The Princeton Gun Club was a small shack on the side of the runway of the Princeton airport and was purportedly used for skeet shooting at one time. It was an ideal location for a classified meeting of Project Sherwood in 1955. The Robin Hood connection came from one of the participants, James Tuck (Friar Tuck) of Los Alamos. Representatives of the four US laboratories working on fusion (Livermore, Oak Ridge, Los Alamos, and Princeton) fit into the small room. Edward Teller was there. After hearing about our trying to hold a plasma with a magnetic field, he exclaimed, “It’s like holding jello with rubber bands!” Indeed, the jello would squeeze out between rubber bands, exchanging places with an equal volume of rubber, so that the rubber bands were on the inside and the jello on the outside.

A solution to the basic interchange instability was formulated: weave the rubber bands into a mesh. In a toroidal magnetic field, this is done by magnetic shear. Figure 5.9 shows several magnetic surfaces in a torus, each containing magnetic field lines that are twisted. The twist angle, however, changes from surface to sur­face, so if a ripple starts on one surface and is aligned with the field lines there, as in Fig. 5.8, it finds itself misaligned with the field on the next surface. The differ­ence in pitch angle from one surface to another has been greatly exaggerated. It does not take a very fine mesh of field lines to kill the interchange instability; in fact, we will see later that the amount of twist is limited by another instability.

A graphic picture of how shear stabilization works was provided by an experi­ment by Mosher and Chen [4]. The plasma in Fig. 5.10 was in a straight cylinder with a magnetic field up out of the page. The shaded circle in the center represents a thick rod inside the plasma carrying a current into the page and creating a “poloidal” magnetic field that gives the field lines a helical twist. At the left, a bump on a magnetic surface is shown which might represent an instability getting started.3 In successive views to the right, the current in the rod is increased, twisting

Fig. 5.9 A torus with a sheared helical field

the field lines more and more. Finally, at the right, the measurements show that the bump has been twisted into a thin spiral, so thin that the charges that create the electric field in the Rayleigh-Taylor instability (Fig. 5.8) can leak across the spiral, short-circuiting the electric field and killing the instability. In addition, short-circuiting by electrons moving in the toroidal direction (perpendicular to the page) also happens, and in fact is the main stabilizing effect of shear on the inter­change instability.

To summarize this first of many instabilities, we saw that a plasma cannot fall out of its magnetic container the way water falls out of a bottle because a plasma weighs practically nothing. Its gas pressure, however, is always pushing against the magnetic field. With the slightest perturbation, the plasma organizes itself to create an electric field that causes a tongue of plasma to leak out and a bubble of field to leak in. Being wise to the plasma’s tricks, we can thwart the plasma’s moves by short-circuiting its self-generated electric fields with magnetic shear.

As we now turn our attention from fusion physics to fusion energy, we have to introduce Big Q, as distinct from little q. Little q, as you remember, is the “quality”

factor in toruses like tokamaks and stellarators. It is the reciprocal of the rotational transform, which is the number of times a helical field line encircles the minor axis each time it goes around the whole torus. The variation of q with radius r, or q(r), is perhaps the most important feature in the design of toroidal magnetic bottles. Big Q, on the other hand, has to do with how much energy a fusion reactor will produce. It is the ratio of the fusion energy produced to the energy required to make the plasma:

q _ Fusion energy Input energy

In Chap. 3, we showed this equation for the DT reaction:

D + T ® a + n +17.6 MeV,

where a is an alpha particle (a helium nucleus) and n is a neutron. Most of the 17.6 MeV of energy released is carried by a 14.1 MeV neutron, and the other 3.5 MeV is carried by the alpha particle.11 The neutron energy is the part used to produce the electrical output of the power plant, and the alpha energy is used to keep the plasma hot. Since the a’s are charged, they are confined by the magnetic field, and the hope is to hold them long enough that they can transfer their energies to the DT plasma, keeping it at a steady temperature. But since the a’s have only one-fifth of the fusion energy, Q has to be at least 5 for this to happen. This is called ignition. The plasma is “burning” by itself. The reaction cannot run away as in fission because some instability will quench the plasma as soon as the operational limits are exceeded.

