It can be easily seen that the inverse correlation time is the sum over the inverse correlation times for each shell corresponding to the particular jump mechanism. A similar approach was introduced by Chudley and Elliott [ 26 ] for liquids. Nickel and platinum, both of The composition of the sample was analysed by energy-dispersive X-ray spectroscopy EDX showing 3.
Prior experience has shown that the CCD camera placed close to the sample about 0. Therefore measurements at arbitrary azimuthal angles are equivalent. The temperature was stabilized by a PID temperature controller within a range of 0. Due to the limited intensity of the synchrotron source and low scattering intensity at large scattering angles in the diffuse regime it is beneficial to make the data collected by the CCD camera subject to the so-called droplet algorithm [ 27 ] to detect single photon events. Our implementation of the droplet algorithm is described in [ 24 ].
All measurements were carried out in the diffuse regime, away from Bragg peaks. Therefore such effects do not influence the intensities obtained in an aXPCS experiment. Therefore the total scattering power of the vacancies and their induced displacements is negligible. This contribution can therefore also be neglected. As a result the diffuse intensity is only due to the stochastic occupations of lattice sites by Ni or Pt, and the measured correlation decay therefore corresponds to exchanges of atoms. Small absolute values of reciprocal vectors correspond to very long correlation times.
Instabilities of the experimental setup result in additional intensity variations. This can lead to fault when overlapping with the slowly decaying autocorrelation function caused by the sample itself but is not important if the time-scale of both processes is clearly different. It is vitally important for aXPCS measurements to be performed in thermal equilibrium.
To ensure that order relaxation is negligible, multiple measurements were made repeatedly at different detector positions for each sample. Usually the sample relaxes faster when starting from a higher temperature and shows relaxation times of about one hour. After that time the single crystal stayed stable during the whole experiment. Note that the correlation time as measured in an aXPCS experiment quantifies the timescale over which the atomic arrangement changes by a sizeable amount.
Even though effects such as enhanced grain boundary diffusion can affect the mass transport over macroscopic distances significantly, the contribution of the grain boundaries to the scattered signal and therefore to the correlation decay is negligible. As a consequence, aXPCS is inherently sensitive only to bulk diffusion. To investigate whether there were crystallite growth effects in the polycrystalline sample during the experiment, the sample was studied with electron microscopy.
Unfortunately the results were inconclusive. Therefore we assumed that such effects can be neglected as long as no relaxation in the correlation time is detectable. To simulate a simple diffusion model, Monte Carlo simulation of diffusion is a valuable approach. One vacancy was introduced by randomly emptying a lattice site. A unit time MC step was defined as a number of trials equal to the number of lattice sites. After equilibrating the system, the actual diffusion simulation was started. Finally, using Siegerts relation, the intensity autocorrelation function was calculated and compared with the experimental correlation time.
SRO must be actually measured for exactly the same alloy composition and temperature as the sample under study. It is very difficult to deduce short-range order data from experimentally measured intensity data. A detailed description of the diffuse scattering theory can be found in several articles and books see e.
The factor which actually makes the analysis of diffuse scattering so difficult and time consuming is the separation of SRO scattering which is of interest to us and atomic displacement scattering. Simulations showed that in a system with a very high scattering contrast and low solute atom concentration like Ni 97 Pt 3 , the magnitude of those two terms can be of the same order. Also terms of higher order in the expansion of diffuse intensity, e.
Note that our main goal remains an atomistic diffusion mechanism and the details of the diffuse scattering intensity are not vitally important. Therefore we choose two simple models: m1 a model with no interaction between the atoms and m2 a model with strong Pt-Pt repulsing force. The latter was chosen because a L1 2 ordering type tendency resulting from strong Pt-Pt nearest-neighbour repulsion was assumed to be more likely than from Pt-Pt next-nearest-neighbour attraction or further ranging interactions 1 1 1 A more thorough investigation of the interaction parameters in the system Ni 97 Pt 3 based on ab initio calculation will be subject to a following paper.
