By the end of the 19th century, spectroscopic measurements of sunlight had revealed that the Sun contains a large amount of hydrogen and a small amount of helium. Scientists were fully aware of this in the first decades of the 20th century, but with relativity only recently having been introduced and quantum physics still in its earliest development, there was no possibility of applying this observation to the problem of how stars produced energy. This was a complete mystery until the early 1920s, when British physicist Francis Aston discovered that the combined mass of four hydrogen atoms was slightly greater than the mass of a single helium atom. Einstein’s theory predicted that this difference in mass would be converted into energy, and Aston therefore hypothesized that stars produced energy by fusing hydrogen atoms into helium. This hypothesis was validated over the next 20 years, and the theory of stellar fusion is now regarded as one of the triumphs of modern physics.

It was also quickly realized that fusion reactions could produce tremendous amounts of useful energy. Not only that, but the fuel (hydrogen) that it would require is so abundant in the Earth as to effectively be limitless, and the only waste product is helium, which is not toxic and does not contribute to global warming.

This article will discuss what nuclear fusion is and what its implications are as a power source.

Mass-energy conversion

Unlike in chemistry, mass is not conserved in a nuclear reaction. One will always find that the mass of the products of the reaction is different from the mass of the reactants. This mass difference is called the mass defect, which we write as ∆m. The mass seems to vanish because the mass defect is transformed into energy by Einstein’s equation. The energy obtained from the reaction is E=∆mc². To obtain useful energy, we need ∆m to be positive. In a fusion reaction, this means that we want the mass of the product to be slightly less than the mass of the reactants, such as a helium atom being slightly lighter than four hydrogen atoms. In fission, it means that we want the mass of the products to be less than the mass of the reactant, such as a uranium atom being slightly more massive than the combined mass of the neutrons and the krypton and barium atoms that the reaction produces. It would require more energy to perform the reactions in the opposite directions than would be released: it is possible in principle to split a helium atom into hydrogen, but this process would consume more energy than would be released.

Binding energy

Even though the number of nucleons stays the same in the reaction, why is a helium atom lighter than four hydrogen atoms, and why is a uranium atom heavier than the combined mass of a krypton and a barium atom? Where exactly is the extra mass? To start answering this question, let’s write the energy conservation equation for the reaction. Let E-p be the mass-energy of a proton (which is almost exactly equal to the mass-energy of a hydrogen atom, we are neglecting the electron since its mass is ~1/2,000 that of the proton), E-n the mass energy of a neutron, E-He the mass-energy of a helium atom, and ∆E the energy released by the reaction. The energy equation is:

This tells us that there two terms to the total energy stored in the nucleus of a helium atom. The first is the mass-energy of its four nucleons (two protons and two neutrons, we are treating their mass-energies as approximately equal since the mass-energy of a proton is about 999/1000 that of a neutron) and the second is a negative term with an absolute value of ∆E. This negative energy is called the binding energy. It corresponds to the total potential energy of the interaction in which the strong nuclear force holds all of the nucleons together minus the electric potential energy of the repulsive Coulomb force between charged particles. The binding energy is negative because a particle would have to do work (lose kinetic energy) in order to escape from the nucleus. The binding energy per nucleon is a characteristic property of atoms of a given element, and this energy is depicted in the following chart:

Note: Depicts absolute value of binding energy. Source: Wikimedia Commons

An important rule is that if the product nuclei of a reaction have lower (greater in absolute value and therefore higher on the chart, but lower in the sense of being more negative) binding energy per nucleon than the reactants, then energy will be released. To see why this is the case, imagine an intermediate state after the reaction (either fusion or fission) in which a product nucleus exists for a single instant as an unbound state consisting of a jumble of non-interacting protons and neutrons. In order to become a nucleus, the pile of nucleons must become bound by interacting via the strong nuclear force. The energy of this interaction is the binding energy, which is negative, so the total energy of the system consisting of the pile of nucleons is lowered when it turns into a proper nucleus. But energy must be conserved, so for the system to lower its internal energy, it must have expelled some energy out into its surroundings.

You can also see on the chart that elements heavier than iron release energy when they are split and elements lighter than iron release energy when they are fused. Iron is the most stable element and there is no reaction that can split or fuse iron while also releasing energy.