The first milestone is to achieve Q = 1, which is called scientific breakeven, which assumes that the whole 17.6 MeV is equal to the input energy. The next milestone is to get to ignition at Q=5. To produce net energy, you have to count also the energy needed to make the magnetic fields and the plasma currents, as well as all the electric­ity needed to run the power plant (even the lights!) and the energy used to transmit the power to where it is used. This means that Q has to be at least 10. Figure 8.20 is a Lawson diagram (Chap. 5) plotting nrE vs. T and showing what different tokamaks have achieved in DD and DT plasmas. The heavy curve is for Q = 1 in DT, and we see that this has been reached in JET. The yellow region is ignition at Q greater than 5. The diagonal dashed lines are for constant values of the triple product. The obvious next significant step is to get to ignition, and that is the story of ITER.

Stellarators like the Wendelsteins are large machines with large aspect ratios R/a, where R is the major radius of the ring and a is the radius of the cross section. There is a movement to build smaller, more economical machines by shrinking R to get aspect ratios of 3-5 instead of 10 or more. Proposed compact stellarators have been designed with different magnetic-field configurations to see which would work better. This freedom of design is not available for tokamaks, but it also means that it is harder to converge on the optimal design. The National Compact Stellarator Experiment (NCSX) was funded and under construction at the Princeton Plasma Physics Laboratory, but the project was canceled during the 2009 worldwide eco­nomic depression. Figure 10.8 shows the NCSX and its coil structure. There are only 18 coils of three different shapes. Although this machine was well designed and would have complemented the Wendelstein 7-X nicely, its discontinuation was reasonable. Tokamaks are far ahead in development, and to get a fusion reactor working the fastest way is to give them the highest priority.

Stellarators are second-generation confinement devices. They are probably better suited for reactors than tokamaks, but we need much more experience with how they run. An obvious question is: Where do you put the blankets in a DT stellarator

reactor? The problem is that the magnet coils are not circular but have small twists and bends. The coils have to be close to the plasma for these fine features to be felt; too far, and the details will be smeared out. That’s why the vacuum chamber has to be shaped to fit the coils. In a reactor, one still has to leave room for the tritium­breeding blanket, and the only way to do this is to scale the whole machine larger. There have been several reactor studies from Germany, Japan, and the USA. The ARIES-CS design is shown in Fig. 10.9 and the overall view in Fig. 10.10. It was found possible to place the blanket modules between the plasma and the vacuum wall and superconducting coils.

A solar cell is an electronic device made of semiconductors in layers, just as com­puter chips are, but much larger and simpler. Since each cell produces less than 1 V, cells have to be connected in a series to give a useful voltage, like 12 V. Flashlight batteries generate 1.5 V, and we use two of them in series to get the 3 V required

by the bulb. Solar panels, about half a square meter in size, contain many cells connected together by transparent wires. The difference among conductors (like metals), insulators (like glass), and semiconductors arises from quantum mechanics, which mandates that energy levels in a solid are quantized. That means that electrons cannot have any old energy but must have an energy on one of the allowed levels. Furthermore, no two electrons can be on the same level. This situa­tion is shown in Fig. 3.30. Energy levels occur in bands, two of which are shown, each containing seven energy levels. There are, of course, zillions of levels in actuality. In an insulator, the levels in the lower band are all filled, one electron in each level. This material cannot conduct electricity, because the electrons cannot move. To move, they would have to gain a little energy, but there is no level close enough for them to move up to. In a conductor, the lower band is filled, but the material has some electrons in the upper band, which is not full. Those electrons can conduct electricity because there are levels above that they can move up to. In a semiconductor, the lower band is full, but the bandgap is small, so if the topmost electron gets a big enough kick (from sunlight, for instance), it can jump up to the upper band, where it can move. So a semiconductor conducts sometimes.

