Some Very Large Numbers

The last chapter had a lot of information in it, so let us recapitulate. To get energy from the fusion of hydrogen into helium as occurs in the sun and other stars, we have to make a plasma of ionized hydrogen and electrons and hold it in a magnetic bottle, since the plasma will be too much hot to be held by any solid material. The way a magnetic field holds plasma particles is to make them turn in tight circles, called Larmor orbits, so that they cannot move sideways across magnetic field lines. However, the ions and electrons can move along the field lines in their thermal motions without restraint. Consequently, the magnetic container has to be shaped like a doughnut, a torus, so that the field lines can go around and around without ever running into the walls. The field lines also have to be twisted into helices to avoid a vertical drift of the particles that occurs in a torus but not in a straight cylinder. Ideally, each field line will trace out a magnetic surface as it goes around many times without ever coming back exactly on itself. The plasma is then confined on nested magnetic surfaces which never touch the wall. This ideal picture will be modified in this and later chapters as we understand more about the nature of these invisible, nonmaterial containers.

We’ve gotten an idea of what a magnetic bottle looks like, but how large, how strong, or how precise does it have to be? The sun, after all, has a tremendous gravi­tational force to hold its plasma together; but we on earth have much more limited resources. The size of a fusion reactor will be large if it is to produce backbone power. The torus itself may be 10 meters in diameter, and the reactor with all its components will fill a large four-story building. A better picture will be given in the engineering section later in this book. For experiments on plasma confinement, however, much smaller machines have been used. The figure-8 stellarators, for instance, were only about 3 meters long. Modern torus experiments are about half or a quarter the size of a reactor.

The temperature of the plasma in the interior of the sun is about 15,000,000 (1.5 x 107) degrees, but a fusion reactor will need to be about ten times hotter, or ‘Numbers in superscripts indicate Notes and square brackets [] indicate References at the end of this chapter.

F. F. Chen, An Indispensable Truth: How Fusion Power Can Save the Planet,

DOI 10.1007/978-1-4419-7820-2_5, © Springer Science+Business Media, LLC 2011 150,000,000 (1.5 x 108) degrees. We can use the electron-volt (eV) to make these numbers easier to deal with. Remember that 1 eV is about the amount of energy that holds a molecule together. Remember also that the temperature of a gas is related to the average energy of the molecules in the gas. It turns out that 1 eV is the average energy of particles in a gas at 11,600 K or roughly 10,000 K. So instead of saying 150,000,000°, we can say that the temperature is 15,000 eV or 15 keV. By that we mean that the particle energies in the gas are of the order of 15 keV. When we say degrees, do we mean Fahrenheit, Centigrade, or Kelvin (absolute)? For this discussion, it doesn’t matter, since Fahrenheit and Centigrade degrees dif­fer by less than a factor of 2, and Centigrade and Kelvin differ by only 273°. We do not really care whether the sun is at 10 million or 20 million degrees! It makes a difference to scientists, who use degrees Kelvin, but not for this general overview.

Why do we need a plasma temperature as high as 10 keV? This is because posi­tive ions repel one another with their electric fields, and they must have enough energy to crash through the so-called Coulomb barrier before they can get close enough to fuse together. In Chap. 3, we discussed why a hot plasma is a better solu­tion than the beams of fast ions. Here, we give more details. Figure 5.1 shows a graph of the probability of deuterium-tritium fusion plotted against the temperature of the ions in keV.1 Note that the probability peaks at around 60 keV, but the ion temperature does not have to be that high because the ions have a Gaussian distribu­tion of energies. When the ions are at 10 keV, there are enough ions in the tail of the distribution (Fig. 3.3), near 40 keV, which fuse rapidly enough. Note that at the sun’s 2 keV, the reactivity is very low; so low that ions stick around for millions of years before they undergo fusion. But on earth we do not have that kind of time!