From the atomic configuration we calculated the Warren-Cowley parameters [ 35 ]. These parameters can be used to calculate the short-range order intensity in Laue units according to. The main objective of this paper is to discuss the mechanism which governs diffusion in a Ni-Pt intermetallic alloy.
An essential advantage of a microscopic atomistic method in diffusion investigation is the possibility to identify such a diffusion mechanism. In order to verify which diffusion mechanism operates in a specific intermetallic phase, it is necessary to develop a mathematical model for the diffusion process.
This mathematical idea is generated from the knowledge of the system or it is simply a right guess. A model is determined by the jump probability parameters P n. Needless to say it is not particularly difficult to predict the atomistic mechanism of diffusion in Ni-Pt. It is simply an exchange of a nickel or platinum atom with a vacancy. As the scattering length of platinum considerably exceeds that of nickel and vacancies are very rare in the alloy at the temperatures of measurement, the measurable speckle fluctuations result predominantly from the exchanges of platinum atoms with nickel atoms.
This very simple picture enables studies of further subtleties of the diffusion mechanism, such as the influence of interaction between solute atoms or between solute atoms and vacancies. A simple and useful tool when discussing a diffusion model involving a vacancy is the so-called encounter approximation [ 36 ]. An encounter is here defined as the sum of all exchanges of one and the same atom with one and the same vacancy. As the number of vacancies is much smaller than the number of atoms, the time between one and the same vacancy jumping twice is much smaller than the time between consecutive jumps of one and the same atom.
One can assume that encounters are mutually independent and calculate e. The nearest-neighbour shell can be reached by only one exchange between a solute atom and a vacancy. The second, third and fourth shells can, however all be reached after two exchanges with a vacancy. This encounter model m1 , however, describes only systems with negligible interactions between matrix atoms, solvent atoms and vacancies. We use this as a second model for the description of Pt diffusion and call it modified encounter model m2. MC simulations sites for different interaction energies between platinum atoms were carried out.
Note that the only adjustment parameter is the jump frequency, which is simply a multiplicative constant. The figure shows that the MC simulation with no interaction potential s1 complies with the encounter model for a system with no SRO m1. Unfortunately the quality of the measured data was insufficient to draw sound conclusions about solute-vacancy interaction. It can therefore not be assumed that a sufficient number of small crystallites was covered by the beam during the measurement. For averaging over a small number of crystallites with unknown orientation the Chudley-Elliott model [ 26 ] is, however, still the best guess.
The difference to the single crystal value arises most likely from a difference in sample temperature as a different furnace with a thermocouple at a different position with respect to the sample was used for this measurement. Again a model with only next-nearest neighbour jumps m3 can obviously not appropriately describe the data measured. The relationship between diffusivity, temperature and activation energy can be described fairly accurately by the Arrhenius law [ 1 ]. These scattering methods generally use monochromatic X-rays, which are restricted to a single wavelength with minor deviations.
A broad spectrum of X-rays that is, a blend of X-rays with different wavelengths can also be used to carry out X-ray diffraction, a technique known as the Laue method. This is the method used in the original discovery of X-ray diffraction. Laue scattering provides much structural information with only a short exposure to the X-ray beam, and is therefore used in structural studies of very rapid events Time resolved crystallography. However, it is not as well-suited as monochromatic scattering for determining the full atomic structure of a crystal and therefore works better with crystals with relatively simple atomic arrangements.
The Laue back reflection mode records X-rays scattered backwards from a broad spectrum source. This is useful if the sample is too thick for X-rays to transmit through it. The diffracting planes in the crystal are determined by knowing that the normal to the diffracting plane bisects the angle between the incident beam and the diffracted beam. A Greninger chart can be used  to interpret the back reflection Laue photograph. Other particles, such as electrons and neutrons , may be used to produce a diffraction pattern.