How to cause fusion

We’ve established what happens during nuclear fusion, but we also need to know how to cause two atomic nuclei to fuse.

Atomic nuclei, consisting as they do of uncharged neutrons and positively-charged protons, are all positively charged and therefore they repel each other. However, when the separation between two nuclei is comparable to the nuclear diameter, a new force called the strong nuclear force becomes active. Unlike the electrostatic force, which has infinite range, the strong nuclear force has finite range and strong nuclear interactions will therefore not occur between nuclei that are separated by a distance greater than that range. However, unlike the electrostatic force, the strong force is attractive and holds protons and neutrons together against the repulsive electric force. Two nuclei will fuse if we can bring them close enough together for the strong nuclear force to overpower the electrostatic force.

Rather than thinking about forces here, the picture will be clearer if we think in terms of potential energy, and for a first pass take a naive classical approach that ignores quantum mechanics. A positively charged particle of charge q, like the nucleus of a hydrogen atom (a proton), produces an electrical potential field given by:

Units of joules/Coulomb

Where ε0 is a physical constant called the permittivity of free space. What this potential field tells us is that if two charges Q and q are separated by a distance r, then the potential energy associated with their mutual interaction is:

You can see that this energy becomes greater as the distance r becomes smaller. Therefore, in order to bring the two charges closer together, we need to perform work on the system of the two charges. Imagine trying to force the north poles of two bar magnets together. It’s possible, but it requires some effort. The amount of work we need to do to induce fusion of two protons is therefore the amount of work that we must do to bring two charges of charge q=Q=~1.6 × 10^–19 Coulombs to the distance at which the strong force dominates, r=1.7 femtometers (1fm = 10^-15 meters). Therefore U=1.35×10^-13 Joules, or about 843 keV (1 keV = 1000 electron volts).

To understand the reasoning here in a more tangible way, imagine trying to kick a ball of mass m so that it rolls up to the top of a hill of height h. Near the surface of the Earth, the potential energy of a weight at height h is U=mgh (The nature of potential functions allows us to arbitrarily assert that the potential is zero at the bottom of the hill regardless of the altitude above sea level). If we assume that the shape of the hill is given by some function y(x) then we can envision the hill as a spatial potential barrier U(x) = mgy(x) that the ball must must have kinetic energy greater than mgh in order to cross, otherwise it is blocked. The shape of the hill is arbitrary as long as we ignore air resistance and friction.

This diagram tells us the behavior of the ball for three different conditions on its kinetic energy. If the kinetic energy of the ball is less than mgh then the ball reaches a height less than h and then rolls back down. If the kinetic energy is precisely equal to mgh then the ball rolls up to the top of the hill and stays there. If the kinetic energy is greater than mgh then the ball rolls up to the top of the hill and then rolls down the other side. Let’s look at a diagram that illustrates the situation for two protons as they approach each other.

Note: Vertical axis is not to scale.

This diagram shows the total energy of the interaction of the two protons. If the potential energy is positive, then the protons must do work to reduce their separation and therefore the interaction will tend to cause the protons to repel each other. If the potential energy is negative, then the protons would have to do work to increase their separation and so the interaction will tend to be attractive.

In the section of the curve labelled by A, only the electrostatic interaction is active and the potential is positive. At a distance of about 1.7 fm, labelled by point B, the strong interaction “switches on” and immediately overpowers the electrostatic interaction. The energy at point B is referred to as the height of the barrier and, if the proton starts to the right of the barrier and has energy less than the barrier height, then we refer to the region to the left of the barrier as the classical forbidden region. At distances of less than about 0.7 fm, noted by point C, the strong interaction switches from being positive to being repulsive, so a particle on the section of the curve labelled by D will be pushed back to C.

The section of the potential curve where the electrostatic interaction dominates, V(x) for x>1.7 fm, is called the electrostatic, or Coulomb, barrier. We discussed earlier that the energy of the Coulomb barrier is ~843 keV. In the classical picture, if the incoming proton has kinetic energy less than this amount, then it is not able to cross the Coulomb barrier, analogous to the situation in which a ball must be kicked with a sufficient amount of kinetic energy to make it over the hill.

