Nuclear fusion
Nuclear fusion is the reaction through which 2 or more light nuclei collide with each other to
form a heavier nucleus. This reaction takes place with elements that have a low atomic number
such as hydrogen. As shown in the name, it is the opposite of nuclear fission, which is the
diffusion of heavy elements to form lighter elements. Both nuclear fusion and fission produce
a massive amount of energy. The main difference between fission and fusion is that fission does
not happen naturally while the universe is full of instances of nuclear fusion reactions.
An atom is formed with an atomic nucleus consisting of protons and neutrons and an electron
bound around it. An atom-dependent electron is separated one after another from the edge when
it receives external energy, which is called a free electron. If the amount of energy applied to
an atom is sufficiently large, so that all electrons subordinate to the atom can be separated, the
atom will emit electrons and have a positively charged atomic nucleus alone. This separation
of the nucleus and electrons is called plasma.
If the energy is low, an electromagnetic force acts between these nuclei and cannot be combined.
However, when the atomic nuclei are heated to an ultra-high temperature and the energy of the
nuclei is very high, the distance between the nuclei is narrowed, and when the atoms are close
enough, strong nuclear forces are applied to each other. This combination reaction is called
fusion. Some nuclei have increased binding energy per nucleus, resulting in a smaller mass per
nucleus and a smaller mass-produced than the combined mass of the two nuclei before colliding,
resulting in energy according to mass-energy equivalence. Usually, this energy is a byproduct
of fusion reactions.
An important fusion process is stellar nucleosynthesis that gives power to stars. It starts with a
proton-proton chain reaction, which occurs outside the core of the primordial star, the main
sequence star, and the centre of the red giant. Six protons participate to create the He-4 nucleus
in this phase.
The next step is the triple-alpha process. It happens inside the centre of the red giant. 3 He-4
nucleus fuses to make Carbon-12 nucleus. After this, the next steps depend on the weight of
the stars. CNO cycle happens in stars which have an initial mass of 1.3 times the mass of the
sun. Carbon, Nitrogen, Oxygen repeats being fusion and beta decay.
Stars with an initial mass of 8 times the sun start the carbon burning process which 2 C-12
nuclei fusion to make Ne-20, He-4 nucleus. 12 times start the Oxygen burning process, which
2 O-16 nucleus fusion to make P-31 nucleus. In this step, the Neon burning process happens at
the same time, which Ne-20 nucleus fusion to make Ne-21 nucleus.
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 quantum effect in which
nuclei can tunnel through coulomb forces.
When a nucleon such as a proton or a neutron is added to a nucleus, the nuclear force attracts
it to all the other nucleons of the nucleus (if the atom is small enough), but primarily to its
immediate neighbours due to the short range of the force. The nucleons in the interior of a
nucleus have more neighbouring 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
nucleons are quantum objects. So, for example, since two neutrons in a nucleus are identical to
each other, the goal of 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 therefore 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 atomic number grows.
The net result of the opposing electrostatic and strong nuclear 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 Ni(62), Fe(58), Fe(56), Ni(60). Even though the nickel isotope, Ni(62), is more
stable, the iron isotope Fe(56) is an order of magnitude more common. This is because there is
no easy way for stars to create Ni(62) 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 the same
state. Each proton or neutron's 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 (it is a doubly magic nucleus), 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 quantum mechanical 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 come close
enough for long enough so the strong nuclear force can take over (by way of tunnelling) is the
repulsive electrostatic force overcome. 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 the 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 He(5) 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 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: 𝐹 = 𝑛1 𝑛2 ⟨𝜎𝑣⟩
If a species of nuclei is reacting with a nucleus like itself, such as the DD reaction, then the
product n1n2 must be replaced by
𝑛2
2
.
⟨𝜎𝑣⟩ 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 ⟨𝜎𝑣⟩ 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 advanced technical state.
In a classical picture, nuclei can be understood as hard spheres that repel each other through
the Coulomb force but fuse once the two spheres come close enough for contact. Estimating
the radius of atomic nuclei as about one femtometer, the energy needed for the fusion of two
hydrogens is
This would imply that for the core of the sun, which has a Boltzmann distribution with a
temperature of around 1.4 keV, the probability hydrogen would reach the threshold is
10−290 that is, fusion would never occur. Boltzmann distribution or Maxwell-Boltzmann
distribution is a probability distribution used in thermal-statistics mechanics. It is about the
function of temperature in the physical system but since it is not time for thermal mechanics,
we will not go further. Back to the current topic, fusion in the sun does occur due to quantum
mechanics.
