# Nuclear Fusion

For a given mass of fuel, a fusion reaction like
* $_{1}^{2}{\rm {H}}+_{1}^{3}{\rm {H}}\longrightarrow_{2}^{4}{\rm {He}}+n+;17.59\,{\rm {MeV}}\$.
* (15.5)
yields several times more energy than a fission reaction. This is clear from the curve given in Fig. 15.3. Indeed, a change of the binding energy (per nucleon) is much more significant for a fusion reaction than for a fission reaction. Fusion is, therefore, a much more powerful source of energy. For example, 10 g of deuterium which can be extracted from 500 litres of water and 15 g of tritium produced from 30 g of lithium would give enough fuel for the lifetime electricity needs of an average person in an industrialised country.
But this is not the only reason why fusion attracted so much attention from physicists. Another, more fundamental, reason is that the fusion reactions were responsible for the synthesis of the initial amount of light elements at primordial times when the universe was created. Furthermore, the synthesis of nuclei continues inside the stars where the fusion reactions produce all the energy which reaches us in the form of light.

## Thermonuclear reactions

If fusion is so advantageous, why is it not used instead of fission reactors? The problem is in the electric repulsion of the nuclei. Before the nuclei on the left hand side of Eq. (15.5) can fuse, we have to bring them somehow close to each other to a distance of about a femtometer. This is not an easy task! They both are positively charged and refuse to approach each other. What we can do is to make a mixture of the atoms containing such nuclei and heat it up. At high temperatures the atoms move very fast. They fiercely collide and loose all the electrons. The mixture becomes plasma, i.e. a mixture of bare nuclei and free moving electrons. If the temperature is high enough, the colliding nuclei can overcome the electric repulsion and approach each other to a fusion distance. When the nuclei fuse, they release much more energy than was spent to heat up the plasma. Thus the initial energy investment pays off. The typical temperature needed to ignite the reaction of the type (15.5) is extremely high. In fact, it is the same temperature that our sun has in its center, namely, about 15 megakelvins. This is why the reactions (15.3), (15.5), and the like are called thermonuclear reactions.

## Human-made thermonuclear reactions

The same as with fission reactions, the first application of thermonuclear reactions was in weapons, namely, in the hydrogen bomb, where fusion is ignited by the explosion of an ordinary (fission) plutonium bomb which heats up the fuel to solar temperatures while compressing it immensely.
In an attempt to make a controllable fusion, people encounter the problem of holding the plasma. It is relatively easy to achieve a high temperature (with laser pulses, for example). But as soon as plasma touches the walls of the container, it immediately cools down. To keep it from touching the walls, various ingenious methods are tried, such as strong magnetic field and laser beams directed to plasma from all sides. In spite of all efforts and ingenious tricks, no fusion reactor has demonstrated that it can produce more energy than it consumed creating and containing the plasma. A giant thermonuclear reactor is being planned by the international community to be built in Cadarache, France. It will be the largest of its kind, and theoreticians predict that it may be within the parameters needed to sustain a fusion reaction and extract energy.

