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In
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...
, 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 In physics, the phase problem is the problem of loss of information concerning the phase that can occur when making a physical measurement. The name comes from the field of X-ray crystallography, where the phase problem has to be solved for the de ...
, where phase information is lost during a diffraction measurement. Direct methods provides a method of estimating the phase information by establishing
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
relationships between the recorded
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...
information and phases of strong reflections.


Background


Phase Problem

In
electron diffraction Electron diffraction refers to the bending of electron beams around atomic structures. This behaviour, typical for Wave (physics), waves, is applicable to electrons due to the wave–particle duality stating that electrons behave as both particle ...
, a diffraction pattern is produced by the interaction of the electron beam and the
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macr ...
potential. The real space and
reciprocal space In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fu ...
information about a
crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns t ...
can be related through the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
relationships shown below, where f(\textbf) is in real space and corresponds to the crystal potential, and F(\textbf) is its Fourier transform in reciprocal space. The vectors \textbf and \textbf are position vectors in real and reciprocal space, respectively. : f(\textbf) = \int_^ F(\textbf) e^ dk : F(\textbf) = \int_^ f(\textbf) e^ dr F(\textbf), also known as the
structure factor In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation ...
, is the Fourier transform of a
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to d ...
(i.e. the periodic crystal potential), and it defines the
intensity Intensity may refer to: In colloquial use * Strength (disambiguation) *Amplitude *Level (disambiguation) *Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) * Field strength of electric, ma ...
measured during a diffraction
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs wh ...
. F(\textbf) can also be written in a polar form F(\textbf), where \textbf is a specific reflection in reciprocal space. F(\textbf) has an amplitude term (i.e. , F(\textbf), ) and a phase term (i.e. e^). The phase term contains the position information in this form. : F(\textbf) = , F(\textbf), e^ During a diffraction experiment, the intensity of the reflections are measured as I(\textbf): : I(\textbf) = , F(\textbf), ^2 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 Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a gr ...
, the exit
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
of the electron beam from the crystal in real and reciprocal space can be written respectively as: : \psi(\textbf) = a(\textbf) e^ : \Psi(\textbf) = A(\textbf) e^ Where a(\textbf) and A(\textbf) are amplitude terms, the
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value * Exp ...
terms are phase terms, and \textbf is a reciprocal space vector. When a diffraction pattern is measured, only the intensities can be extracted. A measurement obtains a statistical
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
of the moduli: : I(\textbf) = \langle , \psi(\textbf), ^2 \rangle = \langle , a(\textbf), ^2 \rangle : I(\textbf) = \langle , \Psi(\textbf), ^2 \rangle = \langle , A(\textbf), ^2 \rangle 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 In crystallography, the Sayre equation, named after David Sayre who introduced it in 1952, is a mathematical relationship that allows one to calculate probable values for the phases of some diffracted beams. It is used when employing direct meth ...
, 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 ''Acta Crystallographica'' is a series of peer-reviewed scientific journals, with articles centred on crystallography, published by the International Union of Crystallography (IUCr). Originally established in 1948 as a single journal called ''Act ...
'',
Cochran ''For the history of the surname, see Cochrane.'' Cochran is a surname of Scottish (and most likely of Cumbric) origin. The earliest known appearance is in Dumbartonshire (14th cent). The definition is unclear, however the name may be derived from ...
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 The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfre ...
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- Nano (symbol n) is a unit prefix meaning "one billionth". Used primarily with the metric system, this prefix denotes a factor of 10−9 or . It is frequently encountered in science and electronics for prefixing units of time and length. ;Examp ...
and
micron The micrometre ( international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer ( American spelling), also commonly known as a micron, is a unit of length in the International System of Un ...
-sized
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, fro ...
s,
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bio ...
s, and
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respon ...
s. 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 Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas ...
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 F(\textbf) in the following form, where f(\textbf) is the atomic scattering factor for each atom at position \textbf, and \textbf is the position of atom l: : F(\textbf) = \sum_l f(\textbf) e^ This can be converted to the unitary structure factor U(\textbf) by dividing by N (the number of atoms) and f(\textbf): : U(\textbf) = \frac \sum_l e^ This can be alternatively rewritten in real and reciprocal space as: : u(\textbf) = \frac \sum_l \delta (\textbf-\textbf_) = N u(\textbf)^2 : U(\textbf) = N \sum_h U(\textbf) U(\textbf) This equation is a variation of the Sayre equation. Based on this equation, if the phases of U(\textbf) and U(\textbf) are known, then the phase of U(\textbf) 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: : N \langle U(\textbf) U(\textbf) \rangle = \frac \sum_l e^ = U(\textbf) Meaning that if: : U(\textbf) \approx N \langle U(\textbf) U(\textbf) \rangle Then: : , U(\textbf) - N U(\textbf) U(\textbf), ^2 = , U(\textbf), ^2 + N^2 , U(\textbf)U(\textbf), ^2 - 2 N , U(\textbf) U(\textbf) U(\textbf), \times cos(\phi(\textbf) - \phi(\textbf) - \phi(\textbf)) In the above equation, \langle , U(\textbf) - N U(\textbf) U(\textbf), ^2 \rangle = 0 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 In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables thems ...
can be applied here, which establishes that distributions tend to be
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
in form. By combining the terms of the known moduli, a distribution function can be written that is dependent on the phases: : P(U(\textbf) - N U(\textbf) U(\textbf)) \approx C e^ : P(U(\textbf) - N U(\textbf) U(\textbf)) \approx D e^ This distribution is known as the Cochran distribution. The standard deviation for this
Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It i ...
scales with the reciprocal of the unitary structure factors. If they are large, then the sum in the cosine term must be: : \phi(\textbf) - \phi(\textbf) - \phi(\textbf) \approx 2n\pi,~~n = 0, 1, 2... : \phi(\textbf) \approx \phi(\textbf) - \phi(\textbf) This is called the triplet phase relationship (\Sigma_2). If the phases \phi(\textbf) and \phi(\textbf) are known, then the phase \phi(\textbf) 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. : \prod_h P(U(\textbf) - N U(\textbf) U(\textbf)) \approx 2 N C e^ The most probable value of \phi(\textbf) can be found by taking the derivative of the above equation, which gives a variant of the tangent formula: : tan(\varphi(\textbf)) \approx \frac