Exactly how much time do we have? A magnetic bottle cannot hold a plasma forever because a plasma will always find a way to escape. From Fig. 5.1, we see that the lower the temperature, the slower is the fusion rate, so the plasma contain­ment time has to be longer. The relation among plasma density (n for short), ion temperature (T for short), and confinement time t was originally worked out by J. D. Lawson and is commonly known as the Lawson Criterion. A modified form of this

is shown in Fig. 5.2. The criterion says that the product of density and confinement time — that is, n x t — has to be higher than a value that varies with T. There are two curves. The lower one, marked BREAKEVEN, stands for scientific breakeven, in which the fusion energy just balances the energy needed to create the plasma. Real breakeven would include all the power needed to run the rest of the plant, requiring higher nt. The upper curve, labeled IGNITION, is the nt required for a self-sustaining plasma, in which the plasma heats itself without additional energy. That happens because one of the products of a DT reaction (see Fig. 3.2) is a charged a-particle (a helium nucleus), which is trapped by the magnetic field and stays in the plasma keeping the D’s and T’s hot with its share of the fusion energy. Clearly, the goal of fusion research is to reach ignition, and present plans are to build an experiment that can generate enough a-particles to see how they thermalize.

Now we can answer the question as to how long we must hold the plasma. The breakeven curve in Fig. 5.2 says that nt must be at least 1014 sec/cm3 (marked as 1E+14 on the graph). A reasonable value for the plasma density n is 1014/cm3 (100 trillion ion-electron pairs per cubic centimeter). Therefore, t is of the order of 1 sec. We have to hold the plasma energy in a magnetic bottle for at least 1 sec, not a million years, as in the sun. This has already been achieved, albeit not at such a high density. The progress in fusion can be appreciated when one recalls that the confinement in figure-8 stellarators was about 1 microsecond. Our work has paid off a million-fold.

To confine the plasma in a stellarator, the magnetic field has to be carefully made. Figure 5.3 shows the average distance an ion travels, in kilometers, before it makes a fusion collision.2 The curve is lowest at ion energies of around 60 keV, since the fusion probability in Fig. 5.1 peaks there. At the more normal energy of 40 keV, as explained above, an ion covers about the circumference of the earth as it goes around and around a torus! One might think that a magnetic bottle cannot be made this accurately, but it turns out that confining single particles is not a problem.

After all, storage rings in atom smashers can hold protons for hours or even days. Toroidal fusion experiments do not use focusing magnets as in particle accelerators, but electrons have been shown to be confined for millions of turns even in a primi­tive stellarator [3]. To hold a plasma is much harder because the ions and electrons can cooperate with one another to form their own escape paths. The accuracy of the magnetic field is not the problem.

So far we have considered the shape of the magnetic field but not its strength. A hot gas like a plasma exerts a lot of pressure, and the magnetic bottle has to be strong enough to hold this pressure. How does this pressure compare with our everyday experience? Pressure is density times temperature. Let’s first talk about temperature. Room temperature is about 300 K. Expressed in electron-volts, this is 300/11,600=0.026 eV. A fusion plasma has a temperature of, say, 15 keV, about 600,000 times higher. Fortunately, the density is much lower. Atmospheric density is about 3 x 1019 molecules/cm3, while a fusion plasma has about 2 x 1014 particles/ cm3, about 150,000 times fewer. So the net result is that the magnetic field has to hold a pressure about 600,000/150,000 times higher than normal: roughly 4 atmo­spheres (atm). This is the pressure at which water comes out of a faucet or that felt by a diver at 40 m depth, as one can figure out from the well-known fact that atmo­spheric pressure is about 1 kg/cm2 or 15 lbs./sq. in.2. Four atmospheres is not a huge number, but the pressure has to be exerted by a massless magnetic field! The strength of a magnetic field is measured in Teslas (T), each Tesla being 10,000 gauss (G), which may be a more familiar unit to old-timers. A magnetic field can exert a pressure of about 4 atm/T. Thus, the field strength required to hold a fusion plasma is about 1 T (10,000 G). This is a conservative number, because actual machines go up above 3 T. This is to be compared with the earth’s magnetic field, which is about 0.5 G, or with the strength of a memo-holding refrigerator magnet, about 40 G. However, MRI (Magnetic Resonance Imaging) machines need about 1 T. You can hear the field during an MRI exam because the field has to be oscillated, causing parts of the machine to rattle and hum. To create a 1-T field requires large, heavy “coils” consisting of copper windings or superconductors imbedded in a solid material to hold them in place. This is not a problem and is routinely done in fusion experiments. Though it is the magnetic field that applies pressure to the plasma, the field is held in place by the current-carrying coils; and it is the coils that ultimately bear the pressure. This is also not a problem because the coils have to be made quite sturdily in any case.