After electricity, the form of energy we would miss most is that in gasoline. Our dependence on oil leads us into wars in the Middle East. The price of oil disrupts our economy. The oil crisis of 1973 was so severe that a speed limit of 55 miles per hour was legislated in the USA. (But it had the beneficial effect of increasing government funding for controlled fusion research!) Train buffs will remember the times when trains carried coal, and this was shoveled into steam engines to drive the huffing and puffing trains across the country. Nowadays liquid fuel is at a premium. Gasoline, diesel fuel, and liquefied natural gas are used for transport by cars, buses, trucks, trains, airplanes, and ships. Half of all the world’s oil is used for transportation. How can this be replaced by clean energy? Wind and solar produce electricity, which is not easy to carry around. We cannot all drive nuclear submarines.

Hydrogen has been hyped as a promising candidate for a nonpolluting fuel. It is surprising how many people still think that hydrogen is a source of energy! In fact, it takes a lot of energy to produce hydrogen. Water is one of the most stable elements on earth, which is why we have a blue planet. To take H2O apart into hydrogen and oxygen requires a large energy source to supply the world’s trans­portation needs. Cars run on hydrogen emit only water, but hydrogen is currently produced from natural gas. This not only depletes our precious reserves, but also carbon dioxide is emitted in the process. Even though we still use fossil energy to make hydrogen, transportable hydrogen still has a role to play in reducing pollu­tion. To clear up popular conceptions on hydrogen, we will consider this topic first.

Large amounts of energy can be measured in, say, millions of barrels of oil equiva­lent or kilotons of TNT equivalent. A more familiar household unit is the kilowatt- hour (1,000 Wh), which is used in our electric bills. A 100-W bulb will use 100 W h of electricity every hour. Since there are 3,600 seconds in an hour, a watt-second (which is called a joule) is 1/3,600 of a watt-hour, or the energy used by a 1-W cell phone in 1 s. These are units that we use on a human scale. When we talk about atoms, however, we have to use much smaller units because atoms are very small. There are some 100,000,000,000,000,000,000,000 atoms in a teaspoon of water. So the energy of an atom would be that much smaller than the energy units, like a watt-second, that we encounter in real life.

First, let’s find a way to avoid writing all those zeroes. Scientific notation is an easy shorthand to do this. The large number above has 23 zeroes and is written as 1023, where the superscript, called an exponent, tells how many zeroes follow the 1. A thousand (1,000) would be written as 103 and pronounced “ten to the third power” (or ten cubed in this case). Three thousand would be 103 multiplied by 3, written as 3 x 103. 3,600 would be written 3.6 x 103, and so forth. This works also for fractions if we use negative exponents. One thousandth (1/1,000) would be 10-3. Two hundredths would be 2 x 10-2. The only thing to note is that if we write decimals, 1/1000 would be 0.001, and the number of zeroes is one less than the exponent. But you need not worry about that; just remember that 10-3 is a thousandth, 10-6 is a millionth, 10-9 is a billionth, and so forth.

How much energy is released in fusing two hydrogen atoms? It is approximately 3 x 10-18 J. Joules are too large when dealing with atoms. A more convenient unit of energy is in order. The unit used is the electron-volt, or eV, which is more like the size of the energies of atomic particles. One electron-volt is 1.6 x 10-19 J. Now we can use eVs and stop counting zeroes. Since we will be talking about atoms in the next few chapters, we will use eVs and not worry about changing to more familiar units until we have to design reactors.

