A detector tilt does NOT appear as a cos(2 phi) modulation of the ring radius, as
the previous comment claimed. To first order a misalignment beta gives
r(phi) = R + (R^2 / F) (beta_x cos phi + beta_y sin phi)
which is a cos(phi) term - the same harmonic a wrong beam centre produces. What
separates them is the radius dependence: the centre's amplitude is the same on
every ring, the tilt's grows as R^2. So they are told apart across rings, not
within one, and on a single ring they are exactly degenerate. Measured on a powder
standard the true cos(2 phi) term is of order R^3 beta^2 / F^2 - hundredths of a
pixel, at the noise floor - so it carries nothing usable.
Also add the tilted round trip, which was missing. It doubles as a check that
RingOptimizer's open-coded rotation agrees with DiffractionGeometry's: the fitter
applies Rx(-rot2) Ry(+rot1) by hand rather than going through the geometry's
Rz(-rot3) Rx(-rot2) Ry(+rot1), and those had never been held against each other.
They agree - 0.020 / -0.015 rad recovered as 0.0197 / -0.0148. Dropping rot3 is
right rather than an omission, since rings cannot constrain in-plane roll.
The tilted case yields fewer ring points than the centred one, which is expected
and worth knowing: the extractor searches a window centred on where each ring is
EXPECTED, so a large enough geometry error carries part of a ring out of it.
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
leonarski_fandClaude Opus 5
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