The foil is bent, the Bragg condition is met somewhere along it, and that somewhere is a line. Tilt, and the line walks.
Pick a crystal, a zone axis and how the foil is bent. The page computes the many-beam rocking surface of the crystal (Bloch waves, thermal-diffuse absorption) and paints it on the bent foil, so you see the bright-field and dark-field bend contours, the pairs, the higher orders, the fringes inside them and the bend centre. Move the holder tilt and watch the contours slide. Change the radius of curvature and watch them spread. Read the radius back from the spacing.
Bright-field intensity against the tilt of the beam from the zone axis, at the thickness set by the slider. Every point of the bent foil takes its intensity from one point of this map; the outline shows the part of the map the image on the other tab uses. The dashed lines are the Bragg conditions of the reflections.
R1 is the radius of curvature along the bend direction (measured from the reflection on the right of the image, which points along +x), R2 across it. A negative radius bends the foil the other way. Leave R2 empty for a cylinder.
A thin foil in the microscope is rarely flat. Where it bends, the lattice planes turn with it, and the angle between the beam and the planes changes from place to place. The intensity that comes through a crystal depends on that angle. So the image of a bent foil is a picture of the rocking curve: the intensity against the angle of the beam, drawn on the specimen. This page computes that rocking curve for the whole zone axis at once, with all the reflections that matter, using the same Bloch-wave (scattering matrix) code as the SAED simulator, with thermal-diffuse absorption. It then maps the tilt at every point of a bent foil onto the rocking surface and shows the bright-field and dark-field images. The map is exact for a foil bent to a smooth curve: tilt = holder tilt + distance / radius along each bend direction. Because the rocking surface is computed once, the radius of curvature, the bend direction, the holder tilt, the field of view and the thickness all update at once. The rocking surface is thousands of independent many-beam calculations, one per tilt, so on a browser with WebGPU they run on the graphics card as one compute shader (the same scattering-matrix exponential and thickness stepping, in double-single arithmetic so that the result agrees with the JavaScript code to a part in a million); without WebGPU the same calculation runs in JavaScript workers and takes a few seconds.
A bend contour is not a thing in the specimen. It is a condition. Wherever the lattice planes (hkl) sit at the Bragg angle to the beam, the crystal scatters strongly into the reflection g. Less intensity goes straight through, so the bright-field image is dark there, and the dark-field image made with g is bright there. On a bent foil that condition is met along a line, and that line is the contour. The same planes meet the Bragg angle a second time when the foil is tilted the other way by the same amount: that is the contour of −g. The two are 2θBR apart, with R the radius of curvature of the foil. Between them, where the zone axis points along the beam, all the contours of the zone cross at the bend centre. Further out sit the 2g and 3g contours at two and three times the distance, fainter and narrower because the higher-order reflections are weaker. The old name for a bend contour is an equal-inclination fringe: every point on it has the same inclination of the planes to the beam.
Bright field on the left, dark field with the chosen reflection on the right; each image is scaled from black at zero to white at its own maximum, given in the caption. With label the contours on, the page draws the Bragg condition of every reflection it computed as a dashed line with its index: this is where the kinematic theory puts each contour. The darkest line of a real contour does not sit exactly on that position. The many-beam rocking curve has fringes, and which fringe is deepest changes with thickness, so the dark line wanders by a fraction of the contour width as you move the thickness slider. That is why a radius read from the spacing of two dark lines is good to about 20 percent, and why the readout also gives the radius from the centre of the dark-field peak system, which sits on the Bragg position much better.
The profile is taken along the bend direction through the centre of the image and carries the absolute intensities with the predicted contour positions marked. The Tilt map tab shows the raw rocking surface: intensity against beam tilt, with the part used by the current image outlined. A cylindrical bend uses one line of that map, a dome uses a square of it, and a holder tilt only moves the outline. That is the whole mechanism in one picture: bend contours are the rocking surface painted on the foil, and tilting the specimen slides the paint.
