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It is applied to materials characterization to reveal the atomic scale structure of various substances in a variety of states. The book deals with fundamental properties of X-rays, geometry analysis of crystals, X-ray scattering and diffraction in polycrystalline samples and its application to the determination of the crystal structure. The reciprocal lattice and integrated diffraction intensity from crystals and symmetry analysis of crystals are explained.
To learn the method of X-ray diffraction crystallography well and to be able to cope with the given subject, a certain number of exercises is presented in the book to calculate specific values for typical examples. This is particularly important for beginners in X-ray diffraction crystallography. One aim of this book is to offer guidance to solving the problems of 90 typical substances. For further convenience, supplementary exercises are also provided with solutions. Some essential points with basic equations are summarized in each chapter, together with some relevant physical constants and the atomic scattering factors of the elements.
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. The most common type of goniometer is the "kappa goniometer", which offers three angles of rotation: 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.
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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 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. 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 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.
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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. 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. 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: 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.
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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. 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 structures.
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.
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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. Let's 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.
X-ray crystallography
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.