Precession electron diffraction (PED) is a specialized method to collect electron diffraction patterns in a transmission electron microscope (TEM). By rotating (precessing) a tilted incident electron beam around the central axis of the microscope, a PED pattern is formed by integration over a collection of diffraction conditions. This produces a quasi-kinematical diffraction pattern that is more suitable as input into direct methods algorithms to determine the crystal structure of the sample.
Overview
Geometry Precession electron diffraction is accomplished utilizing the standard instrument configuration of a modern TEM. The animation illustrates the geometry used to generate a PED pattern. Specifically, the beam tilt coils located pre-specimen are used to tilt the electron beam off of the optic axis so it is incident with the specimen at an angle, φ. The image shift coils post-specimen are then used to tilt the diffracted beams back in a complementary manner such that the direct beam falls in the center of the diffraction pattern. Finally, the beam is precessed around the optic axis while the diffraction pattern is collected over multiple revolutions.
The result of this process is a diffraction pattern that consists of a summation or integration over the patterns generated during precession. While the geometry of this pattern matches the pattern associated with a normally incident beam, the intensities of the various reflections approximate those of the kinematical pattern much more closely. At any moment in time during precession, the diffraction pattern consists of a Laue circle with a radius equal to the precession angle, φ. These snapshots contain far fewer strongly excited reflections than a normal zone axis pattern and extend farther into reciprocal space. Thus, the composite pattern will display far less dynamical character, and will be well suited for use as input into direct methods calculations.Notes from Advanced Electron Microscopy course at Northwestern University. Prepared by Professor Laurie Marks.
Advantages
PED possesses many advantageous attributes that make it well suited to investigating crystal structures via direct methods approaches:
Quasi-kinematical diffraction patterns: While the underlying physics of the electron diffraction is still dynamical in nature, the conditions used to collect PED patterns minimize many of these effects. The scan/de-scan procedure reduces ion channeling because the pattern is generated off of the zone axis. Integration via precession of the beam minimizes the effect of non-systematic inelastic scattering, such as Kikuchi lines. Few reflections are strongly excited at any moment during precession, and those that are excited are generally much closer to a two-beam condition (dynamically coupled only to the forward-scattered beam). Furthermore, for large precession angles, the radius of the excited Laue circle becomes quite large. These contributions combine such that the overall integrated diffraction pattern resembles the kinematical pattern much more closely than a single zone axis pattern. Broader range of measured reflections: The Laue circle (see Ewald sphere) that is excited at any given moment during precession extends farther into reciprocal space. After integration over multiple precessions, many more reflections in the zeroeth order Laue zone (ZOLZ) are present, and as stated previously, their relative intensities are much more kinematical. This provides considerably more information to input into direct methods calculations, improving the accuracy of phase determination algorithms. Similarly, more higher order Laue zone (HOLZ) reflections are present in the pattern, which can provide more complete information about the three-dimensional nature of reciprocal space, even in a single two-dimensional PED pattern. Practical robustness: PED is less sensitive to small experimental variations than other electron diffraction techniques. Since the measurement is an average over many incident beam directions, the pattern is less sensitive to slight misorientation of the zone axis from the optic axis of the microscope, and resulting PED patterns will generally still display the zone axis symmetry. The patterns obtained are also less sensitive to the thickness of the sample, a parameter with strong influence in standard electron diffraction patterns. Very small probe size: Because x-rays interact so weakly with matter, there is a minimum size limit of approximately 5 μm for single crystals that can be examined via x-ray diffraction methods. In contrast, electrons can be used to probe much smaller nano-crystals in a TEM. In PED, the probe size is limited by the lens aberrations and sample thickness. With a typical value for spherical aberration, the minimum probe size is usually around 50 nm. However, with Cs corrected microscopes, the probe can be made much smaller.
Practical considerations
Precession electron diffraction is typically conducted using accelerating voltages between 100-400 kV. Patterns can be formed under parallel or convergent beam conditions. Most modern TEMs can achieve a tilt angle, φ, ranging from 0-3°. Precession frequencies can be varied from Hz to kHz, but in standard cases 60 Hz has been used. In choosing a precession rate, it is important to ensure that many revolutions of the beam occur over the relevant exposure time used to record the diffraction pattern. This ensures adequate averaging over the excitation error of each reflection. Beam sensitive samples may dictate shorter exposure times and thus, motivate the use of higher precession frequencies.