The readout gives the extinction distance and Bragg angle of the chosen reflection, the predicted spacing 2θBR, the spacing measured from the profile and the radius it implies, the width of a first-order contour compared with the two-beam estimate R / (g ξg), and the lever arm: how far every contour moves per milliradian and per degree of tilt. The sweep the tilt button rocks the holder by a few milliradians so you can watch the contours cross the field while the labels follow them.
| Feature | Tilt the holder by half a degree | Why |
|---|---|---|
| Bend contour | Sweeps a long way (R times the tilt: 175 nm for R = 20 µm) | It marks an orientation, and the orientation condition moves to a new place |
| Thickness fringes | Stay at the wedge or edge; spacing and contrast change | They mark equal thickness; the excitation error changes their period |
| Dislocation line | Stays; contrast changes, may vanish at g·b = 0 | It is a real object; its visibility depends on g and s |
| Grain or phase boundary | Stays; the fringes on an inclined boundary change | Real object |
| Precipitate, void, particle | Stays; a coherent particle changes its strain contrast | Real object |
| Dark grain next to a bright grain | The dark one goes bright, another goes dark | Orientation contrast, the same physics as a bend contour, one grain at a time |
| Fresnel fringe | Does not move with tilt; changes with focus | Phase contrast at an edge |
Phrased the way people ask them on ResearchGate, in facility FAQs and in teaching labs. Short answers; the interactive above shows each one.
They are dark lines in bright field, bright in dark field, in the image of a thin crystal that is not flat. Where the foil bends, the lattice planes turn with it. Along one line the planes sit at exactly the Bragg angle to the beam. There the crystal sends much of the beam into a diffracted spot, so less goes straight through and bright field is dark. The line follows the places with the same lattice tilt. That is why the old name is equal-inclination fringe. If your foil were perfectly flat you would see none. Almost no foil is.
A bend contour marks an orientation, not a place. Tilt the holder by a small angle δ and the Bragg condition is met somewhere else on the curved foil, a distance Rδ away, where R is the radius of curvature. With R = 20 µm one milliradian (0.06°) moves the contour 20 nm and one degree moves it 350 nm, across most of a field of view. Dislocations, boundaries and particles are objects, so they stay where they are; only their contrast changes. This is also why a two-beam condition holds over only a small patch of a bent foil: walk 200 nm and the excitation error has changed. Use the holder-tilt sliders above and watch the labels follow the lines.
Thickness fringes sit at the edge of the hole or along a wedge and run parallel to the lines of equal thickness. When you tilt, they stay attached to the edge; their spacing and contrast change, but they do not sweep across the foil. Bend contours can be anywhere, are usually broader and curved, come in pairs with a bend centre between them, and sweep across the foil when you tilt. The quick test is a tilt of half a degree: what runs away is a bend contour. Both are complementary between bright and dark field, so that does not separate them.
Most of the time it is orientation, the same physics as a bend contour, one grain at a time. A grain whose planes sit near the Bragg angle diffracts strongly and looks dark in bright field; its neighbour, a few degrees away in orientation, does not and looks bright. Tilt a little and they swap. Thickness and heavy elements also make a region darker, but that darkness does not change with a small tilt. A single bent particle can even show a bend contour across it, a dark band that is not a defect and not a second phase.
Planes (hkl) meet the Bragg angle twice: with the foil tilted by +θB one way (reflection g) and by θB the other way (reflection −g). On a bent foil that is two lines 2θBR apart, with the zone axis between them. The fainter, narrower lines at two and three times the distance are the 2g and 3g contours: higher-order reflections have longer extinction distances, so their contours are weaker and thinner. The fine fringes inside a contour are the thickness oscillations of the rocking curve, the same fringes you see across a CBED disc; their number grows with thickness. At certain thicknesses the middle of a contour is bright and the dark line splits in two.
That is the bend centre, the point where the zone axis points exactly along the beam. Every contour of the zone passes through it, so the number of arms shows the symmetry of the zone: four for [001] of a cubic crystal, six for [111], two for a low-symmetry zone. Williams and Carter call it a real-space zone-axis pattern. It tells you where on the foil you are exactly on axis, which is where you want a high-resolution image or a CBED pattern taken. In a crystal without a centre of symmetry the contrast of the arms is not the same on both sides of the centre, and that has been used to read the polarity of the crystal.