Let’s get an idea of how big 1 eV of energy is. Molecules, CO2 for instance, are held together with an energy of about 1 eV. An atom is a nucleus surrounded by electrons, equal in number to the protons in the nucleus. The outermost electron in an atom is bound to the nucleus with about 10 eV. A fusion reaction yields about 10 million eV or 10 MeV. A fission reaction yields about 100 MeV. The advantage of nuclear power is now obvious. Chemical reactions involve molecules and atoms, as in the burning of gasoline. These reactions yield eVs of energy each, and therefore a large number of molecules (read tankfuls of gasoline) are needed in normal use. Chemical energy is already very efficient. Witness monarch butterflies going 2,000 miles from Canada to Mexico or demoiselle cranes going from Russia to India over the Himalayas with no food or stopping. But chemical energy is infinitesimal compared with nuclear energy. Nuclear reactions yield tens to hundreds of millions of eVs each, so that the fuel needed for even a large power plant occupies a relatively small volume. Some think of hydrogen fusion as “burning” water. To do this in a chemical sense means that you first have to separate the hydrogen from H2O and then ignite the hydrogen. The energy you get is relatively small, since it is a chemical reaction. In any case, you can’t get any more energy out than it took to separate the hydrogen from the oxygen in the first place. But “burning” the hydrogen in a nuclear sense yields many million times more energy than in chemical burning.

The first attempts at fusion were carried out with a simple device called a “pinch,” which we will describe first. This was a tube filled with a low-pressure gas in which a large pulsed current was driven by a voltage applied to electrodes at either end. As shown in Fig. 7.19, the current first ionizes the gas into a plasma and then gener­ates a magnetic field surrounding the plasma. If the cylinder were turned into a torus, the current would be in the toroidal direction, and the field in the poloidal direction. This is like the current in a tokamak, but in a pinch there is no toroidal

field from external coils. The magnetic pressure of the “poloidal” field in Fig. 7.19 then compresses the plasma to a smaller diameter, whereupon the magnetic field gets stronger, compressing the plasma even more. Since compression heats the plasma, the hope was that the heating would reach fusion temperatures. Of course, the system suffered from the kink instability described in Chap. 6, and the kinks drove the plasma into the walls.

The Ware pinch [6] is a more subtle effect occurring in tokamaks and affecting mostly particles which move in banana orbits. The mechanism is illustrated in Fig. 7.20. In tokamaks in which the toroidal current J is driven by a toroidal electric field E, the poloidal-field component Bp which J produces is in the direction shown in diagram. This is the field that gives the necessary twist to the magnetic lines. Crossed electric and magnetic fields give rise to a perpendicular E x B drift of the guiding centers, as shown in Fig. 5.4. This drift is always toward the center of the cross section regardless of where the particle is in its banana orbit, and the drift has the same direction and magnitude for both ions and directions. Note that Bp is small compared to the toroidal component Bt, but Bt is parallel to E, not crossed with it, so it does not give an E x B drift. Thus, the principal fields in a tokamak generate a drift that counteracts the outward diffusion of the plasma, at least for particles trapped in bananas. The Ware pinch effect was invented to explain observations of oscillations occurring when the pinching reached its limits and started over again. This effect has been observed in other tokamaks and is not an artifact of neoclassical theory. It is another of Mother Nature’s gifts.

This method heats ions by pushing them with a rotating electric field whose direction follows the ions’ cyclotron motion as they revolve in their nearly circular Larmor orbits. It is sometimes more efficient to heat a minority species, such as helium-3 rather than deuterium or tritium, because of the way the energy is coupled into the plasma. The cyclotron frequency depends on the magnetic field strength, so the applied electric field has to be of a specific frequency, depending on magnetic field at the location where the ions are to be heated. In ITER, this frequency is in the range around 50 MHz. This is too low a frequency to be transmitted through a pipe, so an antenna has to be placed inside the vacuum chamber. The antenna is outside the field lines leading to the divertor (see Fig. 9.4), but it is so close to the plasma that it will be bombarded by ions. These ions will sputter antenna material into the plasma, and such contamination usually cools the plasma. ITER is to have 20 MW of ion cyclotron heating. The power is not the main problem; the problem here is to design antennas which will not affect the plasma deleteriously.

