Inertial electrostatic confinement

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Inertial electrostatic confinement (often abbreviated as IEC) is a concept for retaining a plasma using an electrostatic field. The field accelerates charged particles (either ions or electrons) radially inward, usually in a spherical but sometimes in a cylindrical geometry. Ions can be confined with IEC in order to achieve controlled nuclear fusion.

1 Approaches to IEC
2 Maximum pressure achievable
3 References
4 External links

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Approaches to IEC

Image:US3386883 - fusor.png

The best-known IEC device is the Farnsworth-Hirsch Fusor.Template:Ref This system consists largely of two concentric spherical electrical grids inside a vacuum chamber into which a small amount of fusion fuel is introduced. Voltage across the grids causes the fuel to ionize around them, and once ionized (and thus charged) it is accelerated towards the center of the chamber due to the voltage. This constant "inrush" of fuel ions keeps the hot plasma confined in the center of the system. Fusors can also use ion guns rather than electric grids.

The fusor's popularity is largely due to the fact that simple versions can be built for as little as $500 to $4000 (in 2003 US dollars), making it accessible to hobbyists, science fair contestants and small universities. Even these simple devices can reproducibly and convincingly produce fusion reactions, but no fusor has ever come close to producing a significant amount of fusion power. They can be dangerous if proper care is not taken because they require high voltages and can produce harmful radiation (neutrons, gamma rays and x-rays).

Two newer approaches both try to solve a problem found in the fusor, pollution of the plasma area due to collisions on the grids which spray high-mass ions into the reaction chamber and cool the fuel. The Polywell uses a magnetic field to trap a quantity of electrons, fuel ions are then accelerated directly into the middle where they are trapped by the electron cloud that forms a "virtual electrode". Another modern approach uses a Penning trap to trap electrons in a system otherwise similar to the Polywell.Template:Ref Template:Ref

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Maximum pressure achievable

Although portable neutron sources based on the IEC concept work and are comercially available, most experts are skeptical that the IEC concept can ever be used for power production. Most discussions of IEC consider the behavior of a small number of ions in potential structures imposed by electrodes. A potential well for ions, however, is a potential hill for electrons, so it is not possible to contain a neutral plasma with any set of electrodes. There must be at least some regions where the charge density of one species or the other dominates. As the density in these regions is raised, at some point the net charge density will destroy the potential well.

In pure IEC, the pressure gradient will be balanced by the electric force on the net charge density. In one dimension this is

<math> -p' + \rho E= 0 </math>,

where p is plasma pressure and ρ is charge density. Gauss's law relates ρ to E as

<math> \epsilon_0 E' = \rho </math>.

Together these give

<math> -p' + \epsilon_0 E' E= 0 </math> or <math> (p-p_0) = (1/2)\epsilon_0 E^2 </math>,

where p₀ is a constant of integration, equal to zero if the electric field vanishes when the density does (which minimizes the electric fields and potential drops for a given density).Template:Ref Note the similarity to the concept of magnetic pressure, B²/2μ₀, from magnetic confinement fusion. This arises from the symmetry of the Maxwell stress tensor with respect to E and B, with the change of sign being due to the fact that the gradients are parallel to E but perpendicular to B. For comparison, a D-T tokamak reactor would operate at about n = 10²⁰ m^-3 and T = 10 keV, which gives an ion pressure of p = (3/2)nkT = 0.24 MPa. Reaching the same pressure in an IEC reactor would require an electric field at the electrode of

<math> E = (2p/\epsilon_0)^{1/2} = 230\,\mathrm{MV/m} </math>

If we assume very generously that, say, 1 MV/m could be maintained at the surface of an electrode in a fusion environment, then an IEC reactor would be a factor of 230² worse than a tokamak in terms of both power density and Lawson criterion.

To find the spatial dependence of the pressure outside the electrode we need to relate the pressure to the charge density. The simplest case is to take a single species (ions or electrons) at a uniform temperature, ρ = nq = pq/kT, and to take p₀ = 0. The result is p(x) ~ x^-2.

Aside from the achievable electric field strength another factor limiting the density in an IEC device will be the fact that the ions must pass through holes in the electrode, and these holes must be smaller than the Debye length. Otherwise, the potential of the electrode will be dropped in the Debye sheath around the hole and will not be available to confine the ions. If the scale of the holes is δ, then we have

<math> \delta \le \lambda_D = \sqrt{\frac{\epsilon_0 k T}{n q^2}} </math>.

<math> p = nkT \le \epsilon_0 (kT/q)^2 / \delta^2 </math>.

The result has the same form as the previous result, but with the electric field at the electrode replaced by (kT/q)/δ. To achieve 1 MV/m with T = 100 keV would require δ no larger than 10 cm. To achieve confinement comparable to a tokamak would require a value 230 times smaller, namely 0.4 mm. Survival of a material grid in contact with a fusion plasma would be a tremendous problem anyway but is unthinkable if it must be structured on a sub-millimeter scale.

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References

Template:Note R. Hirsch, "Inertial-Electrostatic Confinement of Ionized Fusion Gases," Journal of Applied Physics 38, 4522 (1967).
Template:Note R.W. Bussard, "Some Physics Considerations of Magnetic Inertial-Electrostatic Confinement: A New Concept for Spherical Converging-flow Fusion," Fusion Technology 19, 273 (1991).
Template:Note D.C. Barnes, R.A. Nebel, and L. Turner, "Production and Application of Dense Penning Trap Plasmas," Physics of Fluids B 5, 3651 (1993).
Template:Note L.P. Block, "Potential Double Layers in the Ionosphere", Cosmic Electrodynamics 3, 349 (1972)