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Calculators for X-ray crystallography — the arithmetic that comes up over and over at a diffractometer and in a structure report. Each one shows the assumptions it rests on.
They are meant to be useful two ways. One is as a working tool, for when you already know what you are after. The other is as somewhere to see what a number actually means, while you are still learning. Every result has its own address, so an answer can be bookmarked, sent to a colleague or linked from a course page.
These are teaching tools for small-molecule and powder X-ray crystallography: selected concepts, and the arithmetic that goes with them. The nav reads like an end-to-end determination and is not one. Nothing here does specimen preparation, instrument alignment or calibration, detector geometry, frame integration or scaling, least-squares refinement, disorder modelling, restraints and constraints, publication validation, or anything macromolecular. For those you need the real software, and the reading page says which.
Every tool below carries a label saying which measurement its statements are about, and so does every page:
If you are here to learn rather than to look something up, start with a route rather than with the list. Each one is a handful of these pages in a recommended order, with something to do on each. There is a question to answer at the end.
All the paths, with the steps and the exercises. Or carry on down to the full list of calculators.
The continuous spectrum of an X-ray tube: the short-wavelength limit from the tube voltage, and which characteristic lines that voltage can excite.
Powder or single crystal
a copper tube at 40 kV · molybdenum at 50 kV · silver at 20 kV — no lines · the same tube at 50 kV
The characteristic K lines of any element from Moseley's law, fitted to six measured anodes, and where each radiation crosses a K absorption edge.
Powder or single crystal
nickel, the filter for copper · zirconium, the filter for molybdenum · tungsten, far outside the fit
The mass absorption coefficient, also called the mass attenuation coefficient, and the linear coefficient of a formula at any X-ray wavelength.
Powder or single crystal
NaCl · a nickel complex · C6H4Br2 just above the bromine K edge · lead sulfide, 0.2 mm thick, where the crystal decides
The Friedif value of Flack and Shmueli: how much resonant-scattering contrast a composition offers, and so how favourable its absolute structure is.
Single-crystal data
a nickel complex · an organic cation · C6H5SeCH3 just above the selenium K edge
Elemental analysis percentages from a chemical formula: C, H, N, S and every other element in it, with the molar mass and the uncertainty on each.
No diffraction data
The isotope pattern a mass spectrometer would see for a chemical formula, peak by peak, with the exact mass and the nuclides that make up each peak.
No diffraction data
a nickel complex · dibromomethane · tin tetrachloride · a sulfate anion
Find the peaks in a measured powder diffraction scan: position, d-spacing, height, FWHM and area for every peak, with the background taken out.
Powder data
albite – 200 reflections, and the scan cannot resolve them all · rutile at 5 nm – the peaks merge, and nothing can undo that · a 0.1° step – the width that comes back is the step
Crystallite size from the Scherrer equation and a Williamson-Hall separation of size from microstrain, with the instrumental width taken out.
Powder data
size only, simulated at 15 nm · strain only, simulated at 0.30% · both at once, simulated at 25 nm and 0.15%
Take the Lorentz-polarisation factor apart, see what a monochromator does to it, and run a powder pattern backwards from measured intensity to |F| squared.
Powder data
quartz – a reflection lying fifteen places from where the chart puts it · rutile with no monochromator – a heavy absorber, ten micrometres deep · quartz at Cr Kα – three times the absorption of the same crystal at Cu
Bragg's law both ways: the d-spacing from a 2-theta angle, or the angle from a d-spacing, for six characteristic wavelengths, recalculated as you type.
Powder or single crystal
Cell volume, d-spacing, 2-theta, the symmetry-equivalent reflections of a Miller index, and the powder stick pattern of the cell.
Powder or single crystal
cubic – NaCl, Fm3m (200) · cubic – Cu, Fm3m (111) · cubic – α-Fe, Im3m (110) · cubic – CsCl, Pm3m (110) · cubic – ZnS, F43m (111) · hexagonal – quartz, P3221 (101) · tetragonal – cristobalite, P41212 (101) · hexagonal – berlinite, P3121 (101) · tetragonal – rutile, P42/mnm (110) · orthorhombic – aragonite, Pmcn (111) · monoclinic – ZrO2, P21/c (111) · triclinic – albite, C1 (111) · tetragonal – urea, P421m (110)
Index a powder pattern: a list of 2-theta positions back to a cubic, tetragonal or hexagonal unit cell, with hkl for every line.
