Looking at a globe, we see lines that do not end. The latitude lines go in circles around the earth and end on themselves (Fig. 4.11). The longitude lines go north and south until they reach the poles, where they continue over to the other side of the earth (Fig. 4.12). Why can’t we make a magnetic bottle shaped like a sphere with magnetic field lines that go either north-south or east-west? Here’s why. If we look down at the north pole, say, in Fig. 4.11, we see that the field lines go around in smaller and smaller circles. As one gets closer to the pole, the magnetic field must get weaker and weaker, since the fields on opposite sides of the circle are in opposite directions and tend to cancel each other. Exactly at the pole, the field must be zero, since it cannot be in two directions at the same time. This is called an

O-type null. The plasma will leak out at the poles, since there is no magnetic field there to confine it. If we now look at a configuration in which the field lines are like longitude lines, Fig. 4.12 shows that the field lines point toward (or away from) one another at the poles, or cross one another at an angle. Again, the field at the pole must be zero, since it cannot be in two directions at once. This is called an X-type null. A simple shape that is topologically equivalent to a sphere cannot be made into a magnetic bottle. It will have a big leak at the poles, where there is no mag­netic field to hold the plasma.

The simplest shape that will work is a torus, a three-dimensional volume like a tire or a doughnut, with a hole in it, as shown in Fig. 4.13. Mathematicians would call it a doubly connected space. Field lines that have no ends can be imbedded in such a chamber in such a way that ions and electrons cannot find a way out by moving along the field lines. Such closed field lines are of two types. Toroidal field lines, of which one is shown in Fig. 4.13a, go around the torus in the long way, encircling the hole. Poloidal field lines, shown in Fig. 4.13b, go around the short way and do not encircle the hole. Remember that field lines are just a graphic way to show the direction of the magnetic field. There is an infinite number of field lines. The torus is entirely filled with magnetic field, so that plasma placed inside will not, in principle, escape. The ion and electron guiding centers simply move along the field lines and never hit the wall, as long as the field lines they’re on do not wander out of the torus.

Now imagine combining toroidal and poloidal fields into the same torus. A toroidal field line going around the long way will also bend the short way, like an old-fashioned barber pole or the stripes on a candy cane. The field line will look like a Slinky® toy stretched around a lamppost; it is a helix bent into a circle. The generic toroidal and poloidal types of magnetic field will not work. Combining them into a helix is the beginning of the art of making magnetic bottles. All this is necessary because magnetic fields do not stop particles from moving longitudinally, and therefore they must not end on a material wall.

The achievement of reversed shear led to an even more important discovery: internal transport barriers or ITBs (another acronym that I shall avoid). These are like the H-mode pedestal but can be created inside the plasma, away from the walls. They effectively stop the fast transport of plasma to the walls caused by instabilities and turbulence. At the radius where the q profile is at a minimum, the shear in both the poloidal magnetic field and the poloidal E x B drift is so strong that most instabili­ties are quenched, and anomalous diffusion comes to a stop, as if there were a wall in the middle of the plasma. This is another unexpected benefit discovered only by painstaking experiment on large tokamaks. Figure 7.28 shows how an internal transport barrier should be designed. If it is placed close to the axis (dashed line), the hot, dense plasma will be limited to the small volume inside the barrier. Furthermore,

it turns out that a barrier placed further out, as shown by the solid lines, is more consistent with the current profiles achievable with bootstrap current. The width of the barrier also makes a difference. It has to match the size of the turbulent eddies to be suppressed. Since the large eddies are more dangerous, the barrier should not be too narrow.

To create a good internal barrier, the current profile has to be manipulated so that it does not peak at the center, as it tends to do. This is done by adjusting the current in the ohmic heating coils and using waves to drive additional currents (noninduc­tive current drive). The wave used is primarily the so-called lower-hybrid wave, but electron cyclotron waves are also used. The radial location of the currents driven by waves can be adjusted by changing their frequencies. In the most intense discharges produced to date, the bootstrap current can make a significant contribution. Internal transport barriers have been produced in all four of the largest tokamaks in operation: the ASDEX Upgrade in Germany; the DIII-D in General Atomics of San Diego, California; the JT-60U in Japan; and JET, the European tokamak in England. A fifth large machine, the TFTR in Princeton, New Jersey, has been decommissioned and scrapped as a result of budget cuts by the US Congress. The example shown here is from the DIII-D [11].

