Ronchigram

A ronchigram is the diffraction pattern formed when you focus a convergent electron beam on a sample with the beam stationary at a single position. The name comes from the Ronchi test in optics. In 4D-STEM, every scan position produces its own ronchigram, and the full 4D dataset is essentially a collection of ronchigrams across all scan positions.

What it shows

When you form a convergent probe on your sample, the electrons pass through and create a diffraction pattern on the detector. This pattern contains several key features:

The center disk (also called the bright field disk) shows electrons that passed through with minimal scattering. The radius of this disk is set by the convergence angle of your probe. If you have a convergence angle of 20 mrad and the camera length gives you a certain magnification, the center disk will have a corresponding radius in pixels on your detector.

Outside the center disk, you see scattered electrons. For crystalline samples, you observe Bragg disks at specific scattering vectors corresponding to the crystal lattice. For amorphous samples or at high angles, you see a smooth decrease in intensity as the scattering angle increases.

The center of mass of the entire pattern tells you about beam deflection, which relates to electric and magnetic fields in the sample. This is the basis for differential phase contrast (DPC) imaging.

Why ronchigrams are useful

Ronchigrams provide direct access to multiple physical quantities without requiring iterative reconstruction algorithms. You can extract information immediately from the raw diffraction patterns.

For phase imaging, the center of mass shift of the ronchigram is proportional to the phase gradient \(\nabla\phi(x,y)\). Integrate these gradients across your scan positions and you reconstruct the phase map. This is center of mass (CoM) imaging, one of the simplest 4D-STEM techniques.

For strain mapping, the positions of Bragg disks in each ronchigram tell you the local lattice vectors. Track how these positions shift as you scan across the sample, and you map out strain fields with nanometer resolution.

For thickness and composition, the intensity distribution in the ronchigram (especially the ratio between center disk and outer regions) relates to specimen thickness and atomic number. Thicker regions show more scattering into higher angles.

Connection to ptychography and PCTF

Ptychographic reconstruction uses the full information content of ronchigrams at overlapping scan positions. Instead of just extracting the center of mass, you use the complete intensity pattern \(I(k_x,k_y)\) at every scan position. The redundancy from overlapping positions allows you to solve for both the object and probe simultaneously.

The PCTF is often evaluated by comparing reconstructed phases to simulated ground truth. However, you can also evaluate PCTF directly from experimental ronchigrams if you have a known reference sample. Take ronchigrams from your reference, reconstruct the phase, and compare to the expected phase from simulation or prior knowledge. The PCTF tells you which spatial frequencies in your reconstruction are reliable.

When you apply PCTF analysis “directly on the ronchigram,” what you are really doing is:

  1. Taking the measured ronchigram intensity at each position

  2. Reconstructing the exit wave phase using your algorithm (ptychography, SSB, etc.)

  3. Comparing the reconstructed phase Fourier transform to the expected phase Fourier transform

  4. Computing the ratio to get PCTF as a function of spatial frequency

The ronchigram is your raw data input. The PCTF quantifies how well your reconstruction algorithm converts that raw ronchigram data into an accurate phase map.

Practical considerations

The quality of your ronchigrams directly determines what you can extract. Several factors matter:

Camera dynamic range limits your ability to see both the bright center disk and weak high angle scattering simultaneously. Modern direct electron detectors help by providing high dynamic range and fast readout.

Convergence angle determines the size of your center disk. Larger angles give smaller probes (better spatial resolution) but also larger center disks that might obscure Bragg disks for crystalline samples. You need to balance these trade-offs based on your sample and what information you want to extract.

Dose is always a concern, especially for beam-sensitive samples. Each ronchigram requires sufficient electrons to overcome shot noise, but too many electrons damage the sample. 4D-STEM naturally uses more dose than conventional imaging because you need enough signal in every diffraction pattern, not just in the final integrated image.

The detector size and pixel count set your maximum scattering angle and angular resolution. Larger detectors capture more of the diffraction pattern, giving you access to higher spatial frequencies in your reconstruction.

Relation to other techniques

Conventional STEM imaging is essentially a 2D projection of 4D-STEM data. When you form a bright field (BF) image, you are integrating the intensity within the center disk of each ronchigram and mapping that integrated intensity to each scan position. Dark field imaging integrates specific regions outside the center disk. Annular dark field (ADF) integrates a ring at medium angles. High angle annular dark field (HAADF) integrates the outer ring at large scattering angles.

The advantage of 4D-STEM is that you record the full ronchigram, so you can create all these different contrast modes in post-processing. You are not locked into a single detector geometry during acquisition. You can also extract information that no conventional detector geometry can provide, such as the complete phase map from ptychography.

Example applications

In semiconductor device characterization, ronchigrams from different positions across a transistor show varying Bragg disk positions due to strain in the silicon lattice. Map these positions and you reconstruct 2D strain fields that reveal stress around gates and contacts.

In magnetic materials, the center of mass shift has contributions from both electrostatic and magnetic fields. By taking ronchigrams with the beam in opposite directions (using a deflector), you can separate electric and magnetic contributions. This enables mapping of magnetic domains at high spatial resolution.

In biological samples, ronchigrams show weak scattering with no Bragg disks. The center of mass shift reveals the phase gradient from light element distributions. Ptychographic reconstruction of these ronchigrams gives you phase contrast imaging of unstained proteins, cellular structures, and molecular assemblies.

For beam-sensitive materials like metal-organic frameworks (MOFs), you need minimal dose. Take sparse ronchigrams (fewer scan positions, fewer electrons per position) and use compressed sensing or other advanced algorithms to reconstruct from incomplete data. The PCTF helps you understand what spatial frequencies are reliably recovered despite the limited dose.

References