A pinch is a plasma carrying a current so large that the surrounding magnetic field that it generates confines and compresses it. It is basically unstable to the kink instability (Fig. 6.2). A toroidal pinch has the current running around in a torus, so that it is also subject to the gravitational interchange instability (Fig. 5.7). A reversed-field pinch (RFP) adds a toroidal field imposed by external coils, as in a tokamak, and has special properties. The Zeta machine at Harwell, England, one of the first fusion experiments revealed to the world at the 1958 Geneva Atoms-for — Peace Conference, was an RFP (cf. Chap. 8). That machine suffered from a misin­terpretation of the neutrons it generated and was abandoned, but research on RFPs has continued since that time.

The Zeta experiment showed that, after an initial period, the plasma settled into a quiescent, stable state. This was explained by Bryan Taylor’s theory [9], which predicted that the plasma would self-organize into a minimum-force, maximum inductance state. In an RFP, this state has a current distribution that reverses the direction of the helical field lines, as shown in Fig. 10.30. This looks like the toka­mak field of Fig. 5.9, but notice that the outermost field lines are going backwards in the toroidal direction compared to those near the center. Hence the name reversed-field pinch.

In spite of the seeming quiescence of the Taylor state, the RFP suffers from magnetic fluctuations caused by the basic hydromagnetic instabilities, especially the tearing mode (Chap. 6). A conducting shell close to the plasma is needed to

control the resistive wall mode, and active feedback stabilization is needed also. If it can be made to work, the RFP has the great advantage of self-generated magnetic field, requiring the addition of only a small toroidal field from external coils. These coils need not be superconducting, since they consume little power. The relatively weak magnetic field means that very high beta values can be achieved. However, for a reactor, the conducting shell makes the design of the blanket and first wall problematical. The bootstrap current is small, so the large toroidal current has to be driven inductively with a transformer. That means that the plasma has to be pulsed. There is some evidence that a DC current can be created with a oscillating drive [30], but this is at a primitive stage.

In spite of doubts about its reactor relevance, considerable progress has been made in understanding the physics of RFPs. This research is also of interest to space scientists, since processes like reconnection also occur in space. New results come mainly from the RFX machine in Padua, Italy [31], and the MST in the University of Wisconsin [30]. At low power, the RFP does not self-organize suffi­ciently, and magnetic surfaces of many helicities are all tangled up. When the domi­nant mode exceeds 4% at a current of 1.5 MA, however, the plasma snaps into a single helix, whose cross section is shown in Fig. 10.31. The plasma moves off — center into a helical shape, the magnetic surfaces are no longer jumbled, and con­finement is much improved. The electron temperature is seen to increase a factor of 2 to about 850 eV.

In the MST, magnetic chaos is suppressed actively by shaping the current profile with pulsed poloidal current drive. Figure 10.32 is a computer simulation of the magnetic surfaces before and after the current drive is imposed. Plasma confine­ment time is increased ten times by this, up to 12 ms, which is comparable to that in small tokamaks. Electron temperature reaches 2 keV, and ion temperature

1.3 keV. Beta values of order 26% have been achieved. The plasma density sur­passes the Greenwald limit (Chap. 8) by 20% [30]. In spite of the weak magnetic

field, energetic ions have been found to be well contained. This was found by injecting 20 keV neutral beams, which turn into 20 keV deuterons. However, ions in RFPs are not heated by neutral beams because they are naturally heated by reconnection. This is a process in which magnetic-field lines merge, destroying some B-field and converting its magnetic energy into plasma energy. This phenom­enon also occurs in the earth’s magnetic field, so that RFP research, as well as spheromak research, has relevance to other fields of science.

Electric cars will be necessary when oil becomes scarce. Electricity to drive them can come from fossil fuel plants or from carbon-free sources like fission or fusion reactors. Even if fossil fuels are used, GHG emissions are greatly reduced if the fuels are burned at a central utility rather than in vehicles. The main problem is the lack of a suitable battery. Recognizing this urgent need, the Obama administration in the USA has allocated $1.5 billion to the development of advanced batteries. This will greatly expedite this field of research, which was previously hampered by the lack of funding.