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 Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
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 In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal. Etymology The term comes from the Greek language ...
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 In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of that distribution: : D(\phi(\textbf)-\phi(\textbf)-\phi(\textbf)) = A(\textbf) \sqrt[] \times cos (\phi(\textbf)-\phi(\textbf)-\phi(\textbf)) : D(\phi(\textbf)-\phi(\textbf)-\phi(\textbf)) = B(\textbf) cos(\phi(\textbf)-\phi(\textbf)-\phi(\textbf)) In the above distribution, A(\textbf) contains
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
terms, I(\textbf) terms are the experimental intensities, and B(\textbf) contains both of these for simplicity. Here, the most probable phases will maximize the function D(\textbf). If the intensities are sufficiently high and the sum in the cosine term remains \approx 0, then B(\textbf) will also be large, thereby maximizing D(\textbf). 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: : I(\textbf) > I(\textbf)~~ iff~, F(\textbf), > , F(\textbf), Let T(\textbf) be a function that relates the intensity to the phase for the same beam, where N(\textbf) contains normalization terms: : T(\textbf) = e^ \sqrt[]/N(\textbf) Then, the distribution of T(\textbf) values will be directly related to the values of F(\textbf). That is, when the product , F(\textbf)F(\textbf)F(\textbf), is large or small, , T(\textbf)T(\textbf)T(\textbf), 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 Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is cal ...
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: : \Psi(\textbf) = A(\textbf) \sum_l e^ : I(\textbf) = , A(\textbf), ^2 \bigg, \sum_l e^ \bigg, ^2 A(\textbf) is the Fourier transform of a(\textbf), which is
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
and represents the shape of an atom, given by the channeling states (e.g. 1s, 2s, etc.). A(\textbf) is real in reciprocal space and complex in the object plane. If B(\textbf), a conjugate symmetric function, is substituted for A(\textbf), then it is feasible to retrieve atom-like features from the object plane: : B(\textbf) = S(\textbf) , A(\textbf), , where ~S(\textbf) = \pm 1 In the object plane, the Fourier transform of B(\textbf) will be a real and symmetric pseudoatom (b(\textbf)) at the atomic column positions. b(\textbf) 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 In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
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 length In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. It is a transferable property of a bond between atoms of fixed types, relatively independent of the rest ...
s, symmetry, and interference. Constraints may also be statistical in origin, as shown earlier with the Cochran distribution and triplet phase relationship (\Sigma_2). According to Combettes, image recovery problems can be considered as a convex feasibility problem. This idea was adapted by Marks ''et al.'' to the crystallographic phase problem. With a
feasible set In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potent ...
approach, constraints can be considered
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
(highly convergent) or non-convex (weakly convergent). Imposing these constraints with the algorithm detailed earlier can converge towards unique or non-unique
solutions Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...
, depending on the convexity of the constraints.


