You are probably wondering how we can heat a plasma to 100 million degrees (10 keV). We can do that because a plasma is not a collisionless superconductor after all! Although much of the theory of instabilities is done with the approxima­tion of collisionlessness, we now have to take into account collisions between electrons and ions, infrequent though they are. First of all, plasmas can be made only inside a vacuum chamber because its heat would be snuffed out by air. Vacuum pumps create a high vacuum inside the torus. Then a gas such as hydrogen, deute­rium, or helium is bled in up to a pressure that is only three parts in a million (3 x 10-6) times as high as atmospheric pressure. These atoms are then ionized into ions and electrons by applying an electric field, as we will soon show. Although the plasma is heated to millions of degrees, it is so tenuous that it does not take a lot of energy to heat the plasma particles to a million degrees (100 eV) or even 100 million degrees (10 keV). This is the reason a fluorescent tube is cool enough to touch even though the electrons in it are at 20,000°. The density of electrons inside is much, much lower than that of air.

Once we have the desired gas pressure in the torus, we can apply an electric field in the toroidal direction with a transformer (this will be explained later). There are always a few free electrons around due to cosmic rays, and these are accelerated by the E-field so that they strip the electrons off gas atoms that they crash into, freeing more electrons. These then ionize more atoms, and so on, until there is an ava­lanche, like a lightning strike, which ionizes enough atoms to form a plasma. This takes only a millisecond or so. The E-field then causes the electrons to accelerate in the toroidal direction, making a current that goes around the torus the long way. The ions move in the opposite direction, but they are heavy and move so slowly that we can assume that they stay put in this discussion. If the plasma were really col­lisionless, the electrons would “run away” and gain more and more energy while leaving the ions cold. However, there are collisions, and this is the mechanism that heats up the whole plasma.

Running an electric current through a wire heats it because the electrons in the wire collide with the ions, transferring to them the energy gained from the applied voltage. According to Ohm’s law, the amount of heating is proportional to the wire’s resistivity and to the square of the electric current. In toasters, a high — resistance wire is used to create a lot of heat. High resistance is hard to get in a plasma because it is almost a superconductor. The number of ions that electrons collide with may be 10 orders of magnitude (1010 or 10 billion times) smaller than in a solid wire. Nonetheless, heating according to Ohm’s Law (“ohmic heating”) is effective because very large currents can be driven in a plasma, currents above 100,000 A (100 kA), and even many megamperes (MA). This is the most conve­nient way to heat a plasma in a torus, but when the resistance gets really low at fusion temperatures, other methods are available.

Calculating the resistance of a plasma is not easy because the collisions are not billiard-ball collisions. The transfer of energy between electrons and ions occurs through many glancing collisions as they pass by at a distance, pushing one another with their electric fields. This problem was first solved by Spitzer and Harm [5], and their formula for plasma resistivity (“Spitzer resistivity”) allows us to compute exactly how to raise a plasma’s temperature by ohmic heating.

This resistivity formula allows us to calculate something of even more interest; namely, the rate at which plasma collisions can move plasma across magnetic field lines. Every time an electron collides with an ion, both their guiding centers shift more or less in the same direction, so both of them move across the field lines. The plasma, then, spreads out (diffuses) across the magnetic field the way an ink drop diffuses in a glass of water until the ink reaches the wall. This is a slow process, but nonetheless it limits how long a magnetic bottle can hold a plasma. There is, how­ever, a big difference between ordinary diffusion and plasma diffusion in a mag­netic field. In ordinary diffusion, collisions slow up the diffusion rate by making the ink molecules, for instance, undergo a random walk. The more the collisions, the slower the diffusion. A magnetically confined plasma, on the other hand, does not diffuse at all unless there are collisions. Without collisions, the particles would just stay on the same field line, as in Fig. 4.5. Collisions cause them to random walk across the B-field, and the collision rate actually speeds up the diffusion. Since a

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hot plasma makes very few collisions, being almost a superconductor, this “classical” diffusion rate is very slow. This is called “classical” diffusion because it is the rate predicted by standard, well-established theory and applies to normal, “dumb” gases. Unfortunately, plasma can diffuse out rapidly by generating its own electric fields; and it leaks out much faster than at the classical rate.

Figure 5.11 shows the classical confinement time of a hot plasma as a function of magnetic field. We have assumed fusion-like electron and ion temperatures of 10 keV and a plasma diameter of 1 m — a large machine, but smaller than a full reactor. What is shown is the time for the plasma density to drop to about one-third of its initial value. This is similar to the “half-life” of a radioisotope used in medicine, a concept most people are familiar with. We see that at a field of 1 T (10,000 G), which we found before to be necessary to balance the plasma pressure, the time is about 90 secs — a minute and a half. This is much longer than what the Lawson criterion requires, which, we recall, is about 1 sec. It was this prediction of very good confinement that gave early fusion researchers the optimistic view that con­trolling the fusion reaction was a piece of cake. It did not happen, of course. Numerous unanticipated instabilities caused the confinement times to be thousands of times shorter than classical, and it is the understanding and control of these instabilities that has taken the last five decades to solve.

Notes

1. The data are from Bosch and Hale [1]. The vertical axis is actually reactivity in units of 1016 reactions/cm3/sec.

2. Such data were originally given by Post [2] and have been recomputed using more current data.

3. What is actually shown here is an equipotential of the electric field, which is the path followed by the guiding centers in an E xB drift. The short-circuiting occurs when the spacing becomes smaller than the ion Larmor radius, so that the ions can move across the field lines to go from the positive to the negative regions on either side of the equipotential. The curves are measured, not computed.