Beamline

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

ApertureGrid

classsource

The apertures of an accelerator grid, each emitting a copy of one beamlet.

slopes

$$\mathbf s = -\frac{\mathbf c}{\mathbf f} + \mathbf s_\mathrm{steer}, \qquad \mathbf c = (c_y, c_z), \ \mathbf f = (f_y, f_z)$$

rotation

$$R = I + \sin\vartheta\, K + (1 - \cos\vartheta)\, K^2, \qquad R\, \hat{\mathbf x} = \frac{(1, s_y, s_z)}{|(1, s_y, s_z)|}$$

copy

$$\mathbf x' = R\, (\mathbf x - \bar x \hat{\mathbf x}) + (\bar x, c_y, c_z)$$

Assumes

• $K$ the cross-product matrix of the unit rotation axis $\hat{\mathbf x} \times \hat{\mathbf d}$ (Rodrigues); the identity below a sine of $10^{-15}$
• every aperture carries the whole beamlet power; a flat grid has $f = \infty$; $\bar x$ the mean $x$ of the beamlet's markers

beamlet_markers

functionsource

The 3D markers of a beamlet traced in 2D, from its crossings of a plane.

power

$$P = \frac{I}{z e} \cdot \tfrac12 m v^2 \ \text{(cylindrical)}, \qquad P = \frac{I}{z e} \cdot \tfrac12 m v^2\, \ell \ \text{(planar, slit length } \ell)$$

states

$$P_s = P\, F_s / n_\mathrm{copies}$$

copies

$$\text{cylindrical: rotated about } x \text{ by } \varphi \sim U(0, 2\pi); \qquad \text{planar: } z \sim U(-\ell/2, \ell/2)$$

Assumes

• only the forward crossings of the plane, of one beam species if given; $z$ the charge the current counts, in e
• one marker per trajectory and state with a share, repeated copies times

beamline_summary

functionsource

The power balance of a beamline as a dataset.

balance

$$P_\mathrm{port} + \sum_\mathrm{parts} P_\mathrm{wall} = P_\mathrm{in} + W \quad \text{(to a relative } 10^{-9})$$

halo

$$P_\mathrm{halo} = \sum_{|\mathbf r_k - \bar{\mathbf r}| > r_\mathrm{halo}} P_k, \qquad \bar{\mathbf r} = \langle (y, z) \rangle_P$$

Assumes

• the balance is checked, a mismatch raises; $W$ the work of the fields (RegionResult.work)

Calorimeter

classsource

A flat calorimeter of tiles in the plane x = position, facing the beam.

map

$$p_{ab} = \frac{1}{A_{ab}} \sum_{k \in \text{tile } ab} P_k \quad [\mathrm{W/m^2}]$$

Assumes

• the markers drift on straight lines to the face (backwards for one already past it); those outside the face pass by
• tiles of equal area on the face extent, closed bounds

divergence_from_widths

functionsource

The source width and the divergence from beam widths at several distances.

widths

$$w^2(L) = w_0^2 + (L \theta)^2$$

Assumes

• linear least squares of $w^2$ on the columns $(1, L^2)$, at least two distances; negative $w_0^2$ or $\theta^2$ clipped to 0

Duct

classsource

The duct from the end of the traced region to the port.

depth

$$\tau(x) = \sum_t \sigma_t^\mathrm{out} \int_{x_0}^{x} n_t\, dx', \qquad \sigma_t^\mathrm{out} = \sum_{j \ne 0} \sigma_t(0 \to j)$$

birth

$$\tau(x_\mathrm{born}) = \tau(x_0) + \xi\, \frac{v_x}{|\mathbf v|}, \qquad \xi \sim \mathrm{Exp}(1)$$

product

$$P(j) \propto \sum_t n_t(x_\mathrm{born})\, \sigma_t^\mathrm{out}\, \sigma_t(0 \to j)$$

walls

$$\mathbf x_\mathrm{end} = \mathbf x_0 + \mathbf v \frac{x_\mathrm{port} - x_0}{v_x}, \qquad \text{the first wall crossing of the segment, else the port}$$