Examples

Direct methods with electron diffraction datasets have been used to solve for a variety of structures. As mentioned earlier, surfaces are one of the cases in electron diffraction where scattering is kinematical. As such, many surface structures have been solved for by both X-ray and electron diffraction direct methods, including many of the
silicon Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ...
,
magnesium oxide Magnesium oxide ( Mg O), or magnesia, is a white hygroscopic solid mineral that occurs naturally as periclase and is a source of magnesium (see also oxide). It has an empirical formula of MgO and consists of a lattice of Mg2+ ions and O2− ...
,
germanium Germanium is a chemical element with the symbol Ge and atomic number 32. It is lustrous, hard-brittle, grayish-white and similar in appearance to silicon. It is a metalloid in the carbon group that is chemically similar to its group neighbo ...
,
copper Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkish ...
, and
strontium titanate Strontium titanate is an oxide of strontium and titanium with the chemical formula Sr Ti O3. At room temperature, it is a centrosymmetric paraelectric material with a perovskite structure. At low temperatures it approaches a ferroelectric pha ...
surfaces. More recently, methods for automated three dimensional electron diffraction methods have been developed, such as automated
diffraction tomography Diffraction tomography is an inverse scattering In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the i ...
and rotation electron diffraction. These techniques have been used to obtain data for structure solution through direct methods and applied for
zeolites Zeolites are microporous, crystalline aluminosilicate materials commonly used as commercial adsorbents and catalysts. They mainly consist of silicon, aluminium, oxygen, and have the general formula ・y where is either a metal ion or H+. These ...
, thermoelectrics, oxides, metal-organic frameworks,
organic compounds In chemistry, organic compounds are generally any chemical compounds that contain carbon-hydrogen or carbon-carbon bonds. Due to carbon's ability to catenate (form chains with other carbon atoms), millions of organic compounds are known. The ...
, and
intermetallics An intermetallic (also called an intermetallic compound, intermetallic alloy, ordered intermetallic alloy, and a long-range-ordered alloy) is a type of metallic alloy that forms an ordered solid-state compound between two or more metallic elem ...
. 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 Cryogenic electron microscopy (cryo-EM) is a cryomicroscopy technique applied on samples cooled to cryogenic temperatures. For biological specimens, the structure is preserved by embedding in an environment of vitreous ice. An aqueous sample sol ...
technique Microcrystal Electron Diffraction (MicroED). MicroED has been used for a variety of materials, including crystal fragments, proteins, and enzymes.