In the following example, a double barrier was actually achieved, consisting of an H-mode edge barrier in addition to an internal barrier. The q profiles are shown in Fig. 7.29, one with the internal barrier alone and one with a double barrier. In neither case does the q-value drop below the Kruskal-Shafranov limit of 1.

The effect of the barriers on the plasma is shown in Fig. 7.30. Both curves show the high temperatures and density inside the internal barrier, and the solid curve shows the large increase in temperature when the edge barrier is added.

That the transport barriers dramatically reduce the losses and increase the energy containment in a tokamak is shown in Fig. 7.31. What is shown is the radial variation of the thermal diffusivity % (chi) of ions and electrons; that is, the rate at which their energies are being transported outwards at each radius in the discharge. A low value is good; a high value is bad. The dotted curves show % when there are no transport barriers. As before, the dashed curves show the case with an internal barrier alone and the solid one with both barriers. These dip well below the barrier-less curve inside the barriers. At the bottom of the % plot, are thin lines showing what % should be according to neoclassical theory; that is, if there were no instabilities. Normally, the turbulence level from microinstabilities makes % much larger than the ideal theoretical value. Here, we see that the internal barrier has brought % down to the ideal level for the first time, at least in the inside part of the plasma.

These results were obtained in a powerful tokamak, with 1.3 MA (megamperes) of toroidal current, a toroidal field of 2 T (20,000 G), and a bootstrap-current frac­tion of 45%. For best barrier formation, it was found that it was better to heat the

plasma with neutral beams injected opposite to the direction of the tokamak current than along it, as in the usual case. This could be done without moving the large beam injectors by simply reversing the polarity of the current in the ohmic heating coils. In the larger JET tokamak, running with DT instead of pure deuterium, ion temperatures up to 40 keV, maintained for almost 1 s, were achieved with an inter­nal transport barrier. The magnetic field there was 3.8 T, and the plasma current was 3.4 MA [12]. Taken together, the data from the large tokamaks, especially those with large bootstrap fraction, give credence to the hope that the Advanced Tokamak scenarios can be used in the design of a practical reactor.

Although the possibility that reversed shear and internal transport barriers could reduce the plasma loss rate was predicted theoretically [13, 14], turning the idea into reality depended on the availability of machines large enough to produce this effect and on hands-on twiddling of these machines to attain the right conditions. The diagnostics needed to quantify these results with detailed measurements inside the plasma also required major equipment and advanced technique. For instance, to get the q(r) profiles a sophisticated method called Motional Stark Effect was used which actually measures the pitch of the field lines at every radius.

Fusion has suffered from the reputation that it is always promised to be available in 25 years. This was because the difficulties were not initially known. They have been overcome, but it took time and funding to build the necessarily large research machines, to train a generation of plasma physicists, and to develop the diagnostic tools to be able to see what we are doing. The underlying physics is now understood well enough that more accurate estimates of what it takes to make magnetic fusion work can be made. Thousands of dedicated physicists and engineers labored for decades to bring fusion within the foreseeable future. There are still a few physics problems to be solved, as described in the next chapter. Engineering is another matter. What has to be done to make fusion reactors practical is the subject of Chap. 9.

Notes

1. Courtesy of Roscoe White.

2. An acute reader would ask, “Why don’t we just let those non-gyrating particles go and con­fine the rest?” The reason is that those particles which have leaked out would be quickly regenerated by the plasma in what is called a velocity-space instability. It is another of a plas­ma’s tricks to bring itself to thermal equilibrium without waiting for collisions to do so.

3. A nice treatment of this is given by Jeff Freidberg, in Plasma Physics and Fusion Energy, Cambridge University Press, 2007.