So how do we give the proton enough kinetic energy? The simplest and most efficient way is to make it very “hot”. Of course, temperature is not defined for single atoms, but we can define the temperature for a large sample of hydrogen atoms, call it T. The average kinetic energy for a sample of monatomic gas at temperature T is ⟨K⟩ = (3/2)kT where k is Boltzmann’s Constant. We find that the temperature required is absurdly high: 6.5 billion Kelvin. Not only is this orders of magnitude above anything that could reasonably be achieved on Earth, but the core of the Sun has an estimated temperature of “only” 15 million Kelvin, which is about 0.23% of the temperature we obtained through our naive approach. So how is it that stellar fusion could possibly occur, and how could we ever hope to do fusion here on Earth?

Barrier penetration

The answer is in the phenomenon of barrier penetration, also known as quantum tunneling. We all know that position is not well-defined for particles at atomic and subatomic distance scales. If we take one measurement of the position of the incoming proton and find that it is to the right of the Coulomb barrier, with energy less than the barrier height, then there is a nonzero probability of a second measurement finding the proton in the classically forbidden region for any finite value of the barrier height. The calculation, which uses the WKB approximation, is too advanced and involved for the level of this article, but ultimately we can find that the equation giving the probability is:

For the case of stellar fusion of two protons, we find that, assuming the proton at x = 0 does not move much during the collision process, with the incoming proton having average energy given ⟨K⟩ = (3/2)kT so that E = 1935 eV, the probability of barrier penetration is about 1.2×10^-17. This may seem like an extremely small number, but keep in mind that we are dealing with macroscopic quantities of hydrogen atoms. If one gram of hydrogen atoms are incident on one gram of stationary atoms, then 7.2 million fusion events could be expected to occur.

In the specific case of stellar fusion, we should note that the fusion of two protons is only the very first step in what is called the proton-proton cycle. The two hydrogen nuclei fuse and become an extremely unstable bound state called a diproton, which will decay with a half-life estimated to be ~10^-22 seconds. To become a stable deuterium nucleus (which will then be fused into Helium-3, and then ultimately into Helium-4), one of the protons must decay into a neutron by emitting a positron and an electron neutrino. This process is even more unlikely, but nonetheless stars are able to produce enough energy because there are just so many hydrogen atoms present. This situation is particular to the case of stellar fusion, and furthermore would require a long digression into nuclear interactions, so we won’t spend much more time on it in this article.

Regardless of which fusion process we are trying to induce, whether it be two regular hydrogen atoms, or two atoms of deuterium, deuterium and tritium, or anything else, this is the basic approach: a gas of atoms is heated up to the point where the kinetic energy of their random thermal motion is great enough to give them a sufficiently high change of tunneling, and therefore fusing, when they collide. In stellar fusion, the heat to first ignite the reaction is produced by friction and pressure when all of the gas atoms collapse inwards as the star forms, and from there the required heat is produced by the chain reaction. In artificial fusion, we have to be a bit more creative. There are three main techniques currently being researched. The first is called neutral beam injection, and this process produces the heat by shooting extremely high-energy particles into the plasma. The second uses rapidly-oscillating magnetic fields to pump energy into the plasma. Third is Ohmic heating, which exploits the tendency of a conductor (such as a plasma) to heat up when a high current is passed through it. A major outstanding problem is figuring out how to set up the reaction so that the fusion reactions themselves contribute to keeping the plasma at the needed temperatures. Efficient heating remains one of the central concerns of fusion research, especially since artificial fusion, which requires a faster reaction rate than stellar fusion, requires temperatures in excess of 100 million Kelvin.

Types of reactors

So far, this article has been rather abstract and some may find that to be somewhat tedious. But now we are in a position to start making this more concrete by talking about some of the different types of fusion reactors that are being researched today, which hopefully be more interesting. Note that unlike with stellar fusion, nearly all artificial reactors producing helium by fusing deuterium and tritium, either in the D-D cycle (two deuterium atoms to produce one helium) or the D-T cycle (one deuterium atom and one tritium to produce on helium).

The Tokamak

The Tokamak reactor is probably the most instantly recognizable of the technologies in this section. The name is Russian and is the acronym for the Russian words for “toroidal chamber with magnetic coils”, or alternatively “toroidal chamber with axial magnetic field”. Developed in the former Soviet Union 1950s, the Tokamak is the most thoroughly researched and developed style of fusion reactor and remains a leading candidate for large-scale fusion power production.