Powder data
quartz, twelve lines – one answer · α-iron, three lines – six cells, and no way to choose · NaCl – one lattice, two descriptions
Reciprocal cell axes and angles from a unit cell, with both metric tensors, and the angle between a lattice direction and a lattice plane.
Powder or single crystal
the normal to (100) is 30° away from [100] – quartz · and in a cubic cell they coincide – NaCl, [111] and (111) · the zone law: [110] lies in (110), so hu+kv+lw = 0 · NaCl – halite (rock salt) · Cu – copper, face-centred cubic · α-Fe – ferrite, body-centred cubic · CsCl – caesium chloride · ZnS – sphalerite (zinc blende) · quartz – α-quartz, SiO2 · cristobalite – α-cristobalite, SiO2 · berlinite – berlinite, AlPO4 · rutile – rutile, TiO2 · aragonite – aragonite, CaCO3 · ZrO2 – baddeleyite, monoclinic zirconia · albite – low albite, NaAlSi3O8 · urea – urea, CO(NH2)2
The Niggli reduced cell of any unit cell, the matrix that produces it, the true Bravais lattice type, and the twinning a near-higher-symmetry cell permits.
Powder or single crystal
Cu, the primitive cell – secretly cubic F · α-Fe, the primitive cell – secretly cubic I · a cell with β = 90.02° – orthorhombic, or not · quartz – already reduced, and unchanged
The Ewald sphere drawn to scale for a cell and a wavelength, the limiting sphere, and how many reflections each anode can reach.
Powder or single crystal
NaCl (220), further out on the sphere · copper (111) with Mo radiation · quartz (600), too fine for Cu Kα
Systematic absences of any of the 230 space groups in any setting, with a reflection tester and the groups that share each pattern of absences.
Powder or single crystal
P21/c – the commonest space group there is · P21/n – the same group, a different setting · P43212 – screw axes and nothing else · R3c – rhombohedral, in hexagonal axes · Fd3m – diamond, and (2 0 0) missing · C2/c – a glide inside a centred net · Pnma – every other reflection, in one zone
Which space groups are consistent with the reflection conditions you observe, what to measure next to narrow the list, and what absences can never decide.
Powder or single crystal
P21/c – the commonest determination there is · P21/c – the same data, empty classes too · Pbca – one pattern, two systems · P4cc – four groups, one reflection apart · Fd3m – diamond
The Wilson plot of a named structure, the temperature factor read off its slope, and the statistics that say whether the crystal has a centre of inversion.
Single-crystal data
albite, 52 atoms in the cell — centrosymmetric · quartz, no centre of inversion · cristobalite — the same, from different symmetry · albite with an overall B of 3 Ų · rutile — where the test does not apply · berlinite — where the test is confidently wrong
The structure factor of any reflection from a cell, a space group and an atom list, with each atom's contribution and an Argand diagram.
Powder or single crystal
cubic – NaCl, Fm3m (200) · cubic – Cu, Fm3m (111) · cubic – α-Fe, Im3m (110) · cubic – CsCl, Pm3m (110) · cubic – ZnS, F43m (111) · hexagonal – quartz, P3221 (101) · tetragonal – cristobalite, P41212 (101) · hexagonal – berlinite, P3121 (101) · tetragonal – rutile, P42/mnm (110) · orthorhombic – aragonite, Pmcn (111) · monoclinic – ZrO2, P21/c (111) · triclinic – albite, C1 (111) · tetragonal – urea, P421m (110)
The Patterson function of a named structure, transformed from its intensities alone, with every interatomic vector derived from the atom positions beside it.
Single-crystal data
zirconia — the zirconium read straight off · aragonite, calcium at a quarter · quartz at 6 terms — the vectors merge · quartz at 30 terms — and separate · rock salt — where there is nothing to read
The electron density of a named structure summed back from its structure factors, and the same sum with the phases thrown away, as a measurement leaves it.
