<h1>Physical Principle of Tokamaks</h1>
<p>In a nucleus, strong force interaction overcomes repulsive electromagnetic interaction of positively-charged protons to keep the nucleus intact. For two nuclei to fuse into a single nucleus (and to release an excessive amount of energy in this process), they have to approach each other to a very small distance about 0.3 10<sup>-15</sup> m, within the reach of this interaction. However, in order for nuclei to get this close, a high potential barrier of repulsive electromagnetic interaction has to be overcome first. Term thermonuclear fusion describes situation where fusion fuel is heated to such temperatures that nuclei from the end of velocity distributive function (the fastest ones) have sufficient kinetic energy for overcoming this barrier (with just a little help from quantum tunneling effects). However, reaching temperatures necessary for ignition of fusion medium is not a trivial matter and it is necessary to prevent the fusion medium from escaping (expanding) to the walls and cooling itself upon contact with the wall of reactor. Before reaching thermonuclear temperatures, fuel gas becomes fully ionized and enters fourth state of matter called plasma. Since plasma consists of charged particles, it is possible to confine it in magnetic field. Currently, the most successful concept of magnetic confinement fusion reactor is tokamak.</p>
<img src="images/figs/tokamak_overall.jpg"/>
<p>The principal scheme of this device can be seen in fig. 1. Plasma is generated from a neutral gas (usually H<sub>2</sub> or its isotopes D<sub>2</sub> or T<sub>2</sub>) by a fast transformer action when a fast discharge is driven through primary transformer circuit coil. This coil can be seen in fig. 1, located in the middle of the tokamak. This discharge will induce a non-zero time derivation of magnetic flux through the central iron transformer core. This will result in generation non-zero rotation of electric field in toroidal direction (see fig. 1). As this electric field is generated, it ionizes the originally (mostly) neutral gas into plasma. This event is also characterized as <em>plasma brakdown</em>. Generated plasma inside of reactor chamber is then confined by a strong magnetic field. This is possible thanks to the fact that charged particles of plasma are forced to gyrate around the magnetic field lines by Lorentz force. This confining field is of toroidal direction (similarly to the breakdown electric field) and is generated by series of magnetic coils placed around the toroidal chamber of reactor (see fig. 1).</p>
<img src="images/figs/tokamak_above.jpg"/>
<p>However, such a field cannot be homogeneous in radial direction (outwards from the center). Fig. 2 represents tokamak and toroidal field coils with their currents as seen from above. Ampere’s law in form:</p>
<img src="images/eqs/ampere_eq_1.JPG" width='150'/>
<p>implies that magnitude of B decreases with 1/R. Such an inhomogeneous field gives rise to <em>gradient B drift</em>, accordingly to equation:</p>
<img src="images/eqs/grad_B_eq.JPG" width='150'/>
<p>Since previous equation depends on the charge polarity of particles, positively charged ions will drift to the top of the tokamak, while electrons to the opposite side. This charge separation would induce an additional electric field - see fig. 3. This electric field, together with primary toroidal magnetic field, will induce another drift of charged particles of plasma, as described in following equation:</p>
<img src="images/eqs/ExB_drift_eq.JPG" width='65'/>
<p>This drift is not a function of particle polarity anymore. Moreover, it is directed outwards from the center of toroid (see fig. 3). Plasma is thus driven towards outer wall (also called <em>low field side</em> or <em>LFS</em>) of the tokamak and hence the confinement is lost.</p>
<img src="images/figs/tokamak_poloidal.jpg"/>
<p>To prevent this, it is necessary to modify the abovementioned magnetic field inside of tokamak. The discharge is thus driven through primary transformer coil once again. However, unlike in the case of plasma breakdown, this time the discharge has slower time evolution and longer duration in order to keep the existence of toroidal electric field induced by this transformer action as long and steady as possible. Presence of this electric field will force plasma particles to propagate in the toroidal direction, while gyrating around magnetic field lines. Resultantly, an electric current will be flowing in the plasma column. Such a current will induce a magnetic field of poloidal direction (likewise a current flowing through conducting wire) and by superposition of this field with primary toroidal magnetic field, there is obtained a field of helical character (see fig. 1). As particles are driven in toroidal direction by the electric field, helical character of this magnetic field enables positively charged particles, accumulated on the top, to reach the negatively charged particles, which are accumulated on the bottom (and vice-versa). The accumulated charges on the opposite sides of the tokamak chamber will be thus short-circuited in this process and resultantly, induction of an unwanted electric field depicted in fig. 3 will be hindered.</p>
<p>It should be noted that particles can not be accelerated by toroidal electric field to infinite energies. As particle follows its trajectory, a possibility of its collision with adjacent particles exist. In their collision, the particles trade their energies and thus plasma is thermalized and gets one step closer to becoming a Maxwellian-distributed medium. Additionally, these collisions store energy obtained from electric field into thermal energy of plasma and thus heat the fusion medium. For purposes of quantifying this phenomenon, Joule-Lenz’s law could be expressed by following equation:</p>
<img src="images/eqs/joule_lenz_eq.JPG" width='70'/>
<p>where <em>j</em> is current density and <em>η</em> is resistivity of plasma. Since following expression applies for plasma resistivity:</p>
<img src="images/eqs/resistivity_eq.JPG" width='125'/>
<p>it is evident that Ohmic heating mechanism described above cannot heat plasma to temperatures necessary for ignition of fusion medium all by itself. Thus, other means of heating have to be implemented, such as neutral beam injection, cyclotron frequency heating, lower hybrid frequency heating etc.</p>
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