As shown in Chap. 8, disruptions are disasters. Magnetic containment is suddenly lost, and the plasma drifts vertically into the walls, depositing all its thermal energy. The tokamak current tries to keep itself going as the plasma goes away, so very high voltages are generated. Runaway electrons of MeV energies are created by the high voltages, and these electrons crash into the walls, generating high-energy X-rays. The plasma current is used to generate the poloidal magnetic field, and as this field decays with the current, large forces are applied to the magnetic coils and conducting parts of the tokamak structure. The entire energy in the plasma, magnetic field, and tokamak current is something like 500 MJ, and in a disruption this is all dumped into the structure of ITER in 1/30th of a second [18]. This is like an explosion of 120 kg (260 lbs.) of TNT. Disruptions are expected in ITER, and its parts are designed to withstand them. Disruptions have to be eliminated in reactors, which would be so heavily damaged as to require lengthy shutdowns for repair.

There is a possible scenario of how a change in the magnetic structure of the tokamak discharge, such as a coalescing of magnetic islands, can cause a disrup­tion. It has been confirmed in experiment that staying well below the known stability limits, such as the density limit, can avoid disruptions. A reactor, however, needs to operate at the highest level to lower the cost of electricity (COE). Since a disruption is now known to be a vertical displacement of the plasma, there are ideas on stopping these displacements with a coil or coils inside the chamber. Such a coil is included in Fig. 9.27. Though it is not possible to stop a disruption once it starts, there are ways to mitigate the damage. Disruptions have magnetic precursors which can be detected, and fast action can be taken. Injection of liquid jets or solid pellets of a frozen gas have been tried, but these have led to creation of too many runaway electrons. A large puff of a gas like argon can be driven well into the plasma, be ionized into high-Z ions, and increase the resistivity so that the current dies more gently. Fast gas valves have been developed for this purpose. There is then a smaller tendency to induce currents elsewhere, lower forces on the structure, and fewer runaway electrons. After a disruption, there is only gas left in the vacuum chamber. This has to be pumped out and the discharge started all over again.

Laser fusion would require a separate book to describe. Here it is treated briefly as an alternative to tokamak fusion. The idea is to put a deuterium-tritium mixture

into a capsule about 2 mm (~1/16 in.) in diameter and hit it from all sides with a huge amount of laser power for several billionths of a second. The power from NIF in one short pulse is 500 times the total electric power capacity of the USA. This is what is supposed to happen.

In Fig. 10.39a, the laser or ion beam energy impinges on a capsule uniformly from all sides. The capsule contains fuel in the form of solid DT, covered by a sacrificial layer called an ablator. The ablator is immediately ionized into a dense plasma, which expands violently away from center. The capsule is compressed as if jet engines had taken off on all sides. Figure 10.39b shows the expanding plasma compressing the capsule. With sufficient laser power, the DT fuel is compressed to a density of 1,000 g/cm3, approximately 100 times the density of lead; and the temperature reaches 10 keV. The breakeven condition equivalent to the Lawson criterion (Chap. 5) is pR > 1 g/cm2, where p is the density in g/cm3 and R is the final radius in centimeters. Fusion occurs, and there is a miniature explosion releasing the helium and neutron products. The energy of an NIF pulse is about 1.8 MJ, and the energy generated could be as much as 100 MJ, equivalent to 24 kg of TNT. To produce 1,000 MW of thermal power would require ten explosions per second. Glass lasers, however, can pulse only once every few hours.

The Nature of Sunlight

If we take a solar cell a square meter in size, put it on top of the atmosphere, and face it directly toward the sun, it would receive solar radiation energy of 1.366 kW/m2. Take it down to the surface of the earth, and the light will be attenuated by the air’s absorption and scattering. The net result is the convenient figure of 1 kW/m2. Over the whole earth, there is enough sunlight in an hour to supply all the energy use in the world for a year! If you find this hard to believe, as I did, we can do a back-of-the- envelope calculation in Box 3.1.

Box 3.1 How Much Sunlight Does the Earth Get in 1 hour?

The radius of the earth is about 6,400 km (4,000 miles). Replace the earth with a disk 6,400 km in radius, and the disk would get the same amount of sunlight. We do not count the back side of the disk, and that takes care of the fact that there is no sunlight at night (Fig. 3.19).