Although electron, neutron, and X-ray scattering are based on different physical processes, the resulting diffraction patterns are analyzed using the same coherent diffraction imaging techniques. As derived below, the electron density within the crystal and the diffraction patterns are related by a simple mathematical method, the Fourier transform , which allows the density to be calculated relatively easily from the patterns.
However, this works only if the scattering is weak , i. Weakly scattered beams pass through the remainder of the crystal without undergoing a second scattering event. Such re-scattered waves are called "secondary scattering" and hinder the analysis. Any sufficiently thick crystal will produce secondary scattering, but since X-rays interact relatively weakly with the electrons, this is generally not a significant concern. Since this thickness corresponds to the diameter of many viruses , a promising direction is the electron diffraction of isolated macromolecular assemblies , such as viral capsids and molecular machines, which may be carried out with a cryo- electron microscope.
Moreover, the strong interaction of electrons with matter about times stronger than for X-rays allows determination of the atomic structure of extremely small volumes. The field of applications for electron crystallography ranges from bio molecules like membrane proteins over organic thin films to the complex structures of nanocrystalline intermetallic compounds and zeolites. Neutron diffraction is an excellent method for structure determination, although it has been difficult to obtain intense, monochromatic beams of neutrons in sufficient quantities.
Traditionally, nuclear reactors have been used, although sources producing neutrons by spallation are becoming increasingly available. Being uncharged, neutrons scatter much more readily from the atomic nuclei rather than from the electrons. Therefore, neutron scattering is very useful for observing the positions of light atoms with few electrons, especially hydrogen , which is essentially invisible in the X-ray diffraction. Neutron scattering also has the remarkable property that the solvent can be made invisible by adjusting the ratio of normal water , H 2 O, and heavy water , D 2 O.
The oldest and most precise method of X-ray crystallography is single-crystal X-ray diffraction , in which a beam of X-rays strikes a single crystal, producing scattered beams. When they land on a piece of film or other detector, these beams make a diffraction pattern of spots; the strengths and angles of these beams are recorded as the crystal is gradually rotated. For single crystals of sufficient purity and regularity, X-ray diffraction data can determine the mean chemical bond lengths and angles to within a few thousandths of an angstrom and to within a few tenths of a degree , respectively.
The atoms in a crystal are not static, but oscillate about their mean positions, usually by less than a few tenths of an angstrom. X-ray crystallography allows measuring the size of these oscillations. The technique of single-crystal X-ray crystallography has three basic steps.
The first—and often most difficult—step is to obtain an adequate crystal of the material under study. The crystal should be sufficiently large typically larger than 0. In the second step, the crystal is placed in an intense beam of X-rays, usually of a single wavelength monochromatic X-rays , producing the regular pattern of reflections.
The angles and intensities of diffracted X-rays are measured, with each compound having a unique diffraction pattern. Multiple data sets may have to be collected, with each set covering slightly more than half a full rotation of the crystal and typically containing tens of thousands of reflections.
In the third step, these data are combined computationally with complementary chemical information to produce and refine a model of the arrangement of atoms within the crystal. The final, refined model of the atomic arrangement—now called a crystal structure —is usually stored in a public database. As the crystal's repeating unit, its unit cell, becomes larger and more complex, the atomic-level picture provided by X-ray crystallography becomes less well-resolved more "fuzzy" for a given number of observed reflections.
Two limiting cases of X-ray crystallography—"small-molecule" which includes continuous inorganic solids and "macromolecular" crystallography—are often discerned. Small-molecule crystallography typically involves crystals with fewer than atoms in their asymmetric unit ; such crystal structures are usually so well resolved that the atoms can be discerned as isolated "blobs" of electron density. By contrast, macromolecular crystallography often involves tens of thousands of atoms in the unit cell. Such crystal structures are generally less well-resolved more "smeared out" ; the atoms and chemical bonds appear as tubes of electron density, rather than as isolated atoms.
In general, small molecules are also easier to crystallize than macromolecules; however, X-ray crystallography has proven possible even for viruses and proteins with hundreds of thousands of atoms, through improved crystallographic imaging and technology. Although crystallography can be used to characterize the disorder in an impure or irregular crystal, crystallography generally requires a pure crystal of high regularity to solve the structure of a complicated arrangement of atoms.
