The Physics of Energy, page 77
In steady state the power deposited in the plasma by α particles and by external heating must balance the power that the plasma loses to the surroundings,
(19.20)
The potentially useful thermal power output of the plasma fusion reactor receives contributions from , and . The ratio of the net thermal power output to the power supplied by external heating is known as the fusion gain factor Q,
(19.21)
where we have used the steady-state condition (19.20). The fusion gain factor is thus the total power density produced by fusion compared to the heating that must be externally supplied .
The contributions to the plasma fusion reactor power balance can be described in more detail as follows:
Power added by α particles The power density produced in and delivered to the plasma by α particles is obtained by multiplying the fusion reaction rate by the α particle energy . According to eq. (19.4), the fusion reaction rate per unit volume in a gas consisting of deuterons and tritons with densities and equals . The fusion process depends upon tunneling and therefore the cross section increases rapidly with energy. The product is averaged over the Boltzmann distribution at temperature T, as denoted by and is dominated by the Gamow window (§18.4.1). The resulting dependence of on is shown in Figure 19.13 for the dt reaction as well as for dd and d. The power density added to the plasma by α particles is then
(19.22)
where we have set . Note that grows rapidly with temperature between and 25 keV, which is the temperature range relevant for MCF.
Figure 19.13 for dt, d, and dd fusion as a function of (in keV). After [100].
Neutron power density The power density of the neutrons produced by dt fission is four times as large as . This power escapes the plasma and is deposited on the material that surrounds the reactor.
Power lost by radiation The radiative power lost by the plasma is dominated not by thermal radiation, but instead by the radiation emitted when electrons accelerate during collisions with the positive ions in the plasma. This form of radiation, known as bremsstrahlung (German for braking radiation), is dominantly in the ultraviolet or soft X-ray part of the spectrum, and it escapes from the plasma. Bremsstrahlung is described further in §20.2.2. This radiation is absorbed in the material that surrounds the reactor.
Because it originates in two-body collisions, , like and , is proportional to the plasma density squared. Explicit calculation shows that grows with temperature proportional to ,
(19.23)
where Wm3/(keV) [90]. In the temperature range of interest, 5 keV 25 keV, grows much more slowly with temperature than .
Power loss by conduction The energy lost to the surroundings via conduction can be parameterized as a fraction of the energy density in the plasma, which is given by in the ideal gas approximation, so
(19.24)
The (reciprocal) coefficient of proportionality is known as the energy confinement time, and measures the time over which the plasma would lose an appreciable fraction of its thermal energy by conduction. The parameters and must be regarded as phenomenological because, among other reasons, conductive losses do not scale with volume.
External heating Finally, in order to maintain its high temperature, some additional power may be supplied to the plasma. In present experiments this power is supplied by external sources, although in principle it would ideally be obtained from the energy delivered to the surroundings by neutrons, conduction, and radiation. The power density supplied by this external heating is parameterized by .
dt Fusion Power Balance
In a steady-state fusion reaction, the fusion gain factor is defined as the ratio of the net thermal power output of the system to the power supplied to heat the plasma,
is termed breakeven and defines ignition. A fusion reactor need not reach ignition to provide useful thermal power. must, however, be much greater than one in order to allow for losses and for reactor power systems.
Equation (19.21) describes the fusion gain factor as the ratio of fusion power output to power input. In a real fusion power system, the net useful thermal output would be further reduced by losses and by the power required to operate the system that confines the plasma. Thus the fusion gain factor is an upper limit on the engineering gain factor , which would include these other effects. The performance of experiments in controlled fusion is nevertheless usually gauged by the value of the fusion gain factor. is (somewhat arbitrarily) termed breakeven and the limit , in which no external power is needed to sustain the fusion reaction, is known as ignition. Note that it is not necessary for a controlled fusion device to reach ignition in order to provide useful thermal power. In an ideal world without losses, useful power could even be extracted from a system with . In practice, however, would have to be much greater than one in order to allow for losses and for the power necessary to operate the reactor systems.
As of 2017 no controlled fusion device has yet reached breakeven. The most ambitious MCF experiment, the Joint European Torus, or JET, in Great Britain has reached a fusion gain factor of for a brief period [101]. The new International Thermonuclear Experimental Reactor, or ITER, project under construction in France aims for [101].
19.6.2 Fusion Performance Criteria
Plasma density and temperature must be high to produce nuclear fusion. To sustain a fusion reaction, the rate of thermal energy transfer out of the plasma, parameterized by , must also be kept low. The Lawson criterion, first defined by British engineer and physicist John D. Lawson in 1955, is a combination of these parameters that is relatively easy to measure, and which serves as a useful benchmark on the path to a practical fusion reactor. More recently the Lawson criterion has been at least partially superseded by another closely related benchmark known as the triple product or fusion product. Although they can be adapted to any fusion reaction, we describe these performance criteria for the case of the dt fusion reaction.
