[ESSSANS] Apply the SANS beam center as a beam direction

[SimonHeybrock]

Fixes #404.

Why

The beam center was applied as a fixed transverse shift of every detector pixel. That is wrong on an instrument with banks at different distances, which is what #404 is about, and it also leaked into places that have nothing to do with the beam.

The information in #404 is largely superseded. The current understanding is that the rear-detector rail calibration at Loki will be embedded in the positions in the file, that the sample is where it claims to be, and that the beam center therefore describes the direction of the beam, set by the collimation slits. A given angular deviation of the beam shows up as a proportionally smaller transverse offset on a bank closer to the sample, so a single offset cannot be applied unchanged to all banks.

What changed

Three independent steps, one commit each.

fix(sans): measure beam center relative to sample position — the center-of-mass finder projected the center-of-mass onto the plane normal to the beam without first subtracting the sample position, so the offset was measured from the origin of the lab frame rather than from the beam axis through the sample. Latent only: every currently supported instrument puts the sample on the beam axis at the origin, and the results are bit-for-bit unchanged on the Loki-at-Larmor and SANS2D test data. Fixed because the next commit makes the sample position load-bearing.

refactor(sans): apply beam center to the scattered beam, not to pixel positions — the beam center describes where the beam is, not where the detector is, so it now overrides scattered_beam in the coordinate transformation graph instead of being turned into a DetectorPositionOffset. The pixel positions keep the values from the file. Callers no longer have to undo the shift to recover the real geometry, which removes the raw_pos = position + beam_center workarounds in the SANS2D masks.

two_theta, phi, Q and the wavelength are unchanged bit for bit; I verified this by patching the pre-refactor code so that only its solid angle sees unshifted positions, which reproduces the new results exactly. The one thing that does change is the solid angle, which was computed from beam-center-shifted positions and so carried a smooth spurious tilt across every bank. Solid angle is a property of the real geometry and must not depend on the beam. On the real Loki geometry the size of that artefact is:

Spurious change in solid angle against pixel position projected on the beam-center direction. Each bank is a straight line through zero, with a slope set by its distance from the sample.

banks	distance	spurious solid-angle error
rear, loki_detector_0	5.05 m	-0.28% to +0.27%
mid frames, loki_detector_1-4	2.96 - 3.38 m	-0.77% to +0.75%
front frames, loki_detector_5-8	1.21 - 1.63 m	-3.52% to +3.63%

It is a tilt rather than scatter: each bank is a straight line through zero when plotted against the pixel position projected on the beam-center direction, with a slope set by its distance from the sample, so it biases I(Q) systematically rather than averaging out. The knock-on effect on the single-bank Loki-at-Larmor test data is only 0.041% in I(Q) and 0.22% in I(Qx,Qy), which is why the reference results move at all.

feat(sans): treat the beam center as a beam direction — every pixel is now shifted in proportion to its own distance from the sample, which maps the actual beam onto the nominal axis. This is the actual fix for #404, and it covers the depth of a bank without any special treatment.

refactor(sans): apply the beam center as an incident beam direction — the shear above is replaced by pointing incident_beam along the beam center, which is what a beam center is. Exact everywhere rather than exact along the beam and first order elsewhere, and it deletes the shear provider.

This requires scipp/scippneutron#725 (stacked on scipp/scippneutron#724), which makes the beam-aligned coordinate system follow the incident beam instead of its projection onto the horizontal plane. Without it, phi and Qx/Qy silently drop the vertical component of the beam center. This PR therefore cannot be merged until that is released.

For the real Loki geometry (geometry-loki-2026-04-13.nxs, sample at the origin), taking a 34.3 mm offset determined on the rear bank:

Transverse shift applied to pixels against distance from the sample. Before is a flat line at 34.3 mm for all nine banks; after is a straight line through the sample position. Thick segments mark each bank's depth extent.

bank	z range / m	shift before / mm	shift after / mm	over-correction
loki_detector_0 (rear)	5.005 – 5.099	34.3	34.0 – 34.7	1.00x
loki_detector_1	2.891 – 3.029	34.3	19.7 – 20.6	1.71x
loki_detector_2	3.321 – 3.441	34.3	22.6 – 23.4	1.49x
loki_detector_3	2.891 – 3.029	34.3	19.7 – 20.6	1.71x
loki_detector_4	3.321 – 3.441	34.3	22.6 – 23.4	1.49x
loki_detector_5	1.044 – 1.370	34.3	7.1 – 9.3	4.19x
loki_detector_6	1.502 – 1.760	34.3	10.2 – 12.0	3.10x
loki_detector_7	1.112 – 1.370	34.3	7.6 – 9.3	4.07x
loki_detector_8	1.502 – 1.759	34.3	10.2 – 12.0	3.10x

The front window frames were over-corrected by more than a factor of four. Worth noting for the review: the banks are much deeper than one might assume, up to 327 mm for loki_detector_5, which is 27% of its own distance from the sample. Doing the correction per pixel rather than per bank costs nothing here and removes the question entirely.

