Generator

The count is the volume, to within a surface

Every cell's reflections enumerated at a stated resolution, beside the estimate that the number of lattice points in a sphere is its volume divided by the reciprocal cell volume — which is (4π/3)d⁻³ times the cell volume. The two agree to within 1.7 per cent, and the discrepancy is the surface term: it is the points near the boundary, and it shrinks as the sphere grows. The last column is what an experiment actually collects, after symmetry has been used to discard reflections whose intensities are equal.

6 essays call resolution. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

Everything measurable at λ = 1.54 Å, and it is a finite set. A section through the reciprocal lattice of a cubic cell of 10 Å, with the limiting sphere drawn. Bragg's law reaches a reflection only if its spacing is at least half the wavelength, so the measurable reflections are the lattice points inside a sphere of radius 2/λ and the points outside it are not merely unmeasured — no experiment at this wavelength can reach them. In this section 516 points lie inside; in the full sphere there are 9,092.

How it is known

How many reflections there are

The reciprocal lattice is infinite and a measurement is not. A wavelength cuts a sphere out of it, the number of points inside is the sphere's volume times the cell's, and symmetry then removes all but a fraction — though never quite the fraction a division would give.

Three atoms, four resolutions. A one-dimensional Fourier synthesis of the same three atoms, cut at four different resolutions. Nothing is approximate except the edge: every amplitude and every phase used is exact, and the only information withheld is the reflections outside the sphere. The peaks broaden as the cut-off comes in, and beside every peak sits a negative ripple that the coarsest map cannot distinguish from a real absence of density. Both effects are the transform of the sphere rather than anything about the structure.

How it is known

As sharp as the sphere is wide

A map made from a truncated sum is not a blurred picture of the structure. It is the structure convolved with the transform of the sphere — so peaks acquire a width proportional to the resolution, and a negative ripple of twenty-two per cent that no improvement in the data ever reduces.

Observations per unknown, against resolution. Unique reflections divided by refinable parameters, for a triclinic cell, no angle a right angle, with three coordinates and six displacement parameters for every atom and one non-hydrogen atom per 18 Å³, in a molecular crystal. The scale is logarithmic because the fall is a cube: 16.3 at 0.8 Å and 0.30 at 3 Å. The line at one is where a determination stops being over-determined, and it is crossed at about 2.0 Å.

How it is known

The unknowns against the observations

A structure determination is a fit of some number of parameters to some number of measurements, and both counts can be worked out before any data exist. The ratio turns out not to depend on how large the crystal's cell is, or on how symmetric it is — only on the resolution, and on that as a cube.

$the most consistent answer has 100 per cent of the signs. 60 runs of the sign procedure from 60 different random starts, each plotted at its self-consistency — a figure computed without any knowledge of the answer — against the fraction of its signs that are in fact right. Throwing out the 1 run that reached the uniform solution — every sign the same, perfectly consistent and physically a single peak — the highest consistency belongs to a run with 100 per cent of the signs right. The ranking works here, and the reason it works is that the cell is small. Nothing in the plot's horizontal axis knows the answer, which is the only reason a procedure of this kind is a procedure at all.$ How it is known

The formula that has the answer already

The tangent formula rebuilds each phase from all the others, and the true phase set is very nearly a fixed point of it — hand it the answer and it hands the answer back. Start it anywhere else and it does not arrive. Having a fixed point and finding it are different problems, and the second is where the subject spent twenty years.

What each Laue class buys, in measurements per reflection. The eleven Laue classes, with how many distinct reflections a block of indices holds under each and how many times a data set measures the average one. The redundancy is always below the order of the class and the gap is the special reflections. This is the number an experiment is planned around: repeated measurements of what symmetry says must agree are the only estimate of precision that does not come from a model, so a triclinic crystal has to be turned through far more of the sphere than a cubic one to be measured as well.

How it is known

Every reflection, several times over

A diffraction experiment does not measure each reflection once. Symmetry relates a reflection to the others of its orbit, and those are the same reflection seen from another direction — so a hundred thousand measurements may contain twelve thousand reflections, each observed eight times.

$66 reflections that no rotation reaches. Every reflection inside the limiting sphere of an orthorhombic cell, 7 × 11 × 13 Å at 1.4 ångström, plotted by its distance from the rotation axis against its height along it. The ones marked are those a rotation about that axis can never bring into diffracting position: turning the crystal moves a point on a circle at fixed height, so a point too close to the axis can never acquire the component along the beam that the Ewald condition demands. The blind region is a cusp about the axis, it is 4.3 per cent of the sphere here, and nothing but remounting the crystal removes it.$ How it is known

What one turn of the crystal reaches

Every reflection inside the limiting sphere is measurable by some orientation. A crystal on a spindle has one axis, and a region around it never reaches the Ewald sphere at all — however patiently the crystal is turned.

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