection). The following table shows estimates of the optimum
temperature and the power ratio at that temperature for several
reactions.
[20]
fuel |
Ti (keV) |
Pfusion/PBremsstrahlung |
2
1D-3
1T |
50 |
140 |
2
1D-2
1D |
500 |
2.9 |
2
1D-3
2He |
100 |
5.3 |
3
2He-3
2He |
1000 |
0.72 |
p+-6
3Li |
800 |
0.21 |
p+-11
5B |
300 |
0.57 |
The actual ratios of fusion to Bremsstrahlung power will likely be
significantly lower for several reasons. For one, the calculation
assumes that the energy of the fusion products is transmitted completely
to the fuel ions, which then lose energy to the electrons by
collisions, which in turn lose energy by Bremsstrahlung. However,
because the fusion products move much faster than the fuel ions, they
will give up a significant fraction of their energy directly to the
electrons. Secondly, the ions in the plasma are assumed to be purely
fuel ions. In practice, there will be a significant proportion of
impurity ions, which will then lower the ratio. In particular, the
fusion products themselves
must remain in the plasma until they have given up their energy, and
will
remain some time after that in any proposed confinement scheme.
Finally, all channels of energy loss other than Bremsstrahlung have been
neglected. The last two factors are related. On theoretical and
experimental grounds, particle and energy confinement seem to be closely
related. In a confinement scheme that does a good job of retaining
energy, fusion products will build up.
If the fusion products are
efficiently ejected, then energy confinement will be poor, too.
The temperatures maximizing the fusion power compared to the
Bremsstrahlung are in every case higher than the temperature that
maximizes the power density and minimizes the required value of the
fusion triple product. This will not change the optimum operating point for
2
1D-
3
1T very much because the Bremsstrahlung fraction is
low, but it will push the other fuels into regimes where the power
density relative to
2
1D-
3
1T is even lower and the required confinement even more difficult to achieve. For
2
1D-
2
1D and
2
1D-
3
2He, Bremsstrahlung losses will be a serious, possibly prohibitive problem. For
3
2He-
3
2He,
p+-
6
3Li and
p+-
11
5B the Bremsstrahlung losses appear to make a fusion
reactor using these fuels with a quasineutral, isotropic plasma
impossible. Some ways out of this dilemma are considered—and rejected—in
Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium by Todd Rider.
[22]
This limitation does not apply to non-neutral and anisotropic plasmas;
however, these have their own challenges to contend with.