Pure, regular crystals can sometimes be obtained from natural or synthetic materials, such as samples of metals , minerals or other macroscopic materials. The regularity of such crystals can sometimes be improved with macromolecular crystal annealing    and other methods. However, in many cases, obtaining a diffraction-quality crystal is the chief barrier to solving its atomic-resolution structure. Small-molecule and macromolecular crystallography differ in the range of possible techniques used to produce diffraction-quality crystals.
Small molecules generally have few degrees of conformational freedom, and may be crystallized by a wide range of methods, such as chemical vapor deposition and recrystallization.
Atomic Diffusion in Glasses Studied with Coherent X-Rays
By contrast, macromolecules generally have many degrees of freedom and their crystallization must be carried out so as to maintain a stable structure. For example, proteins and larger RNA molecules cannot be crystallized if their tertiary structure has been unfolded ; therefore, the range of crystallization conditions is restricted to solution conditions in which such molecules remain folded. Protein crystals are almost always grown in solution.
The most common approach is to lower the solubility of its component molecules very gradually; if this is done too quickly, the molecules will precipitate from solution, forming a useless dust or amorphous gel on the bottom of the container. Crystal growth in solution is characterized by two steps: nucleation of a microscopic crystallite possibly having only molecules , followed by growth of that crystallite, ideally to a diffraction-quality crystal.
The crystallographer's goal is to identify solution conditions that favor the development of a single, large crystal, since larger crystals offer improved resolution of the molecule. Consequently, the solution conditions should disfavor the first step nucleation but favor the second growth , so that only one large crystal forms per droplet. If nucleation is favored too much, a shower of small crystallites will form in the droplet, rather than one large crystal; if favored too little, no crystal will form whatsoever.
Other approaches involves, crystallizing proteins under oil, where aqueous protein solutions are dispensed under liquid oil, and water evaporates through the layer of oil. It is extremely difficult to predict good conditions for nucleation or growth of well-ordered crystals.
The various conditions can use one or more physical mechanisms to lower the solubility of the molecule; for example, some may change the pH, some contain salts of the Hofmeister series or chemicals that lower the dielectric constant of the solution, and still others contain large polymers such as polyethylene glycol that drive the molecule out of solution by entropic effects.
It is also common to try several temperatures for encouraging crystallization, or to gradually lower the temperature so that the solution becomes supersaturated. These methods require large amounts of the target molecule, as they use high concentration of the molecule s to be crystallized. Due to the difficulty in obtaining such large quantities milligrams of crystallization-grade protein, robots have been developed that are capable of accurately dispensing crystallization trial drops that are in the order of nanoliters in volume. This means that fold less protein is used per experiment when compared to crystallization trials set up by hand in the order of 1 microliter.
Several factors are known to inhibit or mar crystallization. The growing crystals are generally held at a constant temperature and protected from shocks or vibrations that might disturb their crystallization. Impurities in the molecules or in the crystallization solutions are often inimical to crystallization.
Conformational flexibility in the molecule also tends to make crystallization less likely, due to entropy. Molecules that tend to self-assemble into regular helices are often unwilling to assemble into crystals. Having failed to crystallize a target molecule, a crystallographer may try again with a slightly modified version of the molecule; even small changes in molecular properties can lead to large differences in crystallization behavior.
The crystal is mounted for measurements so that it may be held in the X-ray beam and rotated. There are several methods of mounting. In the past, crystals were loaded into glass capillaries with the crystallization solution the mother liquor. Nowadays, crystals of small molecules are typically attached with oil or glue to a glass fiber or a loop, which is made of nylon or plastic and attached to a solid rod. Protein crystals are scooped up by a loop, then flash-frozen with liquid nitrogen.