The area of the disk is nr2, as you well remember. That works out to be some 130,000,000 km2. In square meters, the area of the disk is a million times that, which is 130,000,000,000,000 m2. (Those who are meter-chal­lenged can think of a square meter as a square yard.) Each square meter gets 1 kW, so the total power over the earth is that large number of kilowatts. The number is too long to write, but we can use shorthand and write it as 1.3 x 1014 kilowatts, where the 14 stands for the number of decimal places after the “1.” (This is scientific notation, which was explained in Chap. 2.)

To compare this with our energy consumption, we have to convert kilo­watts into Quads per year. We can use Table 2.1 in Chap. 2 to make this conversion. It takes several steps, but 1.3 x 1014 kW is the same as 440 Quads per hour. Also in Chap. 2, we found that our civilization consumes about 500 Quads per year, almost the same number. So indeed, sunlight hitting the earth every hour carries about the same energy as we consume in a year!

Figure 3.20 shows the annual variation of sunlight. This shows that the earth is tipped relative to the plane of its orbit. Consider a location in the northern hemisphere, on the upper red line, say. In the summer, the sun would be on the left, so as the earth rotates, more of that red line is in the sunlit region, and less in the blue night region. Days are longer than nights. In the southern hemisphere, the opposite is true. When the earth moves to the opposite side of its orbit, the sun appears to come from the right. The blue region is then sunlit, and less time is spent in there than in the yellow night region. Days are shorter then nights in the northern hemisphere. Furthermore, the sun never gets high above the horizon at high latitudes. Since we cannot easily store solar energy from summer to winter, solar power is inequitably distributed.

(continued)

Box 3.1 (continued)

So why aren’t we all fried by the sun? First, there’s the factor of 2. Except at the equator, sunlight comes in at an angle, not from overhead, so that the power is spread out over a larger area. To figure out how many kW/m2 there are at any given latitude and longitude is a long exercise in spherical trigonometry, but we can average. The area of a hemisphere is 2pr2, happily just twice that of the disk. So the average inso­lation over the earth is only 0.5 kW/m2. Since the earth’s axis is tilted with respect to its orbit, there are seasons; and people living at high latitudes have a bigger dif­ference between winter and summer. They also get less sun altogether. Figures 3.21 and 3.22 show this. The number 0.5 kW/m2 is averaged over latitude and seasons. Then there are clouds and storms and smog which prevent the sun from shining. That cuts the average to below 250 W/m2, and it is not available everywhere. Even so, it is a lot of energy, if we could only learn how to capture it efficiently. The average person in the USA uses about 500 W of electricity, averaged over 24 hours. Two square meters of solar cells in a good location could generate this if they were 100% efficient. Right now, it is hard to get 10% except in the laboratory.

Fresh uranium is mostly U238, with only 0.7% of U235. Unless neutrons are very carefully preserved, there are not enough of them to sustain a chain reaction without increasing the amount of U235. Normally, uranium has to be enriched to 3-5% U235 by separating out the U235 and adding it to normal uranium. Because the two iso­topes differ in mass by only 1.3%, separation is slow; and large installations are needed to fuel power plants. The two main methods are gas diffusion, used in the USA and France, and gas centrifuge, used in Russia and the rest of Europe [41]. In gas diffusion, uranium hexafluoride (UF6) is passed multiple times through porous barriers through which U235 passes 0.43% faster than U238. Gas centrifuges are tall cylinders spinning at high speeds in vacuum. The centrifugal force pushes the heavier isotope out faster. Though gas centrifuges are more efficient, using only 0.09% of the energy generated by the plant compared with 3.6% for gas diffusion, it is a newer technology and it would be costly for the USA to convert to it. The operative word here is not “convert” but “covert.” Centrifuges are discussed further in the Nuclear Proliferation section.