How wrong was that? Geometry alone answers it, using the real pixel positions and comparing the scattering angle each pixel gets under the two treatments:

Error in Q per bank from applying the rear-bank offset unchanged. Dot is the median pixel, bars are the 90th and 99th percentile.

bank	z / m	2theta range	median |dQ|/Q	p90	p99
loki_detector_0 (rear)	5.052	0.01 - 10.9 deg	0.02%	0.06%	0.20%
loki_detector_1	2.960	5.3 - 17.1 deg	1.41%	2.58%	3.47%
loki_detector_2	3.381	3.7 - 10.6 deg	2.15%	3.40%	4.29%
loki_detector_3	2.960	4.9 - 16.8 deg	1.29%	2.68%	3.67%
loki_detector_4	3.381	3.1 - 10.0 deg	2.23%	3.80%	5.08%
loki_detector_5	1.207	15.1 - 51.4 deg	1.31%	2.99%	4.41%
loki_detector_6	1.631	10.5 - 41.9 deg	1.67%	3.49%	5.41%
loki_detector_7	1.241	14.8 - 45.2 deg	1.70%	3.32%	4.61%
loki_detector_8	1.631	9.9 - 42.0 deg	1.64%	3.61%	5.70%

A systematic, bank-wide 1.3% to 2.2% error in Q on the window frames, reaching 5.7% in the tail, against 0.02% on the bank the beam center was measured on. That is the size of the bug.

On the single-bank Loki-at-Larmor test data there is, correctly, almost nothing to see: with one bank the correction changes by at most 0.58 mm, far below the pixel size. What little change appears is binning jitter, pixels crossing Q-bin boundaries, rather than a shift of the curve. Rebinning shows this directly:

Q bins	mean change	rms change	|mean|/rms	sign flips between adjacent bins
100	+0.004 sigma	0.283 sigma	0.01	65%
25	+0.010 sigma	0.117 sigma	0.09	54%
10	+0.026 sigma	0.095 sigma	0.27	56%

A smooth systematic survives rebinning; this collapses as the bins coarsen while the mean stays at zero, and adjacent bins disagree in sign more often than chance, which is what one bin losing what its neighbour gains looks like. So for existing single-bank reductions this commit is a no-op to within the noise, and the reference results move by 0.14 sigma median and 0.63 sigma at most.

The Loki-at-Larmor reference results are regenerated in the two commits that change them.

User-facing change

BeamCenter is now the position of the beam center relative to the sample, sc.vector([x, y, L_ref], unit='m'): the transverse offset together with the distance from the sample at which it was determined. That distance is what turns an offset into a direction, so it has to be part of the value.

Only the direction enters the correction; the magnitude cancels. A value determined on one bank therefore applies to all banks, and, if the beam direction really is set by the collimation alone, also to a detector moved to a different distance along its rail — which was not true before, since the old code would have applied the wrong offset. Conversely the value is tied to the collimation that produced it, so it has to be re-determined when the slits change. Nothing in the files records which collimation a given value came from, so that one is documented rather than checked.

A zero vector still means "no correction", so existing wf[BeamCenter] = sc.vector([0, 0, 0], unit='m') keeps working unchanged. A non-zero offset with no distance is rejected with a message explaining what is missing, rather than being silently misapplied as before. Both beam-center finders report the distance they used, so values obtained from them need no change beyond being carried around whole. The workflow GUI parameter moves from Vector2dParameter to the existing Vector3dParameter.

Accuracy and known limitation

The correction is now a rotation of the incident beam, so it is exact for every pixel and there is no approximation left to quote.

Earlier revisions of this PR applied it as a shear of the scattered beam, on the estimate that the resulting error in two_theta stayed below 1e-5. That estimate was too optimistic: measured across the Loki-at-Larmor detector, with the beam center the finder returns, the relative error in two_theta has a median of 3.8e-4 and a maximum of 1.2e-3. It grows for pixels close to the sample, which the single-pixel estimate happened to miss. That is what moves the reference results in the last commit.

The distance reported by the center-of-mass finder is the intensity-weighted mean depth of the bank, not its geometric center, because layers closer to the sample see more counts. That is consistent rather than wrong: the transverse position of the beam is linear in the distance from the sample, so any intensity-weighted average of points on the beam is again a point on the beam, and the reported distance says where it ended up. The premise holds only approximately, though, and on Loki-at-Larmor the per-layer estimates disagree by more than the geometry allows, so the uncertainty on the distance is of the order of the depth of the bank itself. That is irrelevant for extrapolating to other banks, a few centimetres out of several metres, but it means the variation of the correction within one bank is not resolved better than the noise on it. This is written up in the docstring of beam_center_from_center_of_mass.

Test plan

New tests/beam_center_test.py covers how the correction is applied: zero scattering angle at the beam center and everywhere else along the beam, phi measured around the beam rather than around the nominal axis (both horizontally and vertically), correct behaviour with an off-axis sample and with gravity, and the rejection of a beam center without a distance. New tests/beam_center_finder_test.py covers what the finders return, on small synthetic detectors that run in under a second. tests/loki/workflow_test.py gains the two workflow-level guarantees that the beam center moves neither the pixel positions nor the solid angle; both fail on the previous code.

The I(Q)-based finder is only exercised by tests that are currently skipped as too slow, so I ran it by hand on the SANS2D tutorial data: it reproduces the verified Mantid result to within the 2e-3 the skipped test asserts. The beam-center-finder notebook needed updating, since the internal _iofq_in_quadrants now takes the plane in which the offsets are given, and I executed all of its cells to confirm.

Note for anyone running the tests locally: if esssans is installed non-editable, pytest will silently pick up the installed copy rather than the working tree.