In Chap. 4, we saw that the guiding centers of ions and electrons gyrating in a tor­oidal magnetic field have vertical drifts because the field is nonuniform; that is, it varies horizontally. The reason is that the particle feels a different magnetic field on

each side of its Larmor orbit. A similar effect occurs in the presence of an electric field. This is shown in Fig. 5.4. There, the magnetic field (B-field) is coming out of the paper, and the electric field (E-field) points from left to right. Consider first the positive ion. It tries to follow its usual circular path, but it is pushed to the right by the E-field. Having higher energy, its orbit becomes larger. As it cycles back to the left, it moves against the E-field and is slowed up, so its orbit is smaller on the left side. This clearly causes the center of the orbit, the guiding center, to drift down­wards. Now consider the electron on the left. Since it has opposite charge, it gyrates counterclockwise instead of clockwise, and is pushed to the left instead of to the right by the E-field. The result is that it also drifts downwards. Furthermore, since it is lighter and moves faster than the ion, it executes more orbits in the same time interval, ending up with exactly the same downward drift! The result is that parti­cles have an E x B (E-cross-B) drift that is perpendicular to both the B-field and the E-field, and which has the same speed and same direction for ions and electrons regardless of their energies.

It may seem strange that when you push in one direction, the particle goes in a perpendicular direction, but this effect is the same as that in a toy gyroscope. When the gyroscope tips down from vertical, gravity pulls it downwards, but the gyro­scope precesses horizontally. If you follow a point on the rotating ring, you will see that under gravitational pull the whole ring will move sideways, just as do the orbits in Fig. 5.4. The same effect causes a rolling hoop to go a long way before falling over. When the hoop starts to lean over to the left, say, gravity will pull the hoop downwards, and the gyroscopic effect will turn the hoop to the left, so that it travels in a direction that will straighten it up. The front wheel of a bicycle also benefits from this effect, but only in a small way. There are stronger stabilizing forces in a bicycle.

Ever since the early days of tokamak research, it has been noticed that the plasma density could never be raised above a certain limit. Sometimes this limit was blamed on a loss of confinement via an unspecified instability, sometimes on exces­sive energy loss by radiation, and sometimes the plasma suffered a disruption. In 1988, Greenwald et al. [28] put together the data from different machines to see what the density limit depended on. They came up with a surprisingly simple answer: roughly speaking, the density limit depended only on the tokamak current per unit area! For those who would rather have a formula, the one for the Greenwald density nG2 is given in Note 8 hrs.8 This limit has been found to be obeyed in all tokamaks regardless of what mechanism causes the problem at high densities. No one has yet found a theory that explains this; the Greenwald limit is purely empirical. Figure 8.18 shows how well the Greenwald limit is obeyed in two large tokamaks. In almost all shots, the measured density cannot be raised above the straight line,

which is the Greenwald limit. This unexplained law is so universal that it is used in the design of future machines. The design would be to achieve, say, 85% of nG, or 95%, depending on how adventurous one wants to be.

A more ambitious tokamak for technology tests has been proposed by a team at General Atomics in San Diego, California [24]. This machine is shown in Fig. 9.32. Note that this depicts only one side of the torus; the major axis is at

the left edge of the diagram. The dominant feature is the huge copper toroidal field coil surrounding machine. It will produce a field of 6 T (60,000 G). As seen by the size of the human figure compared to that in Fig. 8.23, FDF is actually smaller than JET. Yet the machine produces 250 MW of fusion power and can run continuously for two weeks at a time. The neutron flux is the required 1-2 MW/m2, and the fluence is 3-6 MW-years/m2 over a life of ten years.

Though FDF is much smaller than ITER, it can produce the neutrons for tech­nological testing because it does not reach ignition. It runs steadily at Q=5, where Q is the fusion power divided by the power input to the plasma. For ignition Q > 10 is necessary, and that is much more difficult. Nonetheless, FDF needs all the features of advanced tokamaks: high bootstrap current, internal transport barriers, radiofre­quency current drive, and so forth. Remote handling will be developed, with replacement components lowered from the top, where the upper part of the toroidal field coil can be removed. Initially, blanket modules will be tested. Then, after a 2-year shutdown, a full solid ceramic blanket will be installed and tested. In the third stage, after another 2-year shutdown, a Pb-Li blanket will be installed. Only a machine with a full blanket can test such quantities as thermal stress, nuclear waste and disposal, radiation damage, and material lifetimes.