However, untreated protein crystals often crack if flash-frozen; therefore, they are generally pre-soaked in a cryoprotectant solution before freezing. Generally, successful cryo-conditions are identified by trial and error. The capillary or loop is mounted on a goniometer , which allows it to be positioned accurately within the X-ray beam and rotated. An older type of goniometer is the four-circle goniometer, and its relatives such as the six-circle goniometer.
Small scale can be done on a local X-ray tube source, typically coupled with an image plate detector. These have the advantage of being relatively inexpensive and easy to maintain, and allow for quick screening and collection of samples. However, the wavelength light produced is limited by anode material, typically copper.
Further, intensity is limited by the power applied and cooling capacity available to avoid melting the anode. X-rays are generally filtered by use of X-ray filters to a single wavelength made monochromatic and collimated to a single direction before they are allowed to strike the crystal. The filtering not only simplifies the data analysis, but also removes radiation that degrades the crystal without contributing useful information. Collimation is done either with a collimator basically, a long tube or with a clever arrangement of gently curved mirrors.
Mirror systems are preferred for small crystals under 0. Rotating anodes were used by Joanna Joka Maria Vandenberg in the first experiments   that demonstrated the power of X rays for quick in real time production screening of large InGaAsP thin film wafers for quality control of quantum well lasers. Synchrotron radiation sources are some of the brightest lights on earth.
It is the single most powerful tool available to X-ray crystallographers. It is made of X-ray beams generated in large machines called synchrotrons. These machines accelerate electrically charged particles, often electrons, to nearly the speed of light and confine them in a roughly circular loop using magnetic fields. Synchrotrons are generally national facilities, each with several dedicated beamlines where data is collected without interruption.
Synchrotrons were originally designed for use by high-energy physicists studying subatomic particles and cosmic phenomena. The largest component of each synchrotron is its electron storage ring. This ring is actually not a perfect circle, but a many-sided polygon. At each corner of the polygon, or sector, precisely aligned magnets bend the electron stream.
As the electrons' path is bent, they emit bursts of energy in the form of X-rays. Using synchrotron radiation frequently has specific requirements for X-ray crystallography.
The intense ionizing radiation can cause radiation damage to samples, particularly macromolecular crystals. Recently, free-electron lasers have been developed for use in X-ray crystallography. The intensity of the source is such that atomic resolution diffraction patterns can be resolved for crystals otherwise too small for collection. However, the intense light source also destroys the sample,  requiring multiple crystals to be shot. As each crystal is randomly oriented in the beam, hundreds of thousands of individual diffraction images must be collected in order to get a complete data-set.
This method, serial femtosecond crystallography, has been used in solving the structure of a number of protein crystal structures, sometimes noting differences with equivalent structures collected from synchrotron sources. When a crystal is mounted and exposed to an intense beam of X-rays, it scatters the X-rays into a pattern of spots or reflections that can be observed on a screen behind the crystal. A similar pattern may be seen by shining a laser pointer at a compact disc.
The relative intensities of these spots provide the information to determine the arrangement of molecules within the crystal in atomic detail. The intensities of these reflections may be recorded with photographic film , an area detector such as a pixel detector or with a charge-coupled device CCD image sensor. The peaks at small angles correspond to low-resolution data, whereas those at high angles represent high-resolution data; thus, an upper limit on the eventual resolution of the structure can be determined from the first few images. Some measures of diffraction quality can be determined at this point, such as the mosaicity of the crystal and its overall disorder, as observed in the peak widths.
Some pathologies of the crystal that would render it unfit for solving the structure can also be diagnosed quickly at this point. One image of spots is insufficient to reconstruct the whole crystal; it represents only a small slice of the full Fourier transform. The rotation axis should be changed at least once, to avoid developing a "blind spot" in reciprocal space close to the rotation axis.
Atomic diffusion studied with coherent X-rays.
It is customary to rock the crystal slightly by 0. Multiple data sets may be necessary for certain phasing methods. For example, MAD phasing requires that the scattering be recorded at least three and usually four, for redundancy wavelengths of the incoming X-ray radiation. A single crystal may degrade too much during the collection of one data set, owing to radiation damage; in such cases, data sets on multiple crystals must be taken.
