Beam optics

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

`BeamOptics`

Steady ion beams in planar or cylindrical geometry.

poisson: $\nabla^2 \phi = -\frac{\rho_b + \rho_p}{\varepsilon_0}$
plasma: $\rho_p = s\, \rho_0\, e^{-s (\phi - U_p) / T}, \qquad \rho_0 = \sum_\mathrm{beams} \frac{J}{v_0}$
motion: $\frac{d\mathbf x}{dt} = \mathbf v, \qquad \frac{d\mathbf v}{dt} = \frac{q}{m}\left(\mathbf E + \mathbf v \times \mathbf B\right), \qquad \mathbf E = -\nabla\phi$
space charge: $\rho_b = \frac{1}{V_\mathrm{node}} \sum_\text{steps} \operatorname{sign}(q)\, I\, \Delta t \times (1 - f)$

Assumes

steady: the beam is the fixed point of field solve, trace and deposit (time is always 0)
planar $(x, y)$ per unit depth or cylindrically symmetric $(x, r)$ ; the solenoids and a measured axial profile only in cylindrical geometry
an imposed static $\mathbf E$ and/or $\mathbf B$ from a field map (beamtracer.field_map) in the frame of the grid, added to the solved field and the solenoids
a Boltzmann plasma species: electrons ( $s = -1$ ) for positive-ion extraction, positive ions ( $s = +1$ ) for negative-ion extraction; the extracted species are traced beams
configuration in e, u and eV, converted to C, kg and J; with MPI the trajectories are shared between the ranks and every rank solves the field

Built from