Precession electron diffraction

Precession Electron Diffraction new article content ...

Precession Electron Diffraction (PED) is a specialized method to collect electron diffraction patterns in a transmission electron microscope (TEM). By rotating (precessing) the incident electron beam around the optic 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 a direct methods algorithm to solve the original crystal structure of the sample.

Overview of Technique

Geometry of Procession Electron Diffraction

Procession electron diffraction is accomplished utilizing the standard instrument configuration of a modern TEM. Figure 1 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 some semi-angle, φ. The image shift coils post-specimen are then used to tilt the diffracted beams back in a complimentary 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.

Fig 1: Geometry of electron beam in precession electron diffraction (Currently Placeholder)

The result of this process is a diffraction pattern that consists of a summation or integration over the patterns generated over precession angles. While the geometry of this pattern matches that corresponding to a normally incident beam, the intensities of the various reflections approximate those of the kinematical pattern much more closely. At any given angle, the diffraction pattern consists of a Laue circle with a radius equal to the precession angle, φ. It is crucial to note that 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 (see section 1.3).

Direct Methods

Direct methods in crystallography are a collection of mathematical techniques that seek to determine crystal structure based on measurements of diffraction patterns and potentially other a priori knowledge (constraints). The central challenge of inverting measured diffraction intensities (i.e. applying an inverse Fourier Transform) to determine the original crystal potential is that phase information is lost in general since intensity is a measurement of the square of the modulus of the amplitude of any given diffracted beam. This is known as the phase problem of crystallography.

If the diffraction can be considered kinematical, constraints may be used to probabilistically relate the phases of the reflections to their amplitudes, and the original structure can be solved via direct methods (see Sayre equation). This is often the case in x-ray diffraction, and is one of the primary reasons that technique has been so successful at solving crystal structures. However, in electron diffraction, the probing wave interacts much more strongly with the electrostatic crystal potential, and complex dynamical diffraction effects can dominate the measured diffraction patterns. This makes application of direct methods much more challenging without a priori knowledge of the structure in question.

Benefits of Procession Electron Diffraction

PED possesses many advantageous attributes that make it well suited to investigating crystal structures via direct methods approaches:

1. 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.

Figure 2: Effect of precession angle on observed diffraction pattern (Currently Placeholder)

1. Broader Range of Measured Reflections:

The Laue circle that is excited at any given moment during precession extends farther into reciprocal space, so 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 extension procedures. 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 2-D PED pattern.

1. 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, which is a strong parameter in standard electron diffraction patterns.

1. Very Small Probe Size:

Because x-rays interact so weakly with matter, there is a minimum size limit of approximately 5µm 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. With a typical value for spherical aberration, the minimum probe size is usually around 50nm. 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 keV. Patterns can be formed under parallel or convergent beam conditions. Most modern TEMs can achieve a tilt angle ranging from 0-3°.

Precession frequencies can be varied from Hz to kHz, but in standard cases 60 Hz has been used. The important condition on precession rate is 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.

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 spherical aberration, Cs, the probe diameter, dprobe varies with convergence angle, α, and precession angle, φ, as :

d ~ 4Csφ2α

Thus, if the specimen of interest is quite small, the condition of independent illumination places limitations on the permissible precession angle. This is most significant for conditions of convergent beam illumination. 50 mRad is a general upper limit on precession angle on standard TEMs, but can be surpassed in Cs corrected instruments.

If the precession angle is made too large, further complications due to the overlap of the ZOLZ and HOLZ can occur, complicating the indexing of the diffraction pattern and sometimes confusing the relative intensities of reflections, thereby reducing the usefulness of the collected pattern for direct methods calculations.

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. Gjonnes (Oslo), and Migliori (Bologna), and L. Marks (Northwestern).

In 2004, NanoMEGAS developed the first commercial procession system capable of being retrofit to any modern TEM (figure 3). This hardware solution enabled more widespread implementation of the technique and spurred its more mainstream adoption into the crystallography community [7]. Software methods have also been developed to achieve the necessary scanning and descanning using the built-in electronics of a modern TEM.

As of June, 2015, more than 200 publications have relied on the technique to solve or corroborate crystal structures; many on materials that could not be solved by other conventional crystallography techniques like x-ray diffraction. According to NanoMEGAS, their retrofit hardware system is used in more than 75 laboratories across the world.

Theoretical Considerations

While it is clear that precession reduces many of the dynamical diffraction effects that plague other forms of electron diffraction, the resulting patterns cannot generally be considered purely kinematical in nature. 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. In purely kinematical diffraction, the intensities of various reflections, Ig, are related to the square of the amplitude of the structure factor, Fg: I_g^kinematical=|F_g |^2

This relationship is generally far from true for experimental dynamical electron diffraction and when many reflections have a strong excitation error, sz. 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 :

I_g^kinematical∝I_g^experimental×g√(1-(g/(2R_o )) )

where g is the amplitude 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 diffraction 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 ]:

I_g^kinematical∝I_g^experimental×(g√(1-(g/(2R_o )) )×A_g/(∫_0^(A_g)▒〖J_0 (2x)dx〗)) where A_g=(2πtF_g)/k, t is the sample thickness, and k is the wave-vector of the electron beam. J0 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). For a more complete treatment of these theoretical correction factors that have been shown to adjust measured intensities into better agreement with kinematical patterns, 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.

External links

• www.example.com