The probability that fusion occurs is greatly increased compared to the classical picture, thanks
to the smearing of the effective radius as the DeBroglie wavelength as well as quantum
tunnelling through the potential barrier. To determine the rate of fusion reactions, the most
interesting value is the cross-section. Cross-section is a physical quantity that describes the
probability that particles will fuse by giving a characteristic area of interaction. An estimation
of the fusion cross-sectional area is often broken into three pieces:
In this equation, 𝜎𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 is the geometric cross section, T is the barrier transparency and R
is the reaction characteristics of the reaction. 𝜎𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 is of the order of the square of the
ℎ
1
DeBroglie wavelength 𝜎𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 ≈ 𝜆2 = (𝑚 𝑣)2 ∝ ∈. 𝑚𝑟 is the reduced mass of the system
𝑟
and ∈ is the centre of mass energy of the system. T can be approximated by the Gamow
transparency which has the form of e exponential of minus root gamow factor and comes
from estimating the quantum tunneling probability through the potential barrier. R contains
all the nuclear physics of the specific reaction and takes very different values depending on
the nature of the interaction. However, for most reactions, the variation of R(∈) is small
compared to the variation from the Gamow factor and so is approximated by a function called
the Astrophysical S-factor, S(∈) which is weakly varying in energy. Putting these
dependencies together, one approximation for the fusion cross section as a function of energy
takes the form:
From all these ideas of nuclear fusion, we can produce energy from nuclear fusion caused
artificially. This is called nuclear fusion power generation. Different to fusion in stars, fusion
power generation aims for deuterium-tritium fusion reaction which is easier.
Just like nuclear fission, fusion can develop massive energy. However, we need to go through
many walls to make it happen. One of the walls is that we need very high temperature or
pressure. For the easiest reaction deuterium-tritium fusion, which already has been said, we
need at least 136 million kelvin and 500 million kelvins for practical application, For
deuterium-deuterium, at least 150 million kelvins, at least 580 million kelvins for deuteriumHelium-3. For this reason, DT reaction is treated as the ideal main model for fusion power
development.
Why do people try this reaction so hard if it is this hard? It is because fusion power generation
holds greater advantages and fewer disadvantages than current nuclear power. The efficiency
of fusion power is massive. A gram of hydrogen can produce 638GJ of energy. The mass loss
ratio of the protium fusion reaction is about 0.71%, whereas the mass loss ratio of the uranium235 fission reaction is about 0.1%. This means that protium fusion produces 7 times more
energy than uranium fission. A gram of hydrogen can substitute 21 tons of coal and 12500 litres
of oil. Theoretically, if we have 50kg of hydrogen, we can operate a 1GW fusion development
plant for a year. If we do some mathematics here, with 1000t of hydrogen, the whole world
does not have to worry about energy.
Fuel reservation of energy is another one of the advantages. Hydrogen, which is the fuel of
fusion power can be obtained from the ocean just by distilling it. Hydrogen is considered
‘infinite’ in our world. By comparison, the confirmed reservation of uranium is about 5.5
million tons, and 909 billion tons. This number can be seen so much, but we must focus on the
point that fission and combustion ‘break down’ to gain energy while fusion ‘sums up’ to gain
energy.
Compared to thermal/fission power plant, fusion plant has no chance of massive disaster. A
common misunderstanding that people have is that accidents at fusion plants will be more
serious than those at fission plants. However, fusion power has nothing to do with chain
reactions or nuclear explosions, unlike fission power. Exaggeratedly, an oven in people’s house
is more dangerous than fusion plants Fusion power is generated by adding a small amount of
hydrogen to the fusion reactor whenever needed, and even if there is a problem controlling the
nuclear reaction in the reactor, there is no fuel enough to cause an explosion. Hydrogen, a fuel
used in fusion power generation, is trapped in a reactor in a very thin plasma state, which,
unlike solids, has a very low density and thus a very low heat energy capacity per volume. For
this reason, even if the plasma hits the inner wall of the reactor, the reactor does not melt, but
rather the plasma cools and stops the nuclear reaction. In short, even if nuclear fusion reaction
control fails, it cools down on its own, so no disasters can happen. So why can't fission be done
safely by adding a little uranium? Because in fission, something called critical mass exists for
a chain reaction to occur. Since it is a method of collecting more than a certain amount and
slowly controlling chain reactions using neutron decelerators, it cannot cause fission little by
adding 0.1g of uranium. To keep chain reaction, We need to provide a constant supply of power,
but if the amount of uranium is small, the chain reaction will eventually stop. Of course, it
would be theoretically possible if there was a continuous neutron resource, but neutron
resources are very expensive. A gram of californium, for example, is 27,730,749 dollars. The
worst disaster that can be happened in a fusion plant is the leakage of tritium.
Lastly, fusion power produces very few harmful substances. Unlike conventional nuclear
power generation that generates high-level radioactive waste and requires permanent disposal
and isolation of major structures and components, fusion power generation does not generate
high-level radioactive waste and the amount of radioactivity is minimal. Numerical radiation
might come out more than fission power, but unlike fission power which produces radioactive
material, the reaction products of fusion are helium, which is not radioactive.
So why is it not done yet if it is this good? Despite the numerous advantages described above,
the actual development level is very low and slow. To make fusion happen, we need to maintain
high temperature and high pressure. The required temperature for this is at least 100 million
degrees, which already sounds very hard. KSTAR, a fusion development plant operated in
South Korea managed to keep 100 million degrees for 20 seconds and this is considered a huge
achievement. The goal stated by the institute is to maintain it for 300 seconds until 2025. We
can see how difficult it is just by looking at this. But to maintain the plant, it should be kept for
365 days a year.
