# Inertial electrostatic confinement

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.

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## Approaches to IEC

The most well-known IEC device is the Farnsworth-Hirsch Fusor.Template:Ref Its popularity is largely due to the fact that simple versions can be built for as little as $500 to$4000 (in 2003 U.S. dollars), making it accessible to hobbyists, science fair contestents 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). Fusors can also use ion guns rather than simply electric grids.

Other approaches replace the physical cathode with a virtual cathode formed by a concentration of electrons confined by magnetic fields.Template:RefTemplate:Ref

## Maximum pressure achievable

Most experts are skeptical that the IEC concept can ever be used for power production, and some are even skeptical of the basic concept. 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

[itex] -p' + \rho E= 0 [itex],

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

[itex] \epsilon_0 E' = \rho [itex].

Together these give

[itex] -p' + \epsilon_0 E' E= 0 [itex] or [itex] (p-p_0) = (1/2)\epsilon_0 E^2 [itex],

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). 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

[itex] E = (2p/\epsilon_0)^{1/2} = 230\,\mathrm{MV/m} [itex]

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

[itex] \delta \le \lambda_D = \sqrt{\frac{\epsilon_0 k T}{n q^2}} [itex].
[itex] p = nkT \le \epsilon_0 (kT/q)^2 / \delta^2 [itex].

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 submillimeter scale.

## 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).

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