Crystallography & Symmetry
patterns derived from their groups
Series

Resolution — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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 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.

    part 1 · diffraction
  2. 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.

    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.

    part 2 · diffraction
  3. 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 ų, 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.

    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.

    part 3 · diffraction
  4. 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.

    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.

    part 4 · diffraction
  5. 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.

    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.

    part 5 · diffraction
  6. Completeness against sweep, for four spindles. Completeness against rotation range for one tetragonal crystal, class 4/mmm, on four different spindles. The four-fold spindle rises fastest at first, then stops at 88.9 per cent at forty degrees and never improves. The two-fold spindles reach completion near 87 degrees. The general spindle is highest at forty-five degrees and reaches completion at the same 87.

    The same measurements, and a hole that stays

    How much of the sphere must be swept for a given redundancy? The question has two answers, because it has two parts. The mean number of observations depends only on the sweep, to within two per cent whichever axis the crystal turns on. Completeness depends entirely on the axis, and on one kind of axis no sweep ever finishes: when the Laue group keeps the spindle's line, every copy of a blind reflection is blind too, and fifteen of a tetragonal crystal's 135 reflections are never measured.

    part 6 · diffraction

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