Beam tilt and precession#

from de_twin import DigitalTwin, ManualClock

twin = DigitalTwin("Dense Au on holey C", camera="DESim", clock=ManualClock())
col = twin.column
col.set_beam_tilt_mrad(2.0, 0.0)  # tilt the illumination
col.set("Precession", True)  # sweep the tilt round a cone...
col.set("PrecessionAngle", 20.0)  # ...of this half-angle, mrad
col.set("PrecessionFrequency", 100.0)  # Hz
col.set("PrecessionDescan", True)  # descan brings the pattern back (the default)

Beam tilt#

Beam tilt tilts the incident beam.

  • TEM imaging. The tilt enters the objective transfer as χ(k + k_t) − χ(k_t), with the envelopes tilted the same way. This gives axial coma, tilt-induced defocus and astigmatism, and an image shift of about defocus × tilt. The shift is what the autofocus beam-tilt pairs measure: the displacement between +τ and −τ is 2·Δf·τ.

  • Diffraction (SAED and CBED / 4D-STEM). A tilted beam moves the whole pattern by tilt / (mrad per pixel), and it tilts the Ewald sphere relative to the crystal. Tilting the beam towards −g by the Bragg angle excites +g.

Precession#

Precession is the beam-tilt coils driven round a cone, with descan below the specimen bringing the pattern back. The twin models it that way.

  • What is averaged. SAED and kinematic 4D-STEM patterns (PED at every scan point) average over the tilts swept during each frame. The cone is sampled at 24 tilts around the static beam tilt.

  • Short frames. A frame shorter than one precession period covers only part of the cone, as on a real instrument. The phase advances with time, and it is quantised to the 24 cone tilts so partial frames reuse cached patterns.

  • Effect on the pattern. Reflections further out are excited, and the intensities move towards kinematic values.

  • Without descan. The pattern rides a ring of radius θ/λ. Only the Bragg spots and the direct beam sweep; rings and the diffuse background stay centred, which is a simplification.

  • Imaging is not affected. TEM imaging does not precess, and its caches are unaffected. Hollow-cone imaging is not modelled.

  • Coherent STEM. The coherent (multislice) STEM model does not sweep its probe. Only its kinematic Bragg add-on sees precession.

  • Noiseless summaries. Renderer.datacube, virtual_image and ground_truth show the average over the whole cone.