The most common semiconductor is silicon. The bandgap in silicon is 1.1 eV. It is not important at this point to know how much energy an eV is; it will be explained amply in Chap. 4. The “kick” that the electrons get from sunlight to cross the bandgap depends on the color of the light that hits it. Sunlight contains a range of colors, as we know by separating them with a prism (Fig. 3.31), giving rise to the proverbial sequence violet, indigo, blue, green, yellow, orange, and red. Light can be considered as a stream of photons, which are particles with energy but no mass. No, they do not follow E=mc2! Each color corresponds to photons of a certain energy. Those at the blue end of the spectrum have more energy, and those at the red end have less. For a photon to make a semiconductor conduct, it must have an energy of at least 1.1 eV. That means that the part of sunlight redder than that will be lost. For silicon PV, the idea is to add semiconductors with other bandgaps that can capture the other parts of the solar spectrum.

After a photon kicks an electron into the conduction band, what happens next? This is shown in Fig. 3.32. This is the semiconductor part of Fig. 3.30, but showing only the electrons on the top level. After an electron is kicked into the conduction band, it leaves a hole in the valence band. What we have not shown is that the electrons actually belong to atoms consisting of a positive nucleus surrounded by enough electrons to make the whole atom uncharged. These atoms are locked into a crystal lattice. In Fig. 3.32a, an electron has been knocked out of one atom into the conduction band. It leaves behind an atom with a missing electron and therefore

has a charge +1. That atom, shown in white, has a “hole” in it; that is, a place where an electron should fit but is missing. An electron can then jump from a neighboring atom, thus filling the hole but leaving a hole in the neighboring atom. As shown in B, the hole can move like a positive electron! If an electric field is applied, the electron in the conduction band will move one way, and the hole in the valence band will move the opposite way. These electron-hole pairs will conduct electricity, and now we have to see how the current is collected.

The electrons and holes cannot be collected directly with a copper plate con­nected to a wire because these charges cannot cross the interface between these very different materials. A buffer layer has to be added. These buffer layers are made of “doped” silicon. Here, doping is legal. By adding a few “impurities,” which are specially chosen atoms with one more or one less electron than silicon has, we can make n-type or p-type highly conductive semiconductors. The former has a net nega­tive charge, and the latter a net positive charge. We can then make a sandwich of three layers to form the basic unit of a solar cell (Fig. 3.33). Opposite charges attract, so when solar photons create electron-hole pairs in the silicon, the electrons are attracted to the p-type layer at the bottom, and the holes to the n-type layer at the top. Since they are negative, the electrons carry a current in the opposite direction to their motion. The buffer layer allows them to flow into wires carrying the current to the load (the appliance or battery that uses the juice). When the electrons reach the n-type layer, they fill the holes that had migrated there. The voltage generated is the bandgap voltage. The larger the bandgap, the higher the voltage. This makes sense, since only the energetic photons can push an electron across a large bandgap.

This subject is logically treated here because of the radioactive waste problem of fission reactors. However, fusion reactors have not yet been described. This section can be best understood if Chap. 9 on fusion engineering is read first. The reason for combining fusion with fission is that it could benefit both systems. Fission reactors can be run subcritically for better safety, and their high-level wastes can be trans­muted into fuel and a much smaller amount to be sequestered. Fusion reactors, on the other hand, can be run subcritically also, without producing all the energy of the reactor, greatly accelerating the time for their development. Many plasma theorists have advocated fission-fusion hybrids, notably Jeffrey Freidberg at M. I.T. and Wallace Manheimer at the Naval Research Laboratory in the USA. The idea was first proposed by none other than Hans Bethe. However, their arguments do not include specifics on how a hybrid reactor might be designed. A group at the University of Texas has proposed a reactor based on a spherical torus (see Chap. 10), a new fusion device that has not been extensively tested. The most detailed engi­neering design has been done by a group at the Georgia Institute of Technology (Georgia Tech) under the leadership of W. M. Stacey. Their subcritical advanced burner reactor [45] will be described here. A diagram of it appears in Fig. 3.61.

