In crystallography, direct methods is a set of techniques used for structure determination using diffraction data and a priori information. It is a solution to the crystallographic phase problem, where phase information is lost during a diffraction measurement. Direct methods provides a method of estimating the phase information by establishing statistical relationships between the recorded amplitude information and phases of strong reflections.
Background Phase Problem
In electron diffraction, a diffraction pattern is produced by the interaction of the electron beam and the crystal potential. The real space and reciprocal space information about a crystal structure can be related through the Fourier transform relationships shown below, where is in real space and corresponds to the crystal potential, and is its Fourier transform in reciprocal space. The vectors and are position vectors in real and reciprocal space, respectively.
, also known as the structure factor, is the Fourier transform of a three-dimensional periodic function (i.e. the periodic crystal potential), and it defines the intensity measured during a diffraction experiment. can also be written in a polar form , where is a specific reflection in reciprocal space. has an amplitude term (i.e. ) and a phase term (i.e. ). The phase term contains the position information in this form.
During a diffraction experiment, the intensity of the reflections are measured as :
This is a straightforward method of obtaining the amplitude term of the structure factor. However, the phase term, which contains position information from the crystal potential, is lost.
Analogously, for electron diffraction performed in a transmission electron microscope, the exit wave function of the electron beam from the crystal in real and reciprocal space can be written respectively as:
Where and are amplitude terms, the exponential terms are phase terms, and is a reciprocal space vector. When a diffraction pattern is measured, only the intensities can be extracted. A measurement obtains a statistical average of the moduli:
Here, it is also clear that the phase terms are lost upon measurement in an electron diffraction experiment. This is referred to as the crystallographic phase problem.
History
In 1952, David Sayre introduced the Sayre equation, a construct that related the known phases of certain diffracted beams to estimate the unknown phase of another diffracted beam. In the same issue of Acta Crystallographica, Cochran and Zachariasen also independently derived relationships between the signs of different structure factors. Later advancements were done by other scientists, including Hauptman and Karle, leading to the awarding of the Nobel Prize in Chemistry (1985) to Hauptman and Karle for their development of direct methods for the determination of crystal structures.
Comparison to X-Ray Direct Methods
The majority of direct methods was developed for X-ray diffraction. However, electron diffraction has advantages in several applications. Electron diffraction is a powerful technique for analyzing and characterizing nano- and micron-sized particles, molecules, and proteins. While electron diffraction is often dynamical and more complex to understand compared to X-ray diffraction, which is usually kinematical, there are specific cases (detailed later) that have sufficient conditions for applying direct methods for structure determination.
Theory Unitary Sayre Equation
The Sayre equation was developed under certain assumptions taken from information about the crystal structure, specifically that all atoms considered are identical and there is a minimum distance between atoms. Called the "Squaring Method," a key concept of the Sayre equation is that squaring the electron-density function (for X-ray diffraction) or crystal potential function (for electron diffraction) results in a function that resembles the original un-squared function of identical and resolved peaks. By doing so, it reinforces atom-like features of the crystal.
Consider the structure factor in the following form, where is the atomic scattering factor for each atom at position , and is the position of atom :
This can be converted to the unitary structure factor by dividing by N (the number of atoms) and :
This can be alternatively rewritten in real and reciprocal space as:
This equation is a variation of the Sayre equation. Based on this equation, if the phases of and are known, then the phase of is known.
Triplet Phase Relationship
The triplet phase relationship is an equation directly relating two known phases of diffracted beams to the unknown phase of another. This relationship can be easily derived via the Sayre equation, but it may also be demonstrated through statistical relationships between the diffracted beams, as shown here.
For randomly distributed atoms, the following holds true:
Meaning that if:
Then:
In the above equation, and the moduli are known on the right hand side. The only unknown terms are contained in the cosine term that includes the phases. The central limit theorem can be applied here, which establishes that distributions tend to be Gaussian in form. By combining the terms of the known moduli, a distribution function can be written that is dependent on the phases:
This distribution is known as the Cochran distribution. The standard deviation for this Gaussian function scales with the reciprocal of the unitary structure factors. If they are large, then the sum in the cosine term must be:
This is called the triplet phase relationship (). If the phases and are known, then the phase can be estimated.