With full blankets, FDF as currently designed can demonstrate a closed fuel cycle, breeding as much tritium as it uses, reaching a TBR of 1.2. In fact, if operated at 400 MW of fusion power, it could actually breed tritium at the rate of 1 kg per year to be stored for use in DEMO. This is a very ambitious goal. In this sense, FDF is comparable to ITER in what it will accomplish. ITER will push superconducting technology, test alpha particle effects, and aim for ignition, but FDF will tackle the harder problems of technology with a smaller machine. FDF will not be cheap at perhaps one-third the cost of ITER; but since it will be a direct replacement for DIII-D, much of the expertise is already in place; and, importantly, the politics of an international project can be avoided. After the cancelation of TFTR, the USA needs to regain its position at the forefront of fusion research.

Even though a suitable driver has not yet been found, reactor studies can still be made, especially for the simpler direct drive case [42]. The energy released from each cap­sule is equivalent to tens of kilograms of TNT, but the blast waves of TNT cannot be produced because there is so little material involved [43]. The only ions are from the tiny capsule and the DT fuel, plus the helium that is produced. Much of the energy comes out as radiation, and the first wall has to withstand that. The neutrons, as usual, go through the first wall into breeding blankets. The first wall has to withstand the radiation, mostly in the form of X-rays. Inertial fusion has the advantage over toka — maks in that there is a much larger distance between the energy source and the wall. The main candidates for the first wall are (1) a solid material like the SiC/SiC com­pounds proposed for tokamaks, (2) a wall thinly wetted with a liquid, and (3) a water­fall of liquid FliBe (Chap. 9) covering a solid wall. In laser fusion, solid walls would suffer from repeated thermal cycling, which greatly shortens their life.

In direct drive, 71% of the fusion energy comes out as neutrons, 27% as ions, and only 1.4% as X-rays. In indirect drive, 69% comes out as neutrons, 5.8% as

ions, and a whopping 25% as X-rays since the hohlraums are designed to produce X-rays [47]. The ions and X-rays deposit their energy in a very thin layer on a dry wall [48], which must be well cooled to take the heat. A more serious problem is the deposition of the fast alpha particles into the wall, forming helium bubbles that cause the wall to exfoliate. A method to avoid this is to impose a cusp magnetic field (Fig. 7.8) to protect the wall and lead the ions into divertors. However, this requires strong superconducting coils as in magnetic fusion.

A wetted wall can be a thin layer of FliBe injected through small holes in the first wall and protecting the wall from ions and X-rays. The liquid is collected at the bottom, re-processed, and re-injected at the top of the chamber. A thick liquid wall [43] is a cylindrical waterfall of FliBe or PbLi between the target and the solid wall. The waterfall intercepts the fusion products, goes into a tank below the cham­ber, and is re-processed and re-injected at the top. In this case, the target has to enter from the top or bottom. Sethian et al. [42] have compared direct-drive reactors based on diode-pumped glass lasers and KrF lasers. Both kinds have been shown to withstand repeated pulsing at 5-10 Hz at low power. They have similar wall-plug efficiencies: 10% and 7%, respectively; they are compared in case high-power pulsing can be developed.

In inertial fusion, there is the problem of restoring the vacuum in the 100 ms between shots. The remaining gas must not be ionized by the laser. The laser beams have to strike the target with 20-pm accuracy from 10 to 20 m away, and a small amount of gas will deflect the beams. A “glint” system has been tested to overcome this [42]. As the cap­sule nears the center of the chamber, a small laser is fired to illuminate it. The direction of the reflection is detected, and mirrors are moved to keep the beam on target. To do this with 48 beams, however, is a daunting task, and only spherical targets in direct drive can be used. There is no clear path to fusion energy with lasers.