Fusion Spin Dependence

Jamie Ray
March 18, 2011

Submitted as coursework for Physics 241, Stanford University, Winter 2011

Fig. 1: The Stern-Gerlach setup, which originally used silver atoms. Notice that each spot corresponds to an allowed spin value, compared to the spectrum of spins expected without quantum mechanics. Source: Wikimedia Commons


Oil, coal, and natural gas are the fuels that currently power the Earth. But it's hydrogen that powers the stars, particularly our Sun. It isn't burned in a fuel cell with the help of oxygen. Instead, it's fused together to make other elements, in the massive heat and pressure of the Sun's core. The process of nuclear fusion is the ultimate source of all our energy use, much of which is derived from sunlight - even fossil fuels wouldn't be around without light from the Sun. When the nucleus of one atom fuses with another nucleus, the total mass of the system decreases and is released as energy, according to Einstein's famous equation E = mc2. Just a little mass corresponds to a lot of energy, so researchers around the world have been attempting to bring the Sun to Earth by building fusion reactors since the 1950s. Although the barriers are high, considerable progress has been made, and an understanding of fusion's spin dependence may bring fusion power closer to reality.

Nuclear Spin

Quantum mechanics gives us a model to describe the properties of fundamental particles, as well as atomic nuclei. The word 'quantum' refers to a discrete quantity that cannot be split into smaller pieces - for example, one can never find a half-electron's worth of charge. All particles also have what is called 'spin,' another quantized property that helps to explain their behavior. Spin is related to the phenomenon of magnetism, and the value of a particle's spin corresponds to a tiny magnet aligned in the same direction. While the term 'spin' was initially used as an analogy to larger objects, like a bicycle wheel, that spin around their own axis, the particles so described aren't actually spinning in this way. Instead, like charge, spin is another fundamental property of particles that has been observed, but can't be explained as having a 'source.' Confusingly, however, it does contribute to the total angular momentum in a quantum system, which must be conserved just like in classical systems.

Fig. 2: Reaction rates modeled by smooth curves as a function of the energy (expressed as temperature) of the reacting nuclei. Source: Wikimedia Commons

The famous Stern-Gerlach experiment demonstrates the existence of spin by using magnetic fields to produce a force on the nuclei according to their spin. For the purpose of fusion, it is important to understand that a particle (in our case, a nucleus) can only take on quantized values of spin. Moreover, while different particles can have different allowed values for their spin, it is is difficult to control the spins of a big group of particles - they tend to just be randomly distributed. [1] Finally, if we do manage to manipulate spin, we must remember that the total value of angular momentum (like charge) is conserved, so increasing one particle's spin requires a decrease somewhere else. As it turns out, these quantum-mechanical rules play a role in the way that nuclei fuse together.

Nuclear Cross-Sections

To put it simply, making two nuclei fuse into one is a difficult process even for the most common fusion reactions. There are two main forces involved - the electric force and the strong force. Nuclei are composed of protons, which have a positive electric charge, and neutrons, which have no charge. Thus, the protons in one nucleus repel the protons in the other through the electric force. Obviously that can't be the whole story or nuclei would just fly apart all the time. Instead, they are held together by what is called the 'strong force,' which acts only at small distances between these particles. Once they get close enough, the strong force begins to attract them and they get pulled closer together. The problem for fusion is that 'close enough' is very close (almost touching) and requires the nuclei to be moving fast enough to defeat the electric repulsion, also known as the Coulomb barrier.

The ability to penetrate the Coulomb barrier can be measured for many different fusion reactions, and is known to the 'fusion cross-section.' We can compare fusion to the process of shooting an arrow at a target - one nucleus is the arrow, and the bulls-eye is the other. If it is easy to get 'close enough' to fuse, then the bulls-eye is large. The twist is that the faster we shoot the arrow, the easier the target is to hit, because we are more likely to penetrate the Coulomb barrier! There are many combinations of nuclei that can fuse to create energy, but the most favorable are those that can occur easily (large cross-section) and release a lot of energy per reaction. Currently, the best candidates are processes that fuse a deuterium nucleus (deuterium is a form of hydrogen with an extra neutron) with either a tritium (two extra neutrons - this process is also known as D-T) or another deuterium (D-D) nucleus. [2]

Measuring Cross Sections

In developing potential fusion 'power plants,' mathematical models of fusion cross sections are often used. This is because it is difficult to measure the relevant cross sections directly. As explained above, the cross section depends on the speed (or energy) of the nuclei, and experiments to measure it often use beams of nuclei so that the energy is precisely known. Either a target containing the reactant nuclei is prepared (this target may be either a solid or gas containing deuterium) or two opposing beams are used. A particle accelerator is used to fire the nuclei, which as a result are usually charged. [3] Some of these particles will fuse, and the number of them doing so can be measured by detecting the products of the reaction (protons or other charged particles). For the beam-target case, modern experiments fire nuclei of a specified energy at a stationary target, then measure both how many particles pass through and how many extra charged particles are created to try to determine the likelihood of a fusion reaction occurring at that energy. [3]

While it allows for calibration and more precise measurements of cross sections, this setup is not practical to actually generate energy because it is too power-intensive. Furthermore, to cause and measure a significant number of fusion events, the energies of the accelerated nuclei are usually quite high. In a true fusion reactor, the energies will be distributed differently and will generally also be lower, and so fusion researchers extrapolate down from the measured cross sections using complicated models to try to predict the relevant values. Recently, lower energies have been used in the beams, which has allowed for refinement of the theoretical models since there is more data that can be used to 'fit the curve' predicted by theory. [2] The problem remains, however, that these measurements can still be imprecise and only give cross sections for a few chosen energies - for example, one might interpolate between energies of 10 and 50 keV (kilo electron-volts) to get the cross section at 30. The range that is likely to be used in the first successful fusion reactions is 15-60 keV. [2]

