Magnetically confined D-T fusion

Figure 1.3: Schematic of a toroidal magnetic confinement system showing the basic magnetic geometry. The major and minor radii ($R$ and $a$) are labelled together with the poloidal and toroidal magnetic field ${B_{\mathrm{pol}}}$ and ${B_{\mathrm{tor}}}$. A toroidal plasma current $I_{\mathrm{pl}}$ may also be present

A scheme is needed for containing a D-T plasma well enough that the required temperatures for fusion can be sustained whilst simultaneously facilitating the extraction of excess energy on a continuous basis. The fact that charged particles moving in a magnetic field tend to gyrate in helical orbits about lines of magnetic force can be exploited to confine the reacting plasma by means of specially shaped magnetic fields. It became clear from the earliest attempts at confining plasmas with ``magnetic bottle'' configurations, that particle and energy losses are unacceptably high due to motion along non-closed field lines. Also, the plasma rapidly becomes polluted with eroded particles from any material surface in direct contact with it, further increasing energy loss due to atomic radiation, which scales with the square of the atomic charge.

In order to avoid both the losses due to particles moving along open field lines and contact with container walls, the field lines are arranged such that they occupy a toroidal volume in which they close upon each other. In this way, particles of mass $m$ and charge $q$ are free to move parallel to field lines but are constrained to follow helical orbits of Larmor or gyro-radius $\rho_\mathrm{L} = m v_\perp / q B$, with $B$ the magnetic field strength and $v_\perp$ the speed perpendicular to the field lines. The frequency of this perpendicular movement is known as the gyro-, Larmor or cyclotron frequency $\omega = e B / m$. Collisions lead to radial diffusion by transferring particles from one orbit to another; this occurs with a step-size of $\rho_\mathrm{L}$ according to classical diffusion theory and leads to inevitable leakage of plasma species and thus of energy. The problem of magnetically confining fusion plasmas has been likened to attempting to contain molasses with rubber bands.

The products of the D-T reaction are an $\alpha$-particle and an energetic neutron. The $\alpha$-particle carries away 20% of the energy and, being charged, remains confined by the magnetic field. It thus heats the plasma since its energy is quickly re-imparted to the bulk of the plasma through collisions with cooler particles. However, the magnetic field is transparent to the neutron, which escapes from the plasma but can be thermalized in a lithium blanket surrounding the plasma and its energy thus captured. This energy can subsequently be harnessed to generate steam much like in any other power station. The lithium blanket could also be used to breed tritium fuel using the escaping neutron, via the reactions:

$$\mathrm{n}_0^1 + \mathrm{Li}_3^6 \rightarrow \mathrm{He}_2^4 + \mathrm{T}_1^3 + 4.8 \mathrm{MeV}$$ (1.5)

$$\mathrm{n}_0^1 + \mathrm{Li}_3^7 \rightarrow \mathrm{He}_2^4 + \mathrm{T}_1^3 + \mathrm{n}_0^1 -2.5 \mathrm{MeV},$$ (1.6)

thus closing the fuel cycle. Lithium is a common element occurring stably in two isotopes as $\mathrm{Li}_3^6$ (7.5% abundance) and $\mathrm{Li}_3^7$ (92.5%). It is a soft, chemically reactive metal at room temperature whose low boiling point of roughly $180^\circ$ C renders it usable as a liquid metal coolant in a reactor.

To be of practical use, the fusion reaction must be self-sustaining, thus energy losses due to escaping particles and radiation etc. must be compensated by the $\alpha$-particles depositing their energy back into the plasma. This power balance is summarized in the so-called Lawson criterion [2]:

$$n \tau_\mathrm{E} > 1.5 \times 10^{20}~\mathrm{m}^{-3}\mathrm{s},$$ (1.7)

where $n$ is the particle density of the D-T fuel and $\tau_\mathrm{E}$ is the energy confinement time (the ratio of plasma stored energy to the power loss).

A containment device generating a simple solenoidal magnetic field (where field lines are circular and toroidally closed) would at first glance appear to be sufficient to confine a plasma. In fact, closer consideration of the particle orbits reveals that this is not so. In this configuration, the toroidal field varies as the inverse major radius, $1/R$. The Larmor radii of the particles thus differ as they gyrate, being always smaller in the region of higher field (towards the centre of the torus). This causes a charge-dependent particle drift, whereby electrons and ions move oppositely in the $\mathbf{B} \times \nabla B$ (i.e. vertical) direction. The resulting charge separation then gives rise to an electric field $\mathbf{E}$ which causes the plasma to drift out of the torus in the $\mathbf{E} \times \mathbf{B}$ direction.

This effect can be negated by supplying an additional poloidal field such that the field lines twist helically about the plasma column. A schematic drawing of such a toroidal confinement system is shown in figure 1.3. Plasma particles traveling along the field lines thus experience a $\mathbf{B} \times \nabla B$ drift velocity which changes sign as they move around the torus, the net drift effect thus averaging to zero. The way in which the magnetic field `twist' is generated forms a broad categorization of confinement devices.