The case for electrons
You have spent the last rung building diffraction with X-rays — the reciprocal lattice, the Ewald sphere, the structure factor. So why swap to a beam of electrons at all? Because electrons interact with matter roughly 10^3 to 10^4 times more strongly than X-rays do. An X-ray beam sails through a millimetre of crystal barely noticing it is there; an electron beam is stopped dead by a fraction of a micron. That ferocity is precisely the point: a beam you can focus down to a nanometre, and that scatters hard from a vanishingly small volume, lets you get a full diffraction pattern from a single grain, a tiny precipitate, even one nanocrystal — objects a laboratory X-ray source would never see. This is why the transmission electron microscope became crystallography's tool for the very small.
The same strength comes with a short-wavelength bonus. Accelerate electrons through 200 kV and their (relativistic) wavelength is about 2.5 pm — that is 0.0251 angstrom, some sixty times shorter than the 1.54 angstrom of Cu K-alpha. A short wavelength is no obstacle to diffraction (only a wavelength longer than the spacing is), and here it pays a spectacular dividend. The Ewald sphere has radius 1/lambda; at 2.5 pm that radius is about 40 per angstrom, dwarfing the roughly 0.5 per angstrom spacing between reciprocal points. So near the origin the sphere is almost a flat plane, and instead of grazing one reciprocal point at a time — the X-ray predicament, with its tightly curved sphere — it slices clean through a whole two-dimensional layer of them at once.
SAED: a photograph of one reciprocal-lattice plane
Start with the simplest electron pattern, made with a parallel beam. In the TEM you drop a physical aperture into an image plane so that only electrons from one chosen patch of the specimen — a single grain, say — carry on to form the pattern. That is the 'selected area', and the technique is selected-area electron diffraction (SAED). Switch the lenses from imaging to diffraction mode and the screen fills with a bright central spot — the undiffracted 000 beam, the part that went straight through — surrounded by a regular net of diffracted spots. Because the near-flat Ewald sphere cuts a whole plane of the reciprocal lattice, that net is a near-undistorted photograph of one layer of it: the old promise that a diffraction pattern IS the reciprocal lattice made literal — a single page of the book laid flat on the detector.
One honest subtlety keeps this from being magic. Strictly, only points exactly on the Ewald sphere diffract, and even a flat sphere touches points in just one plane. Two things conspire to light up a full net anyway. First, the specimen is a thin slab, and a thin object in real space is stretched into a long shape in reciprocal space — the inverse-size relationship you met as peak broadening. Each reciprocal point smears into a short rod (a 'rel-rod') poking along the beam direction. Second, the sphere, though nearly flat, keeps a whisker of curvature. Between them the sphere intersects a rod even where it would miss the ideal point, so a broad two-dimensional net blazes at once. The beam runs straight down a crystal direction [uvw] — the zone axis — and every plane whose spots you see contains that direction. As a feel for scale, at 200 kV with a camera length L = 1 m, a plane of spacing d = 2 angstrom throws its spot to a radius R = lambda times L / d = about 12.5 mm from the centre — a hand-measurable distance on the old photographic plates.
WHY ELECTRONS SEE A WHOLE PLANE OF SPOTS AT ONCE
X-rays (lambda = 1.54 A): Ewald radius 1/lambda ~ 0.65 A^-1
tightly curved --> touches only a FEW reciprocal points
o o o o o reciprocal-lattice plane
. . ( ) . . . curved sphere grazes 1-2 points
Electrons (lambda = 2.5 pm): 1/lambda ~ 40 A^-1 (~60x bigger)
almost FLAT --> slices a whole plane at once
o o o o o o o
-----o---o---o---O---o---o---o----- flat sphere = a 2D NET
: : : : : : : (thin foil stretches each
: : : : : : : point into a rod, so many
spots light up together)
O = transmitted (000) beam ; o = diffracted spot
Look straight DOWN a zone axis [uvw]: the net is one page of the
reciprocal-lattice book, photographed almost undistorted.- Insert the selected-area aperture to pick one region — often a single grain — and switch the microscope to diffraction mode.
- Find the bright central 000 transmitted spot and the surrounding net of diffracted spots.
- Measure R, the distance from 000 to a chosen spot.
- Use the camera equation R times d = lambda times L (lambda times L is the camera constant) to convert R into that plane's d-spacing.
- Measure the angles and length ratios between spots and match them to a candidate crystal to assign (hkl) indices to each spot.
- The direction you are looking down is the zone axis [uvw]; the symmetry of the whole net hints at the crystal's own symmetry.
