H- beamlet

The equations of beamtracer.optimization.beamlet, declared on the code that implements them and checked by the tests.

`NegativeIonBeamlet`

The fixed settings of a round H- beamlet, and its problems.

objective: $\theta_\mathrm{rms} \text{ of the H}^- \text{ at the exit plane, half the drift behind the GG}$
loss: $\text{loss} = \max\left(0, 1 - \frac{I_{\mathrm H^-}}{I_\mathrm{emit}}\right) - \text{max\_loss}, \qquad I_\mathrm{emit} = \pi (r_\mathrm{PG} - \delta)^2 J$
surface field: $\text{surface\_field} = \frac{\max(E_\mathrm{EG}, E_\mathrm{GG})}{\text{field\_limit}} - 1$
gap field: $\text{gap\_field} = \frac{\max\left(V_\mathrm{EG} / g,\ (V_\mathrm{GG} - V_\mathrm{EG}) / g_\mathrm{acc}\right)}{\text{gap\_field\_limit}} - 1$
divergence limit: $\text{divergence\_limit} = \frac{\theta_\mathrm{rms}}{\text{divergence\_limit}} - 1$

Assumes

the constraints are met when $\le 0$ ; the PG's knife-edge field is reported, not constrained
the H $^-$ current density is an input (the source sets it); $\delta$ = emission_margin
an unconverged Vlasov iteration (a limit cycle where the beam grazes the EG) keeps its objective but ranks behind every converged design

Built from

propertysource

Return the Child-Langmuir current density [A/m^2] of H- over the PG-EG gap.

child langmuir: $J_\mathrm{CL} = \frac{4 \varepsilon_0}{9} \sqrt{\frac{2 e}{m}}\, \frac{V^{3/2}}{d^2}$

Assumes

a planar diode of H $^-$ (1.00849 u) starting at rest; $V$ the extraction voltage, $d$ from the PG's upstream face to the EG's
a diagnostic: perveance_ratio $= J / J_\mathrm{CL}$

References