**2. Limits to high-current beam extraction**

The maximum current that can be extracted in an ion beam is determined by the space-charge limit, which for a cylindrical beam is approximately:

$$
\Pi = I \sqrt{\frac{\mathbf{M}}{2q}} = \left(\frac{\mathbf{4}\varepsilon\_0}{\mathfrak{P}}\right) V^{\frac{\mathbf{3}}{2q} \left(\frac{\mathbf{2}}{q^2}\right)}\tag{1}
$$

*DC Parallel Ribbon Ion Beams for High-Dose Processes DOI: http://dx.doi.org/10.5772/intechopen.111487*

where I is the ion current, M the ion mass, q the ion charge, ε<sup>0</sup> is the permittivity of free space, V the ion beam extraction voltage, g the gap across which this voltage is applied, and r the beam radius. The equation is exact for a beam originating on one plane surface and being accelerated to a second plane surface at a potential V. Since the beam passes through a hole in the second electrode, the potential at the center of the hole is less than V, so in in practice the beam radius r cannot exceed g. In this case, the equation is not exact, since r/g must be <1 in practice, and so it provides an upper limit to the maximum current. The quantity Π is usually called the beam poissance when used for heavy ions. However, for a ribbon beam this equation becomes:

$$
\Pi = I \sqrt{\frac{\overline{\mathbf{M}}}{2q}} = \left(\frac{4\varepsilon\_0}{9}\right) V^{\frac{\mathfrak{A}}{2}\left(\frac{\omega}{\varepsilon^2}\right)}\tag{2}
$$

where a is the beam thickness and b the ribbon breadth at the source aperture. Now the maximum is set by the condition a/g < 1. It is now possible for b to greatly exceed a without the potential at the center of the electrode being significantly different from V, and as a result the perveance from a ribbon-shaped ion beam system can exceed that from a cylindrical system by a factor of the order of b/a. Thus a ribbon beam system in the case of the Nova NV-10 high-current system can carry �44/3 times more current than a cylindrical beam.