The recorded series of two-dimensional diffraction patterns, each corresponding to a different crystal orientation, is converted into a three-dimensional model of the electron density; the conversion uses the mathematical technique of Fourier transforms, which is explained below.
- Encyclopedia of Physical Science and Technology, 3e, Chemical Engineering.
- ID10 Coherent Scattering Science Page.
- DJ Shadows Endtroducing... (33 1/3 Series).
- Atomic Diffusion in Glasses Studied with Coherent X-Rays;
Each spot corresponds to a different type of variation in the electron density; the crystallographer must determine which variation corresponds to which spot indexing , the relative strengths of the spots in different images merging and scaling and how the variations should be combined to yield the total electron density phasing. Data processing begins with indexing the reflections. This means identifying the dimensions of the unit cell and which image peak corresponds to which position in reciprocal space.
A byproduct of indexing is to determine the symmetry of the crystal, i. Some space groups can be eliminated from the beginning. For example, reflection symmetries cannot be observed in chiral molecules; thus, only 65 space groups of possible are allowed for protein molecules which are almost always chiral. Indexing is generally accomplished using an autoindexing routine. This converts the hundreds of images containing the thousands of reflections into a single file, consisting of at the very least records of the Miller index of each reflection, and an intensity for each reflection at this state the file often also includes error estimates and measures of partiality what part of a given reflection was recorded on that image.
A full data set may consist of hundreds of separate images taken at different orientations of the crystal. The first step is to merge and scale these various images, that is, to identify which peaks appear in two or more images merging and to scale the relative images so that they have a consistent intensity scale. Optimizing the intensity scale is critical because the relative intensity of the peaks is the key information from which the structure is determined. The repetitive technique of crystallographic data collection and the often high symmetry of crystalline materials cause the diffractometer to record many symmetry-equivalent reflections multiple times.
This allows calculating the symmetry-related R-factor , a reliability index based upon how similar are the measured intensities of symmetry-equivalent reflections, [ clarification needed ] thus assessing the quality of the data. The data collected from a diffraction experiment is a reciprocal space representation of the crystal lattice. The position of each diffraction 'spot' is governed by the size and shape of the unit cell, and the inherent symmetry within the crystal. The intensity of each diffraction 'spot' is recorded, and this intensity is proportional to the square of the structure factor amplitude.
The structure factor is a complex number containing information relating to both the amplitude and phase of a wave. In order to obtain an interpretable electron density map , both amplitude and phase must be known an electron density map allows a crystallographer to build a starting model of the molecule.
The phase cannot be directly recorded during a diffraction experiment: this is known as the phase problem. Initial phase estimates can be obtained in a variety of ways:. Having obtained initial phases, an initial model can be built. The atomic positions in the model and their respective Debye-Waller factors or B -factors, accounting for the thermal motion of the atom can be refined to fit the observed diffraction data, ideally yielding a better set of phases.
A new model can then be fit to the new electron density map and successive rounds of refinement is carried out. This interative process continues until the correlation between the diffraction data and the model is maximized. The agreement is measured by an R -factor defined as. Both R factors depend on the resolution of the data. Chemical bonding features such as stereochemistry, hydrogen bonding and distribution of bond lengths and angles are complementary measures of the model quality.
Phase bias is a serious problem in such iterative model building. Omit maps are a common technique used to check for this. It may not be possible to observe every atom in the asymmetric unit. In many cases, Crystallographic disorder smears the electron density map. Weakly scattering atoms such as hydrogen are routinely invisible. It is also possible for a single atom to appear multiple times in an electron density map, e.
In still other cases, the crystallographer may detect that the covalent structure deduced for the molecule was incorrect, or changed. For example, proteins may be cleaved or undergo post-translational modifications that were not detected prior to the crystallization. A common challenge in refinement of crystal structures results from crystallographic disorder.