Software


DIRDIF

DIRDIF is a
computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer progra ...
for structure determination through using the
Patterson function The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function i ...
and direct methods applied to difference structure factors. It was first released by Paul Beurkens and his colleagues at the University of Nijmegen in 1999. It is written in Fortran and was most recently updated in 2008. It can be used for structures with heavy atoms, structures of molecules with partly known geometries, and for certain special case structures. Detailed information can be found at its website: http://www.xtal.science.ru.nl/dirdif/software/dirdif.html.


EDM

Electron Direct Methods is a set of programs developed at
Northwestern University Northwestern University is a private research university in Evanston, Illinois. Founded in 1851, Northwestern is the oldest chartered university in Illinois and is ranked among the most prestigious academic institutions in the world. Chart ...
by Professor Laurence Marks. First released in 2004, its most recent release was version 3.1 in 2010. Written in C++, C, and Fortran 77, EDM is capable of performing image processing of high resolution electron microscopy images and diffraction patterns and direct methods. It has a standard GNU license and is free to use or modify for
non-commercial A non-commercial (also spelled noncommercial) activity is an activity that does not, in some sense, involve commerce, at least relative to similar activities that do have a commercial objective or emphasis. For example, advertising-free community ...
purposes. It uses a feasible set approach and
genetic algorithm In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gen ...
search for solving structures using direct methods, and it also has
high-resolution transmission electron microscopy High-resolution transmission electron microscopy is an imaging mode of specialized transmission electron microscopes that allows for direct imaging of the atomic structure of samples. It is a powerful tool to study properties of materials on the ...
image simulation capabilities. More information can be found at the website: http://www.numis.northwestern.edu/edm/index.shtml.


OASIS

OASIS was first written by several scientists from the
Chinese Academy of Sciences The Chinese Academy of Sciences (CAS); ), known by Academia Sinica in English until the 1980s, is the national academy of the People's Republic of China for natural sciences. It has historical origins in the Academia Sinica during the Repub ...
in Fortran 77. The most recent release is version 4.2 in 2012. It is a program for direct methods phasing of protein structures. The
acronym An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in '' NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, a ...
OASIS stands for two of its applications: phasing One-wavelength Anomalous Scattering or Single Isomorphous Substitution protein data. It reduces the phase problem to a sign problem by locating the atomic sites of anomalous scatterers or heavy atom substitutions. More details can be found at the website: http://cryst.iphy.ac.cn/Project/IPCAS1.0/user_guide/oasis.html.


SIR

The SIR ( seminvariants representation) suite of programs was developed for solving the crystal structures of small molecules. SIR is updated and released frequently, with the first release in 1988 and the latest release in 2014. It is capable of both ''
ab initio ''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative singular of ''initium'' ("beginning"). Etymology Circa 1600, from Latin, literally "from the beginning", from a ...
'' and non-''ab-initio'' direct methods. The program is written in Fortran and C++ and is free for academic use. 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. More information can be found at its website: http://www.ba.ic.cnr.it/softwareic/sir2014/.


See also

*
Crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...
*
Transmission electron microscopy Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a gr ...
* Diffraction * Precession electron diffraction * Dynamical diffraction *
Electron crystallography Electron crystallography is a method to determine the arrangement of atoms in solids using a transmission electron microscope (TEM). Comparison with X-ray crystallography It can complement X-ray crystallography for studies of very small crystal ...
*
Electron diffraction Electron diffraction refers to the bending of electron beams around atomic structures. This behaviour, typical for Wave (physics), waves, is applicable to electrons due to the wave–particle duality stating that electrons behave as both particle ...
*
Microcrystal Electron Diffraction Microcrystal electron diffraction, or MicroED, is a CryoEM method that was developed by the Gonen laboratory in late 2013 at the Janelia Research Campus of the Howard Hughes Medical Institute. MicroED is a form of electron crystallography where t ...


References

{{reflist Electron microscopy