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CBED Calculation

CBED (convergent-beam electron diffraction) applies the dynamical core to many incident-beam directions and then places the results into diffraction disks. SAED has one incident direction; CBED treats each point inside the objective aperture as a partial incident plane wave and solves the Bloch-wave problem for each one.

Convergent-Beam Representation

At the entrance surface, the convergent probe can be written as a sum of plane waves using the probe position \(\mathbf R_0\), lens phase \(\chi(\mathbf K)\), and aperture function \(A(\mathbf K)\):

\[\psi_{\mathrm{in}}(\mathbf R,0)=\sum_{\mathbf K\in\mathrm{aperture}} A(\mathbf K)\, \exp(-2\pi i\,\mathbf K\cdot\mathbf R_0)\, \exp[-i\chi(\mathbf K)]\, \exp(2\pi i\,\mathbf K\cdot\mathbf R)\]

Here \(\mathbf K\) is the component of the incident wavevector parallel to the specimen surface. For an ideal circular aperture with convergence semi-angle \(\alpha\) and electron wavelength \(\lambda\),

\[A(\mathbf K)= \begin{cases} 1 & (|\mathbf K|\leq \sin\alpha/\lambda)\\ 0 & (|\mathbf K|> \sin\alpha/\lambda) \end{cases}\]

A representative lens phase, using defocus \(\Delta f\) and spherical aberration \(C_s\), is

\[\chi(\mathbf K)=\pi\lambda|\mathbf K|^2\Delta f+\frac{\pi}{2}C_s\lambda^3|\mathbf K|^4+\cdots\]

In ReciPro this expression is controlled by the aberration, aperture, and convergence-angle settings.

Dynamical Calculation for Each Direction

For CBED, each \(\mathbf K\) inside the aperture is treated as one parallel incident beam. The conceptual workflow is:

  1. Determine the refracted reference wavevector \(\mathbf k_0(\mathbf K)\) from \(\mathbf K\) and the specimen surface normal.
  2. Select the reflected beams using the ranking quantity \(R_{\mathbf g}=|\mathbf g|Q_{\mathbf g}^2\).
  3. Build the structure matrix \(\mathbf A\) and calculate the transmission coefficients \(T_{\mathbf g}(t;\mathbf K)\) at thickness \(t\).

This is the transmission-coefficient calculation from the dynamical core, repeated for every sampled incident direction. For a thickness series, the eigensolution for a given direction can be reused and only the propagation factors need to be updated.

Diffraction-Disk Assembly

Placing the exit waves from all \(\mathbf K\) directions into the diffraction plane gives the intensity inside the transmitted disk and the diffracted disks. If \(\mathbf Q\) is the diffraction-plane coordinate, position-averaged CBED or low-coherence conditions can be approximated as an incoherent intensity sum:

\[I_{\mathrm{CBED}}(\mathbf Q)= \sum_{\mathbf K\in\mathrm{aperture}} \left|\psi_{\mathbf K}(\mathbf Q,t)\right|^2\]

For LACBED-like modes where phase coherence across a wider region matters, the amplitudes must be summed first and the intensity taken afterwards.

What CBED Shows

CBED makes the thickness dependence of the Bloch-wave solution visible as intensity structure inside diffraction disks.

  • Changing the thickness changes disk-interior oscillations, HOLZ lines, and Kossel-Mollenstedt fringes.
  • Changing the incident orientation changes which reflections are strongly excited.
  • Increasing the convergence angle broadens the disks and can reveal overlap and higher-order Laue-zone information.

CBED is therefore the most direct way to view the Bloch-wave result as a disk pattern in the diffraction plane. In ReciPro it is best understood as the combination of convergent-beam discretisation, one dynamical solution per direction, and rearrangement into disk arrays.

Practical Parameters

  • Beam count: Strong zone-axis conditions and HOLZ-line detail require many reflected beams. Check how the disk interiors change as the maximum Bloch-wave count is increased.
  • Angular sampling: If the \(\mathbf K\) sampling inside the aperture is too coarse, the disk intensity becomes granular. Larger convergence angles require finer sampling.
  • Thickness: Thickness series benefit from the eigenvalue method because one eigensolution can be reused for many thicknesses.
  • Coherence: Distinguish conditions where an incoherent intensity sum is sufficient from those where coherent amplitude summation is needed.

See also