The Importance of Spin

With respect to the two forces involved, it seems strange that the spin would have any impact on the fusion cross-section, unless their magnetic moments were to interact in some way. As it turns out, this isn't the case; what is called the 'spin coupling' isn't strong enough to change the behavior noticeably. Instead, conservation laws provide a framework within which fusion is much more likely for certain spin arrangements. Deuterium nuclei, or deuterons, are part of the family of 'spin-1 particles.' This means that they can have spins of +1, 0, or -1. Tritium nuclei, on the other hand, are 'spin-1/2' so they can have spins of +1/2 or -1/2. Spin comes into play when we consider the product of D-T fusion, a He-5 nucleus that is very unstable and quickly breaks up into He-4 and a neutron. [4] According to quantum mechanics the nucleus so produced must have certain quantized characteristics. In its most common form, it has a total angular momentum - which is the sum of regular angular or 'orbital' motion and the spin value - of 3/2. [4] Fusion reactions are unlikely to occur with the regular kind of angular momentum, so it can be a good approximation to assume that only spin will contribute to the 3/2.

As it turns out, the most favorable way to create this 'excited state' of He-5 is to fuse D and T nuclei spinning in the same direction, so that their spins have the same sign and add to a total magnitude of 3/2. The spins of these nuclei will normally be random, but theory predicts that using 'spin-polarized' nuclei - which all spin in the same direction, e.g. +1 (D) and +1/2 (T) - the cross section can be improved by 50%! The case of D-D fusion isn't as well-understood, but researchers have suggested that in this case spin-polarization can also help to increase the probability of the desired reaction (by up to 100%!) and decrease the rate of those that create rogue neutrons, which are one of the primary problems facing this type of fusion. [4] Another interesting effect of spin-polarization is that it can change the direction of the particles that are produced, which are normally protons or neutrons that fly away with the energy created by the fusion. The direction that they travel is obviously important, both for measuring the presence of spin-polarization and for controlling where the produced energy will go!

Applying Nuclear Spin Dependence

The theoretical prediction, of course, is based on assumptions and idealities (for example, that the He-5 must have spin 3/2). To determine whether spin-polarization is actually worth using, it is important to measure the actual dependence of cross sections on the spin of the nuclei. Even gains much smaller than 50% could be useful because there is a kind of 'critical mass' for fusion reactions - once the fuel reaches a certain rate of fusion, the energy will increase and thereby provide positive feedback to increase the rate further. There are several different development paths towards fusion power, the most popular of which are inertial confinement fusion (ICF) and magnetic confinement fusion (MCF). In ICF, a small pellet containing the nuclei to be fused (usually D-T) is imploded using high-power lasers to create a quick burst of fusion. In contrast, MCF slowly heats a plasma that is contained by high-field superconducting magnets until it gets hot enough to fuse. While spin-polarization was really only a theoretical suggestion in the early 1980s, recent technological developments may have brought it into the range of practicality, and experiments using spin-polarized fuel may soon be conducted. [5] These tests of spin-polarization are crucial in answering several questions.

First, theorists predict that depolarization will occur slowly enough that fusion can be sustained, but this relies on models of the nucleus' spin and the ways it can be changed that haven't been tested in a reactor. [4,6] Furthermore, the ability to polarize fuel in the first place has not been fully tested. Current procedures require cooling atoms down to extremely low temperatures (less than 5 K) and subjecting them to intense magnetic fields. [7] This allows the spins of the electrons to align with the magnetic field - at higher temperatures, the thermal energy of the electrons overpowers the effect of the magnet and prevents significant polarization. Then, over a long period of time (and often with the help of some other quantum mechanical coaxing) this spin can be transferred to the nuclei of deuterium atoms. It is easy to predict the extent of the polarization based on theory, but actually measuring how many nuclei end up with spins in the same direction is quite difficult. Again, this issue is one that can partially be solved by actually testing a nuclear reaction using spin-polarized fuel. Depending on how polarized the fuel is, the reaction should both increase, and the direction and other properties of emitted particles will change. [4] In considering the latter, one might imagine the analogy of playing pool: a cue ball that is spinning will result in the balls that it hits flying off in different directions (or with different spins) than one that is not spinning. Again, there is no direct connection because 'spin' doesn't come from spinning in the usual sense, but the example can give some insight into the ways spin-polarization might be detected from fusion reactions.

Finally, the true effect of spin-polarization on cross-sections in a practical fusion situation is not fully understood. Researchers are optimistic about the 50% increase in D-T fusion in particular, especially because this is the easiest reaction to use for experimental reactors. Again, the effect on D-D fusion (which often is discussed in conjunction with D-He3 fusion) is less well understood. Some recent research claims that the suppression of neutrons, one of the most attractive potential features of spin-polarization, may not be significant at realistic energies. However, others have proposed an experimental test that they claim should produce a 15% enhancement in the reaction rate! [5] This and other tests can shed light on our ability to spin-polarize fuels, the effect of such polarization on fusion cross sections, and the rate at which depolarization occurs. If research indicates favorable answers to these issues, it may pave the way to successful, sustained fusion power based on spin polarization - and in so doing, provide the energy source of the future!

© Jamie Ray. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


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[7] G. Rouille et al., "A Polarized HD Target Factory in Europe," Nucl. Instr. A 464, 428 (2001).