## Cold fusion

To visualize the struggle of the nuclei approaching each other, imagine yourself pushing a metallic ball towards the top of a slope shown in Fig. 15.5. The more kinetic energy you give to the ball, the higher it can climb. Your purpose is to make it fall into the narrow well that is behind the barrier.
In fact, the curve in Fig.15.5 shows the dependence of relative potential energy $V_{\rm {eff}}$ between two nuclei on the distance $R$ separating them. The deep narrow well corresponds to the strong short-range attraction, and the $\sim 1/R$ barrier represents the Coulomb (electric) repulsion. The nuclei need to overcome this barrier in order to touch each other and fuse, i.e. to fall into the narrow and deep potential well. One way to achieve this is to give them enough kinetic energy, which means to rise the temperature. However, there is another way based on the quantum laws.
As you remember, when discussing the motion of the electron inside an atom (see What the atom is made of), we said that it formed a cloud of probability around the nucleus. The density of this cloud diminishes at very short and very long distances but never disappears completely. This means that we can find the electron even inside the nucleus though with a rather small probability.
The nuclei moving towards each other, being microscopic objects, obey the quantum laws as well. The probability density for finding one nucleus at a distance R from another one also forms a cloud. This density is non-zero even under the barrier and on the other side of the barrier. This means that, in contrast to classical objects, quantum particles, like nuclei, can penetrate through potential barriers even if they do not have enough energy to go over it! This is called the tunneling effect.
The tunneling probability strongly depends on thickness of the barrier. Therefore, instead of lifting the nuclei against the barrier (which means rising the temperature), we can try to make the barrier itself thinner or to keep them close to the barrier for such a long time that even a low penetration probability would be realized.
How can this be done? The idea is to put the nuclei we want to fuse, inside a molecule where they can stay close to each other for a long time. Furthermore, in a molecule, the Coulomb barrier becomes thinner because of electron screening. In this way fusion may proceed even at room temperature.
This idea of cold fusion was originally (in 1947) discussed by F. C. Frank and (in 1948) put forward by A. D. Sakharov, the father of Russian hydrogen bomb, who at the latest stages of his career was worldwide known as a prominent human rights activist and a winner of the Nobel Prize for Peace. When working on the bomb project, he initiated research into peaceful applications of nuclear energy and suggested the fusion of two hydrogen isotopes via the reaction (15.5) by forming a molecule of them where one of the electrons is replaced by a muon.
The muon is an elementary particle (see Elementary particles) which has the same characteristics as an electron. The only difference between them is that the muon is 200 times heavier than the electron. In other words, a muon is a heavy electron. What will happen if we make a muonic atom of hydrogen, that is a bound state of a proton and a muon? Due to its large mass the muon would be very close to the proton and the size of such atom would be 200 times smaller than that of an ordinary atom. This is clearly seen from the formula for the atomic Bohr radius
$R_{\mbox{Bohr}}={\frac {{\bar {h}}^{2}}{me^{2}}}$
where the mass is in the denominator.
Now, what happens if we make a muonic molecule? It will also be 200 times smaller than an ordinary molecule. The Coulomb barrier will be approximately 200 times thinner and the nuclei 200 times closer to each other. This is just what we need! Speaking in terms of the effective nucleus-nucleus potential shown in Fig. 15.5, we can say that the muon modifies this potential in such a way that a second minimum appears. Such a modified potential is (schematically) shown in Fig. 15.6
The molecule is a bound state in the shallow but wide minimum of this potential. Most of the time, the nuclei are at the distance corresponding to the maximum of the probability density distribution (shown by the thin curve). Observe that this density is not zero under the barrier (though is rather small) and even at $R=0$. This means that the system can (with a small probability) jump from the shallow well into the deep well through the barrier, i.e. can tunnel and fuse.
Unfortunately, the muon is not a stable particle. Its lifetime is only ≈ 10$^{−6}$ s. This means that a muonic molecule cannot exist longer than 1 microsecond. As a matter of fact, from a quantum mechanical point of view, this is quite a long interval.
The quantum mechanical wave function (that describes the probability density) oscillates with a frequency which is proportional to the energy of the system. With a typical binding energy of a muonic molecule of about 300 eV this frequency is 10$^{17}$ Hz or 100 PHz.
This means that the particle hits the barrier with this frequency and during 1 microsecond it makes $10^{11}$ attempts to jump through it. The calculations show that the penetration probability is ≈ 10$^{−7}$. Therefore, during 1 microsecond nuclei can penetrate through the barrier 10000 times and fusion can happen much faster than the decay rate of the muon. Cold fusion via the formation of muonic molecules was done in many laboratories, but unfortunately, it cannot solve the problem of energy production for our needs. The obstacle is the negative efficiency, i.e. to make muonic cold fusion we have to spend more energy than it produces. The reason is that muons do not exist like protons or electrons. We have to produce them in accelerators. This takes a lot of energy. Actually, the muon serves as a catalyst for the fusion reaction. After helping one pair of nuclei to fuse, the muon is liberated from the molecule and can form another molecule, and so on. It was estimated that the efficiency of the energy production would be positive only if each muon ignited at least 1000 fusion events. Experimentalists tried their best, but by now the record number is only 150 fusion events per muon. This is too few. The main reason why the muon does not catalyze more reactions is that it is eventually trapped by a $^{4}$He nucleus which is a by-product of fusion. Helium captures the muon into an atomic orbit with large binding energy, and it cannot escape.
Nonetheless, the research in the field of cold fusion continues. There are some other ideas of how to keep nuclei close to each other. One of them is to put the nuclei inside a crystal. Another way out is to increase the penetration probability by using molecules with special properties, namely, those that have quantum states with almost the same energies as the excited states on the compound nucleus. Scientists try all possibilities since the energy demands of mankind grow continuously and therefore the stakes in this quest are high.