Within the D-shaped toroidal-field coils is the plasma of a fusion reactor, shown in yellow. Surrounding that is the fission fuel core, which is divided into four

Central solenoid Vacuum vessel Blanket and shield Reactor core Plasma

Plasma first wall Toroidal field magnets

concentric rings (gray). Surrounding both is a neutron absorbing blanket which breeds tritium from Li4SiO4 for DT fuel. The fission part is an LMFBR designed at Argonne National Laboratory. The fuel is 36 tons of transuranic waste from LWRs consisting of 40% Zr, 10% Am, 10% Np, and 40% Pu. It is in the shape of 7.3 mm diameter fuel pins, 271 of which form a fuel assembly. The fuel pins include a channel for the liquid sodium coolant. Their complete design and manufacturing process have been specified [46]. The fuel rings (batches) contain 918 assemblies. The tokamak part is a scaled-down ITER operating with conservative parameters lower than the maximum values needed for energy production. These include factors which will be explained in Chap. 9: the Greenwald limit, normalized beta, big Q, and the bootstrap current fraction.

The operating characteristics of this reactor have been extensively calculated. The fission part will generate 3 GWth (gigawatts thermal). It runs subcritically, generating fewer neutrons than is necessary to maintain a chain reaction. The missing neutrons are generated by the fusion part. Since its mission is not to generate power, it can be designed to contribute only 250-500 MWth of energy. The fission fuel is burned in 750-day cycles. Each batch spends one cycle in each position, for a total exposure of four cycles or 3,000 days. After that, it is removed to storage, and its decay heat over the next million years has been reduced by a factor of 2, and thus the storage facility requirements have been halved. The total time of exposure is limited by the life of the fuel cladding under neutron bombardment, set at 200 dpa (displacements per atom).

This amount of burnup of actinides can be greatly improved by reprocessing. If the fuel from the hybrid after four burn cycles is reprocessed, then mixed with
“fresh” waste from LWRs and sent through the hybrid again, the decay heat of the ultimate product can be reduced by 99%. High-level storage facilities can be reduced by a factor of 100. If the 200-dpa limit on neutron damage can be relaxed so that the fuel can be burned for four 3,000-day burn cycles for a total of 12,000 days (25 years), 91.2% of the transuranic waste can be removed after only once through the hybrid reactor. Such a fission-fusion hybrid can treat the waste from four 1,000-MW LWRs.

e

It is possible for the fission reactor to go critical. Zirconium is added to the fuel so that there is negative feedback: when the temperature rises; the reaction slows down. However, if this does not work and there is a runaway reaction, there is less time available for control rods to be inserted than in a normal LWR. Fortunately, there is a simple solution. The reaction cannot run without neutrons from the fusion reactor. The plasma producing these neutrons can be shut off within a second or so by a massive injection of gas.

Proponents of hybrids see that they can make fission safer and at the same time let fusion get online faster. Skeptics see that these would be extremely expensive and difficult reactors to design and construct and would detract from the main objective of developing pure fusion. In any case, this subject is still in its infancy compared with Generation III fission reactors or with tokamak fusion reactors.

In every tokamak discharge, there is a magnetic surface where q = 1. Inside that surface, where q is less than 1, the plasma is unstable to kinks, according to the Kruskal-Shafranov limit. Therefore, it is turbulent and a jumble of oscillations, and

there is no magnetic confinement. Only when plasma gets outside the q = 1 surface and enters the nested magnetic surface and island structure, does it get restrained by the magnetic field and diffuse slowly to the wall.

Very early in tokamak research, experimenters using a synchrotron-radiation method to detect changes in electron temperature observed regular oscillations near the q = 1 surface. These were observed in all tokamaks and always had a sawtooth shape, rising slowly and falling sharply each time, as seen in Fig. 7.5. Since the current is largest inside the q = 1 surface, near the center, the plasma gets hotter there. Higher temperature means less resistivity, and that makes the current even larger and more peaked. When the shape of the current profile changes, so does the whole island structure, as seen in Fig. 7.3. Finally, the magnetic structure is so disturbed that the steady state can no longer be maintained, and the plasma has to change. What the tokamak does is to eject the overly hot plasma in outward bursts, thus cooling the center back to normal. This explanation was for a long time only a conjecture, but recent advances in instrumentation have enabled actual movies of these sawtooth bursts to be taken in real time. These movies show that the tempera­ture actually oscillates several times before the big crash, when hot plasma is shot out and replaced by cooler plasma. Still frames from the movie by H. K. Park of the Princeton Plasma Physics Laboratory are shown in Fig. 7.6, but they do not do justice to the actual product [3].