Single-crystal data
zirconia — 7 of 12 signs negative · aragonite, alternating signs · quartz at 6 terms — the silicons merge · quartz at 30 terms — and separate · caesium chloride — where the demonstration is empty
How often the sign relation of direct methods really holds for a named structure, counted against the probability Cochran's formula predicts.
Single-crystal data
caesium chloride — every relation holds · zirconia — the strong ones hold, the weak ones do not · quartz at 12 terms · quartz at 36 terms — more relations, same rule · albite — where no relation is certain
Solve a structure from its amplitudes alone: random starting phases, a few hundred cycles of flipping the low density, and the atoms appear.
Single-crystal data
cristobalite · rock salt — a score worth nothing · berlinite — nothing to show
The (Fo-Fc) difference map: delete one atom from a known structure and see whether the map puts its strongest peak back where it was.
Single-crystal data
Calculated crystal density from a unit cell, a chemical formula and Z, with the cell volume and the volume per non-hydrogen atom.
No diffraction data
NaCl – halite (rock salt) · Cu – copper, face-centred cubic · α-Fe – ferrite, body-centred cubic · CsCl – caesium chloride · ZnS – sphalerite (zinc blende) · quartz – α-quartz, SiO2 · cristobalite – α-cristobalite, SiO2 · berlinite – berlinite, AlPO4 · rutile – rutile, TiO2 · aragonite – aragonite, CaCO3 · ZrO2 – baddeleyite, monoclinic zirconia · albite – low albite, NaAlSi3O8 · urea – urea, CO(NH2)2
Upload a CIF and read it back as an HTML or LaTeX table: unit cell, space group, atom sites, and the powder pattern the file implies.
Powder or single crystal
Interatomic distances and angles from a cell, a space group and an atom list, each with the uncertainty its own parameters support.
No diffraction data
Quartz, with uncertainties supplied · cubic – NaCl, Fm3m · cubic – Cu, Fm3m · cubic – α-Fe, Im3m · cubic – CsCl, Pm3m · cubic – ZnS, F43m · hexagonal – quartz, P3221 · tetragonal – cristobalite, P41212 · hexagonal – berlinite, P3121 · tetragonal – rutile, P42/mnm · orthorhombic – aragonite, Pmcn · monoclinic – ZrO2, P21/c · triclinic – albite, C1 · tetragonal – urea, P421m
Which components of an anisotropic displacement parameter a site allows, the principal displacements a tensor describes, and whether it is possible at all.
Single-crystal data
a sphere: one value, isotropic · stretched along c: only U33 raised · the same three displacements, tilted by U23 alone · not a possible atom: U23 pushed too far
What R1, wR2, the goodness of fit, Rint and the Flack parameter each measure, with the resolution and the data-to-parameter ratio worked out.
Single-crystal data
a Flack parameter that decides nothing · data that stop early · a refinement held together by restraints
Two of these are the place to start, and they answer the same question from opposite ends.
The Bragg calculator is the relationship the whole technique rests on. A set of lattice planes a distance d apart reflects a wavelength λ at one particular angle. That is the angle at which waves from successive planes come away in step. Put in an angle you measured, and it gives you the spacing that caused it. That angle is where a perfect, endless lattice puts the maximum. A real crystal is neither, so a measured line has a width — that is line broadening.
The HKL calculator goes the other way. Give it a unit cell and a set of Miller indices, which is the label for one family of planes. It works out how far apart those planes are and where the reflection will appear. It also names the other reflections that the crystal’s symmetry makes identical to it.
Work one of the examples above through both, and the connection becomes concrete. The d that the HKL calculator derives from the cell is the same d that the Bragg calculator turns into an angle.
That is the first half of the beginner path. It carries the same crystal on through where the reflections live, and why some are missing.
The standard atomic weights behind the mass, CHN and absorption calculators are IUPAC’s, published by the Commission on Isotopic Abundances and Atomic Weights (CIAAW).
The isotope‑pattern calculator uses a different quantity. Its natural abundances are the ones tabulated by the National Institute of Standards and Technology (NIST).
The CIF parser reads the Crystallographic Information File format, defined by the International Union of Crystallography (IUCr). Their Online Dictionary of Crystallography is a good reference for the terms these tools use.