Disorder can take many forms but in general involves the coexistence of two or more species or conformations. Failure to recognize disorder results in flawed interpretation. Pitfalls from improper modeling of disorder are illustrated by the discounted hypothesis of bond stretch isomerism. In structures of large molecules and ions, solvent and counterions are often disordered.
The use of computational methods for the powder X-ray diffraction data analysis is now generalized. It typically compares the experimental data to the simulated diffractogram of a model structure, taking into account the instrumental parameters, and refines the structural or microstructural parameters of the model using least squares based minimization algorithm. Most available tools allowing phase identification and structural refinement are based on the Rietveld method ,   some of them being open and free software such as FullProf Suite,   Jana,  MAUD,    Rietan,  GSAS,  etc.
Most of these tools also allow Le Bail r efinement also referred to as profile matching , that is, refinement of the cell parameters based on the Bragg peaks positions and peak profiles, without taking into account the crystallographic structure by itself. More recent tools allow the refinement of both structural and microstructural data, such as the FAULTS program included in the FullProf Suite,  which allows the refinement of structures with planar defects e. Once the model of a molecule's structure has been finalized, it is often deposited in a crystallographic database such as the Cambridge Structural Database for small molecules , the Inorganic Crystal Structure Database ICSD for inorganic compounds or the Protein Data Bank for protein and sometimes nucleic acids.
Many structures obtained in private commercial ventures to crystallize medicinally relevant proteins are not deposited in public crystallographic databases. The main goal of X-ray crystallography is to determine the density of electrons f r throughout the crystal, where r represents the three-dimensional position vector within the crystal. To do this, X-ray scattering is used to collect data about its Fourier transform F q , which is inverted mathematically to obtain the density defined in real space, using the formula.
The three-dimensional real vector q represents a point in reciprocal space , that is, to a particular oscillation in the electron density as one moves in the direction in which q points. The corresponding formula for a Fourier transform will be used below. To obtain the phases, full sets of reflections are collected with known alterations to the scattering, either by modulating the wavelength past a certain absorption edge or by adding strongly scattering i.
Combining the magnitudes and phases yields the full Fourier transform F q , which may be inverted to obtain the electron density f r. Crystals are often idealized as being perfectly periodic. In that ideal case, the atoms are positioned on a perfect lattice, the electron density is perfectly periodic, and the Fourier transform F q is zero except when q belongs to the reciprocal lattice the so-called Bragg peaks.
In reality, however, crystals are not perfectly periodic; atoms vibrate about their mean position, and there may be disorder of various types, such as mosaicity , dislocations , various point defects , and heterogeneity in the conformation of crystallized molecules. Therefore, the Bragg peaks have a finite width and there may be significant diffuse scattering , a continuum of scattered X-rays that fall between the Bragg peaks. An intuitive understanding of X-ray diffraction can be obtained from the Bragg model of diffraction.
In this model, a given reflection is associated with a set of evenly spaced sheets running through the crystal, usually passing through the centers of the atoms of the crystal lattice. The orientation of a particular set of sheets is identified by its three Miller indices h , k , l , and let their spacing be noted by d. Such indexing gives the unit-cell parameters , the lengths and angles of the unit-cell, as well as its space group. Since Bragg's law does not interpret the relative intensities of the reflections, however, it is generally inadequate to solve for the arrangement of atoms within the unit-cell; for that, a Fourier transform method must be carried out.
The incoming X-ray beam has a polarization and should be represented as a vector wave; however, for simplicity, let it be represented here as a scalar wave. We also ignore the complication of the time dependence of the wave and just concentrate on the wave's spatial dependence.
At position r within the sample, let there be a density of scatterers f r ; these scatterers should produce a scattered spherical wave of amplitude proportional to the local amplitude of the incoming wave times the number of scatterers in a small volume dV about r. Consider the fraction of scattered waves that leave with an outgoing wave-vector of k out and strike the screen at r screen. From the time that the photon is scattered at r until it is absorbed at r screen , the photon undergoes a change in phase.
