At the simplest level, a mirror configuration consists of a pair of Helmholtz coils with currents flowing in the same direction (as shown in Figure B.6).
Magnetic field intensity varies along B with a minimum value Bmin at the middle and a maximum value Bmax at the coil location. Confinement in the simplest mirror configuration is described by the conservation of energy E = mv2/2 and of the first adiabatic invariant (the magnetic moment p = mv?/2B, v? being the particle velocity perpendicular to B) of a particle of mass m moving in a weakly inhomogeneous magnetic field B. Charged particles spiral around the B field lines at a distance called "the Larmor radius''. These conservation laws imply that a particle moving along the field (with velocity V||) is reflected at the plasma location where mv|/2 — E — pB = 0.
Therefore, upon producing a magnetic field configuration such as that shown in Figure B.6, particles will be trapped provided that the ratio p/E is larger than 1 /Bmax.
It can be shown that, in the case of an isotropic particle distribution function in velocity space, the fraction fT of plasma particles satisfying the trapping condition is given by fT « (1 - 1/R)1/2 with R = Bmax/Bmin, the so-called "mirror ratio''. Particles not satisfying this condition will be promptly lost, with the result of producing an anisotropic distribution function characterized by a "loss-cone" in velocity space. For large values of the mirror ratio, the fraction of unconfined particles is given by 1 - fT « (1/2)R. Obviously, the fraction of unconfined particles can be made smaller if they are injected in the configurations with small parallel velocity (e.g., by perpendicular neutral beam injection). On the other hand, collisions tend to restore isotropy and the loss-cone is continuously populated by scattering in velocity space.
Since electrons have a higher collision frequency than ions, they are scattered in the loss-cone (and therefore lost) at a higher rate. As a consequence, the plasma tends to be positively charged. Its potential, is determined by the condition that transport must be ambipolar (i.e., that overall charge neutrality must be maintained), yielding values in the range ~ 4 - 8Te. The effect of ambipolar potential is that of decreasing the loss of low-energy electrons and increasing ion loss.
As a result, in such a simple configuration confinement is maintained on the ionion collision timescale rii (the timescale for the scattering of a trapped ion into the loss-cone). The ion-ion collision time is proportional to E3/2, with Et the ion energy; therefore higher values of the confinement are achieved by increasing Ei. On the other hand, fast ions tend to transfer their energy by Coulomb-driven collisions preferentially to electrons if Ei > 15Te. If the electron temperature Te is too low, the slowing down of injected ions by the electrons (electron drag) occurs on a fast timescale rSD / TJ2/ne. Thus, electrons must be kept at sufficiently high temperature.
To achieve high electron temperature in an open-ended configuration might appear at first sight a very difficult task. Simple considerations based on classical fluid transport theory would predict very large electron thermal conduction (and therefore very high heating power to keep the electrons at a sufficiently high temperature). However, in experiments characterized by low collisionality (i.e., a mean free path longer than the mirror distance), the electron thermal conductivity along the magnetic field is much lower than the classical estimate. This result is a consequence of the presence of the ambipolar potential ^ that confines the electrons inside the mirror. Only supra-thermal (nonequilibrium) electrons can escape the barrier and contribute to thermal conduction. This has the effect of a dramatic reduction in electron thermal conductivity, at the expense of low plasma density and thus large size of the device (at fixed power).
The nr parameter can be estimated by solving the Fokker-Planck equation accounting for the presence of the ambipolar potential and the electron drag. It can be shown [Post, 1987] that the confinement parameter is approximately given by nr « 2 . 5 x 1016Ei(keV)3/2 log10(R) m-3 s (B . 28)
Note that dependence on the mirror ratio R is only logarithmic, and that the above expression is independent of size and magnetic field. In order to obtain a significant gain, values of Ei in the range of a few hundred kiloelectronvolts are needed. However, above a certain energy fusion cross-sections tend to decrease (at 100 keV for the D-T, and 400 keV for D-3He in the center of the mass frame): therefore, an optimal value exists for ion energy.
All these constraints limit efficient energy production by the simple mirror configuration. Indeed, at the simplest level a mirror reactor works as an energy amplifier. Power is injected through high-energy neutral beams and fusion power is recovered with gain Q = Pfus/Pinj, with Q given by
with Efus the energy released by the fusion reaction. The nr scaling above implies for a simple mirror configuration (using D-T, R = 10, Ei « 300 keV) values of Q « 1, too low even employing advanced techniques for electricity production such as direct conversion. Even lower values (Q « 0.3) are obtained for the D-3He reaction.
In addition to its low gain, the simple mirror configuration has limited MHD stability properties due to the presence of "interchange" instabilities in the region between the mirrors: indeed the exchange of a plasma flux tube with a vacuum flux tube is energetically favorable if the local magnetic field curvature K (K = b-Vb), with b = B/B) is parallel to the pressure gradient, as in the central part of the mirror cell (the opposite occurs near the mirror points; see Figure B.6). The instability is suppressed by superimposing a multipolar field to produce a so-called "minimum-B'' configuration in which a "magnetic well" is produced around the symmetry axis. The demonstration of the stability of minimum-B configurations was achieved in modified mirror systems called "baseball", or Ying-Yang, coils (shown in Figure B.7). Unfortunately, breaking axial symmetry, a multipolar component superimposed to the axisymmetric mirror field has a detrimental effect on radial particle
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