Why the window functions over-report¶

wavelength_windows on the BIFROST train reports two bands:

1.1454 to 3.0515 AA
37.9630 to 37.9950 AA

The second is not a band the instrument delivers. This is why it appears.

The relaxation¶

chopper_inverse_velocity_windows builds, for each disc, the set of inverse velocities that could pass it — and it lets the emission time range over the whole pulse while doing so. In the C:

double wiv_min = (t_open[opening] + n_tau - latest_emission) / path;
double wiv_max = (t_close[opening] + n_tau) / path;

wiv_min uses emission at the end of the pulse and wiv_max emission at the start. So each disc's set is its slice of the transmitted region projected onto the inverse velocity axis, with the emission time integrated away.

Then those sets are intersected across discs.

Why that is not the same question¶

Write A_i for the region disc i passes, and π for projection onto the inverse velocity axis. What the code computes is

⋂ᵢ π(Aᵢ)

What was asked for is

π( ⋂ᵢ Aᵢ )

and these are not equal. An inverse velocity survives the first if each disc admits it at some emission time — the discs being allowed different times. It survives the second only if one emission time gets it through all of them.

Intersection of projections always contains the projection of the intersection, so the computed answer is a superset: it can name bands that nothing passes, and never miss one that something does. The error is one-signed.

Where it bites¶

The relaxation is tight when the discs leave wide, overlapping emission windows: then "some time for each" and "one time for all" coincide, and the two expressions agree.

It fails when the windows are narrow and disjoint. At 38 Å on BIFROST, with the beam at its real width:

Every disc permits something, so every disc's projection contains 38 Å and the intersection keeps it. But the frame-overlap pair's slivers do not overlap each other, so no single emission time gets through both. The intersection of the regions is empty.

Three checks that it is empty¶

Independent of each other and of the polygon code:

1. A brute-force scan over 200 001 emission times across the pulse: every one blocked.
2. An exact interval intersection — each disc's permitted emission times as intervals, intersected: empty.
3. inverse_velocity_time_mask, which keeps both coordinates, over a fine grid spanning 37.90–38.05 Å: 0 allowed cells out of six million.

And Region, which projects the intersection rather than intersecting projections, reports one band.

It is not only the spurious band¶

The same relaxation widens the real band slightly. The window function's low edge is 1.145358 Å; the region's is 1.145836 Å — 0.48 mÅ of inverse velocities that no emission time delivers.

What to do about it¶

Nothing, usually. If you are asking "roughly what band does this train pass", the answer is right to a fraction of a milliångström, and the spurious bands appear at wavelengths no spectrometer is looking at. Take wavelength_windows(...)[0] and get on with it.

If a number is going into a calculation — an acceptance, an area, a source's sampling bounds — use Region, which never separates the two coordinates and so cannot make this error.