Three routes. First, tilt off the zone axis to a two-beam condition, set with the Kikuchi lines, and work in the patch between contours where the excitation error is small and nearly constant; the contours are still there, just moved out of the way. Second, use a thicker or flatter part of the foil: a thick region bends less, and every spacing on this page scales with the radius of curvature. Third, image in STEM with a convergent probe: a convergence and a collection semi-angle of a few milliradians average over the rocking curve and wash the contours out while dislocations keep their contrast (Zhu, Ophus, Toloczko and Edwards 2018); precession does the same in TEM. Remember that the contours are also useful: they show you exactly where the two-beam condition is.
Yes. The spacing of the +g and −g pair gives the local radius of curvature, R = Δx / 2θB; the card above does the arithmetic, and the readout of the simulation shows how good it is: about 20 percent from the darkest lines, better from the centre of the dark-field peaks, because the deepest fringe of a contour wanders with thickness. The sweep of a contour per degree of tilt gives the same radius. The width of a first-order contour scales with R / (g ξg), so a very narrow contour means a long extinction distance or a gentle bend. The number of fringes across a contour grows with thickness, like the fringes of a CBED disc.
Thin things bend easily. Ion damage and redeposition leave stress in a FIB lamella, and a thin electropolished metal foil sags under its own weight and its own residual stress. Radii of curvature of 10 to 100 micrometres are ordinary, which puts the contour pair a few hundred nanometres apart and several orders inside one field of view. Ductile metals are the worst; a stiff ceramic or silicon often bends less. Thicker lamellae, a supporting window or frame, and a low-voltage final polish reduce the bending. Once you are in the microscope, you cannot flatten the foil, so tilt to move the contours away from the feature instead.
They are cousins. Both mark the Bragg condition of one set of planes. Kikuchi lines live in the diffraction pattern: they mark the directions in which diffusely scattered electrons meet the Bragg angle. Bend contours live in the image: they mark the places on the foil where the incident beam meets the Bragg angle. Put the selected-area aperture on a contour and the corresponding Kikuchi line runs through the diffraction spot: you are in the exact two-beam condition there. Tilting moves both by the same angle, the Kikuchi line across the pattern and the contour across the foil.
In bright-field and low-angle annular dark-field STEM, yes, it is the same physics, until the convergence and collection angles are large enough to average over the rocking curve. In HAADF they are weak, but the intensity still changes by a noticeable amount near a zone axis because of channelling, so a HAADF image of a bent foil is not uniform either. In HRTEM the lattice fringes change their contrast and their apparent positions across a contour, which is why you put the bend centre on the feature before recording. Channelling also changes the X-ray and energy-loss yields across a contour, so quantify a composition away from a strong Bragg condition or on a tilted, averaged setting.
The rocking surface is a many-beam Bloch-wave calculation with the beams chosen by their excitation strength over the tilt range (15, 25 or 35 beams; the zero-order zone and whatever higher-order reflections come close to the sphere). Absorption is the thermal-diffuse absorptive potential in the Einstein model with room-temperature Debye-Waller factors, or a fixed ratio, or none. The foil has one thickness everywhere and is bent to a smooth paraboloid: a pure rotation of the lattice with no strain gradient through the thickness, which is the usual approximation and is good when the radius of curvature is many times the thickness, as it is for every case this page can show. Lattice rotation is treated as a beam tilt in the column approximation; the small change of the column direction through the bent foil is ignored. There is no background from inelastic scattering, and the calculation is for a parallel beam: a convergent STEM probe or precession averages this surface over a disc of tilts, which is exactly what removes the contours. A CIF you load is treated the same way as a library structure.
@misc{tripathy2026bendcontours,
author = {Tripathy, Manisha},
title = {Bend Contours},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tem/bend-contours}},
note = {Interactive web tool}
}