ElSeparator

PASS.commands.element.elseparator.ElSeparator models an electrostatic septum with a circulating-beam field-free region, a thin separating electrode, a field region and a high-voltage electrode. The registered command is ElSeparator. CPU and GPU implement a relativistic uniform electrostatic body map with hard-edge potential matching for positive length. Zero length uses a single effective impulse evaluated with each particle’s incident longitudinal velocity. Entering the field region does not itself lose a particle. Survivors continue into the downstream lattice, where the actual later apertures determine later losses.

The fields below configure PASS.para.schema.elements.ElSeparatorItem. The key in Sequence.add(name, item) supplies the element name; it is not a configuration-model field.

Input parameters

Python configuration field	JSON key	Type	Unit	Default	Description
`s`	`S (m)`	`float`	m	`Required`	Longitudinal position of the element exit or zero-length action point.
`length`	`Length (m)`	`float`	m	`0.0`	Nonnegative supplied length, independent of roll.
`voltage`	`V (V)`	`float \| None`	V	`None`	Signed interplate voltage difference; choose V or VL. Nonzero V requires positive length.
`voltage_length`	`VL (V m)`	`float \| None`	V m	`None`	Signed longitudinal voltage integral; supports positive or zero tracking length.
`gap`	`Gap (m)`	`float`	m	`Required`	Positive clear electrode gap.
`septum_position`	`Septum position (m)`	`float`	m	`Required`	Surface d facing the circulating-beam field-free region, along u.
`septum_thickness`	`Septum thickness (m)`	`float`	m	`0.0`	Nonnegative effective material thickness.
`tilt`	`Tilt (rad)`	`float`	rad	`0.0`	Roll used in the projection and kick equations above.
`aperture_type`	`Aperture type`	`str`	—	`'off'`	Existing vacuum aperture geometry.
`aperture_value`	`Aperture value`	`list`	m / rad	`[]`	Existing vacuum aperture geometry.
`num_slices`	`Num slices`	`int`	1	`1`	Strict positive integer; sets internal SC scheduling. The isolated analytic body does not require subdivision.
`space_charge`	`Space charge`	`ElementSpaceCharge \| None`	—	`None`	Optional internal SC configuration; requires positive length.

Input configuration

Specify V or VL, gap and septum geometry for the modeled device. This synthetic example uses a manual polygon with -0.05 < x < 0.0201 and |y| < 0.02. The internal septum remains absorbing. Replace the values with device parameters; the selected strength input is constant on every pass.

{
  "Command": "ElSeparator",
  "S (m)": 1.0,
  "Length (m)": 1.0,
  "V (V)": 1000.0,
  "Gap (m)": 0.01,
  "Septum position (m)": 0.01,
  "Septum thickness (m)": 0.0001,
  "Tilt (rad)": 0.0,
  "Num slices": 16,
  "Aperture type": "polygon",
  "Aperture value": [[-0.05, -0.02], [0.0201, -0.02], [0.0201, 0.02], [-0.05, 0.02]]
}

To use the same integrated strength in this 1 m element, replace V (V) with "VL (V m)": 1000.0. To use a thin model at S=1 m, also set Length (m) to 0. The same input field integral is used at that plane, with the incident particle speed. Its effective impulse need not equal the full thick map exactly; it omits the thick model’s transverse displacement and flight time. For example, (x,y)=(0.005,0) survives without a kick, (0.01005,0) is lost in septum material, and (0.015,0) survives the entry check and receives the field kick.

Injection timing is described in Multi-turn injection and transverse painting. New particles specified at the ES exit start tracking at that plane. Injection generates or loads their coordinates without an additional geometric acceptance cut. ElSeparator evaluates deflection and losses when particles subsequently pass through the element, according to its thick or thin model.

Length, orientation and time

Length (m) is the supplied longitudinal transport length L. S (m) is the exit; the entrance is S-L. Tilt (rad) is a roll about the longitudinal axis, not a pitch or yaw of the element axis. Never multiply or divide L by cos(tilt) for transport, field integration, loss positions or reference time. Horizontal, vertical and inclined separators share one implementation.

With the existing PASS roll convention, temporary geometric coordinates are

\[u=x\cos\theta-y\sin\theta,\qquad v=x\sin\theta+y\cos\theta.\]

The particle arrays remain in the beam frame. A local normal kick is applied as

\[\Delta p_x=K_u\cos\theta,\qquad \Delta p_y=-K_u\sin\theta.\]

Thus tilt=0 selects horizontal deflection and tilt=+/-pi/2 selects vertical deflection; the sign of the voltage and the particle charge determine polarity. Adding pi to the roll reverses the geometric normal. The sign of the septum position does not independently select a field side.