As doubly heavy hydrogen, tritium has two extra neutrons, which do not sit well with a single proton. So tritium decays by emitting an electron, a process known as beta-decay. This loss of a negative charge changes one of the neutrons into a
positively charged proton and converts tritium into helium-3, a helium isotope with two protons and a single neutron instead of the usual two. This decay makes tritium radioactive, and it has to be handled carefully in a fusion plant.

Fortunately, the radioactivity is mild. The electron that is emitted has very low energy, about 19 keV. It cannot penetrate the skin, and even in air can go only 6 mm (1/4 in.) [12]. However, it can be harmful if ingested and must be carefully kept out of the water supply. Unlike fission products, tritium has a short half-life of only 12.3 years. This means that 5.47% of it decays into harmless helium each year. Because of its short life, very little tritium exists naturally. Cosmic rays make about 200 g of tritium a year, and there are only about 4 kg of natural tritium at any one time in the earth’s atmosphere. Man-made tritium raises this to about 40 kg. Compared with this, it will take 1 kg of tritium just to get ITER running on DT, and a reactor may use up 100 kg per year.

Also called the dense plasma focus (DPF), this is one of the oldest devices invented to create fusion. Because of its simplicity, it is used in small laboratories over the world for instructional research. A diagram is shown in Fig. 10.38. A plasma is formed by discharging a large capacitor between the center electrode and the outer cylinder. An ionization front, shown by the white curve, travels rapidly to the end

at the right. There, the current flows between the electrodes in the crown-shaped plasma consisting of streamers. In the center of the crown is a dense Z-pinch which can reach fusion conditions for a brief instant.

Intense X-rays are generated, and with deuterium for DT, neutrons are produced for 10-20 ns [39]. Both diagnostics and theory are difficult for the DPF, and it is not well understood. Nonetheless, some groups are proposing the DPF for p-B11 fusion. There is interesting physics to be studied in the DPF; but, as with all single­pulse machines, it is not suitable as an energy source.

The heart of a hydrogen car is the fuel cell, whose parts are illustrated in Fig. 3.51. Hydrogen is forced into the channels in the anode plate and is then spread out uniformly in the diffusion layer. This layer has been described as a wet rag whose moisture content must be carefully controlled to keep the proton exchange

ANODE BIPOLAR PLATE

HYDROGEN IN

CHANNELS

OXYGEN IN

CHANNELS

CATHODE BIPOLAR PLATE

Fig. 3.51 Schematic of a fuel cell. It is not to scale. The catalyst layers and the PEM are only 10’s of microns thick, while the diffusion layers are 100’s of microns thick. The bipolar plates are of macroscopic dimensions
membrane (PEM) from drying out without dripping. The PEM is a plastic layer like plastic wrap made of a special material called Nafion® made by Dupont Chemical. It has the magical property of allowing hydrogen ions (H+) to pass through but not electrons. It is the platinum catalyst layer that dissociates hydrogen gas (H2) and ionizes it into two hydrogen ions (H+). This is an even more magical property. The catalyst layer consists of platinum nanoparticles thinly deposited on carbon paper which has to be rough to present a large surface area and porous to let the water through. The electrons, being blocked by the PEM, are drained off into a wire to form the electric current that is the output of the cell. When the H+ ions reach the other side, they encounter another catalyst layer, which could be platinum or iridium. Meanwhile, oxygen (O2) from air is pushed into the cathode plate and diffusion layer to meet the hydrogen ions in the catalyst layer. Therefore, the O2 is dissociated into atoms (O) and picks up electrons from the wire that has gone through the load to become negative ions (O-). Each O — then combines with two H+s to form H2O. Hydrogen and oxygen have been combined to form water and electricity. All in all, the fuel cell is a serendipitous invention, but it has problems.