The fusion energy gain coefficient still has a long way to go. Additional energy supplies from
outside are needed to maintain the fusion reactor at 100 million degrees. The energy gained in
this situation should be more than the input. The ratio of energy to energy generated is called
Q ratio, and KSTAR is less than 1. In other words, the energy that goes in is greater. The number
should be higher than 1 to have a meaning at least. For commercial development, the ratio
should be more than 10. To completely substitute thermal/nuclear energy, Q should be higher
than 22.
The most notable methods of fusion power are magnetic confinement and inertial confinement.
The basic structure of magnetic confinement is very simple. First, add deuterium and tritium
to the fusion reactor. Then hold the magnetic field in the fusion reactor to confine plasma
consisted of deuterium and tritium. If we heat the plasma to ultra-high temperature, a fusion
reaction will cause spontaneously. From this reaction, we can gain steam from the energy
obtained. The turbine is activated from this steam.
If you look more closely at the fusion reaction phase, deuterium and tritium fuse to create highenergy alpha rays and neutrons. First, alpha rays cannot escape the plasma because they have
a charge and are affected by magnetic fields, and their energy is converted to heat and
consumed to heat the plasma, and alpha rays that lose energy are converted into ordinary helium
nuclei and released out of the fusion furnace. Next, since neutron rays have no charge, they
escape the plasma without being affected by the electromagnetic field, collide with the outer
lithium blanket and produce new tritium. In addition, the energy of the neutron beam is
transferred to the lithium blanket, which heats the blanket, which then turns the turbine to
produce electricity.
The biggest problem is how to store hundreds of millions of degrees of plasma, but scientists
thought it was not a problem because the theoretical method of locking it into a magnetic field
had been in place since the early days of fusion development, and the prospects for
commercialization of fusion power were very encouraging. The question is what shape the
magnetic field should be. The basis of magnetic field trapping is doughnut-shaped trapping. A
ring-shaped magnetic coil is placed in the shape of a doughnut and plasma is locked in it.
However, the structure of the doughnut-shaped magnetic field trap also weakens the outer
magnetic field outside the doughnut, causing ions and electrons in the plasma to move up and
down and create internal electric fields depending on the potential car. Then again, because of
this electric field, the plasma bends outwards of the doughnut and collapses against the wall.
To prevent this, the plasma band itself needs to be twisted. A tokamak is a doughnut-type device
that uses magnetic fields to trap plasma in the process of fusion generation. The structure of
the tokamak requires another magnetic field to flow current inside such trapped plasma and
create plasma instability so that the plasma does not deviate, and the key is to predict and
prevent the plasma's movement.
Inertial confinement uses a laser or pinch effect, which is initiated by shooting a laser or ion
beam into the inner wall of a fuel pellet composed of a mixture of deuterium and tritium, or by
applying heat and pressure to the fuel pellet using a pinch effect. One way of this is using a
laser beam. Load a 2mm wide fuel pellet into a gold hohlraum, and shoot a 500TW laser at the
target, causing plasma to erupt from the laser target surface, compressing the pellets and
creating fusion at the centre of the fuel target. Due to the limitations of technology, only 20%
of the lasers surveyed on the actual pellets have suffered greatly from experiments. As a result,
expectations for an inertial fusion method were fading, including being audited by its parent
agency, the Ministry of Energy, in 2012, but finally passed the energy milestone in September
2013. This is not considered fusion ignition, but it demonstrates the theory that alpha particles
should be used to increase the temperature of pellets by force, rather than to leave them out.
Through alpha-particle self-heating, the pellet eventually succeeded in releasing more energy
into fusion than it absorbed. Another popular form of inertial confinement is Z-pinch. It uses a
cage-shaped hohlraum surrounding pellets with metal wires, which creates a very strong pinch
effect by leaking huge currents into metal wires, which allows fuel to be ionized and
compressed.
We have looked at what is nuclear fusion, some examples, mathematical meaning and what
can be done by it. There are way more mountains to climb to use fusion as an energy source,
but if we can, this will be one of the most important research of human being. It is currently
impossible and takes a lot of money. However, no one thought we can hear the voices of each
other from another country 100 years, or even 50 years ago. Science is beautiful when it is in
the process of development. Hope one day, we do not have to worry about energy in the world.
References
https://www.iaea.org/fusion-energy/what-is-fusion-and-why-is-it-so-difficult-to-achieve
https://dothemath.ucsd.edu/2012/01/nuclear-fusion/
http://www.bcamath.org/documentos_public/archivos/actividades_cientificas/TRAN.pdf
https://en.wikipedia.org/wiki/Nuclear_fusion
https://en.wikipedia.org/wiki/Magnetic_confinement_fusion
https://www.britannica.com/science/nuclear-fusion
https://www.kistep.re.kr/boardDownload.es?bid=0031&list_no=34990&seq=12230
https://www.mk.co.kr/news/it/view/2020/01/9808/