Charge changes

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

AxialProfile

classsource

A target density (m^-3) along x (m), linear between the points.

profile

$$n(x) = n_i + \frac{x - x_i}{x_{i+1} - x_i} (n_{i+1} - n_i), \quad x_i \le x \le x_{i+1}; \qquad n(x) = n_0,\ n_\mathrm{last} \text{ beyond}$$

line density

$$\int_a^b n\, dx = \text{the trapezoidal rule on } a, \text{ the points inside } (a, b), b \quad \text{(exact)}$$

Assumes

• at least two points, increasing $x$, densities $\ge 0$
• in the tracer the density is taken at the midpoint of each step, as if uniform over it

ChargeChanging

classsource

The targets a beam changes charge on, and how its trajectories follow the changes.

master

$$\frac{d\mathbf F}{ds} = \sum_t n_t(x)\, M_t\, \mathbf F, \qquad (M_t)_{js} = \sigma_t(s \to j), \quad (M_t)_{ss} = -\sum_{j \ne s} \sigma_t(s \to j)$$

force

$$\bar q = e \sum_s z_s F_s \ \text{(fractions)}, \qquad q = e\, z_s \ \text{(analog)}$$

current

$$N_k = \frac{|I_k|}{e} \ \text{in every state}, \qquad \Delta Q = N_k\, e \sum_s z_s \bar F_s\, \Delta t$$

Assumes

• $\mathbf F$ the fractions of a trajectory over the states $s$ of charge $z_s$ (default -, 0, +), $s$ the path length, $M_t$ from plasmacoll.ChargeStateTransitions
• fractions mode: a trajectory feels the force of its mean charge, so states do not part where the field would separate them (fine in a weak field, as in a neutralizer; not in a residual-ion dump or inside an accelerator)
• analog mode: one state per trajectory, changed at sampled events; the mean over trajectories is the fractions mode
• a trajectory starting with $|z| \ne 1$ counts $N_k = |I_k| / e$ as if $|z| = 1$

References

• plasmacoll ChargeStateTransitions, charge_state_fractions

Built from

• advance
• sample
• ChargeTallies

advance

functionsource

The fractions (states, N) after the steps of line densities lines, third order.

taylor

$$\mathbf F \leftarrow \mathbf F + M \mathbf F + \tfrac12 M^2 \mathbf F + \tfrac16 M^3 \mathbf F, \qquad M = \sum_t n_t(x_{1/2})\, \Delta s\, M_t$$

conservation

$$\sum_s (M_t)_{s j} = 0 \quad \Rightarrow \quad \sum_s F_s = \text{const}$$

Order

3

Assumes

• the third-order Taylor polynomial of $e^{M}$ per step; the densities at the step's midpoint, uniform over the step
• the order is in the line density per step $n \Delta s$

sample

functionsource

Use up the optical depth of one step and sample the events where it runs out.

rate

$$\lambda_s(x) = \sum_t n_t(x) \sum_{j \ne s} \sigma_t(s \to j)$$

depth

$$\tau \sim \mathrm{Exp}(1), \qquad \tau \leftarrow \tau - \lambda_s \Delta s, \qquad \text{an event where } \tau \le 0, \text{ at the share } 1 + \frac{\tau}{\lambda_s \Delta s} \text{ of the step}$$

channel

$$P(t, j) = \frac{n_t\, \sigma_t(s \to j)}{\lambda_s}$$

splitting

$$\text{at each of the first events: } k \text{ trajectories of } I / k, \text{ drawing their new states independently}$$

Assumes

• the probability of no event over a path is $\exp(-\int \lambda_s\, ds)$, so the depth of the event is $\mathrm{Exp}(1)$ in $\int \lambda_s\, ds$
• several events per step followed in turn, each new state using up the rest of the step; the force changes from the next step on
• random numbers from seed and the MPI rank, afresh on every trace, so the Vlasov iteration has a fixed point

ChargeTallies

classsource

What the beam did to its targets, summed over trajectories (see the module docstring).

path

$$\text{path}_s = \sum_k N_k \int F_{s,k}\, ds$$

changes

$$\text{changes}_{t, s \to j} = \sigma_t(s \to j) \sum_k N_k \int F_{s,k}\, n_t\, ds$$

ionization

$$\text{ionization}_{t, s} = \sigma^\mathrm{ion}_{t, s} \sum_k N_k \int F_{s,k}\, n_t\, ds$$

Assumes

• in particles per second (per metre of depth in planar geometry); the integrals summed step by step with the mean fractions $\bar F$ of each step
• the ionization cross sections are tallied only: they do not change the beam
• analog mode also sums the sampled events, whose mean is changes

conductance_profile

functionsource

Return the gas density through a stack of grid apertures in free-molecular flow.

drops

$$\Delta n_i = (n_\mathrm{up} - n_\mathrm{down}) \frac{1 / (a_i^2 K_i)}{\sum_j 1 / (a_j^2 K_j)}, \qquad K_i = \frac{1}{1 + 3 l_i / (8 a_i)}$$

flow

$$Q = C_i\, \Delta n_i, \qquad C_i = \frac{\bar v}{4} \pi a_i^2 K_i$$

Assumes

• free-molecular flow through round apertures of radius $a_i$ and length $l_i$; one flow $Q$ through all of them, so the mean speed $\bar v$ cancels
• uniform density in the gaps between the grids (no pumping across them), linear through each aperture
• $K_i$ Dushman's approximation of Clausing's transmission probability

References

• S. Dushman, Scientific Foundations of Vacuum Technique (Wiley)