A Tokamak reactor has a toroidal (donut-shaped) chamber. Magnetic fields are produced by the green coils in the figure and by an electric current conducted by the plasma itself. The resultant magnetic field is helical and indicated by the dark purple arrows in the figure. It is therefore classified as a magnetic confinement reactor, that is, it uses magnetic fields to heat and contain the plasma.

Schematic diagram of a Tokamak and its magnetic fields. Source: CCFE

These are the most common type of experimental reactor, with some three dozen or so currently active in the world. When the ITER Tokamak in France is finished in 2025 it will be the largest Tokamak in the world.

Plasma inside the MAST reactor in the UK. Source: ITER.

The Stellerator

The Stellerator pattern is another magnetic confinement device that follows the same basic operating principle as the Tokamak, but with a key difference. In order to contain the plasma, the Tokamak produces a helical field. This requires a large current to be passed through the plasma itself. This causes the plasma to become less stable, increasing the chances that the magnetic confinement will fail, halting the reaction and potentially damaging the reactor. The Stellerator avoids this by twisting the plasma and the reactor itself rather than creating a twisting magnetic field.

Schematic of the plasma (yellow) and a magnetic field line (green) in the planned Wendelstein 7-X reactor. Source: Wikimedia Commons

This basic problem with Tokamaks was noted by Enrico Fermi and his colleagues very shortly after the Tokamak design was first proposed. However, designing a reactor in this way requires extremely precise computer simulations and drafting, as well as extremely powerful magnetic fields produced by precisely-manufactured superconducting coils, all of which were not available in Fermi’s time. This technology was not available until the 1990s and therefore it has only been fairly recently that Stellerators could be seriously proposed. The Wendelstein 7-X in Germany, completed in 2015, is currently the largest Stellerator in operation and is expected to achieve continuous operation — an important milestone in fusion research — in 2021.

First plasma ignition in Wendelstein 7-X. Source: Max Planck Institute.

Direct Drive

This approach is completely different from the two that we’ve just discussed. A direct drive reactor is classified as an inertial confinement device. In inertial confinement, extremely high amounts of energy are delivered to a pellet of solid fuel, heating the pellet to extreme temperature. The outer layer of the pellet vaporizes and explodes outward with great force, and therefore a reaction force pushes back in, creating a shockwave. This shockwave is responsible for the energy and compression used to heat and confine the resulting plasma. Nearly all recent devices have used lasers.

Simplified depiction of the inertial confine process. Source: Wikimedia Commons.

You can see this process illustrated in the diagram. In step 1, lasers heat the outer layer of the pellet. In step 2, the outer layer vaporizes and produces a shockwave, resulting in forces directed inwards and outwards. In step three, the shockwaves force the pellet to collapse inwards, inducing fusion in step 4.

This approach is currently being researched at the National Ignition Facility in the United States.

Inertial confinement fusion has been criticized by some who allege that it’s a front for nuclear weapons research masquerading as energy research. This may very well be the case with some specific government actors (the NIF in particular is funded by the same government body that manages the nuclear stockpile) but the entire field of inertial confinement fusion is very broad and ICF is still an important and active area of research.

Beam preamplifiers at the NIF. The laser system was used to produce a power of 500 terawatts, although for only a tiny instant of time. Source: Wikimedia Commons

A fuel pellet for the NIF system. Source: Wikimedia Commons.

The Farnsworth Fusor

The projects that we’ve discussed so far have been enormous undertakings that could only ever hope to be accomplished by some of the greatest minds in the world, working at institutions with the financial support of entire nation-states. Here’s one you can try at home!

Well, not quite. You’ll still need a very solid grounding in basic physics and electronics. The project is appropriate for someone with at least a Bachelor’s degree in physics, ideally working in a team, with a budget of a few thousand dollars. As a gauge of the level you’d likely want to be at, it is not unheard of for physics majors to build these for senior projects.

The Farnsworth fusor, or simply fusor, is different from most experimental fusion devices in that its purpose is not to generate useful power. Fusors are hopelessly inefficient. However, they do have some utility as compact and easily-controllable sources of neutron radiation. They also make some very neat pictures.

A reactor built by physicists at University of Wisconsin-Madison. The characteristic “star in a jar” pattern is visible. Image source: UWM.

