Direct methods (electron microscopy)

In crystallography, direct methods is a set of techniques used for structure determination using diffraction data and a priori information.

However, the phase term, which contains position information from the crystal potential, is lost.

[1] In the same issue of Acta Crystallographica, Cochran and Zachariasen also independently derived relationships between the signs of different structure factors.

Electron diffraction is a powerful technique for analyzing and characterizing nano- and micron-sized particles, molecules, and proteins.

The central limit theorem can be applied here, which establishes that distributions tend to be Gaussian in form.

[5] The standard deviation for this Gaussian function scales with the reciprocal of the unitary structure factors.

Likewise, amplitude errors will not severely impact the accuracy of the structure determination.

While most electron diffraction is dynamical, which is more difficult to interpret, there are instances in which mostly kinematical scattering intensities can be measured.

Achieving kinematical conditions is difficult in most cases—it requires very thin samples to minimize dynamical diffraction.

Even though most cases of electron diffraction are dynamical, it is still possible to achieve scattering that is statistically kinematical in nature.

This is what enables the analysis of amorphous and biological materials, where dynamical scattering from random phases add up to be nearly kinematical.

Furthermore, as explained earlier, it is not critical to retrieve phase information completely accurately.

The magnitude of their intensities will then have to be related to the amplitude of their corresponding scattering factors by the relationship: Let

For example, the exit wave and intensity of a sample dominated by channeling can be written in reciprocal space in the form:

In order to successfully solve for a structure, several algorithms have been developed for direct methods.

The Gerchberg-Saxton algorithm was originally developed by Gerchberg and Saxton to solve for the phase of wave functions with intensities known in both the diffraction and imaging planes.

As illustrated in image to the right,[6] one can successively impose real space and reciprocal constraints on an initial estimate until it converges to a feasible solution.

For instance, the fact that the data is produced by a scattering experiment in a transmission electron microscope imposes several constraints, including atomicity, bond lengths, symmetry, and interference.

Constraints may also be statistical in origin, as shown earlier with the Cochran distribution and triplet phase relationship (

Direct methods with electron diffraction datasets have been used to solve for a variety of structures.

These techniques have been used to obtain data for structure solution through direct methods and applied for zeolites, thermoelectrics, oxides, metal-organic frameworks, organic compounds, and intermetallics.

[14] In some of these cases, the structures were solved in combination with X-ray diffraction data, making them complementary techniques.

In addition, some success has been found using direct methods for structure determination with the cryo-electron microscopy technique Microcrystal Electron Diffraction (MicroED).

[15] MicroED has been used for a variety of materials, including crystal fragments, proteins, and enzymes.

Electron Direct Methods is a set of programs developed at Northwestern University by Professor Laurence Marks.

It uses a feasible set approach [10] and genetic algorithm search for solving structures using direct methods, and it also has high-resolution transmission electron microscopy image simulation capabilities.

OASIS was first written by several scientists from the Chinese Academy of Sciences in Fortran 77.

The acronym OASIS stands for two of its applications: phasing One-wavelength Anomalous Scattering or Single Isomorphous Substitution protein data.

The SIR (seminvariants representation) suite of programs was developed for solving the crystal structures of small molecules.

SIR can be used for the crystal structure determination of small-to-medium-sized molecules and proteins from either X-ray or electron diffraction data.