One of the most effective means of plasma confinement to date involves the use of magnetic fields. A particle of charge q and mass m located in a magnetic field of local flux density В is constrained to move according to the Lorentz force given by

dts

m— = g(vxB) (4.9)

dt

with v the velocity vector.

For the case of a solenoidal В-field produced by a helical electrical current with density j, Fig. 4.2a, ions and electrons will move (as will be shown in Ch. 5)-depending upon the initial particle velocity-either parallel or antiparallel to the В-field lines and spiral about them with a radius of gyro-motion of

(4.10)

Scattering reactions may, however, transport them out of these uniform spiral orbits with two consequences: they may be captured into another spiral orbit or they may scatter out of the magnetic field domain. In the absence of scattering they are essentially confined as far as directions perpendicular to В are concerned, traveling helically along В with an unaffected velocity component parallel to B. Subsequently, they will eventually depart from the region of interest.

Solenoidal fields belong to the oldest and most widely used magnetic confinement devices used in plasma physics research. The ion and electron densities may clearly be enhanced by increasing the magnetic field thereby also providing for a smaller radius of gyration. There exists, however, a dominant property which renders their use for fusion energy purposes most detrimental: for

solenoidal dimensions and magnetic fields generally achievable, fuel ion leakage through the ends is so great that these devices provide little prospect for use as fusion reactors.

b) Mirror B-field

If, however, the magnetic field strength is increased specifically at each end of the cylindrical region, i. e. the B-field lines appear to be substantially squeezed together at the ends, Fig. 4.2b, the number of leaking particles is considerably reduced. Such a squeezed field configuration is referred to as a magnetic mirror since it is able to reflect charged particles, as will be shown in Sec. 9.3. We note that ions and electrons possessing excessive motion along the magnetic axis will still penetrate the magnetic mirror throat.

The attractiveness of the mirror concept not withstanding, a magnetic mirror can thus not provide complete confinement and-as in all open-ended configurations-is associated with unacceptably high particle losses through the ends. Hence, to avoid end-leakage entirely, the obvious solution is to eliminate
the ends by turning a solenoidal field into a toroidal field, Fig. 4.2c. The resultant toroidal magnetic field topology has spawned several important fusion reactor concepts; the most widely pursued of such devices is known as the tokamak, which will be discussed in Ch. 10.

At first consideration, the charged particles could be viewed as simply spiraling around the circular field lines in Fig. 4.2c, not encountering an end through which to escape. Any losses would have to occur by scattering or diffusion across field lines in the radial direction causing leakage across the outer surface. Further, we mention that collective particle oscillations may occur, thereby destabilizing the plasma.

One important plasma confinement indicator is the ratio of kinetic particle pressure

Pkin= NikTi+ NekTe (4.11a)

to the magnetic pressure

<4»b)

with |i0 the permeability of free space. This ratio is defined as the beta parameter, P, and is a measure of how effectively the magnetic field constrains the thermal motion of the plasma particles. A high beta would be most desirable but it is also known that there exists a system-specific PmaX at which plasma oscillations start to destroy the confinement. That is, for confinement purposes, we require

Thus, the maximum plasma pressure is determined by available magnetic fields thereby introducing magnetic field technology as a limit on plasma confinement in toroids.

In order to assess the fusion energy production possible in such a magnetically confined, pressure-limited deuterium-tritium plasma, we introduce Eq.(4.12), with the equality sign, into the fusion power density expression

Pfu~ Nd Nt Gv >dt Qdt ^4 J 3)

€a*B4 64 ft

to determine-for the case of Nd = Nt = N/2, Ni = Ne and T, = Te the magnetic pressure-limited fusion power density

which is displayed in Fig. 4.3 as a function of the plasma temperature.

Another overriding consideration of plasma confinement relates to the ratio of total energy supplied to bring about fusion reactions and the total energy generated from fusion reactions. Commercial viability demands

Е/и» Emppiy • (4-15)

Most toroidal confinement systems currently of interest are expected to operate

in a pulsed mode characterized by a bum time Xb. Assuming the fusion power to be constant during xb, or to represent the average power over that interval, the fusion energy generated during this time is given by

E fu= P fU’ Tb=<cTV >dt Nd NtQdtTb (4-16)

for d-t fusion. For perfect coupling of the energy supplied to an ensemble of deuterons, tritons, and electrons, we have

Esupply = jNdkTd+lNtkTt +1NekTe. (4.17)

One may therefore identify an ideal energy breakeven of Efu = ESUppiy as defined

by

< CTV >dt Nd N, Qdt tb = j NdkTd +1 N, kTt + f NekTe (4.18)

12kT <<*>d, Qdt

Imposing Nd = N, = N/2, Ni = Ne, and Td = T, = Te = T we may simplify the relation to write for an ideal breakeven condition:

Since Q* = 17.6 MeV and <Gv>d, is known as a function of kT, we readily compute the product
for kT = 12 keV. Note that for fusion bum to be assured over the period xb, it is required that the actual magnetic confinement time, Tmc > Xb. The incorporation of other energy/power losses as well as energy conversion efficiencies suggests that this ideal breakeven is an absolute minimum, and operational pulsed fusion systems need to possess a product much higher. For instance, the accelerated motion of charged particles inevitably is accompanied by electro-magnetic radiation emission, as described by Eq.(3.39), which constitutes power loss from a confined plasma.

There exist various versions of Eq. (4.19)-depending upon what energy/power terms and conversion processes are included; collectively they are known as Lawson criteria in recognition of John Lawson who first published such analyses.

Problems

4.1 Verify the expression for the sonic speed in a compressed homogeneous deuterium-tritium ICF pellet, given in Eq. (4.4), by evaluating the defining equation v2 = dp/dp with p denoting the mass density of the medium in which the sonic wave is propagating. Assuming the sound propagation is rapid so that heat transfer does not occur, the adiabatic equation of state, pp’Y = C, is applicable to the variations of pressure and density. Determine the constant C using the ideal gas law. To quantify y, which represents the ratio of specific heats at fixed pressure and volume, respectively, c/cy, use the relations cp — су = R (on a per mole basis) and су = fR/2 known from statistical thermodynamics, where R is the gas constant and f designates the number of degrees of freedom of motion (f = 3 for a monatomic gas, f = 5 for a diatomic gas,…).

4.2 Compute vs of Eq.(4.4) for typical hydrogen plasmas and compare to sound propagation in other media.

4.3 Re-examine the specification of a mean-time between fusions, Sec. 4.5, for the more general case of steady-state fusion reactions. How would you define, by analogy, the mean-free-path between fusions, X*?

4.4 If 10% bumup of deuterium and tritium is achieved in a compressed sphere with Rb = 0.25 mm and fulfilling the bum condition of Eq. (4.8) with Ni>0Rb = 5 x 10‘28 m’2, at what rate would these pellets have to be injected into the reactor chamber of a 5.5 GWt laser fusion power plant? The pellets contain deuterium and tritium in equal proportions. Power contributions from neutron — induced side reactions such as in lithium are to be ignored here.

4.5 Assess the minimum magnetic field strength required to confine a plasma having Nj = Ne = 1020 m"3 and Tj = 0.9 Te = 18 keV, when [3max = 8%.

4.6 Compute the maximum fusion power density of a magnetically confined d-h fusion plasma (Nd = Nh, Td = Th = Te, no impurities) as limited by an assumed upper value of attainable field strength, В = 15 Tesla, dependent on the plasma temperature. Superimpose the result for pmax = 2% in Fig. 4.3.