The net radiation arriving at r screen is the sum of all the scattered waves throughout the crystal. The measured intensity of the reflection will be square of this amplitude. For every reflection corresponding to a point q in the reciprocal space, there is another reflection of the same intensity at the opposite point - q. This opposite reflection is known as the Friedel mate of the original reflection.
This symmetry results from the mathematical fact that the density of electrons f r at a position r is always a real number. As noted above, f r is the inverse transform of its Fourier transform F q ; however, such an inverse transform is a complex number in general. The equality of their magnitudes ensures that the Friedel mates have the same intensity F 2. This symmetry allows one to measure the full Fourier transform from only half the reciprocal space, e. In crystals with significant symmetry, even more reflections may have the same intensity Bijvoet mates ; in such cases, even less of the reciprocal space may need to be measured.
The function f r is real if and only if the second integral I sin is zero for all values of r. In turn, this is true if and only if the above constraint is satisfied. Each X-ray diffraction image represents only a slice, a spherical slice of reciprocal space, as may be seen by the Ewald sphere construction.
Both k out and k in have the same length, due to the elastic scattering, since the wavelength has not changed.
Therefore, they may be represented as two radial vectors in a sphere in reciprocal space , which shows the values of q that are sampled in a given diffraction image. Since there is a slight spread in the incoming wavelengths of the incoming X-ray beam, the values of F q can be measured only for q vectors located between the two spheres corresponding to those radii.
In practice, the crystal is rocked by a small amount 0. A well-known result of Fourier transforms is the autocorrelation theorem, which states that the autocorrelation c r of a function f r. Therefore, the autocorrelation function c r of the electron density also known as the Patterson function  can be computed directly from the reflection intensities, without computing the phases. In principle, this could be used to determine the crystal structure directly; however, it is difficult to realize in practice.
Given the inevitable errors in measuring the intensities, and the mathematical difficulties of reconstructing atomic positions from the interatomic vectors, this technique is rarely used to solve structures, except for the simplest crystals. In principle, an atomic structure could be determined from applying X-ray scattering to non-crystalline samples, even to a single molecule. However, crystals offer a much stronger signal due to their periodicity.
A crystalline sample is by definition periodic; a crystal is composed of many unit cells repeated indefinitely in three independent directions. Such periodic systems have a Fourier transform that is concentrated at periodically repeating points in reciprocal space known as Bragg peaks ; the Bragg peaks correspond to the reflection spots observed in the diffraction image. Since the amplitude at these reflections grows linearly with the number N of scatterers, the observed intensity of these spots should grow quadratically, like N 2. In other words, using a crystal concentrates the weak scattering of the individual unit cells into a much more powerful, coherent reflection that can be observed above the noise.
Atomic Diffusion in Glasses Studied with Coherent X-Rays | SpringerLink
This is an example of constructive interference. In a liquid, powder or amorphous sample, molecules within that sample are in random orientations. Such samples have a continuous Fourier spectrum that uniformly spreads its amplitude thereby reducing the measured signal intensity, as is observed in SAXS. More importantly, the orientational information is lost. Although theoretically possible, it is experimentally difficult to obtain atomic-resolution structures of complicated, asymmetric molecules from such rotationally averaged data. An intermediate case is fiber diffraction in which the subunits are arranged periodically in at least one dimension.
X-ray diffraction has wide and various applications in the chemical, biochemical, physical, material and mineralogical sciences. Laue claimed in that the technique "has extended the power of observing minute structure ten thousand times beyond that given us by the microscope". X-ray diffraction, electron diffraction, and neutron diffraction give information about the structure of matter, crystalline and non-crystalline, at the atomic and molecular level.
In addition, these methods may be applied in the study of properties of all materials, inorganic, organic or biological. Due to the importance and variety of applications of diffraction studies of crystals, many Nobel Prizes have been awarded for such studies. Forensic examination of any trace evidence is based upon Locard's exchange principle.