CBED: cones that unlock symmetry and thickness
SAED uses a parallel beam, so each reflection is a sharp dot. Now converge the beam instead — focus it into a fine cone landing on a nanometre of specimen — and something far richer happens: every sharp dot swells into a filled DISC. Why? A cone is a fan of incident directions, and each direction inside the fan is diffracted at its own slightly different angle, so a single reflection is sampled over a small range of orientations all at once. That is convergent-beam electron diffraction (CBED), and each disc is a tiny two-dimensional rocking curve — a map of how that reflection brightens and dims as the incident angle sweeps. The fine structure inside those discs is where CBED earns its keep.
The symmetry of the whole pattern — and of the fine structure inside each disc — mirrors the symmetry of the crystal itself, and reading it off pins down the point group. Better still, CBED cracks a distinction ordinary diffraction cannot. Kinematical intensities obey Friedel's law, I(hkl) = I of the opposite reflection, which makes a centrosymmetric crystal and its non-centrosymmetric twin look identical — a close cousin of the phase problem. But dynamical scattering is sensitive to the very phase Friedel's law hides, so a CBED pattern from a non-centrosymmetric crystal visibly breaks that symmetry. The multiple scattering that muddied our intensities becomes, here, the tool that reveals whether a centre of symmetry is present at all.
Two more prizes fall out of a CBED disc. Push a strong reflection into a clean two-beam condition and its disc fills with parallel bright-and-dark fringes; count their spacing and you read the specimen thickness to within a nanometre or two — a genuine local ruler. And screw axes and glide planes, whose only fingerprint in X-rays is a set of missing spots (systematic absences you cannot always tell apart), announce themselves in CBED as sharp dark bars crossing the forbidden discs — the Gjonnes-Moodie lines. Those dark bands let CBED resolve space-group ambiguities that defeat X-rays, teasing apart symmetry elements like the glide plane from the raw pattern. Honest caveat: full space-group assignment this way is expert craft, not a push-button result.
Kikuchi lines: the crystal's built-in compass
Not every electron passes through cleanly. A fraction scatter inelastically — losing a little energy and spraying off in all directions inside the crystal, like light diffusing inside fog. These wayward electrons then meet the lattice planes from every angle, and those that happen to strike a set of planes at the Bragg angle get diffracted, sweeping out a pair of cones for each plane. On the flat detector each cone pair prints as two almost-straight lines — one bright, one dark — straddling the trace of that plane: a Kikuchi line pair, or Kikuchi band. Unlike the SAED spots, these are not made by the sharp incident beam but by the diffuse glow of scattered electrons already inside the crystal.
Here is the beautiful part. SAED spots stay put on the screen as you tilt the specimen — they only wink brighter or dimmer. Kikuchi lines, by contrast, are locked rigidly to the crystal's orientation, so they sweep bodily across the screen as you tilt, exactly like a map sliding under a fixed compass needle. Tilt a degree and the whole band pattern glides the matching amount. That makes Kikuchi lines the microscopist's road map: each band is the highway of one set of planes, band intersections are zone-axis poles, and to drive the crystal to a precise orientation you simply steer the chosen pole under the centre spot. They read orientation far more finely than the spots alone ever could.
Honest limits, and choosing your probe
One caveat to hold onto through all of this. Because electrons scatter many times over, the intensities in an electron pattern are not the clean |F|^2 of kinematical theory. Spot POSITIONS still faithfully give the lattice — the size, shape and symmetry of the unit cell come through cleanly — but you cannot read spot BRIGHTNESSES as a direct list of structure factors and solve the atom positions the naive X-ray way. Dynamical scattering even lights up reflections that ought to be strictly forbidden: a beam diffracted once by plane A and again by plane B can land exactly where a kinematically absent spot sits, faking a reflection that single-scattering rules forbid. So electron diffraction is unmatched for geometry, orientation and symmetry, yet for quantitative structure solution the X-ray and neutron workhorses still rule.
So which probe for which question? Reach for electrons in a TEM when the thing you care about is tiny or local — a single nanocrystal, a thin precipitate, the orientation relationship across one interface, or a defect you can image and diffract from in the same instrument. Reach for X-rays when you have a decent single crystal or a powder and want an accurate, quantitative structure with trustworthy intensities. Reach for neutron diffraction when you must locate light atoms like hydrogen, separate neighbouring or isotope-labelled elements, or map magnetic order — jobs electrons and X-rays do poorly. No probe is best at everything; each owns a window, and matching probe to question is half the craft of structure characterization.
This guide stayed in reciprocal space, reading the crystal from the spots, discs and bands it throws. But an electron microscope can do something X-rays cannot: with a short enough wavelength and a good lens, it can form a direct IMAGE of the lattice, with columns of atoms resolved as dots on the screen — real space, not its shadow. Turning a diffraction instrument into an atomic camera, through HRTEM and atomic-resolution STEM, is exactly where the next guide, Imaging Atoms with HRTEM, picks up.