Assumes

• neutrals on straight lines along $+x$; a reionized neutral stops where it is born, transitions back to the neutral state are not followed
• $\int n_t\, ds = \int n_t\, dx\, |\mathbf v| / v_x$ along the slanted path; $\tau$ tabulated by the trapezoidal rule on 1025 nodes plus the profile points and inverted by linear interpolation
• the product is drawn with the weights $n_t \sigma_t^\mathrm{out} \sigma_t(0 \to j)$ as coded; $\propto \sum_t n_t \sigma_t(0 \to j)$ would be the rate of each channel, the same for one target
• walls are simgeometry shapes (simgeometry.crossings.crossings: analytic for boxes, spheres and slanted tubes, sampled otherwise); a marker starting inside a wall stops at once

References

• simgeometry crossings

FieldRegion

classsource

Markers through static electric and magnetic fields, to the walls or an exit plane.

kick

$$\mathbf v^- = \mathbf v^{n-1/2} + \eta \mathbf E, \qquad \eta = \frac{q \Delta t}{2m}$$

rotation

$$\mathbf t = \eta \mathbf B, \quad \mathbf s = \frac{2\mathbf t}{1 + t^2}, \quad \mathbf v' = \mathbf v^- + \mathbf v^- \times \mathbf t, \quad \mathbf v^+ = \mathbf v^- + \mathbf v' \times \mathbf s$$

update

$$\mathbf v^{n+1/2} = \mathbf v^+ + \eta \mathbf E, \qquad \mathbf x^{n+1} = \mathbf x^n + \mathbf v^{n+1/2} \Delta t, \qquad \Delta t = \frac{\ell}{|\mathbf v|}$$

Order

2

Assumes

• the fields at $\mathbf x^n$; the leapfrog starts from the velocity pushed back half a step; neutrals move straight
• steps of path length $\ell$ = step, each tested against the walls and the exit plane as a straight segment, so $\ell$ must be well below the radius of curvature
• the rotation is exact for the angle $2 \arctan |\mathbf t|$ and keeps $|\mathbf v|$ without an electric field
• the order is in the step, for the position on the Larmor circle
• a marker keeps its particles per second, so its power follows its kinetic energy, $P = P_0\,|\mathbf v|^2 / |\mathbf v_0|^2$ with the last half-step velocity: the work of the electric field reaches the walls

References

• J. P. Boris, Proc. Fourth Conf. Numerical Simulation of Plasmas, 3 (1970)

fit_footprint

functionsource

Fit a 2D Gaussian (axes along y and z) to a calorimeter map.

gaussian

$$p(y, z) = p_0 \exp\left(-\frac{(y - c_y)^2}{2 w_y^2} - \frac{(z - c_z)^2}{2 w_z^2}\right)$$

Assumes

• an unweighted least-squares fit (scipy.optimize.curve_fit) on the tile centres, started from the map's power-weighted centre and widths and its maximum
• no bounds and no fallback: a failed fit raises

gaussian_beam

functionsource

A round Gaussian beam along +x, starting in the plane x (m).

sampling

$$y, z \sim \mathcal N(0, w^2), \qquad s_y, s_z \sim \mathcal N(0, \theta^2), \qquad \mathbf v = v \frac{(1, s_y, s_z)}{|(1, s_y, s_z)|}$$

width

$$w(L) = \sqrt{w^2 + (L \theta)^2}$$

Assumes

• a round beam along $+x$; every marker carries $P / N$

Markers

classsource

Beam markers in 3D.

weight

$$w = \frac{P}{\tfrac12 m |\mathbf v|^2} \quad [\mathrm{s^{-1}}]$$

rms angle

$$\theta_{y,z} = \sqrt{\left\langle (s - \langle s \rangle)^2 \right\rangle_P}, \qquad s = \frac{v_y}{v_x}, \ \frac{v_z}{v_x}$$

drift

$$\mathbf x \leftarrow \mathbf x + \mathbf v\, \frac{x_\mathrm{plane} - x}{v_x}$$

placed

$$\mathbf x' = R\, (\mathbf x - x_0 \hat{\mathbf x}) + \mathbf o, \qquad R = [\hat{\mathbf e}_x, \hat{\mathbf e}_z \times \hat{\mathbf e}_x, \hat{\mathbf e}_z]$$

Assumes

• markers are streams of particles on straight lines (no field acts on them outside a FieldRegion); $\langle\cdot\rangle_P$ weighted by the power
• placed: $\hat{\mathbf e}_x$ along direction, $\hat{\mathbf e}_z$ the part of up normal to it
• the weight (particles per second) is what freeofcharge reads as a neutral beam