Both the Lawson and triple product criteria estimate the conditions required for ignition, when the plasma increases in temperature or can be maintained in steady state (eq. (19.20)) with no external heating,
(19.25)
Substituting from eqs. (19.22)–(19.24), and using the ideal gas law, , to eliminate n, we obtain
(19.26)
Figure 19.14 shows a plot of the right-hand side of eq. (19.26) as a function of temperature. The red curve includes the contribution from bremsstrahlung losses – the second term in the denominator – the purple curve does not. Ignition is not possible until the temperature exceeds . The minimum value of on the curve occurs at approximately 15 keV, where atm s. Thus a plasma confined at a pressure of 8.3 atm with an energy confinement time of approximately one second can in principle achieve ignition.
Figure 19.14 The lower limit on the product of pressure energy confinement time for ignition (19.26), with and without the contribution of bremsstrahlung losses. After Freidberg [90].
From Figure 19.14 it is apparent that bremsstrahlung losses can be ignored for keV. In this case eq. (19.26) simplifies to
(19.27)
where we again have used the ideal gas law to replace p by . Equation (19.27) is the triple product criterion. The Lawson criterion is obtained by canceling the common factor of T on both sides of the equation,
(19.28)
Clearly the two criteria (19.27) and (19.28) are equivalent – if one is satisfied, so is the other. The triple product owes its ascendence in usage to the convenient fact that is approximately proportional to over the interval keV, so that the right hand side of eq. (19.27) is roughly constant over that interval. Also, the maximum pressure achievable in an MCF reactor, , is roughly constant even though n and T can separately vary significantly [90].
Using the data of Figure 19.13, the triple product is plotted in Figure 19.15 for dd and as well as for dt fusion. The relative difficulty of reaching ignition for these three different fusion fuels can be seen in this figure. Ignition in a dd fusion reactor, for example, would require a product of pressure energy confinement time roughly two orders of magnitude greater than that required for ignition of a dt plasma.
Figure 19.15 The triple product as a function of temperature for dt, dd, and d fusion. (Credit: Dstrozzi reproduced under CC-BY-SA 3.0 license via Wikimedia Commons)
Fusion Performance Criteria
To reach ignition, the product of density n, temperature T, and energy confinement time must be large enough to sustain the plasma without external heating,
This triple product criterion is a benchmark for the performance of plasma fusion reactors. The triple product criterion has largely supplanted the older, closely related Lawson criterion.
Comparison of the value of the triple product for dt, dd, and (Figure 19.15) shows why deuterium–tritium fusion is the reaction of choice for research into controlled nuclear fusion.
19.6.3 Magnetic Confinement Fusion
A hot plasma consists of negatively charged electrons and positively charged nuclei. If the plasma is to reach the temperatures needed for dt fusion ( ), it must be kept out of contact with any normal matter. The goal of magnetic confinement fusion (MCF) devices is to confine the charged particles in a hot plasma with the use of suitably shaped magnetic fields. In this section we explore how this can be accomplished for individual particles moving in fixed fields. We ignore interactions among particles, as well as radiation and all of the other systems required to confine and stabilize a burning plasma. A thorough overview of MCF at a level not far beyond the level of this book can be found in [90].
The Lorentz force law (3.45),
(19.29)
dictates that the magnetic force on a particle of charge q is perpendicular both to its velocity and to the magnetic field. In a constant magnetic field, directed in the z-direction the force is in the xy-plane, causing the particle to move in a circular orbit with radius in the xy-plane (Problem 19.13) as it travels steadily in the -direction. The result is a helical trajectory, winding along or opposite to the direction of B. In a strong (multi-Tesla) magnetic field, particles with energies of order 10 keV move in orbits with radii r that are small compared to the macroscopic dimensions of a fusion reactor (Problem 19.13). Particles of opposite charge circulate along helices of opposite handedness, as shown in Figure 19.16. Thus, a constant magnetic field does an excellent job of confining both positively charged nuclei and electrons in the directions perpendicular to the field lines. The charged particles are not, however, confined in the direction parallel to B.
Figure 19.16 Charged particles move in helical orbits in a constant magnetic field. Positive charges (red) orbit opposite to negative charges (blue) and the radii of the orbits are proportional to the square root of the particle's energy. The vectors v and F are shown at one point of the particle's trajectory.
In the early days of MCF research, many options were considered to confine the plasma motion in the direction along the magnetic field lines. A Russian design in which the magnetic field lines are wrapped into a torus, or donut shape, has emerged as the most promising option. Known as a tokamak (a Russian acronym for toroidal chamber with magnetic coils), this design is characterized by a plasma and toroidal magnetic field that are axisymmetric around the torus (Figure 19.18(a)). A cutaway diagram of the JET, the largest tokamak operated to date, is shown in Figure 19.17. Coils wound around the torus produce a strong magnetic field – 3.45 T at JET, 10 T planned at ITER – that forms a closed ring.
