1
Get a calibration first
Everything below is a division by a constant you have to measure. Skip this step and every d-spacing you report carries whatever error the microscope's nominal setting happens to have that day.
In a diffraction pattern, the distance R from the direct beam to a spot and the spacing d of the planes that made it are related by one equation:
The wavelength is known to as many digits as you want from the accelerating voltage. The camera length is not. What the microscope displays is a nominal value for a lens setting, and the real one drifts with the lens history, the specimen height and the eucentric position. A 5% error in L is ordinary and it lands whole on every d you report.
Record a ring pattern from a polycrystalline standard (evaporated gold is the usual one) at every camera length you use, in the same session, at the same specimen height. Fit the ring radii to the known spacings and you have K directly, without ever needing L or λ separately. The ring calibration tool does the fit, and gives you the elliptical distortion and the image-to-diffraction rotation at the same time, both of which also move a measured spacing.
For a lattice image the equivalent calibration is the pixel size, in nanometres per pixel, at the magnification you used. It has the same problem: the nominal magnification is a setting. Calibrate it on a cross-grating replica or on a known lattice in the same image, and note that it changes with the objective lens current, so a focus change is a calibration change.
2
Measure it, both ways
Put in what you measured and its uncertainty. The number is easy; the error bar is the part that decides what you are allowed to conclude from it.
Why the two routes are not equally precise
The diffraction route measures one distance across a pattern, and the fractional error in d is the fractional error in that distance plus the fractional error in the camera constant. A spot close to the centre is the problem: at R = 3 mm a 0.15 mm error is 5%, while at R = 30 mm the same absolute error is 0.5%. So the low-order reflections, which are the ones you most want, are the ones you measure worst. The fix is the same one used everywhere: measure across the pattern, from a spot to its opposite number through the centre, and halve. That doubles the distance and removes the centre-finding error at the same time.
The FFT route is a discrete transform, so the spot sits on a grid of spacing 1/(N·pixel size). Reading it to the nearest pixel gives a fractional error of one over the number of grid steps out to the spot, which is why a bigger FFT window on the same lattice measures better: the spot moves further from the origin in grid units. Sub-pixel fitting of the peak, which the HRTEM lattice tool does, buys back roughly another factor of five, and after that the pixel-size calibration is what limits you.
Both routes share a trap the arithmetic hides: they assume the pattern or the image is undistorted. An elliptical diffraction pattern gives a d that depends on the azimuth you measured along, and a sheared image does the same. Two members of the same {hkl} family must give the same spacing; when they do not, stop and fix the distortion.
3
Does it name a plane?
A d-spacing on its own identifies nothing. It becomes an identification only when the error bar is narrow enough to exclude the neighbours, and in a crowded region of the spacing axis it usually is not.
| Phase | hkl | d (Å) | Off by | Inside the error bar |
|---|
Lattice parameters are room-temperature values for the pure phases. A solid solution moves them: about 0.5% for a few atomic percent of a substitutional solute in aluminium, more for interstitials. If your candidates are separated by less than that, a d-spacing will not choose between them, and you need a second reflection, an angle between two reflections, or a different technique.
Two spacings and the angle between them is a much stronger constraint than one spacing, because it tests the geometry as well as the size. The zone-axis indexer does that fit and gives you the zone axis with it; the pattern simulator then draws what the answer predicts so you can overlay it on what you recorded. If the two disagree, the disagreement is the result.
4
Where the error comes from
Four contributions, and they are not equal. The list is ordered the way it usually turns out in practice, which is not the order people worry about them in.
| Source | Typical size | What it does | What fixes it |
|---|---|---|---|
| Camera constant, or pixel size | 1% to 5% if uncalibrated, under 0.5% if calibrated in the same session | Scales every spacing by the same factor, so it never shows up as inconsistency between your own measurements. Invisible without a standard. | A standard in the same session at the same height. Nothing else. |
| Reading the distance | 0.5% to 5%, worst for the low-order spots you care most about | Random between measurements, so it shows up as scatter and can be beaten down by averaging. | Measure through the centre and halve. Average several equivalent reflections. Fit the spot rather than clicking it. |
| Elliptical distortion | 1% to 3% on many instruments | Makes the answer depend on the azimuth, so two members of one family disagree. Often mistaken for a real anisotropy. | Measure it once from a ring pattern and correct for it. The ring calibration tool reports it. |
| Specimen height and tilt | 1% to 2% per 100 µm off eucentric | Changes the effective camera length between specimens, so a calibration from yesterday's grid does not apply to today's. | Work at eucentric height every time, and recalibrate when you change holder or specimen. |
The camera constant or pixel size you used and how you got it, the number of reflections you averaged, the uncertainty you are quoting and whether it includes the calibration, and the distortion correction if you applied one. A d-spacing quoted to four figures without any of that is quoting the arithmetic, not the measurement.
Sources
Show the references
- D. B. Williams, C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science, 2nd ed., Springer (2009): the camera constant, the small-angle approximation and the calibration chapter behind every step on this page.
- J. W. Edington, Practical Electron Microscopy in Materials Science, Philips (1976): the measure-across-the-pattern practice and the standard error sources.
- Lattice parameters used here are room-temperature values from standard tables: Al 4.0495, Cu 3.6149, Au 4.0782, Ni 3.5240, α-Fe 2.8665, W 3.1652, Si 5.4309 ångströms.