Each fuel cell generates only 0.6-0.7 V, so as many as 100 of them have to be connected in a series to form a stack with a useful voltage output. Platinum is a precious metal used in jewelry and in catalytic converters. Its price drives the price of fuel cells to about $73/kW, twice the commercially viable value.49 Cyclic opera­tion of PEMs degrades their performance. PEMs have to be heated to at least 60°C from a battery before they can even start, and they need about 100°C to operate reliably. The water in the cell must not boil or freeze under all driving conditions. Corrosion of the bipolar plates is a problem; they cannot be made of a metal that can corrode and contaminate the system with iron or chromium. A carbon com­pound has to be used. Besides the electric motor, the car has to have a system to pressurize the gases. And the fuel cell has to last for 300,000 miles.

Currently, the whole shebang is too large to fit inside a car but can be used in trucks. No large-scale production and testing has been done. What can be gained is a fuel-cell efficiency of 80% times another 80% efficiency of the electric motor, giving a maximum efficiency of 64% in the conversion of hydrogen energy to mechanical energy. This compares favorably with the efficiency of gasoline-driven cars, about 15%, but the energy in producing the hydrogen has not yet been counted. If that part is 40% efficient, the net efficiency is 64 x 40= 26%, still higher than burning natural gas in a gas engine. However, the real gain will be when hydrogen is pro­duced in fission or fusion plants with no use of fossil fuels or emission of GHGs.

So we have found that the best way to produce fusion reactions in a continuous manner is to make a very hot plasma, so hot that it cannot be held in place by any material container. We also decided that of all the forces that we can use to make a wall-less container, only the magnetic force would work. What would a magnetic bottle look like? Actually, it looks like a bagel; but before we get to this, we have to review what we know about magnetic fields. Most people know that the earth has a magnetic field, as shown in Fig. 4.6. The lines with arrows show the direction of the field. A compass needle aligns itself with the field line that passes through it on the earth’s surface, and therefore points toward the magnetic pole, which is close to the geographic pole. The earth’s field is already a magnetic bottle, but an imperfect one. Protons and electrons coming from the sun in the solar wind3 get trapped in this field because charged particles tend to move along field lines, not across them. But the trap has large leaks at the north and south poles where the field lines run into the ionosphere, bringing the particles with them. When electrons strike oxygen atoms in our atmosphere, visible light is emitted which we call the Aurora Borealis. Since the plasma particles can travel in either direction, the same thing happens in the southern hemisphere. The Aurora Australis is not as well known because few people stay out on a winter’s night in Antarctica to watch it, and penguins have other agenda.

Magnetic field lines are, of course, only a mathematical construct. Electric or magnetic fields can be detected only by the forces that they exert. It was the great Scottish physicist James Clerk Maxwell4 who invented the concept of a “field” to describe action at a distance. Once a field at a given position is known, one can calculate the forces which that field would exert on an object there. To depict the shape of a field, one can draw any number of lines. A visual display of magnetic field lines is commonly given in textbooks, where the pattern of iron filings traces the field lines around a horseshoe magnet, as in Fig. 4.7.

Magnetic field lines are sometimes called “lines of force,” but this is a misnomer. The magnetic force is actually perpendicular to the lines! A compass needle points north-south because, when it is not aligned, the north pole of the needle is pushed one way by the magnetic field of the earth, and the south pole the other way, until the needle is aligned with it. Similarly, each elongated iron filing in the horseshoe demonstration acts like a miniature compass needle and points in the direction of the field at its location. It is important to understand what a field line represents, because how a magnetic bottle works depends critically on how these lines are shaped.

The problem with permanent magnets is that the strongest magnetic field it generates is inside the iron of the magnet, where we cannot put any plasma. Fortunately, we can create magnetic fields with electromagnets. In Fig. 4.8a, we show the field around a bar magnet, which is a magnetized iron cylinder; it has basically the same shape as the earth’s field. In Fig. 4.8b, we have replaced the iron bar with a glass tube of the same length and diameter, and we have wound many turns of wire around the tube. When we hook the wire up to a DC voltage source, such as a battery, the current in the wire generates a magnetic field of the same shape as that of the bar magnet! But now we can put plasma inside the glass tube, where the field is much stronger, as you can tell because the lines are closer together.

Fig. 4.8 The magnetic field around (a) a bar magnet and (b) an electromagnet of the same size

Now we can move on to see how to make a leak-proof magnetic bottle for plasma using cleverly shaped wire coils to produce field shapes that will plug all the leaks.