For every traversed reference segment ds, including empty bunches,

\[t_0\leftarrow t_0+\frac{ds}{\beta_0 c},\qquad z_{\mathrm{rel}}=\beta_0c(t_0-t_i).\]

The design reference clock retains the field-free reference transit convention. Individual flight-time differences are computed by the exact field body map or the standard drift in the circulating-beam field-free region. There is no additional tilt time correction or arrival state. A zero-length instance checks geometry and momentum at its plane, applies a VL kick to surviving particles in the field region, then checks momentum validity again. It changes neither x, y, z nor reference time. It cannot locate collisions that would occur inside the omitted physical length; use the thick model to resolve those trajectories and loss positions. Num slices does not repeat the thin kick, and internal space charge requires positive length.

Geometry and voltage

Let d be Septum position (m), t be Septum thickness (m) and g be Gap (m). The high-voltage electrode’s inner surface is uc=d+t+g. These positions are relative to the reference orbit, not the ring’s geometric center. The regions, for every local v, are

circulating-beam field-free region: u < d;
septum material: d <= u <= d+t;
field region: d+t < u < uc;
high-voltage electrode material: u >= uc.

The electrode surfaces are absorbing material boundaries. A zero-thickness septum is a closed plane: touching or crossing it still loses a particle. This is an effective material-loss model, not a discrete-wire or multiple-scattering simulation.

For entry contact tests, projected coordinates within 4*epsilon*(abs(x*cos(tilt))+abs(y*sin(tilt))) of a material surface count as touching; epsilon is the particle dtype’s machine precision. This prevents rotation/storage roundoff from letting a point on a zero-thickness septum pass. Future ray intersections still use the specified surfaces without shifting them.

The plates and field are idealized as covering all local v. Use the separate vacuum aperture for the opposite wall and transverse clearance, including symmetric limits in v. This assumes the modeled acceptance lies within the electrodes’ good-field coverage; it does not model finite-height fringe fields. The polygon is an outer acceptance boundary, not a replacement for the internal septum material. Aperture type: off disables only the vacuum-wall check.

V (V) is the signed septum-minus-high-voltage-electrode potential difference. VL (V m) is its longitudinal integral, not the electric-field integral:

\[V=\phi_{\mathrm{septum}}-\phi_{\mathrm{HV}},\qquad VL=\int V(s)\,ds,\qquad \int E_u(s)\,ds=\frac{VL}{g}.\]

For a thick uniform-field model, V input gives Eu=V/g and VL=V*L; VL input gives Eu=VL/(g*L). For a thin model, VL is an independent input representing the real device integral, even though the tracking length is zero. For zero length the effective normalized impulse is

\[K_u=\operatorname{sgn}(q)\frac{VL/g}{\beta_0 c B\rho} \frac{A}{p_s},\qquad A=\sqrt{\gamma_0^{-2}+\beta_0^2(1+\delta)^2},\qquad p_s=\sqrt{(1+\delta)^2-p_x^2-p_y^2}.\]

Here beta0*c*ps/A is the incident longitudinal speed. The incident state is frozen during this lumped impulse. It keeps dp fixed and is not the exact map of a finite electrode. PASS stores px=Px/P0 and py=Py/P0, so this is a normalized mechanical momentum increment, not an angular kick. Do not divide it by an additional (1+delta). Finite length uses the energy-consistent map below.

PASS stores the positive reference rigidity magnitude. The charge sign is applied explicitly. The current bunch beta and rigidity are read each execution; RF updates them through set_reference_energy. A constant applied voltage therefore does not imply a constant kick during acceleration.

Specify exactly one non-null V or VL; zero is valid. Both or neither are errors. For zero length, nonzero V is rejected with a request to use VL; V=0 permits a pure geometry check. Gap and septum position remain required for either input. The old Voltage (V), electrode-height/center and EX/EY/EXL/EYL fields are not accepted. Material and vacuum checks remain active at zero strength.

Propagation and collisions

The septum and external field-free regions have zero potential. Within the gap, with uf=d+t, the ideal body potential is Phi=-Eu*(u-uf). The high-voltage plate therefore has potential -V. Define W=energy/(P0*c), mu=mc/P0 and k=q*Eu/(P0*c). The entrance edge changes mechanical energy to W+k*(u-uf), keeping transverse momenta and coordinates fixed. The exit edge subtracts k*(u-uf). These are ideal zero-width longitudinal edge impulses, not measured fringe profiles.