Figure 19.17 A cutaway diagram of the Joint European Tokamak (JET). (Credit: Contains public sector information licensed under the Open Government License v2.0)
Figure 19.18 Charged particle motion in a toroidal magnetic field. (a) Magnetic field circulates around the toroid and decreases in strength with distance from the axis. (b) With increasing toward the center of the torus, orbits have smaller radius of curvature toward the toroid axis and larger toward the outside, as a result positive (negative) particles drift down (up). (c) In the resulting perpendicular E and B fields, the radius of curvature at the top (bottom) of the orbit is larger for positive (negative) charge, so particles of both charges drift to the right.
Unfortunately, once it has been bent into a toroidal shape, the magnetic field can no longer be uniform. Instead, as dictated by Ampere's law (3.48), the strength of the field falls like , where ρ is the distance from the central axis of the torus (Figure 19.18(a)). In the presence of only the non-uniform toroidal field, the orbit of a particle in the plane perpendicular to the magnetic field would have a smaller radius of curvature () where is stronger, close to the axis of the toroid, and a larger radius of curvature toward the outside of the toroid. As a result, in a toroidal magnetic field that goes clockwise when viewed from above, the positive ions in the plasma would drift down and the electrons would drift up, as illustrated in Figure 19.18(b). The resulting separation of charge would give rise to an electric field in the vertical direction, which opposes the drift. This is not the end of the story, however. A positive charge orbiting in the plane perpendicular to B would then speed up, due to the electric field, near the top of its orbit when it is moving radially outward and slow down near the bottom where it is moving inward. The result, shown in Figure 19.18(c), is that positive ions would drift toward the outer edge of the torus. This phenomenon is known as drift (see Box 27.2). As the name indicates, it is driven by crossed electromagnetic fields and its direction is perpendicular to both E and B.5 Electrons would drift the same way as the positive ions (see the figure). Thus, both ions and electrons would drift outward, allowing the plasma to leak out of the device.
To prevent drift and stabilize the plasma, tokamaks must also have an induced current that runs around the torus. This current generates a further contribution to the magnetic field, known as the poloidal magnetic field. This field circulates around the smaller cross-sectional circle of the torus, wrapping around the toroidal current, as shown in Figure 19.19. The two components of the magnetic field, toroidal and poloidal, combine to form a magnetic field whose field lines wind helically around the center of the torus (Figure 19.19). Because of this helical twist, the sidewards drift experienced by the charged particles, as they spiral around the field lines, is alternately directed toward and away from the axis of the torus, and cancels out on average. This can be understood by considering the motion of a charged particle in the poloidal field alone – with an initial outward velocity, the particle will follow a helical path centered about a circular poloidal field line. The charge separation that would generate the vertical electric field and spell the death of the plasma in the case of a purely toroidal field thus never develops.
Figure 19.19 Schematic diagram showing the toriodal and poloidal magnetic fields in a tokamak, and the resulting helical field in the plasma. Also shown are the coils that produce the toriodal field and the transformer that induces the plasma current, which in turn gives rise to the poloidal field. (Credit: Image adapted from EUROfusion, www.euro-fusion.org)
Since the plasma has a non-vanishing resistivity, the current that produces the poloidal field also serves to heat the plasma, which is a desirable side-effect. Resistive losses, however, cause the current to die out if it is not constantly replenished. In present plasma fusion experiments, the current is primarily generated inductively by a toroidal EMF produced by ramping the current I in the windings of a central transformer; by Lenz's law, . The plasma current and the poloidal magnetic field that it generates can be sustained only as long as the current in the transformer is changing. This forces present tokamaks to operate in a pulsed mode, in which the transformer current is ramped from a maximum in one direction to a maximum in the opposite direction. At the end of the pulse the poloidal field vanishes, and plasma confinement is completely lost. The plasma must be re-established and the transformer current must be reset before the next pulse.
Steady-state operation of a magnetic confinement fusion reactor would require that the plasma current be maintained in other ways, several of which are actively under investigation. Certain antenna structures can launch waves from the plasma edge that propagate toward the core and transfer a net momentum to the electrons, resulting in the generation of current. Alternatively, complex interactions between the ions and electrons, as well as asymmetries in the particle drift directions, can give rise to an inherent plasma current, though these features have yet to be used effectively to drive a significant current.
In a burning plasma, fusion itself supplies the energy to keep the plasma hot. Below ignition, however, some heat must be supplied from other sources. Resistive heating from the plasma current is one source. Others include injecting high-energy beams of neutral particles, and bathing the plasma in intense microwave electromagnetic radiation.
In addition to all the challenges already described, tokamak plasmas are susceptible to internal instabilities, some of which are potentially disastrous for large machines where the engineering tolerances are already extreme. Many of these issues are related to the plasma response in the presence of a very energetic particle population, such as the particles created by the fusion reactions. Finally, particularly at the relatively low pressures used in current practice, the energy density of the dt fuel is about times smaller compared, for example, to natural uranium fuel for a fission reactor. Thus very large systems or high throughput of the fuel would be needed to achieve power at the gigawatt scale.