Tangent Formula
The tangent formula was first derived in 1955 by Jerome Karle and Herbert Hauptman. It related the amplitudes and phases of known diffracted beams to the unknown phase of another. Here, it is derived using the Cochran distribution.
The most probable value of can be found by taking the derivative of the above equation, which gives a variant of the tangent formula:
Practical Considerations
The basis behind the phase problem is that phase information is more important than amplitude information when recovering an image. This is because the phase term of the structure factor contains the positions. However, the phase information does not need to be retrieved completely accurately. Often even with errors in the phases, a complete structure determination is possible. Likewise, amplitude errors will not severely impact the accuracy of the structure determination.
Sufficient Conditions
In order to apply direct methods to a set of data for successful structure determination, there must be reasonable sufficient conditions satisfied by the experimental conditions or sample properties. Outlined here are several cases.
Kinematical Diffraction One of the reasons direct methods was originally developed for analyzing X-ray diffraction is because almost all X-ray diffraction is kinematical. While most electron diffraction is dynamical, which is more difficult to interpret, there are instances in which mostly kinematical scattering intensities can be measured. One specific example is surface diffraction in plan view orientation. When analyzing the surface of a sample in plan view, the sample is often tilted off a zone axis in order to isolate the diffracted beams of the surface from those of the bulk. Achieving kinematical conditions is difficult in most cases—it requires very thin samples to minimize dynamical diffraction.
Statistical Kinematical 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. Errors in the phase information are tolerable.
Recalling the Cochran distribution and considering a logarithm of that distribution:
In the above distribution, contains normalization terms, terms are the experimental intensities, and contains both of these for simplicity. Here, the most probable phases will maximize the function . If the intensities are sufficiently high and the sum in the cosine term remains , then will also be large, thereby maximizing . With a narrow distribution such as this, the scattering data will be statistically within the realm of kinematical consideration.
Intensity Mapping
Consider two scattered beams with different intensities. The magnitude of their intensities will then have to be related to the amplitude of their corresponding scattering factors by the relationship:
Let ) be a function that relates the intensity to the phase for the same beam, where contains normalization terms:
Then, the distribution of values will be directly related to the values of . That is, when the product is large or small, will also be large and small. So, the observed intensities can be used to reasonably estimate the phases for diffracted beams. The observed intensity can be related to the structure factor more formally using the Blackman formula.
Other cases to consider for intensity mapping are specific diffraction experiments, including powder diffraction and precession electron diffraction. Specifically, precession electron diffraction produces a quasi-kinematical diffraction pattern that can be used adequately in direct methods.
Dominated Scattering In some cases, scattering from a sample can be dominated by one type of atom. Therefore, the exit wave from the sample will also be dominated by that atom type. For example, the exit wave and intensity of a sample dominated by channeling can be written in reciprocal space in the form:
is the Fourier transform of , which is complex and represents the shape of an atom, given by the channeling states (e.g. 1s, 2s, etc.). is real in reciprocal space and complex in the object plane. If , a conjugate symmetric function, is substituted for , then it is feasible to retrieve atom-like features from the object plane:
In the object plane, the Fourier transform of will be a real and symmetric pseudoatom () at the atomic column positions. will satisfy atomistic constraints as long as they are reasonably small and well-separated, thereby satisfying some constraints required for implementing direct methods.
Implementation
Direct methods is a set of routines for structure determination. In order to successfully solve for a structure, several algorithms have been developed for direct methods. A selection of these are explained below.
Gerchberg-Saxton
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. However, it has been generalized for any information in real or reciprocal space. Detailed here is a generalization using electron diffraction information. As illustrated in image to the right, one can successively impose real space and reciprocal constraints on an initial estimate until it converges to a feasible solution.
Constraints
Constraints can be physical or statistical. 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 ().