Advanced technology has not overtaken these brute-force methods. Accelerating uranium ions in beams in which the isotopes would have different momenta was tried initially. During WWII, a plasma discharge was tried in the USA, but instabili­ties arose. This was the origin of Bohm diffusion (see Chap. 6). In the 1970s, a laser method was developed at the Livermore laboratory in California in which a laser beam could preferentially put U235 into an excited state, and this could allow it to be extracted separately. At the same time, another laser method was applied to UF6 at the Los Alamos laboratory in New Mexico. A scheme by John Dawson to use two-ion hybrid cyclotron waves in a uranium plasma was implemented at TRW Inc. in Redondo Beach, California [42]. Though this produced palpable amounts of U235, the project was canceled in favor of the Livermore project for political reasons.

A toroidal plasma current serves two purposes: it generates the necessary twist in the magnetic field, and it can also raise the plasma temperature by ohmic heating. However, there is a limit to how much current can be driven because of yet another instability: the kink instability. Figure 6.2 shows an initially straight current path in the plasma that has bent itself into a kink. The circles show the field lines of the poloidal field that the current generates (the toroidal field is from left to right). Note that the lines are closer together on the inside of the kink than on the outside, indicating that the field is stronger on the inside. The magnetic pressure, therefore, is stronger at the bottom of this picture than at the top, and the kink is pushed further out. The bigger the kink, the larger the pressure difference; and the instability grows rapidly and disrupts the current. Remember that the poloidal field shown here is not the main (toroidal) field that supports the plasma pressure; it is the relatively small field that provides the twist. The toroidal field has a stabilizing influence, since it resists being pushed around by the plasma current. The onset of instability, therefore, depends on
how strong the toroidal field is relative to the current. Conversely, onset of instability depends on how much current there is for a given toroidal field strength.

The limiting current for stable operation is called the Kruskal-Shafranov limit, and it is conveniently expressed in terms of the rotational transform, which is the number of times a field line goes around a torus the short way for each time it goes around the long way (Chap. 4). The critical rotational transform is exactly ONE! The critical current is that which creates a poloidal field large enough to twist the field lines just enough to give unity rotational transform, taking into account the strength of the main toroidal field. Transforms larger than 1 are unstable to kinks; transforms smaller than 1 are stable. The criterion for kink stability is actually quite complicated, since it depends on how the current varies across the plasma, but we can give a rough picture of why a rotational transform of 1 is a magic number.

The kink shown in Fig. 6.2 is in a straight plasma, but the current channel actu­ally flows around the torus and joins back on itself. Figure 6.3 shows the largest unstable kink, which is actually an off-center displacement of the plasma. The plasma has been made unrealistically thin in order to have room to show the effect. In the top view (a), the dashed lines indicate the cross sections viewed in panel (b). Let us assume that the rotational transform is exactly 1. On the right-hand side of either view, the plasma has been displaced toward the outer wall. On the left-hand side, half-way around the torus, the field lines have rotated half-way around the cross section, so the plasma is now close to the inside wall. If the transform is

Fig. 6.3 A large kink distortion of the plasma in a torus: (a) top view and (b) cross-sectional view

exactly 1, when the field lines come back to the right-hand side, they will be in the same place where they started, so the current can flow in a closed path. Remember that the plasma is almost a superconductor; so, without collisions, the electrons carrying the current must stay on the same field line. Now let us assume that the rotational transform is less than 1. Then, upon coming back to the right-hand cross section, the current channel is in the position shown by the cross-hatched circle, which does not match up with its initial position. Since current must flow in a con­tinuous path, this distortion of the current channel is not possible, and this kink cannot form. The plasma is stable for rotational transforms less than 1. In this simple picture, the plasma would also be stable if the transform is greater than 1, as long as it is not exactly 1. However, in that case, the current is strong enough to drive other shapes of kinks, and the plasma is kink-unstable in a way that is not easy to explain.

Since small rotational transform is good while large transform is bad, the recip­rocal of the transform is used in tokamak lore. This is the quality factor q (“little q”), which is high when the plasma is kink-stable and low when it is kink-unstable. If the rotational transform is larger than 1, q is less than 1, and the plasma iskink — unstable. If the rotational transform is smaller than 1, q is larger than 1, and the plasma is kink-stable. What if q is a rational fraction so that the current channel joins up to itself after several trips around the torus? Then very interesting things happen, which we will get to.