In the uniform body, ps and pv are constant. For a forward distance ds and a=k*ds/ps, the analytic map is

\[ \begin{align}\begin{aligned}p_{u,2}=p_{u,1}\cosh a+W_1\sinh a,\qquad W_2=W_1\cosh a+p_{u,1}\sinh a,\\u_2=u_1+\frac{W_2-W_1}{k},\qquad v_2=v_1+\frac{p_v}{p_s}ds,\qquad c\Delta t=\frac{p_{u,2}-p_{u,1}}{k},\\\delta_2=\sqrt{W_2^2-\mu^2}-1,\qquad z_2=z_1+ds-\beta_0c\Delta t.\end{aligned}\end{align} \]

The implementation evaluates cancellation-free increments and the zero-field limit, rather than subtracting nearly equal hyperbolic functions. The short series used near zero is a floating-point evaluation technique, not a paraxial approximation. Across the complete static element, surviving particles recover their entry mechanical energy when no other energy-changing interaction acts. Particles without a positive forward momentum at an edge are removed at that edge; this represents leaving the forward tracking model, not material absorption.

Entrance and exit matching occur once per element, never at internal slice or SC boundaries. A center SC callback receives the state after half a body slice; a boundary callback receives the full body-slice state. Both see the correct design reference time and mechanical particle momenta in the beam frame. Survivors in the circulating-beam field-free region also participate. See Element Slicing and Internal Space Charge for node placement and weights. With no internal SC, one analytic body propagation suffices: Num slices does not change the external-field result. With SC, convergence of the split collective map must still be checked.

Electrode contacts are solved from the body’s energy-position relation, retaining both momentum branches so that turns and grazing contacts are included. Vacuum wall contacts use conservative interval bounds on the analytic curve. Line and coordinate extrema include interior turning points; ellipse bounds enclose the whole interval. This detects non-convex excursions even with both endpoints inside. The earliest candidate is isolated to a longitudinal interval of 1e-12*max(1, segment length) metres, with a separate 64-double-epsilon scaled geometry tolerance. Near tangencies that spatial tolerance can produce a larger uncertainty in the longitudinal contact location. Stored loss positions use the existing float32 array. Field-free motion retains analytic straight intersections.

Particles stop at first contact and keep their mechanical state at that point; an exit edge is not applied to a particle lost inside the body. Subsequent calls preserve the first loss record. Material surfaces and vacuum walls compete for the earliest contact. All existing aperture types remain supported in the beam frame: off, default, rectangle, circle, ellipse, rectcircle, rectellipse, racetrack, octagon and polygon. Tilt rotates the electrodes and field, not the separately specified vacuum chamber.

For a manually entered local polygon, transform each vertex into beam coordinates:

\[x=u\cos\theta+v\sin\theta,\qquad y=-u\sin\theta+v\cos\theta.\]

Enter vertices in boundary order without repeating the first vertex to close the polygon. Self-intersections, non-adjacent edge contacts, repeated vertices and malformed aperture dimensions are rejected during schema validation as well as input preflight.

Tilt=0 puts the high-voltage electrode on local +x, while tilt=pi reverses the layout. Changing the voltage sign changes force polarity, not the geometry.

Drift and approximation limits

ES and Drift share the CPU factors and CUDA inline drift function. With T=px^2+py^2 and ps=sqrt((1+delta)^2-T), define

\[A=\sqrt{\gamma_0^{-2}+(1-\gamma_0^{-2})(1+\delta)^2}, \qquad \frac{\Delta z}{ds}= \frac{\delta(2+\delta)\gamma_0^{-2}-T}{p_s(p_s+A)}.\]

This rationalized form retains small longitudinal slips in single precision. Nonpositive total or longitudinal momentum and nonfinite longitudinal momentum are rejected; positive longitudinal momentum is not artificially clamped. Tracking never wraps the stored z coordinate.

The thick map is exact for the declared ideal uniform body and hard edges, up to floating-point and collision-location tolerances. It does not include a three-dimensional fringe map, finite-height fields, discrete septum wires, scattering or secondary particles. Thin tracking additionally omits the actual body trajectory, energy exchange and transit-time change, and freezes the incident longitudinal speed during the effective impulse. A thin VL model is not asserted to be the exact L-to-zero limit of the finite hard-edge model.

CPU and CUDA implementations reside in the same source file. GPU kernels use double precision for analytic field trajectories and curved collision bounds; particle arrays retain their configured float32 or float64 storage. Internal SC nodes materialize the particle state; compiled kernels are cached per element, device and storage dtype. No particle coordinate array is rotated in place.

Without internal SC nodes, thick GPU tracking fuses entrance, body and exit into one kernel. Field-free particles use the exact straight trajectory with continuous material and aperture checks, including an endpoint check after storage rounding. A surviving field-free trajectory cannot enter the gap without first contacting an absorbing electrode. Particles in the field retain the analytic hard-edge stages and their configured storage-precision rounding. Internal SC nodes retain the split tracking path so that their kicks act on the intermediate state.