One of the most significant parameters affecting the diffraction pattern obtained is the precession angle, φ. In general, larger precession angles result in more kinematical diffraction patterns, but both the capabilities of the beam tilt coils in the microscope and the requirements on the probe size limit how large this angle can become in practice. Because PED takes the beam off of the optic axis by design, it accentuates the effect of the spherical aberrations within the probe forming lens. For a given spherical aberration, Cs, the probe diameter, d, varies with convergence angle, α, and precession angle, φ, as
Thus, if the specimen of interest is quite small, the maximum precession angle will be restrained. This is most significant for conditions of convergent beam illumination. 50 nm is a general lower limit on probe size for standard TEMs operating at high precession angles (>30 mrad), but can be surpassed in Cs corrected instruments. In principle the minimum precessed probe can reach approximately the full-width-half-max (FWHM) of the converged un-precessed probe in any instrument, however in practice the effective precessed probe is typically ~10-50x larger due to uncontrolled aberrations present at high angles of tilt. For example, a 2 nm precessed probe with >40 mrad precession angle was demonstrated in an aberration-corrected Nion UltraSTEM with native sub-Å probe (aberrations corrected to ~35 mrad half-angle).
If the precession angle is made too large, further complications due to the overlap of the ZOLZ and HOLZ reflections in the projected pattern can occur. This complicates the indexing of the diffraction pattern and can corrupt the measured intensities of reflections near the overlap region, thereby reducing the effectiveness of the collected pattern for direct methods calculations.
Theoretical considerations
For an introduction to the theory of electron diffraction, see part 2 of Williams and Carter's Transmission Electron Microscopy text
While it is clear that precession reduces many of the dynamical diffraction effects that plague other forms of electron diffraction, the resulting patterns cannot be considered purely kinematical in general. There are models that attempt to introduce corrections to convert measured PED patterns into true kinematical patterns that can be used for more accurate direct methods calculations, with varying degrees of success. Here, the most basic corrections are discussed. In purely kinematical diffraction, the intensities of various reflections, , are related to the square of the amplitude of the structure factor, by the equation:
This relationship is generally far from accurate for experimental dynamical electron diffraction and when many reflections have a large excitation error. First, a Lorentz correction analogous to that used in x-ray diffraction can be applied to account for the fact that reflections are infrequently exactly at the Bragg condition over the course of a PED measurement. This geometrical correction factor can be shown to assume the approximate form:
where g is the reciprocal space magnitude of the reflection in question and Ro is the radius of the Laue circle, usually taken to be equal to φ. While this correction accounts for the integration over the excitation error, it takes no account for the dynamical effects that are ever-present in electron diffraction. This has been accounted for using a two-beam correction following the form of the Blackman correction originally developed for powder x-ray diffraction. Combining this with the aforementioned Lorentz correction yields:
where , is the sample thickness, and is the wave-vector of the electron beam. is the Bessel function of zeroeth order.
This form seeks to correct for both geometric and dynamical effects, but is still only an approximation that often fails to significantly improve the kinematic quality of the diffraction pattern (sometimes even worsening it). More complete and accurate treatments of these theoretical correction factors have been shown to adjust measured intensities into better agreement with kinematical patterns. For details, see Chapter 4 of reference.
Only by considering the full dynamical model through multislice calculations can the diffraction patterns generated by PED be simulated. However, this requires the crystal potential to be known, and thus is most valuable in refining the crystal potentials suggested through direct methods approaches. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing.
Historical development
The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in 1994. Preliminary investigation into the Er2Ge2O7 crystal structure demonstrated the feasibility of the technique at reducing dynamical effects and providing quasi-kinematical patterns that could be solved through direct methods to determine crystal structure. Over the next ten years, a number of university groups developed their own precession systems and verified the technique by solving complex crystal structures, including the groups of J. Gjønnes (Oslo), Migliori (Bologna), and L. Marks (Northwestern).Own, C. S.: PhD thesis, System Design and Verification of the Precession Electron Diffraction Technique, Northwestern University, 2005, http://www.numis.northwestern.edu/Research/Current/precession.shtmlJ.Gjonnes, V.Hansen, BS Berg, P.Runde, YF Gheng, K.Gjonnes, DL Dorset, C.Gilmore Acta Crystallogr (1998) A54, 306-319BS Berg, V.Hansen, PA Midgley, J Gjonnes Ultramicroscopy 74 (1998) 147-157M Gemmi, L.Righi, G.Calestani, A.Migliori, A.Speghini, M.Santarosa, M.Bettinelli Ultramicroscopy 84 (2000)133-142M.Gemmi, X.Zou, S.Hovmoller, A.Migliori, M.Vennstrom, Y.Anderson Acta Crystallogr A (2003) A59, 117-126