Introduction

With the information they have gathered from the public media, most people who have heard of fusion consider fusion energy to be a pipedream. Their information is out of date. As we have shown in the last two chapters, great advances have been made in fusion physics, and our knowledge of plasma behavior in toroidal magnetic bottles is good enough for us to push on to the next step. This does not mean, however, that fusion is not a pipedream. There is a large chasm between the under­standing of the physics and the engineering of a working reactor. There are problems in the technology of fusion so serious that we do not know if they can be solved. But the payoff is so great that we have to try.

The situation can be compared — or contrasted — with that of the Apollo program to put a man on the moon. In that program, the physics was already known: Newton’s laws of motion covered all the physics that was needed. In the case of fusion, it took over 50 years to establish the science of plasma physics, to develop fast computers, and to understand the physics of magnetic confinement; but we have done it. In the Apollo case, there were engineering problems whose solutions could not be fully tested. Could the nose cone material stand up to the heat of reentry? Can humans survive long periods without gravity and then the stress of reentry? Will micromete­orites puncture the space suits of the astronauts? It was a dangerous experiment, but President Kennedy pushed ahead, and it succeeded marvelously. In the case of fusion, we do not know yet how to build each part of a reactor, but the only way to get this ideal source of energy is to push on ahead. The expense will be comparable to Apollo’s, but at least no human lives are endangered.

The path to a commercial fusion reactor has been studied intensely in the past decade. There are three or four steps: (1) the ITER experiment now being built, (2) one or more large machines for solving engineering problems, (3) DEMO, a proto­type reactor built to run like a real reactor but not producing full power, and (4) FPP, fusion power plant, a full-size reactor built and operated by the utilities industry.

‘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_9, © Springer Science+Business Media, LLC 2011

Step 2 is being hotly debated. Some think that experiments on ITER will give enough information to design DEMO. Others propose intermediate machines designed to solve specific problems such as the tokamak wall material or the breed­ing of tritium. These problems are described in the main part of this chapter. The time it will take to reach the FPP stage might look something like this (Fig. 9.1). Any additional machines for engineering testing before designing DEMO are shown in Fig. 9.1, although they may not be necessary. Although this timeline is called the “fast track” to fusion, it still will take until 2050 before fusion power becomes a reality. The economic downturn at the turn of this decade has already delayed the construction of ITER. Shortening this timeline can be done only with greatly increased funding. In the meantime, expansion of the other renewable energy sources listed in Chap. 3 is still necessary.

The two toughest engineering problems are the material of the “first wall” and the breeding of tritium. These will be discussed in detail. We also mentioned some physics problems that are not completely solved. One concerns “disruptions” which kill the plasma and must be avoided in a reactor. The best known way to avoid them is to operate safely below the tokamak’s limits, and this means less output power. Otherwise, injection of a large puff of gas can stop an incipient disruption; this is a crude solution. A second problem concerns the edge-localized modes (ELMs), instabilities that dump plasma energy into places not designed to absorb it. Currently, internal correction coils are to be inserted inside the plasma chamber to suppress ELMs as well as resistive wall modes (RWMs). This is another crude solu­tion which would not be suitable in a reactor. A third problem concerns the alpha particles (the helium nuclei) which are the products of the D-T fusion reaction. These fast ions can, in theory, excite Alfven waves, and these electromagnetic waves could disrupt plasma confinement. This instability cannot be studied until we can ignite a plasma to produce these alpha particles.

Although these seem to be formidable problems, there will be a learning curve when ITER and DEMO are built. Once industry gets serious about fusion, progress will be rapid. We will go from Model-T Fords to Mercedes-Benzes. We will go from

DC-3s to Airbus A380s. We may even get lucky with more help from Mother Nature and find that fast alpha particles are stabilizing. Where there’s a will, there’s a way. With a positive attitude, the fusion community can continue to achieve and live up to its track record of the last 50 years. Further in the future, in the second half of this century, a second generation of fusion reactors will look quite different from the tokamak as described here. There are other magnetic configurations, simpler than the tokamak, that have not been fully developed for lack of funding. These are described in Chap. 10. Better yet, there are fuel cycles that do not require tritium, thus avoiding almost all of the fuel breeding and radioactivity problems of the first generation of fusion reactors. These advanced fuel cycles can run only with hotter and denser plasmas than we can now produce, but which may be possible once we have learned how to control plasma better. Advanced fuels are also presented in Chap. 10. The engineering problems described here are not the end of the story.