Fusors work by inertial electrostatic confinement. This process is similar to inertial confinement, but it uses an electric field instead of a pressure wave. It is also possibly the simplest approach to achieving fusion. Atoms to be fused (most designs use deuterium since it is comparatively inexpensive) are ionized and therefore become charged. An electric field is created by two concentric, oppositely charged spherical grids. The atoms are flung by the field into the center of the reactor, where they collide and have a small chance to fuse. The following diagram illustrates this process, although only in one dimension.

Simplified diagram of inertial electrostatic confinement.

The positively charged deuterium nuclei fall into the regions where the electric field is present by random thermal motions. It’s a reasonable approximation that the electric field is contained entirely within this region. The field accelerates them towards the center. They miss the anode grids and their momentum carries them forward. The nuclei may fuse when they collide in the center.

Fusors are unlikely to ever have a role in energy production, but because they are small, comparatively inexpensive, and because they can be built and operated by someone without a PhD in plasma physics, they nonetheless are the subject of intense professional and amateur research. A small but thriving community of “Fusioneers” has grown online, drawing from a diverse background including professional physicists, science hobbyists, and the occasional child prodigy.

Cold fusion and other hoaxes

If and when fusion is harnessed as a viable energy source, it will rightly be regarded as one of humanity’s greatest scientific achievements, and fame and riches surely await the scientists and engineers who finally solve the problem. An unfortunate side effect of this is that the history of fusion research is tarnished by well-meaning but over-hyped projects that ultimately fail, hoaxes, outright frauds, and free energy conspiracy theorists.

Foremost among these is so-called “cold fusion”, that is, a fusion reactor that supposedly produces net power at or near room temperature. We described earlier in the article why extremely high temperatures are necessary for fusion to occur. There is no way around this fact that is currently known to science, regardless of the many, many claims made by proponents over the decades. Any claims that fusion has been achieved at or near room temperature, or really any temperature below 10 million degrees Celsius, should be treated with intense skepticism. The only exception to this is muon-catalyzed fusion, a highly speculative but valid process that involves reactions taking place near absolute zero.

Unfortunately there are too many of these floating around the internet for me to ever hope to be able to refute them all. RationalWiki has two fantastic articles on the subject:

When evaluating media claims about any very cutting-edge technology, it’s best to be optimistic but appropriately skeptical, and in the current situation there is actually good reason to be optimistic. Still, always be careful of falling into the trap of media hype and wishful thinking, and never trust anyone who’s trying to convince you of something that sounds too good to be true.

Where do we go from here?

There is good reason to believe that fusion power is possible and could be a key component of our energy supply within our lifetimes. The question is no longer one of technical and scientific feasibility, it is a question of economics and politics. In the United States, we currently have a government that is increasingly disinterested in funding research and that remains in the thrall of the fossil fuel industry. On the global stage, nationalist and reactionary movements threaten the progress of international efforts to collaborate and develop new and sustainable technology. For profit-motivated energy companies, the cold economic calculation simply leaves no incentive to upset the technological status quo. If we want to have fusion power, and with the the threat of climate changing worsening every day we need fusion power, then it will require political action.

There is reason to be hopeful. The developments that have taken place in the European Union have brought nuclear fusion out of the realm of speculation and fusion power is now a near-term prospect. A young and energetic progressive movement has stirred and is now winning elections and aggressively agitating for both scientific and environmental progress. The fossil fuel industry is at last beginning to lose its grip on society as alternatives become more viable and the geopolitics of the oil and coal supply become more unstable. Progress will be slow and steady but there is every reason to believe that fusion will be powering our homes within our lifetimes.

Concluding remarks/rambling

If you made it this far, then thanks a lot for reading. I’ve just been completely neglecting this blog lately and I apologize for that. On the positive side, I did figure out how to use for loops in LaTeX while making the graphics for this article, which was neat. Hopefully I’ll soon be able to start devoting more attention to this. I keep saying that I’m going to try to put out at least one article per week but things have a habit of getting in the way. I’m now planning to start my Essence of Quantum Mechanics series back up now that I’ve had some good thoughts about the style and approach I should be using and the direction I should be taking it in.

As always, I take full responsibility for any errors present and I appreciate any corrections.