Weak-Beam Tilt Simulator

What you are doing, step by step

Weak-beam dark field is a tilt recipe. The image at the end is only as narrow as the excitation error you set up on the way, and the excitation error is set by tilting the crystal (with the holder) and the beam (with the dark-field deflectors) in a fixed order. The screen shows two kinds of thing: spots, which move when the beam tilts and when the crystal tilts, and Kikuchi lines, which are fixed to the crystal and move only when the crystal tilts. That difference is the whole trick, so the simulator draws both, with the deficient (dark) and excess (bright) sides shown the way a real pattern shows them: the line nearer the transmitted beam is dark, the one farther away is bright, and the pair is symmetric only when the beam lies in the plane itself, which is anywhere along the band, the zone axis included.

1. Align a zone axis. The screen opens on the whole map around the [011] zone of copper, with every zone axis to index 3 marked and labelled, the way the Kikuchi Map Navigator draws them, and the dashed patch showing what the holder can reach from this mounting. Drag any crossing to the centre, or click its dot and the holder drives there. The working zone follows whichever axis you align, and the g list under Crystal and row refreshes to the rows through it; click a Kikuchi line to pick that row instead. A zone beyond the holder's tilt range is not off limits: the foil is remounted with that zone at zero tilt (the reach patch jumps to it), which is what you would do at the microscope, and the same happens mid-drag when the holder runs out of tilt. If you mis-set the foil, the crossing you want is off-centre and you find it with α and β: it moves with the crystal, not the beam.
2. Walk off the zone along the band. Zoom in to the pattern (the Pattern button, or scroll) and tilt so that the beam slides along the chosen Kikuchi band, away from the zone axis, by five to ten degrees. Rotating about the band's own normal keeps its plane edge-on, so the (200) row stays at Bragg-ish while every other reflection in the zone loses its excitation and fades. This is the tilt most people do by feel; here the target α and β are printed, and the systematic-row table shows the other beams dying.
3. Set two-beam. Now tilt across the band, perpendicular to it, until the deficient (dark) line of g passes through 000 and the excess (bright) line passes through the g spot. That is s_g = 0: the Ewald sphere passes through g. The table reads s_g = 0 and s_2g, s_3g negative (outside the sphere).
4. Tilt the beam, not the crystal. Switch to centred dark field: the deflectors tilt the incident beam by 2θ_B so the g beam runs down the optic axis. Watch the spots jump left by one spacing while the Kikuchi lines stay put. The beam now makes 3θ_B with the (200) planes, so 3g is at the Bragg condition, and the bright 3g line passes through the 3g spot. That is the g(3g) condition, and s_g = λg² came for free.
5. Aperture around g, image. The axial beam is g, weakly excited. Its image of the dislocation is a narrow line close to the core, at a few percent of the incident intensity, which is why the exposures are long. For g(4g) or g(5g), keep the beam tilted and tilt the crystal further across the band until the 4g or 5g line reaches its spot; the table and badge follow.

Why alpha and beta both matter

The rod tilt α moves the whole pattern up and down the screen; the cradle tilt β moves it left and right, about an axis that is carried by the rod, so its leverage falls as cos α at large rod tilts and the set of orientations you can reach is a curved patch on the sphere rather than a rectangle. A Kikuchi band rarely runs exactly along either axis, so walking along a band and then across it takes a combination of the two, and the combination changes as you go. Dragging the pattern with the mouse solves that combination for you and shows the α and β it implies; the knobs and arrow keys let you do it one axis at a time, which is what the microscope makes you do. The compass in the corner of the screen shows which way each knob moves the pattern from where you are now.

Reading the numbers

The excitation error s is the distance from the reciprocal-lattice point to the Ewald sphere, positive when the point lies inside the sphere. For the systematic row, s_ng = −n g_z − n²λg²/2, where g_z is the component of g along the incident beam; the whole row is set by one tilt angle. The dimensionless w = sξ_g is what the image cares about: the effective extinction distance is ξ_g/√(1 + w²), the background intensity in dark field falls roughly as 1/(1 + w²), and the dislocation line width scales with that effective distance, which is why a large s narrows the line. Extinction distances are computed from the structure factor with Peng's 1996 scattering factors and the relativistic mass correction: Al 111 at 100 kV comes out at 55.6 nm, the classic table value; copper values run 10 to 20 percent above the 1965 table, which used older scattering factors.

What the model leaves out

The image is a two-beam Howie-Whelan calculation in the column approximation for a screw dislocation with isotropic elasticity, with mild absorption (ξ'₀ = ξ'_g = 10ξ_g); it uses the s of the imaging reflection only, so the weak influence of the other row beams (systematic-row dynamical effects, which matter for g(3g) in practice) is not included. Kikuchi line intensity is schematic: geometry is exact, the dark-bright asymmetry follows the beam-to-plane angle, but there is no diffuse-scattering model, no band profile, and no HOLZ line. Spots come from the same stitched map as the Kikuchi Map Navigator: every zone axis in view contributes its zero-order net, out to a few times its nearest spot, and each reflection is drawn where it really diffracts, sized by |F|² and faded by a two-beam envelope in its excitation error, except within about a degree and a half of a zone axis, where the whole net is lit the way many-beam scattering lights it (switch the fade off to see the bare nets). That is a display model: it places the beams and shows which ones are excited, not how bright a many-beam calculation would make them. The holder is an ideal double-tilt stage with no backlash and no specimen drift; the beam tilt is an ideal deflection with no image shift. Camera rotation and flips are not modelled: on your microscope the pattern may be rotated relative to the holder axes, and you should calibrate that once with a known tilt.

References

1. D. J. H. Cockayne, I. L. F. Ray and M. J. Whelan, Investigations of dislocation strain fields using weak beams, Phil. Mag. 20, 1265-1270 (1969). The g(3g) recipe.
2. D. B. Williams and C. B. Carter, Transmission Electron Microscopy, 2nd ed., Springer (2009), chapters 19 (Kikuchi lines), 24 (thickness and bending), 25 (planar defects) and 26 (weak-beam dark field), including the beam-tilt derivation of the g-3g condition.
3. P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron Microscopy of Thin Crystals, Butterworths (1965). Two-beam theory, the column approximation, and the extinction-distance table.
4. A. Howie and M. J. Whelan, Diffraction contrast of electron microscope images of crystal lattice defects, Proc. R. Soc. A 263, 217-237 (1961).
5. L.-M. Peng, G. Ren, S. L. Dudarev and M. J. Whelan, Robust parameterization of elastic and absorptive electron atomic scattering factors, Acta Cryst. A52, 257-276 (1996).
6. J. W. Edington, Practical Electron Microscopy in Materials Science, Macmillan (1976), Monograph 3, on setting up two-beam and weak-beam conditions at the microscope.