The COE depends on some factors that are independent of the power core and others that are specific to fusion. These factors appear in the following formula for COE,

which at first seems rather daunting. However, it is not necessary to know what the formula means in detail; it is used here just as a convenient way to show what affects the cost. The COE is proportional to the quantities in the parenthesis times those in the

rL Л °’6 1

AJ <5pe0A bN4 n03 denominator of the fraction following it. Inside the parenthesis, r is the discount rate, a financial factor similar to interest rate that will be explained later. L is a learning factor which takes into account that the first of a kind is always more expensive than the tenth one made. L starts at 1 and gets smaller with learning, so COE drops. A is the availability, which is the fraction of time the plant is running rather than shut down for repairs. Larger A means lower costs. The fusion reactor designs have A’s ranging from 60 to 80%.

The first two quantities in the denominator at the right have to do with the whole plant, and the last two concern the quality of the plasma in the tokamak. Eta — thermal (pth) is the efficiency of converting heat into electricity. Pe is the size of the plant in terms of electrical power produced. The larger the better because of econ­omy of scale. Beta-normalized (BN) expresses the efficiency with which the plasma current can confine a large amount of hot plasma by creating the right amount of twist in the magnetic field. Finally, N is the ratio of the plasma density to that pre­dicted by Greenwald limit (Chap. 8) for a stable plasma. In the different reactor models, r varies from 5 to 10%, L from 0.5 to 0.7, A from 0.6 to 0.8, Hth from 35 to 60%, Pe from 1 to 2.5 GW, and N from 0.7 (safe) to 1.4 (speculative). Most impor­tantly, BN varies from 2.5 to 5.5, representing the progression from well-established data to hopefully achievable advanced tokamak operation. Figure 9.42 shows the COE predicted from the PPCS models A-D as a function of the learning factor L.

As an example of how sensitive the COE is to assumptions made in the models, Fig. 9.43 shows how the availability factor A changes with the lifetime of the materials

in the first wall and blankets. The lifetime is expressed as the neutron fluence that the materials stand before they have to be replaced. The fluence is measured in years at an equivalent neutron energy flux of 1 MW/m2. The shorter the lifetime, the more often the blankets will have to be replaced, and hence the lower the availability. This then increases the cost (the higher blue points at the left).

An electric field pushes ions and electrons in opposite directions, and it makes sense that a steady electric field cannot confine a plasma. However, Bob Hirsch, who later headed the AEC’s fusion division, proposed a machine which has become popular with amateur fusioneers because of its simplicity. The device has two spherical grids one inside the other, the outer one grounded and the inner one at a large negative potential [51]. Gas is ionized between the grids, and ions are accelerated toward the center, where they accumulate and create a large posi­tive potential. Subsequent ions are repelled by this “virtual anode” and bounce

away back to the grids. They then oscillate inside the sphere and can collide with one another for an occasional fusion. This suffers from the original reason for a thermal plasma, as explained in Chap. 4. Streaming ions fuse only once in 10,000 collisions. The other collisions degrade their energies so that they no longer can fuse and eventually diffuse out of the system. Grids are OK for small experiments, but they will melt at fusion densities. Furthermore, Debye shield­ing at these densities will prevent the applied voltage from reaching the center of the holes in the mesh.

Migma

Early in the game, colliding accelerator beams were proposed, and several mig — matrons were built. With accelerators, it is easy to get ions up to the energy of the peak of the DT reaction, nearly 80 keV, or even the p-B11 reaction, nearly 300 keV. The beams are of low density, but they can be put into storage rings to circulate past the collision point many times. Elastic scattering, however, gener­ates bremsstrahlung radiation, and there are always instabilities with streaming particles. A comprehensive stability theory has never been worked out for migmas.