Trajectories

`beam_with_energy`

functionsource

Seed a beam emitted from a line with a given energy; return it and J / v.

positions: $\mathbf x_k = \mathbf a + \frac{k + 1/2}{N} (\mathbf b - \mathbf a), \qquad k = 0, \dots, N - 1$
velocities: $\mathbf v = v_0 \hat{\mathbf n} + \sigma_\perp g_1 \hat{\mathbf t} + \sigma_\parallel g_2 \hat{\mathbf n} \ (+ \sigma_\perp g_3 \hat{\mathbf z} \text{ cylindrical}), \qquad \hat{\mathbf n} = (t_y, -t_x)$
speed: $v_0 = \sqrt{\frac{2 E e}{m}}, \qquad \sigma_{\parallel,\perp} = \sqrt{\frac{T_{\parallel,\perp}\, e}{m}}, \qquad \rho_0 = \frac{J}{v_0}$
ring: $I_k = \frac{2\pi J s\, r_k}{N} \text{ (cylindrical, a ring)}, \qquad I_k = \frac{J s}{N} \text{ [A/m] (planar)}$

Assumes

$s$ the length of the emission line from $\mathbf a$ to $\mathbf b$ , $\hat{\mathbf t}$ its direction; the beam moves to its right
$g_{1,2,3}$ standard normal: an unscrambled 3D Sobol sequence (origin skipped) through the inverse normal, the same beam on every call, so the Vlasov iteration has a fixed point
the ring currents sum to $J \pi R^2$ up to the midpoint rule's $O(N^{-2})$ for a line along $r$

Trace particles through a static electric (and magnetic) field.

rk4: $\mathbf y^{n+1} = \mathbf y^n + \frac{\Delta t}{6}(\mathbf k_1 + 2\mathbf k_2 + 2\mathbf k_3 + \mathbf k_4), \qquad \mathbf y = (\mathbf x, \mathbf v), \quad \dot{\mathbf y} = (\mathbf v, \mathbf a(\mathbf x, \mathbf v))$
step: $\Delta t = f \min\left(\frac{h}{|\mathbf v|}, \sqrt{\frac{2h}{|\mathbf a|}}\right), \qquad 0 < f \le 1$

Order

4

Assumes

$f$ = step_fraction (0.5 by default); a non-finite estimate (a particle at rest without force) falls back to $f h$
each particle has its own step; the field at the stage points
the order is that of the integrator in $\Delta t$ ; the plane crossings are interpolated linearly along a step

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functionsource

Return the acceleration (3, N) at positions x and velocities v.

motion: $\frac{d\mathbf x}{dt} = \mathbf v, \qquad \frac{d\mathbf v}{dt} = \frac{q}{m}\left(\mathbf E + \mathbf v \times \mathbf B\right)$
planar: $\mathbf E = (E_x, E_y, 0)(x, y) + \mathbf E_\mathrm{imp}(x, y), \qquad \mathbf B = \mathbf B(x, y)$
cylindrical: $r = \sqrt{y^2 + z^2}, \qquad \mathbf E = E_x \hat{\mathbf x} + E_r \hat{\mathbf r} + \mathbf E_\mathrm{imp}, \qquad \mathbf B = B_x \hat{\mathbf x} + B_r \hat{\mathbf r} + B_\theta \hat{\boldsymbol\theta}, \qquad \hat{\boldsymbol\theta} = \hat{\mathbf x} \times \hat{\mathbf r}$
mean charge: $q = \bar q = e \sum_s z_s F_s \quad \text{(charge changes in fractions mode)}$

Assumes

non-relativistic; 3D Cartesian motion in fields of $(x, y)$ or $(x, r)$ , so a trajectory crossing the axis continues at negative $y$
cylindrical: a trajectory stands for a ring of charge; on the axis $\hat{\mathbf r} = 0$ , so $E_r$ and $B_r$ do not act there
the imposed $\mathbf E_\mathrm{imp}$ and $\mathbf B$ (solenoids, a field map) are static and given in the plane: (x, y) or (x, r), the cylindrical ones axial, radial (and azimuthal)

functionsource

Trace particles until they leave the mesh or hit an electrode.

charge: $\Delta Q = \operatorname{sign}(q)\, I\, \Delta t \quad \text{or, with charge changes,} \quad \Delta Q = N_k\, e \sum_s z_s \bar F_s\, \Delta t$
midpoint: $\mathbf x_{1/2} = \tfrac12 (\mathbf x + \mathbf x') + \frac{\Delta t}{8} (\mathbf v - \mathbf v')$
weights: $w_x = \frac{x - x_i}{h}, \qquad w_r = \frac{r^2 - r_j^2}{r_{j+1}^2 - r_j^2} \ \text{(cylindrical)}, \quad w_y = \frac{y - y_j}{h} \ \text{(planar)}$
volumes: $\rho_{ij} = \frac{Q_{ij}}{V_j}, \qquad V_j = h^2 \ \text{(planar)}, \quad V_j = \pi h (r_{j+1/2}^2 - r_{j-1/2}^2) = 2\pi r_j h^2, \quad V_0 = \pi h^3$
path: $\Delta s = \tfrac12 \Delta t\, (|\mathbf v| + |\mathbf v'|) \times \text{the share of the step inside the mesh}$

Assumes

a steady trajectory of current $I$ is a line charge $I / v$ , so a step holds $I \Delta t$ ; currents are magnitudes, the sign is the charge's
$\mathbf x_{1/2}$ the cubic Hermite interpolant at the half step; charge outside the mesh goes to the boundary nodes, whose densities are doubled (half cells)
a trajectory ends transmitted ( $x \ge x_\mathrm{max}$ ), backward ( $x < x_\mathrm{min}$ ), on the wall, on an electrode (tested in the cells touching one) or after max_steps, tested at the end of each step
$\bar F$ the mean of the fractions before and after the step; with MPI the node charges are summed over the ranks

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